  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.
