Steven Roman

Fundamentals of Group Theory

An Advanced Approach Steven Roman Irvine, CA USA

ISBN 978-0-8176-8300-9 e-ISBN 978-0-8176-8301-6 DOI 10.1007/978-0-8176-8301-6 Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2011941115

Mathematics Subject Classification (2010): 20-01

© Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhäuser Boston, c/o Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Printed on acid-free paper

Birkhäuser Boston is part of Springer Science+Business Media (www.birkhauser.com) To Donna

Preface

This book is intended to be an advanced look at the basic theory of groups, suitable for a graduate class in group theory, part of a graduate class in abstract algebra or for independent study. It can also be read by advanced undergraduates. Indeed, I assume no specific background in group theory, but do assume some level of mathematical sophistication on the part of the reader.

A look at the table of contents will reveal that the overall topic selection is more or less standard for a book on this subject. Let me at least mention a few of the perhaps less standard topics covered in the book:

1) An historical look at how Galois viewed groups. 2) The problem of whether the commutator subgroup of a group is the same as the set of commutators of the group, including an example of when this is not the case. 3) A discussion of xY-groups, in particular, a) groups in which all subgroups have a complement b) groups in which all normal subgroups have a complement c) groups in which all subgroups are direct summands d) groups in which all normal subgroups are direct summands. 4) The subnormal join property, that is, the property that the join of two subnormal subgroups is subnormal. 5) Cancellation in direct sums: A group K is cancellable in direct sums if EK¸FLßK¸LÊE¸F{{ (The symbol { represents the external direct sum.) We include a proof that any finite group is cancellable in direct sums. 6) A complete proof of the theorem of Baer that a nonabelian group K has the property that all of its subgroups are normal if and only if KœEF where E is a quaternion group, is an elementary abelian group of exponent #F and is an abelian group all of whose elements have odd order.

vii viii Preface

7) A somewhat more in-depth discussion of the structure of : -groups, including the nature of conjugates in a :: -group, a proof that a -group with a unique subgroup of any order must be either cyclic (for :# ) or else cyclic or generalized quaternion (for :œ# ) and the nature of groups of order ::88 that have elements of order " . 8) A discussion of the Sylow subgroups of the symmetric group (in terms of wreath products). 9) An introduction to the techniques used to characterize finite simple groups. 10) Birkhoff's theorem on equational classes and relative freeness.

Here are a few other remarks concerning the nature of this book.

1) I have tried to emphasize universality when discussing the isomorphism theorems, quotient groups and free groups. 2) I have introduced certain concepts, such as subnormality and chain conditions perhaps a bit earlier than in some other texts at this level, in the hopes that the reader would acclimate to these concepts earlier. 3) I have also introduced group actions early in the text (Chapter 4), before giving a more thorough discussion in Chapter 7. 4) I have emphasized the role of applying certain operations, namely intersection, lifting, quotient and unquotient to a “group extension” LŸK .

A couple of random notes: Unless otherwise indicated, any theorem not proved in the text is an invitation to the reader to supply a proof. Also, sections marked with an asterisk are optional, meaning that they can be skipped without missing information that will be required later.

Let me conclude by thanking my graduate students of the past five years, who not only put up with this material in manuscript form but also put up with the many last-minute changes that I made to the manuscript during those years. In any case, if the reader should find any errors, I would appreciate a heads-up. I can be contacted through my web site www.romanpress.com.

Steven Roman Contents

Preface, vii 1 Preliminaries, 1 Multisets, 1 Words, 1 Partially Ordered Sets, 2 Chain Conditions and Finiteness, 5 Lattices, 8 Equivalence Relations, 10 Cardinality, 12 Miscellanea, 15 2 Groups and Subgroups, 19 Operations on Sets, 19 Groups, 19 The Order of a Product, 26 Orders and Exponents, 28 Conjugation, 29 The Set Product, 31 Subgroups, 32 Finitely-Generated Groups, 35 The Lattice of Subgroups of a Group, 37 Cosets and Lagrange's Theorem, 41 Euler's Formula, 43 Cyclic Groups, 44 Homomorphisms of Groups, 46 More Groups, 46 *An Historical Perspective: Galois-Style Groups, 56 Exercises, 58 3 Cosets, Index and Normal Subgroups, 61 Cosets and Index, 61 Quotient Groups and Normal Subgroups, 65 Internal Direct Products, 72 Chain Conditions and Subnormality, 76

ix x Contents

Subgroups of Index # , 78 Cauchy's Theorem, 79 The Center of a Group; Centralizers, 81 The Normalizer of a Subgroup, 82 Simple Groups, 84 Commutators of Elements, 84 Commutators of Subgroups, 93 *Multivariable Commutators, 95 Exercises, 98 4 Homomorphisms, Chain Conditions and Subnormality, 105 Homomorphisms, 105 Kernels and the Natural Projection, 108 Groups of Small Order, 108 A Universal Property and the Isomorphism Theorems, 110 The Correspondence Theorem, 113 Group Extensions, 115 Inner Automorphisms, 119 Characteristic Subgroups, 120 Elementary Abelian Groups, 121 Multiplication as a Permutation, 123 The Frattini Subgroup of a Group, 127 Subnormal Subgroups, 128 Chain Conditions, 136 Automorphisms of Cyclic Groups, 141 Exercises, 144 5 Direct and Semidirect Products, 149 Complements and Essentially Disjoint Products, 149 Product Decompositions, 151 Direct Sums and Direct Products, 152 Cancellation in Direct Sums, 157 The Classification of Finite Abelian Groups, 158 Properties of Direct Summands, 161 xY-Groups, 164 Behavior Under Direct Sum, 167 When All Subgroups Are Normal, 169 Semidirect Products, 171 The External Semidirect Product, 175 *The Wreath Product, 180 Exercises, 185 6 Permutation Groups, 191 The Definition and Cycle Representation, 191 A Fundamental Formula Involving Conjugation, 193 Parity, 194 Generating Sets for WE88 and , 195 Contents xi

Subgroups of WE88 and , 197 The Alternating Group Is Simple, 197 Some Counting, 200 Exercises, 202 7 Group Actions; The Structure of : -Groups, 207 Group Actions, 207 Congruence Relations on a K -Set, 211 Translation by K , 212 Conjugation by K on the Conjugates of a Subgroup, 214 Conjugation by K on a Normal Subgroup, 214 The Structure of Finite : -Groups, 215 Exercises, 229 8 Sylow Theory, 235 Sylow Subgroups, 235 The Normalizer of a Sylow Subgroup, 235 The Sylow Theorems, 236 Sylow Subgroups of Subgroups, 238 Some Consequences of the Sylow Theorems, 239 When All Sylow Subgroups Are Normal, 240 When a Subgroup Acts Transitively; The Frattini Argument, 243 The Search for Simplicity, 244 Groups of Small Order, 250 On the Existence of Complements: The Schur–Zassenhaus Theorem, 252 *Sylow Subgroups of W8 , 258 Exercises, 261 9 The Classification Problem for Groups, 263 The Classification Problem for Groups, 263 The Classification Problem for Finite Simple Groups, 263 Exercises, 271 10 Finiteness Conditions, 273 Groups with Operators, 273 HH-Series and -Subnormality, 276 Composition Series, 278 The Remak Decomposition, 282 Exercises, 288 11 Solvable and Nilpotent Groups, 291 Classes of Groups, 291 Operations on Series, 293 Closure Properties of Groups Defined by Series, 295 Nilpotent Groups, 297 Solvability, 305 Exercises, 313 xii Contents

12 Free Groups and Presentations, 319 Free Groups, 319 Relatively Free Groups, 326 Presentations of a Group, 336 Exercises, 350 13 Abelian Groups, 353 An Abelian Group as a ™ -Module, 355 The Classification of Finitely-Generated Abelian Groups, 355 Projectivity and the Right-Inverse Property, 357 Injectivity and the Left-Inverse Property, 359 Exercises, 363 References, 367 List of Symbols, 371 Index, 373 Chapter 1 Preliminaries

In this chapter, we gather together some basic facts that will be useful in the text. Much of this material may already be familiar to the reader, so a light skim to set the notation may be all that is required. The chapter then can be used as a reference. Multisets The following simple concept is much more useful than its infrequent appearance would indicate.

Definition Let WQW be a nonempty set. A multiset with underlying set is a set of ordered pairs  QœÖÐ=ß8ѱ=−Wß8−33 3 3™ ß=Á= 3 4 for 3Á4×

 where ™ œ Ö"ß #ß á ×. The positive integer 83 is referred to as the multiplicity of the element =Q3 in . A multiset is finite if the underlying set is finite. The size of a finite multiset Q is the sum of the multiplicities of its elements.

For example, Q œ ÖÐ+ß #Ñß Ð,ß $Ñß Ð-ß "Ñ× is a multiset with underlying set W œÖ+ß,ß-×. The element + has multiplicity # . One often writes out the elements of a multiset according to their multiplicities, as in Q œÖ+ß+ß,ß,ß,ß-× Two multisets are equal if their underlying sets are equal and if the multiplicities of each element in the multisets are equal. Words We will have considerable use for the following concept.

Definition Let \ be a nonempty set. A finite sequence = œÐB"8 ßáßB Ñ of elements of \\ is called a word or string over and is usually written in the form

S. Roman, Fundamentals of Group Theory: An Advanced Approach, 1 DOI 10.1007/978-0-8176-8301-6_1, © Springer Science+Business Media, LLC 2012 2 Fundamentals of Group Theory

= œBâB"8 The number of elements in AAÐ is the length of , denoted by len AÑ . There is a unique word of length ! , called the empty word and denoted by % . The set of all words over \\\ is denoted by ‡‡ and is called the alphabet for \ . A subword or substring of a word == is a subsequence of consisting of consecutive elements of = . The empty word is considered a subword of all words.

The set \\‡ of words over has an algebraic structure. In particular, the operation of juxtaposition (also called concatenation) is associative and has identity % . Any nonempty set with an associative operation that has an identity is called a monoid . Thus, \‡ is a monoid under juxtaposition.

It is customary to allow the use of exponents other than " when writing words, where BœBâB8 î 8 factors for 8! . Note, however, that this is merely a shorthand notation. Also, it does not affect the length of a word; for example, the length of BCD#$ is ' . Partially Ordered Sets We will need some basic facts about partially ordered sets.

Definition A partially ordered set is a pair ÐT ß Ÿ Ñ where T is a nonempty set andŸ is a binary relation called a partial order, read “less than or equal to,” with the following properties: 1)( Reflexivity ) For all +−T , +Ÿ+ 2)( Antisymmetry) For all +ß , − T , +Ÿ,ß ,Ÿ+ Ê +œ, 3)( Transitivity ) For all +ß ,ß - − T , +Ÿ,ß ,Ÿ- Ê +Ÿ- Partially ordered sets are also called posets .

Sometimes partially ordered sets are more easily defined using strict order relations.

Definition A strict order T on a nonempty set is a binary relation that satisfies the following properties: Preliminaries 3

1)( Asymmetry ) For all +ß , − T , +, Ê ,y+ 2)( Transitivity ) For all +ß ,ß - − T , +,ß ,- Ê +- Theorem 1.1 If ÐT ß Ÿ Ñ is a partially ordered set, then the relation +,if +Ÿ,ß+Á, is a strict order on T . Conversely, if is a strict order on T , then the relation +Ÿ,if +, or +œ, is a partial order on T .

It is customary to use a phrase such as “Let T be a partially ordered set” when the partial order is understood. Also, it is very convenient to extend the notation a bit and define WŸ+ for any subset WT of to mean that =Ÿ+ for all =−W . Similarly, +ŸW means that +Ÿ= for all =−W and WŸX means that =Ÿ> for all =−W and >−X.

Note that in a partially ordered set, it is possible that not all elements are comparable. In other words, it is possible to have Bß C − T with the property that BŸ±C and CŸ±B.

Here are some special kinds of partially ordered sets.

Definition Let ÐT ß Ÿ Ñ be a partially ordered set. 1) The order Ÿ is called a total order or linear order if every two elements of TÐ are comparable. In this case, TßŸÑ is called a totally ordered set or linearly ordered set. 2) A nonempty subset of TT that is totally ordered is called a chain in . The family of chains of T is ordered by set inclusion. 3) A nonempty subset of T for which no two elements are comparable is called an antichain in T . 4) A nonempty subset HT of a partially ordered set is directed if every two elements of HH have an upper bound in .

Definition Let ÐT ß Ÿ Ñ be a poset and let +ß , − T . 1) The closed interval Ò+ß ,Ó is defined by Ò+ß,ÓœÖ:−T ±+Ÿ:Ÿ,×

2) The open interval Ð+ß ,Ñ is defined by Ð+ß,ÑœÖ:−T ±+:,× 4 Fundamentals of Group Theory

3) The half open intervals are defined by Ð+ß,ÓœÖ:−T ±+:Ÿ,×and Ò+ß,ÑœÖ:−T ±+Ÿ:,× Here are some key terms related to partially ordered sets.

Definition () Covering Let ÐT ß Ÿ Ñ be a partially ordered set. If +ß , − T , then ,+ covers , written +¡,+Ÿ, , if and if there are no elements of T between + and , , that is, if +ŸBŸ, Ê Bœ+ or Bœ, Definition () Maximum and minimum elements Let ÐT ß Ÿ Ñ be a partially ordered set. 1) A maximal element is an element 7−T with the property that there is no larger element in T , that is :−Tß7Ÿ: Ê 7œ: A maximum ()largest or top element 7−T is an element for which TŸ7 2) A minimal element is an element 8−T with the property that there is no smaller element in T , that is :−Tß:Ÿ8 Ê :œ8 A minimum ()smallest or bottom element 8T in is an element for which 8ŸT Definition () Upper and lower bounds Let ÐT ß Ÿ Ñ be a partially ordered set. Let WT be a subset of . 1) An element ?−T is an upper bound for W if WŸ? The smallest upper bound ?W for , if it exists, is called the least upper bound or join of and is denoted by lub or . Thus, has the WÐWÑ1W? property that WŸ? and if WŸB then ?ŸB . The join of a finite set W œÖ+"8 ßáß= × is also denoted by lubÖ+"8" ßáß+ × or + ”â”+ 8. 2) An element j−T is a lower bound for W if jŸW The largest lower bound jW for , if it exists, is called the greatest lower bound or meet of and is denoted by glb or . Thus, has the WÐWÑ3Wj property that jŸW and if BŸW then BŸj . The meet of a finite set W œÖ+"8 ßáß+ × is also denoted by glbÖ+"8" ßáß+ × or + •â•+ 8. Preliminaries 5

Note that the join of the empty set g is, by definition, the least upper bound of the elements of gT . But every element of is an upper bound for the elements of gT and so the least upper bound is the minimum element of , if it exists. Otherwise g has no join. Similarly, the meet of the empty set is the greatest lower bound of gTg and since all elements of are lower bounds for , the meet of gT is the maximum element of , if it exists.

Now we can state Zorn's lemma, which gives a condition under which a partially ordered set has a maximal element.

Theorem 1.2 ()Zorn's lemma If T is a partially ordered set in which every chain has an upper bound, then T has a maximal element.

Zorn's lemma is equivalent to the axiom of choice. As such, it is not subject to proof from the axioms of ZF set theory. Also, Zorn's lemma is equivalent to the well-ordering principle. A well ordering on a nonempty set \ is a total order on \\ with the property that every nonempty subset of has a least element.

Theorem 1.3 ()Well-ordering principle Every nonempty set has a well ordering. Order-Preserving and Order-Reversing Maps A function 0ÀT Ä between partially ordered sets is order preserving (also called monotone or isotone ) if BŸC Ê 0BŸ0C and an order embedding if BŸC Í 0BŸ0C Note that an order embedding is injective, since 0B œ 0C implies both 0B Ÿ 0C and 0CŸ0B , which implies that BŸC and CŸB , that is, BœC . A surjective order-embedding is called an order isomorphism.

Similarly, a function 0ÀT Ä is order reversing (also called antitone ) if BŸC Ê 0B 0C and an order anti-embedding if BŸC Í 0B 0C An order anti-embedding is injective and if it is surjective, then it is called an order anti-isomorphism. Chain Conditions and Finiteness The chain conditions are a form of finiteness condition on a poset. 6 Fundamentals of Group Theory

Definition Let T be a poset. 1)( T has the ascending chain condition ACC) if it has no infinite strictly ascending sequences, that is, for any ascending sequence

:Ÿ:Ÿ:Ÿâ"#$

there is an index 8:œ:5 ! such that 85 8 for all . 2)( T has the descending chain condition DCC) if it has no infinite strictly descending sequences, that is, for any descending sequence

: : : â"#$

there is an index 8:œ:5 ! such that 85 8 for all . 3)( TT has both chain conditions BCC) if has the ACC and the DCC.

The following characterizations of ACC and DCC are very useful.

Definition Let T be a poset. 1) TT has the maximal condition if every nonempty subset of has a maximal element. 2) TT has the minimal condition if every nonempty subset of has a minimal element.

Theorem 1.4 Let T be a poset. 1) T has the ACC if and only if it has the maximal condition. 2) T has the DCC if and only if it has the minimal condition. Proof. Suppose TW has the ACC and let ©T= be nonempty. Let ""−W . If = is maximal we are done. If not, then we can pick =−W## such that ==" . Continuing in this way, we either arrive at a maximal element in W or we get a strictly increasing ascending chain that does not become constant, which contradicts the ACC. Hence, TT has the maximal condition. Conversely, if has the maximal condition then any ascending sequence in T has a maximal element, at which point the sequence becomes constant. The proof of part 2) is similar.

A poset can express “infinitness” by spreading vertically, via an infinite chain or by spreading horizontally, via an infinite antichain. The next theorem shows that these are the only two ways that a poset can express infiniteness. It also says that if a poset has an infinite chain, then it has either an infinite ascending chain or an infinite descending chain. This theorem will prove very useful to us as we explore chain conditions on subgroups of a group.

Theorem 1.5 Let T be a poset. 1) The following are equivalent: a) T has no infinite chains. b) T has both chain conditions. Preliminaries 7

If these conditions hold, then for any +, in T , there is a maximal finite chain from +, to . 2) The following are equivalent: a) T has no infinite chains and no infinite antichains. b) T is finite. Proof. It is clear that 1a) implies 1b). For the converse, suppose that T has BCC and let be an infinite chain. The ACC implies that has a maximal element BÏ", which must be maximum in since is totally ordered. Then ÖB"× is an infinite chain and we may select its maximum element BB#" . Continuing in this way gives an infinite strictly descending chain, a contradiction to the DCC. Hence, 1a) and 1b) are equivalent.

If 1a) and 1b) hold, then since Ð+ß ,Ó is nonempty, it has a minimal member +" , whence +¡+"" is a maximal chain from + to + . If +" , , then Ð+ß,Ó" has a minimal member +#" and so +¡+ ¡+# is a maximal chain from + to +# . This cannot continue forever and so must produce a maximal finite chain from +, to .

For part 2), assume that TT has no infinite chains or infinite antichains but that is infinite. Using the ACC, we will create an infinite descending chain, in contradiction to the DCC. Since T has the maximal condition, it has a maximal element. Let

` œÖ7±3−M×3 be the set of all maximal elements of TÆ. Denote by B the set of all elements of TBB that are less than or equal to . (This is read: down .) Since ` is a nonempty antichain, it must be finite. Moreover, the ACC implies that

Tœ ÐÆ7Ñ . 3 and so one of the sets, sayÆ73" , must be infinite. The infinite poset

TœÐÆ7ÑÏÖ7×"33"" also has no infinite antichains and no infinite chains. Thus, we may repeat the above process and select an element 7−T3"# such that

TœÐÆ7ÑÏÖ7×#33##

is infinite. Note that 7733"# . Continuing in this way, we get an infinite strictly descending chain.

The presence of a chain condition on a poset T has consequences for meets and joins.

Theorem 1.6 1) Let TT be a poset in which every nonempty finite subset has a meet. If has the DCC, then every nonempty subset of T has a meet. 8 Fundamentals of Group Theory

2) Let TT be a poset in which every nonempty finite subset has a join. If has the ACC, then every nonempty subset of T has a join. Proof. For part 1), let W©T be nonempty. The family of all meets of finite subsets of W7 has a minimal member in T7 and the minimality of implies that 7•+œ7 for all +−T , that is, 7Ÿ+ for all +−T . Hence, +œ7−T 4+−W We leave proof of part 2) to the reader. Lattices Many of the partially ordered sets that we will encounter have a bit more structure.

Definition 1) A partially ordered set ÐT ß Ÿ Ñ is a lattice if every two elements of T have a meet and a join. 2) A partially ordered set ÐT ß Ÿ Ñ is a complete lattice if every subset of T has a meet and a join.

Thus, a complete lattice has a maximum element (the join of T ) and a minimum element (the meet of T ).

Note that if 0ÀT Ä is an order isomorphism of the lattices T0 and , then preserves meets and joins, that is,

0:œ0:and 0:œ0: Š‹4433Š‹ 22 33 However, an order embedding need not preserve these operations.

We will often encounter partially ordered sets T for which every subset of elements has a meet. In this case, joins also exist and T is a complete lattice.

Theorem 1.7 Suppose that ÐT ß Ÿ Ñ is a partially ordered set for which every subset of TÐ has a meet. Then TßŸÑ is a complete lattice, where the join of a subset WT of is the meet of all upper bounds for W . Proof. First, note that the meet of the empty set is the maximum element of T and the meet of TT is the minimum element of and so T is bounded. In particular, the join of g exists.

Let WT be a nonempty subset of . The family YW of upper bounds for is nonempty, since it contains the maximum element. We need only show that the meet is the join of . Since for any , that is, any is 7œ3 Y W =ŸY =−W =−W a lower bound for Y , it follows that =Ÿ7 , that is, 7 W . Moreover, if 8 W then 8−Y and so 7Ÿ8 . Hence, 7 is the least upper bound of W . Preliminaries 9

The previous theorem is very useful in many algebraic contexts. In particular, suppose that \\ is a nonempty set and that Y is a family of subsets of that contains both g\ and and is closed under intersection. (Examples are the subspaces of a vector space, the subgroups of a group, the ideals in a ring, the subfields of a field, the sublattices of a latttice and so on.) Then Y is a complete lattice where the join of any subfamily ZY of is the intersection of all members of YZ containing the members of . Note that this join need not be the union of ZY, since the union may not be a member of . However, if the union of Z is a member of YZ , then it will be the join of .

Example 1.8 1) The set ‘ of real numbers, with the usual binary relation Ÿ , is a partially ordered set. It is also a totally ordered set. It has no maximal elements. 2) The set  œ Ö!ß "ß á × of natural numbers, together with the binary relation of divides, is a partially ordered set. It is customary to write 8±7 to indicate that 87 divides . The subset W of  consisting of all powers of # is a totally ordered subset of  , that is, it is a chain in . The set T œ Ö#ß %ß )ß $ß *ß #(× is a partially ordered set under± . It has two maximal elements, namely ) and #( . The subset œ Ö#ß $ß &ß (ß ""× is a partially ordered set in which every element is both maximal and minimal. The partially ordered set  is a complete lattice but the set of all positive integers under division is a lattice that is not complete. 3) Let WÐ be any set and let c WÑW be the power set of , that is, the set of all subsets of WÐWÑ . Then c , together with the subset relation © , is a complete lattice. Sublattices The subject of sublattices requires a bit of care, since a nonempty subset W of a lattice PP inherits the order of but not necessarily the meets and joins of P . That is, the meet of a subset XW of may be different when X is viewed as a subset of WX than when is viewed as a subset of P .

For example, let P œ Ö"ß #ß $ß 'ß "#× under division and let W œ P Ï Ö'×. Then PW and are both lattices under the same partial order. However, in P we have #”$œ' and in W we have #”$œ"# . Let us use the term P -meet to refer to the meet in P , and similarly for join.

Definition Let PQ be a lattice and let ©PP be a nonempty subset of . 1) QPQ is a sublattice of if the -meet of any finite nonempty subset W©Q exists and is the same as the PW -meet of , and similarly for join, that is, if Wœ W and Wœ W 44QP 22 QP 2) If PQ is a complete lattice, then is a complete sublattice of P if theQ - meet of any subset W©Q exists and is the same as the PW -meet of , and 10 Fundamentals of Group Theory

similarly for join, that is, if

Wœ W and Wœ W 44QP 22 QP Theorem 1.9 1) A nonempty subset TP of a lattice is a sublattice of P if and only if the P - meet and the PE -join of any finite nonempty subset ©T are in T . 2) A nonempty subset TP of a complete lattice is a complete sublattice of P if and only if the PP -meet and the -join of any subset E©TT are in . Equivalence Relations The concept of an equivalence relation plays a major role in mathematics.

Definition Let Wµ be a nonempty set. A binary relation on W is called an equivalence relation on W if it satisfies the following conditions: 1)( Reflexivity ) For all +−W , +µ+ 2)( Symmetry ) For all +ß , − W , +µ, Ê ,µ+ 3)( Transitivity ) For all +ß ,ß - − W , +µ,ß,µ- Ê +µ- Definition LetµW be an equivalence relation on . For +−W , the set of all elements equivalent to + is denoted by Ò+ÓœÖ,−W±,µ+× and is called the equivalence class of + .

Theorem 1.10 LetµW be an equivalence relation on . Then 1) , − Ò+Ó Í + − Ò,Ó Í Ò+Ó œ Ò,Ó 2) For any +ß , − W , we have either Ò+Ó œ Ò,Ó or Ò+Ó ∩ Ò,Ó œ g.

Definition A partition of a nonempty set Wœ is a collection c ÖE3 ±3−M× of nonempty subsets of W , called the blocks of the partition, for which 1) E∩Eœg34 for all 3Á4 2) Wœ-3−M E3 A system of distinct representatives, abbreviated SDR , for a partition c is a set consisting of exactly one element from each block of c . In various contexts, a system of distinct representatives is also called a transversal for c or a set of canonical forms for c .

The following theorem sheds considerable light on the concept of an equivalence relation. Preliminaries 11

Theorem 1.11 1) Let µW be an equivalence relation on a nonempty set . Then the set of distinct equivalence classes with respect toµ are the blocks of a partition of W. 2) Conversely, if c is a partition of Wµ , the binary relation defined by +µ, if + and , lie in the same block of c is an equivalence relation on W , whose equivalence classes are the blocks of c. This establishes a one-to-one correspondence between equivalence relations on WW and partitions of .

The most important problem related to equivalence relations is that of finding an efficient way to determine when two elements are equivalent. Unfortunately, in most cases, the definition does not provide an efficient test for equivalence and so we are led to the following concepts.

Definition Let µW be an equivalence relation on a nonempty set . A function 0ÀW Ä X, where X is any set, is called an invariant of the equivalence relation if it is constant on the equivalence classes, that is, if +µ, Ê 0Ð+Ñœ0Ð,Ñ

A function 0ÀW Ä X is called a complete invariant if it is constant and distinct on the equivalence classes, that is, if +µ, Í 0Ð+Ñœ0Ð,Ñ

A collection Ö0"8 ßá ß0 × of invariants is called a complete system of invariants if

+µ, Í 0Ð+Ñœ0Ð,Ñ33 for all 3œ"ßáß8 Definition Let µW be an equivalence relation on a nonempty set . A subset G©W is said to be a set of canonical forms for the equivalence relation if G is a system of distinct representatives for the partition consisting of the equivalence classes, that is, if for every =−W , there is exactly one -−G such that -µ=.

A set of canonical forms determines equivalence since +ß , − W are equivalent if and only if their corresponding canonical forms are equal. Of course, this will be a practical solution to the problem of equivalence only if there is a practical way to identify the canonical form associated with each element of W . Often, canonical forms provide more of a theoretical tool than a practical one. 12 Fundamentals of Group Theory

Cardinality Two sets WX and have the same cardinality , written WœX kk kk if there is a bijective function (a one-to-one correspondence) between the sets. If WX is in one-to-one correspondence with a subset of , we write WŸX . If W kk kk is in one-to-one correspondence with a proper subset of XX but not with itself, then we write WX . The second condition is necessary, since, for instance, kk kk ™ is in one-to-one correspondence with a proper subset of and yet  is also in one-to-one correspondence with ™™ itself. Hence, œ . kk kk This is not the place to enter into a detailed discussion of cardinal numbers. The intention here is that the cardinality of a set, whatever that is, represents the “size” of the set. It is actually easier to talk about two sets having the same, or different, cardinality than it is to define explicitly the cardinality of a given set (a cardinal number is a special kind of ordinal number).

For us, it is sufficient simply to associate with each set W a special kind of set known as a cardinal number, denoted by WÐWÑ or card , that is intended to measure the size of the set. In the case of finitekk sets, the cardinality is the integer that equals the number of elements in the set.

Definition 1) A set is finite if it can be put in one-to-one correspondence with a set of the form ™8 œ Ö!ß "ß á ß 8  "×, for some nonnegative integer 8 . A set that is not finite is infinite . 2)( The cardinal number of the set  of natural numbers is i! read “aleph nought”) , where i is the first letter of the Hebrew alphabet. Hence, ™œœœi kk kk kk !

3) Any set with cardinality i! is called a countably infinite set and any finite or countably infinite set is called a countable set. An infinite set that is not countable is said to be uncountable.

Theorem 1.12 1)( Schroder¨ – Bernstein Theorem) For any sets WX and , WŸX and X ŸWÊWœX kk kk kk kk kk kk 2)( Cantor's theorem) If cÐWÑ denotes the power set of W then Wc ÐWÑ kk k k Preliminaries 13

3) If c!ÐWÑ denotes the set of all finite subsets of WW and if is an infinite set, then Wœc ÐWÑ kk k! k Cardinal Arithmetic If WX and are sets, the cartesian product W‚X is the set of all ordered pairs W‚X œÖÐ=ß>ѱ=−Wß>−X× If two sets \] and are disjoint, their union is called a disjoint union and is denoted by \“] More generally, the disjoint union of two arbitrary sets WX and is the set W “ X œ ÖÐ=ß !Ñ ± = − W× ∪ ÖÐ>ß "Ñ ± > − X × This is just a scheme for taking the union of WX and while at the same time assuring that there is no “collapse” due to the fact that the intersection of W and X may not be empty.

Definition Let ,- and denote cardinal numbers. Let WX and be sets for which Wœ,- and Xœ . kk kk 1) The sum ,-W is the cardinal number of the disjoint union “X . 2) The product ,- is the cardinal number of W‚X . 3) The power ,- is the cardinal number of the set of all functions from X to W.

We will not go into the details of why these definitions make sense. (For instance, they seem to depend on the sets WX and , but in fact they do not.) It can be shown, using these definitions, that cardinal addition and multiplication are associative and commutative and that multiplication distributes over addition.

Theorem 1.13 Let ,- , and . be cardinal numbers. Then the following properties hold: 1)( Associativity ) ,-.,-.,-.,-.ÐÑœÐÑ and Ð ÑœÐÑ 2)( Commutativity) ,--,œ and ,--, œ 3)( Distributivity ) ,-ÐÑœ . ,-  ,. 14 Fundamentals of Group Theory

4)( Properties of Exponents) a) ,,,-. œ - . b) ÐÑœ,,-. -. c) ÐÑœ,-... , -

On the other hand, the arithmetic of cardinal numbers can seem a bit strange, as the next theorem shows.

Theorem 1.14 Let ,- and be cardinal numbers, at least one of which is infinite. Then ,-,-œ œmax Öß× ,- It is not hard to see that there is a one-to-one correspondence between the power set cÐWÑ of a set W and the set of all functions from W to Ö!ß "× . This leads to the following theorem.

Theorem 1.15 For any cardinal , 1) If Wœ,c then ÐWÑœ#, kk k k 2) , #,

We have already observed that  œi!! . It can be shown that i is the smallest infinite cardinal, that is, kk

,,i0 Ê is a natural number It can also be shown that the set ‘ of real numbers is in one-to-one correspondence with the power set cÐÑ of the natural numbers. Therefore,

‘ œ#i! kk The set of all points on the real line is sometimes called the continuum and so #-i! is sometimes called the power of the continuum and denoted by .

The previous theorem shows that cardinal addition and multiplication have a kind of “absorption” quality, which makes it hard to produce larger cardinals from smaller ones. The next theorem demonstrates this more dramatically.

Theorem 1.16 1) Addition applied a positive countable number of times or multiplication applied a finite number of times to the cardinal number ii!! yields . Specifically, for any nonzero 8− , we have 8 i†iœi!! !and iœi! ! Preliminaries 15

2) Addition and multiplication applied a positive countable number of times to the cardinal number ##ii!! yields . Specifically,

ii!! iii!!! i†#œ#! and Ð#Ñ œ# Using this theorem, we can establish other relationships, such as

iiiii!!!!! # Ÿ Ði! Ñ Ÿ Ð# Ñ œ # which, by the Schro¨der–Bernstein theorem, implies that

ii!! Ði! Ñ œ # We mention that the problem of evaluating ,- in general is a very difficult one and would take us far beyond the scope of this book.

We conclude with the following reasonable-sounding result, whose proof is omitted.

Theorem 1.17 Let ÖE5 ± 5 − O× be a collection of sets with an index set of cardinality Oœ,- . If E Ÿ for all 5−O , then kk k5 k

EŸ-, . 5 »»5−O

Miscellanea The following section need not be read until it is referenced much later in the book. If : is prime, then we will have occasion to write an integer α satisfying α ´"mod : in the form α œ",:> where :±,y . However, the case where :œ# and α œ"#. with . odd is exceptional. In this case, we will need to write α œ",#>, where , is odd and > # . This will ensure that > # when :œ#. Accordingly, it will be useful to introduce the following terminology.

Definition Let α ´"mod :. 1) If :œ# and αα œ"#. where . is odd, that is, if ´$mod %, then the :- standard form of α is α œ",:ß> :±,y and > #

2) In all other cases, the :-standard form of α is α œ",:ß> :±,y

Theorem 1.18 Let :. be a prime and let " . Let 9:Ð8Ñ denote the largest exponent /:8 for which / divides . 16 Fundamentals of Group Theory

1) For "Ÿ5Ÿ:. , :. 9œ.9Ð5Ñ:: ”•Œ5 In particular, :. :.5" º Œ5 and if :# and 5 # or if :œ# and 5 $ , then :. :.5# º Œ5 2) If the : -standard form of α is α œ/,:> then for any . ! ,

.. α::œ/ A: .> where :±Ay . Proof. For part 1), write :.... :Ð:"ÑÐ:?Ñ:Ð5"Ñ . œââ Œ55" ? 5" where "Ÿ?Ÿ5". Now, if "Ÿ3Ÿ. , then :33 ±? if and only if : ±:. ? and so

. 8 : :Í8Ÿ.9Ð: 5Ñ ¹ Œ5 The rest follows from the fact that :±5@ implies @Ÿ5" and if :# and 5 # or if :œ# and 5 $ , then :@ ±5 implies @Ÿ5# .

For part 2), if the :œ -standard form for αα is /,:> , then

. : . .... : . α: œÐ/,:Ñ> : œ/ : / :" ,: .>  /:55 , : >5 5œ# Œ5 where the terms in the final sum are !.œ!:# if . If , then part 1) implies that the 5: th term in the final sum is divisible by to the power .5#>5œ.>"Ò">Ð5"Ñ5Ó .>" If :œ# , then > # and so the 5 th term in the final sum is divisible by : to the power Preliminaries 17

.5">5œ.>"Ò>Ð5"Ñ5Ó .>"

Hence, in both cases, the final sum is divisible by :.>" and so

... .. α:œ/ : / : " ,: .> @: .>" œ/ : : .> Ð/ : " ,@:Ñ where :±Ð/y :". ,@:Ñ.

Chapter 2 Groups and Subgroups

Operations on Sets We begin with some preliminary definitions before defining our principal object of study. For a nonemtpy set \8 , the -fold cartesian product is denoted by \œ\‚â‚\8 ðóóóñóóóò 8 factors Definition Let \8 be a nonempty set and let be a natural number. 1) For 8 " , an 8-ary operation on \ is a function 0À\8 Ä \ 2) A "0 -ary operation À\Ä\ is called a unary operation on \ . 3) A #0 -ary operation À\‚\Ä\ is called a binary operation on \ . 4) A nullary operation on \\ is an element of . An 88 -ary operation, for any natural number , is referred to as a finitary operation.

It is often the case that the result of applying a binary operation is denoted by juxtaposition, writing +, in place of 0Ð+ß ,Ñ .

Definition If 0À\8 Ä \ is an 8 -ary operation on \] and if is a nonempty 88 subset of \0 , then the restriction of to ]0 is a map l] 8 À]Ä\. We say that 8 ]0 is closed under the operation if 0l] 8 maps ] into ] . For a nullary operation B−\ , this means that B−] . Groups We are now ready to define our principal object of study.

Definition A group is a nonempty set K , called the underlying set of the group, together with a binary operation on K , generally denoted by juxtaposition, with the following properties:

S. Roman, Fundamentals of Group Theory: An Advanced Approach, 19 DOI 10.1007/978-0-8176-8301-6_2, © Springer Science+Business Media, LLC 2012 20 Fundamentals of Group Theory

1)( Associativity ) For all +,-−K , , , Ð+,Ñ- œ +Ð,-Ñ

2)( Identity ) There exists an element "−K , called the identity element of the group, for which "+ œ +" œ + for all +−K . 3)( Inverses ) For each +−K , there is an element +" −K , called the inverse of + , for which ++"" œ + + œ " Two elements +ß , − K commute if +, œ ,+ A group is abelian , or commutative, if every pair of elements commute. A group is finite if the underlying set K is a finite set; otherwise, it is infinite . The order of a group is the cardinality of the underlying set K9 , denoted by ÐKÑ or K . kk It is customary to use the phrase “KK is a group” where is the underlying set when the group operation under consideration is understood.

We leave it to the reader to show that the identity element in a group K is unique, as is the inverse of each element. Moreover, for +ß , − K , Ð+Ñœ+" " and Ð+,Ñœ,+" " " In a group K , exponentiation is defined for integral exponents as follows: "8œ!if ÝÚ Ý +â+if 8  ! +œ8 î Û 8 factors Ý Ð+8 Ñ " if 8  ! Ü When K is abelian, the group operation is often (but not always) denoted by and is called addition , the identity is denoted by ! and called the zero element of the group and the inverse of an element +−K is denoted by + and is called the negative of + . In this case, exponents are replaced by multiples: !8if œ! ÝÚ Ý +â+if 8! 8+ œ ðóóñóóò Û 8 terms Ý Ð8+Ñif 8  ! Ü Groups and Subgroups 21

The Order of an Element Definition Let K be a group. 1) If +−K , then any integer 8 for which +œ"8 is called an exponent of + . 2) The smallest positive exponent of +−K , if it exists, is called the order of + and is denoted by 9Ð+Ñ . If + has no exponents, then + is said to have infinite order. An element of finite order is said to be periodic or torsion . 3) An element of order # is called an involution .

Theorem 2.1 Let K+ be a group. If −K has finite order 9Ð+Ñ , then the exponents of +9 are precisely the integral multiples of Ð+Ñ . Proof. Let 8œ9Ð+Ñ . Any integral multiple of 8 is clearly an exponent of + . Conversely, if +7 œ" , then 7œ;8< where !Ÿ<8 . Hence, "œ+7;8<;8<< œ+ œ+ + œ+ and so the minimality of 8< implies that œ!7 , whence œ;8 is an integral multiple of 8 .

Involutions arise often in the theory of groups. Proof of the following result is left as an exercise.

Theorem 2.2 A group in which every nonidentity element is an involution is abelian.

As we will see in a moment, a group may have elements of finite order and elements of infinite order.

Definition A group KK is said to be periodic or torsion if every element of is periodic. A group that has no periodic elements other than the identity is said to be aperiodic or torsion free.

It is not hard to see that every finite group is periodic. On the other hand, as we will see, there are infinite periodic groups, that is, an infinite group need not have any elements of infinite order. Examples Here are some examples of groups.

Example 2.3 The simplest group is the trivial group K œ Ö"× , which contains only the identity element. All other groups are said to be nontrivial . 22 Fundamentals of Group Theory

Example 2.4 The integers ™ form an abelian group under addition. The identity is ! . The rational numbers  form an abelian group under addition and the nonzero rational numbers ‡ form an abelian group under multiplication. A similar statement holds for the real numbers ‘‚ and the complex numbers (and indeed for any field J ).

Example 2.5 ( Cyclic groups) If +K is a formal symbol, we can define a group to be the set of all integral powers of + :

3 GÐ+ÑœÖ+±3−×∞ ™ where the product is defined by the formal rules of exponents: ++34 œ+ 34

This group is also denoted by GØ+Ù∞ or and is called the cyclic group generated by +Ø . The identity of +Ù is "œ+! .

We can also create a finite group GÐ+Ñ8 of positive order 8 by setting !# 8" G8 Ð+Ñ œ Ö" œ + ß +ß + ß á ß + × where the product is defined by addition of exponents, followed by reduction modulo 8: ++34 œ+ Ð34Ñmod 8 This defines a group of order 88 , called a cyclic group of order . The inverse 5Ð5Ñ8mod of + is + . The group G88 Ð+Ñ is also denoted by G or Ø+Ù . Note that for 5 any integer 5+ , the symbol refers to the element of G8Ð+Ñ obtained by multiplying together 5+ copies of . Hence, +œ+558mod and so for any integers 54 and , ++545848 œ+mod + mod œ+ 5848 mod mod œ+ Ð54Ñ8mod œ+ 54

Thus, we can feel free to represent the elements of GÐ+Ñ8 using all integral powers of + and the rules of exponents will hold, although we must remember that a single element of GÐ+Ñ8 has many representations as powers of + . The groups G8∞ Ð+Ñ or G Ð+Ñ are called cyclic groups.

Example 2.6 The set

™8 œ Ö!ß á ß 8  "× of integers modulo a positive integer 88 is a cyclic group of order under addition modulo 8" , generated by the element , since 5−5™8 is the sum of ones. The notation ™™Î8 is preferred by some mathematicians since when : is a Groups and Subgroups 23

prime, the notation ™: is also used for the : -adic integers. However, we will not use it in this way.

If : is a prime, then the set ‡ ™: œ Ö"ß á ß :  "× of nonzero elements of ™: is an abelian group under multiplication modulo : . Indeed, by definition, the set JJ‡ of nonzero elements of any field is a group under multiplication. It is possible to prove (and we leave it as an exercise) that JJ‡ is cyclic if and only if is a finite field.

More generally, the set ‡ of units of a commutative ring with identity is a multiplicative group. In the case of the ring ™8 , this group is ‡ ™™8 œÖ+−8 ±Ð+ß8Ñœ"× ‡ To see directly that this is a group, note that for each +−™8 , there exists ‡ integers BC and such that B+C8œ"B+´"8. Hence, mod , that is, B−™8 is ‡‡ the inverse of + in ™™88 . The group is abelian and it is possible to prove ‡ (although with some work; see Theorem 4.43) that ™8 is cyclic if and only if 8 œ #ß %ß :// or #: , where : is an odd prime.

Example 2.7 ( Matrix groups) The set `7ß8ÐJ Ñ of all 8 ‚ 7 matrices over a field JK is an abelian group under addition of matrices. The set PÐ8ßJÑ of all nonsingular 8‚8 matrices over J is a nonabelian (for 8" ) group under multiplication, known as the general linear group. The set WPÐ8ßJÑ of all 8‚8 matrices over J with determinant equal to " is a group under multiplication, called the special linear group.

Example 2.8 ( Functions ) Let K\ be a group and let be a nonempty set. The set K\\ of all functions from to K is a group under product of functions, defined by Ð01ÑÐBÑ œ 0ÐBÑ1ÐBÑ for all B−\ . The identity in K\ is the function that sends all elements of \ to the identity element "−K . This map is often referred to as the zero map . (We cannot call it the identity map!) Also, the set Y of all bijective functions on K is a group under composition.

Example 2.9 The set

KœÖÐ//ßáѱ/"ß # 3 −™ # × of all infinite binary sequences, with componentwise addition modulo # , is an infinite abelian group that is periodic, since 24 Fundamentals of Group Theory

#Ð/"ß / # ßáÑœÐ!ß!ßáÑ and so every nonidentity element has order # . The External Direct Product of Groups One important method for creating a new group from existing groups is as follows. If KßáßK"8 are groups, then the cartesian product

TœK‚â‚K"8 is a group under componentwise product defined by

Ð+"8"8""88 ßáß+ ÑÐ, ßáß, ÑœÐ+ , ßáß+ , Ñ where +ß,33 −K 3 . The group T is called the external direct product of the groups KßáßK"8 . Although the notation ‚ is often used for the external direct product of groups, we will use the notation

KâK"8}} to distinguish it from the cartesian product as a set .

As an example, the direct product

Z œ G## Ð+Ñ} G Ð,Ñ œ ÖÐ"ß "Ñß Ð+ß "Ñß Ð"ß ,Ñß Ð+ß ,Ñ× of two cyclic groups of order # is called the (Klein ) %-group (and was called the Vierergruppe by Felix Klein in 1884). We will generalize the direct product construction to arbitrary (finite or infinite) families of groups in a later chapter. Symmetric Groups Let \\ be a nonempty set. A bijective function from to itself is called a permutation of \W . The set \ of all permutations of \ is a group under composition, with order \x when \ is finite. Also, W is nonabelian for kk \ \ $. The group W is called the symmetric group or permutation group kk \ on the set \ . The group of permutations of the set M8 œ Ö"ß á ß 8× is denoted by W88 and has order x .

We will study permutation groups in detail in a later chapter, but we want to make a few remarks here for use in subsequent examples. (Proofs will be given later.) If +ßáß+"5 are distinct elements of \ , the expression

Ð+"5 â+ Ñ denotes the permutation that sends ++33" to for 3œ"ßáß5" and sends the last element ++5" to the first element . All other elements of \ are held fixed. This permutation is called a 5W -cycle in \ . For example, in WÖ"ß#ß$ß%× the permutation Ð"$%Ñ sends " to $$, to %%, to " and # to itself. A #-cycle Ð+,Ñ is Groups and Subgroups 25 called a transposition, since it simply transposes +, and , leaving all other elements of \ fixed. We can now see why a permutation group with at least three elements +, , and - is nonabelian, since for example Ð+ ,ÑÐ+ -Ñ Á Ð+ -ÑÐ+ ,Ñ (Composition is generally denoted by juxtaposition as above.)

The support of a permutation 5 −W\ is the set of elements of \ that are moved by 5 , that is, suppÐ55 ÑœÖB−\± BÁB×

Two permutations 57ß−W\ are disjoint if their supports are disjoint. In particular, two cycles Ð+"5 â+ Ñ and Ð, "7 â, Ñ are disjoint if the underlying sets Ö+"5 ßáß+ × and Ö, "7 ßáß, × are disjoint. It is not hard to see that disjoint permutations commute, that is, if 57 and are disjoint, then 5775œ .

It is also not hard to see that every permutation 5 is a product (composition) of pairwise disjoint cycles, the product being unique except for the order of factors and the inclusion of " -cycles. In fact, this is a direct result of the fact that the relation B´Cif5™5 BœC for some 5− is an equivalence relation and thereby induces a partition on M8 . (The reader is invited to write a complete proof at this time or to refer to Theorem 6.1.) This factorization is called the cycle decomposition of 55 . The cycle structure of is the number of cycles of each length in the cycle decomposition of 5 . For example, the permutation 5 œÐ"#$ÑÐ%&'ÑÐ()ÑÐ*Ñ has cycle structure consisting of two cycles of length $# , one cycle of length and one cycle of length " .

It is easy to see that for \ # , any cycle in W\ is a product of transpositions, since kk

Ð+" â+ 8 Ñ œ Ð+ "8 + ÑÐ+ "8" + ÑâÐ+ "$ + ÑÐ+ "# + Ñ and Ð+Ñ œ Ð+ ,ÑÐ+ ,Ñ Hence, the cycle decomposition implies that every permutation is a product of (not necessarily disjoint) transpositions. Although such a factorization is far from unique, we will show that the parity of the number of transpositions in the factorization is unique. In other words, if a permutation can be written as a product of an even number of transpositions, then all factorizations into a 26 Fundamentals of Group Theory product of transpositions have an even number of transpositions. Such a permutation is called an even permutation. For example, since Ð" $ %Ñ œ Ð" %ÑÐ" $Ñ the permutation Ð"$%Ñ is even. Similarly, a permutation is odd if it can be written as a product of an odd number of transpositions. For example, the equation above for Ð+"8 â+ Ñ shows that a cycle of odd length is an even permutation and a cycle of even length is an odd permutation.

One of the most remarkable facts about the permutation groups is that every group has a “copy” that sits inside (is isomorphic to a subgroup of) some permutation group. For example, the Klein %W -group sits inside % as follows: Z œ Ö+ ß Ð" #ÑÐ$ %Ñß Ð" $ÑÐ# %Ñß Ð" %ÑÐ# $Ñ× This is the content of Cayley's theorem, which we will discuss later. Thus, if we knew “everything” about permutation groups, we would know “everything” about all groups! The Order of a Product One must be very careful not to jump to false conclusions about the order of the product of elements in a group. For example, consider the general linear group KPÐ#ß‚ Ñ of all nonsingular #‚# matrices over the complex numbers. Let !" !" Eœ and Fœ Œ"! Œ" " We leave it to the reader to show that EF and have finite order but that their product EF has infinite order. On the other extreme, we have 9Ð++" Ñ œ " regardless of the value of 9Ð+Ñ . Thus, the order of a product of two nonidentity elements can be as small as " or as large as infinity.

On the other hand, the following key theorem relates the order of a power +5 of an element +−K to the order of + . It also tells us something quite specific about the order of the product of commuting elements.

Theorem 2.10 Let K+ be a group and ß,−K . 1) If 9Ð+Ñ œ 8 , then for " Ÿ 5  8 , 8 9Ð+5 Ñ œ gcdÐ8ß 5Ñ In particular, Ø+Ù œ Ø+5 Ù Ígcd Ð9Ð+Ñß 5Ñ œ " Groups and Subgroups 27

2) If 9Ð+Ñ œ 8 and . ± 8 , then 8 9Ð+Ñœ.5 Í 5œ< , where gcdÐ<ß.Ñœ" . 3) If +, and commute, then lcmÐ9Ð+Ñß 9Ð,ÑÑ 9Ð+,Ñlcm Ð9Ð+Ñß 9Ð,ÑÑ gcdÐ9Ð+Ñß 9Ð,ÑÑ ¹¹ In particular, gcdÐ9Ð+Ñß 9Ð,ÑÑ œ " Ê 9Ð+,Ñ œ 9Ð+Ñ9Ð,Ñ

Proof. For part 1), Theorem 2.1 implies that Ð+57 Ñ œ " if and only if 8 ± 57 . But 8578 8±57 Í Í 7 gcdÐ8ß5ѹ¹ gcd Ð8ß5Ñ gcdÐ8ß5Ñ and so the smallest positive exponent 7+ of 5 is 8ÎÐ8ß5Ñgcd . For part 2), according to part 1), the equation 9Ð+5 Ñ œ . is equivalent to 8 œ. gcdÐ8ß 5Ñ which is equivalent to gcdÐ8ß 5Ñ œ 8Î., or 88 gcdÐ. ß 5Ñ œ .. But this holds if and only if 5œ<Ð8Î.Ñ with gcd Ð.ß<Ñœ".

For part 3), let 9Ð+Ñ œα" and 9Ð,Ñ œ . Then Ð+,Ñ"" œ + has order α 9ÐÐ+,Ñ"" Ñ œ 9Ð+ Ñ œ gcdÐßÑα"

But 9ÐÐ+,Ñ" Ñ divides 9Ð+,Ñ and so α 9Ð+,Ñ gcdÐßÑα" ¹ A symmetric argument shows that " 9Ð+,Ñ gcdÐßÑα" ¹ and since these divisors are relatively prime, we have 28 Fundamentals of Group Theory

α" 9Ð+,Ñ gcdÐßÑα"# ¹ But α"α"ÎÐßÑœ# lcm ÐßÑÎÐßÑ α" gcd α".

Corollary 2.11 Let K be a group and let +−K . If 9Ð+Ñœ?@ , where Ð?ß@Ñœ" , then

+œ++"# where +"# ß + − Ø+Ù and 9Ð+" Ñ œ ? and 9Ð+# Ñ œ @ . Proof. Since Ð?ß @Ñ œ " , there exist integers = and > for which =?  >@ œ " . Hence, +œ+=?>@ œ+ =? + >@ Then Ð=?ß@Ñœ" implies that 9Ð+=? Ñœ@ and similarly, 9Ð+>@ Ñœ? . Orders and Exponents Let K+ be a group. We have defined an exponent of −K to be any integer 5 for which +œ"5 . Here is the corresponding concept for subsets of a group.

Definition If W©K is nonempty, then an integer 5 for which =5 œ" for all =−W is called an exponent of WW . If has an exponent, we say that W has finite exponent.

Note that many authors reserve the term exponent for the smallest such positive integer 5W . As with individual elements, if the subset has an exponent, then all exponents of WW are multiples of the smallest positive exponent of .

Theorem 2.12 Let KW be a group. If a nonempty set of K has finite exponent, then the set of all exponents of W is the set of all integer multiples of the smallest positive exponent of W .

For a finite group KÐ , the smallest exponent minexpKÑ is equal to the least common multiple lcmordersÐKÑ of the orders of the elements of K . Thus, if maxorderÐKÑ denotes the maximum order among the elements of K , then maxorderÐKÑ ± minexp ÐKÑ and there are simple examples to show that equality may or may not hold. (The reader is invited to find such examples.) However, in a finite abelian group, equality does hold. Groups and Subgroups 29

Theorem 2.13 1) If K is a finite group, then maxorderÐKÑ ± minexp ÐKÑ œ lcmorders ÐKÑ 2) If K is a finite abelian group, then all orders divide the maximum order and so maxorderÐKÑ œ minexp ÐKÑ œ lcmorders ÐKÑ and KÐ is cyclic if and only if minexp KÑœ9ÐKÑ. Proof. For the proof of part 2), let +−K have maximum order 7 . Suppose to the contrary that there is a ,−K for which 9Ð,ѱ7y . We will find an element of K7 of order greater than , which is a contradiction. Since 9Ð,Ñy±7 , there is a prime : for which 9Ð+Ñœ7œ:@43and 9Ð,Ñœ:? where :±?yy , :±@ and 34 . Then

4 7 9Ð+:? Ñ œand 9Ð, Ñ œ :3 :4 Since K is abelian and these orders are relatively prime, we have

4 9Ð+:? , Ñ œ 7: 34  7 as promised. The last statement of the theorem follows from the fact that a finite group K9 is cyclic if and only if it has an element of order ÐKÑ . Conjugation Let K+ be a group. If ß,−K , then the element ,œ+,++" is called the conjugate of ,+ by . The conjugacy relation is the binary relation on K defined by +´,if ,œ+B for some B−K and if +´, , we say that + and , are conjugate . Conjugacy is an equivalence relation on K+ , since an element is conjugate to itself and if œ,BB then ,œ+" and finally, if ,œ+BC and -œ, , then -œ,CBC œ+ Note that some authors define the conjugate of ,++,+ by as " , so care must be taken when reading other literature.

The function #+ÀK ÄK defined by + #+BœB 30 Fundamentals of Group Theory is called conjugation by + . Conjugation is very well-behaved: It is a bijection and preserves the group operation, in the sense that

###+++ÐBCÑ œ Ð BÑÐ CÑ

Thus, in the language of a later chapter, #+ is a group automorphism. The maps #+ are called inner automorphisms and the set of inner automorphisms is denoted by InnÐKÑ . We will have more to say about InnÐKÑ in later chapters.

Theorem 2.14 Let K be a group and let +ß,ßB−K . Then 1) Conjugation is a bijection that preserves the group operation, that is, ÐB+" Ñ œ ÐB "+ Ñand ÐBCÑ+ œ B + C + for all Bß C − K . 2) The conjugation map satisfies ÐB,+ Ñ œ B +, for all B−K . 3) Conjugacy is an equivalence relation on K . The equivalence classes under conjugacy are called conjugacy classes.

We can also apply conjugation to subsets of KW©K+−K . If and , we write ++ #+WœW œÖ=±=−W× The previous rules generalize to conjugation of sets. In fact, for any WßX © K and +ß , − K , we have ÐWÑ+, œW ,+ and W+ œX + ÍWœX

Conjugation in the Symmetric Group In general, it is not always easy to tell when two elements of a group are conjugate. However, in the symmetric group, it is surprisingly easy.

Theorem 2.15 Let W8 be the symmetric group. 1) Let 5 −W8" . For any 5 -cycle Ð+â+Ñ5 , we have 5 Ð+"5 â+ Ñ œ Ð55 + " â + 5 Ñ

Hence, if 77œ-â-"5 is a cycle decomposition of , then 555 7 œ-"5 â- is a cycle decomposition of 7 5 . 2) Two permutations are conjugate if and only if they have the same cycle structure. Groups and Subgroups 31

Proof. For part 1), we have

5 5+353" Ð+"5 â+ Ñ Ð5 + 3 Ñ œ œ 5+3œ5"

" Also, if ,Á55 +33 for any 3 , then ,Á+ and so 5 " " Ð+â+Ñ,œÐ+â+ÑÐ,ÑœÐ,Ñœ,"55555 "5

5 Hence, Ð+"5 â+ Ñ is the cycle Ð55 +" â + 5 Ñ . For part 2), if 7œ -"7 â- is a cycle decomposition of 7 , then 555 7 œ-"7 â- 5 5 and since --3 is a cycle of the same length as 3 , the cycle structure of 7 is the same as that of 7 .

For the converse, suppose that 57 and have the same cycle structure. If 57 and are cycles, say

57œÐ+â+"8 Ñand œÐ,â,"8 Ñ

- then any permutation -5 that sends +,33 to satisfies œ7 . More generally, if

57œ-â-"7and œ.â."7 are the cycle decompositions of 57 and , ordered so that -5 has the same length as .55 for all , we can define a permutation - that sends the element in the 3 th - position of -3.œ55 to the element in the th position of . Then 57 . The Set Product It is convenient to extend the group operation on a group KK from elements of to subsets of KWXK . In particular, if and are subsets of , then the set product WX (also called the complex product, since subsets of a group are called complexes in some contexts) is defined by WXœÖ=>±=−Wß>−X× As a special case, we write Ö+×W as +W , that is, +W œ Ö+= ± = − W× The set product is associative and distributes over union, but not over intersection. Specifically, for WßXßY © K, WÐXYÑœ ÐWXÑY WÐX∪YÑœWX∪WYand ÐX∪YÑWœXW∪YW WÐX∩YÑ©WX∩WYand ÐX∩YÑW©XW∩YW Also, 32 Fundamentals of Group Theory

+W œ +X Í W œ X Of course, we may generalize the set product to any nonempty finite collection WßáßW"8 of subsets of K by setting

WâWœÖ=â=±=−W×"8 "833 On the other hand, if 5 is a positive integer, then it is customary to let WœÖ=±=−W×55 Thus, in general, WW# is a proper subset of the set product W . Subgroups The substructures of a group are defined as follows.

Definition A nonempty subset LK of a group is a subgroup of K , denoted by LŸK, if L is a group under the restricted product on K . If LŸK and LÁK, we write LK and say that L is a proper subgroup of K . If LßáßL"8 are subgroups of K , we write LßáßL"8 ŸK.

For example, ™Ÿ , since ™ is an abelian group under addition. However, ™: is not a subgroup of ™™ , although it is a subset of and it is a group as well: The issue is that ™: is not a group under ordinary addition of integers.

However, if LL is a subgroup under the first definition above, then satisfies the second definition. To see this, multiplying the equation ""LL œ" L by the " inverse of "KLL in gives "œ"K . Thus, for all 2−L , we have 22œL " " " "œ22KK and so 2L œ2 .

There is another criterion for subgroups that involves checking only closure. Proof is left to the reader.

Theorem 2.16 1) A nonempty subset \K of a group is a subgroup of K if and only if \ is closed under the operations of taking inverses and products, that is, if and only if B−\ Ê B" −\ and Bß C − \ Ê BC − \ 2) A nonempty finite subset \K of a group is a subgroup of K if and only if it is closed under the taking of products. Groups and Subgroups 33

Theorem 2.17 The intersection of any nonempty family of subgroups of a group KK is a subgroup of .

Example 2.18 The set EWW88 of all even permutations in is a subgroup of 8 . To see that E8 is closed under the product, if 57 and are even, then they can each be written as a product of an even number of transpositions. Hence, 57 is also a product of an even number of transpositions and so is in E8 . The subgroup EW88 is called the alternating subgroup of .

For 8 # , the alternating subgroup E88 has order 8xÎ# , that is, E is exactly half the size of W−88 . To see this, note that 55WÐ is odd if and only if "#Ñ is even and 775 is even if and only if Ð" #Ñ is odd. Hence, the map È Ð" #Ñ5 is a self-inverse bijection between E8 and the set of odd permutations.

A group has many important subgroups. One of the most important is the following.

Definition The center ^ÐKÑ of a group K is the set of all elements of K that commute with all elements of K , that is, ^ÐKÑœÖ+−K±+,œ,+ for all ,−K×

A group K is centerless if ^ÐKÑ œ Ö"×. A subgroup LK of is central if L is contained in the center of K .

Two subgroups of a group K can never be disjoint as sets, since each contains the identity of K . However, it will be very convenient to introduce the following terminology and notation.

Definition Two subgroups LO and of a group K are essentially disjoint if L ∩ O œ Ö"× We introduce the notation LìO to denote the set product of two essentially disjoint subgroups and refer to this as the essentially disjoint product of LO and .

Note that if 9ÐLÑ œ 8 and 9ÐOÑ œ 7 , then 9ÐL ì OÑ œ 87

The Dedekind Law The following formula involving the intersection and set product is very handy and we will use it often. The simple proof is left to the reader. 34 Fundamentals of Group Theory

Theorem 2.19 Let K be a group and let EßFßG ŸK with EŸF . 1)( Dedekind law ) EÐF ∩ GÑ œ F ∩ EG 2) E∩GœF∩G and EGœFG Ê EœF Proof. We leave proof of the Dedekind law to the reader. For part 2), E œ EÐE ∩ GÑ œ EÐF ∩ GÑ œ F ∩ EG œ F ∩ FG œ F We leave it to the reader to find an example to show that the condition that EŸF is necessary in Dedekind's law, that is, EÐF∩GÑ is not necessarily equal to EF ∩ EG unless E Ÿ F . Subgroup Generated by a Subset If K+ is a group and −K , then the cyclic subgroup generated by + is the subgroup Ø+Ù œ Ö+5 ± 5 −™ ×

If 9Ð+Ñœ8∞, then +558 œ+ mod and so Ø+Ù œ Ö"ß +ß á ß +8" × Note that Ø+Ù is the smallest subgroup of K containing + , since any subgroup of K+ containing must contain all powers of + .

More generally, if \K is a nonempty subset of , then the subgroup generated by \Ø , denoted by \Ù , is defined to be the smallest subgroup of K containing \\ and is called a generating set for Ø\Ù . Such a subgroup must exist; in fact, Theorem 2.17 implies that Ø\Ù is the intersection of all subgroups of K containing \ . The following theorem gives a very useful look at the elements of Ø\Ù.

Theorem 2.20 Let \K be a nonempty subset of a group and let \" œ ÖB" ± B − \×. Let [œÐ\∪\" Ñ ‡ be the set of all words over the alphabet \∪\" . 1) If we interpret juxtaposition in [K as the group product in and the empty word as the identity in KØ\Ùœ[ , then and so

/" /8 Ø\Ù œ ÖB" âB8 ± B33 − \ß / −™ ß 8  !× ∪ Ö"× Groups and Subgroups 35

2) If K is abelian, then we can collect like factors and so

/" /8 Ø\Ù œ ÖB" âB8 ± B334 − \ß B Á B for 3 Á 4ß / 3 −™ ß 8  !× ∪ Ö"× Proof. It is clear that [©Ø\Ù . It is also clear that [ is closed under product. /""/88"// As to inverses, if AœB"" âB88 −[ then A œB âB −[. Hence, [ŸØ\Ù. However, since \©[ , it follows that Ø\ÙŸ[ and so [œØ\Ù.

Although the previous description of Ø\Ù is very useful, it does have one drawback: Distinct formal words in the set [ may be the same element of the group Ø\Ù , when juxtaposition in [ is interpreted as the group product. For instance, the distinct words BB#" and B are the same group element of Ø\Ù . We will discuss this issue in detail when we discuss free groups later in the book: The matter need not concern us further until then. Finitely-Generated Groups A group KK is finitely generated if œØ\Ù\ for some finite set . If K has a generating set of size 8K , then is said to be 88-generated or to be an - generator group. The Burnside Problem There is a fascinating set of problems revolving around the following question. A finite group K is obviously finitely generated and periodic. In 1902, Burnside [39] asked about the converse: Is a finitely-generated periodic group finite? This is the general Burnside problem.

A negative answer to the general Burnside problem took 62 years, when Golod [40] showed in 1964 that there are infinite groups that are $ -generated and whose elements each have order a power of a fixed prime : (the power depending upon the element). However, this still leaves open some refinements of the general Burnside problem. For example, the Golod groups have elements of arbitrarily large order :8 , that is, they do not have finite exponent, as do finite groups.

The Burnside problem is the problem of deciding, for finite integers 87 and , whether every 87 -generated group of exponent is finite. This problem has been the subject of a great deal of research since Burnside first formulated it in 1902. For example, it has been shown that there are infinite, finitely-generated groups of every odd exponent 7  ''& (Adjan [38], 1979) and of every exponent of the form #75 , where 5 %) (Ivanov [42], 1992).

The restricted Burnside problem, formulated in the 1930's, asks whether or not, for integers 87 and , there are a finite number (up to isomorphism) of finite 87-generated groups of exponent . In 1994, Zelmanov answered this question 36 Fundamentals of Group Theory in the affirmative. For more on the Burnside problem, we refer the reader to the references located at the back of the book. Subgroups of Finitely-Generated Groups A far simpler question related to finitely-generated groups is whether every subgroup of a finitely-generated group is finitely generated. We will show when we discuss free groups that for arbitrary groups this is false: There are finitely- generated groups with subgroups that are not finitely generated. However, in the abelian case, this cannot happen.

Theorem 2.21 Any subgroup of an 8E -generated abelian group is also 8 - generated. In particular, a subgroup of a cyclic group is cyclic. Proof. Let LŸE . The proof is by induction on 8 . If 8œ" , then EœØ+Ù is cyclic. Let 5+ be the smallest positive exponent for which 5 −L . Then Ø+57 Ù Ÿ L. However, if + −L , then 7œ;5< where !Ÿ<5 and so +œ+<7;575; œ+Ð+Ñ −L which can only happen if <œ! , whence +75;5 œÐ+Ñ−Ø+Ù. Thus, LœØ+Ù5 is also cyclic and so the result holds for 8œ" .

Assume the result is true for any group generated by fewer than 8 # elements. Let EœØB" ßáßB 8 Ù and let E8" œØB " ßáßB 8" Ù. By assumption, every subgroup of EÐ8"Ñ8" is -generated, in particular, there exist 2−L3 for which

L ∩E8" œØ2 " ßáß2 8" Ù

/ Now, every 2−L has the form 2œ+B8 where +−E8" and /−™ . If /œ! for all 2−L , then LŸE8" and the inductive hypothesis implies that L is at most Ð8  "Ñ -generated. So let us assume that / Á ! for some 2 − L and let = = be the smallest positive integer for which 2œ,B−L888 and ,−E" .

/ For an arbitrary 2œ+B8 −L, where +−E8" , write /œ;=< where !Ÿ<=. Then ; = ; / ; /;= ; < 28888 2 œ Ð,B Ñ Ð+B Ñ œ +, B œ +, B 8 ; ; Since 28 2−L and +, −E8", the minimality of = implies that <œ! and ; so 22−E8 8", that is, 2−ØEß2Ù8" 8 . Hence,

L ©ØE8" ß2 8 ÙœØ2 " ßáß2 8" ß2 8 Ù©L and so

L œØ2"8"8 ßáß2 ß2 Ù is 8-generated. Groups and Subgroups 37

The Lattice of Subgroups of a Group Let KÐ be a group. The collection sub KÑK of all subgroups of is ordered by set inclusion. Moreover, subÐKÑ is closed under arbitrary intersection and has maximum element KÐ . Hence, Theorem 1.7 implies that sub KÑ is a complete lattice, where the meet of a family Y œÖL±3−M×3 is the intersection Y œLœL 4433 , 3−M 3−M and the join of Y is the smallest subgroup of K that contains all of the subgroups in Y , that is, Y œLœÖWŸK±LŸW for all 3× 2233 , 3−M It is also clear that the join of Y is the subgroup generated by the union of the " LL33's. Specifically, since 3 œL , it follows that the subgroup generated by Y is the set of all words over the union : -L3 ‡ Lœ L œÖ+â+±+−Lß8 !× 2.333333"85 5 3−M 3−M

The join of YY is also denoted by ØÙ and ØL±3−MÙ3 .

Note that, in general, the union of subgroups is not a subgroup. (The reader may find an example in the group of integers.) However, if

LŸLŸâ"# is an increasing sequence of subgroups of K (generally referred to as an ascending chain of subgroups), then it is easy to see that the union is a -L3 subgroup of K . More generally, the union of any directed family of subgroups is a subgroup. (A family WW of subgroups of KE is directed if for every ßF− , there is a G−W for which EŸG and FŸG .)

Theorem 2.22 Let KÐ be a group. Then sub KÑ is a complete lattice, where meet is intersection and join is given by

Lœ L 2.33 3−M¢£ 3−M Also, subÐKÑ is closed under directed unions. Hasse Diagrams For a finite group of small order, we can sometimes describe the subgroup lattice structure using a Hasse diagram, which is a diagram of a partially ordered set that shows the covering relation. 38 Fundamentals of Group Theory

Example 2.23 The Hasse diagrams for the subgroup lattices of the group #$ G% œ Ö"ß +ß + ß + × and the % -group Z œ Ö"ß +ß ,ß +,× are shown in Figure 2.1.

C4 V

{1,a2} {1,a} {1,b} {1,ab}

{1} {1}

S(C4) S(V)

Figure 2.1

Maximal and Minimal Subgroups of a Group The maximal and minimal subgroups of a group play an important role in the theory. At this point, we simply give the definitions.

Definition Let KL be a group and let ŸK . 1) L is minimal if it is minimal in the partially ordered set of all nontrivial subgroups of K () under set inclusion . 2) L is maximal if it is maximal in the partially ordered set of all proper subgroups of K () under set inclusion .

We emphasize that the term maximal subgroup means maximal proper subgroup. Without the restriction to proper subgroups, K would be the only “maximal” subgroup of K . A similar statement can be made for minimal subgroups. Subgroups and Conjugation Since the conjugate of a subgroup is also a subgroup, conjugation sends subÐKÑ to subÐKÑ . In fact, it is an order isomorphism of subÐKÑ .

Theorem 2.24 Let K+ be a group and let −K . The conjugation map + ##++Àsub ÐKÑ Ä sub ÐKÑ defined by L œ L is an order isomorphism on subÐKÑ. Hence, #+ preserves meet and join, that is, + LœL+ ’“,,3 3 and + LœL+ ’“223 3 Groups and Subgroups 39

Proof. It is clear that EŸF Í E++ ŸF and so #+ is an order embedding of KE into itself. But any subgroup of K has the form Eœ##++ Ð+" EÑ and so # is also surjective. The Set Product of Subgroups If LO and are subgroups of a group K , then the set product LO is not necessarily a subgroup of K , as the next example shows.

Example 2.25 In the symmetric group W$ , let L œ Ö++ ß Ð" #Ñ×and O œ Ö ß Ð" $Ñ× Then LO œÖ+ ßÐ"#ÑßÐ"$ÑßÐ"$#Ñ×

# However, Ð"$#Ñ œÐ"#$Ñ is not in LO and so LO is not a subgroup of W$ .

If LO is a subgroup of K , then since LO contains both LO and , we have OL © LO. Conversely, if OL © LO, then LO Ÿ K , since Ð25Ñ" œ5 " 2 " −OL©LO and

Ð25ÑÐ25Ñœ2Ð52Ñ5œ2Ð25Ñ5−LO"" ## ""## "$$# where 2−L33 and 5−O . Thus, LO Ÿ K Í OL © LO Moreover, if OL © LO, then equality must hold, since every B − LO has the form BœÐ25Ñ" œ5 " 2 " ©OL for some 2−L and 5−O . Thus LOœOL.

Theorem 2.26 If LßO Ÿ K , then the following are equivalent: 1) LO Ÿ K 2) OL © LO 3) OL œ LO, that is, L and O permute . In this case, LO œ L ” O is the join of L and O in sub ÐKÑ . The Size of LO There is a very handy formula for the size of the set product LO of two subgroups of a group KL , which holds even if OK is not a subgroup of . Consider the surjective map 0ÀL ‚O Ä LO defined by 40 Fundamentals of Group Theory

0Ð2ß5Ñ œ 25 The inverse map 0L" induces a partition c on ‚O whose blocks are the sets 0" ÐBÑ for B − LO . Hence, there are LO blocks. kk To detemine the size of these blocks, let Bœ25 . Then any element of L‚O can be written in the form Ð2.ß /5Ñ for . − L and / − O and 0Ð2.ß /5Ñ œ B Í 2./5 œ 25 Í ./ œ " Í / œ ." and so 0ÐBÑœÖÐ2.ß.5ѱ.−L∩O×" " Hence, 0ÐBÑœL∩O" ¸¸kk and so LOœL‚OœLOL∩O kkkk k k k kk k as cardinal numbers.

Theorem 2.27 If KL is a group and ßOŸK then LOœLOL∩O kkkk k kk k as cardinal numbers. If LO and are finite subgroups, then LO LO œ kkkk kkL∩O kk The largest proper subgroups of a finite group K are the subgroups of order 9ÐKÑÎ#. The formula in Theorem 2.26 tells us something about how these large subgroups interact with other subgroups of K .

Theorem 2.28 Let KL be a finite group and let ŸK have order 9ÐKÑÎ# . Then any subgroup WK of is either a subgroup of L or else W∩L œ WÎ# kkkk In words, WL lies either completely in or half-in and half-out of L . Also, if +−WÏL, then WœÐW∩LÑ“+ÐW∩LÑ

Proof. If WL+ is not contained in , then there is an −WÏL and so WL ª L “ +L But the latter has size 9ÐKÑ and so WL œ K . Then Groups and Subgroups 41

LWœKL∩W kkkk kkk k implies that Wœ#L∩W. The rest follows easily. kk k k Cosets and Lagrange's Theorem Let LŸK . For +−K , the set +L is called a left coset of LK in . Similarly, the set L+ is called a right coset of LK in . The set of all left cosets of LK in is denoted by KÎL and the set of all right cosets is denoted LÏK . We will refer to left cosets simply as cosets, using the adjectives “left” and “right” only to avoid ambiguity.

The map 0ÀKÎL Ä LÏK defined by 0Ð+LÑ œ L+" is easily seen to be a bijection and so KÎL œ LÏK kkkk Since the multiplication map ..++ÀL Ä+L defined by 2 œ+2 is a bijection, all cosets of a subgroup L have the same cardinality: +L œ L kkkk To see that the distinct left cosets of LK form a partition of , we define an equivalence relation on K by +´,if +Lœ,L Now, +Lœ,L implies that ,−+L . Conversely, if ,−+L , then ,œ+2 for some 2−L and so ,Lœ+2Lœ+L. Hence, + ´ , Í +L œ ,L Í , − +L and so the equivalence class containing ++ is precisely the coset L . Thus, the distinct cosets KÎL form a partition of K . In particular, KœL†KÎL kk kkk k as cardinal numbers. When K is finite, this is the content of Lagrange's theorem.

Theorem 2.29 Let KL be a group and let ŸK . 1) The set KÎL of distinct left cosets of LK in forms a partition of K , with associated equivalence relation satisfying + ´ , Í +L œ ,L Í , − +L This equivalence relation is called equivalence modulo L and is denoted by +´,mod L 42 Fundamentals of Group Theory

or just +´, when the subgroup is clear. Each element ,−+L is called a coset representative for the coset +L , since ,L œ +L . 2) All cosets have the same cardinality and KœL†KÎL kk kkk k as cardinal numbers. 3)( Lagrange's theorem) If K is finite, then K KÎL œ kk kkL kk and so 9ÐLÑ ± 9ÐKÑ In particular, the order of an element +−K divides the order of K .

The converse of Lagrange's theorem fails: We will show later in the book that the alternating group E"#% has order but has no subgroup of order ' .

Note also that +´,mod L Í ," +−L and so, in particular, +´,mod Land +´,mod O Ê +´,mod ÐL∩OÑ

Lagrange's Theorem and the Order of a Product Lagrange's theorem tells us something about the order of certain products 25 in a group even when 25 and do not commute.

Theorem 2.30 Let KL be a group and let and OK be finite subgroups of , with LO Ÿ K. 1) If 2−L and 5−O , then 9Ð25Ñ ± 9ÐLÑ9ÐOÑ In particular, if Ø2ÙØ5Ù Ÿ K is finite, then 9Ð25Ñ ± 9Ð2Ñ9Ð5Ñ 2) If R Ÿ K and 9ÐRÑ is relatively prime to both 9ÐLÑ and 9ÐOÑ , then LR œ OR Ê L œ O Proof. For part 1), Lagrange's theorem implies that 9Ð25Ñ ± 9ÐLOÑ ± 9ÐLÑ9ÐOÑ Groups and Subgroups 43

For part 2), if 2−L then 2œ5+ where 5−O and +−R . Hence 5" 2 œ + − R and so 9Ð+Ñ ± 9ÐOÑ9ÐLÑ and 9Ð+Ñ ± 9ÐRÑ , whence + œ " , that is, 2œ5−O . Hence, LŸO and a symmetric argument shows that LœO. Euler's Formula The Euler phi function 99 is defined, for positive integers 8Ð , by letting 8Ñ be the number of positive integers less than 88 and relatively prime to . The Euler phi function is important in group theory since the cyclic group G88 of order has exactly 9Ð8Ñ generators.

Theorem 2.31 () Properties of Euler's phi function 1) The Euler phi function is multiplicative, that is, if 78 and are relatively prime, then 999Ð78Ñ œ Ð7Ñ Ð8Ñ 2) If :8 is a prime and " , then 9Ð:88" Ñ œ : Ð:  "Ñ These two properties completely determine 9 . Proof. For part 1), consider the 7‚8 matrix " 7" #7" â Ð8"Ñ7" Ô×# 7# #7# â Ð8"Ñ7# ÖÙ ÖÙã EœÖÙ ÖÙ< 7< #7< â Ð8"Ñ7< ÖÙ ÖÙã ÕØ7#7$7â 78 Note that the entries in the

For part 2), 9Ð:8 Ñ is the number of positive integers less than or equal to :8 that are not divisible by :: . The positive integers less than or equal to 8 that are divisible by : are 44 Fundamentals of Group Theory

Ö:†"ß:†#ßáß:†:8" × which is a set of size :Ð8" . Hence, 9 :8Ñœ: 8: 8".

Some simple group theory yields two famous old theorems from number theory.

Theorem 2.32 ()Euler's theorem Let 8+ be a positive integer. If is an integer relatively prime to 8 , then +´"89Ð8Ñ mod This formula is called Euler's formula. ‡ Proof. Since ™98 is a multiplicative group of order Ð8Ñ , Lagrange's theorem ‡ implies that Euler's formula holds for +−™8 and since Ð+58Ñ99Ð8Ñ ´+ Ð8Ñ ´" mod 8 Euler's formula holds for all integers that are relatively prime to 8 .

Corollary 2.33 ()Fermat's little theorem If :Ð is a prime and +ß:Ñœ" , then +´+: mod :

Cyclic Groups Let us gather some facts about cyclic groups.

Theorem 2.34 () Properties of cyclic groups 1)( Prime order implies cyclic) Every group of prime order is cyclic. 2)( Smallest positive exponent) A finite abelian group K is cyclic if and only if minexpÐKÑ œ 9ÐKÑ. 3)( Subgroups ) Every subgroup of a cyclic group is cyclic. a)( Lattice of subgroups: infinite case) If Ø+Ù is infinite, then each power +5 !55 with generates a distinct subgroup Ø+Ù and this accounts for all subgroups of Ø+Ù . b)( Lattice of subgroups: finite case) If 9Ð+Ñ œ 8 then for each . ± 8 , the group Ø+Ù has exactly one subgroup W œ Ø+8Î. Ù of order . and exactly 9Ð.Ñ elements of order . , all of which lie in W . This accounts for all subgroups of Ø+Ù . It follows that for any positive integer 8 ,

8œ9 Ð.Ñ .±8

4)( Characterization by subgroups) If a finite group K8 of order has the property that it has at most one subgroup of each order .±8 , then K is cyclic () and therefore has exactly one subgroup of each order .±8 . Groups and Subgroups 45

5)( Direct products) A direct product

KœK"7}} â K

of finite order is cyclic if and only if each K3 is cyclic and the orders of the factors K3" are pairwise relatively prime. Moreover, if . ßáß.8 are pairwise relatively prime positive integers, then the following hold: a)( Composition ) If Ø+33 Ù is cyclic of order . , then

Ø+"8"8 Ù}} â Ø+ Ù œ ØÐ+ ß á ß + ÑÙ

b)( Decomposition) If KœØ+Ù has order .œ.â."8 , then

Ø+Ù œ Ø+"8 ÙâØ+ Ù

where 9Ð+33 Ñ œ . and Ø+ Ù ∩ Ø+ Ù œ Ö"× 34$ 4Á3

for all 3 . Proof. Part 2) follows from Theorem 2.12. For part 3b), the generators of the cyclic groups of order .. are the elements of order . However, Theorem 2.9 implies that the elements of K. of order are Ö+<Ð8Î.Ñ ± Ð<ß .Ñ œ "× and these all lie in the one cyclic subgroup Ø+8Î. Ù of order . . Hence, there can be no other cyclic subgroups of order . . For the final statement, the sum .±8 9Ð.Ñ simply counts the elements of K by their order.

To prove 4), let HK be the set of all orders of elements of . If +−K has order 9Ð+Ñ œ ., then Ø+Ù is the unique subgroup in K. of order and so all elements of order .Ø must be in +Ù . It follows that there are exactly 9Ð.ÑK elements in of order .−H. Hence, 8 œ99 Ð.Ñ Ÿ Ð.Ñ œ 8 .−H .±8 and so equality holds, whence H8 is the set of all divisors of . In particular, 8−H and so K is cyclic.

For part 5), if each KœØ+Ù33 is cyclic of order ..3 , where the 3 's are pairwise relatively prime, then

9ÐÐ+ ßáß+ ÑÑœlcm Ð. ßáß. Ñœ . œ9ÐKÑ "8 "8$ 3 and so K œ ØÐ+"8 ß á ß + ÑÙ is cyclic. (This also proves part 5a).) Conversely, suppose that K+ is cyclic and for each 33−K , let 46 Fundamentals of Group Theory

s+33 œÐ"ßáß"ß+ß"ßáß"Ñ where +33 is in the th position. The subgroups

Ð3Ñ K œ Ö"×}} â Ö"× }} K3 Ö"× }} â Ö"×

Ð3Ñ where K333 is in the th position are cyclic and if KœØ+Ù.s has order 3 , then KœØ+Ù33 is also cyclic of order .3 . Moreover, . œ 9ÐKÑ œminexp ÐKÑ œ lcm Ð. ß á ß . Ñ $ 3"8 and so the orders .3 are pairwise relatively prime.

For part 5b), since KœØ+Ù is abelian, the product TœØ+ÙâØ+Ù"8 is a subgroup of K+â+−T and since "8 has order

9Ð+"8 â+ Ñ œlcm Ð. " ß á ß . 8 Ñ œ . œ 9ÐØ+ÙÑ it follows that TœK . Finally, if α −Ø+Ù∩ Ø+Ù 34$ 4Á3 then divides as well as the product , which are relatively prime 9Ðα Ñ .34#4Á3. and so α œ" . Homomorphisms of Groups We will discuss the structure-preserving maps between groups in detail in a later chapter, but we wish to introduce a few definitions here for immediate use.

Definition Let KL and be groups. A function 5ÀKÄL is called a group homomorphism () or just homomorphism if 555Ð+,Ñ œ Ð +ÑÐ ,Ñ for all +ß , − K . A bijective homomorphism is an isomorphism. When 55ÀK ÄL is an isomorphism, we write ÀK ¸L or simply K ¸L and say that KL and are isomorphic .

A property of a group is isomorphism invariant if whenever a group K has this property, then so do all groups isomorphic to K . For example, the properties of being finite, abelian and cyclic are isomorphism invariant. More Groups Let us look at a few classes of groups. As we have seen, some groups are given names, for example, the cyclic groups, the symmetric groups, the quaternion group and the dihedral groups (to be defined below). Actually, these are really isomorphism classes of groups. For instance, if K¸W8 , then we might Groups and Subgroups 47 reasonably refer to K as a symmetric group as well. After all, it is the algebraic structure that is important and not the labeling of the elements of the underlying set. Cyclic Groups Theorem 2.33 describes the subgroup lattice structure of a cyclic group. Let us take a somewhat closer look at this structure.

