Selected Titles in This Series

68 David A. Cox and Sheldon Katz , Mirror symmetry and algebraic geometry, 1999 67 A. Borel and N . Wallach, Continuous cohomology, discrete subgroups, and

representations of reductive groups, Second Edition, 1999 66 Yu. Ilyashenko and Weigu Li, Nonlocal bifurcations, 1999 65 Carl Faith, Rings and things and a fine array of twentieth century associative algebra,

1999 64 Rene A. Carmona and Boris Rozovskii , Editors, Stochastic partial differential

equations: Six perspectives, 1999 63 Mark Hovey, Model categories, 1999 62 Vladimir I. Bogachev, Gaussian measures, 1998 61 W . Norrie Everitt and Lawrence Markus, Boundary value problems and symplectic

algebra for ordinary differential and quasi-differential operators, 1999 60 Iain Raeburn and Dana P. Wil l iams, Morita equivalence and continuous-trace

C*-algebras, 1998 59 Paul Howard and Jean E. Rubin , Consequences of the axiom of choice, 1998 58 Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov, Jr., Lectures on

representation theory and Knizhnik-Zamolodchikov equations, 1998 57 Marc Levine, Mixed motives, 1998 56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum

groups: Part I, 1998 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 Wil l iam 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 , Integer-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 S tephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and A d a m 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 Preese, Jaroslav Jezek, and J. B. Nat ion , Free lattices, 1995 41 Hal L. Smith , Monotone dynamical systems: an introduction to the theory of

competitive and cooperative systems, 1995 40.4 Daniel Gorenste in , Richard Lyons, and Ronald Solomon, The classification of the

finite simple groups, number 4, 1999 40.3 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the

finite simple groups, number 3, 1998 40.2 Daniel Gorenste in , Richard Lyons, and Ronald Solomon, The classification of the

finite simple groups, number 2, 1995 (Continued in the back of this publication)

http://dx.doi.org/10.1090/surv/040.4

The Classification of the Finite Simple Groups, Number 4

MATHEMATICAL Surveys and Monographs

Volume 40, Number 4

The Classification of the Finite Simple Groups, Number 4

Part II, Chapters 1-4: Uniqueness Theorems

Daniel Gorenstein Richard Lyons Ronald Solomon

AttEMjJ/

| American Mathematical Society Providence, Rhode Island

Editorial Board Georgia M. Benkar t Tudor Stefan Ra t iu , Chair Pe ter Landweber Michael Renardy

T h e au thors were suppor ted in pa r t by NSF grant # D M S 97-01253.

1991 Mathematics Subject Classification. P r i m a r y 20D05; Secondary 20B20.

ABSTRACT. In this volume several basic uniqueness theorems underlying the classification of the finite simple groups are proved for a minimal counterexample G to the classification; these include and generalize the Bender-Suzuki Strongly Embedded Theorem and the Aschbacher Component Theorem.

Library of Congress Cataloging-in-Publication D a t a ISBN 0-8218-1379-X (number 4) ISBN 0-8218-1391-3 (number 3) ISBN 0-8218-1390-5 (number 2) T h e first vo lume 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

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© 1999 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights

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10 9 8 7 6 5 4 3 2 1 04 03 02 01 00 99

To the memory of Michio Suzuki

Contents

Preface xiii

PART II, CHAPTERS 1-4: UNIQUENESS THEOREMS

Chapter 1. General Lemmas 1 1. Uniqueness Elements 1 2. Permutation Groups 3 3. Two Theorems on Doubly Transitive Groups 7 4. Miscellaneous Lemmas 12

Chapter 2. Strongly Embedded Subgroups and Related Conditions on Involutions 19

1. Introduction and Statement of Results 19 2. The Subsidiary Theorems 25 3. The Basic Setup and Counting Arguments 27 4. Reduction to the Simple Case: Theorem 1 33 5. p-Subgroups Fixing Two or More Points: Theorem 2 38 6. A Reduction: Corollary 3 42 7. Double Transitivity and I a ^ z : Theorem 4 44 8. Reductions for Theorem SE 50 9. Good Subgroups of V Exist 54

