Abelian Groups Derived From Hypergroups

Abstract

We introduce a new strongly regular relation α on a given group G and show that α is a congruence relation on G, with respect to module the commutator subgroup of G. Then we show that the composition of this relation with the fundamental relation β* is equal to the fundamental and γ are is equal to the relation α. and we conclude that if ρ is an arbitrary strongly regular relation on the hypergroup H, then the effect of α on ρ, results in a strongly regular relation such that its quotients is an abelian group.