Abstract. In a connected group of finite Morley rank, if the Sylow 2-subgroups are finite then they are trivial. The proof involves a combination of model theoretic ideas with a device originating in black box group theory.
Let G be a simple K * -group of finite Morley rank of odd type which is not algebraic. Then G has Prüfer 2-rank at most two. This improves on an earlier result of the second and third authors for the tame case. In the critical case G is minimal connected simple of odd type with a proper definable strongly embedded subgroup. The bulk of the analysis relies on the first author's theory of 0-unipotence and the related 0-Sylow subgroup theory, as well as the so-called Bender method adapted to this context.
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