论文标题
委托人，中心化和强烈的简洁群体
Commutators, centralizers, and strong conciseness in profinite groups
论文作者
论文摘要
据说$ g $的组限制了中心化，如果每个$ g \ in g $中的centralizer $ c_g（g）$是有限的，或者在$ g $中具有有限指数。 Shalev表明，一个具有限制性中心化的小组实际上是Abelian。我们对具有统一换向器的限制中心化的概述组感兴趣，即$ [x_1，\ dots，x_k] $的元素，其中$π（x_1）=π（x_2）= \ dots = \ dots =π（x_k）$。这里$π（x）$表示$ x \ in g $的命令的主要除数集。结果表明，这样的组必然具有开放的nilpotent子组。我们使用此结果推断出$γ_k（g）$是有限的，并且仅当$ g $中的一组均匀$ k $ step换向器的基数小于$ 2^{\ aleph_0} $时
A group $G$ is said to have restricted centralizers if for each $g \in G$ the centralizer $C_G(g)$ either is finite or has finite index in $G$. Shalev showed that a profinite group with restricted centralizers is virtually abelian. We take interest in profinite groups with restricted centralizers of uniform commutators, that is, elements of the form $[x_1,\dots,x_k]$, where $π(x_1)=π(x_2)=\dots=π(x_k)$. Here $π(x)$ denotes the set of prime divisors of the order of $x\in G$. It is shown that such a group necessarily has an open nilpotent subgroup. We use this result to deduce that $γ_k(G)$ is finite if and only if the cardinality of the set of uniform $k$-step commutators in $G$ is less than $2^{\aleph_0}$
