J London Math Soc Page1of13 C 2007 London Mathematical Society doi 10 1112/jlms/jdm088 A NEW TRICHOTOMY THEOREM FOR GROUPS OF FINITE MORLEY RANK

Abstract:  There is a longstanding conjecture, of Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. Here we will conclude that a simple K *-group of finite Morley rank and odd type either has normal rank of at most 2, or else is an algebraic group over an algebraically closed field of characteristic not 2. To this end, it suffices to produce a proper 2-generated core in groups with Pr?fer rank 2 and normal rank at least 3, which is what is proved here. Our final conclusion constrains the Sylow 2-subgroups available to a minimal counterexample and, finally, proves the trichotomy theorem in the nontame context.