  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 $\aleph_\omega$. We introduce
$\aleph_0$-strongly homogeneous p-groups, show that there is a universal
abelian p-group for purity which is $\aleph_0$-strongly homogeneous, and
completely characterize the countable $\aleph_0$-strongly homogeneous p-groups.
