# 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?

• There are torsion-free Tarski monsters. – YCor Jun 9 '18 at 23:54
• PS: The existence of such groups (and actually continuum non-isomorphic many) is another theorem of Olshanski, proved in the early 80's, and significantly easier than the version with $p$-torsion (not easy still... say, the version with $C_p$ is significantly harder). – YCor Apr 12 at 20:20