MATHEMATICAL Surveys and Monographs

Volume 40, Number 8

The Classification of the Finite Simple Groups, Number 8

Daniel Gorenstein Richard Lyons Ronald Solomon 10.1090/surv/040.8

The Classification of the Finite Simple Groups, Number 8

Part III, Chapters 12 –17: The Generic Case, Completed

MATHEMATICAL Surveys and Monographs

Volume 40, Number 8

The Classification of the Finite Simple Groups, Number 8

Part III, Chapters 12 –17: The Generic Case, Completed

Daniel Gorenstein Richard Lyons Ronald Solomon Editorial Board Robert Guralnick Benjamin Sudakov Michael A. Singer, Chair Constantin Teleman MichaelI.Weinstein The authors gratefully acknowledge the support provided by grants from the Na- tional Security Agency (H98230-07-1-0003 and H98230-13-1-0229), the Simons Foundation (425816), and the Emeritus Academy. 2010 Mathematics Subject Classification. Primary 20D05, 20D06, 20D08; Secondary 20E25, 20E32, 20F05, 20G40.

For additional information and updates on this book, visit www.ams.org/bookpages/surv-40.8

The ISBN numbers for this series of books includes

ISBN 978-1-4704-4189-0 (number 8) ISBN 978-0-8218-4069-6 (number 7) ISBN 978-0-8218-2777-2 (number 6) ISBN 978-0-8218-2776-5 (number 5) ISBN 978-0-8218-1379-9 (number 4) ISBN 978-0-8218-0391-2 (number 3) ISBN 978-0-8218-0390-5 (number 2) ISBN 978-0-8218-0334-9 (number 1)

Library of Congress Cataloging-in-Publication Data 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

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Contents

Preface ix

Chapter 12. Introduction 1 C C∗ 1. Theorems 7 and 7 1

Chapter 13. Recognition Theory 9 1. Curtis-Tits Systems and Phan Systems 9 2. The Gilman-Griess Theorem for Groups in Chev(2) 14 3. The Wong-Finkelstein-Solomon Method 15 C∗ Chapter 14. Theorem 7: Stage 4b+. A Large Lie-Type Subgroup G0 for p =2 29 1. Introduction 29 ± 2. A 2-Local Characterization of L4 (q), q Odd 33 ∼ ± 3. The Case K = L3 (q)39 ∼ 3 4. The Case K = G2(q)or D4(q)41 5. The Non-Level Case 47 6. The Other Exceptional Cases 58 ∼ a 7. The Case K = PSp2n(q)=Cn(q) 62 8. The Spinn(q)Cases,n ≥ 774 9. The Sp2n Cases 80 10. The Linear and Unitary Cases 87 11. The Orthogonal Case, Preliminaries 112 12. The Orthogonal Case, Completed 122 13. The Cases in Which G0 is Exceptional 141 14. Summary: p = 2 154 C∗ Chapter 15. Theorem 7: Stage 4b+. A Large Lie-Type Subgroup G0 for p>2 155 1. Introduction 155 2. A Choice of p 156 3. The Weyl Group 157 4. The Field Automorphism Case 160 5. Some General Lemmas 167 6. The Case m (B) = 4 178 p ∼ 7. The Case Aut (B) = W (BC )orW (F ) 187 K ∼ n 4 8. The Case AutK (B) = W (Dn), n ≥ 4 193 9. Some Exceptional Cases 206 ∼ ± 10. The Case K/Z(K) = PSLm(q) 207 ∼ 11. The Final Case: K/Z(K) = E6(q) 219

vii viii CONTENTS

12. Identification of G0: Setup 224 ∼ −q 13. G0 = Sp2n+2(q)orAn+1(q) 225 ∼ ± 14. G0 = Dn+1(q) 232 ∼ q 15. G = L (q) 248 0 ∼ k 16. G0 = E8(q) 252 ∼ q 17. G0 = E6 (q)andE7(q) 257 18. The Remaining Cases for G0 264 19. ΓD,1(G) Normalizes G0 277 C∗ Chapter 16. Theorem 7: Stage 5+. G = G0 291 1. Introduction and Generalities 291 2. The Alternating Case 293 3. The Lie Type Case, p = 2: Part 1 294 4. The Lie Type Case, p = 2: Part 2 298 5. The Lie Type Case, p>2 303 Chapter 17. Preliminary Properties of K-Groups 321 1. Weyl Groups and Their Representations 321 2. Toral Subgroups 335 3. Neighborhoods 355 4. CTP-Systems 376 5. Representations 383 6. Computations in Groups of Lie Type 387 7. Outer Automorphisms, Covering Groups, and Envelopes 406 8. p-Structure of Quasisimple K-groups 408 9. Generation 415 10. Pumpups 423 11. Small Groups 455 12. Subcomponents 464 13. Acceptable Subterminal Pairs 470 14. Fusion 474 15. Balance and Signalizers 480 16. Miscellaneous 481 Bibliography 485 Index 487 Preface

