Strong Gelfand pairs of symmetric groups

A strong Gelfand pair is a pair [Formula: see text], of finite groups such that the Schur ring determined by the [Formula: see text]-classes [Formula: see text], is a commutative ring. We find all strong Gelfand pairs [Formula: see text]. We also define an extra strong Gelfand pair [Formula: see text], this being a strong Gelfand pair of maximal dimension, and show that in this case [Formula: see text] must be abelian.

CI-Property for Decomposable Schur Rings over an Abelian Group

A Schur ring over a finite group is said to be decomposable if it is the generalized wreath product of Schur rings over smaller groups. In this paper we establish a sufficient condition for a decomposable Schur ring over the direct product of elementary abelian groups to be a CI-Schur ring. By using this condition we offer short proofs for some known results on the CI-property for decomposable Schur rings over an elementary abelian group of rank at most 5.

Groups whose proper Schur rings are commutative

A Schur ring [Formula: see text] is called Dedekind if the formal sum of every [Formula: see text]-subgroup is in the center of [Formula: see text]. In this paper, we find all finite groups [Formula: see text] such that every proper Schur ring over [Formula: see text] is Dedekind. As a corollary of our main theorem, we find all finite groups [Formula: see text] such that every proper Schur ring over [Formula: see text] is commutative.

On non-abelian Schur groups

A finite group G is called Schur, if every Schur ring over G is associated in a natural way with a regular subgroup of sym (G) that is isomorphic to G. We prove that any non-abelian Schur group G is metabelian and the number of distinct prime divisors of the order of G does not exceed 7.

Recognizing Circulant Graphs of Prime Order in Polynomial Time

A circulant graph $G$ of order $n$ is a Cayley graph over the cyclic group ${\bf Z}_n.$ Equivalently, $G$ is circulant iff its vertices can be ordered such that the corresponding adjacency matrix becomes a circulant matrix. To each circulant graph we may associate a coherent configuration ${\cal A}$ and, in particular, a Schur ring ${\cal S}$ isomorphic to ${\cal A}$. ${\cal A}$ can be associated without knowing $G$ to be circulant. If $n$ is prime, then by investigating the structure of ${\cal A}$ either we are able to find an appropriate ordering of the vertices proving that $G$ is circulant or we are able to prove that a certain necessary condition for $G$ being circulant is violated. The algorithm we propose in this paper is a recognition algorithm for cyclic association schemes. It runs in time polynomial in $n$.