On the Automorphism Groups of Quasiprimitive Almost Simple Graphs

AbstractLet 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 Γ̄ on which G is also locally primitive, and for which in addition AutΓ̄ is quasiprimitive on vertices (that is, every nontrivial normal subgroup of AutΓ̄ is vertex-transitive). It is proved that AutΓ̄ is also an almost simple group. In the general case in which AutΓ is not quasiprimitive on vertices, we show that either every intransitive minimal normal subgroup of AutΓ centralizes L, or L is of Lie type and AutΓ involves an explicitly known same characteristic module for L