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 $f = \overline f \circ \pi$.

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

Proof
Note that Group Homomorphism is Invariant under Congruence Modulo Kernel.

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