Linear algebraic groups and finite groups of Lie type:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2011
|
| Edition: | 1. publ. |
| Series: | Cambridge studies in advanced mathematics
133 |
| Subjects: | |
| Online Access: | Klappentext Inhaltsverzeichnis |
| Physical Description: | XII, 309 S. graph. Darst. |
| ISBN: | 1107008549 9781107008540 |
Staff View
MARC
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| 245 | 1 | 0 | |a Linear algebraic groups and finite groups of Lie type |c Gunter Malle ; Donna Testerman |
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| 300 | |a XII, 309 S. |b graph. Darst. | ||
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Record in the Search Index
| _version_ | 1804148365727891456 |
|---|---|
| adam_text | Contents
Preface
page ix
List of tables
xiii
Notation
xiv
PART I LINEAR ALGEBRAIC GROUPS
1
Basic concepts
3
1.1
Linear algebraic groups and morphisms
3
1.2
Examples of algebraic groups
6
1.3
Connectedness
9
1.4
Dimension
13
Jordan decomposition
15
2.1
Decomposition of endomorphisms
15
2.2
Unipotent groups
18
Commutative linear algebraic groups
20
3.1
Jordan decomposition of commutative groups
20
3.2
Tori, characters and cocharacters
22
Connected solvable groups
26
4.1
The Lie-Kolchin theorem
26
4.2
Structure of connected solvable groups
27
G-spaces and quotients
30
5.1
Actions of algebraic groups
30
5.2
Existence of rational representations
33
Borei
subgroups
36
6.1
The
Borei
fixed point theorem
36
6.2
Properties of
Borei
subgroups
39
vi
Contents
7
The Lie algebra of a linear algebraic group
44
7.1
Derivations and differentials
44
7.2
The adjoint representation
49
8
Structure of reductive groups
51
8.1
Root space decomposition
51
8.2
Semisimple groups of rank
1 53
8.3
Structure of connected reductive groups
57
8.4
Structure of semisimple groups
59
9
The classification of semisimple algebraic groups
63
9.1
Root systems
63
9.2
The classification theorem of Chevalley
68
10
Exercises for Part I
74
PART II SUBGROUP STRUCTURE AND
REPRESENTATION THEORY OF
SEMISIMPLE ALGEBRAIC GROUPS
81
11
BN-pairs and Bruhat decomposition
83
11.1
On the structure of
В
83
11.2
Bruhat decomposition
90
12
Structure of parabolic subgroups, I
95
12.1
Parabolic subgroups
95
12.2
Levi
decomposition
98
13
Subgroups of maximal rank
104
13.1
Subsystem subgroups
104
13.2
The algorithm of
Borei
and
de Siebenthal 107
14
Centralizers and conjugacy classes
112
14.1
Semisimple elements
112
14.2
Connectedness of centralizers
116
15
Representations of algebraic groups
121
15.1
Weight theory
121
15.2
Irreducible highest weight modules
125
16
Representation theory and maximal subgroups
131
16.1
Dual modules and restrictions to
Levi
subgroups
131
16.2
Steinberg s tensor product theorem
134
Contents
vii
17
Structure
of parabolic subgroups, II
140
17.1
Internal modules
140
17.2
The theorem of
Borei
and Tits
145
18
Maximal subgroups of classical type simple algebraic
groups
149
18.1
A reduction theorem
149
18.2
Maximal subgroups of the classical algebraic groups
155
19
Maximal subgroups of exceptional type algebraic groups
166
19.1
Statement of the result
166
19.2
Indications on the proof
168
20
Exercises for Part II
172
PART III FINITE GROUPS OF LIE TYPE
179
21
Steinberg endomorphisms
181
21.1
Endomorphisms of linear algebraic groups
181
21.2
The theorem of Lang-Steinberg
184
22
Classification of finite groups of Lie type
188
22.1
Steinberg endomorphisms
188
22.2
The finite groups GF
193
23
Weyl group, root system and root subgroups
197
23.1
The root system
197
23.2
Root subgroups
200
24
A BN-pair for GF
203
24.1
Bruhat decomposition and the order formula
203
24.2
BN-pair, simplicity and automorphisms
209
25
Tori and Sylow subgroups
218
25.1
F-stable tori
218
25.2
Sylow subgroups
225
26
Subgroups of maximal rank
229
26.1
Parabolic subgroups and
Levi
subgroups
229
26.2 Semisimple conjugacy
classes
232
27
Maximal subgroups of finite classical groups
236
27.1
The theorem of Liebeck and Seitz
237
27.2
The theorem of
Aschbacher 240
viii Contents
28
About the classes C[
,...
