Character tables of p-groups with derived subgroup of prime order I

Two character tables of finite groups are isomorphic if there exist a bijection for the irreducible characters and a bijection for the conjugacy classes that preserve all the character values. We give necessary and sufficient conditions for two finite groups to have isomorphic character tables. In the case of finite p-groups with derived subgroup of order p, we show that the character tables can be classified by equivalence classes of certain homomorphisms of abelian p-groups.