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55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998 54 Casper Goffman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in analysis, 1997 53 Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis, 1997 52 V. A. Kozlov, V. G. Maz'ya? and J. Rossmann, Elliptic boundary value problems in domains with point singularities, 1997 51 Jan Maly and William P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, 1997 50 Jon Aaronson, An introduction to infinite ergodic theory, 1997 49 R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, 1997 48 Paul-Jean Cahen and Jean-Luc Chabert, -valued polynomials, 1997 47 A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May (with an appendix by M. Cole), Rings, modules, and algebras in stable homotopy theory, 1997 46 Stephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and Adam O. Hausknecht, Cogroups and co-rings in categories of associative rings, 1996 44 J. Amoros, M. Burger, K. Corlette, D. Kotschick, and D. Toledo, Fundamental groups of compact Kahler manifolds, 1996 43 James E. Humphreys, Conjugacy classes in semisimple algebraic groups, 1995 42 Ralph Freese, Jaroslav Jezek, and J. B. Nation, Free lattices, 1995 41 Hal L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, 1995 40.3 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 3, 1998 40.2 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 2, 1995 40.1 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 1, 1994 39 Sigurdur Helgason, Geometric analysis on symmetric spaces, 1994 38 Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, 1993 37 Leonard Lewin, Editor, Structural properties of polylogarithms, 1991 36 John B. Conway, The theory of subnormal operators, 1991 35 Shreeram S. Abhyankar, Algebraic geometry for scientists and engineers, 1990 34 Victor Isakov, Inverse source problems, 1990 33 Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, 1990 32 Howard Jacobowitz, An introduction to CR structures, 1990 31 Paul J. Sally, Jr. and David A. Vogan, Jr., Editors, and harmonic analysis on semisimple Lie groups, 1989 30 Thomas W. Cusick and Mary E. Flahive, The Markoff and Lagrange spectra, 1989 29 Alan L. T. Paterson, Amenability, 1988 28 Richard Beals, Percy Deift, and Carlos Tomei, Direct and inverse scattering on the line, 1988 27 Nathan J. Fine, Basic hypergeometric series and applications, 1988 26 Hari Bercovici, Operator theory and arithmetic in H°°, 1988 25 Jack K. Hale, Asymptotic behavior of dissipative systems, 1988 24 Lance W. Small, Editor, Noetherian rings and their applications, 1987 23 E. H. Rothe, Introduction to various aspects of degree theory in Banach spaces, 1986 22 Michael E. Taylor, Noncommutative harmonic analysis, 1986 (Continued in the back of this publication) The Classification of the Finite Simple Groups, Number 3 MATHEMATICAL Surveys and Monographs

Volume 40, Number 3

The Classification of the Finite Simple Groups, Number 3

Part I, Chapter A: Almost Simple X- Groups

Daniel Gorenstein Richard Lyons Ronald Solomon

American Mathematical Society Providence, Rhode Island The authors were supported in part by NSF grant #DMS 94-01852 and by NSA grant #MDA-904-95-H-1048.

1991 Subject Classification. Primary 20D06, 20D08; Secondary 20D05, 20E32, 20G40.

ABSTRACT. This volume examines in detail the internal structure of the finite simple groups of Lie type, the finite alternating groups and the twenty-six sporadic finite simple groups, as well as their almost simple analogues. Emphasis is on the structure of local and their relationships with one another. The foundation is laid for the development of the many specific properties of 3C-groups to be used in the inductive proof of the classification theorem.

Library of Congress Cataloging-in-Publication Data ISBN 0-8218-0391-3 (number 3) ISBN 0-8218-0390-5 (number 2) The first volume was catalogued as follows: Gorenstein, Daniel. The classification of the finite simple groups / Daniel Gorenstein, Richard Lyons, Ronald Solomon. p. cm. — (Mathematical surveys and monographs; v. 40, number 1-) Includes bibliographical references and index. ISBN 0-8218-0334-4 [number 1] 1. Finite simple groups. I. Lyons, Richard, 1945- . II. Solomon, Ronald. III. Title. IV. Series: Mathematical surveys and monographs; no. 40, pt. 1-. QA177.G67 1994 512/.2-dc20 94-23001 CIP

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 (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to [email protected].

© 1998 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 03 02 01 00 99 98 For Lisa and Myriam Contents

