Fixed point ratios for finite primitive groups and applications

Let G be a finite primitive permutation group on set Ω and recall that the fixed point ratio of an element x∈G, denoted fpr(x), is proportion points in by x. Fixed ratios this setting have been studied for many decades, finding wide range applications. In paper, we are interested comparing fpr(x) with order Our main theorem provides classification triples (G,Ω,x) as above property x has prime r fpr(x)>1/(r+1). There several Firstly, extend earlier work Guralnick Magaard determining groups degree m minimal at most 2m/3. Secondly, our result plays key role recent joint Moretó Navarro commuting probability p-elements groups. Finally, use to investigate index group, which allows us answer question Bhargava.