## Exercises in group theory

 E S Lyapin, A Ya Aizenshtat and M M Lesokhin published Exercises in group theory (Russian) in 1967. Eugene Schenkman, himself the author of a famous group theory textbook, has given the following translation of the chapter and section headings:

I. Sets
1.2 Introductory notions
1.2 Mappings
1.3 Binary relations
1.4 Multiplication

II. General algebraic operations
2.1 The notion of algebraic operation
2.2 Basic properties of operations
2.3 Multiplication of subsets
2.4 Homomorphisms
2.5 Semigroups
2.6 Beginning notions of group theory

III. Composition of maps
3.1 General properties
3.2 Maps with inverses
3.3 Maps with inverses on finite sets
3.4 Endomorphisms
3.5 Groups of motions
3.6 Some particular map

IV. Groups and subgroups
4.1 Coset decomposition
4.2 Conjugacy
4.3 Normal subgroups and factor groups
4.4 Subgroups of finite groups
4.5 Commutators
4.6 Solvable groups
4.7 Nilpotent groups
4.8 Automorphisms of groups
4.9 Transitive permutation groups

V. Defining sets of relations
5.1 In semigroups
5.2 In groups
5.3 Free groups
5.4 Groups given by defining sets of relations
5.5 Free products of groups
5.6 Direct products of groups

VI. Abelian groups
6.1 Simplest properties
6.2 Finite abelian groups
6.3 Finitely generated abelian groups
6.4 Infinite abelian groups

VII. Group representations
7.1 Of general type
7.2 Of transformation groups
7.3 Of matrices
7.4 Groups of homomorphisms of abelian groups
7.5 Characters of groups

VIII. Topological and ordered groups
8.1 Metric spaces
8.2 Groups of continuous transformations of metric space
8.3 Topological spaces
8.4 Topological groups
8.5 Ordered groups.

JOC/EFR August 2007