k-Homogeneous groups

Theorem 1. Let G be a group k-homogeneous but not k-transitive on a finite set f2 of n points, where n>=2k. Then, up to permutation isomorphism, one of the following holds: (i) k = 2 and G < AFL(1, q) with n = q-= 3 (mod 4); (ii) k = 3 and PSL(2, q)<<_G<_PFL(2, q), where n 1 = q = 3 (rood 4); (iii) k = 3 and G=AGL(1, 8), AFL(1, 8) or AFL(1, 32); or (iv) k = 4 and G = PSL(2, 8), PFL(2, 8) or PFL(2, 32).