ℤ_3^8 is not a CI-group
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Date
2024
Authors
Morris, Joy
Journal Title
Journal ISSN
Volume Title
Publisher
University of Primorska
Abstract
A Cayley graph Cay(G, S) has the CI (Cayley Isomorphism) property if for every isomorphic graph Cay(G, T), there is a group automorphism α of G such that Sα = T. The DCI (Directed Cayley Isomorphism) property is defined analogously on digraphs. A group G is a CI-group if every Cayley graph on G has the CI property, and is a DCI-group if every Cayley digraph on G has the DCI property. Since a graph is a special type of digraph, this means that every DCI-group is a CI-group, and if a group is not a CI-group then it is not a DCI-group.
In 2009, Spiga showed that ℤ38 is not a DCI-group, by producing a digraph that does not have the DCI property. He also showed that ℤ35 is a DCI-group (and therefore also a CI-group). Until recently the question of whether there are elementary abelian 3-groups that are not CI-groups remained open. In a recent preprint with Dave Witte Morris, we showed that ℤ310 is not a CI-group. In this paper we show that with slight modifications, the underlying undirected graph of order 38 described by Spiga is does not have the CI property, so ℤ38 is not a CI-group.
Description
Open access article. Creative Commons Attribution 4.0 International license (CC BY 4.0) applies
Keywords
Cayley graphs , Elementary abelian groups , CI graphs , CI groups , Isomorphism
Citation
Morris, J. (2024). ℤ_3^8 is not a CI-group. Ars Mathematica Contemporanea, 24(4), Article #P4.04. https://doi.org/10.26493/1855-3974.3087.f36