DOCSLIB.ORG

The Automorphism Tower Problem Simon Thomas

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

The Automorphism Tower Problem Simon Thomas Rutgers University, New Brunswick, New Jersey 08903 E-mail address: [email protected] Contents Preface vii Introduction ix Chapter 1. The Automorphism Tower Problem 1 1.1. Automorphism towers 1 1.2. Some fundamental results 6 1.3. Some examples of automorphism towers 10 1.4. The infinite dihedral group 19 1.5. Notes 23 Chapter 2. Wielandt's Theorem 25 2.1. Automorphism towers of finite groups 25 2.2. Subnormal subgroups 31 2.3. Finite p-groups 35 2.4. Notes 39 Chapter 3. The Automorphism Tower Theorem 41 3.1. The automorphism tower theorem 41 3.2. τκ is increasing 47 3.3. A better bound 51 3.4. The automorphism tower problem revisited 57 3.5. Notes 60 Chapter 4. The Normaliser Tower Technique 61 4.1. Normaliser towers 62 4.2. Coding structures in graphs 68 4.3. Coding graphs in fields 71 4.4. A technical lemma 78 iii iv CONTENTS 4.5. Notes 88 Chapter 5. Hamkins' Theorem 89 5.1. Automorphism towers of arbitrary groups 89 5.2. Two examples and many questions 94 5.3. Notes 98 Chapter 6. Set-theoretic Forcing 101 6.1. Countable transitive models of ZFC 102 6.2. Set-theoretic forcing 108 6.3. Preserving cardinals 116 6.4. Nice P-names 125 6.5. Some observations and conventions 127 6.6. τ(G) is not absolute 131 6.7. An absoluteness theorem for automorphism towers 139 6.8. Iterated forcing 141 6.9. Notes 149 Chapter 7. Forcing Long Automorphism Towers 151 7.1. The nonexistence of a better upper bound 152 7.2. Realising normaliser towers within infinite symmetric groups 158 7.3. Closed groups of uncountable degree 164 7.4. τκ can be strictly increasing 170 7.5. Two more applications 172 7.6. The main gap 176 7.7. Notes 177 Chapter 8. Changing The Heights Of Automorphism Towers 179 8.1. Changing the heights of automorphism towers 180 8.2. More normaliser towers 186 8.3. Rigid trees 196 8.4. Notes 209 Chapter 9. Bounding The Heights Of Automorphism Towers 211 9.1. The overspill argument 211 CONTENTS v 9.2. Conservative extensions 217 9.3. The reflection argument 221 9.4. Notes 230 Index 231 Bibliography 235 Preface This is a preliminary version of an unpublished book on the automorphism tower problem, which was intended to be intelligible to beginning graduate students in both logic and algebra. Simon Thomas vii Introduction If G is a centreless group, then there is a natural embedding of G into its automorphism group Aut(G), obtained by sending each g 2 G to the corresponding inner automorphism ig 2 Aut(G). It is easily shown that the group Inn(G) of inner automorphisms is a normal subgroup of Aut(G) and that CAut(G)(Inn(G)) = 1. In particular, Aut(G) is also a centreless group. This enables us to define the automorphism tower of G to be the ascending chain of centreless groups G = G0 E G1 E G2 E :::Gα E Gα+1 E ::: such that for each ordinal α (a) Gα+1 = Aut(Gα); and S (b) if α is a limit ordinal, then Gα = Gβ. β<α (At each successor step, we identify Gα with Inn(Gα) via the natural embedding.) The automorphism tower is said to terminate if there exists an ordinal α such that Gα+1 = Gα. Of course, this occurs if and only if there exists an ordinal α such that Aut(Gα) = Inn(Gα). In this case, the height τ(G) of the automorphism tower is defined to be the least ordinal α such that Gα+1 = Gα. In 1939, Wielandt proved that if G is a finite centreless group, then its automorphism tower terminates after finitely many steps. However, there exist natural examples of infinite centreless groups whose automorphism towers do not terminate in finitely many steps. For example, it can be shown that the automorphism tower of the infinite dihedral group D1 terminates after exactly ! +1 steps. The classical version of the automorphism tower problem asks whether the automorphism tower of an arbitrary centreless group G eventually terminates, perhaps after a transfinite number of steps. In the 1970s, a number of special cases of the automorphism tower problem were solved. For example, Rae and Roseblade proved that the automorphism tower of a centreless Cernikovˇ group terminates after finitely many steps and Hulse proved ix x INTRODUCTION that the automorphism tower of a centreless polycyclic group terminates after countably many steps. In each of these special cases, the proof depended upon a detailed understanding of the groups Gα occurring in the automorphism towers of the relevant groups G. But the problem was not solved in full generality until 1984, when I proved that the automorphism tower of an arbitrary centreless group + G terminates after at most 2jGj steps. The proof is extremely simple and uses only the most basic results on automorphism