A sufficient condition for nilpotency of the nilpotent residual of a finite group

Abstract

LetGbe a finite group with the property that if{a,b}are powers of{\delta_{1}^{*}}-commutators such that{(|a|,|b|)=1}, then{|ab|=|a||b|}. We show that{\gamma_{\infty}(G)}is nilpotent.

Coprime commutators in finite groups

“…In particular, Theorem A applies to γ ∞ (G), showing that the nilpotent residual γ ∞ (G) of a finite group G is nilpotent if and only if |ab| = |a||b| whenever a and b are γ * k -commutators of coprime orders, for some k ≥ 2. Moreover, since γ * 2 -commutators and δ * 1 -commutators coincide, Theorem A shows that it is enough to impose the condition on the orders of δ * 1 -commutators without considering their powers, and that a group satisfying such a condition is necessarily soluble, improving both results in [8] and [6].…”

“…Every element of G is both a γ * 1 -commutator and a δ * 0 -commutator. Now let k ≥ 2 and let X be the set of all elements of G that are powers of γ Coprime commutators have been studied by many authors (for example see [1,2,6,8,10]). In [8] it is established that the nilpotent residual γ ∞ (G) of a finite group G is generated by commutators of primary elements of coprime orders, where a primary element is an element of prime power order.…”

“…Coprime commutators have been studied by many authors (for example see [1,2,6,8,10]). In [8] it is established that the nilpotent residual γ ∞ (G) of a finite group G is generated by commutators of primary elements of coprime orders, where a primary element is an element of prime power order.…”

“…More precisely they proved that the nilpotent residual γ ∞ (G) of a finite group G is nilpotent provided that |ab| = |a||b| whenever a, b are powers of δ * 1 -commutators of coprime orders. In [6], they also posed the following question.…”

On nilpotency of higher commutator subgroups of a finite soluble group

“…We end this short introduction by mentioning that there are several other recent results related to the theorem of Baumslag and Wiegold (see in particular [4,7,9,[11][12][13]).…”