Fixed point ratios for finite primitive groups and applications

Abstract

Let $G$ be a finite primitive permutation group on a set $\Omega$ and recall that the fixed point ratio of an element $x\in G$, denoted $\mathrm{fpr}(x)$, is the proportion of points in $\Omega$ 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 $\mathrm{fpr}(x)$ with the order of $x$. Our main theorem classifies the triples $(G,\Omega ,x)$ as above with the property that $x$ has prime order $r$ and $\mathrm{fpr}(x)>1/(r+1)$. 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.

arXiv comment66 pages; to appear in Adv. Math