10. The Structures of V and D 58 11. Proof of Theorem 6 (and Theorem SE) 62 12. The Group L = (z,t, [02>(D),t]) 65 13. The Unitary Case 72 14. The Suzuki Case 74 15. The Linear Case 79 16. The Structure of D and Cx(u) 82 17. The Proof of Theorem ZD 88 18. The Weak 2-Generated 2-Core 91 19. The J2 Case 95 20. Bender Groups as 2-Components 98 21. Theorem SU: The 2-Central Case 101 22. The Ji Case 103 23. The 2A9 Case 105 24. Theorem SA: Strongly Closed Abelian 2-Subgroups 109 25. Theorem SF: Terminal Bender Components 112 26. Theorem SF: Product Disconnection 119 27. Theorem ,49 127

x CONTENTS

Chapter 3. p-Component Uniqueness Theorems 131 1. Introduction 131 2. Theorem PUi: The Subsidiary Results 137 3. Exceptional Triples and 3C-Group Lemmas 138 4. M-Exceptional Subgroups of G 142 5. Centralizers of Elements of XP(K*) 147 6. The Proofs of Theorems 1 and 2 150 7. Theorem 3: The Nonsimple Case 153 8. Theorem 3: The Simple Case 163 9. Theorem 4: Preliminaries 165

10. Normalizers of p-Subgroups of K 168 11. The Case K £ S£(p) 171 12. The S£{p) case 172 13. Theorem 5: The M-Exceptional Case 179 14. The Residual M-Exceptional Cases 188 15. Completion of the Proof of Theorem PUi: The p-Rank 1 Case 193 16. Reductions to Theorem PUi 194 17. Further Reductions: The Non-Normal Case 201 18. Theorem PU2: The Setup 206 19. The Case r < p: A Reduction 208 20. The Case r = 1: Conclusion 211 21. The Case r = p = 2: A. Reduction 214 22. The Case r = p = 2, ~K ^ L2(q) 216 23. Theorem PU3: The Simple Case 220 24. The Residual Simple Cases 223 25. The Nonsimple Case 224 26. Theorem PU4 226 27. Corollaries PU2 and PU4 227 28. Aschbacher's Reciprocity Theorem 228 29. Theorem PU5 232

Chapter 4. Properties of K-Groups 237 1. Automorphisms 237 2. Schur Multipliers and Covering Groups 238 3. Bender Groups 245 4. Groups of Low 2-Rank 253 5. Groups of Low p-Rank, p Odd 256 6. Centralizers of Elements of Prime Order 257 7. Sylow 2-Subgroups of Specified DC-Groups 268 8. Disconnected Groups 271 9. Strongly Closed Abelian Subgroups 276

10. Generation 283 11. Generation and Terminal Components 293 12. Preuniqueness Subgroups and Generation 302 13. 2-Constrained Groups 328 14. Miscellaneous Results 331

Background References 333

Expository References 334

C O N T E N T S xi

Errata for Number 3, Chapter 1^: Almost Simple X-Groups 335

Glossary 338

Index of Terminology 340

Preface

With three preliminary volumes under our belt, we at last turn directly to the proof of the Classification Theorem; that is, we begin the analysis of a minimal counterexample G to the theorem. Thus G is a DC-proper simple group. Specifically, the heart of this volume provides a collection of uniqueness and pre-uniqueness theorems for G, proved in Chapters 2 and 3.

Chapter 2 is primarily dedicated to the proof of a fundamental 2-uniqueness theorem of Michael Aschbacher, which we call Theorem ZD. Historically there is a sequence of basic papers leading up to Theorem ZD. First is a fundamental paper of Michio Suzuki [Su4] on 2-transitive permutation groups; the result of that paper is one of our Background Results [PI, Part II]. Building on that, Helmut Bender [Be3] established the Strongly Embedded Subgroup Theorem, which we call Theorem SE. Finally Aschbacher [A3] proved Theorem ZD and its corollaries, based on Bender's theorem. In both Bender's and Aschbacher's papers, the first major step is to establish that the group under investigation acts 2-transitively on a certain set. Aschbacher's arguments for this step are modelled closely on Bender's and so it seems natural to combine these proofs. This unified proof constitutes Sections 3-7 of Chapter 2, and follows the original arguments of Bender and Aschbacher closely. However, our treatment of Section 7 benefits considerably from unpublished notes of David Goldschmidt. After this point the proofs by Bender and Aschbacher diverge both in the originals and here. Indeed we follow the originals quite closely at most points, and we have made use of Peterfalvi's revision of this proof [PI] in Sections 8-11 of Chapter 2.