towers, together with some elemen- tary properties of the infinite cardinal numbers. Of course, this still leaves open the problem of determining the best possible upper bound for the height τ(G) of the automorphism tower of an infinite centreless group G. For example, it is natural to ask whether there exists a fixed cardinal κ such that τ(G) ≤ κ for every infinite centreless group G. This question turns out to be relatively straightforward. For each ordinal α, it is possible to construct an infinite centreless group G such that τ(G) = α. However, if we ask whether it is true that τ(G) ≤ 2jGj for every infinite centreless group G, then matters become much more interesting. For example, it is independent of the classical ZFC axioms of set theory whether τ(G) ≤ 2@1 for every centreless group G of cardinality @1; i.e. this statement can neither be proved nor disproved using ZFC. This book presents a self-contained account of the automorphism tower prob- lem, which is intended to be intelligible to beginning graduate students in both logic and algebra. There are essentially no set-theoretic prerequisites. The only requirement is a basic familiarity with some of the fundamental notions of algebra. The first half of the book presents those results which can be proved using ZFC; and also includes an account of the necessary set-theoretic background, such as the notions of a regular cardinal and a stationary set. This is followed by a short intro- duction to set-theoretic forcing, which is aimed primarily at algebraists. The final three chapters explain why a number of natural problems concerning automorphism towers are independent of ZFC. In more detail, the book is organised as follows. In Chapter 1, we introduce the notion of the automorphism tower of a centreless group and illustrate this notion by computing the automorphism towers of a number of groups. In Chapter 2, we present a proof of Wielandt's theorem that the automorphism tower of a finite INTRODUCTION xi centreless group terminates after finitely many steps. In Chapter 3, we prove that the automorphism tower of an infinite centreless group G terminates after strictly + less than 2jGj steps. This chapter also contains an account of the basic theory of regular cardinals and stationary sets. Much of the remainder of this book is concerned with the problem of constructing centreless groups with extremely long automorphism towers. Unfortunately it is usually very difficult to compute the successive groups in the automorphism tower of a centreless group. In Chapter 4, we introduce the normaliser tower technique, which enables us to almost entirely bypass this problem. Instead, throughout most of this book, we only have to deal with the much easier problem of computing the successive normalisers of a subgroup H of a group G. As a first application of this technique, we prove that if κ is an infinite cardinal, then for each ordinal α < κ+, there exists a centreless group G of cardinality κ such that the automorphism tower of G terminates after exactly α steps. In Chapter 5, we present an account of Hamkins' work on the automorphism towers of arbitrary (not necessarily centreless) groups. Chapter 6 contains an introduction to set-theoretic forcing, which is intended to be intelligible to beginning graduate students. In Chapter 7, we show that it is impossible to find + a better bound in ZFC than 2jGj for the height of the automorphism tower of an infinite centreless group G. For example, it is consistent that there exists a centreless group G of cardinality @1 such that the automorphism tower of G has height strictly greater than 2@1 . On the other hand, in Chapter 9, we show that it is consistent that the height of the automorphism tower of every centreless group G of @1 cardinality @1 is strictly less than 2 . In Chapter 8, we consider the relationship between the heights of the automorphism towers of a single centreless group G computed in two different models M ⊂ N of ZFC. I am very grateful to David Nacin and Steve Warner for carefully reading the manuscript of this book and for supplying me with a substantial number of corrections. Thanks are due to the many friends and colleagues with whom I have discussed the material in this book, especially Greg Cherlin, Warren Dicks, Ulrich Felgner, Ed Formanek, Ken Hickin, Wilfrid Hodges, Otto Kegel, Felix Leinen, Peter Neumann, Dick Phillips and John Wilson. Particular thanks are due to my coauthors Joel xii INTRODUCTION Hamkins and Winfried Just. Of course, like so many logicians, my greatest debt of gratitude is owed to Saharon Shelah, who I would like to thank for all that he has taught me over the last twenty years.
Recommended publications
  • Complete Objects in Categories

