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which group is infinite but all of its subgroups are finite?

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4  
If you mean "proper subgroup" then see e.g. en.wikipedia.org/wiki/Tarski_monster_group –  Yemon Choi Oct 27 '12 at 11:28
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There are much easier examples than Tarski Monsters! For example, fix a prime $p$, and take the multiplicative group of complex $p^n$-th roots of unity for all $n \ge 0$. –  Derek Holt Oct 27 '12 at 12:10
    
Of course, the group mentioned by Derek is just the Prufer group. –  Salvatore Siciliano Oct 27 '12 at 22:42

1 Answer 1

Yes but the Tarski monsters were the first known finitely generated examples. It is open whether there exist finitely presented groups with this property.

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As an elaboration to Jack's answer, see some of the remarks here: mathoverflow.net/questions/78410/… –  Yemon Choi Oct 27 '12 at 12:40
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Yes but there was no request for a finitely generated example! I suspect that you are just trying to make an easy question more interesting (which is of course not unreasonable)! –  Derek Holt Oct 27 '12 at 13:11

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