Unit groups of integral group rings

Abstract

Let U ( Z G ) U(\mathbb {Z}G) be the unit group of the integral group ring Z G \mathbb {Z}G . A group G G satisfies ( ∗ ) ({\ast }) if either the set T ( G ) T(G) of torsion elements of G G is a central subgroup of G G or, otherwise, if x ∈ G x \in G does not centralize T ( G ) T(G) , then for every t ∈ T ( G ) , x − 1 t x = t − 1 t \in T(G),\,{x^{ - 1}}tx = {t^{ - 1}} . This property appears quite frequently while studying U ( Z G ) U(\mathbb {Z}G) . In this paper we investigate why one encounters this property and we have also given a "unified proof" for some known results regarding this property. Further, some additional results have been obtained.