The Classification of Finite Simple Groups

Mathematical Surveys and Monographs Volume 172

The Classiﬁ cation of Finite Simple Groups Groups of Characteristic 2 Type

Michael Aschbacher Richard Lyons Stephen D. Smith Ronald Solomon

American Mathematical Society

surv-172-smith3-cov.indd 1 2/4/11 1:15 PM http://dx.doi.org/10.1090/surv/172

The Classification of Finite Simple Groups

Mathematical Surveys and Monographs Volume 172

The Classification of Finite Simple Groups Groups of Characteristic 2 Type

Michael Aschbacher Richard Lyons Stephen D. Smith Ronald Solomon

American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair MichaelA.Singer Eric M. Friedlander Benjamin Sudakov MichaelI.Weinstein

2010 Mathematics Subject Classiﬁcation. Primary 20D05; Secondary 20C20.

Abstract. We complete an outline, aimed at the non-expert reader, of the original proof of the Classification of the Finite Simple Groups. The first half of such an outline, namely Volume 1 covering groups of noncharacteristic 2 type, had been published much earlier by Daniel Gorenstein in his very detailed 1983 work [Gor83]. Thus the present book, which we regard as “Volume 2” of that project, aims at presenting a reasonably detailed outline of the second half of the Classification: namely the treatment of groups of characteristic 2 type. Aschbacher was supported in part by NSF DMS 0504852 and subsequent grants.

Lyons was supported in part by NSF DMS 0401132, NSA H98230-07-1-0003, and subsequent grants.

Smith was supported in part by NSA H98230-05-1-0075 and subsequent grants.

Solomon was supported in part by NSF DMS 0400533, NSA H98230-07-1-0014, and subsequent grants.

For additional information and updates on this book, visit www.ams.org/bookpages/surv-172

Library of Congress Cataloging-in-Publication Data The classiﬁcation of ﬁnite simple groups : groups of characteristic 2 type / Michael Aschbacher ... [et al.]. p. cm. — (Mathematical surveys and monographs ; v. 172) Includes bibliographical references and index. ISBN 978-0-8218-5336-8 (alk. paper) 1. Finite simple groups. 2. Representations of groups. I. Aschbacher, Michael, 1944– QA177.C53 2011 512.2—dc22 2010048011

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 is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2011 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 http://www.ams.org/ 10987654321 161514131211 To the memory of Danny Gorenstein

Contents

Preface xi

Background and overview 1 Chapter 0. Introduction 3 0.1. The Classification Theorem 3 0.2. Principle I: Recognition via local subgroups 4 0.3. Principle II: Restricted structure of local subgroups 7 0.4. The finite simple groups 16 0.5. The Classification grid 19 Chapter 1. Overview: The classification of groups of Gorenstein-Walter type 25 The Main Theorem for groups of Gorenstein-Walter type 25 1.1. A strategy based on components in centralizers 26 1.2. The Odd Order Theorem 28 1.3. (Level 1) The Strongly Embedded Theorem and the Dichotomy Theorem 29 1.4. The 2-Rank 2 Theorem 33 1.5. (Level 1) The Sectional 2-Rank 4 Theorem and the 2-Generated Core Theorem 35 1.6. The B-Conjecture and the Standard Component Theorem 41 1.7. The Unbalanced Group Theorem, the 2An-Theorem, and the Classical Involution Theorem 44 1.8. Finishing the Unbalanced Group Theorem and the B-Theorem 48 1.9. The Odd Standard Component Theorem and the Aschbacher-Seitz reduction 53 1.10. The Even Standard Component Theorem 55 Summary: Statements of the major subtheorems 59 Chapter 2. Overview: The classification of groups of characteristic 2 type 63 The Main Theorem for groups of characteristic 2 type 63 2.1. The Quasithin Theorem covering e(G) ≤ 265 2.2. The trichotomy approach to treating e(G) ≥ 366 2.3. The Trichotomy Theorem for e(G) ≥ 469 2.4. The e(G) = 3 Theorem (including trichotomy) 75 2.5. The Standard Type Theorem 77 2.6. The GF (2) Type Theorem 77 2.7. The Uniqueness Case Theorem 78 Conclusion: The proof of the Characteristic 2 Type Theorem 80

vii viii CONTENTS

Outline of the classification of groups of characteristic 2 type 83 Chapter 3. e(G) ≤ 2: The classification of quasithin groups 85 3.1. Introduction: The Thompson Strategy 86 3.2. Preliminaries: Structure theory for quasithin 2-locals (SQTK-groups) 88 3.3. More preliminaries: Some general techniques 90 3.4. The degenerate case: A Sylow T in a unique maximal 2-local 98 3.5. The Main Case Division (Possibilities for a suitable group L and module V ) 100 n 3.6. The Generic Case—where L = L2(2 ) with n>1 103 3.7. Reducing to V an FF-module for L 106 3.8. Cases with L over F2n for n>1 109 3.9. Cases with L over F2 (but not L3(2)) 111 3.10. Cases with L = L3(2), and analogues for L2(2) 117 3.11. The final case where Lf (G, T ) is empty 120 3.12. Bonus: The Even Type (Quasithin) Theorem for use in the GLS program 123 Chapter 4. e(G) = 3: The classification of rank 3 groups 127 4.1. The case where σ(G) contains a prime p ≥ 5 128 The Signalizer Analysis 128 The Component Analysis 130 4.2. The case σ(G)={3} 133 The Signalizer Analysis 134 The Component Analysis 142 Chapter 5. e(G) ≥ 4: The Pretrichotomy and Trichotomy Theorems 149 5.1. Statements and Definitions 149 5.2. The Signalizer Analysis 152 5.3. The Component Analysis (leading to standard type) 159 Chapter 6. The classification of groups of standard type 173 6.1. The Gilman-Griess Theorem on standard type for e(G) ≥ 4 173 Identifying a large Lie-type subgroup G0 174 The final step: G = G0 177 6.2. Odd standard form problems for e(G) = 3 (Finkelstein-Frohardt) 180 Chapter 7. The classification of groups of GF (2) type 183 Introduction 184 7.1. Aschbacher’s reduction of GF (2) type to the large-extraspecial case 185 7.2. The treatment of some fundamental extraspecial cases 188 7.3. Timmesfeld’s reduction to a list of possibilities for M 192 7.4. The final treatment of the various cases for M 199 7.5. Chapter appendix: The classification of groups of GF (2n) type 204 CONTENTS ix

Chapter 8. The ﬁnal contradiction: Eliminating the Uniqueness Case 213 8.1. Prelude: From the Preuniqueness Case to the Uniqueness Case 215 8.2. Introduction: General strategy using weak closure and uniqueness theorems 223 8.3. Preliminary results and the weak closure setup 226 8.4. The treatment of small n(H) 230 8.5. The treatment of large n(H) 234

Appendices 249 Appendix A. Some background material related to simple groups 251 A.1. Preliminaries: Some notation and results from general group theory 251 A.2. Notation for the simple groups 254 A.3. Properties of simple groups and K-groups 256 A.4. Properties of representations of simple groups 261 A.5. Recognition theorems for identifying simple groups 262 A.6. Transvection groups and transposition-group theory 264 Appendix B. Overview of some techniques used in the classiﬁcation 267 B.1. Coprime action 267 B.2. Fusion and transfer 269 B.3. Signalizer functor methods and balance 272 B.4. Connectivity in commuting graphs and i-generated cores 280 B.5. Application: A short elementary proof of the Dichotomy Theorem 287 B.6. Failure of factorization 290 B.7. Pushing-up, and the Local and Global C(G, T ) Theorems 292 B.8. Weak closure 299 B.9. Klinger-Mason analysis of bicharacteristic groups 302 B.10. Some details of the proof of the Uniqueness Case Theorem 305

References and Index 313 References used for both GW type and characteristic 2 type 315 References mainly for GW type (see [Gor82][Gor83] for full list) 317 References used primarily for characteristic 2 type 321

