Theses and Dissertations

May 2016

Dissertation

Degree Name

Doctor of Philosophy

Mathematics

Allen D. Bell

Committee Members

Jay H. Beder, Yi Ming Zou, Ian M. Musson, Jeb F. Willenbring

Keywords

Finiteness Conditions, Group Rings, Group Theory

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.

COinS