Restricted-finite groups with some applications in group rings

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 quasi-cyclic 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} $.