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 s s ⁎ 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 ∈ S is distributive. (A scheme is said to b...

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Bibliographic Details
Published in:Journal of algebra 2010-12, Vol.324 (12), p.3572-3578
Main Author: Zieschang, Paul-Hermann
Format: Article
Language:English
Subjects:
Abstract theory of association schemes
Distributive normal subgroup lattice
Group theory
Schurian schemes
Thin radical
Thin residue
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Summary:In this article, we investigate association schemes S (on finite sets) in which each element s satisfies s s ⁎ 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 ∈ 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.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2010.04.013