有限群理论以在论述上简明、但在论证上简单而引人注目。并且以基础的方式应用于数学的多个分支，例如数论。
本书给出了有限群简明、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。
- 目录
- Preface
- Conventions and Notation
- 1 Preliminaries
- 1.1 Group actions
- 1.2 Normal subgroups, automorphisms, characteristic subgroups, simple groups
- 1.3 Filtrations and Jordan-Hölder 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 Frattini subgroup
- 3.10 Characterizations using subgroups generated by two elements
- 3.11 Exercises
- 4 Group extensions
- 4.1 Cohomology 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 of relatively prime orders
- 4.7 Liftings of homomorphisms
- 4.8 Application to p-adic liftings
- 4.9 Exercises
- 5 Hall subgroups
- 5.1 π-subgroups
- 5.2 Preliminaries: permutable subgroups
- 5.3 Permutable families of Sylow subgroups
- 5.4 Proof of theorem 5.1
- 5.5 Sylow-like properties of the π-subgroups
- 5.6 A solvability criterion
- 5.7 Proof of theorem 5.3
- 6 Frobenius groups
- 6.1 Union of conjugates of a subgroup
- 6.2 An improvement of Jordan’s theorem
- 6.3 Frobenius groups: definition
- 6.4 Frobenius 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
- 10 Small Groups
- 10.1 Small groups and their isomorphisms
- 10.2 Embeddings of A4, S4 and A5 in PGL2(Fq)
- Bibliography
- Index of names