Expository references mentioned 329 Index 333

Preface

The present book, “The Classification of Finite Simple Groups: Groups of Characteristic 2 Type”, completes a project of giving an outline of the proof of the Classification of the Finite Simple Groups (CFSG). The project was begun by Daniel Gorenstein in 1983 with his book [Gor83]—which he subtitled “Volume 1: Groups of Noncharacteristic 2 Type”. Thus we regard our present discussion of groups of characteristic 2 type as “Volume 2” of that project. The Classification of the Finite Simple Groups (CFSG) is one of the premier achievements of twentieth century mathematics. The result has a history which, in some sense, goes back to the beginnings of proto-group theory in the late eighteenth century. Many classic problems with a long history are important more for the mathematics they inspire and generate, than because of interesting consequences. This is not true of the Classification, which is an extremely useful result, making possible many modern successes of finite group theory, which have in turn been applied to solve numerous problems in many areas of mathematics. A theorem of this beauty and consequence deserves and demands a proof accessible to any mathematician with enough background in finite group theory to read the proof. Unfortunately the proof of the Classification is very long and complicated, consisting of thousands of pages, written by hundreds of mathematicians in hundreds of articles published over a period of decades. The only way to make such a proof truly accessible is, with hindsight, to reorganize and rework the mathematics, collect it all in one place, and make the treatment self-contained, except for some carefully written and selected basic references. Such an effort is in progress in the work of Gorenstein, Lyons, and Solomon (GLS) in their series beginning with [GLS94], which seeks to produce a second-generation proof of the Classification. However in the meantime, there should at least be a detailed outline of the existing proof, that gives a global picture of the mathematics involved, and explicitly lists the papers which make up the proof. Even after a second-generation proof is in place, such an outline would have great historical value, and would also provide those group theorists who seek to further simplify the proof with the opportunity to understand the approach and ideas that appear in the proof. That is the goal of this volume: to provide an overview and reader’s guide to the huge literature which makes up the original proof of the Classification. Soon after the apparent completion of the Classification in the early 1980s, Daniel Gorenstein began a project aimed at giving an outline of the original proof. He provided background in a substantial Introduction [Gor82], in particular dis- cussing the partition of simple groups into groups of odd characteristic and groups of characteristic 2 type. Then in Volume 1 [Gor83] he described the treatment of

xi xii PREFACE the groups of odd characteristic in detail. However he did not complete the rest of his project, in part because the proof for groups of characteristic 2 type remained incomplete, specifically that part of the proof treating the quasithin groups un- dertaken by Mason [Mas]. This gap was recently filled by the Aschbacher-Smith classification of the quasithin groups [AS04b]. Hence it is now possible to fin- ish Gorenstein’s project by outlining the proof for the groups of characteristic 2 type. We accomplish that goal here, adopting his title, and regarding the work as “Volume 2” in the series. While we recommend that the interested reader consult Gorenstein’s books, we also intend that our treatment should be sufficiently self-contained that those works will not be a prerequisite. Therefore in Chapter 1, we supply an overview of the treatment of the groups of odd characteristic, which is much briefer than Gorenstein’s detailed treatment. In fact, throughout our exposition, we will be less detailed than Gorenstein, since we believe that a briefer outline of the main steps will be more accessible and useful to most readers. On the other hand, we are careful to honor the important fundamental goal of explicitly listing those works in the literature which make up the proof that all simple groups of characteristic 2 type are known. Mathematics, particularly the proof of a complex theorem, is hierarchical. We will list the results on groups of characteristic 2 type at the top of that hierarchy, which we refer to as “level 0” results. These are the papers containing subtheorems whose union affords the classification of the groups of characteristic 2 type. We also discuss the papers at level 1: the principal subsidiary results used in the proofs of subtheorems at level 0. We will not usually attempt an analysis through levels 2 and beyond; that is, as a rule we do not discuss those papers used to establish the subsidiary results, and so on, down to first principles and the level of textbooks. But our outline could be used as a starting point for such a deeper analysis of the proof. Finally we will typically assume that the reader has some familiarity with con- cepts, terminology, notation, and results from elementary group theory, such as might be standard in a first year graduate algebra course. Beyond that, we will try to give more advanced definitions when they arise in our discussion. In addition we provide in Chapter A of the Appendix a review of some intermediate material on simple groups and their properties. The Index should be helpful when encoun- tering new terminology and notation; normally the index entry given in boldface indicates either the definition, or the most fundamental page reference. Acknowledgments. We would like to thank various colleagues for helpful comments on early stages of this work; especially Rebecca Waldecker. (And thanks as usual to the referee.) Smith is grateful to All Souls College Oxford for a Visiting Fellowship during Hilary Term 2009.

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Index

Page locations for deﬁnitions, as well as for references which are particularly fundamental, are indicated in boldface.

∗, central product A ∗ B, 251 characterization of M12 [Asc03a] , 112, I(A), p-subgroups invariant under 118 p-group A, 138, 220 characterization of U3(3) [Asc02a] , 118 classical involution theorem abstract minimal parabolic, 87, 87, 94, 102, [Asc77a, Asc77b] , 40, 46, 170, 247, 235, 294, 295 259 AG(X), automizer NG(X)/CG(X), 174 minor correction [Asc80a], 46 algebraic groups (as approach to Lie type condition for strongly embedded [Asc73] groups), 17, 257 , 180 almost e(G) = 3 classiﬁcation -special groups, 104, 106, 110, 112–114, part I [Asc81b], 75, 128, 145, 180, 116, 118, 122, 124 216, 247 strongly p-embedded, see also embedded part II [Asc83a], 75, 133, 180, 216, Alperin, J. 219, 247 -Brauer-Gorenstein, 2-rank 2 ﬁnite group theory (book) [Asc00] , 267, [ABG70, ABG73b, ABG73a] , 34, 268, 276, 290, 292 35 GF (2)-representations [Asc82] , 93, 96, -Gorenstein, transfer and fusion [AG67, 146, 170, 231, 242, 247, 291, 292, 301, p 243], 178 302 Alperin-Goldschmidt conjugation family, -Gorenstein-Lyons, uniqueness theorems 270, 270, 271, 288 [AGL81], 79, 133, 141, 145, 149, 214, Alperin-Goldschmidt Fusion Theorem, 98, 215, 215 270, 288 large extraspecial (unitary) [Asc77], 77, alternating simple groups, 254 185, 188, 189, 199, 207 Alternating Theorem, 49,53 large symplectic not extraspecial Alward, L. [Asc76a], 77, 185, 205 − standard Ω8 (2) [Alw79],59 Lie type and odd characteristic [Asc80b] amalgam, 86, 96, 104, 105, 116, 118, 226, , 247, 259 263, 296 Local C(G, T )theorem[Asc81a], 207, Goldschmidt —, 97, 113, 115, 119, 122, 293 n 219 L2(2 ) standard blocks [Asc81d], 298 method, 96, 97, 104, 106, 110, 111, odd transpositions [Asc72] , 175, 180, 113–118, 120, 121, 170, 223 188, 207, 208, 265 leading to “small” modules, 97 pushing-up results [Asc81d] , 233, 247, parameter b for —, 97 299 Am-block, 293 pushing-up theorem [Asc78a] , 207, 298 Andrilli, S. -Segev, uniqueness of J4 [AS91] , 108, uniqueness of ON [And80] , 260 260 Aschbacher, M., 42, 45, 54, 68, 115, 187, -Seitz, involutions in characteristic 2 220, 279, 280, 307, 309 [AS76a] , 54, 133, 146, 170, 179, 180, characterization of G2(3) [Asc02b] , 119 187, 189, 199, 201, 258 characterization of HS [Asc03b] , 119 minor correction , 54

