# C-closed implies local 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., local 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 local powering-invariant subgroup of $G$: for any natural number $n$ and any element $h \in H$ such that the equation $x^n = h$ has a unique solution $x \in G$, we must have $x \in H$.

## Proof

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