A theorem of Hertweck on $p$-adic conjugacy of $p$-torsion units in group rings

Abstract

A proof of a theorem of M. Hertweck presented during a seminar in January 2013 in Stuttgart is given. The proof is based on a preprint given to me by Hertweck. Let $R$ be a commutative ring, $G$ a finite group, $N$ a normal $p$-subgroup of $G$ and denote by $RG$ the group ring of $G$ over $R$. It is shown that a torsion unit $u$ in $\mathbb{Z}G$ mapping to the identity under the natural homomorphism $\mathbb{Z}G \rightarrow \mathbb{Z}G/N$ is conjugate in the unit group of $\mathbb{Z}_pG$ to an element in $N$. Here $\mathbb{Z}_p$ denotes the $p$-adic integers. The result is achieved proving a result in the context of the so-called double action formalism for group rings over $p$-adic rings. This widely generalizes a theorem of Hertweck and a related theorem by Caicedo-Margolis-del R\'io and has consequences for the study of the Zassenhaus Conjecture for integral group rings.