On countable coverings of word values in profinite groups

TL;DR

This paper investigates conditions under which certain subgroups generated by multilinear commutator words in profinite groups are locally finite or of finite rank, based on countable coverings by specific types of subgroups.

Contribution

It establishes new results linking countable coverings of w-values in profinite groups to the local finiteness and finite rank of the verbal subgroup w(G).

Findings

w(G) is locally finite if w-values are covered by countably many periodic subgroups.

w(G) has finite rank if w-values are covered by countably many subgroups of finite rank.

In virtually soluble profinite groups, w(G) is locally finite with finite exponent when all w-values have finite order.

Abstract

We prove the following results. Let w be a multilinear commutator word. If G is a profinite group in which all w-values are contained in a union of countably many periodic subgroups, then the verbal subgroup w(G) is locally finite. If G is a profinite group in which all w-values are contained in a union of countably many subgroups of finite rank, then the verbal subgroup w(G) has finite rank as well. As a by-product of the techniques developed in the paper we also prove that if G is a virtually soluble profinite group in which all w-values have finite order, then w(G) is locally finite and has finite exponent.