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Compact groups in which all elements have countable right Engel sinks

Published online by Cambridge University Press:  13 November 2020

E. I. Khukhro
Affiliation:
Charlotte Scott Research Centre for Algebra, University of Lincoln, Lincoln, UK Sobolev Institute of Mathematics, Novosibirsk630090, Russia (khukhro@yahoo.co.uk)
P. Shumyatsky
Affiliation:
Department of Mathematics, University of Brasilia, DF 70910-900, Brasilia, Brazil (pavel@unb.br)

Abstract

A right Engel sink of an element g of a group G is a set ${\mathscr R}(g)$ such that for every xG 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.

Type
Research Article
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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