333 334 INDEX

-Seitz, standard known type with respect to A, 278 [AS76b, AS81], 52, 54, 133, 146, 260 Baumann, B. n -Smith, preliminaries for quasithin pushing-up L2(2 )[Bau79] , 147, 170, [AS04a], 87, 294, 299 294 -Smith, quasithin classiﬁcation [AS04b], Baumann’s Lemma, 219, 294, 294 xii, 65, 85, 198, 299 B-Conjecture, 19, 42, 43, 48, 53 sporadic groups book [Asc94] , 105, 112, B-Theorem, 42, 53,60 113, 116, 118, 190, 192, 259, 260 Bp-Property (odd analogue), 22, 162, standard alternating [Asc08], 49,54 165 standard component theorem [Asc75a], Beisiegel, B. 43, 54, 123, 188, 190, 299 semi-extraspecial p-groups [Bei77] , 209, standard F3 components [Asc82a],52 210 2 standard Tits F4(2) [Asc82b],56 Bender, H., 7, 253 thin groups [Asc78b], 65, 85, 86, 204, dihedral revision [Ben81],34 299 -Glauberman, dihedral revision [BG81], 3-transpositions book [Asc97] , 133, 247, 34 259, 260, 264 -Glauberman, odd order local revision tight embedding [Asc76b] , 188, 205, [BG94],29 297, 299 normal p-group in p-solvable [Ben67], 2-components [Asc75b] , 188 268 2-generated core [Asc74],40 proof of odd order uniqueness theorem Uniqueness Case [Ben70],29 part I [Asc83b], 74, 76, 80, 94, 133, Signalizer Functor Theorem [Ben75], 214, 214, 223, 299 275 part II [Asc83c], 74, 80, 94, 133, 214, strongly embedded subgroups [Ben71], 214, 299 31 weak closure [Asc81e] , 94, 207, 227, Bender groups, BN-rank 1 in characteristic 299, 302, 307, 308 two, 6 Aschbacher-Goldschmidt functor, 155, 170 Bender-Suzuki Theorem, see also Strongly Aschbacher-Seitz Reduction Theorem, 54 Embedded Theorem Aschbacher symplectic not extraspecial Bender-Thompson Signalizer Lemma, 133, theorem, 185, 188, 203, 208 145, 160, 170, 220, 269, 303 Aschbacher unitary extraspecial theorem, Bennett, C. 189, 190–193, 202, 206 -Shpectorov, revision of Phan [BS04], a2 (Suzuki type for involution), 189, 189, 47, 263 193–196, 199, 202, 209–211 block Am- —, 293 B(−), product of non-quasisimple Aschbacher- — in C(G, T )Theorem,293 2-components, 41 χ-—,293 balance, 12, 12, 13, 14, 21, 44, 124, 129, χ0-—,93 n 130, 136, 154, 155, 157, 169, 170, L2(2 )- —, 293 272–278, 278, 279, 280, 287 Bloom, D. and Θ-signalizers, 275 subgroups of PSL3(q)[Blo67] , 133, and uniqueness subgroups, 278 146, 258 k-—,278, 279, 280 Bmax(G; p), elementary groups exhibiting 1- —, 155, 278 m2,p(G), 70 2- —, 154, 155, 278 BN weak —, 170, 280 -pair, 258 with respect to A, 278 weak — of rank 2, 96 1 k + 2 -balanced functor, 280 -rank, 257 L-—,21, 23, 154, 161 bootstrapping Lp -—,see also L-balance between p-uniqueness and 2-uniqueness, local (1)- —, 129, 154, 278 213, 215 in K-groups, 279 Borel, A. strong —, 154, 154, 155, 156, 164, 218 -Tits, Borel-Tits Theorem [BT71] , 18, 3 local 2 - —, 129, 155 164, 180 obstructions to —, 12, 14, 44, 136, 170, Borel subgroup in Lie type group, 257 278 Bourbaki, N. INDEX 335

root systems [Bou68] , 169 -Held, standard L3(4) [CH81, CH85], Brauer, R., 5, 35 52 Alperin- — -Gorenstein, 2-rank 2 Chermak, A., 220 [ABG70, ABG73b, ABG73a],35 Chev(p), Lie type groups in characteristic -Fowler, finite possibilities given a fixed p, 254 involution centralizer [BF55],5 Chevalley involution centralizer approach [Bra57], construction of Lie type groups, 17, 257 5 group, see also Lie type group -Suzuki, quaternion Sylows [BS59],34 χ-block, see also block -Suzuki-Wall, characterization of L2(q) χ0-block, see also block [BSW58] , 34, 264 classical Brauer-Suzuki Theorem, 6, 34, 34,35 involution, 46 building, 4, 17, 258 matrix groups (Lie type), 255 Burgoyne, N. Classical Involution Theorem, 40, 46, 48, -Griess-Lyons, Chevalley groups 50, 51, 53, 55, 187, 203, 280 [BGL77] , 169, 170, 180, 190 Classification of the Finite Simple Groups, Thompson reduction [Bur77],48 see also CFSG 3-centralizers in Chev(2) [Bur83], 171 Clifford’s theorem, 254, 308 -Williamson, on Borel-Tits theorem Collins, M. [BW76] , 169 Sylow of type L3(q)[Col73] , 182 -Williamson, semisimple Chevalley commuting graph, 32 classes [BW77] , 169, 180, 258 disconnected — Burnside, W., 5, 34 and signalizer functors, 37, 280 finite groups book [Bur55] , 146 and strong embedding, 32, 281 Burnside Fusion Theorem, 269 complement Burnside Transfer Theorem, 271, 311 Frobenius —, 251 complete Campbell, N., 294 signalizer functor, 13, 275 pushing-up result in thesis [Cam79], completion 170, 219, 222, 239, 294 of a signalizer functor, 153, 275 Cartan subgroup in Lie type group, 257 of an amalgam, 96 Carter, R. component, 253 simpleLietypebook[Car89] , 169, 180, locally k-unbalanced —, 279 201, 255, 256 locally unbalanced —, 278 C-component, 89 maximal —, 42 central product, 251 p-—,see also p-component centric standard —, 43 p- —, 270 odd —, 70 CFSG, xi, 3, 81, 223 3- —, see also 3-component original proof (completed 2004), xi 2-—,see also 2-component second effort of type, 10 Gorenstein-Lyons-Solomon, xi connectedness, see also commuting graph C(G, T ), 293 constrained C(G, T )-Theorem p—, 268 Global —, see also Global control C(G, T )-Theorem of fusion, 269 Local —, see also Local of transfer, 271 C(G, T )-Theorem Conway, J. ∗ C (G, T ), 132 construction of Co1 [Con69] , 260 characteristic lectures on sporadic groups [Con71], p (group of —), 9 146, 169, 259 local —, 9 -Wales, construction of Ru [CW73] , 260 p type, 9 Cooperstein, B., 92 subgroup, 73 -Mason, unpublished FF-module analysis 2type,9,63 [CM80] , 146, 201, 209, 210, 291 classification of simple groups of —, coprime action, 11, 267 64,80 core Cheng, Kai Nah, 52 k-generated p-—(Γk,P (G)), 31 336 INDEX

O2 of involution centralizer, 44 e(G) = 3 Theorem, 67, 69, 70, 76, 80, 85, cover 127, 213, 223, 298 double —, 260 Egawa, Y. triple —, 260 standard M24 [Ega81],58 + critical subgroup, 267 standard Ω8 (2) [Ega80],59 c2 (Suzuki type for involution), 187, 187 + -Yoshida, standard 2Ω8 (2) [EY82],59 Curtis, C. embedded -Kantor-Seitz, 2-transitive Chevalley strongly —, 31 groups [CKS76] , 180 strongly p-— lectures on Chevalley groups [Cur71], almost —, 78, 79, 79, 150, 213–215, 262 217–219, 221, 223, 310 Lietypepresentations[Cur65] , 263 tightly —, 43 Curtis-Tits Theorem, 28, 47, 132, 133, 145, Epn ,elementaryp-subgroup of rank n,50 176, 181, 182, 263 equivariant function, 36 Dade, E., 29 signalizer functor, 13, 272 Davis, S. even -Solomon, some standard sporadics case (characteristic 2 type) for CFSG, 9 [DS81] , 58, 59 characteristic, 66,85 Delgado, A. type, 66, 85, 123, 168 -Goldschmidt-Stellmacher, theory of Even Standard Component Theorem, 56, amalgams [DGS85] , 96, 105 56,57 Δ (D), 278 G Even Type (Quasithin) Theorem in Dempwolff, U. [AS04b] for use in GLS, 66, 123 characterization of L (2) [Dem73b], n exceptional groups of Lie type, 255 182, 192 existence problem for simple groups, 6 characterization of Ru [Dem74] , 260 extraspecial p-group, 252 second cohomology of L (2) [Dem73a], n large —, 113, 164, 181, 185, 185, 186, 192 188–193, 198–202 -Wong, characterization of L (2) n classification, see also GF (2) Type [DW77a] , 192 Theorem -Wong, large extraspecial reducible I extremal conjugate, 269 [DW77b], 77, 191 -Wong, large extraspecial reducible II [DW78], 191, 191 F (−), Fitting subgroup, 253 ∗ Dempwolff-Wong Theorem, 191, 194, 195, F (−), generalized Fitting subgroup, 253 202, 209, 211 failure of factorization, 291 diagonal automorphism of Lie type group, determining groups and modules, 91, 92, 258 146, 246, 291, 292 Dichotomy Theorem, 11, 25, 27, 32, 63, 67, methods, 91, 92, 146, 218, 219, 229, 262, 81, 223, 287, 304 290, 290, 294 Dickson, L. E. module (FF-module), 91, 92, 95, 102, linear groups [Dic58] , 258 103, 107–111, 115, 116, 201, 239, 261, Dickson’s Theorem, 133, 170, 258 291, 291, 292, 294 Dieudonné, J. ratios q andq ˆ,91 geometry of classical groups [Die55], solvable groups exhibiting —, 221, 292 169 Feit, W., 5 dihedral group, 252 -Thompson, odd order theorem [FT63], Dihedral Sylow Theorem, 34 5, 28 direct product, 251 -Thompson, self-centralizing order 3 disconnectedness, see also commuting [FT62] , 182 graph, disconnected Fendel, D. double cover, 260 characterization of Co3 [Fen73] , 260 doubly transitive, 30 FF-modules, see also failure of factorization Dynkin diagram for Lie type group, 257 field automorphism of Lie type group, 258 Finkelstein, L. E(−), product of components, 253 centralizer with cyclic Sylows [Fin77c], e(−), maximum odd p-rank in 2-locals, 20 55, 58 INDEX 337

