Universal Property of Quotient Groups

Universal Property of Quotient Groups: Let $G$ be a group and let $N\trianglelefteq G$ be a normal subgroup . Let $\pi:G\to G/N$ be the canonical projection to the quotient group . If $K$ is a group and $f:G\to K$ is a group homomorphism such that $N\subseteq \ker(f)$ (where $\ker(f)$ is the kernel of $f$), then there exists a unique homomorphism $\bar f:G/N\to K$ with $f = \bar f \circ \pi.$

Equivalently: giving a homomorphism $G/N\to K$ is the same as giving a homomorphism $G\to K$ that sends every element of $N$ to the identity of $K$.

Proof sketch: Define $\bar f(gN)=f(g)$. This is well-defined because if $gN=g'N$ then $g^{-1}g'\in N\subseteq\ker(f)$, so $f(g)=f(g')$. Homomorphism and uniqueness follow because $\pi$ is surjective and $\bar f$ is forced by $\bar f(\pi(g))=f(g)$.