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 ∼ = GL(V ) and the action is the natural action. The same result holds also under the assumption of Prüfer 2-rank of G being equal to the Morley rank of V .
We prove that if G is a group of finite Morley rank that acts definably and generically sharply n-transitively on a connected abelian group V of Morley rank n with no involutions, then there is an algebraically closed field F of characteristic 2 such that V has the structure of a vector space of dimension n over F and G acts on V as the group GL n (F) in its natural action on F n .
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