Some Examples of a Hypergroup (ℤₙ ,∘ₘ ℤₙ)

Abstract

Let G be a group and N a normal subgroup of G. If the hyperoperation ∘ₙ is defined by

x ∘ₙ y = (xy)N for all x, y ∈ G,

then (G, ∘ₙ) is a hypergroup. Since mℤₙ is a normal subgroup of ℤₙ, (ℤₙ, ∘ₘℤₙ) is a hypergroup.

In this paper, we let G∣ₙH = {g ∘ₙ H ∣ g ∈ G} and give some examples that G∣ₘℤₙ kℤₙ equals ℤₙ/kℤₙ.

We take a hyperoperation ∘ₙ to construct cosets of any subgroup H of G instead of coset multiplication by the binary operation of G and study some examples of this new structure of cosets.