Pure subgroup

Definition
A subgroup $$H$$ of an abelian group $$G$$ is termed a pure subgroup, isolated subgroup, or serving subgroup if it satisfies the following condition: for any $$h \in H$$ and $$n \in \mathbb{N}$$ such that the equation $$nx = h$$ has a solution for $$x \in G$$, the same equation has a solution $$x \in H$$.

The analogous notion for arbitrary (possibly non-abelian) groups is discussed under the name local divisibility-closed subgroup.

Facts

 * Pure subgroup of torsion-free abelian group not implies direct factor