Internal Direct Products Are Isomorphic To External Direct Products

We need to build a map from one group to the other, and then prove that the map is an isomorphism (a surjective homomorphism with trivial kernel). The map that takes an element $(h,k)\in H\times K$ and sends it to the element $hk\in HK=G$ should be what we need. It should be clear that the map is surjective (why? Start with something in $G=HK$ and produce an element $(h,k)$ that maps to it.) To prove that this map is a homomorphism, we'll have to show that $hk=kh$. To prove that the map is injective, show that the kernel is trivial. The three properties of being an internal direct product will help.