We prove the following results. Let to 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 to-values have finite order, then w(G) is locally finite and has finite exponent.
Eloisa Detomi, Marta Morigi, Pavel Shumyatsky (2015). On countable coverings of word values in profinite groups. JOURNAL OF PURE AND APPLIED ALGEBRA, 219, 1020-1030 [10.1016/j.jpaa.2014.05.030].
On countable coverings of word values in profinite groups
MORIGI, MARTA;
2015
Abstract
We prove the following results. Let to 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 to-values have finite order, then w(G) is locally finite and has finite exponent.| File | Dimensione | Formato | |
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