Topics in Finite Groups

1 Introduction
1.1 Preliminary Definitions
1.2 Examples of Groups
1.3 Some Basic Group Concepts and Terminology
1.4 Cyclic Groups
1.5 Direct and Semidirect Product of Groups
1.5.1 Direct Product
1.5.2 Examples of Direct Product
1.5.3 Semidirect Product
1.5.4 Examples of Semidirect Product
1.6 Isomorphism, Homomorphism, and Automorphism
1.6.1 Isomorphism and Homomorphism
1.6.2 Examples of Isomorphic Groups
1.6.3 Homomorphism and Kernel of Homomorphism
1.6.4 Examples of Quotient Groups
1.6.5 Homomorphism/Isomorphism Theorems
1.6.6 Abelian p-groups
1.6.7 Composition Series of Finite Groups
1.6.8 Jordan–Hölder Theorem
1.6.9 Automorphism group of a group
1.6.10 The Group Z∗n

2 Permutation Groups and Sylow Theorems
2.1 Preliminaries
2.2 Multiply Transitive Groups
2.3 Regular Normal Subgroups and Multiple Transitivity
2.4 Sylow Theorems I
2.5 Sylow Theorems II
2.6 Nilpotent and Solvable Groups

3 Representations and Characters
3.1 Representations
3.2 Characters
3.3 Tensor Product of Two Matrices
3.4 Product of Two Characters
3.5 Induced Representations and Induced Characters
3.6 Burnside’s paqb-theorem

4 Solvable Groups
4.1 Generalization of Sylow Theorems
4.2 Schur–Zassenhaus Theorem

5 Frobenius Groups
5.1 Introduction
5.2 Fixed-Point-Free Automorphisms
5.3 Characters of Some Finite Groups

Appendix 1. Nonabelian Groups of Order pq (p > q are Prime Numbers)
Appendix 2. Nonabelian Groups of Order p3, p an Odd Prime
Appendix 3. A Group Having a Class Preserving Outer Automorphism
Appendix 4. Outer Automorphism of S6

Problems
Hints or Solutions to Problems
Glossary
Credits and Sources Acknowledged
Bibliography
Index