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Does an infinite non-abelian 3-group of exponent greater than or equal to 9 have an infinite abelian subgroup?

I know that every 2-group or 3-group of exponent 3 has an infinite abelian subgroup. I wonder whether the result holds or not for 3-groups of higher exponent.

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    $\begingroup$ In the Burnside group with two generators and exponent $3^n$, for sufficiently large $n$, every abelian group is cyclic, hence finite. This is pointed out in the answer to the same question in math.stackexchange.com/questions/496086/… $\endgroup$ – Ashot Minasyan Nov 14 '13 at 10:03
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    $\begingroup$ The existence of an infinite abelian subgroup is shared by all infinite locally finite groups. For infinite finitely generated torsion groups, it's not always true, as given by Ashot's comment. $\endgroup$ – YCor Jul 21 at 10:55
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(cw answer, copied from Ashot Minasyan's comment.) In the Burnside group with two generators and exponent $3^n$, for sufficiently large $n$, every abelian group is cyclic, hence finite. This is pointed out in the answer to the same question in this MathSE post.

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