Compact groups with countable Engel sinks

Khukhro, Evgeny and Shumyatsky, Pavel (2020) Compact groups with countable Engel sinks. Bulletin of Mathematical Sciences . ISSN 1664-3607

Full content URL: https://doi.org/10.1142/S1664360720500150

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Compact groups with countable Engel sinks
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Abstract

An Engel sink of an element $g$ of a group $G$ is a set ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$. (Thus, $g$ is an Engel element precisely when we can choose ${\mathscr E}(g)=\{ 1\}$.) It is proved that if every element of a compact (Hausdorff) group $G$ has a countable (or finite) Engel sink, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent. This settles a question suggested by J.~S.~Wilson.

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:40619
Deposited On:22 May 2020 13:37

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