A combinatorial approach to doubly transitive permutation groups.
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.
scientific article; zbMATH DE number 5280742
scientific article