The Engel graph of a finite group

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 $[x{,}_{n}y]=1$ 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.