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      A Generic Identification Theorem for Groups of Finite Morley Rank, Revisited

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          Abstract

          This paper contains a stronger version of a final identification theorem for the `generic' groups of finite Morley rank.

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          The root subgroups for maximal tori in finite groups of Lie type

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            On central extensions of algebraic groups

            In this paper the following theorem is proved regarding groups of finite Morley rank which are perfect central extensions of quasisimple algebraic groups. T heorem 1. Let G be a perfect group of finite Morley rank and let C 0 be a definable central subgroup of G such that G/C 0 is a universal linear algebraic group over an algebraically closed field; that is G is a perfect central extension of finite Morley rank of a universal linear algebraic group. Then C 0 = 1. Contrary to an impression which exists in some circles, the center of the universal extension of a simple algebraic group, as an abstract group, is not finite in general. Thus the finite Morley rank assumption cannot be omitted. C orollary 1. Let G be a perfect group of finite Morley rank such that G/Z(G) is a quasisimple algebraic group. Then G is an algebraic group. In particular, Z(G) is finite ([4], Section 27.5). An understanding of central extensions of quasisimple linear algebraic groups which are groups of finite Morley rank is necessary for the classification of tame simple K*-groups of finite Morley rank, which constitutes an approach to the Cherlin-Zil’ber conjecture. For this reason the theorem above and its corollary were proven in [1] (Theorems 4.1 and 4.2) under the assumption of tameness , which simplifies the argument considerably. The result of the present paper shows that this assumption can be dropped. The main line of argument is parallel to that in [1]; the absence of the tameness assumption will be countered by a model-theoretic result and results from K -theory. The model-theoretic result places limitations on definability in stable fields, and may possibly be relevant to eliminating certain other uses of tameness.
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              Quasithin groups

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                Author and article information

                Journal
                25 November 2011
                Article
                1111.6037
                51a9fb61-08a6-44b3-b9bc-e1814f63a937

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                math.GR

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