Centralizers of commutators in finite groups

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 $\pi (x)\cap \pi (y)=\varnothing$. 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 $|x^G|\le n$ whenever $x$ is a coprime commutator, then $G$ has a nilpotent subgroup of $n$-bounded index. -- If $|x^G|\le n$ 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.