Isomorphisms of Cayley graphs on nilpotent groups
نویسندگان
چکیده
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(G1;S1) and Cay(G2;S2) are connected Cayley graphs of finite valency on two nilpotent groups G1 and G2, then every isomorphism from Cay(G1;S1) to Cay(G2;S2) factors through to a well-defined affine map from G1/N1 to G2/N2, where Ni is the torsion subgroup of Gi. For the special case where the groups are abelian, these results were previously proved by A. A. Ryabchenko and C. Löh, respectively.
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