A sufficient condition for nilpotency of the commutator subgroup

Abstract

Let G be a finite group with the property that if a, b are commutators of coprime orders, then |ab| = |a||b|. We show that G ′ is nilpotent.The following criterion of nilpotency of a finite group was established by B. Baumslag and J. Wiegold [1].Theorem 1. Let G be a finite group in which |ab| = |a||b| whenever the elements a, b have coprime orders. Then G is nilpotent.

A criterion for metanilpotency of a finite group

“…Here the symbol |x| stands for the order of the element x in a group G. In [1] a similar sufficient condition for nilpotency of the commutator subgroup G ′ was established.…”

A Nilpotency Criterion for Some Verbal subgroups

“…As remarked in [2], by a result of Kassabov and Nikolov [9], this is not true in general (see Example 4.3). Two easier counterexamples are given in Section 4.…”

“…Two easier counterexamples are given in Section 4. On the other hand, the answer to the above question is positive when w is a lower central word [1,2]. Motivated by this, we prove the following nilpotency criterion for w(G), where w is a simple commutator word without any repetition of the first variable.…”

Coprime commutators in finite groups

“…Here the symbol |x| stands for the order of the element x in a group G. In this direction, in 2016, R. Bastos and P. Shumyatsky established a characterization for the nilpotency of the commutator subgroup of a finite group [4].…”

“…Theorem 2 ( [4]). Let G be a finite group in which |ab| = |a||b| whenever the elements a, b are commutators of coprime orders.…”