Coprime commutators in finite groups

2018/11/25 by Monetta, Carmine, Bastos, Raimundo
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1811.10025

Abstract

Let G be a finite group and let k ≥ 2. We prove that the coprime subgroup γ_k^*(G) is nilpotent if and only if |xy|=|x||y| for any γ_k^*-commutators x,y ∈ G of coprime orders (Theorem A). Moreover, we show that the coprime subgroup δ_k^*(G) is nilpotent if and only if |ab|=|a||b| for any powers of δ_k^*-commutators a,b∈ G of coprime orders (Theorem B).

Related

• Centralizers of commutators in finite groups
• The Nilpotency Criterion for the Derived Subgroup of a Finite Group
• On finite groups in which coprime commutators are covered by few cyclic\n subgroups
• Conjugacy classes, characters and products of elements
• Strong conciseness of coprime and anti-coprime commutators