Paper accepted by “Proceedings of the American Mathematical Society”

The paper “On finite groups containing an element whose Engel sink is small” by Evgeny Khukhro and Pavel Shumyatsky has been accepted for publication in “Proceedings of the American Mathematical Society”. This research is a product of collaboration between group-theorists from Lincoln and Brasilia. The Lincoln algebraist Evgeny Khukhro was partially supported by the International Center for Mathematics at the Southern University for Science and Technology in Shenzhen, China, during his visit in April–May of 2026.

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.