On the existence of uncountable Hopfian and co-Hopfian abelian groups

TL;DR

This paper investigates the existence and properties of uncountable Hopfian and co-Hopfian abelian groups, establishing non-existence results for certain classes and constructing examples of absolutely Hopfian groups.

Contribution

It proves non-existence of uncountable co-Hopfian reduced abelian groups under specific conditions and constructs torsion-free abelian groups that are absolutely Hopfian.

Findings

No co-Hopfian reduced abelian groups of size less than  with infinite _tor(G)

Uncountable abelian groups of certain sizes are not co-Hopfian

Existence of torsion-free abelian groups that are absolutely Hopfian

Abstract

We deal with the problem of existence of uncountable co-Hopfian abelian groups and (absolute) Hopfian abelian groups. Firstly, we prove that there are no co-Hopfian reduced abelian groups $G$ of size $< \mathfrak{p}$ with infinite $\mathrm{Tor}_p(G)$, and that in particular there are no infinite reduced abelian $p$-groups of size $< \mathfrak{p}$. Secondly, we prove that if $2^{\aleph_0} < \lambda < \lambda^{\aleph_0}$, and $G$ is abelian of size $\lambda$, then $G$ is not co-Hopfian. Finally, we prove that for every cardinal $\lambda$ there is a torsion-free abelian group $G$ of size $\lambda$ which is absolutely Hopfian, i.e., $G$ is Hopfian and $G$ remains Hopfian in every forcing extensions of the universe.