Cocentral implies centralizer-dense

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., cocentral subgroup) must also satisfy the second subgroup property (i.e., centralizer-dense subgroup)
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Definitions used

Cocentral subgroup

Further information: cocentral subgroup

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

Proof

Given:A group $G$ with center $Z (G)$ , a cocentral subgroup $H$ . In other words, $H Z (G) = G$ .

To prove: $C_{G} (H) = Z (G)$ , where $C_{G} (H)$ denotes the centralizer of $H$ in $G$ .

Proof: Suppose $K = C_{G} (H)$ . Then $H \leq C_{G} (K)$ . Also, $Z (G) \leq C_{G} (K)$ . Thus, $H Z (G) \leq C_{G} (K)$ , so $G \leq C_{G} (K)$ , forcing $C_{G} (K) = G$ , forcing $K \leq Z (G)$ . On the other hand, $Z (G) \leq K$ since anything in the center of $G$ centralizes $H$ . Thus, $K = Z (G)$ , completing the proof.