Graduate Texts in Mathematics 80 Editorial Board

J.H. Ewing F.W. Gehring P.R. Halmos

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Graduate Texts in Mathematics

TAKEunlZARING. Introduction to 33 HIRSCH. Differential Topology. Axiomatic Set Theory. 2nd ed. 34 SPITZER. Principles of Random Walk.

2 OXTOBY. Measure and Category. 2nd ed. 2nd ed. 3 SCHAEFER. Topological Vector Spaces. 35 WERMER. Banach Algebras and Several 4 HILTON/STAMMBACH. A Course in Complex Variables. 2nd ed.

Homological Algebra. 36 KELLEy/NAMIOKA et al. Linear 5 MAC LANE. Categories for the Working Topological Spaces.

Mathematician. 37 MONK. Mathematical Logic. 6 HUGHES/PiPER. Projective Planes. 38 GRAUERT/FIuTZSCHE. Several Complex 7 SERRE. A Course in Arithmetic. Variables. 8 TAKEUTIlZARING. Axiomatic Set Theory. 39 ARVESON. An Invitation to C*-Algebra~ .

9 HUMPHREYS. Introduction to Lie Algebras 40 KEMENy/SNELLlKNAPP. Denumerable and Representation Theory. Markov Chains. 2nd ed.

10 COHEN. A Course in Simple Homotopy 41 APOSTOL. Modular Functions and Theory. Dirichlet Series in Number Theory.

II CONWAY. Functions of One Complex 2nd ed. Variable I. 2nd ed. 42 SERRE. Linear Representations of Finite

12 BEALS. Advanced Mathematical Analysis. Groups. 13 ANDERSON/fuLLER. Rings and Categories 43 GILLMAN/JERISON. Rings of Continuous

of Modules. 2nd ed. Functions. 14 GOLUBITSKy/GUILLEMIN. Stable Mappings 44 KENDIG. Elementary Algebraic Geometry.

and Their Singularities. 45 LOEVE. Probability Theory I. 4th ed. 15 BERBERIAN. Lectures in Functional 46 LOEVE. Probability Theory II. 4th ed.

Analysis and Operator Theory. 47 MOISE. Geometric Topology in 16 WINTER. The Structure of Fields. Dimensions 2 and 3. 17 ROSENBLATT. Random Processes. 2nd ed. 48 SACHSlWu. General Relativity for 18 HALMOS. Measure Theory. Mathematicians. 19 HALMos. A Hilbert Space Problem Book. 49 GRUENBERGIWEIR. Linear Geometry.

2nd ed. 2nd ed. 20 HUSEMOLLER. Fibre Bundles. 3rd ed. 50 EDWARDS. Fermat's Last Theorem. 21 HUMPHREYS. Linear Algebraic Groups. 51 KLINGENBERG. A Course in Differential 22 BARNEs/MACK. An Algebraic Introduction Geometry.

to Mathematical Logic. 52 HARTSHORNE. Algebraic Geometry. 23 GREUB. Linear Algebra. 4th ed. 53 MANIN. A Course in Mathematical Logic. 24 HOLMES. Geometric Functional Analysis 54 GRAVERIW ATKINS. Combinatorics with

and Its Applications. Emphasis on the Theory of Graphs. 25 HEWITT/STROMBERG. Real and Abstract 55 BROWN/PEARCY. Introduction to Operator

Analysis. Theory I: Elements of Functional 26 MANES. Algebraic Theories. Analysis. 27 KELLEY. General Topology. 56 MASSEY. Algebraic Topology: An 28 ZARlSKilSAMUEL. Commutative Algebra. Introduction.

Vol.I. 57 CROWELL!FOX. Introduction to Knot 29 ZAR!SKIlSAMUEL. Commutative Algebra. Theory.

Vol.II. 58 KOBLITZ. p-adic Numbers. p-adic 30 JACOBSON. Lectures in Abstract Algebra I. Analysis. and Zeta-Functions. 2nd ed.

Basic Concepts. 59 LANG. Cyclotomic Fields. 31 JACOBSON. Lectures in Abstract Algebra 60 ARNOLD. Mathematical Methods in

II. Linear Algebra. Classical Mechanics. 2nd ed. 32 JACOBSON. Lectures in Abstract Algebra

III. Theory of Fields and Galois Theory. continued after index

Derek J.S. Robinson

A Course in the Theory of Groups

Second Edition

With 40 Illustrations

Springer

Derek J.S. Robinson Department of Mathematics University of Illinois at

Urbana-Champaign Urbana, IL 61801 USA

Editorial Board

IH. Ewing Department of

Mathematics Indiana University Bloomington, IN 47405 USA

F.W. Gehring Department of

Mathematics University of Michigan Ann Arbor, MI 48109 USA

Mathematics Subject Classification (1991): 20-01

Library of Congress Cataloging-in-Publication Data Robinson, Derek John Scott.

