Highly transitive imprimitivities

Marston Conder ^{∗}

Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand m.conder@auckland.ac.nz

Vaughan Jones ^{†}

Mathematics Department, University of California Berkeley, Berkeley, CA 94720, U.S.A. vfr@math.berkeley.edu

November 8, 2005

Abstract

Some new observations are made about imprimitive permutation groups associated with subfactors of von Neuman algebras. Of particular interest are examples of a group G containing two maximal subgroups H and K such that G ≠ HK, and such that the action of G on the space of cosets of H ∩ K has small rank (few suborbits). The rank 6 case turns out to correspond to the action of the collineation group on flags of a Desarguesian projective plane, and a special case of interest for rank 7 corresponds to the action of a 4-transitive group on ordered pairs of distinct points. Some other new (and unexpected) fundamental properties of groups are described along the way.

1 Introduction

Organising groups by the transitivity of their actions is as old as group theory itself. The idea that highly transitive group actions are scarce is basic to the discovery and classification of finite simple groups.

Subfactors began as an infinite dimensional non-commutative extension of Galois theory. A subgroup H of a finite group G gives rise to a subfactor by choosing an (outer) aciton of G on a II_{1} factor R and letting N ⊆ M be the pair of fixed point algebras R^{G} ⊆ R^{H}. These are the simplest and best understood subfactors.

∗Research supported in part by the N.Z. Centres of Research Excellence Fund (grant UOA 201)

†Research supported in part by NSF Grant DMS93–22675, the N.Z. Centres of Research Excellence Fund (grant UOA 201), and the Swiss National Science Foundation

To a subfactor of finite index is associated a ‘standard invariant’ or planar algebra, which is primarily a sequence P_{n} (for n ≥ 0) of finite-dimensional vector spaces (obtained by decomposing tensor powers ⊗_{N}^{n}M as a bimodule). The planar algebra for R^{G} ⊆ R^{H} is well understood. The vector space P_{n} is naturally the space of functions on X^{n} (where X is the coset space (G : H), which we will denote by G/H), invariant under the action of G. Thus dim P_{n} is the number of orbits of G on X^{n}.

Before proceeding further, we should point out that (a) for the simplest II_{1} factor R, an outer action of a finite group is unique up to conjugacy [13], and that (b) Izumi has shown in [11] that if the action of G on X is primitive, that is, if H is maximal in G, then one can reconstruct both G and H from the subfactor R^{G} ⊆ R^{H}.

Clearly the number of orbits for the action of the symmetric group S_{X} on X^{n} is a lower bound for dim P_{n}, and to say an action is k-transitive is just to say that dim P_{k} is equal to that lower bound. For |X| > n this number of orbits is equal to the number of partitions of a set of size n, called the nth Bell number, also equal to the coefficient of x^{n}/n! in the exponential generating function E(E(x)) + 1, where E(x) = e^{x} − 1 (see [1] or [8]).

For a subfactor not necessarily of the form R^{G} ⊆ R^{H}, planar partitions still make sense, and the universal lower bound on dim P_{n} is the Catalan number \frac{1}{n+1} \binom{2n}{n}, for all but small values of n. Accordingly, subfactors can be “more transitive” than group actions. This observation led the second author to the notion of supertransitivity, and the beginnings of the study of subfactors from this point of view in [14] and [15].

The planar algebra encodes all the algebraically accessible data for a subfactor N ⊆ M. The most obvious piece of extra structure that can arise for a subfactor is the existence of an intermediate subfactor N ⊂ T ⊂ M. In [3] it is shown that in the presence of such a T, planar partitions can be enriched, and the necessary loss of supertransitivity is reflected in a lower bound of \frac{1}{2n+1} \binom{3n}{n} for dim P_{n}, for all but small n. An intermediate subfactor for R^{G} ⊂ R^{H} is necessarily of the form R^{K} for some subgroup K with H < K < G. This leads to the following (somewhat vague) group-theoretic question:

Question 1.1 What are the most transitive imprimitive actions of finite groups?

In a sense, the most highly transitive imprimitive permutation groups of given composite degree d are the wreath products S_{a} wr S_{b} where ab = d, with 1 < b ≤ a < d. Each such group G is a semi-direct product of the direct product of b copies of the symmetric group S_{a} (acting independently on b copies of a set of size a, called the blocks) by a single copy of the symmetric group S_{b} which permutes the b blocks. We may take the stabilizer of a point as the subgroup H, and the setwise stabilizer of the block containing that point as the intermediate subgroup K.

