Finite 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., finite 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 finite subgroup of $G$. Then, $H$ is a local powering-invariant subgroup of $G$: for any $g\in H$ and $n\in \mathbb {N}$ such that the solution to $x^{n}=g$ is unique for $x\in G$, we must have $x\in H$.

Related facts

• Finite implies powering-invariant
• Finite not implies divisibility-closed