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Construction of some hypergroups from combinatorial structures.

Ali Reza Ashrafi¹, Ahmad Reza Eslami-Harandi

¹Department of Mathematics, Faculty of Science, University of Kashan, Kashan, Iran. ashrafi@kashanu.ac.ir

Journal of Zhejiang University. Science

|March 27, 2003

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Researchers defined a new hyperoperation on the group Syme(G) using Jajcay's rotary product. This structure forms a hypergroup, with its subhypergroups creating a lattice whose minimal elements relate to Aut(G).

Area of Science:

Group Theory
Abstract Algebra
Combinatorial Group Theory

Background:

Jajcay introduced the rotary product of groups in studies from 1993 and 1994.
The study focuses on the automorphism groups of Cayley maps.

Purpose of the Study:

To define a hyperoperation on the group Syme(G), the stabilizer of the identity element in Sym(G).
To establish that (Syme(G), [symbol: see text]) forms a hypergroup.
To characterize the subhypergroups of this hypergroup and analyze the lattice structure they form.

Main Methods:

Utilizing Jajcay's rotary product to define a novel hyperoperation, denoted by [symbol: see text].
Applying group theory and hypergroup theory to analyze the properties of Syme(G) under this hyperoperation.

Related Experiment Videos

Investigating the lattice formed by the subhypergroups of Syme(G) under join and meet operations.

Main Results:

The set Syme(G) equipped with the defined hyperoperation [symbol: see text] is proven to be a hypergroup.
The subhypergroups of (Syme(G), [symbol: see text]) are characterized.
The set of all subhypergroups of Syme(G) forms a lattice under ordinary join and meet.
The minimal elements of order two within this lattice are shown to be a subgroup of Aut(G).

Conclusions:

The novel hyperoperation provides a new algebraic structure on Syme(G).
The characterization of subhypergroups and the resulting lattice structure offer insights into the group's properties.
The connection between the lattice's minimal elements and Aut(G) highlights a significant structural relationship.