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 $\text {1996}$, Springer

ISBN 0-387-94461-3

Subject Matter

• Group Theory

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