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As part of my homework I was given a list of descriptions of groups and I need to determine which pairs are isomorphic. Here are two I am not sure about:

  1. The group $(\mathbb{Q}_+,\cdot)$ of positive rationals with multiplication
  2. The group $(\{\frac{n}{m}\mid m\text{ is odd}\},+)$ of rationals with odd denominators, with addition

Both groups are countable infinite. In both of them, only the identity is of finite order. I don't have other ideas how to distinguish them. As for trying to prove they're isomorphic, my intuition was to maybe look at prime factorizations of integers, but I couldn't make it work.

An answer or a hint would be appreciated.

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1 Answer 1

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Hint: Consider divisibility of elements. For a group element $g$ and integer $n$, when can you solve $g=nh$?

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  • $\begingroup$ Take $n=3$. In the second group - I can always divide. In the first group, I cannot divide $2$. That's it? $\endgroup$
    – user91963
    Commented Aug 27, 2013 at 22:04
  • $\begingroup$ That is right. You have shown there is no element in the first that can match with $2$ in the second. $\endgroup$
    – Ross Millikan
    Commented Aug 27, 2013 at 22:15
  • $\begingroup$ @user91963 Yep, that's exactly what I had in mind! $\endgroup$
    – Chris Culter
    Commented Aug 27, 2013 at 22:16

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