On 2-step solvable groups of finite Morley rank

Abstract

We prove the following results: Theorem 1. Let G G be a connected, centerless, solvable group of class 2 and of finite Morley rank. Then we can interpret in G G finitely many connected, solvable of class 2 and centerless algebraic groups G ~ 1 , … , G ~ n {\tilde G_1}, \ldots ,{\tilde G_n} over algebraically closed fields K i {K_i} in such a way that G G interpretably imbeds in G ~ = G ~ 1 ⊕ ⋯ ⊕ G ~ n \tilde G = {\tilde G_1} \oplus \cdots \oplus {\tilde G_n} . Furthermore, G ′ = ( G ~ ) ′ G’ = (\tilde G)’ . Let F ( G ) F(G) denote the Fitting subgroup of G G . Theorem 2. Let G , G ~ , G ~ i G,\tilde G,{\tilde G_i} be as in Theorem 1. Then (i) F ( G ) = F ( G ~ ) ∩ G F(G) = F(\tilde G) \cap G . (ii) F ( G ) F(G) has a complement V V in G : G = F ⋊ V G:G = F \rtimes V . (iii) Elements of F ( G ) F(G) are unipotent elements of G G in G ~ \tilde G . (iv) If the characteristic of each base field K i {K_i} of G ~ i {\tilde G_i} is different from 0, then V V is definable and its elements are semi-simple in G ~ \tilde G .