The strongly Hopfian abelian groups

Abstract

Let A be an abelian group, A is said strongly hopfian if for all f ∈ End(A) we have the chain ker(f) ⊆ ker(f²) ⊆ ker(f³) ⊆ ker(f⁴) ⊆ ... is stationary. In this paper, we characterize strongly hopfian abelian group in the class of torsion abelian group. After we show that the p-component of strongly hopfian abelian group is strongly hopfian. But for the torsion part, we construct strongly hopfian abelian group whose the torsion part is not strongly hopfian.