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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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