On the Automorphism Groups of Quasiprimitive Almost Simple Graphs

Let Γ be a graph and let G be a subgroup of automorphisms of Γ. Then G is said to be locally primitive on Γ if, for each vertex v, the stabilizer Gv induces a primitive group of permutations on the set of vertices adjacent to v. This paper investigates pairs (Γ, G) for which G is locally primitive on Γ, G is an almost simple group (that is, L ≤ G ≤ Aut(L) for some nonabelian simple group L), and the simple socle L is transitive on vertices. Each such graph is a cover of a possibly smaller graph on which G is also locally primitive, and for which in addition Aut is quasiprimitive on vertices (that is, every nontrivial normal subgroup of Aut is vertex-transitive). It is proved that Aut is also an almost simple group. In the general case in which Aut Γ is not quasiprimitive on vertices, we show that either every intransitive minimal normal subgroup of Aut Γ centralizes L, or L is of Lie type and Aut Γ involves an explicitly known same characteristic module for L.