I'm asked the following question:
Let $G$ be a group, $K\unlhd G$, $N\leq K$, and $N\unlhd G.$ Show that $K/N\unlhd G/N$.
My work is as follows. Define the canonical map $\pi:G\to G/N$. Then since $N\leq K\leq G$, we have $K\mapsto K/N$.
If $\pi$ preserves normal subgroups under mapping, then we are done. However, I can't find this statement in Artin, so I'm not sure if it's valid.
Is the statement below true, and if so why is it true?
If $\pi:G\to H$ is a canonical map and $N\unlhd G$, then $\pi(N)\unlhd H$.