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

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 $\text{PSL}(2,p^f)$.

16 pages, LateX. Thoroughly revised. Added references. Changed presentation of modular method. Lemma 5.6 replaced by Proposition 2.2

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• Kimmerle conjecture for the Held and O'Nan sporadic simple groups
• Integral group ring of Rudvalis simple group
• Unit groups of integral finite group rings with no noncyclic abelian finite subgroups
• Integral group ring of the Mathieu simple group M12