Groups of finite Morley rank with a pseudoreflection action

Abstract

In this work, we give two characterisations of the general linear group over an algebraically closed field as a group G of finite Morley rank acting on an abelian connected group V of finite Morley rank definably (in the sense that G (sic) V is a group of finite Morley rank in which G and V are definable), faithfully and irreducibly. We prove that if the pseudoreflection rank of G is equal to the Morley rank of V, then V has a definable vector space structure over an algebraically closed field, G congruent to GL(V) and the action is the natural action. The same result holds also under the assumption of Prufer 2-rank of G being equal to the Morley rank of V. (c) 2012 Elsevier Inc. All rights reserved.