Finite groups with Engel sinks of bounded rank

TL;DR

This paper investigates finite groups where each element's Engel sink generates a subgroup of bounded rank, proving such groups have a large nilpotent quotient with a bounded-rank normal subgroup.

Contribution

It establishes a link between the rank of Engel sinks and the structural properties of finite groups, specifically the existence of a large nilpotent quotient.

Findings

Finite groups with bounded-rank Engel sinks have a large nilpotent quotient.

Existence of a normal subgroup of bounded rank in such groups.

Characterization of group structure based on Engel sink properties.

Abstract

For an element $g$ of a group $G$, an Engel sink is a subset ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$. A~finite group is nilpotent if and only if every element has a trivial Engel sink. We prove that if in a finite group $G$ every element has an Engel sink generating a subgroup of rank~$r$, then $G$ has a normal subgroup $N$ of rank bounded in terms of $r$ such that $G/N$ is nilpotent.