On profinite groups with commutators covered by countably many cosets

Let w be a group-word. Suppose that the set of all w-values in a profinite group G is contained in a union of countably many cosets of subgroups. We are concerned with the question to what extent the structure of the verbal subgroup w(G) depends on the properties of the subgroups. We prove the follo...

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Bibliographic Details
Published in:Journal of algebra 2018-08, Vol.508, p.431-444
Main Authors: Detomi, Eloisa, Morigi, Marta, Shumyatsky, Pavel
Format: Article
Language:English
Subjects:
Commutators
Cosets
Profinite groups
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Summary:Let w be a group-word. Suppose that the set of all w-values in a profinite group G is contained in a union of countably many cosets of subgroups. We are concerned with the question to what extent the structure of the verbal subgroup w(G) depends on the properties of the subgroups. We prove the following theorem. Let C be a class of groups closed under taking subgroups, quotients, and such that in any group the product of finitely many normal C-subgroups is again a C-subgroup. If w is a multilinear commutator and G is a profinite group such that the set of all w-values is contained in a union of countably many cosets giGi, where each Gi is in C, then the verbal subgroup w(G) is virtually-C. This strengthens several known results.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2018.05.006