Infinitely generated and isomorphism

  • Thread starter doongly
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  • #1
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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.
 

Answers and Replies

  • #2
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What you said for [tex]\mathbb{Q}^+[/tex] is too sketchy for me to guess whether you have the right idea or not.

For [tex]\mathbb{Q}^\times[/tex], 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.
 

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