scholarly journals An answer to Hirasaka and Muzychuk: Every p-Schur ring over Cp3 is Schurian

2008 ◽  
Vol 308 (9) ◽  
pp. 1760-1763 ◽  
Author(s):  
Pablo Spiga ◽  
Qiang Wang
Keyword(s):  
Schur Ring

scholarly journals SCHUR RING AND QUASI-SIMPLE MODULES

2009 ◽  
Vol 28 (2) ◽  
Author(s):  
Pedro Domínguez-Wade
Keyword(s):  
Simple Modules ◽  
Schur Ring

Strong Gelfand pairs of symmetric groups

2020 ◽  
pp. 2150054
Author(s):  
Gradin Anderson ◽  
Stephen P. Humphries ◽  
Nathan Nicholson
Keyword(s):  
Commutative Ring ◽  
Finite Groups ◽  
Symmetric Groups ◽  
Maximal Dimension ◽  
Gelfand Pair ◽  
Gelfand Pairs ◽  
Schur Ring

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.

scholarly journals Recognizing Circulant Graphs of Prime Order in Polynomial Time

10.37236/1363 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Mikhail E. Muzychuk ◽  
Gottfried Tinhofer
Keyword(s):  
Adjacency Matrix ◽  
Prime Order ◽  
Circulant Matrix ◽  
Recognition Algorithm ◽  
Association Schemes ◽  
Necessary Condition ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Coherent Configuration ◽  
Schur Ring

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 $n$.

Centraliser rings of multiplication groups on quasigroups

1976 ◽  
Vol 79 (3) ◽  
pp. 427-431 ◽  
Author(s):  
J. D. H. Smith
Keyword(s):  
Group Theory ◽  
Vector Space ◽  
Symmetric Group ◽  
Binary Operation ◽  
Schur Ring ◽  
The Third ◽  
Permutation Matrices ◽  
Finite Set ◽  
Ring Method

Let (Q,.) be a finite quasigroup, i.e. a finite set Q with a binary operation. called multiplication such that in the equation x.y = z any two elements determine the third uniquely. Then the mappings R(x): Q → Q; q ↦ q.x and L(x): Q → Q; q ↦ x.q are permutations of Q. The multiplication group G of Q is the subgroup of the symmetric group on Q generated by {R(x), L(x) | x ∈ Q}. If S is a field, G has a faithful representation Ḡ by permutation matrices acting on the S-vector space with Q as basis. The set of matrices commuting with Ḡ forms an S-algebra (under the usual operations) called the centraliser ring V(G, Q) of G on Q. The purpose of this note is to show how the permutation-theoretic object ‘centraliser ring’ may be expressed in terms of the quasigroup structure of Q, both to prepare one tool for the long-term programme of classifying finite quasigroups by means of their multiplication groups, and for comparison with the Schur ring method of group theory.

Groups whose proper Schur rings are commutative

2018 ◽  
Vol 17 (11) ◽  
pp. 1850206
Author(s):  
Eun-Kyung Cho
Keyword(s):  
Finite Groups ◽  
Schur Ring

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 Simply Transitive Primitive Permutation Groups

1969 ◽  
Vol 21 ◽  
pp. 1062-1068 ◽  
Author(s):  
R. D. Bercov
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Prime Power ◽  
Direct Decomposition ◽  
Permutation Groups ◽  
Prime Power Order ◽  
Schur Ring ◽  
Abelian Subgroups ◽  
Prime 2

In (1) we considered finite primitive permutation groups G with regular abelian subgroups H satisfying the following hypothesis:(*) H = A × B × C, where A is cyclic of prime power order pα ≠ 4, B has exponent pβ < pα, and C has order prime to p.We remark that an abelian group fails to satisfy (*) (apart from the minor exception associated with the prime 2) if and only if it is the direct product of two subgroups of the same exponent.We showed in (1) that such a group G is doubly transitive unless it is the direct product of two or more subgroups each of the same order greater than 2. This was done by showing that (in the terminology of (3)) the existence of a non-trivial primitive Schur ring over H implies such a direct decomposition of H.

scholarly journals CI-Property for Decomposable Schur Rings over an Abelian Group

2019 ◽  
Vol 26 (01) ◽  
pp. 147-160 ◽  
Author(s):  
István Kovács ◽  
Grigory Ryabov
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Wreath Product ◽  
Direct Product ◽  
Elementary Abelian Group ◽  
Sufficient Condition ◽  
Schur Ring ◽  
Generalized Wreath Product ◽  
A Finite Group ◽  
Ci Property

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.

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