Consider the groups Q+ and Q* (rational under addition and ration under multiplication). Prove that neither of these groups is finitely geneated by using the fact that there are infinitely many primes. And prove that Q+ is not isomorphic to Q*.
2. The attempt at a solution
I know that for Q+ if we pick coprimes to be the denominators; then 1/w cannot be generated by rationals. Is this a proof? But I have no idea about Q* and the isomorphism.