Universal Property of Quotient Group

Theorem
Let $G$ and $H$ be groups.

Let $N \trianglelefteq G$ be an normal subgroup.

Let $\pi: G \to G / N$ be the quotient epimorphism.

Let $f: G \to H$ be a group homomorphism with $N \subset \ker f$.

Then there exists a unique group homomorphism $\overline f: G / N \to H$ such that $f = \overline f \circ \pi$.

$\xymatrix{ G \ar[d]^\pi \ar[r]^{\forall f} & H \\ G/N \ar[ru]_{\exists ! \bar f} }$

Existence
Let $\sim$ denote congruence modulo $N$, which is an equivalence relation on $X$ by Congruence Modulo Subgroup is Equivalence Relation.

For all $g \in G$, let $[g]$ denote the equivalence class of $g$ under $\sim$.

Note that Group Homomorphism is Invariant under Congruence Modulo Kernel.

In particular, since $N \subseteq \ker f$, we have that $f$ is $\sim$-invariant.

Hence, by Universal Property of Quotient Set, there exists a unique such mapping $\overline f: G/\sim{} \to H$, which is given by:
 * $\forall g \in G \quad \overline f([g]) = f(g)$

The fact that this is well-defined is shown in Universal Property of Quotient Set.

By definition of coset space, $G/N = G/\sim$.

Also, by Left Cosets are Equal iff Product with Inverse in Subgroup, for all $g \in G$ the equivalence class $[g]$ is the coset of $N$ containing $g$, which is $gN$.

Thus $\overline f$ is given by:
 * $\forall g \in G \quad \overline f(gN) = f(g)$

We have that $\overline f$ is a group homomorphism.

Indeed, for all $x, y \in G$ we have:

Thus we have shown the existence of such a group homomorphism.

Uniqueness
By Universal Property of Quotient Set, there exists a unique such mapping $\overline f$.

A fortiori, there exists at most one such group homomorphism.

Also see

 * First Isomorphism Theorem for Groups
 * Universal Property of Quotient of Topological Group