# 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

 Documents
 Compact groups in which all elements have countable right Engel sinks Authors' Accepted Manuscript Format: PDFSize: 377kBLicensed under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International [Download]
 Preview
PDF
khu-shu201.pdf - Whole Document
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 G Mathematical and Computer Sciences > G110 Pure Mathematics College of Science > School of Mathematics and Physics 42617 17 Nov 2020 12:03