Association schemes and permutation groups
type: Item
instance of: scholarly article
title: Association schemes and permutation groups
zbMATH Open document ID: 1021.20004
DOI: 10.1016/S0012-365X(02)00798-7
author: Priscila P. Alejandro
author: R. A. Bailey
author: Peter J. Cameron
published in: Discrete Mathematics
publication date: 2003-06-09T00:00:00Z
review text: A coherent algebra is an algebra of \(n\times n\) complex matrices which has a basis \(\{B_0,B_1,\dots,B_t\}\) consisting of matrices with entries 0 and 1 satisfying the following conditions: (a) \(B_0+B_1+\cdots+B_t=J_n\); (b) there is a subset of \(\{B_0,B_1,\dots,B_t\}\) with sum \(I_n\); (c) the set \(\{B_0,B_1,\dots,B_t\}\) is closed under transposition. The algebra is called homogeneous if the subset referred to in (b) contains just one element, which we take to be \(B_0=I_n\). We call a transitive permutation group AS-free if it preserves no non-trivial association scheme. A transitive permutation group \(G\) is called AS-friendly if there is a unique finest \(G\)-invariant association scheme. Primitive groups which are not AS-friendly do exist; the smallest example is \(\text{PSL}(2,13)\), acting on 78 points (Cameron). There are AS-free groups which are not stratifiable: \(\text{PSL}(3,3)\), degree 234; \(M_{12}\), degree 1320; \(J_1\), degree 1463, 1540 or 1596; and \(J_2\), degree 1800 (Soicher). Corollary 6. If \(G\) has a regular Abelian subgroup \(A\), then \(G\) is stratifiable; and if \(G\) contains the holomorph of \(A\), then \(G\) is geneously transitive. For a finite group \(G\), the following conditions are equivalent: (1) the regulear action of \(G\) is AS-friendly; (2) the regular action of \(G\) is stratifiable; (3) either \(G\) is Abelian or \(G\simeq Q_8\times A\), where \(Q_8\) is the quaternion group of order 8 and \(A\) is an elementary Abelian 2-group (Theorem 7). Theorem 9. Let \(G\) be an AS-free permutation group. Then \(G\) is primitive and basic. Moreover, \(G\) is 2-homogeneous, of diagonal type, or almost simple. Among other things it is shown that a diagonal group with non-Abelian base group cannot be stratifiable if it has ten or more factors, or generously transitive if it has nine or more.
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 05E30
zbMATH DE Number: 1924755
zbMATH Keywords: association schemes
zbMATH Keywords: coherent configurations
zbMATH Keywords: transitive permutation groups
zbMATH Keywords: primitive permutation groups
reviewed by: Alexandre A. Makhnev
describes a project that uses: GAP
MaRDI profile type: Publication
cites work: Q4272607
cites work: Suprema and infima of association schemes
cites work: Q4828542
cites work: Q4882944
cites work: Q4284633
cites work: Endliche Gruppen I
cites work: Q3901672
full work available at URL: s0012-365x(02)00798-7
OpenAlex ID: W2155873710
description: scientific article; zbMATH DE number 1924755