Compact groups with countable Engel sinks

2019/08/30 by E. I. Khukhro, Khukhro, E. I., Pavel Shumyatsky +1
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.1908.11637

openalex publication_date 2019/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

An Engel sink of an element g of a group G is a set \mathscr E(g) such that for every x∈ G all sufficiently long commutators [...[[x,g],g],… ,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.

Related

• On profinite groups with automorphisms whose fixed points have countable Engel sinks
• Finite groups with Engel sinks of bounded rank
• Compact groups in which commutators have finite right Engel sinks
• Almost Engel compact groups
• Compact groups all elements of which are almost right Engel