Orbital diameters of the symmetric and alternating groups

For a primitive group G acting on a ﬁnite set (cid:2) , we deﬁne the orbital diameter to be the maximum of the diameters of all orbital graphs of G . In this paper, we study the orbital diameters of symmetric and alternating groups. We give necessary numerical conditions for the orbital diameter to be bounded by some constant c and give precise descriptions of the actions for which the orbital diameter is bounded by 5. For each primitive action, we also either determine all orbital graphs of diameter 2 or give descriptions of inﬁnite families of orbital graphs of diameter 2.


Introduction
Let G be a finite transitive permutation group acting on a set . We will denote such a group by (G, ). The orbitals of (G, ) (or just G) are the orbits of the natural action of G on × . The diagonal orbital is 0 := {(α, α) : α ∈ }. To each orbital , we associate the paired orbital * := {(β, α) : (α, β) ∈ )}; is called self-paired if = * . For a given orbital , the orbital graph ∪ * is defined to be the undirected graph whose vertex set is and edge set is ∪ * . Henceforth we will assume that = 0 and write if is self-paired. Note that orbital graphs belong to a wider family of graphs known as edge-transitive graphs. By a theorem of D. G. Higman, a permutation group (G, ) is primitive if and only if all the orbital graphs are connected [see [5], (1.12)]. For a graph , we will use V ( ) to denote the vertex set of . The valency of α ∈ V ( ) is the number of edges that are connected to α; if all vertices of have the same valency v, then v is defined to be the valency of . For a connected graph , the distance between two vertices u and v is the length of a shortest path between u and v and will be denoted by d (u, v); we will always assume that u = v and use the notation d (u, v) if it is clear which graph is in question. The diameter of a connected graph is the greatest distance between any two vertices and will be denoted by diam( ). The orbital diameter of a primitive permutation group (G, ) is the maximum of the diameters of the orbital graphs of (G, ) and will be denoted by diam O (G, ). If C is a class of finite primitive permutation groups, then C is said to be bounded if there exists a positive integer d such that diam O (G, ) ≤ d for all (G, ) in C. In this paper, we study the orbital diameters of finite symmetric and alternating groups in their primitive actions. The study is motivated by a paper [7] by Liebeck, Macpherson and Tent in which the authors describe infinite bounded classes of finite primitive permutation groups. Let C be such an infinite class. The main theorem of [7] gives tight structural information about the primitive groups in C depending on the type of the primitive group as classified in the O'Nan-Scott theorem [8]. In particular, if k is a fixed integer and C is an infinite class of finite symmetric groups S n , or alternating groups A n , with natural action on I {k} , the set of k-subsets of I := {1, . . . , n}, then C is bounded. Moreover, this is the only primitive action of S n and A n which gives rise to a bounded infinite class.
In this paper, we present versions of the results in [7] for finite symmetric and alternating groups with precise numerical bounds on the orbital diameters in question (see Theorem 1.1). As a consequence, we classify the primitive actions of the symmetric and alternating groups for which the orbital diameter is at most 5 (see Theorem 1.2). In addition, we prove results classifying orbital graphs of diameter 2 (see Theorem 1.3).
We would like to mention here that the results in this paper are part of a more general programme of classifying bounded infinite families of finite primitive permutation groups. Such results of the author on primitive actions of simple groups of Lie type will be forthcoming. We will use the notation (G : H ) to denote the set of right cosets of a subgroup H in a group G.
Let G := S n or A n and I := {1, . . . , n}. Henceforth we will assume that n ≥ 5. If H is a maximal subgroup of G then one of the following holds: 1. H is intransitive on I , and hence H ∼ = (S k × S n−k ) ∩ G, where 1 ≤ k < n/2. Note that for k = 1, there is only one orbital graph n 0 which has diameter 1; henceforth, we will assume that k > 1. Lastly, if n = 2k, then H := S k × S k is not a maximal subgroup of S n ; however, the orbital graphs of the action of S n on the set (S n : H ) are still n i , for i ∈ {0, . . . , k − 1}, and n i is connected for all i = 0.
