The Classification of the Finite Simple Groups, Number 6, Part IV: The Special Odd Case

545 Pages·2004·57.919 MB·English

by  Daniel Gorenstein, Richard Lyons, Ronald Solomon

http://dx.doi.org/10.1090/surv/040.6
MATHEMATICAL
Surveys and Monographs
Volume 40, Number 6
The Classification
of the Finite Simple Groups,
Number 6
Part IV: The Special Odd Case
Daniel Gorenstein
Richard Lyons
Ronald Solomon
American Mathematical Society
Providence, Rhode Island
Editorial Board
Jerry L. Bona Peter S. Landweber, Chair
Michael G. Eastwood Michael P. Loss
J. T. Stafford
The authors gratefully acknowledge the support of the National Science Foundation
grants DMS0070801 and DMS0400533.
2000 Mathematics Subject Classification. Primary 20D06, 20D08; Secondary 20D05,
20E32, 20G40.
ABSTRACT. The Special Odd Case of the proof of the classification of finite simple groups is given,
as outlined in the first number of this series.
For additional information and updates on this book, visit
www.ams.org/bookpages/surv-40.6
Library of Congress Cataloging-in-Publication Data
ISBN 0-8218-2777-4 (number 6)
ISBN 0-8218-2776-6 (number 5)
ISBN 0-8218-1379-X (number 4)
ISBN 0-8218-1391-3 (number 3)
ISBN 0-8218-1390-5 (number 2)
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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10 9 8 7 6 5 4 3 2 1 10 09 08 07 06 05
To the memory of
Walter Feit (1930-2004)
and
Leonard D. Solomon (1916-2000)
This page intentionally left blank
Contents
Preface ix
PART IV: THE SPECIAL ODD CASE
Chapter 1. General Introduction to the Special Odd Case 1
1. The Goals: Theorems C and C 1
2 3
2. Theorems C and 6^ 2
2
3. General Comments on the Proof of Theorem C 6
2
4. Theorem C and Its Proof 9
3
Chapter 2. General Lemmas 13
1. The Bender/Glauberman Method 13
2. p-Groups and Coprime Action 18
3. Transfer and Fusion 28
4. 2-Components and 2-Groups 32
5. Pumpups and 2-Terminality 35
6. Terminality 37
7. Semirigidity 38
8. Preuniqueness Subgroups 43
9. Signalizer Functors and Balance 48
10. Ordinary Character Theory 48
11. Modular Character Theory 51
12. £N-Pairs 56
13. Number Theory 58
14. Miscellaneous 58
Chapter 3. Theorem 6^: Stage 1 61
1. Introduction 61
2. The 2-Rank 2 Case: Alperin's Theorem 64
3. Theorem 2: 2-Terminal 2-Components of 2-Rank 1 67
4. The Fusion of z and Structure of Q 67
5. Involutory /-Automorphisms 71
6. Elimination of /-Automorphisms 75
7. Theorem 3: The Nonfused Case 77
8. Theorem 3: The Fused Case 80
9. Theorem 4: 2-Terminal 2-Components of 2-Rank > 2 86
10. The O'N Case 90
11. Theorem 5 93
12. Groups of 2-Rank 3 94
13. 2-Groups of Type M 99
12
CONTENTS
14. The Proof of Theorem 6 102
Chapter 4. Theorem Q* : Stage 2 105
2
1. Introduction 105
2. The Structure of Involution Centralizers 107
3. Other X-Subgroups of G 113
4. Odd Order Signalizers 117
5. An Equivariant if °°-Theorem 123
6. Theorem 1: Uniqueness Properties of M 126
7. Theorem 1: The Case ra (S) > 3 131
2
8. Theorem 1: The Case m (S) = 2 133
2
9. Theorems 2 and 3: Normalizers of z-Invariant Subgroups of O (M) 135
p
10. Theorem 2: {2,p}-Uniqueness Properties of M 143
11. Theorem 2: Reduction to the Ly Case 146
12. Theorem 2: The Ly Case 148
13. Theorem 3: The 2-Rank 2 Case 150
14. The Identification of L (q) 157
3
15. Reduction to Ls(q)-Levi Form 160
16. The L (g)-Levi Form Case 164
3
17. Theorems 4 and 5: The Structure of M 165
18. Theorems 4 and 5: Uniqueness Properties of M 167
19. The Strong ^-Property 172
20. Theorem 6 173
Chapter 5. Theorem C : Stage 3 181
2
1. Introduction 181
2. Theorem 1: The Embedding of F 183
3. Character Theory for the Dihedral Case 188
4. Coherence 190
5. The Case |5| > 8 194
6. Theorem BG : The Case \S\ > 8 200
2
7. Character Theory for the Case \S\ = 4 203
8. Elimination of the Extruded Case 208
9. The Identification of A 217
7
10. The Identification of L (7) and A 221
2 6
11. Reduction to the Brauer-Suzuki-Wall Case 223
12. The Identification of L (q) 224
2
13. Theorem 3: The Case M = QS 230
14. Theorem 3: The Structure of C 233
15. Theorem 3: The Final Contradiction 240
16. Theorem 4: Local Analysis 245
17. Regularity and \G\: Stage 1 252
18. Regularity: The Case B ^ 1 259
19. Regularity: The Case Q = 1 264
20. G&U {q) 266
3
Chapter 6. Theorem C : Stage 4 273
2
1. Introduction 273
2. Theorem 1: Field Automorphisms 275
3. The Characteristic of K 286
CONTENTS vii
4. G^2G (q) 290
2
5. Theorem 2: The Weyl Group 293
6. Theorem 2: The Tor ally Singular Case 297
7. Theorem 2: The Case q = 3n > 9 300
8. Theorem 2: The Case q = 9 304
9. Theorem 3: The Structure of a Sylow p-Subgroup of G 309
