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Abelian subgroup in an infinite non-abelian 3-group

MathOverflow Asked by D. N. on November 3, 2021

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.

One 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

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