The author studies the normalizer \(N\) of a group \(G\) in its group ring \(RG\). More precisely, let \(G\) be a not necessarily finite group and let \(R\) be any commutative ring. Then, conjugation by \(N\) gives automorphisms on \(G\) and the corresponding group of automorphisms is denoted by \(\operatorname{Aut}_R(G)\). This group has very interesting features. Especially, if it is not just the group of inner automorphisms of \(G\), it can be used to construct two non-isomorphic groups with isomorphic group rings over \(R\). The first examples of two non-isomorphic finite groups \(G_1\) and \(G_2\) with isomorphic integral group rings \(\mathbb{Z} G_1\) and \(\mathbb{Z} G_2\) are given recently in ground breaking work by Martin Hertweck. Another very interesting property of \(\operatorname{Aut}_R(G)\) is that its elements operate trivially on the cohomology ring \(H^*(G,R)\).
The author studies the group \(\operatorname{Aut}_R(G)\) especially for infinite, mainly FC-groups, but very interesting consequences are also given for finite groups. The paper contains a wealth of information on this problem and the results are too numerous to be presented here.