Let $D = G \times H$ be the direct product of groups $G$ and $H$. Prove that $D$ has a normal subgroup $N$, such that $N$ is isomorphic to $G$ and $D/N$ is isomorphic to $H$.
Here's where I stand...I know what a direct product is and I know what a normal subgroup is but I have no idea how to prove that the direct product of two such arbitrary groups has a normal subgroup.
And then with the isomorphisms. I know how to show basic isomorphisms, show that a mapping is homomorphic and then show that it is onto and one to one, but I don't know how you would show isomorphisms from completely arbitrary subgroups to groups.