If H is imprimitive on I , then the action of G := S n or A n on (G : H ) is equivalent to the action of G on the set I (k,l) , of (k, l)-partitions of I ; here a (k, l)-partition of I is a partition of I into l sets A 1 , . . . , A l , each of size k. Henceforth such a (k, l)partition of I will be denoted by A 1 | . . . |A l . Let A := A 1 | . . . |A l and B := B 1 | . . . |B l be two (k, l)-partitions of I . We will denote by I AB the l × l matrix with entries (I AB ) i j := |A i ∩ B j |. Note that the row and columns sums of I AB are k, and for every l × l matrix M whose entries are non-negative integers, and whose row and column sums are k, there exists a pair (A, B) of (k, l)-partitions of I such that I AB = M. Let A, B, C, D ∈ I (k,l) . Then the pairs (A, B) and (C, D) lie in the same orbital of S n if and only if there exist l × l permutation matrices P and Q such that I C D = P I AB Q, i.e. I AB and I C D are equal up to permutation of rows and columns (see Proposition 4.1). Let N denote the set of all l × l matrices with non-negative integer entries, and row and column sums equal to k. Define an equivalence relation ∼ on N , where M ∼ M if and only if M can be obtained from M by a permutation of rows and columns. The orbitals of (G, I (k,l) ) can then be described as . If M T = M, then we will denote the corresponding orbital graph by [M] .
In the case where H is primitive on I and G := S n , we will assume that H = A n , since diam O (G, (G : A n )) = 1, for any n.
We now state our main results.    Remark Tables 2 and 3 include: 1. the rank of the action, the diameter of the orbital graphs and their valencies in each case, 2. four cases with n = 14, 18 or 24, marked with question marks, where we have been unable to compute the orbital diameters. The proofs of Theorem 1.2(1), (2) and (3) can be found on Pages 9, 25, and 29, respectively.
We also analyse orbital graphs of diameter 2. For the case where = I (k,l) , we find all orbital graphs of diameter 2. For = I (k,l) , we find an example of an infinite family of orbital graphs of diameter 2; there could be more infinite families of orbital graphs of diameter 2, but classifying all of them could take significantly more effort.   Tables 4 and 5 specify the precise pairs (n, H ) for which (G, ) has an orbital graph of diameter 2 (the case n = 24 is still just a possibility; see the remark below). Remark 1. Tables 4 and 5 include the rank of the action, the size of the set D 2 , of orbital graphs of diameter 2 and the valencies of the orbital graphs in each case. 2. Table 5 contains the case n = 24, marked with a question mark, where we have been unable to determine or disprove the existence of an orbital graph of diameter 2. 3. The notation x [y] has been used in the 'valencies' column of Table 5 to denote that the valency x appears y times.
The proofs of the main results require some analysis of the three types of primitive actions of G := S n or A n . In the first case, namely the action of G on where := I {k} , some work has been done in [2] on finding the diameters of the corresponding orbital graphs. However, we believe that the main result in [2] has an error and state the correct result below.  [3] , 11,232 [25] 15 P SL 4 (2) 1687 ≥8 4 0 ,320 [8] , . . . 16 2 3 : P SL 2 (7) 151 79 40,320 [1] , 80,640 [2] , 53,760 [1] , 64,512 [1] , 161,280 [7] , 107,520 [3] , The error occurs in line 2 on p. 6648 of [2]. According to [2], for any i > k/2 , diam( n i ) = 3, and in Theorem 7, a construction for a path of length 3 is specified for any two vertices A and B with m := |A ∩ B| < 2i − k. This involves constructing a set Z = Z (n, k, i, m) which is, however, impossible to construct for many values of n, k, i and m; for example, let (n, k, i, m) : Note that in order to prove Theorem 1.3(1), we do not require Theorem 1.4 in its entirety; however, we give a proof of the whole theorem for completeness. The proof of Theorem 1.4(1) and (2c) can be found on Page 14, and the proofs of (2a), (2b) and (3) can be found on Pages 9, 12, and 16, respectively.
