Structures induced on transversals to a subgroup of a group and hypergroups over the group

TL;DR

This paper introduces hypergroups over a group, generalizing quotient groups by defining structural mappings on transversals of a subgroup, and explores their categorical relationships with groups, linear spaces, and fields.

Contribution

It presents a new concept of hypergroups over a group, extending the structure of quotient groups and unifying various algebraic categories.

Findings

Hypergroups over a group generalize quotient groups.

The category of hypergroups over a group includes groups, linear spaces, and fields.

Structural mappings on transversals form the basis of hypergroup definitions.

Abstract

On the transversals of a subgroup of a group, using the binary operation of the group, structural mappings are defined. Based on these mappings, the notion of the hypergroup over the group is introduced, which generalizes the notion of the quotient-group of the given group with respect to the normal subgroup. The category of the hypergroups over the group contains as subcategories the category of groups, the category of linear spaces, the category of fields.