On modules over group rings of groups with restrictions on the system of all proper subgroups

TL;DR

This paper investigates modules over group rings with restrictions on their proper subgroups, proving that under certain conditions, the group is isomorphic to a quasi-cyclic q-group for some prime q.

Contribution

It establishes conditions under which groups with specific module restrictions are necessarily quasi-cyclic q-groups, extending understanding of module-group interactions.

Findings

G is isomorphic to a quasi-cyclic q-group for some prime q

Results apply to modules over rings of integers, p-adic integers, and associative rings

Conditions involve classes of artinian, minimax, or finite modules

Abstract

We consider the class $\mathfrak M$ of $\bf R$--modules where $\bf R$ is an associative ring. Let $A$ be a module over a group ring $\bf R$$G$ where $G$ is a group and let $\mathfrak L(G)$ be a set of all proper subgroups of $G$ such that if $H \in \mathfrak L(G)$ then $A/C_{A}(H)$ belongs to $\mathfrak M$. We study an $\bf R$$G$--module $A$ such that $G \not = G'$, $C_{G}(A) = 1$, $A/C_{A}(G) \not \in \mathfrak M$, and $\mathfrak M$ is one of the classes: artinian $\bf R$--modules, minimax $\bf R$--modules, finite $\bf R$--modules. We consider the cases: 1) $\mathfrak M$ is a class of all artinian $\bf R$--modules, $\bf R$ is either a ring of integers or a ring of $p$--adic integers; 2) $\mathfrak M$ is a class of all minimax $\bf R$--modules, $\bf R$ is a ring of integers, G is a locally soluble group; 3) $\mathfrak M$ is a class of all finite $\bf R$--modules, $\bf R$ is an associative ring. In these cases we prove that $G$ is isomorphic to a quasi--cyclic $q$--group for some prime $q$.