-Frohardt, odd standard Ln(2) [FF81a], Finkelstein- —, odd standard Ln(2) 182 [FF81a] , 182 -Frohardt, standard 3-components Finkelstein- —, standard 3-components [FF84, FF79, FF81b], 77, 180, 181 [FF84, FF79, FF81b], 77, 180, 181 maximals of Co3 and McL [Fin73] , 260 trilinear form for J3 [Fro83] , 118 -Rudvalis, maximals of J2 [FR73] , 146, FSU, see also Fundamental Setup 260 functor -Rudvalis, maximals of J3 [FR74] , 146, signalizer —, 12, 273 260 equivariant —, 13, 272 -Solomon, odd standard Sp2n(2) Fundamental Setup (FSU) for Quasithin [FS79b] , 182 Theorem, 101 -Solomon, standard M12,Co3 [FS79a], Fundamental Weak Closure Inequality 58 (FWCI) for Quasithin Theorem, see standard J1–J4,Ree also weak closure [Fin75, Fin76b, Fin77b],58 fusion, 5 standard M22,M23 [Fin77a, Fin76a], control of —, 269, 271 58 theorems, 269 Finkelstein-Frohardt Theorem, 70, 144, 181 FWCI, see also weak closure Fischer, B., 187 3-transpositions [Fis71] , 133, 247, 260, Γk,P (G), k-generated p-core, 31 0 264 Γ2,P (G), weak 2-generated 2-core, 37 Fischer’s Theorem, 47, 132, 133, 146, 187, Gaschütz, W., 190 190, 194, 203, 247, 264, 265 generalized ∗ − Fitting subgroup F (−), 253 Fitting subgroup F ( ), 253 generalized — F ∗(−), 253 self-centralizing property of —, 7, 254 Fletcher, L. quaternion group, 252 -Stellmacher-Stewart [FSS77] , 182 generation properties for simple groups, F 1-modules, 292 132, 146, 150, 154, 156, 169, 170, 179, 217, 218, 247, 258 Fong, P., 39 generic, in sense of large-engough, 20 -Seitz, split BN-pairs of rank 2 [FS73], geometry from subgroups of simple group, 7 97 G-equivariant -Wong, characterization of rank 2 groups function, 36 [FW69] , 170 G-equivariant Foote, R., 297 signalizer functor, 13, 272 Aschbacher blocks [Foo82] , 207 getting started functor, 128, 130, 135, 154, expository paper on blocks [Foo80], 155 292, 296 GF (2) type, 73 standard L (q)[Foo78],50 2 GF (2) Type Theorem, 73, 78, 80, 106, 124, form 143, 147, 150, 168, 181–183, 184, 206, standard —, see also standard form 223 odd —, see also standard form GF (2n)type,204 4-group (elementary of rank 2), 15 GF (2n) Type Theorem, 137, 147, 205, 223, Fowler, K. 226, 229, 247 Brauer- —, finite possibilities given a Gilman, R. fixed involution centralizer [BF55],5 constrained components [Gil76], 296 Frame, J. S. -Gorenstein, class 2 Sylow 2-subgroups properties U4(2), Sp6(2) [Fra51] , 190 [GG75], 298 Frattini argument, 252 — -Griess, standard type classification Fritz, F. [GG83], 77, 130, 173 small components [Fri77a, Fri77b], 51, on standard component theorem [Gil76] 52 ,43 Frobenius -Solomon, unbalancing reduction complement, 251 [GS79a],50 group, 251 Gilman-Griess Presentation Theorem, 174, kernel, 251 176, 177 Frobenius, G., 5, 30, 34 Gilman-Griess Theorem (Standard Type), Frohardt, D. 173 338 INDEX

Glauberman, G., 13, 294 strongly closed (product fusion) [Gol75] Bender- —, dihedral revision [BG81], , 170, 188, 198 34 trivalent graphs [Gol80], 97, 170, 219 Bender- —, odd order local revision weakly embedded 2-locals [Gol72] , 170 [BG94],29 Goldschmidt amalgam, see also amalgam, global and local [Gla71] , 180 Goldschmidt lectures on factorizations [Gla77], 140, Goldschmidt Fusion Theorem, see also 147, 170 Goldschmidt, strongly closed (2-fusion rank 3 Solvable Signalizer Functor theorem) Theorem [Gla76], 275 Gomi, K. revisions to Brauer-Suzuki [Gla74],34 2-locals with class 2 Sylows [Gom75], —’s Argument, 295 298 n solvable failure of factorization[Gla73], standard Sp4(2 ),U4(2) 170, 292 [Gom78a, Gom78b],56 solvable signalizer functor theorem standard Sp6(2) [Gom80],59 [Gla76] , 170 Gorenstein, D., xi, 7, 12, 13, 15, 42, 44, 64, Sylow normalizers controlling transfer 67, 68, 253, 272 [Gla70], 171 Alperin- —, transfer and fusion [AG67, -Thompson, normal p-complement p 243], 178 theorem [Gla68] , 180 Alperin-Brauer- —, 2-rank 2 Z∗-theorem[Gla66], 269 [ABG70, ABG73b, ABG73a],35 ZJ-Theorem [Gla68],29 Aschbacher- — -Lyons, uniqueness Glauberman’s Argument, 295 theorems [AGL81], 79, 133, 141, 145, Glauberman-Niles Theorem, 170, 219, 221, 149, 214, 215, 215 294 finite groups textbeook [Gor80a] , 252, Glauberman Triple Factorization, 140 267, 271 Glauberman Z∗-Theorem, see also Gilman- —, class 2 Sylow 2-subgroups Z∗-Theorem [GG75], 298 Global C(G, T )-Theorem, 98, 99, 132, 133, -Harada, characterization of J2,J3 145, 147, 158, 207, 224, 226, 236, 237, [GH69],39 239, 245, 247, 292, 296 -Harada, low 2-rank and Lie families GLS [GH71a] , 180 Gorenstein-Lyons-Solomon project, xi, -Harada, sectional 2-rank 4 [GH74] , 38, 66, 223 298 no. 1: overview, outline [GLS94], xi, 98, -Harada, Sylow of type 2An [GH71b], 99, 253 45, 180 no. 2: general group theory [GLS96], 12, introduction to CFSG [Gor82],xi,25, 13, 110, 124, 190, 251, 267–272, 280, 254, 267, 271, 290, 292, 299 286, 288, 290–292, 295, 300, 309, 311 -Lyons, functors and nonconnectedness no. 3: properties of simple groups [GL82], 285 [GLS98], 89, 190, 201, 254, 256, 259, -Lyons, on Local C(G, T )-Theorem 262 [GL93] , 295 no. 4: uniqueness theorems [GLS99], 98, -Lyons, nonsolvable signalizer functors 115, 123 [GL77] , 169 no. 5: the generic case, balance -Lyons, trichotomy for e(G) ≥ 4[GL83], [GLS02], 155 133, 145, 150, 180, 214, 216, 247, 256 no. 6: the special odd case [GLS05], 35, -Lyons-Solomon, second effort CFSG, see 39, 51 also GLS Goldschmidt, D., 12, 13, 45, 47, 279 outline of GW type classification Delgado- — -Stellmacher, theory of [Gor83] , xi, 25, 63, 287 amalgams [DGS85] , 96, 105 signalizer functors [Gor69b] , 272, 275 -O’Nan pairs, [GLS96, 14.2] , 110, 124 -Walter, balance [GW75], 21, 169 rank 3 Signalizer Functor Theorem -Walter, dihedral Sylows [GW65a],34 [Gol72a], 275 -Walter, layer [GW71] , 169 rank 4 Solvable Signalizer Functor Gorenstein-Walter Alternative, 27, 128, Theorem [Gol72b], 170, 274, 275 134, 135, 153, 161 strongly closed (2-fusion theorem) Gorenstein-Walter type, see also GW type [Gol74] , 45, 98, 175, 180, 198, 207 Gramlich, R. INDEX 339

