Infinite non cyclic group with proper subgroups being cyclic

Question

I was thinking about the question regarding infinite groups with all of its proper subgroups being cyclic but the group itself is not cyclic. The finite case is a well known fact $(K_4)$. I have found this beautiful counterexample for infinite groups here about $$\left\langle \frac{a}{2^k} \mid a\in \mathbb{Z}, k \in \mathbb{N} \right\rangle.$$ Any prime $p$ would have done the job besides $2$. Now my question is the following-

Are there any other groups satisfying this property?

Answer 1

Yes. Look up Tarski monster group. Given a prime $p$, this is an infinite simple group all of whose proper subgroups are finite cyclic of order $p$. They exist for all large enough $p$.

Answer 2

For instance the Tarski monster group has the property that it is finitely generated, infinite and every proper subgroup is a cyclic group of a fixed prime order $p$.