Onto group homomorphism How many onto group homomorphism is possible from $(\mathbb{Q},+)$ onto $(\mathbb{Q}\setminus \lbrace 0\rbrace,\cdot)$?

None.
Let $\phi\colon \mathbb Q\to\mathbb Q^\times$ be a homomorphism.
Then only finitely many primes occur as factor (in positive or negative power) in $f(1)$.
On the other hand, for $x\in \mathbb Q$ with $x>0$, we have $\underbrace{x+\ldots+x}_n=\underbrace{1+\ldots +1}_m$ for some $n,m\in\mathbb N$, hence $f(x)^n=f(1)^m$. We conclude that $f(x)$ does not involve any primes beyond those occuring in $f(1)$. For $x<0$, the situation is the same as $f(x)=\frac1{f(-x)}$.
Since there are infinitely primes, $\phi$ is not onto.
