Internal central product

Definition

Suppose $G$ is a group and $H$ and $K$ are subgroups of $G$. We say that $G$ is an internal central product of $H$ and $K$ if both the following conditions are satisfied:

1. Every element of $H$ commutes with every element of $K$, i.e., the subgroups centralize each other.
2. $G=HK$, i.e., $G$ is the product of the two subgroups.

Note that in this case, both $H$ and $K$ are central factors of $G$.

There is a corresponding external notion: external central product.