Isomorphisms of modular group algebras: An algorithm and its application to groups of order \(2^ 6\)
For finite \(p\)-groups \(G\) and \(H\) denote by \(F(G)\) and \(F(H)\) their group rings over the field \(F = Z/pZ\). It is known that \(F(G)\cong F(H)\) implies \(G \cong H\) when \(|G|\leq p^ 5\) but in general the isomorphism problem is unresolved. The author resolves the situation for \(|G|= 2^ 6\) (and shows that the ring isomorphism does imply the group isomorphism) by developing an algorithm for the lifting of homomorphisms based on an idea of \textit{K. W. Roggenkamp} and \textit{L. L. Scott} [to appear] and implementing it for groups of order \(2^ 6\). The author notes that the implementation uses the system SISYPHOS which does symbolic computations in modular group rings of \(p\)-groups. CAYLEY was used to calculate the group invariants.
scientific article; zbMATH DE number 370652