Two Generalizations of Hopfian Abelian Groups

TL;DR

This paper introduces and studies the concepts of relatively and weakly Hopfian Abelian groups, establishing their properties, characterizations, and differences from traditional Hopfian groups, especially in the context of p-groups and torsion-free groups.

Contribution

It generalizes Hopfian groups by defining relative and weak variants, providing key properties, characterizations, and examples distinguishing these from classical Hopfian groups.

Findings

For reduced Abelian p-groups with finite p^ωG, relative and ordinary Hopficity coincide.

If p^ωG is bounded and G/p^ωG is Hopfian, then G is relatively Hopfian.

Existence of a reduced, relatively Hopfian Abelian p-group with infinite p^ωG that is not Hopfian.

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 $p^{\omega}G$ 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 $p^{\omega}G$ is bounded and $G/p^{\omega}G$ is Hopfian, then $G$ is relatively Hopfian. This allows us to construct a reduced relatively Hopfian Abelian $p$-group $G$ with $p^{\omega}G$ 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.