Abstract
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 $\varphi$ of $G$ with $\varphi(S)=S'$. 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)CI$_f$-group if the same condition holds under the restricted assumption that $S$ is finite; and an infinite group is a (D)CI$_f$-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 locally-finite. We find infinite families of groups that are (D)CI$_f$-groups but not strongly (D)CI$_f$-groups, and that are strongly (D)CI$_f$-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 $\mathbb Z^n$. We suggest several open problems related to these ideas, including the question of whether or not any infinite (D)CI-group exists.
Highlights
There has been considerable work done on the Cayley Isomorphism problem for finite groups and graphs, little attention has been paid to its extension to the infinite case
A Cayleygraph Γ = Cay(G; S) is a (D)CI-graph if whenever φ : Γ → Γ is an isomorphism, with Γ = Cay(G; S ), there is a group automorphism α of G with α(S) = S
We prove that no infinite abelian group is a (D)CI-group, and that any (D)CI-group must be a torsion group that is not locally finite
Summary
There has been considerable work done on the Cayley Isomorphism problem for finite groups and graphs, little attention has been paid to its extension to the infinite case. If one wishes to study this problem in the context of locally-finite (infinite) (di)graphs, it is necessary to consider disconnected as well as connected (di)graphs For this reason, we give two new definitions. In this paper we will construct examples of groups that are (D)CIf -groups but not strongly (D)CIf -groups (despite being finitely generated) and groups that are strongly (D)CIf -groups but not (D)CI-groups, so these definitions are interesting We further study these classes, in the case of infinite abelian groups, including a complete characterisation of the locally-finite graphs on Zn that are (D)CI-graphs. In the only prior work that we are aware of that is aimed at solving the CI problem for infinite graphs, Ryabchenko [29] uses the standard definition (the same one we gave above) for a CI-group, and claims to have proven that every finitely-generated free abelian group is a CI-group. Has exactly two connected components, and the connection set S is invariant under some automorphism of order 3 of S
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