[Math] Difference between external and internal direct product

Question

What is the difference between external and internal direct product ?? I think both of them boil down to the same thing.

Answer 1

They are two different ways of looking at the same thing, but the definitions are basically equivalent.

Every internal direct product $G$ is naturally isomorphic to the external direct product (of its direct factors).

and

Every external direct product is naturally realized as an internal direct product.

The biggest distinction I've seen is that if $A,B \subset G $, and $A\times B \cong G$, we say $G$ is the internal direct product of $A$ and $B$. However, if $A,B$ are not subgroups of $G$ (rather, they are isomorphic to direct factors of $G$), we would say $G \cong A \times B$ is an external direct product.