List of Tables xi Preface xiii PART I, CHAPTER A: ALMOST SIMPLE K-GROUPS Chapter 1. Some Theory of Linear Algebraic Groups 1 1.1. Fundamental Notions 2 1.2. Jordan Decomposition 4 1.3. Unipotent Groups; One-Parameter Groups 5 1.4. Tori 6 1.5. Solvable Algebraic Groups 6 1.6. Borel Subgroups 7 1.7. Radical; Reductive, Semisimple and Simple Groups 7 1.8. Abstract Root Systems 8 1.9. Weights, Roots and Root Subgroups in Reductive Groups 13 1.10. Classification of Semisimple Groups; Isogenics and Versions 15 1.11. £iV-Structure 16 1.12. Chevalley Generators and Relations 17 1.13. Parabolic Subgroups 22 1.14. Weights and Representations 23 1.15. , and Endomorphisms 27 1.16. Notes 30 Chapter 2. The Finite Groups of Lie Type 31 2.1. Steinberg Endomorphisms and Lang's Theorem 31 2.2. The Finite Groups of Lie Type 36 2.3. Bruhat Structure 40 2.4. Chevalley Relations for Finite Groups of Lie Type 45 2.5. Automorphisms 53 2.6. Subsystem Subgroups and Parabolic Subgroups 63 2.7. The Classical Groups 67 2.8. Equicharacteristic Representations 73 2.9. Presentations; Curtis-Tits Theorem 79 2.10. Standard Notation 89 Chapter 3. Local Subgroups of Groups of Lie Type, I 91 3.1. A Theorem of Borel and Tits 91 3.2. Structure of Parabolic Subgroups 95 3.3. Equicharacteristic Sylow Structure 106 x CONTENTS

Chapter 4. Local Subgroups of Groups of Lie Type, II 119 4.1. Semisimple Automorphisms of Algebraic Groups 119 4.2. Semisimple Automorphisms of Finite Groups of Lie Type 134 4.3. Centralizers of Involutions in Auto (if): Statements 143 4.4. Centralizers of Involutions in Auto(K): Proofs 153 4.5. Centralizers of Involutions in Groups in £jie(r), r Odd: Statements 170 4.6. Centralizers of Involutions in Groups in £ie(r), r Odd: Proofs 186 4.7. Inner-Diagonal and Graph Automorphisms of Exceptional Odd Prime Orders 204 4.8. Some Other p-Local Subgroups 219 4.9. The Remaining Types of Non-Inner Automorphisms 227 4.10. Cross-Characteristic Sylow Structure 236

Chapter 5. The Alternating Groups and the Twenty-Six Sporadic Groups 249 5.1. Schur Multipliers and Coverings of Simple Groups 249 5.2. The Alternating Groups 252 5.3. Structure of the Twenty-Six Sporadic Groups 259 5.4. Justification of the Tables: The Simple Case 288 5.5. Justification of the Tables: The Non-Simple Case 294 5.6. Ranks of Sporadic Groups 302

Chapter 6. Coverings and Embeddings of Quasisimple DC-Groups 311 6.1. The Schur Multipliers and Covering Groups of the Simple X-Groups 311 6.2. Some Embeddings of Quasisimple Groups 313 6.3. The Action of Out (if) on M(K) 317 6.4. Splitting and Stable Involutions 324 6.5. Subgroups of Small Simple Groups 328

Chapter 7. General Properties of X-Groups 335 7.1. Verification of the Fundamental X- Properties 335 7.2. Generation of Semisimple Groups: Generalities 343 7.3. Generation for Groups in Qhev: Statements of Main Results 346 7.4. Generation for Groups in Qhev: Proofs 349 7.5. Generation for Groups in Alt U Spor 380 7.6. Strongly p-Embedded Subgroups and Proper fc-Generated Cores 383 7.7. Balance and Signalizers 385 7.8. Fusion 399 Background References 403 Expository References 404 Errata for Numbers 1 and 2 409 Glossary 410

Index 414 List of Tables

Chapter 1 1.8. Irreducible reduced crystallographic root systems 12 1.12.5. Centers of universal Chevalley groups 19 1.12.6. Generators of Z(K) 20 1.15.2d. Graph automorphisms and their centralizers 28 Chapter 2 2.2. Orders of the finite groups of Lie type 39 2.3.2d. Twisted root systems 42 2.4. Structure of root groups X& 46 2.4.7. Elements of Cartan subgroups 51 2.5.12c. Structure of Outdiag(if) 58 Chapter 3 3.3.1. Equicharacteristic ranks of groups of Lie type 108 3.3.2. Cardinalities of abelian sets of roots 112 Chapter 4 4.3.1. Inn (if) -conjugacy classes of involutions in Auto (if), an<^ their cen­ tralizers 145 4.3.2. Centers of centralizers of involutions in Auto (if) 149 4.3.3. Connected centralizers of involutions in Auto (if) 151 4.3.4. Action of elements of V-^ on centralizers of involutions 152 4.5.1. Inner-diagonal and graph involutions for adjoint groups in Lieir), r odd 172 4.5.2. Inner-diagonal and graph involutions for universal groups in £ie(r), r odd 178 4.5.3. Inner-diagonal and graph involutions for ^|m(g) = D^q), q odd 182 4.5.5. Fundamental involutions 185 4.7.1. Inn(if)-conjugacy classes of subgroups of p in exceptional groups Auto (if), their centralizers and normalizers 206 4.7.2. Action of elements of T-^ on centralizers in exceptional groups if 208 4.7.3A. Conjugacy classes and centralizers of subgroups of order 3 for ex­ ceptional groups in Lie(r) 210 4.7.3B. Conjugacy classes and centralizers of subgroups of order 5 in E$(q) 211 4.10.6. Maximal commuting sets S(P) of fundamental subgroups, K G ehev(r), r odd 244 xii LIST OF TABLES

