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Recognizing Cartesian decompositions for transitive permutation groups

For a given permutation group G 6 Sym Ω, the class of maximal subgroups of Sym Ω that contain G is, in general, difficult to describe. In this paper we address the problem of describing a class of maximal subgroups of Sym Ω or Alt Ω that are full stabilisers of Cartesian decompositions of Ω and contain a given permutation group G with a transitive minimal normal subgroup M . We find that for a fixed ω ∈ Ω the set of G-invariant Cartesian decompositions of Ω is in one-to-one correspondence with the set of Gω -invariant Cartesian systems of subgroups of M . This result enables us to give a complete description of G-invariant Cartesian decompositions in the case where M is a non-abelian simple group.