Centralizers of commutators in finite groups

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 is an element of G such that pi(x) & cap; pi(y) = theta. Here pi(g) denotes the set of prime divisors of the order of the element g is an element of 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 Ix(G)I <= n whenever x is a coprime commutator, then G has a nilpotent subgroup of n-bounded index. If Ix(G)I <= n for every anti-coprime commutator x is an element of G, then G has a subgroup H of nilpotency class at most 4 such that [G : H] and I gamma(4)(H)I are both n-bounded. We also consider finite groups in which the centralizers of coprime, or anti-coprime, commutators are of bounded order. (C) 2022 Elsevier Inc. All rights reserved.