A generalization of Sylow's theorems on finite groups to association schemes

Let (X,G) denote an association scheme (or briefly, a scheme) in the sense of [5], and let p denote a prime number. An element g in G will be called p-valenced if ng is a power of p. A subset of G will be called p-valenced if each of its elements is p-valenced. A p-valenced subset F of G will be called a p-subset if nF is a power of p. Let us assume (X,G) to be finite. Then, by definition, nG is finite. We shall denote by Sylp(G) the set of all closed p-subsets H of G such that p does not divide nG//H . It is the purpose of the present note to prove the following theorem.