Phan theory [Gra04] , 264 Gorenstein- —, low 2-rank and Lie graph families [GH71a] , 180 commuting —, 32 Gorenstein- —, sectional 2-rank 4 graph automorphism of Lie type group, 258 [GH74] , 38, 298 grid (of major subdivisions in the CFSG), Gorenstein- —, Sylow of type 2An 21 [GH71b] , 45, 180 Griess, R. nonconnected Sylow revision [Har81], Burgoyne- — -Lyons, Chevalley groups 40, 284 [BGL77] , 169, 170, 180, 190 on Yoshida transfer [Har78] , 175 friendly giant (construction of M) properties of HN [Har76], 203, 260 [Gri82] , 260 self-centralizing E8 [Har75], 50, 51, 180 -Lyons, automorphisms of Tits group short chains of subgroups [Har68] , 209 [GL75] , 169 -Solomon, standard Mathieu [HS08],58 -Mason-Seitz, standard Bender standard 2M22 [Har],58 [GMS78],54 Harris, M. -Meierfrankenfeld-Segev, uniqueness of odd Lie type [Har81b], 51 M [GMS89] , 260 -Solomon, 2-component dihedral type I multipliers for known groups I [Gri72], [HS77],51 198, 261 2-component dihedral type II [Har77], multipliers for known groups II [Gri80], 51 169, 261 standard L3(3),U3(3) multipliers for known groups III [Gri85] [Har80a, Har81a],52 , 261 Held, D. multipliers for Lie type [Gri73] , 146, Cheng- —, standard L3(4) 169, 180 [CH81, CH85],52 multipliers for sporadic groups [Gri74], simple groups related to M24 [Hel69], 169 146, 192, 260 properties of M [Gri76] , 201 Higman, D. G. -Solomon, unbalancing L3(4),He -Sims, construction of HS [HS68] , 260 [GS79b] 52,58 Higman, G. n standard 4M22 [Gri],58 condition for splitting of SL2(2 )action, Griess-Mason-Seitz Theorem, 54,56 205 Guralnick, R. fixed-point-free action [Hig57] , 146 -Malle, FF–modules for simple groups Hall- —, p-length of p-soluble group [GM02, GM04] , 92, 146, 201, 209, [HH56] , 170 210, 291, 292 -McKay, existence and uniqueness of J3 GW type, 10 [HM69] , 260 classification of simple groups of —, 25, unpublished “Some p-local Conditions” 63, 64 [Hig72] , 182 unpublished “Odd Characterizations” half-splitting prime, 151 [Hig68] , 182, 205 Hall, J. Holt, D. blocks with alternating sections [Hal82] 2-central involution fixing unique point , 147 [Hol78], 299 Hall, M. Holt’s Theorem, 145, 174, 177, 182, 200, -Wales, existence and uniqueness of J2 201, 210, 299 [HW68] , 260 independent proof of F. Smith [Smi79a] Hall, P., 252, 271 , 299 -Higman, p-length of p-soluble group H∗(T,M) in proof of Quasithin Theorem, [HH56] , 170 87 Sylow π-subgroups for solvable groups, 253 involution, 5 -Wielandt Transfer Theorem, 175, 180 centralizer approach to simple groups, 5 Hall π-subgroup of solvable group, 253 classical —, 46 Harada, K. isolated vertex in commuting graph, 36, 286 blocks of orthogonal type[Har80b] , 147 Gorenstein- —, characterization of J2,J3 J(−), Thompson subgroup, 291 [GH69],39 James, G. 340 INDEX

modules for Mathieu groups [Jam73], TI-subgroup, 204 247, 262 classification, see also GF (2n)Type Janko, Z., 39, 77, 185 Theorem all 2-locals solvable [Jan72] , 185 layer discovery, properties of J1 [Jan66] , 124, p- —, 170, 253 260 2- —, 253 discovery, properties of J2,J3 [Jan69], L-balance, see also balance, 278 39, 260 L-Balance Theorem, 21 discovery, properties of J4 [Jan76] , 146, Leon, J. 203, 247, 260 -Sims, existence and uniqueness of B -Wong, characterization of HS [JW69], [LS77] , 260 187, 188, 260 levels 0, 1, ... of dependency for results Jones, W. quoted, xii -Parshall, 1-cohomology for Lie type Levi [JP76] , 133, 146, 262 complement in decomposition of Jordan, C., 30 parabolic, 257 decomposition of parabolic subgroup, 257 Kantor, W. Lie Curtis- — -Seitz, 2-transitive Chevalley rank, see also BN-rank groups [CKS76] , 180 type groups, 16, 257 kernel local Frobenius —, 251 characteristic p,9 k-generated p-core Γ (G), 31 k,P group theory, 3 K-group hypothesis, 21, 48, 63, 74, 77, subgroup, 3, 251 181–183 Local C(G, T )-Theorem, 26, 93, 99, 113, Klein 4-group (elementary of rank 2), 15 158, 170, 206, 218, 219, 221, 222, 293, Klinger, K. 293, 295, 297, 301 -Mason, characteristic 2,p type [KM75], locally unbalanced p-component, 278 159, 160, 168, 268, 302 locally k-unbalanced p-component, 279 Klinger-Mason argument, 68, 143, 160, 168, L (G), p-layer, 253 302 p L (2n) standard block theorem, 298 Klinger-Mason Dichotomy, 27, 68, 168, 302 2 L (G), 2-layer, 253 Weak —, 303, 304, 305 2 Lundgren, J. Richard Konvisser, M. all 2 locals solvable [Lun73] , 185 3-groups, theorem on 3-groups, 170 -Wong, large extraspecial solvable Korchagina, I., 304 [LW 76] , 202, 203 K-proper, see also K-group hypothesis Lyons, R., 68 Ku, C. Aschbacher-Gorenstein- — , uniqueness characterization of M22 [Ku97] , 106 theorems [AGL81], 79, 133, 141, 145, n L2(2 )-block, 293 149, 214, 215, 215 Λi(G), 4, 32 Burgoyne-Griess- —, Chevalley groups commuting graph on rank-ip-subgroups, [BGL77] , 169, 170, 180, 190 32 discovery, properties of Ly [Lyo72] , 180, ◦ Λi(G) ,4 260 Λ1(G), 32 Gorenstein- —, functors and Λ2(G), 36 nonconnectedness [GL82], 285 Landazuri, V. Gorenstein- —, on Local -Seitz, minimal dimensions for modules C(G, T )-Theorem [GL93] , 295 [LS74] , 262 Gorenstein- —, nonsolvable signalizer Lang’s Theorem, 180 functors [GL77] , 169 large Gorenstein- —, trichotomy for e(G) ≥ 4 extraspecial subgroup, see also [GL83], 133, 145, 150, 180, 214, 216, extraspecial 247, 256 symplectic-type subgroup, 73, 185, 202 Gorenstein- — -Solomon, second effort classification, see also GF (2) Type CFSG, see also GLS Theorem Griess- —, automorphisms of Tits group width-2 classification, 188, 189, 192, [GL75] , 169 193, 199 Sylow of U3(4) [Lyo72],35 INDEX 341 m(−), rank (of abelian group), 252 -Stellmacher, pushing-up rank 2 [MS93] mp(−), p-rank, 252 , 93, 222 m2,p(−), 2-local p-rank, 20 -Stellmacher, qrc-lemma, 91 MacWilliams (Patterson), A. -Stellmacher-Stroth, local characteristic p no normal abelian of rank ≥ 3[Mac70], project [MSS03] , 170 38 minimal parabolic Magliveras, S. abstract —, see also abstract minimal subgroups of HS [Mag71] , 146, 260 parabolic Main Theorem Mitchell, H. CFSG, classifying all finite simple on small-dimensional linear groups groups, 3, 81, 223 [Mit14] , 133 for GW Type Groups, 25, 81, 223 Miyamoto, I. n n 2 n for Characteristic 2 Type Groups, 64, standard U4(2 ),U5(2 ), F4(2 ) 81, 223 [Miy79, Miy80, Miy82],56 Malle, G. Moufang polygons, 97 moving around functor, 129, 130, 140, 142, Guralnick- —, FF–modules for simple 154, 156, 157 groups [GM02, GM04] , 92, 146, 201, 209, 210, 291, 292 Nah, see also Cheng, Kai Nah Manferdelli, J. neighbor (of a triple (B,x,L)inS∗(G; p)), standard Co2 [Man79],58 71, 150–152, 159, 162, 165–167, 174, Martineau, R. P. 176, 177 representations of Sz(2n)[Mar72] , 205 N-group (roughly, minimal simple group), Maschke’s Theorem, 303 20 Mason, D., 39 Niles, R., 294 Griess- — -Seitz, standard Bender noncharacteristic 2 type, 25 [GMS78],54 noncomponent type, 33 Mason, G., 92 nonconnectedness, see also commuting Cooperstein- —, unpublished FF-module graph, disconnected analysis [CM80] , 146, 201, 209, 210, Nonconnectedness Theorem, 39,40 291 Nonsolvable Signalizer Functor Theorem, Klinger- —, characteristic 2,p type 147, 170 [KM75], 159, 160, 168, 268, 302 normal p-complement, 253 quasithin groups, incomplete manuscript Norton, S [Mas] , xii, 65, 85 existence of J4 [Nor80] , 260 maximal O(−), largest normal odd-order subgroup component, 42 O (−) (core), 252 2-component, 49 2 Op(−), largest normal subgroup of p-power unbalancing triple, 50 index, 252 McBride, P., 13 Oπ(−), largest normal subgroup of K∗-conditions, 276 π-index, 252 Nonsolvable Signalizer Functor Theorem Op(−), largest normal p-subgroup, 252 [McB82b, McB82a], 147, 170, 275 Oπ(−), largest normal π-subgroup, 252 McClurg, P., 92 Op,q(G), preimage of Oq G/Op(G) , 252 thesis on FF-modules for almost-simple O’Nan, M., 35 groups [McC82] , 246, 291 characterizations by centralizers of McKay, J. 3-elements [O’N76a] , 170 Higman- —, existence and uniqueness of discovery, properties of ON [O’N76b], J3 [HM69] , 260 146, 180, 260 McLaughlin, J. Goldschmidt- — pairs, [GLS96, 14.2] , construction of McL [McL69a] , 260 110, 124 transvection groups [McL69b], 264 unpublished tables on sporadic groups, McLaughlin’s Theorem, 132, 146, 156, 170, 169 180, 191, 196, 206, 264, 308 odd Meierfrankenfeld, U. case (GW type) for CFSG, 10 A2n+1-blocks, 297 standard Griess- — -Segev, uniqueness of M component, see also standard [GMS89] , 260 component 342 INDEX

