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