Alice Devillers, Cai Heng Li, Geoff Pearce, Cheryl Praeger and I have just uploaded to the arxiv a preprint of our recently submitted paper `On imprimitive rank 3 permutation groups‘.

A permutation group ${G}$ on a set ${\Omega}$ also acts on the set ${\Omega\times\Omega}$ via ${(\omega_1,\omega_2)^g=(\omega_1^g,\omega_2^g)}$. If ${G}$ is transitive on ${\Omega}$ then we cannot expect it to be transitive on ${\Omega\times\Omega}$ as the subset ${\{(\omega,\omega)\mid\omega\in\Omega\}}$ is an orbit. The best that we can hope for is that ${G}$ has two orbits on ${\Omega\times\Omega}$ and this is equivalent to ${G}$ acting 2-transitively on ${\Omega}$, that is, ${G}$ acts transitively on the set of ordered pairs of distinct elements of ${G}$. The rank of a permutation group ${G}$ is the number of orbits of ${G}$ on ${\Omega\times\Omega}$. Thus 2-transitive groups have rank 2.

The number of orbits of ${G}$ on ${\Omega\times \Omega}$ is equal to the number of orbits of the point stabiliser ${G_{\omega}}$ on ${\Omega}$. Indeed, given an orbit ${\Delta}$ of ${G_{\omega}}$ on ${\Omega}$ the set ${\{(\omega,\delta)^g\mid \delta\in\Delta,g\in G\}}$ is an orbit of ${G}$ on ${\Omega\times\Omega}$. Conversely, given an orbit ${\Delta}$ of ${G}$ on ${\Omega\times\Omega}$, the set ${\{\delta\in\Omega\mid (\omega,\delta)\in\Delta\}}$ is an orbit of ${G_{\omega}}$ on ${\Omega}$. In this set up, ${\{\omega\}}$ corresponds to ${\{(\omega,\omega)\mid\omega\in\Omega\}}$.