Finite primitive permutation groups and regular cycles of their elements

TL;DR

This paper proposes a conjecture about the cycle structure of elements in finite primitive groups, reducing the problem to cases involving almost simple groups with classical socles.

Contribution

It introduces a conjecture linking cycle lengths to group structure and reduces the problem to almost simple classical groups.

Findings

Conjecture relates element cycles to group structure in primitive groups.

Reduction of the conjecture to almost simple groups with classical socles.

Abstract

We conjecture that if $G$ is a finite primitive group and if $g$ is an element of $G$, then either the element $g$ has a cycle of length equal to its order, or for some $r,m$ and $k$, the group $G\leq S_m\wr S_r$, preserving a product structure of $r$ direct copies of the natural action of $S_m$ or $A_m$ on $k$-sets. In this paper we reduce this conjecture to the case that $G$ is an almost simple group with socle a classical group.