Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X39120130301The Quasi-morphic Property of Group175185339ENQ.WangDepartment of Mathematics and Systems Science, National University of Defense Technology, P.R.China 410073,Changsha, China.K.LongDepartment of Mathematics and Systems Science, National University of Defense Technology ,P.R.China 410073, Changsha, China.L.FengDepartment of Mathematics and Systems Science, National University of Defense
Technology, P.R.China 410073, Changsha, China.Journal Article20110528A 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.http://bims.iranjournals.ir/article_339_e1fa74b090c7cf943c21d3c24a31908a.pdf