Units of twisted group rings and their correlations to classical group rings

TL;DR

This paper develops a new generic construction for units in integral group rings of finite groups, especially when classical simple components are present, and applies it to resolve conjectures on the structure of the unit group.

Contribution

It introduces a novel generic construction of units in group rings with simple components, expanding understanding of their structure and properties.

Findings

Constructed units generate large free subgroups.

Resolved conjectures on rank and periodic elements of the unit group's abelianization.

Provided a decomposition of twisted group algebras over fraction fields.

Abstract

This paper is centered around the classical problem of extracting properties of a finite group $G$ from the ring isomorphism class of its integral group ring $\mathbb{Z} G$. This problem is considered via describing the unit group $\mathcal{U}( \mathbb{Z} G)$ generically for a finite group. Since the $`90s$ several well known generic constructions of units are known to generate a subgroup of finite index in $\mathcal{U}(\mathbb{Z } G)$ if $\mathbb{Q} G$ does not have so-called exceptional simple epimorphic images, e.g. $M_2 (\mathbb{Q})$. However it remained a major open problem to find a {\it generic} construction under the presence of the latter type of simple images. In this article we obtain such generic construction of units. Moreover, this new construction also exhibits new properties, such as providing generically free subgroups of large rank. As an application we answer positively for several classes of groups recent conjectures on the rank and the periodic elements of the abelianisation $\mathcal{U}(\mathbb{Z} G)^{ab}$. To obtain all this, we investigate the group ring $R \Gamma$ of an extension $\Gamma$ of some normal subgroup $N$ by a group $G$, over a domain $R$. More precisely, we obtain a direct sum decomposition of the (twisted) group algebra of $\Gamma$ over the fraction field $F$ of $R$ in terms of various twisted group rings of $G$ over finite extensions of $F$. Furthermore, concrete information on the kernel and cokernel of the associated projections is obtained. Along the way we also launch the investigations of the unit group of twisted group rings and of $\mathcal{U}( R\Gamma)$ via twisted group rings.