Normal subgroup

A normal subgroup $N$ of group $G$ is one which is closed under conjugation: for all $h\in G$, it is the case that $\{hn{h}^{-1}:n\in N\}=N$. In shorter form, $hN{h}^{-1}=N$.

Since conjugacy is equivalent to "changing the worldview", a normal subgroup is one which "looks the same from the point of view of every element of $G$".

A subgroup of $G$ is normal if and only if it is the kernel of some group homomorphism from $G$ to some group $H$. (Proof.)