On torsion units in integral group rings of Frobenius groups

TL;DR

This paper proves that certain torsion units in the integral group ring of a Frobenius group are conjugate to actual group elements, extending understanding of units in group rings.

Contribution

It establishes that torsion units mapping trivially to the quotient are conjugate to group elements in the semilocalized integral group ring of Frobenius groups.

Findings

Torsion units in the group ring are conjugate to group elements.

The result applies to units mapping to the identity in the quotient.

It advances the understanding of the structure of units in integral group rings.

Abstract

For a finite group $G$, let $\tilde{\mathbb{Z}}$ be the semilocalization of $\mathbb{Z}$ at the prime divisors of $|G|$. If $G$ is a Frobenius group with Frobenius kernel $K$, it is shown that each torsion unit in the group ring $\tilde{\mathbb{Z}} G$ which maps to the identity under the natural ring homomorphism $\tilde{\mathbb{Z}} G \rightarrow \tilde{\mathbb{Z}} G/K$ is conjugate to an element of $G$ by a unit in $\tilde{\mathbb{Z}} G$.