On verbal subgroups in finite and profinite groups

TL;DR

This paper investigates how the properties of verbal subgroups in finite and profinite groups are influenced when all their $w$-values are contained within finitely or countably many subgroups with specific properties, extending understanding of their structure.

Contribution

The paper demonstrates that if all $w$-values in a profinite group are contained in subgroups with certain properties, then the verbal subgroup $w(G)$ inherits those properties, advancing the theory of verbal subgroups.

Findings

Verbal subgroups inherit properties from subgroups containing all $w$-values.

Results apply to subgroups that are periodic or of finite rank.

Extends known results to countably many subgroups cases.

Abstract

Let $w$ be a multilinear commutator word. In the present paper we describe recent results that show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely (or in some cases countably) many subgroups with a prescribed property, then the verbal subgroup $w(G)$ has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank.