Partial augmentations and Brauer character values of torsion units in group rings

TL;DR

This paper explores the relationship between partial augmentations and Brauer character values of torsion units in integral group rings, providing new insights especially for non-solvable groups like $S_5$ and $ ext{PSL}(2,p^f)$.

Contribution

It establishes a novel relation between partial augmentations and Brauer character values for torsion units, extending understanding to non-solvable groups.

Findings

Relation between partial augmentations and Brauer characters established

Implications for rational conjugacy in non-solvable groups discussed

Examples include $S_5$ and $ ext{PSL}(2,p^f)$ groups

Abstract

For a torsion unit $u$ of the integral group ring $\mathbb{Z} G$ of a finite group $G$, and a prime $p$ which does not divide the order of $u$ (but the order of $G$), a relation between the partial augmentations of $u$ on the $p$-regular classes of $G$ and Brauer character values is noted, analogous to the obvious relation between partial augmentations and ordinary character values. For non-solvable $G$, consequences concerning rational conjugacy of $u$ to a group element are discussed, considering as examples the symmetric group $S_{5}$ and the groups $\text{\rm PSL}(2,p^{f})$.