On profinite groups in which commutators are covered by finitely many subgroups

2011/12/26 by Acciarri, Cristina, Shumyatsky, Pavel · 1 citation
#20E18 #20F50 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1112.5879

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 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 G₁,G₂,...,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 G₁,G₂,...,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.

Cited by

• Coprime commutators in profinite groups

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