Is there a surjective homomorphism from $(\Bbb{Q},+)$ to $(\Bbb{Z},+)$

Question

Consider the groups $G = (\Bbb{Q},+)$ and $H = (\Bbb{Z},+)$. Is there a surjective homomorphism from $G$ to $H$? If not, how can I prove there isn't?

I considered a homomorphism that rounds up or down but I saw these operations are not "friendly" with the addition.

Answer 1

Let $\varphi: G\to H$ be a homomorphism.

If $\varphi(g)=1$, then

$$\begin{align} 1&=\varphi\left(\frac{g}{2}+\frac{g}{2}\right)\\ &=\varphi\left(\frac{g}{2}\right)+\varphi\left(\frac{g}{2}\right)\\ &=2\varphi\left(\frac{g}{2}\right), \end{align}$$

but then $\varphi\left(\frac{g}{2}\right)\notin\Bbb Z$.