Separability of Schur Rings over Abelian p-Groups

摘要

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 × C p k $$ D={C}_pimes {C}_{p^k} $$ , 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 |.