Finite primitive groups and regular orbits of group elements

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 $G\le \mathrm{Sym}(m)\text{wr}\mathrm{Sym}(r)$ 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.

21 pages