Central factor is centralizer-closed

Verbal statement
The fact about::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 $$G = C_G(H)$$.