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 Ohio State University 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 Copying and reprinting. 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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 https://www.ams.org/ 10987654321 232221201918 For Sofia and Larry 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.