Quotient Group is Group/Corollary

Corollary to Quotient Group is Group
Let $G$ be a group.

Let $N$ be a normal subgroup of G.

If $G$ is finite, then:
 * $\index G N = \order {G / N}$

Proof
From Quotient Group is Group, $G / N$ is a group.

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

From the definition of quotient group:
 * $\order {G / N} = \dfrac {\order G} {\order N}$

Hence the result.