Divisibility-closed subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed divisibility-closed or divisibility-invariant if it satisfies the following equivalent conditions:


 * 1) For every prime $$p$$ such that $$G$$ is $$p$$-divisible (i.e., for every $$g \in G$$, there exists $$x \in G$$ such that $$x^p = g$$), $$H$$ is also $$p$$-divisible.
 * 2) For every natural number $$n$$ such that $$G$$ is $$n$$-divisible (i.e., for every $$g \in G$$, there exists $$x \in G$$ such that $$x^n = g$$), $$H$$ is also $$n$$-divisible.

Note that we do not require that ''all the $$n^{th}$$ roots in $$G$$ of an element of $$H$$ must be in $$H$$.