The Quasi-morphic Property of Group

Document Type: Research Paper

Authors

1 Department of Mathematics and Systems Science, National University of Defense Technology, P.R.China 410073,Changsha, China.

2 Department of Mathematics and Systems Science, National University of Defense Technology ,P.R.China 410073, Changsha, China.

3 Department of Mathematics and Systems Science, National University of Defense Technology, P.R.China 410073, Changsha, China.

Abstract

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.

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