Let GÐ+Ñ88 be cyclic of finite order 8 and let H be the lattice (under division) of positive integers less than or equal to 88 that divide . Then the map 8Î. 55ÀH88 Äsub ÐGÐ+ÑÑ defined by .œG. œØ+ Ù is an order isomorphism from HÐGÐ+ÑÑ88 to sub , since 5 is surjective and

."# ± . Í Ð8Î.#".. Ñ ± Ð8Î. Ñ Í G"# Ÿ G

Thus, subÐG88 Ð+ÑÑ is order isomorphic to H . For example, Figure 2.2 shows the lattices HÐGÐ+ÑÑ#% and sub #% .

24 G

8 12 C8 C12

4 6 C4 C6

2 3 C2 C3 1 {1}

Figure 2.2 For the infinite case, we must settle for an order anti-isomorphism. Let ™ be the lattice of nonnegative integers under division. If KœØ+Ù is an infinite cyclic group, then the map 5™ÀÄÐKÑ5sub defined by 5 5œØ+Ù is an order anti- isomorphism, since 5 is surjective and 5 ± 8 Í Ø+85 Ù Ÿ Ø+ Ù Thus, subÐKÑ is order anti-isomorphic to ™ . The Quaternion Group Let KPÐ#ß‚ Ñ be the general linear group of all nonsingular #‚# matrices over the complex numbers. Let "! 3 ! ! " !3 M œ ßEœ ßF œ ßG œ Œ!" Œ !3 Œ "! Œ 3! Then it is easy to see that EœFœGœEFGœM### 48 Fundamentals of Group Theory and ÐMÑ\ œ \ÐMÑ œ \ for \œEßF or G . These equations are sufficient to determine all products in the set WœÖ„Mß„Eß„Fß„G× In particular, EF œ Gß FG œ Eß EG œ EEF œ F and FE œ FFG œ G GFœEFFœE GEœEFFGœEGœF Thus, WK is a subgroup of PÐ#ß‚Ñ) of order . Any group isomorphic to W is called a quaternion group.

Note that we have defined the quaternion group in such a way that it is clearly a group, since we have defined it as a subgroup WK of the known group PÐ#ß‚Ñ . However, the quaternion group is often defined without mention of matrices. In this case, it becomes necessary to verify that the definition does indeed constitute a group.

On the other hand, we can leverage our knowledge of existing groups (in particular W0 ) by the following simple device: A bijection ÀKÄ\ from a group K\ to a set can be used to transfer the group product from K\ to by setting 0Ð+Ñ0Ð,Ñ œ 0Ð-Ñif +, œ - This makes \0 a group and an isomorphism from K\ to .

Now, the quaternion group is often defined as the set œ Ö"ß 3ß 4ß 5ß "ß 3ß 4ß 5× with multiplication defined so that " is the identity and 3œ4œ5œ345œ"## # (2.35) Ð"ÑB œ BÐ"Ñ œ B for Bœ3ß4ß5 . As with the set W defined above, these rules are sufficient to define all products of elements of . Rather than show directly that this forms a group, that is, rather than verifying directly the associative, identity and inverse properties, we can simply observe that the map 0ÀW Ä defined by Groups and Subgroups 49

0ÐMÑ œ "ß 0ÐMÑ œ " 0ÐEÑ œ 3ß 0ÐEÑ œ 3 0ÐFÑ œ 4ß 0ÐFÑ œ 4 0ÐGÑ œ 5ß 0ÐGÑ œ 5 is a bijection and that 0Ð\Ñ0Ð]Ñ œ 0Ð^Ñ Í \] œ ^ for all \ß] ß^ − W. Hence, the product in is the image of the product in W and so W is a group because is a group. Note that the main savings here is in the fact that we do not need to verify that the product in is associative. Of course, someone had to verify that matrix multiplication is associative, and we do appreciate that effort very much.

Equations (2.35) are equivalent to 3œ4œ5œ"## # 34 œ 5ß 45 œ 3ß 53 œ 4 Ð"ÑB œ BÐ"Ñ œ B for Bœ3ß4ß5 and many authors use these equations to define the quaternion group.

In order to describe the quaternion group, it is not necessary to mention explicitly all four elements " , 3 , 4 and 5 , since " œ 3# and 5 œ 34 . In fact, can also be defined as the group satisfying the following conditions: œØ3ß4Ùß 9ÐÑœ)ß 3 œ"ß%##$ 3 œ4ß 43œ34 To see this, if " ³ 3#, 5 ³ 34and B ³ Ð"ÑB for Bœ"ß3ß4ß5, then œÖ"ß3ß3ß3ß4ß34ß34ß34×#$ # $ œ Ö"ß 3ß "ß 3ß 4ß 5ß 4ß 5× Moreover, since 43 œ 3$ 4 , it follows that 5#$ œ 3434 œ 33 44 œ " and 345 œ 3434 œ 33$# 44 œ 4 œ " Note that the conditions 50 Fundamentals of Group Theory

œØ3ß4Ùß 3Á4ß 9Ð3Ñœ%ß 3 œ4ß## 343œ4 imply that 9ÐKÑ œ ) and so they provide perhaps the most succinct definition of the quaternion group. Subgroups of the Quaternion Group The quaternion group has a simple subgroup lattice. We leave it as an exercise to verify that the Hasse diagram for subÐÑ is given by Figure 2.3. The subgroup Ø"Ù has order # and the other nontrivial proper subgroups have order %.

Q

<-1>

{1}

Figure 2.3

The Dihedral Groups Many groups come from geometry. Here is one of the most famous. First, we need a bit of terminology. A rigid motion of the plane is a bijective distance- preserving map of the plane. If T is a nonempty subset of the plane, then a symmetry of TT is a rigid motion of the plane that sends onto itself.

Now, for 8 # , let T be a regular 8 -gon in the plane, whose center passes through the origin. Figure 2.4 shows the cases 8œ% and 8œ& .

1 1

5 2 4 2

4 3 3 Figure 2.4 (For 8œ# , T is a line segment.) Label the vertices "ßáß8 in clockwise order, with vertex "ZT on the positive vertical axis. Let be the set of vertices of . Groups and Subgroups 51

The symmetry group KT8 of consists of all symmetries of T . It is possible to prove that the symmetries of T are the same as the symmetries of the vertex set ZK and so we may regard 8 as the set of all symmetries of Z .

In fact, each symmetry of ZZ is a permutation of and is uniquely determined by that permutation. Indeed, it is customary to think of the elements of K8 as permutations of the labeling set M88 œ Ö"ß á ß 8×, that is, as elements of W . Here are a few simple facts concerning K8 #8 , for .

1) Since any $$−K8 preserves adjacency, if BœC then $ ÐB"ÑœC" or $ÐB  "Ñ œ C  ". We denote this by writing $À ÐBß B  "Ñ È ÐCß C  "Ñ or $À ÐBß B  "Ñ È ÐCß C  "Ñ

In the former case, we say that $ preserves orientation in the pair ÐBß B  "Ñ and in the latter case, $ reverses orientation. Note that addition and subtraction are performed modulo 8! , but then is replaced by 8 . Hence, for example, ""œ8 and so " and 8 are adjacent. 2) Any $ −K8 is uniquely determined by its value on two adjacent elements ÐBß B  "Ñ of M8 . To see this, we may assume that 8 $ . If $ preserves orientation on ÐBß B  "Ñ , that is, if $À ÐBß B  "Ñ È ÐCß C  "Ñ then since $$ÐB#Ñ is adjacent to ÐB"ÑœC", we have $ ÐB#ÑœC or $$$ÐB  #Ñ œ C  #. But ÐB  #Ñ Á B œ C and so $À ÐB  "ß B  #Ñ È ÐC  "ß C  #Ñ Repeating this argument gives $ÀÐB5ßB5"ÑÈÐC5ßC5"Ñ for all 5 . A similar argument holds if $ reverses orientation, showing that $ÀÐB5ßB5"ÑÈÐC5ßC5"Ñ Note also that if $ preserves orientation for one adjacent pair, then it preserves orientation for all adjacent pairs and so we can simply say that $ either preserves orientation or reverses orientation.

The group K88 contains both the -cycle 3 œÐ"#â8Ñ which represents a clockwise rotation through $'!Î8 degrees, and the permutation 5 that represents a reflection across the vertical axis. If 8 is even, 52 Fundamentals of Group Theory then 88 5 œÐ#8ÑÐ$8"Ñâ # Š‹## and if 8 is odd then 8" 8" 5 œÐ#8ÑÐ$8"Ñâ " Œ##

To see that K−8 is generated by the rotation 35 and the reflection , let $K8 . If $ preserves orientation, then $À Ð8ß "Ñ È Ð5ß 5  "Ñ and so $3œ−ØßÙ5 35. On the other hand, if $ reverses orientation, then $À Ð8ß "Ñ È Ð5ß 5  "Ñ and so the following hold: $5À Ð#ß "Ñ È Ð5ß 5  "Ñ $5À Ð"ß #Ñ È Ð5  "ß 5Ñ $5À Ð8ß "Ñ È Ð5  #ß 5  "Ñ

5" 5" Hence, $5œœ− 3 and so $ 3 5Ø 3ß 5ÙK. Thus 8 œØ5ß 3Ù .

To examine the group structure of K8 , we have 9Ð35 Ñ œ 8and 9Ð Ñ œ # Also, 5353À Ð"ß 8Ñ È Ð"ß 8Ñ and so 53 is an involution, which gives the commutativity rule 35œœ 53" 53 8"

/5 and so every element of KK88 can be written in the form 53 . Thus, can be described succinctly by

K8 œ Ø53 ß Ùß 9Ð 3 Ñ œ 8ß 9Ð 5 Ñ œ 9Ð 53 Ñ œ # It is easy to see that the #8 elements Öß+ 3 ßáß 38" ß 5 ß 53 ßáß 538" × are distinct and so 9ÐK8 Ñ œ #8 .

Another description of KØ8 can be gleaned from the fact that 53ßÙœØ 553ßÙ and so K8 is generated by a pair of involutions whose product has finite order, that is, if 153œ , then Groups and Subgroups 53

K8 œ Ø51 ß Ùß 9Ð 5 Ñ œ 9Ð 1 Ñ œ #ß 9Ð 51 Ñ œ 8 Without reference to geometry, a finite group KK is a dihedral group if is isomorphic to the symmetry group K8 #K8 , for some . Thus, a group is a dihedral group if any of the following equivalent descriptions hold (here 3 and 5 are just symbols):

1) (Common description) KœÖ"œ3!8 ß 3 ßáß 3" ß 5 ß 53 ßáß 53 8" × has order #8 and 358#œ"ß œ"and 3553 œ 8" 2) (Succinct description 1) KœØßÙß9ÐÑœ8 #53 3 and 9ÐÑœ9ÐÑœ#5 53 3) (Succinct description 2) K œ Ø51 ß Ùß 9Ð 5 Ñ œ 9Ð 1 Ñ œ #ß 9Ð 51 Ñ œ 8 # Note that ÐÑÐÑœ5355 53 553 55 3 œ" and so all elements of the form 535 are involutions.

Unfortunately, the dihedral group K#8H of order is denoted by 8 by some authors (reflecting the fact that K8 consists of symmetries of vertices) and by HK#8 by other authors (reflecting the fact that is a group of order #8 ). We will use the notation HH#8 . Also, in view of the development of #8 , we will often refer to 35 as a rotation and as a reflection, even if 3 and 5 are not actually maps.

For 8œ" , the reflection 53 is the identity and the rotation is the transposition Ð" #Ñ and so H## œ Ö+ ß Ð" #Ñ× is the cyclic group G . For 8 œ # , the dihedral group HGG%## is } . For 8 $ , we have proved that the dihedral groups exist, namely, as certain subgroups of WH8# . Note that 8 is nonabelian for 8 $ . Ubiquity of the Dihedral Group Succinct description 2 of the dihedral group shows that dihedral groups occurs quite often and reinforces the idea that a dihedral group need not consist specifically of symmetries of the plane.

Theorem 2.36 Let K+ be a group. If ß,−K are distinct involutions for which 9Ð+,Ñ œ 8  ∞, then the subgroup Ø+ß ,Ù is dihedral of order #8 . 54 Fundamentals of Group Theory

Subgroups of the Dihedral Groups

Let us determine the subgroups of HWŸH#8 . If #8 , then Theorem 2.28 implies that there are two possibilities. If WŸØ33 Ù , then WœØ8Î. Ù for some .±8 . Otherwise, W∩Ø33 ÙœØ8Î. Ù for some .±8 and if 5 is the smallest positive integer for which 535 −W , then WœÐW∩ØÙÑ“35335 ÐW∩ØÙÑ œØ35338Î. Ù“ 5 Ø 8Î. Ù œØ5358Î. ß 3 Ù

Note that since 535 and 53 5 3 8Î.œ9 53 58Î. are involutions and Ð38Î.Ñœ. , it follows that W# is dihedral of order . , generated by the “reflection” 535 and the 8Î. “rotation” 3 . Thus, the subgroups of H#8 fall into two categories: subgroups of ØÙ3 , which are cyclic and the rest, which are dihedral.

Also, for distinct values of 5 in the range !Ÿ58Î., the sets

5 8Î. 8Î. 5 8Î. Wœ.ß5 53 Ø 3 Ù“ØÙœØß 3 53 3 Ù are distinct subgroups of H#8 and this accounts for all of the dihedral subgroups of H#8 . We can now summarize.

Theorem 2.37 The subgroups of the dihedral group H#8 are of two types. For each .±8 , we have 1) the cyclic subgroup ØÙ38Î. of order . and 2) for each !Ÿ58Î., the dihedral subgroup

5 8Î. 5 8Î. 8Î. WœØß.ß5 53 3 ÙœØ 53 3 Ù“ØÙ 3 of order #. . The Symmetric Groups

The lattice of subgroups of the symmetric group W8 is rather complicated. Indeed, a famous theorem of Arthur Cayley [7] from 1854 (to be discussed in detail later), says that for any group KW , the symmetric group K contains a subgroup that is an “exact copy” of K . Thus, a complete description of the subgroup lattice of the symmetric groups would constitute a complete description of all groups, which does not yet exist! The Additive Rationals Let us examine the subgroup lattice of the additive group  of rational numbers. Let ™ denote the set of positive integers. We say that +Î, − is reduced if + and ,, are relatively prime and ! . Let LŸM be nontrivial. Let be the set of integers in LR , let be the set of numerators of the reduced elements of L and let HL be the set of positive denominators of the reduced elements of . Groups and Subgroups 55

If +−R , then +Î,−L for some integer , and so +−M . Thus, MœR . Moreover, if 3M is the smallest positive integer in , then every +−M is an integral multiple of 3 , for if +œ;3< where !Ÿ<3 , then <œ+;3−M and the minimality of 3<œ!RœMœ3 implies that . Thus, ™ .

If +3Î, − L is reduced, then there exist integers ? and @ for which ?+  @, œ ", whence 3Ð?+@,Ñ3?+3 œœ@3−L ,, , Thus, +3 3 −L Í −L ,, for +ß , −™, ,  !. Thus, +3 Lœ +−™ ß,−H œ, ¹ It follows that H3 is closed under products, since if Î, and 3Î- in L are reduced, then 3Î,-# is also reduced and in L . Also, if .−H and /±. , then /−H . Thus, HH is closed under factors as well. It follows that we can describe by describing which prime powers lie in H .

If :H is prime, let :: be the set of all powers of :HH that lie in . Then has one of three forms.

1) If : ± 3 , then H: œ Ö"× since if : − H then 3Î: − L is an integer, contradicting the fact that 3L is the smallest positive integer in . 2) If :±3y and there is a largest integer 7Ð:Ñ for which :7Ð:Ñ −H , then

7Ð:Ñ H: œ Ö"ß á ß : ×

3) If :±3y and there is no largest integer 7Ð:Ñ for which :7Ð:Ñ −H , then

# H: œ Ö"ß :ß : ß á × In case 1) we set 7Ð:Ñ œ ! and in case 3) we set 7Ð:Ñ œ ∞ and so 7Ð:Ñ is defined for all primes. Let :ß:ßá"# be the sequence of all primes and let

7ÐLÑ œ Ð7Ð:"# Ñß 7Ð: Ñß á Ñ

For convenience, we say that a sequence Ð+"# ß+ ßáÑ where +3 is a nonnegative integer or +œ∞35 is acceptable for 3 if : ±3 implies that +5 œ! . Thus, 7ÐLÑ is acceptable for 3 . 56 Fundamentals of Group Theory

Let us adopt the notation Ð+"# ß+ ßáÑŸÐ, "# ß, ßáÑ to mean that +5 Ÿ, 5 for all 5,−H. If , we may write

//"# ,œ::â"# with the understanding that all but a finite number of exponents /5 are zero. Let

/Ð,ÑœÐ/ß/ßáÑ"# Then for any positive integer , , ,−H Í /Ð,ÑŸ7ÐLÑ and so the pair Ð3ß 7ÐLÑÑ completely determines L since +3 Lœ +−™™ ß,− ß/Ð,ÑŸ7ÐLÑ œ, º On the other hand, let 3−™ and let

=œÐ7"# ß7 ßáÑ be acceptable for 3 . Then the set

+3  LœÐ3ß=Ñ +−ß,−ß/Ð,ÑŸ=™™ œ, º is a nontrivial subgroup of  for which 7ÐLÑ œ = . It is clear that L is closed under negatives. Also, if +3Î, − L and -3Î. − L are reduced, then +3 -3 ?3 œ −L ,.lcm Ð,ß.Ñ is reduced and since /Ðlcm Ð,ß .ÑÑ Ÿ = it follows that LL is closed under addition. Thus, the subgroups Ð3ß=Ñ of  correspond bijectively to the pairs Ð3ß =Ñ , where 3 − ™ and where

=œÐ7"# ß7 ßáÑ is acceptable for 3 . *An Historical Perspective: Galois-Style Groups Let us conclude this chapter with a brief historical look at groups. Evariste Galois (1811–1832) was the first to develop the concept of a group, in connection with his research into the solutions of polynomial equations. However, Galois' version of a group is quite different from the modern version we see today. Here is a brief look at groups as Galois saw them (using a bit more modern terminology than Galois used). Groups and Subgroups 57

Consider a table in which each row contains an ordered arrangement of a set \ of distinct symbols (such as the roots of a polynomial), for example +,-./ -+,./ ,-+./ +,-/. -+,/. ,-+/. where \ œÖ+ß,ß-ß.ß/×. Then each pair of rows defines a permutation of \ , that is, a bijective function on \ . Galois considered tables of ordered arrangements with the property that the set E3 of permutations that transform a given row <<3 into the other rows (or into itself) is the same for all rows 3 , that is, EœE34 for all 3ß4 . Let us refer to this type of table, or list of ordered arrangements, as a Galois-style group.

In modern terms, it is not hard to show that a list of ordered arrangements is a Galois-style group if and only if the corresponding set EœE (3 ) of permutations is a subgroup of the symmetric group W\ . To see this, let the permutation that transforms row <<343 to row be 1 ß4 . Then Galois' assumption is that the sets

EœÖ33ß"3ß811 ßáß × are the same for all 33 . This implies that for each ß? and 4 , there is a @ for which 113ß?œ 4ß@. Hence,

113ß? 3ß4œœ−E 11 4ß@ 3ß4 1 3ß@ 3 and so E3 is closed under composition and is therefore a group.

Conversely, if E" is a permutation group, then since " 111113ß4œœÐÑ−E "ß4 3ß" "ß4 "ß3 " it follows that EœE3" for all 3 and so the ordered arrangement that corresponds to E" is a Galois-style group.

Galois appears not to be entirely clear about a precise meaning of the term group, but for the most part, he uses the term for what we are calling a Galois- style group. Galois also worked with subgroups and recognized the importance of what we now call normal subgroups (defined in the next chapter), although his definition is quite different from what we would see today.

When Galois' work was finally published in 1846, fourteen years after he met his untimely death in a duel at the age of 21, the theory of finite permutation groups had already been formalized by Augustin Louis Cauchy (1789–1857), 58 Fundamentals of Group Theory who likewise required only closure under product, but who clearly recognized the importance of the other axioms by introducing notations for the identity and for inverses.

Arthur Cayley (1821–1895) was the first to consider, in 1854, the possibility of more abstract groups and the need to axiomatize associativity. He also axiomatized the identity property, but still assumed that each group was a finite set and so had no need to axiomatize inverses (only the validity of cancellation). It was not until 1883 that Walther Franz Anton von Dyck (1856–1934), in studying the relationship between groups and geometry, made explicit mention of inverses.

It is also interesting to note that Cayley's famous theorem (to be discussed in Chapter 4), to the effect that every group is isomorphic to a permutation group, completes a full circle back to Galois (at least for finite groups)! Exercises 1. Let K be a group. a) Prove that K has exactly one identity. b) Prove that each element has exactly one inverse. 2. Prove that any group K# in which every nonidentity element has order is abelian. 3. Let LŸK and let 1−K have order 8 . Show that if 15 −L for Ð5ß8Ñœ" then 1−L. 4. Show that a finite subset WK of a group is a subgroup if and only if it is closed under products. 5. Show that if +ß , − K then 9Ð+,Ñ œ 9Ð,+Ñ. 6. Let K^ be a group with center KK . Prove that if every element of that is not in ^K has finite order, then is periodic. 7. Show that the center of is equal toÖ"ß "× . 8. Show that in a group of even order, there is an element of order # . (Do not use Cauchy's theorem, if you know it.) Hint : such an element is equal to its own inverse. 9. Find an example of a group K: and three distinct primes ß; and < for which K has elements + and , satisfying 9Ð+Ñ œ : , 9Ð,Ñ œ ; and 9Ð+,Ñ œ < . 10. Prove that a group with only finitely many subgroups must be finite. 11. Let K7 be a finite group of order and let Ð8ß7Ñœ" . Show that every element 1K of has a unique 8 th root 2 , where 2œ18 . 12. Prove that the group  of rational number has no minimal subgroups. 13. a) Find a group KLOL and subgroups and for which O is not a subgroup of K . b) If KœLOP where LßO and P are subgroups of K , does it follow that LßO and P commute (under set product)? 14. Let JJ be a field and let ‡ be the multiplicative group of nonzero elements of J . Groups and Subgroups 59

a) If JJ is a finite field, show that ‡ is cyclic. Hint : Use the fact that a polynomial equation of degree 88 has at most distinct solutions in J . b) Prove that if JJ is an infinite field, then ‡ is not cyclic. Hint : What are the orders of the nonidentity elements? 15. Let W be a nonempty set that has an associative binary operation, denoted by juxtaposition. Show that W is a group if and only if +WœWœW+ for all +−W . 16. Let K be a nonempty set with an associative binary operation. Assume that there is a left identity ""+œ++−KPP , that is, for all and that each element +++ has a left inverse PPP , that is, +œ"K . Prove that is a group " under this operation and that "œ"PP and +œ+ . 17. Let K8 be a finite abelian group of order . Show that the product of all of the elements of KK is equal to the product of all involutions in (or " if K ‡ has no involutions). Apply this to the multiplicative group ™: where : is prime to deduce that Ð:  "Ñx ´ "mod : which is known as Wilson's theorem. 18. Draw a Hasse diagram of the subgroup lattice of a) the symmetric group W$ b) the dihedral group H) . 19. (Dihedral group) Show that 535 is the same as counterclockwise rotation 3". 20. (Dihedral group) Let T# be a regular 8 -gon. Show that we get the same dihedral group if we use an axis of symmetry that goes through two vertices or through the midpoint of opposite sides of T . 21. (Dihedral group) Find the center of the dihedral group H#8 . 22. Let K+ be a group and suppose that ß,−K satisfy +##$œ" and +,+œ, . Prove that ,œ"& . 23. Let  be the additive group of rational numbers. Let Ð3ß Ð85 ÑÑ describe LŸ and let Ð4ßÐ7ÑÑ5 describe OŸ . a) Under what conditions is OŸL ? b) Under what conditions is L cyclic? 24. Let KW be a finite group and and XK be subsets of . Prove that either WXŸK or KœWX . kk kk kk 25. A group K is locally finite if every finitely-generated subgroup is finite. a) Prove that a locally finite group is periodic. b) Prove that if K is abelian and periodic, then it is locally finite. 26. A group K is said to be locally cyclic if every finitely-generated subgroup of K is cyclic. a) Prove that KK is locally cyclic if and only if every pair of elements of generates a cyclic subgroup. b) Prove that a locally cyclic group is abelian. 60 Fundamentals of Group Theory

c) Prove that any subgroup of a locally cyclic group is locally cyclic. d) Prove that a finitely-generated locally cyclic group is cyclic. e) Find an example of a finitely-generated group that is not locally cyclic. f) Show that the subgroup K of the nonzero complex numbers (under multiplication) defined by

5 Kœ /#38Î:1 ±8ß5−™ š› where : is a fixed prime is locally cyclic but not cyclic. g) Prove that if K is locally cyclic, then all nonidentity elements have infinite order or else all elements have finite order. h) Prove that the additive group of rational numbers is locally cyclic. i) Show that any locally cyclic group whose nonidentity elements have infinite order is isomorphic to a subgroup of the additive group  . j) The distributive laws are E”ÐF∩GÑœÐE”FÑ∩ÐE”GÑ E∩ÐF”GÑœÐE∩FÑ”ÐE∩GÑ for Eß Fß G Ÿ K. Prove that each distributive law implies the other. A lattice that satisfies the distributive laws is said to be a distributive lattice. (It is possible to prove that subÐKÑ is a distributive lattice if and only if K is locally cyclic. This is difficult. A complete solution can be found in Marshall Hall's book The Theory of Groups [16].) Ascending Chain Condition A group K satisfies the ascending chain condition () ACC on subgroups if every ascending sequence

LŸLŸâ"# of subgroups must eventually be constant, that is, if there is an 8! such that LœL85 8 for all 5! .

27. A group K satisfies the maximal condition on subgroups if every nonempty collection of subgroups has a maximal member. Prove that a group K satisfies the maximal condition on subgroups if and only if it satisfies the ascending chain condition on subgroups. 28. A group K satisfies the ACC on subgroups if and only if every subgroup of K is finitely generated. Chapter 3 Cosets, Index and Normal Subgroups

We begin this chapter with a more careful look at subgroups, cosets and indices. Cosets and Index The number of cosets of a subgroup plays an important role in group theory.

Definition Let KL be a group. The index of ŸK , denoted by ÐKÀLÑ , is the cardinality of the set KÎL of all distinct left cosets of LK in , that is, ÐK À LÑ œ KÎL kk Recall that LÏK œ KÎL and so the index is also the cardinality of the set of kkkk right cosets of LK in .

It is convenient to extend the quotient and index notation as follows: If LŸK and if \K is any nonempty subset of , then we write \ÎLœÖBL±B−\× and denote the cardinality of \ÎL by Ð\ À LÑ . This is not entirely standard notation, but it is useful. For example, if OKL is a subgroup of , but O is not a subgroup, we may still want to consider the set LÎO œ LOÎO and the index ÐLO À OÑ. We will also have use for the following concept.

Definition Let LŸK . 1) A set consisting of exactly one element from each coset in KÎL is called a left transversal for LK in () or for KÎL . 2) A set consisting of exactly one element from each right coset in LÏK is called a right transversal for LK in () or for LÏK .

Now we can give some important properties of the index.

S. Roman, Fundamentals of Group Theory: An Advanced Approach, 61 DOI 10.1007/978-0-8176-8301-6_3, © Springer Science+Business Media, LLC 2012 62 Fundamentals of Group Theory

Theorem 3.1 Let K be a group. 1) If LŸK , then ÐK À LÑ œ " Í K œ L 2) If \©K is a union of cosets of OŸK , then \œO†Ð\ÀOÑ kk kk Hence, if LßO Ÿ K , then LO œ O † ÐLO À OÑ kkkk In particular, KœO†ÐKÀOÑ kk kk and so if K is finite, then K ÐK À OÑ œ kk O kk 3)( Multiplicativity) If LŸOŸK then ÐKÀLÑœÐKÀOÑÐOÀLÑ as cardinal numbers. Hence, LŸOand ÐKÀLÑœÐKÀOÑ∞ Ê LœO 4) Let LßO Ÿ K . If K is finite or if LO Ÿ K , then ÐKÀOÑœÐLOÀOÑ∞ Ê KœLO 5) If LßO Ÿ K , then ÐLO À OÑ œ ÐL À L ∩ OÑ

6)( Poincaré's theorem) Let LßáßL"8 ŸK and ÐKÀLÑ∞3 for all 3 and let MœL∩â∩L5" 5. Then Poincaré's inequality holds:

ÐK À L"8 ∩ â ∩ L Ñ Ÿ ÐK À L " ÑâÐK À L 8 Ñ

and so, in particular, ÐKÀL"8 ∩â∩L Ñ is also finite. a) The inequality above can be replaced by division if

ML55" ŸK ()3.2 for all 5œ"ßáß8". b) Equality holds in Poincaré's inequality if and only if

ÐMLÀLÑœÐKÀLÑ5 5" 5" 5" for all 5œ"ßáß8". Hence, if K is finite or if () 3.2 holds for all 5 , then equality holds in Poincaré's inequality if and only if Cosets, Index and Normal Subgroups 63

ML55" œK for all 5œ"ßáß8". 7) If a finite group KLOÐ has subgroups and for which KÀLÑ and ÐK À OÑ are relatively prime, then K œ LO and equality holds in Poincaré's inequality, that is, ÐKÀL∩OÑœÐKÀLÑÐKÀOÑ

Proof. We leave proof of part 1) and part 2) to the reader. For part 3), let M be a left transversal for KÎO and let N be a left transversal for OÎL . If + − K , then +œ35 for some 5−O and 5œ42 for some 4−N , whence +Lœ35Lœ342Lœ34L Moreover, 34L œ 3ww 4 L Ê 3O ∩ 3w O Á g Ê 3 œ 3w and so 4Lœ4Lww , whence 4œ4 . Thus, the set Ö34±3−Mß4−N× is a left transversal for KÎL . We leave proof of part 4) for the reader.

For part 5), let MœL∩O and consider the function 0ÀLÎMÄLOÎO defined by 0Ð2MÑ œ 2O. This map is well defined and injective since if " 2ß2−L"# , then 2# 2−L" implies that " " 2Mœ2M"# Í 2## 2" −M Í 2 2" −O Í 2Oœ2O"# Also, 02 is surjective since for any −L and 5−O , 0Ð2MÑ œ 2O œ 25O Hence, ÐL À L ∩ OÑ œ ÐLO À OÑ.

For part 6), we proceed by induction on 88œ# . For , we have

ÐKÀL"# ∩LÑœÐKÀLÑÐL """# ÀL ∩LÑ œÐKÀLÑÐLLÀLÑ""## ŸÐKÀLÑÐKÀLÑ"# and the last inequality can be replaced by division if LL"# ŸK . Moreover, equality holds if and only if ÐLLÀLÑœÐKÀLÑ"# # #. Assume the result is true for L" ßáßL 8" and let M8" œL " ∩â∩L8". Then

ÐKÀM8" ∩L 8 ÑœÐKÀM8" ÑÐM 8" ÀM 8" ∩L 8 Ñ ŸÐKÀLÑâÐKÀL" 8" ÑÐM 8" ÀM 8" ∩L 8 Ñ œÐKÀLÑâÐKÀL"8"8 ÑÐM"88 L ÀL Ñ ŸÐKÀLÑâÐKÀL"8 Ñ and both inequalities can be replaced with division signs if 64 Fundamentals of Group Theory

ML55" ŸK for all 5œ"ßáß8". Moreover, equality holds if and only if

ÐM5" L 5 ÀLÑœÐKÀLÑ 5 5 for all 5 .

For part 7), since each of ÐK À LÑ and ÐK À OÑ divides ÐK À L ∩ OÑ and since these factors are relatively prime, we have ÐKÀLÑÐKÀOѱÐKÀL∩OÑ Hence, Poincaré's inequality is an equality: ÐKÀL∩OÑœÐKÀLÑÐKÀOÑ and so the finiteness of KKœL implies that O .

We have seen (Theorem 2.21) that any subgroup of a finitely-generated abelian group is finitely generated. We can now prove that if K is a finitely-generated group, then any subgroup of finite index is also finitely generated.

Theorem 3.3 Let KL be a finitely-generated group. If is a subgroup of K of finite index, then L is also finitely generated. Proof. Let X œÖ>"7 ßáß> × be a left transversal for KÎL , with >" œ" . Let ÖB"8 ßáßB × be a generating set for K and let " " [ œÖB"8 ßáßB ×∪ÖB"8 ßáßB × Thus, if +−K , then

+œA:" âA for some A−[3"" . But Aœ>= for some >−X and =−L" and so

+œA:#" âA>= We are now prompted to consider how the >A 's and 's commute. For each >−X and A−[ , there exist unique >w −X and 2−L for which A> œ >w 2 LetW©L be the finite set of all such 2 's, as A varies over [ and > varies over X. Note that ="" −W since A>""" œA œ>=. By continually moving the element belonging to X+ forward in the product expression for , we get ww +œ>=:" â= −>ØWÙ

www for some > −X and =3 −W . But if +−L , then +−>ØWÙ©>L implies that >w œ". Hence, LŸØWÙ and since W©L , we have LœØWÙ . Cosets, Index and Normal Subgroups 65

Quotient Groups and Normal Subgroups Let KL be a group and let ŸK . We have seen that the equivalence relation corresponding to the partition KÎL is equivalence modulo L : +´,mod L if +Lœ,L Now, there seems to be a natural way to “raise” the group operation from K to KÎL by defining +L ‡ ,L œ +,L (3.4) Of course, for this operation to make sense, it must be well defined, that is, we must have for all +ß +"" ß ,ß , − K, " " " +"" +−Land , ,−L Ê Ð+,Ñ"" Ð+,Ñ−L

Taking +œ""" and ,œ, gives the necessary condition +−L Ê ," +,−L (3.5) However, this condition is also sufficient, since if it holds, then

" " " " " " ,"" + +, œ Ð, "" + ,"" ÑÐ, " +, ÑÐ, " ,Ñ − L Note that (3.5) is equivalent to each of the following conditions:

1) +L+" © L for all + − K 2) +L+" œ L for all + − K 3) +L œ L+ for all + − K .

Definition A subgroup LK of is normal in K , written LKü , if +L œ L+ for all +−K . If Lü K and LÁK , we write L–K . The family of all normal subgroups of a group KÐ is denoted by nor KÑ .

Thus, the product (3.4) is well defined if and only if LKü . Moreover, if LKü , then for any +ß,−K , +,L œ +,LL œ +L,L that is, +L ‡ ,L œ +L,L In particular, the set product of two cosets of LL is a coset of . Moreover, if the set product of cosets is a coset, that is, if +L,L œ -L for some - − K , then +, − -L and so -L œ +,L , that is, 66 Fundamentals of Group Theory

+L,L œ +,L Let us refer to this as the coset product rule. Finally, if the coset product rule holds, then LKü , since +L+" © +L+ " L œ L for all +−K . Thus, the following are equivalent:

1) The binary operation +L ‡ ,L œ +,L is well defined on KÎL . 2) LKü . 3) The set product of cosets is a coset. 4) The coset product rule holds.

Moreover, if these conditions hold, then KÎL is actually a group under the set product, for it is easy to verify that the set product is associative, KÎL has identity element L+ and that the inverse of L is +"L . Thus, we can add a fifth equivalent condition to the list above:

5) KÎL is a group under set product.

Before summarizing, let us note that the following are equivalent: LKü +L © L+ for all + − K ,−+LÊ,−L+ for all +ß,−K ,−+LÊ," −+ " L for all +ß,−K +´,mod LÊ+" ´, " mod L for all +ß,−K Also, the following are equivalent: The coset product rule holds +L,L © +,L for all +ß, − K +ww −+Lß, −,LÊ+, ww −+,L for all +ß,−L +ww ´+mod Lß, ´, mod LÊ+, ww ´+,mod L Now we can summarize.

Theorem 3.6 Let LŸK . The following are equivalent: 1) The set product on KÎL is a well-defined binary operation on KÎL . 2) The coset product rule +L,L œ +,L holds for all +ß , − L . Cosets, Index and Normal Subgroups 67

3) LK is a normal subgroup of . 4) KÎL is a group under set product, called the quotient group or factor group of KL by . 5) The inverse preserves equivalence modulo L , that is, +´,mod L Ê +" ´, " mod L for all +ß , − K . 6) The product preserves equivalence modulo L , that is, +ww ´+mod Lß , ´,mod L Ê +, ww ´+,mod L for all +ß +ww ß ,ß , − K.

When we use a phrase such as “the group KÎL ” it is the with the tacit understanding that LK is normal in . Note finally that statements 5) and 6) say that equivalence modulo LK is a congruence relation on . A congruence relation ) on an algebraic structure, such as a group, is an equivalence relation that preserves the (nonnullary) algebraic operations. Thus, a congruence relation ) on a group K must satisfy the conditions +,)) Ê +" , " and +)) ,ß - . Ê Ð+-Ñ ) Ð,.Ñ Theorem 3.6 shows that these two conditions are actually equivalent for groups. More on Normal Subgroups There are several slight variations on the definition of normality that are often useful. We leave proof of the following to the reader.

Theorem 3.7 Let LŸK . The following are equivalent: 1) LKü 2) L©L+ for all +−K 3) LªL+ for all +−K 4) Every right coset of L is a left coset, that is, for all +−Kß there is a ,−K such that L+ œ ,L 5) Every left coset is a right coset. 6) For all +ß , − K , +, − L Ê ,+ − L 7) If +−K and 2−L , then +2œ2+ww for some 2 −L .

Theorem 3.7 implies that a normal subgroup permutes with all subgroups of K . Hence, the normality of either factor guarantees that the set product LO is a subgroup. 68 Fundamentals of Group Theory

Theorem 3.8 Let LßO Ÿ K . 1) If either LO or is normal in K , then L and O permute and LO œ L ” O In this case, we refer to LO as the seminormal join of LO and . 2) If both LO and are normal in K , then LO is also normal in K and we refer to LO as the normal join of LO and . 3) The fact that LOü K does not imply that either subgroup need be normal. Proof. For part 3), let KœW% . Let L œ Ö+, Ð"#ÑÐ$%Ñß Ð"$ÑÐ#%Ñ, Ð"%ÑÐ#$Ñß Ð#%Ñß Ð"$Ñß Ð"#$%Ñß Ð"%$#Ñ× and

O œ W$ œ Ö+ ß Ð"#Ñß Ð"$Ñß Ð#$Ñß Ð"#$Ñß Ð"$#Ñ× Then L ∩ O œ Ö+ ß Ð"$Ñ× has size # and so LO œ Ð) ‚ 'ÑÎ# œ #% œ W , kk kk% which implies that LO œ W% . But neither subgroup is normal: L is not normal since Ð"%ÑÐ"$ÑÐ"%Ñ œ Ð%$Ñ Â L and O is not normal since Ð"%ÑÐ"#ÑÐ"%Ñ œ Ð%#Ñ Â W$.

Example 3.9 ( The normal subgroups of H#8) We have seen that for .±8 , the subgroups of the dihedral group H#8 are

1) the cyclic subgroup ØÙ38Î. of order . , 2) for each !Ÿ58Î., the dihedral subgroup Wœ535 Ø 3 8Î. Ù“Ø 3 8Î. ÙœØ 53 5 ß 3 8Î. Ù of order #. .

Subgroups of type 1) are normal, since conjugation gives

533ÐÑ 3 <8Î. 3 3 5 œ 3 <8Î. −ØÙ 3 8Î. Let

WœØ5358Î. ß 3 Ù be a subgroup of type 2). Then since ØÙW3ü8Î. , it follows that W is normal if and only if the conjugates of the other generator 535 are in W . Conjugation by 3 gives 3533ÐÑ5" œ 53 #5 which is in Wœ if and only if 33#5 578Î. for some integer 7 , that is, if and only if Cosets, Index and Normal Subgroups 69

8 # ´ 7mod 8 . Multiplying both sides of this by .#.´!88±#. gives mod , that is, and since .±8 , we must have 8œ. or 8œ#. . Conversely, if 8œ. or 8œ#. , then the congruence holds and W is closed under conjugation by 3 .

If 8œ. , then WœH#8 . If 8œ#. , then 9ÐWÑœ8 and 5œ! or 5œ" . If 5œ!, then WœØ53 ß## Ù and if 5œ" , then WœØ533 ß Ù. Moreover, since ÐH#8 À WÑ œ #, both subgroups are normal, as we will prove in Theorem 3.17. Thus, the proper normal subgroups of HØ#8 are the subgroups of 3Ù and, for 8 even, the two subgroups Øß53## Ù and Ø 533 ß Ù of order 8 . Special Classes of Normal Subgroups There are two very important special classes of normal subgroups. Note that LKü# if and only if L is invariant under all inner automorphisms + of K .

Definition Let K be a group. 1) A subgroup LK of is characteristic in K if it is invariant under all automorphisms of KL . If is characteristic in K , we write L«K . ( This is not a standard notation, there being none.) We also write LKL«Kö if and LÁK. 2) A subgroup LK of is fully invariant in K if it is invariant under all endomorphisms of K .

Some of the most important subgroups of a group are characteristic. For example, the center ^ÐKÑ of a group K is characteristic in K . The Lattice of Normal Subgroups of a Group If KÐ is a group, then nor KÑÐ is a subfamily of sub KÑ and is partially ordered by set inclusion as well. Moreover, the intersection of any family Y of normal subgroups of KK is normal in and so the meet of Y in norÐKÑ is the same as the meet of Y in subÐKÑ .

As to join, if Y œÖR±3−M×3 is a nonempty family of normal subgroups of K , then the join of Y in the lattice subÐKÑ is the subgroup

RœÖ+â+±+−Rß8 !× 23−M 33333"85 5 Actually, Theorem 3.7 implies that we can collect factors from the same subgroup R3 and so R œ Ö+ â+ ± + − R ß 3 Á 3 for 5 Á 4ß 8 !× 23−M 3333354"85 5

In particular, if Y œÖRßáßR"7 × is a finite family, then the join takes the particularly simple form 70 Fundamentals of Group Theory

Y œÖ+â+ ±+ −R× 2 "73 3 Now, if +−K , then + Rœ Rœ+ R Š‹2223−M 333−M 3 3−M and so the join of YY in subÐKÑ is normal in K . It follows that the join of in subÐKÑ is equal to the join of Y in norÐKÑ . Thus,

YYœœand YY ,,norÐKÑ subÐKÑ 22norÐKÑ subÐKÑ which holds also when Y is the empty family. Hence, norÐKÑ is a complete sublattice of subÐKÑ .

Theorem 3.10 Let K be a group. 1) The subgroups Ö"× and K are normal in K . 2) If ÖR3 ± 3 − M× is a family of normal subgroups of K , then RRand ,23−M 333−M are normal subgroups of KÐ . Hence, nor KÑ is a complete sublattice of subÐKÑ.

The maximal and minimal normal subgroups of a group play an important role in the theory. We state the definitions here for future use.

Definition Let KL be a group and let ü K . 1) L is minimal normal if it is minimal in the partially ordered set of all nontrivial normal subgroups of K () under set inclusion . 2) L is maximal normal if it is maximal in the partially ordered set of all proper normal subgroups of K () under set inclusion . The Quasicyclic Groups For each prime :Ð , we can now describe an infinite abelian group ™ :∞Ñ , called the :-quasicyclic group that has a very interesting subgroup lattice. Specifically, the lattice of proper subgroups of ™Ð:∞ Ñ consists entirely of a single ascending chain of finite cyclic subgroups

Ö"×Ø+ÙØ+Ùâ"# We begin by looking at the quotient group  7 œ™™ 7ß8− ß!Ÿ78ßÐ7ß8Ñœ" ™ š›8 ¹ The set Cosets, Index and Normal Subgroups 71

7 Wœ 7ß8−™ ß!Ÿ78ßÐ7ß8Ñœ" š›8 ¹ is a left transversal for ™ÎÎ and so we can simply identify ™ with W , under addition modulo "7 . Note that the order of Î8−W is 8 .

Let :Ð be a prime and let ™ :∞ÑW be the subgroup of consisting of those elements of order a power of : , that is, 7 ™™Ð:∞5 Ñœ 7ß5− ß!Ÿ7: ß:±7y œ:5 ¹ If L™ Ð:Ñ∞5 and 7Î:−L for 7! , then there are integers + and , for which +7  ,:5∞ œ " and so in ™ Ð: Ñ , "+7,:+75 œœ−L ::55 : 5 Hence, ! Á 7Î:554 − L Í "Î: − L Í "Î: − L for all 4 Ÿ 5 Thus, since L is proper, there must be a largest integer 8 for which "Î:8 − L . Then "Î:55 − L Í 5 Ÿ 8 Í "Î: − Ø"Î:8 Ù and so L œ Ø"Î:88 Ù is cyclic of order : . Hence, the proper subgroups of ™Ð:∞ Ñ are the subgroups Ö!×  Ø"Î:Ù  Ø"Î:# Ù  â In a later chapter, we will ask the reader to prove that the quasicyclic groups ™Ð:∞ Ñ are the only infinite groups (up to isomorphism) with the property that their proper subgroups consist entirely of a single ascending chain

Ö!×W"# W â

The Normal Closure of a Set If \K is a subset of a group , then the smallest normal subgroup of K that contains \K is the intersection of all normal subgroups of that contain \ . This subgroup is called the normal closure of \K in and we will find it useful to use the following notations for this subgroup:

K \ß Ø\Ùnor and nc Ð\ßKÑ The normal closure has a simple characterization as follows. 72 Fundamentals of Group Theory

Theorem 3.11 If \K is a nonempty subset of a group , then the normal closure of \ is the subgroup ncÐ\ß KÑ œ ØB+ ± B − \ß + − KÙ generated by the conjugates of \K in .

We can extend the notation and define for any \ß] © K , \]C œØB ±B−\ßC−]Ù This allows us to describe the join of two subgroups as a set product.

Theorem 3.12 If LßO Ÿ K , then ØLß OÙ œ LO O

Proof. Since LOOO and permute, it follows that LOŸK . Since LŸLOO and OŸLOOO , it follows that ØLßOÙŸLO. The reverse inclusion is clear. Internal Direct Products

Strong Disjointness of a Family of Normal Subgroups

If Y œÖL±3−M×3 is a nonempty family of normal subgroups of a group K , we write

L³ ÖL±3−Mß4Á3× Ð3Ñ 2 4 for the join of all members of YY except L3 . The members of such a family can enjoy two levels of disjointness. The members of Y can be pairwise essentially disjoint, that is,

L34 ∩ L œ Ö"× for all 3Á4 . Note that Y is pairwise essentially disjoint if and only if 2234 œ" for 233 −L and 2 44 −L with 3Á4 imply that 234 œ2 œ" . Also, the members of a pairwise essentially disjoint family Y commute elementwise, that is, 2234 œ22 43 for all 2 3 −L 3 and 2 4 −L 4 , where 3Á4 .

A stronger level of disjointness comes when each L3 is essentially disjoint from the join of the other members of the family. The following useful definition is not standard in the literature.

Definition We will say that a nonempty family Y œÖL±3−M×3 of normal subgroups of a group K is strongly disjoint if Cosets, Index and Normal Subgroups 73

L3 ∩ LÐ3Ñ œ Ö"× for all 3−M .

The property of being strongly disjoint can be characterized as follows.

Theorem 3.13 Let Y œÖL±3−M×3 be a nonempty family of normal subgroups of a group K . Then the following are equivalent: 1) Y is strongly disjoint 2) If

2â2œ"33"8

where 23344 −L and 3 45 Á3 for 4Á5 , then 234 œ" for all 4 . 3) Every nonidentity can be written, in a unique way except for the +−1Y order of the factors, as a product

+œ233"8 â2

where "Á23344 −L and 3 45 Á3 for 4Á5 . The element 234 is called the 34 th component of +3−MÏÖ3ßáß3×3 . For "8, the th component of +" is .

Proof. If Y is strongly disjoint and 2â2œ"33"8, where the factors are from different subgroups and 8 # , then

" 2333"#8 œ Ð2 â2 Ñ − L 3 " ∩ LÐ3" Ñ œ Ö"×

and so 2œ"33"4 . Repeating this argument gives 2œ" for all 4 and so 1) implies 2).

If 2) holds, then the L3 's commute elementwise. To see that 3) holds, it is clear that every nonidentity has such a product representation. Moreover, if +−1Y + has two such product representations, then we may include additional factors equal to " so that

+œ233"8 â2 œ5 33 "8 â5

where 2ß5−L3344 3 4 and 3Á3 4 5 for 4Á5 and at least one factor on each side is not equal to the identity. Then

" " Ð533"8 233"8 ÑâÐ5 2 Ñ œ "

and so 2) implies that 2œ53344 for all 4 . Hence 2) implies 3). Finally if 3) holds and

+−L3 ∩LÐ3Ñ for some 3+ , then the uniqueness condition implies that œ" . 74 Fundamentals of Group Theory

The next theorem says that strong disjointness is a finitary condition and that if YY is strongly disjoint, then it is relatively easy to check that “ÖO× is also strongly disjoint.

Theorem 3.14 Let Y œÖL±3−M×3 be a nonempty family of normal subgroups of a group K . 1) YY is strongly disjoint if and only if every nonempty finite subset of is strongly disjoint. 2) Let O− nor ÐKÑÏYY. If is strongly disjoint, then Y“ÖO× is strongly disjoint if and only if

Y ∩ O œ Ö"× Š‹2 The following result can be quite useful.

Theorem 3.15 Let Y œÖL±3−M×3 be a nonempty family of normal subgroups of a group KOKN©M . For any ü , there is a that is maximal with respect to the property that the family

YN4œÖL±4−N×∪ÖO× is strongly disjoint. Proof. Write

\YœÖN©M± N is strongly disjoint× Then \\ is nonempty and the union of any chain in is in \ . Hence, Zorn's lemma implies that \ has a maximal member. Internal Direct Products We have already discussed the external direct product KL} of two groups K and L . The internal direct product is defined as follows.

Definition A group K is the ()internal direct product of two normal subgroups LOKœLìO and if . We use the notation KœLO to denote the internal direct product.

The internal direct product KœLO is a decomposition of K into an essentially disjoint product of normal subgroups. Since the factors LO and commute elementwise, the product in K takes the form

Ð2"" 5 ÑÐ2 ## 5 Ñ œ Ð2 "# 2 ÑÐ5 "# 5 Ñ where 2−L33 and 5−O . Thus, the groups LO and have the same level of independence as the factors in an external direct product. Indeed, the map 25 È Ð2ß 5Ñ is an isomorphism from LO to L{ O . Cosets, Index and Normal Subgroups 75

Definition A nontrivial group KK is said to be indecomposable if cannot be written as an internal direct product of two proper subgroups, that is, KœLO Ê LœK or OœK The internal direct product can easily be generalized to arbitrary nonempty families of normal subgroups.

Definition A group K is the ()internal direct sum or ()internal direct product of a family YYœÖL±3−M×3 of normal subgroups if is strongly disjoint and . We denote the internal direct product of by Kœ1YY L3 or Y or when Y œÖLßáßL"8 × is a finite family,

LâL"8

Each factor LK3 is called a direct summand or direct factor of . We denote the family of all direct summands of K,CWf ÐKÑ.

Theorem 3.13 implies the following.

Theorem 3.16 Let Y œÖL±3−M×3 be a nonempty family of normal subgroups of a group K . Then the following are equivalent: 1) Kœ Y 2) Every nonidentity +−K can be written, in a unique way except for the order of the factors, as a product

+œ233"8 â2

where "Á23344 −L and 3 45 Á3 for 4Á5 .

A note on terminology is also in order. If Y œÖL±3−M×3 is a family of normal subgroups of a group , to say that the join is direct in or to say KLK1 3 that the direct sum L3 exists in K is the same as saying that Y is strongly disjoint and that the join is the direct sum. Projection Maps Associated with an internal direct product

Kœ ÖL±3−M×3 is a family of projection maps. Specifically, the 3À th projection map 333KÄL is defined by setting 3333Ð+Ñ to be the 3 th component of + . In this case, can be thought of as an endomorphism of K and then the following hold:

1) ÐÑlœ3+3L33 L 76 Fundamentals of Group Theory

2) 3334œ! if 3Á4 , where ! is the zero map # 3) 3333 is idempotent , that is, 3 œ 3 .

Note also that if 3Á4 , then the images LœÐÑ3344im 33 and LœÐÑim commute elementwise.

We will have much more to say about the direct product in a later chapter. Chain Conditions and Subnormality

Sequences of Subgroups and the Chain Conditions Ordered sequences of subgroups play a key role in group theory. For infinite ascending and descending sequences of subgroups, the issue centers around the chain conditions. Generally speaking, chain conditions are considered a form of finiteness condition on a group and we will study the consequences of the chain conditions on various families of subgroups, such as the family of all subgroups, all normal subgroups or all subnormal subgroups throughout the book. For the record, here is the definition, which will be repeated later.

Definition Let KK be a group and let f be a family of subgroups of . 1)( A group K satisfies the ascending chain condition ACC) on f if every ascending sequence

LŸLŸâ"# of subgroups in f must eventually be constant, that is, if there is an 8! such that LœL85 8 for all 5 ! . In this case, we also say that f has the ACC. 2)( A group K satisfies the descending chain condition DCC) on f if every descending sequence

L L â"# of subgroups in f must eventually be constant, that is, if there is an 8! such that LœL85 8 for all 5 ! . In this case, we also say that f has the DCC. 3)( A group KK satisfies both chain conditions BCC) on f if has the ACC and the DCC on ff . In this case, we also say that has BCC. Finite Series and Subnormality Finite ordered sequences of subgroups of a group are just as important as infinite sequences, but rather than conveying any finiteness condition about a group, they convey structural information about the group and are used to classify groups via this structure. For example, a group K that has a finite sequence

Ö"× œ L!"üüü L â L 8 œ K Cosets, Index and Normal Subgroups 77

of subgroups for which each quotient LÎL5" 5 is abelian is called a solvable group. Solvable groups play a key role in the Galois theory of fields and we will study them in detail later in the book.

Unfortunately, the terminology surrounding finite ordered sequences of subgroups is not at all standardized. For example, consider the following types of finite sequences of subgroups of a group K :

1) An arbitrary nondecreasing sequence of subgroups of K :

KŸKŸâŸK!" 8 2) A sequence of subgroups of K of the form

KKâK!"üüü 8 in which each subgroup is normal in its immediate successor. 3) A sequence of subgroups of K of the form

KŸKŸâŸK!" 8 in which each subgroup is normal in the parent group K .

Some authors refer to 1) as a series, 2) as a subnormal series and 3) as a normal series. Some authors refer to 2) as a series and 3) as a normal series. Some authors refer to 2) as a normal series and 3) as an invariant series.

Since arbitrary finite nondecreasing sequences of subgroups are a bit too general to be really useful, it seems reasonable not to give them a special name and simply refer to them as sequences, thus reserving the term series for more useful types of sequences. Accordingly, we choose the following terminology.

Definition Let KK be a group and let !8ßKŸKK. A series in from K! to K 8 is a sequence of subgroups of K of the form

KKâK!"üüü 8 where KK5K55 is normal in " for each . Each group 5 is a term in the series; KK!8 is the lower endpoint of the series and is the upper endpoint. Each extension KK55"ü is a step in the series. A series is proper if each inclusion is proper. The length of a series is the number of proper inclusions. 1) A normal series in KK is a series in in which each term is normal in the parent group K . 2) A characteristic series in KK is a series in in which each term is characteristic in the parent group K . 3) A fully-invariant series in KK is a series in in which each term is fully invariant in the parent group K . 78 Fundamentals of Group Theory

The following generalization of normality is extremely important and we shall have much to say about it in this book.

Definition A subgroup LŸK is subnormal in KLK , written üü , if there is a series

LLüüü"8 âLœK from LKL to . If üü K and LK , we write L––K . The family of subnormal subgroups of KÐ is denoted by subn KÑ .

Thus, LKüü if there is a sequence of “normal steps” from LK to . Subgroups of Index # The largest proper subgroups of a group are the subgroups of index # . We can now improve upon Theorem 2.28.

Theorem 3.17 Let LK# be a subgroup of of index . 1) LKü . 2) If +−K , then +# −L . 3) If KW is finite, then any subgroup of KL is either a subgroup of or else W∩L œ WÎ# kkkk In words, WLWL lies completely in or else lies half-in and half-out of . Also, if +−WÏL , then WœÐW∩LÑ“+ÐW∩LÑ where “ is the disjoint union. Proof. For part 1), if +  L , then ÖLß+L× and ÖLßL+× are both partitions of K+LœL++−KLK and so for all . Hence, ü . For part 2), if +ÂL then +LÁL has order # in KÎL and so +## LœÐ+LÑ œL, which implies that +−L# . For part 3), if W is not contained in L , the normality of L implies that WL œ K and so ÐW À L ∩ WÑ œ ÐLW À LÑ œ ÐK À LÑ œ #

Example 3.18 For 8 # , the alternating group E#8 has index in the symmetric group WEW888 and so ü .

We can now show that the converse of Lagrange's theorem fails.

Example 3.19 The alternating group E%% has order xÎ#œ"# , but has no subgroups of order 'LŸE'L . For if % has order , then has index # and so # 55−L for all −E%% . But there are eight $ -cycles 5 in E and each one is a square: Cosets, Index and Normal Subgroups 79

55œœÐÑ−L%## 5 Since these )' elements cannot fit into a subgroup of size , there can be no such subgroup. Cauchy's Theorem We have seen that the converse of Lagrange's theorem fails to hold. However, there are some partial converses to Lagrange's theorem. For example, there are certain classes of groups for which the converse of Lagrange's theorem does hold. In particular, we will see later in the book that if K is a finite abelian group or if 9ÐKÑ œ :8 where : is prime, then the converse of Lagrange's theorem holds, that is, K5 has a subgroup of any order that divides 9ÐKÑ .

On the other hand, it is true that if a prime :9ÐKÑK divides , then has an element of order :: and hence a subgroup of order . This key theorem is called Cauchy's theorem.

Cauchy's theorem has a very colorful history, beginning with its discovery by Cauchy in 1845 as the main conclusion of a 101-page paper ([6]). Here is a quotation from the abstract of an article on the history of Cauchy's theorem by M. Meo [24]:

The initial proof by Cauchy, however, was unprecedented in its complex computations involving permutational group theory and contained an egregious error. A direct inspiration to Sylow’s theorem, Cauchy’s theorem was reworked by R. Dedekind, G. F. Frobenius, C. Jordan, and J. H. McKay in ever more natural, concise terms.

The proof we give below is essentially the proof of J. H. McKay [23], which first appeared in 1959 and reminds us that we should never stop looking for “better” proofs of even the most basic results.

Theorem 3.20 ()Cauchy's theorem Let K: be a finite group. If is a prime dividing 9ÐKÑ , then K has an element of order : . Proof. The key to the proof is to examine the set

\œÖÐ+ßáß+ѱ+":3": −Kß+â+œ"× which is nonempty, since Ð"ßáß"Ñ−\. In fact, the size of \ is easily computed by observing that the first :" coordinates any B−\ can be assigned arbitrarily and this uniquely determines the final coordinate. Hence,

\œK:" kk kk which is divisible by : . 80 Fundamentals of Group Theory

Now, if we can find a constant : -tuple Ð+ß á ß +Ñ in \ where + Á " , then +: œ " and so 9Ð+Ñœ: , which proves the theorem. Note that a : -tuple BœÐ+ßáß+Ñ": is constant if and only if rotation one position to the right (with wrap around) has no effect, that is, if and only if

Ð+:" ß+ ßáß+ :" ÑœÐ+ " ßáß+ : Ñ

We can put this in group-theoretic language as follows. Let each 5 −W: act on \ by permuting the coordinates. Thus, if 5 is the : -cycle Ð"#â:Ñ , then

5Ð+"::":" ßáß+ ÑœÐ+ ß+ ßáß+ Ñ and so BB is constant if and only if 55œBB , that is, if and only if fixes .

Let us consider how the powers of 55 act on an element B−\ . The : -tuple 5 B comes from B5 by a -fold rotation. The orbit of B−\ is the collection b55ÐBÑ œ ÖBß Bß á ß:" B× of the various rotated versions of B: . Now, the primeness of implies that the elements of b5ÐBÑ are either all distinct or all the same. For if 34B œ5 B for 34, then 534 BœB where !34:. If 5œ34 , then Ð5ß:Ñœ" implies that ?5  @: œ " for some ?ß @ − ™ and so 55Bœ?5@: Bœ 55 ?5 ?: BœÐ 5 5 Ñ ? BœB whence 5b3BœB for all 3 . Thus, ÐBÑ has size " or : for all B−\ .

Now, the orbits bÐBÑ are the equivalence classes of the equivalence relation on \ defined by B´Cif Cœ5™5 B for some 5− and so the distinct orbits form a partition of \, . If is the number of blocks of size "- and is the number of blocks of size : , then \œ,-: kk where , " , since ÖÐ"ßáß"Ñ× is a block of size ":\ . Hence, divides kk implies that :±, and so, in particular, , : , which implies that there are at least :" constant : -tuples Ð+ßáß+Ñ with +Á" . Application to : -Groups The following types of groups play a key role in the study of the structure of finite groups.

Definition Let K: be a nontrivial group and let be a prime. 1) An element +−K is called a :-element if 9Ð+Ñœ:5 for some 5 ! . Cosets, Index and Normal Subgroups 81

2) KK is a :-group if every element of is a : -element. 3) A nontrivial subgroup WK of is called a :-subgroup of KW if is a : - group.

Lagrange's theorem and Cauchy's theorem combine to describe finite : -groups quite succinctly.

Theorem 3.21 A finite group K: is a -group if and only if the order of K is a power of : . Proof. If 9ÐKÑ œ :5 , then Lagrange's theorem implies that every element of K is a :K:K: -element and so is a -group. Conversely, if is a finite -group but ;±9ÐKÑ where ;Á: is prime, then Cauchy's theorem implies that K has an element of order ;9 , which is false. Hence, ÐKÑœ:5 for some 5 " . The Center of a Group; Centralizers We briefly mentioned the following concept earlier.

Definition The center ^ÐKÑ of a group K is the set of all elements of K that commute with all elements of K , that is, ^ÐKÑœÖ+−K±+,œ,+ for all ,−K×

A group K is centerless if ^ÐKÑ œ Ö"×. A subgroup LK of is central if L is contained in the center of K .

It is easy to see that the center of KK is a normal subgroup of .

Example 3.22 a) The center of the quaternion group is^ÐÑ œ Ö"ß "× . b) For 8 $ odd, the dihedral group H#8 is centerless and for 8 $ even,

8Î# ^ÐH#8 Ñ œ Ö+3 ß × c) For 8 $ , the symmetric group W8 is rather large and it should come as no surprise that W−88 is centerless. To see this, suppose that 5 W is not the identity. If the cycle decomposition of 5 contains a 55 $ -cycle with , that is, if 5 œâÐ+,-âÑâ

then 55Ð+ ,Ñ ÁÐ+,ÑÁÐ+,Ñ and so 55. On the other hand, if the cycle decomposition of 5 is a product of disjoint transpositions: 5 œÐ+,Ñâ

then 55Ð, -Ñ ÁÐ and so ,-ÑÁÐ 55,-Ñ-ÂÖ for +ß,× . In either case, 5 Â^ÐWÑ88 and so W is centerless. 82 Fundamentals of Group Theory

d) We will prove in a later chapter that for 8Á% , the alternating group E8 is simple, that is, EE88 has no nontrivial proper normal subgroups. Hence, is centerless for 8 % . The Number of Conjugates of an Element Definition Let K, be a group. The centralizer of an element −K is the set of all elements of K, that commute with :

GK Ð,ÑœÖ+−K±+,œ,+× The centralizer of a subgroup LŸK is the set GÐLÑœ GÐ+Ñ KK,+−L of all elements of KL that commute with every element of .

It is easy to see that centralizers are subgroups of the parent group.

Let K+ be a group and let −K . Then

BC CB" " + œ + Í + œ + Í C B − GKKK Ð+Ñ Í BG Ð+Ñ œ CG Ð+Ñ and so the element +Ð has precisely KÀGKÐ+ÑÑ distinct conjugates. This formula is of considerable importance in the study of finite groups.

Theorem 3.23 Let K+ be a group and let −KÐ . Let conjK +Ñ denote the set of conjugates of +K in . Then conj Ð+Ñ œ ÐK À G Ð+ÑÑ kkKK which divides 9ÐKÑ when K is finite. The set conjKÐ+Ñ is called a conjugacy class in K . The Normalizer of a Subgroup Suppose that LŸK . Then of course L is normal in itself. But it may also be normal in a larger subgroup of K .

Definition Let KL be a group and let ŸKR . The largest subgroup KÐLÑ of K for which LRÐLÑü K is called the normalizer of LK in . A subset \©K is said to normalize L\©RÐLÑ if K .

Theorem 3.24 Let KL be a group. The normalizer of ŸK is + RK ÐLÑ œ Ö+ − K ± L œ L× Will see in the chapter on free groups (Theorem 12.21) that it is possible to have + L§L, in which case +ÂRÐLÑK . However, since Cosets, Index and Normal Subgroups 83

" LœL+++ Í L©Land L ©L if a subset W©K is closed under inverses and has the property that L= ©L for all =−W , then W©RK ÐLÑ. In particular, since WœÖ+−K±L+ ©L× is closed under products, if WW is finite then is a subgroup of KW and so is closed under inverses, whence WœRK ÐLÑ. In particular, if K is finite, then WœRK ÐLÑ.

The normalizer of LL should not be confused with the normal closure of , which is the smallest normal subgroup of KL that contains . Note that if OŸK normalizes LŸK , then LO is also a subgroup of K .

Theorem 3.25 If KL is a group and ŸK , then

GKK ÐLÑü R ÐLÑ

Elementwise Commutativity It is clear that LO and commute elementwise if and only if

LŸGÐOÑKKand OŸGÐLÑ For essentially disjoint subgroups, we need only check that each subgroup is contained in the normalizer of the other. Proof of the following is left to the reader.

Theorem 3.26 Let LO and be essentially disjoint subgroups of a group K . 1) Then LO and commute elementwise if and only if each subgroup is contained in the normalizer of the other, that is,

LŸRÐOÑKK and OŸRÐLÑ In particular, if LßOü K , then L and O commute elementwise. 2) If KœLìO , then L and O commute elementwise if and only if LßOü K. The Number of Conjugates of a Subgroup Let KL be a group and let ŸK . Then

BC CB" " L œ L Í L œ L Í C B − RKKK ÐLÑ Í BR ÐLÑ œ CR ÐLÑ Thus, we obtain a count of the number of conjugates of a subgroup (the analog of Theorem 3.23).

Theorem 3.27 Let KL be a group and let ŸKÐ . Let conjK LÑ denote the set of conjugates of LK in . Then 84 Fundamentals of Group Theory

conj ÐLÑ œ ÐK À R ÐLÑÑ kkKK which divides 9ÐKÑ when K is finite. The set conjKÐLÑ is called a conjugacy class in subÐKÑ . Simple Groups Nontrivial groups with no nontrivial proper normal subgroups are, in some sense, simple.

Definition A nontrivial group K is simple if it has no normal subgroups other than Ö"× and K.

Of course, if K is an abelian group, then all of its subgroups are normal and so an abelian group is simple if and only if its only subgroups are Ö"× and K . This is not easy for a group.

Theorem 3.28 An abelian group K is simple if and only if it is a cyclic group of prime order. Proof. If KK is cyclic of prime order, then Lagrange's theorem implies that has no subgroups of order different from "9ÐKÑK and and so is simple. Conversely, if K1 is simple, then every nonidentity element of K must generate KKœØ1Ù, that is, is cyclic. However, if K is infinite, then Ø1Ù# is a nontrivial proper subgroup of KK , a contradiction. Hence, is finite and cyclic. But if :±9ÐKÑ where : is prime, then Cauchy's theorem implies that K has a subgroup W:KœW of order and so has prime order.

We will show later in the book that the alternating group E8 is simple for all 8Á%. Also, a famous result of Feit–Thompson (1963, [11]), whose proof runs 255 pages, says that every nonabelian finite simple group has even order. Thus:

1) The abelian simple groups are the cyclic groups of prime order. 2) All nonabelian finite simple groups have even order. Commutators of Elements The elements of a group of the form +,+" , " play a special role.

Definition Let K+ be a group. The commutator of ß,−K is the element Ò+ß ,Ó œ +,+" , " We denote the set of commutators of KÐKÑ by . The subgroup generated by all commutators of KK is usually denoted () unfortunately by w and is called the commutator subgroup, or derived subgroup of KK . A group is perfect if KœKw. Cosets, Index and Normal Subgroups 85

We should note that authors who define the conjugate of two elements by +œ,+,," also define the commutator by Ò+ß,Óœ+,+,"".

It is easy to see that KKKww is fully invariant in ; in particular, «K . Also, we leave it as an exercise to prove that if RKü , then KKRw w œ ŒRR Commutators have some very nice algebraic properties. Here are the most basic of these properties.

Theorem 3.29 Let K be a group and let +ß,ß-−K . 1) Ò+ß ,Ó œ " Í +, œ ,+ 2) Ò+ß ,Ó" œ Ò,ß +Ó 3) Ò+ß ,Ó--- œ Ò+ ß , Ó 4) If LKü , then in KÎL , Ò+Lß ,LÓ œ Ò+ß ,ÓL 5) If +ß , − K commute with -ß . − K and vice versa, then Ò+ß ,ÓÒ-ß .Ó œ Ò-+ß .,Ó Note that since Ò+ß ,Ó" œ Ò,ß +Ó, every element of Kw is a product of commutators. The following characterization of commutator subgroups explains why these subgroups are so important.

Theorem 3.30 () R. Dedekind , c. 1880 Let K be a group. Then for any subgroup LŸK, LKü and KÎL is abelian ÍKŸLw In particular, KKw is the smallest normal subgroup of whose quotient is abelian. Proof. If LKü and KÎL is abelian, then in KÎL , we have Ò+ß ,ÓL œ Ò+Lß ,LÓ œ L and so Ò+ß ,Ó − L , whence Kww Ÿ L . Conversely, if KŸLL , then is normal in K2, since for any −L+ and −K , 2+ œ Ò+ß 2Ó2 − L Also, KÎL is abelian since Ò+Lß ,LÓ œ Ò+ß ,ÓL œ L Example 3.31 We leave it to the reader to show that the commutator subgroup of the quaternion group œ is Ö"ß "×w , which is also the set of 86 Fundamentals of Group Theory commutators, that is, œw ÐÑ

w# Similarly, the commutator subgroup of the dihedral group HHœØÙ#8 is #8 3 and so again the commutator subgroup of H#8 is also the set of commutators: w HœÐHÑ#8 #8

Example 3.32 For the symmetric group W8 $8 on symbols, we have w WœEœÐWÑ8 88 w Since all commutators are even permutations, we have WŸE8 8 .

Now we make a few observations. First the product of any finite number of disjoint commutators is a commutator, since if α"33 and α" 44 are disjoint, then

ÒßÓÒßα"α"33 44 ÓœÒ αα"" 4343 ß Ó

Second, if 5.ß−W8 , then 55. " œÒ .5 ß Ó and so if 57 and have the same cycle structure, then the product 75 is a commutator.

Hence, we need only show that any 5 −E8 can be written as a product of disjoint permutations, each of which is a product α"33 where α 3 and " 3 have the same cycle structure.

But since a cycle of odd length is even and a cycle of even length is odd, the cycle decomposition of 5 must have an even number of cycles of even length. In other words, the cycle decomposition of 5 is a product of odd cycles and pairs of even cycles.

Now, every odd cycle 5 œÐ+â+"#7" Ñ can be written as a product of two equal-length cycles as follows:

5 œÐ+â+" #7" ÑœÐ+â+ " 7" ÑÐ+ 7" â+ #7" Ñ Similarly, every pair of disjoint even cycles can be written as a product of two equal-length cycles as follows (where 5 7 ):

5 œ Ð+"#7"#5"#7"57"#757"#5 â+ ÑÐ, â, Ñ œ Ð+ â+ , â, ÑÐ+ , â, Ñ Hence, 5 can be written in the form

5α"α"œÐ"" ÑâÐ 77 Ñ Cosets, Index and Normal Subgroups 87

where α"33 and are equal-length cycles for each 3 and α"α"3344 and are disjoint for 3Á4. Hence, 5 − ÐWÑ8 . When is ÐKÑ œ Kw? We have seen that for the quaternion, dihedral and symmetric groups, Kœw ÐKÑ Despite these examples, however, this is not always true. In fact, the question of precisely when Kœw ÐKÑ is an active area of current research. We give an example of a group for which KÁw ÐKÑ and then discuss a few results in this area. Readers interested in more details may wish to consult the survey article of Kappe and Morse [19].

Example 3.33 (Cassidy [5]) Let K be the set of all matrices of the form "0ÐBÑ2ÐBßCÑ 7Ð0ß 1ß 2Ñ ³ Ô×!" 1ÐCÑ ÕØ!! " where 0ÐBÑ , 1ÐBÑ and 2ÐBß CÑ are polynomials with rational coefficients. A straightforward calculation shows that

7Ð0ß1"" ß2 " Ñ7Ð0ß1 ## ß2 # Ñœ7Ð0 " 0 #" ß1 1 # ß2 " 2 # 0 "# 1 Ñ and 7Ð0ß 1ß 2Ñ" œ 7Ð0ß 1ß 2  01Ñ from which it follows that K is a group. Another computation shows that the commutators are given by

Ò7Ð0"""### ß1 ß2 Ñß7Ð0 ß1 ß2 ÑÓœ7Ð!ß!ß0"# 1 0 #" 1 Ñ Thus, the commutator subgroup Kw is contained in the subgroup of all matrices in K of the form 7Ð!ß!ß2Ñ where 2 − ÒBßCÓ . Moreover, any such matrix is a product of commutators, since

7 !ß!ß + BC34 œ 7Ð+ Bß!ß!Ñß7Ð!ßCß!Ñ3 4 $3ß4 ‘3ß4 3ß4 3ß4 which shows that Kw œÖ7Ð!ß!ß2ѱ2 − ÒBßCÓ× Thus, the matrix E œ 7Ð!ß !ß "  BC  B## C Ñ is in Kw . However, it is not a commutator, for if 88 Fundamentals of Group Theory

## "  BC  B C œ 0"# ÐBÑ1 ÐCÑ  0 #" ÐBÑ1 ÐCÑ

3 for some 0ß0−"#  ÒBÓ and 1ß1−"# ÒCÓ, then equating coefficients of B shows that

3 Cœ01ÐCÑ01ÐCÑ"ß3 # #ß3 "

3 for 3œ!ß" and # , where 0?ß3 is the coefficient of B in 0? ÐBÑ . But this implies that the two-dimensional vector subspace of ÒCÓ spanned by 1"# ÐCÑ and 1 ÐCÑ contains the three independent vectors "C , and C# , which is not possible. Hence, not all members of Kw are commutators.

Here is a small sampling of known results concerning the issue of when Kœw ÐKÑ. Many more results are contained in Kappe and Morse [19].

Theorem 3.34 ()Speigel [31], 1976 If a group K contains a normal abelian subgroup EK whose quotient ÎEK is cyclic, then w œÐKÑ . Proof. If we can find a normal subgroup FKü for which F©ÐKÑKÎF and is abelian, then KŸF©ÐKÑ©Kww whence Kœw ÐKÑ . To this end, let KÎEœØBEÙ and let F œ ÖÒBß +Ó ± + − E× œ Ö+B" + ± + − E× © ÐKÑ To see that FKü , we have Ð+B + " ÑÐ, B , " Ñ œ + B , B + " , " œ Ð+,Ñ B Ð+,Ñ " − F and Ð++ÑœÐ+Ñ+−FB "" "B and for normality,

3333 Ð+B + " Ñ B + œ Ð+ B + " Ñ B œ Ð+ B Ñ B Ð+ B Ñ " − F To see that KÎF is abelian, we have ÒBFß +FÓ œ ÒBß +ÓF œ F and so BF and +F commute for all + − E , which implies that B34 +F and B ,F commute for all +ß , − E .

For small groups, we also have Kœw ÐKÑ .

Theorem 3.35 Let K be a group. The following conditions imply that Kœw ÐKÑ. 1)( Guralnick [15], 1980) Cosets, Index and Normal Subgroups 89

a) Kww is abelian and either 9ÐKÑ  "#) or 9ÐK Ñ  "' . b) K9ww is nonabelian and either ÐKÑ*' or 9ÐKÑ#% . 2)( Kappe and Morse [20], 2005) a) 9ÐKÑœ:8, where : is an odd prime and 8Ÿ& . b) 9ÐKÑ œ #8 , where 8 Ÿ '.

On the other hand, if the center of KK is large compared to the size of w , then there will be elements of Kw that are not commutators.

Theorem 3.36 ()MacDonald [22], 1986 Let K^ be a group with œ^ÐKÑ . If ÐKÀ ^Ñ#w  9ÐK Ñ then KÁw ÐKÑ. Proof. Since DßA − ^ implies that Ò+Dß ,AÓ œ Ò+ß ,Ó it follows that the number of distinct commutators is at most the number of commutators of KÎ^ and that is certainly at most ÐKÀ ^Ñ# . Additional Properties of Commutators Here are some additional properties of commutators. The reader may wish simply to read the statement of the theorem and move on, referring to the theorem as needed at later times. (The proof is not particularly enlightening.)

Theorem 3.37 Let K be a group and let +ß,ß-−K . 1) Ò,ß +Ó œ Ò+" ß ,Ó + œ Ò+ß , " Ó , Ò+ß ,Ó œ Ò+" ß , " Ó ,+ 2) Ò+ß ,-Ó œ Ò+ß ,ÓÒ+ß -Ó, Ò+,ß-Ó œ Ò,ß-Ó+ Ò+ß-Ó

3) Let +"8"7 ßáß+ ß, ßáß, −K. Then 87 Ò+ â+ ß , â, Ó œ Ò+ ß , Óα3ß4 "8"7$$ 34 3œ" 4œ"

where α3ß4−Ø+ " ßáß+ 8 ß, " ßáß, 7 Ù. Also, the order of the factors on the right can be chosen arbitrarily, although the exponents depend on that order. In particular, if 78 and are positive integers, then 90 Fundamentals of Group Theory

87 Ò+87 ß , Ó œ Ò+ß ,Óα3ß4 $$ 3œ" 4œ"

where α3ß4 −Ø+ß,Ù. 4) If Ò+ß ,Ó commutes with both + and , , then for any integers 7 and 8 , a) Ò+78 ß , Ó œ Ò+ß ,Ó 78 777 7 b) Ð+,Ñ œ + , Ò,ß+Ó# Hence, if 9ÐÒ+ß ,ÓÑ divides 7 , then ˆ‰# Ð+,Ñ777 œ + , Proof. The proofs of part 1) and part 2) are straightforward calculations. Part 3) is proved by a double induction. For 7œ" , we induct on 8 . If 8œ" , then the result is clear. If the result holds for 8" , then part 2) implies that

+â+"8" Ò+"8" â+ ß , Ó œ Ò+ 8" ß , Ó Ò+ "8"" â+ ß , Ó and the inductive hypothesis completes the proof for 7œ" . Assume the result is true for 7" and let +œ+"8 â+ . Then

,â,"7" Ò+ß ,"7 â, Ó œ Ò+ß , "7"8 â, ÓÒ+ß , Ó and the inductive hypothesis completes the proof. As to the statement about the order, we can rearrange the factors using the identity BC œ CB B .

Part 4a) follows from part 3) when 7ß 8 ! . The result then follows for negative exponents using part 1). For part 4b), note first that -œÒ,ß+Ó also commutes with +, and and that ,+œ-+,œ+,- Now, Ð+,Ñ7 œ Ð+,ÑâÐ+,ÑÐ+,Ñ ðóóóóóñóóóóóò 7 factors and if we move the rightmost + all the way to the left, the result is Ð+,Ñ77 œ + Ð+,ÑâÐ+,Ñ ,- " ðóóñóóò 7" factors Repeating this process gives Ð+,Ñ7# œ + Ð+,ÑâÐ+,Ñ ,2 -7#7" - ðóóñóóò 7# factors Continuing until all of the + s are on the left gives

7 7" 7" "#âÐ7"Ñ 7 7 7 Ð+,Ñ œ + Ð+,Ñ, - œ + , -# Cosets, Index and Normal Subgroups 91 which proves the result for 7 ! . For 7! we have, using part 4b) for positive exponents along with part 4a) and part 1), Ð+,Ñ7 œ Ð, " + " Ñ 7 7 7 " " 7 œ, + Ò+ ß, Ó# 7 7 7 7 " " 7 œ + , Ò, ß + ÓÒ+ ß , Ó# 7 7 7# " " 7 œ+ , Ò,ß+Ó Ò+ ß, Ó# 7 7 7# 7 œ+ , Ò,ß+Ó Ò+ß,Ó# 7 7 7#  7 œ+ , Ò,ß+Ó Ò,ß+Ó # 7 7 7 œ+ , Ò,ß+Ó# Here is one application of Theorem 3.37 that will be useful later in the book.

Theorem 3.38 If K is a nonabelian group with the property that all cyclic subgroups are normal, then 1) K is periodic 2) Any nonabelian subgroup of K contains a quaternion subgroup 3) If +ß , − K and 9Ð+Ñ  9Ð,Ñ  ), then +, œ ,+ . Proof. To see that KB is periodic, we first show that any Â^ÐKÑ has finite order. Let C−K satisfy -³ÒBßCÓÁ" Then the normality of ØBÙ and ØCÙ imply that - − ØBÙ ∩ ØCÙ and so -œB87 œC for some 8ß 7 Á ! . Thus, - commutes with BC and and so " œ Ò-ß -Ó œ ÒB8 ß C 7 Ó œ ÒBß CÓ87 œ - 87 Hence, -B has finite order and therefore so does . Now let B−^ÐKÑ . If CÂ^ÐKÑ, then BCÂ^ÐKÑ and so BC has finite order and therefore so does B . Hence, K is periodic.

Now suppose that HK is a nonabelian subgroup of . Let BßC−H be chosen so that 9ÐBÑ  9ÐCÑ is minimal among all pairs of noncommuting elements in H . We will show that œØBßCÙ is a quaternion subgroup of H . To see that B and C%: have order , let be a prime dividing 9ÐBÑ and let -œÒBßCÓœB87 œC as shown above. Since 9ÐB:: Ñ  9ÐBÑ, it follows that BC and commute and so Theorem 3.37 implies that " œ ÒB:: ß CÓ œ ÒBß CÓ whence 9Ð-Ñ œ : . Hence, : is the only prime dividing 9ÐBÑ and similarly 9ÐCÑ 92 Fundamentals of Group Theory and so 9ÐBÑ œ :3" and 9ÐCÑ œ :4" for some 3ß 4 ! . Moreover, :3" :œ9Ð-Ñœ9ÐBÑœ8 Ð:3" ß 8Ñ which shows that 8œ:?3 where :±?y . A similar argument for C gives

34 Bœ-œC:? :@ where :±?yy and :±@.

To eliminate ?@ and from the equation above, if α œ?:" mod and "αœ@" mod : , then taking the " power gives

34 Bœ-œC::"α"α and so

34 ÐB""αα Ñ:: œ ÒB ß C Ó œ ÐC Ñ where 9ÐB"α Ñ œ :3" and 9ÐC Ñ œ :4". Thus, replacing BBCC"α by and by gives

34 Bœ-œC:: (3.39) where "Á-œÒBßCÓ and 9ÐBÑ œ :3" ß 9ÐCÑ œ :4" and 9Ð-Ñ œ : We may also assume that 3 4 " , for if 4œ! , then Cœ-œÒBßCÓ, which is false.

The next step is to show that :œ# and that 3œ4œ" . The key observation here is that for any integer 5B , the elements 5CB and do not commute and so 9ÐB54 CÑ 9ÐCÑ œ :". But, if : is odd or if : œ # and 4  " , then :4 9Ð-Ñ œ : ¹ Œ# and so Theorem 3.37 implies that

4444 "ÁÐBCÑ5: œB 5:: C œB 5: - and taking 5œ:34 gives "Á" . Hence, :œ# and 3œ4œ" . Thus, 9ÐBÑ œ 9ÐCÑ œ %ß 9Ð-Ñ œ #and B## œ - œ C Cosets, Index and Normal Subgroups 93 where -BC commutes with and and since BCÐBCÑ œ -CBÐBCÑ œ - it follows that œØBßCÙ is quaternion. Commutators of Subgroups Commutators can be defined for subsets as well as for elements.

Definition The commutator Ò\ß]Ó of two subsets \ß] ©K is the subgroup generated by the commutators: Ò\ß]ÓœØÒBßCÓ±B−\ßC−]Ù Note that, by definition, KœÒKßKÓw for any group K . Here are some of the basic properties of these commutators.

Theorem 3.40 Let LßO Ÿ K . 1) ÒLß OÓ œ ÒOß LÓ 2) LO and commute elementwise if and only if ÒLß OÓ œ Ö"×, in particular, L Ÿ ^ÐKÑ Í ÒLß KÓ œ Ö"× 3) LKÍÒLßKÓŸLü 4) If LßOü K , then ÒLß OÓ Ÿ L ∩ Oand ÒLß OÓü K Also, LßO « K Ê ÒLßOÓ « K 5) LÒLßOÓ normalizes and so ÒLß OÓü ØLß OÙ 6) LÒLßOÓ œ LOOü L O œ ØLßOÙ In particular, ncÐLßKÑ œ LÒLßKÓ 94 Fundamentals of Group Theory

Proof. We prove only part 5). Theorem 3.37 implies that if 2−L" , then

2" " Ò2ß 5Ó œ Ò2"" 2ß 5ÓÒ2 ß 5Ó − ÒLß OÓ Hence, L normalizes ÒLßOÓ . Similarly, O normalizes ÒLßOÓ and so ØLßOÙ also normalizes ÒLß OÓ .