P.R. Halmos Department of

Mathematics Santa Clara University Santa Clara, CA 95053 USA

A course in the theory of groups / Derek J.S. Robinson. - 2nd ed. p. cm. - (Graduate texts in mathematics ; 80)

Inc1udes bibliographical references (p. ) and index. ISBN 978-1-4612-6443-9 ISBN 978-1-4419-8594-1 (eBook) DOI 10.1007/978-1-4419-8594-1 1. Group theory. 1. Title. II. Series.

QA174.2.R63 1995 512'.2-dc20

Printed on acid-free paper.

© 1996 by Springer Science+Business Media New York Originally published by Springer-VerlagNew York, IDc. in 1996 Softcover reprint ofthe hardcover 2nd edition 1996

95-4025

Ali rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC), 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 of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as under­stood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Production coordinated by Brian Howe and managed by Bill Imbornoni; manufacturing supervised by Jeffrey Taub. Typeset by Asco Trade Typesetting Ltd., Hong Kong.

98765432

ISBN 978-1-4612-6443-9

For Judith

Preface to the Second Edition

In preparing this new edition I have tried to keep the changes to a mini­mum, on the principle that one should not meddle with a relatively success­ful text. Thus the general form of the book remains the same. Naturally I have taken the opportunity to correct the errors of which I was aware. Also the text has been updated at various points, some proofs have been im­proved, and lastly about thirty additional exercises are included.

There are three main additions to the book. In the chapter on group extensions an exposition of Schreier's concrete approach via factor sets is given before the introduction of covering groups. This seemed to be desir­able on pedagogical grounds. Then S. Thomas's elegant proof of the auto­morphism tower theorem is included in the section on complete groups. Finally an elementary counterexample to the Burnside problem due to N.D. Gupta has been added in the chapter on finiteness properties.

I am happy to have this opportunity to thank the many friends and col­leagues who wrote to me about the first edition with comments, suggestions and lists of errors. Their efforts have surely led to an improvement in the text. In particular I thank J.C. Beidleman, F.B. Cannonito, H. Heineken, L.c. Kappe, W. M6hres, R. Schmidt, H. Snevily, B.A.F. Wehrfritz, and J. Wiegold. My thanks are due to Yu Fen Wu for assistance with the proofreading. I also thank Tom von Foerster of Springer-Verlag for making this new edition possible, and for his assistance throughout the project.

University of Illinois at Urbana-Champaign, Urbana, Illinois

Derek Robinson

vii

Preface to the First Edition

"A group is defined by means of the laws of combinations of its symbols," according to a celebrated dictum of Cayley. And this is probably still as good a one-line explanation as any. The concept of a group is surely one of the central ideas of mathematics. Certainly there are few branches of that science in which groups are not employed implicitly or explicitly. Nor is the use of groups confined to pure mathematics. Quantum theory, molecular and atomic structure, and crystallography are just a few of the areas of science in which the idea of a group as a measure of symmetry has played an important part.

The theory of groups is the oldest branch of modern algebra. Its origins are to be found in the work of Joseph Louis Lagrange (1736-1813), Paulo Ruffini (1765-1822), and Evariste Galois (1811-1832) on the theory ofalge­braic equations. Their groups consisted of permutations of the variables or of the roots of polynomials, and indeed for much of the nineteenth century all groups were finite permutation groups. Nevertheless many of the funda­mental ideas of group theory were introduced by these early workers and their successors, Augustin Louis Cauchy (1789-1857), Ludwig Sylow (1832-1918), Camille Jordan (1838- 1922) among others.

The concept of an abstract group is clearly recognizable in the work of Arthur Cayley (1821-1895), but it did not really win widespread acceptance until Walther von Dyck (1856-1934) introduced presentations of groups.

The stimulus to study infinite groups came from geometry and topology, the influence of Felix Klein (1849-1925), Sophus Lie (1842-1899), Henri Poincare (1854-1912), and Max Dehn (1878-1952) being paramount. Thereafter the standard of infinite group theory was borne almost single­handed by Otto Juljevic Schmidt (1891-1956) until the establishment of the Russian school headed by Alexander Gennadievic Kuros (1908-1971).

viii

Preface to the First Edition ix

In the meantime the first great age of finite group theory had reached its climax in the period immediately before the First World War with the work of Georg Frobenius (1849-1917), William Burnside (1852-1927), and Issai Schur (1875-1936). After 1928, decisive new contributions were made by Philip Hall (1904-1982), Helmut Wielandt, and, in the field of group repre­sentations, Richard Dagobert Brauer (1901-1977). The subsequent intense interest in the classification of finite simple groups is very largely the legacy of their work.

This book is intended as an introduction to the general theory of groups. Its aim is to make the reader aware of some of the main accomplishments of group theory, while at the same time providing a reasonable coverage of basic material. The book is addressed primarily to the student who wishes to learn the subject, but it is hoped that it will also prove useful to special­ists in other areas as a work of reference.

An attempt has been made to strike a balance between the different branches of group theory, abelian groups, finite groups, infinite groups, and to stress the unity of the subject. In choice of material I have been guided by its inherent interest, accessibility, and connections with other topics. No book of this type can be comprehensive, but I hope it will serve as an intro­duction to the several excellent research level texts now in print.

The reader is expected to have at least the knowledge and maturity of a graduate student who has completed the first year of study at a North American university or of a first year research student in the United Kingdom. He or she should be familiar with the more elementary facts about rings, fields, and modules, possess a sound knowledge of linear alge­bra, and be able to use Zorn's Lemma and transfinite induction. However, no knowledge of homological algebra is assumed: those homological methods required in the study of group extensions are introduced as they become necessary. This said, the theory of groups is developed from scratch. Many readers may therefore wish to skip certain sections of Chapters 1 and 2 or to regard them as a review.

A word about the exercises, of which there are some 650. They are to be found at the end of each section and must be regarded as an integral part of the text. Anyone who aspires to master the material should set out to solve as many exercises as possible. They vary from routine tests of comprehen­sion of definitions and theorems to more challenging problems, some theo­rems in their own right. Exercises marked with an asterisk are referred to at some subsequent point in the text.

Notation is by-and-Iarge standard, and an attempt has been made to keep it to a minimum. At the risk of some unpopularity, I have chosen to write all functions on the right. A list of commonly used symbols is placed at the beginning of the book.

While engaged on this project I enjoyed the hospitality and benefited from the assistance of several institutions: the University of Illinois at

x Preface to the First Edition

Urbana-Champaign, the University of Warwick, Notre Dame University, and the University of Freiburg. To all of these and to the National Science Foundation I express my gratitude. I am grateful to my friends John Rose and Ralph Strebel who read several chapters and made valuable comments on them. It has been a pleasure to cooperate with Springer-Verlag in this venture and I thank them for their unfailing courtesy and patience.

Contents

Preface to the Second Edition Preface to the First Edition Notation

CHAPTER 1 Fundamental Concepts of Group Theory

1.1. Binary Operations, Semigroups, and Groups 1.2. Examples of Groups 1.3. Subgroups and Cosets 1.4. Homomorphisms and Quotient Groups 1.5. Endomorphisms and Automorphisms 1.6. Permutation Groups and Group Actions

CHAPTER 2 Free Groups and Presentations

2.1. Free Groups 2.2. Presentations of Groups 2.3. Varieties of Groups

CHAPTER 3 Decompositions of a Group

3.1. Series and Composition Series 3.2. Some Simple Groups 3.3. Direct Decompositions

vii viii xv

1 4 8

17 25 31

44

44 50 56

63

63 71 80

xi

xii

CHAPTER 4

Abelian Groups

4.1. Torsion Groups and Divisible Groups 4.2. Direct Sums of Cyclic and Quasicyclic Groups 4.3. Pure Subgroups and p-Groups 4.4. Torsion-Free Groups

CHAPTER 5

Soluble and Nilpotent Groups

5.1. Abelian and Central Series 5.2. Nilpotent Groups 5.3. Groups of Prime-Power Order 5.4. Soluble Groups

CHAPTER 6

Free Groups and Free Products

6.1. Further Properties of Free Groups 6.2. Free Products of Groups 6.3. Subgroups of Free Products 6.4. Generalized Free Products

CHAPTER 7 Finite Permutation Groups

7.1. Multiple Transitivity 7.2. Primitive Permutation Groups 7.3. Classification of Sharply k-Transitive Permutation Groups 7.4. The Mathieu Groups

CHAPTER 8

Representations of Groups

8.1. Representations and Modules 8.2. Structure of the Group Algebra 8.3. Characters 8.4. Tensor Products and Representations 8.5. Applications to Finite Groups

CHAPTER 9

Finite Soluble Groups

9.1. Hall n-Subgroups 9.2. Sylow Systems and System Normalizers 9.3. p-Soluble Groups 9.4. Supersoluble Groups 9.5. Formations

Contents

93

93 98

106 114

121

121 129 139 147

159

159 167 174 184

192

192 197 203 208

213

213 223 226 235 246

252

252 261 269 274 277

Contents

CHAPTER 10

The Transfer and Its Applications

10.1. The Transfer Homomorphism 10.2. Griin's Theorems 10.3. Frobenius's Criterion for p-Nilpotence 10.4. Thompson's Criterion for p-Nilpotence 10.5. Fixed-Point-Free Automorphisms

