Restricted-finite groups with some applications in group rings

TL;DR

This paper introduces restricted-finite groups, characterizes their structure, especially for finitely generated and abelian cases, and explores their applications in group rings, including conditions for semisimplicity and specific group decompositions.

Contribution

It defines restricted-finite groups, provides their structural classification, and applies these results to analyze properties of group rings and their modules.

Findings

Infinite restricted-finite abelian groups are isomorphic to Z×K or Z_{p^∞}×K.

Infinitely generated restricted-finite groups have a specific subgroup decomposition.

Conditions under which group rings are semisimple are established.

Abstract

We carry out a study of groups $G$ in which the index of any infinite subgroup is finite. We call them restricted-finite groups and characterize finitely generated not torsion restricted-finite groups. We show that every infinite restricted-finite abelian group is isomorphic to $\mathbb{Z}\times K$ or $\mathbb{Z}_{p^\infty}\times K$, where $K$ is a finite group and $p$ is a prime number. We also prove that a group $G$ is infinitely generated restricted-finite if and only if $G = AT$, where $A$ and $T$ are subgroups of $G$ such that $A$ is normal quasicyclic and $T$ is finite. As an application of our results, we show that if $G$ is not torsion with finite $G'$ and the group-ring $RG$ has restricted minimum condition then $R$ is a semisimple ring and $G\cong T\rtimes\mathbb{Z} $, where $T$ is finite whose order is unit in $R$. The converse is also true with certain conditions including $G = T\times \mathbb{Z}