Chapter 5 5.3. The twenty-six sporadic groups 262 5.3a. The group Mn 262 5.3b. The group M\2 263 5.3c. The group M22 264 5.3d. The group M23 265 5.3e. The group M24 266 5.3f. The group J\ 267 5.3g. The group J2 = H J 268 5.3h. The group J3 = H JM 269 5.3i. The group J4 270 5.3j. The group C03 = -3 271 5.3k. The group Co2 = -2 272 5.31. The group Co\ — -1 273 5.3m. The group HS 274 5.3n. The group Mc 275 5.3o. The group Suz 276 5.3p. The group He = HHM = F7 277 5.3q. The group Ly — LyS 278 5.3r. The group Ru 279 5.3s. The group O'N = 0'S 280 5.3t. The group Fi22 = M(22) 281 5.3u. The group Fi2s = M(23) 282 f ; 5.3v. The group Fi 24 = M(24) 283 5.3w. The group F5 = HN 284 5.3x. The group F3 = Th 285 5.3y. The group F2 = BM = B 286 5.3z. The group F1 = M 287 .6.1. The p-ranks of the sporadic groups 303 .6.2. The 2-local p-ranks of the sporadic groups 309 Chapter 6 6.1.2. Canonical parts MC(K) of Schur multipliers, K £ Ghev, K simple 312 6.1.3. Elementary divisors Me of the exceptional parts Me(K) of Schur multipliers, K e X, K simple 313 6.2.2. Some embeddings of quasisimple %-groups 315 6.3.1. Action of Out (if) on Me(K) 318 Preface

At every stage the proofs of the major classification results concerning finite simple groups have been inductive in nature. Each major theorem rests on a body of results concerning the subgroups, automorphisms, coverings and representations of the simple groups which arise in the conclusion of that theorem or which are permitted by the hypotheses to arise as significant sections of a minimal counterex­ ample. It should come as no surprise then that the proof of the entire Classification Theorem rests on a large body of such results concerning all of the finite simple groups. The purpose of this volume is to establish fundamental facts about the finite simple groups—those of Lie type, the alternating groups and the sporadic groups—as well as to set up a framework for viewing these groups in which all of the results needed in the ensuing volumes of this series can be derived. More de­ tailed specific results are postponed to the volumes in which they are first required. The appropriate type of group to be analyzed here is an almost X, that is, one for which F*(X) is quasisimple. As in the case of the preceding volume in this series, on general , we cannot and have not attempted an encyclopedic treatment. Our utilitarian focus on the proof of the Classification Theorem has dictated our choice of topics, so that our attention is devoted almost exclusively to local structure, automorphisms and covering groups. In particular the vast and lively of ordinary and modular representations of the finite simple groups is barely touched. We do not even broach the topic, important for us, of failure-of-factorization modules, quadratic modules and related issues; indeed U. Meierfrankenfeld and G. Stroth are preparing a monograph on this subject which we intend to add to our Background Results. Furthermore, the extensive theory of the maximal subgroups of the finite simple groups developed since 1980 is hardly represented here, since we need only a few classical results concerning low rank groups of Lie type and a smattering of results for sporadic simple groups. This reflects the fact that (luckily) extensive knowledge of the subgroup structure of finite simple groups is necessary only for the smallest of these groups. The finite groups of Lie type get most of our attention, for their theory is both richer than that of the alternating groups and better developed than that of the sporadic groups. Our Background References for properties of these groups are R. W. Carter's Simple Groups of Lie Type [Cal], R. Steinberg's Yale Lecture Notes [Stl] and Steinberg's Memoir Endomorphisms of Linear Algebraic Groups [St2]. In addition the Gorenstein-Lyons Memoir [GL1] is occasionally useful in this connection. As the underpinning of our development we use the classification theory of semisimple linear algebraic groups. Accordingly our opening chapter reviews the main features of that theory which are important for our subsequent computations, but omits proofs. This omission we make comfortably because of

