Quotient Groups

A quotient group lets us regard elements as the same whenever they differ by an element of a chosen subgroup.

Why normality matters

The multiplication must not depend on the representatives we choose. If $g'=gn$ and $h'=hm$, with $n,m\in N$, then

Normality puts $h^{-1}nh$ in $N$, so $g'h'N=ghN$.

A small example

For the additive group $\mathbb Z$ and subgroup $n\mathbb Z$, the quotient is $\mathbb Z/n\mathbb Z$: integers identified by their remainder modulo $n$.

The First Isomorphism Theorem makes this idea precise for any homomorphism.