Groups reluctant to have infinite subgroup

Question

Is there a group with only one infinite subgroup‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌?

Really I do not remember any example. But it may be trivial! btw, clearly it's torsion if any and it's not product of infintely many finite groups. — user69453, Jul 10 '14 at 14:40

score 5 · Accepted Answer · edited Apr 13 '17 at 12:21

5

Yes.

If you want a group with no proper infinite subgroups then take a Tarski monster group.
If you want a group with precisely one infinite proper subgroup then take a Tarski monster group $T$ and consider the direct product with a cyclic group of prime order, for example, $G=T\times C_2$.

A Tarski monster group is a finitely generated, infinite simple group where every proper, non-trivial subgroup is cyclic of order a fixed prime $p$. Hence, when you form $T\times C_2$ the only infinite subgroup you obtain is the copy of $T$. The proof that Tarski monster groups exist is highly non-trivial.

edited Apr 13 '17 at 12:21

Community

1

answered Jul 10 '14 at 14:45

user1729

31,015

By a subgroup I do not mean a proper one! – user69453 Jul 10 '14 at 14:48
1

Then take a Tarski monster by itself... – user1729 Jul 10 '14 at 14:48

score -3 · Answer 2 · answered Jul 10 '14 at 14:44

-3

The circle group $\mathbb{T}$ has only one infinite subgroup. The torsion subgroup of $\mathbb{T}$ is the $n$th roots for unity for all $n$.

answered Jul 10 '14 at 14:44

MRicci

1,646

1

I don't think so! why!? I think it has a countable subgroup. – user69453 Jul 10 '14 at 14:45
Let $H=\left< e^{qi}\mid q\in \Bbb Q\right>$. Isn't it a proper subgroup of $\Bbb T$? – user69453 Jul 10 '14 at 14:52

Groups reluctant to have infinite subgroup

2 Answers2