Normal subgroup

by Patrick Stevens Jun 17 2016 updated Jun 18 2016

Normal subgroups are subgroups which are in some sense "the same from all points of view".

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]