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 a large groups, to the specilized methods which must be used to identify PSL2, PSL3, PSp 4, 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 Algebraicit
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