Centralizers of commutators in finite groups

TL;DR

This paper investigates the structure of finite groups based on properties of coprime and anti-coprime commutators, establishing bounds on their sizes and implications for the group's nilpotency and subgroup structure.

Contribution

It introduces new bounds on group structure derived from the sizes of conjugacy classes of specific types of commutators, linking commutator properties to nilpotency and subgroup indices.

Findings

Groups with bounded conjugacy class sizes of coprime commutators have a nilpotent subgroup of bounded index.

Groups with bounded conjugacy class sizes of anti-coprime commutators have a subgroup of nilpotency class at most 4 with bounded index and fourth term.

The paper also characterizes groups where centralizers of these commutators are of bounded order.

Abstract

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 $\pi(x)\cap\pi(y)=\emptyset$. Here $\pi(g)$ 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 $\pi(x)=\pi(y)$. 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 $|\gamma_4 (H)|$ are both $n$-bounded. We also consider finite groups in which the centralizers of coprime, or anti-coprime, commutators are of bounded order.