Universitext Editorial Board (North America): S. Axler F.W. Gehring K.A. Ribet Springer New York Berlin Heidelberg Hong Kong London Milan Paris Tokyo This page intentionally left blank Hans Kurzweil Bernd Stellmacher The Theory of Finite Groups An Introduction Hans Kurzweil Bernd Stellmacher Institute of Mathematics Mathematiches Seminar Kiel University of Erlangen-Nuremburg Christian-Albrechts-Universität 1 Bismarckstrasse 1 /2 Ludewig-Meyn Strasse 4 Erlangen 91054 Kiel D-24098 Germany Germany [email protected] [email protected] Editorial Board (North America): S. Axler F.W. Gehring Mathematics Department Mathematics Department San Francisco State University East Hall San Francisco, CA 94132 University of Michigan USA Ann Arbor, MI 48109-1109 [email protected] USA [email protected] K.A. Ribet Mathematics Department University of California, Berkeley Berkeley, CA 94720-3840 USA [email protected] Mathematics Subject Classification (2000): 20-01, 20DXX Library of Congress Cataloging-in-Publication Data Kurzweil, Hans, 1942– The theory of finite groups: an introduction / Hans Kurzweil, Bernd Stellmacher. p. cm. — (Universitext) Includes bibliographical references and index. ISBN 0-387-40510-0 (alk. paper) 1. Finite groups. I. Stellmacher, B. (Bernd) II. Title. QA177.K87 2004 512´.2—dc21 2003054313 ISBN 0-387-40510-0 Printed on acid-free paper. © 2004 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 con- nection 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 in the United States of America. 987654321 SPIN 10940884 A Typesetting: Pages created by the authors using L TEX. www.springer-ny.com Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH Preface Since its early beginnings in the nineteenth century the theory of finite groups has grown to be an extensive and diverse part of algebra. In the beginning of the 1980s, this development culminated in the classification of the finite simple groups, an impressive and convincing demonstration of the strength of its methods and results. In our book we want to introduce the reader—as far as an introduction can do this—to some of the developments in this area that contributed to this success or may open new perspectives for the future. The first eight chapters are intended to give a fast and direct approach to those methods and results that everybody should know who is interested in finite groups. Some parts, like nilpotent groups and solvable groups, are only treated as far as they are necessary to understand and investigate finite groups in general. The notion of action, in all its facets, like action on sets and groups, coprime action, and quadratic action, is at the center of our exposition. In the last chapters we focus on the correspondence between the local and global structure of finite groups. Our particular goal is to investigate non- solvable groups all of whose 2-local subgroups are solvable. The reader will realize that nearly all of the methods and results of this book are used in this investigation. At least two things have been excluded from this book: the representation theory of finite groups and—with a few exceptions—the description of the finite simple groups. In both cases we felt unable to treat these two themes in an adequate way within the framework of this book. For the more important results proved or mentioned in this book we tried to give the original papers as references, and in a few cases also some with alternative proofs. In the Appendix we state the classification theorem of v vi Preface the finite simple groups and also some of the fundamental theorems that are related to the subject of the last chapters. The first eight chapters are accompanied by exercises. Usually they are not ordered by increasing difficulty and some of them demand serious thinking and persistence. They should allow the reader to get engaged with group theory and to find out about his or her own abilities. The reader may want to postpone and revisit later some of the apparently more difficult exercises using the greater experience and insight gained from following chapters. It should be pointed out here that—with the exception of the first chapter— all groups under consideration are meant to be finite. Our special thanks go to our colleague H. Bender. Without him this book would not have been written, and without his encouraging support it would have taken a different shape. We would like to thank J. Hall for reading the entire manuscript and A. Chermak for reading parts of it. We are also grateful to B. Baumann, D. Bundy, S. Heiss, and P. Flavell for their helpful comments and sugges- tions. A German version of this book has been published in 1998 as a Springer- Lehrbuch. Erlangen, Germany Hans Kurzweil Kiel, Germany Bernd Stellmacher February 2003 Contents Preface v List of Symbols xi 1 Basic Concepts 1 1.1 Groups and Subgroups ...................... 