Structure of group rings and the group of units of integral group rings: an invitation

Abstract

During the past three decades fundamental progress has been made on constructing large torsion-free subgroups (i.e. subgroups of finite index) of the unit group $\mathcal {U}(\mathbb {Z}G)$ of the integral group ring $\mathbb {Z}G$ of a finite group G. These constructions rely on explicit constructions of units in $\mathbb {Z}G$ and proofs of main results make use of the description of the Wedderburn components of the rational group algebra $\mathbb {Q}G$. The latter relies on explicit constructions of primitive central idempotents and the rational representations of G. It turns out that the existence of reduced two degree representations play a crucial role. Although the unit group is far from being understood, some structure results on this group have been obtained. In this paper we give a survey of some of the fundamental results and the essential needed techniques.