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arXiv · 2008.11569

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

Eric Jespers
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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 $\U (\Z G)$ of the integral group ring $\Z G$ of a finite group $G$. These constructions rely on explicit constructions of units in $\Z G$ and proofs of main results make use of the description of the Wedderburn components of the rational group algebra $\Q G$. 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.

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Eric Jespers. 2020-08-26. Structure of group rings and the group of units of integral group rings: an invitation. https://arxiv.org/abs/2008.11569

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