xiii xiv PREFACE the several beautiful expositions of this theory already existing: the pioneering Seminaire Chevalley as well as books by A. Borel, J. E. Humphreys and T. A. Springer [Ch3, Borl, Hum2, Spl]. The next three chapters develop, with proofs, the consequences for the local structure of the finite groups of Lie type. The main theme of these chapters is computation, but we also include proofs of the Curtis-Tits theorem and related recognition theorems. The bulk of the computation in these chapters is not new, the principles having been set forth in general and executed in detail by A. Borel, R. W. Carter, C. Chevalley, C. W. Curtis, N. Iwahori, R. Ree, T. A. Springer, and R. Steinberg, as well as H. Azad, N. Burgoyne and C. Williamson, B. Cooperstein and G. Mason, D. I. Deriziotis, M. W. Liebeck, G. M. Seitz and many others drawn to the beautiful detail of these groups. Rounding out the picture of the simple DC-groups is a chapter on the alter­ nating and sporadic groups. The basic properties of the sporadic simple groups which we need—primarily the centralizers of automorphisms of prime order—have been assumed as Background Results and are transcribed for the most part from [GLl] and M. Aschbacher's Sporadic Groups [A2]. Additional consequences of these are derived in this volume. Furthermore, we have taken a couple of well- established facts from the Atlas of Finite Groups [CCNPW1] and Aschbacher's recent 3-Transposition Groups [A19]; these references are all clearly labelled, and our inclusion of these two books as Background References is limited to these par­ ticular citations in Section 5.3. Our sixth chapter discusses Schur multipliers and investigates some minute details about certain exceptional covering groups of simple groups which will be of later use, for one thing because of their connection with certain sporadic groups. As discussed in our first volume, it is only by including [GLl] that we can keep our list of Background References as short as it is. There are in fact precisely two important cases where the results quoted from [GLl] depend on a number of further references to the literature: the local properties of sporadic groups not found in [A2] and the structure of the Schur multipliers of all the finite simple groups, beyond those covered by the general theory for groups of Lie type found in [Stl] and for alternating groups in, say, [Sul]. Our point of view is that it is a separate task to assemble careful and complete expositions of these two sizable theories, as indeed Aschbacher has been doing in the case of sporadic groups with his two books. As promised in our preceding volume, we apply the basic structural results developed in the first six chapters of this volume to verify in our final chapter that every DC-group has properties (5), (5P), (Gp), {Bp), (Cp) and (Mp) for all primes p, properties required for some basic general results such as Lv>-balance. We also set the stage for the signalizer functor method by establishing several basic generation and balance theorems for DC-groups, many due originally to Gary Seitz. In this chapter we proceed under the assumption that all the simple sections of the simple group K under inspection are known simple groups. Our investigation of maximal subgroups of small simple groups is also made with the benefit of this assumption. In the later applications of these results K will be a proper section of our DC-proper simple group G and so the hypothesis will be justified. Certainly it could be avoided and "clean" proofs given (as they were originally) for many, if not all, of the results, but often only at a considerable cost in effort and space. Indeed this is another way in which we have been able to limit the number of Background References. PREFACE xv

A note about labelling: a reference such as 2.9.3 is to Theorem 2.9.3 (or Def­ inition 2.9.3, etc., as the case may be). The numbers for displayed equations and statements, however, are put in parentheses, so that (4.2.12) refers to Equation (4.2.12), for example. Our thanks go to Ellen Scott, who prepared substantial chunks of this manu­ script. In addition, we thank Jonathan Alperin, Paul Flavell, Chao Ku and John Thompson, who have pointed out errors in the first two volumes of this series; we are taking the opportunity here to include a page of errata for those volumes. We are indebted to John Thompson, Peter Sin and Gary Seitz for their numerous use­ ful comments on large portions of this manuscript. Their vigilance has prevented a number of errors from appearing in print. It is a pleasure to acknowledge our debt to some of those from and with whom we learned much of what is contained herein: J. Alperin, M. Aschbacher, N. Burgoyne, R. W. Carter, B. Cooperstein, R. Gilman, G. Glauberman, R. L. Griess, Jr., P. Hewitt, G. Mason, L. Scott, Jr., G. M. Seitz, T. A. Springer, J. G. Thompson, and most importantly, M. E. O'Nan and R. Steinberg. And as ever, a toast to Danny. July, 1997

Richard Lyons and Ronald Solomon

Author addresses:

DEPARTMENT OF MATHEMATICS, RUTGERS UNIVERSITY, NEW BRUNSWICK, NEW JERSEY 08903 E-mail address: [email protected]. edu

DEPARTMENT OF MATHEMATICS, THE OHIO STATE UNIVERSITY, COLUM­ BUS, OHIO 43210 E-mail address: [email protected] Background References

