The classification of 3 2-transitive permutation groups and 1 2-transitive linear groups

A linear group G ≤ GL(V ), where V is a finite vector space, is called 1 2 -transitive if all the G-orbits on the set of nonzero vectors have the same size. We complete the classification of all the 1 2 -transitive linear groups. As a consequence we complete the determination of the finite 3 2 -transitive permutation groups - the transitive groups for which a point-stabilizer has all its nontrivial orbits of the same size. We also determine the (k + 1 2 )-transitive groups for integers k ≥ 2.