form, see also standard form 1-cohomology of linear groups [Pol71], transposition, 265 180 Odd Lie Type Theorem, 52,53 p-radical, 270 Odd Order Theorem, 5, 6, 29, 33, 35 p-rank, 252 Odd Standard Component Theorem, 54, sectional —, 252 54,55 Pretrichotomy Theorem, 149, 213 opposite Preuniqueness Case, 74, 127, 213 root groups in Lie type group, 257 for GW type, 37 original proof of CFSG, xi Preuniqueness-implies-Uniqueness Theorem, 74, 78, 127, 133, 149, 152, 214, 223 Page, D. for GW type, 37 Oxford Ph.D. thesis 1969 [Pag69] , 182 Prince, A. parabolic 5-element on 2-group [Pri77] , 182 abstract minimal —, see also abstract Principle I (Recognition via local minimal parabolic subgroups), 4, 4 subgroup in Lie type group, 257 Principle II (Restricted structure of local parameters subgroups), 4,7 weak closure —, see also weak closure product parameters central —, 251 Parrott, D. direct —, 251 characterization of Ru [Par76] , 146 semdirect —, 251 characterization of Th [Par77] , 202 wreath —, 251 characterizations of Fischer groups Proper 2-Generated Core Theorem, 33, 39, [Par81] , 203 40, 287 Parshall, B. pumpup, 151, 170 Jones- —, 1-cohomology for Lie type p-Uniqueness Theorem, 214 [JP76] , 133, 146, 262 pushing-up, 93, 96, 99, 102, 103, 105, 109, Patterson, A. MacWilliams —, see also 110, 112, 113, 115, 121, 132, 135, 138, MacWilliams (Patterson), A. 142, 147, 158, 170, 206, 215, 218–222, Patterson, N. 228, 233, 239, 246, 247, 292, 292 characterization of Co1 [Pat72] , 190, and strong p-embedding, 218 192, 198, 200, 260, 264 rank-2 groups (Meierfrankenfeld and -Wong, characterization of Suz [PW76] Stellmacher), 93, 108, 110 , 190, 192, 194, 200 p-centric, 270 q(G, V ), parameter for quadratic action, 91 p-complement qˆ(G, V ), parameter for cubic action, 91 normal —, 253 qrc-Lemma (Meierfrankenfeld and p-component, 253 Stellmacher), 92, 102, 121 locally unbalanced —, 278 QTKE-group, 66 locally k-unbalanced —, 279 classiﬁcation, see also Quasithin type, 67 Theorem p-Component Theorem, 68 quasi-dihedral, see also semi-dihedral quasisimple, 253 p-constrained, 268 quasithin groups, 20 Peterfalvi, T. incomplete manuscript [Mas]ofG. odd order Chapter VI revision [Pet84], Mason, xii 29 list of simple —, 88 odd order character revision [Pet00],29 treatment by Aschbacher-Smith, xii, 85 revision of Suzuki 2-transitive [Pet86], Quasithin Theorem, 66, 68, 69, 80, 85, 223 30, 281, 282 quaternion group, 252 Phan,K.W. generalized —, 252 unitary presentations [Pha77a, Pha77b] , 47, 263 radical Phan’s Theorem, 132, 133, 145, 181, 263 p- —, 270 p-layer, 253 rank p-local subgroup, 3, 251 BN- —, 257 p-nilpotent, 34, 253 p- —, 252 Pollatsek, H. k functor, 272 INDEX 343

