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

The group $(\mathbb{Q}, +)$ is divisible, and so is every homomorphic image of it. But $(\mathbb{Q} - \{0 \}, \cdot)$ is not divisible: $\sqrt{2}$ is irrational. Hence there can be no surjective homomorphism between the two groups given. 
Note that any rational number that is not $1$ has an irrational $n$th root for some $n$, so with similar reasoning you can prove a stronger result: any homomorphism $(\mathbb{Q}, +) \rightarrow (\mathbb{Q} - \{0\}, \cdot)$ must be trivial.
