Universal property of quotient group to get epimorphism

Question

I know there are already answers to the question of the universal property of quotient groups. For example here: Universal property of quotient group.

My question is now: If I have the homomorphism: $\phi: G \to G'$ and let H be $H\subset ker(\phi)$. Moreover I have the canonical projection $\pi: G \to G/H$. Due to the univeraly property of quotient groups there exists a unique homomorphism: $\bar{\phi}: G/H \to G'$.

Can I now say that: I know that $\pi$ is surjective. The homormophism $\phi$ is surjective iff $\bar{\phi}$ is surjective?

Answer 1

In the situation displayed in the diagram below we have that $\varphi$ is surjective if and only if $\varphi'$ is surjective.

If $\varphi'$ is surjective, then $\varphi'\circ\pi$ is surjective because composition of surjective functions.

Conversely, if $\varphi$ is surjective, then $\varphi'$ is surjective as well. This doesn't depends on the surjectivity of $\pi$. For if $y'\in G'$ then there exists $x\in G$ such that $y'=\varphi(x)$, hence $y=(\varphi'\circ\pi)(x)\in\operatorname{Im}\varphi'$.

Answer 2

Yes. In fact, this has nothing to do with group theory. If you have a surjective map $f\colon A\longrightarrow B$ and you have a map $g\colon B\longrightarrow C$, then $g$ is surjective if and only if $g\circ f$ is surjective.