# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Ádám, A., Research problem 2-10, J. combin. theory, 2, 393, (1967)  Babai, L., Isomorphism problem for a class of point-symmetric structures, Acta math. acad. sci. hungar., 29, 3-4, 329-336, (1977) · Zbl 0378.05035  Conder, M.; Li, C.H., On isomorphisms of finite Cayley graphs, European J. combin., 19, 8, 911-919, (1998) · Zbl 0916.05034  Dixon, J.D.; Mortimer, B., ()  Dobson, E., On the Cayley isomorphism problem for ternary relational structures, J. combin. theory ser. A, 101, 2, 225-248, (2003) · Zbl 1028.05069  E. Dobson, On transitive ternary relational structures of order a prime-squared, Ars Combin. (in press). · Zbl 1240.20002  Dobson, E., Isomorphism problem for Cayley graphs of $$\mathbb{Z}_p^3$$, Discrete math., 147, 1-3, 87-94, (1995) · Zbl 0838.05081  Dobson, E.; Witte, D., Transitive permutation groups of prime-squared degree, J. algebraic combin., 16, 1, 43-69, (2002) · Zbl 1027.20001  E. Dobson, P. Spiga, Some new groups which are not CI-groups with respect to ternary relational structures (submitted for publication). · Zbl 1290.05074  Elspas, B.; Turner, J., Graphs with circulant adjacency matrices, J. combin. theory, 9, 297-307, (1970) · Zbl 0212.29602  Gorenstein, D., Finite groups, (1968), Harper & Row Publishers New York · Zbl 0185.05701  Kalužnin, L.A.; Klin, M.H., Some numerical invariants of permutation groups, Latviĭsk. mat. Ežegodnik, 81-99, Vyp 18, 222, (1976)  Li, C.H., On isomorphisms of finite Cayley graphs—a survey, Discrete math., 256, 1-2, 301-334, (2002) · Zbl 1018.05044  Meldrum, J.D.P., Wreath products of groups and semigroups, () · Zbl 0833.20001  Muzychuk, M., On the isomorphism problem for cyclic combinatorial objects, Discrete math., 197/198, 589-606, (1999), 16th British Combinatorial Conference (London, 1997) · Zbl 0938.05005  Passman, D., Permutation groups, (1968), W. A. Benjamin, Inc. New York, Amsterdam · Zbl 0179.04405  Pálfy, P.P., Isomorphism problem for relational structures with a cyclic automorphism, European J. combin., 8, 1, 35-43, (1987) · Zbl 0614.05049  Sabidussi, G., Vertex-transitive graphs, Monatsh. math., 68, 426-438, (1964) · Zbl 0136.44608  Spiga, P., On the Cayley isomorphism problem for a digraph with 24 vertices, Ars math. contemp., 1, 1, 38-43, (2008) · Zbl 1163.05058  H. Wielandt, Permutation groups through invariant relations and invariant functions, lectures given at The Ohio State University, Columbus, Ohio, 1969.  Wielandt, H., Mathematische werke/mathematical works, vol. 1, (1994), Walter de Gruyter & Co. Berlin, Group theory, with essays on some of Wielandt’s works by G. Betsch, B. Hartley, I. M. Isaacs, O.H. Kegel and P.M. Neumann, Edited and with a preface by Bertram Huppert and Hans Schneider  Wielandt, H., Finite permutation groups, () · Zbl 0138.02501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.