$\mathbb{Q} \simeq \mathbb{Q}^*_+$ isomorphism [duplicate]

Question

Is it true that $(\mathbb{Q},+) \simeq (\mathbb{Q}^*_+, \times)$? If yes then is there any constructive isomorphism?

Answer 1

We have $(-1)\cdot(-1)=1$ and $-1\ne 1$. Is there any $a\in\mathbb Q$ with $a+a=0$ and $a\ne 0$?

Before you ask what happens if we consider only $(\mathbb Q_{>0}^*,\times)$: For any $a,b\in\mathbb Q\setminus\{0\}$ there exist nonzero integers $n,m$ with $n\cdot a=m\cdot b$. If $(\mathbb Q_{>0}^*,\times)$ were isomorphic, then for any $a,b\in\mathbb Q_{>0}^*\setminus\{1\}$, there would exist nonzero integres $n,m$ with $a^n=b^m$. But there are no such $n,m$ for $a=2,b=3$.

Answer 2

No, it isn't true. Hint: one of these groups has a non-trivial elements of finite order, but the other doesn't.