The CI Problem for Infinite Groups

A finite group G is a DCI-group if, whenever S and S′ are subsets of G with the Cayley graphs Cay(G,S) and Cay(G,S′) isomorphic, there exists an automorphism φ of G with φ(S) = S′. It is a CI-group if this condition holds under the restricted assumption that S = S−1. We extend these definitions to infinite groups, and make two closely-related definitions: an infinite group is a strongly (D)CIf -group if the same condition holds under the restricted assumption that S is finite; and an infinite group is a (D)CIf -group if the same condition holds whenever S is both finite and generates G. We prove that an infinite (D)CI-group must be a torsion group that is not locallyfinite. We find infinite families of groups that are (D)CIf -groups but not strongly (D)CIf -groups, and that are strongly (D)CIf -groups but not (D)CI-groups. We discuss which of these properties are inherited by subgroups. Finally, we completely characterise the locally-finite DCI-graphs on Zn. We suggest several open problems related to these ideas, including the question of whether or not any infinite (D)CIgroup exists.