NOTE. The previous numbers of this series are referenced as follows. [Ii] D. Gorenstein, R. Lyons, and R. M. Solomon, The Classification of the Finite Simple Groups, Number 1. Chapter 1: Overview, Amer. Math. Soc. Surveys and Monographs 40, #1 (1995), 1-78. [I2] D. Gorenstein, R. Lyons, and R. M. Solomon, The Classification of the Finite Simple Groups, Number 1. Chapter 2: Outline of Proof, Amer. Math. Soc. Surveys and Monographs 40, #1 (1995), 79-139. [IQ] , The Classification of the Finite Simple Groups, Number 2. Chapter G: Gen­ eral Group Theory, Amer. Math. Soc. Surveys and Monographs 40, #2 (1996). The full list of Background References appears in the first book of this series. The list below contains all Background References to which we refer in this book. The numbering of the Background and the Expository References is consistent with that in the earlier books. [Al] M. Aschbacher, Theory, Cambridge University Press, Cambridge, 1986. [A2] , Sporadic Groups, Cambridge University Press, Cambridge, 1994. [BeGll] H. Bender and G. Glauberman, Local Analysis for the Odd Order Theorem, London Math. Soc. Lecture Notes Series 188, Cambridge University Press, Cambridge, 1994. [Cal] R. W. Carter, Simple Groups of Lie Type, Wiley and Sons, London, 1972. [Dl] J. Dieudonne, La Geometrie des Groupes Classiques, Springer-Verlag, Berlin, 1955. [Fl] W. Feit, The Representation Theory of Finite Groups, North Holland, Amsterdam, 1982. [FT1] W. Feit and J. G. Thompson, Solvability of groups of odd order (Chapter V, and the supporting material from Chapters II and III only), Pacific J. Math. 13 (1963), 775-1029. [Gl] D. Gorenstein, Finite Groups, 2nd edition, Chelsea, New York, 1980. [GL1] D. Gorenstein and R. Lyons, The local structure of finite groups of characteristic 2 type (Part I only), Memoirs Amer. Math. Soc. 276 (1983). [Hul] B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, 1967. [Isl] I. M. Isaacs, Character Theory of Finite Groups, Academic Press, New York, 1976. [Stl] R. Steinberg, Lectures on Chevalley Groups, Notes by J. Faulkner and R. Wilson, Mimeographed notes, Yale University Mathematics Department (1968). [St2] , Endomorphisms of linear algebraic groups, Memoirs Amer. Math. Soc. 80 (1968). [Sul] M. Suzuki, Group Theory I, II, Springer-Verlag, Berlin, 1982; 1986. The following two references are cited for a half-dozen specific facts about spo­ radic groups in the tables of Section 5.3. They are thus to be added as Background References, but strictly limited to these citations. [A19] M. Aschbacher, 3-Transposition Groups, Cambridge University Press, Cambridge, 1997. [CCNPWl] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985.

403 Expository References

[Anl] S. Andrilli, On the uniqueness of O'Nan's sporadic simple group, Ph.D. Thesis, Rut­ gers University (1980). [Arl] E. Artin, The orders of the classical simple groups, Comm. Pure and Appl. Math. 8 (1955), 455-472. [Ar2] , Geometric Algebra, Wiley-Interscience, New York, 1957. [Al] M. Aschbacher, Finite Group Theory, Cambridge University Press, Cambridge, 1986. [A2] , Sporadic Groups, Cambridge University Press, Cambridge, 1994. [A6] , Tightly embedded subgroups of finite groups, J. Algebra 42 (1976), 85-101. [A9] , A characterization of Chevalley groups over fields of odd order. I, II, Ann. of Math. 106 (1977), 353-468; Correction, Ann. of Math. Ill (1980), 411-414. [A15] , GF'(2) -representations of finite groups, Amer. J. Math. 104 (1982), 683-771. [A19] , 3-Transposition Groups, Cambridge University Press, Cambridge, 1997. [A20] , On finite groups of Lie type and odd characteristic, J. Algebra 66 (1980), 400-424. [A21] , The 27-dimensional module for EQ, IV, J. Algebra 131 (1990), 23-39. [AFGLOS1] M. Aschbacher, W. Feit, D. Gorenstein, R. Lyons, M. E. O'Nan and C. C. Sims (eds.), Proceedings of the Rutgers Group Theory Year, 1983-1984, Cambridge University Press, Cambridge, 1984. [ASe2] M. Aschbacher and G. M. Seitz, Involutions in Chevalley groups over fields of even order, Nagoya Math. J. 63 (1976), 1-91; Correction, Nagoya Math. J. 72 (1978), 135-136. [Azl] H. Azad, Semisimple elements of order 3 in finite Chevalley groups, J. Algebra 56 (1979), 481-498. [AzBaSel] H. Azad, M. Barry and G. Seitz, On the structure of parabolic subgroups, Comm. Alg. 18 (1990), 551-562. [Be3] H. Bender, Transitive Gruppen gerader Ordnung, in dene jede Involution genau einen Punkt festldfit, J. Algebra 17 (1971), 527-554. [Borl] A. Borel, Linear Algebraic Groups, Second Edition, Springer-Verlag, Berlin, 1991. [BCCISSl] A. Borel, R. W. Carter, C. W. Curtis, N. Iwahori, T. A. Springer, and R. Stein­ berg, Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes in Mathematics #131, Springer-Verlag, Berlin, 1970. [BT1] A. Borel and J. Tits, Elements unipotents et sousgroupes paraboliques des groupes reductifs L, Inv. Math. 12 (1971), 97-104. [Borol] A. Borovik, The structure of finite subgroups of simple algebraic groups, Algebra and Logic 28 (1989), 249-279. [Boul] N. Bourbaki, Groupes et Algebres de Lie, Chs. IV-VI, Hermann, Paris, 1968. [BrSal] R. Brauer and C.-H. Sah (eds.), Finite Groups, A Symposium, Benjamin, New York, 1969. [BroMal] M. Broue and G. Malle, Theoremes de Sylow generiques pour les groupes reductifs sur les corps finis, Math. Ann. 292 (1992), 241-262. [BuGrLyl] N. Burgoyne, R. L. Griess and R. Lyons, Maximal subgroups and automorphisms of Chevalley groups, Pacific J. Math 71 (1977), 365-403. [BuWil] N. Burgoyne and C. Williamson, On a theorem of Borel and Tits for finite Chevalley groups, Arch. Math. 27 (1976), 489-491. [BuWi2] , Centralizers of semisimple elements in Chevalley-type groups, Pacific J. Math. 70 (1977), 83-100. [Cal] R. W. Carter, Simple Groups of Lie Type, Wiley and Sons, London, 1972.

