THE QUASI-MORPHIC PROPERTY OF GROUP

A group is called morphic, if for each normal endomorphism α element of end(G), there exists β element of end(G) such that ker(α) = G beta and G α = ker(β). Here, we consider the case that there exist normal endomorphisms beta and γ such that ker(α) = G β and G α = ker(γ). We call G quasi-morphic, if this happens for any normal endomorphism alpha element of end(G). We get the following results: G is quasi-morphic if and only if, for any normal subgroup K such that G/K approximately equal to N triangleleft G, there exist T.H triangleleft G such that G/T approximately equal to K and G/N approximately equal to H. Furthermore, 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.