2024/09/02 by Timothy C. Burness, Burness, Timothy C., Marco Fusari +1
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2409.01043
openalex publication_date 2024/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G \leqslant \rm Sym(Ω) be a finite transitive permutation group and recall that an element in G is a derangement if it has no fixed points on Ω. Let Δ(G) be the set of derangements in G and define δ(G) = |Δ(G)|/|G| and Δ(G)2 = \ xy : x,y ∈ Δ(G)\. In recent years, there has been a focus on studying derangements in simple groups, leading to several remarkable results. For example, by combining a theorem of Fulman and Guralnick with recent work by Larsen, Shalev and Tiep, it follows that δ(G) \geqslant 0.016 and G = Δ(G)2 for all sufficiently large simple transitive groups G. In this paper, we extend these results in several directions. For example, we prove that δ(G) \geqslant 89/325 and G = Δ(G)2 for all finite simple primitive groups with soluble point stabilisers, without any order assumptions, and we show that the given lower bound on δ(G) is best possible. We also prove that every finite simple transitive group can be generated by two conjugate derangements, and we present several new results on derangements in arbitrary primitive permutation groups.