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It is well known that there are finite non-abelian groups whose every proper subgroups are abelian. $S_{3}, Q_{8}$ are such examples. My query is what would be the case if the group is infinite. That is, are there any infinite non-abelian groups whose every nontrivial proper subgroups are infinite abelian?

I was trying taking infinite direct products of non-abelian groups but not being able to find all subgroups as infinite abelian.

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  • $\begingroup$ I'd start with the group generated by letters $x$ and $y$, with the relation $xy=y^nx$ for some $n\ge 2$. $\endgroup$
    – ajotatxe
    Aug 5, 2019 at 9:03
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    $\begingroup$ Such a group is necessarily finitely generated. Indeed any non-commuting pair has to generate the group. In addition it has to be torsion-free. So basically the only examples are the torsion-free "Tarski monsters" (first constructed by Olshanskii). By the way such groups are much easier to construct (using Gromov's methods) than finite exponents analogues. $\endgroup$
    – YCor
    Aug 5, 2019 at 20:58

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Such a group exists: the Tarski monster is an example.

https://en.wikipedia.org/wiki/Tarski_monster_group

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  • $\begingroup$ Thanks for the comment. But I want every subgroup infinite also. Is it possible to find? $\endgroup$
    – Sudipta Purkait
    Aug 5, 2019 at 9:24
  • $\begingroup$ @SudiptaPurkait There are "infinite cyclic" Tarski monster groups too; finitely generated groups where every proper, non-trivial subgroup is infinite cyclic. However, all constructions of Tarski monster groups are highly non-trivial. $\endgroup$
    – user1729
    Aug 5, 2019 at 10:19
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    $\begingroup$ (Also, I think you have a typo in your question title. Tsemo here has answered the question in the title, which differs from the question in the question body.) $\endgroup$
    – user1729
    Aug 5, 2019 at 10:19
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    $\begingroup$ By the way there's no "the Tarski monster". There's no preferred choice of construction. Actually there are uncountably many (torsion-free, or of large prime exponent). $\endgroup$
    – YCor
    Aug 5, 2019 at 20:59

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