Submitted exclusively to the London Mathematical Society DOI: 10.1112/S0000000000000000 ORBIT-HOMOGENEITY IN PERMUTATION GROUPS PETER J. CAMERON and ALEXANDER W. DENT Abstract This paper introduces the concept of orbit-homogeneity of permutation groups: a group G is orbit- t-homogeneous if two sets of cardinality t lie in the same orbit of G whenever their intersections with each G-orbit have the same cardinality. For transitive groups, this coincides with the usual notion of t-homogeneity. This concept is also compatible with the idea of partition transitivity introduced by Martin and Sagan. Further, this paper shows that any group generated by orbit-t-homogeneous subgroups is orbit- t-homogeneous, and that the condition becomes stronger as t increases up to ⌊n/2⌋, where n is the degree. So any group G has a unique maximal orbit-t-homogeneous subgroup Ω t (G), and Ωt (G) ≤ Ω t-1 (G). Some structural results for orbit-t-homogeneous groups and a number of examples are also given. A permutation group G acting on a set V is said to be t-homogeneous if it acts transitively on the set of t-element subsets of V . Informally, this means that all t-element subsets of V are “alike” with respect to the action of G. If the action of G is intransitive, it cannot be t-homogeneous, since the intersections of different t-subsets with orbits of G may be different. We define a more general condition to cover this situation: we say that G is orbit-t-homogeneous on V if two t-sets which meet each orbit in the same number of points are equivalent under the action of G. We give a similar extension of the notion of partition transitivity introduced by Martin and Sagan [8]. As a result of the classification of the finite simple groups [4], all t-homogeneous permutation groups G on sets V with 1 <t< |V |− 1 are known. (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. A permutation group G acting on a set V is said to be orbit-t-homogeneous, or t-homogeneous with respect to its orbit decomposition, if whenever S 1 and S 2 are t-subsets of V satisfying |S 1 ∩ Δ| = |S 2 ∩ Δ| for every G-orbit Δ, there exists g ∈ G with S 1 g = S 2 . Thus, a group which is t-homogeneous in the usual sense is orbit-t-homogeneous; every group is orbit-1-homogeneous; and the trivial group is orbit-t-homogeneous for every t. Furthermore, a group is orbit-2-homogeneous if and only if it is 2-homogeneous on each orbit and, for every α ∈ V , the point stabiliser G α acts transitively on each orbit not containing α. It is also clear that a group of degree n is orbit-t-homogeneous if and only if it is orbit-(n − t)-homogeneous; so, in these cases, we may assume t ≤ n/2 without loss of generality. 2000 Mathematics Subject Classification 20B10.