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

Authors: Kathryn Enochs and Ali Nesin
Journal: Proc. Amer. Math. Soc. 110 (1990), 479-489
MSC: Primary 03C45; Secondary 03C60, 20F16
MathSciNet review: 984788
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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$.

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