C-closed implies powering-invariant

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., c-closed subgroup) must also satisfy the second subgroup property (i.e., powering-invariant subgroup)
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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.

Related facts

Applications

• Powering-invariance is centralizer-closed

Facts used

1. c-closed implies local powering-invariant
2. Local powering-invariant implies powering-invariant

Proof

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