Darstellungstheorietage and Nikolaus Conference 2021

The Isomorphism Problem for group rings with coefficients in a ring \(R\) asks whether the isomorphism type of a group \(G\) is determined by its group ring \(RG\). In general, it has a negative solution if no assumption is made about the ring or the group. For example, for abelian groups it has a positive solution if \(R\) is the field \(\mathbb{Q}\) of rational numbers, but it has a negative solution in case \(R\) is the field of complex numbers. For metabelian groups it has a negative solution for every field, but a positive solution for \(R=\mathbb{Z}\), the ring of integers. With the aim to understand which property in between abelian and metabelian suffices for a positive solution in the case \(R=\mathbb{Q}\), we discuss the Isomorphism Problem for rational group rings of metacyclic groups. We prove a positive result under the assumption that \(G\) is nilpotent.