Groups of finite Morley rank:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford
Clarendon Press
1994
|
| Series: | Oxford logic guides
26 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVII, 409 S. |
| ISBN: | 0198534450 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Contents
1 Basic Group Theory 1
1.1 Generalities and Notation 1
1.2 Semidirect Products 10
1.3 Abelian Groups 12
1.4 Permutation Groups 14
2 Definability 18
3 Interpretability 29
3.1 Definitions 30
3.2 Solvable Groups 33
3.2.1 Finding a Ring in a Solvable Group 33
3.2.2 Heisenberg Groups 37
3.2.3 Some Other Nilpotent Groups 38
3.3 Projective Spaces 40
3.3.1 Projective Planes 40
3.3.2 Automorphism Groups of Projective Planes 42
3.3.3 Projective Spaces 44
3.4 Algebraic Groups 46
3.4.1 Algebraic Geometry 46
3.4.2 Affine Algebraic Groups 48
4 Ranked Universe 52
4.1 Definitions 52
4.1.1 Universe 52
4.1.2 Rank 56
4.2 Basic Properties of Rank 58
4.3 Uniform Families 66
5 Basic Properties 68
5.1 Descending Chain Condition 69
5.2 Connected Component 74
5.3 Uniformly Definable Families of Subgroups 80
5.4 Zil ber s Indecomposability Theorem 84
xiv Contents
5.5 Definable Closure 88
6 Nilpotent Groups 95
6.1 Minimal Subgroups 96
6.2 Structure of Nilpotent Groups 98
6.3 Nilpotent Groups of Morley Rank 2 103
6.4 Locally Finite p Subgroups 104
7 Semisimple Groups 109
7.1 Semisimple Groups 109
7.2 Fitting Subgroup and the Radical 111
7.3 Socle 113
7.4 Generalized Fitting Subgroup 117
8 Fields and Rings 122
8.1 Fields 122
8.2 Regular Subgroups of GLn(K) 126
8.3 Bachmann s Theorem 134
9 Solvable Groups 140
9.1 Fundamental Results 141
9.2 Derived Subgroup 147
9.3 Splitting 148
9.4 Solvable Groups of Class 2 150
9.5 Groups of Morley Rank 2 153
9.5.1 Nonnilpotent Groups of Morley Rank 2 153
9.5.2 Nilpotent Groups of Morley Rank 2, Revisited 154
9.6 Fitting Subgroup 155
9.7 7T* Subgroups of Solvable Groups 155
9.8 Schur Zassenhaus Theorem 159
9.8.1 Connectedness of Sylow /^ Subgroups 161
9.8.2 Maschke s Theorem 162
9.8.3 Schur Zassenhaus Theorem 163
9.8.4 Conjugacy of Complements 165
9.9 Hall Theorem 168
10 2 Sylow Theory 172
10.1 Properties of Involutions 173
10.2 Actions of Involutions 175
10.3 2 Sylow Theorem 175
10.3.1 Reminiscences from Finite Group Theory 176
Contents xv
10.3.2 2 Sylow Theorem for Finite Groups 176
10.3.3 Proof of the 2 Sylow Theorem 177
10.3.4 Sylow 2 Subgroups of Algebraic Groups 182
10.4 Baer Suzuki Theorem 183
10.5 Strongly Embedded Subgroups 185
10.6 Conjugacy Families and Conjugacy Functors 187
10.6.1 Fusion 187
10.6.2 Alperin Goldschmidt Theorem 188
10.6.3 Conjugacy Families 190
10.6.4 Every Inductive Family Is a Conjugation Family 192
10.6.5 Proof of Alperin Goldschmidt Theorem 195
11 Permutation Groups 198
11.1 Clifford s Theorem 200
11.2 Frobenius Groups 203
11.2.1 Generalities and Questions 204
11.2.2 Definability 206
11.2.3 When T is Finite 209
11.2.4 Solvable Frobenius Groups 211
11.2.5 Minimal Counterexamples 215
11.3 Split Doubly Transitive Groups 219
11.4 Sharply 2 Transitive Groups 222
11.4.1 Generalities 222
11.4.2 Connectedness of H 231
11.4.3 Some Easy Classification Results 232
11.4.4 A Few Rank Inequalities 234
11.4.5 When H Is Nilpotent 236
11.4.6 A Few Interesting Facts 237
11.5 Zassenhaus Groups 238
11.5.1 Generalities 238
11.5.2 Definability of the Action 240
11.5.3 Sharply 3 Transitive Groups 244
11.5.4 When T Has an Involution 245
11.5.5 When U Has a Central Involution 245
11.5.6 Discussion 250
11.6 Groups Acting on Strongly Minimal Sets 251
12 Geometries 255
12.1 Generalized n Gons 255
12.1.1 Generalities 255
12.1.2 Ranked Generalized n Gons 259
12.2 Sharply Flag Transitive Groups 265
xvi Contents
12.2.1 ABA Groups 265
12.2.2 When G Is Solvable of Finite Morley Rank 268
12.3 fl/V Pairs of Tits Rank 3 271
12.3.1 Generalities 271
12.3.2 fi/V Pairs 272