    Complete Objects in Categories

    Complete objects in categories James Richard Andrew Gray February 22, 2021 Abstract We introduce the notions of proto-complete, complete, complete˚ and strong-complete objects in pointed categories. We show under mild condi- tions on a pointed exact protomodular category that every proto-complete (respectively complete) object is the product of an abelian proto-complete (respectively complete) object and a strong-complete object. This to- gether with the observation that the trivial group is the only abelian complete group recovers a theorem of Baer classifying complete groups. In addition we generalize several theorems about groups (subgroups) with trivial center (respectively, centralizer), and provide a categorical explana- tion behind why the derivation algebra of a perfect Lie algebra with trivial center and the automorphism group of a non-abelian (characteristically) simple group are strong-complete. 1 Introduction Recall that Carmichael [19] called a group G complete if it has trivial cen- ter and each automorphism is inner. For each group G there is a canonical homomorphism cG from G to AutpGq, the automorphism group of G. This ho- momorphism assigns to each g in G the inner automorphism which sends each x in G to gxg´1. It can be readily seen that a group G is complete if and only if cG is an isomorphism. Baer [1] showed that a group G is complete if and only if every normal monomorphism with domain G is a split monomorphism. We call an object in a pointed category complete if it satisfies this latter condi- arXiv:2102.09834v1 [math.CT] 19 Feb 2021 tion.
  • Group Theory

    Group Theory

    Appendix A Group Theory This appendix is a survey of only those topics in group theory that are needed to understand the composition of symmetry transformations and its consequences for fundamental physics. It is intended to be self-contained and covers those topics that are needed to follow the main text. Although in the end this appendix became quite long, a thorough understanding of group theory is possible only by consulting the appropriate literature in addition to this appendix. In order that this book not become too lengthy, proofs of theorems were largely omitted; again I refer to other monographs. From its very title, the book by H. Georgi [211] is the most appropriate if particle physics is the primary focus of interest. The book by G. Costa and G. Fogli [102] is written in the same spirit. Both books also cover the necessary group theory for grand unification ideas. A very comprehensive but also rather dense treatment is given by [428]. Still a classic is [254]; it contains more about the treatment of dynamical symmetries in quantum mechanics. A.1 Basics A.1.1 Definitions: Algebraic Structures From the structureless notion of a set, one can successively generate more and more algebraic structures. Those that play a prominent role in physics are defined in the following. Group A group G is a set with elements gi and an operation ◦ (called group multiplication) with the properties that (i) the operation is closed: gi ◦ g j ∈ G, (ii) a neutral element g0 ∈ G exists such that gi ◦ g0 = g0 ◦ gi = gi , (iii) for every gi exists an −1 ∈ ◦ −1 = = −1 ◦ inverse element gi G such that gi gi g0 gi gi , (iv) the operation is associative: gi ◦ (g j ◦ gk) = (gi ◦ g j ) ◦ gk.
  • Representation Growth

