论文标题
有限群体中的通勤者的中心化
Centralizers of commutators in finite groups
论文作者
论文摘要
令$ g $为有限的组。 $ g $中的codrime换向器是任何可以写为换向器$ [x，y] $的元素，适合$ x，y \ in g $，使得$π（x）\capπ（y）= \ emptyset $。这里$π（g）$表示元素$ g \ in G $的prime分隔线集。抗prime换向器是可以写为换向器$ [x，y] $的元素，其中$π（x）=π（y）$。本文的主要结果如下。 - 如果$ | x^g | \ leq n $每当$ x $是副委员会时，则$ g $具有$ n $ and的nilpotent子组。 - 如果每个反选换换向器$ x \ in G $中的$ | x^g | \ leq n $，则$ g $在最多$ 4 $的情况下具有$ 4 $的nilpotency类的亚组$ h $，因此$ [g：h] $ and $ | |γ_4（h）| $都是$ n $ bunged。 我们还考虑有限群体，其中企业或反派遣者的中央位置是有限的秩序。
Let $G$ be a finite group. A coprime commutator in $G$ is any element that can be written as a commutator $[x,y]$ for suitable $x,y\in G$ such that $π(x)\capπ(y)=\emptyset$. Here $π(g)$ denotes the set of prime divisors of the order of the element $g\in G$. An anti-coprime commutator is an element that can be written as a commutator $[x,y]$, where $π(x)=π(y)$. The main results of the paper are as follows. -- If $|x^G|\leq n$ whenever $x$ is a coprime commutator, then $G$ has a nilpotent subgroup of $n$-bounded index. -- If $|x^G|\leq n$ for every anti-coprime commutator $x\in G$, then $G$ has a subgroup $H$ of nilpotency class at most $4$ such that $[G : H]$ and $|γ_4 (H)|$ are both $n$-bounded. We also consider finite groups in which the centralizers of coprime, or anti-coprime, commutators are of bounded order.
