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.