finite subgroup

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

-
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

1 Answer

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

-
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
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