Separability of Schur rings over Abelian $p$-groups

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 $p\in\{2,3\}$ and $k\ge1$, 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 $|D|$.