These are rank 3 permutation groups (see [6] or [7] for example), having three orbits on X × X: one the diagonal, one containing ordered pairs of distinct points from the same block, and the other containing ordered pairs of points from different blocks. Moreover, every imprimitive permutation group of degree d = ab having b blocks of imprimitivity (each of size a), is a subgroup of S_{a} wr S_{b}, and so these examples give the highest amount of transitivity.

A generic lower bound on the number of orbits on X^{n} is the coefficient of x^{n}/n! in the exponential generating function E(E(E(x))) + 1, with E(x) = e^{x} − 1 as before.

The next natural step in considering subfactor structure from this point of view is to investigate more complicated intermediate subfactor lattices, following earlier work by Watatani [18] and Watatani and Sano [17].

The simplest situation beyond a chain of intermediate subfactors is that of a ‘quadrilateral’ (M; S, T; N) where S, T and N are subfactors of M such that N ⊂ S ∩ T. In this case we might as well always suppose that S ∩ T = N and that S and T generate M, for otherwise transitivity is decreased. Further, we may suppose that ST = M and that S and T ‘commute’, in the sense that the orthogonal projections e_{S} and e_{T} (onto S and T) commute. Planar partitions can now be further enriched to give a generic lower bound of [\frac{1}{n+1} \binom{2n}{n}]^{2} for dim P_{n}.

In the group-theoretic context, this case involves intermediate subgroups K_{1} and K_{2} of G such that K_{1} ∩ K_{2} = H is core-free in G and also

K_{1}K_{2} = G,  (1)

which leads naturally to the following:

Question 1.2 What are the most transitive imprimitive group actions with two intermediate subgroups as in condition (1)?

It is not difficult to see that in this case the group G must have at least four orbits on X × X: one the diagonal, one containing ordered pairs of distinct points from the same orbit of K_{1} but different orbits of K_{2}, another like this but with the roles of K_{1} and K_{2} reversed, and another containing ordered pairs of points from different orbits of both K_{1} and K_{2}. Hence the rank of G on the coset space X = G/H is at least 4.

One family of examples is as follows: For any integer k > 1, let G be the direct product S_{k} ×C_{2}, of order 2k!, and in this group take K_{1} and K_{2} as the natural subgroups S_{k−1} ×C_{2} (of index k in G) and S_{k} (of index 2 in G), so that H = K_{1} ∩ K_{2} = S_{k−1}. The rank of the action of G on the coset space X = G/H is 4, with orbits on X × X of lengths 2k, 2k, 2k(k − 1) and 2k(k − 1).

Similarly, the factor C_{2} can be replaced by S_{l} for any l > 1, and the rank of the resulting action is still 4. In fact it may not be difficult to prove that these are essentially the only examples of rank 4 (or at least the largest possible examples for given indices |G : K_{i}| and |K_{i} : H|), but in any case, because they are direct products, the resulting subfactors are simply tensor products of the subfactors corresponding to the coset spaces G/K_{1} and G/K_{2}, and are therefore not particularly interesting.

The next case in the subfactor situation is to disallow the possibility that ST = M, thereby decreasing transitivity.

One important point we have not mentioned so far is that for the action of G on G/H to be as transitive as possible, we want the ‘elementary’ actions of G on G/K_{1} and G/K_{2} and the actions of K_{1} and K_{2} on K_{1}/H and K_{2}/H to be highly transitive. For subfactors generally, we make the assumption (called “no extra structure”) that the lower bounds for supertransitivity of the elementary inclusions are attained. Thinking about this concept led to the surprising result in [10] that there are in fact only two possibilities for N ⊂ M if there is no extra structure: one where N is the fixed point algebra under an outer action of S_{3}, and the other where the index [M : N] is 6 + 4\sqrt{2}. For groups we are led to this:

Question 1.3 Among the imprimitive group actions with two intermediate subgroups K_{1} and K_{2} such that K_{1} ∩ K_{2} = H is core-free in G, and

K_{1}K_{2} ≠ K_{2}K_{1},  (2)

which are the most transitive?

