On Graphs Admitting Arc-Transitive Actions of Almost Simple Groups

Abstract

Let Γ be a finite connected regular graph with vertex setVΓ, and letGbe a subgroup of its automorphism group AutΓ. 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 socG=S; that is,Sis a nonabelian simple group andS⊴G≤AutS. We study nonbipartite graphs Γ which areG-locally primitive, such thatShas trivial centralizer in AutΓ andSis not semiregular on vertices. We prove that one of the following holds: (i)S⊴AutΓ≤Aut(S), (ii)G<Y≤AutΓ withYalmost simple and socY≠S, or (iii)Sbelongs to a very restricted family of Lie type simple groups of characteristicp, say, and AutΓ contains the semidirect productZdp:G, whereZdpis a known absolutely irreducibleG-module. Moreover, in certain circumstances we can guarantee thatS⊴AutΓ≤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≤AutΓ≤Aut(G).