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 Problem 5.1 of [Fuc15] below ${\mathrm{\aleph }}_{\omega }$. We introduce ${\mathrm{\aleph }}_0$-strongly homogeneous p-groups, show that there is a universal abelian p-group for purity which is ${\mathrm{\aleph }}_0$-strongly homogeneous, and completely characterize the countable ${\mathrm{\aleph }}_0$-strongly homogeneous p-groups.