Rank bound and extremal classification for primitive groups
Derive the bound rk(G) ≤ n(n + 2) for connected primitive permutation groups (G,X) of finite Morley rank with rk(X) = n, and classify the extremal and near-extremal cases: (a) rk(G) = n(n + 2) with G ≃ PGL_{n+1}(K) acting naturally on the projective space P^n(K) and b(G,X) = n + 2; (b) rk(G) = n(n + 1) with G ≃ AGL_n(K) acting naturally on the affine space A^n(K) and b(G,X) = n + 1; (c) rk(G) = n with G ≃ ASL_n(K) acting naturally on A^n(K) and b(G,X) = n.
References
Conjecture 7. Let (G,X) be a connected primitive permutation group and rkX = n. Then rkG 6 n(n + 2). Moreover, (a) If rkG = n(n+2) then G ≃ PGL n+1(K) for some algebraically closed field K, and the action of G on X is the natural action of PGL n+1(K) on the projective space P (n). In that case, G is generically sharply (n + 2)-transitive on X and
b(G,X) = n + 2. We further conjecture that there are only two other possibilities for the case rkG > n in which K is again an algebraically closed field. (b) rkG = n(n + 1) and G ≃ AGL (K). Tne action of G on X is the natural action of AGL (K) on the affine space A (K); G is n n generically sharply (n+1)-transitive on X and b(G,X) = n+1. (c) rkG = n and G ≃ ASL (K). Tne action of G on X is the natural action of ASL (K) on the affine space A (K); G is n n generically sharply n-transitive on X and b(G,X) = n.