Here we have some very interesting answers. First, as will be shown later, we can use an elementary but not well known observation (with an elegant proof due to David Goldschmidt) that gives G ≠ K_{1}K_{2}∪K_{2}K_{1} in this case, and it then follows that there must be at least 6 orbits of G on X × X. Moreover, for the case where this bound is attained, we find the following:

Theorem 1.4 Let K_{1} and K_{2} be maximal subgroups of the finite group G such that K_{1}K_{2} ≠ K_{2}K_{1}, and the action of G on the coset space X = G/H (where H = K_{1} ∩ K_{2}) is faithful and has rank 6. Then up to isomorphism either

(a) G = S_{3}, |K_{1}| = |K_{2}| = 2 and |H| = 1, or

(b) |G : K_{1}| = |G : K_{2}| = q^{2}+q+1 and |K_{1} : H| = |K_{2} : H| = q+1 for some prime-power q, and there is a 1-to-1 correspondence between points of X and the flags (incident point-line pairs) of a Desarguesian projective plane Π of order q, under which cosets of K_{1} correspond to points of Π and cosets of K_{2} correspond to lines of Π, respectively, and G corresponds to a flag-transitive collineation group of Π.

In a sense, this theorem provides an analogy to doubly-transitive permutation groups. For transitive groups of degree > 1, the minimum rank is 2, attained only when the group is 2-transitive. For groups satisfying the condition given by (2), the minimum rank is 6, attained only by flag-transitive collineation groups of Desarguesian finite projective planes. The corresponding analogue for primitive group actions of rank 3 would be a full classification of examples satisfying (2) with 7 orbits on X × X (where X = G/H). We do not have that, but we do have a complete list in a non-trivial special case:

Theorem 1.5 Let K_{1} and K_{2} be maximal subgroups of the finite group G such that K_{1}K_{2} ≠ K_{2}K_{1}, and |G : K_{1}| ≥ |G : K_{2}|, and such that the action of G on the coset space X = G/H (where H = K_{1} ∩K_{2}) is faithful and has rank 7, with two orbits on X ×X of length |X|. Then

(a) |X| = |G : H| = s(s + 1) for some integer s, and

(b) |G : K_{2}| = s + 1, and |G : K_{1}| = s + 1 or s(s + 1)/2, and

(c) the action of G on G/K_{2} is 4-transitive, and the action of G on X is equivalent to its action on ordered pairs of distinct points of G/K_{2}, while the action of G on G/K_{1} is either equivalent to that on G/K_{2} or equivalent to the action of G on unordered pairs of distinct points of G/K_{2}, and

(d) G is isomorphic to the alternating group A_{s+1} or the symmetric group S_{s+1}, or to one of the Mathieu groups M_{11}, M_{12}, M_{23} or M_{24}.

Conversely, if G is one of the Mathieu groups M_{11}, M_{12}, M_{23} or M_{24}, or the alternating group A_{k} (for k ≥ 5) or the symmetric group S_{k} (for k ≥ 4), then the action of G on ordered pairs of distinct points (in its natural action) gives rise to subgroups K_{1} and K_{2} satisfying the above conditions.

We arrived at these theorems following observations made about examples investigated with the help of the Magma system [5].

2 Preliminaries

Before proving Theorems 1.4 and 1.5, we prove two preliminary facts that are fundamental to the study of imprimitive group actions of the types discussed in the Introduction, and introduce some additional notation.

Proposition 2.1 If K and L are maximal subgroups of the finite group G for which KL ≠ G, and the actions of G on the coset spaces G/K and G/L are both 2-transitive, then |G : K| = |G : L|.

Note that H = K ∩ L does not appear in the statement of this result, and so there is no need for K ∩ L to have any particular properties.

The proposition can be proved in several ways. One way is to use Schur’s lemma from group representation theory (see [9] or [12] for example).

Let V be the vector space of functions from G to a field ℂ of zero characteristic. Let ρ denote right translation on V, and define p = ∑_{x∈K} ρ(x) and q = ∑_{y∈L} ρ(y). Then qp takes right K-invariant functions to right L-invariant functions, and commutes with the left action of G. But since KL ≠ LK, the composite qp is non-zero on the orthogonal complement of constant functions, and so by 2-transitivity and Schur’s lemma, qp is an isomorphism. Hence the dimensions of the spaces of functions from G/K and G/L to ℂ are the same.

Another proof (pointed out to us independently by Peter Cameron and Geoff Robinson) goes as follows: Let {χ_{i} : 1 ≤ i ≤ m} be the irreducible characters of G over ℂ, with χ_{1} trivial, and let χ_{K} and χ_{L} be the permutation characters for the actions of G on the coset spaces G/K and G/L. Then by 2-transitivity, χ_{K} = χ_{1} + χ_{i} and χ_{L} = χ_{1} + χ_{j} for some i, j > 1, with inner product (χ_{K}, χ_{L}) = 1 + (χ_{i}, χ_{j}) = 1 unless i = j. On the other hand,

(χ_{K}, χ_{L}) = \frac{1}{|G|} ∑_{g∈G} χ_{K}(g)\overline{χ_{L}(g)} = \frac{1}{|G|} ∑_{g∈G} Fix_{G/K}(g)Fix_{G/L}(g) = \frac{1}{|G|} ∑_{g∈G} Fix_{G/K×G/L}(g),

which by Burnside’s lemma is the number of orbits of G on G/K × G/L. As G ≠ KL, this number of orbits is at least two, so (χ_{K}, χ_{L}) > 1, and hence i = j, so χ_{K} = χ_{L} and therefore |G : K| = |G : L| (and in fact K and L are conjugate subgroups in G).

