On the diameter of Engel graphs

2023/11/08 by Lucchini, Andrea, Spiga, Pablo
#Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2311.04486

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,_ n y] = 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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