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∈G such that π(x)∩π(y)=∅. Here π(g) denotes the set of prime divisors of the order of the element g∈G. An anti-coprime commutator is an element that can be written as a commutator [x,y], where π(x)=π(y). The main results of the paper are as follows.If |xG|≤n whenever x is a coprime commutator, then G has a nilpotent subgroup of n-bounded index.If |xG|≤n for every anti-coprime commutator x∈G, then G has a subgroup H of nilpotency class at most 4 such that [G:H] and |γ4(H)| are both n-bounded.We also consider finite groups in which the centralizers of coprime, or anti-coprime, commutators are of bounded order.