On p-covalenced association schemes

Let ( X , S ) denote an association scheme where X is a finite set. For a prime p we say that ( X , S ) is p-covalenced (p-valenced) if every multiplicity (valency, respectively) of ( X , S ) is a power of p. In the character theory of finite groups Itoʹs theorem states that a finite group G has a normal abelian p-complement if and only if every character degree of G is a power of p. In this article we generalize Itoʹs theorem to p-valenced association schemes, i.e., a p-valenced association scheme ( X , S ) has a normal p-covalenced p-complement if and only if ( X , S ) is p-covalenced.