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 KœEF 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-×.