Here are some additional properties of commutators of subgroups.

Theorem 3.41 Let Eß Lß O Ÿ K. 1) ÒEß LOÓ œ ÒEß LÓÒEß OÓL where ÒEßOÓL2 œØÒ+ß5Ó ±+−Eß5−Oß2−LÙ

2) If EKüY and if œÖL±3−M×3 is a family of normal subgroups of K , then

ÒEß L Ó œ ÒEß L Ó 2233 This equation still holds even if one member of Y is not normal. In particular, if Eß Lü K and O Ÿ K , then ÒEß LOÓ œ ÒEß LÓÒEß OÓ 3) If RKü , then ÒLRß ORÓ Ÿ ÒLß OÓR and so ÒLRß ORÓR œ ÒLß OÓR 4) If RKü , then LR OR ÒLß OÓR ßœ ”•RR R Proof. For part 1), if +−E , 2−L and 5−O , then Ò+ß 25Ó œ Ò+ß 2ÓÒ+ß 5Ó2L − ÒEß LÓÒEß OÓ and so ÒEß LOÓ Ÿ ÒEß LÓÒEß OÓL . For the reverse inclusion, we have ÒEß LÓ Ÿ ÒEß LOÓ and since Ò+ß 5Ó2 œ Ò+ß 2Ó" Ò+ß 25Ó − ÒEß LOÓ we also have ÒEß OÓL Ÿ ÒEß LOÓ. Cosets, Index and Normal Subgroups 95

For part 2), note that part 1) and the fact that ÒEß LÓü K imply that ÒEß LOÓ Ÿ ÒEß LÓÒEß OÓ and the reverse inclusion is evident. For the general case, since for all , it follows that ÒEß L53 Ó Ÿ ÒEß1 L Ó 5 ÒEß L Ó Ÿ ÒEß L Ó 2253 For the reverse inclusion, each generator of belongs to a commutator ÒEß1 L3 Ó subgroup of the form ÒEß L âL Ó, which is equal to ÒEß L Ó , which in turn 33"8 1 34 is contained in and so 1ÒEß L3 Ó ÒEß L Ó Ÿ ÒEß L Ó 2233 Part 3) follows from the fact that ÒLRßORÓ œ ÒLRßOÓÒLRßRÓO œÒOßLÓÒOßRÓLO ÒLRßRÓ ŸÒOßLÓR Part 4) follows from part 3). *Multivariable Commutators We can extend the definition of commutators as follows. If +ß ,ß - − K , then Ò+ß,ß-ÓœÒ+ßÒ,ß-ÓÓ and in general, if +ßáß+"8 −K, then

Ò+"8"#8 ßáß+ ÓœÒ+ ßÒ+ ßáß+ ÓÓ with Ò+Ó ³ + . Note that some authors define Ò+ß ,ß -Ó to be ÒÒ+ß ,Óß -Ó and, in general, these are not the same.

Theorem 3.42 Let K be a group. The following are equivalent: 1)( Associativity ) For all +ß ,ß - − K , ÒÒ+ß ,Óß -Ó œ Ò+ß Ò,ß -ÓÓ

2)( Distributivity ) For all +ß ,ß - − K , Ò+ß ,-Ó œ Ò+ß ,ÓÒ+ß -Ó 3)( Commutator subgroup is central) KŸ^ÐKÑw 96 Fundamentals of Group Theory

Proof. Since Ò+ß ,-Ó œ Ò+ß ,ÓÒ+ß -Ó, it follows that 2) holds if and only if Ò+ß-Ó, œÒ+ß-Ó for all +ß,ß-−K , which holds if and only if Ò+ß -Ó − ^ÐKÑ for all +ß - − K , that is, if and only if KŸ^ÐKÑw . Hence, 2) and 3) are equivalent.

It is clear that 3) implies 1). Conversely, 1) is Ò+ß ,Ó-Ò+ß ,Ó" - " œ +Ò,ß -Ó+" Ò,ß -Ó " which is equivalent to Ò+ß ,Ó-Ò,ß +Ó-" œ +Ò,ß -Ó+" Ò,ß -Ó " Now, as to the last factor Ò,ß -Ó" , note that 1) with , œ - implies that ÒÒ+ß ,Óß ,Ó œ " for all +ß , − K . Hence, Ò,ß -Ó" œ Ò-ß ,Ó œ ,Ò," ß -Ó, " œ Ò, " ß -Ó and so 1) is equivalent to Ò+ß ,Ó-Ò,ß +Ó-" œ +Ò,ß -Ó+" Ò, " ß -Ó Expanding the commutators gives +,+,-,+,+-" " " " " œ+,-,-+,-,- " " " " " and cancelling gives +" , " -,+, " + " œ -, " - " + " , " -, Moving the first factor on the right side to the left side and the last factor on the left side to the right side gives -+,-,+,œ,-+,-,+" " " " " " " " which is equivalent to Ò-ß+,Ó,œ,Ò-ß+,Ó" " " " " " " " which shows that Ò-" ß+ " , " Ó commutes with ," , but - " ß+ " , " and , " represent arbitrary elements of KÒBßCÓD and so commutes with for all Bß Cß D − K, that is, Kw Ÿ ^ÐKÑ .

Theorem 3.43 ()Hall–Witt Identity Let +ß ,ß - − K . Then Ò+ß," ß-Ó , Ò,ß- " ß+ÓÒ-ß+ - " ß,Ó + œ" Cosets, Index and Normal Subgroups 97

Proof. For the first factor we have Ò+ß ," ß -Ó , œ ,Ò+ß Ò," ß -ÓÓ, " œ,+Ò,ß-Ó+Ò,ß-Ó," " " " " œ,+,-,-+-,-,," " " " " " œ,+," -,- " + " -, " - " œ+,, - + " Ð, " Ñ -

By cycling +, , and - , we get Ò,ß-" ß+Ó - œ, - - + , " Ð- " Ñ + and Ò-ß+" ß,Ó + œ- + + , - " Ð+ " Ñ , The product is easily seen to self-destruct.

If LßO and P are subgroups of K , then we define ÒLßOßPÓœÒLßÒOßPÓÓ and more generally, if LßáßL"8 ŸK, then

ÒL"8"#8 ßáßL ÓœÒL ßÒL ßáßL ÓÓ with ÒLÓ ³ L.

The subgroup ÒLß ÒOß PÓÓ is generated by elements of the form Ò2ß -"8 â- Ó , where 2−L and -3333 œÒ5ßjÓ for 5 −O and j 3 −P . Hence, Theorem 3.37 implies that

Ò2ß - â- Ó œ Ò2ß - Óα3 "8$ 3 where α3"8−Ø2ß-ßáß- ÙŸØLßOßPÙ.

Corollary 3.44 ()Three subgroups lemma Let KL be a group. Let , OP and be subgroups of K . Then if any two of the commutators ÒLßOßPÓ , ÒPßLßOÓ or ÒOßPßLÓ are contained in a normal subgroup RK of , then so is the third commutator. Proof. We assume that ÒLßOßPÓŸR and ÒPßLßOÓŸR and use the Hall– Witt Identity. Since Ò2ß5" ßjÓ−ÒLßOßPÓ and Òjß2" ß5Ó−ÒPßLßOÓ for 2−L, 5−O and j−P , it follows from the Hall-Witt identity that Ò5ßj" ß2Ó j œÐÒ2ß5" ßjÓ 5 Òjß2 " ß5Ó 2 Ñ " −R and since R is normal in K , we deduce that Ò5ßj" ß2Ó−R. Hence, ÒOßPßLÓŸR. 98 Fundamentals of Group Theory

Exercises 1. Let KL be a group and let ü KL . Prove that if and KÎL are centerless, then K is centerless. 2. Let KœW$#} G. Show that K has two subgroups LO' and of order that are centerless but that KœLO is not centerless. Thus, the set product of centerless subgroups may be a group that is not centerless. What about the direct product of centerless groups? 3. If a group KK has the property that all of its subgroups are normal, must be abelian? 4. Let K be a finite group and let R"7 ßáßR be normal subgroups of K . Show that RâR divides R âR . kkkkkk"7 " 7 5. A subgroup LK of a group is permutable if LOœOL for all subgroups OK of . Prove that a maximal subgroup that is permutable is normal. 6. Show that if a normal subgroup LK of a group contains no nonidentity commutators, then LK is central in . 7. Use the Feit–Thompson Theorem to prove that a nontrivial group K of odd order is not perfect, that is, KKw . 8. Suppose that KK is a finite group and that w K . Prove that K has a normal subgroup O of prime index. 9. Show that the set product of subnormal subgroups need not be a subgroup. Hint: Check the dihedral group H) . 10. Let KLÐ be a finite group and let be a subgroup for which KÀLÑ is prime. Show that if there is at least one left coset +L of L other than L itself that is also equal to some right coset L, , then Lü K . 11. a) Prove that the commutator subgroup of the quaternion group is Ö"ß "×. b) Prove that the commutator subgroup of the dihedral group H#8 is w# HœØÙ#8 3 . 12. For which values of 8H is it true that the dihedral group #8 has a pair of proper normal subgroups LO and for which

H#8 œ LOand L ∩ O œ Ö"×

13. Let K be a group and let L Ÿ K . Prove that GKK ÐLÑü R ÐLÑ. w 14. Prove that WœE8 88 for 8 % in the following manner. Use the fact that E w is simple for all 8Á% . Show that W8 ∩E88ü E . Deduce that ww w W∩EœÖ×8888++ or W∩EœE8. Show that W∩EœÖ× 8 8 is impossible. 15. Prove that if RKü , then KKRw w œ ŒRR 16. a) Find an infinite group that is periodic. Hint : Look at the quotient groups of the additive group of rationals. b) Prove that if LKü and L and KÎL are periodic, then so is K . Cosets, Index and Normal Subgroups 99

17. Let RŸ^ÐKÑ , the center of K . Show that Rü K and that if KÎR is cyclic, then K is abelian. 18. Prove that a maximal subgroup QK of a group is normal if and only if KŸQw . 19. Let L–K have prime index. Let B−L have the property that GÐBÑGÐBÑLK. Prove that an element C−L is conjugate to BK in if and only if it is conjugate to BL in . 20. Let 1 œÖ:ßáß:"7 × be a nonempty set of primes. A 1-group is a group whose order 88 has the property that all primes dividing lie in the set 1 . For example, a group of order #$# † & † ( is a Ö#ß &ß (× -group. Let K be a finite group and let LßáßL"7 be normal subgroups such that KÎL3 is a -group for all . Prove that is also a -group. 113KÎL+ 3 21. A subgroup LK of a group is abnormal if +−ØLßL+ Ù for all +−K . Prove that L is abnormal if and only if it satisfies the following conditions: a) If LŸOŸK , then RK ÐOÑœO . b) LO is not contained in distinct conjugate subgroups, that is, if ŸK and OÁO++, then L©yO∩O . 22. Let LßO Ÿ K . a) Prove that LOO œ LÒLßOÓü ØLßOÙ œ L O b) If KœØ\Ù for some subset \K of , show that w K œØÒBßCÓ±BßC−\Ùnor 23. Let WK be a nonempty subset of a group and let LßOŸK . Show that WœÐWÑLO O L 24Þ Let EßLßO ŸK. Prove that ÒELß EOÓ œ ÒELß EÓÒELß OÓ 25. Let LßOßP Ÿ K. a) Show that if ÒLßOßPÓœÖ"× and ÒPßLßOÓœÖ"×, then ÒOßPßLÓœÖ"×. b) Show that if LO , and P are normal in K , then ÒLßOßPÓŸÒPßLßOÓÒOßPßLÓ 26. Let KR be a group and let be a normal subgroup. Let BßC−K and suppose that B8 C 7 −R and CBC>+7 B "+8 −R. Prove that CBC> B " − R. 27. Let KL be a group and let ŸKLK . Prove that if has finite index in , then ÐKÀLÑœÐKÀLB Ñ for any B−K . 100 Fundamentals of Group Theory

28. Let K^ be a nonabelian group. Show that ÐKÑ is a proper subgroup of any centralizer GÐ1ÑK . 29. Let L Ÿ K . Prove that GKK ÐLÑü R ÐLÑÞ 30. Let KL be a group and let ŸK . a) Prove that RŸK is normal in K if and only if all subsets of K normalize R. b) Prove that if WL normalizes the subgroups and O of K , then W normalizes their join L”O , that is,

RÐLÑ∩RÐOÑŸRÐL”OÑKK K c) Prove that if WX and normalize L , then the set product WX normalizes L. d) Prove that a subgroup LŸK normalizes all supergroups of itself. ++ + 31. Prove that if LŸK , then RKK ÐLÑœR ÐLÑ. In particular, GK Ð2Ñœ + GÐ2ÑK . 32. Let K be a group and let LŸOŸK with Lü K . Prove that

LRÐLÑK RœK ŒOO 33. a) Show that the subgroup L œ Ö+ ß Ð" #ÑÐ$ %Ñß Ð" $ÑÐ# %Ñß Ð" %ÑÐ# $Ñ× of W% is normal. Hint : Show that the conjugate of any transposition is another transposition, in fact, Ð+ ,Ñ5 œ Ð55 + ,Ñ. b) Show that the subgroup O œ Ö3ß Ð" #ÑÐ$ %Ñ× is normal in L but not normal in W% . c) Conclude that normality is not transitive. 34. Let K be a group. a) Let QK be the intersection of all subgroups of that have finite index in KQ . Show that is normal in K . b) Let LŸK with finite index. Show that there is a normal subgroup RKü for which RŸLŸK where R also has finite index in K . 35. Let KB be a group generated by two involutions and C . Show that K has a normal subgroup of index # . 36. Let K be a finite group of odd order. Show that the product of all of the elements of KK , taken in any order, is in the commutator subgroup w . 37. (P. Hall) Let Eß F Ÿ K . Show that if ÒEß Eß FÓ œ Ö"×, then ÒEß FÓ is an abelian group. Hint : One possible proof is as follows: Show that E commutes with ÒEß FÓ and with ÒFß EÓ and that F commutes with Ew . Then use a direct computation to show that Ò+ß ,ÓÒBß CÓ œ ÒCß B" ÓÒ+ß ,Ó where +ß B − E and ,ß C − F . Finally, use Theorem 3.37. 38. Let :Á# be prime and consider the group ∞ Kœ™{™ Ð: Ñ # where ™Ð:∞ Ñ is the : -quasicyclic group. Show that K has a unique Cosets, Index and Normal Subgroups 101

maximal subgroup QQ , but that is not maximum in the lattice of all proper subgroups of K . 39. A group KK is said to be metabelian if its commutator subgroup w is abelian. (Some authors define a group KK to be metabelian if w is central in K, which is stronger than our definition.) a) Prove that KK is metabelian if and only if has a normal abelian subgroup EKÎE for which is also abelian. b) Prove that the dihedral group H#8 is metabelian. c) Let J: be a finite field, such as ™: where is prime. Let +ß,−J where +Á!. The map 5+ß,ÀJÄJ defined by

5+ß,ÀBÈ+B, is called an affine transformation of JÐ . Show that the set aff JÑ of all affine transformations of JJÐ is a subgroup of J . Show that aff JÑ is metabelian. d) Prove that if KœEF where E and F are abelian, then K is ,+"" +,"" metabelian. Hint : Show that Ò+ß,Ó œÒ+ß,Ó for +ß+" −E and ,ß," − F. Use the fact that EF œ FE . 40. Let K: be a group of order 8 , where : is prime. Suppose that for each +−K, the centralizer G+K œG Ð+Ñ has index " or : . a) Prove that GK++ü . Hint : Assume ÐKÀGÑœ:. Let Mœ G1 ŸG , + + 1−K

Show that MK is normal in . Then show that the elements of KÎM are actually permutations of KÎG+++ , where 1MÐBG Ñ œ 1BG . Show that KÎM is a subgroup of WKÎL . b) Prove that KŸ^ÐKÑw . 41. We have seen that the family subÐKÑ of all subgroups of K is a complete lattice. Let norÐKÑ be the family of normal subgroups of K . a) Show that the join of two normal subgroups EF and is the set product EF. b) Show that norÐKÑ is a complete sublattice of subÐKÑ . c) Prove that the distributive laws E”ÐF∩GÑœÐE”FÑ∩ÐE”GÑ E∩ÐF”GÑœÐE∩FÑ”ÐE∩GÑ

for EßFßG ŸK imply the modular law: For EßFßG ŸK with EŸF, E ” ÐF ∩ GÑ œ F ∩ ÐE ” GÑ A lattice that satisfies the modular law is said to be a modular lattice. d) Prove that norÐKÑ is a modular lattice. 102 Fundamentals of Group Theory

e) Is norÐKÑ necessarily distributive, that is, do the distributive laws necessarily hold? Hint : Consider the % -group Z œ Ö"ß +ß ,ß +,×. f) Prove that subÐKÑ need not be modular. Hint : Consider the alternating group E% of order "# . Let E œ ØÐ" #ÑÐ$ %ÑÙß F œ ØÐ" #ÑÐ$ %Ñß Ð" $ÑÐ# %ÑÙ and G œØÐ"#$ÑÙ. g) Prove the Dedekind law: For Eß Fß G Ÿ K with E Ÿ F , EÐF ∩ GÑ œ F ∩ ÐEGÑ Find an example to show that for arbitrary subgroups Eß Fß G of K , EÐF ∩ GÑ is not necessarily equal to EF ∩ EG and so the condition that EŸF is necessary. h) Let Eß F and G be subgroups of KEŸF with . Prove that if E∩G œF∩G and EG œ FG then EœF. 42. Let KR be a group with subgroups and LR and suppose that LŸK . Prove that if ÐK À LÑ and R are finite and relatively prime, then RŸL . kk 43. Let KL be a group and ŸK . a) Let L©\©K. Show that the relation B´C if B" C−L is an equivalence relation on \ . What do the equivalence classes of this relation look like? Let Ð\ À LÑ be the cardinality of the set of equivalence classes. b) If L©\©]©K, where \L©\ . Show that if Ð\ÀLÑœ Ð] À LÑ  ∞, then \ œ ] . 44. Prove that RKü and KÎR both have the ACC on subgroups if and only if K has the ACC on subgroups. 45. Sometimes it is useful to relate the commutator subgroup ÒLß OÓ of subgroups LO and to the commutator subgroup Ò\ß]Ó of generating sets for LO\] and . Let and be nonempty subsets of a group K . a) Show that

Ò\ß Ø] ÙÓ œ Ò\ß Ø] ÙÓØ] Ù œ Ò\ß ] ÓØ] Ù b) Show that

ÒØ\Ùß Ø] ÙÓ œ Ò\ß ] ÓØ\ÙØ] Ù œ Ò\ß ] ÓØ] ÙØ\Ù

Complex Groups Let KK be a group. Let Z be a nonempty family of subsets of that forms a group under set product. Such a group has been called a complex group based on (see Allen [1]). Let be the union of the subsets in . We denote the K -ZZ identity of ZZ by IE . Also, we denote the inverse of − by E" . In the following exercises, let Z be an arbitrary complex group based on the group K . Cosets, Index and Normal Subgroups 103

46. Let RKü . Show that KÎR is a complex group in which the members form a partition of K . Thus, quotient groups are complex groups. 47. Show that if  is the multiplicative group of all positive rational numbers and if d‘œÖÐ<ß∞ѱ<− × then dd is a complex group. Do the members of form a partition of  ? What is the identity of d ? Does it contain the identity "−  ? Compare the sizes of d and . Could d be a quotient group of  ? 48. Let ™ be the additive group of integers and let 5 be a positive integer. Show that the set

m™5 œ ÖÖ8× ∪ Ò8  5ß ∞Ñ ± 8 − ×

is a complex group. What is the identity of m5 ? What is the negative of the element E œ Ö"× ∪ Ò"  5ß ∞Ñ? Is this the set of negatives in ™ of the elements of E ? 49. Show that all members of Z have the same cardinality. 50. Show that for Eß F − Z , E©F Ê F" ©E " 51. Show that for Eß F − Z , IŸKßEFœI Ê E∪F©Ior ÐE∪FÑ∩Iœg 52. Show that for Eß F − Z , /−EF" ∩FE " Ê EœF and EF" ∪ FE " © I Ê E œ F 53. Suppose that the members of ZZ form a partition of K . Prove that is a quotient group of K , that is, Z is the set of cosets of some normal subgroup of K" . Hint : Show that −II and that ü K . 54. Prove that Z is a quotient group of KI if and only if ŸK . Hint : Prove that if and only if the members of are pairwise disjoint and . IŸK ZZ- ŸK 55. Prove that EÁF−Z Ê E∩Fœgor E∩F is infinite

Chapter 4 Homomorphisms, Chain Conditions and Subnormality

Homomorphisms The structure-preserving functions between two groups are referred to as homomorphisms. Before giving a formal definition, let us make a few remarks about functions.

We denote the action of a function 0ÀW Ä X on = − W by either 0= or 0Ð=Ñ , depending on readability. If k denotes the power set, then the induced map 0ÀkÐWÑ Ä kÐXÑ is defined by 0ÐYÑ œ Ö0Ð?ѱ ? − Y× and the induced inverse map 0ÀkÐXÑÄkÐWÑ" is defined by 0ÐZÑœÖ=−W±0Ð=Ñ−Z×"

If 0ÀW Ä W , then a subset E © W is invariant under 0 , or 0 -invariant , if 0ÐEÑ© E. If YY is a family of functions from WW to , then a subset E © W is - invariant if E0 is -invariant for all 0−Y .

Now we can define homomorphisms.

Definition Let KL and be groups. A function 5ÀKÄL is called a group homomorphism () or just homomorphism if 555Ð+,Ñ œ Ð +ÑÐ ,Ñ The set of all homomorphisms from KL to is denoted by homÐKßLÑ . The following terminology is employed: 1) A surjective homomorphism is an epimorphism, which we denote by 5ÀKq» L.

S. Roman, Fundamentals of Group Theory: An Advanced Approach, 105 DOI 10.1007/978-0-8176-8301-6_4, © Springer Science+Business Media, LLC 2012 106 Fundamentals of Group Theory

2) An injective homomorphism is a monomorphism or embedding , which we denote by 5äÀK L . If there is an embedding from KL to , we say that K can be embedded in LKL and write ä . 3) A bijective homomorphism is an isomorphism, which we denote by 5ÀK ¸L. If there is an isomorphism from KL to , we say that KL and are isomorphic and write K¸L . 4) A homomorphism of K into itself is an endomorphism. The set of all endomorphisms of KÐ is denoted by End KÑ . 5) An isomorphism of K onto itself is an automorphism. The set of all automorphisms of KÐ is denoted by Aut KÑ .

If 5ÀK ÄL is a homomorphism, then it is easy to see that 555"œ"and Ð+" ќР+Ñ" for any +−K . Also, if 55 ÀK¸L , then the inverse map "ÀL¸K is an isomorphism from LK to . The map that sends every element of K to the identity "−L is called the zero map . (We cannot call it the identity map!)

In general, induced inverse maps are more well behaved than induced direct maps. Here is an example.

Theorem 4.1 Let 5ÀK ÄL be a group homomorphism. 1))( a Image preserves subgroups) WŸK Ê5 WŸL b)( Surjective image preserves normality) If 5 is surjective, then WKÊü5ü WL 2)( Inverse image preserves subgroups and normality) XŸL Ê5" XŸK and XLÊü5ü" XK While it is true that isomorphic groups have essentially the same group– theoretic structure, one must be careful in applying this notion to the subgroup structure of a group. For example, in the group ™ of integers, the subgroups #™™™™ œ Ö#8 ± 8 − × and $ œ Ö$8 ± 8 − × are isomorphic but the quotients ™™Î# and ™™ Î$ have different sizes. Thus, isomorphic subgroups need not have the same index.

Example 4.2 Let H' œØßÙ35 be the dihedral group, where 9ÐÑœ$3 and 9Ð5 Ñ œ #. Let W$' be the symmetric group of order ' . The map 0À H Ä W$ defined by Homomorphisms, Chain Conditions and Subnormality 107

0Ð5335 Ñ œ Ð# $Ñ 3 Ð" # $Ñ 5 is an isomorphism and so H¸W'$ . This tells us that the group of symmetries of the triangle is the group of permutations of the vertices.

Definition Let KÀ be a group and let 5 KÄKK be an endomorphism of . 1) 5 is nilpotent if 58 œ! for some 8! , where ! is the zero map. 2) 5 is idempotent if 55# œ We leave it as an exercise to show that the zero map is the only endomorphism that is both nilpotent and idempotent. Sums of Homomorphisms If 57ßÀKÄL are homomorphisms, then the map 5ÀKÄL 7 defined by Ð57  ÑÐ+Ñ œ Ð 57 +ÑÐ +Ñ is a homomorphism if and only if the images imÐÑ57 and im ÐÑ commute elementwise.

Definition Let KL and be groups. The sum of a family 55"8ßáßÀKÄL of homomorphisms is the function defined by

Ð5555"8"8  â  ÑÐ+Ñ œ Ð +ÑâÐ +Ñ for all +−K .

If the images imÐÑ553" commute elementwise, then the sum â58 is a homomorphism and in this case, the sum itself is commutative, that is,

5555"8"8â œ11 â for any 1 −W8 . Moreover, composition distributes over addition,

55ÐâÑœâ"8"8 5 55 55 and

ÐâÑœ5555555"8"8 â

Hence, if the images imÐÑ53 commute elementwise, then the binomial formula holds, 108 Fundamentals of Group Theory

7 744" 8 ÐâÑœ55"8 55" â8 4ßáß4"8 4â4œ7"8Œ for any 7! .

Theorem 4.3 Let 55"8ßáß −End ÐKÑ have the property that the images imÐÑ53 commute elementwise. If each 53 is nilpotent, then so is the sum

75œâ"8 5 Proof. For any 7! ,

87 87 4" 48 75œâ" 58 4ßáß4"8 4â4œ87"8Œ

7 But if each 553 is nilpotent, then there is a positive integer 7œ for which 3 ! for all 3 and so each term in the sum above is the zero map, since one of the exponents 473 is greater than or equal to . Kernels and the Natural Projection The kernel of a homomorphism 5ÀK ÄL is the subgroup kerÐ55 ÑœÖ+−K± +œ"× consisting of all elements of KL that are sent to the identity of . This is easily seen to be a normal subgroup of K . Conversely, any normal subgroup is a kernel.

Theorem 4.4 Let RKü1 . The map R ÀKÄKÎR defined by

1R Ð+Ñ œ +R is an epimorphism with kernel R and is called the natural projection or canonical projection modulo R .

It is possible to tell whether a homomorphism is injective from its kernel.

Theorem 4.5 A group homomorphism 5ÀK ÄL is injective if and only if kerÐ5 Ñ œ Ö"×. Groups of Small Order One of the most important outstanding problems of group theory is the problem of classifying various types of groups up to isomorphism. This problem is called the classification problem and is, in general, very difficult. The classification problem for all groups is unsolved. The classification problem for finitely- generated abelian groups is solved (see Theorem 13.4). The classification Homomorphisms, Chain Conditions and Subnormality 109 problem for finite simple groups seems to have been solved, and we discuss this in more detail in a later chapter.

Groups of relatively small order have been fully classified up to isomorphism. For example, one can find such a classification of groups of order &! or less in Weinstein [36]. Of course, Lagrange's theorem gives a simple solution to the classification problem for groups of prime order, since all such groups are cyclic.

A bit later in the book, we will solve the classification problem for groups of order "& or less. For now, we can solve the classification problem for groups of order ) or less, which are easily analyzed by looking at the possible orders of the elements. In fact, since groups of prime order are cyclic, we need only look at groups of order %' , and ) . Groups of Order % Let K% be a group of order . If K% has an element of order , then K is cyclic and K¸G%. Otherwise, all nonidentity elements have order # , which implies that K is abelian. In this case, if +ß , − K are distinct nonidentity elements, then K œ Ö"ß +ß ,ß +,× is the Klein % -group. Thus, the groups of order % are (up to isomorphism):

1) G% , the cyclic group 2) Z¸G##} G, the Klein 4-group. Groups of Order ' If 9ÐKÑ œ ' , then Cauchy's theorem implies that there exist +ß , − K with 9Ð+Ñ œ $ and 9Ð,Ñ œ # . Since Ø+Ù has index # , it is normal in K and since conjugation preserves order, we have +œ+,,#or +œ+ If +, œ + , then K is abelian. In this case, since 9Ð+Ñ and 9Ð,Ñ are relatively prime, 9Ð+,Ñ œ 9Ð+Ñ9Ð,Ñ œ ' and so K is cyclic. If +,#" œ+ œ+ , then K œØ+ß,Ùß9Ð+Ñœ$ß9Ð,Ñœ#ß,+œ+" , and so KHH is the dihedral group '' . Also, since ¸WK$ , the group is also a symmetric group. Thus, the groups of order ' are (up to isomorphism):

1) G' , the cyclic group 2) H¸W'$ , the nonabelian dihedral (and symmetric) group. 110 Fundamentals of Group Theory

Groups of Order )

Let 9ÐKÑ œ ) . If K has an element of order ) , it is cyclic and K ¸ G) . If every nonidentity element of K#K has order , then is abelian. In this case, let +ß, and -K be distinct elements of with -Á+, . It is easy to see that K œ Ö"ß+ß,ß-ß+,ß+-ß,-ß+,-× and that K¸GGG###}}.

Now suppose that K+% has an element of order but no elements of order ) . Then Ø+Ùü K . For any ,  Ø+Ù , we have K œ Ø+ß ,Ù and since +, − Ø+Ù has order % , we must have ,+," œ +or ,+," œ + $ If ,+," œ + , then K is abelian. Moveover, K has an element -# of order that is not in Ø+Ù . To see this, note that if ,  Ø+Ù and 9Ð,Ñ œ % , then ,# − Ø+Ù has order # and so ,## œ+ . Hence, Ð+,Ñ ###% œ+ , œ+ œ" and so -œ+, . Thus,

K œ Ø+Ù ì Ø-Ù ¸ G%#} G On the other hand, suppose that ,+,"$" œ + œ + . If there is a involution ,ÂØ+Ù, then +, is also an involution and so Theorem 2.36 implies that Ø,ß+,Ù is dihedral of order #9Ð+Ñœ) , that is, KœØ,ß+,Ù¸H).

Finally, if all elements ,  Ø+Ù have order % , then ,# − Ø+Ù and so K œØ+ß,Ùß+Á,ß9Ð+Ñœ%ß,##"$ œ+ ß,+, œ+ which is the quaternion group. Thus, the groups of order ) are (up to isomorphism):

1) G) , the cyclic group 2) GG%#} , abelian but not cyclic 3) GGG###}}, abelian but not cyclic 4) H) , the (nonabelian) dihedral group 5) , the (nonabelian) quaternion group. A Universal Property and the Isomorphism Theorems The following property is key to many other properties of group homomorphisms.

Definition Let KO be a group and let üYK . Let ÐKàOÑ be the family of all pairs ÐLß55 À K Ä LÑ, where À K Ä L is a group homomorphism and O©ker Ð5 Ñ. Homomorphisms, Chain Conditions and Subnormality 111

u G S

∃!τ σ

H

Figure 4.1 Referring to Figure 4.1, a pair ÐWß?ÀKÄWÑ−Y ÐKàOÑ is universal in Y5ÐKàOÑ if for any pair ÐLß ÀK ÄLÑ−Y ÐKàOÑ, there is a unique group homomorphism 7ÀW ÄL for which the diagram in Figure 4.1 commutes , that is, for which 75‰?œ The map 75 is called the mediating morphism for and we say that 5 can be factored uniquely through ?W or that 5 can be lifted uniquely to .

Existence and uniqueness (up to isomorphism) of universal pairs is given by the following theorem.

Theorem 4.6 () Universal pairs Let KO be a group and let ü K . 1)( Existence ) The pair

ÐKÎOß1O À K Ä KÎOÑ

where 1YO is the canonical projection modulo OÐ is universal in KàOÑ . The mediating morphism for 57ÀK ÄL is the map ÀKÎOÄL defined by 75Ð1OÑ œ 1 Also, imÐÑœ75 im ÐÑand ker ÐÑœ 7 ker ÐÑÎO 5 2)( Uniqueness ) Referring to Figure 4.2

π G K G/K

∃!σ ∃!τ u

S

Figure 4.2 112 Fundamentals of Group Theory

if ÐWß?ÀKÄWÑ is also universal in YÐKàOÑ , then the mediating morphisms 57 and are inverse isomorphisms, whence W¸KÎO . Proof. For part 1), a mediating morphism for 5ÀK ÄL , if it exists, must satisfy 75Ð1OÑ œ 1 for all 1−K and so must be unique. But the condition O©ker Ð5 Ñ implies that 7 is a well defined map and it is easy to see that it is a homomorphism. Finally,

imÐ77 Ñ œ ÐKÎOÑ œ 71 ‰O ÐKÑ œ 5 ÐKÑ œim Ð 5 Ñ and

77Ð+OÑ œ ! Í ‰15O Ð+Ñ œ ! Í + œ ! Í + −ker Ð 5 Ñ and so kerÐÑœ75 ker ÐÑÎO.

For part 2), since ÐKÎOß1O Ñ and ÐWß ?À K Ä WÑ are both universal, ? can be factored uniquely through 1O , that is,

71‰œ?O and 1O can be factored uniquely through ? , that is,

51‰?œ O Hence, Љщ?œ?75 But ?‰ can be factored uniquely through itself and since + ?œ? , it follows that 75‰œ +. Similarly, 57‰œ + and so 5 and 7 are inverse isomorphisms.

The following well-known results are direct consequences of the universal property of the pair ÐKÎLß1L Ñ .

Theorem 4.7 () The isomorphism theorems Let K be a group. 1)( First isomorphism theorem) Every group homomorphism 5ÀK ÄL induces an embedding 55ä5ÀKÎker Ð Ñ L of KÎker Ð Ñ defined by 555Ð1ker Ð ÑÑ œ 1 and so K ¸ÐÑim 5 kerÐÑ5 Homomorphisms, Chain Conditions and Subnormality 113

2)( Second isomorphism theorem) If LßO Ÿ K with Oü K , then L∩Oü L and LO L ¸ OL∩O 3)( Third isomorphism theorem) If LŸOŸK with LßOü K , then OÎLü KÎL and KO K ¸ LL„ O Hence ÐKÀOÑœÐKÎLÀOÎLÑ

Proof. For part 1), since ÐKÎOß1O Ñ is universal, there is a unique homomorphism 75715ÀKÎOÄim Ð Ñ for which ‰O œ . The rest follows from the fact that imÐ75 Ñ œ im Ð Ñ and ker Ð 7 Ñ œ ker Ð 5 ÑÎO œ Ö"×.

For part 2), the map 55ÀL ÄLOÎO defined by Ð2Ñœ2O is clearly an epimorphism with kernel L∩O and the first isomorphism theorem completes the proof. Proof of part 3) is left to the reader.

Theorem 4.8 Let KK"# and be groups and let LK3œ"ß#33ü for . Then KK} K K "#¸ "} # LL"#} L " L #

Proof. Let 7}ÀK" K # ÄÐK "" ÎL Ñ } ÐK ## ÎL Ñ be defined by

7Ð+" ß + # Ñ œ Ð+ "" L ß + ## L Ñ We leave it to the reader to show that 7 is an epimorphism with kernel LL"#} . The first isomorphism theorem then completes the proof. The Correspondence Theorem The following correspondence theorem has a great many uses. We use the notation subÐRà KÑ to denote the lattice of all subgroups of K that contain R .

Theorem 4.9 ()Correspondence theorem Let KR be a group, let ü K and let 1ÀK ÄKÎR be the natural projection. Referring to Figure 4.3, 114 Fundamentals of Group Theory

G π G/N K π K/N H π H/N N π {N}

Figure 4.3 let 1Àsub ÐRàKÑ Ä sub ÐKÎRÑ be the map defined by 1ÐLÑ œ LÎR 1) 11 is an order isomorphism, that is, is a bijection for which LŸO Í LÎRŸOÎR for all LßO −sub ÐRàKÑ. In particular, every subgroup of KÎR has the form LÎR for a unique L −sub ÐRàKÑ. 2) Normality is preserved in both directions, that is, LOÍLÎROÎRüü for all LßO −sub ÐRàKÑ. Moreover, the corresponding factor groups are isomorphic, that is, OOL ¸ LRR„ and so 1 also preserves index: ÐO À LÑ œ ÐOÎR À LÎRÑ It follows that subnormality is also preserved in both directions: LOÍLÎROÎRüü üü 3) If R«K , then the property of being characteristic is preserved in only one direction, specifically, LK «ÊL«K RR for any L−sub ÐRàKÑ, but the converse fails. Proof. For the surjectivity of 1 , any subgroup of KÎR has the form WœÖBR±B−\× Homomorphisms, Chain Conditions and Subnormality 115 for some index set . The set is a subgroup of since \ L œ- ÐBRÑ K Bß C − L implies that BRß CR − W and so B" R and BCR are in W , whence BßBC−L" . Also, 1LœLÎRœÖBR±B−LלW and so 1 is surjective.

For part 3), if 51−Aut ÐKÑ , then R ‰5 À K Ä KÎR is an epimorphism with kernel 57"RœR . Hence, the map ÀKÎRÄKÎR defined by 75Ð+RÑ œ Ð +ÑR is an automorphism of KÎR and so 7 ÐLÎRÑ Ÿ LÎR. Thus, for 2−L, Ð2ÑRœÐ2RÑ−LÎR57 and so 52−L , which shows that L«K . As to the converse, let KœOœ # H), LœØ33 Ù and Rœ^ÐKÑœØ Ù. Then L and R are characteristic in H) but LÎ^ÐKÑ ¸ G#) is not characteristic in H Î^ÐKÑ ¸ G#} G#. The rest of the proof is left to the reader. Group Extensions It will be convenient to make the following definition.

Definition Let KL be a group. We refer to subgroups and O of K for which LŸO as an extension . We also refer to LKü as a normal extension. The index of an extension LŸO is ÐOÀLÑ .

This use of the term extension is consistent with its use in field theory, where if J©O are fields, then O is referred to as an extension of J . The term extension has another meaning in group theory, which we will encounter later in the book: An extension of a pair ÐRßÑ of groups is a group K that has a normal subgroup RRKww isomorphic to and for which ÎR¸ . However, since LŸK is an extension whereas ÐRßÑ has an extension, there should be no ambiguity in adopting the current definition.

Various operations can be performed on a group extension to yield another extension.

Definition Let K be a group. Let EŸFŸK and GŸK . 1) The intersection of EŸF with G is E∩G ŸF∩G 2) If GKü , the normal lifting of EŸFG by is EG Ÿ FG 116 Fundamentals of Group Theory

3) If RŸEŸF with Rü F , then the quotient of EŸF by R is EF Ÿ RR and () for want of a better term the unquotient of EÎR Ÿ FÎR is EŸF. Inheritance of Group Properties Let c be a property of groups, such as being cyclic, being finite or being abelian. (Technically, c can be thought of as a subclass of the class of all groups.) We write K−cc to denote the fact that the group K has property . A group property c is isomorphism invariant if K−cc ß L¸K Ê L− We will say that a normal extension EFüc has property if the quotient FÎE has property cü . For example, to say that EF is cyclic is to say that FÎE is cyclic. A property c of groups is inherited by subgroups if K−cc ß LŸK Ê L− and c is inherited by quotients if K−cü ß L K Ê KÎL−c

Preservation of Group Properties The second isomorphism theorem implies that intersection preserves normality, that is, EFßGŸKÊE∩GF∩Güü and that the quotient satisfies F∩G F∩G EÐF∩GÑ F œ¸Ÿ E∩G E∩ÐF∩GÑ E E Hence, any isomorphism-invariant property of EFü that is inherited by subgroups, such as being finite, abelian or cyclic, is inherited by intersections. For example, if EFüü is cyclic, then so is E∩GF∩G. Similar statements can be made about normal liftings, quotients and unquotients, as follows.

Theorem 4.10 Let K be a group with EŸFŸK. Let c be an isomorphism- invariant property of groups. 1)( Intersection ) Normality is preserved by intersection, that is, for GŸK , EFÊE∩GF∩Güü Also, if cc is inherited by subgroups, then is preserved by intersection. Homomorphisms, Chain Conditions and Subnormality 117

2)( Normal lifting) Normality is preserved by normal lifting, that is, for RKü , EFÊERFRüü Also, if cc is inherited by quotients, then is preserved by normal lifting. 3)( Quotient and unquotient) Normality is preserved by quotient and unquotient, that is, for RFü and RŸEŸF, EFÍEÎRFÎRüü Also, c is preserved by quotient and unquotient. Proof. For part 2), to see that ERü FR , note that F normalizes both ER and and so FE normalizes RRŸE . Also, RRE implies that normalizes R and so FR normalizes ER , that is, ERü FR . Since FR œ EFR œ ERF, it follows that FR ERF F F F EÐF ∩ RÑ œ¸ œ ¸ ER ER F ∩ ER EÐF ∩ RÑ E‚ E

Centrality Another very important property of extensions EŸF in a group K that involves more than just the quotient FÎE alone is centrality .

Definition Let EKüü . The extension EFK in is central in K if FK Ÿ^ EEŒ Note that centrality is not a property of FÎE alone, since it depends on KÎE as well, which is why we use the phrase central in K . Theorem 3.40 implies that an extension EFü is central in K if and only if ÒFß KÓ Ÿ E Thus, for RKü and RŸE , EF K EFüü central in KÍ central in RR R and so centrality is preserved by quotient and unquotient. Also, for any LŸK , ÒF ∩ Lß LÓ Ÿ E ∩ L and so EFü central in KÊE∩LŸF∩L central in L Finally, for any RKü , 118 Fundamentals of Group Theory

ÒFRß KÓ œ ÒFß KÓÒRß KÓ Ÿ ER and so EFü central in KÊERŸFR central in K Thus, centrality is preserved by intersection and normal lifting as well.

Theorem 4.11 Let K be a group. The property of an extension being central in K is preserved by intersection, normal lifting, quotient and unquotient. Projections and the Zassenhaus Lemma Combining intersection with lifting allows us to project one normal extension in KE into another. Specifically to project üüF into LO , we first intersect EFü with O to get ÐE ∩ OÑü ÐF ∩ OÑ and then lift by L to get LÐE∩ OÑü LÐF ∩ OÑ

This extension is the projection of EFüü into LO and is denoted by ÐEüü FÑ Ä ÐL OÑ We leave it to the reader to show that the same extension is obtained by first lifting EFü by L and then intersecting with O .

As to the factor group of the projection, the isomorphism theorems give LÐF ∩ OÑ LÐE∩ OÑÐF ∩ OÑ œ LÐE∩ OÑ LÐE∩ OÑ F∩O ¸ ÐF ∩ OÑ ∩ LÐE ∩ OÑ F∩O œ ÐE ∩ OÑÒÐF ∩ OÑ ∩ LÓ F∩O œ ÐE ∩ OÑÐF ∩ LÑ But the last quotient remains unchanged if we reverse the roles of the two extensions EŸF and LŸO . Hence, the reverse projection ÐLüü OÑ Ä ÐE FÑ has an isomorphic quotient, that is, LÐF ∩ OÑ EÐO ∩ FÑ ¸ LÐE∩ OÑ EÐL ∩ FÑ Homomorphisms, Chain Conditions and Subnormality 119

This result was proved by Zassenhaus in 1934 and was given the name butterfly lemma by Serge Lang because of the shape of a certain figure associated with an alternate proof.

Theorem 4.12 ()Zassenhaus lemma [37], 1934 Let K be a group and let EFüüand LO be normal extensions in K . Then the reverse projections ÐEFÑÄÐLOÑüüand ÐLOÑÄÐEFÑ üü have isomorphic factor groups, that is, LÐF ∩ OÑ EÐO ∩ FÑ ¸ LÐE∩ OÑ EÐL ∩ FÑ Let us make a few very simple observations about projections:

1) If EFGüü, then the projections of the contiguous extensions EFü and FGüü into LO are also contiguous, that is, LÐE∩ OÑüü LÐF ∩ OÑ LÐG ∩ OÑ 2) If EFüü is projected into LO where F O , then the projection has top group subgroup O . 3) If EFüü is projected into LO where EŸL , then the projection has bottom subgroup L .

Thus, we can project a series

EEâE"#üüü 8 in KL into an extension ü O to get a new series

LÐE"# ∩ OÑüüü LÐE ∩ OÑ â LÐE 8 ∩ OÑ

In particular, if EŸL"8 and OŸE , then the series runs from LO to . Inner Automorphisms If KÀ is a group, then the map # KÄAutÐKÑ defined by

##+œ + is a group homomorphism, since ##+,œ^ # +, . The kernel of # is ÐKÑ and the image of #5 is InnÐKÑ , which is normal in AutÐKÑ since for any −Aut ÐKÑ and 1−K,

" " " " Ð5#++ 5 Ñ1 œ 5 Ò+Ð 5 1Ñ+ Ó œ Ð5 +Ñ1Ð 5 +Ñ œ #5 1 and so 120 Fundamentals of Group Theory

5 ##+ œ 5+ An automorphism of K that is not an inner automorphism is called an outer automorphism of K and the factor group AutÐKÑÎ Inn ÐKÑ is called the outer automorphism group of K , even though its elements are not automorphisms.

Theorem 4.13 Let K be a group. 1)A The map #ÀK Äut ÐKÑ defined by

##+œ + is a group homomorphism with image InnÐKÑü Aut ÐKÑ and kernel ^ÐKÑ . Hence, K InnÐKÑ ¸ ^ÐKÑ In particular, if KK is centerless, then ¸InnÐKÑ. 2) If LŸK , then

+−RK+ ÐLÑ Í# −Aut ÐLÑ

Moreover, the restricted map #À RKK ÐLÑ ÄAut ÐLÑ has kernel G ÐLÑ and so GKK ÐLÑü R ÐLÑ and RÐLÑ K ä AutÐLÑ GÐLÑK

In particular, if LKüü , then GÐLÑKK and K ä AutÐLÑ GÐLÑK

Characteristic Subgroups Normality is not transitive: If RKü , then the normal subgroups of R are not necessarily normal in KW . Examples can be found in the symmetric group % . However, the property of being characteristic is transitive. Here are some of the basic properties associated with characteristic subgroups.

Theorem 4.14 Let K be a group. 1)A L«K if and only if 55LœL for all −ut ÐKÑ . 2)( Transitivity ) E«F«G Ê E«G and E«Füü G Ê E G Homomorphisms, Chain Conditions and Subnormality 121

3) Any subgroup of a cyclic group is characteristic and so LŸØ+Ùüü K Ê L K

4)( Extension property) For any RŸLŸK, LK R«Kß « Ê L«K RR Proof. For part 3), a finite cyclic group K has only one subgroup of each size and so it must be invariant under any automorphism of K . On the other hand, if KœØ+Ù is infinite cyclic and 55−Aut ÐKÑ , then + must be a generator of K and so 55+œ+ or +œ+" . Hence, 5 is either the identity map or the map that sends any element to its inverse. In either case, any subgroup of K is 5 - invariant.

Definition A nontrivial group K is characteristically simple if it has no nontrivial proper characteristic subgroups.

As an example, the % -group Z œ Ö"ß +ß ,ß +,× is characteristically simple but not simple. Elementary Abelian Groups The simplest type of abelian group is a cyclic group of prime order. Perhaps the next simplest type of abelian group is an external direct product of cyclic groups of the same prime order.

Definition An elementary abelian group K is an abelian group in which every nonidentity element has the same finite order.

Theorem 4.15 Let K be an elementary abelian group. 1) Every nonidentity element of K: has prime order . 2) Writing the group product additively, K is a vector space over ™: ,where if α™−+−K+: and , then α is the sum of α copies of + . The subgroups of K are the same as the subspaces of KK and the group endomorphisms of are the same as linear operators on K . 3) If K is finite, then

K¸™}:: â }™ (This also holds when K is infinite, but we have not yet defined infinite direct products.) Proof. We use additive notation for K9 . For part 1), if Ð+Ñœ787 where " , then 9Ð8+Ñ œ 7 and so 78 œ 7 , that is, 8 œ " . Thus, 9Ð+Ñ œ : is prime for all nonidentity +−K . For part 2), let Š and  denote addition and multiplication in ™™: , respectively. If 8− is nonnegative, let 8‡+ be the sum of 8 copies of +ß−. Then for α" ™: , there exists an integer 8 for which 122 Fundamentals of Group Theory

α"ŠœÐÑ8: α" and so ЊÑ+œÒÐÑ8:Ó‡+œÐч+œ+,α" α" α" α " Similarily, Ðα  " Ñ+œÒ α" 8:Ó‡+œÐα" ч+œ α Ð " +Ñ The other requirements of scalar multiplication are also met, namely, "+ œ + and αααÐ+  ,Ñ œ +  , We leave proof of the remaining part of part 2) to the reader. For part 3), K is vector-space isomorphic to a direct sum of a certain number of one-dimensional subspaces of K+ , that is, subspaces of the form ™: œØ+Ù for +−K . These subspaces are cyclic subgroups of K and the vector space isomorphism is also a group isomorphism.

Theorem 4.16 The following are equivalent for an abelian group K : 1) K is elementary abelian 2) K is characteristically simple and has at least one nonidentity torsion element. Proof. If KK is an elementary abelian group, then the automorphisms of are the linear automorphisms of KW . It follows that if ŸK is nontrivial and proper, then for any nonzero +−W and nonzero B−KÏW , there is an automorphism sending +B to and so W is not characteristic in KK . Hence, is characteristically simple. Of course, K has a nonidentity torsion element.

For the converse, assume that K: is characteristically simple and let be a prime dividing the order of some element +−E . Then (using additive notation)

E: œ Ö+ − E ± :+ œ !× is nontrivial and characteristic in EEœE , whence : is an elementary abelian group.

Note that the :Ð -quasicyclic group ™ :∞Ñ , which has proper subgroup lattice Ö!×  Ø"Î:Ù  Ø"Î:# Ù  â is characteristically simple, since there is at most one subgroup of any given size. However, it is not an elementary abelian group since it has no nonzero torsion elements. Homomorphisms, Chain Conditions and Subnormality 123

Multiplication as a Permutation If K+ is a group, then multiplication by −K is a permutation of K ; specifically, the multiplication map .+ÀK ÄK defined by

.+Bœ+B is bijective. In this context, multiplication is also referred to as (left ) translation.

The map ..ÀK ÄWK+ that sends + to provides a representation of the elements of the group KK as permutations of the set . Moreover, the representation . is a group homomorphism, since

...+,œ + , This is not the only way to represent the elements of a group as permutations of some set. For example, if LŸK , then the elements of K can also be represented as permutations of a quotient set KÎL via the multiplication map 5+ÀK ÄKÎL defined by

5+Ð1LÑ œ +1L

It is clear that 55+ is a permutation of KÎL and that the map ÀKÄWKÎL sending + to 5+ is a group homomorphism.

The representation map .ÀK ÄWK is described by saying that K acts on itself by ()left translation and the representation map 5ÀK ÄKÎL is described by saying that KKÎL acts on by ()left translation . Both of these representations fit the pattern of the following definition.

Definition An action of a group K\ on a nonempty set is a group homomorphism -ÀK ÄW\ , called the representation map for the action. Thus, -+-Ð"Ñ œ ß Ð+" Ñ œ Ð - +Ñ" and -Ð+,Ñ œ -- Ð+Ñ Ð,Ñ for all +ß , − K . When -- is a group action, we say that K\ acts on by . The permutation --++ is usually denoted by + , or simply by itself.

We should mention that translation is not the only important action of a group on a set. We will see in a later chapter that conjugation is also an important group action.

Since a representation map is assumed only to be a homomorphism, it is possible for two distinct elements of KW to have the same representation in \ . Although it may seem at first that this is not a particularly desirable quality, we will see that such representations can yield important results. An injective representation, that is, an embedding -äÀK W\ is said to be faithful . In this 124 Fundamentals of Group Theory

case, KW is isomorphic to a subgroup of \ . The action of K on itself by translation is faithful but the action of KKÎL on by translation need not be faithful.

Here is one interesting application of left translation.

Theorem 4.17 Let LK be a proper subgroup of a group . Then there are distinct group homomorphisms 57ßÀKÄO into some group O that agree on L. Proof. For BÂKÎL , we construct two distinct group actions of K on the set \œKÎL∪ÖB× that agree on LÀKÄW . Let 5 \ be left translation on KÎL that also leaves B fixed, that is, for any +−K ,

55++Ð,LÑ œ +,Land B œ B and let 7ÀK ÄW\ be defined by

75++œÐLBÑ ÐLBÑ

If 2−L , then 5522 fixes both LB and and so and ÐLBÑ are disjoint permutations, whence 5722œ2−L for all . On the other hand, 7 and 5 are distinct since if +ÂL , then

75++LœLbut LÁL

The Left Regular Representation: Cayley's Theorem

The action .ÀK ÄWK of K on itself by left translation is called the left regular representation of K . Since the left regular representation is faithful, it follows that every group is isomorphic to a subgroup of some symmetric group!

Theorem 4.18 ()Cayley's theorem [7], 1854 Every group K is isomorphic to a subgroup of the symmetric group WKK , via the action of on itself by left translation. Multiplication by KKÎL on ; the Normal Interior

If LK has finite index, the action 5ÀKÄWKÎL of K on KÎL by left translation yields a variety of remarkable consequences, mainly due to the nature of the kernel of this action, which is the intersection of all conjugates of L: Homomorphisms, Chain Conditions and Subnormality 125

kerÐÑœÖB−K±55+B œ× œÖB−K±B+Lœ+L for all +−K× œÖB−K±B+−+L for all +−K× œÖB−K±B−+" L+ for all +−K× œL+ , +−K Thus,

O³ L+ ,+−K is both normal in KLR and contained in . Moreover, if is any normal subgroup of KL that is contained in , then RœR++ ŸL for all +−K and so RŸO . In other words, O is the largest subgroup of L that is normal in K .

Theorem 4.19 Let LŸK . The largest subgroup of L that is normal in K is called the normal interior or core of LL , which we denote by ‰ . The core of L is the kernel Lœ‰+ L , +−K of the action of KKÎL on by left translation.

Thus, left translation 5ÀK ÄWKÎL induces an embedding K ä W L‰ KÎL ‰ of KÎL into WKÎL and so ÐK À L‰ Ñ ± ÐK À LÑx In particular, if KL is simple, then ‰ is trivial and so 9ÐKÑ ± ÐK À LÑx These facts have some rather interesting consequences relating to the existence of normal subgroups and subgroups of small index of a group.

Theorem 4.20 Let KL be a group and let ŸK have finite index. Then ‰ KÎLä WKÎL and so 126 Fundamentals of Group Theory

ÐK À L‰ Ñ ± ÐK À LÑx In particular, ÐK À L‰ Ñ is also finite and ÐL À L‰ Ñ ± ÐÐK À LÑ  "Ñx 1) Any of the following imply that LKü : a) LÐ is periodic and KÀLÑœ: is equal to the smallest order among the nonidentity elements of L . b) K9 is finite and ÐLÑÐ and ÐKÀLÑ"Ñx are relatively prime, that is, for all primes : , : ± 9ÐLÑ Ê : ÐK À LÑ which happens, in particular, if ÐK À LÑ is equal to the smallest prime dividing 9ÐKÑ. 2) If KK is finitely generated, then has at most a finite number of subgroups of any finite index 7 . 3) If K is simple, then 9ÐKÑ ± ÐK À LÑx a) If KK is infinite, then has no proper subgroups of finite index. b) If K9 is finite and ÐKÑy±7x for some integer 7K , then has no subgroups of index 7 or less. Proof. For part 1), first note that :; is a prime. If is a prime dividing ÐLÀL‰Ñ , then Cauchy's theorem implies that there is an 2−L for which ; œ 9Ð2L‰ Ñ ± 9Ð2Ñ, whence ; : , a contradiction to the fact that ; ± ÐL À L‰ Ñ ± Ð:  "Ñx Hence, ÐL À L‰‰ Ñ œ " and L œ Lü K.

For part 2), ÐL À L‰ Ñ divides both ÐÐK À LÑ  "Ñx and 9ÐLÑ and so must equal " ÐL À L‰ Ñ ± Ð7  "Ñx But ÐL À L‰ Ñ also divides 9ÐLÑ , which is relatively prime to Ð:  "Ñx and so ÐL À L‰ Ñ œ ".

For part 3), suppose that the result is true for all normal subgroups. Then there are only finitely many possibilities for normal interiors of subgroups of index 77, since such a normal interior must have index dividing x . But each normal subgroup R of finite index can be the normal interior for only finitely many subgroups, since there are only finitely many subgroups containing R .

Now, normal subgroups correspond to kernels of homomorphisms. In particular, if ÐK À RÑ œ 5, then the homomorphism Homomorphisms, Chain Conditions and Subnormality 127

51‰ÀKÄKÎRWR5ä where 5 is the left regular representation of KÎR in W5 , has kernel R . Thus, distinct normal subgroups R5 of index yield distinct homomorphisms from K into WK5 . However, since is finitely generated, there are only a finite number of such homomorphisms and so there are only a finite number of normal subgroups of K5 of index .

Example 4.21 If 9ÐKÑ œ :8 ? with : prime and :  ? , then we will prove in a later chapter that KL::K has a subgroup of order 8 (a Sylow -subgroup of ). Since :Ð?"Ñx8 and are relatively prime, it follows that LKü . The Frattini Subgroup of a Group The following subgroup of a group is most interesting.

Definition The Frattini subgroup FÐKÑ of a finite group K is the intersection of the maximal subgroups of K .

Definition Let K+ be a group. An element −KK is called a nongenerator of if whenever the subset W©K generates K, then so does the set WÏÖ+× . Thus, a nongenerator is an element that is not needed in any generating set.

Here are the basic properties of the Frattini subgroup of a finite group. But first a definition.

Definition Let KO be a group and let ŸKL . A subgroup of K is called a supplement of OKœLO if .

Theorem 4.22 Let K be a finite group. 1) FÐKÑ is the set of all nongenerators of K . 2) FÐKÑ « K. 3) The following are equivalent: a) Every maximal subgroup of K is normal. b) KÎF ÐKÑ is abelian. 4) If O–K , then OŸF ÐKÑ if and only if O has no proper supplements. Proof. For part 1), if W©K does not generate KW , then is contained in a maximal subgroup QÐ , which also contains FFKÑ and so W∪ÐKÑ©Q does not generate KÐ . Hence, the elements of F KÑ are nongenerators. Conversely, if BK is a nongenerator of and BÂQQ for some maximal subgroup , then Q∪ÖB× generates K and so Q generates K , which is false. Thus, all nongenerators of KÐKÑ are in F .

For part 2), if 5`−ÐKÑAut , then the induced map is a bijection on the family of maximal subgroups of K and so 128 Fundamentals of Group Theory

5FÐ ÐKÑÑ œ 5 ` œ 5` œ ` œ F ÐKÑ Š‹,,, For part 3), assume that every maximal subgroup QK of is normal. Then KÎQ has no proper nontrivial subgroups and so is abelian. Hence, KŸQw and so KŸwwFF ÐKÑ . Conversely, if KŸ ÐKÑ and Q is a maximal subgroup of K , then KŸQw and so Qü K .

For part 4), assume first that OŸF ÐKÑ . If LK , then LŸQ for some maximal subgroup Q of K and so LOŸQK. Conversely, if O has no proper supplements, then every maximal subgroup QQŸQOK satisfies and so QœQO , whence OŸQ and so OŸF ÐKÑ . Subnormal Subgroups We wish now to take a closer look at the concept of subnormality. As to the existence of subnormal subgroups, we have the following examples to show that all possibilities may occur, and that a subnormal subgroup need not be normal.

Example 4.23 1) A simple group has no nontrivial proper subnormal subgroups. 2) In the dihedral group HœØßÙ) 53 , we have # Ø553 Ù–Ø ß Ù–H)

and so ØÙ5 is subnormal in H)) but not normal in H . In fact, all subgroups of H) are subnormal. 3) In the symmetric group WL$ , the subgroup œØÐ"#ÑÙ is maximal and so the only sequence of subgroups from LW to $$ is LŸW . Since L is not normal in W$$ , it follows that L is not subnormal in W . Of course, ØÐ"#$ÑÙ is subnormal in WW$$ , being normal in .

The following theorem outlines the simplest properties of subnormality and is a direct consequence of Theorem 4.10.

Theorem 4.24 Let KL be a group. Let ßOŸK . 1)( Transitivity ) LOüü and OKÊLKüü üü 2)( Intersection ) If PŸK , then LOÊL∩PO∩Püü üü In particular, LŸOŸKß Lüü K Ê Lüü O Homomorphisms, Chain Conditions and Subnormality 129

and LKßOKÊL∩OKüü üü üü 3)( Normal lifting) If RKü , then LOÊLRORüü üü 4)( Quotient/unquotient) If RŸL and Rü O , then LOÍLÎROÎRüü üü

Intersections and Subnormality While the intersection of two, and hence any finite number, of subnormal subgroups is subnormal, this is not the case for an arbitrary family of subnormal subgroups.

Example 4.25 Let K be the infinite dihedral group, KœÖ35333 ß ±3− ™ × where 9Ð53 Ñ œ # , 9Ð Ñ œ ∞ and 3553 œ " . If

#3 LœØß3 53 Ù then and so . But, is self-normalizing (equal LL3"üü 3 LK3 ü+ LœØÙ3 5 to its own normalizer) and so is not subnormal in K .

However, since the family subnÐKÑ is closed under finite intersection, Theorem 1.6 does imply the following.

Theorem 4.26 If a group KÐ has the DCC on subn KÑ , then the intersection of any family of subnormal subgroups is subnormal. The Sequence of Normal Closures of a Subgroup

If LKüü , then the first step down in a series for L is an extension LK" ü . Moreover, since ncÐLß KÑ Ÿ L", the largest possible first step down in a series from LK to is ncÐLß KÑü K By repeatedly taking normal closures of LL , we get a series from to K that descends more rapidly than any other series from LK to .

Definition Let KL be a group and let ŸK . The sequence of normal closures of LK in is defined by 130 Fundamentals of Group Theory

LœKßLœÐLßLÑÐ!Ñ Ð"Ñ nc Ð!Ñ and LÐ3"Ñ œÐLßLÑnc Ð3Ñ for 3 !.

To see that the sequence of normal closures descends more rapidly than any other proper series

LœL<<" –L –â–L ! œK it is clear that LŸLÐ!Ñ !3 and if LŸLÐ3Ñ , then

LÐ3"Ñ œnc ÐLß LÐ3Ñ Ñ Ÿnc ÐLß L33" Ñ Ÿ L

Hence, LŸLÐ3Ñ 3 for all 3 .

Theorem 4.27 A subgroup LK of is subnormal in K if and only if the sequence of normal closures of LK in reaches L , in which case this sequence is a series of shortest length from LKL to . If üü K , then the length of the sequence of normal closures is called the subnormal index of LK in , which we denote by =ÐLß KÑ .

We gather a few facts about subnormal indices.

Theorem 4.28 Let KL be a group and let üüK have sequence of normal closures

LœLÐ=Ñ –L Ð="Ñ –â–LÐ!Ñ œK 1)( Triangle inequality) Lüü O üü K Ê =ÐLß KÑ Ÿ =ÐLß OÑ  =ÐOß KÑ 2)Aut If 5 −ÐKÑ , then

55ÐLÐ3Ñ Ñ œ Ð LÑÐ3Ñ for all 3 ! . a) If 55LœL , then LÐ3Ñ œL Ð3Ñ for all 3 . In particular, = RÐLÑ© RÐLÑ KK, Ð3Ñ 3œ!

and so if OŸK normalizes L , then O normalizes every LÐ3Ñ . b) The sequence of normal closures for 5LK in is

55L œ ÐLÐ=Ñ Ñ– 5 ÐL Ð="Ñ Ñ–â– 5 ÐLÐ!Ñ ÑœK and so =ÐLß KÑ œ =Ð5 Lß KÑ. Homomorphisms, Chain Conditions and Subnormality 131

Proof. For part 2), we have 55ÐLÐ!Ñ Ñ œ Ð LÑÐ!Ñ and if 55 ÐLÐ3Ñ Ñ œ Ð LÑÐ3Ñ, then

555555ÐLÐ3"Ñ Ñœnc Ð Lß ÐLÐ3Ñ ÑÑœnc Ð LßÐ LÑÐ3Ñ ÑœÐ LÑÐ3"Ñ

*Joins and Subnormality The question of whether the join of subnormal subgroups is subnormal is much more involved than the same question for intersection. Of course, if O is subnormal in KL and if is normal in K , then we may lift the sequence of normal closures of OK in

OœOÐ>Ñ –O Ð>"Ñ –â–OÐ!Ñ œK (4.29) by LL to get a series for the join O . Thus, LKßOüüüüü KÊLOK Actually, the process of lifting by L is the same as projecting the sequence (4.29) into the extension LKü .

If neither factor LO or is normal, we are tempted to project (4.29) into each extension L–LÐ3Ñ Ð3"Ñ and concatenate the resulting series. This almost works, the problem being that it produces a series from

LÐO∩LÐ=Ñ Ð>Ñ Ð="Ñ ÑœLO∩LÐ="Ñ to KL , rather than from O to K . So a slight modification is in order. Note that the unwanted LÐ="Ñ comes from the upper endpoint of the extension L–LÐ=Ñ Ð="Ñ.

Thus, if we assume that OLOL3 normalizes , then normalizes Ð3Ñ for all and so LOÐ3Ñ is a subgroup of KLLO and Ð3Ñü Ð3"Ñ . If we now project (4.29) into the extension LLOÐ3Ñü[ Ð3"Ñ , the result is a series 3 with lower endpoint

LÐO∩LÐ3Ñ Ð>Ñ Ð3"Ñ OÑœLOÐ3Ñ and upper endpoint LOÐ3"Ñ , that is,

[üü3ÀLÐ3Ñ O â LÐ3"Ñ O

Moreover, since [[Ð3"Ñ and Ð3Ñ are contiguous, the concatenation

[[Ð=Ñ Ð="Ñâ [ Ð"Ñ is a series from LO to K and so LOüü K .

Note also that each of the series [3 has length at most > and so =ÐLOß KÑ Ÿ =ÐLß KÑ=ÐOß KÑ 132 Fundamentals of Group Theory

Theorem 4.30 If LßOüü K and if O normalizes L , then LO œ ØLßOÙüü K and =ÐØLß OÙß KÑ Ÿ =ÐLß KÑ=ÐOß KÑ

*The Subnormal Join Property Theorem 4.30 implies the following useful characterization of the join question.

Theorem 4.31 Let KL be a group and let ßOüüK. The following are equivalent: 1) ØLß OÙüü K 2) LKO üü 3) ÒLß OÓüü K. Proof. Recall from Theorem 3.40 that ÒLßOÓüü LÒLßOÓœLOO L O œØLßOÙ Thus, 1)ÊÊ 2) 3). On the other hand, if 3) holds, then since LKüü and ÒLß OÓüü K and L normalizes ÒLß OÓ , Theorem 4.30 implies that LO œ LÒLßOÓüü K and so 2) holds. If 2) holds, then 1) holds since OL normalizes O .

Definition If a group K has the property that the join of any two subnormal subgroups is subnormal, then K is said to have the subnormal join property, or SJP.

Theorem 4.31 gives us one simple criterion for the SJP. If the commutator subgroup KKw of has the property that all of its subgroups are subnormal, then ÒLßOÓ is subnormal in Kw and therefore also in K , for all LßO ŸK and so Theorem 4.31 implies that KK has the SJP. In particular, this occurs when w is abelian, that is, when K is metabelian .

Theorem 4.32 If the commutator subgroup KKw of has the property that all of its subgroups are subnormal, in particular, if KK is metabelian, then has the subnormal join property.

On a different line, if Kw has the ACC on subnormal subgroups (a finiteness condition), then K has the subnormal join property.

Theorem 4.33 ()Robinson Let KK be a group for which w has the ACC on subnormal subgroups. Then K has the subnormal join property. Homomorphisms, Chain Conditions and Subnormality 133

Proof. Let LO and be subnormal in K . The proof is by induction on =œ=ÐLßKÑ. If =Ÿ" , the result is clear, so assume that = # . If +−K then ++ LÐ"Ñ œnc ÐL ß KÑ œ nc ÐLß KÑ œ LÐ"Ñ

+ and so LL and are subnormal subgroups of LLÐ"Ñ . But Ð"Ñ also has the property that its commutator subgroup has the ACC on subnormal subgroups. www To see this, note that L«LKÐ"Ñ Ð"Ñ üü implies that LKÐ"Ñ and so subn ÐLÑÐ"Ñ w is contained in subnÐK Ñ . Thus, since =ÐLß LÐ"Ñ Ñ œ =  ", the inductive ++ hypothesis implies that ØLß L Ùüü LÐ"Ñ and so ØLß L Ùüü K.

Moreover, this can be extended to more than one conjugate of L . For example, if , − K , then ØLßL+, Ù œ ØL , ßL ,+ Ù and so the join of ØLßL+ Ù and ØLßL+, Ù is the subnormal subgroup ØLßL+, ßL Ù. Note also that we can replace L by any conjugate of L and so ØLßL++"8 ßáßL Ùüü K for all +ßáß+"8 −K.

Next, we show that

ØÒLß+"8 ÓßáßÒLß+ ÓÙüü K for all +ßáß+"8 −K. First, note that ØLß ÒLß +ÓÙ œ ØLß L+ Ù and so

++"< ØLßÒLß+"< ÓßáßÒLß+ ÓÙœØLßL ßáßL Ùüü K

But Theorem 3.37 implies that LÒ normalizes each Lß+3Ó and so

ØÒLß+"< ÓßáßÒLß+ ÓÙüü ØLßÒLß+ "< ÓßáßÒLß+ ÓÙü K Now we can complete the proof. The ACC on subnÐKw Ñ implies that there is a finite subset M œÖ+"8 ßáß+ שO for which

ÒLßOÓœQM" ³ØÒLß+ ÓßáßÒLß+ 8 ÓÙ for if not, then there is a strictly increasing sequence

QQâMM"# of subnormal subgroups of KÒLßOÓKw . Hence, üü .

We refer the reader to Robinson [26], page 389, for an example of a group that does not have the SJP. 134 Fundamentals of Group Theory

*The Generalized Subnormal Join Property It is not necessarily the case that a group with the SJP also has the property that the join of any family of subnormal subgroups of K is subnormal. This is called the generalized subnormal join property or GSJP.

Theorem 4.34 ()Wielandt [35], 1939 Let K be a group. 1) If KÐ has the ACC on subn KÑ , then K has the GSJP. 2) If K has BCC on subnÐKÑ , in particular, if K is finite, then subnÐKÑ is a complete sublattice of subÐKÑ . Proof. For part 1), since KKw also has the ACC on subnormal subgroups, has the SJP. Hence, subnÐKÑ is closed under finite join and so Theorem 1.6 implies that K has the GSJP.

Robinson has provided the following characterization of the GSJP, whose proof uses ordinal numbers.

Theorem 4.35 ()Robinson A group K has the generalized subnormal join property if and only if the union of every chain of subnormal subgroups is subnormal. Proof. If K has the GSJP, then the union of a chain of subnormal subgroups is the join of that chain and so is subnormal. For the converse, we first show that K has the SJP by induction on the smaller of the two subnormal indices. Let LßOüü K and let =œmin Ö=ÐLßKÑß=ÐOßKÑ× If =Ÿ" , then ØLßOÙüü K. Assume that = # and that the result holds when the subnormal index is less than = .

Well order the set O so that

OœÖ5α ±α$  × where $ is an ordinal number and 5œ"! . Let

5α PœØL±" α"  Ù be the join of the first " conjugates of LO by elements of . Then

O P"$ ŸLÐ"Ñ and P œL œLÒLßOÓ

To show that PK$ is subnormal in , we use transfinite induction on " .

If "üœ! , then P"" œLü K. For successor ordinals, if PKPüü , then " 55"" and LL are contained in Ð"Ñ and =ÐLßLÐ"ÑÑœ=". Also, LÐ"Ñ has the property that the union of any chain of subnormal subgroups is subnormal. Hence, the induction hypothesis implies that Homomorphisms, Chain Conditions and Subnormality 135

5" PœØPßLÙL""" üüÐ"Ñ ü K Finally, if - is a limit ordinal, then Pœ ÖP±α-  × -α. is the union of a chain of subnormal subgroups, which is subnormal in K by hypothesis. Hence, the transfinite induction is complete and PK$ üü , whence ØLß OÙüü K by Theorem 4.31. Thus, K has the SJP.

Now, let ff©ÐKÑsubn . The join is not affected by including in the join of every finite subset of ff , so we may assume that is closed under finite joins. For convenience, we refer to a family cc©ÐKÑsubn as good if contains f and is closed under finite joins and chain joins. Then subnÐKÑ is good and the intersection ffw of all good families is the smallest good family containing . Let Nœ ffand Nww œ 22 Then N©Nww . But if N§N , then there is an W− wfww for which W©yN and so if gfœÖW−w ±W©N× it follows that fg©§ fww. But g is also good and so the minimality of f implies that NœNww . Finally, since f is closed under chain join, Zorn's lemma implies that ffww has a maximal member QQ§N . If , then there is an R−w for which R©yQ and so Q”R−fw, which contradicts the maximality of Q , whence QœNw and ffœœQ−©ÐKÑww fsubn 22 Thus, K has the GSJP. When All Subgroups Are Subnormal There are a variety of ways to characterize finite groups in which all subgroups are subnormal. (We will characterize groups in which all subgroups are normal in Theorem 5.20.) To explore this further, we need some additional terminology.

By definition, the normalizer RÐLÑK of a subgroup LK of is the largest subgroup of KL in which is normal. Thus, a subgroup that is equal to its own normalizer is as “unnormal” as possible, since it is normal only in itself.

Definition Let K be a group. 1) A subgroup LŸK is self-normalizing if LœRK ÐLÑ. 2) A group KK has the normalizer condition if has no proper self- normalizing subgroups, that is, if

LK Ê LRK ÐLÑ 136 Fundamentals of Group Theory

Note that the term self-normalizing is a bit misleading, since every subgroup normalizes itself. (We would prefer the term unnormal .)

A proper subnormal subgroup L cannot be self-normalizing, since the first step in a series shows that L is normal in some subgroup larger than itself. Hence, if all subgroups of KK are subnormal, then has the normalizer condition. The converse is also true if KK is finite (or more generally, if has the ACC on subgroups), since the proper series of normalizers

L – RKKK ÐLÑ – R ÐR ÐLÑÑ – â must eventually reach K .

Theorem 4.36 The following are equivalent for a finite group K : 1) Every subgroup of K is subnormal 2) K has the normalizer condition.

If KQ has the normalizer condition, then any maximal subgroup must be normal in KQ , for we have RKÐQÑQ and the maximality of implies that RÐQÑœKK . Thus, with respect to the following conditions on a finite group K:

1) Every subgroup of K is subnormal 2) K has the normalizer condition 3) Every maximal subgroup of K is normal 4) KÎF ÐKÑ is abelian we have proved (see Theorem 4.22) that 1)ÍÊÍ 2) 3) 4) For finite groups, it is our eventual goal to prove not only that these four conditions are equivalent, but that they are equivalent to several other conditions, one of which is that K satisfy a strong converse of Lagrange's theorem, namely, if 8±9ÐKÑ , then K has a normal subgroup of order 8 . Chain Conditions Let us take a closer look at chain conditions for groups, beginning with the definition.

Definition Let KK be a group and let f be a family of subgroups of . 1)( A group K satisfies the ascending chain condition ACC) on f if every ascending sequence

LŸLŸâ"# of subgroups in f must eventually be constant, that is, if there is an 8! Homomorphisms, Chain Conditions and Subnormality 137

such that LœL85 8 for all 5 ! . In this case, we also say that f has the ACC. 2)( A group K satisfies the descending chain condition DCC) on f if every descending sequence

L L â"# of subgroups in f must eventually be constant, that is, if there is an 8! such that LœL85 8 for all 5 ! . In this case, we also say that f has the DCC. 3)( A group KK satisfies both chain condition BCC) on f if has the ACC and the DCC on ff . In this case, we also say that has the BCC.

Theorem 1.5 implies that K has the BCC on ff if and only if has no infinite chains.

Our main interest will center on the case where KRK is a group, üf and is one of the following families: subÐRà KÑß nor ÐRà KÑß subn ÐRà KÑ of all subgroups, normal subgroups and subnormal subgroups, respectively, of KR that contain .

The ACC and DCC are, in general, independent of each other, that is, all four combinations are possible. For example, a finite group has both chain conditions on subgroups. An infinite cyclic group (such as the integers) has the ACC on subgroups but not the DCC on subgroups. The :Ð -quasicyclic group ™ :∞Ñ has the DCC on subgroups but not the ACC. Finally, the group of rational numbers  has neither chain condition on subgroups.

The chain conditions can be characterized as follows.

Definition Let K© be a group and let f subÐKÑ . 1) K has the maximal condition on ff if every nonempty subfamily of contains a maximal element. 2) K has the minimal condition on ff if every nonempty subfamily of contains a minimal element.

Theorem 4.37 Let K© be a group and let f subÐKÑ . 1) KK has the maximal condition on ff if and only if has the ACC on . 2) KK has the minimal condition on ff if and only if has the DCC on . Proof. Suppose K satisfies the maximal condition on f and that

LŸLŸâ"# is an ascending sequence of members of f . Then the subgroups L5 have a maximal member LL88 , which implies that 5ŸL8 for all 5 ! . Conversely, 138 Fundamentals of Group Theory suppose K satisfies the ACC on f` and let be a nonempty subfamily of f . If L−"#`` is not maximal, then there is an L− for which LL"# . If L# is not maximal, then there is an L$ for which L "#$ L L . This must stop after a finite number of steps and so ` must have a maximal member. The proof of part 2) is analogous.

In certain cases, the ACC can be characterized in another important way. If f ©ÐKÑsub is closed under arbitrary intersections, then every subset \K of is contained in a smallest element of ff , called the -closure of \ , which is Ø\Ù œ ÖL −f ± \ © L× f , Note that the set on the right is nonempty, since the assumption that f is closed under arbitrary intersections implies that f contains the empty intersection, which is KØ itself. We say that \Ù\f is f-generated by and if \ is a finite set, we say that Ø\Ùf is finitely f-generated by \ .

Theorem 4.38 Let K© be a group and let f subÐKÑ be closed under arbitrary intersections and closed under unions of ascending sequences. Then K has the ACC on ff if and only if every L− is finitely f -generated. Proof. Suppose KL satisfies the ACC on ff and let − . If L is not finitely f - generated, then for any 2−L" , we have

Ø2" Ùf  L

Hence, there is an 2−LÏØ2Ù#"f for which

Ø2""# Ùff  Ø2 ß 2 Ù  L We can continue to choose elements to produce an infinite strictly ascending sequence, in contradiction to the ACC on ff . Hence, L is finitely -generated.

Conversely, suppose every element of ff is finitely -generated and let

LŸLŸâ"# be an ascending sequence of subgroups in . Then and so ffLœ- L5 − LœØ\Ùf for some finite set \ . Hence, there is an index 7 for which \©L777 and so LœL . But then L5 œL for all 5 ! .

Note that in the preceeding theorem, f can be subÐKÑ , nor ÐKÑ , the family of all characteristic subgroups of K or the family of all fully-invariant subgroups of K.

We next describe how the chain conditions are inherited.

Theorem 4.39 Let KR be a group and let üYK . Let ÐRàKÑ be one of the following families of subgroups of K : Homomorphisms, Chain Conditions and Subnormality 139

subÐRà KÑß nor ÐRà KÑß subn ÐRà KÑ and let YYÐKÑ œ ÐÖ"×ß KÑ. Write Y − ACC to denote the fact that Y has the ACC, and similarly for the DCC . 1)( Quotients ) YYÐRà KÑ −ACC Í ÐKÎRÑ − ACC 2)( Extension ) YYÐRÑß ÐKÎRÑ −ACC ÊY ÐKÑ − ACC 3)( Direct products)

YYÐK"# Ñß ÐK Ñ −ACC ÍY} ÐK"# K Ñ − ACC Similar statements hold for the DCC in place of the ACC . Proof. Part 1) follows from the correspondence theorem. For part 2), let

KŸKŸâ"# be an ascending chain in YÐKÑ . The sequences

K∩RŸK∩RŸâ"# and

KRŸKRŸâ"# are ascending chains in YYÐRÑ and ÐRà KÑ , respectively. Since part 1) implies that YÐRà KÑ has the ACC, each sequence is eventually constant and so there is an index 7 for which

K73 ∩RœK7 ∩Rand K73 RœK 7 R for all 3 ! . Hence, Theorem 2.18 implies that K73 œK 7 for all 3 ! and so YÐKÑ − ACC. A similar argument holds for the DCC.

For part 3), let YYÐK"# Ñ , ÐK Ñ − ACC and let

T œ K"#}} K ß R " œ K " Ö"×ß R # œ Ö"× { K #

Then YYÐKÑ−33 ACC implies that ÐRÑ− ACC for 3œ"ß# . Also, TÎR " ¸R# and so YYÐTÎRÑ−" ACC. Hence, ÐTÑ− ACC by the extension property. Conversely, if YYYYÐT Ñ − ACC, then ÐR33 Ñ © ÐT Ñ implies that ÐR Ñ − ACC and so YÐK3 Ñ − ACC for 3 œ "ß # . Chain Conditions and Homomorphisms The presence of a chain condition can have a significant impact on homomorphisms. For example, if K has the ACC on normal subgroups, then any surjective endomomorphism 5ÀK ÄK is also injective. To see this, consider the kernel sequence of 5 : 140 Fundamentals of Group Theory

kerÐÑŸ55 ker Ð#$ ÑŸ ker Ð 5 ÑŸâ

Since this must eventually be constant, we have kerÐÑœÐ5588 ker 5 Ñ for all 5 !. Now, if +−ker Ð55 Ñ , then +œ8 , for some ,−K and so "œ55 +œ8" , and so ,−ker Ð558" Ñœ ker Ð8 Ñ, whence +œ 58 ,œ". Thus, ker Ð 5 Ñ is trivial.

Let us refer to the subgroups ÖKßKßKá×55#$ 5 as the higher images of K and the sequence K 55 K # K â as the image sequence of 5 . If K has the DCC on all subgroups, then any injective endomomorphism 5ÀK ÄK is also surjective, since the image sequence of 55 must eventually be constant and so 88"Kœ5 K for some 8 . Hence, the injectivity of 555 implies that Kœ K , whence is surjective. Of course, if K has the DCC on normal subgroups only, then we can draw the same conclusion provided that 5 has normal higher images.

Theorem 4.40 Let K− be a group and let 5 EndÐKÑ. 1) If K has the ACC on normal subgroups, then 55 surjectiveÊ injective 2) If KK has the DCC on all subgroups or if has the DCC on normal subgroups and 5 has normal higher images, then 55 injectiveÊ surjective Note that 55 has normal higher images if preserves normality in general.

Definition Let KÀ be a group. A homomorphism 5 KÄL is normality preserving if RKÊü5ü RL The composition of two normality-preserving homomorphisms is normality preserving. For endomorphisms, a stronger condition than that of preserving normality is the following.

Definition An endomorphism 55ÀK ÄK of a group K is normal if commutes with all inner automorphisms #1 of K+ , that is, for any −K , 55Ð+11 Ñ œ Ð +Ñ for all 1−K . Homomorphisms, Chain Conditions and Subnormality 141

Of course, the composition of normal maps is normal. Also, it is easy to see that a normal endomorphism is normality preserving. One of the advantages of normal maps over other normality-preserving maps is that if 5ÀK ÄK is normal and if LKü5 is -invariant, then 5lÀLÄLL is also normal. Fitting's Lemma If K− has both chain conditions on normal subgroups and if 5 EndÐKÑ has normal higher images, then both the kernel and image sequences of 5 must eventually be constant and so there is an 7! for which Oœker Ð557 Ñœ ker Ð73 Ñand Lœ 557 Kœ 73 K for all 3 ! . There is much that can be said about the subgroups LO and .

First, 55LœL and OŸO and so L and O are 5 -invariant. Also, if +−L∩O, then +œ5577 , and +œ" . Hence, 5#7,œ" and so , −ker Ð55#7 Ñ œ ker Ð7 Ñ, whence + œ " . Thus, L ∩ O œ Ö"×. To show that KœLO, if +−K , then there is a ,−K for which 557#7+œ , and so +Ð557" Ð, ÑÑ −ker Ð 7 Ñ , whence + œ Ò+557"7 Ð, ÑÓ Ð,Ñ − LO Thus, Kœim Ð5577 Ñker Ð Ñ

7 Finally, since 55ÐOÑ œ Ö"×, the map lO is nilpotent and since 5ÐLÑ œ L , the map 5lL is surjective. We have just proved the important Fitting's lemma, which we can make a bit more general than the previous argument.

