On graphs admitting arc-transitive actions of almost simple groups

Let Gamma be a finite connected regular graph with vertex set V Gamma, and let G be a subgroup of its automorphism group Aut Gamma. Then Gamma is said to be G-locally primitive if, for each vertex alpha, the stabilizer G(alpha) is primitive on the set of vertices adjacent to alpha. In this paper we assume that G is an almost simple group with socle soc G = S; that is, S is a nonabelian simple group and S left-pointing triangle with bar underneath G less than or equal to Aut S. We study nonbipartite graphs Gamma which are G-locally primitive, such that S has trivial centralizer in Aut Gamma and S is not semiregular on vertices. We prove that one of the following holds: (i) S left-pointing triangle with bar underneath Aut Gamma less than or equal to Aut(S), (ii) G < Y less than or equal to Aut Gamma with Y almost simple and soc Y not equal S, or (iii) S belongs to a very restricted family of Lie type simple groups of characteristic p, say, and Aut Gamma contains the semidirect product Z(p)(d):G, where Z(p)(d) is a known absolutely irreducible G-module. Moreover, in certain circumstances we can guarantee that S Aut Gamma less than or equal to Aut(S). For example, if Gamma is a connected (G,2)-arc transitive graph with sz(q) less than or equal to G less than or equal to Aut(Sz(q)) (q = 2(2n+1) greater than or equal to 8) or G = Ree(q) (q = 3(2n+1) greater than or equal to 27), then G less than or equal to Aut Gamma less than or equal to Aut(G). (C) 1998 Academic Press.