Association schemes in which the thin residue is a finite cyclic group

In this article we investigate association schemes S (on finite sets) in which each element s satisfies ss*s = {s} It is shown that these schemes are schurian if the partially ordered set of the intersections of the closed subsets s*s of S with s is an element of S is distributive (A scheme is said to be schurian if it arises (in a well-understood way) from a transitive permutation group) It is also shown that if these schemes are schurian the transitive permutation group from which they arise have subnormal one-point stabilizers As a consequence of the first result one obtains that schemes are schurian if their thin residue is thin and has a distributive normal closed subset lattice (normal subgroup lattice) This implies for instance that schemes are schurian if their thin residue is a cyclic group (C) 2010 Elsevier Inc All rights reserved