On the Cayley isomorphism problem for ternary relational structures (Q1869760)

The Cayley isomorphism problem asks for which groups \(G\) it is true that every two Cayley objects of \(G\) are isomorphic if and only if they are isomorphic by a group isomorphism of \(G\). Such groups are called CI-groups with respect to the underlying class of combinatorial objects. The objects considered here are ternary relational structures. A list of all possible CI-groups with respect to color ternary relational structures is provided. Of those, the groups of order \(2n\) and \(4n\) with a cyclic Sylow 2-subgroup are either shown to be CI-groups as is the case for most of them, or, in the unresolved cases necessary and sufficient conditions for the group to be a CI-group with respect to ternary relational structures are provided. Since a CI-group with respect to ternary relational structures is also a CI-group with respect to binary relational structures, some of the examined groups provide new examples of CI-groups with respect to graphs and digraphs.