Groups reluctant to have infinite subgroup

Question

Is there a group with only one infinite subgroup‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌?

Answer 1

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.

Answer 2

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 $n$.