Splitting in solvable groups of finite Morley rank

Abstract

We exhibit counterexamples to a Conjecture of Nesin, since we build a connected solvable group with finite center and of finite Morley rank in which no normal nilpotent subgroup has a nilpotent complement. The main result says that each centerless connected solvable group G of finite Morley has a normal nilpotent subgroup U and an abelian subgroup T such that G = U o T , if and only if, for any field K of finite Morley rank, the connected definable subgroups of K∗ are pseudo-tori. Also we build a centerless connected solvable group G of finite Morley rank with no definable representation over a direct sum of interpretable fields. 2000 Mathematics Subject Classification 03C45, 20A15 (primary); 03C60 (secondary)