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$\fbox{1}$ Prove that any finitely generated subgroup of $(\mathbb{Q},+)$ is cyclic.

$\fbox{2}$ Prove that $\mathbb{Q}$ is not isomorphic to $\mathbb{Q}\times \mathbb{Q}$.

Any hints would be appreciated.

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    A hint that almost always applies: organize and present your own thoughts on the question. If you don't figure out the answer yourself in the process, the information you provide will not only allow others to write more helpful answers, but it will give them confidence you're looking to learn how to solve problems rather than to copy answers.2012-07-20
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    I don't think the downvote is appropriate here. The votes should represent the quality of the question, not of the person who asked it. If the downvote came for a different reason, please explain. There are many questions on the site whose quality is more or less as this one without any downvotes.2012-07-20
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    @PZZ: One of the hover legends on the downvote button _is_ "this question does not show any research effort", and that certainly does describe this question. It's rather uncharitable to assume that the downvoter voted out of a dislike of the asker rather than because of the actual problems that do exist with the question.2012-07-20
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    @HenningMakholm Fair enough. However, I still stand by the fact that there are many questions showing just as little research effort that are not downvoted.2012-07-20
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    @PZZ I try to hold off on downvoting if the user is very new, and to leave a message explaining the general sentiment if I have time. In felipeuni's case I think that Zev has done this a few times. I do hope that he or she does not take any downvotes too hard — it's just a signal that one could do better.2012-07-20

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