Isomorphisms of Cayley graphs on nilpotent groups

Let S be a finite generating set of a torsion-free, nilpotent group G. We show that every automorphism of the Cayley graph Cay(G; S) is affine. (That is, every automorphism of the graph is obtained by composing a group automorphism with multiplication by an element of the group.) More generally, we show that if Cay(G(1); S-1) and Cay(G(2); S-2) are connected Cayley graphs of finite valency on two nilpotent groups G(1) and G(2), then every isomorphism from Cay(G(1); S-1) to Cay(G(2); S-2) factors through to a well-defined affine map from G(1)/N-1 to G(2)/N-2, where is the torsion subgroup of G,. For the special case where the groups are abelian, these results were previously proved by A. A. Ryabchenko and C. Loh, respectively.