Local divisibility-closed subgroup

Definition
Suppose $$G$$ is a group and $$H$$ is a subgroup of $$G$$. We say that $$H$$ is local divisibility-closed or local divisibility-invariant in $$G$$ if the following holds: for any $$h \in H$$ and any natural number $$n$$ such that the equation $$x^n = h$$ has a solution for $$x$$ in $$G$$, the equation $$x^n = h$$ has a solution for $$x \in H$$.

Conjunction with group properties for ambient group

 * If the whole group is an abelian group, a more standard term for local divisibility-closed subgroup is pure subgroup.

Incomparable properties

 * Sylow subgroup: See Sylow not implies local divisibility-closed

Facts

 * Derived subgroup not is local divisibility-closed in nilpotent group