Derangements in non-Frobenius groups

2024/09/05 by Daniele Garzoni, Garzoni, Daniele · 3 citations
Mathematics · #14G05 #20B05 #20G40 #Commutative Algebra and Its Applications #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2409.03305

openalex publication_date 2024/09/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that if G is a transitive permutation group of sufficiently large degree n, then either G is primitive and Frobenius, or the proportion of derangements in G is larger than 1/(2n^1/2). This is sharp, generalizes substantially bounds of Cameron--Cohen and Guralnick--Wan, and settles conjectures of Guralnick--Tiep and Bailey--Cameron--Giudici--Royle in large degree. We also give an application to coverings of varieties over finite fields.

Cited by

• Prime degree irreducible representations of simple algebraic groups and finite simple groups of Lie type
• Minimal generation of finite simple groups of Lie type by regular unipotent elements
• Fixed Point Free Actions of Group Elements in Linear Groups and Finite Group Representations

Related

• Derangements in simple and primitive groups
• Derangements in finite classical groups for actions related to extension field and imprimitive subgroups and the solution of the Boston-Shalev conjecture
• Groups generated by derangements
• On transitive sets of derangements in primitive groups
• Orbits of permutation groups with no derangements