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General Topology
An Engel sink of an element g of a group G is a set ℰ(g)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal E}(g)$$\end{document} such that for every x ∈ G all sufficiently long commutators [⋯[[x, g], g],…, g] belong to ℰ(g)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal E}(g)$$\end{document}. (Thus, g is an Engel element precisely when we can choose ℰ(g)={1}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal E}(g) = \{ 1\} $$\end{document}.) It is proved that if a profinite group G admits an elementary abelian group of automorphisms A of coprime order q2 for a prime q such that for each a ∈ A {1} every element of the centralizer CG(a) has a countable (or finite) Engel sink, then G has a finite normal subgroup N such that G/N is locally nilpotent.
Israel Journal of Mathematics – Springer Journals
Published: Apr 1, 2022
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