quotient group

Definition

todo

Universal property

Let $G$ and $H$ be groups. Let $N⊴G$ be a normal subgroup. Let $f:G\to H$ be a group homomorphism with $N\subset \ker f$.

Then there exists a unique group homomorphims $\tilde{f}:G/N\to H$ such that

$f=\tilde{f}\circ \pi$

where $\pi$ is the quotient map.

Diagram for better understanding: