The Bender method in groups of finite Morley rank

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 Borovik’s program of transferring methods from finite group theory. Borovik’s program has led to considerable progress; however, the conjecture itself remains decidedly open. In Borovik’s program, groups of finite Morley rank are divided into four types, odd, even, mixed, and degenerate, according to the structure of their Sylow 2subgroup. For even and mixed type the algebraicity conjecture has been proven. The present paper is part of the program to bound the Prüfer rank of minimal simple groups of finite Morley rank and odd type. In [CJ04], Cherlin and Jaligot achieved a bound of Prüfer rank two for tame minimal simple groups. Here a group of finite Morley rank is said to be tame if it does not involve a field of finite Morley rank with a proper infinite definable subgroup of it’s multiplicative group. Cherlin, Jaligot, and the present author will bound the Prüfer rank at two in [BCJ05]. Tameness is used in two important ways in [CJ04]. The final number theoretic contradiction of [CJ04] uses tameness in an essential way, and [BCJ05] will completely replace this argument. However, the very first use of tameness in [CJ04] produces the following fact, which shows that intersections of Borel subgroups are abelian.