Quotient group

For a Normal subgroup $N⊴G$, the quotient group (or factor group) $G/N$ is the set of Coset

$$G/N=\{gN:g\in G\},$$

made into a group by the operation

$$(g_{1}N)(g_{2}N)=(g_1{g}_2)N.$$

The well-definedness of this multiplication is exactly the condition that $N$ be normal - see Normal subgroup.

Properties

• Identity: $eN=N$.
• Inverses: $(gN{)}^{-1}=g^{-1}N$.
• Order: $|G/N|=[G:N]=|G|/|N|$ (when $G$ is finite).
• Natural surjection: the map $\pi :G\to G/N$, $\pi (g)=gN$, is a surjective Homomorphism with Kernel $N$.

First Isomorphism Theorem

For any homomorphism $\theta :G\to H$:

$$G/\ker (\theta )\cong \mathrm{Im}(\theta ).$$

So every quotient group is “an image of $G$ under some homomorphism” - and conversely. Quotients and images are dual ways of saying the same thing.

See Isomorphism theorems for the second and third versions.

Examples

• $\mathbb{Z}/n\mathbb{Z}\cong C_n$ - integers mod $n$ form a cyclic group.
• $\mathbb{R}/\mathbb{Z}\cong S^1$ (the circle group).
• $S_n/A_n\cong C_2$ (signature).
• ${\mathrm{GL}}_n(\mathbb{R})/{\mathrm{SL}}_n(\mathbb{R})\cong {\mathbb{R}}^{\ast }$ (determinant).
• $G/G=\{e\}$ (trivial quotient).
• $G/\{e\}=G$ (no information thrown away).

Geometric Interpretation

$G/N$ is what you get by “collapsing each coset of $N$ to a single point” - the algebraic structure that survives after that collapse. If $N$ measures a notion of equivalence, $G/N$ is ”$G$ up to that equivalence”.