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 U (ZG) of the integral group ring ZG of a finite group G. These constructions rely on explicit constructions of units in ZG and proofs of main results make use of the description of the Wedderburn components of the rational group algebra QG. 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.

Central units of integral group rings of monomial groups

“…Although a lot of information is known, yet it is far from being completely understood. We refer to the various surveys including dedicated books on the topic [10,11,15,16,18,19] for its history. Bass units (introduced by Bass [4]) are generic construction of units in ZG which are analogous to cyclotomic units.…”