On the generic family of Cayley graphs of a finite group

The Cayley graphs Gm(G)=Cay(Gm,S) of Cartesian powers the finite group G with respect to canonical symmetric subsets S=S(m)⊂Gm are investigated. It is proved that for each m>1 graph Gm(G) determines up isomorphism. groups automorphisms Aut(Gm(G)) determined. shown if a non-abelian group, then Aut(Gm(G))≃(Gm⋊Aut(G))⋊Dm+1, where Dm+1 dihedral order 2m+2. If an abelian (with some exceptions m=3), Aut(Gm(G))≃Gm⋊(Aut(G)×Sm+1), Sm+1 degree m+1. Relations between and Bergman-Isaacs Theorem on rings fixed-point-free actions discussed.