On an algebra related to orbit-counting

With any permutation group G on an infinite set Ω is associated a graded algebra script A sign^G (the algebra of G-invariants in the reduced incidence algebra of finite subsets of Ω). The dimension of the nth homogeneous component of script A sign^G is equal to the number of orbits of G on n-subsets of Ω. If it happens that G is the automorphism group of a homogeneous structure M, then this is the number of unlabelled n-element substructures of M. Many combinatorial enumeration problems fit into this framework. I conjectured 20 years ago that, if G has no finite orbits on Ω, then script A sign^G is an integral domain (and even has the stronger property that a specific quotient is an integral domain). I shall say that G is entire if script A sign^G is an integral domain, and strongly entire if the stronger property holds. These properties have (rather subtle) consequences for the enumeration problems. The conjecture is still open; but in this paper I prove that, if G is transitive on Ω and the point stabilizer H is (strongly) entire, then G is (strongly) entire.