I would just like verification that the following implications are correct:
If $H$ and $K$ are normal subgroups of an arbitrary group $G$, then $H \cap K$ is a normal subgroup of $G$. But the fact that $H$ and $K$ are normal implies that $[H,K] \subset H \cap K$ (assume this was already proven - not a difficult proof). Then, (this is the key implication that I am questioning) this implies that $[H,K]$ is normal in G.