The remainder of Chapter 2 treats three corollaries of Theorem ZD. The clas­sification of DC-proper simple groups with a 2-uniqueness subgroup, Theorem SU, combines a proof of Aschbacher's theorem [A3] on groups with a proper 2-generated core, but only for DC-proper simple groups, with ideas of Koichiro Harada [HI]. Fi­nally the important Theorems SA and SF of Goldschmidt [Go5] and Holt [Hoi] are proved only for DC-proper simple groups of even type, which circumvents many of the difficulties in the original papers.

Our approach to Theorem ZD and the consequent Theorems SA and SZ in Chapter 2 is to formulate them not just for G but for an arbitrary finite group X. Thus for the major part of this chapter, we depart from our original plan regarding 2-uniqueness theorems, as announced in [Ii] and carried out in preprints preliminary to this chapter. That strategy, which remains workable, was to prove the results only for the DC-proper simple group G; in the case of Theorem SE, for example, making central use of a classification of DC-groups M whose set of involutions is permuted transitively by some subgroup of M of odd order. Indeed, such a group M can be proved either to be solvable or to have a unique composition factor of even order, which is isomorphic to Z^tf) for some q = 3 mod 4. We have returned

xiii

XIV PREFACE

to an approach close to the original one of Bender and Aschbacher, for the sake of efficiency.

Chapter 3 is devoted to five "p-component pre-uniqueness theorems", Theorems PU1-PU5. The common theme is a maximal subgroup M which has a p-component K such that the centralizer W of KjOv> (K) has p-rank at least 2 and such that CG(V) < M for every non-identity p-element of W. This type of situation will arise typically when K is maximal in some ordering on the set of p-components of centralizers of p-elements of G, and the p'-cores O P / ( G G ( X ) ) , X G W, have already been assembled, for example by the signalizer functor method or by an assumption that they are trivial. The generic conclusion is that M is a strongly p-embedded subgroup of G, which yields an immediate contradiction if p = 2. When p is odd and the DC-proper simple group G has even type, it will later be shown that again G does not exist. Thus the eventual import of Chapter 3 will be to establish in general that maximal p-components of centralizers of p-elements have centralizers of p-rank 1. (Of course there are counterexamples to this statement, for example when p = 2 and G is an alternating group.) Historically results of this type were established first for p = 2 by Powell and Thwaites, whose ideas are incorporated into Section 17 of Chapter 3. Shortly thereafter, Aschbacher proved his Component Theorem, a more definitive version for p = 2. Robert Gilman gave a somewhat different proof of Aschbacher's result. Many of the ideas of Aschbacher and Gilman also appear here, but some of their delicate analysis is replaced by a detailed consideration of DC-groups. The results proved here are new in the case that p is odd. We remark that in the proof of the first major pre-uniqueness theorem, Theorem PUi, the case p = 2 is handled fairly quickly thanks to Aschbacher's criterion for a strongly embedded subgroup (Theorem ZD). Thus in Sections 7-15 of Chapter 3 the prime p is odd.

The structure and embedding of p-component uniqueness subgroups when p = 2 and K has 2-rank one is somewhat exceptional. In particular the main assertion of Theorem PU4, that terminal components are standard, is not valid in this situa­tion. In other words, K could commute elementwise with a conjugate. Historically Aschbacher and Richard Foote were able to show that there could be only one such conjugate, and we prove a similar result in Theorem PU5. Thanks are due to Professor Foote, who suggested years ago that such a result ought to be simple to prove.

As noted above, with the exception of Theorem ZD and its corollaries Theorems SE and SZ, all the results are proved for a DC-proper simple group G. The proof thus can and does rely heavily on the theory of almost simple DC-groups extablished in pU]. Those DC-group properties essential for Chapters 2 and 3 are collected in Chapter 4 of this volume and either extend or follow directly from the theory presented in our preceding volume [IA] . A much briefer Chapter 1 similarly extends our second volume [IQ] with some "general" (as opposed to DC-group theoretic) results pertinent to our task. Notable here is some theory of permutation groups underlying the proof of Theorem ZD.

