On countable coverings of word values in profinite groups

2013/09/03 by Detomi, Eloisa, Morigi, Marta, Shumyatsky, Pavel
#20E18 #20F12 #20F14 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1309.0636

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.

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