On association schemes with thin thin residue

Let S be a scheme of finite valency, and assume that O(nu) (S) subset of O(nu) (S). It is known that S is schurian (which means that S arises from a finite group) if the normal closed subsets (normal subgroups) of O(nu) (S) are linearly ordered with respect to set-theoretic inclusion; cf. [M. Hirasaka, P.-H. Zieschang, Sufficient conditions for a scheme to originate from a group, J. Combin. Theory Set. A 104 (2003) 17-27]. In this note, it is shown that S is schurian if O(nu) (S) is direct product of two simple closed subsets (finite simple groups) of different order. (C) 2009 Elsevier Inc. All rights reserved.