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Let $G$ be a finite primitive permutation group on a set $\Omega$ and recallthat the fixed point ratio of an element $x \in G$, denoted ${\rm fpr}(x)$, isthe proportion of points in $\Omega$ fixed by $x$. Fixed point ratios in thissetting have been studied for many decades, finding a wide range ofapplications. In this paper, we are interested in comparing ${\rm fpr}(x)$ withthe order of $x$. Our main theorem classifies the triples $(G,\Omega,x)$ asabove 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 ofGuralnick and Magaard by determining the primitive permutation groups of degree$m$ with minimal degree at most $2m/3$. Secondly, our main result plays a keyrole in recent work of the authors (together with Moret\'{o} and Navarro) onthe commuting probability of $p$-elements in finite groups. Finally, we use ourmain theorem to investigate the minimal index of a primitive permutation group,which allows us to answer a question of Bhargava.

Fixed point ratios for finite primitive groups and applications