To study the orbital graphs for the action of G on , where := I (k,l) and n = kl, we will need Theorem 1.4(2) as well as the following results.
Then the following hold: 2i . The proof of Theorem 1.6(1) and (2a) can be found on Page 22, and the proof of (2b) can be found on Page 24.

Computing diameters using MAGMA
Some of the results in this paper rely on computations which have been done using magma [1]. In this section, we describe the methods used for the computations and their implementation in magma.

Method CAM: using collapsed adjacency matrices
Let G be a transitive permutation group acting on a finite set . Let 1 := {α}, 2 , . . . , r be the distinct orbits of G α on , i.e. the suborbits of G, with representatives α 1 := α, α 2 , . . . , α r , where r is the rank of the action. The suborbits of G are in one-to-one correspondence with the orbitals of G; a suborbit β G α corresponds to the orbital (α, β) G := {(α, β) g : g ∈ G}, where (α, β) g denotes the action of g on (α, β). Let E be a union of orbitals of (G, ). Define an r × r matrix A E where for k, j ∈ {1, . . . , r }, and E(α k ) := {γ ∈ : (α k , γ ) ∈ E}. Note that A E does not depend on the choice of the suborbit representative α k . If E = ∪ * where is an orbital of G, then A E is called the collapsed adjacency matrix for the orbital graph ∪ * (with respect to the ordering of the suborbits). The corresponding collapsed graph is defined to be the undirected graph C ∪ * , with vertex set {1, . . . , r }, and in which (k, j) is an edge if (A ∪ * ) k j > 0. When computing the diameter of an orbital graph, it is advantageous to work with the corresponding collapsed graph since it has significantly fewer vertices than the orbital graph but in fact the same diameter. The following lemma, which is based on Theorem 3.2 in [10], justifies the switch from ∪ * to C ∪ * . Lemma 2.1 Let (G, ) be a finite primitive permutation group and an orbital graph of G. Then diam( ) = diam( C ).
The following result aids the computation of collapsed adjacency matrices; see Proposition 2.2 in [10].

Implementation of CAM in MAGMA
magma does not have an in-built function to calculate collapsed adjacency matrices. We can, however, use an in-built function to define the group G and iterate over the maximal subgroups of G to find the required subgroup H ∼ = G α . We can then use more in-built functions to firstly construct the permutation group (G, ) where := (G : H ), and secondly, find the suborbits of G, with representatives α 1 := α, α 2 , . . . , α r . Let := (α, β) G be an orbital of G. We calculate A by implementing the method described in [10, p. 24]. If is self-paired, then A is the collapsed adjacency matrix for the orbital graph ∪ * . If is not self-paired, then we can use the function trace : → G (constructed whilst implementing the method described in [10]) where trace(γ ) = g means that α g = γ , to find g ∈ G such that β g = α. Then * = (α, α g ) G , and we can again use the method in [10] to compute the matrix A * . The collapsed adjacency matrix for the orbital graph ∪ * is then A + A * by Lemma 2.2. Once we have a collapsed adjacency matrix A E where E := ∪ * , the diameter of C E (and hence the diameter of E by Lemma 2.1) is the least number of times A E has to be multiplied by itself before each non-diagonal entry has taken a value greater than zero at least once.
The implemented method works well when |G : H | is small. For large values of |G : H |, the method fails at the very first step of constructing the permutation group (G, ).

Action of S n on k-subsets of I
In this section, we prove Theorems 1.1(1), 1.2(1), and 1.3(1). We then prove Theorem 1.4 which is used to deduce Theorem 1.6 in Sect. 4.
Recall that in the orbital graph n k−1 , two k-subsets are joined by an edge if they intersect in k − 1 points. Observe that this graph has diameter k, and so k ≤ c. Conversely, suppose k ≤ c and let be an orbital graph of (G, ). For a k-subset A and an integer j ≥ 1 define is a union of non-trivial orbits of G A on . There are k non-trivial orbits of G A on , and therefore diam( ) ≤ k ≤ c.
