On Graphs Admitting Arc-Transitive Actions of Almost Simple Groups☆

Abstract Let Γ be a finite connected regular graph with vertex set V Γ, and let G be a subgroup of its automorphism group Aut Γ. Then Γ is said to be G-locally primitive if, for each vertex α, the stabilizer G α is primitive on the set of vertices adjacent to α. 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  ⊴  G  ≤ Aut  S . We study nonbipartite graphs Γ which are G -locally primitive, such that S has trivial centralizer in Aut Γ and S is not semiregular on vertices. We prove that one of the following holds: (i) S  ⊴ Aut Γ ≤ Aut( S ), (ii) G Y  ≤ Aut Γ with Y almost simple and soc  Y  ≠  S , or (iii) S belongs to a very restricted family of Lie type simple groups of characteristic p , say, and Aut Γ contains the semidirect product Z d p : G , where Z d p is a known absolutely irreducible G -module. Moreover, in certain circumstances we can guarantee that S  ⊴ Aut Γ ≤ Aut( S ). For example, if Γ is a connected ( G , 2)-arc transitive graph with Sz( q ) ≤  G  ≤ Aut(Sz( q )) ( q  = 2 2 n  + 1  ≥ 8) or G  = Ree( q ) ( q  = 3 2 n  + 1  ≥ 27), then G  ≤ Aut Γ ≤ Aut( G ).