ON DEFINABLE GROUPS and D-GROUPS in CERTAIN FIELDS with A GENERIC DERIVATION

We continue our study from [18] of finite dimensional definable groups in models of the theory T, the model companion of an o-minimal L-Theory T expanded by a generic derivation as in [9]. We generalize Buium's notion of an algebraic D-group to Ldefinable D-groups, namely (G, s), where G is a L-definable group in a model of T, and s : G → (G) is an L-definable group section. Our main theorem says that every definable group of finite dimension in a model of T is definably isomorphic to a group of the form (G, s) = {g G : s(g) = g}, for some L-definable D-group (G, s) (where (g) = (g, g)). We obtain analogous results when T is either the theory of padically closed fields or the theory of pseudo-finite fields of characteristic 0.