A counterexample for the isomorphism-problem of polycyclic groups (Q1901025)

The authors construct two non-isomorphic polycyclic groups \(G_1\) and \(G_2\) such that the integral group rings \(\mathbb{Z} G_1\) and \(\mathbb{Z} G_2\) are Morita equivalent. Hence for a suitable ring \(R\) of algebraic integers in a global algebraic number field, \(RG_1\) is isomorphic to \(RG_2\). This result gives an answer to an open problem in the representation theory of groups.