The graph extension theorem

Abstract

A sufficient condition is given that a transitive permutation group G admits a transitive extension G ∗ {G^\ast } . The condition is graph-theoretic and does not involve any direct algebraic properties of the group being extended. The result accounts for a fairly wide class of doubly transitive groups, including the two doubly transitive representations of the groups Sp ( 2 n , 2 ) {\text {Sp}}(2n,2) , and the doubly transitive representations of the Higman-Sims group, and the Conway group (.3).