The Classification of the Finite Simple Groups, Number 8

506 Pages·2018·5.955 MB·English

by  Daniel Gorenstein, Richard Lyons, Ronald Solomon

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.
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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
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For
Sofia
and
Larry
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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
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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
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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
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