We now prove Theorem 1.3 (1). Recall that for i ∈ {0, . . . , k − 1}, the graph n i has vertex set I {k} where I = {1, . . . , n}, and edge set {(A, B) : |A ∩ B| = i}. Henceforth, for two sets A and B, the notation A\B will be used to refer to the complement of B in A. The following lemma follows from Lemmas 1 and 2 in [3] and the fact that n i is a subgraph of the graph K (n, k, i) from [3].

If A and B are connected by a path of length
We now continue proving the rest of Theorem 1.4. We will need to prove the following lemma.
where f and g are as in Tables 6 and 7, for v even and v odd, respectively.
In order to prove Lemma 3.5, we now prove the following sequence of results.
, and t ≥ 3. Suppose the following conditions hold: and for t odd (t = 2r + 1), Remark By Lemma 3.7, if (1) and (3) of Lemma 3.6 hold, then so does (2); hence, after Lemma 3.7 has been proved, we may omit (2) from the statement of Lemma 3.6. Similarly, after Lemma 3.5 has been proved, we may also omit (3).
Proof of Lemma 3. 6 We proceed by induction on t. For the base case, suppose the conditions of the lemma hold for t = 3. Then by Corollary 3.4 and P (3), Therefore the result for t = 3 follows. Now let t = 2r > 3 and suppose the lemma holds for t − 1. Suppose also that the conditions of the lemma hold for t. Then the conditions of the lemma also hold for t − 1, and so by the inductive assumption, Now by P(2r ), we have that and therefore we obtain the result for t = 2r by combining (1) and (2). To complete the induction, it remains to show the result for t + 1 = 2r + 1 which can be done using similar arguments to the above and is left to the reader.
Proof We proceed by induction on t. For the base case, assume that P(3) holds and diam( n i ) ≥ 4. Suppose n ≥ 5(k−i) 2 . Then using Corollary 3.4 and P(3) we see that This completes the base case. Now let t = 2r ≥ 4 and assume that the lemma holds for t − 1.
To complete the induction, it remains to show the result for t + 1 = 2r + 1 which can be done using similar arguments to the above and is left to the reader.
Then by Corollary 3.4, we have d(A, B) > 2, and using the fact that n ≥ 5(k−i) 2 , we see that m ≤ n − 2k + 3i. Now as above, we show that d(A, B) = 3 by constructing a path of length 3 between A and B.
Proof Suppose diam( n 0 ) = 3. By Theorem 1.3(1), we have n < 3k − 1 and by (3) holds. Now using Lemma 3.8 and the fact that n ≥ 5k−1 2 , we see This proves the first statement of the corollary. The statement for i > 0 can be proved using similar arguments to the above and is left to the reader.
We need one final lemma before we can prove Lemma 3.5.
Proof Suppose the conditions in the first part of the lemma hold. Choose a neighbour This proves the first part of the lemma. Now suppose the conditions in the second part of the lemma hold. Choose a neighbour A 1 of A such that |A 1 ∩ B| is maximal. Then (5) implies m > i, and so Therefore using Lemma 3.7 (with t = 2r ) and P(2r ), we have d(A 1 , B) = 2r , and hence d(A, B) = 2r + 1.
Proof of Lemma 3.5 For v = 3, the result is proved in Lemma 3.8. For v ≥ 4, we proceed by induction on v.

By Lemma 3.12 we know that if
We will now show that this inequality is in fact an equality. Proof Suppose d(A, B) = l + 1 and A − C 1 − · · · − C l − B is a shortest path. It can be seen using a short inductive argument that C l can differ from A by at most l(k − i) points. Using this and the fact that |C l ∩ B| ≤ |A ∩ B| + |C l \ A|, we get from which the claim follows.
Hence d(A, B) ≥ 2 ≥ k−m k−i , from which the claim follows.
The following corollary is a consequence of Lemmas 3.12 and 3.13.

