| Morphic endmorphism is an abstraction of structural processes between two mathematical structures,which is of great significance in category theory and basic algebra.This paper is based on the study of morphic and quasi-morphic of groups.First of all,we defined a group homomorphism with some properties,called dual homomorphism,we gave the characterizations of group homomorphisms and dual homomorphisms,and discussed some properties of dual homomorphisms.Secondly,we also gave the relationships between dual homomorphism and group isomorphism,we pointed out that the dual homomorphism is injective if and only if it is surjective,and also illustrated that the composition of dual homomorphism is not necessarily a dual homomorphism.Moreover,this article also gave the definition of quasi dual homomorphism on the basis of quasi-morphic,and we obtained a sufficient and necessary condition that a group homomorphism is a quasi dual homomorphism.Finally,we discussed the morphism and quasi morphism of abelian groups,and proved that finite abelian p-groups are quasi morphic groups. |
