Quotients

Quotient

Let $G$ be a group and $N◃G$ a normal subgroup. Let $G/N$ denote the set of cosets (left or right, they are the same) $G/N=\{gN\mid g\in G\}.$

Proposition

$(G/N,\cdot )$ is a group with composition that $gN\cdot hN:=ghN$ for all $g,h\in G$. And the identity is $N$, while the inverse is given by $g^{-1}N$ for any $gN\in G/N$.

Proposition

$|G/N|=\frac{|G|}{|N|}=[G:N]$ for finite groups $G$ and $N$.

Canonical(Natural) Map

Let $G$ be a group and $N◃G$ a normal subgroup, then the canonical map is defined to be: $\pi :G\to G/N,g\mapsto gN$

Proposition

The canonical map $\pi$ is a surjective homomorphism whose kernel is $N$.