    Representation Growth

    Universidad Autónoma de Madrid Tesis Doctoral Representation Growth Autor: Director: Javier García Rodríguez Andrei Jaikin Zapirain Octubre 2016 a Laura. Abstract The main results in this thesis deal with the representation theory of certain classes of groups. More precisely, if rn(Γ) denotes the number of non-isomorphic n-dimensional complex representations of a group Γ, we study the numbers rn(Γ) and the relation of this arithmetic information with structural properties of Γ. In chapter 1 we present the required preliminary theory. In chapter 2 we introduce the Congruence Subgroup Problem for an algebraic group G defined over a global field k. In chapter 3 we consider Γ = G(OS) an arithmetic subgroup of a semisimple algebraic k-group for some global field k with ring of S-integers OS. If the Lie algebra of G is perfect, Lubotzky and Martin showed in [56] that if Γ has the weak Congruence Subgroup Property then Γ has Polynomial Representation Growth, that is, rn(Γ) ≤ p(n) for some polynomial p. By using a different approach, we show that the same holds for any semisimple algebraic group G including those with a non-perfect Lie algebra. In chapter 4 we apply our results on representation growth of groups of the form Γ = D log n G(OS) to show that if Γ has the weak Congruence Subgroup Property then sn(Γ) ≤ n for some constant D, where sn(Γ) denotes the number of subgroups of Γ of index at most n. As before, this extends similar results of Lubotzky [54], Nikolov, Abert, Szegedy [1] and Golsefidy [24] for almost simple groups with perfect Lie algebra to any simple algebraic k-group G.
  • Groups and Rings

    Groups and Rings

    Groups and Rings David Pierce February , , : p.m. Matematik Bölümü Mimar Sinan Güzel Sanatlar Üniversitesi [email protected] http://mat.msgsu.edu.tr/~dpierce/ Groups and Rings This work is licensed under the Creative Commons Attribution–Noncommercial–Share-Alike License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ CC BY: David Pierce $\ C Mathematics Department Mimar Sinan Fine Arts University Istanbul, Turkey http://mat.msgsu.edu.tr/~dpierce/ [email protected] Preface There have been several versions of the present text. The first draft was my record of the first semester of the gradu- ate course in algebra given at Middle East Technical University in Ankara in –. I had taught the same course also in –. The main reference for the course was Hungerford’s Algebra []. I revised my notes when teaching algebra a third time, in – . Here I started making some attempt to indicate how theorems were going to be used later. What is now §. (the development of the natural numbers from the Peano Axioms) was originally pre- pared for a course called Non-Standard Analysis, given at the Nesin Mathematics Village, Şirince, in the summer of . I built up the foundational Chapter around this section. Another revision, but only partial, came in preparation for a course at Mimar Sinan Fine Arts University in Istanbul in –. I expanded Chapter , out of a desire to give some indication of how mathematics, and especially algebra, could be built up from some simple axioms about the relation of membership—that is, from set theory.
  • The Group of Automorphisms of the Holomorph of a Group

    The Group of Automorphisms of the Holomorph of a Group

    Pacific Journal of Mathematics THE GROUP OF AUTOMORPHISMS OF THE HOLOMORPH OF A GROUP NAI-CHAO HSU Vol. 11, No. 3 BadMonth 1961 THE GROUP OF AUTOMORPHISMS OF THE HOLOMORPH OF A GROUP NAI-CHAO HSU l Introduction* If G = HK where H is a normal subgroup of the group G and where K is a subgroup of G with the trivial intersection with H, then G is said to be a semi-direct product of H by K or a splitting extension of H by K. We can consider a splitting extension G as an ordered triple (H, K; Φ) where φ is a homomorphism of K into the automorphism group 2I(if) of H. The ordered triple (iϊ, K; φ) is the totality of all ordered pairs (h, k), he H, he K, with the multiplication If φ is a monomorphism of if into §I(if), then (if, if; φ) is isomorphic to (iϊ, Φ(K); c) where c is the identity mapping of φ(K), and therefore G is the relative holomorph of if with respect to a subgroup φ(-K) of Sί(ίf). If φ is an isomorphism of K onto Sί(iϊ), then G is the holomorph of if. Let if be a group, and let G be the holomorph of H. We are con- sidering if as a subgroup of G in the usual way. GoΓfand [1] studied the group Sί^(G) of automorphisms of G each of which maps H onto itself, the group $(G) of inner automorphisms of G, and the factor group SIff(G)/$5(G).
  • Minimum Circuit Size, Graph Isomorphism, and Related Problems∗

