  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.
