## An infinite group such that every proper subgroup is finite?

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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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2
A Google search for the exact quote "every proper subgroup is finite" helps. – Jonas Meyer Feb 5 at 4:38

3
For a finitely generated example see Tarski monster. – Benjamin Steinberg Feb 5 at 5:14

4
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

## 1 Answer

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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1
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
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