On $2$-step solvable groups of finite Morley rank

Abstract:

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