Universal Property of Quotient Groups - Imported Knowlpedia Content

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.

Equivalent characterizations

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 $K$ .