Twisted group ring isomorphism problem and infinite cohomology groups (Q2104888)

For a commutative ring \(R\) and finite groups \(G\) and \(H\) write \(G \sim_R H\) if there exists a group isomorphism \(\psi : H^2(G,R^*) \to H2(H,R^*)\) such that for any \([\alpha ] \in H^2(G, R^*)\), \(R^\alpha G \simeq R^{\psi (\alpha )}H\). The authors continue their investigation of the twisted group ring isomorphism problem (TGRIP), which asks to find the \(\sim_R\)-classes for a given \(R\), and also its relation with the classical isomorphism problem. In particular, for which groups \(G \sim_R H\) implies \(G \simeq H\)? They concentrate in this paper on the following question: Are there non-isomorphic groups \(G\) and \(H\) and a field \(F\) with characteristic not dividing \(|G|\) such that \(G \sim_{\mathbb{Q}} H\) but \(G \nsim_F H\)? The following main result leads to a positive answer to this question: Let \(p\) be an odd prime, let \(G\) and \(H\) be the two non-abelian groups of order \(p^3\), let \(F\) be a field of characteristic different from \(p\), and let \(\zeta\) be a primitive \(p\)-th root of unity in an extension of \(F\). Then \(G \sim_F H\) if and only if \(F\) does not contain a primitive \(p\)-th root of unity and \(\zeta\) is in the image of the norm map of the field extension \(F( \sqrt[p]{\lambda}, \zeta)/F(\zeta)\) for all \(\lambda \in F^*\). In particular, \(G \nsim_{\mathbb{Q}} H\). Several related problems are also formulated and studied.