Compact groups in which all elements have countable right Engel sinks

Khukhro, Evgeny and Shumyatsky, Pavel (2020) Compact groups in which all elements have countable right Engel sinks. Proceedings of the Royal Society of Edinburgh Section A: Mathematics . ISSN 0308-2105

Full content URL: https://doi.org/10.1017/prm.2020.81

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Compact groups in which all elements have countable right Engel sinks
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Abstract

A right Engel sink of an element $g$ of a group $G$ is a set ${\mathscr R}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$. (Thus, $g$ is a right Engel element precisely when we can choose ${\mathscr R}(g)=\{ 1\}$.) It is proved that if every element of a compact (Hausdorff) group $G$ has a countable right Engel sink, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent.

Keywords:Compact groups, profinite groups, pro-$p$ groups, finite groups, Lie ring method, Engel condition, locally nilpotent groups
Subjects:G Mathematical and Computer Sciences > G110 Pure Mathematics
Divisions:College of Science > School of Mathematics and Physics
ID Code:42617
Deposited On:17 Nov 2020 12:03

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