Título:	Strong conciseness of coprime commutators in profinite groups
Autor(es):	Heras, Iker de las Pintonello, Matteo Shumyatsky, Pavel
Afiliação do autor:	Heinrich-Heine-Universität, Mathematisches Institut Euskal Herriko Unibertsitatea UPV/EHU, Department of Mathematics University of Brasilia, Department of Mathematics
Assunto:	Grupos profinitos Comutadores
Data de publicação:	2023
Editora:	Elsevier Inc.
Referência:	HERAS, Iker de las; PINTONELLO, Matteo; SHUMYATSKY, Pavel. Strong conciseness of coprime commutators in profinite groups. Journal of Algebra, [S. l.], v. 633, p. 1-19, 1 November 2023. DOI: https://doi.org/10.1016/j.jalgebra.2023.06.003.
Abstract:	Let G be a profinite group. The coprime commutators γ∗ j and δ∗ j are defined as follows. Every element of G is both a γ∗ 1 -value and a δ∗ 0 -value. For j ≥ 2, let X be the set of all elements of G that are powers of γ∗ j−1-values. An element a is a γ∗ j -value if there exist x ∈ X and g ∈ G such that a = [x, g] and (|x|, |g|) = 1. For j ≥ 1, let Y be the set of all elements of G that are powers of δ∗ j−1-values. The element a is a δ∗ j -value if there exist x, y ∈ Y such that a = [x, y] and (|x|, |y|) = 1. In this paper we establish the following results. A profinite group G is finite-by-pronilpotent if and only if there is k such that the set of γ∗ k-values in G has cardinality less than 2ℵ0 (Theorem 1.1). A profinite group G is finite-by-(prosoluble of Fitting height at most k) if and only if there is k such that the set of δ∗ k-values in G has cardinality less than 2ℵ0 (Theorem 1.2).
Unidade Acadêmica:	Instituto de Ciências Exatas (IE) Departamento de Matemática (IE MAT)
DOI:	https://doi.org/10.1016/j.jalgebra.2023.06.003
Versão da editora:	https://www.sciencedirect.com/science/article/pii/S002186932300282X?via%3Dihub#kws0020
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