In references (even within this volume), we shall specify the four chapters of this volume as Hi, II2, II3 and II4, respectively.

We are grateful to Michael O'Nan for his assistance with the proof of Theorem 3.2 of Chapter 1; to Sergey Shpectorov for his enthusiastic support and constructive comments during the preparation of Chapter 2; to Michael Aschbacher and Hel­mut Bender for their support and helpful comments and suggestions; and to Inna

PREFACE xv

Korchagina for rooting out errors in the previous volume. Our deep appreciation goes to John G. Thompson, Peter Sin and Chat-Yin Ho at the University of Florida both for the many corrections and improvements which they have suggested and for the moral support which they have provided.

We also gratefully acknowledge the support of the National Science Foundation grant #DMS 97-01253; and the second author thanks the Ohio State University for its generosity and warm hospitality during his visit in the winter of 1998, during the preparation of this manuscript.

Our joy in preparing this exposition of the beautiful theory of finite simple groups has been tempered for all these years by the haunting memory of Danny Gorenstein. His conception of this project is still a sure guide for us, and we miss his companionship, insight and energy as much as ever.

During the preparation of this volume the world lost another master of finite group theory: Michio Suzuki, who took ill in the winter of 1997-1998 and died on May 31, 1998. Chapter 2 revolves around the classification of finite groups with a strongly embedded subgroup, which as we have noted was pioneered by Suzuki in the 1950's and completed by Bender over ten years later. This theorem has been a cornerstone of the theory of finite simple groups; virtually every general classifi­cation theorem in the subject has relied on it—except the Odd Order Theorem, of course. While Suzuki's fundamental contributions are to be found over many parts of the group-theoretic landscape, this particular one is surely the most far-reaching and typifies his spirited originality and powerful insight. Out of deep respect and affection we have dedicated this book to his memory.

October, 1998

RICHARD LYONS Department of Mathematics Rutgers University New Brunswick, New Jersey 08903 E-mail: lyonsOmath. r u t g e r s . edu

RONALD SOLOMON Department of Mathematics The Ohio State University Columbus, Ohio 43210 E-mail: solomon@math. ohio-state. edu

Background References

N O T E . The previous chapters 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] , 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.

[IG] » The Classification of the Finite Simple Groups, Number 2. Chapter G: General Group Theory, Amer. Math. Soc. Surveys and Monographs 40, # 2 (1996).

[IA] > The Classification of the Finite Simple Groups, Number 3. Chapter A: Almost Simple X-Groups, Amer. Math. Soc. Surveys and Monographs 40, # 3 (1997).

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, Finite Group Theory, Cambridge University Press, Cambridge, 1986. [A2] , Sporadic Groups, Cambridge University Press, Cambridge, 1994. [Gl] D. Gorenstein, Finite Groups, 2nd edition, Chelsea, New York, 1980. [GLl] 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. [PI] T. Peterfalvi, Le Theoreme de Bender-Suzuki (Part II only), Asterisque 142 -143 (1986),

141-296.

333

Expository References

[Al] M. Aschbacher, Finite Group Theory, Cambridge University Press, Cambridge, 1986. [A2] , Sporadic Groups, Cambridge University Press, Cambridge, 1994. [A3] , Finite groups with a proper 2-generated core, Trans. Amer. Math. Soc. 197 (1974),

87-112. [A4] , On finite groups of component type, Illinois J. Math. 19 (1975), 78-115. [A6] , Tightly embedded subgroups of finite groups, J. Algebra 42 (1976), 85-101. [Be3] H. Bender, Transitive Gruppen gerader Ordnung, in dene jede Involution genau einen

Punkt festlafit, J. Algebra 17 (1971), 527-554. [Gil] R. H. Gilman, Components of finite groups, Comm. Alg. 4 (1976), 1133-1198. [Go5] D. M. Goldschmidt, 2-fusion in finite groups, Ann. of Math. 99 (1974), 70-117. [Gl] D. Gorenstein, Finite Groups, Second Edition, Chelsea, New York, 1980. [GLl] D. Gorenstein and R. Lyons, The local structure of finite groups of characteristic 2 type

(Part I only), Memoirs Amer. Math. Soc. 276 (1983). [GL6] , Signalizer functors, proper 2-generated cores, and nonconnected groups, J. Alge­

bra 75 (1982), 10-22. [H2] K. Harada, Groups with nonconnected Sylow 2-subgroups revisited, J. Algebra 70 (1981),

339-349. [Heril] C. Hering, On finite groups operating doubly transitively on their involutions, Arch. Math.

