Real numbers/No group homomorphism/Irrational numbers to 0/Exercise

Show that there does not exist a group homomorphism

$\varphi \colon (\mathbb {R} ,0,+)\longrightarrow G$

to a group ${}G$ with the property that ${}r\in \mathbb {R}$ is irrational if and only if ${}\varphi (r)=0$.

Create a solution