Coprime commutators in finite groups

Abstract

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

References in corpus (2)

• A Sufficient Condition for Nilpotency of the Commutator Subgroup
• A criterion for metanilpotency of a finite group

Cited by in corpus (2)

• Strong conciseness of coprime commutators in profinite groups
• $p$-nilpotency criteria for some verbal subgroups