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Compact groups in which all elements have countable right Engel sinks

Part of: Structure and classification of infinite or finite groups Compact groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  13 November 2020

E. I. Khukhro Open the ORCID record for E. I. Khukhro [Opens in a new window]  and
P. Shumyatsky Open the ORCID record for P. Shumyatsky [Opens in a new window]
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)
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Abstract

A right Engel sink of an element g of a group G is a set ${\mathscr R}(g)$ such that for every x ∈ 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 groupsprofinite groupspro-p groupsfinite groupsLie ring methodEngel conditionlocally nilpotent groups

MSC classification

Primary: 20E18: Limits, profinite groups
Secondary: 20F19: Generalizations of solvable and nilpotent groups 20F45: Engel conditions 22C05: Compact groups
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 6 , December 2021 , pp. 1790 - 1814
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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