论文标题
紧凑型组具有许多有限顺序的元素
Compact groups with many elements of bounded order
论文作者
论文摘要
Lévai和Pyber提出了以下猜想：让$ g $是一个小组组，使得方程$ x^n = 1 $的解决方案集具有正面的措施。然后，$ g $具有开放子组$ h $和一个元素$ t $，使得coset $ th $的所有要素都有订单划分$ n $（请参阅[kourovka Notebook，2019年第19号]的问题14.53）。猜想的有效性已在[拱门。数学。 （巴塞尔）75（2000）1-7]对于$ n = 2 $。在这里，我们研究了紧凑型组$ g $的猜想，这些构想不一定是涂鸦，$ n = 3 $;我们表明，在后一种情况下，组$ g $包含一个开放的正常$ 2 $ - ENGEL子组。
Lévai and Pyber proposed the following as a conjecture: Let $G$ be a profinite group such that the set of solutions of the equation $x^n=1$ 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 Problem 14.53 of [The Kourovka Notebook, No. 19, 2019]). The validity of the conjecture has been proved in [Arch. Math. (Basel) 75 (2000) 1-7] for $n=2$. Here we study the conjecture for compact groups $G$ which are not necessarily profinite and $n=3$; we show that in the latter case the group $G$ contains an open normal $2$-Engel subgroup.