Action of S n on (k, l)-partitions of I
We begin this section by finding the orbitals of the action of S n on I (k,l) . We then prove a result from which Theorem 1.5 is deduced and then use Theorem 1.5 to deduce Theorem 1.1 (2). We continue by proving a sequence of results which lead up to the proof of Theorem 1.2(2), and we conclude the section by proving Theorem 1.3(2). For the converse, suppose I C D = P I AB Q for some l ×l permutation matrices P and Q. Then after permuting the A i s and B j s suitably, we may suppose I AB = I C D . Now since S n is n-transitive, we can pick g ∈ S n such that  Remark If A, B ∈ V ( [M kl ] ) are adjacent, then there exists a transposition in S n sending A to B. Hence every path between A and B can be represented by a sequence of transpositions and the number of transpositions is the length of the path. For a path P between A and B, we will use σ P to denote the product of the corresponding sequence of transpositions. Proof Let A and B be two (k, l)-partitions such that Write A = A 1 | . . . |A l and B = B 1 | . . . |B l . If d(A, B) = r and P is a path of length r between A and B, then σ P sends A to B and for each i ∈ {1, . . . , l} there exists j ∈ {1, . . . , l} such that σ P sends A i to B j . Now |A i ∩ B j | ≤ k 2 and hence σ P must move at least k − k 2 = k 2 elements from each A i . Therefore σ P must move at least l k 2 elements in total. Any permutation which moves x elements cannot be expressed as a product of fewer than x/2 transpositions. Hence we must have r ≥ l Now since k 2 ≥ 1, (13) implies l ≤ 2c, and since l ≥ 2 it also implies k ≤ 2c + 1, and hence n = kl ≤ 2c(2c + 1).
Next we prove Theorem 1.2(2). Suppose G := S n or A n and the orbital diameter of (G, I (k,l) ) is bounded by 5. By Theorem 1.1(2), we must have 1 < l ≤ 10 and 1 < k ≤ 11, and we can make these values more precise using Proposition 4.3. In order to prove Theorem 1.2(2), for each possible pair (k, l), we must either determine that the diameters of all orbital graphs are bounded by 5 or find an orbital graph of diameter greater than 5. In order to reduce the possibilities for the pair (k, l) we prove the following results. In what follows, for a permutation σ , the support of σ is supp(σ ) := {x : x σ = x}. For l = 3, we prove Let d(A, B) = r and suppose that P is a path of length r between A and B. As in the proof of Proposition 4.3, σ P must move at least k 2 elements from each A i . If there exists i such that σ P moves all k elements from A i , then σ P moves at least k + 2 k 2 elements in total, and hence r ≥ k 2 + k 2 = k ≥ 2 k 2 . Now assume that σ P fixes at least one element from each A i . Set X 1 := A 1 ∩ B 1 , Y 1 := A 2 ∩ B 2 and Z 1 := A 3 ∩ B 3 , and let X 2 , Y 2 and Z 2 be their complements in A 1 , A 2 and A 3 , respectively. Then the following information can be read off I AB ( denotes disjoint union): Then A i σ P = B i for all i. Now if α ∈ X 2 and β := α σ P , then β ∈ X 1 Y 2 , so β = α. Also, if β ∈ Y 2 then β σ P ∈ Y 1 Z 2 , so β σ P = α, and if β ∈ X 1 , then β σ P ∈ X 1 Y 2 , so again β σ P = α. Hence α is contained in a cycle of σ P , of length at least 3. Repeating similar arguments, we see that if α ∈ Y 2 or α ∈ Z 2 , then α is contained in a cycle of σ P , of length at least 3. By the remark following Definition 4.2, we have that r is greater then or equal to the minimal length of a factorisation of σ P by transpositions. Using the fact that the minimal length of a factorisation of a permutation σ by transpositions is |supp(σ )|−c σ , where c σ is the number of non-trivial cycles in the disjoint cycle decomposition of σ , we get r ≥ |supp(σ P )|−c σ P . Suppose that there are s disjoint cycles of length at least 3 in the disjoint cycle decomposition of σ P . Let I 3 ⊆ {1, . . . , n} be the union of the supports of these cycles. Then |supp(σ P )| − σ P ≥ |I 3 | − s. Furthermore, s ≤ 1 3 |I 3 | and hence we get r ≥ 2 3 |I 3 |. On the other hand, the preceding discussion implies Combining the two, we obtain r ≥ 2 k Repeating similar arguments to the above, we can see that every element of X 1 ∪ Y 1 ∪ Z 1 must be contained in a cycle of length at least 3, and so as above, we have that Therefore d(A, B) ≥ 2 k 2 , from which the result follows.
