Given a finite group G, the Engel graph of G is a directed graph Γ(G) encoding pairs of elements satisfying some Engel word. Namely, Γ (G) is the directed graph, where the vertices are the non-hypercentral elements of G and where there is an arc from x to y if and only if [x,ny] = 1 for some n ℕ. From previous work, it is known that, except for a few exceptions, Γ (G) is strongly connected. In this paper, we give an absolute upper bound on the diameter of Γ (G), when Γ(G) is strongly connected.
Lucchini, A., Spiga, P. (2025). On the diameter of Engel graphs. JOURNAL OF ALGEBRA AND ITS APPLICATIONS, 24(10) [10.1142/S0219498825502391].
On the diameter of Engel graphs
Spiga P.
2025
Abstract
Given a finite group G, the Engel graph of G is a directed graph Γ(G) encoding pairs of elements satisfying some Engel word. Namely, Γ (G) is the directed graph, where the vertices are the non-hypercentral elements of G and where there is an arc from x to y if and only if [x,ny] = 1 for some n ℕ. From previous work, it is known that, except for a few exceptions, Γ (G) is strongly connected. In this paper, we give an absolute upper bound on the diameter of Γ (G), when Γ(G) is strongly connected.
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