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?

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    $\begingroup$ There are torsion-free Tarski monsters. $\endgroup$ – YCor Jun 9 '18 at 23:54
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    $\begingroup$ 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). $\endgroup$ – YCor Apr 12 '19 at 20:20

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.

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