Description:
<jats:p>In this paper the following theorem is proved regarding groups of finite Morley rank which are perfect central extensions of quasisimple algebraic groups.</jats:p><jats:p>T<jats:sc>heorem</jats:sc>1.<jats:italic>Let G be a perfect group of finite Morley rank and let C</jats:italic><jats:sub>0</jats:sub><jats:italic>be a definable central subgroup of G such that G/C<jats:sub>0</jats:sub>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</jats:italic><jats:sub>0</jats:sub>= 1.</jats:p><jats:p>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.</jats:p><jats:p>C<jats:sc>orollary</jats:sc>1.<jats:italic>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</jats:italic>([4],<jats:italic>Section</jats:italic>27.5).</jats:p><jats:p>An understanding of central extensions of quasisimple linear algebraic groups which are groups of finite Morley rank is necessary for the classification of<jats:italic>tame</jats:italic>simple<jats:italic>K*-groups</jats:italic>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<jats:italic>tameness</jats:italic>, 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<jats:italic>K</jats:italic>-theory. The model-theoretic result places limitations on definability in stable fields, and may possibly be relevant to eliminating certain other uses of tameness.</jats:p>