Finally, we have a third proof, offered to us by Primož Potočnik, using the theory of block designs (see [2]): The coset spaces G/K and G/L can be taken as the point-set and block-set of a block design, with incidence given by non-empty intersection

Kx ∼ Ly if and only if Kx ∩ Ly ≠ ∅.

By 2-transitivity of G on G/K, any two points lie in the same number of blocks, so this is a 2-design, and since G ≠ KL, the design is incomplete. Now by Fisher’s inequality (provable by considering the ranks of the incidence matrix M and the product MM^{T}) there are at least as many blocks as points, so |G : K| ≤ |G : L|. But the same argument applies to the dual design, and so |G : L| ≤ |G : K|; hence equality.

Proposition 2.2 If K and L are subgroups of the finite group G such that KL ∪ LK is a subgroup of G, then also KL is a subgroup of G.

Proof (a simplification of an unpublished one by David Goldschmidt). Let M = LK ∪KL, and define J = {g ∈M : KLg = KL}. Then J is a subgroup of M, with L ⊆ J ⊆ KL, and M = KJ ∪ JK, and KJ = KLJ = KL. Now assume JK ≠ M, and let g be any element of M \JK. Then Lg ∩ JK = ∅ (for otherwise g would be an element of LJK = JK), but M = KJ ∪ JK, and so Lg ⊆ KJ. It follows that KLg ⊆ KJ = KL, so by definition of J (and finiteness of G), we find g ∈ J and hence g ∈ JK, contrary to the hypothesis on g. Thus JK = M, so JK is a subgroup of G, giving KL = KJ = JK, which in turn implies that KL is a subgroup of G (and KL = LK). □

Now suppose that G is a transitive but imprimitive permutation group on a set X of size d, and that H = G_{x}, the stabilizer in G of some point x ∈ X. Also suppose that the stabilizer H is contained in two different maximal subgroups K and L of G, such that KL ≠ LK (or equivalently, such that KL is not a subgroup of G), and that K ∩ L = H.

Observe that the condition KL ≠ LK implies that G ≠ KL, and the fact that H is the stabilizer of a point in X implies that H is core-free in G.

Let k = |G : K| and l = |G : L|, the numbers of (right) cosets of K and L respectively in G, and similarly, let r = |K : H| and s = |L : H|. Then G has a system of imprimitivity with k blocks B_{1}, B_{2}, …, B_{k}, each of size r, such that B_{1} is the orbit of x under the subgroup K (and the remaining k −1 blocks B_{i} are the images of B_{1} under elements of G), and another system with l blocks C_{1}, C_{2}, …, C_{l}, each of size s, such that C_{1} is the orbit of x under the subgroup LK (and so on).

Lemma 2.3 Under the above conditions, the following hold:

(a) d = kr = ls,

(b) rs < d < kl,

(c) k > s and l > r,

(d) If r ≤ s then k ≥ l,

(e) B_{1} ∩ C_{1} = {x}, and

(f) |B_{i} ∩ C_{j}| = 0 or 1 for 1 ≤ i ≤ k and 1 ≤ j ≤ l.

Proof. First note that d = |G|/|H| = (|G|/|K|)(|K|/|H|) = kr and similarly d = (|G|/|L|)(|L|/|H|) = ls. Also |G| > |KL| = |K||L|/|K ∩ L| = |K||L|/|H|, which gives both d = |G|/|H| > (|K|/|H|)(|L|/|H|) = rs and kl = (|G|/|K|)(|G|/|L|) > |G|/|H| = d. The other properties involving k, l, r and s now follow easily. Next, the assumption that K ∩L = H = G_{x} implies that B_{1} ∩ C_{1} = x^{K} ∩x^{L} = x^{H} = {x}. Furthermore, if y ∈ B_{i}∩C_{j}, then choosing g ∈ G such that x^{g} = y gives B_{i} = B_{1}^{g} and C_{j} = C_{1}^{g}, and it follows that B_{i} ∩ C_{j} = B_{1}^{g} ∩ C_{1}^{g} = (B_{1} ∩ C_{1})^{g} = {x}^{g} = {y}, and thus |B_{i} ∩ C_{j}| ≤ 1 for all i and j. □

