Powering-invariant subgroup

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


 * 1) For every prime $$p$$ such that $$G$$ is powered over $$p$$ (i.e., it is uniquely $$p$$-divisible), $$H$$ is also powered over $$p$$.
 * 2) For every prime $$p$$ such that for all $$g \in G$$, there exists unique $$x$$ such that $$x^p = g$$, if we denote $$x = g^{1/p}$$, then the subgroup $$H$$ is invariant under the function $$g \mapsto g^{1/p}$$
 * 3) For every natural number $$n$$ such that for all $$g \in G$$, there exists unique $$x$$ such that $$x^n  = g$$, if we denote $$x = g^{1/n}$$, then the subgroup $$H$$ is invariant under the function $$g \mapsto g^{1/n}$$

Dual property
The dual property to this is quotient-powering-invariant subgroup. A subgroup $$H$$ of a group $$G$$ is termed quotient-powering-invariant in $$G$$ if $$H$$ is a normal subgroup of $$G$$ and for every prime number $$p$$ such that $$G$$ is $$p$$-powered, $$G/H$$ is also $$p$$-powered.

The correspondence we use is subgroup $$\leftrightarrow$$ quotient group.