10. The (J3, 7V)-Pair Structure of G 313
11. The Commutator Relations 314
12. The Action of K H 318
X
13. The Isomorphism Type of G 321
Chapter 7. Theorem C : Stage 5 325
2
1. Introduction 325
2. Theorem Sp: 2-Local Structure 325
3. The p-Locals of G 328
4. The Identification of PSp {q) 332
4
5. SL±\q)-Amal gams 333
6. Theorem LU: Local Structure 340
7. Theorem LU: The General Case 343
8. Theorem LU: The L (5) Case 344
4
Chapter 8. Theorem C : Stage 1 349
3
1. Introduction 349
2. Theorem 1 353
3. Theorems 2 and 3 354
4. Terminally Unbalancing Triples 356
5. Theorem 4: Balance with Respect to Elements of £*(G) 364
6. Theorem 5: £ (G) C e 366
2 2
7. Theorem 5 372
8. Theorem 6 378
9. Theorem 7 382
Chapter 9. Theorem C : Stages 2 and 3 389
3
1. Introduction 389
2. Theorem 1: The L (4) Case 392
3
3. Theorem 1: The L (g), A Cases 394
2 7
4. Fusion in the A Cases 401
n
5. Theorem 2: Elements of V*(G) 408
2
6. The Level of Balance 413
7. Signalizer Functors 424
8. The L-Preuniqueness of M 432
9. Theorems 3 and 4: Reductions 440
10. TheL (4) Case 443
3
11. G^A 444
n
12. Theorem C : Stage 3 446
3
Chapter 10. IV^: Preliminary Properties of X-Groups 451
1. The Groups L (q) and A 451
2 7
2. The Groups L (4) and U (4) 462
3 3
3. Product Structures 467
viii CONTENTS
4. Other Groups of Lie Type 471
5. Alternating Groups and Some Sporadic Groups 477
6. Generation of X- Groups 480
7. Pumpups 484
8. 2-Signalizers and p-Stability 494
9. if-Groups with Prescribed Local Subgroups 509
10. Signalizers and Balance 512
11. Representations 515
Background References 519
Expository References 521
Glossary 523
Index 527
Preface
This volume contains the proofs of Theorems C2 and C3, as stated in the first
volume of this series [I2].
Theorems C and 63 constitute the classification of finite simple groups G of
2
special odd type. This condition requires that no 2-component K of an involution
centralizer of G is of generic type in the sense that KjOyiK) G S2? but G is not of
restricted even type. The latter condition is rather technical, but primarily it entails
that either G has 2-rank 2 or, for some 2-component K of an involution centralizer
of G, K/Z*(K) = L (q) for some odd q or K/0^{K) is a member of a small
2
finite set of additional quasisimple groups. In fact, we prove a strengthened version
of Theorem C2, which we call Theorem Q^, in which the ban on 2-components of
generic type is relaxed. Theorem C permits us to classify all X-proper simple
2
groups G having a 2-Thin Configuration in the sense of [IIIi; 2.1]. This extension
is an essential ingredient in the proof of Theorem 67, the classification of groups
of generic type (begun in the previous volume and to be completed in the next
volume). The simple groups arising as conclusions to our theorems are the finite
simple groups of Lie type in odd characteristic of BN rank 1 or 2 (with some
exceptions) together with L±(q), (q odd, q ^ 1 (mod 8)), A , n G {7,9,10,11},
n
and the five sporadic groups Mu, M12, Mc, Ly and O'N.
The special odd condition represents our measure of smallness for simple groups
which are not of even type. Other measures have been used in the past, namely
2-rank, normal 2-rank or sectional 2-rank. Indeed, our list of conclusions differs
little from the conclusions of the Sectional 2-Rank 4 Theorem of Gorenstein and
Harada [GH1]. The Gorenstein-Harada Memoir depends on a long list of prior
results. Some of these likewise form part of our Background Results, most no
tably the Feit-Thompson Theorem [FT1] on the solvability of groups of odd order,
and the body of results yielding recognition theorems for the split (B, iV)-pairs
of rank 1. Others have been incorporated into our proof of Theorem 62, notably
the classification of finite simple groups of 2-rank 2 by Gorenstein-Walter [GW1],
Alperin-Brauer-Gorenstein [ABG1] and Lyons [LI] and much of the classification
of finite simple groups with an abelian Sylow 2-subgroup by Walter [Wal], together
with involution centralizer recognition theorems for finite simple groups of Lie type
in odd characteristic of £W-rank 2 by Brauer [Br5], Fong and W. J. Wong [FW1],
[Fol].
The classification of finite simple groups of 2-rank at most 2 by Brauer and
Suzuki [BrSul], Feit and Thompson, Gorenstein and Walter, Alperin and Brauer
and Gorenstein, and Lyons was a major accomplishment of the 1960's. During the
late 1960's, the Signalizer Functor Method was developed, primarily by Gorenstein
and Walter, and with particular emphasis on the prime 2. The importance of
2-connectivity for this method (cf. [I<s; Section 22]) again focussed attention on
ix