A note on connected cubic Cayley graphs

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