The Graph Extension Theorem

A sufficient condition is given that a transitive permutation group G admits a transitive extension . 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 <!-- MATH ${\text{Sp}}(2n,2)$ --> , and the doubly transitive representations of the Higman-Sims group, and the Conway group (.3).