Cocentral implies central factor

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., central factor)
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Statement

Any cocentral subgroup of a group is a central factor.

Definitions used

Cocentral subgroup

Further information:

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

Further information: 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 $Z(G)\leq C_{G}(H)$, because any element of the center centralizes every element of $H$. Thus, $HZ(G)\leq HC_{G}(H)$. Since $G=HZ(G)$, we obtain $G=HC_{G}(H)$.