2021/12/07 by Timothy C. Burness, Robert M. Guralnick, Burness, Timothy C. +1
Engineering · Mathematics · #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Limits and Structures in Graph Theory #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2112.03967
openalex publication_date 2021/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a finite primitive permutation group on a set Ω and recall that the fixed point ratio of an element x ∈ G, denoted \rm fpr(x), is the proportion of points in Ω fixed by x. Fixed point ratios in this setting have been studied for many decades, finding a wide range of applications. In this paper, we are interested in comparing \rm fpr(x) with the order of x. Our main theorem classifies the triples (G,Ω,x) as above with the property that x has prime order r and \rm fpr(x) > 1/(r+1). There are several applications. Firstly, we extend earlier work of Guralnick and Magaard by determining the primitive permutation groups of degree m with minimal degree at most 2m/3. Secondly, our main result plays a key role in recent work of the authors (together with Moretó and Navarro) on the commuting probability of p-elements in finite groups. Finally, we use our main theorem to investigate the minimal index of a primitive permutation group, which allows us to answer a question of Bhargava.