The Root Finite Condition on Groups and Its Application to Group Rings

Abstract

A group $G$ is said to satisfy the root-finite condition if for every $g \in G$, there are only finitely many $x \in G$ such that there exists a positive integer $n$ such that $x^n = g$. It is shown that groups satisfy the root-finite condition iff they satisfy three subconditions, which are shown to be independent. Free groups are root-finite. Ordered groups are shown to satisfy one of the subconditions for the root-finite condition. Finitely generated abelian groups satisfy the root-finite condition. If, in a torsion-free abelian group $G$, there exists a positive integer $r$ such that the subgroup $A_r$ of elements of $G$ taken to the $r^{\mathrm{th}}$ power has index less than $r$ in $G$, then $G$ does not satisfy the root-finite condition. Finitely generated finite conjugate groups satisfy the root-finite condition. Infinite groups with finitely many conjugacy classes fail to satisfy the root-finite condition. Torsion-free polycyclic-by-finite groups satisfy two of the subconditions for the root-finite condition. Finitely generated nilpotent groups satisfy the root-finite condition. If $KG$ is a group ring, for every nonidentity element $x$ of $G$, the following left module is defined $\mathcal{M}_x=KG/KG(x-1)$. This module is shown to be faithful if $G$ satisfies the root-finite condition and $x$ has an infinite conjugacy class. If $KG$ is a prime group ring, then $\mathcal{M}_x$ is not faithful if the conjugacy class of $x$ is finite. An analogous problem concerning skew polynomial and skew-Laurent polynomial rings is discussed.