Does there exists a nontrivial group homomorphism from $\Bbb{Q}/\Bbb{Z}$ to $\Bbb{Z}/2^n\Bbb{Z}$ ($n\ge 1$)?
Suppose there exists a group homomorphism $f:\Bbb{Q}/\Bbb{Z}\to \Bbb{Z}/2^n\Bbb{Z}$. For some $a\in \Bbb{Q}/\Bbb{Z}$, $f(a)=b$, $b\in\Bbb{Z}/2^n\Bbb{Z}$.
Then, $f(2^na)=0$. I cannot proceed from here. I want to solve this question by myself, so only just a hint is also appreciated.