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.

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Conjugacy, Normal Subgroups, and Quotient Groups: $\S 47$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 50.4 \ \text{(iii)}$ Quotient groups