On the automorphism groups of quasiprimitive almost simple graphs

Let Gamma be a graph and let G be a subgroup of automorphisms of Gamma. The n G is said to be locally primitive on Gamma if, for each vertex upsilon, t he stabilizer G(upsilon) induces a primitive group of permutations on the s et 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 (t hat is, L less than or equal to G less than or equal to Aut(L) for some non abelian simple group L), and the simple socle L is transitive on vertices. Each such graph is a cover of a possibly smaller graph <(Gamma)over tilde> on which G is also locally primitive, and for which in addition Aut <(Gamma )over tilde> is quasiprimitive on vertices (that is, every nontrivial norma l subgroup of Aut <(Gamma)over tilde> is vertex-transitive). it is proved t hat Aut <(Gamma)over tilde> is also an almost simple group. Tn the general case in which Aut Gamma is not quasiprimitive on vertices, we show that eit her every intransitive minimal normal subgroup of Aut Gamma centralizes L, or L is of Lie type and Aut Gamma involves an explicitly known same charact eristic module for L. (C) 1999 Academic Press.