A nilpotency criterion for some verbal subgroups

Abstract

The word $w=[x_{i_1},x_{i_2},\dots ,x_{i_k}]$ is a simple commutator word if $k\ge 2,i_1\ne i_2$ and $i_j\in \{1,\dots ,m\}$, for some $m>1$. For a finite group $G$, we prove that if $i_1\ne i_j$ for every $j\ne 1$, then the verbal subgroup corresponding to $w$ is nilpotent if and only if $|ab|=|a||b|$ for any $w$-values $a,b\in G$ of coprime orders. We also extend the result to a residually finite group $G$, provided that the set of all $w$-values in $G$ is finite.

References in corpus (4)

• A Sufficient Condition for Nilpotency of the Commutator Subgroup
• A criterion for metanilpotency of a finite group
• A Sufficient Condition for Nilpotency in a Finite Group
• A nilpotency criterion for some verbal subgroups

Cited by in corpus (2)

• A nilpotency criterion for some verbal subgroups
• $p$-nilpotency criteria for some verbal subgroups