有限群导引（英文版）
作者： Jean-Pierre Serre
出版时间：2016年4月
出版社：高等教育出版社
- 高等教育出版社
- 9787040446418
- 1版
- 141165
- 0045175727-2
- 16开
- 2016年4月
- 230
- 181
- 理学
- 数学
- O152.1
- 数学类
- 研究生
《有限群导引》给出了有限群简明、基础的介绍，以最大限度地服务初学者和数学家。本书共10章，每章都配备了一系列的练习。
让—皮埃尔·塞尔(Jean—Pierre Serre)，享有盛誉的法国著名数学家，主要的学术贡献领域是拓扑学、代数几何与数论。他曾获得多项重要的数学奖项，包括1954年的菲尔兹奖、2000年的沃尔夫数学奖与2003年的阿贝尔奖。他被公认为是在数学写作方面世界上最好的数学家之一。
Preface
Conventions and Notation
1 Preliminaries
1.1 Group actions
1.2 Normal subgroups, automorphisms, characteristic subgroups, simple groups
1.3 Filtrations and Jordan-HSlder theorem
1.4 Subgroups of products: Goursat's lemma and Ribet's lemma
1.5 Exercises
2 Sylow theorems
2.1 Definitions
2.2 Existence of p-Sylow subgroups
2.3 Properties of the p-Sylow subgroups
2.4 Fusion in the normalizer of a p-Sylow subgroup
2.5 Local conjugation and Alperin's theorem
2.6 Other Sylow-like theories
2.7 Exercises
3 Solvable groups and nilpotent groups
3.1 Commutators and abelianization
3.2 Solvable groups
3.3 Descending central series and nilpotent groups
3.4 Nilpotent groups and Lie algebras
3.5 Kolchin's theorem
3.6 Finite nilpotent groups
3.7 Applications of 2-groups to field theory
3.8 Abelian groups
3.9 The Prattini subgroup
3.10 Characterizations using subgroups generated by two elements
3.11 Exercises
4 Group extensions
4.1 Cchomology groups
4.2 A vanishing criterion for the cohomology of finite groups
4.3 Extensions,'sections and semidirect products
4.4 Extensions with abelian kernel
4.5 Extensions with arbitrary kernel
4.6 Extensions of groups cf relatively prime orders
4.7 Liftings of homomorphisms
4.8 Application to p-adic liftings
4.9 Exercises
5 Hall subgroups
5.1 п-subgrcups
5.2 Preliminaries: permutable subgroups
5.3 Permutable families of Sylow subgroups
5.4 Proof cf theorem 5.1
5.5 Sylow-like properties of the п-subgroups
5.6 A solvability criterion
5.7 Proof of theorem 5.3
5.8 Exercises
6 Frobenius groups
6.1 Union of conjugates of a subgrcup
6.2 An improvement of Jordan's theorem
6.3 Frobenius groups: definition
6.4 Probenius kernels
6.5 Frobenius complements
6.6 Exercises
7 Transfer
7.1 Definition of Ver : Gab → Hab
7.2 Computation of the transfer
7.3 A two-century-old example of transfer: Gauss lemma
7.4 An application of transfer to infinite groups
7.5 Transfer applied to Sylow subgroups
7.6 Application: groups of odd order < 2000
7.7 Application: simple groups of order ≤ 200
7.8 The use of transfer outside group theory
7.9 Exercises
8 Characters
8.1 Linear representations and characters
8.2 Characters, hermitian forms and irreducible representations
8.3 Schur's lemma
8.4 Orthogonality relations
8.5 Structure of the group algebra and of its center
8.6 Integrality properties
8.7 Galois properties of characters
8.8 The ring R(G)
8.9 Realizing representations over a subfield of C, for instance the field R
8.10 Application of character theory: proof of Frobenius's theorem 6.7
8.11 Application of character theory: proof of Burnside's theorem 5.4
8.12 The character table of A5
8.13 Exercises
9 Finite subgroups of GLn
9.1 Minkowski's theorem on the finite subgroups of GLn(Q)
9.2 Jordan's theorem on the finite subgroups of GLn(C)
9.3 Exercises