    Minimum Circuit Size, Graph Isomorphism, and Related Problems∗

    SIAM J. COMPUT. c 2018 Society for Industrial and Applied Mathematics Vol. 47, No. 4, pp. 1339{1372 MINIMUM CIRCUIT SIZE, GRAPH ISOMORPHISM, AND RELATED PROBLEMS∗ ERIC ALLENDERy , JOSHUA A. GROCHOWz , DIETER VAN MELKEBEEKx , CRISTOPHER MOORE{, AND ANDREW MORGANx Abstract. We study the computational power of deciding whether a given truth table can be described by a circuit of a given size (the minimum circuit size problem, or MCSP for short) and of the variant denoted as MKTP, where circuit size is replaced by a polynomially related Kolmogorov measure. Prior to our work, all reductions from supposedly intractable problems to MCSP/MKTP hinged on the power of MCSP/MKTP to distinguish random distributions from distributions pro- duced by hardness-based pseudorandom generator constructions. We develop a fundamentally dif- ferent approach inspired by the well-known interactive proof system for the complement of graph isomorphism (GI). It yields a randomized reduction with zero-sided error from GI to MKTP. We generalize the result and show that GI can be replaced by any isomorphism problem for which the underlying group satisfies some elementary properties. Instantiations include linear code equivalence, permutation group conjugacy, and matrix subspace conjugacy. Along the way we develop encodings of isomorphism classes that are efficiently decodable and achieve compression that is at or near the information-theoretic optimum; those encodings may be of independent interest. Key words. reductions between NP-intermediate problems, graph isomorphism, minimum circuit size problem, time-bounded Kolmogorov complexity AMS subject classifications. 68Q15, 68Q17, 68Q30 DOI. 10.1137/17M1157970 1. Introduction. Finding a circuit of minimum size that computes a given Boolean function constitutes the overarching goal in nonuniform complexity theory.
  • View This Volume's Front and Back Matter

    View This Volume's Front and Back Matter

    http://dx.doi.org/10.1090/gsm/092 Finite Group Theory This page intentionally left blank Finit e Grou p Theor y I. Martin Isaacs Graduate Studies in Mathematics Volume 92 f//s ^ -w* American Mathematical Society ^ Providence, Rhode Island Editorial Board David Cox (Chair) Steven G. Krantz Rafe Mazzeo Martin Scharlemann 2000 Mathematics Subject Classification. Primary 20B15, 20B20, 20D06, 20D10, 20D15, 20D20, 20D25, 20D35, 20D45, 20E22, 20E36. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-92 Library of Congress Cataloging-in-Publication Data Isaacs, I. Martin, 1940- Finite group theory / I. Martin Isaacs. p. cm. — (Graduate studies in mathematics ; v. 92) Includes index. ISBN 978-0-8218-4344-4 (alk. paper) 1. Finite groups. 2. Group theory. I. Title. QA177.I835 2008 512'.23—dc22 2008011388 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2008 by the American Mathematical Society. All rights reserved. Reprinted with corrections by the American Mathematical Society, 2011.
  • Notes on Finite Group Theory

