Two Generalizations of Hopfian Abelian Groups

2024/08/02 by A. R. Chekhlov, Peter Danchev, Chekhlov, Andrey R. +5
Mathematics · #Advanced Topology and Set Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Group Theory (math.GR) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2408.01277

openalex publication_date 2024/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper targets to generalize the notion of Hopfian groups in the commutative case by defining the so-called \bf relatively Hopfian groups and \bf weakly Hopfian groups, and establishing some their crucial properties and characterizations. Specifically, we prove that for a reduced Abelian p-group G such that p^ωG is Hopfian (in particular, is finite), the notions of relative Hopficity and ordinary Hopficity do coincide. We also show that if G is a reduced Abelian p-group such that p^ωG is bounded and G/p^ωG is Hopfian, then G is relatively Hopfian. This allows us to construct a reduced relatively Hopfian Abelian p-group G with p^ωG an infinite elementary group such that G is \bf not Hopfian. In contrast, for reduced torsion-free groups, we establish that the relative and ordinary Hopficity are equivalent. Moreover, the mixed case is explored as well, showing that the structure of both relatively and weakly Hopfian groups can be quite complicated.

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