(Basel) 22 (1971), 456-458. [Hoi] D. Holt, Transitive permutation groups in which a 2-central involution fixes a unique

point, Proc. Londond Math. Soc. (3) 37 (1978), 165-192. [Hul] B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, 1967. [HuB2] B. Huppert and N. Blackburn, Finite Groups II, Springer-Verlag, Berlin, 1982. [Isl] I. M. Isaacs, Character Theory of Finite Groups, Academic Press, New York, 1976. [J4] Z. Janko, The nonexistence of a certain type of finite simple group, J. Algebra 18 (1971),

245-253. [L2] R. Lyons, Evidence for a new finite simple group, J. Algebra 20 (1972), 540-569. [PI] T. Peterfalvi, Le Theoreme de Bender-Suzuki. I, II, Asterisque 142-143 (1986), 141-296. [Si] R. M. Solomon, Maximal 2-components in finite groups, Comm. Algebra 4 (1976), 561-

594. [Su4] M. Suzuki, On a class of doubly transitive groups. II, Ann. of Math. 79 (1964), 514-589.

334

Erra ta for Number 3, Chapter 1^: Almost Simple x-Groups

Page 55, Line -11: Statement of Lemma 2.5.7: g A u t l W W = W CAuti(R)(K,a) = (a)

Page 57, Line 18: At the end of Definition 2.5.10, add: (f) Auto(K) = image of CAuto(R)(a) in Aut(K).

Page 58, Line -10: If K ~ Am(q), I W r f e ) If K 9* Am(q) (m > 1), A2m+l(<Z) Page 261, Line -11: Fi'24 Fi23

Page 288, Line -11: \M*\ \02(M)*\ Page 290, Line 1: -E(C(2A)) E{C{2B)) Page 297, Line -18: K = Cox K = Co0

Page 299, Line 16: lower bound for P x Q8 is 30 lower bound for a faithful complex representation of P x Qs in which the involution of Z(Qg) acts as —/ is 30 Page 302, Line 12: - E(CK(z)) E(CK(zA)) Page 302, Line 16: becuase because Page 302, Line 19: -{H]^- \H\s Page 304, Line -13: / is a homogeneous /-module VQ is a homogeneous /-module Page 304, Line -8: z e Qf

0 Z(J) < Qf0

Page 308, Line -17: B/Z(B) ~22E6(2)- BZ(B) ^ 2E6(2)// Page 309, Line -6: K e X K e X and K is simple Page 309, Line -1: or K ~ Jx or K ^ 2G 2(3t) , n odd, n>l,oiK^J1

Page 314, Line 13: d im(^ i ) dimfW;) Page 316, Line -2: -Vqa + Vq/3 Fqa + Fq(3 Page 316, Line -2: Vqa Fqa Page 317, Line 11: C(2X2)D4(2)(X) C(2X2)D4(2)(ZI)

Page 317, Line 16: \Sp6(2)\ |Sp6(2)|2

Page 319, Line 12: - Q is abelian Q is abelian Page 319, Line 13: ~Q- Q (twice) Page 319, Line 14: - Q is abelian Q is abelian Page 319, Line 15: ~Qr Q (four times)

335

Page 329, Line 12: (r2a + era + l ) /3 (r2a + era + l ) /d Page 332, Line -3: Theorem 6.5.5a misstates the structure of Borel subgroups of 2G2(3^), n > 1. Tiie assertion should be:

(a) Borel subgroups of if, of order q3(q — 1). If £ = [/if is such a Borel subgroup, with \U\ = q3 and |ii~| = q — 1, and if t is the involution of H, then |Cc/(t)| = q, and the groups 0 2 ( £ ) , Z(U)H and B//Z2(U) are all Frobenius groups.

