Cocentral implies central factor

Statement
Any cocentral subgroup of a group is a central factor.

Cocentral subgroup
A subgroup $$H$$ of a group $$G$$ is a cocentral subgroup if $$HZ(G) = G$$ where $$Z(G)$$ is the center of $$G$$.

Central factor
A subgroup $$H$$ of a group $$G$$ is a central factor if $$HC_G(H) = G$$ where $$C_G(H)$$ is the centralizer of $$H$$ in $$G$$.

Related facts

 * Cocentrality is upward-closed
 * Cocentral implies right-quotient-transitively central factor
 * Cocentral implies join-transitively central factor

Proof
Given: A group $$G$$, a subgroup $$H$$ of $$G$$ such that $$HZ(G) = G$$ where $$Z(G)$$ is the center of $$G$$.

To prove: $$HC_G(H) = G$$ where $$C_G(H)$$ is the centralizer of $$H$$ in $$G$$.

Proof: We have $$Z(G) \le C_G(H)$$, because any element of the center centralizes every element of $$H$$. Thus, $$HZ(G) \le HC_G(H)$$. Since $$G = HZ(G)$$, we obtain $$G = HC_G(H)$$.