I'm reviewing group theory for a comprehensive exam and this question came up.
Suppose I have two groups $G$ and $K$ and $\varphi$, a surjective homomorphism from $G$ to $K$. How can I prove that if $N$ is normal in $K$, then $\varphi^{-1}(N)$ is normal in $G$?
Here's my sketch: suppose that $N$ is a normal subgroup of $K$, then for each $k = \varphi(g) \in K$, we have the following:
$N = kNk^{-1}$
$\varphi^{-1}(N) = \varphi^{-1}(kNk^{-1}) = \varphi^{-1}(k)\varphi^{-1}(N)\varphi^{-1}(k^{-1})) = g\varphi^{-1}(N)g^{-1}$
Which, hopefully, shows that $\varphi^{-1}(N)$ is normal in $G$. I just want to make sure I'm not missing anything major in my proof.
Thank you for your insight.