For k = 3, we have the following result. i=1 supp(a i ), and now a careful analysis, whose details are left to the reader, shows that in all cases, there exists 1 ≤ i ≤ l 2 and 1 ≤ r = s ≤ l such that a i = (x y) for some x ∈ A r and y ∈ A s ; moreover, the supports of a i and a j are disjoint for all 1 ≤ j ≤ l 2 such that j = i, and |supp(σ )∩ A r | = |supp(σ )∩ A s | = 1. In particular, a i commutes with every other transposition and we can write σ P = a i σ P where σ P := a 1 · · · a i−1 a i+1 · · · a l 2 . Thus We also have a small improvement in the lower bound in Proposition 4.3 which is obtained as follows. Since I AB is given by (12), we can Then B i = X i+1 Y i for all 1 ≤ i ≤ l − 1, and B l = X 1 Y l . By the proof of Proposition 4.3, σ P moves at least k 2 elements from each A i . Hence by the assumption, exactly k 2 elements are moved from each A i . Therefore A i σ P = B i , for all i. In particular, σ P sends X i to X i+1 , for i ∈ {1, . . . , l − 1}, and X l to X 1 . Hence if we write σ P = C 1 · · · C r , where the C i are non-trivial disjoint cycles, then we can conclude that each C i has length which is a multiple of l, say t i l, and r ≤ k 2 . Then r i=1 t i l = k 2 l and σ P cannot be expressed as a product of fewer than and P is a path between A and B, then σ P moves at least l k 2 points from A. Now if σ P moves exactly l k 2 points, then by Proposition 4.6, diam( [M kl ] ) ≥ k 2 (l − 1). Otherwise, σ P moves at least l k 2 + 1 points and hence cannot be expressed as a product of fewer than   Table 8. Then diam O (G, ) ≥ 6.
For the case l = 2, we find the diameters of all orbital graphs of (S k2 , I (k,2) ) as stated in Theorem 1.6. Recall that for l = 2, the orbital graphs are Note that the matrix M i is determined by (M i ) 11 and if A, B ∈ V ( [M i ] ), then the matrix I AB is determined by (I AB ) 11 . In order to prove Theorem 1.6 we will need to prove the following criterion which relates distances in the graphs [M i ] to distances in the graphs 2k i from Sect. 3. In what follows, for a k-subset X of I := {1, . . . , 2k}, we will use the notation X c to refer to the complement in I of the set X .
The following lemma follows directly from Lemmas 3.3 and 3.5.
and if 5i ≥ k, then Proof Note that we must have 2i < k (else the diameter is 2), and hence (14) holds. Then the proof follows from Lemmas 4.9,4.10,and (14).
In order to prove Theorem 1.6(2b), we need the following.
Proof We proceed by induction or r . The base case r = 3 has been done in Lemma 4.11. Now suppose d [M i ] (A, B) ≥ r ≥ 4 and that the lemma holds for all s < r . Then using Lemma 4.12 and the inductive hypothesis to determine the pairs of vertices for which the distance is strictly less than r , we obtain that For j as in (17), by Lemmas 4.9 and 4.10, if (2r − 1)i < k, then We obtain the result by combing the inequalities above with (17).
The next corollary is a direct consequence of the proof of Lemma 4.14.
