Separability of Schur Rings over Abelian p-Groups

Abstract

A Schur ring (an S-ring) is said to be separable if each of its algebraic isomorphisms is induced by an isomorphism. Let C_n be the cyclic group of order n. It is proved that all S-rings over groups \( D={C}_p\times {C}_{p^k} \), where p ∈ {2, 3} and k ≥ 1, are separable with respect to a class of S-rings over Abelian groups. From this statement, we deduce that a given Cayley graph over D and a given Cayley graph over an arbitrary Abelian group can be checked for isomorphism in polynomial time with respect to |D|.