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Preface | ix | |
Part IV | The Special Odd Case | |
Chapter 1 | General Introduction to the Special Odd Case | 1 |
1 | The Goals: Theorems [characters not reproducible] and [characters not reproducible] | 1 |
2 | Theorems [characters not reproducible] and [characters not reproducible] | 2 |
3 | General Comments on the Proof of Theorem [characters not reproducible] | 6 |
4 | Theorem [characters not reproducible] and Its Proof | 9 |
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 | BN-Pairs | 56 |
13 | Number Theory | 58 |
14 | Miscellaneous | 58 |
Chapter 3 | Theorem [characters not reproducible]: 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 f-Automorphisms | 71 |
6 | Elimination of f-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 [greater than or equal] 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[subscript 12] | 99 |
14 | The Proof of Theorem 6 | 102 |
Chapter 4 | Theorem [characters not reproducible]: Stage 2 | 105 |
1 | Introduction | 105 |
2 | The Structure of Involution Centralizers | 107 |
3 | Other K-Subgroups of G | 113 |
4 | Odd Order Signalizers | 117 |
5 | An Equivariant K[superscript infinity]-Theorem | 123 |
6 | Theorem 1: Uniqueness Properties of M | 126 |
7 | Theorem 1: The Case m[subscript 2](S) [greater than or equal] 3 | 131 |
8 | Theorem 1: The Case m[subscript 2](S) = 2 | 133 |
9 | Theorems 2 and 3: Normalizers of z-Invariant Subgroups of O[subscript p](M) | 135 |
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[subscript 3](q) | 157 |
15 | Reduction to L[subscript 3](q)-Levi Form | 160 |
16 | The L[subscript 3](q)-Levi Form Case | 164 |
17 | Theorems 4 and 5: The Structure of M | 165 |
18 | Theorems 4 and 5: Uniqueness Properties of M | 167 |
19 | The Strong B-Property | 172 |
20 | Theorem 6 | 173 |
Chapter 5 | Theorem [characters not reproducible]: Stage 3 | 181 |
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 [vertical bar S vertical bar greater than or equal] 8 | 194 |
6 | Theorem BG[subscript 2]: The Case [vertical bar S vertical bar greater than or equal] 8 | 200 |
7 | Character Theory for the Case [vertical bar S vertical bar] = 4 | 203 |
8 | Elimination of the Extruded Case | 208 |
9 | The Identification of A[subscript 7] | 217 |
10 | The Identification of L[subscript 2](7) and A[subscript 6] | 221 |
11 | Reduction to the Brauer-Suzuki-Wall Case | 223 |
12 | The Identification of L[subscript 2](q) | 224 |
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 [vertical bar G vertical bar]: Stage 1 | 252 |
18 | Regularity: The Case B [not equal] 1 | 259 |
19 | Regularity: The Case Q = 1 | 264 |
20 | G [approximate] U[subscript 3](q) | 266 |
Chapter 6 | Theorem [characters not reproducible]: Stage 4 | 273 |
1 | Introduction | 273 |
2 | Theorem 1: Field Automorphisms | 275 |
3 | The Characteristic of K | 286 |
4 | G [approximate superscript 2]G[subscript 2](q) | 290 |
5 | Theorem 2: The Weyl Group | 293 |
6 | Theorem 2: The Torally Singular Case | 297 |
7 | Theorem 2: The Case q = 3[superscript n greater than sign] 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 (B, N)-Pair Structure of G | 313 |
11 | The Commutator Relations | 314 |
12 | The Action of K[subscript 1]H | 318 |
13 | The Isomorphism Type of G | 321 |
Chapter 7 | Theorem [characters not reproducible]: Stage 5 | 325 |
1 | Introduction | 325 |
2 | Theorem Sp: 2-Local Structure | 325 |
3 | The p-Locals of G | 328 |
4 | The Identification of PSp[subscript 4](q) | 332 |
5 | SL[subscript 4 superscript plus or minus 1](q)-Amalgams | 333 |
6 | Theorem LU: Local Structure | 340 |
7 | Theorem LU: The General Case | 343 |
8 | Theorem LU: The L[subscript 4](5) Case | 344 |
Chapter 8 | Theorem [characters not reproducible]: Stage 1 | 349 |
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 [epsilon subscript *](G) | 364 |
6 | Theorem 5: L[subscript 2](G) [subset equals] [characters not reproducible] | 366 |
7 | Theorem 5 | 372 |
8 | Theorem 6 | 378 |
9 | Theorem 7 | 382 |
Chapter 9 | Theorem [characters not reproducible]: Stages 2 and 3 | 389 |
1 | Introduction | 389 |
2 | Theorem 1: The L[subscript 3](4) Case | 392 |
3 | Theorem 1: The L[subscript 2](q), A[subscript 7] Cases | 394 |
4 | Fusion in the A[subscript n] Cases | 401 |
5 | Theorem 2: Elements of V[subscript 2 superscript *](G) | 408 |
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 | The L[subscript 3](4) Case | 443 |
11 | G [approximate] A[subscript n] | 444 |
12 | Theorem [characters not reproducible]: Stage 3 | 446 |
Chapter 10 | IV[subscript K]: Preliminary Properties of K-Groups | 451 |
1 | The Groups L[subscript 2](q) and A[subscript 7] | 451 |
2 | The Groups L[subscript 3](4) and U[subscript 3](4) | 462 |
3 | Product Structures | 467 |
4 | Other Groups of Lie Type | 471 |
5 | Alternating Groups and Some Sporadic Groups | 477 |
6 | Generation of K-Groups | 480 |
7 | Pumpups | 484 |
8 | 2-Signalizers and p-Stability | 494 |
9 | K-Groups with Prescribed Local Subgroups | 509 |
10 | Signalizers and Balance | 512 |
11 | Representations | 515 |
Background References | 519 | |
Expository References | 521 | |
Glossary | 523 | |
Index | 527 |
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