Sylow theory for p = 0 in solvable 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 provides a collection of tools which play a role in the analysis of odd type groups, and may have applications in degenerate type. These tools involve the “0-unipotence” technology introduced and applied in [Bur04b, Bur04a]. In [Bur04b], the 0-unipotence theory is restricted to those results needed for the applications in signalizer functor theory. The tools presented below develop the general theory further. These results will be applied in [Bur05], which deals with minimal simple groups, and again in [BCJ05], where the applications to odd type are given. A central theme of the present paper is “Sylow theory for p = 0”, in solvable groups of finite Morley rank. As we will see below, it would be more accurate to say p = (0, r) here, where the parameter r represents the “reduced rank” of [Bur04b] (cf. §1). So, from our point of view, the case p = 0 splits into infinitely many “primes”. The paper is organized as follows. Section 1 recalls the definition of the “p-unipotent radical” Up(G) and its analogs for p = 0, the operators U0,r, as introduced in [Bur04b] and [Bur04a]. Section 2 shows that, in nilpotent groups, several basic properties of the connected component operator also hold for the U0,r operator, including a variant of the normalizer condition (Lemma 2.4 below). Section 3 provides the first substantial result, a decomposition theorem for nilpotent groups.