Signalizers and Balance in Groups of Finite Morley Rank

There is a longstanding conjecture, due to Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. The most successful approach to this conjecture has been Borovik’s program analyzing a minimal counterexample, or simple K-group. We show that a simple K-group of finite Morley rank and odd type is either aalgebraic of else has Prufer rank at most two. This result signifies a switch from the general methods used to handle large groups, to the specilized methods which must be used to identify PSL2, PSL3, PSp4, and G2. The Algebraicity Conjecture for simple groups of finite Morley rank, also known as the Cherlin-Zilber conjecture, states that simple groups of finite Morley rank are simple algebraic groups over algebraically closed fields. In the last 15 years, the main line of attack on this problem has been the Borovik program of transferring methods from finite group theory. This program has led to considerable progress; however, the conjecture itself remains decidedly open. We divide groups of finite Morley rank into four types, odd, even, mixed, and degenerate, according to the structure of their Sylow 2-subgroups. For even and mixed type the Algebraicity Conjecture has been proven, and connected degenerate type groups are now known to have trivial Sylow 2-subgroups [BBC07]. The present paper is part of the program to analyze a minimal counterexample to the conjecture in odd type, where the Sylow 2-subgroup is divisible-abelian-by-finite. It is the final paper in a sequence proving that such a minimal counterexample, or simple nonalgebraicK-group, has Prüfer 2-rank at most two. High Prüfer Rank Theorem. A simple K-group of finite Morley rank with Prüfer 2-rank at least three is algebraic. This will be a consequence of the following so-called trichotomy, which is proved in the present paper. Here the traditional term “trichotomy” refers to the fact that there is also the Prüfer 2-rank ≤ 2 case, which is largely unexplored at present. Generic Trichotomy Theorem. Let G be a simple K-group of finite Morley rank and odd type with Prüfer 2-rank ≥ 3. Then either 1. G has a proper 2-generated core, or 2. G is an algebraic group over an algebraically closed field of characteristic not 2. The High Prüfer Rank Theorem then follows by applying the next two results. 2000 Mathematics Subject Classification. 03C60 (primary), 20G99 (secondary). Burdges was supported by NSF grant DMS-0100794, and Deutsche Forschungsgemeinschaft grant Te 242/3-1.