Theorem 4.41 ()Fitting's lemma, 1934 Let K be a group with the BCC on normal subgroups and let 5 −ÐKÑEnd have normal higher images. 1) There is an 7! for which Kœim Ð5577 Ñker Ð Ñ where a)im LœÐÑ55577 and OœÐÑker are -invariant b) 55llLO is surjective and is nilpotent. 2) In particular, if K is indecomposable, then 5 is either nilpotent or an automorphism of K . Automorphisms of Cyclic Groups Let us examine the automorphism group of the cyclic groups. It is clear that an automorphism of a cyclic group is completely determined by its value on a generator and that this value also generates the group. 142 Fundamentals of Group Theory

" The only generators of an infinite cyclic group GÐ+Ñ∞ are + and + and so

AutÐG∞ Ñ œ Ö+7 ß ×

" where 7BœB for all B−G∞8 . Now suppose that G œØ+Ù is cyclic of order 8−ÐGÑ. If 7 Aut 8 , then 7+œ+5 5‡ for some "Ÿ58 . But 9Ð+Ñœ9Ð+Ñ and so we must have 5−™8 . Moreover, this condition uniquely determines an automorphism 75 defined by 353 75Ð+ Ñ œ +

‡ for !Ÿ38 . Hence, there is precisely one automorphism 7™5 for each 5− 8 .

‡ Moreover, the map 5™ÀÄ8 Aut ÐGÑ85 defined by 5 5œ 7 is an isomorphism, ‡‡ since it is clear that 5™ is bijective and if 4ß 5 −88 and 45 œ ;8  < with < − ™ , then

33<345343 777<5Ð+Ñœ+ œ+ œ Ð+Ñœ574 Ð+Ñ

‡ Thus, AutÐG8 Ñ ¸ ™8.

Theorem 4.42 For the cyclic groups, we have

" AutÐG∞ Ð+ÑÑ œ Ö+7 ß À + È + × and ‡‡ AutÐG85 Ð+ÑÑ œ Ö7™™ ± 5 −88 × ¸ 5 where 7755 is defined by Ð+Ñ œ + . In particular, the automorphism group of a cyclic group is abelian. ‡ A Closer Look at ™8 ‡ The previous theorem prompts us to take a closer look at the groups ™8 .

/" /7 Theorem 4.43 If 8œ:" â:7 where the :33 are distinct primes and / " , then ‡‡ ‡ ™™}}™8 ¸â//" 5 :" :5

‡// Moreover, ™8 is cyclic if and only if 8 œ #ß %ß : or #: where : is an odd prime. /3 Proof. Let <œ:3 3 . For the direct product decomposition, consider the map ‡‡ defined by 5™ÀÄ8< }™3

5Ð?Ñ œ Ð? mod <"7 ß á ß ? mod < Ñ

This map is a group homomorphism and if 5?œÐ"ßáß"Ñ, then ?´" mod <3 , that is, <±Ð?"Ñ33 for all 3 . Since the < 's are pairwise relatively prime, it Homomorphisms, Chain Conditions and Subnormality 143

‡ follows that 8±Ð?"Ñ , that is, ?œ" in ™58 . Thus, is a monomorphism and since the domain and the range of 55 have the same size (see Theorem 2.30), is an isomorphism.

‡‡ Now, Theorem 2.33 implies that ™™8 is cyclic if and only if each factor /3 is :3 cyclic and the orders

‡ /"3 9Й /3 Ñ œ : Ð:3  "Ñ :3 3

‡ are relatively prime. So let us take a look at ™:/ for : prime.

‡‡ Let :# . To see that ™™::/ is cyclic, first note that if /œ" , then is cyclic ‡ since ™™: is a field. Assume that /" . It is sufficient to find elements in :/ of the relatively prime orders :" and :/" .

‡5 To find an element of order :" , let +−™:/ have order : 7 , where 7±:". Then

5 +œ":Ð:"Ñ:7 mod /" and /" implies that

5 +œ"::7 mod But if +Ð is chosen so that +ß:Ñœ" , then Fermat's little theorem implies that 5 +´+:7mod : and so +´"mod :, whence 7œ:" . Thus, 5 9Ð+Ñ œ :5: Ð:  "Ñ and so 9Ð+ Ñ œ :  ", as desired.

Moreover, since :# , the expression " œ": is in : -standard form and so Theorem 1.18 implies that

/" Ð"  :Ñ:/ œ "  A:

/" ‡ where :±Ay . Hence, ": has order : in ™:/ .

‡‡ Now consider the case :œ# . It is easy to see that ™™# and % are cyclic. For ‡‡ / $, the elements of ™™##// are odd integers. For "#+− , it is easy to see by induction that

#55 # Ð"#+Ñ œ"# B5 In particular,

#//# / Ð"#+Ñ œ"# B/# ´"mod #

‡/# which implies that ™#/ has exponent # and so cannot be cyclic. 144 Fundamentals of Group Theory

‡/ In summary, we can say that ™:/ is cyclic if and only if :# or : œ# or / ‡ :œ%Þ We can now piece together our facts. As mentioned earlier, ™8 is cyclic ‡ if and only if each factor ™ /3 is cyclic and the orders :3

‡ /"3 9Й /3 Ñ œ : Ð:3  "Ñ :3 3

/"3 are relatively prime. Since :Ð:"Ñ3 33 is even unless :œ#/œ" and 3 , there ‡‡ can be at most one factor involving an odd prime or ™™% . Thus, 8 is cyclic if and only if 8 œ #ß %ß :// or #: where :  # is prime. Exercises 1. Let 5ÀK ÄL be a group homomorphism. a) Prove that if WŸK , then 5 WŸL . b) Prove that if 5ü5 is surjective and WK , then WLü . c) Prove that if XŸL , then 5" XŸK . d) Prove that if XLü5ü , then " XK . 2. Let KO and be groups and let LŸKÞ Show that it is not always possible to extend a homomorphism 5ÀL ÄO to K . 3. Show that the :Ð -quasicyclic group ™ :∞Ñ is isomorphic to the subgroup of all complex :8 th roots of unity. 4. Show that if KQ has a unique maximal subgroup , then KÎQ is cyclic of prime order. 5. Let K be a finite group with normal subgroups L and O . If KÎL ¸ KÎO, does it follow that L¸O ? 6. a) Find a property of groups that is inherited by quotient groups but not by subgroups. b) Find a property of groups that is inherited by subgroups but not by quotient groups. 7. Show that a group K+ is abelian if and only if the map È+" is an automorphism of K . 8. Let K be a group. a) Determine all homomorphisms 5ÀG7 Ð+ÑÄK. b) Determine all homomorphisms 5ÀG∞ Ð+ÑÄK. 9. Are all normality-preserving homomorphisms normal? Hint : Use the fact that W$ is simple. 10. Let KR be a group and let be a normal cyclic subgroup of K . Prove that w KŸGÐRÑK . 11. Let KK be a group. Prove that w is central if and only if Inn ÐKÑ is abelian. 12Þ− An endomorphism 5 EndÐKÑ is central if ++−^ÐKÑ"5 for all + − K . This is equivalent to 55+^ÐKÑ œ +^ÐKÑ, that is, acts like the identity on KÎ^ÐKÑ . Prove that a) A normal surjective endomorphism is central. b) A central endomorphism is normal. Homomorphisms, Chain Conditions and Subnormality 145

13. An abelian group E+ (written additively) is divisible if for any −E and any positive integer 8,−E8 , there is a for which ,œ+ . Prove that a characteristically simple abelian group E is divisible. 14. Let K+ be a group and let −KÀ . Is the map 5 KÄK defined by 55,œÒ,ß+Ó an endomorphism of K ? If not, under what conditions on K is an endomorphism? 15. Let K:: be a group of order # where is a prime. a) Prove that if KK is abelian, then is either cyclic or isomorphic to the direct product of two cyclic groups of order : . b) Show that the distinct sets Ö+K × of conjugates of elements in K form a partition of K . Show that ^ÐKÑ Á Ö"×. Show that K must be abelian. 16. Prove that W¸H$' , where W $ is the symmetric group of order 'H and ' is the dihedral group of order ' . 17. Prove that if KLK has a periodic subgroup of finite index, then is periodic. (In loose terms, if “most” of the elements of K have finite order, then all elements of K have finite order.) 18. Let KO be a finite group. Let ü KL . A subgroup of K is called a supplement of OKœLOL if . Let be a minimal supplement of O . Prove that L∩OŸF ÐLÑ. 19. Let Y be a family of groups with the following properties: 1) If K−YY and L¸K , then L− . 2) If K−Yü and R K , then KÎR− Y . 3) If RüYYY K and if R− and KÎR− , then K− . Prove that if R−YY and O− are subgroups of KOK with ü , then RO − Y. 20. If KK is a finite group and 5 is an automorphism of that fixes only the identity element, that is, 1Á" implies 5 1Á1 , show that KœÖ1"5 1±1−K× 21. Prove that if LKOKüü and üü , then it does not necessarily follow that the set product LO is a subgroup of K . Hint : Look at the dihedral group H). 22. Let KE be a finite group. Suppose that ü K is minimal among all normal subgroups of KE and that is abelian. Prove that E is an elementary abelian group. 23. Show that a subgroup LK of a group is characteristic if and only if 55LœL for all automorphisms −Aut ÐKÑ . 24. Find an example of a normal subgroup LKL of a group for which is not characteristic in K . 25. Let K be a group. Prove the following: a) The property of being characteristic is transitive: If E«F and F«G , then E«G. b) If E«F and Füü G , then E G . 26. Show that ^ÐKÑ « K . 146 Fundamentals of Group Theory

27. Let K be a finite group and let LKü . Show that if Ð L ß ÐK À LÑÑ œ ", kk then L«K. 28. Let K be a group and let LŸK . Prove that L«K implies ÒLßKÓ«K . 29. Let KL be a finitely-generated group and let ŸK with ÐKÀLÑ∞. Show that there is a subgroup OŸL that is characteristic in K and has finite index in K . 30. Let KR be a group and let –K . Let LKR . Show that ŸFÐLÑ implies RŸF ÐKÑ. 31. Let KK be a group. Let ww be the commutator subgroup of KK . Let w be the commutator subgroup of Kw . In general, we can continue to take commutator subgroups and KœÐKÑÐ8Ñ Ð8"Ñ w is called the 8th commutator subgroup of KK«K . Prove that Ð8Ñ . 32. Let KÐ be a group. Prove that if Inn KÑ is cyclic, then K is abelian. 33. a) Show that the commutator subgroup of a group is fully invariant. b) Show that the center of a group need not be fully invariant. Hint : consider GÐ+Ñ#$} W. /" /7 34. Let K8 be a finite abelian group of order œ:" â:7 where the :3 's are distinct primes. For each prime :3 , let

KœÖ+−K±+:Ð:3 Ñ is a 3 -element ×

a) Show that K«KÐ:3 Ñ . b) Show that

K œ Kand K ∩ K œ Ö"× 22Ð:333 Ñ Ð: Ñ 4Á3 Ð: Ñ

35. (Chinese remainder theorem) Let 7ßáß7"5 be pairwise relatively prime integers greater than " . Let 0"8 ßáß0 be integers. Prove that the system of congruences

B´0""mod 7 ã B´085mod 7

has a unique solution modulo the product 7â7"5 . : Hint : Use ™73 . 36. Let K: be a group of order 8 where : is a prime. Show that a subgroup L of index :K must be normal in . Hint : Consider the map - ÀKÄWKÎL , where WKKÎL is the symmetric group on ÎL , defined by -Ð1ÑÐ+LÑ œ 1+L. 37. Find the subgroup lattice of . Which subgroups are normal? Is abelian? What is the center of ? What is w ? 38. Prove that the quasicyclic groups ™Ð:∞ Ñ are the only infinite groups with the property that their proper subgroups consist entirely of a single ascending chain

Ö"×W"# W â Homomorphisms, Chain Conditions and Subnormality 147

39. Let K8 be a group and let " be an integer. 8 a) When is the 80 th power map 88ÀKÄK0 defined by Ð+Ñœ+ a homomorphism? Must K be abelian? b) When is the 88 th power map a homomorphism for all ? c) Let K88 œ Ö+ ± + − K× and let KÐ8Ñ œ Ö+ − K ± + 8 œ "×. Show that both of these sets are normal subgroups of K . What is the relationship between these subgroups? 40. Find the automorphism group AutÐZ Ñ of the 4-group Z . 41. Let 5üÀKq » O be an epimorphism. Show that if LKü , then 5üüLO=ÐLßOÑŸ=ÐLßKÑ and 5 . 42. Prove that the subgroup ØÙ5 of the dihedral group

##8 HœØß±œ"œÙ#8" 53 5 3 has subnormal index 8 . 43. If Y œÖL±3−M×3 is a family of subnormal subgroups of a group K and if there is an integer for which for all , show that is ==ÐLßKÑŸ=3−M3 +Y subnormal in K . 44. Prove that any finitely-generated metabelian group has the generalized SJP. 45. A group KK is Hopfian if is not isomorphic to any proper quotient group of itself. A group KK is co-Hopfian if is not isomorphic to any proper subgroup of itself. a) Prove that KK is Hopfian if and only if every endomorphism of is an automorphism of K . b) Prove that KK is co-Hopfian if and only if every monomorphism of is an automorphism. c) Show that  is both Hopfian and co-Hopfian. Hint : To show that is Hopfian, show that ÎL is not torsion free unless L œ Ö"× . To show that  is co-Hopfian, show that any proper subgroup L of is not divisible. An abelian group K+ is divisible if −K8 and −™ imply that there is a ,−K for which +œ8, . d) Show that ™ is Hopfian but not co-Hopfian. e) Show that the quasicyclic group ™Ð:∞ Ñ is co-Hopfian but not Hopfian. f) Show that the additive group of all polynomials in infinitely many variables is neither Hopfian nor co-Hopfian. 46. Find two nonisomorphic groups with isomorphic automorphism groups. 47. Prove that the multiplicative group  of positive rational numbers is isomorphic to the additive group ™ÒBÓ of polynomials over the integers. Hint: Use the fundamental theorem of arithmetic. 48. Let KK be a group. Show that if is centerless, then

GÐÐKÑÑœÖ×AutÐKÑ Inn +

49. Show that AutÐW$$$ Ñ œ Inn ÐW Ñ ¸ W . 50. Prove that if K is a group in which every nonidentity element is an involution, then K has a nontrivial automorphism. 148 Fundamentals of Group Theory

51. Consider a series for K

Ö"לKKKâKœK!"#üüüü 8

for which each factor group KÎK3" 3 is abelian. Show that if L is a subgroup of K , then there is a series of subgroups

Ö"לLLLâLœL!"#üüüü 7 of L whose factor groups are also abelian. 52. ()Robinson Let RKü and suppose that R has the GSJP and KÎR has the ACC on subnormal subgroups. Show that K also has the GSJP. Hint : Apply Theorem 4.35. Let œÖL±3−M×3 be an arbitrary chain in subnÐKÑ and let . To show that , consider the chains Yœ-üYKü RœÖLR±3−M×3 and

RÎR œ ÖLRÎR3 ± 3 − M×

in K and KÎR , respectively. Let L" RÎR be a maximal member of RÎR. Show that Y œ L" ÐY ∩RÑ. Chapter 5 Direct and Semidirect Products

In this chapter, we will explore the issue of the decomposition of groups into “products” of subgroups. To say simply that a group K can be decomposed into a set product of two proper subgroups KœLO leaves something to be desired. The first problem is that the representation of an element of K2 as a product 52 for −L5 and −O need not be unique. The second problem is that, in general, we have no information about how the elements of LO and interact, for example, what is 52 as a member of LO ? The first problem is easily addressed. The second is not as simple. Complements and Essentially Disjoint Products Uniqueness in a set product decomposition is easily characterized.

Theorem 5.1 Let LßO Ÿ K . 1) Every element +−LO has a unique representation as a product +œ25 for 2−L5−O and if and only if LO and are essentially disjoint. 2) Every element +−K can be uniquely represented as an element of LO if and only if KœLìO , that is, if and only if K œ LOand L ∩ O œ Ö"×

In this case, OL is said to be a complement of in K . A subgroup L of K is complemented if it has a complement in K .

Note that since KœLO implies KœOL , the concept of complement is symmetric in LO and . Also, if KœLìO , then KœLìOœLO kk k k kkkk as cardinal numbers.

S. Roman, Fundamentals of Group Theory: An Advanced Approach, 149 DOI 10.1007/978-0-8176-8301-6_5, © Springer Science+Business Media, LLC 2012 150 Fundamentals of Group Theory

We pause for a small clarification of terminology. In the more specialized literature of group theory, devoted to the study of the subgroup lattice subÐKÑ of a group K , such as appears in the book Subgroup Lattices of Groups, by Schmidt [30], the term complement of L−sub ÐKÑ refers to a subgroup O for which K œ L ” Oand L ∩ O œ Ö"× This terminology is consistent with the terminology of lattice theory. Indeed, Schmidt uses the term permutable complement for the concept given in our definition. However, our definition follows the trend in general treatments of group theory (including most textbooks).

Even for normal subgroups, complements need not exist. For example, no nontrivial proper subgroup of ™ has a complement. Moreover, when complements do exist, they need not be unique; for example, in W$ every subgroup of order #E is a complement of the alternating subgroup $ and so, for example,

W$$ œ E ì ØÐ" #ÑÙ œ E$ ì ØÐ" $ÑÙ are two essentially disjoint product representations of W$ .

On the other hand, any complement OR of a normal subgroup is isomorphic to the quotient KÎR , for we have KRìOO O œ¸ œ¸O RRR∩OÖ"×

Theorem 5.2 Let KR be a group. If a normal subgroup of K is complemented, then all complements of RK are isomorphic to ÎR and hence to each other. Complements and Transversals Another way to characterize complements is through the notion of a transversal. Let us recall the following definition.

Definition Let LŸK . 1) A set consisting of exactly one element from each left coset in KÎL is called a left transversal for LK in () or for KÎL . 2) A set consisting of exactly one element from each right coset in LÏK is called a right transversal for LK in () or for LÏK .

It is of interest to know which subgroups of KL are left transversals for . The simple answer is that these are precisely the complements of L .

Theorem 5.3 Let LŸK . The following are equivalent for a subgroup OŸK . 1) OL is a complement of in K . Direct and Semidirect Products 151

2) OL is a left transversal for in K . 3) OL is a right transversal for in K . Proof. Suppose that OL is a complement of in K . If 5"#Lœ5L , then " 5# 5"" −O∩LœÖ"× and so 5 œ5# . Also, since KœOL , every +−K has the form +œ52−5L for some 5−O . Thus, the cosets 5L for 5−O form a left transversal for LK in . Conversely, suppose that OŸK is a left transversal for KÎL . Since L contains a single member of O , it follows that L ∩ O œ Ö"×. Also, since the cosets KÎL partition K , any + − K has the form +œ52, for some 5−O and 2−L and so KœOL . Thus, O is a complement of LK in . A similar argument can be made for the right cosets. Product Decompositions If KœLìO , then every element of K has a unique representation as a product 25 with 2 − L and 5 − O . The problem, however, is that we have no information about how the elements of LO and interact. Without a commutativity rule that expresses a product 52 in the form 2ww 5 , where 2ß 2ww − L and 5ß 5 − O , we have no way to simply a product of the form

Ð2"" 5 ÑÐ2 ## 5 Ñ The simplest commutativity rule, that is, elementwise commutativity 25 œ 52 holds if and only if both factors LO and are normal in K and this essentially reflects the fact that there is no interaction between LO and . Weaker forms of decomposition come by weakening the requirement that both factors be normal. Here are the relevant definitions in one place for comparison purposes.

Definition Let KL be a group and let ßOŸK . 1) If KœLìO then KL is called the essentially disjoint product of and O . In this case, LO is called a complement of in K . 2) If KœLìOß Lü K then KL is called the semidirect product of with O . In this case, L is called a normal complement of OK in . The semidirect product is denoted by KœLz O 3) If KœLìOß Lüü KßO K then KL is called the direct product of and O . In this case, L is called a direct complement of OK in () and vice versa . The direct product is 152 Fundamentals of Group Theory

denoted by KœLO Any of these products is nontrivial if both factors are proper.

The direct product decomposition of a group is a very strong form of decomposition and so is rather specialized. However, the semidirect product is one of the most useful constructions in group theory. For example, for 8Á#mod % , the dihedral groups H#8 have no direct product decompositions, but they do have a semidirect product decomposition

HœØÙØÙ#8 3z5 Nevertheless, all three types of product decompositions are special. To illustrate the point, an infinite cyclic group has no nontrivial essentially disjoint product decompositions at all. Direct Sums and Direct Products

External Direct Sums and Products We have already defined the external direct product of a finite number of groups. The generalization of this product to arbitrary families of groups leads to two important variations. Intuitively speaking, if , is an arbitrary (finite or infinite) cardinal number, we can consider the set of all ordered “, -tuples” as well as the set of all ordered , -tuples that have only a finite number of nonzero coordinates. The notion of an ordered , -tuple is generally described by a function.

Definition Let Y œÖK±3−M×3 be a family of groups. 1) The external direct product of the family Y is the group

}YœKœ0ÀMÄK0Ð3Ñ−K } 333š›. ¹ of all functions 0M from the index set to the union of Y for which the 3 th coordinate 0Ð3Ñ of 0 belongs to K3 for all 3 . The group operation is componentwise product: Ð01ÑÐ3Ñ œ 0Ð3Ñ1Ð3Ñ

The support of 0−} K3 is the set suppÐ0Ñ œ Ö3 − M ± 0Ð3Ñ Á "×

2) The external direct sum of the family Y is the group {YœKœ0−±0 { }Y has finite support 3 ef also under componentwise product. Direct and Semidirect Products 153

Of course, when Y is a finite family, the direct product and direct sum coincide. Note that authors vary on their use of the terms direct sum and direct product. For example, some authors reserve the term sum for abelian groups and some authors use the term cartesian product for direct product. Internal Direct Products For convenience, we repeat the definition of the internal direct product.

Definition A group K is the ()internal direct sum or ()internal direct product of a family YYœÖL±3−M×3 of normal subgroups if is strongly disjoint and . We denote the internal direct product of by Kœ1YY L3 or Y or when Y œÖLßáßL"8 × is a finite family,

LâL"8

Each factor LK3 is called a direct summand or direct factor of . We denote the family of all direct summands of K,CWf ÐKÑ.

We should mention that the notation for direct sums and products varies considerably among authors. For example, some authors use the notation KœL‚O for both the internal and external direct sum (as well as the cartesian product), justifying this on the grounds that the two types of direct sums are isomorphic. While this may be reasonable, in an effort to avoid any ambiguity, we have adopted the following notations:

1) Set product LO 2) Essentially disjoint set product LìO 3) Cartesian product of sets L‚O 4) External direct product

LO}}and L3 5) External direct sum

LO{{and L3 6) Internal direct product (sum)

LOand L3 154 Fundamentals of Group Theory

7) Semidirect product LOz Note that our notation for the internal direct product emphasizes the fact that the two subgroups are normal (a juxtaposition of the symbols —– and used to indicate normality) and is similar to the established notation z for the semidirect product.

To see that internal and external direct sums are isomorphic, suppose that Kœ{ K3. For each 5 , let

LœÖ0−53{ K±0Ð3Ñœ" for 3Á5×

We leave it as an exercise to show that LK55ü and that KœL . On the other hand, if Kœ K33 , then the map 5{À K Ä K3 defined by 50œ Ö0Ð3ѱ3−Mß0Ð3ÑÁ"× $ is an isomorphism. For this reason, many authors drop the adjectives “internal” and “external” when discussing direct sums. The Universal Property of Direct Products and Direct Sums Direct products and direct sums can each be characterized by a universal property. Projection and Injection Maps If

Kœ}{ ÖL±3−M×33or Kœ ÖL±3−M× is an external direct product or external direct sum, we define the 3 th projection map 333ÀK ÄL by

33Ð0Ñ œ 0Ð3Ñ

Note that 33 is an epimorphism and that an element 0−K is uniquely determined by the values 33Ð0Ñ , which can be specified arbitrarily, as long as 333Ð0Ñ − L for all 3 .

The 3À th injection map ,33LÄK is defined by 24œ3if ,3Ð2ÑÐ4Ñ œ œ " otherwise for all 2−L3 . These maps are injective. Note also that + if 3œ4 3,43‰œ œ ! otherwise For an internal direct sum Direct and Semidirect Products 155

Kœ ÖL±3−M×3 we have already defined the projection 3333ÀK ÄL by setting 3Ð+Ñ to be the 3 th component of +À . For the internal direct sum, the injection maps ,33LÄK are just the inclusion maps, defined by ,33Ð2Ñ œ 2 for all 2 − L . Universality of Direct Products

Let ÖL3 ± 3 − M× be a family of groups and let

Tœ} ÖL×3 3−M with projection maps 33 . Then the ordered pair c3œTßÖ× 33−M has a universal property that characterizes the direct product up to isomorphism.

G

fi fj ∃!τ

Hi U Hj ui uj

Figure 5.1 Definition Referring to Figure 5.1, let Y be the family of all ordered pairs Z œKßÖ0ÀKÄL× 333−M where KÖ is a group and 0333ÀKÄL×−M is a family of homomorphisms. We refer to K as the vertex of the pair Z . A pair h œYßÖ?ÀYÄL× 333−M in YYZ is universal for if for any pair − Y , there is a unique homomorphism 7ZÀK ÄY between the vertices, called the mediating morphism for , for which

?‰337 œ0 for all 3−M .

Theorem 5.4 Let ÖL3 ± 3 − M× be a family of groups. 1) The pair c3œ TßÖ ÀT ÄL× 333−M 156 Fundamentals of Group Theory

where TÖ is the direct product of the family L33× and 3 is the 3 th projection map, is universal for Y . 2) If the pair ÐKß Ö033−M × Ñ is also universal for Y , then the mediating morphism 7ÀK ÄT is an isomorphism and so K ¸T . Proof. For part 1), for any pair ZY− , there must exist a unique mediating morphism 7ÀK ÄT satisfying

3733‰œ0 But this is equivalent to

3733Ð+Ñœ0Ð+Ñ for all +−K and this does uniquely define a homomorphism 7 . The proof of part 2) is also to the proof of Theorem 4.5 and we leave the details to the reader. Universality of Direct Sums

Let ÖL3 ± 3 − M× be a family of groups and let

Wœ{ ÖL±3−M×3 with injections ,3 . The ordered pair

f,œÐWßÖ33−M × Ñ also has a universal property that characterizes the direct sum up to isomorphism.

Hi U Hj ui uj

fi ∃!τ fj

G

Figure 5.2 Definition Referring to Figure 5.2, let Y be the family of all ordered pairs Z œKßÖ0ÀLÄK× 33 3−M where KÖ is a group and 033ÀLÄK× 3−M is a family of homomorphisms. We refer to K as the vertex of the pair. A pair hYœYßÖ?ÀLÄY× − 33 3−M is universal for YZ if for any pair − Y , there is a unique homomorphism 7ZÀY ÄK between the vertices, called the mediating morphism for , for which Direct and Semidirect Products 157

7 ‰?33 œ0 for all 3−M .

Theorem 5.5 Let ÖL3 ± 3 − M× be a family of groups. 1) The pair f,œ WßÖ ÀL ÄW× 33 3−M where WÖ is the direct sum of the family L33× and , is the 3 th injection map, is universal for Y . That is, for any pair Z œKßÖ0ÀLÄK× 33 3−M where KÖ is a group and 033ÀLÄK× 3−M is a family of homomorphisms, there is a unique homomorphism 7ÀW ÄK for which

7,‰œ033 for all 3−M , or equivalently,

7lœ0L33 for all 3−M . In other words, a homomorphism 7ÀWÄO is uniquely

determined by its restrictions 7lLL33 to the factors , which may be any homomorphisms from LK3 to . 2) If the pair ÐKß Ö033−M × Ñ is also universal for Y , then the mediating morphism 7ÀW ÄK is an isomorphism and so K ¸W . Proof. For part 1), for any pair ÐOß Ö533 À L Ä O× 3−M Ñ in Y , there must be a unique mediating morphism 7ÀW ÄK satisfying

7,‰œ033 But this specifies how 7 is defined on any element of W that has support of size "W and therefore on any element of that has finite support, that is, on all of W . We leave the details of this and the proof of part 2) to the reader. Cancellation in Direct Sums A group K is cancellable in direct sums (or cancellable for short) if EK¸FLß K¸L Ê E¸F We follow a line similar to Hirshon [18] to prove that any finite group is cancellable in direct sums. On the other hand, Hirshon shows that even infinite cyclic groups are not cancellable in direct sums.

Theorem 5.6 Any finite group K is cancellable in direct sums. Proof. It is sufficient to show that EKœFLß K¸L Ê E¸F 158 Fundamentals of Group Theory and we prove this by induction on 9ÐKÑ . If 9ÐKÑ œ " , the result is clear. Assume that any group of order less than 9ÐKÑ is cancellable and let [ œEKœFLß K¸L First, we observe that if F ∩ K œ Ö"×, then [ œ F  K, for we have FK [ ¸K¸L¸ FF and since these groups are finite, they have the same size and so they are equal, whence FKœ[. It follows that EF and are both direct complements of K in [ and so are isomorphic. A similar argument holds if E ∩ L œ Ö"×.

Thus, we may assume that F∩K and E∩L are nontrivial. Then EK LF œ ÐE∩LÑÐF∩KÑ ÐE∩LÑÐF∩KÑ and Theorem 5.19 implies that EKLF }}¸ E∩L F∩K E∩L F∩K Since L¸K , we have LE KKL F }}¸ }} Ö"× E ∩ L F ∩ K Ö"× E ∩ L F ∩ K and Theorem 4.8 implies that LEK}} LKF}} ¸ Ö"×}} ÐE ∩ LÑ ÐF ∩ KÑ Ö"× }} ÐE ∩ LÑ ÐF ∩ KÑ Splitting this in a different way gives EL KFL K }}¸ }} Ö"× E ∩ L F ∩ K Ö"× E ∩ L F ∩ K Since 9ÐKÎÐF ∩ KÑÑ  9ÐKÑ, the induction hypothesis implies that KÎÐF ∩ KÑ is cancellable. Similarly, LÎÐE∩ LÑ is cancellable and so E ¸ F . The Classification of Finite Abelian Groups We are now in a position to solve the classification problem for finite abelian groups, that is, we can identify a system of distinct representatives for the isomorphism classes of finite abelian groups.

Theorem 5.7 Let E? be a finite abelian group. If −E has maximum order among all elements of EQœØ?Ù , then has a direct complement, that is, there Direct and Semidirect Products 159 is a subgroup Z for which EœQ Z Proof. The proof is by induction on the order of E9ÐEÑœ" . If , the result is clear. Assume the result holds for all abelian groups of order less than that of E . We may also assume that QE . If we find a subgroup \ŸE for which \ ∩ Q œ Ö!×, since then 9Ð?  \Ñ œ 9Ð?Ñ, the inductive hypothesis applied to the quotient KÎ\ gives E Q\ Z QZ œœ \\\\ for some Z ŸE . Hence, EœQZ and since ÐQ\Ñ∩Z œ\, it follows that Q∩Z©Q∩\œÖ!×, whence EœQ Z.

To this end, let \EQ be a minimal subgroup of that is not contained in . If B − \, then ØBÙ is not contained in Q and so \ œ ØBÙ . If 9ÐBÑ œ +, with + and , relatively prime and greater than " , then Ø+BÙ  ØBÙ and so Ø+BÙ Ÿ Q . Similarly, Ø,BÙ Ÿ Q and so B − Ø+BÙ  Ø,BÙ Ÿ Q, a contradiction. Hence, 9ÐBÑ œ : is prime and so \ ∩ Q œ Ö!×.

Thus, if E?E is a finite abelian group and if " has maximum order in , then

EœØ?Ù"" Z for some Z"" Ÿ E . If Z Á Ö!× and ? # has maximum order in Z" , then

EœØ?Ù"## Ø?Ù Z where 9Ð?#" Ñ ± 9Ð? Ñ. Since E is finite, this process must eventually result in a cyclic decomposition

EœØ?Ù"8 â Ø?Ù of Eœ , where if α559Ð?Ñ , then

αα88"±±â± α "

Moreover, if α" has the prime factorization

//"ß" "ß7 α" œ:" â:7 then αα5"± implies that

//5ß" 5ß7 α5 œ:" â:7 where /5ß3 Ÿ/ 5"ß3 for all 3œ"ßáß7. Then Theorem 2.33 implies that

Ø?55ß"5ß7 Ù œ Ø@ Ù â Ø@5 Ù

/5ß3 where 9Ð@5ß3 Ñ œ :3 for all 3 . A subgroup of order a power of a prime : is called 160 Fundamentals of Group Theory a :-primary (or just primary ) subgroup. It is customary when writing E as a direct sum of primary cyclic subgroups to collect the terms associated with each prime.

Theorem 5.8 () The cyclic decomposition of a finite abelian group Let E be a finite abelian group. 1)( Invariant factor decomposition) E is the direct sum of a finite number of nontrivial cyclic subgroups

E œ Ø?"8 Ù â Ø? Ù

where if α55œ9Ð?Ñ , then

αα88"±±â± α "

The orders α3 are called the invariant factors of E . 2)( Primary cyclic decomposition) If

//5ß" 5ß7 α5 œ:" â:7

where /5ß3 Ÿ/ 5"ß3 for all 3œ"ßáß7, then E is the direct sum of primary cyclic subgroups:

E œ ÒØ?"ß"" Ù â Ø?ß5"7 ÙÓ  â ÒØ? 7ß"7 Ù  â Ø?ß5 ÙÓ

//3ß4 3ß4 where 9Ð?3ß4 Ñ œ :33. The numbers : are called the elementary divisors of E.

We note the following:

1) The product of the invariant factors is equal to the product of the elementary divisors and is the order of the group. 2) The invariant factor α5 is equal to the maximum order of the elements of the group

Ø?"5 Ù â Ø? Ù

In particular, the largest invariant factor α" is equal to the maximum order of the elements of E . 3) The multiset of invariant factors of E uniquely determine the multiset of elementary divisors of E and vice versa. In particular, the elementary divisors are determined by factoring the invariant factors and the invariant factors are determined by multiplying together appropriate elementary divisors. Uniqueness Although the invariant factor decomposition and the primary cyclic decomposition are not unique, the multiset of invariant factors and the multiset of elementary divisors are uniquely determined by E . Actually, the proof is Direct and Semidirect Products 161 quite easy if we use the cancellation property of Theorem 5.6. Suppose that

EœØ?Ù"8 â Ø?Ù where α55œ9Ð?Ñ and

αα88"±±â± α " and that

EœØ@Ù"7 â Ø@ Ù where "55œ9Ð@Ñ and

""77"±±â± " "

Then α""" and are both equal to the maximum order of the elements of E and so α"""œØ?Ù¸Ø@Ù . Hence " " and Theorem 5.6 implies that

Ø?#8#7 Ù â Ø? Ù ¸ Ø@ Ù  â Ø@ Ù

Now, α"## and are equal to the maximum orders in the two isomorphic groups above and so α"##œ7 and we may cancel again. Hence, œ8 and α"55œ for all 5.

Theorem 5.9 () Uniqueness Let E be a finite abelian group. The multiset of invariant factors () and elementary divisors for E is uniquely determined by E. Properties of Direct Summands Let us explore some of the properties of direct summands. The following simple facts are worth explicit mention:

1) The direct summand property is transitive: L −Wf ÐOÑß O −Wf ÐKÑ Ê L −Wf ÐKÑ 2) The property of being a direct summand in inherited by subgroups: L−Wf ÐKÑß LŸOŸK Ê L−Wf ÐOÑ In fact, KœLNß LŸOŸK Ê OœLÐN∩OÑ 3) Normal subgroups of direct summands are normal in the group: RLßL−ÐKÑÊRKüWfü 162 Fundamentals of Group Theory

4) The property of being characteristic is preserved by intersection with a direct summand: R«Kß L−Wf ÐKÑ Ê R∩L«L

Good Order Let K be a group and suppose that KœEEww œFF If E©F , then it does not necessarily follow that Eww ªF , as is easily seen in the Klein % -group, for example. For convenience, let us say that an equation of the form KœEEww œFFß E©F is in good order if EªFww .

Theorem 5.10 Let K be a group. If KœEEww œFFß E©F then we can replace either EFww or by another subgroup to get an equation in good order. Specifically, the following are in good order: KœEÒFÐE∩FÑÓœFFww w and KœEEwww œFÐE∩EFÑ Proof. The first equation follows directly from Dedekind's law. For the second equation, F∩ÐE∩EFÑœEwww ∩ÐF∩EFÑ w œEww ∩EÐF∩FÑ œEww ∩E œ Ö"× and FÐEww ∩ EF Ñ œ FEÐE ww ∩ EF Ñ œ FÐEE ww ∩ EF Ñ œ FEF w œ K

Chain Conditions on Direct Summands and Remak Decompositions Direct summands are also special with respect to chain conditions. We have seen that for the families subÐKÑ and nor ÐKÑ , the two chain conditions (ACC and DCC) are independent. However, for the family WfÐKÑ of direct summands of K , the two chain conditions are equivalent. Direct and Semidirect Products 163

Theorem 5.11 A group KÐ has the ACC on Wf KÑ if and only if it has the DCC on WfÐKÑ. Proof. Suppose that K has the DCC on direct summands and let

HŸHŸâ"# be an ascending sequence in WfÐKÑ . Theorem 5.10 implies that there is a descending sequence

I I â"# where KœH33 I for all 3 . This descending sequence must eventually become constant, at which time so does the original sequence of H3 's.

A group with the the BCC on direct summands can be decomposed into a finite direct sum of indecomposable factors.

Definition If K is a group, then any decomposition

Kœ â"8 where each K5 is indecomposible is called a Remak decomposition of .

A minimal direct summand is a minimal member of the family WfÐKÑ Ï Ö"× of all nontrivial direct summands of K . A direct summand is minimal if and only if it is indecomposable.

Theorem 5.12 ()Remak If a group K has either ( and therefore both ) chain condition on direct summands, then K has a Remak decomposition

Kœ â"8

Proof. Let K"" be a minimal direct summand of . If œK , then K is indecomposable and we are done. Otherwise,

Kœ I"" where I"# Á Ö"× also has BCC on direct summands. Let be a minimal direct summand of IK"# , which is also a minimal direct summand of . If œI" , we are done. If not, then

KœI"## Since the sequence

â""# is a strictly increasing sequence of direct summands of K , it must become constant and so this construction must terminate after a finite number of steps, resulting in a decomposition of K into a finite direct sum of minimal direct summands. 164 Fundamentals of Group Theory

A Maximality Property We paraphrase Theorem 3.15.

Theorem 5.13 Let Y œÖL±3−M×3 be a nonempty family of normal subgroups of a group KOKN©M . For any ü , there is a that is maximal with respect to the property that the direct sum

O L4 Œ4−N exists.

One consequence of Theorem 5.13 is the following.

Theorem 5.14 Suppose that Kœ is the join of a family Y ÖL3 ±3−M× of minimal normal subgroups. For any OKü , there is a N©M for which

KœO L4 Œ4−N and so norÐKÑ œWf ÐKÑ

Proof. Let N©M be maximal with respect to the fact that the direct sum

QœORN4 Œ4−N exists. If 3ÂN , then QN3 ∩Rü K implies that QN3 ∩R œÖ"× or R 3 ŸQ N . But if QN3 ∩ R œ Ö"×, then the direct sum QN∪Ö3× exists, contradicting the maximality of N . Hence, R3N ŸQ for all 3−M and so KœQN . xY-Groups It is natural to ask the following types of questions. Let k ÐKÑ be a class of subgroups of a group K , such as the class of all subgroups, all normal subgroups or all characteristic subgroups.

1) For which groups KÐ is it true that all subgroups in k KÑ are complemented? 2) For which groups KÐ is it true that all subgroups in k KÑ have normal complements? 3) For which groups KÐ is it true that all subgroups in k KÑ are direct summands?

There has been much research done on these and related questions and it has become customary to make the following types of definitions in this regard. Direct and Semidirect Products 165

Definition 1) An aC-group is a group in which all subgroups are complemented. 2) An nC-group is a group in which all normal subgroups are complemented. 3) An aD-group is a group in which all subgroups are direct summands. 4) An nD-group is a group in which all normal subgroups are direct summands. 5) An aNC-group is a group in which all subgroups have normal complements.

Note that an nNC-group is the same as an nD-group. We will characterize aD- groups, nD-groups and aNC-groups. We will also describe (without proof) the characterization of aC-groups. The theory of the least restrictive of these conditions—the nC-groups—appears to be much more complicated and we refer the reader to Christensen [8] and [9] for more details. aD-Groups Groups in which all subgroups are direct summands were characterized by Kertész in 1952. This is the strongest of the xY-conditions and is indeed very restrictive.

Theorem 5.15 (: aD-groups Kertész [21], 1952)A group K is an aD-group if and only if it is the direct product of cyclic groups of prime order. Proof. If KG is a direct sum of cyclic subgroups 3 of prime order, then since each G3 is minimal normal, Theorem 5.14 implies that subÐKÑ œ nor ÐKÑ œWf ÐKÑ and so K is an aD-group.

For the converse, suppose that KK is an aD-group. Then any subgroup of is also an aD-group. However, the only cyclic aD-groups Ø+Ù are those of square- free order (that is, order a product of distinct primes). For it is clear that Ø+Ù cannot have infinite order and if :# ± 9Ð+Ñ for any prime : , then Ø+Ù has an element B: of order and so Ø+ÙœØBÙO where :±9ÐOÑ . Hence, O has an element of order : as well, which is too many elements of order :K for a cyclic group. Hence, every element of has square- free order.

Now, consider the family Y of all cyclic subgroups of prime order in K . If +−K has order 9Ð+Ñœ:â:"7, where the factors are distinct primes, then Theorem 2.33 implies that there are subgroups Ø+33 Ù of order : for which

Ø+ÙœØ+ÙâØ+"7 Ù 166 Fundamentals of Group Theory and so . But since each member of is minimal normal, Theorem 5.14 Kœ1YY implies that K is the direct sum of a subfamily of Y . nD-Groups The nD-condition is not as strong as the aD-condition, but is still very restrictive. One reason is that normality is a finitary condition, involving individual elements and so the condition WfÐKÑ œnor ÐKÑ imparts finitary properties to the otherwise global condition of being a direct summand.

In particular, the union of any ascending sequence of normal subgroups is normal and so in an nD-group, the union of any ascending sequence of direct summands is a direct summand. This condition implies a certain measure of finiteness for nD-groups. Specifically, if K is a nontrivial nD-group and

HŸHŸâ"# is an ascending chain of direct summands in R , then

Kœ H Iœ ÐHIÑ Š‹..33 for some IŸK . Hence, if Kœnc ÐWßKÑ is the normal closure of a finite subset W©K , then W©H88 I for some 8 " and so KœH I, which implies that HœH83 8 for all 3 ! . Thus, K has the ACC on direct summands and so K has a Remak decomposition.

Note also that any normal subgroup (direct summand) HK of an nD-group is an nD-group, since WfÐHÑ œ Wf ÐKÑ ∩sub ÐHÑ œ nor ÐKÑ ∩ sub ÐHÑ œ nor ÐHÑ It follows that any nontrivial nD-group contains an indecomposable direct summand.

We can now characterize nD-groups. Suppose that K is a nontrivial nD-group and consider the family Y œÖW±3−M×3 of all indecomposable direct summands of K , along with the trivial subgroup. Theorem 5.13 implies that there is a subset N©M that is maximal with respect to the fact that the direct sum

Wœ W4 4−N exists in K and so the nD-condition implies that KœWL for some LŸK . But if L is not trivial, then it contains an indecomposable direct summand of KN , which contradicts the maximality of . Hence, L is trivial and Direct and Semidirect Products 167

Kœ W4 4−N is a direct sum of indecomposable subgroups. In particular, each W3 is minimal normal in KK and so is the direct sum of minimal normal subgroups.

Conversely, if K is the direct sum of minimal normal subgroups, then Theorem 5.14 implies that norÐKÑ œWf ÐKÑ, that is, K is an nD-group.

Theorem 5.16 The following are equivalent for a group K : 1) K is an nD-group 2) K is the direct sum of minimal normal subgroups 3) K is the direct sum of simple subgroups. aNC-Groups We next show that aNC-groups are the same as aD-groups, by showing that an aNC-group is abelian.

Theorem 5.17 (: aNC-groups Weigold [34], 1960 ) A group is an aNC-group if and only if it is an aD-group, that is, if and only if it is a direct sum of cyclic subgroups of prime order. Proof. An aD-group is an aNC-group. For the converse, we show that an aNC- group KE has trivial commutator subgroup and so is abelian. If ŸK is abelian, then KœRzü E for some R K and so KœÒREßREÓŸRw Hence, KKw is in the normal complement of any abelian subgroup of , including all cyclic subgroups Ø+Ù of K . Hence, Kw is trivial. aC-Groups Finally, we state without proof the following theorem on aC-groups.

Theorem 5.18 () aC-groups 1)( Hall, P. [17], 1937; see also Schmidt [30], p. 122) A finite group K is an aC-group if and only if K is a direct sum of groups of square-free order. 2)( See Schmidt [30], p. 123) A group K is an aC-group if and only if

Kœ N3 z Lj Š‹3−M ŒjJ−

where each RL34 and each is cyclic of prime order and RK33ü for all . Behavior Under Direct Sum The following theorem describes how some basic constructions behave with respect to direct sums and emphasizes the fact that the summands in a direct sum have a great deal of independence. 168 Fundamentals of Group Theory

Theorem 5.19 Let

Kœ L3 3−M Then the following hold: 1)( Center of K )

^ÐKÑ œ  ^ÐL3 Ñ 3−M 2)( Commutator of K) ww KœL3 3−M

3) If RL33ü for all 3 , then

L3 L 3−M ¸ { 3 R333−M R 3−M

4) If L«K3 , then

AutÐKÑ ¸} Aut ÐL3 Ñ 3−M

Proof. For part 1), since the L3 's commute elementwise, it is clear that ^ÐL3" Ñ Ÿ ^ÐKÑ. But if D − ^ÐKÑ , then let D œ 2 â28 with each 23 in a different factor L433 . Then D−^ÐLÑ and 25 −^ÐLÑ3 for 5Á3 and so 2−^ÐLÑ33, whence D−^ÐLÑ3.

For part 2), Theorem 3.41 implies that ww K œÒLßLÓœ3 4 ÒLßLÓœ 34 ÒLßLÓœ 33 L 34 3ß4 3 33

For part 3), the function

L3 5{ÀLÄ3 3−M 3−M R3 defined by

5,Ð+ÑÐ5Ñ œ55R5 Ð+ÑR œ 1,5 ‰ Ð+Ñ

where 1,R55 is the canonical projection map and is the 5 th injection map is an epimorphism. Moreover, 5,+œ" if and only if 55Ð+Ñ−R for all 5 and so kerÐÑœR5 3.

As to part 4), since L«K3L , if 55 −Aut ÐKÑ , then l3 −Aut ÐLÑ3 and so we can define a map 0ÀAut ÐKÑ Ä} 3−MAut ÐL 3 Ñ by Direct and Semidirect Products 169

0Ð55 ÑÐ3Ñ œ lL3 This is a group homomorphism since

0Ð57 ÑÐ3Ñœ Ð 5 ‰ 7 ÑlLLL333 œ Ð 5 l щР7 l Ñ œ 0Ð 5 ÑÐ3щ0Ð 7 ÑÐ3Ñœ Ò0Ð 5 Ñ0Ð 7 ÑÓÐ3Ñ for all 3−M and so 0Ð57 ‰ Ñœ0Ð 5 Ñ0Ð 7 Ñ. Furthermore, 0 is injective since

5+lœL3 for all 3 implies 5+ œ .

To see that 0−Ð is surjective, if 7}3−MAut L 3Ñ, then the universality of the direct sum implies that there is a unique homomorphism 5ÀK ÄK satisfying

57lœL33 , the 3 th coordinate of 7 . But the bijectivity of each 73 implies that 55−ÐKÑ0ÐÑœAut and so 7 . When All Subgroups Are Normal All subgroups of an abelian group are normal, but all subgroups of the quaternion group are normal and yet is not abelian. A Hamiltonian group is a nonabelian group all of whose subgroups are normal. In 1933, Baer [3] published a characterization of Hamiltonian groups. Baer's theorem says that the Hamiltonian groups are actually a special type of abelian group with an additional quaternion direct summand.

Theorem 5.20 ()Baer [3], 1933 A group K is Hamiltonian if and only if KœEF where E is a quaternion group, is an elementary abelian group of exponent # and F is an abelian group all of whose elements have odd order. Proof. First suppose that KœEF and let LŸK . LŸEFIf , then LKüü. Note that if " − L , then Kw œ Ö„"× Ÿ L and so L K . Every 2−L has the form 2œ;+, where ;− +−E , and ,−F has odd order, then 2œ;œ;#9Ð,Ñ #9Ð,Ñ # and so ;# −L . If 9Ð;Ñœ% for some 2−L , then "−L and so Lü K . On the other hand, if 9Ð;ÑŸ# for all 2−L , then LŸÖ„"×EFœ^ÐKÑ and so LKü . Thus, K is Hamiltonian.

For the converse, let KK be Hamiltonian. Theorem 3.38 shows that is periodic and that any nonabelian subgroup of K contains a quaternion subgroup.

If FK is the set of odd-order elements of and Q is the set of elements of order a power of # , then F ∩ Q œ Ö"×. Moreover, for any Bß C − K , the normality of ØBÙ and ØCÙ imply that ØBß CÙ œ ØBÙØCÙ and so 9ÐBCÑ ± 9ÐBÑ9ÐCÑ. It follows that 170 Fundamentals of Group Theory

FQ and are (normal) subgroups of K . Finally, every +−K has order 9Ð+Ñ œ #5 7 where 7 is odd and so Corollary 2.11 implies that + œ ,7 for some ,−F and 7−Q . Thus, KœFQ Since FQ does not contain a quaternion subgroup, it must be abelian and since is therefore nonabelian (since KQ is nonabelian), contains a quaternion subgroup œØBßCÙ .

We are left with showing that QœE, where E is an elementary abelian group of exponent # . The centralizer

GœGQ ÐÑ of œØBßCÙ in Q has exponent # . To see this, note that if D−G has order % , then ØDBÙü Q has order % and so DB$C œ ÐDBÑ œ DBor DB $C$$ œ ÐDBÑ œ D B and since the former is false, we have Dœ"# . Thus, G is an elementary abelian group of exponent #G . Moreover, since is a vector space over ™# , every subgroup (i.e., subspace) has a direct complement and so GœØBÙE# where E# is also an elementary abelian group of exponent .

Consider the quotient QÎG , which contains the four distinct cosets G , BG , CG and BCG . Poincaré's theorem gives

ÐQÀGÑœÐQÀGQQ ÐBÑ∩G ÐCÑÑ Ÿ ÐQ À GQQ ÐBÑÑÐQ À G ÐCÑÑ œBQQ C ¸¸¸¸ œ% and so QÎGœÖGßBGßCGßBCG× which implies that Q œ G œÐØB Ù # EÑ œ E But the only involution in B is # , which is not in E and so ∩ E œ Ö"× . Thus, QœE and KœQFœEF Direct and Semidirect Products 171

Semidirect Products We now turn to semidirect products.

Definition Let K be a group. If KœRìL, Rü K then RL is called a normal complement of in K and K is called the semidirect product of RL by , denoted by KœRz L A semidirect product is nontrivial if both factors are proper.

Example 5.21 The dihedral group H#8 is a nontrivial semidirect product:

HœØÙØÙ#8 3z5 The symmetric group is also a nontrivial semidirect product:

W88 œ Ez ØÐ" #ÑÙ On the other hand, the quaternion group is not a nontrivial semidirect product, since the orders of the factors must be #% and but the only subgroup of of order #% is contained in every subgroup of order .

For an arbitrary semidirect product KœRz L, the commutativity rule is 28 œ Ð282" Ñ2 œ 8 2 2 for 2−L and 8−R and this yields the multiplication rule

2" Ð8"" 2 ÑÐ8 ## 2 Ñ œ 8 " 8# 2 "# 2 for 2−L33 and 8−R . Thus, for the semidirect product, the multiplication rule 2" shows that “cross products” are involved, in the form of conjugates 8" . This has some perhaps unexpected consequences.

For example, if E¸E"# and F¸F "# , then EF¸EF ""##. On the other hand, if Ø+Ù is cyclic of order ' , then Ø+ÙœØ+Ù#$z Ø+ÙœÖ"ß+ß+×Ö"ß+× #%$ But we also have

W$ œ ØÐ" # $ÙÙz+ ØÐ" #ÑÙ œ Ö ß Ð" # $Ñß Ð" $ #Ñ× z+ Ö ß Ð" #Ñ× and so the nonisomorphic groups Ø+Ù and W$ can be written as a semidirect product, where corresponding factors are isomorphic! The reason that this is not a contradiction is that the values of the conjugates are different in each group. For instance, 172 Fundamentals of Group Theory

$ Ð+#+ Ñ œ + # but

Ð"#$ÑÐ" #Ñ œÐ#"$ÑÁÐ"#$Ñ

Semidirect Products and Extensions of Epimorphisms

Let KK and "" be groups with LŸKÀLqK and let 5 » be an epimorphism. The key to describing the possible extensions of 5 to K is to consider the possible kernels for such an extension.

Suppose that KL is a group and that ßOŸK contain a normal subgroup NKü . Then LÎNOÎN and are complements in KÎN if and only if L∩OœNand KœLO It will be convenient to make the following nonstandard definition.

Definition Let KN be a group and let ü K . If LßOŸK satisfy L∩OœNand KœLO we will say that LO and are complements modulo NO . If is a normal complement of LN modulo , we will write KœOz LÒmod NÓ

Now, if 55ÀKq » K"" is an extension of ÀLq » K , then L∩kerÐÑœ5 kerÐÑ5 To see that KœLkerÐÑ55, since is surjective, for any +−K , there is an 2−L for which 555 +œ2œ2 and so 2+−" ker ÐÑ5, whence +œ2Ð2" +Ñ−Lker Ð5 Ñ and so kerÐÑ55 is a normal complement of L modulo ker ÐÑ : Kœker Ð5z Ñ LÒmod ker Ð 5 ÑÓ (5.22) It follows that every extension 55 of is uniquely determined by its kernel kerÐÑ5 .

On the other hand, suppose that KœOz5 LÒmod ker Ð ÑÓ

Then the ignore-O map 5ÀK ÄK" defined by 55Ð25Ñ œ Ð2Ñ Direct and Semidirect Products 173

for 2−L and 5−O is well defined since if 25œ25"" for 2 " −L and 5−O" , then " " 22œ55−L∩OœÐÑ" " ker 5 " and so 55Ð2" 2Ñ œ ", that is, 2 œ5 2" . The normality of O implies that 5 is a group homomorphism, since

2 5555555Ð525"" 2 Ñ œ Ð55" 22"" Ñ œ Ð22 Ñ œ Ð2Ñ Ð2" Ñ œ Ð52Ñ Ð5"" 2 Ñ The kernel of 5 is kerÐ55 ÑœÖ25± 2œ!לker Ð 5 ÑOœO and so 55 is the unique extension of with kernel O .

Theorem 5.23 Let KK and "" be groups, let LŸKÀLK and let 5 q» be an epimorphism. 1) Given any normal complement OL of modulo ker ÐÑ5 , KœOz5 LÒmod ker Ð ÑÓ

the ignore-OÀKÄK map 5O" defined by 55Ð25Ñ œ 2 for all 2−L and 5−O is the unique extension of 5 with kernel O . Moreover, every extension 55 of is an ignore-OOœÐÑ map, where ker 5 . 2) If KœOz5 L, then has a unique extension 5ÀKÄK" for which kerÐÑœO55 ker ÐÑ Proof. For part 2), the Dedekind law implies that L∩Oker Ð5 Ñœker Ð55 ÑÐL ∩ OÑ œker Ð Ñ and so OÐÑker 55 is a normal complement of L modulo ker ÐÑ . Semidirect Products and One-Sided Invertibility Semidirect products are related to one-sided invertibility.

Definition Let 5ÀK ÄL be a group homomorphism. 1) A left inverse of 555 is a homomorphism PPÀL ÄK for which ‰5 œ+ . If 55 has a left inverse, then is said to be left invertible. 2) A right inverse of 55 is a homomorphism ÀK ÄL for which 5‰5 œ+ . If 55 has a right inverse, then is said to be right invertible.

Unlike the two-sided inverse, one-sided inverses need not be unique. A left- invertible homomorphism 5 is injective, since 174 Fundamentals of Group Theory

5+œ 5 , Ê 55PP ‰ +œ 55 ‰ , Ê +œ, and a right-invertible homomorphism 5 is surjective, since if ,−L , then

,œ55 Ð ,Ñ−im Ð 5 Ñ For set functions, the converses of these statements hold: 5 is left-invertible if and only if it is injective and 5 is right-invertible if and only if it is surjective. However, this is not the case for group homomorphisms.

Let 5äÀK K" be injective. Referring to Figure 5.3,

K σ|im(σ) im(σ) (σ|im(σ))-1 G G1

Figure 5.3 the map 55limÐÑ5 ÀK¸im ÐÑ obtained from 5 by restricting its range to imÐÑ5 is an isomorphism and the left inverses of 5 are precisely the extensions of imÐÑ"5 55P"œÐ l Ñ Àim Ð 5 ѸK to K . Hence, Theorem 5.23 implies that there is one left inverse 55P for for each normal complement OÐÑ of im 5 and kerÐÑœO55P . Moreover, this accounts for all left inverses of . In particular, 5 is left-invertible if and only if imÐÑ5 has a normal complement.

Now let 5ÀKq » K" be surjective. Referring to Figure 5.4,

ker(σ)

H σ|H

-1 σR=(σ|H) G G1

Figure 5.4 " If Kœker ÐÑ5z L for some LŸK , then 5 lÀL¸KL"L and ÐlÑ5 is a right inverse of 55 , with image LÀ . Conversely, if "Kä5K is a right inverse of , then

Kœker Ð5z Ñim Ð 5 Ñ

For if +−ker Ð55 Ñ∩im Ð Ñ, then +œ 5 , and 5 +œ" , whence ,œ" and so Direct and Semidirect Products 175

+œ". Also, for any +−K , we have +œ +ÒÐ55 ‰ Ñ+Ó" Ð 55 ‰ Ñ+ −ker Ð 5z5 Ñ im Ð Ñ ˆ‰

Theorem 5.24 Let 5ÀK ÄK" be a group homomorphism. 1) If 55 is injective, then has a unique left inverse 5P with kernel O for each normal complement OÐÑ of im 5 . Hence,

Kœ"Pker Ð5z Ñim ÐÑ 5 This accounts for all left inverses of 5 . " 2) If 55 is surjective, then has a unique right inverse 5L"œÐ5 l Ñ ÀK ÄK for each complement LÐÑLœÐÑ of ker 55 . Hence, im and

Kœker Ð5z Ñim Ð 5 Ñ This accounts for all of the right inverses of 5 .

Here is a nice application of this theorem.

Theorem 5.25 Let LO and be indecomposable groups. Let αÀOÄL and "αÀL ÄO be homomorphisms for which "−Aut ÐLÑ and im Ð" Ñü O . Then α" and are isomorphisms. Proof. The map

" "α"αP œÐ Ñ ‰ ÀOÄL is a left inverse of "" and so is injective and

Lœker Ð""P Ñim Ð Ñ

But since ""P is not the zero map, must be surjective as well. Hence, " and therefore also α are isomorphisms. The External Semidirect Product To see how to externalize the semidirect product, let us review the internal version. If we write the multiplication rule for the semidirect product RLz in the form

Ð8"" 2 ÑÐ8 ## 2 Ñ œ 8 " Ò# 2" Ð8 # ÑÓ2 "# 2 where #z2 is conjugation by 2RL , then is completely described by the subgroups RL and , along with the family \#œÖ±2−L×2 of inner automorphisms. Moreover, since the map ##À2 È 2 is a homomorphism, # is a representation of LÐRÑ in Aut .

In the spirit of externalization, we separate the components by writing 25 as Ð2ß 5Ñ to get 176 Fundamentals of Group Theory

Ð8"" ß 2 ÑÐ8 ## ß 2 Ñ œ Ð8 "2## " Ð8 Ñß 2 "# 2 Ñ But if the factors RL and are to be arbitrary groups, then the inner automorphisms #2 make no sense. However, any representation of )ÀL ÄAut ÐRÑ of L in Aut ÐRÑ can play the role of conjugation and define a group, as we now show.

Theorem 5.26 Let RL and be groups and let )ÀLÄÐRÑAut be a homomorphism. We denote ))Ð2Ñ by 2 . Let R z) L be the cartesian product R‚L together with the binary operation defined by

Ð+ß BÑÐ,ß CÑ œ Ð+)B Ð,Ñß BCÑ

Then KœRz) L is a group; in fact, it is a semidirect product Rzz L œ R ‚ Ö"× Ö"× ‚ L ) Also, Ð+ßBÑœÐ+ß"ÑÐ"ßBÑ and

Ð"ßBÑ Ð+ß"Ñ œÐ)B Ð+Ñß"Ñ which shows that ) does define the inner automorphisms of elements of R ‚ Ö"× by elements of Ö"× ‚ L . The group RLz) is called the external semidirect product of RL by defined by ) . Proof. To see that multiplication is associative, we have

ÒÐ+ß BÑÐ,ß CÑÓÐ-ß DÑ œ Ð+)B Ð,Ñß BCÑÐ-ß DÑ œ Ð+))BBC Ð,Ñ Ð-Ñß BCDÑ and

Ð+ß BÑÒÐ,ß CÑÐ-ß DÑÓ œ Ð+ß BÑÐ,)C Ð-Ñß CDÑ œ Ð+))BC Ð, Ð-ÑÑß BCDÑ œ Ð+))BBC Ð,Ñ Ð-ÑÑß BCDÑ It is easy to check that Ð"ß "Ñ is the identity and

" " " Ð+ßBÑ œÐ)B" Ð+ ÑßB Ñ

Thus, RLz) is a group. For the rest, routine calculation gives Ð+ß "ÑÐ,ß "Ñ œ Ð+,ß "Ñand Ð+ß "Ñ" œ Ð+ " ß "Ñ and Ð"ß BÑÐ"ß CÑ œ Ð"ß BCÑand Ð"ß BÑ" œ Ð"ß B " Ñ Direct and Semidirect Products 177

and so R ‚ Ö"× and Ö"× ‚ L are subgroups of RLz) . To see that R ‚ Ö"× is normal, since Ð+ßBÑœÐ+ß"ÑÐ"ßBÑ we need only check that

Ð"ßBÑ " " Ð+ß "Ñ œ Ð"ß BÑÐ+ß "ÑÐ"ß B Ñ œ Ð"ß BÑÐ+ß B Ñ œ Ð)B Ð+Ñß "Ñ − R ‚ Ö"× and

Ð+ß "ÑÐ,ß"Ñ œ Ð,ß "ÑÐ+ß "ÑÐ," ß "Ñ œ Ð,+," ß "Ñ − R ‚ Ö"×

Clearly, ÐR ‚ Ö"×Ñ ∩ ÐÖ"× ‚ LÑ œ ÖÐ"ß "Ñ× and so Rz) L is the (internal) semidirect product of R ‚ Ö"× by Ö"× ‚ L .

Note that the zero representation ))ÀL ÄAut ÐKÑ defined by Ð2Ñœ+K defines the external direct product KL} .

It is common to write the external version of the semidirect product using the notation of the internal version. Thus, if KL and are groups, we can specify a semidirect product WœKz) L by taking the set of formal products WœÖ12±1−Kß2−L× in place of ordered pairs and specifying the commutativity rule

21 œ)2 Ð1Ñ2 where )ÀL ÄAut ÐKÑ is a homomorphism.

Example 5.27 Given a group K , there is one rather obvious way to create an external semidirect product KLz) , namely, by taking LœÐKÑAut and )ÀL ÄL to be the identity. The group product in this case is Ð+57 ÑÐ, Ñ œ + 557 Ð,Ñ

The group KÐKÑz)Aut is called the holomorph of K .

We may generalize this by taking LÐ to be any subgroup of Aut KÑ and ) to be the inclusion map from LÐKÑKL to Aut . The group z) is called the relative holomorph of KL with respect to .

Finally, if 5z−ÐKÑAut , then the relative holomorph KØÙ) 5 is called the extension of K by 5 . The group product in this case is Ð+5534 ÑÐ, Ñ œ + 55 334 Ð,Ñ

8 As an example, let KœG#8 Ð+Ñ be cyclic of order # , where 8 $ . If 7œ#8", then the power map defined by .+œ+7" is an automorphism of K 178 Fundamentals of Group Theory of order #7" , since is relatively prime to #8 and Ð7"Ñ#8#8 œÐ# "Ñ# "´"mod #8

Thus, the extension of KW by .z is just Hœ8#GÐ8 +ÑG) #ÐBÑ and has group product Ð+34 ß "ÑÐ+ ß "Ñ œ Ð+ 34 ß "Ñ Ð+3 ß BÑÐ+ 4 ß "Ñ œ Ð+3Ð7"Ñ4 ß BÑ Ð+34 ß "ÑÐ+ ß BÑ œ Ð+ 34 ß BÑ Ð+3 ß BÑÐ+ 4 ß BÑ œ Ð+3Ð7"Ñ4 ß "Ñ for all ! Ÿ 3ß 4 Ÿ #83  ". Since Ð+ ß B43 Ñ œ Ð+ß "Ñ Ð"ß BÑ4, setting α œ Ð+ß "Ñ and 0 œÐ"ßBÑ gives

8#8"" WH8 œØα0 ß Ùß 9Ð α Ñœ# ß 9ÐÑœ#ß 0 0α œ α 0

8" The group WH8 is called the semidihedral group of order # .

Example 5.28 Recall that if GÐ is an infinite cyclic group, then Aut GÑœÖ+7ß× " ‡ where 7™À+È+ and if G is cyclic of order 8 , then Aut ÐGѸ8. Let G∞ Ð+Ñ and GÐ,Ñ∞ be infinite cyclic groups. Then a homomorphism

)+À G∞∞ Ð,Ñ ÄAut ÐG Ð+ÑÑ œ Ö ß7 × is completely determined by the value )+,, , which can be either or 7 . If )œ + , then )z) is the zero map and GÐ+ÑGÐ,Ñ∞∞) is direct. If ,œ7 , then the commutativity rule in the group GÐ+ÑGÐ,Ñ∞∞z) is ,+ œ7 Ð+Ñ, œ +" ,

The automorphisms of a finite cyclic group GÐ+Ñ8 are the 5 th power homomorphisms 55 defined by 5 55Ð+Ñ œ +

‡ where 5 − ™)8 . Thus, the representations 5∞À G Ð,Ñ ÄAut ÐG8 Ð+ÑÑ are the homomorphisms defined by )555Ð,Ñ œ and so the possible semidirect products are

34 5 G8∞ Ð+Ñz™)5 G Ð,ÑœÖ+, ±!Ÿ3Ÿ8"ß4− ß,+œ+ ,×

‡ where 5−™8.

Example 5.29 To define a semidirect product GÐ+Ñ$%z ) GÐ,Ñ, we must specify a homomorphism

)+À G%$# Ð,Ñ ÄAut ÐG Ñ œ Ö ß5 ×

# where 5#Ð+Ñ œ + . The zero homomorphism defines the direct product Direct and Semidirect Products 179

GÐ+Ñ$%})5 GÐ,Ñ. If ,# œ , then the semidirect product XœGÐ+Ñ$z) GÐ,Ñ % defined by ) is X œ Ø+ß ,Ùß 9Ð+Ñ œ $ß 9Ð,Ñ œ %ß ,+ œ +# ,

We have not yet encountered this group of order "# . In particular, X ¸y E% since # 9Ð+, Ñ œ ' and X ¸y H"# since 9Ð,Ñ œ % . We will show later that EHX% , "# and are the only nonabelian groups of order "# (up to isomorphism).

Example 5.30 Let us examine the possibilities for an external semidirect product of the form

GÐ+ÑGÐ,Ñ::78z) where :G is prime. The automorphisms of :7 Ð+Ñ are the 5 th power maps 5‡ 5™)5:À + È + for 5 − 7 . The function À G87 Ð,Ñ ÄAut ÐG: Ð+ÑÑ satisfying )5,5œ9 defines a homomorphism if and only if Ð55ѱ9Ð,Ñ , that is, if and only :8 if 5+5 œ , or

Ð:8 Ñ +œ+5 or finally

8 5´":Ð: Ñ mod 7 (5.31) As an example, for 7œ" , Fermat's theorem implies that (5.31) is equivalent to 5´"mod : and since "Ÿ5: , it follows that 5œ" . Hence, the only semidirect product of the form

GÐ+Ñ::z) G8 Ð,Ñ is the direct product.

If 8œ" , then (5.31) is equivalent to 5´":7mod : Any 5 of the form 5œ"?:7" where ?: satisfies this congruence, since 5:7 œÐ"?:" Ñ:7 œ"A: Thus, for each ?: , there is a semidirect product

GÐ+ÑGÐ,Ñ::7 z) where

7" ,+ œ +"?: , 180 Fundamentals of Group Theory

Example 5.32 Let H be a group and let KœH³HHH$ }}. Then each permutation 5)−W$ defines an automorphism 5 of K by permuting the coordinates in K . For example,

)Ð" $ÑÐBßCßDÑœÐDßCßBÑ

Moreover, the map )5ÀW$ ÄAut ÐKÑ sending to )5 is a homomorphism, since ))57œ ) 57. Thus, the semidirect product $ KWœHWzz))$$ exists. To illustrate the product, if 5 œÐ"$Ñ , then Ð+ß,ß-Ñß57 Ñ ÐBßCßDÑß Ñ œ Ð+ß,ß-Ñ )57 ÐBßCßDÑß 5 œ Ð+ß,ß-ÑÐDßCßBÑß57 œÐ+Dß,Cß-BÑß57 We will generalize this example in the next section. *The Wreath Product To generalize Example 5.32, let H be a group, let H be a nonempty set and let Kœ} H =H− be the external direct product of HH copies of H , indexed by . Each kk permutation 5H of defines an automorphism )5 of K by permuting the coordinate positions of any 0−K . Specifically, the = th coordinate of 0 becomes the ÐÑ5= -th coordinate of )55Ð0Ñ , that is, ) Ð0ÑÐÑœ0ÐÑ 5= = , or equivalently,

" Ð0ÑÐÑœ0Ð)=5 5= Ñ Thus,

" )55Ð0Ñ œ 0 ‰

The map )5ÀK ÄK is easily seen to be an automorphism of K , since it is clearly bijective and

" " " )555))55Ð01ÑœÐ01щ œÐ0‰ ÑÐ1‰ Ñœ Ð0Ñ5 Ð1Ñ

Moreover, if ŸW H5 , then the map ))ÀÄAut ÐKÑ defined by 5œ) is a homomorphism, since

" " " " )5775)5))57 0œ0‰Ð Ñ œ0‰Ð ‰ Ñœ7 Ð0щ œ5 7 Ð0Ñ

Hence, the semidirect product Kz ) exists. It is easy to describe the commutativity rule in words: To place the factors in the product 50 in the reverse order, simply permute the coordinate positions of 0 using 5 . Direct and Semidirect Products 181

Note that it is not essential that the second coordinates in the ordered pairs Ð0ß5z Ñ − K) be actual permutations of H as long as they act like permutations, that is, as long as there is a homomorphism -ÀÄW H . As is customary, we denote the permutation -H;; of by itself.

Thus, if ; acts on H) , then for each − , the map ;ÀKÄK defined by " );0œ0‰; is an automorphism of KÀ and the map ))ÄÐAut KÑ; defined by œ); is a homomorphism.

The semidirect product KKz)) of by defined by is one version of the wreath product of K by . The other version comes by replacing the external direct product Kœ}{ H by the external direct sum Kœ H .

Definition Let H be a group, let H be a nonempty set and let be a group acting on H= . We denote the action of ;− on −H by ; = . 1) Let Kœ} H =H−

Define a homomorphism ))ÀÄAut ÐKÑ by ; œ ); where " );0œ0‰;

Then the semidirect product Kz ) is called the complete wreath product of H by with index set H and base K . 2) If we replace K by the external direct sum Kœ{ H =H−

the resulting semidirect product Kz ) is called the restricted wreath product of K by with index set H and base K .

A common notation for the wreath product is H› . To emphasize the index set, we will write H›H . There does not seem to be a standard notion to distinguish the two wreath products, so we use H› for the restricted wreath product and H›› for the complete wreath product.

Note that if H and are finite groups and H is a finite set, then

HHH H››œH œHz œH kk kH k ¸¸¸¸kk kk kk Example 5.33 ( Regular wreath product) Let H and be groups and let Kœ} H ;− 182 Fundamentals of Group Theory be the direct product of H indexed by the group . Let act on itself by left translation, that is, );<œ;< , that is, the action of is the left regular representation. In this case, the complete wreath product H›› is called a regular wreath product, which we denote by H››< . Thus, the product has the form Ð0ß ;ÑÐ1ß <Ñ œ Ð0Ð1 ‰ ;" Ñß ;<Ñ

Wreath Products as Permutations Under certain reasonable conditions, the elements of a wreath product can be thought of as permutations. Specifically, let

[œH››H

Let Kœ}H=H− H and assume that acts faithfully on .

Now suppose that the group H acts faithfully on a nonempty set A . Then the elements of [‚ act on the set AH . In particular, for Ð0ß;Ñ−[ define Ð0ß ;ч ÀAH ‚ Ä AH ‚ by Ð0ß;ч Ð-= ß ÑœÐ0Ð; =- Ñ ß; = Ñ The map Ð0ß;ч‡‡ is injective since Ð0ß;Ñ Ð-= ß ÑœÐ0ß;Ñ Ð -w ß =w Ñ implies that Ð0Ð;=- Ñ ß ; = Ñ œ Ð0Ð; =ww Ñ - ß ; = w Ñ and so ==wwœœÐ and -- . Also, 0ß;Ñ ‡ is surjective since for any Ðß-= Ñ− A ‚ H, we have Ð0ß ;ч"" Ð0Ð=- Ñ ß ; = Ñ œ Ð -= ß Ñ Hence, Ð0ß ;ч is a permutation of AH‚ .

‡ Moreover, the map 55À[ ÄWAH‚ defined by Ð0ß;ÑœÐ0ß;Ñ is a homomorphism, since ÒÐ0ß ;ÑÐ1ß <ÑÓ‡ Ð-= ß Ñ œ Ð0Ð1 ‰ ;" Ñß ;<ч Ð-= ß Ñ œ ÐÒ0Ð1 ‰ ;" ÑÐ;<=- ÑÓ ß ;< = Ñ œ ÐÒ0Ð;<==-= ÑÐ1 ‰ ;" ÑÐ;< ÑÓ ß ;< Ñ œ Ð0Ð;<==-= Ñ1Ð< Ñ ß ;< Ñ œÐ0ß;ÑÐ1Ð<‡ =- Ñ ß< = Ñ œÐ0ß;ч‡ Ð1ß<Ñ Ð-= ß Ñ As to the kernel of 5+ , if Ð0ß ;ч œ , then Direct and Semidirect Products 183

Ð0Ð;=- Ñ ß ; = Ñ œ Ð -= ß Ñ for all -=H−H and − . Since the actions of H on HA and on are faithful, we deduce that ;œ" and 0Ð==5ä Ñœ" for all . Thus, À[ WAH‚ is an embedding of [W into AH‚ .

When [œH›H is the restricted wreath product, we can describe the image of the embedding explicitly. For each .−H and αH − , let .α −K be defined by .œif =α .ÐÑœα = œ "Áif =α

‡ Let \W be the subgroup of AH‚ generated by the permutations Ð.αß"Ñ and Ð"ß ;ч, that is, ‡‡ \ œ ØÐ.α ß "Ñ ß Ð"ß ;Ñ ± . − HßαH − ß ; −Ù Certainly, \©im Ð5 Ñ.

To see that the reverse inclusion holds, we observe that Ð0ß ;ч‡‡ œ ÒÐ0ß "ÑÐ"ß ;ÑÓ œ Ð0ß "Ñ Ð"ß ;ч and so it is sufficient to show that Ð0ß "Ñ − \ for all 0 − K . Since Ð01ß "ч‡‡ œ ÒÐ0ß "ÑÐ1ß "ÑÓ œ Ð0ß "Ñ Ð1ß "ч we have ‡‡‡‡ Ð0"8 â0 ß;Ñ œÐ0 " ß"Ñ âÐ0 8 ß"Ñ Ð"ß;Ñ

Now, any 0−K has finite support suppÐ0ÑœÖ=="8 ßáß × and so

0 œ 0Ð=="8 Ñ=="8 â0Ð Ñ Hence, ‡‡‡‡ Ð0ß"Ñ œÐ0Ð=="8 Ñ=="8 â0Ð Ñ ß"Ñ œÐ0Ð = " Ñ = " ß"Ñ âÐ0Ð = 8 Ñ = 8 ß"Ñ −\ Thus, imÐÑœ\5 .

Theorem 5.34 Let [œH››H be a wreath product and suppose that acts faithfully on HA . Suppose also that H acts faithfully on . Then the map 5À[ ÄWAH‚ defined by 5Ð0ß ;Ñ œ Ð0ß ;ч where Ð0ß ;ч ÀAH ‚ Ä AH ‚ is defined by Ð0ß;ч Ð-= ß ÑœÐ0Ð; =- Ñ ß; = Ñ 184 Fundamentals of Group Theory

is an embedding of [W into AH‚ . When [œH›H is the restricted wreath product, the image of 5 is ‡‡ imÐ5α ÑœØÐ.α ß"Ñ ßÐ"ß;Ñ ±. −Hß −H ß; −Ù It is convenient to drop the ‡ notation and to think of elements of a wreath product H›› as permutations of AH‚ .

Example 5.35 A permutation matrix T8‚8 is an matrix with entries from the set Ö!ß "× with the property that each row contains exactly one " and each column contains exactly one "E . Multiplication of a matrix on the left by a permutation matrix TE permutes the rows of . Similarly, multiplication on the right by TE permutes the columns of . Let c8 be the multiplicative group of all 8‚8 permutation matrices.

Ð4Ñ For T−c83 , let T denote the 3 th row of T and let T denote the 4 th column. The rows of T also define a permutation 1HT of œ Ö"ß á ß 8×, in particular, 1T3Ð3Ñ is the column number of the "T in row . In this way, T8 is isomorphic to the symmetric group W08T8 . Clearly, the map ÀTÈ1c is bijective. If − , Ð4Ñ then ÐT Ñ3ß4 œ T 3 œ " if and only if the column number 5 of the " in 3 is the same as the row number 5" of the in column TÐ4Ñ , that is, if and only if 5 œ111T Ð3Ñ implies that Ð5Ñ œ 4 . Hence, TÐ3Ñ œ 4 implies that 11TÐÐ3ÑÑœ4, that is, 1TT œ 11. Hence, 0 is an anti-isomorphism from c8 to WÀ88 . Since the transpose map 7cÄ c88 is an anti-automorphism of c , it follows that the composite map 0‰7c is an isomorphism from 88 to W .

We can generalize the notion of a permutation matrix as follows. If L is an abelian multiplicative group, define c8ÐLÑ to be the set of 8 ‚ 8 matrices with the property that each row and each column has exactly one entry from L , all other entries being !L . Matrix multiplication is defined using the product in along with !†!œ!†+œ! and !+œ+!œ+ for +−L . This makes ccc888ÐLÑ a group. Note that is a subgroup of ÐLÑ , with the identity of L playing the role of " .

We can describe c88ÐLÑ in terms of wreath products as follows. Let W act on Hcœ Ö"ßáß8× by the usual evaluation and let 8T act on H by TÐ5Ñ œ1 Ð5Ñ.

Let Wcc be the set of all diagonal matrices in 88ÐLÑ . Then every matrix in ÐLÑ is a product HT where H −WcW and T − 8 . Also, is a normal subgroup of ccW888ÐLÑ. Hence, ÐLÑ œ zc.

