Commutators, centralizers, and strong conciseness in profinite groups

TL;DR

This paper investigates profinite groups with restricted centralizers of uniform commutators, establishing conditions under which such groups have open nilpotent subgroups and relating the finiteness of certain derived subgroups to the cardinality of specific commutator sets.

Contribution

It introduces the concept of restricted centralizers of uniform commutators in profinite groups and proves the existence of open nilpotent subgroups under these conditions.

Findings

Profinite groups with restricted centralizers of uniform commutators have open nilpotent subgroups.

The finiteness of _k(G) is equivalent to the set of uniform k-step commutators having cardinality less than continuum.

Abstract

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 $\pi(x_1)=\pi(x_2)=\dots=\pi(x_k)$. Here $\pi(x)$ 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 $\gamma_k(G)$ is finite if and only if the cardinality of the set of uniform $k$-step commutators in $G$ is less than $2^{\aleph_0}$