Orbit - Homogeneity

We introduce the concept of orbit-homogeneity of permutation groups: a group G is orbit thomogeneous 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. We show that any group generated by orbit t-homogeneous subgroups is orbit t-homogeneous, and that the condition becomes stronger as t increases up to bn/2c, where n is the degree. So any group G has a unique maximal orbit t-homogeneous subgroup Ωt(G), and Ωt(G) ≤ Ωt−1(G). We also give some structural results for orbit t-homogeneous groups and a number of examples. 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 . The t-homogeneous groups which are not t-transitive have been classified (see [4, 5, 6]); the classification of t-transitive groups for t > 1 follows from the classification of finite simple groups [3] (the list is given in [1]). 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 S1 and S2 are t-subsets of V satisfying |S1 ∩ Vi| = |S2 ∩ Vi| for every G-orbit Vi, there exists g ∈ G with S1g = S2. 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. It is also clear that a group of degree n is orbit-thomogeneous if and only if it is orbit-(n − t)-homogeneous; so, in these cases, we may assume t ≤ n/2 without loss of generality. If two sets S1 and S2 are subsets of V satisfying |S1 ∩ Vi| = |S2 ∩ Vi| for every G-orbit Vi then S1 and S2 are said to have the same structure with respect to G (or just to have the same structure if the group is obvious). Theorem 4.3.4 of [2] is the following: Theorem 1. If G and H are orbit-t-homogeneous on V , then so is 〈GH〉. Young extended the concept of homogeneous groups by investigating the relationship between permutation groups and partitions [8]. A partition of V , P = (P1, P2, . . . , Pk), is said to have shape |P | = (|P1|, |P2|, . . . , |Pk|) . A group element g ∈ G is said to map the partition P onto a partition Q = (Q1, Q2, . . . , Qk) if Pig = Qi for all i. Obviously, a pre-requisite for this is that P and Q have the same structure with respect to G, i.e. that Pi and Qi have the same 2000 Mathematics Subject Classification 20B10. 2 peter j. cameron and alexander w. dent structure for all i. The permutation group G is said to be orbit-λ-transitive if, for any two partitions of V that have shape λ and the same structure, P and Q say, there exists some g ∈ G that maps P to Q. A permutation group of degree n is orbit-t-homogeneous if and only if it is orbit-λ-transitive, where λ = (n− t, t). The following is a more generalised version of Theorem 1. Theorem 2. If G and H are orbit-λ-transitive on V , then so is 〈GH〉. Proof. Let P and Q be partitions of V that have the same structure with respect to 〈GH〉 and have shape λ. It is sufficient to show that there exists σ ∈ 〈GH〉 such that Pσ = Q when – P1 = S1 ∪ {x1} and P2 = S2 ∪ {x2}, – Q1 = S1 ∪ {x2} and Q2 = S2 ∪ {x1}, – Pi = Qi for all i > 2, for some S1, S2 ⊆ V . Since P and Q have the same structure with respect to 〈GH〉, x1 and x2 must lie in the same 〈GH〉-orbit and so there exists an element σ′ = g1h1 . . . gmhm such that x1σ = x2. Suppose that m = 1 and let y = x1g1. Note that x1 and y lie in the same G-orbit and that y and x2 lie in the same H-orbit. If y = x1 or y = x2 then result is obvious, so assume that is not the case. There are now several cases to deal with. Suppose that y ∈ P1, i.e. S1 = S′ 1 ∪ {y}, and consider the partition R = (R1, R2, . . .) where R1 = S′ 1 ∪ {x1, x2}, R2 = S2 ∪ {y}, Ri = Pi = Qi for all i > 2. The partitions P and R have the same structure with respect to H and both have shape λ. Hence there exists h ∈ H such that Ph = R. Similarly the partitions R and Q have the same structure with respect to G and so there exists g ∈ G such that Rg = Q. Hence the result holds. Suppose that y ∈ P2, i.e. S2 = S′ 2 ∪ {y}, and consider the partition R = (R1, R2, . . .) where R1 = S1 ∪ {y}, R2 = S′ 2 ∪ {x1, x2}, Ri = Pi = Qi for all i > 2. The partitions P and R have the same structure with respect to G and both have shape λ. Hence there exists g ∈ G such that Pg = R. Similarly the partitions R and Q have the same structure with respect to H and so there exists h ∈ H such that Rh = Q. Hence the result holds. If y / ∈ P1 ∪P2 then, without loss of generality, it can be assumed that y ∈ P3, i.e. P3 = S3 ∪ {y} for some S3 ⊆ V . Consider the partitions R = (R1, R2, . . .) where R1 = S1 ∪ {y}, R2 = S2 ∪ {x2}, R3 = S3 ∪ {x1}, Ri = Pi = Qi for all i > 3, and T = (T1, T2, . . .) where T1 = S1 ∪ {x2}, T2 = S2 ∪ {y}, T3 = S3 ∪ {x1}, Ti = Pi = Qi for all i > 3. Note that both partitions have shape λ. The partitions P and R have the same structure with respect to G, hence there exists g ∈ G such that Pg = R. The partitions R and T have the same structure with respect to H, hence there exists h ∈ H such that Rh = T . The partitions T and Q have the same structure with