Homogeneity in infinite permutation groups

It is shown that ifG is a permutation group on an infinite setX, andG is (k?1)-transitive but notk-transitive (wherek ≥ 5), then the following hold:

1. G is not (k + 3)-homogeneous.
2. IfG is (k + 2)-homogeneous, then the group induced byG on ak-subset ofX is the alternating groupA _k .