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Profinite groups with restricted centralizers of powers

Cristina Acciarri, Pavel Shumyatsky

TL;DR

This work studies profinite groups for which the centralizers of $n$th powers are always finite or open, extending Shalev's result on restricted centralizers. The authors prove that such a group $G$ contains an open normal subgroup $T$ with $G/Z(T)$ of finite exponent, and deduce that the commutator subgroup $T'$ has finite exponent, yielding a $( ext{finite exponent})$-by-abelian-by-finite structure. The proof combines probabilistic methods for open centralizers, Zelmanov's theorem on compact torsion groups, and Khukhro's virtually nilpotent theory. This provides tight structural constraints on profinite groups with restricted centralizers of $n$th powers and informs the interplay between centralizer restrictions and exponent-bounded quotients in the profinite setting.

Abstract

A group $G$ is said to have restricted centralizers if for every $x\in G$ the centralizer $C_G(x)$ either is finite or has finite index in $G$. Shalev showed that a profinite group with restricted centralizers is virtually abelian. Here we take interest in profinite groups $G$ for which there is an integer $n$ such that $C_G(x^n)$ is either finite or open whenever $x\in G$. It is shown that such a group $G$ has an open normal subgroup $T$ with the property that $G/Z(T)$ has finite exponent.

Profinite groups with restricted centralizers of powers

TL;DR

This work studies profinite groups for which the centralizers of th powers are always finite or open, extending Shalev's result on restricted centralizers. The authors prove that such a group contains an open normal subgroup with of finite exponent, and deduce that the commutator subgroup has finite exponent, yielding a -by-abelian-by-finite structure. The proof combines probabilistic methods for open centralizers, Zelmanov's theorem on compact torsion groups, and Khukhro's virtually nilpotent theory. This provides tight structural constraints on profinite groups with restricted centralizers of th powers and informs the interplay between centralizer restrictions and exponent-bounded quotients in the profinite setting.

Abstract

A group is said to have restricted centralizers if for every the centralizer either is finite or has finite index in . Shalev showed that a profinite group with restricted centralizers is virtually abelian. Here we take interest in profinite groups for which there is an integer such that is either finite or open whenever . It is shown that such a group has an open normal subgroup with the property that has finite exponent.
Paper Structure (2 sections, 4 theorems, 1 equation)

This paper contains 2 sections, 4 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']}

Key Result

Theorem 1.1

Let $n$ be a positive integer and $G$ a profinite group in which the centralizer of $x^n$ is either finite or open for every $x\in G$. Then $G$ has an open normal subgroup $T$ such that $G/Z(T)$ has finite exponent.

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • proof