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

TL;DR

This paper investigates a special class of association schemes derived from transitive permutation groups with specific normalizer properties, establishing inequalities related to the structure and size of these groups.

Contribution

It introduces new properties of association schemes linked to the orbitals of certain transitive groups with prime-power normalizer conditions.

Findings

Derived inequality: |G:N_G(N_G(H))| ≤ p+1

Characterized association schemes with specific normalizer structures

Connected group-theoretic properties to combinatorial schemes

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 $H <N_G(H)<N_G(N_G(H))\unlhd G$ 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 $|G:N_G(N_G(H))|\leq p+1$.