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:
 * $\left|{G / N}\right| = \dfrac {\left|{G}\right|} {\left|{N}\right|}$

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

From the definition of subgroup index:
 * $\left|{G / N}\right| = \left[{G : N}\right]$

From Lagrange's Theorem, we have:
 * $\left[{G : N}\right] = \dfrac {\left|{G}\right|} {\left|{N}\right|}$