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 subgroups. It is natural to ask in what way the structure of the verbal subgroup w(G) depends on the properties of the covering subgroups. The present article is a survey of recent results related to that question. In particular we survey results on finite and countable coverings of word-values (mostly commutators) by procyclic, abelian, nilpotent, and soluble subgroups, as well as subgroups with finiteness conditions. The last section of the paper is devoted to relation of the described results with Hall’s problem on conciseness of group-words.

Coverings of commutators in profinite groups / Acciarri, C; Shumaytsky, P. - In: RENDICONTI DEL SEMINARIO MATEMATICO DELL'UNIVERSITA' DI PADOVA. - ISSN 0041-8994. - 137:(2017), pp. 237-257. [10.4171/RSMUP/137-14]

Coverings of commutators in profinite groups

Acciarri C;
2017-01-01

Abstract

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 subgroups. It is natural to ask in what way the structure of the verbal subgroup w(G) depends on the properties of the covering subgroups. The present article is a survey of recent results related to that question. In particular we survey results on finite and countable coverings of word-values (mostly commutators) by procyclic, abelian, nilpotent, and soluble subgroups, as well as subgroups with finiteness conditions. The last section of the paper is devoted to relation of the described results with Hall’s problem on conciseness of group-words.
137
237
257
Coverings of commutators in profinite groups / Acciarri, C; Shumaytsky, P. - In: RENDICONTI DEL SEMINARIO MATEMATICO DELL'UNIVERSITA' DI PADOVA. - ISSN 0041-8994. - 137:(2017), pp. 237-257. [10.4171/RSMUP/137-14]
Acciarri, C; Shumaytsky, P
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/1255520
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