Compact groups with countable Engel sinks

E. I. Khukhro; P. Shumyatsky

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arXiv · 1908.11637

Compact groups with countable Engel sinks

E. I. Khukhro·P. Shumyatsky

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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.

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BibTeX RIS

E. I. Khukhro, P. Shumyatsky. 2019-08-30. Compact groups with countable Engel sinks. https://arxiv.org/abs/1908.11637

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