# Internal central product

## Definition

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

- Every element of commutes with every element of , i.e., the subgroups centralize each other.
- , i.e., is the product of the two subgroups.

Note that in this case, both and are central factors of .

There is a corresponding *external* notion: external central product.