The isomorphism problem for rational group algebras of finite metacyclic nilpotent groups (Q6618744)

The isomorphism problem for group rings asks if the isomorphism between the group rings \(RG\) and \(RH\), implies that the groups \(G\) and \(H\) are isomorphic, where \(RG\) denotes the group ring of the group \(G\) over the ring \(R\). The question has been studied in many concrete situations and this paper is another contribution to the problem.\N\NThe authors show that if \(G\) and \(H\) are both metacyclic and \(G\) is nilpotent and the group rings \(\mathbb{Q}G\) and \(\mathbb{Q}H\) are isomorphic, then \(G\) and \(H\) are also isomorphic. Note that there are easy negative examples if one weakens the hypothesis either by asking that only one of the groups is metacyclic or that the ground field can be any field of characteristic \(0\). The main result is a consequence of a more general result, for which we denote for a finite group \(G\) by \(\pi_G\) the set of primes such that \(G\) contains a normal Hall \(p'\)-subgroup. Then, if \(G\) and \(H\) are metacyclic and \(\mathbb{Q}G\) is isomorphic to \(\mathbb{Q}H\) one has \(\pi_G = \pi_H\) and the Hall \(\pi_G\)-subgroups of \(G\) and \(H\) are isomorphic.\N\NFirst the authors prove their main theorem for \(p\)-groups and use their previous classification of finite metacyclic groups [\textit{À. García-Blázquez} and \textit{Á. del Río}, ``A classification of metacyclic groups by group invariants'', Preprint, arXiv:2301.08683] together with the concept of Strong Shoda Pairs to study certain Wedderburn components of the rational group rings in question. They then introduce the concepts of \(p\)-components, which are those Wedderburn components whose degree is a power of \(p\) and whose center embeds in an extension of \(\mathbb{Q}\) by a \(p\)-power root of unity. These components can be used to identify the group ring of the Sylow \(p\)-subgroup of \(G\) in the Wedderburn decomposition of \(\mathbb{Q}G\) for \(p \in \pi_G\). Consequently the result for \(p\)-groups implies the main theorem.