4-Homogeneous groups

Characterization of families of rank 3 permutation groups by the subdegrees I

“…Note. The result stated by KA~TO~ in [8] is proved there only under the additional assumption that GA[ A has order ~: 8, which is quite sufficient for our purposes.…”

The classification of partition homogeneous groups with applications to semigroup theory

“…where q is a prime power congruent to 3 (mod 4), or G = AGL(1, 8), AΓL (1,8) or AΓL (1,32), with n = 8, 8, 32 respectively (Kantor [24]); • if G is 4-homogeneous but not 4-transitive, then G = PGL(2, 8), PΓL (2,8) or PΓL (2,32), with n = 9, 9, 33 respectively (Kantor [23]); • using the Classification of Finite Simple Groups, all the t-transitive groups for t ≥ 2 have been determined; lists are available in [11,15]. In particular, the only 6-transitive groups are the symmetric and alternating groups, and the only 4-or 5-transitive groups are these and the Mathieu groups M 11 , M 12 , M 23 and M 24 .…”

“…Clearly a t-transitive group is thomogeneous. Livingstone and Wagner [32] showed that the converse is true if 5 ≤ t ≤ n/2; the t-homogeneous, but not t-transitive groups for t = 2, 3, 4 were determined by Kantor [23,24].…”

Orbit-Homogeneity in Permutation Groups

“…(We may assume without loss that t |V |/2. Without the classification it can be shown that such a group is, with certain known exceptions, always t-transitive, see [5][6][7]; and the list of the t-transitive groups, which can be found in [1], follows from the classification.) For our more general concept, the determination of orbit-t-homogeneous groups is not complete, but we give a number of results in this direction.…”