Transitive sets of derangements in primitive actions of PSL₂(q)

2025/12/22 by Peter Müller, Müller, Peter
Mathematics · Engineering · #Finite Group Theory Research #graph theory and CDMA systems #Advanced Combinatorial Mathematics

paper · doi:10.48550/arxiv.2512.19500

Abstract

Problem 8.75 of the Kourovka Notebook [10], attributed to John G. Thompson, asks the following: Suppose G is a finite primitive permutation group on Ω, and α, β are distinct points of Ω. Does there exist an element g∈ G such that α^g=β and g fixes no point of Ω? A recent negative example is given in [12], where G is the Steinberg triality group ³D₄(2) acting primitively on 4,064,256 points. At present this is the only negative example known. In this note we show that almost simple primitive permutation groups with socle isomorphic to PSL₂(q) do not give negative examples.

Citations

• On transitive sets of derangements in primitive groups
• Finite subgroups of PGL(2,K)
• Maximal subgroups of almost simple groups with socle \PSL(2,q)

Related

• On transitive sets of derangements in primitive groups
• On derangements in simple permutation groups
• Derangements in simple and primitive groups
• Derangements in non-Frobenius groups
• Derangements in finite classical groups for actions related to extension field and imprimitive subgroups and the solution of the Boston-Shalev conjecture