# 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

 Documents
 Preview
PDF
khu-shu191.pdf - Whole Document
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 G Mathematical and Computer Sciences > G110 Pure Mathematics College of Science > School of Mathematics and Physics 40619 22 May 2020 13:37