Intersections of pure subgroups in abelian groups

Abstract:

It is shown that a subgroup $H$ of an abelian group $G$ is an intersection of pure subgroups of $G$ if and only if, for all primes $p$ and positive integers $n$, ${p^n}g \in H$ and ${p^{n - 1}}g \notin H$ imply that there exists $z \in G$ such that ${p^n}z = 0$ and ${p^{n - 1}}z \notin H$. This solves a problem posed by $L$. Fuchs in [2] and [3].