Abelian subgroup in an infinite non-abelian 3-group

Question

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.

YCor · Answer

(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.

Answered by YCor on November 3, 2021

Abelian subgroup in an infinite non-abelian 3-group

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