Paper Info
Title
The Ci Problem For Infinite Groups
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 phi of G with phi(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)C If-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.
Year
Venue
Field
2016
ELECTRONIC JOURNAL OF COMBINATORICS
Discrete mathematics,Infinite group,Combinatorics,Existential quantification,Automorphism,Cayley graph,Isomorphism,Finite group,Mathematics
DocType
Volume
Issue
Journal
23
4
ISSN
Citations 
PageRank 
1077-8926
0
0.34
References 
Authors
3
1
Name
Order
Citations
PageRank
Joy Morris17816.06