Volumes 5, 7, and 8 of this series form a trilogy treating the Generic Case of the classification proof. An overview of the general strategy for this set of volumes, along with a brief history of the original treatment of these results, is provided in the preface and Chapter 1 of Volume 5. We shall not repeat that here; rather, we refer the reader to Volume 5. By the end of Volume 7, we arrived at the existence in our K-proper simple group G, for a suitably chosen prime p, of one of the following:

∼ (a) (Alternating case) a subgroup G0 ≤ G such that G0 = An for some n ≥ 13, p =2,andΓD,1(G) normalizes G0 for any root 4-subgroup D of G0. Thus for any involution d ∈ G0 which is the product of two disjoint transpositions, CG(d) normalizes G0;or (b) (Lie-type case) an element x of order p whose centralizer CG(x)hasa p-generic quasisimple component K = dL(q) ∈ Chev(r), where p, r are a pair of distinct primes with q apowerofr, such that either p =2andr is odd, or r =2andp divides q2 − 1.

In Chapter 16 of this volume, we shall prove that in case (a), G = G0,sothat G is a K-group, as desired. However, almost all our attention in this volume will be on case (b), where in order to catch up with case (a), we construct a subgroup G ∈ Chev such that Γ (G):=N (Q) | 1 = Q ≤ D normalizes G for a suitable 0 ∼ D,1 G 0 subgroup D = Zp × Zp of G. The results of Volume 7 provide a lot more information in case (b). Thus, we know that CG(K) has a cyclic or quaternion Sylow p-subgroup, and if p> 2, then K itself contains a copy of Zp × Zp × Zp. Significantly, also, a family of centralizers “neighboring” CG(x) also have semisimple p-layers. Considerable additional information is known about these centralizers. All this is spelled out precisely in Theorem 1.2 in Chapter 12 of this volume. [Note: The numbering of chapters in this volume continues that of Volumes 5 and 7. In particular, Chapter 12 is the first chapter of this volume.] Roughly speaking, using the terminology of the initial treatment of the Classification Theorem, we are faced with “standard form problems” for components in Chev whose centralizers have p-rank 1. In this volume we complete the proof of C∗ K  ∅ Theorem 7. Let G be a -proper simple group. Assume that γ(G) = and that G does not possess a p-Thin Configuration for any prime ∼ (7)∗ p ∈ γ(G).ThenG = G0 for some G0 ∈ K . ∗ (See p. 1 and p. 4 for the definitions of the set K(7) of “generic” known simple groups and the set γ(G) of primes associated to G.) In particular, in conjunction with the 2-Uniqueness Theorems in Volume 4 and the main theorems of Volume 6,

ix xPREFACE we have completed our treatment of simple groups of odd type, in the sense that we have proved the following theorem. Theorem O. Let G be a K-proper simple group. Assume that either m2(G) ≤ Lo ∩ T ∪ G  ∅ ∈ C 2 or 2(G) ( 2 2) = . Then either G is an alternating group or G hev(r) ∼  for some odd prime r or G = M11, M12, J1, Mc, Ly,orO N. Put another way, we have the following theorem. Theorem. Let G be a minimal counterexample to the Classification Theorem. Then G is of even type, i.e., the following conditions hold:

(a) m2(G) ≥ 3; (b) O2 (CG(t)) = 1 for all t ∈I2(G);and Lo ⊆ C (c) 2(G) 2. C∗ Theorem 7 also includes the following result when G is of even type. Theorem GE. Let G be a K-proper simple group of even type. Assume that γ(G) contains at least one odd prime and that G does not possess a p-Thin Configuration for any prime p ∈ γ(G).ThenG ∈ Chev(2). Historically, the portion of Theorem O covered in Volumes 5, 7, and 8 was prin- cipally the work of M. Aschbacher, J. H. Walter, D. Gorenstein, J. G. Thompson, and M. E. Harris. In particular, Chapter 14 in this volume relies at many points on arguments originally given by Aschbacher in his Classical Involution Paper [A9]. His work has been a continual source of inspiration for us. Theorem GE was treated originally by the first two authors [GL1]incon- junction with a paper of Gilman and Griess [GiGr1]. Our work here follows a similar outline to this earlier work and at numerous points benefits from arguments found therein. However, although the earlier work relied almost exclusively on the Gilman-Griess Theorem (see Section 2 of Chapter 13), we employ a variety of recog- nition techniques, as discussed also in Sections 1 and 3 of Chapter 13. In particular, the theory of Curtis-Tits and Phan systems, developed in the last several decades, plays a very important role. Also useful is an identification method pioneered by W. J. Wong [Wo1] in the early 1970s for the recognition of classical groups in odd characteristic, and later extended by Finkelstein and the third author to many cases of classical groups in characteristic 2, e.g. [FinS1]. In barest outline, the strategies for the non-alternating generic case of Theo- rem O and for Theorem GE are identical. The main theorems of Volume 7 have produced, for some p ∈ γ(G), an element x ∈Ip(G) and a component K of CG(x) with K ∈ Chev(r)andwithm (C (K)) = 1, having an acceptable subterminal p G ∼ (x, K)-pair (y, L) (defined on p. 7), where D := x, y = Zp × Zp and L is a large component of CK (y). Moreover, we have some u ∈ D −x such that the pumpup Lu of L in CG(u) is quasisimple with L

