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Does there exist an infinite non-abelian group, such that all its nontrivial proper subgroups are isomorphic to $C_\infty$?

It is rater obvious, that if such group exists then it is generated by any non-commuting pair of its elements. However I failed to prove anything else about it.

This question was inspired by Ol’shanski theorem, that states, that for all primes $p > 10^{75}$, there exist continuum-many infinite non-abelian groups, such that all their nontrivial proper subgroups are isomorphic to $C_p$ (Such groups are called Tarski Monster $p$-groups).

Chain Markov
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  • I am pretty sure Ol'shanskii acutally shows this in his book Geometry of Defining Relations in Groups using the same methods to construct Tarski Monsters(also done in that book) –  Apr 12 '19 at 15:26
  • Note: $C_\infty$ denotes the infinite cyclic group. – YCor Apr 12 '19 at 20:17

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