Relationship between Internal direct product and External direct product of groups

Question

Can somemone explain to me why the internal and external direct products are essentially the same thing? Thank you in advance.

Answer 1

Let $G$ be a group and $H,K$ be subgroups of $G$. Then $G$ is the internal direct product of $H$ and $K$ if (i) $H$ and $K$ both are normal subgroups of $G$
(ii)$G=HK$
(iii)$H\cap K=\{e\}$

Let $G_1,G_2$ be groups. Then a group $G$ is the external direct product of $G_1$ and $G_2$ if $G=G_1\times G_2$.

Define $\phi:G\rightarrow H\times K$ by $\phi(hk)=(h,k)$ for every $hk\in G$. It can be checked that $\phi$ is an isomorphism. Hence we can say that the internal direct product of $H$ and $K$ is isomorphic to the external direct product of $H$ and $K$.

On the other hand, given $H\times K$, you can check that $H\times\{e\}$ and $\{e\}\times K$ satisfies the three conditions of internal direct product. In other words, we can say that $H\times K$ is the internal direct product of $H\times\{e\}$ and $\{e\}\times K$. And of course, $H\times\{e\}\cong H$ and $\{e\}\times K\cong K$.