Now, any H−W can be identified with the ordered 8 -tuple of diagonal elements of HL and, in fact, W is isomorphic to 8 , since matrix multiplication in W is elementwise product in L8 . Hence, 8 czcc8888ÐLÑ ¸ L œ L ›HH ¸ L › W Direct and Semidirect Products 185

When L7 is the group of th roots of unity in ‚c , the group 8ÐLÑ is called a generalized symmetric group or monomial group. Exercises 1. Prove that the external direct product is commutative and associative, up to isomorphism, that is, LO¸OL}} and ÐL}} OÑ P ¸ L }} ÐO PÑ Is there an identity for the external direct product? 2. a) Suppose that Kœ{ K3 . For each 5 , let

LœÖ0−53{ K±0Ð3Ñœ" for 3Á5×

Show that LK55ü and that KœL . b) If Kœ K33 , show that K¸{ K . 3. Let LL , 3 and O be groups and let

LœL"=}} â L Suppose that 5ÀL ¸O . For each 3 , let

L33 œ Ö"×}} â Ö"× }} L Ö"× }} â Ö"×

where L33 is in the th component. Prove that

Oœ55 L"= â L

4. Let K œ L"8 âL be a finite group, where L3ü K and Ð9ÐL3 Ñß 9ÐL 4 ÑÑ œ " for . Prove that the join is a direct product, that is, 3Á4 1L3 KœL"8 âL . 5. Suppose that K œ L  O and that Rü K . Prove that if R ∩ L œ Ö"× œ R∩O, then RŸ^ÐKÑ . 6. Let 9ÐKÑœ78 where 7 and 8 are relatively prime. Let LŸK with 9ÐLÑ œ 7. Show that a subgroup OŸK is a complement of L if and only if 9ÐOÑ œ 8 . 7. Prove that all nonabelian groups of order ::$ , prime are indecomposable. You may assume that all groups of order :# are abelian (which is true). 8. Prove that the group  of rational numbers is indecomposible. 9. Prove that H8#8 is indecomposable if and only if ´y#mod %. 10. Let KœØ+Ù be a cyclic group. a) If K is infinite, show that K œ L ì O implies that L œ Ö"× or O œ Ö"×. b) If 9ÐKÑ œ 8 , describe precisely the conditions under which a nontrivial essentially disjoint product representation KœLìO exists. 11. Let Kœ 3−M L 3 and let OŸK . 186 Fundamentals of Group Theory

a) Show that it is not necessarily true that

? Oœ ÐL∩OÑ3 3−M even if OKü . b) Recall that a group that is equal to its own commutator subgroup is called perfect . Prove that if O is normal and perfect, then

Oœ ÐL∩OÑ3 3−M c) Prove that if K is periodic and if for all

B− L4 4Á3

and C − L3 , the orders 9ÐBÑ and 9ÐCÑ are relatively prime, then

Oœ ÐL∩OÑ3 3−M

This holds in particular for finite families if the factors L3 have relatively prime exponents. 12. Let KL be a group and let be a simple subgroup of K# with index . What can you say about any other nontrivial proper normal subgroup of K ? Must such a subgroup exist? (For the latter, you may assume that the alternating group EW8&8 is simple and that is centerless for $ .) 13. (Chinese remainder theorem for groups) Let K be a group and let LßáßL"8 be normal subgroups. Consider the map 5}ÀKÄ KÎL3 defined by

5+œÐ+Lßáß+LÑ"8

a) Show that 55 is a homomorphism with kerÐ ÑœL"8 ∩â∩L . b) Show that if the indices ÐK À L3 Ñ are finite and pairwise relatively prime, then 5 is surjective and so for any +ßáß+"8 −K, there is a for which . This can also be written 1−K 1−+ +L33 1´+""mod L ã 1´+88mod L c) Discuss the uniqueness of the solution 1 in part b). 14. Let KR be a group and let be minimal normal in K . a)A Prove that if KE is characteristically simple, then there is a ©utÐKÑ such that

KœR 5 R Œ5−E Direct and Semidirect Products 187

Also, RK is simple, as is every term in the sum above and so is the direct sum of isomorphic simple subgroups. b) Prove that if KR has the DCC on normal subgroups, then is the direct sum of isomorphic simple groups. 15. To see that infinite groups are not, in general, cancellable in direct products, prove that ™}™}™™}™ÒBÓ ¸ ÒBÓ , where ™ÒBÓ is the abelian group of polynomials over ™™}™™ but ¸y . 16. Let Kœ W33 , where W is simple for all 3 . Prove that the center ^ÐKÑ is the direct sum of those factors WK3 that are abelian. Hence, is centerless if and only if all of the factors W3 are nonabelian. 17. Let

Kœ W3 3−M

be centerless, where each W3 is simple. Prove that the only minimal normal subgroups of KW are the summands 3 . 18. Let

Kœ W3 3−M

be centerless, where each W3 is simple. Prove that the normal subgroups of KWM are precisely the direct sums of the 3 's taken over the subsets of . 19. Let Kœ 3−M L 3 where L3 are simple subgroups and L3 ¸L 4 for all 3ß 4 − M. Prove that K is characteristically simple. Hint : Consider the centers of the L3 . Minimal Normal Subgroups 20. Let KL be a group. Show that if and O are distinct minimal normal subgroups of KLO , then and commute elementwse. 21. Let EK be an abelian minimal normal subgroup of a group . Show that if K œ EL where L  K , then E ∩ L œ Ö"×. 22. Let RßáßR"8 be minimal normal subgroups of K , let QœRâR"8 and let OKü . Prove that there is a subset of RßáßR"8 , say after reindexing RßáßR"7, such that

OQœOR"7 âR 23. Let EŸFŸK and assume that E is a minimal normal subgroup of F and FK is a minimal normal subgroup of . Assume further that the set Ö×EœÖ1E1±1−K×" of conjugates of E is finite. Show that E is simple and that FE is the direct product of conjugates of . 24. Prove that the epimorphic image of a minimal normal subgroup is either trivial or minimal normal. Semidirect Products 25. Let KœLz O. Show that if 9ÐLÑœ# , then KœLO. 26. Let K œ Lz) O. Show that if O is simple, then 9ÐOÑ ± 9ÐAut ÐLÑÑ. 188 Fundamentals of Group Theory

27. Show that for every positive integer of the form 8 œ #Ð#5  "Ñ there is a centerless group of order 8 . 28. Prove that if KœLzz O, then KœL O+ for any +−K . Hence, if O is a complement of a normal subgroup, then so is any conjugate of O . 29. Show that it is not always possible to extend a homomorphism 5ÀL ÄKw from a subgroup LŸK to K . 30. Show that if KœLz O is an internal semidirect product, then K is isomorphic to an external semidirect product K¸Lz) O for some )ÀO ÄAut ÐLÑ . 31. Prove that

™z'$œE Fand W œGz H

where E¸™™$# ¸G and F¸ ¸H, but clearly ™ '¸Wy $ . 32. Prove that WœE88z+ ÖßÐ"#Ñ×. 33. a) Prove that H¸E)"z™™ F, where E¸ %"# and F¸ . b) Prove that H¸G)#z™ F, where G¸Z and F¸## . What does this say about semidirect product decompositions? 34. Prove that H¸EF#8 z™™, where E¸8 and F¸# . 35. What is wrong, if anything, with the following argument? Let KœLz O. Then the projection maps 33EÀK ÄL and O ÀK ÄO defined by 33LOÐ25Ñ œ 2 and Ð25Ñ œ 5 have kernel OL and , respectively, whence both LO and are normal subgroups of K and so KœLO. 36. Recall that KœKPÐ8ßJÑ is the general linear group of all invertible 8‚8 matrices over the field J and WœWPÐ8ßOÑ is the subgroup of matrices with determinant equal to " . a) Prove that WPÐ8ßOÑü KPÐ8ßJÑ. b) Find a complement of WPÐ8ßJÑ in KPÐ8ßJÑ . Hint : How does one get a special matrix from a general one? 37. Let K+ be a group. For any −K , denote left translation by + by j+ . Thus, jÀKÄK+++ is defined by jÐBÑœ+B . Let _ œÖj±+−K×. Let T_œÐKÑAut . Finally, let LœØßÙT be the subgroup of the symmetric group WK generated by _T and . a) Prove that Lœ_zT . b) Prove that GÐÑœL _e, the subgroup of all right translations <ÀBÈB<+ . 38. In this exercise, we describe all groups of order :; where :  ; are primes. We will assume a fact to be proved later in the book: If K is a finite group and if :9 is the smallest prime dividing ÐKÑ: , then any subgroup of index is normal in K . a) Describe the automorphism group AutÐG:: Ñ , where G œ Ø+Ù is cyclic of order a prime : . b) Show that any group of order :; is a semidirect product of a group of order ;: by a group of order . c) Describe the possible external direct products of GG;: by , where : and ; are distinct primes. Direct and Semidirect Products 189

d) “Internalize” the external semidirect products in the previous part to show that up to isomorphism, the groups of order :; are the direct : product GG;: and for each 7Á" satisfying 7´"mod ;, a group described by K œ Øα" ß Ùß 9Ð α Ñ œ :ß 9Ð " Ñ œ ;ß α" œ "7 α 39. Let EœÖ7#8 ±7ß8−™ × be the additive subgroup of and let Fœ™ B be an additive infinite cyclic group. Prove that the group KœEz) F, where )BÐ+Ñ œ #+ , has the ACC on normal subgroups but that the normal subgroup EK of does not have the ACC on normal subgroups. Wreath Products

40. One must be cautious in working with the action of 5 −W8 on the product HH8. Recall that 5 permutes the coordinates of an element of 8 . Suppose 8 that Ð."8 ßáß. Ñ−H . Then

5Ð."8 ßáß. ÑœÐ.55 " ßáß. 8 Ñ

For 77−W8" , compute Ð5 ÑÐ. ßáß.8 Ñ. Are you sure? 41. In this exercise, we take a combinatorial look at the wreath product [œW›W#&, with the help of Figure 5.5.

a1 b1

b a 5 1 2 5 a 5 0 2 b2 4 3 b4 a3 a 4 b3

Figure 5.5 Informally speaking, a graph is a set of vertices (or nodes ) together with a set of edges connecting pairs of vertices. Figure 5.6 is an example of a graph Z . Two vertices are said to be adjacent if there is an edge between them. A bijection of the vertices of a graph that preserves adjacency is called an isomorphism. Show that the automorphisms of Z are isomorphic to the wreath product [œW›W#&. 42. Show that the wreath product is associative. Specifically, let KL act on ? , act on HA and O act on , all actions being faithful. Show that ÐK ››HH L Ñ ››AA O ¸ K ›› ÐL ›› O Ñ. 43. Show that the regular wreath product is not associative. Specifically, if K , LO and are nontrivial finite groups, explain why ÐK›LÑ›O<< cannot possibly be isomorphic to K›<< ÐL› OÑ. 190 Fundamentals of Group Theory

44. Show that the restricted wreath product G›G## is isomorphic to the dihedral group H) . 45. Let \ be a nonempty set of size 85 and let c œÖF"8 ßáßF × be a partition of \5− into equal-sized blocks of size . Call a permutation 5 W\ nice if it also permutes the blocks, that is, for all 34 , there is a such that 5FœF34. Show that the set R of nice permutations is a group isomorphic to W›W58. 46. Referring to Example 5.35, let L œ Ö„"× be the multiplicative group of square roots of unity. The group c8ÐLÑ is called the hyperoctahedral group. Show that c8#ÐLÑ is isomorphic to the subgroup KŸW 8 of all permutations of \ œÖ8ßáß"ß"ßáß8× with the property that 11Ð5Ñ œ  Ð5Ñ. 47. Let [œH››H be a wreath product where H act faithfully on A . Suppose that both actions are transitive , that is, for any ==ß−w H there is a ;− for which; œ==w and similarly for the other action. Prove that the permutation representation of [ is also transitive, that is, for any pair Ðß-= ÑßÐß -ww = Ñ− A ‚ H there is an Ð0ß;Ñ−[ for which Ð0ß ;чw Ð-= ß Ñ œ Ð - ß =w Ñ Chapter 6 Permutation Groups

Permutations are fundamental to many branches of mathematics. In this chapter, we examine permutations from a group-theoretic perspective. The Definition and Cycle Representation A permutation of a nonempty set \\ is a bijective function on . The set of all permutations of \W is denoted by \ . As is customary, when

\ œ M8 ³ Ö"ß á ß 8× we write WW\8 as . As we have seen, the set W\ of permutations of a nonempty set \ forms a group under composition of functions. For the record, we have:

Definition Let \W be a nonempty set. The symmetric group \ on the set \ is the group of all permutations of \ , under composition of functions.

There are various notations for permutations. The notation ++â+ 5 œ "# 8 Œ++â+33"# 3 8 is sometimes used to denote the permutation that sends ++43 in the top row to 4 in the bottom row. This notation is a bit combersome and we prefer the cycle notation described in an earlier chapter. In particular, if +Á+−\34 for 3Á4 , then

5 œÐ+â+Ñ"5 denotes the permutation 5 −W\ defined, for !Ÿ3Ÿ8 , by

+3" for "Ÿ3Ÿ5" 5+œ3 œ +3œ5" for and sending all other elements of \ to themselves. Such a permutation 5 is called a 5W# -cycle in \ . A -cycle

S. Roman, Fundamentals of Group Theory: An Advanced Approach, 191 DOI 10.1007/978-0-8176-8301-6_6, © Springer Science+Business Media, LLC 2012 192 Fundamentals of Group Theory

5 œÐ+,Ñ is called a transposition. Two cycles Ð+"5 â+ Ñ and Ð, "7 â, Ñ are disjoint if +Á,34 for all 3ß4 .

When \Ö is an infinite set containing the elements +3 ±3−™× , then we can define the infinite cycle

5 œÐáß+"!" ß+ß+ßáÑ where +Á+34 for 3Á4 as the permutation 5™ sending +5 to + 5" for all 5− and leaving all other elements of \ fixed.

Theorem 6.1 For a permutation group W\ , the following hold. 1) Disjoint cycles in W\ commute. 2) Every permutation in W\ is a product of disjoint cycles, the product being unique except for the order of factors. A representation of 5 as a product of disjoint cycles () with or without " -cycles is called a cycle representation or cycle decomposition of 55 . The cycle structure of a permutation is the sequence of cycle lengths in a cycle decomposition of 5 , or equivalently, the number of cycles of each length in a cycle decomposition of 5 . Proof. Part 1) we leave to the reader. For part 2), let 5 −W\ and define an equivalence relation on \B´CCœB by if 55 for some integer 5 . For B−\ , the equivalence class containing B is ÒBÓœÖ5™3 B±3− ×

3 Note that the restriction 55lÒÒBÓ is a permutation of BÓB . In fact, if the elements are distinct for all 3−™5 , then lÒBÓ is the infinite cycle

# " # 55555lœÐáßBßBßBßBßBßáÑÒBÓ On the other hand, if 5534Bœ B for 34 , then 543 BœB and if 7 is the 7 smallest positive integer for which 55BœB , then lÒBÓ is the 7 -cycle

#7" 5555lœÐBßBßBßáßÒBÓ BÑ The distinct equivalence classes form a partition of \ and 5 is a product of the disjoint cycles 5lFF as varies over these equivalence classes.

A cycle of length 55W has order in \ . Thus, a transposition is an involution, as is any product of disjoint transpositions. A power of a cycle need not be a cycle; for example Ð"#$%Ñ# œÐ"$ÑÐ#%Ñ

Since disjoint cycles commute, if 55œ-â-"7 is a cycle representation of , then for any integer 5 , Permutation Groups 193

555 5 œ-"7 â- 55 Moreover, 5+œ- if and only if 3 œ3+ for each .

Theorem 6.2 The order of 5 −W\ is the least common multiple of the lengths ()orders of the disjoint cycles in a cycle decomposition of 5 .

The group properties of W\\ do not depend upon the nature of the set , but only upon its cardinality. More formally put, if \œ] , then the groups W and kk kk \ W] are isomorphic. Accordingly, we will feel free to state theorems in terms of either WW\8 or (when \ is finite). A Fundamental Formula Involving Conjugation

When a permutation 7 −W\ is written as a product of disjoint cycles, it is very easy to describe the conjugates 775 of . It is also remarkably easy to tell when two permutations are conjugate.

Theorem 6.3 1) Let 5 −W8" . For any 5 -cycle Ð+â+Ñ5 , 5 Ð+"5 â+ Ñ œ Ð55 + " â + 5 Ñ

Hence, if 77œ-â-"5 is a cycle decomposition of , then 555 7 œ-"5 â- is a cycle decomposition of 7 5 . 2) Two permutations are conjugate if and only if they have the same cycle structure. Proof. For part 1), we have

5 5+353" Ð+"5 â+ Ñ Ð5 + 3 Ñ œ œ 5+3œ5"

" Also, if ,Á55 +33 for any 3 , then ,Á+ and so 5 " " Ð+â+Ñ,œÐ+â+ÑÐ,ÑœÐ,Ñœ,"55555 "5

5 Hence, Ð+"5 â+ Ñ is the cycle Ð55 +" â + 5 Ñ . For part 2), if 7œ -"7 â- is a cycle decomposition of 7 , then 555 7 œ-"7 â- 5 5 and since --3 is a cycle of the same length as 3 , the cycle structure of 7 is the same as that of 7 .

For the converse, suppose that 57 and have the same cycle structure. If 57 and are cycles, say 194 Fundamentals of Group Theory

57œÐ+â+"8 Ñand œÐ,â,"8 Ñ

- then any permutation -5 that sends +,33 to satisfies œ7 . More generally, if

57œ-â-"7and œ.â."7 are the cycle decompositions of 57 and , ordered so that -5 has the same length as .35 , then we can define a permutation - that sends the element in the th - position of -3.œ55 to the element in the th position of . Then 57 .

The previous theorem implies that it is easy to tell when a subgroup L of the symmetric group W\ is normal.

Theorem 6.4 A subgroup LŸW\ is normal if and only if whenever 5 −L , then so are all permutations in W\ with the same cycle structure as 5 . Parity Every cycle is a product of transpositions, to wit

Ð++â+ÑœÐ++ÑÐ++"# 7 "7 "7" ÑâÐ++ÑÐ++Ñ "$ "# and so every permutation is a product of transpositions. While this representation is far from unique, the parity of the number of transpositions is unique.

Theorem 6.5 Let 5 −W8 . 1) Exactly one of the following holds: a) 5 can be written as a product of an even number of transpositions, in which case we say that 5 is even () or has even parity . b) 5 can be written as a product of an odd number of transpositions, in which case we say that 5 is odd () or has odd parity . 2)s The symbol Ð"Ñ5 or g Ð55 Ñ , called the sign or signum of , is equal to " if 55 is even and " if is odd. We have Ð"Ñ5 œ Ð"Ñ85 where 5 is the number of cycles in the cycle decomposition of 5 ( including "-cycles) . Proof. For part 1), if 5 can be written as a product of an even number of transpositions and an odd number of transpositions, say

533œâ"#@"#?" œâ 77 then the identity can be written as a product of an odd number of transpositions

+77œâ"#?"#@" 3 â 3 To show that this is not possible, we choose a representation of + as a product of an odd number of transpositions as follows: Permutation Groups 195

1) Find the smallest odd integer 7" for which + is the product of 7 transpositions. 2) Choose an integer 5 that appears in at least one such representation of + . 3) Among all representations of + as a product of 7 transpositions that contain 55, let 7 be the one whose rightmost appearence of is as far to the left as possible, say

7))œâ">">"7 ÐB5Ñâ ) )

where ))>"â5 7 does not involve .

Note that > # , since otherwise the only appearance of 5 is in )7+" and so Á . However, we can easily move this rightmost occurrence of 5 one transposition to the left by using the following substitutions. Suppose that )>" œÐ+,Ñ . Note that Ð+ ,Ñ Á ÐB 5Ñ since otherwise the two transpositions would cancel, contradicting the definition of 7 .

1) If Ð+ ,Ñ and ÐB 5Ñ are disjoint, then they commute Ð+ ,ÑÐB 5Ñ œ ÐB 5ÑÐ+ ,Ñ 2) If Ð+,ÑœÐB,Ñ for ,Á5 , then ÐB,ÑÐB5ÑœÐB5,ÑœÐ,B5ÑœÐ,5ÑÐ,BÑ 3) If Ð+,ÑœÐ5,Ñ where +Á5 , then write Ð5 ,ÑÐB 5Ñ œ Ð5 B ,Ñ œ ÐB , 5Ñ œ ÐB 5ÑÐB ,Ñ Thus, we can move the rightmost occurrence of 5 to the left, contradicting the construction of 7 and proving part 1).

For part 2), a cycle of length 7 # can be written as a product of 7" transpositions as above. Now suppose that the cycle decomposition of 5 is

5 œ-â-.â."<"= where lenÐ-33 Ñ œ 7 # and len Ð. 3 Ñ œ " . Then 5 can be written as a product of the following number of transpositions: < Ð7 "ÑœÐ8=Ñ<œ85 3 3œ"

Generating Sets for WE88 and

There are a variety of useful generating sets for the symmetric group W8 . For example, we have seen that the set of transpositions generates W8 . 196 Fundamentals of Group Theory

Theorem 6.6 The following sets generate W8 . 1) The set of transpositions. 2) The “transpositions of " ” Ð" #Ñß Ð" $Ñß á ß Ð" 8Ñ 3) The “adjacent transpositions” Ð"#ÑßÐ#$ÑßÐ$%ÑßáßÐ8"8Ñ 4) The 8 -cycle Ð"â8Ñ and the transposition Ð" #Ñ . 5) The cycle Ð#â8Ñ and a single transposition Ð"5Ñ for #Ÿ5Ÿ8 . Proof. For part 2) we have

Ð+ ,Ñ œ Ð" ,ÑÐ" +Ñ and so the subgroup generated by the transpositions of " contains all transpositions. For part 3), if E is the subgroup generated by the adjacent transpositions, then Ð" #Ñ − E and if Ð" 5Ñ − E , then so is Ð" 5  "Ñ œ Ð" 5ÑÐ5 5"Ñ

Hence, part 2) implies that E œ W8 . For part 4), if T œ ØÐ#â8Ñß Ð" #ÑÙ, then Ð" #Ñ − T and if Ð5  " 5Ñ − T , then so is Ð5ß 5  "Ñ œ Ð5  " 5Ñ5 for all 5Ÿ8" . Hence, T œW8 . For part 5), if 5 œÐ#â8Ñ , then Ð" 5Ñ5 œ Ð" 5  "Ñ Ð" 5  "Ñ5 œ Ð" 5  #Ñ ã Ð" 8  "Ñ5 œ Ð" 8Ñ and since .5œœÐ8â#Ñ" , Ð" 5Ñ. œ Ð" 5  "Ñ Ð" 5  "Ñ. œ Ð" 5  #Ñ ã Ð" $Ñ. œ Ð" #Ñ and so we get all transpositions of " .

As to the alternating group, we have the following.

Theorem 6.7 For 8 $ , E8 is generated by the $ -cycles. Proof. Any even permutation is a product of pairs of distinct transpositions. For “overlapping” transpositions, we have Permutation Groups 197

Ð+ ,ÑÐ+ -Ñ œ Ð+ - ,Ñ and for disjoint transpositions, we can arrange for overlaps as follows: Ð+ ,ÑÐ- .Ñ œ Ð+ ,ÑÐ, -ÑÐ, -ÑÐ- .Ñ œ Ð, - +ÑÐ- . ,Ñ

Hence, E$8 is generated by -cycles.

Subgroups of WE88 and

The alternating group EW88 sits very comfortably inside .

Theorem 6.8 5 1) The signum map 95ÀÈÐ"Ñ is a group homomorphism from W8 onto the multiplicative group Ö"ß "× , with kernel EEW#888 . Hence, ü has index . 2) EW88 is the only subgroup of of index # . 3)( Subgroups of W8) If LŸW88 , then either LŸE or L contains an equal number of even and odd permutations and so has even order. 4)( Subgroups of E8) For 8 $ , E8 has no subgroup of index # . Proof. For part 2), let LW# be a subgroup of 8 of index . Theorem 3.17 implies # that 55−L for any −W8 and so the squares of all $ -cycles are in L . But any $$-cycle is the square of another -cycle: Ð+ , -Ñ œ Ð+ - ,Ñ# and so L$ contains all -cycles, which generate EL88 , whence œE . Part 3) # also follows from Theorem 3.17. For part 4), if ÐE8 À LÑ œ #, then 5 − L for all 5 −E8 and so L contains all $ -cycles, a contradiction. The Alternating Group Is Simple

We wish to show that the alternating group E88 is simple for Á% , but not for 8œ%. We will leave proof of the cases 8Ÿ% to the reader and assume that 8 &.

Suppose that RWü 8 is nontrivial. We have seen that any two $ -cycles are conjugate in W$8 . However, it is also true that any two -cycles are conjugate in E8, for 8 & . To see this let α"œÐ++"#$ +Ñ and œÐ,,"#$ ,Ñ be distinct $ - cycles with +Á,"3 for all 3 . If

5 œÐ+,ÑÐ+,ÑÐ+,Ñ"" ## $$

5 where Ð+33 , Ñ is the identity when +3 œ , 3 , then α" œ . Finally, if 5 is not even, Ð+" BÑ5 then we can take BÂÖ+""#$ ß, ß, ß, ×, in which case α"œ .

Thus, since E$88 is generated by -cycles, we can prove that E is simple by showing that any nontrivial normal subgroup of E$8 contains at least one - cycle. 198 Fundamentals of Group Theory

E& is Simple

To see that E& is simple, note that the possible cycle representations of elements of E"& , excluding -cycles, are І††††Ñß Ð†††Ñand І†ÑІ†Ñ If R& contains a -cycle, we may assume that 5 œÐ"#$%&Ñ−R. To shorten the &$ -cycle to a -cycle, we reverse the effects of 5 on "$ and by taking the product 7 œÐ#"%$&ÑÐ"#$%&ÑœÐ#&%Ñ and since

Ð#"%$&ÑœÐ"#$%&ÑÐ" #ÑÐ$ %Ñ −R we deduce that R$ contains the -cycle 7 .

If R contains the product of two disjoint transpositions, we may assume that Ð" #ÑÐ$ %Ñ − R To get a $ -cycle, recall that the product of two distinct nondisjoint transpositions is a $ -cycle. Thus, Ð" #ÑÐ$ %ÑÐ" #ÑÐ% &Ñ œ Ð$ %ÑÐ% &Ñ œ Ð% & $Ñ and since

Ð" #ÑÐ% &Ñ œ ÒÐ" #ÑÐ$ %ÑÓÐ$%&Ñ − R we see that R$ contains a -cycle in this case as well.

E88 is Simple for &

We now examine the general case 8 & . Let Rü5 E8 and let −R . Case 1 Suppose that the cycle decomposition of 5 contains a cycle of length 5 % , say

51œÐ+â+BCDÑ"7 where 7 ". If

- œÐBDC+7" â+Ñ then -1 and are disjoint and

" 7-15³ œ ÐB D C +7""7 â + ÑÐ+ â + B C DÑ œ ÐB +7 DÑ is a $ -cycle. Moreover, Permutation Groups 199

" ÐB C DÑ ÐB C DÑ 1-œ 1 Ð+â+CDBÑœÒÐ+â+BCDÑÓ"7 1 "7 œ5 −R and so -1" −R , which implies that the $ -cycle 7 is in R . Case 2 Suppose that the cycle decomposition of 5 contains two or more $ -cycles, say 51œÐ+,-ÑÐBCDÑ Then R contains the permutation 75³Ð+ C DÑ œ ÐC , -ÑÐB D +Ñ1 and therefore also the & -cycle 57" œ Ð+ , -ÑÐB C DÑÐ+ D BÑÐ- , CÑ œ Ð+ B , D CÑ Hence, case 1) completes the proof. Case 3 If the cycle decomposition of 5 consists of a single $ -cycle, with possibly some additional transpositions,

511œÐ+,-Ñ"7 â then R$ contains the -cycle 5##œÐ+,-Ñ œÐ+-,Ñ

Case 4 If the cycle decomposition of 5 is a product of at least three disjoint transpositions,

51œ Ð+ ,ÑÐB CÑÐD AÑ"7 â1 then R also contains

Ð, BÑÐC DÑ 75³ œ Ð+ BÑÐ, DÑÐC AÑ11"7 â and therefore also 57 œ Ð+ ,ÑÐB CÑÐD AÑÐ+ BÑÐ, DÑÐC AÑ œ Ð+ C DÑÐ, A BÑ and so case 2) completes the proof. Case 5 Suppose that the cycle decomposition of 5 has the form 5 œÐ+,ÑÐ-.Ñ If BÂÖ+ß,ß-ß.×, then R contains 200 Fundamentals of Group Theory

75³Ð+ , BÑ œ Ð, BÑÐ- .Ñ and therefore also the $ -cycle 57 œ Ð+ ,ÑÐ- .ÑÐ, BÑÐ- .Ñ œ Ð+ ,ÑÐ, BÑ œ Ð, B +Ñ

Thus, in all cases R$ contains a -cycle and so RœE8 .

Theorem 6.9 The alternating group E88 is simple if and only if Á% .

As to the normal subgroups of W8 , we have the following.

Theorem 6.10 If 8 Á % , then W88 has no normal subgroups other than Ö"×ß E and W8. Proof. This is easily checked for 8Ÿ# so assume that 8 $ . If Rü W8 , then R∩E88ü+ E. Hence, R∩E8 œÖ× or R∩E88 œE. But if R∩E8 œÖ×+, then RœÖß+5 × where 5 is an odd involution and so has cycle decomposition

5 œ Ð+"" , ÑâÐ+ 55 , Ñ where the transpositions Ð+33 , Ñ are disjoint. Since R is normal, it must contain all permutations with the same cycle structure as 5 , which is not the case if 8 $. Hence, E88 ŸR and so RœE or RœW8 .

Example 6.11 (An infinite simple group) Using the fact that E8 is simple for 8Á%, we can construct an example of an infinite simple group. Let \ be an infinite set and let W\\ be the symmetric group on . Thus, 5lsuppÐÑ5 is a permutation of suppÐÑ55 with no fixed points and is the identity on \Ïsupp Ð5 Ñ. Let WÐ\Ñ be the subgroup of W\\ consisting of all elements of W that have finite support. Let EWÐ\Ñ be the subgroup of Ð\Ñ consisting of those permutations 55−WÐ\Ñ for which lsuppÐÑ5 is an even permutation. We leave it as an exercise to show that EEÐ\Ñ is an infinite simple group. Moreover, Ð\Ñ is the smallest nontrivial normal subgroup in W\ . Some Counting It is sometimes useful to know how many permutations there are with a particular cycle structure.

Theorem 6.12 1) The number of cycles of length 5W in 8 is 88x Ð5  "Ñx œ Š‹5 5Ð8  5Ñx

In particular, the number of 8 -cycles in W8 is Ð8  "Ñx . Permutation Groups 201

2) The number of permutations in W<83 whose cycle structure consists of cycles of length 53 , for 3œ"ßáß7 is 8x <" <7 <"7 xâ< x5" â57

3) Let =Ð8ß 5Ñ be the number of permutations in W8 whose cycle structure has exactly 5" cycles () including -cycles . Then =Ð8ß "Ñ œ Ð8  "Ñx and for 5 # =Ð8ß5Ñœ=Ð8"ß5"ÑÐ8"Ñ=Ð8"ß5Ñ Also, 8 =Ð8ß5ÑB5Ð8Ñ œB ³BÐB"ÑâÐB8"Ñ 5œ" The numbers =Ð8ß 5Ñ are known as the Stirling numbers of the first kind. The expression B8Ð8Ñ is known as the th upper factorial. Proof. We leave proof of part 1) to the reader. For part 2), we write down a template consisting of <533 cycles of length , with dots in place of the elements of M8 œ Ö"ß á ß 8×. For example, if the cycle lengths are $ß $ß #ß " , then the template is І††ÑІ††ÑІ†ÑІÑ

Now, the dots can be replaced by the elements of M8x8 in different ways. However, there are two ways in which the number 8x is an overcount of the desired number. First, for each of the <533 cycle templates of length , a cycle can <3 start at any of its 583 elements, so we must divide by x by 53 , giving 8x <" <7 5â5" 7

Second, for each cycle length 5

For part 3), it is easy to see that =Ð8ß "Ñ œ Ð8  "Ñx

In general, we group the permutations in W58 with exactly cycles into two groups, depending on whether the permutation contains the "Ð8Ñ -cycle . There are =Ð8  "ß 5  "Ñ permutations containing the " -cycle Ð8Ñ . The other permutations are formed by inserting 88 after any of the " elements in the =Ð8"ß5Ñ permutations of 8" elements into 5 cycles. Thus, for 5 # 202 Fundamentals of Group Theory

=Ð8ß5Ñœ=Ð8"ß5"ÑÐ8"Ñ=Ð8"ß5Ñ Now we can verify the formula for =Ð8ß 5Ñ by induction. It is easy to see that the formula holds for 8œ" . Assume the formula is true for =Ð7ß5Ñ where 78 . Then 8 =Ð8ß 5ÑB5 5œ" 88 œÐ8"ÑxB =Ð8"ß5"ÑB55 Ð8"Ñ =Ð8"ß5ÑB 5œ# 5œ# 8" œÐ8"ÑxBB =Ð8"ß5ÑB5 5œ" 8 Ð8"Ñ =Ð8"ß5ÑB5 Ð8#ÑxB –—5œ" œ Ð8  "ÑxB  BÐBÑÐ8"Ñ  Ð8  "ÑÒBÐ8"Ñ  Ð8  #ÑxBÓ œÐB8"ÑBÐ8"Ñ œBÐ8Ñ as desired. Exercises 1. Let 5 œÐ"#$ÑÐ%&ÑÐ'Ñ and 7 œ Ð% & 'ÑÐ" $ÑÐ#Ñ

3 be elements of W−'' . Find a permutation 35Wœ for which 7 . Is 3 unique? 2. Find the smallest normal subgroup of W% containing 5 œ Ð" #ÑÐ$ %Ñ. Is this the smallest subgroup of W% containing 5 ? 3. Let 55−W8 have prime order : . Prove that is a product of disjoint : - cycles. If 5 moves all elements of M:±88 , show that . 4. Show that two even permutations may be conjugate in WE&& but not in . 5. Prove that E#E8 has no subgroup of index without using the fact that 8 is simple for 8Á% . How does this relate to Lagrange's theorem? 6. a) Find all subgroups of E% . b) Find all normal subgroups of EE%% . Is the join of all of its proper normal subgroups? 7. Prove that the $ -cycles Ð"#$Ñß Ð"#%Ñß á ß Ð"#8Ñ generate E8 . 8. Let 8 $ and 5 " . Prove that E8 is generated by the cycles of length #5  ". Permutation Groups 203

9. Prove that for 8 & , E8 is generated by the set of all products of pairs of disjoint transpositions. Does this hold for 8& ? 10. Let 8 & . Prove that the only proper subgroup L of W88 with ÐW ÀLÑ8 is E8. 11. Prove that W88% is centerless for $E . Prove that is centerless. What about E8 in general? 12. A subgroup LW of 8 is transitive if for any 4ß5−M8 , there is a 5 −L for which 54œ5 . Prove that the order of a transitive subgroup LW of 8 is divisible by 8 . 13. A subgroup L of W8" is 5 -ply transitive if for any distinct 3 ßáß35 −M8 and distinct 4ßáß4"58 −M there is a permutation 5 −L for which 53œ4??. Prove that E8 is Ð8#Ñ -ply transitive. Is it Ð8"Ñ -ply or 8 -ply transitive? 14. Let \E be an infinite set. Define the alternating group \ is defined to be the set of all permutations in W8 that can be written as the product of an even number of transpositions. Let LW\\ be the set of all permutations in that fix all but a finite number of elements of \ . a) What is the relationship between EL\\ , and W \ (including normality and index)? b) Prove that E\ is simple. 15. In W8 , for each 5 œ$ßáß8, let

5 ¨©# 5 œÐ353Ñ 5 $ 3œ" for example,

5$ œÐ"#Ñ 5% œÐ"$Ñ 5& œÐ"%ÑÐ#$Ñ 5' œÐ"&ÑÐ#%Ñ 5( œÐ"'ÑÐ#&ÑÐ$%Ñ 5) œÐ"(ÑÐ#'ÑÐ$&Ñ

Show that the permutations 55"8ßáß generate W 8" . 16. What is the largest order of the elements of W"! ? 17. (Determining the parity of a permutation) Let \ œÖB"8 ßáßB ×. Let T be the set of all polynomials in the B3 's with rational coefficients. Then we can apply a 55−W8 to the elements of T by applying to the variables individually. For example, if " :œBB  B$ "$ # # 204 Fundamentals of Group Theory

then " 555:œ ÐBÑÐBÑ 5 ÐBÑ−T$ "$# # a) Show that for :ß ; − T , 555555Ð:  ;Ñ œ Ð:Ñ  Ð;Ñß Ð:;Ñ œ Ð:Ñ Ð;Ñ

and for 57ß−W\ , 57Ð:ќР57 Ñ: b) Consider the polynomial BB :œ:ÐBßáßBÑœ34 −T "8$ 34 34

Show that if 5 œÐBB"+ Ñ is a transposition, then 55ÐB Ñ  ÐB Ñ 5:œ 34œ: $ 34 34

c) Show that for any 5 −W8 , 5:œÐ"Ñ:5

Hence, if 5 − W8 , then since :Ð"ß #ß á ß 8Ñ œ ", we have 55Ð3Ñ  Ð4Ñ 5:Ð"ß #ß á ß 8Ñ œ œ Ð"Ñ5 $ 34 34

d) Let 55−W8 . An inversion in is a pair Ð4ß3Ñ of indices with "Ÿ3ß4Ÿ8 satisfying 43 but 55 Ð4Ñ Ð3Ñ . Prove that the sign of 5 is the parity of the number of inversions in 5 . e) Determine whether the permutation 5 −W* is even or odd, where 5 œÐ(#%*$)'&"Ñ

18. For each 7-−W\\ , we may associate the conjugation map 7 −W defined 7 by -57 ÐÑœ 5. a) Show that the map --ÀK ÄW\ defined by Ð7 Ñœ-7 is a homomorphism. b) Find the kernel of - . c) Suppose that gÁE©\ is invariant under conjugation, that is, 5 +−E implies that + −E . Then we can restrict -5 to a permutation of w EÀ. Find the kernel of the map - HÄWE . 19. Let KW be any normal subgroup of 888 (such as E or W itself). Permutation Groups 205

a) Show that if 573ßß −K , then 73 55œÍ−GÐÑ 735K b) Find a one-to-one correspondence between the set 5K of conjugates of K 55 and the set of cosets of GÐÑK in K . The set 5 is referred to as the conjugacy class of 55 or the orbit of under conjugation. The centralizer GÐÑK 55 is also referred to as the stabilizer of under conjugation. c) Prove the orbit-stabilizer relationship for conjugation in K : For any 5 −K, K 55K œÐKÀG Ð ÑÑœ kk ¸¸ K GÐÑ5 kkK

d) Let αα−W8W be an 8 -cycle. Prove that G8 Ð ÑœØα Ù. Put another way, a permutation 5α−W8 commutes with if and only if it is a power of α . 20. If 57ß−E88 are conjugate in W , it does not necessarily follow that 5 and 7 are conjugate in E−88 . Suppose that 5 E commutes with an odd permutation - −W8 a) Prove that if 57 and are conjugate in W8 , then they are also conjugate in E8. b) Prove that the centralizers of 5 are related as follows:

GÐÑœGÐÑ∪GÐÑW8 55-5EE88 and, in particular, GÐÑœ#GÐÑ55 kkkkW8 E8

Note : It is not hard to find permutations 5 −E8 that commute with an odd permutation. For instance, if 55−E8 does not move either + or , , then commutes with the transposition Ð+ ,Ñ . Thus, for instance, an Ð8  #Ñ -cycle in E−8 commutes with a transposition. As another example, if 5 E8 interchanges +, and , then 5. œÐ+,Ñ .ww where . and . fix +, and . It follows that 5 commutes with Ð+ ,Ñ . 21. Let \− be a nonempty set. Define the support of a permutation 5 W\ to be the set of elements of \ that are moved by 5 : suppÐ55 ÑœÖB−\± BÁB×

Let WWÐ\Ñ be the set of all permutations in \ with finite support, that is, for which suppÐÑ5 is a finite set. This set is called the restricted symmetric group on the set \ and is used in defining the determinant of an infinite matrix. a) Show that suppÐÑœ55 supp Ð" Ñ. b) Show that suppÐÑ©57 supp ÐÑ∪ 5 supp ÐÑ 7 . c) Show that suppÐÑœÐÐÑÑ757" 7supp 5 . d) Show that suppÐÑ∩57 supp ÐÑœg implies that 5775œ . 206 Fundamentals of Group Theory

e) Show that WWÐ\Ñ ü \\ and that WœWÐ\Ñ if and only if \ is finite. f) Show that if \W is infinite, then Ð\Ñ is an infinite group in which every element has finite order and that WÎW\ Ð\Ñ is infinite. g) Let \E be infinite and let Ð\Ñ be the subgroup of WÐ\Ñ consisting of those permutations 55−WÐ\Ñ for which lsuppÐÑ5 is an even permutation. Show that EEÐ\Ñ is an infinite simple group. Moreover, Ð\Ñ is the smallest normal subgroup in W\ . 22. For the Stirling numbers =Ð8ß 5Ñ of the first kind, show that 8 =Ð8ß 5Ñ œ 8x 5œ" 23. (Stirling numbers of the second kind) For the curious, we present a brief discussion of the “other” Stirling numbers. We have seen that the Stirling numbers =Ð8ß 5Ñ of the first kind count the number of ways to partition a set of size 85 into disjoint nonempty “cycles.” The Stirling numbers of the second kind, denoted by WÐ8ß5Ñ , count the number of ways to partition a set of size 85 into nonempty disjoint subsets. It is clear that WÐ8ß"Ñ œ " a) Find WÐ8ß#Ñ . b) Show that for 8! , WÐ8ß5Ñœ5WÐ8"ß5ÑWÐ8"ß5"Ñ c) Show that 8 Bœ8Ð WÐ8ß5ÑB5Ñ 5œ" Chapter 7 Group Actions; The Structure of : -Groups

Group Actions

We have had a few occasions to use group actions -ÀK ÄW\ in the past and we now wish to make a more systematic study of group actions, beginning with the definition.

Definition An action of a group K\ on a nonempty set is a group homomorphism -ÀK ÄW\ , called the representation map for the action. The permutation --++ is often denoted by + , or simply by itself when no confusion can arise. Thus, "B œ Band Ð+,ÑB œ +Ð,BÑ for all B−\ . When -- is a group action, we say that K\ acts on by or that \ is a K-set under - . An action is faithful if it is an embedding. 1) An element +−K is said to fix B−\ if +BœB and move B if +BÁB . 2) An element B−\ is stable if every element of KB fixes . We will denote the set of all stable elements by Fix\ÐKÑ or just Fix ÐKÑ .

Let \\\ denote the set of all functions from to \ . If \ is a finite set, then any \ function --ÀK Ä\ that satisfies "+œ+ and -,+ œ--, is an action of K on \, since

--++Bœ CÊ ----++" + Bœ" + C ÊBœC--++" ++" ÊBœC--"" ÊBœC and so -+ is injective and therefore a permutation of \ . Thus, for finite sets, we do not need to check explicitly that -+ is a permutation of \ .

S. Roman, Fundamentals of Group Theory: An Advanced Approach, 207 DOI 10.1007/978-0-8176-8301-6_7, © Springer Science+Business Media, LLC 2012 208 Fundamentals of Group Theory

Orbits and Stabilizers Two elements Bß C − \ are K-equivalent if there is an + − K for which +B œ C. Since - is a homomorphism, K -equivalence is an equivalence relation on \ . The equivalence classes KB œ Ö+B ± + − K× are called the orbits of the action. We will use the notations KB , orb ÐBÑ and orbKÐBÑ for the orbit of BK under . The distinct orbits form a partition of the set \.

On the group side of an action, we can associate to each element B−\ the set of all element of KB that fix :

stabÐBÑ œ stabK ÐBÑ œ Ö+ − K ± +B œ B× This subgroup of KB is called the stabilizer of .

Definition Let K\ be a group and a nonempty set. 1) An action of K\ on is transitive if every pair of elements of \K are - equivalent, that is, if there is only one orbit in \ . In this case, we also say that K\ is transitive on . 2)s An action of K\ on is regular if it is transitive and if the stabilizer tabÐBÑ is trivial for every B−\ . In this case, we also say that K is regular on \. The Kernel of the Representation Map

The kernel of the representation map -ÀK ÄW\ is

kerÐÑœÖ+−K±--++ œ× As we have remarked, the action -- is faithful if kerÐÑ is trivial.

Theorem 7.1 The kernel of an action -ÀK ÄW\ is the intersection of the stabilizers of all elements of \ , kerÐÑœ- stab ÐBÑ , B−\

The importance of the kernel Oœker Ð-- Ñ of the representation map ÀKÄW\ stems from two facts: OK is a normal subgroup of and there is an embedding of KÎO into the symmetric group W . In particular, if \ œ 8 is finite, then \ kk ÐKÀOѱ8x and if - is faithful, then 9ÐKѱ8x . The Key Relationships A set consisting of precisely one element from each orbit of K\ in is a system of distinct representatives, or SDR for the orbits in \ . Group Actions; The Structure of : -Groups 209

For a given SDR, we will have occasion to separate the representatives of the " - element orbits from the representatives of the orbits of size greater than " . Accordingly, we denote a system of distinct representatives for the orbits of size greater than " by SDR" .

Our immediate goal is to establish some key facts concerning group actions. First, as the various elements of a group KB act on an element −\ , the orbit of BK is described. Of course, different elements of may have the same effect on B. In fact, +B œ ,B Í ," + −stab ÐBÑ Í +stab ÐBÑ œ , stab ÐBÑ Thus, +B œ ,B if and only if + and , are in the same coset of stabÐBÑ in K and so we can think of the cosets themselves as acting on the elements B−\ , describing each element of the orbit of B with no duplications, that is, distinct cosets send B−\ to distinct elements of \ .

Put another way, there is a bijection between KÎstab ÐBÑ and KB . This gives the orbit-stabilizer relationship KB œ ÐK Àstab ÐBÑÑ kk It follows that

\ œ KB œ ÐK Àstab ÐBÑÑ kkB−SDR k k B−SDR We will refer to this equation as the class equation for the action of K\ on . (Actually, this term is traditionally reserved for a specific case of this equation, arising from the specific action of K on itself by conjugation. We will encounter this specific case a bit later in the chapter.)

Another key property of a group action is the following. If Bß C − \ are K - equivalent, say Cœ+B for +−K , then the stabilizers of BC and are related as follows: stabÐ+BÑ œ Ö, − K ± ,+B œ +B× œÖ,−K±+" ,+BœB× œÖ,−K±+" ,+−stab ÐBÑ× œÐBÑstab + Finally, suppose that K\ acts on and that the restriction of this action to a subgroup LŸK is transitive on \ . Then the action of K is duplicated by the action of L , that is, for any 1−K and B−\ , there exists an 2−L for which 2B œ 1B or, equivalently, 1 − 2stabK ÐBÑ. Hence,

KœLstabK ÐBÑ Moreover, if L\ is regular on , then 210 Fundamentals of Group Theory

L∩stabKL ÐBÑœ stab ÐBÑœÖ"× and so

KœLìstabK ÐBÑ Now we summarize.

Theorem 7.2 Let the group K\ act on the set . 1)( Orbit-stabilizer relationship) For any B−\ , KB œ ÐK Àstab ÐBÑÑ kk Thus, for a finite group K , K KB œ kk kk stabÐBÑ kk and KB divides K . 2)( Thekk class equation kk)

\œ ÐKÀstab ÐBÑÑ kk B−SDR where the sum is taken over a system of distinct representatives SDR for the orbits in \ . We can also write this as \ œFix ÐKÑ  ÐK Àstab ÐBÑÑ kk k\ k B−SDR"

3)( The stabilizer relationship) For any B−\ and +−K , stabÐ+BÑ œ stab ÐBÑ+ Thus stabilizers of an orbit in \K form a conjugacy class of and therefore the stabilizers of K -equivalent elements have the same cardinality. 4)( The Frattini argument) If the action of LŸK is transitive on \ , then

KœLstabK ÐBÑ and if L\ is regular on , then

KœLìstabK ÐBÑ

and so stabKÐBÑ is a complement of LK in .

When the group action is transitive, the class equation and orbit-stabilizer relationship become quite simple.

Theorem 7.3 If a group K\ acts transitively on a set , then all stabilizers are conjugate and the orbit-stabilizer relationship () and the class equation is simply Group Actions; The Structure of : -Groups 211

\œÐKÀstab ÐBÑÑ kk for any B−\ . Hence, if K is finite, then \K divides . kk kk Congruence Relations on a K -Set If K\ acts on the elements of a set , then there is a natural way in which K also acts on the power set kÐ\Ñ of \ , namely, +W œ Ö+= ± = − W× for all W©\ and +−K . Let us refer to this action as the induced action of K on kÐ\Ñ.

Now, a K\ -set is a set with some structure, namely, the group action of K on \´ and an equivalence relation on \ is compatible with this action if it satisfies the following definition.

Definition An equivalence relation ´K\ on a -set is called a K-congruence relation on \ if it preserves the group action, that is, if B´C Ê +B´+Cfor all +−K We denote the set of all congruence classes under ´\δ by and the congruence class containing B−\ by ÒBÓ .

Thus, if ´K is a -congruence relation on \ , then the induced action is an action on the partition \Î ´ and +ÒBÓ œ Ò+BÓ for all B−\ . Conversely, suppose that the induced action of K on kÐ\Ñ is an action on a partition ccœÖF±3−M×33 of \ , that is, +F− for all +−K and F−3 cc. Then the equivalence relation ´ associated to is a K -congruence relation on \ .

In other words, the partitions of \K that correspond to the -congruence relations are the partitions c œÖF±3−M×3 that are closed under the induced action of KkÐ\Ñ on .

Moreover, if K\ acts transitively on and if ´ is a K -congruence relation on \K, then also acts transitively on \δ and so \ œKW³Ö+W±+−K× ´ is the orbit of any given conjugacy class W−\δ . Hence, the partitions of \ that correspond to the KK -congruence relations of a transitive group are the partitions of the form KW , where W © \ is nonempty. 212 Fundamentals of Group Theory

But if KW is a partition of \ , then +WœW or +W∩Wœg for all +−K . Conversely, if +W œ W or +W ∩ W œ g for all + − K , then the distinct members of KW form a partition of \ . To see this, suppose that B − +W ∩ ,W. Then +B−W∩+,W" " and so +,WœW" , whence ,Wœ+W . Moreover, the transitivity of KK implies that Wœ\ .

Hence, if W©\ is nonempty, then KW is a partition of \ if and only if +W œ Wor +W ∩ W œ g for all +−K .

Theorem 7.4 Let a group K\ act on a nonempty set . 1) The partitions of \K that correspond to the -congruence relations on \ are the partitions of \K that are closed under the induced action of on kÐ\Ñ. 2) Suppose that K\ acts transitively on . a) The partitions of \K that correspond to the -congruence relations are the partitions of the form KW œ Ö+W ± + − K× where W©\ is nonempty. b) A nonempty subset W\ of is a congruence class under some K - congruence if and only if W has the property that +W œ Wor +W ∩ W œ g for all +−K . Such a subset W\ of is called a block of K .

Thus, W©\ is a block of K if and only if KW is a partition of \ . It follows that if WK+ is a block of , then so is W+ for all −K .

Now that we have established the basic properties of group actions, we can examine a few of the most important examples of group actions. Then we will use group actions to explore the structure of : -groups. In the next chapter, we will use group actions to explore the structure of finite groups and to prove the famous Sylow theorems. Translation by K Earlier in the book, we discussed two fruitful examples of the action of translation by elements of a group KK , namely, on the elements of and on the elements of a quotient set KÎL . The action of K on itself is the left regular representation of KW as a subgroup of the symmetric group K , as described by Cayley's theorem. Group Actions; The Structure of : -Groups 213

Let us review the action of translation by KKÎL on :

-1Ð+LÑ œ 1+L This action is transitive and so all stabilizers are conjugate. Since stabÐ+LÑ œ stab ÐLÑ++ œ L the kernel of the action is the normal closure of L , kerÐÑœ- L+‰ œL , +−K which is the largest normal subgroup of KLÐ contained in . If KÀLÑœ7, then the embedding K ä W¸W L‰ ÐKÀLÑ 7 implies that ÐK À L‰ Ñ ± ÐK À LÑx The consequences of this action were recorded earlier in Theorem 4.20, but we repeat them here for easy reference.

Theorem 7.5 Let KL be a group and let K have finite index. Then ‰ KÎLä WKÎL and so ÐK À L‰ Ñ ± ÐK À LÑx In particular, ÐK À L‰ Ñ is also finite and ÐL À L‰ Ñ ± ÐÐK À LÑ  "Ñx 1) Any of the following imply that LKü : a) LÐ is periodic and KÀLÑœ: is equal to the smallest order among the nonidentity elements of L . b) K9 is finite and ÐLÑÐ and ÐKÀLÑ"Ñx are relatively prime, that is, for all primes : , : ± 9ÐLÑ Ê : ÐK À LÑ This happens, in particular, if ÐK À LÑ is the smallest prime dividing 9ÐKÑ. 2) If KK is finitely generated, then has at most a finite number of subgroups of any finite index 7 . 214 Fundamentals of Group Theory

3) If K is simple, then 9ÐKÑ ± ÐK À LÑx a) If KK is infinite, then has no proper subgroups of finite index. b) If K9 is finite and ÐKÑy±7x for some integer 7K , then has no subgroups of index 7 or less. Conjugation by K on the Conjugates of a Subgroup The elements of KÐ act by conjugation on sub KÑ , + -+ÐLÑ œ L for all LŸK . The orbit of a subgroup L is the conjugacy class conjKÐLÑ and the stable elements are FixÐKÑ œnor ÐKÑ The stabilizer of a subgroup L is its normalizer and so ++ RÐLÑœRÐLÑKK Also, the orbit-stabilizer relationship is conj ÐLÑ œ ÐK À R ÐLÑÑ kkKK which we discussed earlier in the book (Theorem 3.27). Conjugation by K on a Normal Subgroup Let RKü and let K act on the elements of R by conjugation: 1 -1Ð+Ñ œ + for all +−R . The orbits of this action K+ œ +KB ³ Ö+ ± B − K× are called the conjugacy classes of RK under . The stabilizer of +−R is its centralizer GÐ+ÑK and the kernel of the representation - is kerÐÑœ- GÐ+ÑœGÐRÑ , KK +−R The orbit-stabilizer relationship is +K œ ÐK À G Ð+ÑÑ ¸¸ K as we saw in Theorem 3.23.

The stable elements of RR are the elements of that commute with every element of K and so Group Actions; The Structure of : -Groups 215

FixR ÐKÑ œ ^ÐKÑ ∩ R Hence, the class equation is

Rœ^ÐKÑ∩R ÐKÀGÐ+ÑÑ kk k k K +−SDR" When KR acts on itself by conjugation, that is, when œK , the class equation is

Kœ^ÐKÑ ÐKÀGÐ+ÑÑ kk k k K +−SDR" This is the equation to which the name class equation is traditionally applied and is one of the most useful tools in finite group theory. The Structure of Finite : -Groups We now wish to study the structure of a very special type of finite group.

Definition Let K: be a nontrivial group and let be a prime. 1) An element +−K is called a :-element if 9Ð+Ñœ:5 for some 5 ! . 2) KK is a :-group if every element of is a : -element. 3) A nontrivial subgroup WK of is called a :-subgroup of KW if is a : - group.

As we saw earlier in the book, Lagrange's theorem and Cauchy's theorem conspire to give the following result.

Theorem 7.6 A finite group K: is a -group if and only if the order of K is a power of : .

When a :K -group acts on a set \ , the class equation has the property that all of the terms ÐK Àstab ÐBÑÑ that are greater than " are divisible by : . This gives the following simple but useful result.

Theorem 7.7 If a :K -group acts on a set \ , then \´Fix ÐKÑmod : kk k\ k We now turn to the key properties of finite : -groups. The Center-Intersection Property It will be convenient to make the following nonstandard definition.

Definition A group K has the center-intersection property if every nontrivial normal subgroup of KK intersects the center of nontrivially. 216 Fundamentals of Group Theory

Note that a finite group K has the center-intersection property if and only if every nontrivial normal subgroup of K contains a central subgroup of prime order. Any finite :K -group has the center-intersection property, for if RKü is nontrivial, then KR acts on the elements of by conjugation and Theorem 7.7 implies that R´^ÐKÑ∩Rmod : kk k k which shows that ^ÐKÑ ∩ R  ". kk Theorem 7.8 A finite :K -group has the center-intersection property. 1) ^ÐKÑ is nontrivial. 2) K9 is simple if and only if ÐKÑœ: .

The fact that the center of a : -group is nontrivial tells us something very significant about groups of order :# .

Corollary 7.9 If 9ÐKÑ œ :# , then K is abelian. In fact, K is either cyclic or is the direct product of two cyclic subgroups of order : . Proof. We must have ^ÐKÑ œ : or :# . But if ^ÐKÑ œ : , then KÎ^ÐKÑ is kk kk cyclic and so K^ is abelian, which is a contradiction. Hence, ÐKÑœ:# and K kk is abelian. If KK is not cyclic, then is elementary abelian of exponent : and so is the direct product of two cyclic subgroups of order : . :-Series and Nilpotence We next show that : -groups have normal subgroups of all possible orders. But first a couple of definitions.

Definition () Central Series and : -Series Let K be a group. 1) A normal series

LLâL!"üüü 8

in KK is central in if each factor group LÎ5"LK 5 is central in ÎL5 , that is,

LÎLŸ^ÐKÎLÑ5" 5 5 A group is nilpotent if it has a central series starting at the trivial subgroup Ö"× and ending at K . 2) If :L is a prime, then a :-series from to K is a series

LœL–L–â–LœK!" 8

whose steps LL55" have index : .

Definition Let :L be a prime. If O is an extension in K: of index , we refer to OL as a :-cover of . (sOLÐ is a cover of in the lattice ubKÑ .) Group Actions; The Structure of : -Groups 217

Theorem 7.5 implies that if O: is a -cover of L , then L–O .

Theorem 7.10 Let K: be a finite -group. Then every LK:O has a -cover and if L–K , then O can be chosen so that L–O is central in K . 1) There is a :LK -series from to . 2) If LKü , then there is a central :LK -series from to . In particular, K has a normal subgroup containing L9 of every order between ÐLÑ and 9ÐKÑ ()under division . 3) K is nilpotent. Proof. The proof is by induction on 9ÐKÑ . The theorem is true if 9ÐKÑ œ : . Assume 9ÐKÑ  : and that the theorem is true for all groups smaller than K . Let LK and let R be central in K of order : . If LœÖ"× , then R is a : -cover of LL–R and is central in K . Assume that L Á Ö"× . We may also assume that RŸL if L–K . In any case, RŸL or R∩LœÖ"×.

If R ∩ L œ Ö"×, then RL is a : -cover of L and if R Ÿ L , then the induction hypothesis implies that LÎR has a : -cover OÎR and so O is a : -cover of L . Also, if L–K , the inductive hypothesis implies that LÎR – OÎR is central in KÎR for some O Ÿ K and so Theorem 4.11 implies that L–O is central in K. The Normalizer Condition Theorem 7.10 implies that a : -group has the normalizer condition, that is,

LK Ê LRK ÐLÑ and therefore several other nice properties (see the discussion following Theorem 4.35).

Corollary 7.11 The following hold in a finite :K -group : 1) K has the normalizer condition 2) Every subgroup of K is subnormal 3) Every maximal subgroup of K is normal 4) KÎF ÐKÑ is abelian. Maximal and Minimal Subgroups Maximal and minimal subgroups play a key role in the study of finite : -groups. For a finite :K -group , Cauchy's theorem implies that a subgroup L is minimal if and only if 9ÐLÑ œ : and the center-intersection property implies that L is minimal normal if and only if it is central of order : .

As to the maximal subgroups of KL , Theorem 7.10 implies that a subgroup is maximal in KL:L if and only has index and that is maximal normal in K if and only if L: has index . 218 Fundamentals of Group Theory

Theorem 7.12 () Maximal and minimal subgroups Let K: be a finite -group and let LŸK . 1) L: is minimal if and only if it has order . 2) L: is minimal normal if and only if it is central of order . 3) The following are equivalent: a) L is maximal b) L is maximal normal c) ÐK À LÑ œ : . The Frattini Subgroup of a : -Group; The Burnside Basis Theorem The fact that any maximal subgroup Q: of a -group K is normal and has index :+ implies that if −K , then Ð+QÑ: œ Q and so +−Q:: . Thus, K©F ÐKÑ , the Frattini subgroup of K . It follows that KÎF ÐKÑ is an elementary abelian group of exponent : .

Conversely, if KÎO is elementary abelian, then it is characteristically simple and so FÐKÎOÑ œ ÖO×. Hence,

FÐKÑ Ÿ Q Ÿ O , OŸQK Q maximal We have shown that FÐKÑ is the smallest normal subgroup OK of for which the quotient KÎO is elementary abelian.

Theorem 7.13 Let :K be a prime. Let be a group of order :8 , with Frattini subgroup FÐKÑ of order :7 . 1) FFÐKÑ is the smallest normal subgroup of K for which KÎ ÐKÑ is an elementary abelian group. Moreover, KÎF ÐKÑ has exponent : and so is a vector space over ™: , of dimension 87 . 2) FÐKÑ œ Kw: K 3)( The Burnside Basis Theorem) Any generating set for K contains a generating set of size 87 . Proof. Part 1) has been proved. For part 2), since ÖKw: + ± + − K× is a subgroup of KÎKww , it follows that KK: is a normal subgroup of K . In fact, KÎKw K: is elementary abelian of exponent :Ð and so part 1) implies that F KÑŸKw:KŸ FFÐKÑ. Hence, ÐKÑ œ Kw: K .

For part 3), write FFœÐKÑ . We show that

K œØ1"5 ßáß1 Ù Í KÎFF œØ1" ßáß1 5 F Ù One direction is clear and since F is the set of nongenerators of K , we have

KÎFF œØ1" ßáß1 5 F Ù Ê K œØÖ1"5 ßáß1 ×∪F ÙœØ1 "5 ßáß1 Ù Group Actions; The Structure of : -Groups 219

Thus, since KÎF™ is a : -space of dimension 8  7 , any generating set for KÎF contains a generating set of size 87 . Hence, the same holds true for K . Number of Subgroups of a : -Group We now wish to inquire about the number of subgroups of a given size :. in a 8 :K-group of order : . Let sub.. ÐKÑÐKÑ and nor denote the families of subgroups and normal subgroups, respectively, of K:K of size . . Then acts by conjugation on sub..ÐKÑ and the stable set is nor ÐKÑ , whence subÐKÑ ´ nor ÐKÑmod : kkkk.. Our plan is to show that nor ÐKÑ ´ "mod :. kk. Theorem 7.14 Let K:: be a nontrivial -group of order 8 . 1) The number of maximal subgroups of K is :"87 subÐKÑ œ nor ÐKÑ œ ´ "mod : kkkk8" 8" :" 2) For any !Ÿ.Ÿ8 , subÐKÑ ´ nor ÐKÑ ´ "mod : kkkk.. Proof. For part 1), if 9ÐF ÐKÑÑ œ :7, then Theorem 7.13 implies that EœKÎF™ ÐKÑ is a vector space over : of dimension 87 . In general, if Z is a vector space over ™: of dimension 5Z , then the number of subspaces of of dimension . is Ð:55  "ÑÐ:  :ÑâÐ: 5."  : Ñ ZÐ5ß.Ñœ Ð:.. "ÑÐ: :ÑâÐ: .." : Ñ (We ask the reader to supply a proof in the exercises.) Hence, the number of subgroups (subspaces) of Z: of order 5" is Ð:55 "ÑÐ: :ÑâÐ: 55#5 : Ñ : " ZÐ5ß5"Ñœ œ Ð:5"  "ÑÐ: 5"  :ÑâÐ:5"  : 5# Ñ :  " In particular, the number of maximal subgroups of KÎF ÐKÑ is :"87 ZÐ87ß87"Ñœ :" and this is also the number of maximal subgroups of K .

For part 2), let ? ÐKÑ œnor ÐKÑ and let ´ stand for congruence modulo : . ..kk Then part 1) implies that ??" ÐKÑ´". We show that ?. ÐKÑ´" by induction on 9ÐKÑ . If 9ÐKÑ œ : , the result is clear. Assume it is true for : -groups smaller than K. 220 Fundamentals of Group Theory

If Q −nor8" ÐKÑ, then K acts on sub.ÐQÑ by conjugation. The stable set is nor..ÐKÑ ∩ sub ÐQÑ and so the inductive hypothesis implies that norÐKÑ ∩ sub ÐQÑ ´ sub ÐQÑ ´ " kk..kk . Hence, the set

f œ ÖÐRß QÑ ± R  Qß R −nor.8 ÐKÑß Q − nor " ÐKÑ× has size f ´? ÐKÑ´". kk 8"

On the other hand, for each R−nor.8 ÐKÑ, there is one Q−nor " ÐKÑ containing RK for each maximal subgroup of ÎR and since ? ÐKÎRÑ´", we have f ´? ÐKÑ. Thus, ? ÐKÑ´" . 8." kk . . *Conjugates in a : -Group In the study of finite : -groups, it can be useful to examine the conjugates of an element +, by the powers of another element .

Theorem 7.15 Let K: be a finite -group and let +−K9Ð+Ñœ: have order 7 . Let ,−K and suppose that +œ+, α for some integer α ´y"mod :7Ø,Ù. Let + be the set of conjugates of + by the elements of Ø,Ù . Then +œ:Ø,Ù . ¸¸ where . " and :.: is the smallest power of :, for which . commutes with + . 1) If :# or if α ´y$mod %, then

7. +Ø,Ù œÖ+ "5: ±5 œ"ßáß:. ×

where 7. " and +" Â+ Ø,Ù. 2) If :œ# and α ´$mod %, then one of the following holds: a) If . " , then

7. +Ø,Ù œÖ+ /5 5: ±5 œ"ßáß:. ×

" Ø,Ù where 7. " , + Â+ and half of the /"5 's are and half are ". b) If .œ! , then

7" +œÖ+ß+Ø,Ù "# ×or +œÖ+ß+×Ø,Ù " 3)) a If :# , then no element of K: of order can be conjugate to one of its own powers other than the first power. Group Actions; The Structure of : -Groups 221

b) If :œ# , then no element +−K of order # can be conjugate to one of its own powers other than ++ or " . Proof. The conjugates of +Ø,Ù by are

5 ,+,55 œ+α for 5œ"ßáß< , where <œ +Ø,Ù . In fact, < is the smallest positive integer for ¸¸ which ,+++<Ø commutes with . Also, ,Ù is the orbit of under conjugation by Ø,Ù and so <œ:. for some . " .

Note that <+ is also the smallest positive integer for which α< œ+ , that is, the smallest positive integer for which α<7´" mod : and so

3) :±7:α . " 4) :±7:y α ." "

Furthermore, since α:. ´"mod :, Fermat's little theorem implies that αα´"mod : . Thus, if œ/-:> is in : -standard form, then Lemma 1.18 implies that for any ? ! ,

?? α::œ/ A: ?> where :±Ay . In particular,

5) :±/.>"α :." : ." 6) :±/œ".>"y αα :.. : : .

From 3) and 6), we see that 7Ÿ.> .

Now, if /œ":." then 4) and 5) imply that .>Ÿ7 and so 7œ>. . This implies that 7.œ> ". Also, α œ/-:7. and so for "Ÿ5Ÿ:. ,

57.557. α œÐ/-: Ñ œ/ : A5 Hence,

557. +œ+α /: A5 . where no two distinct A:5 's are congruent modulo , since otherwise we would . not get :A distinct conjugates. Thus, we can assume that 5 ranges over the set Ö"ß á ß :. × and so

7. +Ø,Ù œÖ+ /5 5: ±5 œ"ßáß:. ×

Now, if :# or α ´y$mod %, then /œ" and so /5 œ" for all 5 . If :œ# and 222 Fundamentals of Group Theory

. α ´$mod % but ." , then we still have /: œ" but since /œ" , as 5 ranges from ": to .5 , the term / alternates between "" and and so half of the terms /"5 are and half are " .

.7. Also, as 5":/ ranges from to , the exponents 5 5:„ range from ": to „"  :7. But + is conjugate to itself and so one of these exponents must be conjugate to ": modulo 77 . Therefore, the last exponent is ": and since no other exponent is conjugate to " modulo :7 , it follows that +" Â +Ø,Ù .

The case /œ":." occurs precisely when :œ#´$% , α mod and .œ" , in which case +œ has exactly two conjugates and α "-#> with - odd and > #.

> If > 7 , then +α œ+"-# œ+ " and so +Ø,Ù œÖ+ß+ " ×. If >7 , then 7Ÿ"> implies that >œ7" and so α œ"-#7" where - is odd, that is, -œ" . Hence, the two conjugates of + are + itself and

7" +œ+α "# Note finally that the case +œÖ+ß+×Ø,Ù " does occur in the dihedral group 75 " H#7" œØßÙ35 where 9ÐÑœ# 3 and 9ÐÑœ#5 and 3 œ 3 . Also, the case

7" +œÖ+ß+Ø,Ù "# × occurs in the semidihedral group

7#7"" WH7 œØα0 ß Ùß 9Ð α Ñœ# ß 9ÐÑœ#ß 0 0α œ α 0 For part 3), if 9Ð+Ñ œ : , then the number of conjugates of + is both a power of : and at most :" , whence it must equal " . *Unique Subgroups in a : -Group A cyclic :: -group has a unique subgroup of order . If :# , then the converse of this is true: A :: -group that has a unique subgroup of order is cyclic. We begin with a definition.

8 Definition A generalized quaternion group of order #8 # , is a group 8 with the following properties:

8" # #8# " " œ8 Ø+ß ,Ùß 9Ð+Ñ œ # ß 9Ð,Ñ œ %ß , œ + ß ,+, œ +

If 8œ$ , then 8 is a quaternion group.

We will show later in the book that such a group exists: It is a special case of the dicyclic group. We leave it as an exercise to show that Ø,# Ù is the only subgroup =8 of order # in 78 but that for any ###, the group has at least two =5 subgroups of order #B−ÏØ+ÙBœ+, . Also, any 8 has the form , where Group Actions; The Structure of : -Groups 223

7" Ð+5# ,Ñ œ + 5 Ð,+ 5 Ñ, œ , # œ + #

5 and so 9Ð+ ,Ñ œ % . Thus, any element of Ï8 Ø+Ù has order % . It follows that if 8" 8 %, then Ø+Ù is the unique cyclic subgroup of #8 of order and so Ø+Ù « 8.

We will prove that if a :K -group has a unique subgroup of order :K , then is cyclic if :# and K is either cyclic or generalized quaternion if :œ# . First, = let us show that if KL: has a unique subgroup = of any order , where :Ÿ:= 9ÐKÑ, then K must have a unique subgroup of order : .

Since a : -group has subgroups of all orders dividing the order of the group, we have for any subgroup OŸK ,

9ÐL== Ñ Ÿ 9ÐOÑ Ê L Ÿ O Also, since any subgroup OŸK of order less than := is contained in some subgroup of order := , we have

9ÐOÑ Ÿ 9ÐL== Ñ Ê O Ÿ L

Thus, all subgroups of KL either contain == or are contained in L . In this sense, LK= forms a bottleneck in the lattice of subgroups of . It follows that L= is cyclic, for if +ÂL==== , then Ø+ÙŸyL and so L ŸØ+Ù , whence L is cyclic. Since L= is cyclic, the subgroup lattice of K has the form shown in Figure 7.1 and so K: contains exactly one subgroup of each order . with !Ÿ.Ÿ= . In particular, K:K has a unique subgroup of order . Thus, we have shown that has a unique subgroup of some order :== , where :Ÿ: 9ÐKÑ if and only if K has a unique subgroup of order : .

G

......

Hs

{1}

Figure 7.1 224 Fundamentals of Group Theory

Let K: be a noncyclic group of order 8 " . To show that K has more than one subgroup of order :K , it suffices to show contains a nontrivial subgroup E as well as an element of order :E that is not in . We first consider the case :# .

Theorem 7.16 Let :# be prime and let 9ÐKÑœ:8 ". 1) If K+ is noncyclic and if −K is an element of maximum order, then K has an element of order :Ø that is not contained in +Ù . 2) If K: has a unique subgroup of order = for some "Ÿ=8 , then K is cyclic. Proof. We have already seen that part 2) follows from part 1). To prove part 1), assume that KE is noncyclic. Let œØ+Ù+ , where −K: has maximum order 7 . If 7œ" , then all nonidentity elements of K have order : and so we may assume that #Ÿ78 .

Let E–F where 9ÐFÑœ:7". If ,−FÏE , then ,: −E and so ,: œ+ > for some > ! . If >œ! , then 9Ð,Ñœ: and we are done, so let us assume that >!. Since +>: œ, does not have maximum order in K , it follows that :±> and so ,œ+:?: for some !?:7".

Now, if ,+, commutes with , then +? ÂE: has order . Assume now that no ,−FÏE commutes with + . We wish to show that there is still an integer B for which 9Ð,+BB Ñ œ : . If , − F Ï E , then to get a formula for Ð,+ Ñ: , we need a commutativity rule for +, and . But since E–F , there is an α Á" for which +œ+, α and since ,+: commutes with , Theorem 7.15 implies that

7" +Ø,Ù œÖ+ "5: ±5 œ"ßáß:× Moreover, ,ÂE implies that ,3 −FÏE for all "Ÿ3: and so we may replace ,, by an appropriate power of so that α œ":7".

Now, ,+ œ +α , and so ,+BB œ +α , Then an easy induction shows that for 5 " ,

#5 Ð,+B5 Ñ œ + BÐαα âÑ5 α, and so

#: #: Ð,+B Ñ : œ + BÐαα  â α Ñ, : œ + ?:BÐαα  â α Ñ Hence, we want an integer B for which Group Actions; The Structure of : -Groups 225

"α: B´?:Ð:Ñα mod 7 "α Since α œ":7", Theorem 1.18 implies that α:7œÐ":" Ñ: œ"A: 7 where :±Ay and so "α: Bα œ BÐ"  :7" ÑA: ´ BA: Ðmod :7 Ñ "α Hence, we want an integer B for which BA ´ ? Ðmod :7" Ñ

‡" But AB is invertible in ™:7" and so we may take œ?A .

Now we turn to the case :œ# . (The reader may wish to skip the proof upon first reading.)

Theorem 7.17 Let K# be a nontrivial group of order 8 . 1) K# has a unique subgroup of order if and only if K is cyclic or a generalized quaternion group. 2) If K# has a unique subgroup of order = for some "=8 , then K is cyclic. Proof. We have already seen that part 2) follows from part 1). Let K be a noncyclic group of order #K8 with a unique involution. We will show that is a generalized quaternion group. Let +−K be an element of maximum order #7 and let EœØ+Ù . Clearly, we may assume that 7 # .

In one case we will need to be a bit more specific about the choice of the cyclic subgroup E7 . Namely, if œ# , then since the unique subgroup of order # is normal in K#R% , it has a -cover of order , which is cyclic since the % -group has two involutions. In this case, we let EœR and so Eü K . Thus, if 7œ# , we may assume that E–K .

If F œ Ø,ß EÙ is a # -cover of E , then 9ÐFÎEÑ œ # and so ,# − E , which implies that ,#5 œ+ for some 5! . But 9Ð+Ñœ9Ð,Ñ9Ð,ÑŸ9Ð+Ñ5 # and so #±5 , that is,

> ,œ+##? for > " and ? odd. Since Ø+ÙœE?? , we can rename + to + to get

> ,œ+##

> for > " . Since 9Ð,Ñœ9Ð+##7> Ñœ# , we also have 226 Fundamentals of Group Theory

9Ð,Ñ œ #7>" where >Ÿ7" since 9Ð,Ñ#.

>" If ,+, commutes with , then +Â# E is an involution, contrary to assumption. Thus, ,+ does not commute with ; in fact, E is not properly contained in any abelian subgroup of K .

However, since E–F and ,# −E , Theorem 7.15 implies that

7" +œÖ+ß+Ø,Ù "# ×or +œÖ+ß+×Ø,Ù " and so for any #FœØ,ßEÙE -cover of , there is an +−E for which

> ,+," œ +α and ,## œ + where either ααœ"#7" or œ" .

Conjugating the second equation by , and using the first equation gives

>>> ,œ,+,##"## œ+α œ+

>> >+1 and so +œ+## , that is, + # œ" . Hence, #±#7>" , which implies that 7Ÿ>"Ÿ7, that is, >œ7" .

Now, if for any # -cover FœØ,ßEÙ , we have α œ"#7", then α œ"#> and so

> ,+," œ + "# œ , # + " which implies that +,"  E is an involution, a contradiction. Hence, for all # - covers FœØ,ßEÙ of E , we have α œ" and

7" ,+," œ + " and ,## œ + It follows that 9Ð,Ñ œ % and so F can be described as follows:

7" F œ Ø+ß ,Ùß 9Ð+Ñ œ #7## ß 9Ð,Ñ œ %ß , œ + ß ,+," œ + " that is, Fœ 7" is generalized quaternion and so every element of FÏE has order %7 $E«F and if , then .

We want to show that FœK . If not, then F has a # -cover G , that is, E–F–G ŸK where 9ÐGÑ œ :72. Recall that if 7 œ # , then we have chosen E so that EKü and if 7 $ , then E«F and so in either case, E–G . Group Actions; The Structure of : -Groups 227

The quotient group GÎE is either GÐ-EÑ%## or GÐ-EÑGÐ.EÑ. In the latter case, the subgroups Ø-ß EÙ and Ø.ß EÙ are # -covers of E and so +œ+-". œ+ Hence, +œ+-. which implies that EØEß-.Ù is abelian, a contradiction. Hence, # GÎE œ G% Ð-EÑ. But then Ø- ßEÙ is a # -cover of E and so

# +œ+-" However, since EK is not properly contained in an abelian subgroup of , the smallest power of -+ that commutes with is -% −E and so Theorem 7.0 (where .œ#) implies that +" Â+ Ø-Ù , a contradiction.

7 Thus KœFœ 7" is generalized quaternion of order # . *Groups of Order ::88 With an Element of Order " We can use the previous result to take a close look at nonabelian groups K of order ::88 that have an element of order " . We will restrict attention to the case :#.

Let 9Ð+Ñ œ :8" and E œ Ø+Ù . Theorem 7.16 implies that there is a , − K Ï E with 9Ð,Ñ œ :. Then KœØ+Ùz Ø,Ù and it remains to see how +, and interact. Since EKü , we have ,+,œ+" 5 for some 5" and Theorem 7.15 implies that the conjugates of + by Ø,Ù are

8# +Ø,Ù œÖ+ "5: ±5 œ"ßáß:× Since any nonidentity element of Ø,Ù generates Ø,Ù , we can take 5 œ " and write

8# ,+," œ + ": Theorem 7.18 Let :# be a prime. Let K be a nonabelian group of order :8 with an element +: of order 8" . Then KœØ+Ùz Ø,Ù where 9Ð,Ñ œ : and

8# ,+," œ + ": To see that such a group exists, recall from Example 5.30 that there is a semidirect product 228 Fundamentals of Group Theory

GÐ+ÑGÐ,Ñ:8" z) : where

":8# ),Ð+Ñ œ +

*Groups of Order :$ We have seen that groups of order :: are cyclic and that groups of order # are either cyclic or the direct product of two cyclic subgroups of order : (Corollary 7.9). Theorem 7.18 gives us insight into groups of order :$ .

If :œ# , we have seen that, up to isomorphism, the groups of order :$ œ) are

1) G) 2) GG%#} 3) GGG###}} 4) , the (nonabelian) quaternion group 5) H) , the (nonabelian) dihedral group

More generally, we will show that for any prime :: , the groups of order $ are (up to isomorphism):

1) G:$ 2) GG:# } : 3) GGG:::}} 4) YXÐ$ß™: Ñ, the unitriangular matrix group (described below) 5) The group KœØ+ß,Ù where K œ Ø+ß ,Ùß 9Ð+Ñ œ :# ß 9Ð,Ñ œ :ß ,+," œ +": Thus, there are only two nonabelian groups of order :$ (up to isomorphism). We will leave analysis of the abelian groups of order :$ to a later chapter, where we will prove that any finite abelian group is the direct product of cyclic groups.

So let :# be prime and let K be a nonabelian group of order :K$ . If has an element +: of order # , then Theorem 7.18 implies that KœØ+Ùz Ø,Ù where 9Ð,Ñ œ : and ,+,œ+" ": It remains to consider the case where K:^ has exponent . The center œ^ÐKÑ is nontrivial but cannot have order :K# , since then Î^ is cyclic and so K is abelian. Hence, 9Ð^Ñ œ : and so 9ÐKÎ^Ñ œ :#. Hence, KÎ^ is abelian with exponent : and so Group Actions; The Structure of : -Groups 229

KÎ^ œ Ø+^Ù  Ø,^Ù Moreover, since D³Ò,ß+Ó−^, we have ^œØDÙ . Hence, K œØ+ß,ßDÙß9Ð+Ñœ9Ð,Ñœ9ÐDÑœ"ßD−^ÐKÑß,+œD+, To see that this does describe a group, we have the following.

Definition Let Q be a commutative ring with identity. A matrix −KPÐ8ßÑ is unitriangular if it is upper triangular () has ! 's below the main diagonal and has " 's on the main diagonal. We denote the set of all unitriangular matrices by YXÐ8ßÑ .

We will leave it as an exercise to show that

# YXÐ8ß™ Ñ œ :Ð8 8ÑÎ# kk: $ and that for :# , the group YXÐ$ß™: Ñ has order : and exponent : . Also, YXÐ$ß™# Ñ ¸ . Exercises 1. A left action of K\ on is sometimes defined as a map from the cartesian product K‚\ to \ , sending Ð+ßBÑ to an element +B−\ , satisfying a) "B œ B for all B − \ b) Ð+,ÑB œ +Ð,BÑ for all B − \ , + , , − K . A right action of K\ on is a map from the cartesian product \‚K\ to , sending ÐBß +Ñ to an element B+ − \ , satisfying c) B" œ B for all B − \ d) BÐ+,Ñ œ ÐB+Ñ, for all B − \ , + , , − K . Given a left action, show that the map ÐBß1Ñœ1" B is a right action. What about ÐBß1Ñœ1B ? 2. Let -ÀK ÄW\ be an action of K\ on . a) Prove that - is regular if and only if it is transitive and stabÐBÑ œ Ö"× for some B−\ . b) Prove that - is regular if and only if it is transitive and for all distinct 1ß2−K, we have 1BÁ2B for all B−\ . c) Prove that if - is faithful and transitive and if K is abelian, then the action is regular. 3. Let K: be a finite group and let be the smallest prime dividing 9ÐKÑ . Prove that any normal subgroup of order : is central. 4. Let K be an infinite group. Use normal interiors (not Poincaré's theorem) to prove that if LO and have finite index in K , then so does L∩O . 5. Show that the condition that K be finitely generated cannot be removed from the hypotheses of Theorem 7.5. 6. Let KL be a finite simple group and let ŸK have prime index ÐK À LÑ œ :. Prove that : must be the largest prime dividing 9ÐKÑ and that :9# does not divide ÐKÑ . 230 Fundamentals of Group Theory

7. Let KK be a finite group. Prove that a transitive action of on \ is regular if and only if Kœ\ . kk kk 8. Let 9ÐKÑœ#8 where 8 " is odd. Let +−K have order # . Show that under the left regular representation of K+ on itself, the element corresponds to an odd permutation. Show that K is not simple. 9. a) Prove that if KL is a finitely generated infinite group and is a subgroup of finite index in KK , then has a characteristic subgroup O of finite index for which OŸL . b) Show that the condition that K be finitely generated is necessary. 10. The action of a group K\ on a set is #-transitive if for any pairs ÐBß CÑß Ð?ß @Ñ − \ ‚ \ where B Á C and ? Á @ , there is an + − K for which +B œ ? and +C œ @ . Prove that for a # -transitive action, the stabilizer stabÐBÑ is a maximal subgroup of K for all B − \ . Equivalence of Actions

Two group actions -.ÀK ÄW\] and ÀL ÄW are equivalent if there is a pair Ðαα ß0Ñ where ÀK Ä L is a group isomorphism and 0À\ Ä ] is a bijection satisfying the condition 0Ð1BÑœ Ðα 1ÑÐ0BÑ

In this case, we refer to Ðß0Ñα- as an equivalence from to . .

11. a) Show that the inverse of an equivalence is an equivalence. b) Show that the (coordinatewise) composition of two “compatible” equivalences is an equivalence. 12. Let --ÀK ÄW\ be a transitive action and let B−\ . Show that is equivalent to the action of left-translation by KKÎÐBÑ on stab . 13. Suppose that -.ÀK ÄW\] and ÀL ÄW are equivalent transitive actions, under the equivalence Ðß0Ñα . Prove that stabÐBѸLC for any B−\ , where Cœ0B. Conjugacy 14. Let K1 be a group and let −KØ . Show that 1KÙK is a normal subgroup of . 15. Let K1 be a finite group and let −K . Show that GÐ1Ñ KÎKw where kkkkK KKw is the commutator subgroup of . 16. Let K be a finite group and let LŸK with ÒKÀLÓœ# . Suppose that GÐ2ÑŸLK for all 2−L . Prove that KÏL is a conjugacy class of K . 17. Let K: be a -group and let LŸK be a nonnormal subgroup of K and let +−K. Show that the number of conjugates of L that are fixed by every element of L:+ is positive and divisible by . 18. a) Let KL be a finite group and let ü K . Show that

5ÐKÎLÑ œ 5ÐKÑ  5K ÐLÑ "

where 5ÐLÑK is the number of K -conjugacy classes of L . Group Actions; The Structure of : -Groups 231

b) Let KK be a finite nonabelian group such that Î^ÐKÑ is abelian. Show that 5ÐKÑ KÎ^ÐKÑ^ÐKÑ" kkkk 19. a) Find all finite groups (up to isomorphism) that have exactly one conjugacy class. b) Find all finite groups (up to isomorphism) that have exactly two conjugacy classes. c) Find all finite groups (up to isomorphism) that have exactly three conjugacy classes. 20. a) If ;− and 8! , show that there are only finitely many solutions 5ßáß5"8 in positive integers to the equation "" ;œ â 55"8 Hint: Use induction on 8 . Look at the smallest denominator first. b) Show that for any integer 8! , there are only finitely many finite groups (up to isomorphism) that have exactly 8 conjugacy classes. Hint: Use the class equation. 21. a) Let LK be a proper subgroup of a finite group . Show that the set Wœ L1 . 1−K

is a proper subset of K . b) If LK is a proper subgroup of a group and ÐKÀLÑ∞, then the set

Wœ L1 . 1−K

is a proper subset of K . 22. Let \K be a conjugacy class of and let \"œÖB " ±B−\×. a) Show that \K" is also a conjugacy class of . b) Show that if K has odd order, then \ œ Ö"× is the only conjugacy class for which \œ\" . c) Show that if K\ has even order, then there is a conjugacy class other than Ö"× for which \ œ \" . d) Show that if K is finite and 5ÐKÑ is even, then 9ÐKÑ is even. Show by example that the converse does not hold. 23. Let K# be a group of order 7K . Suppose that has a conjugacy class of size 77. Prove that is odd, and that K has an abelian normal subgroup of size 7. 24. Let LK be normal in and suppose that ÐKÀLÑœ:B is a prime. Let −L have the property that there is a 1−KÏL such that 1BœB1 . a) Show that G ÐBÑ œ : G ÐBÑ . kkkkKL b) Show that BœBLK . 232 Fundamentals of Group Theory

::-Groups and -Subgroups 25. a) Let 0ÀK Ä L be a group homomorphism. If K: is a -group, under what conditions, if any, is L: a -group? b) Let 0ÀK Ä L be a group homomorphism. If L: is a -group, under what conditions, if any, is L: a -group? c) Let LKü . If L and KÎL are both :-groups, under what conditions, if any, is K: a -group? 26. Let K: be a finite -group. a) Prove that any cover of LK has index : . b) Prove that a cover of the center ^ÐKÑ is abelian. 27. Let LŸK . Prove that K is a : -group if and only if L and KÎL are : - groups. 28. Let K9 be a finite simple nonabelian group. Show that ÐKÑ is divisible by at least two distinct prime numbers. 29. Prove that the derived group K:w of a -group K is a proper subgroup of K . 30. Let K: be a -group. Show that if LŸKÐKÀLÑ∞ÐKÀLÑ and , then is a power of : . 31. Let Kœ K: be a direct product of : -subgroups for distinct primes : . Show that if LŸK , then Lœ ÐL∩KÑ: . What if the primes are not distinct? 32. Show that the generalized quaternion group

7" # #7# " " 7œ Ø+ß ,Ùß 9Ð+Ñ œ # ß 9Ð,Ñ œ %ß , œ + ß ,+, œ + has only is single involution. 33. Prove that a finite : -group has the normalizer condition using the action of RÐLÑKK on the conjugates conj ÐLÑ of L by conjugation. 34. Let K: be a nonabelian group of order $ , where : is a prime. Determine the number 5ÐKÑ of conjugacy classes of K . Additional Problems 35. Let : be a prime. a) Show that KPÐ8ß™ Ñ œ Ð:88  "ÑÐ:  :ÑâÐ: 88"  : Ñ kk:

b) Show that YXÐ8ß™: Ñ is a : -group, in fact,

# YXÐ8ß™ Ñ œ :Ð8 8ÑÎ# kk:

c) Show that YXÐ8ß™™:: Ñ is a Sylow : -subgroup of KPÐ8ß Ñ . d) For 8œ# or 8"Ÿ: , show that YXÐ8ß™: Ñ has exponent : . e) Show that YXÐ$ß™# Ñ ¸ . 36. Let J; be a finite field of size and let Z be an 8 -dimensional vector space over JZ . Show that the number of subspaces of of dimension 5 is Group Actions; The Structure of : -Groups 233

8 Ð;88  "ÑÐ;  ;ÑâÐ; 85"  ; Ñ ³ Š‹5; Ð;55  "ÑÐ;  ;ÑâÐ; 55"  ; Ñ

8 The expressions ÐÑ5 ; are called Gaussian coefficients. Hint : Show that the number of 5Z -tuples of linearly independent vectors in is Ð;88  "ÑÐ;  ;ÑâÐ; 85"  ; Ñ 37. Let LßO Ÿ K . a) Show that the distinct double cosets E1F , where + − K , form a partition of K . b) Show that E1F œ ÐE À F1 ∩ EÑ. kk

Chapter 8 Sylow Theory

In 1872, the Norwegian mathematician Peter Ludwig Mejdell Sylow [32] published a set of theorems which are now known as the Sylow theorems. These important theorems describe the nature of maximal : -subgroups of a finite group, which are now called Sylow :-subgroups. (For convenience, we will collect the Sylow theorems into a single theorem.) Sylow Subgroups We begin with the definition of a Sylow subgroup.

