On verbal subgroups in finite and profinite groups

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.