$\mathbb{Q}$ is not isomorphic to $\mathbb{Q}^+$ How can we show that $\mathbb{Q}$ as an additive group is not isomorphic to $\mathbb{Q}^+$ as a multiplicative group?
Both have a countable number of elements, neither is cyclic, neither has an element $x\neq e$ such that $x^2=e$, both are abelian ... I don't know what to use.

If there were an isomorphism $f:(Q^+,\times)\to (Q,+)$ then $f(2)$ would be the sum $a+a$ where $a=f(2)/2$, but then $a$ could not have an inverse image under $f$.
