Suppose that every proper subgroup of a group is finite.
Does it imply that the group is finite?
Asked
Active
Viewed 692 times
-1
-
3No, consider $\mathbb Q/\mathbb Z$. – Nishant Dec 16 '14 at 16:27
-
http://mathoverflow.net/questions/120829/an-infinite-group-such-that-every-proper-subgroup-is-finite – Jean-Claude Arbaut Dec 16 '14 at 16:28
-
For what it's worth, the example by @Jean-ClaudeArbaut is called the Prufer $p$-group. – Balarka Sen Dec 16 '14 at 16:30
-
4@Nishant: that group has lots of proper infinite subgroups, e.g. the Prufer $p$-groups for a fixed prime $p$. – Qiaochu Yuan Dec 16 '14 at 16:31
-
@Behaviour It might not be. Notice how that question asks for an abelian group and this is just a group. – Ali Caglayan Dec 16 '14 at 16:36
-
@Alizter An abelian example automatically gives an example for this group. – Balarka Sen Dec 16 '14 at 16:38
-
For a nonabelian example, just take direct limit of all finite groups, as said in the comments on the MO post. – Balarka Sen Dec 16 '14 at 16:39
-
@BalarkaSen Yes but example for non-abelian which is enough reason not to close this question. – Ali Caglayan Dec 16 '14 at 16:39
-
@Alizter Fair enough, withdrew my vote and posted a CW answer. – Dec 16 '14 at 16:45
-
@Nishant: $\mathbb{Q}/\mathbb{Z}$ is not actually an example. It has $\mathbb{Z}[p^{-1}] / \mathbb{Z}$ and $\mathbb{Z}_{(p)}/\mathbb{Z}$ as proper subgroups. In fact, it is their direct sum. – Jan 12 '18 at 16:17
1 Answers
1
No. There are even abelian counterexamples, given in Find an abelian infinite group such that every proper subgroup is finite. More examples are found in MathOverflow post An infinite group such that every proper subgroup is finite?
