Strongly and Uniformly Strongly co-Hopfian Abelian Groups

TL;DR

This paper characterizes strongly and uniformly strongly co-Hopfian Abelian groups, revealing their structural properties, relationships with torsion subgroups, and providing examples illustrating the complexity of their structure.

Contribution

It generalizes previous results by Abdelalim, characterizes strongly co-Hopfian groups as direct sums, and explores their properties related to torsion and cotorsion groups.

Findings

Strongly co-Hopfian groups are direct sums of an sp-group and a divisible group.

A group with strongly co-Hopfian torsion and torsion-free parts is strongly co-Hopfian.

Strongly co-Hopfian groups are cotorsion if and only if they are algebraically compact.

Abstract

We consider the so-called {\it strongly co-Hopfian} and {\it uniformly strongly co-Hopfian} Abelian groups, significantly generalizing some important results due to Abdelalim in the J. Math. Analysis (2015). Specifically, we prove that any strongly co-Hopfian group is a direct sum of an sp-group and a divisible group, both of which are strongly co-Hopfian. We also show that a group whose maximal torsion subgroup and corresponding torsion-free factor are both strongly co-Hopfian will also be strongly co-Hopfian. We provide several examples demonstrating that the converse of this statement does {\it not} generally hold, thus illustrating that the structure of genuinely mixed strongly co-Hopfian groups is rather complicated and does {\it not} entirely depend on the structure of its maximal torsion subgroup. We also establish that a strongly co-Hopfian group is cotorsion exactly when it is algebraically compact and, particularly, a reduced (adjusted) cotorsion group is strongly co-Hopfian only when its maximal torsion subgroup is strongly co-Hopfian. Additionally, we demonstrate that a strongly co-Hopfian group is uniformly strongly co-Hopfian exactly when its maximal torsion subgroup is strongly co-Hopfian.