Difference between external and internal direct product? What is the difference between external and internal direct product ?? I think both of them boil down to the same thing.
 A: 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. 
A: Let $G$ be a group with identity element $e$. Let $H$ and $K$ be normal subgroups of $G$ such that $H \cap K= \{e\}$. Then $H \times K \simeq H \oplus K$ where $H \times K$ is the internal direct product between $H$ and $K$ and $H \oplus K$ is the external direct product between $H$ and $K$. When $H$ and $K$ have nontrivial intersection this may not hold.
A: The difference is is that in the internal direct product is between the subgroups of the same group $G$ but in the external direct product it can be made for any two groups in which they don't have any relation between them
