Prove or disprove: there exists no binary operation $*$ on $\mathbb Q^+$ s.t. $(\mathbb Q^+,*)$ is not isomorphic to $(\mathbb Q^+,×)$

Question

I am a first year college student who just started studying abstract algebra.

I have been discussing the following problem with my friends at another university for a couple of days:

Prove or disprove: there exists no binary operation $*$ on $\mathbb Q^+$ s.t. $(\mathbb Q^+,*)$ is not isomorphic to $(\mathbb Q^+,×)$

I have been trying to wrap my head on this problem, but I could not accept the claim that there cannot be any isomorphism at all. For instance, why would not the identity mapping form the isomorphism between the two groups? If not, could you please pinpoint my wrong intuition and guide me on how to approach this problem?

Answer 1

Let $f: \mathbb Q^+ \to \mathbb Z$ be a bijection between the integers and the positive rationals. Define $$x*y = f^{-1}\left( f(x) + f(y)\right).$$ Then $(\mathbb Q^+, *) \approx(\mathbb Z, +)$. But the integers under addition are not isomorphic to the positive rationals under multiplication.