Compact groups in which commutators have finite right Engel sinks

2024/10/08 by Khukhro, Evgeny, Shumyatsky, Pavel
#20D25 #20E18 #20F45 #22C05 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2410.05840

Abstract

A right Engel sink of an element g of a group G is a subset containing all sufficiently long commutators [...[[g,x],x],… ,x]. We prove that if G is a compact group in which, for some k, every commutator [...[g₁,g₂],… ,g_k] has a finite right Engel sink, then G has a locally nilpotent open subgroup. If in addition, for some positive integer m, every commutator [...[g₁,g₂],… ,g_k] has a right Engel sink of cardinality at most m, then G has a locally nilpotent subgroup of finite index bounded in terms of m only.

Related

• Compact groups with countable Engel sinks
• Compact groups all elements of which are almost right Engel
• Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks
• Finite groups with Engel sinks of bounded rank
• On finite groups with an automorphism of prime order whose fixed points have bounded Engel sinks