Centralizers of commutators in finite groups

2022/05/04 by Detomi, Eloisa, Morigi, Marta, Shumyatsky, Pavel
#20E45 #20F14 #20F24 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2205.01995

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 |x^G|≤ n whenever x is a coprime commutator, then G has a nilpotent subgroup of n-bounded index. -- If |x^G|≤ 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.

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