Does there exists a onto Homomorphism from $\{z\in \mathbb{C}\mid z^n=1 \text{ for some }n\in \mathbb{N}\}$ to $\mathbb{Z}_{2019}$?
I think there does not exist any onto Homomorphism. All prime subgroups whose order does not divide $2019$ must be in the kernel. Intuitively there is a lack of symmetry. But I don't whether this intuition works in infinite groups or not. If works then which property should I use here.
Let $\phi$ be any Homomorphism.
Let $a$ be any element in the larger group G.
Then $a= e^{\frac{2\pi i k}{n}}$ for some k and n.
Now $a' = e^{\frac{2\pi i k}{2019n}}$ is also an element of G.
Now $2019.\phi(a')=0\implies \phi({a'}^{2019})=0\implies \phi(a)=0$.
Hence $\phi$ is trivial.