Groups of Finite Morley Rank with a Pseudoreflection Action

TL;DR

This paper characterizes the general linear group within groups of finite Morley rank acting on abelian connected groups, establishing conditions under which the action is equivalent to the natural linear action.

Contribution

It provides two characterizations of the general linear group acting on a finite Morley rank abelian group, linking pseudoreflection rank and Prufer 2-rank to the natural linear action.

Findings

If pseudoreflection rank equals Morley rank, then G is isomorphic to GL(V).

The same holds if Prufer 2-rank equals Morley rank.

V has a vector space structure over an algebraically closed field.

Abstract

In this work, we give two characterisations of the general linear group as a group $G$ of finite Morley rank acting on an abelian connected group $V$ of finite Morley rank definably, faithfully and irreducibly. To be more precise, we prove that if the pseudoreflection rank of $G$ is equal to the Morley rank of $V$, then $V$ has a vector space structure over an algebraically closed field, $G\cong GL(V)$ 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 $V$.