A countable universal torsion abelian group for purity

We show that there is a countable universal abelian p-group for purity, i.e., a countable abelian p-group U such that every countable abelian p-group purely embeds in U. This is the last result needed to provide a complete solution to [Fuc15, Problem 5.1] below \(\aleph_{\omega}\) $${\mathrm{\aleph }}_{\omega }$$ . We introduce \(\aleph_{0}\) $${\mathrm{\aleph }}_0$$ -strongly homogeneous p-groups, show that there is a universal abelian p-group for purity which is \(\aleph_{0}\) $${\mathrm{\aleph }}_0$$ -strongly homogeneous, and completely characterize the countable \(\aleph_{0}\) $${\mathrm{\aleph }}_0$$ -strongly homogeneous p-groups.