Graduate Texts in Mathematics 80 Editorial Board
J.H. Ewing F.W. Gehring P.R. Halmos
Springer Science+Business Media, LLC
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.