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

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.