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The isomorphism problem for Cayley ternary relational structures for some abelian groups of order $$8p$$. (English) Zbl 1230.05155
Summary: 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 $${\mathbb{Z}}^3_2 \times {\mathbb{Z}}_p$$, $$p\geq 11$$ a prime, are isomorphic if and only if they are isomorphic by a group automorphism of $${\mathbb{Z}}^3_2 \times {\mathbb{Z}}_p$$. This result then implies that any two Cayley digraphs of $${\mathbb{Z}}^3_2 \times {\mathbb{Z}}_p$$ are isomorphic if and only if they are isomorphic by a group automorphism of $${\mathbb{Z}}^3_2 \times {\mathbb{Z}}_p$$, $$p\geq 11$$ a prime.

##### MSC:
 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
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