Group with proper subgroups infinite cyclic
Suppose $G$ is an infinite group such that any proper non-trivial subgroup of $G$ is infinite cyclic. Is $G$ itself then infinite cyclic?
If we would only require the proper subgroups to be cyclic, then the Tarski monster groups would yield some counter-examples. Are there analogous examples of Tarski monsters where proper subgroups are infinite cyclic?
As @YCor has indicated, there are torsion-free Tarski monsters. For a reference, check Theorem 28.3 in Chapter 9, §28.1 of the book "Geometry of Defining Relations in Groups" by Ol’shanskii.