## A Group Homomorphism that Factors though Another Group

## Problem 490

Let $G, H, K$ be groups. Let $f:G\to K$ be a group homomorphism and let $\pi:G\to H$ be a surjective group homomorphism such that the kernel of $\pi$ is included in the kernel of $f$: $\ker(\pi) \subset \ker(f)$.

Define a map $\tilde{f}:H\to K$ as follows.

For each $h\in H$, there exists $g\in G$ such that $\pi(g)=h$ since $\pi:G\to H$ is surjective.

Define $\tilde{f}:H\to K$ by $\tilde{f}(h)=f(g)$.

**(a)** Prove that the map $\tilde{f}:H\to K$ is well-defined.

**(b)** Prove that $\tilde{f}:H\to K$ is a group homomorphism.