$f$ factors through $q$, where $q$ is the quotient map, and $f$ is a homomorphism.

Question

Let $K$ be a normal subgroup of $G$ and $q:G \to G/K$ be the quotient map. Let $f:G \to H$ be a homomorphism with $K \subseteq \ker(f)$. Prove that $f$ factors through $q$, meaning that there exists a homomorphism $\varphi:G/K \to H$ such that $f = \varphi \circ q$.

I am not sure how to begin this problem. My idea was to write $G/K$ as $\bar{G}$, and then to define the map $q$ as $q(a)=\bar{a}$, for some $a\in G$, which means that $q$ is a homomorphism, and then to show that $\varphi:G/K \to H$ is a homomorphism. But I have no strategic map.

Answer 1

Hints:

1. What must be the value of $\varphi(\bar a)$ for $a\in G$ if $\varphi\circ q=f$?
2. Show that it gives a well defined map $G/K\to H$.
3. Verify that it's a homomorphism.