Graduate Texts in Mathematics 80 Editorial Board

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

Graduate Texts in Mathematics

1 TAXI!un/ZARJNG. Introduction to Axiomatic Set Theory. 2nd ed. 2 OXTOBY. Measure and Category. 2nd ed. 3 SCHAEFFER. Topological Vector Spaces. 4 HILTON/STAMMBACH. A Course in Homological Algebra. 5 MAC LANE. Categories for the Working Mathematician. 6 HUGHES/PIPER. Projective Planes. 7 SERRE. A Course in Arithinetic. 8 TAKEUrI/ZAJUNG. Axiometic Set Theory. 9 HuMPHREYS. Introduction to Lie Algebras and Representation Theory.

10 CoHEN. A Course in Simple Homotopy Theory. 11 CONWAY. Functions of One Complex Variable. 2nd ed. 12 BEALS. Advanced Mathematical Analysis. 13 ANDERSON/FULLER. Rings and Categories of Modules. 2nd ed. 14 GOLUBITSKY/GUlLEMlN. Stable Mappings and Their Singularities. 15 BERBERIAN. Lectures in Functional Analysis and Operator Theory. 16 WINTER. The Structure of Fields. 17 ROSENBLATT. Random Processes. 2nd ed. 18 HALMos. Measure Theory. 19 IlALMOS. A Hilbert Space Problem Book. 2nd ed. 20 HUSEMOLLER. Fibre Bundles. 3rd ed. 21 HuMPHREYS. linear Algebraic Groups. 22 BARNES/MACK. An Algebraic Introduction to Mathematical Logic. 23 GREUB. Linear Algebra. 4th ed. 24 HOLMES. Geometric Functional Analysis and Its Applications. 2S HEWITT/STROMBERG. Real and Abstract Analysis. 26 MANES. Algebraic Theories. 27 KELLEY. General Topology. 28 ZARISKJ/SAMUEL. Commutative Algebra. Vol. I. 29 ZARISKJ/SAMUEL. Commutative Algebra. Vol. II. 30 JACOBSON. Lectures in Abstract Algebra I. Basic Concepts. 31 JACOBSON. Lectures in Abstract Algebra II. Linear Algebra. 32 JACOBSON. Lectures in Abstract Algebra III. Theory of Fields and Galois Theory. 33 HIRSCH. Differential Topology. 34 SPITZER. Principles of Random Walk. 2nd ed. 35 WERMER. Banach Algebras and Several Complex Variables. 2nd ed. 36 KELLEY/NAMIOKA et aJ. Linear Topological Spaces. 37 MoNK. Mathematical Logic. 38 GRAUERT/FRITZSCHE. Several Complex Variables. 39 ARVESON. An Invitation to C··Algebras. 40 KEMENY/SNELL/KNAPP. Denumerable Markov Chains. 2nd ed. 41 APOSTOL. Modular Functions and Dirichlet Series in Number Theory. 2nd ed. 42 SERRE. Linear Representations of Finite Groups. 43 GILLMAN/JERISON. Rings of Continuous Functions. 44 KENDIG. Elementary Algebraic Geometry. 45 LoEvE. Probability Theory I. 4th ed. 46 LoEVE. Probability Theory II. 4th ed. 47 MOISE. Geometric Topology in Dimensions 2 and 3.

continued after Index

Derek J. S. Robinson

A Course in the Theory of Groups'

Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest

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

Urbana-Champaign Urbana, IL 61801 USA

Editorial Board

J.R.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.

A course in the theory of groups. (Graduate texts in mathematics; v. 80) Bibliography: p. Includes Index. ISBN-13: 978-0-387-94092-2

P.R. Ralmos Department of

Mathematics Santa Clara University Santa Clara, CA 95053 USA

1. Groups, Theory of. texts in mathematics; 80.

I. Title. II. Series: Graduate

QA171.R568 512'.22

Printed on acid-free paper.

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© 1993 by Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the writ­ten permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, 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 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 managed by Jim Harbison; manufacturing supervised by Vincent Scelta.

