Central subgroup implies join-transitively 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., central subgroup) must also satisfy the second subgroup property (i.e., join-transitively central factor)
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Statement

Statement with symbols

Suppose $G$ is a group and $H$ is a central subgroup of $G$, i.e., $H$ is contained in the center of $G$. Suppose $K$ is a central factor of $G$. Then, the join of subgroups $⟨H,K⟩$, which is also the product of subgroups $HK$, is a central factor of $G$.

Related facts

• Cocentral implies join-transitively central factor
• Right-quotient-transitively central factor implies join-transitively central factor
• Direct factor implies join-transitively central factor
• Central factor is not finite-join-closed