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:

$\index G N = \order {G / N}$

$\blacksquare$

Sources

• 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $1$: Introduction to Finite Group Theory: $1.7$