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1

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)

flag	asked Feb 5 at 4:35 Amudhan 81●4

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A Google search for the exact quote "every proper subgroup is finite" helps. – Jonas Meyer Feb 5 at 4:38

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For a finitely generated example see Tarski monster. – Benjamin Steinberg Feb 5 at 5:14

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I'm fairly sure this question or some slight variation had been asked before with the same answers. – Benjamin Steinberg Feb 5 at 5:20

Colin Reid · Accepted Answer · 2013-02-05 04:37:59Z UTC

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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.

answered Feb 5 at 4:37

Colin Reid
2,337●8●14

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

An infinite group such that every proper subgroup is finite?