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Journal of Zhejiang University SCIENCE A 2003 Vol.4 No.1 P.76-79

http://doi.org/10.1631/jzus.2003.0076


Construction of some hypergroups from combinatorial structures


Author(s):  Ali Reza Ashrafi, Ahmad Reza Eslami-Harandi

Affiliation(s):  Department of Mathematics, Faculty of Science, University of Kashan, Kashan, Iran

Corresponding email(s):   ashrafi@kashanu.ac.ir

Key Words:  Finite group, Rotary closed subgroup, Hypergroup, Sub-hypergroup, Combinatorial structures


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Ali Reza Ashrafi, Ahmad Reza Eslami-Harandi. Construction of some hypergroups from combinatorial structures[J]. Journal of Zhejiang University Science A, 2003, 4(1): 76-79.

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Abstract: 
Jajcay's studies (1993; 1994) on the automorphism groups of Cayley maps yielded a new product of groups, which he called, rotary product. Using this product, we define a hyperoperation ⊙ on the group Syme(G), the stabilizer of the identity e∈G in the group Sym(G). We prove that (Syme(G), ⊙) is a hypergroup and characterize the subhypergroups of this hypergroup. Finally, we show that the set of all subhypergroups of Syme(G) constitute a lattice under ordinary join and meet and that the minimal elements of order two of this lattice is a subgroup of Aut(G).

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Reference

[1]Biggs, N.and White,A.T., 1979. Permutation Groups and Combinatorial Structures. Mathematical Society Lecture Notes 33, Cambridge University Press, Cambridge.

[2]Birkhoff,G., 1967. Lattice Theory(3rd. ed.), AMS Coll. Publ.Providence.

[3]Corsini,P., 1993, Prolegomena of Hypergroup Theory. Second Edition, Aviani Edittore.

[4]Jajcay,R., 1993, Automorphism groups of Cayley maps. J. of Comb. Theory, Ser.B, 59: 297-310.

[5]Jajcay, R., 1994, On a new product of groups. Europ. J. Combinatorics, 15: 251-252.

[6]Madanshekaf,A. and Ashrafi,A.R., 1998. Generalized Action of a hypergroup on a set. Italian J. of Pure and Appl. Math., (3): 127-135.

[7]Marty,F., 1934.Sur une generalization de lanotion de groupe, 8iem Congress Math. Scandinaves, Stockholm. p.45-49.

[8]Vougiouklis,T., 1994. Hyperstructures and their Representations, Hadronic Press, Inc.

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