The Engel graph of almost simple groups

TL;DR

This paper studies the strongly connectivity of Engel graphs in finite non-abelian simple groups, reducing the problem to understanding these graphs in simple groups, and providing new insights into their structure.

Contribution

It advances the understanding of Engel graphs by analyzing their strongly connectivity specifically in finite non-abelian simple groups, building on prior reductions.

Findings

Engel graph strongly connectivity characterized for simple groups

Reduction of general group problem to simple groups case

New structural properties of Engel graphs identified

Abstract

Given a finite group $G$, the Engel graph of $G$ is a directed graph encoding pairs of elements satisfying some Engel word. From the work of Detomi, Lucchini and Nemmi, the strongly connectivity of the Engel graph of an arbitrary group $G$ is reduced to the understanding of the strongly connectivity of the Engel graph of non-abelian simple groups. In this paper, we investigate the strongly connectivity of the Engel graph of finite non-abelian simple groups.