# On finite groups in which commutators are covered by Engel subgroups

## TL;DR

This paper proves that in finite groups where all commutator values are contained within certain Engel subgroups, the associated verbal subgroup exhibits bounded Engel properties depending on the initial parameters.

## Contribution

It establishes a bounded Engel property for verbal subgroups in finite groups under specific conditions involving commutator values and Engel subgroups.

## Key findings

- Verbal subgroup w(G) is s-Engel for some bounded s
- Conditions relate union of subgroups covering all w-values
- Results depend on parameters m, n, w

## Abstract

Let $m,n$ be positive integers and $w$ a multilinear commutator word. Assume that $G$ is a finite group having subgroups $G_1,\ldots,G_m$ whose union contains all $w$-values in $G$. Assume further that all elements of the subgroups $G_1,\ldots,G_m$ are $n$-Engel in $G$. It is shown that the verbal subgroup $w(G)$ is $s$-Engel for some $\{m,n,w\}$-bounded number $s$.