Groups of finite Morley rank with a generically sharply multiply transitive action

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 GLn(F) in its natural action on Fn.