Central factor is centralizer-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., central factor) satisfying a subgroup metaproperty (i.e., centralizer-closed subgroup property)
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Statement

Verbal statement

The Centralizer (?) of a central factor of the whole group is also a central factor of the whole group.

Proof

Given: $H\leq G$ is a subgroup such that $HC_{G}(H)=G$.

To prove: $C_{G}(H)$ is a central factor of $G$: $C_{G}(H)C_{G}(C_{G}(H))=G$.

Proof: Clearly, $H\leq C_{G}(C_{G}(H))$, so $C_{G}(H)H\leq C_{G}(H)C_{G}(C_{G}(H))$. Since $HC_{G}(H)=G$, we have $C_{G}(H)H=G$ (for more, see permuting subgroups), so $G\leq C_{G}(H)C_{G}(C_{G}(H))$, forcing $G=C_{G}(H)$.