Chapter 15. The principal difference is that, when p = 2, the existence of commut- ing involutions in the centers of suitable fundamental SL2(q) subgroups of K and of Lu permits these subgroups to be lined up properly for the verification of suitable Curtis-Tits or Phan relations. An analogue is not available when p is odd, but— following Gilman and Griess—we use a toral subgroup B of exponent p visible both # in CG(x)andallCG(u), u ∈ D , permitting the construction of a Weyl group W for G as a quotient of NG(B). Then G0 may be identified with K, N for suitable N ≤ NG(B) covering W . A version of this strategy of recognizing G0 as K, N was first implemented by W. J. Wong for the characterization of symplectic and or- thogonal groups in odd characteristic. Moreover, in the Classical Involution Paper, M. Aschbacher constructed a toral 2-subgroup of G andanassociatedWeyl-type group (which he calls the Thompson group), building on ideas of J. G. Thompson.

By the end of Chapter 15, we have constructed in case (b) a subgroup G0 of G with G0 ∈ Chev(r)andΓD,1(G) normalizing G0. Finally, in Chapter 16, we prove that G = G0 in both case (a) and case (b) by showing that, if not, then NG(G0)isaproperstrongp-uniqueness subgroup of G of component type, contrary to hypothesis. (Note that for p = 2, the simple groups with a proper strong 2-uniqueness subgroup are classified in Volume 4, using the arguments of Bender and Aschbacher.) In this volume, we have found it of great convenience, and sometimes of neces- sity, to use some known theorems not previously included in our set of Background Results, viz., L. E. Dickson’s classification of smallest-dimensional faithful modules for the symmetric groups [Di2]; J. McLaughlin’s characterization of groups gener- ated by transvections on a finite vector space of odd order [McL2]; R. Gramlich’s exposition of the state of Phan theory [Grm2]; and theorems of R. Blok, C. Hoff- man, and S. Shpectorov on amalgams of Curtis-Tits and Phan types [BHS]. The full set of Background Results consists now of these four references and the Back- ground Results listed in our first volume [I1]. At an appropriate future moment, we shall add to these the Aschbacher-Smith tomes on quasithin groups. We continue the notational conventions established in Volume 2 of this series [IG]. We refer to the chapters of this book as [III12], [III13], [III14], [III15], [III16] and [III17]. As in previous volumes, the last chapter [III17] collects the necessary K-group lemmas for the main chapters [III14]–[III16], and thus logically precedes them. As early as 1999, the authors began a discussion with Curtis Bennett and Sergey Shpectorov concerning the possibility of a modern treatment of the theorems of Phan characterizing the finite unitary groups in odd characteristic. They took up this problem and expanded its scope, later recruiting Corneliu Hoffmann and Rieuwert Blok as collaborators. Also, in the early 2000s, Ralf Gramlich began his dissertation work under the supervision of Aryeh Cohen. Gramlich visited Rutgers during part of the 2001–02 academic year, which the third author spent at Rutgers. Gramlich became a leader in the development of Phan Theory and recruited many others to this project. We extend our warmest thanks to all of these mathematicians for their contributions to the development of this body of recognition theorems. We also thank Bob Griess, Jon Hall, Len Scott, Gary Seitz, and Gernot Stroth for numerous helpful conversations and valuable suggestions. Special thanks once xii PREFACE again go to our wives, Lisa and Rose, for their patience during our many hours of mathematical distraction. And, lastly, we recall with deep regret the passing of our colleague and friend, Kay Magaard, with whom we had many hours of fruitful discussion of some the topics destined for future volumes of this series, along with many lighter moments of laughter and good fellowship. We will miss him often as the years go by. This volume, in tandem with Volume 7, has spent many years in preparation. Both of us wish to thank the National Science Foundation and the National Security Agency for their years of grant support. The third author also extends thanks to the Ohio State University Emeritus Academy for its support and to the Simons Foundation for the current collaborative research grant (Award ID 425816) which has funded several weeks of face-to-face collaboration both in Ohio and in New Jersey. Richard Lyons and Ronald Solomon August, 2018