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The authors are grateful to J. L. Alperin, P. Flavell, C. Ku and J. G. Thompson for pointing out errors in [Ii, I2, IG]- On page 47 and page 140 of [Ii], the correct Background Reference is: [Cal] R. W. Carter, Simple Groups of Lie Type, Wiley and Sons, London, 1972. The same reference should be corrected also on p. 142, along with the following Expository Reference: [Ca2] R. W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Com­ plex Characters, Wiley-Interscience, London, 1985. In Definition 12.1 of [I2], the group (72(8) should be removed from the set C3 and placed in T3. m In the remark on line 6 of page 115 of [IG], the example J = SLn(r ) is appropriate only if n is odd. U On page 117, line -3 of [IG], replace "A < CP9(u)" by R± < CP9(u)"; and on 9 9 the next line replace "1 ^ Rx < A < P n P n Y" by "I ^ Ri < P C\P C\ Y". In Definition 21.1 of [IG], replace "^/-subgroups" by M-invariant ^-subgroups". In Lemma 29.5 of [IG], a hypothesis needs to be added. The following is adequate: Assume that there is a mapping (j> : E —> D such that i for all i G E, and whenever i, j G E with i < j, then (j>(i) < (j>(j).

409 GLOSSARY

PAGE SYMBOL

260 t 32 ~, ~K 260 {-}» 251, 321 2K, 3K, nK, [X)K 148 3*, 6*, n* (n€Z) 8, 14 (a,P) 10 a* 13 a9 13 as 11 ae 260-1 A • B, A#B, A*B,AYB,AYB 11 Am 2 36-37 Am{q), Am(q), Bm{q), ••• , G2(q), ' 252 An 27 Aut(K), Auto (if), Auti(if) 41 B 7 B

11 Bm 42 BCm 74 •B(K,a) 134 s-y* /^l*0 s~io 39 Ehev, &hev(r) 18 Cijafi 11 ^m 271-273 Co\, Co2, Co3 385 AH(E) 11 Dm 4 Dn D*n 260 U8 11 E6l £7, Eg 260 Epn 8 En, E 29 Vq 57 <&K 29 ^ (*7f) + 2 F, F, Fr 284-287 F\, F2, F3, F5

410 GLOSSARY 411

11 F4 281-283 FJ22, Fi_2^Fi'2i, Fi2i 2-3 F{K], F(K) 100 Fy

28 7P, 7P,K 343 rEtk(x), r^k(x), rEk(x), r^pO, r£fe(x), rB>,_fc(x) 57 rK 29 IV 28 r0, ro7? 383 TPtk(X) 11 G2 69-72 GL(V), GU(V), GSp(V), GO(V) 32 H*{a,K) 41, 50 iJ, #a 18 /iQ(t) 50-51 /ia(f) 277 He 274 FS1 10 ht 22 htj 53 ii^ 57 Inndiag^) 67 Isom(V, /) 72 I{V) 267-270 Ji, J2, J3, J4 36 A- 3 r 5 K+ 38 #a 19 Ka 6 fm 312 jCp 36 (K, a)

63 KSo, K^0)W, K^0,w 38 Xu 19 Ku 6, 13 A, A(T) 25 Ay(T) 36 £ie, £ie(r) 39 £iee:cc, £ieea:c(r) 65 Lj 22 Lj 9 £(w) 278 Ly 262-266 Mu, M12, M22, M23, M24 308 m2,p(X), m2p(X) 275 Mc 311 MC(K) 412 GLOSSARY

312 Me(K) 65 Mj 22 Mj 251 M(K) 346 Wx(Y;r) 41 N 14 N 21 No 65 Nj 51 n& 18 na{t) 89 nz 9 N(w) 69, 71 oLWtOi^vfj^oivj) 280 O'N 16 OiV) 16, 69-72 n(v),n^rn(q),n2rn+1(q) 177 Outc*°(L*) 57 Outdiag(if) 130 n*,n,, n*,ns* 260 pa+b 68-69 PGL(V), PGLm+1(F), PTL(V) 69-72 PGO{V), PGO^V), PGO±{q)

70 PGSp(yq,f),PGSp2m(q) 22 Pj 65 Pj 69-72 P£l(V), PCl±(V), Pilfm(q) 4, 68 PSL(V), PEL(V) 69-72 PSO{V), PSO±{V), PSO±(q)

70 PSp(V), PSp2m(q) 68-69 PSU(Vq2), PSUm+1(q) 29 i> 260 QT 35 q{K,a) 345 Qr(X,E)

8 ra 7 R(K), RU(K) 74 X(K,a) 179 Ru 31 a, xa 8 E, E+ 13 S* 41 E, E 44 E+ 44 ^a 45 ±x« 130 Si, E? 22 S./.S} GLOSSARY 413

