# Darstellungstheorietage and Nikolaus Conference 2021

Speaker: Àngel García Blàzquez (Murcia)

Title: The Isomorphism Problem for Rational Group Algebras of Metacyclic Groups

Abstract:

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.