On the isomorphism of semisimple group algebras

Let KG be the group algebra of an abelian p-group G over a field K of the first kind with respect to p and let H be an abelian p-group. Berman and Mollov (1986) have given necessary and sufficient conditions, i.e. a criterion, for the isomorphism of KG and KH as K-algebras when the first Ulm factor G/G(1) is a direct sum of cyclic groups. In this paper we give new and simplified necessary and sufficient conditions for this isomorphism. In the case when G is a direct sum of cyclic groups we correct an essential inaccuracy in the original proof of the criterion.