1 1.2 Homomorphisms and Normal Subgroups ............ 10 1.3 Automorphisms .......................... 15 1.4 Cyclic Groups ........................... 21 1.5 Commutators ........................... 24 1.6 Products of Groups ........................ 27 1.7 Minimal Normal Subgroups ................... 36 1.8 Composition Series ........................ 39 2 Abelian Groups 43 2.1 The Structure of Abelian Groups ................ 43 2.2 Automorphisms of Cyclic Groups ................ 49 3 Action and Conjugation 55 3.1 Action ............................... 55 3.2 Sylow’s Theorem ......................... 62 3.3 Complements of Normal Subgroups ............... 71 vii viii Contents 4 Permutation Groups 77 4.1 Transitive Groups and Frobenius Groups ............ 77 4.2 Primitive Action ......................... 84 4.3 The Symmetric Group ...................... 87 4.4 Imprimitive Groups and Wreath Products ........... 91 5 p-Groups and Nilpotent Groups 99 5.1 Nilpotent Groups ......................... 99 5.2 Nilpotent Normal Subgroups ..................104 5.3 p-Groups with Cyclic Maximal Subgroups ...........108 6 Normal and Subnormal Structure 121 6.1 Solvable Groups .........................121 6.2 The Theorem of Schur-Zassenhaus ...............125 6.3 Radical and Residue .......................128 6.4 π-Separable Groups .......................133 6.5 Components and the Generalized Fitting Subgroup .........................141 6.6 Primitive Maximal Subgroups ..................145 6.7 Subnormal Subgroups ......................156 7 Transfer and p-Factor Groups 163 7.1 The Transfer Homomorphism ..................163 7.2 Normal p-Complements .....................169 8 Groups Acting on Groups 175 8.1 Action on Groups .........................175 8.2 Coprime Action ..........................183 8.3 Action on Abelian Groups ....................190 8.4 The Decomposition of an Action ................197 8.5 Minimal Nontrivial Action ....................203 8.6 Linear Action and the Two-Dimensional Linear Groups ...........................210 Contents ix 9 Quadratic Action 225 9.1 Quadratic Action .........................225 9.2 The Thompson Subgroup ....................230 9.3 Quadratic Action in p-Separable Groups ............239 9.4 A Characteristic Subgroup ....................249 9.5 Fixed-Point-Free Action .....................257 10 The Embedding of p-Local Subgroups 261 10.1 Primitive Pairs ..........................262 10.2 The paqb-Theorem ........................276 10.3 The Amalgam Method ......................281 11 Signalizer Functors 303 11.1 Definitions and Elementary Properties .............304 11.2 Factorizations ...........................311 11.3 The Completeness Theorem of Glauberman ..........325 12 N-Groups 335 12.1 An Application of the Completeness Theorem ..............................338 12.2 J(T )-Components ........................347 12.3 N-Groups of Local Characteristic 2 ...............356 Appendix 369 Bibliography 373 Books and Monographs ........................373 Journal Articles .............................374 Index 381 This page intentionally left blank List of Symbols [x, y], 24 Gα1,...,αn−1 ,85 [X, Y ], 25 G/N,13 ∼ [X, Y, Z], 25 G = H,11 (z,n), 22 GL(V ), 210 <, ≤,4 GLn(q), 212 X,5 Im ϕ,11 g,5 Inn G,16 X ,5 J(G), 236 ×n Ker ϕ,11 Gi,27 i=1 N G,12 A (G), 236 NH (A), 59 A | | V (G), 233 /θ(A), 304 | | AB,6 /θ(A, π), 306 A B,14 o(g), 5 An,89 nZ,21 p Aut G,15 Op(G), O (G), 129 p CH (G), 59 Op (G), O (G), 129 K Cn,21 OK(G), O (G), 129 π ∆, 84 Oπ(G), O (G), 129 E(G), 143 Ω(G), 47 F q, 210 Ωi(G), 47 F (G), 104 P,9 ∗ F (G), 143 Φ(G), 105 |G|,3 π, 129 G ,24 π(G), 9 G,2 π(n), 9 G, g,14 PGL(V ), 222 Gα,57 PSL(V ), 222 Gp,45 r(G), 48 |G : U|,7 S(G), 124 xi xii List of Symbols SΩ,55 U ≤ G,4 Sn,55 U char G,17 SL(V ), 212 W (S), 250 SLn(q), 212 X H,34 ϕ −ϕ Sylp G,63 X , x ,11 ϕ−1 SylπG, 135 Y ,11 θ, 304 Z,21 θ(G), 304 Z(G), 16 Chapter 1 Basic Concepts In this first chapter we introduce some of the basic concepts of finite group theory. Most of these concepts apply to arbitrary groups, whether finite or infinite. For that reason we will make no assumption (as we will in the later chapters) that the objects under consideration are finite. 1.1 Groups and Subgroups A nonempty set G is a group, if to every pair (x, y) ∈ G × G an element xy ∈ G is assigned, the product of x and y, satisfying the following axioms: Associativity: x(yz)=(xy)z for all x, y, z ∈ G .