M ATHEMATICAL Surveys and Monographs Volume 40, Number 8 The Classification of the Finite Simple Groups, Number 8 Daniel Gorenstein Richard Lyons Ronald Solomon Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms 10.1090/surv/040.8 The Classification of the Finite Simple Groups, Number 8 Part III, Chapters 12 –17: The Generic Case, Completed Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms M ATHEMATICAL 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 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms Editorial Board Robert Guralnick Benjamin Sudakov Michael A. Singer, Chair ConstantinTeleman Michael I. Weinstein The authors gratefully acknowledge the support provided by grants from the Na- tionalSecurityAgency(H98230-07-1-0003andH98230-13-1-0229),theSimonsFoundation (425816),and the Ohio State University Emeritus Academy. 2010 Mathematics Subject Classification. Primary 20D05, 20D06, 20D08; Secondary 20E25, 20E32,20F05, 20G40. For additional informationand 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. (Mathematicalsurveysandmonographs: v.40,number1–) Includesbibliographicalreferencesandindex. ISBN0-8218-0334-4[number1] 1. Finitesimplegroups. I.Lyons,Richard,1945–. II.Solomon,Ronald. III.Title. IV.Series: Mathematicalsurveysandmonographs,no.40,pt. 1–;. QA177.G67 1994 512(cid:2).2-dc20 94-23001 CIP Copying and reprinting. 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(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttps://www.ams.org/ 10987654321 232221201918 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms For Sofia and Larry Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms Contents Preface ix Chapter 12. Introduction 1 1. Theorems C and C∗ 1 7 7 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 Chapter 14. Theorem C∗: Stage 4b+. A Large Lie-Type Subgroup G for 7 0 p=2 29 1. Introduction 29 ± 2. A 2-Local Characterization of L (q), q Odd 33 ∼ ± 4 3. The Case K =L (q) 39 ∼ 3 4. The Case K =G (q) or 3D (q) 41 2 4 5. The Non-Level Case 47 6. The Other Exceptional Cases 58 ∼ 7. The Case K =PSp (q)=C (q)a 62 2n n 8. The Spin (q) Cases, n≥7 74 n 9. The Sp Cases 80 2n 10. The Linear and Unitary Cases 87 11. The Orthogonal Case, Preliminaries 112 12. The Orthogonal Case, Completed 122 13. The Cases in Which G is Exceptional 141 0 14. Summary: p=2 154 Chapter 15. Theorem C∗: Stage 4b+. A Large Lie-Type Subgroup G for 7 0 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 ) or W(F ) 187 K n 4 8. The Case Aut (B)∼=W(D ), n≥4 193 K n 9. Some Exceptional Cases 206 10. The Case K/Z(K)∼=PSL±(q) 207 ∼m 11. The Final Case: K/Z(K)=E(cid:2)(q) 219 6 vii Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms viii CONTENTS 12. Identification of G : Setup 224 0 13. G ∼=Sp (q) or A−(cid:2)q(q) 225 0 2n+2 n+1 ∼ ± 14. G =D (q) 232 0 n+1 15. G ∼=L(cid:2)q(q) 248 0 ∼ k 16. G =E (q) 252 0 8 17. G ∼=E(cid:2)q(q) and E (q) 257 0 6 7 18. The Remaining Cases for G 264 0 19. Γ (G) Normalizes G 277 D,1 0 Chapter 16. Theorem C∗: Stage 5+. G=G 291 7 0 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 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms Preface Volumes 5, 7, and 8 of this series form a trilogy treating the Generic Case of theclassificationproof. Anoverviewofthegeneralstrategyforthissetofvolumes, 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 G ≤ G such that G ∼= A for some 0 0 n n ≥ 13, p = 2, and Γ (G) normalizes G for any root 4-subgroup D of D,1 0 G . Thus for any involution d ∈ G which is the product of two disjoint 0 0 transpositions, C (d) normalizes G ; or G 0 (b) (Lie-type case) an element x of order p whose centralizer C (x) has a G p-generic quasisimple component K = dL(q) ∈ Chev(r), where p,r are a pair of distinct primes with q a power of r, such that either p = 2 and r is odd, or r =2 and p divides q2−1. In Chapter 16 of this volume, we shall prove that in case (a), G=G , so that 0 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 suchthatΓ (G):=(cid:7)N (Q)|1(cid:8)=Q≤D(cid:9)normalizesG forasuitable 0 D,1 G 0 subgroup D ∼=Z ×Z of G. p p The results of Volume 7 provide a lot more information in case (b). Thus, we know that C (K) has a cyclic or quaternion Sylow p-subgroup, and if p > G 2, then K itself contains a copy of Z × Z × Z . Significantly, also, a family p p p of centralizers “neighboring” C (x) also have semisimple p-layers. Considerable G 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 Theorem C∗. Let G be a K-proper simple group. Assume that γ(G)(cid:8)=∅ and 7 that G does not possess a p-Thin Configuration for any prime p∈γ(G). Then G∼=G for some G ∈K(7)∗. 0 0 (Seep. 1andp. 4forthedefinitions ofthe setK(7)∗ of“generic”knownsimple 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 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms