Unit groups of integral group rings

Let $U(\mathbb{Z}G)$ be the unit group of the integral group ring $\mathbb{Z}G$. A group $G$ satisfies $({\ast})$ if either the set $T(G)$ of torsion elements of $G$ is a central subgroup of $G$ or, otherwise, if $x \in G$ does not centralize $T(G)$, then for every $t \in T(G),\,{x^{ - 1}}tx = {t^{ - 1}}$. This property appears quite frequently while studying $U(\mathbb{Z}G)$. 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.