Page 333, Line -9: Z2 x Z/2O?2) ^2 x £2(3) Page 338, Line 1: Replace this line by: We proceed in a sequence of lemmas. Page 338, Line 10: Replace this line by: We set Y = KXX, so that X < Or> (Y), and next prove: Page 345, Line 11: T E^X(K)- VE^_^K) Page 345, Line 12: YE,^V(U) <YE^.V(K) Y'E^_X(U) < T'^^K) Page 345, Line 13: r ^ - i f f i ) T'^^K) Page 354, Line 14: In the proof of Theorem 7.3.3, we omitted here a reduction to the case that mp(E) = 2. This reduction is needed to justify the assertion in line 15 that Y = YE,*-i(K). Thus, the following paragraph should be inserted before "We set":

We first reduce the proof to the case mp(E) = 2. Indeed, if the theorem holds in that case, then to complete the proof we must argue that if a noncyclic elementary abelian p-group E acts faithfully on K in such a way that one of the conclusions of 7.3.3 is satisfied by each F G 82(E), then E itself satisfies that same conclusion. This is accomplished by a few observations in the various cases. In case 7.3.3c, Out (if) has order 3 by 2.5.12, so m2(A\it(K)) = m2{K) = 3 and the desired conclusion is obvious. In cases 7.3.3ehijkl, as well as the case K = 2A2(2) of 7.3.3a, it is immediate from 4.10.3 and 2.5.12 that mp(Aut(K)) = 2, with Out (if) a p'-group in case (e) and mp(K) — 1 in cases (h) and (i). Thus the desired conclusions hold in these cases as well. In the remaining cases, it suffices to assume that mp(E) = 3 and derive a contradiction. In cases 7.3.3df, Out(if) is a pf-group by 2.5.12, and 4.10.3ae implies that mp(K) = 3 and that every element of K of order p lies in a conjugate of E. But in these cases of 7.3.3 it is stipulated that certain conjugacy classes of K of order p do not meet E, contradiction. In cases 7.3.3bg, we consider the character of E on the natural if-module, which (since p ^ r) lifts to a complex character \. The conditions of cases (b) and (g) force xix) — — 1 f° r e a c n x £ E#. As (x, 1^) is an integer, %(1) = —1 modp 3 . However, x(l) = 5 or 8, with p = 2 or 3, respectively, a contradiction. Finally, the only remaining case is that 7.3.3a holds and E acts on K = Lp(q) like a subgroup E* < GLp(q), and the preimage F* in E* of any F E 82(E) satisfies (F*y = Vti(Z(K)). But then (E*)' = Q>i(Z(K)), and so CE*(x) is a maximal subgroup of E* for all x G E* — Z(E*). Choosing such an element x and using mp(E) = 3, we find y € E* such that (x, y) is abelian and has a noncyclic image in E, a final contradiction accomplishing our reduction.

Page 354, Line 22: ;/-subgoup ^/-subgroup Page 357, Line -8: —its Lie components. its Lie components. (See also Definitions 4.2.2 and 4.9.3, and Proposition 4.9.4.) Page 358, Line -2: 2i^4(2^) 2F4(2^) /

336

Page 364, Line -15: Then Then if we define Tr~ AK) to be the subgroup

of K generated by all r-elements centralizing some subgroup of E of index 2, we have Page 365, Line -16: t2 and t2 t2 and t2

Page 381, Line -4: tyl3 % Page 381, Line -3: - ^ U Q0 &ij U fi0

Page 382, Line 3: TEt—r(K) T'E,_r{K) Page 382, Line 5: A*.. A*.. Page 382, Line 8: then ^ ^ = §i3', then

Page 382, Line 9: 4 - | ^ [ = | * i j | H i ^ l + |fl0| 4 = l * * i I = lntf I + lfi°l Page 382, Line 15: 02(A*i3) 02(A*..)

