# finite subgroup

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

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If you mean "proper subgroup" then see e.g. en.wikipedia.org/wiki/Tarski_monster_group –  Yemon Choi Oct 27 '12 at 11:28
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