**Authors: **Cristina Acciarri Pavel Shumyatsky

**Publish Date**: 2012/08/03

**Volume:** 274, **Issue:** 1-2, **Pages:** 239-248

## Abstract

For a family of group words w we show that if G is a profinite group in which all wvalues are contained in a union of finitely many subgroups with a prescribed property then the verbal subgroup wG has the same property as well In particular we show this in the case where the subgroups are periodic or of finite rank If G contains finitely many subgroups G 1 G 2 G s of finite exponent e whose union contains all γ k values in G it is shown that γ k G has finite e k sbounded exponent If G contains finitely many subgroups G 1 G 2 G s of finite rank r whose union contains all γ k values it is shown that γ k G has finite k r sbounded rank

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