On finite groups containing an element whose Engel sink is small

TL;DR

This paper proves that in finite groups, the order is bounded by the size of an element's Engel sink, using the classification of finite simple groups, extending earlier involution-specific results.

Contribution

It establishes bounds on the order of finite groups based on the size of an element's Engel sink, generalizing previous involution cases.

Findings

Order of G is bounded by the size of the Engel sink of g.

Uses classification of finite simple groups in the proof.

Extends earlier results from involutions to general elements.

Abstract

For an element $g$ of a group $G$, a right Engel sink of $g$ is a subset of $G$ containing all sufficiently long commutators $[...[[g ,x],x],\dots ,x]$ for all $x\in G$. A left Engel sink of $g$ is a subset of $G$ containing all sufficiently long commutators $[...[[x ,g ],g ],\dots ,g]$ for all $x\in G$. Using the classification of finite simple groups we prove that if a finite group $G$ has an element $g$ such that $G=[G,g]$, then the order of $G$ is bounded in terms of a right Engel sink of $g$, as well as in terms of a left Engel sink of $g$. Earlier Guralnick and Tracey proved this in the case where $g$ is an involution without using the classification.