CHAPTER 11

The Theory of Group Extensions

11.1. Group Extensions and Covering Groups 11.2. Homology Groups and Cohomology Groups 11.3. The Gruenberg Resolution 11.4. Group-Theoretic Interpretations of the (Co)homology Groups

CHAPTER 12 Generalizations of Nilpotent and Soluble Groups

12.1. Locally Nilpotent Groups 12.2. Some Special Types of Locally Nilpotent Groups 12.3. Engel Elements and Engel Groups 12.4. Classes of Groups Defined by General Series 12.5. Locally Soluble Groups

CHAPTER 13

Subnormal Subgroups

13.1. Joins and Intersections of Subnormal Subgroups 13.2. Permutability and Subnormality 13.3. The Minimal Condition on Subnormal Subgroups 13.4. Groups in Which Normality Is a Transitive Relation 13.5. Automorphism Towers and Complete Groups

CHAPTER 14 Finiteness Properties

14.1. Finitely Generated Groups and Finitely Presented Groups 14.2. Torsion Groups and the Burnside Problems 14.3. Locally Finite Groups 14.4. 2-Groups with the Maximal or Minimal Condition 14.5. Finiteness Properties of Conjugates and Commutators

CHAPTER 15

xiii

285

285 292 295 298 305

310

310 326 333 341

356

356 363 369 376 381

385

385 393 396 402 408

416

416 422 429 437 439

Infinite Soluble Groups 450

15.1. Soluble Linear Groups 450 15.2. Soluble Groups with Finiteness Conditions on Abelian Subgroups 455

xiv Contents

15.3. Finitely Generated Soluble Groups and the Maximal Condition on Normal Subgroups 461

15.4. Finitely Generated Soluble Groups and Residual Finiteness 470 15.5. Finitely Generated Soluble Groups and Their Frattini Subgroups 474

Bibliography 479

Index 491

Notation

G,H, ...

X,~, ...

0(, /3, y, ...

x, y, z, ...

[x, y]

H~G

H s;.G, H < G

H<J G

Hsn G

H 1 H2 "'Hn

<X;.IA E A)

<XIR)

d(G)

rp(G), ro(G), r(G)

Sets, groups, rings, etc.

Classes of groups

Functions

Elements of a set

Image of x under 0(

y-1xy

X-1y-1XY

H is isomorphic with G

H is a subgroup, a proper subgroup of the group G.

H is a normal subgroup of G

H is a subnormal subgroup of G

Product of subsets of a group

Subgroup generated by subsets X;. of a group

Group presented by generators X and re­lators R

Minimum number of generators of G

p-rank, torsion-free rank, (Priifer) rank of G

xv

XVI Notation

Gn, nG Subgroup generated by all gn or ng where gEG

G[n] Subgroup generated by all 9 E G such that gn = 1 or ng = o.

I S I Cardinality of the set S

I G : H I Index of the subgroup H in the group G

I x I Order of the group element x

CG(H), NG(H) Centralizer, normalizer of H in G

HG, HG Normal closure, core of H in G

Aut G, Inn G Automorphism group, inner automor-phism group of G

Out G Aut GjInn G, outer automorphism group ofG

HoI G Holomorph of G

Homn(G, H) Set ofO-homomorphisms from G to H

Endn G Set of O-endomorphisms of G

HI x ... x Hn, HI EB ... EB Hn} Set product, direct products, direct sums

Dr H). ).eA

H D< N, N ~ H Semidirect products

Cr H). Cartesian product, Cartesian sum ).eA

H"" K Wreath product

HI * ... * Hn , Fr H). Free products ).eA

H ® K Tensor product

G' = [G, G] Derived subgroup of a group G

Gab GIG'

G(a) Term of the derived series of G

raG, (aG Terms of the lower central series, the

(G

Fit G

Frat G

upper central series of G

Center of G

Fitting subgroup of G

Frattini subgroup of G

Notation

M(G)

O,,(G)

1,,(G)

StG(X), XG

SymX

7L, Q, IR, IC

RG

GL(V)

GL(n, R), SL(n, R)

PGL(n, R), PSL(n, R)

T(n, R), U(n, R)

B(n, e)

max, min

Schur multiplicator of G

Maximal normaln-subgroup of G

n-Iength of G

Stabilizer of X in G

Symmetric group on X

xvii

Symmetric, alternating groups of degree n

Dihedral group of order n

Generalized quaternion group of order 2"

Sets of integers, rational numbers, real numbers, complex numbers

7Ljn7L

Group of units of a ring R with identity

Group ring of a group G over a ring R with identity element

Augmentation ideals

Group of nonsingular linear transforma­tions of a vector space V

General linear and special linear groups

Projective general linear and projective special linear groups

Groups of triangular, unitriangular matri­ces

Free Burnside group with n generators and exponent e

Induced module, induced character

Maximal, minimal conditions

Matrix with (i, j) entry 1 and other en­tries O.