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While reading about the Burnside problem, I thought of the following question:

 If every proper subgroup of G is finite, does it follow that G is also finite?

Despite extensive searching (and thinking), I am unable to find a solution. (I suspect that the answer is no)

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    $\begingroup$ A Google search for the exact quote "every proper subgroup is finite" helps. $\endgroup$
    – Jonas Meyer
    Commented Feb 5, 2013 at 4:38
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    $\begingroup$ For a finitely generated example see Tarski monster. $\endgroup$
    – Benjamin Steinberg
    Commented Feb 5, 2013 at 5:14
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    $\begingroup$ I'm fairly sure this question or some slight variation had been asked before with the same answers. $\endgroup$
    – Benjamin Steinberg
    Commented Feb 5, 2013 at 5:20

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No. The direct limit of the cyclic groups of order $p^n$ is infinite, but every proper subgroup of it is finite.

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    $\begingroup$ I like it. It is the rotation group of the circle of length 1 by elements of Z[1/2]. $\endgroup$
    – Matt Brin
    Commented Feb 7, 2013 at 2:39

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