  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.