Lie —, 257 reduction for standard Chevalley sectional p- —, 252 [Sei79a, Sei79b] 52, 56 -3 groups (e(G)=3),75 some small standard components [Sei81] recognition theorems, 262 , 52, 59 reductive Lie type group, 257 standard linear [Sei77],56 Reifart, A., 201, 202 Seitz Generation Theorem, 132, 146, 180, characterization of Th [Rei76] , 260 259 2 large extraspecial— E6(2), E6(2) semi-dihedral, 252 [Rei78b, Rei78c] , 200 semidirect product, 251 3 large extraspecial— D4(2) [Rei78a], semisimple 202 element in Lie type group, 257 Robinson, D. S∗(G; p), triples (B,x,L) with maximal vanishing of homology [Rob76] , 180 component L, 70 root shadow, 99 involution, 265 Shpectorov, S. groups generated by —s [Tim75a], Bennett- —, revision of Phan [BS04], 265 47, 263 subgroup in Lie type group, 257 Shult, E. Rudvalis, A. fusion theorem, 98, 198 Finkelstein- —, maximals of J2 [FR73], Sibley, D., 29 146, 260 signalizer, 12, 169, 272 Finkelstein- —, maximals of J3 [FR74], functor, 12, 274 146, 260 A-—,273 Aschbacher-Goldschmidt —, 155, 170 Schur, I. balanced —, 12 condition for unique covering [Sch04], complete —, 13 180 completion of —, 275 Schur multiplier, 260 equivariant —, 13 determined for simple groups, 261 for Dichotomy Theorem, O C (−) , Schur’s Lemma, 303 p G 14 second effort, approach to CFSG by GLS, getting started —, 128, 130, 135, 154, xi 155 Sectional 2-Rank 4 Theorem, 38, 39, 40, k + 1 -balanced, 280 48, 50, 58, 175, 188, 203 2 sectional p-rank, 252 method, 12, 27, 36, 36, 41, 44, 45, 47, Segev, Y. 49, 52, 74, 128, 129, 134, 135, 137, 147, 153–155, 159, 161, 169, 272 Aschbacher- —, uniqueness of J4 [AS91] , 108, 260 moving around —, 129, 130, 140, 142, Griess-Meierfrankenfeld- —, uniqueness 154, 156, 157 of M [GMS89] , 260 of rank k, 273 Seitz, G. —s and balance, 278 Aschbacher- —, involutions in vs. A-signalizer functor, 273 characteristic 2 [AS76a] , 54, 133, 146, Signalizer Functor Theorem, 13, 14, 20, 32, 170, 179, 180, 187, 189, 199, 201, 258 129, 137, 138, 147, 155, 170, 274–276, minor correction , 54 276, 279, 288, 304 Aschbacher- —, standard known type simplified standard type, 72 [AS76b, AS81], 52, 54, 133, 146, 260 Simplified Trichotomy Theorem, 75 balance in Lie type [Sei82], 170 Sims, C. Curtis-Kantor- —, 2-transitive Chevalley existence, uniqueness of Ly [Sim73] , 260 groups [CKS76] , 180 Higman- —, construction of HS [HS68] Fong- —, split BN-pairs of rank 2 , 260 [FS73],97 Leon- —, existence and uniqueness of B generation in Lie type [Sei82], 132, 146, [LS77] , 260 169, 170, 179, 180, 217, 247, 258, 259 Smith, F. Griess-Mason- —, standard Bender all 2-locals solvable [Smi75] , 185 [GMS78],54 blocks as uniqueness groups [Smi] , 207 Landazuri- —, minimal dimensions for characterization of Co2 [Smi74] , 190, modules [LS74] , 262 198, 247, 260, 264 344 INDEX

large extraspecial (unitary) [Smi77b], solvable failure of factorization, see also 189, 190, 191 failure of factorization large extraspecial restrictions [Smi76a], special p-group, 206–208, 252 190–192 large —, 185, 204, 205, 208 large extraspecial with full orthogonal splitting prime, 151 [Smi77c], 192, 194, 195, 202 half- —, 151 large symplectic not extraspecial sporadic simple groups, 16, 255 [Smi77a], 188 Springer, T. 2-central involution fixing unique point -Steinberg, conjugacy classes [SS70], [Smi79a], 299 169 Smith, P. SQTK-group, 88 construction of Th [Smi76b] , 260 list of simple —s, 88 Smith, S. standard Aschbacher- —, quasithin classification component, 43 [AS04b], xii, 65, 85, 299 odd —, 70 Aschbacher- —, quasithin preliminaries form, 22, 28, 43, 173, 296 [AS04a], 87, 294, 299 odd —, 70, 130, 144, 181 groups of GF (2n)type[Smi81], 204, problem (for a given L), 43, 49 210 reduction of GW type to —, 53 large extraspecial expository lecture subcomponent, 71 [Smi80], 183 subgroup, see also standard component large extraspecial–orthogonal [Smi80b], type, 152, 173 77, 182, 199, 200, 202, 210 simplified —, 72 large extraspecial–type E [Smi80a], 77, Standard Component Theorem, 28, 42, 43, 200, 201, 210 46, 53, 55, 173 large extraspecial–widths 4, 6[Smi79b, Standard Type Theorem, 77, 80, 130, 166, 3.2], 77, 196, 201 168, 173, 183, 223 Smith orthogonal extraspecial theorem, Standard Form Theorem for Blocks, 296 197, 199, 200, 202, 203 Steinberg, R. Solomon, R., 45, 47, 287 endomorphisms of algebraic groups 2An components [Sol75] , 45, 58 [Ste68a] , 169 alternating components [Sol76b]49 generators, relations, coverings [Ste62], part II [Sol77] , 49, 58 180 signalizers [Sol78a],49 lectures on Chevalley groups [Ste68b], An blocks [Sol81], 297 169, 180, 262 certain 2-local blocks [Sol81] , 207 representations of algebraic groups characterization of Co3 [Sol74],48 [Ste63] , 180 Davis- —, some standard sporadics Springer- —, conjugacy classes [SS70], [DS81] , 58, 59 169 expository paper on blocks [Sol80] , 292 Steinberg relations for Lie type group, 176, Finkelstein- —, standard M12,Co3 180, 182, 263 [FS79a], 58 Stellmacher, B. Finkelstein- —, odd standard Sp2n(2) Delgado-Goldschmidt- —, theory of [FS79b] , 182 amalgams [DGS85] , 96, 105 Gilman- —, unbalancing reduction Fletcher- — -Stewart [FSS77] , 182 [GS79a],50 Meierfrankenfeld- —, pushing-up rank 2 Gorenstein-Lyons- —, second effort [MS93] , 93, 222 CFSG, see also GLS Meierfrankenfeld- —, qrc-lemma, 91 Griess- —, unbalancing L3(4),He Meierfrankenfeld- — -Stroth, local [GS79b] 52,58 characteristic p project [MSS03] , 170 Harada- —, standard Mathieu [HS08], Stewart, W. B. 58 Fletcher-Stellmacher- — [FSS77] , 182 Harris- —, 2-component dihedral type I strongly [HS77],51 closed, 98, 269 maximal 2-components [Sol76a],49 embedded, 31, 281 -Timmesfeld, tightly embedded [ST79], locally 1-balanced, see also balance 207, 297 p-embedded, 79, 170, 221, 222, 224, 256, n -Wong, L2(2 )blocks[SW81], 297, 298 281 INDEX 345

almost —, see also embedded reduction for Unbalanced Group Strongly Embedded Theorem, 6, 15, 26, Theorem, 48 31, 32, 67, 68, 74, 124, 171, 190, 207, Thompson amalgam strategy, see also 285, 287, 289, 294 Thompson strategy Stroth, G., 201, 202 Thompson A × B Lemma, 267 characterization of BM [Str76] , 201 Thompson Dihedral Lemma, 133, 145, 160, extraspecial × elementary [Str78] , 207 165, 268 groups of GF (2n)type[Str80], 208, 210 Thompson factorization, 91, 220, 239, 246, Meierfrankenfeld-Stellmacher- —, local 291, 295, 300 characteristic p project [MSS03] , 170 Thompson order formula, 200 2 standard E6(2) [Str81],59 Thompson Replacement Lemma, 291 Uniqueness Case revision [Str96] , 223 Thompson strategy, 86, 86, 87, 89, 90, 93, subcomponent 95–98, 100, 102, 103, 117, 120, 122, standard —, 71 225, 235, 296, 299 subgroup functor, 272 Thompson subgroup J(−), 291 —ofrankk, 272 Thompson Transfer Lemma, 99, 117, 118, —withK-property, 273 122, 211, 271, 271, 283 balanced —, 273 Thompson Transitivity Theorem, 137, 272 central —, 273 3-component, 253 coprime —, 273 3-transposition, 264 equivariant —, 272 group, 264 locally constant —, 273 theorem (Fischer), 265 solvable —, 273 {3, 4}+-transposition, 265 subnormal, 253 TI-subgroup, 251 Suzuki, M., 35 tightly embedded, 43 Brauer- —, quaternion Sylows [BS59], Timmesfeld, F., 187 34 condition for weakly closed TI-set Brauer- — -Wall, characterization of [Tim79a], 207 L2(q)[BSW58] , 34, 264 elementary abelian TI-subgroups characterization of linear groups [Tim77] , 205 [Suz69a], 181, 192 groups of GF (2n)type discovery, properties of Suz [Suz69b], case division [Tim78b], 204, 204, 209 260 note on 2-groups of — [Tim79c], 208 2-transitive groups [Suz62] , 30, 31 weakly closed case [Tim81], 206 Suzuki type for involution, 187 large extraspecial [Tim78a], 77, 192, Sylow 2-Uniqueness Theorem, 296 199, 200, 202, 262 Sylp(G), set of Sylow p-subgroups of G,3, minor correction [Tim79b], 192 251 root involutions [Tim75a] , 180, 190, symplectic type, 252 191, 202–206, 210, 211, 265, 299 large — subgroup, see also large Solomon- —, tightly embedded [ST79], Syskin, S. 207, 297 standard Th [Sys81],58 {3, 4}+-transpositions [Tim73] , 190, 194, 196, 200, 201, 204, 265 Θ+, 274 weakly closed TI-sets [Tim75b] , 190, Θ-signalizer, 274 195, 205, 206, 247, 297–299, 307 Thomas, G. Tits, J. n characterization of U5(2 )[Thm70], Borel- —, Borel-Tits Theorem [BT71], 190 18, 164, 180 Thompson, J., 5, 12, 15, 29, 41, 42, 45, 47, buildings (ICM 1962 lecture) [Tit63], 48, 185, 199, 294 169 Feit- —, odd order theorem [FT63],5, Lie type presentations [Tit62] , 263 28 -Weiss, Moufang polygons [TW02] , 97, Feit- —, self-centralizing order 3 [FT62] 105, 116, 118 , 182 Tits building, see also building Glauberman- —, normal p-complement Tits system, 258 theorem [Gla68] , 180 torus N-groups [Tho68], 20, 65, 73, 86, 156, inLietypegroup,257 159, 170, 185, 268, 287, 290, 299 nonsplit —, 257 346 INDEX

