On the Cayley isomorphism problem for ternary relational structures

A ternary relational structure X is an ordered pair (V,E), where E⊂V3. A ternary relational structure X is a Cayley ternary relational structure of a group G if the left regular representation of G is contained in the automorphism group of X. A group G is a CI-group with respect to ternary relational structures if whenever X and X′ are isomorphic Cayley ternary relational structures of G, then X and X′ are isomorphic if and only if they are isomorphic by an automorphism of G. In this paper, we will provide a (relatively short) list of all possible CI-groups with respect to color ternary relational structures. All of these groups have order 2dn, 0⩽d⩽5, and n a positive integer with gcd(n,ϕ(n))=1, where ϕ is Eulerʹs phi function. If d=0, it has been shown by Pálfy that Zn is a CI-group with respect to every class of combinatorial objects. We then show that of the possible CI-groups with respect to color ternary relational structures of order 2n and 4n with a cyclic Sylow 2-subgroup, most are CI-groups with respect to ternary relational structures, and in the unresolved cases, give a necessary and sufficient condition for the group to be a CI-group with respect to ternary relational structures. In particular, we determine all cyclic CI-groups with respect to ternary relational structures. Finally, a group G that is a CI-group with respect to ternary relational structures is also a CI-group with respect to binary relational structures (i.e. graphs and digraphs). Some of the groups considered in this paper are not known to be CI-groups with respect to graphs or digraphs, and we thus provide new examples of CI-groups with respect to graphs and digraphs.