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14.53 (1999)

Open

Conjecture: Let $G$ be a profinite group such that the set of solutions of the equation $x^n = 1$ has positive Haar measure. Then $G$ has an open subgroup $H$ and an element $t$ such that all elements of the coset $tH$ have order dividing $n$.

This is true in the case $n = 2$. It would be interesting to see whether similar results hold for profinite groups in which the set of solutions of some equation has positive measure.

Contributor: L. Levai, L. Pyber

Progress

Comment of 2021: This is also proved for $n = 3$ (A. Abdollahi, M. S. Malekan, Adv. Group Theory Appl., 13 (2022), 71–81).

Comment of 2025: Significant progress was obtained in (S. Kionke, N. Otmen, T. Toti, M. Vannacci, T. Weigel, Preprint, 2025, https://arxiv.org/pdf/2507.19086).

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