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 $\{ h n h^{-1} : n \in N \} = N$. In shorter form, $hNh^{-1} = N$.

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.)

%%%knows-requisite(Equaliser (category theory)): From a category-theoretic point of view, the kernel of $f$ is an equaliser of an arrow $f$ with the zero arrow; it is therefore universal such that composition with $f$ yields zero. %%%

[todo: why are they interesting]