Fixed point ratios for finite primitive groups and applications

TL;DR

This paper classifies primitive permutation groups based on fixed point ratios of elements of prime order and explores applications in group theory, including minimal degrees, commuting probabilities, and minimal indices.

Contribution

It provides a classification of triples where fixed point ratio exceeds a certain bound, extending previous work and enabling new applications in finite group theory.

Findings

Classified primitive groups with fixed point ratio > 1/(r+1) for prime order elements.

Determined primitive groups with minimal degree at most 2/3 of degree.

Analyzed minimal index of primitive groups to answer Bhargava's question.

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 ${\rm fpr}(x)$, 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 ${\rm fpr}(x)$ 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 ${\rm fpr}(x) > 1/(r+1)$. 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 Moret\'{o} 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.