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A, 158 when p =2,30 a, superscript, 30 Eq (X), 94 acceptable subterminal pair, 7 even type, 303 agreeable subgroup, 25 restricted, 8 almost strongly p-embedded subgroup, 4 extendible, 13 types of, 4 Alperin, J., 314 F, F(K), 6 Alt, 1 Φm (mth cyclotomic polynomial), 7 anchored component, 453 f, f(K), 6 Aschbacher, M., x, 12 Finkelstein, L., x, 15 classical involution theorem, 29 G0, 29 ∗ B , 157 G0,2 ∈ C B0, 175 G0 hev, construction of, 29–290 Gi Background Results, xi 2-groups, 5 Baer, R., 332 γ(G), 4 Bender, H., xi Generic Case, 4 Bennett, C., 12 Gilman,R.H.,x,15,29 Blok,R.,xi,10,12 Gilman-Griess theory, 14 breather, 303 Gorenstein, D., x brush, 393 Gp,5 Gi p,5 o o C , C(V ) , 15 Gi -groups, 6 C p hev, 1 Gramlich, R., xi, 12  o CI , CI , CI , 16 Griess,R.L.,Jr.,x,15,29 classical involution, 89, 294, 299 G∗-trick, 253 compatible, 14 Cp,5 Harris, M. E., x CT-pairs and CT-systems, 376–383 HD, 18 CT-system, see also weak CT-system Hoffman, C., xi, 10, 12 Curtis-Tits theorem, 9 Holt, D., 318 C(V ), 15 Horn, M., 12 hs, superscript, 30 De, 29, 155 Devillers, A., 12 ignorable, 7 Dickson, L. E., xi, 323 ILo(G), 4 − p + Io Dn (q), Dn (q), 16 p (G), 4 dp, 6 J∗ dp(G), 6 p(G), 7 dp(K), 6 Jp(G), 6 Dunlap, J., 12 Jr, Jr, 171

+ − E6 (q), E6 (q), 16 κ, 15 (7)∗ q K , 1 when p is odd, 16, 155 K(7)+, 3

487 488 INDEX

K¨ohl, R., xi, 12 Stage 4b+, 3 K-proper, 1 Stage 5+, 291 Kp, 5 Thompson, J. G., x, xi, 46, 246 K-preuniqueness subgroup, 4 torally embedded, 339 T ,5 q p L2 (X), 94 triality-free, 475 long root p-element, 474 Lo p(G), 4 u, superscript, 30 L+(q), L−(q), 15 uniqueness subgroup almost strongly p-embedded subgroup Mal’cev, A. I., 337, 405 strongly p-embedded type, 4 McLaughlin, J., xi strongly closed type, 4–5 milestone, 303 strong p-uniqueness subgroup, 4 m p(K), 5 strong p-uniqueness subgroup of M¨uhlherr, B., 12 component type, 4

N, N(x, K, y, L), x, 2,29 VI , VI , 16 Natural, Mr., see also Griess, R. L., Jr. Vκ,p, 16 neighborhood, 2 Nickel, W., 12 W , 158 W ∗, 159 P, 18 W1, 173 P-pairs and P-systems, 376–383 Wagner, A., 322 pair, 4 Walter,J.H.,x acceptable subterminal, 7 weak CT-system, weak Curtis-Tits system, p-component preuniqueness subgroup, 4 9 p-generic, 5 weak CTP-system, 14 Phan system, see also weak P-system weak P-system Phan theorem, 13, 14 extendible, 13 Phan theory, 10–14 weak Phan system, weak P-system, 13

Phan, K.-W., 12 WH , WK , WL, WLu , 158 Phan-able, 11 Wielandt, H., 359 PP-pair, 377 Witzel, S., 12 primitive prime divisor, 147, 386 W (L), 159 p-Thin Configuration, 1 Wong, W., 15 Wong,W.J.,x Q(G), Qp(G), 156 W ≡ W (L), 159 Q∗(G), 156 ≡ L ≡ L W V W ( ), W W ( ), 322 R, 173 Zalesski˘ı, A. E., 322 ranking functions F, 6 F, F(K), 6 τ, τ(K), 6 f, f(K), 6 reflection module, 326 ρ(K, I), 436

Sereˇzkin, V. N., 322 Shpectorov,S.,xi,10,12 Solomon 2-fusion systems, 79, 393 Solomon, R., 15 splitting K ∈ Chev(r), 8 Spor, 1 standard CT-pair, 9 standard Phan pair, standard P-pair, 10 Suzuki, M., 332

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SURV/40.8