Quotient group

Definition

Let $G$ be a group and let $N$ be a normal subgroup of $G$. The quotient group $G/N$ is the set of left cosets $\{gN : g \in G\}$ with the operation $(aN)(bN) = (ab)N$.

Remarks

The operation is well defined because $N$ is normal: if $aN = a'N$ and $bN = b'N$, then $abN = a'b'N$. The identity of $G/N$ is $N = eN$, and the inverse of $aN$ is $a^{-1}N$. When $G$ is finite, $|G/N| = [G : N] = |G| / |N|$. The map $g \mapsto gN$ is a surjective homomorphism from $G$ to $G/N$ with kernel $N$.

Explanations

L1 human

A quotient group treats all elements of one coset as one element. For example, the integers modulo $2$ come from the integers when you treat all even numbers as one element and all odd numbers as another. The normality of $N$ makes sure that the product of two cosets does not depend on the chosen elements.