# Book:Derek J.S. Robinson/A Course in the Theory of Groups/Second Edition

## Derek J.S. Robinson: *A Course in the Theory of Groups (2nd Edition)*

Published $1996$, **Springer**

- ISBN 0-387-94461-3.

### Subject Matter

### Contents

- Preface to the Second Edition

- Preface to the First Edition

- Notation

- CHAPTER 1: Fundamental Concepts of Group Theory
- 1.1. Binary Operations, Semigroups, and Groups
- 1.2. Examples of Groups
- 1.3. Subgroups and Cosets
- 1.4. Homomorphisms and Quotient Groups
- 1.5. Endomorphisms and Automorphisms
- 1.6. Permutation Groups and Group Actions

- CHAPTER 2: Free Groups and Presentations
- 2.1. Free Groups
- 2.2. Presentations of Groups
- 2.3. Varieties of Groups

- CHAPTER 3: Decompositions of a Group
- 3.1. Series and Composition Series
- 3.2. Some Simple Groups
- 3.3. Direct Decompositions

- CHAPTER 4: Abelian Groups
- 4.1. Torsion Groups and Divisible Groups
- 4.2. Direct Sums of Cyclic and Quasicyclic Groups
- 4.3. Pure Subgroups and p-Groups
- 4.4. Torsion-Free Groups

- CHAPTER 5: Soluble and Nilpotent Groups
- 5.1. Abelian and Central Series
- 5.2. Nilpotent Groups
- 5.3. Groups of Prime-Power Order
- 5.4. Soluble Groups

- CHAPTER 6: Free Groups and Free Products
- 6.1. Further Properties of Free Groups
- 6.2. Free Products of Groups
- 6.3. Subgroups of Free Products
- 6.4. Generalized Free Products

- CHAPTER 7: Finite Permutation Groups
- 7.1. Multiple Transitivity
- 7.2. Primitive Permutation Groups
- 7.3. Classification of Sharply /c-Transitive Permutation Groups
- 7.4. The Mathieu Groups

- CHAPTER 8: Representations of Groups
- 8.1. Representations and Modules
- 8.2. Structure of the Group Algebra
- 8.3. Characters
- 8.4. Tensor Products and Representations
- 8.5. Applications to Finite Groups

- CHAPTER 9: Finite Soluble Groups
- 9.1. Hall π-Subgroups
- 9.2. Sylow Systems and System Normalizers
- 9.3. p-Soluble Groups
- 9.4. Supersoluble Groups
- 9.5. Formations

- CHAPTER 10: The Transfer and Its Applications
- 10.1. The Transfer Homomorphism
- 10.2. Gain's Theorems
- 10.3. Frobenius's Criterion for p-Nilpotence
- 10.4. Thompson's Criterion for p-Nilpotence

- 10.5. Fixed-Point-Free Automorphisms

- CHAPTER 11: The Theory of Group Extensions
- 11.1. Group Extensions and Covering Groups
- 11.2. Homology Groups and Cohomology Groups
- 11.3. The Gruenberg Resolution
- 11.4. Group-Theoretic Interpretations of the (Co)homology Groups

- CHAPTER 12: Generalizations of Nilpotent and Soluble Groups
- 12.1. Locally Nilpotent Groups
- 12.2. Some Special Types of Locally Nilpotent Groups
- 12.3. Engel Elements and Engel Groups
- 12.4. Classes of Groups Defined by General Series
- 12.5. Locally Soluble Groups

- CHAPTER 13: Subnormal Subgroups
- 13.1. Joins and Intersections of Subnormal Subgroups
- 13.2. Permutability and Subnormality
- 13.3. The Minimal Condition on Subnormal Subgroups
- 13.4. Groups in Which Normality Is a Transitive Relation
- 13.5. Automorphism Towers and Complete Groups

- CHAPTER 14: Finiteness Properties
- 14.1. Finitely Generated Groups and Finitely Presented Groups
- 14.2. Torsion Groups and the Burnside Problems
- 14.3. Locally Finite Groups
- 14.4. 2-Groups with the Maximal or Minimal Condition
- 14.5. Finiteness Properties of Conjugates and Commutators

- CHAPTER 15: Infinite Soluble Groups
- 15.1. Soluble Linear Groups
- 15.2. Soluble Groups with Finiteness Conditions on Abelian Subgroups
- 15.3. Finitely Generated Soluble Groups and the Maximal Condition on Normal Subgroups
- 15.4. Finitely Generated Soluble Groups and Residual Finiteness
- 15.5. Finitely Generated Soluble Groups and Their Frattini Subgroups

- Bibliography

- Index