252 Zn

34 aQ 36 EfoJ^Efo), E+(g), E-{q) 104 &(Xi *-*a 29 •^To (Sw 11 gn 67 Sim(V,/) 4, 68-69 SL(V), SLm+l(q) 4 SLn(F) 16, 69-72 SO(V), SO^V), SO±(q) 16, 69-70 Sp(V), Sp2m(q) 16, 69-72 Spin(V), Spin^iV), Spinfm(q), Spin2m+i(q), Spin(V,f) 75 St 68-69 SU{Vq2), SUm+1(q) 41 T(K,CT) 7 T 4 Tn 69 e 41 u 65 Uj 95 U) 22 Uj, Ulj 4 un 17 uw 41 V 98 V} 25 K 8 W, W(E) 41 w 13 w0 47 w& 130 Wi,0, wu w* 9 Wj 44 Xa 15 Xa 18 xa(t) 46 X&(t), X&(t,u), X&(t,U,v) 5 x(t)

44 Y& 136 Z 16 ^o Zs INDEX accidental isomorphisms 40 adjoint group 15, 38 adjoint module 26 affine algebra 2 , JF-algebraic group, F- 2 adjoint version 15 connected component of 3 dimension of 3 isogeny of 15 of 2 morphism of 2 radical of 7 reductive 7 semisimple 7, 15 components of 8 simple 7 simply connected 15 solvable 6-7 universal version 15 Alperin-Goldschmidt conjugation family 94 alternating groups 252-259 coverings of 253 natural module for 252 rank of 258 Atlas 259 notation 260 references to 279, 284, 285 automorphism 27, 53-62 diagonal 53, 56 field, graph, graph-field 60, 62, 140 inner-diagonal 59-60 outer 58 semisimple 4, 119-143 of equal-rank, graph, or parabolic type 125 Balance 385-399 local 1-balance 385-388, 390-391 local /c-balance, k > 1 388-390 strong local 1-balance 392-393 weak local 1-balance 389-390

414 INDEX

BN-pa,ir see Tits system Borel subgroup 7, 17, 20 a-invariant 33 unipotent radical of 21 Borel's Theorem 7 Borel-Tits Theorem 91-94 .Bp-property 337 Bruhat Normal Form 17, 43 84 Cartan matrix 24 Cartan subgroup 50-51 centralizer of element of prime order in alternating and symmetric groups 253-258 in classical groups 147, 182-185, 219-221 in exceptional groups 204-211, 222-224 of field or graph-field automorphism 227-228, 337 of graph automorphism 28, 141, 147, 170-186, 229 of involution 94, 143-153, 170-186, 227-229, 255-256, 262-287 in simply connected group 121, 133 in 262-287 character 6 characteristic U-nonsingular 20, 23, 95, 106 Chevalley basis 21 Chevalley Commutator Formula 18, 47-50 Chevalley generators 18-19, 51-52 Chevalley Relations 18, 45-51 16, 26-27, 67-73, 96-98, 185, 219-221 classical vector space 67 Clifford group 16, 314 cocharacter 6 component 8 conjugacy see fusion connected centralizer 134-135 covering, 38, 249-251 of 253 embeddings of 313-317 involutions in 294-297 isogenies of 15 isomorphisms of 321 of simple groups 311-313 of sporadic groups 303-309 ranks of 303, 328 Curtis-Tits Theorem 81 cyclotomic polynomial 237 Dickson invariant 16, 70 direct product 3, 15 10, 25 416 INDEX

extended 10, 125, 130 isomorphism of 10 endomorphism 27 Frobenius endomorphism 34-35 level of 34-35 Steinberg endomorphism 27, 31-38 action on homogeneous space 32-33 in standard form 37 equal rank type element 125 finite 36-39 adjoint 38 compatible 87 isomorphism of 40 order of 39 rank (twisted or untwisted) of 42 p-rank of 108, 239-241 simple 39 Sylow subgroups of 108-118, 236-244 universal 38 fundamental ST2-subgroups 104-105, 185-186, 243-248 fusion in X-groups 129-132, 171, 212, 262-287, 399-402 generation 343-349, 381-384 of alternating groups 381-382 of groups in Ghev 346-349 of sporadic groups 382-383 Gilman-Griess Theorem 87-88 graph type element 125 half- 16, 144, 321 half-spin module 16, 27 height 10 high weight vector 25 internal modules 25, 78, 98-102 involutions centralizers of see centralizer of element of prime order in covering groups 294-297, 322 in spin groups 314 splitting, stable 294, 324-328 isogeny 15, 30 isometry 67 Jordan decomposition 5 Jordan subgroup 224-227 Lang's Theorem, Lang-Steinberg Theorem 31 level 34-35 Levi complement, decomposition, factor 22, 66 Lie component 134-135, 233-236 Lie-Kolchin Theorem 7 INDEX 26, 68-69, 219-221 subgroups of 328-329 local balance see balance maximal torus see torus maximal parabolic subgroup 66, 98-102 modules see representations one-parameter group 5-6 16, 26, 69-72, 182-183 parabolic subgroup 22-23, 65-67, 93-102 in a classical group 96-98 maximal, minimal 66 in untwisted groups 98-100 in Steinberg and Suzuki-Ree groups 100-102 opposite 79 unipotent radical of 22, 65-66 parabolic type element 125 parametrization 5 polynomial functions 2 quasisimple groups 311-328 embeddings of 315 isomorphism of 321 3 r-exact action 142, 345, 352 radical 7 rational mapping 4 rational if-module 24 adjoint 26 examples 26-27 half-spin, spin 16, 27 7, 13-15 Ree-Suzuki group 37 subgroups of 101-102, 107, 332-333 representations 23-27, 73-79, 262-287 basic 74 classification of irreducible 73-75 examples of 26-27, 77-78 high weights of 25, 74 irreducible 74 of defect 0 76 splitting fields of 76 75-76 restricted range 74 root 8 fundamental 8 height of 10 highest, lowest 10, 130, 134 long, short 13 418 INDEX

