Fixed point ratios for finite primitive groups and applications

Abstract

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 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 fpr(x) with the order of x. Our main theorem provides a classification of the triples (G,Ω,x) as above with the property that x has prime order r and 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 joint work 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.