Finite and profinite groups with small Engel sinks of p-elements

TL;DR

This paper investigates the structure of finite and profinite groups where p-elements have small or finite Engel sinks, establishing bounds and structural decompositions related to these properties.

Contribution

It proves that groups with bounded Engel sinks for p-elements have a specific normal subgroup structure, extending to profinite groups with finite Engel sinks.

Findings

Finite groups with small Engel sinks have a normal subgroup with bounded index.

Profinite groups with finite Engel sinks have a normal subgroup that is virtually pro-p.

Structural decompositions relate Engel sink size to group quotients.

Abstract

A (left) Engel sink of an element g of a group G is a subset containing all sufficiently long commutators [...[[x,g],g],...,g], where x ranges over G. We prove that if p is a prime and G a finite group in which, for some positive integer m, every p-element has an Engel sink of cardinality at most m, then G has a normal subgroup N such that G/N is a p'-group and the index [N:O_p(G)] is bounded in terms of m only. Furthermore, if G is a profinite group in which every p-element possesses a finite Engel sink, then G has a normal subgroup N such that N is virtually pro-p while G/N is a pro-p' group.