Solvable Groups

Abstract

Abstract The structure theory of solvable groups of finite Morley rank is in a quite satisfactory state, though by no means as detailed as the theory available in the algebraic case [189). We obtain the conjugacy of Sylow subgroups for arbitrary primes (whereas the general theory of the next chapter works only for the prime 2) as well as the extensions of the Sylow theory due to Philip Hall (Section 9.9). Section 9.1 contains Zil’ber’s very useful technical result on the structure of certain actions of one abelian group on another, which leads to the theorem in Section 9.2 that the derived subgroup of a connected solvable group G of finite Morley rank is nilpotent. In the algebraic case there is also a representation of G as a semidirect product of an abelian group T acting on a nilpotent subgroup U. We would like such a result in our case, with at least U definable, but this problem is open. We discuss it in general in Section 9.3 and in the case of connected solvable groups of class 2 in Section 9.4. Specializing further in Section 9.5 to the case of rank 2, we obtain a detailed structure theory in the nonnilpotent case, and we make some further comments on the nilpotent case, motivated by the observation that in the study of solvable connected groups of finite Morley rank, nilpotent groups often arise in practice as the commutator subgroup of a given group. One candidate for the subgroup U above is the connected component of the Fitting subgroup, F (G)0 In Section 9.6 we show that the quotient G/ F(G)0 is divisible abelian (that it is abelian is already contained in the result of Section 9.2).