By Agnes Heller.

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Assume U has (EI0). Let Q be an ω-constructible set, internal to the Aut(U)-invariant set C. Assume C and Q ∪ C are stably embedded in the sense (SE2). Say all are defined over F , and write dcl for dcl F etc. (1) There exists an ω-constructible group G, and a constructible action of G on Q (both defined over F ), such that G is isomorphic to Aut(Q/C) as a permutation group on Q. (2) G is C-internal. (3) There exists an ω-constructible G-torsor P , an ω-constructible group H (defined over F ∪ C), such that dcl(P, C) = dcl(Q, C), and H = AutG (P ).

We may take e to be a canonical parameter for (V and for) ge . Let P = Aut(U/F, C) e be the orbit of e under Aut(U/F, C). Denote by ≡C the relation of Aut(U/F, C)-conjugacy. 2, ≡C is ω-constructible. P is a class of ≡C , hence is also ω-constructible. If σ ∈ Aut(U/F, C) fixes e, then (as ge is surjective) it must fix Q pointwise. But then for any e ∈ P , the graph of ge , a subset of Q × C k , is also fixed by σ (pointwise, hence as a relation). Since e is a canonical parameter, σ(e ) = e . Thus the stabilizer of e in Aut(U/F, C) fixes all of P .

Carries P bijectively to P , commuting with G. 1. A supplementary lemma. 1 are defined over the base structure. The need for this additional point, and its model theoretic proof, were first seen by Poizat in the original Picard-Vessiot context, and extended by Pillay to a more general context of definable automorphism groups in differentially closed fields. 4 (at least for G itself). 1. 1, assume in addition that every ω-constructible group ˜ ) is existentially closed in C(U). ˜ is constructible.

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