Association schemes with a certain type of $p$-subschemes

Abstract

In this article, we focus on association schemes with some properties derived from the orbitals of a transitive permutation group $G$ with a one-point stabilizer $H$ satisfying $H<N_G(H)<N_G(N_G(H))⊴G$ and $|N_G(N_G(H))|=p^3$ where $p$ is a prime. By a corollary of our main result we obtain some inequality which corresponds to the fact $|G:N_G(N_G(H))|\le p+1$.