On the automorphism groups of quasiprimitive almost simple graphs

Let Gamma be a graph and let G be a subgroup of automorphisms of Gamma. Then G is said to be locally primitive on Gamma if, for each vertex upsilon, the stabilizer G(upsilon) induces a primitive group of permutations on the set of vertices adjacent to upsilon. This paper investigates pairs (Gamma, G) for which G is locally primitive on Gamma, G is an almost simple group (that is, L less than or equal to G less than or equal to 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. Tn the general case in which Aut Gamma is not quasiprimitive on vertices, we show that either every intransitive minimal normal subgroup of Aut Gamma centralizes L, or L is of Lie type and Aut Gamma involves an explicitly known same characteristic module for L. (C) 1999 Academic Press.