Association schemes and permutation groups

Abstract

Every permutation group which is not 2-transitive acts on a non-trivial coherent configuration, but the question of which permutation groups G act on non-trivial association schemes (symmetric coherent configurations) is considerably more subtle. A closely related question is: when is there a unique minimal G-invariant association scheme? We examine these questions, and relate them to more familiar concepts of permutation group theory (such as generous transitivity) and association scheme theory (such as stratifiability). Our main results are the determination of all regular groups having a unique minimal association scheme, and a classification of groups with no non-trivial association scheme. The latter must be primitive, and are 2-homogeneous, or almost simple, or of diagonal type. The diagonal groups have some very interesting features, and we examine them further. Among other things we show that a diagonal group with non-abelian base group cannot be stratifiable if it has ten or more factors, or generously transitive if it has nine or more; and we characterise the quaternion group Q 8 as the unique non-abelian group T such that a diagonal group with eight factors T is generously transitive.