On Graphs Admitting Arc-Transitive Actions of Almost Simple Groups

Let Γ be a finite connected regular graph with vertex setVΓ, and letGbe a subgroup of its automorphism group Aut Γ. Then Γ is said to beG-locally primitiveif, for each vertex α, the stabilizerGαis primitive on the set of vertices adjacent to α. In this paper we assume thatGis an almost simple group with socle soc G = S; that is,Sis a nonabelian simple group andS G ≤ Aut S. We study nonbipartite graphs Γ which areG-locally primitive, such thatShas trivial centralizer in Aut Γ andSis not semiregular on vertices. We prove that one of the following holds: (i)S Aut Γ ≤ Aut(S), (ii)G < Y ≤ Aut Γ withYalmost simple and soc Y ≠ S, or (iii)Sbelongs to a very restricted family of Lie type simple groups of characteristicp, say, and Aut Γ contains the semidirect productZdp:G, whereZdpis a known absolutely irreducibleG-module. Moreover, in certain circumstances we can guarantee thatS Aut Γ ≤ Aut(S). For example, if Γ is a connected (G, 2)-arc transitive graph with Sz(q) ≤ G ≤ Aut(Sz(q)) (q = 22n + 1 ≥ 8) orG = Ree(q) (q = 32n + 1 ≥ 27), thenG ≤ Aut Γ ≤ Aut(G).