Question: I already work a counterexample seems to work for answer my own question here:

[Homotopy category is not Abelian.](https://math.stackexchange.com/questions/3388828/homotopy-category-is-not-abelian?noredirect=1#comment6972009_3388828)

But I need to know if in the following sequence of functions in the category of abelian groups 

$$\mathbb{Z} \xrightarrow[]{f} \frac{\mathbb{Z}}{2 \mathbb{Z}} \xrightarrow[]{g} \mathbb{Z}$$

there is some way to define $f$ and $g$ such $gf=1\_{\mathbb{Z}}$. My intuition says this is not posible but cannot justify why?

Answer: 
*Hint:* $g$ cannot be surjective.
If such maps $f$ and $g$ exist then $f$ must be injective, but that is an absurd because $\mathbb{Z}/2\mathbb{Z}$ is a finite group whereas $\mathbb{Z}$ is an infinite group.