Integral group ring automorphisms without Zassenhaus factorization

Abstract

An automorphism $\alpha$ of an integral group ring $\mathbb{Z} G$, where $G$ is a finite group, is said to have a Zassenhaus factorization if it is the composition of an automorphism of $G$ (extended to a ring automorphism) and a central automorphism. In 1988, Roggenkamp and Scott constructed a group $G$ (of order $2880$) such that $\mathbb{Z} G$ has a normalized (i.e., augmentation preserving) automorphism $\alpha$ which has no Zassenhaus factorization. In this paper, short proofs of the following two results are given. (1) For a group $G$ of order $144$, there is a normalized automorphism $\alpha$ of $\mathbb{Z} G$ which has no Zassenhaus factorization. Moreover, $\alpha$ can be chosen to have finite order. (2) There is a group $G$ of order $1200$, with abelian Sylow subgroups and Sylow tower, such that $\mathbb{Z} G$ has a normalized automorphism which has no Zassenhaus factorization.