The Engel graph of a finite group (Q2679199)

Let \(G\) be a finite group. The commuting graph of \(G\) has as vertices the non-central elements of \(G\) and two distinct vertices are adjacent if and only if they commute in \(G\). The notion was introduced by \textit{R. Brauer} and \textit{K. A. Fowler} in their seminal paper [Ann. Math. (2) 62, 565--583 (1955; Zbl 0067.01004)] showing that only finitely many groups of even order can have a prescribed centralizer. In this work, the authors, as a generalization of the commuting graph, introduce the concept of Engel graph, that is the directed 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 \(n \in \mathbb{N}\). The main results are that \(\Gamma(G)\) is always weakly connected and is strongly connected if \(G/Z_{\infty}(G)\) is neither Frobenius nor almost simple. Another interesting result is that if \(G\) is soluble and \(G/Z_{\infty}(G)\) is not a Frobenius group, then \(\operatorname{diam}\Gamma(G) \leq 4\). Furthermore, there exists a soluble group \(G\) such that \(\operatorname{diam}\Gamma(G) = 4\).