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The Classification of the Finite Simple Groups Book

The Classification of the Finite Simple Groups
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  • The Classification of the Finite Simple Groups
  • Written by author Daniel Gorenstein
  • Published by American Mathematical Society, December 2004
  • This volume contains the proofs of Theorems C2 and C3, which constitute the classification of finite simple groups G of special odd type. The special odd condition is introduced as representing the measure of smallness for simple groups which are not of e
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Prefaceix
Part IVThe Special Odd Case
Chapter 1General Introduction to the Special Odd Case1
1The Goals: Theorems [characters not reproducible] and [characters not reproducible]1
2Theorems [characters not reproducible] and [characters not reproducible]2
3General Comments on the Proof of Theorem [characters not reproducible]6
4Theorem [characters not reproducible] and Its Proof9
Chapter 2General Lemmas13
1The Bender/Glauberman Method13
2p-Groups and Coprime Action18
3Transfer and Fusion28
42-Components and 2-Groups32
5Pumpups and 2-Terminality35
6Terminality37
7Semirigidity38
8Preuniqueness Subgroups43
9Signalizer Functors and Balance48
10Ordinary Character Theory48
11Modular Character Theory51
12BN-Pairs56
13Number Theory58
14Miscellaneous58
Chapter 3Theorem [characters not reproducible]: Stage 161
1Introduction61
2The 2-Rank 2 Case: Alperin's Theorem64
3Theorem 2: 2-Terminal 2-Components of 2-Rank 167
4The Fusion of z and Structure of Q67
5Involutory f-Automorphisms71
6Elimination of f-Automorphisms75
7Theorem 3: The Nonfused Case77
8Theorem 3: The Fused Case80
9Theorem 4: 2-Terminal 2-Components of 2-Rank [greater than or equal] 286
10The O'N Case90
11Theorem 593
12Groups of 2-Rank 394
132-Groups of Type M[subscript 12]99
14The Proof of Theorem 6102
Chapter 4Theorem [characters not reproducible]: Stage 2105
1Introduction105
2The Structure of Involution Centralizers107
3Other K-Subgroups of G113
4Odd Order Signalizers117
5An Equivariant K[superscript infinity]-Theorem123
6Theorem 1: Uniqueness Properties of M126
7Theorem 1: The Case m[subscript 2](S) [greater than or equal] 3131
8Theorem 1: The Case m[subscript 2](S) = 2133
9Theorems 2 and 3: Normalizers of z-Invariant Subgroups of O[subscript p](M)135
10Theorem 2: {2, p}-Uniqueness Properties of M143
11Theorem 2: Reduction to the Ly Case146
12Theorem 2: The Ly Case148
13Theorem 3: The 2-Rank 2 Case150
14The Identification of L[subscript 3](q)157
15Reduction to L[subscript 3](q)-Levi Form160
16The L[subscript 3](q)-Levi Form Case164
17Theorems 4 and 5: The Structure of M165
18Theorems 4 and 5: Uniqueness Properties of M167
19The Strong B-Property172
20Theorem 6173
Chapter 5Theorem [characters not reproducible]: Stage 3181
1Introduction181
2Theorem 1: The Embedding of F183
3Character Theory for the Dihedral Case188
4Coherence190
5The Case [vertical bar S vertical bar greater than or equal] 8194
6Theorem BG[subscript 2]: The Case [vertical bar S vertical bar greater than or equal] 8200
7Character Theory for the Case [vertical bar S vertical bar] = 4203
8Elimination of the Extruded Case208
9The Identification of A[subscript 7]217
10The Identification of L[subscript 2](7) and A[subscript 6]221
11Reduction to the Brauer-Suzuki-Wall Case223
12The Identification of L[subscript 2](q)224
13Theorem 3: The Case M = QS230
14Theorem 3: The Structure of C233
15Theorem 3: The Final Contradiction240
16Theorem 4: Local Analysis245
17Regularity and [vertical bar G vertical bar]: Stage 1252
18Regularity: The Case B [not equal] 1259
19Regularity: The Case Q = 1264
20G [approximate] U[subscript 3](q)266
Chapter 6Theorem [characters not reproducible]: Stage 4273
1Introduction273
2Theorem 1: Field Automorphisms275
3The Characteristic of K286
4G [approximate superscript 2]G[subscript 2](q)290
5Theorem 2: The Weyl Group293
6Theorem 2: The Torally Singular Case297
7Theorem 2: The Case q = 3[superscript n greater than sign] 9300
8Theorem 2: The Case q = 9304
9Theorem 3: The Structure of a Sylow p-Subgroup of G309
10The (B, N)-Pair Structure of G313
11The Commutator Relations314
12The Action of K[subscript 1]H318
13The Isomorphism Type of G321
Chapter 7Theorem [characters not reproducible]: Stage 5325
1Introduction325
2Theorem Sp: 2-Local Structure325
3The p-Locals of G328
4The Identification of PSp[subscript 4](q)332
5SL[subscript 4 superscript plus or minus 1](q)-Amalgams333
6Theorem LU: Local Structure340
7Theorem LU: The General Case343
8Theorem LU: The L[subscript 4](5) Case344
Chapter 8Theorem [characters not reproducible]: Stage 1349
1Introduction349
2Theorem 1353
3Theorems 2 and 3354
4Terminally Unbalancing Triples356
5Theorem 4: Balance with Respect to Elements of [epsilon subscript *](G)364
6Theorem 5: L[subscript 2](G) [subset equals] [characters not reproducible]366
7Theorem 5372
8Theorem 6378
9Theorem 7382
Chapter 9Theorem [characters not reproducible]: Stages 2 and 3389
1Introduction389
2Theorem 1: The L[subscript 3](4) Case392
3Theorem 1: The L[subscript 2](q), A[subscript 7] Cases394
4Fusion in the A[subscript n] Cases401
5Theorem 2: Elements of V[subscript 2 superscript *](G)408
6The Level of Balance413
7Signalizer Functors424
8The L-Preuniqueness of M432
9Theorems 3 and 4: Reductions440
10The L[subscript 3](4) Case443
11G [approximate] A[subscript n]444
12Theorem [characters not reproducible]: Stage 3446
Chapter 10IV[subscript K]: Preliminary Properties of K-Groups451
1The Groups L[subscript 2](q) and A[subscript 7]451
2The Groups L[subscript 3](4) and U[subscript 3](4)462
3Product Structures467
4Other Groups of Lie Type471
5Alternating Groups and Some Sporadic Groups477
6Generation of K-Groups480
7Pumpups484
82-Signalizers and p-Stability494
9K-Groups with Prescribed Local Subgroups509
10Signalizers and Balance512
11Representations515
Background References519
Expository References521
Glossary523
Index527


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