Vol 230, No 3 (2018)

In this paper, we consider the question on determination of an Abelian group by its multiplication group.

In this paper, we consider the question of determinability of an Abelian group by its automorphism group in the class of completely decomposable torsion-free Abelian groups whose direct summands of rank 1 have idempotent types.

Torsion-free quotient divisible groups were introduced by R. Beaumont and R. Pierce in 1961. In 1998, A. A. Fomin and W. Wickless defined quotient divisible mixed groups and proved that categories of mixed quotient divisible groups and finite-rank torsion-free groups with quasihomomorphisms as morphisms are dual. In studying finite-rank torsion-free groups, rank-1 torsion-free groups play an important role, as many problems that are being solved in this class arise from them. This paper is devoted to the study of the homomorphism groups of rank-1 quotient divisible groups. The main achieved result is a full description of the homomorphism group in the language of characteristics.

This paper describes the chief factors of the commutator subgroups of finite nonsolvable groups G with disconnected Gruenberg–Kegel graph having exactly 5 vertices in the case where G/F(G) is an almost simple n-primary group for n ≤ 4.

The concept of the determinant of a generalized matrix of order 2 is introduced, and some properties of such determinants are presented.

A ring K is said to be a unique addition ring (UA-ring) if on its multiplicative semigroup (K, ·) it is possible to set only one binary operation of + turning (K, ·, +) into a ring. We call an Abelian group an End-UA-group if its endomorphism ring is a UA-ring. In this paper, End-UA-groups are found in a class of algebraically compact Abelian groups.

In this work, some aspects of cleanness of endomorphism rings of SP-groups are considered. These groups form one of the classes of mixed Abelian groups. The cleanness of endomorphism rings of self-small SP-groups is proved. Some sufficient conditions are found for the converse proposition to hold.

Some basic theorems on quotient divisible Abelian groups are proved.

We consider various ways of characterizing the additive groups of T-rings.

Abelian groups with commutative commutators of endomorphisms are studied. The structure of these groups is described among torsion, completely decomposable, coperiodic, and split mixed groups.