Order of Quotient Group

Theorem
Let $G$ be a finite group.

Let $N$ be a normal subgroup of $G$.

Let $G / N$ be the quotient group of $G$ by $N$.

Then:
 * $\dfrac {\order G} {\order N} = \order {G / N}$

where $\order G$ denotes the order of $G$.

Proof
From Lagrange's Theorem:
 * $\dfrac {\order G} {\order N} = \index G N$

where $\index G N$ is the index of $N$ in $G$.

By definition of index:
 * $\index G N = \order {G / N}$