Finite Group TheoryCambridge University Press, 26 juin 2000 - 304 pages This second edition develops the foundations of finite group theory. For students already exposed to a first course in algebra, it serves as a text for a course on finite groups. For the reader with some mathematical sophistication but limited knowledge of finite group theory, the book supplies the basic background necessary to begin to read journal articles in the field. It also provides the specialist in finite group theory with a reference on the foundations of the subject. Unifying themes include the Classification Theorem and the classical linear groups. Lie theory appears in chapters on Coxeter groups, root systems, buildings, and Tits systems. 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. |
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 |
| 301 | |
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Expressions et termes fréquents
A-invariant algebra assume G basis central extension chapter char(F character classical groups complement completing the proof conjugacy classes conjugation contained contradiction coset Coxeter groups Coxeter system cyclic define elementary abelian equivalent Exercise exists F-algebra F-space FG-module FG-representation finite group finite simple groups Frobenius Functor Functor Theorem G₁ geometry GL(V group G groups of Lie Hence holds homomorphism hyperbolic hypothesis induction integer involution irreducible isometry isomorphism lemma Let G Let H Lie type linear matrix maximal minimality of G module morphism nilpotent nondegenerate nonsingular nontrivial normal p-complement normal subgroup orbits orthogonal space p-group permutation representation prime Prove remains to show representation of G Schur multiplier semidirect product semisimple Signalizer Functor solvable space over F subgroup of G subset subspace surjective Sylow p-subgroup symplectic totally singular transvection unique V₁ vector space Weyl group