98765432

ISBN-13: 978-0-387-94092-2 DOl: 10.1007/978-1-4684-0128-8

e-ISBN-13: 978-1-4684-0128-8

TO MY PARENTS AND

TO BENEDICT

Preface

" 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 a 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 of algebraic 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 Cflme 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).

vii

viii Preface

In the meantime the first great age of finite group theory had reached its apogee 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, Helmut Wielandt, and, in the field of group representations, Richard Dagobert Brauer (1901-1977). The present 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 UK. 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 large 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 in Urbana-Champaign, the University of Warwick, Notre Dame University, and the University of Freiburg. To all of these and to the National Science

Preface ix

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.

University of Illinois Urbana, Illinois August, 1980

DEREK ROBINSON

Contents

Preface

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

Chapter 4 Abelian Groups

4.1 Torsion Groups and Divisible Groups 4.2 Direct Sums of Cyclic and Quasicyclic Groups

Vll

xv

1

1 4 7

17 25 31

44

44 50 54

61

61 69 77

90

90 95

Xl

xii

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 Halln-subgroups 9.2 Sylow Systems and System Normalizers 9.3 p-soluble Groups 9.4 Supersoluble Groups 9.5 Formations

Contents

102 111

II7

117 125 135 142

153

153 161 167 177

185

185 190 196 201

206

206 215 219 228 239

245

245 254 261 266 269

Contents

Chapter 10 The Transfer and Its Applications

10.1 The Transfer Homomorphism 10.2 Griin's Theorems 10.3 Frobenius' Criterion for p-nilpotence 10.4 Thompson's Criterion for p-nilpotence 10.5 Fixed-point-free Automorphisms

Chapter II 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

xiii

276

276 283 286 289 295

301

301 313 320 328

342

342 350 355 362 367

371

371 379 382 388 394

401

401 407 411 419 422

xiv

Chapter 15 Infinite Soluble Groups

15.1 Soluble Linear Groups 15.2 Soluble Groups with Finiteness Conditions on Abelian Subgroups 15.3 Finitely Generated Soluble Groups and the Maximal Condition on

Normal Subgroups 15.4 Finitely Generated Soluble Groups and Residual Finiteness 15.5 Finitely Generated Soluble Groups and Their Frattini Subgroups

Bibliography

Index

Contents

433

433 438

444 453 457

461

472

Notation

G,H, ...

0(, /3, y, ...

x,y, z, ...

xO( or xfl.

[x, y]

H~G

H ~ G,H < G

H<JG

HsnG

H 1H 2 ···Hn , nH ). ).eA

<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

Products 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, O-rank, (Priifer) rank of G

xv

xvi

G[n]

lSI

IG:HI

Aut G, Inn G

Out G

HoI G

Hom,lG, H)

EndnG

H 1 X .•• x Hn, H 1 EE> •.. EE> Hn}

DrH;. ;'eA

HI>< N, N ><I H

CrH;. ;'eA

H 1 *···*Hn, FrH;. ;'eA

H@K

G' = [G, G]

Index of Notation

Subgroup generated by all gn or ng where g E G

Subgroup generated by all g E G such that gn = 1 or ng = o.

Cardinality of the set S

Index of the subgroup H in the group G

Order of the group element x

Centralizer, normalizer of H in G

Normal closure, core of H in G

Automorphism group, inner auto-morphism group of G

Aut GjInn G, outer automorphism group ofG

Holomorph of G

Set of Q-homomorphisms from G toH

Set of Q-endomorphisms of G

Direct products, direct sums

Semidirect products

Cartesian product, Cartesian sum

Wreath product

Free products

Tensor product

Derived subgroup of a group G

GjG'

Term of the derived series of G

Terms of the lower central series, the upper central series of G

Centre of G

Index of Notation

Fit G

Frat G

M(G)

O,,(G)

l,,(G)

StG(X), XG

SymX

Z,O, IR, IC

Z"

R*

RG

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

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

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

B(n, e)

max,mm

XVII

Fitting subgroup of G

Frattini subgroup of G

Schur multiplicator of G

Maximal normal n-subgroup of G

n-length of G

Stabilizer of X in G

Symmetric group on X

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

Z/nZ

Group of units of a nng 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.