Definition Let K: be a group and let be a prime. A Sylow :-subgroup of K is a maximal :K -subgroup of () under set inclusion . The set of all Sylow : - subgroups of KÐ is denoted by Syl: KÑ: . The number of Sylow -subgroups of a group K8 is denoted by ::ÐKÑ8 , or just when the context is clear.

Of course, if a prime :9ÐKÑK divides , then contains a Sylow : -subgroup; in fact, every :K -subgroup of is contained in a Sylow : -subgroup. Also, if K is an infinite group and if L: is a -subgroup of K , then an appeal to Zorn's lemma shows that K: has a Sylow -subgroup containing L .

Since conjugation is an order isomorphism and also preserves the group order of elements, it follows that if W: is a Sylow -subgroup of K , then so is every conjugate WW+ of .

Note also that if K is finite and 9ÐKÑ œ :8 7 where Ð:ß 7Ñ œ " , then any subgroup of order ::8 is a Sylow -subgroup. We will prove the converse of this a bit later: Any Sylow subgroup of K: has order 8 . The Normalizer of a Sylow Subgroup Let K: be a finite group. If a Sylow -subgroup W of K happens to be normal in KKÎW, then has no nonidentity : -elements. Hence, :±ÐKÀWÑWy and so is the

S. Roman, Fundamentals of Group Theory: An Advanced Approach, 235 DOI 10.1007/978-0-8176-8301-6_8, © Springer Science+Business Media, LLC 2012 236 Fundamentals of Group Theory set of all :K -elements of . It also follows that W«K , since automorphisms preserve order.

Of course, WR is always normal in its normalizer KÐWÑ .

Theorem 8.1 Let KW be a finite group and let −Syl:ÐKÑ. 1) W:R is the set of all -elements of KÐWÑ . 2) Any :+−KÏWW -element moves by conjugation, that is, WÁW+ . 3) W:R is the only Sylow -subgroup of KÐWÑ . 4) :±ÐRÐWÑÀWÑy K . 5) W«RK ÐWÑ.

If W−Syl: ÐKÑ, then ++ WŸRÐWÑK for any +−K . Hence, if + normalizes RK ÐWÑ , then + WŸRÐWÑK

+ and since W:R is also a Sylow -subgroup of KÐWÑ , Theorem 8.1 implies that + WœW. In other words, if + normalizes RÐWÑK , then + also normalizes W and so

RÐRÐWÑÑœRÐWÑKK K

Theorem 8.2 The normalizer RÐWÑK of a Sylow subgroup of K is self- normalizing, that is,

RÐRÐWÑÑœRÐWÑKK K

Soon we will be able to prove that not only is RÐWÑK self-normalizing, but so is any subgroup of KRÐWÑ containing K . The Sylow Theorems

Let KW be a finite group and let −Syl:ÐKÑ:. The fact that any -element +ÂW moves W: by conjugation prompts us to look at the action of a -subgroup O of K by conjugation on the set + conjKÐWÑ œ ÖW ± + − K× of conjugates of WK in . As to the stabilizer of W+ , we have ++ + stabÐWÑœRK ÐWÑ∩OœW ∩O and so orb ÐW++ Ñ œ ÐO À W ∩ OÑ kkO which is divisible by :OŸW unless + , in which case the orbit has size " . Sylow Theory 237

Hence, ++ FixconjKÐWÑÐOÑ œ ÖW ± O Ÿ W × and so conj ÐWÑ ´ ÖW++ ± O Ÿ W ×mod : kkkkK Now if O: is a Sylow -subgroup of KOŸWOœW , then ++ if and only if and so

":O−ÐWÑmod if conjK conjKÐWÑ ´ kkœ !:OÂÐWÑmod if conjK

It follows that OÂconjKK ÐWÑ is impossible and so Syl:ÐKÑœ conj ÐWÑ is a conjugacy class and

8´": mod : Note also that 8 œconj ÐWÑ œ ÐK À R ÐWÑÑ ± 9ÐKÑ :Kkk K Finally, we can determine the order of a Sylow :W -subgroup , since

ÐK À WÑ œ ÐK À RKK ÐWÑÑÐR ÐWÑ À WÑ and neither of the factors on the right is divisible by :W . Hence, the order of is the largest power of :9ÐKÑ dividing . We have proved the Sylow theorems.

Theorem 8.3 () The Sylow theorems [32], 1872 Let K be a finite group and let 9ÐKÑ œ :8 7, where : is a prime and :y ± 7 . 1) The Sylow :K -subgroups of are the subgroups of K: of order 8 . 2)Syl :ÐKÑ is a conjugacy class in subÐKÑ . 3) The number 8:: of Sylow -subgroups satisfies

8:: ´ "mod :and 8 œ ÐK À RK ÐWÑÑ ± 9ÐKÑ

where W−Syl: ÐKÑ. 4)Syl Let W−: ÐKÑ . a) W8 is normal if and only if : œ" . b) W is self-normalizing if and only if 8: œÐKÀWÑœ7, in which case all Sylow :K -subgroups of are self-normalizing. 5) If O: is a -subgroup of K , then ÖW −Syl ÐKÑ ± O Ÿ W× ´ "mod : kk: We will prove later in the chapter that every normal Sylow : -subgroup of a finite group is complemented. This is implied by the famous Schur–Zassenhaus theorem. However, we also have the following simple consequence of Theorem 3.1 concerning supplements of Sylow subgroups. 238 Fundamentals of Group Theory

Theorem 8.4 Let K: be a finite group. Then any Sylow -subgroup of K and any subgroup whose index is a power of : are supplements. Sylow Subgroups of Subgroups Let KL be a finite group and let ŸK . We wish to explore the relationship between Syl::ÐKÑ and Syl ÐLÑ . On the one hand, every W −Syl: ÐLÑ is contained in a X−Syl: ÐKÑ and so the set

Syl::ÐWà KÑ œ ÖX −Syl ÐKÑ ± W Ÿ X × is nonempty. Moreover, since E−Syl: ÐWàKÑ implies that E∩L is a : - subgroup of LW containing , we have

E−Syl: ÐWàKÑ Ê E∩LœW

In particular, the families Syl:ÐWà KÑ are disjoint, that is,

W Á X −Syl::: ÐLÑ ÊSyl ÐWà KÑ ∩ Syl ÐX à KÑ œ g and so

8ÐLÑŸ8ÐKÑ::

On the other hand, if W−Syl: ÐKÑ, then the intersection W∩L need not be a Sylow :L -subgroup of , as can be seen by taking LW and to be distinct Sylow :-subgroups of K . However, if LW is a subgroup of K , then 9ÐWÑ ± 9ÐLWÑ and so LW L∩Wkk œ L kkW kk kk where LW Î W is not divisible by : . Hence, L ∩ W and L are divisible by kkkk k k kk the same powers of :L∩W−ÐLÑ and so Syl: .

Theorem 8.5 Let KL be a finite group and let ŸK . 1)Syl If W−: ÐLÑ , then

E−Syl: ÐWàKÑ Ê E∩LœW and

W Á X −Syl::: ÐLÑ ÊSyl ÐWà KÑ ∩ Syl ÐX à KÑ œ g and so

8ÐLÑŸ8ÐKÑ:: Sylow Theory 239

2)Syl If W−: ÐKÑ and LWŸK , then

W∩L−Syl: ÐLÑ

Some Consequences of the Sylow Theorems Let us consider some of the more-or-less direct consequences of the Sylow theorems. A Partial Converse of Lagrange's Theorem A Sylow :W -subgroup of a group K has subgroups of all orders dividing 9ÐWÑ . This gives a partial converse to Lagrange's theorem.

Theorem 8.6 Let K:: be a finite group and let be a prime. If 5 ±9ÐKÑ , then K has a subgroup of order :5 . More on the Normalizer of a Sylow Subgroup

Recall that the normalizer RÐWÑK of a Sylow subgroup WK of is self- normalizing. Now we can say more.

Theorem 8.7 Let KW be a finite group and let −Syl:ÐKÑ. If

WŸRK ÐWÑŸLŸK then LL is self-normalizing. In particular, if K , then L is not normal in K . Proof. Conjugating by any +−RK ÐLÑ gives ++ W ŸRK ÐWÑ ŸLŸK and so both WW and ++ are Sylow : -subgroups of L . It follows that WW and are conjugate in L2 . Hence, there is an −LW for which 2+ œW , that is, 2+−RKK ÐWÑŸL. Thus, +−L and so R ÐLÑœL.

The normalizer of a Sylow : -subgroup has a somewhat stronger property than is expressed in Theorem 8.7. In the exercises, we ask the reader to prove that RÐWÑK is abnormal. Counting Subgroups in a Finite Group In an earlier chapter, we proved that if K: is a -group and :±9ÐKÑ5 , then the 5 number 8ÐKÑ:ß5 of subgroups of K: of order satisfies

8ÐKÑ´":ß5 mod : (8.8) We have just proved that for any finite group K:±9ÐKÑ for which ,

8ÐKÑ´": mod : To see that (8.8) holds for all finite groups, we count the size of the set 240 Fundamentals of Group Theory

5 Y5 œ ÖÐLß WÑ ± L Ÿ Wß W −Syl: ÐKÑß 9ÐLÑ œ : × modulo :W . On the one hand, for each −Syl::ÐKÑ, there are 8ß5ÐWÑ´" subgroups of W: of order 5 and so Y ´8 ÐKц"´" kk5: On the other hand, for each LŸK of order :5 , Theorem 8.3 implies that ÖW −Syl ÐKÑ ± L Ÿ W× ´ " kk: and so Y ´8 ÐKц"´8 ÐKÑ kk5:ß5 :ß5

Hence, 8ÐKÑ´":ß5 and we have proved an important theorem of Frobenius.

Theorem 8.9 ()Frobenius [13], 1895 Let K9 be a group with ÐKÑœ:87 where Ð7ß:Ñœ" . Then for each "Ÿ5Ÿ8 , the number 8:ß5 ÐKÑ of subgroups of K: of order 5 satisfies

8ÐKÑ´":ß5 mod :

When All Sylow Subgroups Are Normal Several good things happen when all of the Sylow subgroups of a group K are normal. In particular, let K be a finite group. In an earlier chapter (see Theorem 4.22 and Theorem 4.35), we showed that among the conditions:

1) Every subgroup of K is subnormal 2) K has the normalizer condition 3) Every maximal subgroup of K is normal 4) KÎF ÐKÑ is abelian the following implications hold: 1)ÍÊÍ 2) 3) 4) We also promised to show that these four conditions are equivalent, which we can do now, adding several additional equivalent conditions into the bargin.

First, let us speak about arbitrary (possibly infinite) groups. If K is a group, let KKtor denote the set of all torsion (periodic) elements of . If K is abelian, then KKtor is a subgroup of . However, in the nonabelian general linear group KPÐ#ß‚ Ñ, the elements !" !" Eœ and Fœ Œ"! Œ" " Sylow Theory 241

are torsion but their product is not. Hence, KKtor is not always a subgroup of . Note that when KKtor is a subgroup of , then Ktor «K since automorphisms preserve order.

Theorem 8.10 Let K be a group in which every Sylow subgroup is normal. Let the Sylow subgroups of KÖ]±:−× be : c . Then

Kœ]tor : :−c and so KŸKtor . Thus, the product of two elements of finite order has finite order. Proof. Since the Sylow : -subgroups are normal and pairwise essentially disjoint, they commute elementwise. In particular, if +ßáß+"8 come from distinct Sylow subgroups, then

9Ð+"8 â+ Ñ œ 9Ð+ " Ñâ9Ð+ 8 Ñ and so the family of Sylow subgroups is strongly disjoint and

]³]: ©Ktor :−c

/" /7 For the reverse inclusion, if +−Ktor has order 8œ:" â:7 where the primes /5 :3" are distinct, then Corollary 2.11 implies that +œ+â+7, where 9Ð+Ñœ:55 and so +−]5:5 , whence +−] .

Now we turn to finite groups in which all Sylow subgroups are normal.

Theorem 8.11 Let KÖ be a finite group, with Sylow subgroups ]: ±:−c×. The following are equivalent: 1) Every Sylow subgroup of K is normal. 2) K: is the direct product of its Sylow -subgroups

Kœ ]: :−c

3) If LŸK , then

Lœ ÐL∩]Ñ: :−c

4) K: is the direct product of -subgroups. 5)( Strong converse of Lagrange's theorem) If 8±9ÐKÑ , then K has a normal subgroup of order 8 . 6) Every subgroup of K is subnormal. 7) K has the normalizer condition. 8) Every maximal subgroup of K is normal. 9) KÎF ÐKÑ is abelian. If these conditions hold, then K has the center-intersection property. In particular, ^ÐKÑ is nontrivial. 242 Fundamentals of Group Theory

Proof. Theorem 8.10 shows that 1) implies 2) and the converse is clear. If 1) holds, then L]: Ÿ K and so the Sylow : -subgroups of L are ÖL ∩ ]:: ± : −cü ×. Moreover, since L ∩ ] L, it follows that L is the direct product of its Sylow : -subgroups and so 3) holds. It is clear that 3) implies 2) and so 1)–3) are equivalent. Also, it is clear that 2)Ê: 4). If 4) holds and is a prime dividing 9ÐKÑ , then we can isolate the factors in the direct product of K that have exponent : , say KœT where T:; is a direct product of -subgroups and is a direct product of - subgroups for various primes ;Á: . Then T is the set of all : -elements of K and so T: a Sylow -subgroup of KTK . But ü and so 1) holds. Thus, 1)–4) are equivalent.

It is clear that 5)ÊÊ8 1). To see that 2) 5), any divisor of 9ÐKÑ is a product where and since has a normal subgroup of order , the 8œ# .::: . ±9Ð]Ñ ] : .: direct product of these subgroups is a normal subgroup of K8 of order . Thus, 1)–5) are equivalent and we have already remarked that 6)ÍÊÍ 7) 8) 9)

To see that 3)ÊLK 6), if , then one of the factors L∩]:: is proper in ] and so there is a subgroup R: for which

L∩]:: –R Ÿ] : Hence,

L– ÐL∩]ÑR;: ;Á:

Since LKK is an arbitrary proper subgroup of , it follows that all subgroup of are subnormal. To see that 6)Ê] 1), if : is not normal in K , then since RÐ]ÑKK: is subnormal, there is a subgroup L: for which RÐ]Ñ–LŸKK: : . But this contradicts the fact that RÐ]ÑK: is self-normalizing. Hence, ]–K: . Thus, 1)–7) are equivalent and imply 8).

Similarly, if 8) holds but ]K: is not normal in , then there is a maximal subgroup QŸK for which

]ŸRÐ]ÑŸQ–K:K: which contradicts Theorem 8.7. Hence, ]K: ü and 1) holds.

Finally, if 3) holds and RŸK is nontrivial, then R∩];;ü ] is nontrivial for some ;−c and

R ∩ ^ÐKÑ œ  ÐR ∩ ^ÐKÑ ∩ ]: Ñ :−c Sylow Theory 243

But R∩^ÐKÑ∩];; œR∩^Ð]Ñ is nontrivial and therefore so is R∩^ÐKÑ.

The hypotheses of the previous theorems hold for all abelian groups.

Corollary 8.12 Let K be an abelian group. 1)( Primary decomposition) Then K:tor is the direct product of its Sylow - subgroups. 2)( Converse of Lagrange's theorem) If K8 is finite and ±9ÐKÑK , then has a subgroup of order 8 .

We will add one additional characterization (nilpotence) to Theorem 8.11 in a later chapter (see Theorem 11.8). When a Subgroup Acts Transitively; The Frattini Argument The Frattini argument (Theorem 7.2) shows that if a group K acts on a nonempty set \LŸK and if is transitive on \ , then

KœLstabK ÐBÑ and if L\ is regular on , then

KœLìstabK ÐBÑ and so stabKÐBÑ is a complement of LK in . To apply this idea, let K be a finite group and let WŸLü K where W−Syl:K ÐLÑ. Let K act on conj ÐWÑ by conjugation. Since + W−Syl: ÐLÑ for any +−K , it follows that L acts transitively on conjKÐWÑ . Hence,

K œ LstabKK ÐWÑ œ LR ÐWÑ This specific argument is also referred to as the Frattini argument.

Theorem 8.13 Let KL be a finite group and let ŸK . If W−Syl:ÐLÑ, then

KœLRK ÐWÑ and if the action of LÐ by conjugation on conjK WÑ is regular, then

KœLìRK ÐWÑ This theorem can be used to show that the Frattini subgroup of a finite group K has the property that all of its Sylow subgroups are normal in K . 244 Fundamentals of Group Theory

Theorem 8.14 ()Frattini [12], 1885 If K is a finite group, then the Frattini subgroup FÐKÑ has the property that all of its Sylow subgroups are normal in K. Proof. If W−Syl: ÐFFü Ñ, then WŸ K and the Frattini argument shows that

KœF RK ÐWÑ

But if RÐWÑKK , then there is a maximal subgroup QK of for which RÐWÑŸQK and so KŸQ , a contradiction. Hence, RÐWÑœKK and WKü . The Search for Simplicity The Sylow theorems, along with group actions and counting arguments, provide powerful tools for the analysis of finite groups. A key issue with respect to finite groups is the question of simplicity. As we will discuss in a later chapter, the issue of which finite groups (up to isomorphism) are simple appears to be resolved, but the resolution is so complex that some mathematicians may still have questions regarding its completeness and its accuracy.

We have seen that a group of prime-power order :8 has a normal subgroup of each order :±:58 . Accordingly, we will do no further direct analysis of : - groups in this chapter.

Throughout our discussion, :] will denote a prime, : will denote an arbitrary Sylow :8 -subgroup and, as always, : denotes the number of Sylow : -subgroups of K . Recall that

1) 8œ"5:: for some integer 5 ! . 2) 8:K: œ ÐK À R Ð] ÑÑ ± 9ÐKÑ.

8 Note that if 9ÐKÑ œ : 7, where :y ± 7 , then 8:: ± 9ÐKÑ if and only if 8 ± 7 .

The following facts (among others) are useful in showing that a group is not simple:

3) ]8:: is normal if and only if œ" . 4) If ÐK À LÑ is equal to the smallest prime dividing 9ÐKÑ , then L – K . 5) The kernel of any action -ÀK ÄW\ is a normal subgroup of K . 6) If L  K , then ÐK À L‰‰ Ñ ± ÐK À LÑx. Hence, if 9ÐKÑy ± ÐK À LÑx, then L is a nontrivial proper normal subgroup of K .

We will also have use for the fact that if : is prime and "Ÿ/: , then /: is the smallest integer for which :±Ð/:Ñx/ . Sylow Theory 245

The 8:-Argument It happens quite often that for some odd prime :±9ÐKÑ , the integers "5: do not divide 9ÐKÑ unless 5 œ ! , in which case 8:: œ " and ]ü K . Let us refer to the argument

8œ"5:±9ÐKÑÊ: 5œ! as the 8:-argument. Note that the 8:: -argument does not hold if œ# , unless 9ÐKÑ is a power of # , since "  #5 ± 9ÐKÑ for some 5  " .

Example 8.15 If 9ÐKÑ œ **)# œ # † ( † #$ † $", then routine calculation shows that the 8( argument holds: "(5±9ÐKÑ Ê 5œ! and so ]–K( and K is not simple.

Example 8.16 If 9ÐKÑœ:78 for 8 " , 7" and :±7y , then 8œÐ"5:ѱ7: and so if 7: , then 5œ! , whence ]–K: . Thus, groups of order :88 ß#: ßáßÐ:"Ñ:8 for :8 "]–K prime and have : and so are not simple.

A little programming shows that among the orders up to "!!!! (not including prime powers) there are only &'* orders (less than ' %) that are not succeptible to the 8: -argument for some : . Thus, the vast majority of orders up to "!!!! are either prime powers or have the property that groups of that order have a normal Sylow :-subgroup. Counting Elements of Prime Order # If : is a prime and :±9ÐKÑ but :y ±9ÐKÑ , then each of the 8: distinct Sylow : - subgroups of K: has order and so the subgroups are pairwise essentially disjoint. Hence, K8 contains exactly : †Ð:"Ñ: distinct elements of order . Sometimes this simple counting of elements (for different primes : ) is enough to show than one of the Sylow subgroups is normal.

Example 8.17 Let 9ÐKÑœ$!œ#†$†&. Then based on the fact that 8:$ œ "  5: ± 9ÐKÑ, we can conclude only that 8 − Ö"ß "!× and 8& − Ö"ß '× . However, if 8œ"!$& and 8œ' , then K contains at least 8†Ð$"Ñœ#!$ elements of order $#% and elements of order & , totalling %% elements. Hence, one of ]]$& or must be normal in K . 246 Fundamentals of Group Theory

Index Equal to Smallest Prime Divisor 5 If 9ÐKÑœ:; where :; are primes, then ];; –K , because ÐKÀ]Ñœ: is the smallest prime dividing 9ÐKÑ . Moreover, it is clear that

Kœ];:z ]

# Example 8.18 If 9ÐKÑœ$†& œ(&, then ]& –K and

Kœ]$&z ]

Also, "$5±#& holds only for 5œ! or 5œ) and so 8$$ œ" or 8 œ#& . Note that if 8œ"$$ , then Kœ]]& is abelian.

When 9ÐKÑ œ :; , we can give a fairly complete analysis as follows.

Theorem 8.19 Let 9ÐKÑœ:; , with :; primes. Then

KœGÐ,Ñ;:z GÐ+Ñ where ,œ,+5 for some "Ÿ5; and 5: ´"mod ;. Moreover, K is cyclic if and only if :±;"y . Proof. We have seen that

Kœ];:zz ] œGÐ,Ñ ; GÐ+Ñ : Thus, +,+" œ, 5 for some "Ÿ5; and repeated conjugation by + gives

: ,œ+,+::5 œ, : which implies that 5´"mod ;. Moreover, 8±;::: and so 8œ" or 8œ; . But 8œ":: if and only if ]Kü , that is, if and only if K is cyclic and 8œ;: if and only if "5:œ; , that is, if and only if :±;"y .

Example 8.20 Let us return to the case 9ÐKÑ œ $! œ # † $ † &. We saw in Example 8.17 that one of ]]$& or must be normal in K . It follows that ]]$&üü K has order "& and so is cyclic. Hence, ]ß]«]]$ & $& K and so both ]]$& and are normal in K . Using the Kernel of an Action

The kernel of an action -ÀK ÄW\ is normal in K and this can be a useful technique for finding normal subgroups, although they need not be Sylow subgroups.

For example, if KÐ acts on Syl: KÑ by conjugation, then the representation map -ÀK ÄW5:" has kernel Sylow Theory 247

Oœ R Ð]Ñ , K ]−Syl: ÐKÑ which is a normal subgroup of K . The problem is that it may be either trivial or equal to K .

Let 9ÐKÑ œ :7= ? and 9ÐOÑ œ : @ , where 7 "ß ?  " and :y ± ? . Also, let 8œ5:"::. It is clear that OœK is equivalent to 8œ" and implies that =œ7. Conversely, if =œ7 , then O contains a Sylow :WK -subgroup of . But the only Sylow :KR -subgroup of in K:Ð]Ñ]8 is itself and so œ" and OœK. Thus,

OœKÍ]–KÍ=œ7:

As to the nontriviality of OK , the induced embedding of ÎO into W5:" implies that ? :Ð5:"Ñx7= @ ¹ and so :7= ±Ð5:Ñx. Hence, if 5: , then 75Ÿ= . It follows that if 57 , then =! and O is nontrivial. Thus, 5 min Ö7ß :× Ê O Á Ö"×

We note finally that O8 has a somewhat simpler form if : œ? , since then each ]: is self-normalizing and Oœ ] , ]−Syl: ÐKÑ

Theorem 8.21 Let 9ÐKÑœ:7 ? where : is prime, 7 " , ?" and :±?y . Let 8œ"5:: . 1) If 5œ! , then ]: –K . 2) If !5min Ö7ß:×, then

Oœ R Ð]Ñ , K ]−Syl: ÐKÑ

is a nontrivial proper normal subgroup of K:@ of order = , where 75Ÿ=Ÿ7" and @±? . In addition, if 5œÐ?"ÑÎ:, then

Oœ ] , ]−Syl: ÐKÑ

has order := .

Example 8.22 If 9ÐKÑ œ "!) œ $$ † %, then "  $5 ± % and so 5 œ ! or 5 œ " . Thus, this case is not amenable to the 85: -argument. However, if œ" then 248 Fundamentals of Group Theory

Theorem 8.21 implies that

Oœ ] , ]−Syl$ ÐKÑ is a nontrivial proper normal subgroup of K*K of order . Thus, is not simple.

If 9ÐKÑ œ ")* œ $$ † (, then "  $5 ± ( and so 5 œ ! or 5 œ # . If 5 œ # , then Theorem 8.21 implies that

Oœ ] , ]−Syl$ ÐKÑ is a nontrivial proper normal subgroup of K$* of order or .

## If 9ÐKÑ œ $!! œ # † $ † &, then 8& œ "  &5 ± "# and so 5 œ ! or 5 œ " (and 8œ'& ). But if 5œ" , then Theorem 8.21 implies that K is not simple.

Even when O is trivial and the previous theorem does not apply, we learn that KWä 5:", which can sometimes be useful.

Example 8.23 If 9ÐKÑ œ :Ð:  "Ñ, where : is prime. Then 8: œ " or 8œ:": . While the previous theorem does not apply, if 8œ:": , then O œ ] œ Ö"× , ]−Syl: ÐKÑ

Hence, KWä :" . As an example, if 9ÐKÑœ"#œ$†%, then either ]$ –K or Kä W%%. But 9ÐKÑœ9ÐEÑ and so in the latter case, K¸E%% . Thus, if K¸Ey then ]–K$ . We will use this fact later to determine all groups of order "# . The Normal Interior If LK , we have seen that ÐK À L‰ Ñ ± ÐK À LÑx and so if 9ÐKÑy ± ÐK À LÑx, then L‰ is a nontrivial proper normal subgroup of K . Hence, if

/" /7 9ÐKÑ œ :" â:7 where :â:"7 are primes and 7 # , then for any 5 ,

/5 /55 : ßÐKÀLÑ/:55 Ê :5 yy ±ÐKÀLÑx Ê 9ÐKѱÐKÀLÑx and so L–K‰ is nontrivial.

/" /7 Theorem 8.24 Let 9ÐKÑ œ :" â:7 where :"7  â  : are primes and 7 #. Suppose that /55 : for some "Ÿ5Ÿ7 . Sylow Theory 249

‰ 1) If LK has index ÐKÀLÑ/:55, then L is a nontrivial proper normal subgroup of K . ‰ 2) In particular, if "8:5533 / :, then R K: Ð]Ñ is a nontrivial proper normal subgroup of K .

# Example 8.25 Let 9ÐKÑ œ '#!" œ $ † "$ † &$. Then 8$ œ "  $5 ± "$ † &$, which implies that 8$$$$ − Ö"ß "$×. If 8 œ " , then ] – K . If 8 œ "$  &$, then ‰ Theorem 8.24 implies that RÐ]ÑK$ is a nontrivial proper normal subgroup of KK. Thus is not simple. Using the Normalizer of a Sylow Subgroup

Let :Á; be primes dividing 9ÐKÑ and let ]; −Syl; ÐKÑ. Under the assumption that 8";K and so RÐ]ÑK;, suppose that :±8y ;K , that is, :±9ÐRÐ]ÑÑ; and that T −Syl:K; ÐR Ð] ÑÑ. There are various things we can say about 9ÐRK ÐT ÑÑ .

First, if T–RÐ]ÑK; , then RÐ]ÑŸRÐTÑ K; K and so

9ÐRK; Ð] ÑÑ ± 9ÐR K ÐTÑÑ

On the other hand, even if TR is not normal in K;Ð]Ñ , the fact that ]RÐ]Ñ;K;ü implies that T]; is a subgroup of K . Hence, if T]; is abelian, then ]ŸRÐTÑ;K and so

9Ð];K Ñ ± 9ÐR ÐT ÑÑ

‡ In either case, if T:KT is not a Sylow -subgroup of but –T−ÐSyl: KÑ, then ‡ TŸRÐTÑK , whence ‡ 9ÐT Ñ ± 9ÐRK ÐT ÑÑ

These conditions tend to make RÐTÑK large.

Example 8.26 If 9ÐKÑ œ $'(& œ $ † &## † ( , then it is easy to see that # 8(( − Ö"ß "&×. If 8 œ "& , then 9ÐR K Ð]( ÑÑ œ & † (. Let T be a Sylow & - # subgroup of RÐ]ÑK( . The number of such subgroups is "&5±( and so T–RK( Ð]Ñ. Hence, # &†( ±9ÐRK ÐTÑÑ

‡‡ Also, T&T−ÐKÑT–T has index in Syl& and so , whence # &±9ÐRÐTÑÑK and so

## &†(±9ÐRÐTÑÑK

Hence, either RÐTÑœKKK, in which case T–K or else RÐTÑ has index $ in KK and so is normal in . 250 Fundamentals of Group Theory

Suppose that 9ÐKÑ œ :;?, where : Á ; are primes that do not divide ? . If :y ± 8;;, that is, if : ± 9ÐRÐ] ÑÑ, then ]: Ÿ RÐ]; Ñ and so ]: ]; Ÿ K has order :; . Hence, if :±Ð;"Ñy , then ]]:; is abelian (cyclic) and so ];: ŸRÐ]Ñ . Thus, ; ± 9ÐRÐ]:: ÑÑ and so 8 ± 9ÐKÑÎ:;.

Theorem 8.27 If 9ÐKÑ œ :;?, where :  ; are primes that do not divide ? , then 9ÐKÑ :±Ð;"Ñyyand :±8 Ê 8 ;:¹ :; Example 8.28 If 9ÐKÑ œ "()& œ $ † & † ( † "(, then a routine calculation gives

8$" − Ö"ß (ß )&ß &*&×and 8( − Ö"ß $&× But 9ÐKÑ $  "(ß $yy ± Ð"(  "Ñß $ ± 8 Ê 8 œ $& "( $ ¹ $†"( and so 8œ"$$ or 8œ( . Hence, Theorem 8.24 now implies that one of ]$ or ‰ RÐ]$ Ñ is a proper nontrivial normal subgroup of K . Groups of Small Order We have already examined the groups of order %' , and ) . Let us now look at all groups of order "& or less. Of course, all groups of prime order are cyclic. We will again denote an arbitrary Sylow :K] -subgroup of by : . Groups of Order % The groups of order % are (up to isomorphism):

1) G% , the cyclic group 2) Z¸G##} G, the Klein 4-group. Groups of Order ' The groups of order ' are (up to isomorphism):

1) G' , the cyclic group 2) H¸W'$ , the nonabelian dihedral (and symmetric) group. Groups of Order ) The groups of order ) are (up to isomorphism):

1) G) , the cyclic group 2) GG%#} , abelian but not cyclic 3) GGG###}}, abelian but not cyclic Sylow Theory 251

4) H) , the (nonabelian) dihedral group 5) , the (nonabelian) quaternion group. Groups of Order * Theorem 7.9 implies that the groups of order * are (up to isomorphism):

1) G* , the cyclic group 2) GG$$} , abelian but not cyclic. Groups of Order "! If 9ÐKÑ œ "! œ # † &, then Theorem 8.19 implies that K œØ+ß,Ùß9Ð+Ñœ#ß9Ð,Ñœ&ß+,+" œ, 5 where 5´"# mod &, that is, 5œ" or % . In the former case, + and , commute and KK is cyclic. In the latter case, œH""! . Thus, the groups of order ! are (up to isomorphism):

1) G¸GG"! &} #, the cyclic group 2) H"! , the nonabelian dihedral group. Groups of Order "# We have accounted for three nonabelian groups of order "# : the alternating group EH%" , the dihedral group # and the semidirect product XœGÐ+Ñ$%z GÐ,Ñ, where ,+," œ + # We wish to show that this completes the list of nonabelian groups of order "# .

Assume that K¸Ey %$ . Then Example 8.23 shows that ] œGÐ+Ñ$ü K and so KœGÐ+Ñ$z ] #, where ] #% œGÐ,Ñ or ] ## œGÐBÑGÐCÑ #.

If KœGÐ+Ñ$%z GÐ,Ñ, then ,+," œ +or ,+," œ + # In the former case KK is abelian and in the latter case œX . If

K œ G$## Ð+Ñz ÐG ÐBÑ  G ÐCÑÑ then +œ++B# or and +œ++ C# or We consider three cases. If +BC œ+ and + œ+ , then K is abelian. If +B œ+ and +C# œ +, then 9Ð+BÑ œ 9Ð+Ñ9ÐBÑ œ ' and Ð+BCÑ#$ œ +BC+BC œ +CÐB+BÑC œ +C+C œ + œ " 252 Fundamentals of Group Theory

Hence, Ø+BCß CÙ is generated by two distinct involutions whose product has B#C order 'K and so Theorem 2.36 implies that œH"# . Finally, if +œ+œ+, then Ð+BÑ#$B œ+B+Bœ+ œ" and +C# œÐ+ÑC œ+% œ+ and so 9Ð+BCÑœ 9Ð+Ñ9ÐBCÑ œ '. Thus, Ø+Bß CÙ is dihedral and K œ H"# .

Thus, the groups of order "# are (up to isomorphism)

1) G¸GG"# %} $, the cyclic group 2) GGG##$}}, abelian but not cyclic 3) E% , the nonabelian alternating group 4) H"# , the nonabelian dihedral group 5) X , the nonabelian group described above. Groups of Order "% If 9ÐKÑ œ "% œ # † (, then Theorem 8.19 implies that KœØ+ß,Ùß9Ð+Ñœ#9Ð,Ñœ(ß+,+, " œ, 5 where 5´"# mod (, that is, 5œ" or ' . In the former case, K is cyclic. In the latter case, K" is dihedral. Thus, the groups of order % are (up to isomorphism):

1) G¸GG"% (} #, the cyclic group 2) H"% , the nonabelian dihedral group. Groups of Order "& Theorem 8.19 implies that all groups of order "& are cyclic. On the Existence of Complements: The Schur–Zassenhaus Theorem In this section, we use group actions to prove the Schur–Zassenhaus Theorem, which gives a simple sufficient (but not necessary) condition under which a normal subgroup LK of a group has a complement O , thus giving a semidirect decomposition KœLz O.

Definition Let KL be a finite group. A Hall subgroup of K is a subgroup with the property that its order 9ÐLÑ and index ÐK À LÑ are relatively prime.

The Schur–Zassenhaus Theorem states that a normal Hall subgroup L has a complement. The tool that we will use in proving the Schur–Zassenhaus Theorem is the Frattini argument (Theorem 7.2). In particular, we consider the action of left translation by KL on the set e of all right transversals of and show that this action is regular, whence

KœLz stabK ÐÑ for any − e . So let us take a closer look at transversals and their actions. Sylow Theory 253

Transversals and Their Actions Let KL be a finite group and let be a normal Hall subgroup of K , with right cosets

LÏKœÖLœL"7 ßáßL ×

Let ee be the set of all right transversals of LœÖ<ßáß<×− . If "7 where <−L33 for all 3 and if +−K , then

L+<33 œ +L< and so the cosets L+<3 are distinct. Hence,

+ œ Ö+< ß"7 á ß +< × − e and it is clear that K acts on e by left translation.

Although LK need not act transitively on e , we can raise the action of to an action on the congruence classes of an appropriate K -congruence on e so that LK does act transitively. The -congruence condition is ´W Ê +´+W for all +−K , that is,

Ö<"7 ßáß< ×´Ö= "7 ßáß= × Ê Ö+< " ßáß+< 7 ×´Ö+= " ßáß+= 7 ×

This leads us to try the following. Assuming that W and are indexed so that <3 and =L´33 are in the same right coset , define a binary relation by 7 ´Wif <=" œ" $ 3 3 3œ" Letting 7 lW œ < = " $ 3 3 3œ" the definition becomes ´WiflWœ"

" This relation is clearly reflexive. Also, since <=3 3 −L for all 3 , if L is abelian , then for any ß Wß X − e, ÐlWÑ" œ Wl andÐlWÑÐWlX Ñ œ lX and so ´Î is an equivalence relation on ee . Let ´ denote the set of equivalence classes of e and let ÒÓ denote the equivalence class containing . 254 Fundamentals of Group Theory

" Note that if 2−L , then <=3 3 −L implies that 7 Ð2ÑlW œ 2< =" œ 2 ÐlWÑ 7 $ 3 3 3œ"

Moreover, since <=33 and are in the same right coset of L , so are +<+=3 and 3 and so for any +−K ,

77 +l+Wœ Ð+<ÑÐ+=Ñ" œ+ <=" + " œÐlWÑ + $$33 33 3œ" 3œ" Hence, +´+W if and only if ´W . Thus, ´ is Ka -congruence on e and the induced action is +ÒÓ œ Ò+Ó Now, L acts transitively if and only if for all ßW−e , there is an 2−L for which Ò2Ó œ ÒWÓ , that is, for which " œ Ð2ÑlW œ 2 ÐlWÑ7 or equivalently, 2œlW7 But this equation always has a solution in L since 7 œ ÐK À LÑ and 9ÐLÑ are relatively prime.

The action of Lδ on e is also regular, since Ò2ÓœÒÓ if and only if "œÐ2Ñlœ2 ÐlÑœ277 which implies that 2œ" . Hence, if L is a normal abelian subgroup of K , the Frattini argument implies that

K œ Lz stabK ÐÒÓÑ for any − e .

As to the conjugacy statement, any conjugate of a complement of L is also a complement of L . On the other hand, if KœLz O then Theorem 5.3 implies that O−e and so

Lzz OœKœLstabK ÐÒOÓÑ

But OŸstabK ÐÒOÓÑ and so

OœstabK ÐÒOÓÑ Sylow Theory 255

Thus, the set of complements of LK in is precisely the set

ÖstabK ÐÒÓÑ ± − × e Also, since KÎ acts transitively on e ´ , all stabilizers are conjugate and so all complements of L are conjugate. We have proved the abelian version of the Schur–Zassenhaus Theorem.

Theorem 8.29 ()Schur–Zassenhaus Theorem—abelian version If K is a finite group, then any normal abelian Hall subgroup LK of has a complement in K . Moreover, the complements of LK in form a conjugacy class of subÐKÑ . The Schur–Zassenhaus Theorem The condition of abelianness can be removed from the Schur–Zassenhaus Theorem. The proof of the general Schur–Zassenhaus theorem uses the Frattini argument as well, but also uses the Feit–Thompson theorem to establish the conjugacy portion of the theorem. Let us isolate the portion that uses the Feit– Thompson Theorem.

Theorem 8.30 Let K be a nontrivial group of odd order. 1) KKw 2) KO has a normal subgroup of prime index. Proof. Part 1) can be proved by induction on 9ÐKÑ . If 9ÐKÑ œ $ , then K is abelian and Kw œ Ö"×  K. Assume the result holds for groups of odd order less than 9ÐKÑ and let 9ÐKÑ be odd. Then the Feit–Thompson theorem implies that K is either abelian or nonsimple. If K is abelian, then Kw œ Ö"×  K. If K is nonsimple, then there is a Ö"×–O–K and so KÎO has odd order less than 9ÐKÑ. Hence, the inductive hypothesis implies that ÐKÎOÑw  KÎO, which implies that KKww . For part 2), since KK , it follows that K has a maximal normal subgroup OKKÎO containing w . Then is simple and abelian and so has prime order.

Theorem 8.31 ()Schur–Zassenhaus theorem Any normal Hall subgroup L of finite group KK has a complement in . Moreover, the complements of L in K form a conjugacy class in KK . In particular, any normal Sylow subgroup of is complemented. Proof. We may assume that L is nontrivial and proper. The proof of the existence of a complement is by induction on 9ÐKÑ . The result is true if 9ÐKÑ œ ". Assume it is true for all orders less than 9ÐKÑ . Let Lü Hall K denote the fact that LK is a normal Hall subgroup of .

If LO has a proper supplement , that is, if KœLO for some OK , then MœL∩Oü Hall O and so the inductive hypothesis implies that OœMz N , whence 256 Fundamentals of Group Theory

KœLOœLÐMzz NÑœL N and so L is complemented.

But since L:9 is a Hall subgroup, if is a prime dividing ÐLÑ , then any W−Syl: ÐLÑ is also a Sylow : -subgroup of K and the Frattini argument implies that KœLR where RœRK ÐWÑ. Thus, if RK , then L is complemented. On the other hand, if RœK , then W is a normal Sylow : -subgroup of KWK and so ö , whence ^³^ÐWÑöü K. Since LÎ^Hall KÎ^, the inductive hypothesis implies that KLO œ z ^^^ for some OŸK and so O is a supplement of L . But if OK , then L is complemented and if OœK , then Lœ^ is abelian and so L is complemented in this case as well.

We now turn to the statement about conjugacy. Since any conjugate of a complement of LL is also a complement of , we are left with showing that any two complements OOL"# and of are conjugate. The proof is by induction on 9ÐKÑ. Of course, the result is true if 9ÐKÑ œ " . Assume the result is true for groups of order less than 9ÐKÑ .

If LR contains a nontrivial proper subgroup that is normal in K , then O3RÎR is a complement of LÎR in KÎR for each 3 and so by the inductive hypothesis,

OR OR+R OR+ #"œœ" RRŒ R for some +−K . Hence, + ROœROzz# " and so the inductive hypothesis applied to the group ROKz # implies that + OO## and " are conjugates, whence so are OO and " .

If LK does not contain a nontrivial proper subgroup that is normal in , then we can easily dispatch the case 9ÐLÑ odd with the help of Theorem 8.30, which implies that Lwwwö L – K and so L – K , whence L œ Ö"× , that is, L is abelian. Then the abelian version of the Schur–Zassenhaus Theorem completes the proof. So we may assume that 9ÐLÑ is even, which implies that 9ÐKÎLÑ is odd. Sylow Theory 257

Then Theorem 8.30 implies that KE has a normal subgroup for which LŸE–K and ÐKÀEÑœ: is prime. Hence,

EœLzz ÐO∩EÑ"#and EœL ÐO∩EÑ and the inductive hypothesis implies that + O∩EœÐO∩EÑ#" for some +−E . But

ÐO33 À O ∩ EÑ œ ÐO 3 E À EÑ œ ÐK À EÑ œ : and so O∩E33 is a supplement of any ]−Syl: ÐOÑ3, that is,

O33 œ ÐO ∩ EÑ] 3 Conjugating this for 3œ" by + gives ++ O"" œ ÐO# ∩ EÑ] + and so it is clear that we need to relate ]]" to #" . But O∩EOü " implies that

]ŸOŸRÐO∩EÑ""K" and so ++ ]ŸRÐO∩EÑœRÐO∩EÑ" K" K# +,+ Hence, ]]#K and "" are Sylow : -subgroups of RÐO∩EÑ]œ]#, whence # for some ,−RK# ÐO∩EÑ. Conjugating by , gives ,+ ,+ OœÐO∩EÑ]œÐO∩EÑ]œO""#### as desired.

The Schur–Zassenhaus Theorem leads to the following important corollary.

Corollary 8.32 Let 9ÐKÑœ87 where Ð8ß7Ñœ" . If K has a normal () Hall subgroup R8 of order , then any subgroup LK7 of that has order w dividing 7R is contained in some complement of . Proof. The Schur–Zassenhaus Theorem implies that there is a OŸK for which KœRz O. Then RL∩O œ7w and kk RLœRÐRL∩OÑzz Hence, the Schur–Zassenhaus Theorem implies that there exists +−K for which LœÐRL∩OÑŸO++ But OR+ is also a complement of in K . 258 Fundamentals of Group Theory

*Sylow Subgroups of W8

In this section, we determine the Sylow subgroups of the symmetric group W8 in terms of wreath products. First, we need to compute the order of a Sylow : - subgroup of W8 .

Theorem 8.33 Let :8 be a prime dividing x and let 7 8œ+!" + :â+ 7 : be the base-:8! representation of , that is, Ÿ+5 :/ . The largest exponent of ::±8x for which / is 7 PÐ7Ñ œ + QÐ5Ñ 5 5œ" where :"5 QÐ5Ñœ:55" : â"œ :" and so the order of a Sylow :W -subgroup of 8 is 7 :œ:PÐ7Ñ +5 QÐ5Ñ $ 5œ"

In particular, the order of a Sylow :W -subgroup of :5 is :QÐ5Ñ Proof. The number of factors in 8x œ " † #â8 that are multiples of : is 8Î: gh where BB8 is the floor of . Among these Î: factors, there are 8Î:# factors gh g h g h that are multiples of : . Thus, PÐ7Ñ œ 8Î:  8Î:#7  â  8Î: gh¨© g h Using the base-:8 expansion of , we can write this as

7" 7# PÐ7ÑœÐ+"# + :â+ 7 : ÑÐ+ #$ + :â+ 7 : Ñâ+7 7" œ+"# + Ð:"Ñâ+7 Ð: â"Ñ œ+"# QÐ"Ñ+QÐ#Ñâ+ 7 QÐ7Ñ

Let us first determine the Sylow :W -subgroups of the symmetric groups :8 . Since the order of such a Sylow :: -subgroup is QÐ5Ñ , all we need to do is find a subgroup of W:5 of size

55" :œ:QÐ5Ñ : : â" Since Sylow Theory 259

QÐ5"Ñœ :QÐ5Ñ" it follows that : :œ:†:QÐ5"Ñ QÐ5Ñ ˆ‰ and this puts us in mind of the wreath product, since if H is a finite group and œ œ:H , then kk kk H›› œ H †:: kkkkH Now, the cyclic group G: acts faithfully on itself by left translation and so

[³G":G:ä W: ¸W and since the Sylow :W: -subgroups of :: have order , they are isomorphic to G . Since G: acts faithfully on itself by left translation, the regular wreath product

[œG››G#:<:

# acts faithfully on G‚GœG:: : , that is,

## [W¸W# ä G:: and since

::"QÐ#Ñ 9Ð[Ñœ:# †:œ: œ: it follows that the Sylow :W -subgroups of :# are isomorphic to [# .

# Now, [G#: acts faithfully on : and G acts faithfully on itself and so

[$ œ ÐG : ›› <: G Ñ ›› <: G

$ acts faithfully on G: , that is,

$$ [W¸W$ ä G:: and since

:" : :# :" QÐ$Ñ 9Ð[ÑœÐ:$ Ñ †:œ: œ: it follows that the Sylow :W -subgroups of :$ are isomorphic to [$ .

8" In general, if [G88" acts faithfully on : , then the -fold regular wreath product

[88"<::<:<:<: œ [ ›› G œ ÐÐG ›› G Ñ ›› G Ñâ ›› G

8 acts faithfully on G[W: and so 8:ä 8 . Moreover, since 260 Fundamentals of Group Theory

9Ð[Ñœ [: †:œ:QÐ8"Ñ:" œ: QÐ8Ñ 88"kk it follows that the Sylow :W -subgroups of :88 are isomorphic to [ .

To determine the nature of the :W -Sylow subgroups of 8 , note that if

c œÖFßáßF"7 × is a partition of M833 œ Ö"ß á ß 8× and if 5 is a permutation of F , then the map

55œ‚â‚"7 5 defined by 55Ð5Ñ œ44 Ð5Ñ if 5 − F , is a permutation of M8 . It follows that if Fœ8, then kk55

XœW88"7}} â W is isomorphic to a subgroup of W8 .

We would like to find a partition c of M8 œ Ö"ß á ß 8× whose block sizes are powers of :: . We can do this from the base- representation of 8 : 7 8œ+!" + :â+ 7 : by letting

c œÖF5ß4 ±4œ"ßáß+×5

5 be any partition of M+85 consisting of blocks of size : . Each symmetric group

WW−F8F5ß4 is isomorphic to a subgroup of where we simply let 5 W5ß4 be the identity on MÏF85ß45ß4 . If ] is a Sylow : -subgroup of WF5ß4 , then the direct product

]œ} ]5ß4 5ß4 is isomorphic to a subgroup of W8 of order 7 :+QÐ5Ñ5 $ 5œ" and since ]: has the correct order, it is isomorphic to a Sylow -subgroup of W8 .

Theorem 8.34 Let :8 be a prime dividing x and let 7 8œ+!" + :â+ 7 : be the base-:8! representation of , that is, Ÿ+5 : . Sylow Theory 261

1) The Sylow :W -subgroups of :8 are isomorphic to the 8 -fold regular wreath product

[8 œÐÐG››GÑ››GÑâ››G : <: <: <:

2) The Sylow :W -subgroups of 8 are isomorphic to a direct product

++!" +7 ]œÐ]Ñ":}}} Ð]Ñ â Ð]Ñ :7

where ]:::55 is a Sylow -subgroup of W . Exercises 1. Prove that if L: is a normal -subgroup of a finite group KL , then is contained in every Sylow :K -subgroup of . 2. Let the order of K: be a product ;< of three distinct primes, with :;<. Show that if one of 8;< or 8 is equal to " , then K has a normal subgroup of order ;< . 3. Find all Sylow subgroups of W88$#$ . What are and ? 4. Find all Sylow subgroups of E88%#$ . What are and ? 5. Show that E%& has no subgroup of index . 6. Show that no group of order &' is simple. 7 7. Let 9ÐKÑœ: Ð%:"Ñ for 7 " . Show that 8:: œ" or 8 œ%:" or else :œ# and 8: œ$. 8. Show that there are no simple groups of order :;# , where : and ; are primes. 9. Let 8œ:55 7 with Ð7ß:Ñœ" . Show that if :y ±Ð7"Ñx, then there is no simple group of order 85 . This holds if is “sufficiently large” relative to 9ÐKÑÎ:5. 10. Show that there is no simple group of order )&) . 11. Show that there is no simple group of order $#% . 12. Show that there is no simple group of order $$*$ . 13. Show that there is no simple group of order %!*& . 14. Show that any group of order &'" is abelian. 15. Let 9ÐKÑœ'! . Prove that if 8& " , then K is simple. Hint : Use the fact that groups of order "& , #! and $! have a normal subgroup of order & . 16. Let 5 $ be an odd integer. Show that every Sylow : -subgroup of the dihedral group H#5 is cyclic. 17. How many Sylow #E -subgroups does & contain? 18. Let RKüü and suppose that WR is a normal Sylow :R -subgroup of . Prove that WK is normal in . 19. Let KK be a finite group. Prove that is the group generated by all of its Sylow subgroups. 20. Let KR be a finite group and let ü K . Let W be a Sylow :K -subgroup of not contained in RWRÎR: . Show that is a Sylow -subgroup of KÎR . 21. Let L: be a -subgroup of a finite group K+ÂL: . Let be a -element. Is ØLß +Ù necessarily a : -subgroup of K ? Does it help to assume that LKü ? 262 Fundamentals of Group Theory

22. A subgroup LK of a group is abnormal if +−ØLßL+ Ù

for all +−K . Prove that if Oü K and W−Syl:K ÐOÑ, then R ÐWÑ is abnormal in K: . In particular, the normalizer of a Sylow -subgroup of K is abnormal. 23. Let KL be a finite group and let ßOŸK . a) Suppose that LKüü and OK . Show that if W: is a Sylow - subgroup of KWœÐW∩LÑÐW∩OÑ , then . b) Show that if we drop the condition that both subgroups be normal, then the conclusion of the previous part may fail. c) Assume that LKü . Show that for each prime :±9ÐKÑ , there is some Sylow :W -subgroup for which W∩L: is a Sylow -subgroup of L , W∩O is a Sylow : -subgroup of O and W∩L∩O is a Sylow : - subgroup of L∩O . d) Assume that LKü . Show that for each prime :±9ÐKÑ , there is some Sylow : -subgroup W for which W œ ÐW ∩ LÑÐW ∩ OÑ. 24. (S. Abhyankar) Let K:9 be a finite group and let be a prime dividing ÐKÑ . Let :ÐKÑ be the subgroup of K generated by the union of the Sylow : - subgroups of K:ÐKÑKK . Show that ü . A finite group is a quasi :-group if :ÐKÑ œ K . Prove that the following are equivalent: a) K: is a quasi -group. b) K: is generated by all of its -elements. c) K: has no nontrivial quotient group whose order is relatively prime to . Chapter 9 The Classification Problem for Groups

The Classification Problem for Groups One of the most important outstanding problems of group theory is the problem of classifying all groups up to isomorphism. This is the classification problem for groups. More precisely, isomorphism of groups is an equivalence relation. Therefore, a set of canonical forms or a complete invariant constitutes a theoretical solution to the classification problem for groups. Of course, it may not be a practical solution unless some form of “algorithm” is available for determining the canonical form or invariant of any group.

The classification problem for groups is unsolved and seems to be exceedingly difficult. It is even beyond present day ability to classify all finite groups. All finite abelian groups have been classified. Indeed, we shall see in a later chapter that a finite group is abelian if and only if it is the direct sum of cyclic groups of prime power orders. Moreover, the multiset of prime powers (which need not be distinct) is a complete invariant for isomorphism. All finite simple groups seem to have been classified. We will elaborate on this in more detail below. The Classification Problem for Finite Simple Groups The classification problem for finite simple groups is generally believed by experts in the field to have been solved. The classification theorem is the following:

Up to isomorphism, a finite simple group is one of the following: 1) A cyclic group G: of prime order. 2) An alternating group E8 &8 for . 3) A classical linear group. 4) An exceptional or twisted group of Lie type. 5) A sporadic simple group (these include Mathieu groups, Janko groups, Conway groups, Fischer groups, Monster groups and more).

S. Roman, Fundamentals of Group Theory: An Advanced Approach, 263 DOI 10.1007/978-0-8176-8301-6_9, © Springer Science+Business Media, LLC 2012 264 Fundamentals of Group Theory

The effort to solve this problem spanned the years from roughly 1950 to 1980 and involves something on the order of 15,000 pages of mathematics produced by a variety of researchers, some of which is as yet unpublished. As a result, a second effort, led by three group theorists: Daniel Gorenstein, Richard Lyons, and Ron Solomon has been underway to collect this massive effort into a single source, which now spans five volumes and will, when finished, treat most (but not all) of the overall project.

The shear massiveness of this work has prompted some to believe that it is too soon to say categorically that the classification theorem as it is currently formulated is indeed a theorem. This viewpoint is further supported by the fact that, in the ensuing years since 1980 several gaps, some of which were quite serious, have appeared. Fortunately, all of the known gaps have since been filled.

As an example, Michael Aschbacher [2] writes in his 2004 article The Status of the Classification of Finite Simple Groups as follows:

I have described the Classification as a theorem, and at this time I believe that to be true. Twenty years ago I would also have described the Classification as a theorem. On the other hand, ten years ago, while I often referred to the Classification as a theorem, I knew formally that that was not the case, since experts had by then become aware that a significant part of the proof had not been completely worked out and written down. More precisely, the so- called “quasithin groups” were not dealt with adequately in the original proof. Steve Smith and I worked for seven years, eventually classifying the quasithin groups and closing this gap in the proof of the Classification Theorem. We completed the write-up of our theorem last year; it will be published (probably in 2004) by the AMS.

Let us take a very broad look at the approach taken to solve the classification problem for finite simple groups. The abelian case is easily settled, so we will concentrate our remarks on finite nonabelian simple groups.

We have already mentioned the famous result of Feit–Thompson (whose proof runs about 255 pages itself) that says that every nonabelian finite simple group has even order. This implies that any nonabelian finite simple group K contains an involution ,G . Moreover, the centralizer Ð,Ñ is a nontrivial proper subgroup of KK , since the center of is trivial.

Thus, every nonabelian finite simple group has a nontrivial proper involution centralizer GÐ,Ñ . At least this gives us a starting point for an investigation: Perhaps one can relate the structure of the whole group KGÐ,Ñ to that of . The Classification Property for Groups 265

Indeed, we will show that if K is a nonabelian finite simple group with involution centralizer GÐ,Ñ , then GÐ,Ñ " 9ÐKÑ kk x ¹ Œ# The significance of this result is that GÐ,Ñ , and therefore the right-hand side kk above, does not depend on the structure of KG nor on the nature of Ð,Ñ beyond its size. Thus, for a given even number ; , there are only a finite number of possible orders of groups that have an involution centralizer of size ; . But for each finite order, there are only finitely many isomorphism classes of groups of that order, because there are only finitely many multiplication tables of that size. It follows that there are only finitely many isomorphism classes of nonabelian finite simple groups that have an involution centralizer of size ; .

Thus, the size of an involution centralizer does at least restrict the number of possible isomorphism classes of its parent groups. This raises the question of whether more details about the structure of an involution centralizer (and related substructures) might do more than just restrict the number of isomorphism classes for its parent.

Indeed, in 1954, Richard Brauer proposed that for a finite nonabelian simple group KG with involution centralizer Ð,Ñ , the possible isomorphism classes of KG are determined by the isomorphism class of Ð,Ñ . Moreover, during the period of 1950–1965, Brauer and others developed methods for determining the isomorphism classes of all finite simple groups that have an involution centralizer isomorphic to a given group L . However, involution centralizers alone prove not to be sufficient to solve the classification problem for nonabelian finite simple groups.

Note that an involution centralizer GÐ,Ñ is also the normalizer RK ÐØ,ÙÑ of the # - subgroup Ø,Ù . More generally, the normalizer RK ÐLÑ of a : -subgroup L of a group KK is called a :-local subgroup of . Thus, an involution centralizer is a special type of #K -local subgroup of . The search for nonisomorphic nonabelian finite simple groups involves looking at the entire : -local structure of a group and can be roughly described as follows:

1) If the current list of nonisomorphic nonabelian finite simple groups is not complete, let K be a minimal counterexample. Thus, any proper subgroup of K is on the list. 2) Show that the :K -local structure of resembles that of a simple group W that is already on the list. 3) Use this resemblance to show that KW is isomorphic to . If not, then perhaps K must be added to the list.

So let us proceed to the promised theorem. 266 Fundamentals of Group Theory

Involutions Let K8 be an even-order group whose set \ of involutions has size " and let \\w œ ∪ Ö"× If => is an involution, then for any involution , we have > œ =Ð=>Ñ œ =B where Bœ=> has the property that BœB=" This property of B is very important.

Definition Let K be a group. 1) An element B−K is real by +−K if BœB+" Also, B+ is real if it is real by some element . Let e be the set of real elements of K . 2) An element B−K is strongly real by = if BœB=" where =B is an involution. Also, is strongly real if it is strongly real by some involution =K . Let f denote the set of strongly real elements of .

Thus, every pair of involutions is related by a strongly real element. It is not hard to see that \f , and e are each closed under conjugation and so each set is a union of conjugacy classes. Associated with the equation BœB+" are some important sets.

Definition Let K be a group. 1) For B−K , let GÐBÑœÖ+−K±Bw+ œB" ×

be the set of all elements by which B is real. The extended centralizer of B−K is G‡w ÐBÑ œ GÐBÑ ∪ G ÐBÑ where GÐBÑ is the centralizer of B . The Classification Property for Groups 267

2 a) For B−f , let EÐBÑœÖ=−\ ±B=" œB × be the set of involutions by which B is strongly real. b) For =−\ , let fÐ=Ñ œ ÖB − K ± B=" œ B × be the set of strongly real elements by the involution = .

Let us take a look at these sets. Note first that B−K is real if and only if GÐBÑw is nonempty and B−K is strongly real if and only if EÐBÑ is nonempty. GÐBч If B−e and +ß,−GÐBÑw , then BœB+", œB and so +" , − GÐBÑ. Thus, Gw ÐBÑ is a coset of GÐBÑ , Gw ÐBÑ œ +GÐBÑ and so GÐBÑœw GÐBÑ kkkk Thus, for B−e and +−GÐBÑw ,

w ‡ GÐBÑ if B − \ GÐBÑœGÐBÑ∪+GÐBÑœ w œ GÐBÑ“+GÐBÑif B −e\ Ï where if B−e\ Ï w , then ÐGÐBÑÀGÐBÑÑœ#‡

EÐBÑ If B−f , then EÐBÑ©GÐBÑw and so EÐBÑ Ÿ GÐBÑ kkkk But we can do a bit better in some cases. Since EÐ"Ñ œ \ , we have EÐ"Ñ œ 8 kk and if B −\ , then EÐBÑ © GÐBÑ Ï Ö"× and so EÐBÑ Ÿ GÐBÑ  " kkkk Also, for any 1−K , it is easy to see that EÐB11 Ñ œ EÐBÑ 268 Fundamentals of Group Theory and so EÐB1 Ñ œ EÐBÑ kkkk that is, EÐBÑ is constant on conjugacy classes of K . kk WÐ=Ñ If =−\ , then we have seen that \fw ©= Ð=Ñ Conversely, if B−f\ Ð=Ñ for =− , then Ð=BÑ# œ =B=B œ B" B œ " and so >œ=B−\ w . Hence, \fw œ= Ð=Ñ and so ffÐ=Ñœ=Ð=Ñœ8" kkkk The Fundamental Relation Now we can count the size of the set YœÖÐ=ßBÑ−\f ‚ ±B=" œB × in two ways. From the point of view of an =−\ , Yœf Ð=Ñœ8Ð8"Ñ kk k k =−\ From the point of view of an B−f , Yœ EÐBÑ kk k k B−f Thus,

88œ# EÐBÑ (9.1) kk B−f To split up the sum on the right, we choose a system of distinct representatives (SDR)

Q œÖB!" ßB ßáßB > × for the conjugacy classes of K , where

1) ÖB! œ "× 2) ÖB"? ßáßB × is an SDR for the conjugacy classes in \ The Classification Property for Groups 269

w 3) ÖB?" ßáßB @ × is an SDR for the conjugacy classes in f\Ï

Then since EÐBÑ is constant on conjugacy classes and since kk BK œ ÐK À GÐBÑÑ, equation (9.1) can be written ¸¸ @ 8#  8 œ EÐB Ñ ÐK À GÐB ÑÑ 33 3œ! kk Splitting this sum further gives ?@ 8#  8 œ 8  EÐB Ñ ÐK À GÐB ÑÑ  EÐB Ñ ÐK À GÐB ÑÑ 33 33 3œ" kk 3œ?" kk ?@ Ÿ 8  Ð GÐB Ñ  "ÑÐK À GÐB ÑÑ  GÐB Ñ ÐK À GÐB ÑÑ 3333 3œ" kk 3œ?" kk ? Ÿ 8  ? K  ÐK À GÐB ÑÑ  Ð@?ÑK 3 kk 3œ" kk ? œ8@K  BK ¸¸3 kk 3œ" œ@K kk and so 88Ÿ@K# (9.2) kk To get further estimates, note that ? 8œ\ œ ÐKÀGÐBÑÑ (9.3) 3 kk 3œ" Now, if we assume that the center ^ÐKÑ of K has odd order, then it contains no involutions and so GÐB33 Ñ  K for B − \ . Hence, if 7 is the smallest index among all proper subgroups of K , we have 8œ\ 7? (9.4) kk We can make a similar estimate for @ f œ"8 ÐKÀGÐBÑÑ 3 kk 3œ?"

w and since B−3 f\ Ï , the terms in the final sum satisfy ‡‡ ‡ ÐK À GÐB33333 ÑÑ œ ÐK À G ÐB ÑÑÐG ÐB Ñ À GÐB ÑÑ œ #ÐK À G ÐB ÑÑ

‡ Therefore, if we assume that GÐBÑ3 is also proper in K , then 270 Fundamentals of Group Theory

ÐK À GÐB3 ÑÑ #7 and so f "8#7Ð@?Ñ (9.5) kk ‡ The condition that GÐBÑ3 is proper in K is a bit awkward, but is satisfied if we assume that K# has no subgroups of index .

The inequalities (9.4) and (9.5) together imply that f "8 @Ÿkk #7 and so (9.2) implies that f "8 88Ÿ# kk K #7 kk Some elementary algebra, using the fact that f ŸK , gives kk k k "K K KÎ8" 7Ÿ kk kk" œ kk #8ŒŒ 8 Œ# We have proved a key theorem.

Theorem 9.6 () R. Brauer and K. A. Fowler [4], 1955 Let K be a group of even order with exactly 8 " involutions. Assume that ^ÐKÑ has odd order. Then either K# has a subgroup of index () which must be normal or K has a proper subgroup L with KÎ8" ÐK À LÑ Ÿ kk Œ#

Equation (9.3) implies that for any involution ,B , which we can assume is " , we have 8""? œ KGÐBÑGÐ,Ñ kk3œ" k3 k k k that is, K kkŸGÐ,Ñ 8 kk Hence, if K# fails to have a (normal) subgroup of index , then it has a subgroup L for which The Classification Property for Groups 271

GÐ,Ñ " ÐK À LÑ Ÿ kk Œ# Since LK is a proper subgroup of , it follows that the normal interior L‰ is a proper normal subgroup with ÐK À L‰ Ñ ± ÐK À LÑx. In particular, if K is simple, then 9ÐKÑ ± ÐK À LÑx and so GÐ,Ñ " 9ÐKÑ kk x º Œ# Thus, we arrive at our final goal.

Theorem 9.7 Let K, be a finite nonabelian simple group and let −K be an involution. Then the centralizer GÐ,Ñ is a proper subgroup of K and GÐ,Ñ " 9ÐKÑ kk x º Œ# This is the result that we promised at the beginning of this section and is as far as we propose to take our discussion of the classification problem for nonabelian finite simple groups. Exercises 1. Let K be a group. a) Under what conditions does the set W œ\ ∪ Ö"× of elements of K of exponent #K form a subgroup of ? b) Under what conditions does the set WK form a normal subgroup of ? c) If WK is a subgroup of , what can you say about the strongly real elements of the group? 2. Prove that f is closed under conjugation. 3. Prove that if a finite group KK has a nontrivial real element, then has even order. 4. Find the real elements in the symmetric group W8 . Find the strongly real elements. 5. In the alternating group E8 , show that any permutation that is a product of disjoint cycles of length congruent to "% modulo is real. 6. Find the real elements of the dihedral group H#8 . Find the strongly real elements. 7. Find the real elements of the quaternion group . Find the strongly real elements.

Chapter 10 Finiteness Conditions

There are many forms of finiteness that a group can possess, the most obvious of which is being a finite set. However, as we have observed, chain conditions are also a form of finiteness condition. Another type of finiteness condition on a group KK is the condition that has a finite direct sum decomposition

KœH"8 âH that cannot be further refined by decomposing any of the factors H3 into a direct sum; that is, for which each H3 is indecomposable. In this chapter, we explore these finiteness conditions. First, however, we generalize the notion of a group. Groups with Operators As we have seen, a group K has several important families of subgroups, in particular, the families of all subgroups, all normal subgroups, all characteristic subgroups and all fully-invariant subgroups. Each of these families can be characterized as being the family of all subgroups that are invariant under a certain subset of EndÐKÑ . In particular, a subgroup LK of is

1) normal if and only if it is invariant under InnÐKÑ , 2) characteristic if and only if it is invariant under AutÐKÑ , 3) fully invariant if and only if it is invariant under EndÐKÑ .

We can also say that the subgroups of K are invariant under the empty subset of EndÐKÑ.

This point of view leads us to define the concept of groups with operators, which will include all of these special cases. Intuitively speaking, a group with operators is a group KK with a distinguished family X of endomorphisms of . Rather than associate a subgroup of EndÐKÑ with K directly, we use a function 0ÀHH ÄEnd ÐKÑ from a set into EndÐKÑ . Here is the formal definition.

S. Roman, Fundamentals of Group Theory: An Advanced Approach, 2 37 DOI 10.1007/978-0-8176-8301-6_10, © Springer Science+Business Media, LLC 2012 274 Fundamentals of Group Theory

Definition Let H be a set. An H-group is a pair ÐKß 0ÀH ÄEnd ÐKÑÑ where K0 is a group and ÀH ÄÐEnd KÑ is a function. It is customary to denote the endomorphism 0Ð== Ñ simply by and thus write 0Ð == Ñ+ as + ( some authors write +=) . An HH -group is called a group with operators and is called the operator domain. Let K be an H -group. 1) An H-subgroup LK of is an H -invariant subgroup LK of . We use the notations LŸHH K and LüH K to denote an -subgroup and a normal H - subgroup of K , respectively. We also use the notations HH-subÐKÑ and -norÐKÑ to denote the set of all HH -subgroups of K and the set of all normal - subgroups of K , respectively. 2) If KL and are H -groups, an H-homomorphism from KL to is a homomorphism 5ÀK ÄL that is compatible with the group operators, that is, 5=Ð+ÑœÐ+Ñ =5

for all +−K . A bijective H -homomorphism is an H-isomorphism, and similarly for the other types of homomorphisms. We write 5ÀK ¸H L to denote an HH -isomorphism from KL to . The existence of an -isomorphism from KL to is denoted by K¸LH . 3) If L−H -nor ÐKÑ , then the HH-quotient group ( or -factor group) is the quotient group KÎL with operators =H− defined by ==Ð+LÑ œ Ð +ÑL

for all +−K , that is, defined so that the canonical projection 1HL is an - homomorphism.

Let ÐKß 0ÀHH ÄEnd ÐKÑÑ be an -group. Then a subgroup LŸK is an H - subgroup of K if and only if the restricted operators

H==HlœÖ0ÐÑl±−×LL are operators on LL , that is, if and only if is an HlL -group. We will always think of an HH -subgroup LK of as an lL -group, although we will use notation such as OŸH L in place of OŸHlL L . Thus, H -subgroupness is transitive, that is,

LŸHH Kß OŸ L Ê OŸ H K Note that an H -subgroup LK of may be an operator group in other related ways. To illustrate, if HHœÐKÑLInn , then is an lL -group as well as an operator group under its own family InnÐLÑ of inner automorphisms. But in Finiteness Conditions 275

general, InnÐLÑ is a proper subset of InnÐKÑlL , since conjugation by +−KÏL need not be an inner automorphism of L .

If HH is the empty set, then an -group is nothing more than an ordinary group and it is customary to drop the prefix “HH -”. Also, in the most important cases, is a subset of EndÐKÑ and 0ÀH ÄEnd ÐKÑ is the inclusion map, in which case 0 is suppressed. This applies to the cases HHœInn ÐKÑ , œ Aut ÐKÑ and H œÐKÑEnd .

Example 10.1 Let ZJ be a vector space over a field . Each α −J defines an endomorphism of the abelian group Z by scalar multiplication. Thus, a vector space over JJ is a group with operators . An J -subgroup is a subspace and an J -homomorphism is a linear transformation. The Lattice of HH -Subgroups of an -Group Let K be an H -group. Then the intersection and the join of any family YHœÖL±3−M×3 of -subgroups of K is also an H -subgroup of K . Hence, the meet and join in H-subÐKÑ is the same as the meet and join in subÐKÑ . In other words, H -subÐKÑ is a complete sublattice of subÐKÑ .

We leave it to the reader to show that the H -subgroup Ø\ÙH generated by a nonempty subset \K of is

Ø\ÙH œ Ø==H B ± B − \ß − Ù

An HH -subgroup LK of an -group is finitely H-generated if LœØ\ÙH for some finite set \\ . This generalizes the normal closure of a subset of K . The HH-Isomorphism and -Correspondence Theorems The concept of universality given in Theorem 4.5 and the consequent isomorphism theorems have direct generalizations to groups with operators. Let KO be a HY -group and let ŸHHK . Let ÐKàOÑ be the family of all pairs ÐLß55H À K Ä LÑ, where À K Ä L is an -homomorphism and O ©ker Ð5 Ñ . Then the pair

ÐKÎOß1O À K Ä KÎOÑ is universal in Y5HÐKàOÑ , in the sense that for any pair ÐLß ÀK ÄLÑ in YHHÐKà OÑ, there is a unique mediating -homomorphism 7ÀWÄL for which

71‰œO 5 To see this, note that Theorem 4.5 guarantees the existence of a mediating homomorphism 715H . But if O and are -homomorphisms, then for any =H− and +−K, 7=ÐÐ+OÑÑœÐÐ+ÑOÑœÐ+ÑœÐ+Ñœ 7 = 5= =5 =7 Ð+OÑ and so 7H is also an -homomorphism. Also, H -universality enjoys the same 276 Fundamentals of Group Theory uniqueness up to isomorphism as ordinary universality (HHœg ). The - isomorphism theorems now follow in the same manner as before.

Theorem 10.2 () The H-isomorphism theorems Let K be an H -group. 1)( First H-isomorphism theorem) Every H5-homomorphism ÀK ÄL induces an H55ä -embedding ÀKÎker Ð Ñ L defined by 555Ð1ker Ð ÑÑ œ Ð1Ñ and so K ¸ÐÑim 5 kerÐÑ5 H

2)( Second H-isomorphism theorem) If LßO −Hü -sub ÐKÑ with O K , then L∩O−H-nor ÐLÑ and LO L ¸ OL∩OH 3)( Third H-isomorphism theorem) If LŸOŸK with LßO−H -nor ÐKÑ , then OÎL −H-nor ÐKÎLÑ and KO K ¸ H LL„ O 4)( The H-correspondence theorem) Let R−HH-nor ÐKÑ. If - sub ÐRàKÑ denotes the lattice of all H -subgroups of KR that contain , then the map 1Àsub ÐRà KÑ Ä sub ÐKÎRÑ defined by 1ÐLÑ œ LÎR preserves HH -invariance in both directions and so maps -subÐRà KÑ bijectively onto H -subÐKÎRÑ . HH-Series and -Subnormality We can now generalize the notion of series and subnormality to groups with operators. This will provide considerable time savings in our later work.

Definition Let K be an H -group. 1) An H-series in K is a series

ZüüüÀKKâK!" 8 in which every term is an H -subgroup of K . 2) A refinement of an HZH[ -series is an -series obtained from Z by including zero or more additional H -subgroups between the endpoints. A proper refinement is a refinement that includes at least one new subgroup. Finiteness Conditions 277

We review the usual suspects in the context of H -series: a) If HHœg , an -series is just a series. b)Inn If HHœÐKÑ , an -series is a normal series. c)Aut If HHœÐKÑ, an -series is a characteristic series. d)End If HHœÐKÑ, an -series is a fully invariant series.

Of course, if ÖK3 × is a nonproper H -series for K , we can dedup the series by removing any duplicate subgroups to obtain a proper series.

Definition Let K be an HH -group. Two equal-length -series

ZüüüÀKKâK!" 8"8 ü K and

[üüüÀLLâL!" 8"8 ü L in KK with common endpoints !!œL and K 88œL are H-isomorphic ( also called H-equivalent) if there is a bijection 0 of the index set Ö!ß á ß 8  "× for which

KÎK¸L3" 3 H 0Ð3Ñ" ÎL 0Ð3Ñ As usual, when H œg , we use the term isomorphic . Thus, for example, the series

Ö"×–G::; –ÐG G Ñ and

Ö"×–G;:; –ÐG G Ñ are isomorphic. H-Subnormality

H-subnormality of a subgroup LŸH K requires not just that L be subnormal and HH -invariant, but that the entire series that witnesses subnormality be an - series.

Definition Let KL be an HH -group. An -subgroup of KK is H-subnormal in , denoted by LKüüH , if there is an H -series from LK to . We use the notations

subnHHÐKÑ and subn ÐRà KÑ to denote the family of all HH -subnormal subgroups of K and the family of all - subnormal subgroups of KR that contain , respectively.

We can now generalize Theorem 4.24. 278 Fundamentals of Group Theory

Theorem 10.3 Let KL be an HH -group and let ßO−-subÐKÑ. 1)( Transitivity )

LOüüHHHand OKÊLKüü üü 2)( Intersection ) If P−H -sub ÐKÑ, then

LOÊL∩PO∩PüüHHüü In particular,

LŸOŸKß LüüHH K Ê Lüü O and

LKßOKÊL∩OKüüHH üü üüH 3)( Normal lifting) If R−H -nor ÐKÑ, then

LOÊLRORüüHHüü 4)( Quotient/unquotient) If R−H -nor ÐOÑ and RŸLŸO, then

LOÍLÎROÎRüüHHüü

Composition Series

If KßK!8 ŸH K, then refinement is a partial order on the set of all proper H - series from KK!8 to (assuming that this set is nonempty). Maximal proper H - series are particularly important.

Definition Let KK be an H -group and let !8ßKŸH K. An H-composition series from KK!8 to is a proper H -series

K–K–â–K!" 8 that is maximal in the family of all proper H -series from KK!8 to , under refinement. If there is an H -composition series from KK!8 to , we will write

bÐKàKÑbÐKÑKœÖ"×CompSerHH!8 or CompSer 8 when ! The factor groups of an H -composition series are called H-composition factors. 1)( A maximal series H œg) is simply called a composition series and the factor groups are called composition factors. 2)( A maximal normal series H œÐKÑKInn) in is called a chief series or principal series and the factor groups are called chief factors or principal factors.

When the endpoints of a series are clear from the description of the series, we will drop the “from-to” terminology. To characterize maximal series, we use the following concept. Finiteness Conditions 279

Definition A nontrivial H -group KK is H-simple if has no nontrivial proper normal H-subgroups.

The H -correspondence theorem implies the following.

Theorem 10.4 A proper HH -series is an -composition series if and only if its factor groups are H -simple.

Thus, a series

ZÀK!" –K –â–K 8 in KK is a composition series if and only if its factor groups 5"ÎK 5 are simple and Z is a chief series if and only if each factor group KÎK5" 5 is a minimal normal subgroup of KÎK5 .

It is clear from Theorem 10.4 that any HH-series that is -isomorphic to an H - composition series is also an H -composition series. Also, if we remove an endpoint from an HH -composition series, the result is also an -composition series (with different endpoints, of course). Finally, if

K!5 –â–Kand K 58 –â–K are H -composition series in K , then so is their concatenation

K–â–K!8

The Extension Problem An extension of a pair ÐRßÑ of groups is a group K that has a normal subgroup RRKww isomorphic to and for which ÎR is isomorphic to . The extension problem for the pair ÐRßÑ is the problem of determining (up to isomorphism) all possible extensions of ÐRßÑ . Note that any external semidirect product KœRz ) is an extension of ÐRßÑ . However, semidirect products alone do not solve the extension problem. For example, ™ is an extension of Ð#™™ ß# Ñ but ™ is not a semidirect of any of its nontrivial proper subgroups.

The importance of the extension problem can be clearly seen in the light of composition series. Suppose that we can solve the extension problem and that we can determine (up to isomorphism) all simple groups. The simple groups are precisely the groups that have a composition series of length " :

Ö"× œ K!" – K

Next, for each K" , we solve the extension problem for all pairs of the form ÐK"" ß L Ñ, where L" ranges over the simple groups. The solutions K# are precisely the groups that have composition series of length # : 280 Fundamentals of Group Theory

K–K–K!"# where KÎK¸L#" ". Continuing to extend by all possible simple groups produces all possible groups that have composition series, and this includes all finite groups. Thus, in particular, if we can solve the extension problem and if we can determine all finite simple groups, we can determine all finite groups. Unfortunately, a practical solution to the extension problem does not exist at this time. The Zassenhaus Lemma and the Schreier Refinement Theorem Let KK be an HH -group. Our next goal is to show that any two -series in with the same endpoints have refinements that are H -isomorphic. This result is called the Schreier refinement theorem and has two extremely important consequences, as we will see.

First, let us recall that the projection ÐEüü FÑ Ä ÐL OÑ of EFüü into LO is the extension LÐE∩ OÑü LÐF ∩ OÑ Moreover, when EFL , , and O are H -subgroups, then so are LÐE∩OÑ and LÐF ∩ OÑ and the Zassenhaus lemma (Theorem 4.12) generalizes directly to H-subgroups.

Theorem 10.5 ()Zassenhaus lemma [37], 1934 Let K be an H -group and let EFüüand LO where Eß Fß Lß O −H-sub ÐKÑ . Then the reverse projections ÐEFÑÄÐLOÑüüand ÐLOÑÄÐEFÑ üü have H -isomorphic factor groups, that is, LÐF ∩ OÑ EÐO ∩ FÑ ¸ LÐE∩ OÑH EÐL ∩ FÑ Now we can prove the Schreier refinement theorem. Let LŸO be H - subgroups of an HH -group KL and consider a pair of -series from to O :

ZüüüÀLœKKâKœO!" 8 and

[üüüÀLœLLâLœO!" 7 Projecting each of the 78 steps of [Z into each of the steps of creates a new H-series with 78 steps, some of which may be trivial. In view of the preceeding Finiteness Conditions 281 remarks, the new HZ -series is a refinement of . Similarly, projecting each of the 8K steps of into each of the 7 steps of [H creates a new -series with 78 steps. Moreover, since the two sets of projections consist of inverse pairs, the Zassenhaus lemma implies that the resulting series are H -isomorphic.

Theorem 10.6 ()Schreier refinement theorem, Schreier [29], 1928 Let K be an HH -group. Then any two -series in K with the same endpoints have H - isomorphic refinements. Consequences of the Schreier Refinement Theorem The Schreier refinement theorem has two very important consequences. First, suppose that there is an H -composition series in KLO from to . Then any proper H-series

[ÀL–L!" –â–L 8 with HH -subnormal endpoints between LO and can be refined to an - composition series. To see this, note that since LL!8 and are H -subnormal, the series [H can be expanded to an -series ^ from LO to and the Schreier refinement theorem implies that ^ and H have -isomorphic refinements to proper series ^ww and , respectively. But Hwœ is an -composition series and therefore so is ^[w , which contains a refinement of .

The second consequence of the Schreier refinement theorem is that any two H - composition series with the same endpoints are H -isomorphic.

Theorem 10.7 Let KL be an HH -group and let ŸOK be -subgroups of . 1) Suppose that there is an H -composition series from LO to of length 8 . If LŸL!8 are HH -subnormal subgroups of KLO between and , then any - series from LL!8 to can be refined to an H -composition series and so has length at most 8 . 2)( The Jordan–Hölder Theorem) Every two H -composition series from L to O are H -isomorphic. In particular, they have the same length.

The Jordan–Hölder Theorem allows us to make the following definition.

Definition Let KL be an HH -group and let and OK be -subgroups of . If bCompSerH ÐLàOÑ, then the H-composition distance .ÐLßOÑ from L to O is the length of any H -composition series from LO to . The distance .ÐOÑ œ .ÐÖ"×ß OÑ is called the H-composition length of O . For chief series, the terms chief distance and chief length are also employed.

Of course, the composition distance is positive definite (when it is defined), that is, if bÐLOÑ.ÐLßOÑ !CompSerH à , then and .ÐLßOÑœ! Í LœO 282 Fundamentals of Group Theory

The Existence of H -Composition Series Any finite group has an H -composition series. For abelian groups, composition series and chief series coincide and an abelian group has a composition series if and only if it is finite, since each factor group must have prime order. Thus, not all groups have composition or chief series.

The existence of an HH -composition series between -subnormal subgroups L and OK of an HH -group can be characterized in terms of chain conditions on - subnormal subgroups. If bÐLàOÑCompSerH , then any H -series from LO to has length at most .ÐLß OÑ . Hence, Theorem 1.5 gives the following.

Theorem 10.8 Let KL be an H -group and let ßO−subnHÐKÑ. Then

bCompSerHH ÐLàOÑ Ísubn ÐLàOÑ has BCC Proof of the following is left to the reader.

Theorem 10.9 Let K be an H -group. 1) (Subgroup ) If O− subnH ÐKÑ and LŸO , then

bÐLKÑÊbÐLàOÑbÐOàKÑCompSerHHHà CompSer and CompSer and .ÐLß KÑ œ .ÐLß OÑ  .ÐOß KÑ

2) (Quotients ) If R−H -nor ÐKÑ, then

bÐRKÑÍbÐKÎRÑCompSerHHà CompSer and .ÐRß KÑ œ .ÐKÎRÑ

3) (Extensions ) If R−H -nor ÐKÑ, then

bÐRÑßbÐKÎRÑÊbÐKÑCompSerHH CompSer CompSer H 4)( Direct products) If KL and are H -groups, then

bÐKÑbÐLÑÍbÐKLÑCompSerHH and CompSer CompSer H} and .ÐK{ LÑ œ .ÐKÑ  .ÐLÑ

The Remak Decomposition Let us recall Theorem 5.12. Finiteness Conditions 283

Theorem 10.10 ()Remak If a group K has either ( and therefore both ) chain condition on direct summands, then K has a Remak decomposition

Kœ â"8 that is, each 5 is indecomposible.

The question of uniqueness of a Remak decomposition is rather more complicated than the question of existence. Recall that if

KœL"8 âL then the 5À th projection map 3L55 KÄL is defined by

3LL55Ð+Ñ œ Ò+Ó

where Ò+ÓLL5 is the 5 th coordinate of + . Moreover, 3 3 is idempotent and normal as an endomorphism of K . Note also that the sum 33â is projection LL33" 5 onto LâL33"54 and so is an endomorphism of K . The Krull–Remak–Schmidt Theorem Suppose now that

KœL"8 âL (10.11)

with projection maps 11LL"8ßáß and

KœO"7 âO (10.12)

with projection maps ,,OO"7ßáß , where the factors LO5 and 5 are indecomposable and 8Ÿ7 .

