论文标题
紧凑型组的尼尔疗法概率
Nilpotent probability of compact groups
论文作者
论文摘要
让$ k $成为任何正整数和$ g $ a compact（Hausdorff）组。令$ \ mf {np} _k（g）$表示$ k+1 $随机选择元素$ x_1，\ dots，x_ {k+1} $ sapply $ [x_1，x_2，\ dots，x_ dots，x__ {k+1}] = 1 $。我们研究以下问题：如果$ \ mf {np} _k（g）> 0 $，那么，最多有$ k $的班级nilpotent子组是否存在？答案对小组群是积极的，我们给出了新的证据。我们还证明，$ g $的连接组件$ g^0 $ of $ g $是Abelian，并且最多$ k $的nilpotent子组$ n $ of $ k $，因此$ g^0n $在$ g $中开放。
Let $k$ be any positive integer and $G$ a compact (Hausdorff) group. Let $\mf{np}_k(G)$ denote the probability that $k+1$ randomly chosen elements $x_1,\dots,x_{k+1}$ satisfy $[x_1,x_2,\dots,x_{k+1}]=1$. We study the following problem: If $\mf{np}_k(G)>0$ then, does there exist an open nilpotent subgroup of class at most $k$? The answer is positive for profinite groups and we give a new proof. We also prove that the connected component $G^0$ of $G$ is abelian and there exists a closed normal nilpotent subgroup $N$ of class at most $k$ such that $G^0N$ is open in $G$.
