论文标题
紧凑型组具有高通勤概率的一组亚组
Compact groups with high commuting probability of monothetic subgroups
论文作者
论文摘要
如果$ h $是紧凑型组$ g $的子组，则$ h $ comport的随机元素的随机元素的随机元素为$ g $的概率由$ pr（h，g）$表示。令$ \ langle g \ rangle $代表元素$ g \ in g $中生成的一组子组，让$ k $是$ g $的子组。我们证明$ pr（\ langle x \ rangle，g）> 0 $ in K $中的任何$ x \ in k $ in k $ in k $，并且仅当$ g $具有开放的正常子组$ t $，因此$ k/c_k（t）$是扭转。尤其是，$ pr（\ langle x \ rangle，g）> 0 $ in G $中的任何$ x \ in G $ in G $，并且仅当$ g $实际上是按中心的，也就是说，有一个开放的正常亚组$ t $，使得$ g/z（t）$是Torsion。我们还推断出该结果的许多推论。
If $H$ is a subgroup of a compact group $G$, the probability that a random element of $H$ commutes with a random element of $G$ is denoted by $Pr(H,G)$. Let $\langle g\rangle$ stand for the monothetic subgroup generated by an element $g\in G$ and let $K$ be a subgroup of $G$. We prove that $Pr(\langle x\rangle,G)>0$ for any $x\in K$ if and only if $G$ has an open normal subgroup $T$ such that $K/C_K(T)$ is torsion. In particular, $Pr(\langle x\rangle,G)>0$ for any $x\in G$ if and only if $G$ is virtually central-by-torsion, that is, there is an open normal subgroup $T$ such that $G/Z(T)$ is torsion. We also deduce a number of corollaries of this result.
