Morris, Joy

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The CI problem for infinite groups
(Electronic Journal of Combinatorics, 2016) Morris, Joy
A ﬁnite group G is a DCI-group if, whenever S and S0 are subsets of G with the Cayley graphs Cay(G,S) and Cay(G,S0) isomorphic, there exists an automorphism ϕ of G with ϕ(S) = S0. 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.