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Do you know an example of a $2$-locally nipotent group $G$ which is not locally nilpotent?

$2$-locally nilpotent: every subgroup which is generated by $2$ elements is nilpotent.

locally nilpotent: every finitely generated subgroup is nilpotent.

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The groups referred to in YCor's answer to this question are infinite $d$-generator $p$-groups in which every $(d-1)$-generator subgroup is finite, and hence nilpotent since it is a $p$-group. So they provide examples, although it might be hard to find out much about them.

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  • $\begingroup$ Why are these groups not locally nilpotent? (I guess a Burnside group cannot be nilpotent, but cannot see why just now.) $\endgroup$ – user1729 Jun 22 '18 at 19:20
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    $\begingroup$ More generally a finitely generated solvable torsion group is finite. You can prove it by induction on the derived length. $G/G'$ is finitely generated abelian and hence finite, so $G'$ is finitely generated, and the result follows by induction. $\endgroup$ – Derek Holt Jun 22 '18 at 19:40
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    $\begingroup$ @user1729 It's an exercise to show that every infinite finitely generated solvable group has a finite index subgroup having $\mathbf{Z}$ as quotient. $\endgroup$ – YCor Jun 23 '18 at 10:08

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