Let \(f\colon G\to H\) be a homomorphism . Then the kernel \(\ker f\) is normal .
Proof
Let \(K=\ker f\) and \(g\in G\). It is easy to see that \(g^{-1}Kg\subseteq K\).
On the other hand, consider \(k\in K\) and since \(gkg^{-1}\in K\) we may write \(k=g^{-1}(gkg^{-1})g\in g^{-1}Kg\).