Restricted-finite groups with some applications in group rings

Authors : Bijan Taerı, Mohammad Reza Vedadı

Pages : 51-65

Doi:10.24330/ieja.1438622

View : 78 | Download : 88

Publication Date : 2024-07-12

Article Type : Research Paper

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} $.
Keywords : Artinian ring, group ring, locally finite group, maximal condition, minimal condition, restricted finite group