positive, negative 8 T-root 13 root element 103 root involution 252 root 13 root subgroup 13, 45-46, 103-104 of alternating and 252 root four-group 252 long root element, subgroup 103 T-root subgroup 13 root system 8 classification of 12 closed subset of 8 crystallographic 8 dual 13 Dynkin diagram of 10 fundamental system in 8 irreducible 10 isomorphic 10 orthogonal decomposition of 9-10 positive subsystem of 8 abelian, semiabelian subset of 111 ideal in 9 reduced 8 Weyl group of 8 twisted 41-42 cr-conjugacy 32 cr-setup 36 standard 37, 60 Schreier conjecture 335 251 and automorphisms 251, 317-324 canonical part 311-312 exceptional part 312-313 semisimple element or automorphism 4, 119-143 semisimple group 7, 15 of 19-20 similarity 67 simple connectivity 15 Smith's Theorem 78 spin group, module 16, 69, 71, 314 spinorial norm 69 splitting fields 76 sporadic groups 259-294, 297-309 automorphism groups, Schur multipliers and subgroups of 262-287 ranks of 302-309 standard notation 89 Steinberg connectedness theorem 121 INDEX

Steinberg endomorphism see endomorphism Steinberg group 37 Steinberg module 75-76 Steinberg relations 79-80 Steinberg Tensor Product Theorem 75 strongly p-embedded subgroup 383-385 subcomponents 341 subsystem subgroup 63-65 Suzuki- see Ree-Suzuki group Sylow subgroups of groups of Lie type 106-118, 236-244 of sporadic groups 262-287 symmetric bilinear forms 72 symmetric group see alternating group 16, 26, 70 Tits system 16, 43 torus 6 character, cocharacter of 6 in adjoint group 57 maximal torus 6, 53, 236-237 cr-split, cr-invariant 33 subtorus 6 28 twisted group of Lie type 37 twisted rank 42 2-defined expression 81 unipotent element, group, radical 4, 5, 7, 22, 65-66 68-69, 219-221 universal group 15, 38 untwisted group of Lie type 36-37 untwisted rank 42 /7-nonsingular see characteristic version of an algebraic or finite group 15, 38 weak p-signalizer property 394-396 weakly closed subgroup 399 weight 13 basic 74 dominant 24 fundamental 24 high 25, 74 multiplicity of 24 in restricted range 74 weight lattice 13 weight vector 24 Witt's Lemma 72 Zariski topology 2 Selected Titles in This Series (Continued from the front of this publication)

21 Albert Baernstein, David Drasin, Peter Duren, and Albert Marden, Editors, The Bieberbach conjecture: Proceedings of the symposium on the occasion of the proof, 1986 20 Kenneth R. Goodearl, Partially ordered abelian groups with interpolation, 1986 19 Gregory V. Chudnovsky, Contributions to the theory of transcendental numbers, 1984 18 Frank B. Knight, Essentials of Brownian motion and diffusion, 1981 17 Le Baron O. Ferguson, Approximation by polynomials with integral coefficients, 1980 16 O. Timothy O'Meara, Symplectic groups, 1978 15 J. Diestel and J. J. Uhl, Jr., Vector measures, 1977 14 V. Guillemin and S. Sternberg, Geometric asymptotics, 1977 13 C. Pearcy, Editor, Topics in operator theory, 1974 12 J. R. Isbell, Uniform spaces, 1964 11 J. Cronin, Fixed points and topological degree in nonlinear analysis, 1964 10 R. Ayoub, An introduction to the analytic theory of numbers, 1963 9 Arthur Sard, Linear approximation, 1963 8 J. Lehner, Discontinuous groups and automorphic functions, 1964 7.2 A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Volume II, 1961 7.1 A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Volume I, 1961 6 C. C. Chevalley, Introduction to the theory of algebraic functions of one variable, 1951 5 S. Bergman, The function and conformal mapping, 1950 4 O. F. G. Schilling, The theory of valuations, 1950 3 M. Marden, Geometry of polynomials, 1949 2 N. Jacobson, The theory of rings, 1943 1 J. A. Shohat and J. D. Tamarkin, The problem of moments, 1943