The Engel graph of a finite group

TL;DR

This paper studies the Engel graph of finite groups, showing it is always weakly connected and characterizing conditions for strong connectivity based on the group's structure.

Contribution

It introduces the Engel graph for finite groups and proves its connectivity properties, providing new insights into the structure of such groups.

Findings

Engel graph is always weakly connected.

Strong connectivity depends on the group's quotient structure.

Characterization of when the Engel graph is strongly connected.

Abstract

For a finite group $G,$ we investigate the direct graph $\Gamma(G),$ whose vertices are the non-hypercentral elements of $G$ and where there is an edge $x\mapsto y$ if and only if $[x,_ny]=1$ for some $n \in \mathbb N.$ We prove that $\Gamma(G)$ is always weakly connected and is strongly connected if $G/Z_{\infty}(G)$ is neither Frobenius nor almost simple.