Orbit-homogeneity in Permutation Groups

This paper introduces the concept of orbit-homogeneity of permutation groups: a group G is orbitt-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 orbitt-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). 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 S1 and S2 are t-subsets of V satisfying |S1 ∩∆| = |S2 ∩∆| for every G-orbit ∆, 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. 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 Gα 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. 2 peter j. cameron and alexander w. dent If two sets S1 and S2 are subsets of V satisfying |S1 ∩ ∆| = |S2 ∩ ∆| for every G-orbit ∆ 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 [3] 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 [9]. An ordered 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 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 general version of Theorem 1. Theorem 2. If G and H are orbit-λ-transitive on a finite set V , then so is 〈GH〉. Proof outline. We begin by showing that it suffices to prove that there exists a σ ∈ 〈GH〉 that maps one partition to another when the two partitions differ in that two elements, x1 and x2, have “swapped” parts. We note that, since these two elements must lie in the same 〈GH〉-orbit, there must be a finite chain of elements g1h1 . . . gmhm that map one to the other. We show that we can map one partition to the other if m = 1 by splitting the proof into three cases based on whether the intermediate point y = x1g1 lies in the same part of the partition as x1, x2, or neither of these points. We extend these results by induction to cover all values of m using similar arguments. Therefore the theorem holds. Proof. Let P = (P1, P2, . . .) and Q = (Q1, Q2, . . .) be finite partitions of the finite set V that have the same structure with respect to 〈GH〉 and have shape λ. We say that a point x ∈ V is “bad” if x ∈ Pi but x / ∈ Qi for some integer i. Hence, the bad points are the points that need to be “moved” in order to map P to Q. Since V is finite, we may enumerate these bad points x1, ..., xk and assume, without loss of generality, that x1 ∈ P1 \Q1. Since P has the same structure as Q with respect to 〈GH〉, there must exist a bad point y ∈ Pi ∩ Q1, for some i 6= 1, that is in the same orbit as x1. Consider the partition P (1) given by: (i) P (1) 1 = (P1 \ {x1}) ∪ {y}, (ii) P (1) i = (Pi \ {y}) ∪ {x1}, (iii) P (1) j = Pj for all j 6= 1, i. The partitions P and P (1) differ only in that x1 and y have swapped parts, but P (1) has at most k − 1 bad points. Hence (by a simple induction), it is easy to see that there exists a chain of partitions P = P , P , P , . . . , P (l) = Q with the orbit-homogeneity in permutation groups 3 same structure and shape such that the only difference between P (j) and P (j+1) is the swapping of two bad points. Therefore, in order to prove the theorem it is sufficient to show that there exists σ ∈ 〈GH〉 such that Pσ = Q when (i) P1 = S1 ∪ {x1} and P2 = S2 ∪ {x2}, (ii) Q1 = S1 ∪ {x2} and Q2 = S2 ∪ {x1}, and (iii) Pj = Qj for all j > 2, for some distinct x1, x2 ∈ V and S1, S2 ⊆ V \ {x1, x2}. 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 then x1 and x2 lie in the same H-orbit and so, as H is orbit-λ-transitive, there exists an element h ∈ H that maps P onto Q. If y = x2 then x1 and x2 lie in the same G-orbit and so, as G is orbit-λ-transitive, there exists an element g ∈ G that maps P onto Q. We therefore assume that y 6= x1, x2. We split the proof into three cases depending on whether y ∈ P1, y ∈ P2 or y ∈ Pi for some i ≥ 3. 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 respect to G, hence there exists g′ ∈ G such that Tg′ = Q. Hence the result holds when m = 1. Assume, as induction hypothesis, that the theorem holds for a given value of m 4 peter j. cameron and alexander w. dent and consider the case when σ′ = g1h1 . . . gm+1hm+1. Let y = xg1h1 . . . gmhm. If y = x1 or y = x2 then the result is obvious, so we will assume that this is not the case. 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. Since there exists an element gm+1hm+1 ∈ 〈GH〉 such that ygm+1hm+1 = x2 there must exist an element σ1 ∈ 〈GH〉 such that Pσ1 = R. Similarly, since there exists an element g1h1 . . . gmhm ∈ 〈GH〉 such that x1g1h1 . . . gmhm = y there must exist, by induction, an element σ2 ∈ 〈GH〉 such that Rσ2 = Q. Hence Pσ1σ2 = Q. 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. Since there exists an element g1h1 . . . gmhm ∈ 〈GH〉 such that x1g1h1 . . . gmhm = y there must exist, by induction, an element σ1 ∈ 〈GH〉 such that Pσ1 = R. Similarly, since there exists an element gm+1hm+1 ∈ 〈GH〉 such that ygm+1hm+1 = x2 there must exist an element σ2 ∈ 〈GH〉 such that Rσ2 = Q. Hence Pσ1σ2 = Q. 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. Since there exists an element g1h1 . . . gmhm ∈ 〈GH〉 such that x1g1h1 . . . gmhm = y there must exist, by induction, an element σ1 ∈ 〈GH〉 such that Pσ1 = R. Similarly, since there exists an element gm+1hm+1 ∈ 〈GH〉 such that ygm+1hm+1 = x2 there must exist an element σ2 ∈ 〈GH〉 such that Rσ2 = T . Lastly, since there exists an element g1h1 . . . gmhm ∈ 〈GH〉 such that x1g1h1 . . . gmhm = y there must exist, by induction, an element σ3 ∈ 〈GH〉 such that Tσ3 = Q. (It may be supposed that such a σ3 cannot be assumed to exist as g1h1 . . . gmhm maps x1 to y rather than mapping y to x1, and the corresponding element of 〈GH〉 that maps y to x1 is eh−1 m g −1 m h −1 m−1 . . . g −1 1 e ∈ 〈GH〉 which is too long to apply the inductive assumption. However, since g1h1 . . . gmhm maps x1 to y, there exists a σ ∈ 〈GH〉 that maps Qσ = T . Therefore σ3 = σ−1 maps Q to T .) Hence Pσ1σ2σ3 = Q. Therefore, the theorem holds for σ′ = g1h1 . . . gm+1hm+1 and so, by induction, for all values of m. Hence any permutation group G on V has a unique subgroup Ωλ(G) which is maximal with respect to being orbit-λ-transitive. Proposition 3. For any permutation group G that acts on a finite set V , and any shape λ of V , the subgroup Ωλ(G) is normal in G. Proof. Pick g ∈ G and set H = gΩλ(G)g−1. Let P and Q be any two partitions orbit-homogeneity in permutation groups 5 of V with shape λ and the same structure with respect to H. Then Pg and Qg have shape λ and the same structure with respect to G, therefore there exists an element σ ∈ Ωλ(G) such that Pgσ = Qg. Hence P (gσg−1) = Q and so H is orbitλ-transitive. Since Ωλ(G) is a maximal orbit-λ-transitive subgroup of G, we have H ≤ Ωλ(G). However |H| = |Ωλ(G)| and so H = Ωλ(G). Therefore Ωλ(G) is normal in G. If λ = (λ1, . . . , λk) is a shape of a partition of V then, without loss of generality, it can be assumed that λ1 ≥ λ2 ≥ . . . ≥ λk . Furthermore, if μ = (μ1, . . . , μm) is the shape of another partition of V then a partial ordering can be defined where μ dominates λ, written λ E μ, if