Traces of Torsion Units

TL;DR

This paper explores a weaker version of Zassenhaus's conjecture concerning torsion units in group rings, extending the discussion to certain infinite groups and analyzing their conjugacy properties.

Contribution

It introduces a modified conjecture applicable to some infinite groups, broadening the scope of the original finite group conjecture.

Findings

Proposes a weaker form of Zassenhaus's conjecture for infinite groups

Provides conditions under which torsion units are conjugate in the group algebra

Extends the analysis of torsion units beyond finite groups

Abstract

A conjecture due to Zassenhaus asserts that if $\ G$ is a finite group then any torsion unit in $\mathbb{Z}G$ is conjugate in $\mathbb{Q}G$ to an element of $\ G$. We present a weaker form of this conjecture for some infinite groups.