Abstract Lévai and Pyber proposed the following as a conjecture: Let G be a profinite group such that the set of solutions of the equation x n = 1 {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 (see [V. D. Mazurov and E. I. Khukhro, Unsolved Problems in Group Theory. The Kourovka Notebook. No. 19, Russian Academy of Sciences, Novosibirisk, 2019; Problem 14.53]). The validity of the conjecture has been proved in [L. Lévai and L. Pyber, Profinite groups with many commuting pairs or involutions, Arch. Math. (Basel) 75 2000, 1–7] for n = 2 {n=2} . Here we study the conjecture for compact groups G which are not necessarily profinite and n = 3 {n=3} ; we show that in the latter case the group G contains an open normal 2-Engel subgroup.

中文翻译：

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具有许多有界阶元素的紧致群

摘要 Lévai 和 Pyber 提出以下猜想：设 G 是一个超群，使得方程 xn = 1 {x^{n}=1} 的解集具有正 Haar 测度。然后 G 有一个开子群 H 和一个元素 t，使得陪集 tH 的所有元素都具有除数 n（参见 [VD Mazurov 和 EI Khukhro，群论中未解决的问题。库罗夫卡笔记本。第 19 号，俄罗斯科学院，Novosibirisk，2019 年；问题 14.53]）。猜想的有效性已在[L. Lévai 和 L. Pyber，具有许多通勤对或对合的 Profinite 组，Arch。数学。(Basel) 75 2000, 1–7] 对于 n = 2 {n=2} 。这里我们研究紧群 G 的猜想，G 不一定是超限且 n = 3 {n=3} ；我们证明，在后一种情况下，群 G 包含一个开正规 2-Engel 子群。