## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 46, Number 1 (2002), 233-245.

### 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.

#### Article information

**Source**

Illinois J. Math., Volume 46, Number 1 (2002), 233-245.

**Dates**

First available in Project Euclid: 13 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1258136152

**Digital Object Identifier**

doi:10.1215/ijm/1258136152

**Mathematical Reviews number (MathSciNet)**

MR1936087

**Zentralblatt MATH identifier**

1010.20001

**Subjects**

Primary: 20C10: Integral representations of finite groups

Secondary: 16S34: Group rings [See also 20C05, 20C07], Laurent polynomial rings

#### Citation

Hertweck, Martin. Integral group ring automorphisms without Zassenhaus factorization. Illinois J. Math. 46 (2002), no. 1, 233--245. doi:10.1215/ijm/1258136152. https://projecteuclid.org/euclid.ijm/1258136152