Page 384, Line 1: -SL2(5) ^ 2Ai(4), 5L2(5) = 2^(4 ) , (2)2B2(2f), Page 396, Line -11: K locally fe-balanced K is locally fc-balanced Page 399, Line -15: irreducibly on P irreducibly on fii(P) Page 402, Line -4: Theorem 7.8.1 Proposition 7.8.1

337

Glossary

Page Symbol

133 r^i

1 C(X,M,p) 1 C 1 HX,M,p) 1 T 25 G^A9 24 G « Ji 24 G « L 2 ( 2 " ) , 2£2(2?)orC/3(2") 15, 92 [IG; 17.1] rs,2(G), rs,fe(G) 92 r|,2(G) 23, 92 r°Sl2(G) 135 £C§(G) 133 [*] 91-92 A, A(X), Ay 3 fiy 21 V 139-140 S£(p), S(p) 139 (Em x En)o, ( S m ? Dn)o 140 T(P) 1 W(X,M,p) 1 w 2 Uk(X,M,p) 2 Uk(X,M,p) 135 (x,K)<(x1,K1) 20 Y° 19 Z,ZY

Notation special to Theorem ZD and its proof (Chapter 2, Sections 2-17)

25 a 25 D 30 J 29 JK(s) 28 m, rap 25 M 25 ft 67 r(x,y,w) 26 S

338

GLOSSARY

44 0{Y) 25 Z

Notation special to Theorems PU1-PU5 and their proofs (C

137 I(X)=1P(X) 131, 142 K 142 K\, • • •, Kr 134, 142 K* 137 m{X) = mp(X) 131, 142 M 134, 137 M 143 N, N0 (A^=M-exceptional subgroup of G) 142 P 142 Pi,• ••, Pr 131, 142 Q 142 Qo 142 \c£\ , . . . , \c£r

142 r 131, 142 T 142 T\,..., Tr 208 V = V(G;Q) 142 W 206 y = y(G;M;Q) 207 y0 142 Z

Index of Terminology

2-group of type Ag, type X, etc. 268 2-uniqueness subgroup 23 almost p-const rained subgroup 134 almost strongly p-embedded subgroup 132, 134 centralizer of involution pattern 16 control of G-fusion 132

strong control of fusion 2 exceptional triple 138-139 /-automorphism 136 group of even type 23 X^-ngid 2-component 135 intrinsic 2-component 228 if-preuniqueness subgroup 131 if-wreathed subgroup 134 Lp/-balance [IG; Section 5] M-exceptional subgroup 143

type of 143 shape of 146

p-component preuniqueness hypothesis 131 p-component preuniqueness subgroup 131 p-exceptional quasisimple group 302 product-disconnected set of involutions 20-21 restricted M-exceptional subgroup 144 shear 67 simple Bender groups 19 solvable p-component [IG; Section 13] standard component 133, [IG; 18.5] strongly classical involution 136 strongly closed subgroup 16 strongly embedded subgroup [IG; 17.1] strongly p-embedded subgroup [IG; 17.1] strongly ^-embedded subgroup 20 terminal component 133, [IG; 6.1] Theorems

Aschbacher's Reciprocity Theorem 137, 228 Bender's criterion for 2-transitivity 4 Corollaries PU2, PU4, PU5 133, 136 Corollary SA 23 Corollary ZD 22

340

INDEX OF TERMINOLOGY

Hering's Theorem (groups 2-transitive on their involutions) 7 PUi, ...,PU5 134-136 PU1, ...,PUJ 132-133 SA (Goldschmidt) 23 SE (Bender-Suzuki) 20 SF (Holt) 24 SU 23 Suzuki's Theorem (on 2-transitive groups) 5 SZ (Aschbacher) 20 Witt's Lemma (for permutation groups) 4 ZD (Aschbacher) 21

Selected Titles in This Series (Continued from the front of this publication)

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 poly logarithms, 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, Representation theory 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 Na than 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 21 Albert Baernste in , David Drasin, Peter Duren, and Albert Marden, Editors,

The Bieberbach conjecture: Proceedings of the symposium on the occasion of the proof, 1986

20 K e n n e t h 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. T imothy O'Meara, Symplectic groups, 1978 15 J. Dieste l and J. J. Uhl , Jr., Vector measures, 1977 14 V. Guil lemin 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 . Pres ton , The algebraic theory of semigroups, Volume II, 1961 7.1 A. H. Clifford and G. B. Pres ton , 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. B e r g m a n , The kernel 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

(See the AMS catalog for earlier titles)