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Do the positive rationals under multiplication contain a subgroup of finite index?

Similar questions usually rely the fact a subgroup is divisible, however, this is not the case in this question. I have a feeling that the answer should be "no", but have been unable to show this.

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    Please try to make the body of your posts self-contained; don't rely on the title for content. Think of the subject line as the spine of a book; it should be informative, help put it in place, but readers should be be asked to go look at the spine for content.2011-04-12

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