On finite groups containing an element whose Engel sink is small

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.