NoteA note on connected cubic Cayley graphs

Let Γ=Cay(G,S) be a connected cubic Cayley graph, and let X=AutΓ be the automorphism group of Γ. It is proved in this note that either G has non-trivial core ∩x∈XGx in X, or the stabilizer Xu of a vertex u of Γ is non-abelian.