Rank 3 permutation groups

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\}}.

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