The isomorphism problem for Cayley ternary relational structures for some abelian groups of order

A ternary relational structure X is an ordered pair ( V , E ) where V is a set and E a set of ordered 3-tuples whose coordinates are chosen from V (so a ternary relational structure is a natural generalization of a 3-uniform hypergraph). A ternary relational structure is called a Cayley ternary relational structure of a group G if Aut ( X ) , the automorphism group of X , contains the left regular representation of G . We prove that two Cayley ternary relational structures of Z 2 3 × Z p , p ≥ 11 a prime, are isomorphic if and only if they are isomorphic by a group automorphism of Z 2 3 × Z p . This result then implies that any two Cayley digraphs of Z 2 3 × Z p are isomorphic if and only if they are isomorphic by a group automorphism of Z 2 3 × Z p , p ≥ 11 a prime.