Algebraic structure of association schemes of prime order

Abstract

Finite groups of prime order must be cyclic. It is natural to ask what about association schemes of prime order. In this paper, we will give an answer to this question. An association scheme of prime order is commutative, and its valencies of nontrivial relations and multiplicities of nontrivial irreducible characters are constant. Moreover, if we suppose that the minimal splitting field is an abelian extension of the field of rational numbers, then the character table is the same as that of a Schurian scheme.