,Cf and <S
244
28.1
Structure and maximality of groups in
C f
244
28.2
On the class
S
246
29
Exceptional groups of Lie type
250
29.1
Maximal subgroups
250
29.2
Lifting result
254
30
Exercises for Part III
263
Appendix A Root systems
268
A.I Bases and positive systems
268
A.
2
Decomposition of root systems
272
A.3 The length function
276
A.
4
Parabolic subgroups
278
Exercises
281
Appendix
В
Subsystems
282
B.I The highest root
282
B.2 The
affine Weyl
group
285
B.3 Closed subsystems
286
B.4 Other subsystems
290
B.5 Bad primes and torsion primes
292
Exercises
296
Appendix
С
Automorphisms of root systems
297
Exercises
300
References
301
Index
305
у
tne autnors, tnis
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 ofgroups.
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.
Cambridge Studies in Advanced Mathematics
EDITOR] U B<i RD
Béla Bollobás University
of Memphis
William Fulton University of Michigan
Anatole Katok
Pennsylvania State University
francés
Kirvvan University of Oxford
Peter
Sárnak
Princeton
University
Barry Simon California
Insuline
of Technology
Burt
Totaro
University of Cambridge
Cambridge
UNIVERSITY PRESS
w w
w.cambridge.
οι
g
8-1-107-00854-0
ιΌ08540
|
| any_adam_object | 1 |
| author | Malle, Gunter 1960- Testerman, Donna 1960- |
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| classification_tum | MAT 225f MAT 204f |
| ctrlnum | (OCoLC)750950639 (DE-599)GBV656847506 |
| dewey-full | 512.482 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.482 |
| dewey-search | 512.482 |
| dewey-sort | 3512.482 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| spelling | Malle, Gunter 1960- Verfasser (DE-588)1016705093 aut Linear algebraic groups and finite groups of Lie type Gunter Malle ; Donna Testerman 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2011 XII, 309 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 133 Lineare algebraische Gruppe (DE-588)4295326-1 gnd rswk-swf Endliche Gruppe (DE-588)4014651-0 gnd rswk-swf Lineare algebraische Gruppe (DE-588)4295326-1 s Endliche Gruppe (DE-588)4014651-0 s b DE-604 Testerman, Donna 1960- Verfasser (DE-588)143763016 aut Cambridge studies in advanced mathematics 133 (DE-604)BV000003678 133 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024399173&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024399173&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Malle, Gunter 1960- Testerman, Donna 1960- Linear algebraic groups and finite groups of Lie type Cambridge studies in advanced mathematics Lineare algebraische Gruppe (DE-588)4295326-1 gnd Endliche Gruppe (DE-588)4014651-0 gnd |
| subject_GND | (DE-588)4295326-1 (DE-588)4014651-0 |
| title | Linear algebraic groups and finite groups of Lie type |
| title_auth | Linear algebraic groups and finite groups of Lie type |
| title_exact_search | Linear algebraic groups and finite groups of Lie type |
| title_full | Linear algebraic groups and finite groups of Lie type Gunter Malle ; Donna Testerman |
| title_fullStr | Linear algebraic groups and finite groups of Lie type Gunter Malle ; Donna Testerman |
| title_full_unstemmed | Linear algebraic groups and finite groups of Lie type Gunter Malle ; Donna Testerman |
| title_short | Linear algebraic groups and finite groups of Lie type |
| title_sort | linear algebraic groups and finite groups of lie type |
| topic | Lineare algebraische Gruppe (DE-588)4295326-1 gnd Endliche Gruppe (DE-588)4014651-0 gnd |
| topic_facet | Lineare algebraische Gruppe Endliche Gruppe |
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