Non-solvable groups of Morley rank 3
A bad group is a simple group of finite Morley rank with nilpotent Borel subgroups (a Borel subgroup is a maximal solvable definable connected subgroup). In the paper, it is proved that if G has Morley rank 3, then G has no involutions, each proper centralizer is connected and G is the union of the conjugates of a proper centralizer. As a corollary, one obtains the unstability of \(SO_ n({\mathbb{R}})\) for \(n\geq 3\). The proof uses the geometry of involutions developed by Bachmann. The work is a continuation of the work initiated by \textit{G. Cherlin} [Ann. Math. Logic 17, 1-28 (1979; Zbl 0427.20001)]. The same results were proved by \textit{A. Borovik} (preprint) independently. The results were generalized to minimal bad groups by \textit{L. J. Corredor} [J. Symb. Logic 54, No.3, 768- 773 (1989)] and independently by \textit{B. Poizat} and \textit{A. Borovik} [Mauvais groupes, Sib. Mat. Zhurn. (to appear)].
scientific article; zbMATH DE number 4120157
scientific article