Sylow 0-unipotent subgroups in groups of finite Morley rank

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One of the central tools in the classification of simple algebraic groups is the distinction between semisimple subgroups and unipotent subgroups. It is not a priori clear how to make this distinction for torsion-free subgroups of a group of finite Morley rank. We exploit the ``graded'' notion of 0-unipotence to develop a Sylow theory for torsion-free subgroups of a solvable group of finite Morley rank. This Sylow theory provides a robust alternative to the usual theory of Carter subgroups, and will be used in the analysis of intersections of Borel subgroups in minimal simple groups.