12.3.3 Buildings 277
12.3.4 BTV Pairs of Finite Morley Rank 281
13 Bad Groups 287
13.1 Bad Groups 287
13.2 Bad Groups of Morley Rank 3 294
14 CN and CIT Groups 300
14.1 CN Groups 300
14.1.1 Proof of Theorem 14.2 301
14.1.2 Proof of Theorem 14.1 311
14.2 CIT Groups 312
14.2.1 Introduction 312
14.2.2 Solvable CIT Groups 313
14.2.3 Three Propositions 315
14.2.4 When G Has Infinite Disjoint Sylow 2 subgroups 317
14.2.5 Higman s Theorem 317
14.2.6 2 Local Subgroups 323
14.2.7 Classification of Simple CIT Groups 325
14.2.8 Nonsimple CIT Groups 330
A Miscellaneous Results 331
A.I Rings 331
A. 1.1 Definitions and Generalities 331
A. 1.2 Generalities on Rings of Finite Morley Rank 333
A. 1.3 Semisimple Rings of Finite Morley Rank 337
A. 1.4 Lie Algebras 340
A. 1.5 Alternative Division Rings 341
A.2 Irreducible G Modules 343
A.3 Linear Groups of Finite Morley Rank 346
A.4 Generic Normal Subgroups 347
B Open Problems 352
B.I The Four Key Problems 352
B.2 Outline of a Possible Classification 354
B.2.1 2 Sylow Theory 354
B.2.2 Characterizations of the Groups PSL2 (K) 356
Contents xvii
B.2.3 Characteristic 2 Type Groups 359
B.2.4 Odd Type Groups 359
B.2.5 Groups of Mixed Type 363
B.3 Auxiliary Questions 363
B.3.1 Automorphisms of Fields 364
B.3.2 Suzuki 2 Groups 364
B.3.3 Algebraic Groups 365
B.4 Other Problems 367
B.4.1 Different Classes of Groups 368
B.4.2 Permutation Groups and Geometries 369
B.5 Some Model Theoretical Properties 371
C Link with Model Theory 372
C. 1 Ranked Groups Have Finite Morley Rank 372
C.2 Rank on Elementary Extensions 374
D Hints to the Exercises 378
Bibliography 384
Index 399
|
| any_adam_object | 1 |
| author | Borovik, Alexandre Nesin, Ali 1956- |
| author_GND | (DE-588)119247135 |
| author_facet | Borovik, Alexandre Nesin, Ali 1956- |
| author_role | aut aut |
| author_sort | Borovik, Alexandre |
| author_variant | a b ab a n an |
| building | Verbundindex |
| bvnumber | BV010006760 |
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| classification_tum | MAT 203f |
| ctrlnum | (OCoLC)832366916 (DE-599)BVBBV010006760 |
| dewey-full | 512.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.2 |
| dewey-search | 512.2 |
| dewey-sort | 3512.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:44:49Z |
| institution | BVB |
| isbn | 0198534450 |
| language | English |
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| physical | XVII, 409 S. |
| publishDate | 1994 |
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| series | Oxford logic guides |
| series2 | Oxford logic guides |
| spelling | Borovik, Alexandre Verfasser aut Groups of finite Morley rank Alexandre Borovik and Ali Nesin Oxford Clarendon Press 1994 XVII, 409 S. txt rdacontent n rdamedia nc rdacarrier Oxford logic guides 26 Endliche Gruppe (DE-588)4014651-0 gnd rswk-swf Endliche Gruppe (DE-588)4014651-0 s DE-604 Nesin, Ali 1956- Verfasser (DE-588)119247135 aut Oxford logic guides 26 (DE-604)BV000013997 26 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006634168&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Borovik, Alexandre Nesin, Ali 1956- Groups of finite Morley rank Oxford logic guides Endliche Gruppe (DE-588)4014651-0 gnd |
| subject_GND | (DE-588)4014651-0 |
| title | Groups of finite Morley rank |
| title_auth | Groups of finite Morley rank |
| title_exact_search | Groups of finite Morley rank |
| title_full | Groups of finite Morley rank Alexandre Borovik and Ali Nesin |
| title_fullStr | Groups of finite Morley rank Alexandre Borovik and Ali Nesin |
| title_full_unstemmed | Groups of finite Morley rank Alexandre Borovik and Ali Nesin |
| title_short | Groups of finite Morley rank |
| title_sort | groups of finite morley rank |
| topic | Endliche Gruppe (DE-588)4014651-0 gnd |
| topic_facet | Endliche Gruppe |
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