C-closed implies powering-invariant

Statement
Suppose $$G$$ is a group and $$H$$ is a c-closed subgroup of $$G$$, i.e., $$H$$ occurs as the centralizer of some subset (and hence also of some subgroup) of $$G$$. Then, $$H$$ is a powering-invariant subgroup of $$G$$: for any prime number $$p$$ such that $$G$$ is $$p$$-powered, $$H$$ is also $$p$$-powered.

Applications

 * Powering-invariance is centralizer-closed

Facts used

 * 1) uses::c-closed implies local powering-invariant
 * 2) uses::Local powering-invariant implies powering-invariant

Proof
The proof follows directly from Facts (1) and (2).