Coprime commutators in finite groups

Abstract

Let G be a finite group and let $k\ge 2.$ We prove that the coprime subgroup ${\gamma }_k^{*}(G)$ is nilpotent if and only if $\left|\mathit{xy}\right|=\left|x\right|\left|y\right|$ for any ${\gamma }_k^{*}$-commutators $x,y\in G$ of coprime orders (Theorem A). Moreover, we show that the coprime subgroup ${\delta }_k^{*}(G)$ is nilpotent if and only if $\left|\mathit{ab}\right|=\left|a\right|\left|b\right|$ for any powers of ${\delta }_k^{*}$-commutators $a,b\in G$ of coprime orders (Theorem B).

Keywords:

• Coprime commutators
• finite groups
• nilpotency

2010 Mathematics Subject Classification:

• 20D30
• 20D25

Acknowledgment

The authors wish to thank Professor Pavel Shumyatsky for interesting discussions and the anonymous referee for the insightful comments. Moreover, this study was carried out during the second author’s visit to the University of Brasilia. He wishes to thank the Department of Mathematics for the excellent hospitality.

Additional information

Funding

The work of the first author was partially supported by FAPDF/Brazil, while the second author was supported by the “National Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA - INdAM; https://www.altamatematica.it/gnsaga/).