In searching for possible isomorphisms between the LO -factors and the - factors, we recall Theorem 5.25 as it applies to the restricted projection maps

1,LO34lÀOÄL 4 3 and OL43 lÀLÄO 3 4

Namely, if ÐlÑÐlÑ−ÐLÑ1,LO34 OL 43 Aut3 and im ÐlÑO,üOL43 4, then 1 LO34 l and

,OL43l are isomorphisms.

Note however that imÐlÑO,üOL43 4, since if 2−L5−O3 and 4 , then

" " " 5Ò,,,,,,OL43 l Ð2ÑÓ5 œOOO4 Ð5Ñ 4 Ð2Ñ 4 Ð5Ñ œO4 Ð525 Ñ −im ÐOL43 l Ñ Thus, we have the following.

Theorem 10.13 Let

KœL"8"7 âL œO âO where the factors LO55 and are indecomposable. If the composition 284 Fundamentals of Group Theory

α13ß4œÐ L34 l O щР, O 43 l L ÑÀL 3 ÄL 3 of the restricted projection maps is an automorphism of L3 , then the maps

1,LO34lÀO¸L 4 3 and OL43 lÀL¸O 3 4 are isomorphisms.

In attempting to show that a composition is an automorphism, we are reminded of Fitting's lemma. We have assumed that L3 is indecomposable. Also, since the restriction and composition of normal maps is normal, α3ß4 is normal and so has normal higher images. Thus, if we assume that K has BCC on normal subgroups, then Fitting's lemma implies that α3ß4 is either nilpotent or an automorphism for all 3ß 4 .

To see that for each 3 , not all of the maps α3ß4 can be nilpotent, note that for 4Á5,

α3ß4œÐ α 3ß5 1, LO34 1, LO 35 ÑlœÐÑl L 3 1 L 3 , O 4 , O 5 L 3 which is an endomorphism of LÐ33 and so the images imααß4Ñ and imÐ3ß5Ñ commute elementwise. Also,

1,LO3"â 1,LO37 œ 1 L 3 Ð , O " â ,OL7 Ñœ 1 3 and so

αα1+3ß" â3ß7 œ L33 l L œ L 3 which is not nilpotent. Hence, Theorem 4.3 implies that α3ß4 is not nilpotent for some 43 . It follows from Fitting's lemma that for each , there is a 4 for which α3ß4 −ÐLÑAut3 and so

1,LO34lÀO¸L 4 3 and OL43 lÀL¸O 3 4 Now, we can make a significant improvement to this by noticing that if OœL45 for some 5Á3 , then

α13ß4œÐ L34 l O ÑÐ , O 43 l L ќР1 L 35 l L ÑÐ , L 53 l L Ñœ! and so if we delete from the sum all terms indexed by a for which 4α3ß4 4 O4" −ÖL ßáßL 8 ×ÏÖL× 3, the sum remains unchanged and so is not nilpotent. Hence,

1,LO34lÀO¸L 4 3 and OL43 lÀL¸O 3 4 for some 4 for which O4" ÂÖLßáßL 8 ×ÏÖL× 3.

Now suppose that after possible reindexing of the O5 's, there is a " for which Finiteness Conditions 285

KœO"5 âO"58 L âL (10.14) with projections .."8ßáß , where

-,3OL33³lÀL¸O33 for all "Ÿ3Ÿ5". We may also assume that L54 ÁO for all 4 5 or else we can replace LO54 by (reindexed to OÑ5 . This certainly holds for 5œ" . Then there is a 4 5 and we may assume that 4œ5 , for which

1-LO55lÀO¸L 5 5and 5 ³, OL55 lÀL¸O 5 5

Moreover, since 1L55 maps OL isomorphically onto 5 and maps the complement

LœLâLLâLÐ5Ñ "5"5"8 to Ö"× , it follows that

LÐ5Ñ ∩ O5 œ Ö"×

Thus, if we replace LO55 by in (10.14), the result is a direct sum

K"" œO âO 55"8 L âL and our goal is to show that KœK" . To this end, if

..Ð5Ñ œââ"5 ."5 ."8 . then ..5KœÐ5Ñ + and the map

),..œO55 Ð5Ñ is a normal endomorphism of KÐÑ©OL with im ))5 Ð5Ñ. To show that is surjective and so KœO5 LÐ5Ñ, it is sufficient to show that ) is injective.

Now, any +−K has the form +œB2 for B−LÐ5Ñ and 2−L5 . But for any B−LÐ5Ñ,

),..ÐBÑ œO55 ÐBÑÐ5Ñ ÐBÑ œ B and any 2−L5 ,

),..,Ð2Ñ œO555 Ð2ÑÐ5Ñ Ð2Ñ œO Ð2Ñ and so

),ÐB2Ñ œ BO5 Ð2Ñ and since LÐ5Ñ ∩ O5 œ Ö"×, it follows that )ÐB2Ñ implies B œ " and

,,OO55Ð2Ñ œ ". But lL5 À L5 ¸ O5 and so 2 œ "Þ Thus, ) is injective. 286 Fundamentals of Group Theory

It follows that

KœO"55 âO L"8 âL

for all "Ÿ5Ÿ8 and -,5OL55 œ55 l ÀL ¸O and so this holds for all 5œ"ßáß8. In particular, 8œ7 .

Note also that since ,O33 ÐL Ñ œ O 3, the map

--1-1,1,1œ"L"8 â8L œ O" L"8 â OL8 is a surjective normal endomorphism of K−ÐKÑ and so - Aut . Moreover,

-ÐL55 Ñ œ O We can now summarize.

Theorem 10.15 () The Krull–Remak–Schmidt Theorem Let K be a group that has BCC on normal subgroups. Suppose that

KœL"8"7 âL œO âO where all factors LO34 and are indecomposable. Then 8œ7 and there is a reindexing of the OK3 's and a normal automorphism - of for which

-ÀL33 ¸O for all 3œ"ßáß8 and for each "Ÿ5Ÿ8 ,

KœO"55 âO L"8 âL

True Uniqueness The Krull–Remak–Schmidt Theorem gives uniqueness of the terms of a Remak decomposition up to isomorphism. Let us now consider the question of when a group K has an essentially unique Remak decomposition, that is, a Remak decomposition that is unique up to the order of the factors. First suppose that

KœL"8"8 âL œO âO are Remak decompositions of K8" (where ). Then the Krull–Remak– Schmidt Theorem implies that there is a normal automorphism -ÀK ÄK for which -LœO55 (after reindexing). Hence, if L5 is invariant under every normal automorphism of KOœLŸLOœL5 , then 555- and so 55 for all and K has an essentially unique Remak decomposition.

By way of converse, suppose that K has an essentially unique Remak decomposition

KœL"8 âL with projections Ößáß×11"8, but that there is a normal endomorphism - of K Finiteness Conditions 287

for which -1LŸ""yL . Then there is a 5" for which 5-Á! on L " , that is, there is an B−L"55 for which "Á1- ÐBÑ−L .

The set

OœÖ2†"51- Ð2ѱ2−L× " is easily seen to be a normal subgroup of KOÁL and "" , since

B†1-5"" ÐBÑ−O ÏL But it is easy to see that

KœO"# L âL 8

Moreover, if O" is decomposable, then there would be a Remak decomposition of K8 consisting of more than terms, which is false. Hence, this is a Remak decomposition of K that is distinct from the previous decomposition. We have proved the following.

Theorem 10.16 Let K have BCC on normal subgroups and let

KœL"8 âL be a Remak decomposition of K . The following are equivalent: 1) This Remak decomposition of K is essentially unique . 2) LK5 is invariant under all normal endomorphisms of . 3) LK5 is invariant under all normal automorphisms of .

If αÀL3 Ä^ÐKÑ is a nonzero homomorphism, then we can build a normal endomorphism -ÀK ÄK by specifying that α if 5œ3 -lœL 5 œ !5Á3if

The map - is normal since for any +−K , 23355 −L , 2 −L where 5Á3 , ++++ ---Ð2Ñœ35 233 œÐ 2Ñand - Ð2Ñœ"œÐ - 25 Ñ

Thus, if L3 is not - -invariant, then Theorem 10.16 implies that the Remak decomposition of K is not unique.

Conversely, suppose that the Remak decomposition of K is not unique and so there is a normal endomorphism -1 of Kl for which 4L-3 Á!4 for some Á3 . Then for 2−L33 and 2−L 44 ,

24 24 ÒÐ2ÑÓœ---33 Ð2ÑœÐ2Ñ3 which shows that -1lÀLÄ^ÐKÑL333. Hence, 4L3- lÀLÄ^ÐLÑ 4 is a nonzero homomorphism. 288 Fundamentals of Group Theory

Theorem 10.17 Let K have BCC on normal subgroups and let

KœL"8 âL be a Remak decomposition of K . 1) The following are equivalent: a) This Remak decomposition of K is essentially unique. b) Every homomorphism ααÀL33 Ä^ÐKÑ satisfies ÀL Ä^ÐLÑ3. c) There are no nonzero homomorphisms -ÀL34 Ä^ÐLÑ for 3Á4 . 2) If KK is either perfect or centerless, then has an essentially unique Remak decomposition. ww Proof. For part 2), if KœK , then L33 œL3 for all 3 and so if - ÀL Ä^ÐLÑ4 for 4Á3 , then for +ß,−L3 , ---ÐÒ+ß ,ÓÑ œ Ò +ß ,Ó œ "

and so -lœ!KL3 . If is centerless, the result is immediate. Exercises 1. Give an example of an infinite group with a composition series. 2. Find isomorphic refinements of the two series Ö!×–:™™ – and Ö!× – ;™™ – where :; and are distinct primes. 3. Prove that if KL and are groups and if

ZÀÖ"לK!" –K –â–K 8 œK and

[ÀÖ"לL!" –L –â–L 7 œL are composition series for KL and , respectively, then the series

ÐÖ"×{{ Ö"×Ñ – ÐK" Ö"×Ñ – â – ÐK88"87{{ Ö"×Ñ – ÐK L Ñ – â – ÐK { L Ñ is a composition series for KL{ . 4. How many composition series does a cyclic group of order :8 have? 5. Prove that the multiset of composition factors of a group is an invariant under isomorphism, but not a complete invariant. 6. Prove the uniqueness part of the fundamental theorem of arithmetic using the Jordan–Hölder Theorem. 8 7. Let KœG: be the direct product of 8 cyclic groups of prime order : . How many compositions series does KK have? Hint : is a vector space. Finiteness Conditions 289

8. Prove that a subgroup of a group with a composition series need not have a composition series as follows. Let M œ Ö"ß #ß á × and let K œ WÐMÑ be the restricted symmetric group on M . (Recall from an earlier exercise that this is the set of all permutations of M8 that have finite support.) For each " , let

KœÖ−K±BœB8 55 for B8× and let

LœÖ−K±l8855Ö"ßáß8× is even × a) Show that Kœ K . -8" 8 b) Show that has index in and so . Lœ-8" L8 # K Lü K c) Show that L is simple. d) Show that LE contains an infinite abelian subgroup . e) Show that EL has no composition series but that does. 9. Let c be an isomorphic-invariant property of finite groups. Let K be a group that has cüc and for which if RK , then KÎR has . Prove that the following are equivalent: a) K has a normal series

Ö"× œ K!"üüü K â K 8 œ K whose factor groups have c . b) KK has the property that any nontrivial quotient group of has a nontrivial normal subgroup that has c . Such a group K is said to be hyper-c . 10. Prove the following facts: a) For any +ß , − K ,

Ò+,ÓLLL333 œ Ò+Ó Ò,Ó

and so the projection map 355ÀK ÄL is a homomorphism (and an endomorphism of K ). b) For any +−K ,

+ œ Ò+ÓLL"= âÒ+Ó

c) The projection map commutes with any inner automorphism #1 of K and therefore preserves normality. 11. Let KL have BCC on normal subgroups and suppose that is a group for which K}} K¸L L. Show that K¸L .

Chapter 11 Solvable and Nilpotent Groups

Classes of Groups By a class ^ of groups, we mean a subclass of the class of all groups with the following two properties:

1) ^ contains a trivial group 2) ^ is closed under isomorphism, that is, K−^^and L¸K Ê L− For example, the abelian groups form a class of groups. A group of class ^ is called a ^^-group and ^ -group LK that is a subgroup of a group is called a - subgroup of K . A class ^ is a trivial class if it contains only one-element groups. Closure Properties We will be interested in the following closure properties for a class ^ of groups:

1)( Subgroup ) K−^^ ß LŸK Ê L− 2) ()Intersection and Cointersection For LßO Ÿ K, LßO −^^ Ê L ∩ O − KK K ß−^^ Ê − LO L∩O 3)( Quotient and Extension) For RKü , K−^^ Ê KÎR− RßKÎR −^^ Ê K −

S. Roman, Fundamentals of Group Theory: An Advanced Approach, 291 DOI 10.1007/978-0-8176-8301-6_11, © Springer Science+Business Media, LLC 2012 292 Fundamentals of Group Theory

4)( Seminormal Join, Normal Join and Cojoin) For LßO Ÿ K, LßO−^^ ßone normal in K Ê LO− LßO−^^ ßboth normal in K Ê LO− KK K ß−^^ Ê − LO LO 5) (Direct product) LßO−^}^ Ê L O− These properties are not independent.

Theorem 11.1 The following implications hold for a class ^ of groups: 1) subgroupÊ intersection 2) quotientÊ cojoin 3) seminormal joinÊÊ normal join direct product 4) subgroup and direct productÊ cointersection Thus, a class that is closed under subgroup, quotient, seminormal join, extension is closed under all nine properties above. Proof. Part 1) is clear. For part 2), we have KKLO ¸−^ LO L‚ L For part 3), the direct product LO{{ is the seminormal join of LÖ"× and Ö"×{^ O, each of which is in if Lß O − ^ . For part 4), if KÎLß KÎO − ^, then KKK ä}− ^ L∩O L O via the map 5À 1ÐL ∩ OÑ È Ð1Lß 1OÑ.

The following definition will prove very convenient.

Definition Let ^ be a class of groups. 1) A ^^-series is a series whose factor groups belong to the class . 2) A ^^=8-group is a group that has a -series and a ^-group is a group that has a normal ^ -series. 3) The ^^=8-class is the class of all ^= -groups and the -class is the class of all ^8-groups.

Our main interest is in the ^^=8 and classes in which ^ is either the class of cyclic groups or the class of abelian groups. However, we are also interested in a class of groups that is not a ^^=8 or class, namely, the nilpotent groups. Solvable and Nilpotent Groups 293

Definition 1)) a A cyclic series is a series that has cyclic factor groups. b) An abelian series is a series that has abelian factor groups. c) A central series for a group K is a normal series

Ö"× œ K!"üüü K â K 8 œ K for which

KÎKŸ^ÐKÎKÑ5" 5 5 for all 5 . 2) As shown in the table below, we have the following definitions.

Series Normal Series Abelian Solvableœ Solvable Cyclic Polycyclic Supersolvable Central Nilpotent

a) A group is solvable if it has an abelian series. b) A group is polycyclic if it has a cyclic series. c) A group is supersolvable if it has a normal cyclic series. d) A group K is nilpotent if it has a central series.

Note that in previous chapters (see Theorem 7.10), we have found it convenient to use the term central series even when the series does not start at the trivial group, which requires that we distinguish carefully between series in K and series for K.

As the title of this chapter suggests, our primary interest is in nilpotent and solvable groups. It is clear that nilpotent groups are solvable. Also, all abelian groups are nilpotent and Theorem 7.10 shows that all finite : -groups are nilpotent: finite :-group or ÊÊnilpotent solvable abelian

Note, however, that W$ is solvable but not nilpotent. Also, the symmetric groups W88 are solvable if and only if & . In fact, if 8 & , then W8 has only one nontrivial proper normal subgroup E8 , which is not abelian. Operations on Series In order to study the closure properties of series-based classes of groups and of the nilpotent class, we must consider various operations on series. Indeed, the operations of intersection, normal lifting, quotient and unquotient as defined in 294 Fundamentals of Group Theory

Theorem 4.10 can be applied to each step in a series. Specifically, we have the following operations on series. Let

ZüüÀK!8 â K be a series in K .

1) The intersection of Z with LŸK is

Züü∩L ÀK!8 ∩L â K ∩L œL 2) The normal lifting of Zü by RK is

ZüüRÀKR!8 â KR 3) The quotient of Zü by RK is ZRKR KR Àâ!8üü RR R 4) The unquotient of the series KK ZüüÀâ!8 RR

where RŸK! is

ZüüÅRÀK!8 â K

5) For LŸK3 , the concatenation of the series

[üüÀL!7 â L and

^üüÀL77 â L 8 is the series

[^‡ÀLâL!77 üü üü âL8 6) For the series

[üüÀL!8 â L and

ZüüÀK!7 â K in Kµ , the interleaved series [Z is formed as follows. First, if =œmax Ö8ß7× , then we extend whichever series is shorter by repeating the upper endpoint (LK87 or ) an appropriate number of times to make the Solvable and Nilpotent Groups 295

resulting series of equal length = . Then

[Z{ü{ü{üµ ÀÐL!! KÑ ÐL "! KÑ ÐL "" KÑ ÐL#"{ü{üü{ KÑ ÐL ## KÑ â ÐL == KÑ Note that the intersection, normal lifting, quotient, unquotient and interleave of normal series is normal. However, the concatenation of two normal series need not be normal.

These operations are used as follows.

Theorem 11.2 Let KL be a group and let ŸK and Rü K . Let a) Z be a series for K b) [ be a series for L c) a be a series for R d) d be a series for KÎR . Then 1)( Subgroup ) Z ∩L is a series for L 2)( Seminormal join) a[‡R is a series for LR 3)( Quotient ) ZRÎR is a series for KÎR 4)( Extension ) ad‡Ð ÅRÑ is a series for K 5)( Direct product) If ZZ33 is a series for K3œ"ß#µ for , then "Z# is a series for KK"#{ . Closure Properties of Groups Defined by Series Theorem 4.10 and the previous theorems provide the following facts about closure of ^^=8 -classes and -classes.

Theorem 11.3 Let ^ be a class of groups. 1)( Subgroup ) If ^^ is closed under subgroup, then the =8 and ^ classes are closed under subgroup. 2)( Seminormal join) If ^^ is closed under quotient, then the = -class is closed under seminormal join. 3)( Quotient ) If ^^ is closed under quotient, then the =8 and ^ classes are closed under quotient. 4)( Extension ) The ^= -class is closed under extension. 5)( Direct product) The ^^=8 and classes are closed under direct product.

In particular, if ^^ is closed under subgroup and quotient, then the = -class is closed under the following eight operations: 6) subgroup 7) quotient 8) intersection 9) cointersection 10) cojoin 11) direct product 296 Fundamentals of Group Theory

12) seminormal join 13) extension and the ^8 -class is closed under all of these operations except seminormal join and extension. Proof. For 1), if ^^ is closed under subgroup, then (normal) -series are closed under intersection and so the ^^=8 and classes are closed under subgroup. For 2) and 3), if ^^ is closed under quotient, then -series are closed under normal lifting. Hence, a[‡R and Z^RÎR are ^-series. It follows that the =-class is closed under seminormal join and the ^^=8 and classes are closed under quotient. For 5), if Z^33 is a (normal) -series for Kµ , then ZZ"# is a (normal) ^{-series for KK"# .

Thus, since the classes of cyclic groups and abelian groups are closed under subgroup and quotient, we have the following.

Theorem 11.4 The polycyclic, solvable and supersolvable classes are closed under the following eight operations () except where noted : 1) subgroup 2) quotient 3) intersection 4) cointersection 5) cojoin 6) direct product 7)( seminormal join except for supersolvable ) 8)( extension except for supersolvable ) .

Let us now turn to the closure properties of nilpotent groups.

Theorem 11.5 1) Central series are closed under intersection, normal lifting, quotient and unquotient. 2) The class of nilpotent groups has the following seven closure properties: a) subgroup b) quotient c)( normal join this is Fitting's theorem, to be proved a bit later) d) direct product e) intersection f) cointersection g) cojoin but not extension. Proof. Part 1) follows from Theorem 4.10 and Theorem 4.11. The statement about extensions in part 3) can be verified by looking at W$ . Solvable and Nilpotent Groups 297

Nilpotent Groups We now undertake a closer look at nilpotent groups. We will prove that a finite group K: is nilpotent if and only if it is the direct product of -groups and so Theorem 8.11 shows that finite nilpotent groups have many other interesting characterizations. Higher Centers An extension LOKü in is central in K if and only if OK Ÿ^ LLŒ and so the largest subgroup OK of for which LOü is central in K is the subgroup O for which OK œ^ LLŒ

With this in mind, we define a function ''œÐKÑK on nor by KÐRÑ' ^œK ŒRR for RKü . Note also that ' ÐRÑ K K K œ^ « RRRŒ and so 'KÐRÑ « K .

Writing ''55ÐÖ"×Ñ as Ð"Ñ , the proper series 'ö'ö'ö!"#Ð"Ñ Ð"Ñ Ð"Ñ â is called the upper central series for KÐ and each '5 "Ñ is called a higher center of K^ . The first higher center is the center ÐKÑ of K .

To see that the upper central series ascends more rapidly than any other central series of the form

Ö"לKKKâ!"#üüü we have 'ÐK55" Ñ K for all 5 and it is clear that 5 KŸ5 ' Ð"Ñ holds for 5œ! . If this holds for a particular value of 5 , then the monotonicity of ' implies that 298 Fundamentals of Group Theory

55" K5" Ÿ'''' ÐKÑŸ 5 Ð Ð"ÑÑœ Ð"Ñ

5 Hence, KŸ5 ' Ð"Ñ for all 5 .

Theorem 11.6 Let K be a nilpotent group. The upper central series 'ö'ö'ö!"#Ð"Ñ Ð"Ñ Ð"Ñ â for K is characteristic and ascends more rapidly than any other central series for K , that is, if

Ö"לKKKâ!"#üüü is central in K , then

5 KŸ5 ' Ð"Ñ for all 5 ! . 1) KK is nilpotent if and only if the upper central series reaches . 2) If KK is nilpotent, then all central series for have length greater than or equal to the length of the upper central series for K .

The higher centers have some important applications.

Theorem 11.7 Let KÐ be nilpotent, with higher centers '5 "Ñ . 1) If LŸK , then L'ü'55 Ð"Ñ L" Ð"Ñ 2) If RKü , then R ∩''55 Ð"Ñ œ Ö"× Ê R ∩" Ð"Ñ Ÿ ^ÐKÑ As a consequence, 3) K has the property that every subgroup is subnormal 4) K has the center-intersection property 5) Every chief series for K is central. Proof. For part 1), since ÒKß''5" Ð"ÑÓ Ÿ 5 Ð"Ñ, Theorem 3.41 implies that ÒL''55"5 Ð"Ñß L Ð"ÑÓ œ ÒL ' Ð"Ñß LÓÒL '' 55"L Ð"Ñß Ð"ÑÓ

But each factor on the right is contained in L''ü'555 Ð"Ñ and so L Ð"Ñ L" Ð"Ñ. Part 3) follows from part 1), since we may lift the upper central series for K by LL to get a series from to KLK , whence is subnormal in .

For part 2), since ÒÐ"ÑßKÓŸÐ"ÑÒRßKÓŸR''5" 5 and , it follows that ÒR ∩''5" Ð"Ñß KÓ Ÿ R ∩5 Ð"Ñ œ Ö"× and so R∩'5" Ð"ÑŸ^ÐKÑ. For part 4), there is a largest 5 for which Solvable and Nilpotent Groups 299

R∩''55 Ð"ÑœÖ"× and so R∩" Ð"Ñ is a nontrivial subgroup of ^ÐKÑ . For part 5), the factor group KÎK5" 5 of a chief series Z for K is a minimal normal subgroup of the nilpotent group KÎK5 and so the center-intersection property implies that KÎK5" 5 is central. Thus, Z is central.

We can now augment Theorem 8.11 by adding the nilpotent condition.

Theorem 11.8 The following are equivalent for a finite group K : 1) K is nilpotent. 2) Every Sylow subgroup of K is normal. 3) K: is the direct product of its Sylow -subgroups

Kœ ]: :−c

4) If LŸK , then

Lœ ÐL∩]Ñ: :−c

5)( Strong converse of Lagrange's theorem) If 8±9ÐKÑ , then K has a normal subgroup of order 8 . 6) K: is the direct product of -subgroups. 7) Every subgroup of K is subnormal. 8) K has the normalizer condition. 9) Every maximal subgroup of K is normal. 10) KÎF ÐKÑ is abelian. Proof. Theorem 8.11 states that 2)–10) are equivalent. Moreover, Theorem 7.10 implies that a finite : -group is nilpotent and therefore so is any direct product of finite : -groups. Hence, 6) implies 1). Theorem 11.7 shows that 1) implies 7). Lower Centers In terms of commutators, an extension LOKü in is central in K if and only if ÒOß KÓ Ÿ L and so LœÒOßKÓ is the smallest subgroup OKLO of for which ü is central in KÒOßKÓ«O ; in fact, . Thus, if we define the “commutator with K ” function >>œÐKÑK on sub by

>KÐOÑ œ ÒOß KÓ then >>KKÐOÑ is the smallest subgroup of K for which ÐOÑ Ÿ O is central in K. The (possibly infinite) descending central series âÐKÑÐKÑÐKÑœKö>#"! ö> ö> is called the lower central series for KÐ and each >5 KÑ is called a lower center of KÐ . The first lower center is the commutator subgroup > KÑœKw . 300 Fundamentals of Group Theory

An induction argument shows that the lower central series descends more rapidly than any other central series, in the sense that if

âKKKœKüüü#"! is central in K , then

5 > ÐKÑ Ÿ K5

5 for all 5 ! . For if >> ÐKÑŸK5, then the monotonicity of implies that 5" 5 >>>>ÐKÑœ Ð KÑŸ ÐKÑŸK55" Theorem 11.9 Let K be a group. The lower central series âÐKÑÐKÑÐKÑœKö>#"! ö> ö> for KK is characteristic and descends more rapidly than any central series for , that is, if

âKKKœKüüü#"! is central in K , then

5 > ÐKÑ Ÿ K5 for all 5 ! . 1) K is nilpotent if and only if the lower central series reaches Ö"× . 2) If KK is nilpotent, then all central series for have length greater than or equal to the length of the lower central series for K .

The commutator function >K has some simple but useful properties that are consequences of Theorems 3.40 and 3.41 .

Theorem 11.10 Let KL be a group and let ßßOPü K. 1) >üL ÐOÑ K 2) >>LOÐOÑ œ ÐLÑ 3) >>>>>>LLLLÐOPÑ œ ÐOÑ ÐPÑ and OLO ÐPÑ œ ÐPÑ ÐPÑ 4) >L is deflationary, that is,

>L ÐOÑ Ÿ L In fact,

>L ÐOÑ Ÿ L ∩ O Hence, for all 5 " ,

55 >>L ÐLÑ ŸK ÐKÑ Solvable and Nilpotent Groups 301

5) If RKü and RŸL∩O, then for any 5 " , L >5 ÐLÑR >5 œ O OÎRŒRR 6) For 5 ",

>5 ÐLOÑ œ >>â ÐFÑ LO $ EE"5 EßáßEßF−ÖLßO×"5

Proof. Part 5) holds for 5œ" since

L L O ÒLßOÓR>O ÐLÑR >OÎR œß œ œ ŒRRRR ” • R and if it holds for a particular value of 5 , then LLO >>5" œß5 OÎRŒRRR ”OÎRŒ • >5 ÐLÑR O œßO ”•RR ÒÐLÑßOÓR>5 œ O R >5"ÐLÑR œ O R and so this holds for all 5 " .

For part 6), we have

> ÐLOÑ œ>>>> ÐLÑ ÐLÑ ÐOÑ ÐOÑ œ >ÐFÑ LO LOLO $ E EßF−ÖLßO× and an easy induction proves the general formula. Nilpotency Class Theorem 11.9 and Theorem 11.6 imply that for a nilpotent group, the upper and lower central series have the same length.

Definition Let K be a nilpotent group. The common length of the upper and lower central series is called the nilpotency class of K , which we denote by nilclassÐKÑ.

Moreover, if K is nilpotent and

Z ÀÖ"לK!7 –â–K œK is a central series for K7 of length , then 302 Fundamentals of Group Theory

75 5 >'KKÐKÑ Ÿ K5 Ÿ Ð"Ñ for all 5 œ !ß á ß 7, where >'33 ÐKÑ œ Ö"× and Ð"Ñ œ K if 3 nilclass ÐKÑ.

The nilpotent groups of class ! are the trivial groups and the nilpotent groups of class " have ^ÐKÑ œ K or equivalently, Kw œ Ö"× and so are the nontrivial abelian groups. A group K# has nilpotency class if and only if either of the following conditions holds:

1) Ö"×  Kww  K and ÒK ß KÓ œ Ö"×, or equivalently, Ö"×Kw Ÿ^ÐKÑK 2) Ö"×  ^ÐKÑ  K and KK ^œ Œ^ÐKÑ ^ÐKÑ or equivalently, KK is not abelian but Î^ÐKÑ is abelian.

Theorem 3.42 implies that KŸ^ÐKÑw if and only if ÒÒ+ß ,Óß -Ó œ Ò+ß Ò,ß -ÓÓ for all +ß ,ß - − K . Hence, for nonabelian groups, this condition is equivalent to being nilpotent of class # .

Theorem 11.11 Let K be nilpotent. 1) If LŸK , then nilclassÐLÑ Ÿ nilclass ÐKÑ 2) If RKü , then nilclassÐKÎRÑ Ÿ nilclass ÐKÑ 3)( Fitting's theorem) The join of two normal nilpotent subgroups of a group KL is nilpotent. In fact, if ßOü K , then nilclassÐLOÑ Ÿ nilclass ÐLÑ  nilclass ÐOÑ Proof. The first two parts follow from Theorem 11.10. For part 3), Theorem 11.10 implies that

>5 ÐLOÑ œ >>â ÐFÑ LO $ EE"5 EßáßEßF−ÖLßO×"5 for all 5 " . Now, suppose that nilclassÐLÑ œ - and nilclassÐOÑ œ . and let 5œ-.. Then each factor on the right above has the form Solvable and Nilpotent Groups 303

Pœ>>EE"-. â ÐFÑ

Among the subgroups EßáßE"-. ßF, suppose there are 2L 's and 5O 's, where 25œ-.". Then either 2 -" or 5 ." and we may assume without loss of generality that 2 -" . Since >O is deflationary, removing all >O 's from the expression for P results in a possibly larger subgroup

2 >L ÐOÑif F œ O Qœ 2" œ >L ÐLÑif F œ L However, in the former case,

22" 2" 2" >>>>>>LLLLÐOÑœLO ÐOÑœ ÐLÑŸ ÐLÑ and so

2" - P Ÿ Q Ÿ>>LL ÐLÑ Ÿ ÐLÑ œ Ö"× Hence,

-. >LOÐLOÑ œ Ö"× which implies that LO is nilpotent of class at most -  . . An Example We now describe a family of groups showing that for any - ! , there are nilpotent groups of nilpotency class -ÐÑ . Let `8 be the family of all 8‚8 matrices over a commutative ring Y with identity and let œYXÐ8ßÑ be the unitriangular matrices over . (Recall that a matrix is unitriangular if it is upper triangular, with "Ð 's on the main diagonal.) Denote the 3ß4Ñ th entry in Q by Q5 !53ß4 . For , the th superdiagonal QÐ5ÑQ of a matrix are the elements of the form Q3ß35 .

For !Ÿ5Ÿ8", let R5 be the set of all 8‚8 matrices over ! with 's on or below the 5E th superdiagonal, that is, for −`8ÐÑ,

E−R53 Í Eß4 œ! for all 4Ÿ35 It is routine to confirm that

RR57 ©R 57" In particular,

RR55 ©R 5 For 5 !, let

Y55 œ ÖM×  R œ ÖM  E ± E − R 5 × 304 Fundamentals of Group Theory

To see that YY5 is a subgroup of , we have

Y55 Y œ ÐÖM×  R 5 ÑÐÖM×  R 5 Ñ © ÖM×  R 55555  R  R R © Y

? and if E−R5 , then since E œ! for some ? ! , it follows that " # ?" ÐM  EÑ œ M  E  E  â „ E − Y5

As to commutators, if E−R57 and F−R , then ÒM  Eß M  FÓ œ ÐM  EÑÐM  FÑÐÐM  FÑÐM  EÑÑ" œÐMEFEFÑÐMFEFEÑ" œÐM\ÑÐM]Ñ" œMÐ\]ÑÐM]Ñ" where \œEFEFand ] œFEFE Moreover,

\] œEFFE−R57" implies that ÒMEßMFÓ−Y57" and so

ÒY57 ß Y Ó Ÿ Y 57" Taking 7œ! gives

ÒY55 ß Y Ó Ÿ Y"5 Ÿ Y which shows both that YYY–5 is normal in and that 5"YY 5 is central in . Thus, the series

ÖM× œ Y8" – â – Y"! – Y œ Y is central and so Y8 is nilpotent of class at most " .

On the other hand, let I!3ß4 denote the matrix with all s except for a " in the Ð3ß 4Ñthe position. Then for 3 Á 4  " , an easy calculation shows that

ÒMI3ß4 ßMI 4ß4" ÓœMI3ß4" In particular, if 8 $ , then

ÒMI"ß# ßMI #ß$ ÓœMI"ß$ and so >ÐKÑ œ ÒKß KÓ Á ÖM×. Also, if 8 % , then

ÒMI"ß$ ßMI $ß% ÓœMI"ß% and so >>#ÐKÑ œ Ò ÐKÑß KÓ Á ÖM×. More generally, Solvable and Nilpotent Groups 305

ÒMI"ß8" ßMI 8"ß8 ÓœMI"ß8 ÁM and so >8#ÐKÑ Á ÖM×, which shows that Y is nilpotent of class 8  " . Solvability We now turn to a discussion of solvable groups. Perspective on Solvability Solvable groups have played an extremely important role in the study of the location of the roots of polynomials over a field J . Let us pause to describe this role in general terms. For more details, we refer to reader to Roman, Field Theory [27].

If JIJ is a subfield of a field , we say that ŸI is a field extension. Associated to each field extension JŸI is a group KJ ÐIÑ , called the Galois group of the extension and defined as the group of all (ring) automorphisms 5 of IJ+ that fix the elements of , that is, for which 5 œ+ for all +−J . It turns out that the properties of the “simpler” Galois group can often shed considerable light on properties of the “more complex” field extension.

To illustrate, one of the principal motivations for the development of abstract algebra since, oh say 3000 B.C., has been the desire (expressed in one form or another) to find the roots of a polynomial :ÐBÑ with coefficients from a given base field J .

In fact, we now know that for any nonconstant polynomial :ÐBÑ of degree . , there is an extension JJ of , called an algebraic closure of J , that contains a full set of .:ÐBÑ roots for . Moreover, lying between the fields JJ and is the smallest field I containing these roots of :ÐBÑ , called a splitting field for :ÐBÑ . The desire to express the roots of :ÐBÑ by arithmetic formula (similar to the quadratic formula) or to show that this could not be done is what motivated Galois to first define some version of what we now know as a group.

The idea of expressing the roots of a polynomial :ÐBÑ by formula means that, starting with the elements of the base field J , we can “capture” all of the roots of :ÐBÑ through a finite number of special types of extensions of J . In particular, for each extension, we are allowed to include an 8 th root of an existing element and only whatever else is required in order to make a field.

Specifically, for the first extension, we may choose any ?−J! and any root <œ8! ? of the polynomial B8! ? where 8 " . Then the first extension is !!È !! JŸJÐ8! ?Ñ È ! where JÐ8! ? Ñ is the smallest subfield of J containing J and < . Repeated È !! 306 Fundamentals of Group Theory extensions produce a tower of fields of the form

J Ÿ JÐ88!! ? Ñ Ÿ JÐ ? ß8"5 ? Ñ Ÿ â Ÿ JÐ8! ? ßáß8 ? Ñ ÈÈ!!"!5È ÈÈ where each ?3 is an element of the immediately preceeding field. This type of tower of fields is called a radical series. If all of the roots of a polynomial :ÐBÑ lie within a radical series over J , then we say that the polynomial equation :ÐBÑ œ ! is solvable by radicals. (For simplicity, we assume that J has characteristic !.)

It is not hard to show that a polynomial equation :ÐBÑ œ ! is solvable by radicals if and only if the roots of :ÐBÑ can be captured within a finite tower of fields

J Ÿ JÐααααα""#"7 Ñ Ÿ JÐ ß Ñ Ÿ â Ÿ JÐ ßáß Ñ where each field has prime degree over the previous field, that is, the dimension of each field as a vector space over the previous field is a prime number.

Now let I: be the splittting field for ÐBÑ in J . If the radical series above does capture the roots of :ÐBÑ , that is, if I Ÿ JÐαα"7 ß á ß Ñ, then taking Galois groups (which reverses inclusion) gives the descending sequence

ZÀKJ ÐIÑ KJÐαα"" Ñ ÐIÑ â KJÐ ßáßα7 Ñ ÐIÑ But

KÐIÑKÐIÑœÖ×JÐαα"7 ßáß Ñ I + and so the sequence Z reaches the trivial group. Galois showed that (in modern terminology) Z is an abelian series and so KÐIÑJ is solvable. Thus, Galois proved that if :ÐBÑ œ ! is solvable by radicals, then its Galois group KJ ÐIÑ is solvable. He also proved the converse.

Now, an element 5 − KJ ÐIÑ of the Galois group fixes the coefficients of :ÐBÑ , since they lie in J< . Hence, 5 must send a root of :ÐBÑ in I to another root of :ÐBÑ in I , that is, the elements of KJ ÐIÑ are permutations of the set of roots of :ÐBÑ in I . Moreover, since I is generated by these roots, each 5 − KJ ÐIÑ is uniquely determined by its behavior on these roots. Thus, one often thinks of KÐIÑJ simply as a subgroup of the permutation group W8 , where 8 œdeg Ð:ÐBÑÑ .

In fact, there are many cases in which KÐIÑJ8 , thought of as a subgroup of W , is actually W:8 itself. For example, it can be shown for any prime , the Galois group of the splitting field of any irreducible polynomial 0ÐBÑ − ÒBÓ of degree :W with exactly two nonreal roots is : . Solvable and Nilpotent Groups 307

However, W888 is not solvable for & , since E is simple and so the only nontrivial series for W888 is Ö+ ×–E –W , which is not abelian. Thus, the roots of the polynomials described above cannot be captured within a radical series, that is, these polynomials are not solvable by radicals.

Note that this shows that there are individual polynomials that are not solvable by radicals. Thus, not only is there no general formula, similar to the quadratic, cubic and quartic formulas, for the solutions of arbitrary quintic equations, but there are even individual quintic equations whose solutions are not obtainable by formula!

Galois used his remarkable theory in his paper Memoir on the Conditions for Solvability of Equations by Radicals of 1831 (but not published until 1846!), to show that the general equation of degree & or larger is not solvable by radicals. (Proofs that the & th degree equation is not solvable by radicals were offered earlier: An incomplete proof by Ruffini in 1799 and a complete proof by Abel in 1826.)

Thus, the notion of solvability arose through the desire to settle the question of whether we could solve all polynomial equations by simple formula. Of course, solvable groups are important for other reasons. In fact, we will see that the class of solvable groups has a sort of super-Sylow theorem, to wit, if K is solvable of order 78 where Ð7ß 8Ñ œ " , then K has a () Hall subgroup of order 77 and all subgroups of order are conjugate. The Derived Series For any solvable group, there is an abelian series that descends more rapidly than any other abelian series. Moreover, this series is also characteristic in K . A series

Ö"× œ K!"üüü K â K 7 œ K is abelian if and only if w KŸK5" 5 for all 5œ!ßáß7", where the prime denotes commutator subgroup.

Definition Let K be a group. The subgroups defined by KœKKœKÐ!Ñ , Ð"Ñ w and, in general for 8 " , KœÐKÑÐ8Ñ Ð8"Ñ w are called the higher commutators of KK . The group Ð8Ñ is called the 8 th commutator subgroup of K . The series of higher commutators: 308 Fundamentals of Group Theory

âKKKKööööÐ$Ñ Ð#Ñ Ð"Ñ is called the derived series for K .

The monotonicity of the commutator operation implies that the derived series descends from K more rapidly than any other abelian series. Moreover, since K«KÐ5"Ñ Ð5Ñ, the derived series is characteristic.

Theorem 11.12 Let K be a group. 1) The derived series

âKKKKööööÐ$Ñ Ð#Ñ Ð"Ñ is the abelian series of steepest descent, in the sense that if the series

âKKKKœK$#"!üüü is abelian, then

Ð5Ñ KŸK5 for all 5 . 2) K is solvable if and only if its derived series reaches the trivial group, that is, if and only if there is a 5 " for which KÐ5Ñ œ Ö"× . The smallest integer 8 for which KÐ8Ñ œ Ö"× is called the derived length of K , which we denote by derlenÐKÑ. 3) A group K is solvable if and only if it has a normal abelian series. 4) The length of any abelian series for K is greater than or equal to the derived length of K .

We will have use for the following fact about higher commutators of quotient groups.

Theorem 11.13 Let KR be a group and let ü K . Then

KKRÐ8Ñ Ð8Ñ œ ŒRR for all 8 ! . Proof. For any RŸLü K, we have Lw L L ÒLßLÓR Lw R œß œ œ ŒRRRR ” • R In particular, for LœK , we have KKKÒKßKÓRKRw w œß œ œ ŒRRRRR ” • Solvable and Nilpotent Groups 309 and so the result holds for 8œ" . Assuming that the result holds for an arbitrary 5L, we have with œKÐ8ÑR , ww KKKRÐÐ8"Ñ Ð8Ñ Ð8Ñ KRÐ8ÑÑ wR œœœ ŒRR–— Œ RR Finally, Theorem 3.41 implies that

ÐKRÑRœÒKRßKRÓRœÒKßKÓRœKÐ8Ñ w Ð8Ñ Ð8Ñ Ð8Ñ Ð8Ñ Ð8"Ñ R and so the result follows.

Theorem 11.14 If K is solvable, EßFßL ŸK and RßOü K , then 1) derlenÐLÑ Ÿ derlen ÐKÑ 2) derlenÐKÎRÑ Ÿ derlen ÐKÑ 3) derlenÐKÑ Ÿ derlen ÐRÑ  derlen ÐKÎRÑ 4) derlenÐE  FÑ Ÿmax Öderlen ÐEÑß derlen ÐFÑ×. Proof. For part 1), since LŸKÐ5Ñ Ð5Ñ for all 5 !8œ , if derlen ÐKÑ then LÐ8Ñ Ÿ K Ð8Ñ œ Ö"×. Thus, derlenÐLÑ Ÿ 8. For part 2), Theorem 11.13 implies that if KÐ8Ñ œ Ö"× , then ÐKÎRÑÐ8Ñ œ ÖR×. For part 3), if KÎR has derived length .KRÎRœÖR×KŸR , then Ð.Ñ and so Ð.Ñ . If the derived length of R is /, then KÐ./Ñ Ÿ R Ð/Ñ œ Ö"× and so the derived length of K./ is at most . Part 4) follows from the fact that ÐE  FÑwww œ E  F . Properties of Solvable Groups If RK is a minimal normal subgroup of a group and if

ZüüÀÖ"לK.! â K œK is any normal series in K , then R Ÿ K33 or else R ∩ K œ Ö"× for all 3 and so there is an index 5 for which

R Ÿ K55and R ∩ K" œ Ö"× Therefore, if KK is solvable and if Z is the derived series for , then R Ÿ KÐ5Ñ and R ∩ KÐ5"Ñ œ Ö"× and so RwÐ Ÿ R ∩ K5"Ñ œ Ö"×, whence R is abelian.

Theorem 11.15 Let KR be solvable. Any minimal normal subgroup of K is abelian. Moreover, if RR contains a nontrivial element of finite order, then is elementary abelian. 310 Fundamentals of Group Theory

Proof. For the final statement, R: has an element of prime order and since : R³ÖB−R±Bœ"×R: ü it follows that RœR: is an elementary abelian group.

If K is solvable and has a composition series, then the factor groups of the composition series are both simple and solvable and therefore cyclic of prime order.

Theorem 11.16 The following are equivalent for a group K that has a composition series. 1) K is solvable. 2) Every composition series for K has prime order factor groups. 3) K has a cyclic series in which each factor group has prime order. 4) KK has a cyclic series, that is, is polycyclic. 5) Every chief series for K has factor groups that are elementary abelian. Proof. We have seen that 1) implies 2) and it is clear that 2) implies 3), that 3) implies 4) and that 4) implies 1). Thus, 1)–4) are equivalent.

It is clear that 5) implies 1). If KK is solvable, the factor groups 5"ÎK 5 of a chief series are minimal normal in the solvable group KÎK5 and so are elementary abelian by Theorem 11.15.

The following theorem contains some sufficient (but not necessary) conditions for solvability. The proof of the Feit–Thompson Theorem is quite involved, running almost 300 pages. For a proof of the Burnside result, we refer the reader to Robinson [26].

Theorem 11.17 1)( Feit– Thompson Theorem) Any group of odd order is solvable; equivalently, every finite nonabelian simple group has even order. 2)( Burnside :; Theorem) Every group of order :;78 where : and ; are primes, is solvable. Proof. The equivalence in part 1) is left as an exercise. Hall's Theorem on Solvable Groups Let KL be a finite group. Recall that a Hall subgroup of K is a subgroup with the property that its order 9ÐLÑ and index ÐK À LÑ are relatively prime. The Schur–Zassenhaus Theorem tells us that every normal Hall subgroup has a complement and that all such complements are conjugate.

As to the existence of Hall subgroups, the Sylow theorems tell us that if 9ÐKÑ œ :55 7 with : prime and Ð: ß 7Ñ œ " , then K has a Hall (Sylow) subgroup of order :5 and that all such subgroups are conjugate. In 1928, Philip Solvable and Nilpotent Groups 311

Hall showed that for a finite solvable group, this result applies not just to prime power factors.

Theorem 11.18 ()Hall's theorem, 1928 Let K be a finite solvable group with 9ÐKÑœ+,, where Ð+ß,Ñœ" . Then K has a Hall subgroup of order + and all subgroups of order + are conjugate. Proof. We may assume that +ß ,  " . The proof is by induction on 9ÐKÑ . If 9ÐKÑ œ ", the result holds trivially. Assume that it holds for all groups of order less than 9ÐKÑ . If K does not have a minimal normal subgroup, then K is simple and solvable and therefore cyclic of prime order and so +ß ,  " is false. Thus, KR has a minimal normal subgroup , which as we have seen, is an elementary abelian group of prime power order :7 . There are cases to consider, based on whether :±+:±,77 or . Case 1: 9ÐRÑœ:7 œ, In this case, RK is a normal Hall subgroup of . Hence, the Schur–Zassenhaus Theorem implies that R has a complement and all such complements are conjugate. But the complements of R+ are precisely the subgroups of order . Case 2: 9ÐRÑ œ :77 ± , and :  , If :77 ±, but : Á, , then 9ÐKÎRÑœ+Ð,Î:7 Ñ+, and so the inductive hypothesis implies that KÎR has a subgroup OÎR of order + . Hence, 9ÐOÑ œ +:7  +, and the inductive hypothesis applied to O shows that O (and hence KL+ ) has a subgroup of order .

As to conjugation, if 9ÐLÑœ9ÐLÑœ+"#, then L 3 ∩RœÖ"× and so 9ÐL3 RÎRÑ œ +. Hence, the inductive hypothesis implies that LRBR LR #"œ ŒRR for some B−K and so B LRœLR# " B But 9ÐL"" RÑ œ 9ÐKÑ and L and L# are Hall subgroups of L" R of order + . B Hence, the induction hypothesis implies that LL"" and # are conjugate in LR , whence in K . Case 3: 9ÐRÑ œ :7 ± + If 9ÐRÑ œ :77 ± +, then 9ÐKÎRÑ œ Ð+Î: Ñ,  +, and the inductive hypothesis implies that KÎR has a subgroup OÎR of order +Î:7 , whence 9ÐOÑ œ + .

As to conjugacy, if 9ÐLÑ œ + , then R Ÿ L , for if not, then 9ÐLRÑ ± 9ÐLÑ9ÐRÑ œ +:7 and so RL is a subgroup of K of order greater than + but relatively prime to , , 312 Fundamentals of Group Theory

which contradicts Lagrange's theorem. Therefore, if 9ÐL"# Ñ œ 9ÐL Ñ œ +, then 7 L"# ÎR and L ÎR are Hall subgroups of KÎR of order +Î: and so L" ÎR and LÎR#" are conjugate in KÎR , whence L and L# are conjugate in K .

A sort of converse of the previous theorem also holds. The proof uses the Burnside :; theorem (Theorem 11.16).

Definition Let K:8 be a finite group and let be a prime for which œ:57 , where Ð7ß :Ñ œ " and 5 " . Then a Hall :w-subgroup of K is a subgroup L of order 7 .

Theorem 11.19 If a finite group K: has a Hall w -subgroup for every prime : dividing 9ÐKÑ , then K is solvable. Proof. Assume that the theorem is false and let K be a counterexample of smallest order. If KR is not simple, then let be a nontrivial proper normal subgroup of KR . We leave it as an exercise to show that ∩L is a Hall :w - subgroup of R and LRÎR is a Hall :w -subgroup of KÎR . Hence, R and KÎR are solvable and therefore so is KK , a contradiction. Hence, is simple.

/" /8 Now suppose that 9ÐKÑ œ :" â:8 , where the :3 's are distinct primes and w / "3 . The Burnside :; theorem implies that 5 $ . If K3 is a Hall :3 -subgroup /3 of K9ÐKÑœ9ÐKÑÎ: , then 3 3 and the Poincaré theorem and the fact that the indices ÐK À K3 Ñ are pairwise relatively prime imply that for any 5 of the groups K3,

/34 ÐKÀK33 ∩â∩K Ñœ : "5$4 34 and so 9ÐKÑ K∩â∩Kœ33"5/ kk: 34 #434 Also, for any 3 , 9ÐKÑ 9ÐKÑ Kì K œK K œ œ9ÐKÑ ¹¹¹¹3434,,4Á3 kk 4Á3 :/3 /4 3 #4Á3: 4 and so

Kì KœK 34,4Á3

//"# If LœK$8 ∩â∩K, then 9ÐLÑœ:"# : , which is solvable by the Burnside :; theorem. If L is simple, then L is abelian and so L is cyclic of prime order, which is false. Hence, let RL be a minimal normal subgroup of . Then R is elementary abelian of exponent, say ::"" and so is contained in any Sylow - /" subgroup of LK∩L: . But ## has order " and so RK∩Lü . Now, Solvable and Nilpotent Groups 313

RŸRK∩L" and so

KKÐK∩LÑK#" # RŸR ŸRŸK# and so the normal closure RKK is a proper nontrivial normal subgroup of , a contradiction. Exercises 1. Show that the classes of cyclic groups, abelian groups and nilpotent groups do not have the extension property. 2. Show that if E–F is abelian or cyclic, then any refinement ELFüü is also abelian or cyclic. Nilpotent Groups 3. Can a nontrivial centerless group be nilpotent? 4. a) Prove that any finite nilpotent group is supersolvable. b) Find an example to show that not every finite supersolvable group is nilpotent. 5. Let KK be a finite group. Prove that is nilpotent if and only if every nontrivial quotient group of K has a nontrivial center. 6. Let KE be nilpotent but not abelian. Let ŸK be maximal with respect to being normal in KE and abelian. Prove that œGKÐEÑ . Hint : Show that GÎEü KÎE and that GK ÐEÑÎE ∩ ^ÐKÎEÑ Á Ö"×. 7. For any even positive integer 88 , prove that every group of order is nilpotent if and only if 8# is a power of . 8. Prove that a nilpotent group is supersolvable if and only if it satisfies the ascending chain condition on subgroups. 9. If L- is nilpotent of class and O.L is nilpotent of class , prove that } O is nilpotent of class maxÐ-ß .Ñ . Solvable Groups

10. Prove that W% is solvable but not supersolvable. 11. Prove that W88 is solvable for Ÿ% . 12. Assuming that E& is the smallest nonabelian simple group (which it is), prove that every group of order less than '! is solvable. 13. Let RK be a nontrivial proper normal subgroup of a group . Let : be a prime and :±9ÐRÑ . Let L be a Hall :w -subgroup of K . Prove that R∩L and RLÎR are Hall :w -subgroups of R and KÎR , respectively. 14. Prove that the following are equivalent for a finite group K : a) K is solvable. b) Every nontrivial normal subgroup of K has a nontrivial abelian quotient group. c) Every nontrivial quotient group of K has a nontrivial abelian normal subgroup. 314 Fundamentals of Group Theory

/" /7 15. Let K8 be a finite group of order œ:" â:7 . Prove that K is solvable if and only if the composition length of is . K-œ/ 3 16. Prove that a solvable group with a composition series must be finite. 17. Let K be a nontrivial finite solvable group. a) Prove that b:ÐKÑ is nontrivial for some prime :K , that is, has a nontrivial normal : -subgroup. b) Prove that b;ÐKÑ is nontrivial for some prime ;K , that is, has a normal subgroup LKÎL; for which is a nontrivial -group. 18. Let K be a finite group. Prove directly that an abelian series can always be refined into a cyclic series with prime order factor groups. 19. Prove that the following are equivalent: a) Any finite group of odd order is solvable. b) Any finite nonabelian simple group has even order. 20. a) Prove that a finite group KW is solvable if and only if w ÁW for all subgroups W Á Ö"× of K. b) Prove that if KW contains a nonabelian simple subgroup , then K is not solvable. w c) Show that WÁW for all subgroups of the dihedral group H#8 , showing that HR#8 is solvable. Hint : Find an abelian subgroup of index #R . How do subgroups interact with ? 21. A subgroup LK of a group is abnormal if +−ØLßL+ Ù for all +−K . Prove that the normalizer of a Hall subgroup of a solvable group is abnormal. Polycyclic Groups 22. a) Let K8 be a polycyclic group with a cyclic series of length . Prove that K8 is -generated. b) Let E8 be an -generated abelian group for 8 "E . Prove that is polycyclic. 23. Prove that the following are equivalent: a) K is polycyclic. b) Every subgroup of K is finitely generated and solvable. c) Every normal subgroup of K is finitely generated and solvable. 24. Prove that a group K is polycyclic if and only if it is solvable and satisfies the maximal condition on subgroups, that is, if and only if every nonempty collection of subgroups of K as a maximal member. 25. a) Let EF have an infinite cyclic factor group. Let

EœLLâLœF!" 7 be a proper refinement of EF . Describe the factor groups of this refinement. Solvable and Nilpotent Groups 315

b) Let K be polycyclic. Prove that the number of steps whose factor group is infinite is the same for all cyclic series for K . Hint : Any two cyclic series have isomorphic refinements. Supersolvable Groups 26. Prove that any supersolvable group is countable. 27. Prove that if KÎR is supersolvable and R is cyclic, then K is supersolvable. 28. Prove that a group is supersolvable if and only if it has a series in which each factor group is cyclic of prime order or cyclic of infinite order. Hint : Recall that E«F and Füü K implies E K . 29. Let KLK be supersolvable. Prove that if is a maximal subgroup of , then ÐK À LÑ is prime. Hint : First assume that L–K and look at KÎL . Then assume that LK is not normal in and factor by the normal interior L‰ . Conclude that it is sufficient to consider L‰ œ Ö"× . Consider the subgroup E in the first step Ö"×  E in a normal cyclic series and how it interacts with L. 30. Prove that if KK is supersolvable, then w is nilpotent. Hint : Let

Ö"לK!8 âK œK be a normal cyclic series for K and consider the series www Ö"לK!8 ∩K âK ∩K œK

which is normal and cyclic as well. Let FœK5" and EœK5 . To show that the series is central, it is sufficient to show that \EÐF∩KÑEKww ³Ÿ^ EEŒ E But \ÎE is cyclic and normal in KÎE . What can be said about ÐKÎEÑw ? Radicals and Residues Definition Let ^ be a class of groups. Let K be a group. 1) If the partially ordered set ^ü^ÐKÑ œ ÖL K ± L − ×

has a top element, it is called the ^-radical for K and is denoted by b^ÐKÑ. 2) If the partially ordered set KÎ^ü œ ÖL K ± KÎL − ^ ×

has a bottom element, it is called the ^-residue for K and is denoted by b^ÐKÑ. 316 Fundamentals of Group Theory

^ 31. Show that the ^b -radical ^ÐKÑ and the ^ -residue bÐKÑ are characteristic subgroups of K () if they exist . 32. Let ^ be a class of groups closed under subgroup, quotient and join if at least one factor is normal. Let KL be a group and let ü K . Assume that all mentioned radicals and residues exist. Prove the following: a) If LŸb^ ÐKÑ , then

b^ÐKÑ K Ÿ b^ LLŒ and the inclusion may be proper. b) If LŸb^ ÐKÑ , then KÐKÑb^ b^ œ ŒLL 33. Let ^^ be a class with the extension property: R− ßKÎR−O implies K−^. Prove that the following hold for any group K : a) The ^b -radical of KÎ^ ÐKÑ is trivial, that is, K bb^^œ Ö ÐKÑ× Œb^ÐKÑ b) The ^bb -residue of ^^ÐKÑ is ÐKÑ , that is, bb^^ÐÐKÑÑœÐKÑ b ^ 34. Let ^ be the class of finite groups. a) Show that there are groups with no ^ -radical. b) Show that there are groups with no ^ -residue. 35. Let K be a nontrivial finite group. Prove that the following are equivalent: a) K is solvable. b) For every proper normal subgroup O–K , the factor group KÎO has a nontrivial :ÐKÎOÑ:KÎO -radical b: for some prime , that is, has a nontrivial normal : -subgroup. c) For every nontrivial characteristic subgroup OK of , the ; -residue b;ÐOÑ of O is proper in O for some prime ; , that is, there is a proper normal subgroup EO of such that OÎE is a nontrivial a ; -group. Additional Problems

36. Let Kœ be a group. Let >>K . Ð5Ñ 5 a) Prove that KŸ>K ÐKÑ . b) Prove that derlenÐKÑ Ÿ nilclass ÐKÑ. 37. Let Kœ be a group. Let >>K . Prove the following: a) ÒÐKÑßÐKÑÓŸ>>54 > 54" ÐKÑ Hint : Use induction. Use the three subgroups lemma on Ò>>5" ÐKÑß 4" ÐKÑÓ œ ÒÒ >5 ÐKÑß KÓß >4" ÐKÑÓ. 554" b) >>>4ÐKÑÐKÑ Ÿ ÐKÑ for 5ß 4 " . Solvable and Nilpotent Groups 317 c) Ò>'54 ÐKÑß ÐKÑÓ Ÿ ' 45" ÐKÑ for 4 5  " . Ð5Ñ j5 5 d) KŸ> ÐKÑ, where jœ#"5 . Hence, derlenÐKÑ is less than or equal to the smallest integer 5j -³ÐKÑ for which 5 nilclass and so derlenÐKÑ Ÿlog Ð-  "Ñ ij#

Chapter 12 Free Groups and Presentations

Throughout this chapter, \\ denotes a nonempty set of formal symbols and " denotes the set of formal symbols ÖB" ± B − \×. Further, we assume that \ and \\" are disjoint and write w œ\“\". Free Groups

The idea of a free group J\\\ on a nonempty set is that J should be the “most general” possible group containing \\ , that is, the elements of should generate JJ\\ but have no relationships within . In this case, \ is referred to as a set of free generators or a basis for the free group J\ .

To draw an analogy, if ZZ is a vector space, then a subset of is a basis for Z if and only if for any vector space [[ and any assignment of vectors in to the vectors in , there is a unique linear transformation from Z[ to that extends this assignment. This property and the analogous property that defines free groups are best described using univerality.

Definition Let \Ð be a nonempty set. A pair Jß,À\ÄJÑ where J is a group has the universal mapping property for \\ () or is universal for if, as pictured in Figure 12.1, for any function 0À\ Ä K from \ to a group K , there is a unique group homomorphism 70 ÀJ ÄK for which

7,0 ‰œ0

The map 70 is called the mediating morphism for 00 . In this case, we say that can be factored through , or that 0JJ can be lifted to . The group is called a free group on \\ and is called a set of free generators or a basis for J . The map ,, is called the universal map for the pair ÐJ ß Ñ . We use the notation J\ to denote a free group on \ .

S. Roman, Fundamentals of Group Theory: An Advanced Approach, 319 DOI 10.1007/978-0-8176-8301-6_12, © Springer Science+Business Media, LLC 2012 320 Fundamentals of Group Theory

κ X F

∃!τf f

G

Figure 12.1

It is clear that the universal map ,, is injective. Moreover, \J generates \ , for if Ø\ÙJ, \ , then Theorem 4.17 implies that there are distinct group homomorphisms 57ßÀJÄK\ into some group K that agree on Ø\Ù, . Therefore, if 0œ57 l\\ œ l , then the uniqueness condition of mediating morphisms is violated. Hence, ,\J generates . For these reasons, it is common to suppress the map , and think of \J as a subset of \ .

The following theorem says that the universal property characterizes groups up to isomorphism.

Theorem 12.1 Let \Ð be a nonempty set. If Jß,-Ñ and ÐKßÑ are universal for \À, then there is an isomorphism 5 J¸K connecting the universal maps, that is, for which 5,‰œ -

Proof. There are unique mediating morphisms 77-,ÀJ ÄK and ÀK ÄJ for which

7,--,‰œand 7-, ‰œ and so

77---,‰‰œand 77,,,- ‰‰œ

But the identity maps ++KJ and are the unique mediating morphisms for which

+--KJ‰œand +,, ‰œ and so

77-,‰œ +KJand 77,- ‰œ + which shows that the maps 77-, and are inverse isomorphisms.

w Definition Let \ be a nonempty set. A word AÐB"8 ßáßB Ñ over \ ( or the equation AÐB"8 ß á ß B Ñ œ ") is a law of groups if

AÐ+"8 ß á ß + Ñ œ " for all groups K+−K and all 3 . Free Groups and Presentations 321

The following theorem says that what's true in a free group is true in all groups.

Theorem 12.2 Let \ be a nonempty set and let AÐB"8 ßáßB Ñ be a word over w \ÐOßÑ. If \ , is universal on \ , then the following are equivalent: 1) AÐ,, B"8 ß á ß B Ñ œ " in J \ 2) AÐB"8 ß á ß B Ñ is a law of groups. Proof. Let K+ be a group and let 3 −K for 3œ"ßáß8 . The function sending B+33 to can be lifted to a unique homomorphism 5ÀJÄK\ for which 5,Bœ+33 and so

AÐ+"8 ß á ß + Ñ œ5, AÐ B " ß á ß , B 8 Ñ œ 5 " œ " Cauchy's theorem says that any group is isomorphic to a subgroup of a symmetric group. There is an analog for quotients of free groups, but first we require a definition.

Definition Let Y œÖK±3−M×3 be a family of groups. A subgroup O of the direct product }Y is called a subdirect product of the family Y if the restricted projection maps 33OlÀOÄK 3 are surjective for all 3−M .

If KÀ is a group, then the identity map + KÄK can be lifted to an epimorphism 5ÀJK q» K and so K is isomorphic to a quotient of the free group JK . More generally, if \Ð is a set for which card\Ñ cardÐKÑ, then any surjection 0À\q» K can be lifted to an epimorphism 5ÀJ\ q » K and the induced map 5ÀJ\ ÎO¸K shows that K is isomorphic to a quotient group of the free group JÐ\. Moreover, we have freedom to choose the values of 7 BOÑ for B−\ arbitrarily, but O depends on that choice.

More generally, if Y œÖK±3−M×3 is a nonempty family of groups and if \ is a set for which cardÐ\Ñ card ÐK3 Ñ for all 3 − M , then there are isomorphisms 773\ÀJ ÎO 3 ¸K 3 for all 3−M , where 3 ÐBOÑ 3 can be specified arbitrarily, but O3 depends on that choice. Now, the “Chinese” map

5}ÀJ\\3 Ä J ÎO 3−M defined by for has kernel . Hence, if is the 55ÐAÑÐ3Ñ œ AO3\ A − J M œ+ O3 induced embedding, then the composition

}73\‰ÀJÎM 5 ä } 3−M3 K shows that JÎM\ is isomorphic to a subdirect product of the family Y . Moreover, we can specify that the element BM be sent to any element of }K3 , for all B−\ (again at the expense of O3 ).

Theorem 12.3 Let Kœ be a group, let Y ÖK3 ±3−M× be a nonempty family of groups and let \ be a set. 322 Fundamentals of Group Theory

1) If cardÐ\Ñ card ÐKÑ, then there is an isomorphism 7À J\ ÎR ¸ K where we can choose the elements 7BB−\ for arbitrarily, but R depends on that choice. 2) Suppose that cardÐ\Ñ card ÐK3 Ñ and that we have specified isomorphisms for all . Let . Then is isomorphic to a 733ÀK ¸J \ ÎO 3 3−M M œ+ O3 J \ ÎM subdirect product of the family Y7 and the isomorphisms 3 can be used to specify the elements BM for B − \ arbitrarily in }KM3 , but depends on that choice. Construction of the Free Group The notion of a free group can be defined constructively, without appeal to the universal mapping property. When a constructive approach is taken, one usually hastens to verify the universal mapping property, since this is arguably the most useful property of free groups. On the other hand, since we have chosen to define free groups via universality, we should hasten to give a construction for free groups.

Let [œÐ\Ñw‡ be the set of all words over the alphabet \w . As a shorthand, we allow the use of exponents, writing BâBif 8  ! ÝÚ î Ý 8 factors Bœ8 B" âB " if 8! Û ðóñóò Ý 8 factors Ý % if 8œ! Ü where % is the empty word. It is important to keep in mind that this is only a shorthand notation . Thus, for example, BB%# and B ' are both shorthand for BBBBBB and so B%#' B œ B . However, BB% # is shorthand for BBBBB" B " but BB#% is shorthand for BB and so B#ÁB# .

Since the operation of juxtaposition on [ is associative and since the empty word % is the identity, the set [ is a monoid under juxatpostion. In an effort to form a group, we also want to require that BB" œ% œ B" B for all B − \ . More specifically, consider the following rules that can be applied to members of [ :

1) Removal rules: For =.ß−[ and B−\ , =.=.BB" Ä =.=.BBÄ" BB" Ä % BBÄ" %

where one of =. or may be missing. Free Groups and Presentations 323

2) Insertion rules: For =.ß−[ and B−\ , =.ÄBB =" . =.ÄBB =" . % ÄBB" % ÄB" B where one of =. or may be missing.

Let us refer to a finite sequence =ßáß="5 of applications of these rules as a reduction of ?@ to of length 5 (even though @ may have greater length than ? ). The trivial reduction is an application of no rules and so ? is obtained from itself by the trivial reduction. Since the removal and insertion rules come in inverse pairs, reduction defines an equivalence relation ´[ on . Let [δ denote the set of equivalence classes of [ÒAÓ , with denoting the equivalence class containing A .

Since equivalent words must represent the same group element, it is really the equivalence classes that are the candidates for the elements of the free group J\ on \ . Moreover, since ?´<ß @´= Ê ?@´<= the equivalence relation ´[ is a monoid congruence relation on and so we may raise the operation of juxtaposition from [[δ to , that is, the operation Ò?ÓÒ@Ó œ Ò?@Ó is well-defined on [δ and makes [δ into a group, with identity ÒÓ% and for which

/""/55" // ÒB"" âB55 Ó œ ÒB âB Ó However, since it is easier to work with elements of [ rather than equivalence classes, we prefer to use a system of distinct representatives for [δ . A desirable choice would be the set consisting of the unique word of shortest length from each equivalence class, and so we must prove that such words exist.

Let us say that a word A is reduced if it is not congruent to a word of shorter length. It is clear that a removal rule can be applied to a word A−[ if and only if AB contains a subword of the form B" or B "B for B−\ . Further, a reduced word A has no such subword and so no removal rules can be applied to AA. We want to prove that the converse also holds, that is, a word is reduced if and only if no removal rules can be applied to A . Then we can show that the set of reduced words form a system of distinct representatives for [δ . 324 Fundamentals of Group Theory

Theorem 12.4 Let <−[ be a word that does not contain a subword of the form BB" or B " B for B − \ . If A ´ < , then there is a reduction from A to < that involves only removal rules. 1) A word is reduced if and only if it does not contain a subword of the form BB" or B " B for B − \ . /" /8 2) A word is reduced if and only if it can be written in the form BâB" 8 with BÁB33"3 and /Á! for all 3 . 3) The set of reduced words is a system of distinct representatives for [δ . Proof. Among all reductions from A< to , select a reduction with the fewest number of steps and suppose that there is at least one insertion step. Denote the steps by =ß=ßáß="# 7 and suppose that step =5 results in the word ?5 . Let = 5 be the last insertion step, say

?œ5" α" " ?œ5 α" ÐBBÑ where we have marked B with an overbar to distinguish it from any other occurrences of the symbol B . (A similar argument will work for the insertion of BB" .)

Since there are no further insertion steps, the pair BB" is never separated during the remaining steps, but must be altered at some point by a subsequent " removal rule. Suppose that BB is unaltered until step =54 and so the intermediate steps are

?œ5" α" " ?œ5 α" ÐBBÑ " ?œÐBBÑ5"+1 α" " ã " ?œÐBBÑ54"α" 4" 4"

" There are three possibilities for step =BB54 . First, if is removed, that is, if

?œ54α" 4" 4" then the insertion (and subsequent deletion) of BB" could have been omitted from the reduction process, in which case steps ==554 and do nothing can be removed from the reduction, resulting in a shorter reduction, which is a contradiction.

" The other possibilities involve an interaction of BB with either α"4" or 4" . w" One possibility is that αα4" œB4" and w"" ?œBÐBBÑ54" α"4" 4" w" ?œ54 α"4" ÐBÑ4"

" But since ?œ54α" 4 4, step = 54 can be replaced by the removal of BB , Free Groups and Presentations 325

w resulting in the same reduction as in the first case. Similarly, if ""4" œB 4" and

" w ?œÐBBÑB54"α" 4" 4" w ?œ54α" 4" ÐBÑ4" then since ?œ54α" 4" 4", again we can replace this reduction by the first reduction. Hence, a shortest reduction of A< to cannot have any insertion steps.

For part 1), suppose that no removal rules can be applied to <−[ and that A´<. Then there is a reduction from A to < that involves only removal steps and so lenÐ<Ñ Ÿ len ÐAÑ. Hence, < is reduced. The converse is clear.

/" /8 For part 2), if AœB" âB8 is reduced but B33"3 œB where / and / 3" have opposite signs, then we can apply a removal rule to A to produce a shorter equivalent word, which is not possible. Thus, if BœB33" , we may add the /" /8 corresponding exponents. Conversely, if AœB" âB8 is a word for which BÁB33"3 and /Á! , then no removal rule can be applied to A and so part 1) implies that A is reduced.

For part 3), any A−[ can be reduced using only removal rules to a word < for which no removal rules apply and so < is reduced. Thus, every word is congruent to a reduced word. Moreover, if ?Á@ are congruent reduced words, then there must be a reduction consisting of zero or more removal steps that brings ?@ to , but no removal steps can be applied to ? and so ?œ@ .

Thus, each word A−[ is equivalent to a unique reduced word =< and we may use the bijection ÒAÓ Ç A< to transfer the group structure from [ Î ´ to the set J?\\ of reduced words, specifically, if ß@−J , then ?@ œ Ð?@Ñ<

w It will be convenient to refer to the set \\ of reduced words on by the following name (which is not standard terminology).

Definition Let \ be a nonempty set. The concrete free group \ on \ is the set

/" /8 œÖ×∪ÖBâB±BÁB−\ß/Á!×\3% " 8 3"3 of all reduced words over the alphabet \w , under the operation of juxtaposition followed by reduction. The rank rkÐÑ\\ of is the cardinality of \ .

Note that the use of the term “concrete” is not standard. Most authors would refer to \\ simply as the free group on , which is justified by the following. 326 Fundamentals of Group Theory

Theorem 12.5 Let J\\ be the concrete free group on a set and let 4À\ÄJ\ be the inclusion map. Then the pair ÐJ\\ ß 4Ñ is universal for \J and so is a free group on \ . Proof. Let 0À\ Ä K . If 77 ÀJ\ Ä K is defined by %œ " and

/" ///8"8 7ÐB" âB8 Ñ œ 0ÐB"8 Ñ â0ÐB Ñ

/" /8 for BâB−J" 8 \, then 7 ‰4œ0 on \ . Also, it is clear that reduction can take place before or after application of 7 without affecting the final result. However, reduction has no effect in K?‡@ and so if denotes the operation of J\ , then 77Ð? ‡ @Ñ œ ÐÐ?@Ñ<< Ñ œ Ò 7777 Ð?@ÑÓ œ Ð?@Ñ œ Ð?Ñ Ð@Ñ which shows that 7 is a mediating morphism for 0 . As to uniqueness, if ww w 77‰4œ ‰4, then 77 and agree on \ , which generates J\ and so 77 œ . Relatively Free Groups Freedom can also come in the context of some restrictions. For example, the free abelian group on a set \ is the most “universal” abelian group generated by \ . More generally, if ^ is any class of groups, we can ask if there is a most universal ^ -group generated by a nonempty set \ . If so, such a group is referred to as a free ^^ -group or a relatively free group. For the class of abelian groups, free ^ -groups do exist, but this is not the case for all classes of groups, as we will see.

The definition of a free ^- group generalizes that of a free group.

Definition Let ^ be a class of groups and let \ be a nonempty set. A pair ÐO\\\ ß,^, À \ Ä O Ñ where O is a -group generated by \ , has the ^- universal property mapping for \\ () or is ^-universal for if for any function 0À\ Ä K from \ to a ^ -group K , there is a unique group homomorphism 70\ÀO ÄK for which

70 ‰4œ0

The map 70 is called the mediating morphism for 0O . The group \ is called a free ^-group on \\ with free generators or basis and ,^ is called the - universal map for the pair ÐO\ ß, Ñ .

Note that if ÐO\ ß,^ Ñ is -universal, then the ^ -universal map , is injective. For this reason, one often thinks of \O as a subset of \ . Proof of the following theorems is left to the reader.

Theorem 12.6 Let \Ð be a nonempty set. If Oß,-^Ñ and ÐKßÑ are -universal for \À , then there is an isomorphism 5 O¸K for which 5,‰œ - Free Groups and Presentations 327

Definition Let ^ be a class of groups and let \ be a nonempty set. A word w AÐB"8 ßáßB Ñ over \ () or the equation AÐB"8 ßáßB Ñœ" is a law of ^ - groups if

AÐ+"8 ß á ß + Ñ œ " for all ^ -groups O and all +3" −O . If ÖB ßB# ßáש\, then we denote the set of all laws of ^_ -groups in JÐÑÐÑ\\ by ^ or _^ .

Theorem 12.7 Let ^ be a class of groups and let \ be an infinite set. Then _^\\ÐÑ is a fully invariant subgroup of the free group J . Proof. It is clear that the product of two laws of ^^ -groups is a law of -groups, as is the inverse of a law of ^5 -groups. Also, if −ÐJÑEnd \ , then for any AÐB"8 ß á ß B Ñ −_^ Ð Ñ,

555AÐB"8 ß á ß B Ñ œ AÐ B " ß á ß B 8 Ñ −_ Ð^ Ñ and so _^ÐÑ is fully invariant in J\ .

Theorem 12.8 Let \ be a nonempty set and let AÐB"8 ßáßB Ñ be a word over w \ÐOßÑ. If \ ,^ is -universal on \ , then the following are equivalent: 1) AÐ,, B"8 ß á ß B Ñ œ " in O \ 2) AÐB"8 ß á ß B Ñ is a law of ^ -groups.

We have said that there are classes of groups for which relatively free groups do not exist. For example, the class of all finite groups is such a class, as we will see later. There is one very important type of class for which relatively free groups do exist, however.

Definition Let \œ be a nonempty set and let _ ÖA3 ±3−M× be a subset of the concrete free group J\ . The equational class () or variety with laws _ is the class X_ÐÑ of all groups K for which each A−3 _ is identically "K on .

Note that an equational class is closed under subgroup, quotient and direct product. For equational classes, we can construct relatively free groups using our construction of the concrete free group.

Theorem 12.9 Let ^X_œÐÑ be an equational class of groups with laws _ . 1) For any group K , the verbal subgroup

__ÐKÑœØAÐ+"8 ßáß+ ѱA− ß+ 3 −KÙ is fully invariant in K . 2) If J\\ is the concrete free group on , then the pair

ÐO\\ œ J Î_,1 ÐJ \ Ñß œ ‰ 4Ñ 328 Fundamentals of Group Theory

where 1_ is projection modulo ÐJ\\ Ñ and 4À \ Ä J is inclusion, is ^ - universal for \ . We refer to ^\ as the concrete ^-free group. Proof. Write JœJ\3 . For part 1), if 5 −End ÐKÑ, then for any + −K ,

555AÐ+"8 ß á ß + Ñ œ AÐ + " ß á ß + 8 Ñ −_ ÐKÑ and so _^ÐKÑ is fully invariant in K . For part 2), to see that O œ O\ is a - group, if AÐB"8 ß á ß B Ñ − _, then for any +3 − J ,

AÐ+"8__ ÐJÑßáß+ ÐJÑÑœAÐ+ "8 ßáß+ Ñ __ ÐJÑœ ÐJÑ and so O\ satisfies the laws in _ .

To see that Ð߉4Ñ^1\ is ^ -universal, referring to Figure 12.2, let 0À\ÄL , where L is a ^ -group.

j π X FX KX=F/L(F)

σ f τ

H Figure 12.2 Then 0À can be lifted uniquely to a homomorphism 5 JÄL satisfying 5_‰4œ0. But if AÐBßáßB"8 Ñ− and ? 3\ −J , then

555AÐ?"8 ß á ß ? Ñ œ AÐ ? " ß á ß ? 8 Ñ œ "

Hence, AÐ?"8 ß á ß ? Ñ −ker Ð5_ Ñ and so ÐJ Ñ Ÿker Ð 5 Ñ. Thus, the universality of quotients implies that 5 can be lifted uniquely to a homomorphism 7_ÀJÎ ÐJÑÄL for which 715‰ œ and so 71‰‰4œ‰4œ0 5 As to uniqueness, if 71ww‰‰4œ‰‰4œ0 7w 1 then the uniqueness of 57171 implies that www‰œ ‰ and the uniqueness of 7 implies that 77wwwœ . More on Equational Classes If ^X is a class of groups, then the equational class ÐÐÑÑ_^ is the class of all groups that satisfy the laws of ^^ -groups. Of course, a -group satisfies the laws of ^^X -groups and so ©ÐÐÑÑ_^. The interesting issue is that of equality, that is, for which classes ^^^ is it true that the laws of -groups hold only for - groups? The answer is that this happens if and only if ^ is an equational class. Free Groups and Presentations 329

For if ^X`œÐ Ñ is the equational class for a set ` of laws over \w , then `_^©ÐÑ and so X_^ÐÐÑÑ©ÐÑœ©ÐÐÑÑ X` ^ X_^ whence X_^ÐÐÑÑœ ^.

Theorem 12.10 A class ^ of groups satisfies ^X_^œÐÐÑÑ that is, the laws of ^^ -groups hold only for -groups, if and only if ^ is an equational class.

Example 12.11 Let ^ be the class of all finitely-generated groups. If AÐB"8 ß á ß B Ñ is a law of ^ , then AÐB"8 ß á ß B Ñ œ " in all finitely-generated groups and therefore in all groups. Hence, X_^ÐÐÑÑ is the class of all groups and so ^ is not an equational class.

The following theorem makes it relatively easy to tell when a class ^ is an equational class.

Theorem 12.12 ()Birkhoff The following are equivalent for a class ^ of groups: 1) ^ is an equational class 2) ^ is closed under subgroup, quotient and direct product 3) ^ is closed under quotient and subdirect product. Proof. It is clear that 1) implies 2), which implies 3). To show that 3) implies 1), we show that X_^ÐÐÑÑ© ^. Let K−ÐÐÑÑ X_^ , that is, K satisfies the laws of ^-groups. For the proof, we work with quotients of free groups.

Now, if cardÐ] Ñ card ÐKÑ, there is an Rü K for which K ¸ J] ÎR and so J]" ÎR satisfies the laws of ^_ -groups. Hence, if AÐC ßáßC8 Ñ−] Ð^ Ñ, then AÐC"8 ß á ß C Ñ − R and so _^] Ð Ñ Ÿ R. Hence, JJ R K¸]] ¸ RÐÑÐÑ_^]]‚ _^ and the proof would be complete if we knew that JÎ]]_^ Ð Ñ was a ^ -group. In fact, the same argument works for any MŸ_^] Ð Ñ , since then MŸR and so JJR K¸]] ¸ RMM‚

Thus, we only need to find an MŸ_^]] Ð Ñ for which J ÎM is a ^ -group.

Now, if A−J] is not a law of ^^ -groups, then there is a -group OA that violates A , that is, for which 330 Fundamentals of Group Theory

AÐ5Aß" ß á ß 5 Aß8A Ñ Á " for some 5−OAß3 A . Moreover, if cardÐ]Ñ ÐOÑ card A , then there is an isomorphism 77A]ÀJ ÎR A ¸O A for which A3AÐCR Ñœ5 Aß3 for all 3 . Hence, JÎR]A also violates A in the sense that

AÐC"8 ß á ß CA Ñ Â R A and so

M³ R Ÿ_^ Ð Ñ , A] AÂ_^ Ð Ñ

But Theorem 12.3 implies that the quotient JÎM] is isomorphic to a subdirect product of Y^ and is therefore a -group.

We can now relate equational classes and existence of relatively free groups.

Theorem 12.13 ()Birkhoff The following are equivalent for a nontrivial class ^ of groups: 1) ^X is an equational class, that is, ÐÐÑÑœ_^^ 2) ^^ is closed under quotient and for every nonempty set \ , there is a - universal pair ÐO\ ß, Ñ . Proof. We have seen that 1) implies 2). For the converse, we show that 2) implies X_^ÐÐÑÑ© ^. Let K−ÐÐÑÑ X_^ .

j σ G FG G

κ λ τ

KG

Figure 12.3

Referring to Figure 12.3, let 4À K Ä JK be the inclusion map into the concrete free group JK . We lift two maps. The ^, -universal map can be lifted to a unique homomorphism --ÀJKK ÄO for which 4œ,- . Moreover, is surjective since ,+KO generates K . Also, the identity map ÀKÄK can be lifted to a unique epimorphism 55ÀJK q » K for which 4œ+ .

w To see that kerÐ-5 ÑŸ ker Ð Ñ, if AÐB"8 ßáßB Ñ is a word over \ and if AÐ4+"8 ßáß4+ Ñ−ker Ð- Ñ for + 3 −K , then

AÐ+ßáß,,-"8 +Ñœ AÐ4+ßáß4+Ñœ" "8 in OK" and so Theorem 12.8 implies that AÐB ßáßB8 Ñ−_^ Ð Ñ. Hence in K , Free Groups and Presentations 331

55AÐ4+"8 ßáß4+ ÑœAÐ 4+ " ßáß5 4+ 8 ÑœAÐ+ "8 ßáß+ Ñœ" and so AÐ4+"8 ßáß4+ Ñ−ker Ð5-5 Ñ, whence ker Ð ÑŸ ker Ð Ñ.

Hence, 57 induces an epimorphism ÀOKK q » K defined for each 5 −O by " 75-5œ Ð5Ñ and so K¸OK Îker Ð7 Ñ and since the latter is a ^ -group, so is K. Free Abelian Groups The class of abelian groups is an equational class, with law ÒBß CÓ œ " . Thus, free abelian groups exist. In fact, if K is a group, then the verbal subgroup is _ÐKÑœØÒ+ß,Ó±+ß,−KÙ which is just the commutator subgroup KKw of . Hence, Theorem 12.9 implies that if J\\ is the concrete free group on , then the group w EœJÎJ\\\ is free abelian.

Definition Let \J be a nonempty set. If \ is the concrete free abelian group on w \EœJÎJ, then \\\ is the concrete free abelian group on \ . It is customary to think of EJ\\ as the group with the additonal condition that the elements of \ commute.

Theorem 12.14 If \E is a nonempty set, then the free abelian group \ satisfies

E¸\ {{™ ØBÙ¸ B−\ B−\ Proof. The function 0À\ Ä{ ØBÙ defined by BCœBif 0ÐBÑÐCÑœ œ "CÁBif for all C−\ can be lifted uniquely to a homomorphism 7{ÀE\ Ä ØBÙ for which

/" ///8"8 7ÐB" âB8 Ñ œ 0ÐB"8 Ñ â0ÐB Ñ and so /5 if /" /8 BCœB5 5 7ÐB" âB8 ÑÐCÑ œ œ "CÂÖBßáßB×if "8

Now, 7α is surjective, since if −{ ØBÙ has support ÖB"8 ßáßB × and /5 /" /8 ααÐB5 Ñ œ B5 , then œ7 ÐB" âB8 Ñ. Also, 7 is injective, since if /" /8 /5 77ÐB" âB8 Ñœ!, then B5 œ" for all 5 and so /5 œ! for all 5 . Thus, is an isomorphism. 332 Fundamentals of Group Theory

The following theorem explains the term basis used for the set \ in the free abelian group E\ .

