the quasi-morphic property of group

a group is called morphic if for each normal endomorphism α in end(g),there exists β such that ker(α)= gβ and gα= ker(β). in this paper, we consider the case that there exist normal endomorphisms β and γ such that ker(α)= gβ and gα = ker(γ). we call g quasi-morphic, if this happens for any normal endomorphism α in end(g). we get the following results: g is quasi-morphic if and only if, for any normal subgroup k and n such that g/k≌n, there exist normal subgroup t and h such that g/t≌k and g/n≌h. further, we investigate the quasi-morphic property of finitely generated abelian group and get that a finitely generated abelian group is quasi-morphic if and only if it is finite.