Abstract:
The paper is devoted to abelian groups containing at least one endomorphism whose kernel coincides with its image. Note that the condition $\ker\varphi=\mathrm{Im} \varphi$ implies the equality $\varphi^2=0$, that is, $\varphi$ is a nilpotent endomorphism of nilpotency index $2$.
The main technical result of the paper is Theorem 1, in which a criterion for the existence of an endomorphism of an abelian group whose kernel coincides with its image is obtained in the language of subgroups.
In this paper, the existence of an endomorphism whose kernel coincides with its image is completely solved for Abelian groups from the classes of cyclic and cocyclic groups, elementary $p$-primary Abelian groups, and finitely generated Abelian groups.
The main result of the paper is Theorem 12, which proves that a finitely generated Abelian group $A$ has an endomorphism whose image coincides with its kernel if and only if either $A$ is a finite group whose order is a perfect square, or $A=F \oplus K$, where $F$ is a free Abelian group of even rank and $K$ is an arbitrary finite Abelian group.
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