Recognizing PGL$_3$ via generic 4-transitivity

Abstract

Abstract. We show that the only transitive and generically 4-transitive action of a group of finite Morley rank on a set of Morley rank 2 is the natural action of PGL3 on the projective plane.

Permutation groups of finite Morley rank

“…The conjecture is known for n = 1 [43] (actually the full classification of transitive permutation groups acting on "strongly minimal" sets, viz. sets of rank and degree 1) and more recently n = 2 [3]. (Parenthetically, Gropp [39] had shown that the degree of sharp generic transitivity on a set of Morley rank 2 is at most 6.…”

Simple Groups of Morley Rank 5 Are Bad

“…Proof of Proposition 3.6. Let X = {(i, j) ∈ I (G) 2 : ij = ji}, which G definably permutes by conjugation. Notice that X has rank 4.…”

Rank 3 Bingo

“…More material. The main ingredients in this section are Hrushovski's Theorem on strongly minimal actions, the analysis of bad groups, and Wiscons' analysis of groups of rank 4 • -group and we shall freely use uniqueness principles in Claims 2 and 3.…”