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: $H \le G$ 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, $H \le C_G(C_G(H))$, so $C_G(H)H \le C_G(H)C_G(C_G(H))$. Since $HC_G(H) = G$, we have $C_G(H)H = G$ (for more, see permuting subgroups), so $G \le C_G(H)C_G(C_G(H))$, forcing $G = C_G(H)$.