    Notes on Finite Group Theory

    Notes on finite group theory Peter J. Cameron October 2013 2 Preface Group theory is a central part of modern mathematics. Its origins lie in geome- try (where groups describe in a very detailed way the symmetries of geometric objects) and in the theory of polynomial equations (developed by Galois, who showed how to associate a finite group with any polynomial equation in such a way that the structure of the group encodes information about the process of solv- ing the equation). These notes are based on a Masters course I gave at Queen Mary, University of London. Of the two lecturers who preceded me, one had concentrated on finite soluble groups, the other on finite simple groups; I have tried to steer a middle course, while keeping finite groups as the focus. The notes do not in any sense form a textbook, even on finite group theory. Finite group theory has been enormously changed in the last few decades by the immense Classification of Finite Simple Groups. The most important structure theorem for finite groups is the Jordan–Holder¨ Theorem, which shows that any finite group is built up from finite simple groups. If the finite simple groups are the building blocks of finite group theory, then extension theory is the mortar that holds them together, so I have covered both of these topics in some detail: examples of simple groups are given (alternating groups and projective special linear groups), and extension theory (via factor sets) is developed for extensions of abelian groups. In a Masters course, it is not possible to assume that all the students have reached any given level of proficiency at group theory.
  • Groups with All Subgroups Subnormal

    Groups with All Subgroups Subnormal

    Note di Matematica Note Mat. 28 (2008), suppl. n. 2, 1-149 ISSN 1123-2536, e-ISSN 1590-0932 NoteDOI 10 Mat..1285/i128590(2008)0932v28n suppl.2supplp1 n. 2, 1–149. doi:10.1285/i15900932v28n2supplp1http://siba-ese.unisalento.it, © 2008 Università del Salento Groups with all subgroups subnormal Carlo Casolo Dipartimento di Matematica “Ulisse Dini”, Universit`adi Firenze, I-50134 Firenze Italy [email protected] Abstract. An updated survey on the theory of groups with all subgroups subnormal, in- cluding a general introduction on locally nilpotent groups, full proofs of most results, and a review of the possible generalizations of the theory. Keywords: Subnormal subgroups, locally nilpotent groups. MSC 2000 classification: 20E15, 20F19 1 Locally nilpotent groups In this chapter we review part of the basic theory of locally nilpotent groups. This will mainly serve to fix the notations and recall some definitions, together with some important results whose proofs will not be included in these notes. Also, we hope to provide some motivation for the study of groups with all subgroups subnormal (for short N1-groups) by setting them into a wider frame. In fact, we will perhaps include more material then what strictly needed to understand N1-groups. Thus, the first sections of this chapter may be intended both as an unfaithful list of prerequisites and a quick reference: as such, most of the readers might well skip them. As said, we will not give those proofs that are too complicate or, conversely, may be found in any introductory text on groups which includes some infinite groups (e.g.
  • ENGEL ELEMENTS in GROUPS 2 Positive Integer N, a Group Is Called N-Engel If All of Whose Elements Are Left N-Engel Elements

    ENGEL ELEMENTS in GROUPS 2 Positive Integer N, a Group Is Called N-Engel If All of Whose Elements Are Left N-Engel Elements

    ENGEL ELEMENTS IN GROUPS ALIREZA ABDOLLAHIDepartment of Mathematics, University of Isfahan, Isfahan 81746-73441, IRAN. Email: [email protected] Abstract We give a survey of results on the structure of right and left Engel elements of a group. We also present some new results in this topic. 1 Introduction Let G be any group and x,y ∈ G. Define inductively the n-Engel left normed commutator [x,n y] = [x,y,...,y] n of the pair (x,y) for a given non-negative integer| {z }n, as follows: −1 −1 −1 y [x,0 y] := x, [x,1 y] := [x,y]= x y xy =: x x , and for all n> 0 [x,n y] = [[x,n−1 y],y]. An element a ∈ G is called left Engel whenever for every element g ∈ G there exists a non-negative integer n = n(g, a) possibly depending on the elements g and a such that [g,n a] = 1. For a positive integer k, an element a ∈ G is called a left k-Engel element of G whenever [g,k a] = 1 for all g ∈ G. An element a ∈ G is called arXiv:1002.0309v1 [math.GR] 1 Feb 2010 a bounded left Engel element if it is left k-Engel for some k. We denote by L(G), Lk(G) and L(G), the set of left Engel elements, left k-Engel elements, bounded left Engel elements of G, respectively. In definitions of various types of left Engel elements a of a group G, we observe that the variable element g appears on the left hand side of the element a.
  • Some Topics on Permutable Subgroups in Infinite Groups

