Special endomorphisms of finitely generated Abelian groups
https://doi.org/10.22405/2226-8383-2025-26-3-247-256
Abstract
The paper is devoted to abelian groups containing at least one endomorphism whose kernel coincides with its image. Note that the condition ker𝜙 = Im 𝜙 implies the equality 𝜙^2 = 0, that is, 𝜙 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 ⊕ 𝐾, where 𝐹 is a free Abelian group
of even rank and 𝐾 is an arbitrary finite Abelian group.
About the Author
Asadullah Sarwary
Moscow Pedagogical State University
Russian Federation
Russian Federation
postgraduate student
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Review
For citations:
Sarwary A. Special endomorphisms of finitely generated Abelian groups. Chebyshevskii Sbornik. 2025;26(3):247-256. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-3-247-256
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