# A nilpotency criterion for some verbal subgroups

## TL;DR

This paper establishes a criterion for nilpotency of certain verbal subgroups in finite and residually finite groups based on the order properties of their $w$-values, specifically for simple commutator words.

## Contribution

It provides a new nilpotency criterion for verbal subgroups generated by specific simple commutator words in finite and residually finite groups.

## Key findings

- Verbal subgroup is nilpotent iff $|ab|=|a||b|$ for coprime $w$-values.
- Criterion applies when $i_1
eq i_j$ for all $j
eq 1$ in the word.
- Extension of results to residually finite groups with finite $w$-values.

## Abstract

The word $w=[x_{i_1},x_{i_2},\dots,x_{i_k}]$ is a simple commutator word if $k\geq 2, i_1\neq i_2$ and $i_j\in \{1,\dots,m\}$, for some $m>1$. For a finite group $G$, we prove that if $i_{1} \neq i_j$ for every $j\neq 1$, 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.