Finite primitive groups and regular orbits of group elements

TL;DR

This paper characterizes finite primitive permutation groups, showing that elements either have a cycle matching their order or the group preserves a specific product structure, resolving longstanding conjectures.

Contribution

It proves a structural classification of elements in finite primitive groups, answering open questions and confirming conjectures about their cycle structures and group actions.

Findings

Elements either have a cycle of length equal to their order or the group preserves a product structure.

Provides a classification that resolves a question by Siemons and Zalesski.

Confirms a conjecture by Giudici, Praeger, and the second author.

Abstract

We prove that if $G$ is a finite primitive permutation group and if $g$ is an element of $G$, then either $g$ has a cycle of length equal to its order, or for some $r$, $m$ and $k$, the group $G \leq \mathrm{Sym}(m) \textrm{wr} \mathrm{Sym}(r)$ preserves the product structure of $r$ direct copies of the natural action of $\mathrm{Sym}(m)$ on $k$-sets. This gives an answer to a question of Siemons and Zalesski and a solution to a conjecture of Giudici, Praeger and the second author.