Linear Algebraic Groups and Finite Groups of Lie Type

✍ Scribed by Gunter Malle, Donna Testerman

• Publisher: Cambridge University Press
• Year: 2011
• Tongue: English
• Leaves: 325
• Series: Cambridge Studies in Advanced Mathematics 133
• Edition: 1
• Category: Library

✦ Synopsis

Originating from a summer school taught by the authors, this concise treatment includes many of the main results in the area. An introductory chapter describes the fundamental results on linear algebraic groups, culminating in the classification of semisimple groups. The second chapter introduces more specialized topics in the subgroup structure of semisimple groups, and describes the classification of the maximal subgroups of the simple algebraic groups. The authors then systematically develop the subgroup structure of finite groups of Lie type as a consequence of the structural results on algebraic groups. This approach will help students to understand the relationship between these two classes of groups. The book covers many topics that are central to the subject, but missing from existing textbooks. The authors provide numerous instructive exercises and examples for those who are learning the subject as well as more advanced topics for research students working in related areas.

✦ Table of Contents

Cover......Page 1
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
Tables......Page 15
Notation......Page 16
PART I LINEAR ALGEBRAIC GROUPS......Page 17
1.1 Linear algebraic groups and morphisms......Page 19
1.2 Examples of algebraic groups......Page 22
1.3 Connectedness......Page 25
1.4 Dimension......Page 29
2.1 Decomposition of endomorphisms......Page 31
2.2 Unipotent groups......Page 34
3.1 Jordan decomposition of commutative groups......Page 36
3.2 Tori, characters and cocharacters......Page 38
4.1 The Lie–Kolchin theorem......Page 42
4.2 Structure of connected solvable groups......Page 43
5.1 Actions of algebraic groups......Page 46
5.2 Existence of rational representations......Page 49
6.1 The Borel fixed point theorem......Page 52
6.2 Properties of Borel subgroups......Page 55
7.1 Derivations and differentials......Page 60
7.2 The adjoint representation......Page 65
8.1 Root space decomposition......Page 67
8.2 Semisimple groups of rank 1......Page 69
8.3 Structure of connected reductive groups......Page 73
8.4 Structure of semisimple groups......Page 75
9.1 Root systems......Page 79
9.2 The classification theorem of Chevalley......Page 84
10 Exercises for Part I......Page 90
PART II SUBGROUP STRUCTURE AND REPRESENTATION THEORY OF SEMISIMPLE ALGEBRAIC GROUPS......Page 97
11.1 On the structure of B......Page 99
11.2 Bruhat decomposition......Page 106
12.1 Parabolic subgroups......Page 111
12.2 Levi decomposition......Page 114
13.1 Subsystem subgroups......Page 120
13.2 The algorithm of Borel and de Siebenthal......Page 123
14.1 Semisimple elements......Page 128
14.2 Connectedness of centralizers......Page 132
15.1 Weight theory......Page 137
15.2 Irreducible highest weight modules......Page 141
16.1 Dual modules and restrictions to Levi subgroups......Page 147
16.2 Steinberg’s tensor product theorem......Page 150
17.1 Internal modules......Page 156
17.2 The theorem of Borel and Tits......Page 161
18.1 A reduction theorem......Page 165
18.2 Maximal subgroups of the classical algebraic groups......Page 171
19.1 Statement of the result......Page 182
19.2 Indications on the proof......Page 184
20 Exercises for Part II......Page 188
PART III FINITE GROUPS OF LIE TYPE......Page 195
21.1 Endomorphisms of linear algebraic groups......Page 197
21.2 The theorem of Lang–Steinberg......Page 200
22.1 Steinberg endomorphisms......Page 204
22.2 The finite groups......Page 209
23.1 The root system......Page 213
23.2 Root subgroups......Page 216
24.1 Bruhat decomposition and the order formula......Page 219
24.2 BN-pair, simplicity and automorphisms......Page 225
25.1 F-stable tori......Page 234
25.2 Sylow subgroups......Page 241
26.1 Parabolic subgroups and Levi subgroups......Page 245
26.2 Semisimple conjugacy classes......Page 248
27 Maximal subgroups of finite classical groups......Page 252
27.1 The theorem of Liebeck and Seitz......Page 253
27.2 The theorem of Aschbacher......Page 256
28.1 Structure and maximality of groups in CFi......Page 260
28.2 On the class S......Page 262
29.1 Maximal subgroups......Page 266
29.2 Lifting result......Page 270
30 Exercises for Part III......Page 279
A.1 Bases and positive systems......Page 284
A.2 Decomposition of root systems......Page 288
A.3 The length function......Page 292
A.4 Parabolic subgroups......Page 294
Exercises......Page 297
B.1 The highest root......Page 298
B.2 The affine Weyl group......Page 301
B.3 Closed subsystems......Page 302
B.4 Other subsystems......Page 306
B.5 Bad primes and torsion primes......Page 308
Exercises......Page 312
Appendix C Automorphisms of root systems......Page 313
Exercises......Page 316
References......Page 317
Index......Page 321