Given a finite group [removed]G, the Engel graph of [removed]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 [removed]G and where there is an arc from [removed]x to [removed]y if and only if [removed][x,ny] = 1 for some [removed]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. (2024). On the diameter of Engel graphs. JOURNAL OF ALGEBRA AND ITS APPLICATIONS [10.1142/S0219498825502391].
On the diameter of Engel graphs
Spiga P.
2024
Abstract
Given a finite group [removed]G, the Engel graph of [removed]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 [removed]G and where there is an arc from [removed]x to [removed]y if and only if [removed][x,ny] = 1 for some [removed]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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