split —, 257 noncomponent —, 33 transfer standard —, 152 control of, 271 simpliﬁed —, 72 theorems, 271 Suzuki — for involution, 187 transposition symplectic —, 252 methods for identifying groups, 186, 187, twisted Lie — groups, 255 189–191, 193, 194, 196, 199–202, 204–208, 210, 211, 264 unbalanced odd —, 265 locally —, 278 3- —, 264 locally k- —, 279 {3, 4}+-—,265 Unbalanced Group Theorem, 42, 44, 52, transvection, 264 55, 56, 60 groups generated by —s, see also unbalancing triple, 44 McLaughlin’s Theorem unipotent Trichotomy Theorem, 67, 68, 72, 80, 127, element in Lie type group, 257 133, 150, 173, 183, 184, 213, 223, 304 radical of parabolic subgroup, 257 for GW type, 33 uniqueness Simpliﬁed —, 75 case, 79, 213, 223 Weak —, 67, 68, 160 for GW type, 31 triple cover, 260 odd order — theorem, 29 twisted groups of Lie type, 255 problem for simple groups, 7 subgroup, 26, 29, 31, 86, 90, 100 2An, double cover of alternating group, 260 systems, 105 2An Theorem, 45, 48, 49, 53 2-component theorems, 31, 94, 96, 103, 122, 138, 215, type, see also component type 219, 221–227, 233–236, 238, 239, 241, 2-connected, 142, 154 242, 244–246, 295, 300, 301 2-constrained, 268 2- — subgroup, 29 2-generated p-core, 31 Uniqueness Case Theorem, 68, 74, 78, 80, Aschbacher’s theorem on proper — 86, 127, 133, 214, 299 2-core, 40 universal weak —, 37 form of Lie type group, 261 2-layer, 253 Volume 1, Gorenstein’s odd case outline 2-local subgroup, 251 [Gor83],xi 2-nilpotent, 34, 253 2-Preuniqueness Case, 37 Waldecker, R., xii 2-Preuniqueness Theorem (Odd Case), 40 Wales, D. 2 -component, 253 Conway- —, construction of Ru [CW73] 2-rank, 252 , 260 2-rank 2 Theorem, 35, 35, 38, 51, 55, 99, embedding of J2 in G2(4) [Wal69a], 175, 180, 186, 187 247, 262 2-reduced, 229 Hall- —, existence and uniqueness of J2 2-transitive, 30 [HW68] , 260 2-uniqueness subgroup, 29 Wall, G. E. 2-Uniqueness Theorem, see also Strongly Brauer-Suzuki- —, characterization of Embedded Theorem L2(q)[BSW58] , 34, 264 type Walter, J., 7, 42, 44, 48, 253, 272 a2 for involution, 189 abelian Sylow 2-subgroups [Wal69b], characteristic p —, 9 182 characteristic 2 —, 9 characterization of Chevalley groups component —, 10 [Wal86],51 c2 for involution, 187 Gorenstein- —, balance [GW75], 21, 169 even —, 66 Gorenstein- —, dihedral Sylows GF (2) —, 73 [GW65a],34 GF (2n)—,204 Gorenstein- —, layer [GW71] , 169 Gorenstein-Walter —, see also GW type weak GW —, 10 BN-pair of rank 2, 96 Lie — groups, see also Lietypegroups closure, 96, 300 noncharacteristic 2 —, 25 fundamental — inequality FWCI, 95 INDEX 347

generalized — Wi, 300 Yamaki, H. methods, 90, 94–96, 106–116, 121, 122, characterization of Sp6(2) [Yam69] , 180 218, 219, 223, 225–228, 230–235, Yoshida, T. 238–243, 246, 292, 299 character-theoretic transfer [Yos78], parameters, 94, 95, 96, 107, 108, 110, 146, 182, 271 + 124, 225, 227, 231, 232, 241, 242, Egawa- —, standard 2Ω8 (2) [EY82],59 262, 300, 301 ∗ Z (G), preimage of Z G/O2 (G) , 15, 253 k-balance, 280 ∗ S-blocks, 218 Z -Theorem (Glauberman), 72, 167, 171, 2-generated p-core, 37 180, 185, 190, 198, 253, 269 Weak Trichotomy Theorem, see also Zassenhaus, H., 30, 264 Trichotomy Theorem ZJ-theorem (Glauberman), 29 weakly closed, 269 Weir, A. Sylow subgroups of classical groups [Wei55] , 169 Weiss, R. Tits- —, Moufang polygons [TW02], 97, 105, 116, 118 Weyl group in Lie type group, 131, 132, 143, 144, 174–177, 182, 257 width of extraspecial group, 252 Wielandt, H., 7, 271 Hall- — Transfer Theorem, 175, 180 Williamson, C. Burgoyne- —, on Borel-Tits theorem [BW76] , 169 Burgoyne- —, semisimple Chevalley classes [BW77] , 169, 180, 258 Wong,S.K. Dempwolff- —, characterization of Ln(2) [DW77a] , 192 Dempwolff- —, large extraspecial reducible I [DW77b], 77, 191 Dempwolff- —, large extraspecial reducible II [DW78], 191, 191 Janko- —, characterization of HS [JW69] , 187, 188, 260 Lundgren- —, large extraspecial solvable [LW 76] , 202, 203 Patterson- —, characterization of Suz [PW76] , 190, 192, 194, 200 n Solomon- —, L2(2 )blocks[SW81], 297, 298 Wong, W., 39 Fong- —, characterization of rank 2 groups [FW69] , 170 wr, see also wreath product wreath product A wr B of groups, 251 wreathed 2-group, 252

Yamada, H., 56 standard n 3 n 2 2n+1 G2(2 ), D4(2 ),U5(2), F4(2 ) [Yam79a, Yam79b, Yam79c, Yam85],56 standard U6(2) [Yam79d],59

The book provides an outline and modern overview of the classification of the finite simple groups. It primarily covers the “even case”, where the main groups arising are Lie-type (matrix) groups over a field of characteristic 2. The book thus completes a project begun by Daniel Gorenstein’s 1983 book, which outlined the classification of groups of “noncharacteristic 2 type”. However, this book provides much more. Chapter 0 is a modern overview of the logical structure of the entire classification. Chapter 1 is a concise but complete outline of the “odd case” with updated references, while Chapter 2 sets the stage for the remainder of the book with a similar outline of the “even case”. The remaining six chapters describe in detail the fundamental results whose union completes the proof of the classification theorem. Several important subsidiary results are also discussed. In addition, there is a comprehensive listing of the large number of papers referenced from the literature. Appendices provide a brief but valuable modern introduction to many key ideas and techniques of the proof. Some improved arguments are developed, along with indica- tions of new approaches to the entire classification—such as the second and third generation projects—although there is no attempt to cover them comprehensively. The work should appeal to a broad range of mathematicians—from those who just want an overview of the main ideas of the classification, to those who want a reader’s guide to help navigate some of the major papers, and to those who may wish to improve the existing proofs.

For additional information and updates on this book, visit AMS on the Web www.ams.org/bookpages/surv-172 www.ams.org

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