    Some Topics on Permutable Subgroups in Infinite Groups

    SOME TOPICS ON PERMUTABLE SUBGROUPS IN INFINITE GROUPS TESI DI DOTTORATO SCIENZE MATEMATICHE E INFORMATICHE - XXIX CICLO DIPARTIMENTO DI MATEMATICA E APPLICAZIONI RENATO CACCIOPPOLI ROBERTO IALENTI UNIVERSITÀ DEGLI STUDI DI NAPOLI FEDERICO II CONTENTS Introduction 1 Preliminaries 1 1.1 Permutable subgroups . 1 1.2 Generalized permutable subgroups . 5 2 Permutability conditions on subgroups of infinite rank 9 2.1 S-permutable subgroups of infinite rank . 11 2.2 Semipermutable subgroups of infinite rank . 15 2.3 Nearly and almost permutable subgroups of infinite rank . 20 3 Some further problems on permutability 47 3.1 Polycyclic groups with permutability conditions on finite ho- momorphic images . 47 3.2 Groups with finite abelian section rank factorized by mutu- ally permutable subgroups . 54 INTRODUCTION A subgroup H of a group G is said to be permutable (or quasinormal) if HK = KH for every subgroup K of G. This concept has been introduced by Ore [54] and the condition HK = KH is equivalent to the requirement that the set HK is a subgroup. As a consequence, if H is a permutable subgroup of a group G, then for any subgroup K, hH, Ki is just the set of all elements hk, where h is in H and k is in K. With an easy argument based on set equality, it can be proved that if H is permutable, then HK \ L = (H \ L)K for any subgroups K and L such that K ≤ L. This property is known as Dedekind identity (or modular law) and note that, since K \ L = K, the equality HK \ L = (H \ L)(K \ L) is a form of associative law.
  • 9 Automorphism Groups

    9 Automorphism Groups

    9 Automorphism groups For any group G the set of automorphisms of G forms a group under com- position. This is the automorphism group of G denoted Aut(G). Nonabelian groups have inner automorphisms Ág 2 Aut(G) given by ¡1 Ág(x) = gxg and the assignment g 7! Ág gives a homomorphism G ! Aut(G) with kernel Z(G). The image of this homomorphism is evidently the group of inner automorphisms of G which we denote Inn(G). Thus: Inn(G) »= G=Z(G): Proposition 9.1. Inn(G) E Aut(G). Proof. Suppose that à 2 Aut(G) and Ág 2 Inn(G). Then ¡1 ¡1 ¡1 ¡1 ÃÁgà (x) = Ã(gà (x)g ) = Ã(g)xÃ(g ) = ÁÃ(g)(x) ¡1 so ÃÁgà = ÁÃ(g). The automorphisms of G which are not inner automorphisms are called outer automorphisms. However, the outer automorphism group of G is the quotient group: Aut(G) Out(G) := = coker Á Inn(G) Automorphisms of abelian groups Let A denote an additive group. Since A has no (nontrivial) inner automor- phisms we have: Out(G) = Aut(G). The endomorphisms of A form a ring End(A) with pointwise addition [(® + ¯)(x) = ®(x) + ¯(x)] and multiplication given by composition. Proposition 9.2. The automorphism group of an additive group A is the group of units of the ring End(A): Aut(A) = End(A)£: If A = Z=n is cyclic then an endomorphism ® of Z=n is given by multi- plication by ®(1) 2 Z=n. Thus: Corollary 9.3. End(Z=n) = Z=n is a commutative ring and Aut(Z=n) = (Z=n)£ Everyone knows that for n = p is prime, (Z=p)£ is cyclic of order p ¡ 1.