Abstract
For a family of group words w we show that if G is a profinite group in which all w-values are contained in a union of finitely many subgroups with a prescribed property, then the verbal subgroup w(G) has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank. If G contains finitely many subgroups G 1, G 2, . . . , G s of finite exponent e whose union contains all γ k -values in G, it is shown that γ k (G) has finite (e, k, s)-bounded exponent. If G contains finitely many subgroups G 1, G 2, . . . , G s of finite rank r whose union contains all γ k -values, it is shown that γ k (G) has finite (k, r, s)-bounded rank.
Highlights
A covering of a group G is a family {Si}i∈I of subsets of G such that G = i∈I Si
If {Hi}i∈I is a covering of G by subgroups, it is natural to ask what information about G can be deduced from properties of the subgroups Hi
The first result in this direction is due to Baer, who proved that G admits a finite covering by abelian subgroups if and only if it is central-by-finite
Summary
If the set of all w-values in a group G can be covered by finitely many subgroups, one could hope to get some information about the structure of the verbal subgroup w(G) In this direction we mention the following result that was obtained in [17]. The corresponding verbal subgroups γk(G) are the terms of the lower central series of G Another distinguished sequence of outer commutator words are the derived words δk, on 2k variables, which are defined recursively by δ0 = x1, δk = [δk−1(x1, . Let w be an outer commutator word and G a profinite group that has finitely many periodic subgroups G1, G2, . Let w be an outer commutator word and G a profinite group that has finitely many subgroups G1, G2, . Throughout the paper the expression “(a, b, . . .)-bounded” stands for “bounded from above by a function depending only on the parameters a, b, . . .”
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