Definition:Quotient Group

Theorem
Let $$G$$ be a group.

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

Then the left coset space $$G / N$$ is a group, where the group product is defined as:
 * $$\left({a N}\right) \left({b N}\right) = \left({a b}\right) N$$

$$G / N$$ is called the quotient group (or factor group) of $$G$$ by $$N$$.

Corollary
It follows that if $$G$$ is finite, then $$\left|{G / N}\right| = \frac {\left|{G}\right|} {\left|{N}\right|}$$.

Proof
The operation has been shown in Product of Cosets to be well-defined.

Now we need to demonstrate that $$G / N$$ is a group.

G0: Closure
This follows from the fact that the product of cosets is well-defined.

As $$a b \in G$$, it follows that $$\left({a b}\right) N$$ is a left coset.

Alternatively, we have:

$$ $$ $$

G1: Associativity
The associativity of product of cosets follows directly from Subset Product of Associative is Associative:

$$ $$ $$ $$ $$

G2: Identity
The left coset $$e N = N$$ serves as the identity:

$$ $$ $$

Similarly $$\left({x N}\right) N = x N$$.

G3: Inverses
We have $$\left({x N}\right)^{-1} = x^{-1} N$$:

$$ $$ $$

Similarly $$\left({x^{-1} N}\right) \left({x N}\right) = N$$.

Thus all the group axioms are fulfilled, and $$G / N$$ has been shown to be a group.

Proof of Corollary
From the definition of Subgroup Index, $$\left|{G / N}\right| = \left[{G : N}\right]$$.

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