A combinatorial approach to doubly transitive permutation groups.
type: Item
instance of: scholarly article
title: A combinatorial approach to doubly transitive permutation groups.
zbMATH Open document ID: 1157.20003
author: Izumi Miyamoto
published in: Discrete Mathematics
publication date: 2008-05-29T00:00:00Z
review text: Let \(G\) be a finite doubly transitive but not triply transitive permutation group acting on a set \(X\). Then the orbit structure of \(G\) on \(X^3=X\times X\times X\) is not the trivial one induced by the symmetric group. The approach of the present paper is to obtain the orbits of \(G\) on \(X^3\) by studying certain combinatorial objects called superschemes (related to the concept of association schemes). As a starting point the orbits of a point stabilizer \(G_\alpha\) on \(X^3\) which define a superscheme are used to create a superscheme of \(G\) on \(X^3\) where \(G\) is considered as a transitive extension of \(G_\alpha\). The algorithm obtained allows for computing the transitive extension \(G\) itself. The algorithm has been implemented and tested successfully for the groups \(G=\text{PSL}(m,q)\) and \(\text{Sp}(2m,2)\), where \(m\geq 3\). The paper contains a short discussion of the superscheme concept in relation to transitive extensions of transitive permutation groups.
Mathematics Subject Classification ID: 20B20
Mathematics Subject Classification ID: 05E20
Mathematics Subject Classification ID: 05E30
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 20B40
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 5280742
zbMATH Keywords: finite permutation groups
zbMATH Keywords: doubly transitive permutation groups
zbMATH Keywords: orbits
zbMATH Keywords: point stabilizers
zbMATH Keywords: actions on ordered triples
zbMATH Keywords: transitive extensions
zbMATH Keywords: association schemes
zbMATH Keywords: superschemes
zbMATH Keywords: algorithms
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: j.disc.2007.08.032
OpenAlex ID: W2089913722
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DOI: 10.1016/J.DISC.2007.08.032
description: scientific article; zbMATH DE number 5280742