# Dashkova O. Yu.

### Locally soluble AFA-groups

Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 459-469

Let $A$ be an $\mathbf{R}G$-module, where $\mathbf{R}$ is a ring, $G$ is a locally solvable group, $C_G (A) = 1$, and each proper subgroup $H$ of $G$ for which $A/C_A(H)$ is not an Artinian $\mathbf{R}$-module is finitely generated. It is proved that a locally solvable group $G$ that satisfies these conditions is hyperabelian if R is a Dedekind ring. We describe the structure of $G$ in the case where $G$ is a finitely generated solvable group, $A/C_A(H)$ is not an Artinian $\mathbf{R}$-module and $\mathbf{R}$ is a Dedekind ring.

### On modules over group rings of nilpotent groups

Ukr. Mat. Zh. - 2012. - 64, № 1. - pp. 13-23

We study an $\mathbf{R}G$-module $A$, where $\mathbf{R}$ is a ring, $A/C_A(G)$ is not a minimax $\mathbf{R}$-module, $C_A(G) = 1$, and $G$ is a nilpotent group. Let $\mathfrak{L}_{nm}(G)$ be the system of all subgroups $H \leq G$ such that the quotient modules $A/C_A(G)$ are not minimax $\mathbf{R}$-modules. We investigate a $\mathbf{R}G$ - module $A$ such that $\mathfrak{L}_{nm}(G)$ satisfies either the weak minimal condition or the weak maximal condition as an ordered set. It is proved that a nilpotent group $G$ that satisfies these conditions is a minimax group.

### On modules over integer-valued group rings of locally soluble groups with rank restrictions imposed on subgroups

Ukr. Mat. Zh. - 2011. - 63, № 9. - pp. 1206-1217

We study the $ZG$-module $A$ such that $Z$ is the ring of integers, the group $G$ has infinite section $ p$-rank (or infinite 0-rank), $C_G(A) = 1$, $A$ is not a minimax $Z$-module, and, for every proper subgroup $H$ of infinite section $p$-rank (or infinite 0-rank, respectively), the quotient module $A/C_A(H)$ is a minimax $Z$-module. It is proved that if the group $G$ under consideration is locally solvable, then $G$ is a solvable group. Some properties of a solvable group of this type are obtained.

### On one class of modules over integer group rings of locally solvable groups

Ukr. Mat. Zh. - 2009. - 61, № 1. - pp. 44-51

We study a $Z G$-module $A$ in the case where the group $G$ is locally solvable and satisfies the condition min–naz and its cocentralizer in A is not an Artinian $Z$-module. We prove that the group G is solvable under the conditions indicated above. The structure of the group $G$ is studied in detail in the case where this group is not a Chernikov group.

### Groups of finite non-Abelian sectional rank

Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1324–1331

We study non-Abelian locally finite groups and non-Abelian locally solvable groups of finite non-Abelian sectional rank and prove that their (special) rank is finite.

### Locally almost solvable groups of finite non-Abelian rank

Ukr. Mat. Zh. - 1990. - 42, № 4. - pp. 477-482

### Solvable groups of finite non-Abelian rank

Ukr. Mat. Zh. - 1990. - 42, № 2. - pp. 159–164