Finite Group Theory

Couverture
Cambridge University Press, 26 juin 2000 - 304 pages
During the last 40 years the theory of finite groups has developed dramatically. The finite simple groups have been classified and are becoming better understood. Tools exist to reduce many questions about arbitrary finite groups to similar questions about simple groups. Since the classification there have been numerous applications of this theory in other branches of mathematics. Finite Group Theory develops the foundations of the theory of finite groups. It can serve as a text for a course on finite groups for students already exposed to a first course in algebra. It could supply the background necessary to begin reading journal articles in the field. For specialists it also provides a reference on the foundations of the subject. This second edition has been considerably improved with a completely rewritten Chapter 15 considering the 2-Signalizer Functor Theorem, and the addition of an appendix containing solutions to exercises.
 

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Table des matières

Preliminary results
1
Categories
6
Graphs and geometries
7
Abstract representations
9
Permutation representations
13
Sylows Theorem
19
Representations of groups on groups
22
Characteristic subgroups and commutators
25
Coxeter groups
141
Root systems
148
The generalized Fitting subgroup
156
The generalized Fitting subgroup
157
Thompson factorization
162
Central extensions
166
Linear representations of finite groups
177
Characters in coprime characteristic
178

Solvable and nilpotent groups
27
Semidirect products
29
Central products and wreath products
32
Linear representations
35
The general linear group and special linear group
42
The dual representation
46
Permutation groups
53
Rank 3 permutation groups
59
Extensions of groups and modules
64
Coprime action
70
Spaces with forms
75
Witts Lemma
81
Spaces over finite fields
85
The classical groups
88
pgroups
105
Coprime action on pgroups
112
Change of field of a linear representation
117
Representations over finite fields
123
Minimal polynomials
127
Presentations of groups
138
Characters in characteristic O
181
Some special actions
192
Transfer and fusion
197
Alperins Fusion Theorem
200
Normal pcomplements
202
Semiregular action
205
The geometry of groups of Lie type
209
Buildings
215
BNpairs and Tits systems
218
Signalizer functors
229
Finite simple groups
242
Involutions in finite groups
243
Connected groups
245
The finite simple groups
249
An outline of the Classification Theorem
260
Appendix
269
References
297
List of symbols
299
Index
301

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