Abstract
We show that a generically sharply t-transitive permutation group of finite Morley rank on a set of rank r satisfies t <= r + 2 provided the pointwise stabilizer of a generic (t - 1)-tuple is an L-group, which holds, for example, when this stabilizer is solvable or when r <= 5. This makes progress towards establishing the natural bound on t implied by the Borovik-Cherlin conjecture that every generically (r + 2)-transitive permutation group of finite Morley rank on a set of rank r is of the form PGLr+1(F) acting naturally on Pr(F). Our proof is assembled from three key ingredients that are independent of the main theorem - these address actions of Alt(n) on L-groups of finite Morley rank, generically 2-transitive actions with abelian point stabilizers, and simple groups of rank 6.