Definition A subset WEE of the free abelian group \\ is independent in if

/" /8 =33 −Wß =" â=8 œ% ß = Á=4 Ê /3 œ! for all 3 Theorem 12.15 Let \ be a nonempty subset of an abelian group A. The following are equivalent: 1) E\ is a free abelian group with basis 2) \EE is independent in and generates 3) Except for the order of the factors, every nonidentity element +−E has a unique expression of the form

/" /8 +œB" âB8 for B345 ÁBß/ Á!ß8 " Proof. If 1) holds, we have seen that \E generates and if

/" /8 AœB" âB8 œ% for BÁB34 , then /œ! 3 for all 3 , since otherwise A would be a reduced word equivalent to the shorter word % . Hence, 2) holds. If 2) holds, then every element of E+ has at least one such expression. But if −E has two distinct expressions:

/" /0870" +œB"" âB87 œC âC where we may assume that BÁC87 (or we can cancel), then

/" /087 0" BâBC""87 âC œ% violates independence. Hence, 3) holds.

Finally, to see that 3) implies 1), suppose that 0À\ Ä K , where K is an abelian group. If a mediating morphism 77 does exist, then ÐBÑ œ 0ÐBÑ for all B − \ and so 77 is unique, since \E generates . Define a map ÀEÄK by

/" ///8"8 7ÐB" âB8 Ñ œ 0ÐB"8 Ñ â0ÐB Ñ

/" /8 which is well defined since the expressions BâB" 8 are unique up to order but K is abelian. The map 7 is easily seen to be a homomorphism.

The next theorem says that, up to isomorphism, there is only one free (or free abelian) group of each cardinal rank.

Theorem 12.16 Let \] and be nonempty sets. 1)) a E¸EÍ\œ] \]kk kk b) If WEW \ generates , then \ kk k k 2)) a J¸JÍ\œ] \]kk kk Free Groups and Presentations 333

b) If WJW \ generates , then \ kk k k Proof. Suppose first that \œ] and let 0À\Ä] be a bijection. Extend the kk kk range of 00À\ÄJ0À\ÄE so that ]] (or ). Then there is a unique mediating morphism 77ÀJ\] ÄJ (or ÀE \ ÄE ]) for which 7ÐBÑ œ 0ÐBÑ

/" /8 for all B−\ . To see that 7 is injective, if AœB" âB8 and 0ÐBÑœC33 , then

//""////8"88 77ÐAÑ œ ÐB"" âB88 Ñ œ 0ÐB"8 Ñ â0ÐB Ñ œ C âC

Hence, 7%ÐAÑ œ implies /3 œ ! for all 3 and so A œ % . Also, 7 is surjective since 0À\ Ä ] is surjective. Hence, J\] ¸ J (and E \ ¸ E ] ).

For the converse of part 1), let EEE represent either \] or . We use additive notation. The quotient EÎ#E is elementary abelian of exponent # and is therefore a vector space over ™# . Moreover, if WE generates the group , then WÎ#E œ Ö=  #E ± = − W× generates the vector space EÎ#E over ™# and if W is independent in E , then WÎ#E is linearly independent in EÎ#E , since an equation of the form

Ð="8 #EÑâÐ= #EÑœ#E for 8! and =34 Á= in W implies that

="8" â= œ#Ð/>"7 â/ >7 Ñ for >−W33 and /−™ , which is not possible.

It follows that \Î#E\\ is a basis for E Î#E\ and so dimÐE Î#E Ñ œ \ \\kk and similarly for ]Þ Thus, since E\] ¸ E implies E \\]] Î#E ¸ E Î#E , we have \ œdim ÐE Î#E Ñ œdim ÐE Î#E Ñ œ ] kk \\ ] ]kk and if WE generates \ , then W WÎ#E \ kk k\ k k k This completes the proof of part 1).

For part 2), if J¸J\] , then ww E¸JÎJ¸JÎJ¸E\\\] ] ] and so part 1) implies that \œ] . Finally, if WJ generates , then kk kk \ 334 Fundamentals of Group Theory

ww w WÎJœÖ=J±=−W×\\ generates EœJÎJ\\\ and so W WÎJ \ÎJœ\ww kk k\\ k k k k k

Theorem 12.17 Let E\\ be free abelian on . Then all independent sets have cardinality at most \ . kk Proof. It is sufficient to prove the result for Eœ {™B−\ B where ™™B œ for all BZœ. The set {B−\ B is a vector space over the rational field  and it is easy to see that a subset

œÖ@±3−MשE3 is dependent in E if and only if  is linearly dependent over . But in the vector space Z\ , all sets of cardinality greater than are linearly dependent over  kk and therefore also over ™ .

The Nielsen–Schreier Theorem says that every subgroup of a free group is free and so the subgroups of free groups are very restricted. (Nielsen proved this result for finitely-generated groups in 1921 and Schreier generalized it to all groups in 1927.)

Theorem 12.18 1) Any subgroup of a free group is free. 2) Any subgroup WE of a free abelian group \ is free abelian and rkÐWÑ Ÿ rk ÐEÑ. Proof. We omit the difficult proof of part 1) and refer the interested reader to Robinson [26]. For part 2), we may assume that Eœ\B−\{ ØBÙ and that \ is well ordered. Since the elements of W0 have finite support, for −W , we can let 3Ð0Ñ be the largest index B for which 0ÐBÑ Á " .

For each B−\ , consider the set

MœÖ0ÐBѱ0−Wß3Ð0ÑŸB×B

Then MŸØBÙBB and so MœØ0ÐBÑÙBB for some 0−WB . We show that W is free on the set

œÖ0BB ±B−\ß0ÐBÑÁ"× If does not span WW , among those elements of not in the span of , choose an element 1C for which œ3Ð1Ñ is the smallest possible. Since 5 1ÐCÑ − MCC œ Ø0 ÐCÑÙ, it follows that 1ÐCÑ œ 0C ÐCÑ for some nonzero 5 − ™ . Then

5 5 Ð10ÑÐBÑœ1ÐBÑ0ÐBÑœ"CC for all B C

5 5 and so 3Ð10CC Ñ  C, which implies that 10 is in the span of . But then Free Groups and Presentations 335

5 5 1œÐ10CC Ñ0 is also in the span of , a contradiction. Thus, spans W .

Also, is independent, since if

//"8 0â0œ"BB"8 where BB34 for 34 , then applying this to B8 gives

/8 0ÐBÑœ"B8 8 and so /œ!83 . Similarly, /œ! for all 3 and so is independent. Hence, Theorem 12.15 implies that WŸ is free over . Also, it is clear that \ and kk k k so rkÐWÑ Ÿ rk ÐEÑ. Applications of Free Groups Sometimes free groups can help produce complements.

Theorem 12.19 If 5ÀKq» J\\ is an epimorphism, where J\ is free on , then Oœker Ð5 Ñ is complemented in K . Proof. Define a function 77À\ ÄK by letting B be a fixed member of 5"ÐBÑ . Since J\\ is free on , there is a unique homomorphism 7ÀJ\ ÄK that extends 7 on \B . Since for any −\ , 57‰ÐBÑœB it follows that 57‰œ + . In other words, 7 is a right inverse of 5 and so Theorem 5.23 implies that kerÐÑ5 is complemented in K .

With the help of free groups, we can provide an example of a finitely-generated nonabelian group with a subgroup that is not finitely generated. We have already proved (Theorem 2.21) that if K8 is an -generated abelian group, then every subgroup of K8 can be generated by or fewer elements.

Theorem 12.20 Let \œÖBßC× and let J\ be the # -generated free group on \KœØWÙ. Let , where WœÖCBC55 ±5!×

Then KJ is isomorphic to the free group ^ on a countably infinite set ^ œÖD"# ßD ßá× and so is not finitely generated. 55 Proof. Consider the function 0À^ Ä K defined by 0ÐD5 Ñ œ C BC . Then there 55 is a unique mediating morphism 77ÀJ^5 ÄK for which ÐD ÑœC BC . It is clear that 7 is surjective. 336 Fundamentals of Group Theory

In addition, if / 7%ÐD3" âD/37 Ñ œ 33"7

where 3Á355"3 , /5 Á! and 7 " , then 3 / 3 3 / 3 3 / / 3 3 / 3 CBCBCBâBC"3"#$7 "#3 #$3 3" 7"73 BCœ7 7 % in KŸJ\ . Since the left-hand side can be reduced to % using only removal steps, it follows that 3œ355" for all 5 and so 7œ" and 3/ 3 CB"3" C " œ% But the left-hand side of this equation reduces to % by removal steps if and only if /œ!" , which is false. Hence, 7 is injective and therefore an isomorphism.

We can also provide an example of a group KL with a subgroup ŸK for which +L+"  L.

Theorem 12.21 Let J\\\ be the free group on œÖBßC× . Then J has a subgroup LBLBL for which " . Proof. Let L consist of the empty word % and the set of all words of the form

AœBCBCâBCB8585"" ## 858 << <" with , , for and . Note < " 53"<"3 Á!ß8 !ß8 Ÿ! 8 Á! #Ÿ3Ÿ< 83 œ! that C−L . We leave it to the reader to show that L is a subgroup of J\ . It is clear that

BAB" œ B 8""##<<<" " C 5 B 8 C 5 âB 8 C 5 B 8 " − L and so BLB" Ÿ L. However, BAB" Á C for all A − L since if A Á % , then BAB" has length at least $ . Hence, BLB"  L. Presentations of a Group One way to define a group is to list all of the elements of the group and then give the group's multiplication table, which shows explicitly how to multiply all pairs of elements of the group. Then it is necessary to verify the defining axioms of a group: associativity, identity and inverses. For example, the cyclic group GÐ+Ñ$ is # G$ œ Ö"ß +ß + × with multiplication table Free Groups and Presentations 337

"++# ""++# +++"# ++"+##

Of course, we generally abbreviate this description by writing

334Ð34)mod$ G$ œ Ö+ ± ! Ÿ 3 Ÿ #×ß + + œ + It is routine in this case to check the group axioms.

On the other hand, it is tempting to define a group by giving a set of generators for the group along with some properties satisfied by these generators. The issue then becomes one of deciding whether there is a group that has these generators with these properties.

To illustrate, consider the following description of a group K : K œ Ø+ß ,Ùß +#% œ "ß , œ "ß +,+, œ " This description gives a nonempty set \œÖ+ß,× of generators for K and certain relations on KA , that is, equations of the form œ" where A is a word over the generators and their inverses. Note that there is nothing in the description above that precludes the possibility that +œ, .

In order to guarantee that such a group exists, we require that all relations must have the form Aœ" . The left-hand side A is referred to as a relator . Expressions such as AÁ" are not permitted. However, it is customary to take the liberty of writing a relation in the form Aœ@ as a more intuitive version of A@" œ ". For example, the last relation above can be written ,+ œ +" , " or ,+ œ +,$ In view of the precise nature of relations, given any nonempty generating set \ and any set e of relations, there is always one group that is generated by \ and satisfies the relations e : It is the trivial group, where each generator is taken to be the identity.

On the other hand, the best hope for getting a nontrivial group generated by \ and satisfying the relations e is to start with the concrete free group J\ and factor out by the smallest normal subgroup R required in order to satisfy the given relations, that is, the normal closure of the relators of e . 338 Fundamentals of Group Theory

With respect to the example above, the quotient group

JÖBßC× Kœ" #% ØB ßC ßBCBCÙnor

#% has generators + œ BR and , œ CR , where R œ ØB ß C ß BCBCÙnor that satisfy the relations given for a group K .

Thus, we are lead to the concept of a free presentation of a group, given in the next definition.

Definition Let J\\\ be free on . An epimorphism 5 ÀJKq» is called a free presentation of K\ . The set is called a set of generators for the presentation and if kerÐÑœØÙ5enor so that J K¸ \ ØÙe nor the set e is called a set of relators of the presentation. In this case, we write K¸Ø\±e Ù

If <−e is a relator, then the equation <œ" is called a relation .

We will often refer to a free presentation of KK simply as a presentation of . It is common to say that the group KØ itself has presentation \±eÙ when J K¸ \ ØÙe nor and that K is defined by generators and relations. Note, however, that it is the set 5e\KØ that actually generates . A presentation \±Ù\ is finite if ∪e is a finite set. Finally, we will often blur the distinction between a relator and the corresponding relation, using whichever is more convenient at the time.

We can form a more concrete version of the group defined by generators \ and relations e as follows. If KœØ\Ù and if J\ is the concrete free group on \ , w then a word over \J has two contexts: as an element of \ and as an element of KÀ. Moreover, there is a unique epimorphism 5 J\ q» K defined by specifying that 5BœB for all B−\ . This map can be thought of simply as a change of context and it is convenient to give it this name officially.

Definition Let KœØ\Ù and let J\ be the concrete free group on \ . 1)» We call the unique epimorphism 55ÀJ\ q K defined by BœB for all B−\ the change of context map associated to K\ and . 2)» If the change of context map 5eÀJ\ q K has kernel R œØ Ùnor , so that the induced map 7ÀJ\ ÎRÄK defined by Free Groups and Presentations 339

7ÐBRÑ œ B

is an isomorphism, we say that KØ has concrete presentation \±eÙ and write KœØ\±e Ù.

It is clear that if KœØ\±ee Ù with RœØ Ùnor and if AÐBßáßBÑ"8 is a word over \w, then

AÐB"8 ß á ß B Ñ œ " in K Í AÐB" Rß á ß B 8 RÑ œ R in J \ ÎR Í AÐB"8 ßáßB Ñ−R in J \

Every Group Has a Presentation

The change of context map 5ÀJK q » K shows that every group has a concrete presentation. Moreover, the kernel of this presentation is essentially the multiplication table for K . To be more specific, if +ß,ß-−K and - œ+, , then " " +,- œ " in K and so +,- must be factored out of JK . So let " R œØ+,- ±+ß,ß-−Kß-œ+, in KÙŸJK

It is easy to see that RJ is a fully invariant subgroup of K and that RŸkerÐ5Ñ . For the reverse inclusion, if AÐ+"8 ßáß+ Ñ−ker Ð5 Ñ, then AÐ+"8 ßáß+ Ñœ" in K and so AÐ+"8 ßáß+ Ñ−R. Thus, R œker Ð5e Ñ and so K œØK± Ù, where e œ Ö+,-" ± +ß ,ß - − Kß - œ +, in K× Note also that if K is finite, then so is e .

Theorem 12.22 Every group KØ has concrete presentation K±eÙ , where e œ Ö+,-" ± +ß ,ß - − Kß - œ +, in K× Moreover, if KØ is finite, then K±Ùe is a finite presentation of K .

The concrete presentation ØK ±e Ù is rather large and we can improve upon it in general.

Theorem 12.23 If K¸Ø\±ef Ù, then KœØ]± Ù, where ] Ÿ \ and kk kk efŸK. In particular, has a finite presentation if and only if it has a finite concretekk kk presentation. Proof. Let .eÀJ\ q» K be a free presentation of K with kernel R œØ Ùnor . Then the set ]œ.5 \ generates K . Let ÀJq] » K be the change of context map. If fe is the set of relators in ] obtained from by replacing each 340 Fundamentals of Group Theory occurrence of B−\ by . B , then the following are equivalent:

AÐ.. B"8 ß á ß B Ñ −ker Ð 5 Ñ AÐ.. B"8 ß á ß B Ñ œ " in K AÐB"8 ß á ß B Ñ − Øe Ùnor AÐ..f B"8 ß á ß B Ñ − Ø Ùnor and so kerÐ5f ÑœØ Ùnor and KœØ]±f Ù. Finitely Presented Groups A group is finitely presented if it has a finite presentation.

Theorem 12.24 If K\K has a finite presentation and if is a generating set for , then K has a finite presentation of the form

ØB"8"7 ßáßB ±< ßáß< Ù where B−\3 . Proof. Let Ø] ±f Ù be a finite concrete presentation of K , with

] œÖC"? ßáßC ×and f œÖ=ÐC3"? ßáßC ѱ3œ"ßáß@×

Then there is a finite subset \!" œÖB ßáßB 8 ש\ for which ] ©Ø\! Ù and so \K! generates and we can write

B33"? œ0 ÐC ßáßC Ñà 3œ"ßáß8 and

C44"8 œ- ÐB ßáßB Ñà 4œ"ßáß? Let

L œØB"8 ßáßB ±e Ù where e is the set of relators formed from the relations

=Ð3""-- ÐB ßáßB 8 Ñßáß ?" ÐB ßáßB 8 ÑÑœ"à 3œ"ßáß@ and

B44"" œ0- Ð ÐB ßáßB 8 Ñßáß- ?" ÐB ßáßB 8 ÑÑà 4œ"ßáß8

Since each of these relations holds in KØ , the subgroup eÙnor is contained in the kernel of the change of context epimorphism 55ÀJ\! q » K and so induces an ww epimorphism 55ÀLq » K defined by ÐBÑœB33 .

To see that 5w is injective, if w 5 ÐAÐB"8 ß á ß B ÑÑ œ " then Free Groups and Presentations 341

AÐB"8 ß á ß B Ñ œ " in K, whence

AÐ00"" ÐC ßáßC ? Ñßáß 8" ÐC ßáßC ? ÑÑœ" in K and so

AÐ00f"" ÐC ßáßC ? Ñßáß 8" ÐC ßáßC ? ÑÑ−Ø Ùnor in J]4 . Replacing each C by -4 ÐB" ßáßB8 Ñ implies that

AÐB"8 ß á ß B Ñ − Øe Ùnor

w in L , that is, AÐB"8 ßáßB Ñœ" in L . Hence, 5 is an isomorphism and so K has presentation ØB"8 ßáßB ±e Ù.

Theorem 12.25 Let KR be a group and let ü K . If R and KÎR are finitely presented, then so is K . Proof. Let

R œØB"8"? ßáßB ±< ßáß< Ù and let

KÎRœØC"7"@ RßáßC R ±= ßáß= Ù

If \ œÖB"8 ßáßB × and ] œÖC"7 ßáßC × and Eœ\∪] , then K is generated by E . As to relators, we have

=ÐC3" RßáßC 8 RÑœ" Í =ÐC3" ßáßC 8 ÑœAÐB 3" ßáßB 8 Ñ

" where AÐBßáßBÑ3" 8 is a word in the BBR3 's and 3 's. Also, is normal if and only if

" " CBC4344 œ@ 3ß4" ÐBßáßBÑ 8 and C BC34 œD 3ß4" ÐBßáßBÑ 8 for words @D3ß4 and 3ß4 . The following set e of relations captures the relations <3 and =R4 as well as the fact that ü K :

<ÐB3" ßáßB 8 Ñœ" for 3œ"ßáß? =ÐC3" ßáßC 8 ÑœAÐB 3" ßáßB 8 Ñ for 3œ"ßáß@ " C43 BC4 œ@ 3ß4" ÐB ßáßB 8 Ñ for 3œ"ßáß8à4œ"ßáß7 " CBCœDÐBßáßBÑ4 34 3ß4" 8 for 3 œ "ß á ß 8à 4 œ "ß á ß 7

ww www w www Let \ œÖB"8 ßáßB ×, ] œÖC "7 ßáßC × and E œ\ ∪] and let ww LœØE±eEÄEw Ù ww w w w If R œØB"8 ßáßB ÙŸL, then the relators in eü show that RL . 342 Fundamentals of Group Theory

ww Let 0ÀJEw Ä K be the unique epimorphism for which 0B34 œ B34 and 0C œ C . w Since ØÙe nor Ÿker Ð0Ñ, there is a unique epimorphism 1ÀLqK » for which ww ww 1Bœ0B33 œB33and 1Cœ0C33 œC Moreover, the restriction 1À Rw q » R is an isomorphism, since if ww 1ÐAÐB"8 ß á ß B ÑÑ œ " then

AÐB"8 ß á ß B Ñ œ "

w in R and so AÐB"8 ßáßB Ñ−Øe Ùnor in J\ . It follows that ww w AÐB"8 ß á ß B Ñ − Øe EÄEw Ùnor ww w in JEw and so AÐB"8 ßáßB Ñœ" in R . Hence, if O œker Ð1Ñ , then O ∩ Rw œ Ö"× Our goal is to show that O œ Ö"× .

Since 1ÐRw Ñ œ R , the epimorphism 1À L q » K induces an epimorphism 1ÀLÎRqww» KÎR for which www w 1ÐCRÑœ1ÐCÑRœCR444 for all 41 . Moreover, w is an isomorphism, since if wwwww 1 ÐAÐC"7 R ß á ß C R ÑÑÑ œ AÐC"7 Rß á ß C RÑÑ œ R in KÎR, then

AÐC"7 RßáßC RÑÑ−Ø= ""7?"7 ÐC RßáßC RÑßáß= ÐC RßáßC RÑÙnor in the free group on ÖC"7 Rß á ß C R×. Hence, ww w w ww w w ww w w AÐC"7 R ßáßC R ÑÑ−Ø="? ÐC "7 R ßáßC R Ñßáß= ÐC "7 R ßáßC R ÑÙnor ww w w in the free group on ÖC"7 R ß á ß C R ×, which implies that ww w AÐC"7 ß á ß C Ñ − R that is, ww w w w AÐC"7 R ßáßC R ÑÑœR in LÎRww . Thus, 1 is an isomorphism. Finally, if " Á 5 − O , then 5  Rw and so 5Rww Á R , whence Rœ1Ð5ÑRœ1Ð5RÑÁRww Free Groups and Presentations 343

It follows that O œ Ö"× , that is, 1À L ¸ K is an isomorphism and so K is finitely presented. Combinatorial Group Theory Before looking at other examples, let us return briefly to the question of whether a given presentation Ø\ ±e Ù defines a nontrivial group. From one point of view, this question has a rather surprising answer. It can be shown that no algorithm can ever exist that determines whether or not an arbitrary set of generators and relations defines a nontrivial group! Nor is there any algorithm that determines whether the group defined by an arbitrary finite presentation is finite or infinite.

Definition A decision problem is a problem that has a yes or no answer, such as whether or not a given word in JK\ is the identity in a group . 1) A decision problem is decidable or solvable if there is an algorithm, called a decision procedure, that stops after a finite number of steps and returns “yes” when the answer is yes and “no” when the answer is no. A decision problem is undecidable or unsolvable if it is not decidable. 2) A decision problem is semidecidable or semisolvable if there is an algorithm that stops after a finite number of steps and returns “yes” when the answer is yes. However, the algorithm need not stop if the answer is no.

The word problem for a group KØ with presentation \±eÙ , first formulated in 1911 by Max Dehn, is the problem of deciding whether or not an arbitrary word over \Kw is the identity element of (or, equivalently, whether or not two arbitrary words over \Kw are the same element of ). It has been shown that there exist individual groups K with finite presentations for which the word problem is unsolvable.

On the other hand, there are large classes of groups for which the word problem is solvable. For example, the word problem is solvable for all free groups (in view of Theorem 12.4), for all finite groups and for all finitely-generated abelian groups. In fact, it is an active area of current research in group theory to study classes of groups for which the word problem can be solved.

On the other hand, the word problem for finitely-presented groups is semidecidable. For if Ø\ ±e Ù is a finite presentation of a group K and if A−K, then since e is a finite set, there is an algorithm that checks all of the elements of ØÙe nor one-by-one looking for A . If Aœ"K in , then this algorithm will eventually encounter A . The problem is that the algorithm will not terminate if AÁ" and so this is not a decision procedure. One way to mitigate this problem is to intermix the steps of this algorithm with the steps of another algorithm that stops if AÁ" , assuming that such an algorithm exists. 344 Fundamentals of Group Theory

For example, the following defines a class of group for which such an algorithm does exist.

Definition A group K" is residually finite if for any Á+−K , there is a normal subgroup RKü for which +ÂRKÎR and is finite.

Theorem 12.26 The word problem is solvable for the class of all finitely- presented residually finite groups. Proof. Let KœØ\±e Ù be a finite presentation of a residually finite group K and let A−K . It is possible to enumerate all finite groups by constructing multiplication tables. Also, for a given finite group J , there are only a finite number of group homomorphisms from KJ to , since all such homomorphisms are uniquely determined by the functions 0À\ Ä J . Consider the following algorithm:

1) Compute the next finite group J . 2) For each group homomorphism 55ÀK ÄJ , stop the algorithm if AÁ" .

Now, if AÁ" in K , then since K is residually finite, there is an RKü for which A Â R and KÎR is finite. Hence, the canonical projection 1À K Ä KÎR, which is encountered in step 2) above, satisfies 1AÁ" and so the algorithm will stop if AÁ" . Thus, we can intermix this algorithm with the aforementioned algorithm to get a decision procedure for the word problem for K.

The issues discussed above fall under the auspicies of an area of algebra known as combinatorial group theory. On the Order of a Presented Group Since a relation cannot be a nonequality , there is no way to specify the order of an element or subgroup of a group by relations. Thus, for example, among the following descriptions of a group, only the first description is a presentation:

#% 1) K" œØ+ß,±+ œ"ß, œ"ß+,+,œ"Ù

#% 2) K## œ Ø+ß ,Ù, 9ÐK Ñ œ )ß + œ "ß , œ "ß +,+, œ "

3) K$ œ Ø+ß ,Ùß 9Ð+Ñ œ #ß 9Ð,Ñ œ %ß +,+, œ "

Since the relation +,+, œ " is equivalent to the commutativity relation ,+ œ +,$ each of the groups above has underlying set W œ Ö+34 , ± ! Ÿ 3 Ÿ "ß ! Ÿ 4 Ÿ $× Free Groups and Presentations 345

and so 9ÐK5# Ñ Ÿ ) for 5 œ "ß #ß $ . Moreover, since 9ÐK Ñ œ ) implies 9Ð+Ñ œ # and 9Ð,Ñ œ % , any group satisfying 2) also satisfies 3). Conversely, 9ÐK$ Ñ œ % # or )9ÐKÑœ%KœØ,Ù+œ, , but if $$ , then and , which does not satisfy +,+, œ ". Hence, 9ÐK$ Ñ œ ) and so 2) and 3) describe the same group.

Thus, if we show that a group KK fitting description 2) or 3) exists, then has order ) and satisfies the relations ee given by 1). Hence, is contained in the kernel of the change of context map 5ÀJÖ+ß,× ÄK and so ) ÐJ ÀØe5 Ù Ñ ÐJ Àker Ð ÑÑœK œ) Ö+ß,× nor Ö+ß,× kk

It follows that kerÐÑœØÙ5enor and so K has presentation Ø\±e Ù .

Theorem 12.27 Suppose that a group with presentation Ø\ ±e Ù has order at most 8∞ . Then any group K of order 8 generated by \ and satisfying the relations in ee has presentation Ø\ ± Ù . Proof. Let 5ÀJ\ q » K be the change of context epimorphism. Since ØÙe5nor Ÿker ÐÑ , we have 8 ÐJ ÀØe5 Ù Ñ ÐJ Àker Ð ÑÑœK œ8 \\nor kk from which it follows that kerÐÑœØÙ5enor and so J K¸ \ ØÙe nor

Referring to our previous example, since the dihedral group H) fits description 2) and has order ) , we have

#% H) ¸Ø+ß,±+ œ"ß, œ"ß+,+,œ"Ù More generally, one of the simplest presentations with two generators is L œØBßC±B87 œ"ßC œ"ßCBœBCÙ > for some !>7 . The commutativity relation shows that LœÖBC34 ±!Ÿ38 and !Ÿ47× and so 9ÐLÑŸ78. Moreover, one can prove by induction that for !Ÿ57 and !Ÿ48,

4 CB54 œBC 45> and so

4 ÐB35 C ÑÐB 4j C Ñ œ B 345>j C (12.28) However, we can define a group K7 whose underlying set consists of the 8 distinct formal symbols 346 Fundamentals of Group Theory

KœÖBC34 ±!Ÿ38 and !Ÿ47× with product defined by (12.28). This product is associative, since

44? ÒÐB35 C ÑÐB 4j C ÑÓÐB ?@ C Ñ œ ÐB 345>j C ÑÐB ?@ C Ñ œ B 34?Ð5>jÑ>@ C and

?4?? ÐB3 C 5 ÑÒÐB 4 C j ÑÐB ? C @ ÑÓ œ B 3 C 5 ÐB 4? C j> @ Ñ œ B 34? C 5> j> @ and inverses exist, since

83 ÐB35" C Ñ œ B 8375> C Hence, K8 is a group of size 7 that satisfies e and so K has presentation Ø\ ±e Ù and 9ÐLÑ œ 78.

Theorem 12.29 1) The presentation Ø\ ±e Ù œ Ø+ß , ± +87 œ "ß , œ "ß ,+ œ +, > Ù where !>7 defines the group KœÖ+,34 ±!Ÿ38 and !Ÿ47× where 9ÐKÑœ789Ð+Ñœ89Ð,Ñœ7, , and

4 Ð+35 , ÑÐ+ 4j , Ñ œ + 345>j , Moreover, any group of order 78 that is generated by \ and satisfies the relations ee has presentation Ø\ ± Ù . 2) The presentation Ø] ±f Ù œ Ø-ß . ± -87 œ "ß . œ "ß .- œ - = .Ù defines the group LœÖ-.34 ±!Ÿ38 and !Ÿ47× where 9ÐLÑœ789Ð-Ñœ89Ð.Ñœ7, , and

5 Ð-35 . ÑÐ- 4j . Ñ œ - 34=5j . Moreover, any group of order 78 that is generated by ] and satisfies the relations ff is defined by Ø] ± Ù .

Let us now consider some examples of presentations. Free Groups and Presentations 347

Dihedral Groups

Since the dihedral group HœØßÙ#8 53 of order #8 satisfies the relations e5œÖ#8 œ"ß 3 œ"ß 3553 œ 8" ×

Theorem 12.29 implies that HØ#8 has presentation Ö53ß×± eÙ and multiplication table

4 ÐÑÐÑœ5335 53 4j 5 345Ð8"Ñj 3

We leave it as an exercise to show that H#8 is also presented by Ø] ±f Ù œ ØBß C ± B## œ "ß C œ "ß ÐBCÑ 8 œ "Ù Thus, two rather different looking presentations can be equivalent , that is, can present the same group. Quaternion Group To find a presentation for the quaternion group, note that œØ3ß4Ù satisfies the relations e œÖ3œ"ß3œ4ß43œ34×%##$ and so we need only show that any group presented by Ø\ ±e Ù has order at most )K . If has presentation Ø\±e ÙœØBßC±B%##$ œ"ßC œBßCBœBCÙ then K œ ÖB=> C ± ! Ÿ =ß > Ÿ $× However, since CœB## , we see that K œ ÖB=> C ± ! Ÿ = Ÿ $ß ! Ÿ > Ÿ "× and so 9ÐKÑ Ÿ ) . Thus, ¸ Ø\ ±e Ù . Dicyclic Groups Consider the presentation Ø\±e ÙœØBßC±B#8 œ"ßC # œB 8 ßCBœB " CÙ If LœØ\±e Ù then using the fact that CœB#8 , we have 348 Fundamentals of Group Theory

L œ ÖB34 C ± ! Ÿ 3 Ÿ #8  "ß ! Ÿ 4 Ÿ "× and so 9ÐLÑ Ÿ %8 .

A double induction shows that for !Ÿ5% and !Ÿ4#8 ,

45 5 BC 5 even CB54 œB Ð"Ñ45 C œ œ BC5#84 5 odd and so

5 ÐB35 C ÑÐB 4j C Ñ œ B 3Ð"Ñ45j C (12.30) However, we can define a group KB by choosing two distinct symbols and C and setting K œ ÖB34 C ± ! Ÿ 3  #8ß 4 œ !ß "× with product defined by (12.30). This product is associative:

555j ÒÐB3 C 5 ÑÐB 4 C j ÑÓÐB ? C @ Ñ œ ÐB 3Ð"Ñ 4 C 5j ÑÐB ? C @ Ñ œ B 3Ð"Ñ 4Ð"Ñ ?C 5j@ and

j5j ÐB3 C 5 ÑÒÐB 4 C j ÑÐB ? C @ ÑÓ œ B 3 C 5 ÐB 4Ð"Ñ ? C j@ Ñ œ B 3Ð"Ñ Ð4Ð"Ñ ?ÑC 5j@ and inverses exist:

5 ÐB3 C 5 Ñ " œ B Ð"Ñ Ð#83Ñ C %5 Hence, K% is a group of size 8 that satisfies the relations e and so K¸Ø\±ee Ù and 9ÐLÑœ%8 . Any group with presentation Ø\± Ù is called a dicyclic group of order %8 .

A special case of the dicyclic group is when #8 has the form #8" , in which case the presentation is

8" 8# Ø\±e ÙœØBßC±B### œ"ßC œB ßCBœB" CÙ A group with this presentation is called a generalized quaternion group. When 8œ$, this is Ø\±e ÙœØBßC±B%##" œ"ßC œBßCBœB CÙ which is the presentation for the quaternion group . The Symmetric Group

Recall from Theorem 6.5 that the symmetric group W88 is generated by the " adjacent transpositions >5 œ Ð5 5  "Ñ for 5 œ "ß á ß 8  ". Note also that the >5's satisfy the relations Free Groups and Presentations 349

# >>5" 5 >5 œ"ß >55" œ> ß >54 > œ>> 45 for 45Á„" Now let

\ œÖB"8" ßáßB × and let e be the relations

# B5 œ"ß B5" BB 5 5" œBB 5 5" Bß 5 BB 5 4 œBB 4 5 for 45Á„"

Let KœØ\±e Ù. Since W85 is generated by the elements > œÐ55"Ñ and satisfies the relations e with B>55 replaced by , Theorem 12.27 implies that if 9ÐKÑŸ8x, then W8 ¸Ø\±e Ù. We prove the former by induction on 8 .

# If 8œ# , then KœØBÙ"" where B" œ" and so KœÖ"ßB× has order # . Assume that the result holds for the subgroup L œØB"8# ßáßB Ù with relations e . We show that every +ÂL can be written in the form

+œB55" B âB 8" A for "Ÿ5Ÿ8", where A−L . Let us refer to a substring BB33" âB 8" as being in proper order. Then

+œ?ÐBB33" âB 8" Ñ@ for some 3Ÿ8" , where @−L . If ? is the empty string, then we are done. w Otherwise, let ?œ?B4 . Here are the possibilities:

1) If 4Ÿ3# , then B4 commutes with all factors to its right and so w +œ?ÐBB33" âB 8"4 ÑB@

where B@−L4 . 2) If 4œ3" , then w +œ?ÐBBBâBÑ@3" 3 3" 8" 3) If 4œ3 , then ww +œ?BÐBB3 3 3" âB 8" Ñ@œ?ÐB3" âB 8" Ñ@ 4) If 4 3" , then w +œ?BÐBB433"8" âB Ñ@ w œ ? ÐB344"48" âB B B âB Ñ@ w œ ? ÐB3 âB 4" B 4 B 4" âB 8" Ñ@ w œ?ÐBâB3 4" BâB 4 8" ÑB 4" @

where B@−L4" . 350 Fundamentals of Group Theory

Thus, in all cases, we can reduce the length of the substring appearing to the left of the substring in proper order by one symbol. Repeated application brings + to the desired form

+œB55" B âB 8" A It follows that 9ÐKÑ œ 8 † 9ÐLÑ Ÿ 8x.

Theorem 12.31 The symmetric group WØ8 has presentation \±eÙ , where

\ œÖB"8" ßáßB × and let e consist of the relations

# B5 œ"ß B5" BB 5 5" œBB 5 5" Bß 5 BB 5 4 œBB 4 5 for 45Á„" We close by noting that the relations above are equivalent to

#$# B5 œ"ß ÐB5" BÑ 5 œ"ß ÐBBÑ5 4 œ" for 4Ÿ5#

Exercises 1. An equational class ^ is abelian if all members of the class are abelian groups. Characterize abelian equational classes. 2. Let K% be the dicyclic group of order 8 , 8" , with presentation Ø\±e ÙœØBßC±B#8 œ"ßC # œB 8 ßCBœB #8" CœB " CÙ a) KD has exactly one involution . b) ^ÐKÑ œ ØDÙ c) KÎ^ÐKÑ ¸ H#8 3. Let \Ð be a nonempty set and let J\ß,Ñ be universal for \ . Let w AÐB"8 ßáßB Ñ be a word over \ and let 0ÀÖB"8 ßáßB ×ÄÖC "8 ßáßC × be an injection, where C−\3 . Prove that

AÐ,, B"8 ß á ß B Ñ œ " Í AÐ ,, C"8 ß á ß C Ñ œ "

4. Let J\ be the free group on the set œÖB"8ßáßB×. Show that J has a subgroup of index 7 for all "Ÿ7Ÿ8 . 5. Let J\ be the free group on œÖBßC× and let KœØ+Ù} Ø,Ù be the direct product of two infinite cyclic groups. The function 0À\ Ä K defined by 0BœÐ+ß"Ñß0CœÐ"ß,Ñ induces a unique mediating morphism 7ÀJ ÄK . What is the kernel of 7 ? 6. Let J\\ be the free group on . Prove the following: a) If ]©\ is nonempty, then J]\ ŸJ . b) If ] © \ is nonempty, then J] ∩ J\Ï] œ Ö"×. In particular, if B−\Ï], then BÂJ] . 7. Characterize the abelian groups that are free groups. 8. Let JL be a free group and let ŸJL have finite index. Show that intersects every nontrivial subgroup of J nontrivially. Free Groups and Presentations 351

9. Let J\ be free on the disjoint union ∪] of nonempty sets. Prove that JÎØ]Ùnor is free on \ . 10. Show that if Rüz K and KÎR is free, then KœR L for some LŸK . 11. Prove that if \" , then the free group J is centerless. kk \ 12. Let JÀ be a free group. Suppose that 5 Eq »F is an epimorphism and that 7-ÀJ ÄF is a homomorphism. Prove that there is a ÀJ ÄE for which 5-‰œ 7. This is called the projective property of free groups. 13. If ^ is a class of groups, then a group K is residually ^^ or a residually - group if for any "Á+−K , there is a normal subgroup RK+ ü for which +ÂR++ and KÎR is a ^ -group. a) Prove that a group K is residually ^ if and only if it is isomorphic to a subdirect product of ^ -groups. b) Prove that if a œÖR±3−M×3 is a family of normal subgroups of a group and if is a -group for all , then is KKÎR33^ 3−MKÎR+ residually ^. 14. Prove that the presentation Ø] ±f Ù œ Ø-ß . ± -87 œ "ß . œ "ß .- œ - = .Ù defines the group LœÖ-.34 ±!Ÿ38 and !Ÿ47× where 9ÐLÑœ789Ð-Ñœ89Ð.Ñœ7, , and

5 Ð-35 . ÑÐ- 4j . Ñ œ - 34=5j .

15. Show that H#8 is presented by Ø] ±f Ù œ ØBß C ± B## œ "ß C œ "ß ÐBCÑ 8 œ "Ù 16. Let 53 and be distinct symbols. Let HœÖ35333 ß ±3− ™ × with product defined by the properties 53553#œ"ß œ ". Thus, 3533œ 53 a) Show that HH is a group and that is presented by

#" T" œØBßC±B œ"ßCBœBC Ù b) Show that H is also presented by

## T# œØBßC±B œ"ßC œ"Ù

Any group presented by TT"# or is called an infinite dihedral group. 17. Let KœØ\±e Ù be a finite presentation of K , where \œÖBßáßB×"8 w and e œÖ<ßáß<"7 × and 78 . Show that K is an infinite group as follows. 352 Fundamentals of Group Theory

a) Reduce the problem to the abelian case as follows. Let E\ be the free abelian group on \ . Show that there is an epimorphism from ww JÎØÙEÎØÙ\\eenor to . w b) Show that EÎØÙ\ e is infinite. Chapter 13 Abelian Groups

In this chapter, we study abelian groups. We will write abelian groups using additive notation. One of our main goals is to provide a complete solution to the classification problem for finitely-generated abelian groups. That is, we will describe all finitely-generated abelian groups up to isomorphism.

Perhaps the most natural place to begin is to observe that the elements of finite order in an abelian group EE form a subgroup of , since 9Ð+,Ñ œlcm Ð9Ð+Ñß 9Ð,ÑÑ Let us remind the reader that this is not the case in a general group. For example, in KPÐ#ß‚ Ñ , let !" !" Eœ and Fœ Œ"! Œ" " Then EF and have finite order but EF has infinite order.

Definition Let E+ be an abelian group. An element −E that has finite order is called a torsion element. The subgroup EEtor of all torsion elements in is called the torsion subgroup of E . A group that has no nonzero torsion elements is said to be torsion free and a group all of whose elements are torsion elements is said to be torsion .

Of course, a finite group is a torsion group, but the converse is not true: i! Consider the direct product ™™# of an infinite number of copies of # .

The quotient EÎEtor is easily seen to be torsion free and it would be nice if Etor was always a direct summand of E , that is, if

EœEtor  F for some FßE , since then F ¸ EÎEtor would be torsion free and we would

S. Roman, Fundamentals of Group Theory: An Advanced Approach, 353 DOI 10.1007/978-0-8176-8301-6_13, © Springer Science+Business Media, LLC 2012 354 Fundamentals of Group Theory have a nice decomposition of any abelian group. Unfortunately, this is not the case. To show this, we require a definition.

Definition Let E+ be an abelian group. An element −E is divisible by an integer 8, if there is an element −E+ for which œ8,E . A group is divisible if every element is divisible by every nonzero integer.

Theorem 13.1 The torsion subgroup of an abelian group need not be complemented. Proof. Let Eœ }™: be the external direct product of the abelian groups ™: , taken over all primes :+−E . For , we use the notation +: in place of +Ð:Ñ . The torsion subgroup Etor is the subgroup of all elements with finite support. If EœEtor  F, then F¸EÎEtor , so this prompts us to look for an isomorphism-invariant property that holds in EÎEtor but not in F . This property is divisibility.

Specifically, we will show the following:

1) No nonzero element of EF (and hence ) is divisible by all primes : . 2) There are nonzero elements of EÎEtor that are divisible by all primes : .

Since the only element of ™: that is divisible by :! is , if +−E is divisible by :, then +: œ! . Hence, if + is divisible by all primes, it follows that +œ! .

Now let +−E be the element for which +: œ" for all : . For a given prime : , to say that +œ:, for some ,−E is to say that :,; œ" for all primes ; . But if ;Á:, then : has an inverse <:; in the field ™ . Hence, if , is defined by ,: œ<: and ,œ!: , then

Ð+:,Ñ:: œ+ :, : œ" and for all ;Á: ,

Ð+:,Ñ;; œ+ :, ; œ! and so +:,−Etor.

Despite the negative nature of the previous result, we will show that if E is a finitely-generated abelian group, then Etor is complemented. This is a key to the structure theorem for finitely-generated abelian groups. Abelian Groups 355

An Abelian Group as a ™ -Module An abelian group E has a natural scalar multiplication defined upon it, namely, multiplication by the integers: If α™−+−E and , we set !œif α ! ÝÚ Ý +â+if α ! α+œ ðóóñóóò (13.2) Û α terms Ý Ðαα Ñ+if  ! Ü Under this operation, an abelian group E is a ™ -module, as defined below.

Definition Let be a commutative ring with identity. An -module ( or a module over Q) is an abelian group , together with a scalar multiplication, denoted by juxtaposition, that assigns to each pair Ð<ß ?Ñ − ‚ Q , an element <@ − Q. Furthermore, the following properties must hold for all <ß = − and ?ß @ − Q: <Ð?@Ñœ

The ring Q is called the base ring of and the elements of are called scalars.

Note that an abelian group E is a ™™ -module and, conversely, a -module is nothing more than an abelian group, since the scalar multiplication of a ™ - module Q must be the operation defined in (13.2). Moreover, the subgroups of the abelian group QQ are the submodules of the module and the group homomorphisms between the abelian groups QR to are the linear (module) maps between the ™ -modules QR and . The Classification of Finitely-Generated Abelian Groups We solved the classification problem for finite abelian groups in Theorem 5.7. Also, Theorem 12.14 solves the classification problem for free abelian groups. Using these theorems, we can now solve the classification problem for finitely- generated abelian groups. The first step is to note the following.

Theorem 13.3 A finitely-generated abelian group E is torsion free if and only if it is free. Proof. We leave proof that if E is free, then it is torsion free as an exercise. For the converse, let W œÖ@"8 ßáß@ × be a generating set for the torsion-free abelian group EE . The proof is based on the fact that since is torsion free, it is a torsion-free ™ -module. Moreover, for any +−E , the multiplication map ..™++ÀEÄE defined by Bœ+B is a -module automorphism of E . 356 Fundamentals of Group Theory

Let Ö?"5 ßáß? × be a maximal linearly independent subset of W . Of course, if 5œ8, then W is a basis for EE and so is free. Assume otherwise and let

W œÖ?"5"85 ßáß? ß@ ßáß@ ×

For each @Ö?ßáß?ß@×3"5 , the set 3 is linearly dependent and so there exist +−3 ™ for which

+@33 −Ø?ßáß?Ù " 5

If +œαα"85 â , then

.+"5"8Eœ+Ø? ßáß? ß@ ßáß@5"5 Ù©Ø? ßáß? Ù and since the latter is a free abelian group, Theorem 12.18 implies that .+E is also free and therefore so is E .

If EE is a finitely-generated abelian group , then Etor is a subgroup of E and the quotient EÎEtor is torsion-free and finitely generated and so is free. Since the canonical projection map 1ÀEÄEÎEtor is an epimorphism, Theorem 12.19 implies that Etor has a complement:

EœJ Etor where J¸EÎEtor is free and finitely-generated. Moreover, Theorem 12.16 implies that JE has finite rank and since tor is finitely generated (Theorem 2.21), torsion and abelian, it is finite.

As to uniqueness of this decomposition, if EœJw  X w where JX is free and is torsion, then clearly, XŸE+tor . But if −Etor and +œ>0 where >−X and 0−Jw and so 0œ+> is torsion, whence 0œ! w and +−X . Thus, X œEtor . It follows that JJ and are both complements of Etor and hence are isomorphic.

We can now state the fundamental theorem of finitely-generated abelian groups.

Theorem 13.4 (The fundamental theorem of finitely-generated abelian groups) Let E be a finitely-generated abelian group, with torsion subgroup Etor. 1) Then

EœJ Etor

where J< is free of finite rank and Etor is finite. As to uniqueness, if EœJw  X Abelian Groups 357

w w where JX is free and is torsion, then XœEÐtor and rkJќРrk JÑ. The number <œrk ÐJÑ is called the free rank of E . 2)( Invariant factor decomposition) E is the direct sum of a finite number of cyclic subgroups

E œ ØB"<"8 Ù â ØB Ù  Ø? Ù  â Ø? Ù

where 9ÐB33 Ñ œ ∞ and 9Ð? Ñ œα3 # and

αα88"±±â± α "

The orders α3 are called the invariant factors of E . 3)( Primary cyclic decomposition) If

//5ß" 5ß7 α5 œ:" â:7 then

E œ ØB"< Ù â ØB Ù

ÒØ?"ß"" Ù â Ø?ß5"7 ÙÓ â ÒØ? 7ß"7 Ù â Ø?ß5 ÙÓ

/3ß4 where 9ÐB33 Ñ œ ∞ and 9Ð?ß4 Ñ œ :3 and

/ / â / "3ß" 3ß# 3ß53

/3ß4 The numbers :E3 are called the elementary divisors of . /3ß4 3) The multiset Ö×α3 of invariant factors and the multiset Ö:×3 of elementary divisors are uniquely determined by the group E . Projectivity and the Right-Inverse Property A diagram of the form 57 EFGÒÒ where EF , and G are groups and 57 and are group homomorphisms is exact if imÐÑœ57ker ÐÑ It is customary to regard the figure 5 EF!ÒÒ as exact and to omit the second homomorphism, since it must be the zero map. Thus, this figure says precisely that 5 is surjective. Similarly, the figure 5 !EFÒÒ says precisely that 5 is injective.

According to Theorem 5.23, a group homomorphism 5ÀK ÄKw has a right inverse 55 if and only if it is surjective and kerÐÑ is complemented: 358 Fundamentals of Group Theory

Kœker Ð5z Ñ O for some OŸK and in this case,

Kœker Ð5z Ñim Ð 5 Ñ Let us say that an abelian group T has the right-inverse property if every epimorphism 5ÀEq » T , where E is abelian, has a right inverse. This is illustrated in Figure 13.1.

P σ R ι

A P 0 σ

Figure 13.1 An apparently stronger property is given in the following definition.

P λ τ

A B 0 σ

Figure 13.2 Definition An abelian group T is projective if, referring to Figure 13.2, for any epimorphism 5ÀEq» F of abelian groups and any homomorphism 7-ÀT ÄF, there is a homomorphism ÀT ÄE for which 5-‰œ 7 In this case, we say that 7- can be range-lifted to .

While the projective property appears to be stronger than the right-inverse property, the two properties are actually equivalent. The following theorem is the main result on projective abelian groups.

Theorem 13.5 Let K be an abelian group. The following are equivalent: 1) K is a free abelian group. 2) K is projective. 3) KÀ has the right-inverse property, that is, any surjection 7 EÄK , where E is abelian, has a right inverse. 4)» If 55ÀEq K , where E is abelian, then kerÐÑ is a direct summand of E . Proof. We have seen that 3) and 4) are equivalent. Assume that 1) holds and let K œJ\\ be free on \ . Let 57 ÀEÄF be surjective and let ÀJ ÄF . Then for Abelian Groups 359

each B−\ , there is an +BB −E for which 57+ œ B . Define a function 0À\ Ä E by 0ÐBÑœ +B\ . Since J is free, there is a unique homomorphism --ÀJ\B ÄE for which Bœ+ . Then

5-‰ÐBÑœ+œB 5B 7 and so 5-‰œ 7 on \ and therefore on J\\ . Hence, KœJ is projective and 2) holds. It is clear that 2) implies 3).

Finally, suppose that 3) holds. The identity map +ÀK ÄK can be lifted to a homomorphism 5ÀJKK ÄK where J is the free abelian group with basis K . Of course, 55 is surjective and so 3) implies that has a right inverse 5KÀK ÄJ . Hence, kerÐÑ5 is complemented, that is,

JœK ker ÐÑ5  W But 5ÀW ÄK is an isomorphism and so K is isomorphic to a direct summand of a free abelian group and is therefore free abelian by Theorem 12.18. Hence, 1) holds. Injectivity and the Left-Inverse Property

We have seen that a monomorphism 5äÀE F has a left-inverse 5P if and only if imÐÑ55 has a complement OF in , in which case O¸ker ÐP Ñ. Dual to the right-inverse property is the left-inverse property: An abelian group I has the left-inverse property if every monomorphism 5äÀI F to an abelian group F has a left inverse.

Dual to the projective property is the injective property.

Definition An abelian group I is injective if, referring to Figure 13.3, for any embedding 5äÀE F and any homomorphism 7ÀEÄI there is a homomorphism -ÀEÄI for which -5‰œ 7 In this case, we say that 7-ÀEÄI can be domain-lifted to ÀF ÄI by 5ÀEÄF.

σ 0 A B τ λ E

Figure 13.3 Baer [3] proved that if this condition holds in the special case where Fœ™ is the group of integers and EŸ™5 and ÀEÄ ™ is the inclusion map, then the condition holds in general and I is injective. 360 Fundamentals of Group Theory

Theorem 13.6 An abelian group I is injective if and only if it satisfies Baer's criterion: Any homomorphism 7ÀM ÄI , where M is a subgroup of the integers ™- can be extended to a homomorphism ÀÄI™ on ™ . Proof. We wish to show that any homomorphism 7ÀEÄI can be domain- lifted to -5ÀF ÄI by any monomorphism ÀEä F . Note that 7 can be domain-lifted by 5755 to ‰" À EÄI, since 5 ÀEÄ 5 E is an isomorphism. This is shown in Figure 13.4.

σ 0 A σ(A) X B τ τσ−1 λ E

Figure 13.4 Suppose we have domain-lifted 75- by to À\ ÄF , where 5 EŸ\ ŸF, that is, 7-5œ‰ If \F , then for any +−\ÏF , we can lift -5 by to a map on \Ø+Ù as follows. We need only define -™- on Ø+Ù œ + . But is already defined on ™-+∩\œM+ for some MŸ^ . So if " ÀMÄI is defined by

-α"ÐÑœÐ+Ñ -α then Baer's criterion implies that --"# can be extended to ÀÄI™ . Then the map -À\Ø+ÙÄI defined by

---ÐB  <+Ñ œ ÐBÑ # Ð<Ñ+ for any <−™ is well defined since if B<+œC=+, then BCœÐ=<Ñ+ and so =<−M , which implies that

--"Ð=  <Ñ œ ÐÐ=  <Ñ+Ñ œ - ÐB  CÑ and so

------ÐB  <+Ñ œ ÐBÑ ## Ð<Ñ+ œ ÐCÑ  Ð=Ñ+ œ ÐC  =+Ñ Moreover, since -5‰œ 7 and im ÐÑŸ 5 dom ÐÑ -, it follows that -5‰œ 7 .

This discussion prompts us to apply Zorn's lemma. Let f be the collection of all pairs Ð\ß-5 Ñ , where ÐEÑ © \ © F and - is a lifting of 75 by to a map on \ . Then f5 is nonempty since we may take \œ E . Order f by setting Ð\ß.- ÑŸÐ]ß Ñ if \©] and -. l] œ . Abelian Groups 361

If is a chain in , let . If , then one of .œÖÐ\ß33 Ñ× f Y œ- \3 B−\ 3 ∩\ 4 .3 and ..43 is an extension of the other and so Bœ..4 B . Hence, we may define by ..Bœ33 B for any 3 satisfying B−\ . Then ÐYß. Ñ is an upper bound for and so Zorn's lemma implies that f- has a maximal element ÐQß Ñ . But if QF, then there is a further lifting of - , which contradicts the maximality of ÐQß- Ñ and so Q œ F .

We can now present our main theorem on injective groups.

Theorem 13.7 Let K be an abelian group. The following are equivalent: 1) K is injective. 2) K is divisible. 3) KÀ has the left-inverse property, that is, every monomorphism 5äKF to an abelian group F has a left inverse. 4)i If 5äÀK E , where E is abelian, then mÐÑ5 is a direct summand of E . Proof. We know that 3) and 4) are equivalent. Assume first that K is injective. For 1−K and 8−™7 we seek 2−K for which 1œ82 . The map ÀØ8ÙÄI defined by 7™Ð<8Ñ œ <1 for all < − can be domain-lifted by the inclusion map 4À Ø8Ù Ä ™, that is, -7‰4œ Hence, 1œ7- Ð8Ñœ ‰4Ð8Ñœ - Ð8Ñœ8Ð"Ñ - and so 2œ- Ð"Ñ . Thus, K is divisible and 1) implies 2).

To see that 2) implies 1), we show that 2) implies the Baer criterion. Let 7™7™ÀØ5ÙÄK where Ø5ÙŸ . To show that can be extended to , define -™++À Ä K by - Ð8Ñ œ 8+ , for + − K . Then -+ extends 7 if - +Ð5Ñ œ 7 Ð5Ñ, that is, if 7Ð5Ñ œ 5+ . But since K is divisible, there is an + − K for which this holds. Hence, 2) implies 1).

It is clear that 1) implies 3). Finally, suppose that I has the left-inverse property. We wish to show that I is injective. The story of the proof is shown in Figure 13.5. 362 Fundamentals of Group Theory

σ 0 A B λ τ κB µB G GBúû κG

πS µG GBúû S

Figure 13.5 We seek the map --ÀF ÄK for which ‰5 œ7 . A first attempt might be to consider the direct sum KF , with canonical injections ,,KF and . Since ,,KKÀK ÄK F is an injection, it has a left inverse ÐÑP and we can take -,œÐKP Ñ ‰ , F. However, it may not be the case that -57‰ œ .

On the other hand, perhaps we can factor the direct sum KF by a subgroup W, with projection map 1W in such a way that the compositions

.1,FWFœ‰ÀFÄÐKFÑÎW and

.1,KWKœ‰ÀKÄÐKFÑÎW satisfy

.5.7FK‰œ ‰ where .-KK is left-invertible. In this case, if œÐ. ÑP ‰.F ÀFÄK, then

-,‰œÐщ .KP . F ‰œÐщ 5 . KP . K ‰œ 7 7 as desired. But the condition .5.7FK‰œ ‰ is

1,51,7WF‰‰œ‰‰ WK that is, Ð!ß57 +ÑWœÐ +ß!ÑW for all +−E and so if WœÖÐ+ß+ѱ+−E×75 then this equation holds. Also, ..KK is injective since 1−ker Ð Ñ implies that Ð1ß!Ñ−W and so Ð1ß!ÑœÐ+ß75 +Ñ. Hence, 5 +œ! and so +œ! , whence 1œ!. Abelian Groups 363

Exercises 1. Let E+ be a torsion-free abelian group. Prove that if −E is divisible by 8−™, then the “quotient” ,−E is unique. 2. Prove that a subset of an abelian group E is a basis if and only if for every +−E , there are unique elements ,"8 ßáß, − and unique scalars αα™"8ßáß − for which

+œαα"" , â88 , 3. Let E be a free abelian group. Show that it is not necessarily true that any linearly independent set of size rkÐEÑ is a basis for E . 4. Let EE be a free abelian group of finite rank and let œF G . Prove that rkÐEÑ œ rk ÐFÑ  rk ÐGÑ. 5. Prove that any abelian group E is isomorphic to a quotient of a free abelian group. 6. A subgroup LŸK of an additive abelian group K is pure if for any 2−L and 7−™ , 2œ71 for 1−K Ê 2œ72ww for some 2 −L a) Prove that if K is an abelian group, then the set of periodic elements of K is pure. b) Prove that any direct summand of an abelian group is pure. c) Find a nonpure subgroup of the cyclic group ™8# . 7. Prove that an abelian group E is finitely generated if and only if it is isomorphic to a quotient of a free abelian group of finite rank. 8. Let E be a free abelian group of finite rank 8 . Let W œÖ="8 ßáß= × be a generating set for EW . Prove that is a basis for E . Hint : Let \ œÖB"8 ßáßB × be a basis for E and define the map 7ÀEÄE by 7ÐB33 Ñ œ = and extending to a surjective homomorphism. What about kerÐÑ7 ? 9. Let J8 be a free abelian group of rank . Let L be a subgroup of J of rank 58. Prove that KÎL contains an element of infinite order. 10. Show that, in general, a basis for a subgroup of a free abelian group cannot be extended to a basis for the entire group. 11. Prove that any free abelian group is torsion free. 12. Let KK be a torsion-free abelian group. Suppose that has a subgroup J that is free and has finite index. Prove that K is free abelian. 13. Let E be a finite abelian group. a) Prove that if :E œ Ö!× , then E is a vector space over the field ™™™: œÎ:. b) Prove that for any subgroup WE of the set WÐ:Ñ œ Ö@ − W ± :@ œ !× 364 Fundamentals of Group Theory

is also a subgroup of EEœWX and if  , then EœWÐ:Ñ Ð:Ñ X Ð:Ñ 14. Let J8 be a free abelian group of rank . Show that AutÐJÑ is isomorphic to the group of all 8‚8 matrices with determinant equal to „" . 15. Let  be the multiplicative group of positive rational numbers. a) Show that ™ is isomorphic to the additive group ÒBÓ of polynomials over the integers. Hint : Use the fundamental theorem of arithmetic. b) Show that the multiplicative group  of positive rationals is a free abelian group of countably infinite rank. 16. Prove that the quasicyclic group ™Ð:∞ Ñ is divisible. 17. Prove that a free abelian group is not divisible. 18. a) Show that the rational numbers  are not finitely generated as an abelian group under addition. b) Show that  is divisible and therefore not free by an earlier exercise. c) Show that  is torsion free. Thus, a torsion-free abelian group need not be free. 19. Prove that if Eœ} E33 , where E and E are abelian groups, then E is injective if and only if E33 is injective for all . 20. Let EE be an abelian group. Show that the set div of all divisible elements is a subgroup of EE . Show that div is a direct summand of E . 21. Let HH be a divisible group. Prove that tor is divisible. 22. Let EW be a finitely generated abelian group. For any subgroup of E let

EœÖ+−E±+:Ð:Ñ is a -element ×

Show that E«EÐ:Ñ . Describe the order of EÐ:Ñ in terms of the elementary divisors of E . 23. How can one tell from the elementary divisors of a finite abelian group when that group is cyclic? 24. Use one of the decomposition theorems to prove that a finite abelian group E5 has a subgroup of order for every 5±9ÐEÑ . 25. Find, up to isomorphism, all finite abelian groups of order "!!! . Which are cyclic? 26. Prove that every abelian group of order 426 is cyclic. 27. Let : be a prime. Let

E œ Ø?"7 Ù â Ø? Ù

03 where 9Ð?3 Ñ œ : and let

F œ Ø@"8 Ù â Ø@ Ù

/3 where 9Ð@3 Ñ œ : . Assume that EŸF . a) Prove that 7Ÿ8. Abelian Groups 365 b) Prove that

0Ÿ/ß0787"8""87" Ÿ/ßáß0Ÿ/

References

[1] Allen, W., Complex Groups, The American Mathematical Monthly, Vol. 67, No. 7 (1960) 637–641. [2] Aschbacher, M., The Status of the Classification of Finite Simple Groups, Notices of the AMS, Volume 51, Number 7, 2004. [3] Baer, R., Situation der Untergruppen und Struktur der Gruppe, S.-B. Heidelberg. Akad. Math.-Nat. Klasse 2 (1933) 12–17. [4] Brauer, R. and Fowler, K. A., On Groups of Even Order, Annals of Mathematics, 2nd Ser., Vol. 62, No. 3 (1955) 565–583. [5] Cassidy, P. J., Products of commutators are not always commutators: An example, The American Mathematical Monthly, Vol. 86, No. 9 (1979) 772. [6] Cauchy, A.-L., Mémoire sur les arrangements que l’on peut former avec des lettres données et sur les permutations ou substitutions à l’aide desquelles on passe d’un arrangement à un autre. Exercises d’analyse et de physique mathématique 3 (1845) 151–252. (Reprinted in: Oeuvres complètes, vol. 13, second series. Gauthier-Villars, Paris (1932) pp. 171– 282.) [7] Cayley, A., On the theory of groups as depending on the symbolic equation )8 œ" , Phil. Mag., 7 (4) (1854), 40–47. [8] Christensen, C., Complementation in groups, Math. Zeitschr. 84 (1964) 52–69. [9] Christensen, C., Groups with complemented normal subgroups, J. London Math. Soc., 42 (1967) 208–216. [10] Feit, W. and Thompson, J. G., A Solvability Criterion for Finite Groups and Some Consequences, Proc. Nat. Acad. Sci. USA 48 (1962) 968–970. [11] Feit, W. and Thompson, J. G., Solvability of Groups of Odd Order, Pacific J. Math 13 (1963) 775–1029. [12] Frattini, G., Intorno alla generazione dei gruppi di operazioni, Rend. Atti Accad. Lincei, (4) 1 (1885) 281-285, 455–457. [13] Frobenius, G., Berliner Sitzungsberichte, (1895) 995. [14] Fuchs, L., Abelian Groups, Pergammon Press, 1960. [15] Guralnick, R., Expressing group elements as commutators, Rocky Mountain J. Math., Vol. 10, No. 3 (1980), 651–654. [16] Hall, M., The Theory of Groups, AMS Chelsea, 1976.

S. Roman, Fundamentals of Group Theory: An Advanced Approach, 367 DOI 10.1007/978-0-8176-8301-6, © Springer Science+Business Media, LLC 2012 368 References

[17] Hall, P., Complemented groups, J. London Math. Soc., 12, (1937), 201– 204. [18] Hirshon, R., On Cancellation in Groups, American Mathematical Monthly, Vol. 76, No. 9 (1969) 1037–1039. [19] Kappe, L.-C. and Morse, R., On commutators in groups, Proceedings of Groups St. Andrews 2005 LMS Lecture Series, also available online at http://faculty.evansville.edu/rm43/publications/commutatorsurvey.pdf. [20] Kappe, L.-C. and Morse, R., On commutators in p-groups, J. Group Theory, Vol. 8, No. 4 (2005), 415–429. [21] Kertész, A., On groups every subgroup of which is a direct summand, Publ. Math. Debrecen, 2 (1952), 74–75. [22] MacDonald, I. D., Commutators and their products, Amer. Math. Monthly, Vol. 93, No. 6 (1986), 440–444. [23] McKay, J. H., Another proof of Cauchy’s group theorem, American Mathematical Monthly 66, 1959, 119. [24] Meo, M., The mathematical life of Cauchy's group theorem, Historia Mathematica 31 (2004) 196–221. [25] Miller, G. A., The regular substitution groups whose orders is less than 48, Quarterly Journal of Mathematics, 28, (1896), 232–284. [26] Robinson, D., A Course in the Theory of Groups, Springer, 1996. [27] Roman, S., Field Theory, Second Edition, Springer, 2006. [28] Roman, S., Advanced Linear Algebra, Third Edition, Springer, 2007. [29] Schreier, O., Über den Jordan-Hölderschen Satz, Abh. math. Sem. Univ. Hamburg 6 (1928), 300-302. [30] Schmidt, R., Subgroup Lattices of Groups, De Gruyter Expositions in Mathematics 14, 1994. [31] Spiegel, E., Calculating commutators in groups. Math. Mag. , Vol. 49, No. 4 (1976), 192–194. [32] Sylow, L., Théorèmes sur les groupes de substitutions, Math. Ann. 5 (1872), 584–594. [33] Weber, H., Lehrbuch der Algebra, Vol 2. Braunschweig, second edition, 1899. [34] Weigold, J., On Direct Factors in Groups, J. London Math Soc., 35 (1960), 310–320. [35] Wielandt, H., Eine Verallgemeinerung der invarianten Untergruppen, Math. Z., 45 (1939), 209–244. [36] Weinstein, M., Examples of Groups, Second Edition, Polygonal Publishing, 2000. [37] Zassenhaus, H. J., Qum Satz von Jordan-Hölder-Schreier, Abh. Math. Semin. Hamb. Univ. 10 (1934) 106–108. References on the Burnside Problem [38] Adjan, S.I., The Burnside problem and identities in groups, Translated from the Russian by John Lennox and James Wiegold, Ergebnisse der References 369

Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 95 (1979), Springer-Verlag. [39] Burnside, W., On an Unsettled Question in the Theory of Discontinuous Groups, Quart. J. Pure Appl. Math. 33 (1902) 230–238. [40] Golod, E. S., On Nil-Algebras and Residually Finite -Groups, Isv. Akad. Nauk SSSR Ser. Mat. 28 (1964) 273–276. [41] Hall, M., Solution of the Burnside Problem for Exponent Six, Ill. J. Math. 2 (1958) 764–786. [42] Ivanov, S., On the Burnside Problem on Periodic Groups, Bulletin of the American Mathematical Society, Volume 27, Number 2 (1992) 257–260. [43] Levi, F. and van der Waerden, B. L., Über eine besondere Klasse von Gruppen, Abh. Math. Sem. Univ. Hamburg 9 (1933) 154–158. [44] Novikov, P. S. and Adjan, S. I., Infinite Periodic Groups I, II, III., Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968) 212–244, 251–524, and 709–731. [45] Sanov, I. N., Solution of Burnside's problem for exponent four, Leningrad State Univ. Ann. Math. Ser. 10 (1940) 166–170.

List of Symbols

Ÿ : subgroup ¡+¡,: means that , covers + ³ the item on the left is defined by the item on the right œÀ the item on the right is defined by the item on the left “ : disjoint union of sets ‚ : cartesian product Š: internal direct sum; in the abelian case }: external direct product {: external direct sum z : semidirect product OLzümod NOKßKœLOßL∩OœN : «L«K: means that L is characteristic in K öö: LK means that L is characteristic and proper in K ä : denotes an embedding (injective map) Ò?Ó: the equivalence class containing ? Ð5ß 8Ñ: If 5 and 8 are integers, this is gcdÐ5ß 8Ñ ACC: ascending chain condition BCC: both chain conditions DCC: descending chain condition WfÐKÑ: the set of direct summands in K GÐ+Ñ8 : the cyclic group Ø+Ù of order 8 ncÐ\ß KÑ: the normal closure of \K in +: identity map M8 œ Ö"ß á ß 8× ]: for Sylows so we do not conflict with symmetric group. FÐKÑ: the Frattini subgroup of K ÐK À LÑ: index of L in K gcdÐ+ß ,Ñ: greatest common divisor of + and , lcmÐ+ß ,Ñ: least common multiple of + and , ÒLß OÓ: the commutator subgroup of LO and LL‰: the normal interior of LìO: The set product LO where L and O are essentially disjoint. LÀÐ3Ñ 1 ÖL±4Á3×4

371 372 List of Symbols

LÀÐ5Ñ a term in the sequence of normal closures of LŸK +œÖ+±7−Q×Q7 conjKÐLÑ: the set of conjugates of LK by Fix\ÐKÑ : the set of elements of \ fixed by the action of K LLK: the normal closure of in K >KÐLÑ œ ÒLß KÓ 'KÐLÑ œ \ where \ÎL œ ^ÐKÎLÑ ÒKÓ: the set of commutators of K ^^ÐKÑ: the set of all normal subgroups of K that belong to class KÎ^: the set of all normal subgroups R of K for which KÎR belongs to class ^ _^ÐÑÀThe set of laws of ^ groups subÐKÑ: the lattice of subgroups of a group K . sub.ÐKÑ: the subgroups of a finite :K: -group of order HH-subÐKÑ : the lattice of -subgroups of a group K norÐKÑ: the lattice of normal subgroups of a group K . nor.ÐKÑ: the normal subgroups of a finite :K: -group of order HH-norÐKÑ : the lattice of normal -subgroups of a group K subÐRà KÑÀ the family of all subgroup of KR containing subnHÐLà KÑ: the family of all HH -subnormal subgroups of an -group K that contain L subnHÐKÑ: the family of all HH -subnormal subgroups of an -group K Syl:ÐKÑ: the Sylow : -subgroups of K Syl:ÐWà KÑ: the Sylow : -subgroups of K that contain W suppÐ0Ñ : the support of 0 =ÐLß KÑÀ the length of the sequence of normal closures of LK in . f.ÐKÑ: for a : -group K , the set of subgroups of order : . a.ÐKÑ: for a : -group K , the set of normal subgroups of order : kÐWÑ À the power set of a set W bÐKÑKCompSerH : has an H -composition series bÐLàOÑKCompSerH : has a composition series from LO to ‡ ™™8: Ö+ −8 ± Ð+ß 8Ñ œ "× SDR: system of distinct representatives homÐKß LÑ: The set of all homomorphisms from KL to › : the restricted wreath product ›› : the complete wreath product ›› <: the regular wreath product ü : normal – : normal and proper üü: subnormal ––: subnormal and proper \œ\“\w", where \ is a nonempty set Index

#-transitive, 230 base, 181 %-group, 24 basis, 319, 326 BCC, 6, 76, 137 abelian series, 293 Bernstein Theorem, 12 abelian, 20 binary operation, 19 abnormal, 99, 262, 314 Birkhoff, 329, 330 ACC, 6, 60, 76, 136 block, 10, 212 acceptable, 55 both chain condition, see BCC aC-group, 165 bottom, 4 action, 123, 207 Brauer, 270 acts, 123, 207 Burnside Theorem, 310 addition, 20 Burnside Basis Theorem, 218 aD-group, 165 Burnside problem, 35 adjacent, 189 butterfly lemma, 119 affine transformation, 101 algebraic closure, 305 cancellable 7, 157 alphabet, 2 canonical forms, 10, 11 alternating group, 33, 203 canonical projection, 108 aNC-group, 165 Cantor's theorem, 12 antichain, 3 cardinal number, 12 antisymmetry, 2 cardinality, 12 antitone, 5 cartesian product, 13 aperiodic, 21 Cassidy, 87 8:-argument, 245 Cauchy's theorem, 79, 124 ascending chain condition, see center, 33, 81 ACC center-intersection property, 215 associativity, 13, 20 centerless, 33, 81 asymmetry, 3 central in, 117 automorphism, 106 central series, 293 central, 33, 81, 144, 216 Baer, 169 centralizer, 82 Baer's criterion, 360 chain, 3 base field, 305 change of context map, 338 base ring, 355 characteristic series, 77

373 374 Index characteristic, 69 H-composition series, 278 characteristically simple, 121 concatenation, 294 chief distance, 281 concrete free abelian group, 331 chief factors, 278 concrete free group, 325 chief length, 281 concrete -free group, 328 chief series, 278 concrete presentation, 339 Chinese remainder theorem, 146, K-congruence relation, 211 186 congruence relation, 67 class equation, 209, 210 conjugacy class, 30, 82, 84, 205, class, 291 214 ^=-class, 292 conjugate, 29 ^8-class, 292 conjugation by, 30 classification problem, 108, 263 continuum, 14 closed interval, 3 coordinate, 152 closed, 19 core, 125 closure, 138 correspondence theorem, 113 co-Hopfian, 147 H-correspondence theorem, 276 cointersection, 291 coset product rule, 66 cojoin, 292 coset representative, 42 combinatorial group theory, 343, countable, 12 344 countably infinite, 12 commutative, 20 :-cover, 216 commutativity rule, 151 covers, 4 commutativity, 13 cycle decomposition, 25, 192 commutator subgroup, 84, 146, 307 cycle representation, 192 commutator, 84, 93 cycle structure, 25, 192 commute elementwise, 72 cycle, 24, 191 commute, 20, 111 cyclic group, 22 complement, 149, 151 cyclic series, 293 complemented, 149 cyclic subgroup, 34 complements modulo, 172 complete invariant, 11 DCC, 6, 76, 137 complete lattice, 8 decidable, 343 complete sublattice, 9 decision problem, 343 complete system of invariants, 11 decision procedure, 343 complete wreath product, 181 Dedekind law, 34 complex group, 102 Dedekind, 85 complex product, 31 defined by generators and relations, complexes, 31 338 component, 73 derived length, 308 componentwise product, 24 derived series, 308 H-composition distance, 281 derived subgroup, 84 composition factors, 278 descending chain condition, see H-composition factors, 278 DCC H-composition length, 281 dicyclic group, 53, 348, 351 composition series, 278 direct complement, 151 Index 375 direct factor, 75, 153 external direct product, 24, 152 direct product, 74, 75, 151, 153 external direct sum, 152 direct sum, 75, 153 external semidirect product, 176 direct summand, 75, 153 directed, 3, 37 H-factor group, 274 disjoint union, 13 factor group, 67 disjoint, 25, 192 factored through, 319 distributive lattice, 60 factored uniquely, 111 distributive laws, 60 faithful, 123, 207, 208 distributivity, 13 Feit, 310 divisible, 145, 147, 354 Feit–Thompson, 84 domain-lifted, 359 Fermat's little theorem, 44 double cosets, 233 field extension, 305 finitary operation, 19 edges, 189 finite exponent, 28 :-element, 80, 215 finite, 1, 12, 20, 338 elementary abelian group, 121 finitely f-generated, 138 elementary divisors, 160, 357 finitely H-generated, 275 embedded, 106 finitely generated, 35 embedding, 106 finitely presented, 340 empty word, 2 first isomorphism theorem, 112 endomorphism, 106 first H -isomorphism theorem, 276 endpoint, 77 Fitting's Lemma, 141 epimorphism, 105 Fitting's Theorem, 302 equational class, 327 fix, 207 equivalence class, 10 Fowler, 270 equivalence modulo, 41 Frattini argument, 210 equivalence relation, 10 Frattini subgroup, 127 equivalence, 230 Frattini, 244 K-equivalent, 208 free generators, 319, 326 H-equivalent, 277 free group, 319 equivalent, 230, 347 free ^-group, 326 essentially disjoint, 33 free presentation, 338 essentially disjoint product, 33, 151 free rank, 357 essentially unique, 286 Frobenius, 240 Euler phi function, 43 fully invariant, 69 Euler's formula, 44 fully-invariant series, 77 Euler's theorem, 44 fundamental theorem of finitely- even parity, 194 generated abelian even permutation, 26 groups, 356 even, 194 exact, 357 K-congruence relation, 211 exponent, 21, 28 K-equivalent, 208 extended centralizer, 266 K-set, 207 extension problem, 279 Galois group, 305 extension, 115, 177, 279, 291 Galois-style group, 57 376 Index

Gaussian coefficients, 233 idempotent, 76, 107 general Burnside problem, 35 identity, 20 general linear group, 23 ignore-^ map, 172 generalized quaternion group, 222, image sequence, 140 348 indecomposable, 75 generalized subnormal join independent, 332 property, 134 index set, 152, 181 generalized symmetric group, 185 index, 61, 115 8-generated, 35 induced action, 211 f-generated, 138 induced inverse map, 105 generating set, 34 induced map, 105 8-generator group, 35 infinite cycle, 192 generators, 338 infinite dihedral group, 129 good order, 162 infinite order, 21 graph, 189 infinite, 12, 20 greatest lower bound, 4 inherited, 116 group homomorphism, 46, 105 injection map, 154 group with operators, 274 injective, 359 group, 19 inner automorphisms, 30 ^-group, 291 insertion rules, 323 ^=-class, 292 interleaved series, 294 ^8-group, 292 intersection, 115, 294 H-group, 274 invariant factor decomposition, 1-group, 99 160, 357 :-group, 81, 215 invariant factors, 160 %-group, 24 invariant factors, 357 GSJP, 134 invariant, 11, 105, 204 Guralnick, 88 inverse, 20 inversion, 204 half open intervals, 4 involution, 21 Hall subgroup, 252, 310 is direct, 75 Hall :w-subgroup, 312 isomorphic, 46, 106 Hall, P., 167 H-isomorphic, 277 Hall–Witt Identity, 96 isomorphism invariant, 46, 116 Hall's theorem, 311 isomorphism theorems, 112 Hamiltonian group, 169 H-isomorphism theorems, 276 Hasse diagram, 37 isomorphism, 46, 106, 189 higher center, 297 H-isomorphism, 274 higher commutators, 307 isotone, 5 higher images, 140 holomorph, 177 join, 4 homomorphism, 46, 105 Jordan–Ho¨ lder Theorem, 281 H-homomorphism, 274 Hopfian, 147 ^-group, 291 hyper-c , 289 ^-radical, 315 hyperoctahedral group, 190 ^-residue, 315 Index 377

^-subgroup, 291 mediating morphism, 111, 155, ^-series, 292 156, 319, 326 ^-universal property mapping, 326 meet, 4 ^-universal, 326 metabelian, 101, 132 ^8-class, 292 minimal condition, 6, 137 ^8-group, 292 minimal direct summand, 163 ^=-class, 292 minimal element, 4 ^=-class, 292 minimal normal, 70 Kappe, 89 minimal, 38 kernel sequence, 139 minimum, 4 kernel, 108 modular lattice, 101 Kertész, 165 modular law, 101 Klein, 24 module, 355 Krull–Remak–Schmidt Theorem, -module, 355 286 monoid, 2 monomial group, 185 Lagrange's theorem, 42 monomorphism, 106 largest, 4 monotone, 5 lattice, 8 Morse, 89 law, 320, 327 move, 207 least upper bound, 4 multiplicative, 43 left action, 229 multiplicity, 1 left coset, 41 multiset, 1 left inverse, 173 left invertible, 173 8-ary operation, 19 left regular representation, 124, 212 8:-argument, 245 left-inverse property, 359 8-generated, 35 length, 2, 77, 323 8-generator group, 35 lifted, 111, 319 natural projection, 108 linear order, 3 nC-group, 165 linearly ordered set, 3 nD-group, 165 :-local subgroup, 265 negative, 20 locally cyclic, 59 nilpotency class, 301 locally finite, 59 nilpotent, 107, 216, 293 lower bound, 4 nodes, 189 lower center, 299 nongenerator, 127 lower central series, 299 nontrivial, 21 normal closure, 71 MacDonald, 89 normal complement, 151, 171 maximal condition on subgroups, normal extension, 115 60 normal interior, 125 maximal condition, 6, 137 normal join, 68 maximal element, 4 normal lifting, 115, 294 maximal normal, 70 normal series, 77 maximal, 38 normal, 65, 140 maximum, 4 normality preserving, 140 3 87 Index normalize, 82 :-standard form, 15 normalizer condition, 135 :-subgroup, 81, 215 normalizer, 82 pairwise essentially disjoint, 72 nullary operation, 19 partial order, 2 partially ordered set, 2 H-composition distance, 281 partition, 10 H-composition factors, 278 perfect, 84, 186 H-composition length, 281 periodic, 21 H-composition series, 278 permutable complement, 150 H-correspondence theorem, 276 permutable, 98 H-equivalent, 277 permutation group, 24 H-factor group, 274 permutation matrix, 184 H-group, 274 permutation, 24, 191 H-homomorphism, 274 permute, 39 H-isomorphic, 277 ply transitive, 203 H-isomorphism theorems, 276 Poincaré's inequality, 62 H-isomorphism, 274 Poincaré's theorem, 62 H-quotient group, 274 polycyclic, 293 H-series, 276 posets, 2 H-simple, 279 positive, 54 H-subgroup, 274 power of the continuum, 14 H-subnormal, 277 power, 13 odd parity, 194 preserves orientation, 51 odd, 26, 194 primary cyclic decomposition, 160, open interval, 3 357 operator domain, 274 primary decomposition, 243 orbit, 80, 205, 208 primary, 160 orbit-stabilizer relationship, 205, :-primary, 160 209, 210 principal factors, 278 order anti-embedding, 5 principal series, 278 order anti-isomorphism, 5 product, 13 order embedding, 5 projection map, 75, 154 order isomorphism, 5 projection, 118 order preserving, 5 projective property, 351 order reversing, 5 projective, 358 order, 20, 21 proper refinement, 276 outer automorphism group, 120 proper subgroup, 32 outer automorphism, 120 proper, 77 pure, 363 1-group, 99 :-cover, 216 quasi -group, 262 :-element, 80, 215 :-quasicyclic group, 70 :-group, 81, 215 quaternion group, 48 :-local subgroup, 265 H-quotient group, 274 :-quasicyclic group, 70 quotient group, 67 :-series, 216 quotient, 116, 294 Index 379

-module, 355 self-normalizing, 135 radical series, 306 semidecidable, 343 ^-radical, 315 semidihedral group, 178 range-lifted, 358 semidirect product, 151, 171 rank, 325 seminormal join, 68 real, 266 semisolvable, 343 reduced, 54, 323 sequence of normal closures, 129 reduction, 323 :-series, 216 refinement, 276 H-series, 276 reflexivity, 2, 10 ^-series, 292 regular wreath product, 182 series, 77 regular, 208 set product, 31 relation, 338 K-set, 207 relative holomorph, 177 sign, 194 relators, 338 signum, 194 Remak decomposition, 163 H-simple, 279 Remak, 163, 283 simple, 84 removal rules, 322 size, 1 representation map, 123, 207 SJP, 132 residually finite, 344 smallest, 4 residually -group, 351 solvable by radicals, 306 residually, 351 solvable, 77, 293, 343 ^-residue, 315 special linear group, 23 restricted Burnside problem, 35 Speigel, 88 restricted symmetric group, 205 splitting field, 305 restricted wreath product, 181 stabilizer relationship, 210 reverse projection, 118 stabilizer, 205, 208 reverses orientation, 51 stable, 207 right action, 229 :-standard form, 15 right coset, 41 step, 77 right inverse, 173 Stirling numbers of the first kind, right invertible, 173 201 right-inverse property, 358 Stirling numbers of the second rigid motion, 50 kind, 206 Robinson, 132, 134, 148 strict order, 2 string, 1 f-generated, 138 strongly disjoint, 72 scalars, 355 strongly real, 266 Schmidt, 167 subdirect product, 321 Schreier refinement theorem, 281 subgroup generated by, 34 Schreier, 281 ^-subgroup, 291 Schro¨der–Bernstein, 12 :-subgroup, 81, 215 Schur–Zassenhaus theorem, 255 H-subgroup, 274 SDR, 10, 208 subgroup, 32 second isomorphism theorem, 113 sublattice, 9 second H-isomorphism theorem, 276 subnormal index, 130 380 Index subnormal join property, 132 unitriangular, 229, 303 H-subnormal, 277 universal map, 319, 326 subnormal, 78 universal mapping property, 319 substring, 2 ^-universal property mapping, 326 subword, 2 universal, 111, 155, 156, 319 sum, 13, 107 ^-universal, 326 superdiagonal, 303 unquotient, 116, 294 supersolvable, 293 unsolvable, 343 supplement, 127, 145 upper bound, 4 support, 25, 152, 205 upper central series, 297 Sylow :-subgroup, 235 upper factorial, 201 Sylow theorems, 237 symmetric group, 24, 191 variety, 327 symmetry group, 51 verbal subgroup, 327 symmetry, 10, 50 vertex, 155, 156 system of distinct representatives, vertices, 189 10, 208 Vierergruppe, 24 term, 77 Weigold, 167 third isomorphism theorem, 113, well ordering, 5 276 well-ordering principle, 5 Thompson Theorem, 310 Wielandt, 134 Three subgroups lemma, 97 Wilson's theorem, 59 top, 4 word problem, 343 torsion element, 353 word, 1 torsion free, 21 torsion free, 353 Zassenhaus lemma, 119, 280 torsion subgroup, 353 zero map, 23, 106 torsion, 21, 353 zero, 20 total order, 3 Zorn's lemma, 5 totally ordered set, 3 transitive, 203, 208, 230 #-transitive, 230 transitivity, 2, 3, 10 translation, 123 transposition, 25, 192 transversal, 10, 61, 150 triangle inequality, 130 trivial class, 291 trivial group, 21 trivial reduction, 323 unary operation, 19 uncountable, 12 undecidable, 343 underlying set, 1, 19