On the torsion units of the integral group ring of finite projective special linear groups

TL;DR

This paper investigates the structure of torsion units in the integral group ring of finite projective special linear groups, using the HeLP Method to analyze potential counterexamples to Zassenhaus's conjecture.

Contribution

It applies the HeLP Method to describe partial augmentations of hypothetical counterexamples in projective special linear groups, advancing understanding of Zassenhaus's conjecture.

Findings

Partial augmentations of potential counterexamples characterized

HeLP Method effectively applied to these groups

Provides evidence supporting or refuting the conjecture in this context

Abstract

H. J. Zassenhaus conjectured that any unit of finite order and augmentation one in the integral group ring of a finite group $G$ is conjugate in the rational group algebra to an element of $G$. One way to verify this is showing that such unit has the same distribution of partial augmentations as an element of $G$ and the HeLP Method provides a tool to do that in some cases. In this paper we use the HeLP Method to describe the partial augmentations of a hypothetical counterexample to the conjecture for the projective special linear groups.