Abstract
Let Γ be a graph and let G be a group of automorphisms of Γ. The graph Γ is called G-normal if G is normal in the automorphism group of Γ. Let T be a finite non-abelian simple group and let G=Tl with l≥1. In this paper we prove that if every connected pentavalent symmetric T-vertex-transitive graph is T-normal, then every connected pentavalent symmetric G-vertex-transitive graph is G-normal. This result, among others, implies that every connected pentavalent symmetric G-vertex-transitive graph is G-normal except T is one of 57 simple groups. Furthermore, every connected pentavalent symmetric G-regular graph is G-normal except T is one of 20 simple groups, and every connected pentavalent G-symmetric graph is G-normal except T is one of 17 simple groups.
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