Proof of Theorem 1.6(2b) Suppose diam( [M i ] ) = 2. It suffices to show that for r ≥ 3, We proceed by induction on r . The base case r = 3 is done in Proposition 4.13. Now let r ≥ 4 and suppose for the inductive hypothesis that To complete the proof of Theorem 1.2(2), we need to establish which of the values in Table 8 in Proposition 4.8 actually yield diam O (G, I (k,l) ) ≤ 5. To do this, we have used magma. We have devised two codes; the first takes as inputs the values k and l and returns the diameters of all orbital graphs of (G, I (k,l) ). In order to be able to deal with as high values of n = kl as possible, the code is designed to calculate the collapsed adjacency matrix of each orbital graph as outlined in Sect. 2. As mentioned in Sect. 2, this code has limitations when the index gets large. In this case, we have devised a second code; fix two (k, l)-partitions A and B (usually with I AB is as in (12)), and let σ ∈ G := S kl be a permutation sending A to B. Then any permutation sending A to B is of the form wσ , where w ∈ G A ∼ = S k S l . Let R be the set of permutations sending A to B. The code takes as inputs the values of k, l and σ and calculates for each τ ∈ R the minimal number of transpositions T (τ ) that τ can be a product of. The code returns the value min τ ∈R T (τ ), which is (by the remark following Definition 4.2) the value of d (A, B) d(A, B) for all pairs (A, B), but for the results in this paper, only a lower bound is required.
Proof of Theorem 1.2 (2) Table 8 lists the values of k and l which remain to be considered. For the case where l = 2, using Theorem 1.6 it can be seen that diam O (G, ) ≤ 5 if and only if 2 ≤ k ≤ 11. For l ≥ 3, magma has been used to verify the result as outlined in the paragraph after the proof of Theorem 1.6(2b). The data obtained can be found in Appendix 1.
Although as n = kl tends to infinity the diameters of the corresponding orbital graphs are unbounded, there do exist infinite families of orbital graphs with small diameters. We will now present one infinite family of diameter 2 graphs and prove Theorem 1.3 (2). In order to do this, we will need the following preliminary definitions and lemma.

Definition 4.16
Let A ∈ M n (R ≥0 ) be an n × n matrix with non-negative real entries, and let k ≥ 1 be an integer. We say that A is a k-doubly stochastic matrix if each row sum and each column sum is k. We say that A is doubly stochastic if A is 1-doubly stochastic. (a 1 , . . . , a n ) of A is a sequence of n entries of A such that no two entries lie in the same row or column of A. A diagonal D is called positive if a i > 0 for all i.

Definition 4.17 Let
The following lemma can be found as Lemma 7.4.4 in [6]. . . |B k . We will show that there exists a vertex C such that I AC ∼ M ∼ I C B . We will do this by describing an explicit construction for C. Define I AB := 1 k I AB . Then I AB is a doubly stochastic matrix. By Lemma 4.18, I AB , and therefore k I AB , has a positive diagonal. Moreover, the latter diagonal has positive integer entries, in particular at least 1. Denote this diagonal by D = (a i 1 j 1 , . . . , a i k j k ), where for r ∈ {1, . . . , k}, a i r j r := |S i r j r | > 0 and S i r j r := A i r ∩ B j r . Let C 1 be a k-subset which consists of exactly one element from each intersection S i r j r , for r ∈ {1, . . . , k}. Now |C 1 | = k and |A i ∩ C 1 | = 1 = |C 1 ∩ B i |, for all i ∈ {1, . . . , k}. Note that when we subtract 1 from each of the entries a i r j r in I AB then we are left with a (k − 1)-doubly stochastic matrix I (1) AB where (I (1) AB ) i r j r = |(A i r ∩ B j r ) \ C 1 |, for r ∈ {1, . . . , k}, and (I (1) AB ) xy = (I AB ) xy , otherwise. As before we can use Lemma 4.18 to deduce that I (1) AB has a positive integer diagonal, say D = (b t 1 u 1 , . . . , b t k u k ), where for r ∈ {1, . . . , k}, either b t r u r = |(A t r ∩ B u r )\C 1 | or b t r u r = |A t r ∩ B u r |. In the former case, define T t r u r := (A t r ∩ B u r )\C 1 , and in the latter case, define T t r u r := A t r ∩ B u r . Note that in either case T t r u r ∩ C 1 = ∅. We let C 2 be the set which consists of exactly one element from each of the sets T t r u r , for r ∈ {1, . . . , k}. Then |C 2 | = k, |A i ∩ C 2 | = 1 = |C 2 ∩ B i | for all i ∈ {1, . . . , k}, and the matrix I (2) AB is (k − 2)-doubly stochastic. We may repeat the above process to construct C j at which stage we are left with a (k − j)-doubly stochastic matrix. Then we can construct C j+1 as long as k − j ≥ 1, i.e. j + 1 ≤ k. So we may construct sets C 1 , . . . , C k such that where Y := β H . In particular (since |Y | ≤ |H |), we have that Proof Let v denote the valency of ∪ * . If is self-paired, then v = |β H |, and if is not self-paired, then v = 2|β H |. Now the result follows from the following inequality: We will also need the following result (Corollary 1.2 in [9]) concerning the orders of primitive subgroups of S n and A n .

Lemma 5.2
If H is a primitive subgroup of S n not containing A n , then |H | < 3 n . Moreover, if n > 24, then |H | < 2 n .
Next we prove Theorem 1.2(3). By Lemma 5.1, if (G, ) has an orbital graph of diameter d, then So either n ≤ 24 or, using Lemma 5.2, Using (22) with d = 5, we see that n ≤ 173 for G = S n and n ≤ 174 for G = A n . We can then use in-built functions in magma to run through values of n ≤ 174 and for each such n, iterate through the maximal subgroups of G and output only the primitive subgroups H which satisfy (21) with d = 5. This way we limit the number of possible pairs (n, H ) for which (G, (G : H )) has an orbital graph of diameter d ≤ 5; to 22 and 30 for G = S n and G = A n , respectively. Now if diam O (G, ) ≤ 5, then there must exist an orbital graph of diameter d where d ≤ 5, and hence we also obtain all possibilities of (G, (G : H )) for which the orbital diameter is bounded by 5. In order to rule out some of these possibilities, we prove the following result.
Now k = |G α : G α ∩ g −1 G α g| and |G α ∩ g −1 G α g| ≥ |X |. Hence k ≤ |H | |X | . Using this in (23), we obtain the result.  In light of Proposition 5.3, picking a large subgroup X ≤ H such that N G (X ) > N H (X ) can determine the existence of an orbital graph of large diameter. We have used magma to devise a code which, given G and H , finds the largest such X and calculates the lower bounds in Proposition 5.3. This code has been used with the possibilities of (G, (G : H )) obtained using (21) with d ≤ 5, to obtain the following result. To deal with the remaining cases in Tables 9 and 10, we have again used magma to devise a code which takes as input a value of n and outputs each possible primitive group H from Tables 9 and 10, the diameters of the corresponding orbital graphs, and  (7) 8 PG L 2 (7) 9 AG L 2 (3) 10 P L 2 (9) 12 PG L 2 (11)  Tables 9 and 10. Finally, magma has been used to verify (by implementing method CAM as in Sect. 2) that out of the possible cases, the only cases for which diam O (G, ) ≤ 5 are the ones listed in the theorem. The data obtained is given in Appendix 1.
Proof of Theorem 1.3(3) Using (22) with d = 2, we obtain n ≤ 24 for both G = S n and G = A n . We then use in-built functions in magma to run through values of n ≤ 24 and for each such n, iterate through the maximal subgroups of G to obtain only the primitive ones H which satisfy (21) with d = 2. We conclude that if there exists an orbital graph of diameter 2, then n and H must be in Tables 11 or 12. Finally, magma has been used to verify (by implementing method CAM as in Sect. 2) that the only pairs (n, H ) for which diameter 2 graphs occur are the ones stated in the theorem.