Infinitely generated and isomorphism

[doongly]

Homework Statement

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.

 

[ystael]

What you said for $$\mathbb{Q}^+$$ is too sketchy for me to guess whether you have the right idea or not.

For $$\mathbb{Q}^\times$$, think about how to generate the primes.

For the isomorphism: one way to see that two groups are not isomorphic is to show that they have different subgroup structure.