which group is infinite but all of its subgroups are finite?
$\begingroup$
$\endgroup$
3
-
4$\begingroup$ If you mean "proper subgroup" then see e.g. en.wikipedia.org/wiki/Tarski_monster_group $\endgroup$– Yemon ChoiOct 27, 2012 at 11:28
-
15$\begingroup$ 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 $n \ge 0$. $\endgroup$– Derek HoltOct 27, 2012 at 12:10
-
$\begingroup$ Of course, the group mentioned by Derek is just the Prufer group. $\endgroup$– Salvatore SicilianoOct 27, 2012 at 22:42
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
2
Yes but the Tarski monsters were the first known finitely generated examples. It is open whether there exist finitely presented groups with this property.
-
$\begingroup$ As an elaboration to Jack's answer, see some of the remarks here: mathoverflow.net/questions/78410/… $\endgroup$ Oct 27, 2012 at 12:40
-
7$\begingroup$ 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)! $\endgroup$ Oct 27, 2012 at 13:11