Without loss of generality we may assume that r ≤ s. Let us now label the points of some of the blocks for K and L as follows:

B_{1} = {x, y_{2}, y_{3}, …, y_{r}}  C_{1} = {x, z_{2}, z_{3}, z_{4}, …, z_{s}}
B_{2} = {z_{2}, u_{22}, u_{23} …, u_{2r}}  C_{2} = {y_{2}, v_{22}, v_{23}, v_{24}, …, v_{2s}}
B_{3} = {z_{3}, u_{32}, u_{33} …, u_{3r}}  C_{3} = {y_{3}, v_{32}, v_{33}, v_{34}, …, v_{3s}}
. . . . . .  . . . . . .
. . . . . .  C_{r} = {y_{r}, v_{r2}, v_{r3}, v_{r4}, …, v_{rs}}
B_{s} = {z_{s}, u_{s2}, u_{s3} …, u_{sr}}  . . . . . .
. . . . . .

We can think of the blocks B_{i} as cosets of K in G, and the blocks C_{j} as cosets for L in G, respectively. Note that x and the points y_{2}, y_{3}, …, y_{r} of block B_{1} lie in r different L-blocks C_{j}, while x and the points z_{2}, z_{3}, …, z_{s} of block C_{1} lie in s different K-blocks B_{i}, by the last part of Lemma 2.3. These observations help give us a partial classification of the sub-orbits of the given group action, as follows.

Lemma 2.4 Each of the following sets is a union of orbits of H = G_{x} on X:

X_{1} = {x},
X_{2} = B_{1}\X_{1} = {y_{2}, y_{3}, …, y_{r}},
X_{3} = C_{1}\X_{1} = {z_{2}, z_{3}, …, z_{s}},
X_{4} = (B_{2} ∪ B_{3} ∪ ⋯ ∪ B_{s})\X_{3} = {u_{ji} : 2 ≤ i ≤ r, 2 ≤ j ≤ s},
X_{5} = (C_{2} ∪ C_{3} ∪ ⋯ ∪ C_{r})\X_{2} = {v_{ij} : 2 ≤ i ≤ r, 2 ≤ j ≤ s},
X_{6} = X \(X_{1} ∪ X_{2} ∪ X_{3} ∪ X_{4} ∪ X_{5}).

Proof. The first is obvious; the second and third follow from the fact that B_{1}^{g} = B_{1} and C_{1}^{g} = C_{1} for all g ∈ G_{x}; the fourth and fifth follow from the fact that G_{x} must preserve the union of blocks B_{i} containing a point of X_{3} and the union of blocks C_{j} containing a point of X_{2}; and the last now follows easily. □

This immediately gives us a lower bound on the number of sub-orbits:

Lemma 2.5 The sets X_{4} and X_{5} are distinct, so the union X_{4} ∪ X_{5} contains at least two orbits of H = G_{x}, and hence the number of orbits of G_{x} on X is at least five.

Proof. Assume that X_{4} = X_{5}. Then _{1≤i≤s}^{∪}B_{i} = _{1≤j≤r}^{∪}C_{j}. Call this common union U. Next, the subgroup K preserves the block B_{1} and therefore preserves the set {C_{1}, C_{2}, …, C_{r}} of all blocks C_{j} containing a point of B_{1}, and similarly the subgroup L preserves the set {B_{1}, B_{2}, …, B_{s}} of all blocks B_{i} containing a point of C_{1}, and hence the common union U is preserved by both K and L. This, however, implies that U is preserved by ⟨K, L⟩ = G (the latter following since K and L are distinct maximal subgroups of G), and so U = X, and therefore d = |X| = |U| = rs, contradicting part (b) of Lemma 2.3. □

But further, we have the following, as a consequence of Proposition 2.2:

Lemma 2.6 The set X_{6} is non-empty, and hence the number of orbits of H = G_{x} on X must be at least six.

Proof. First, observe that since {x}^{L} = C_{1} = {x, z_{2}, z_{3}, z_{4}, …, z_{s}}, every element L takes the block B_{1} to one of B_{1}, B_{2}, …, B_{s}, and therefore {x}^{KL} = B_{1}^{L} = _{1≤i≤s}^{∪}B_{i}. Similarly {x}^{LK} = C_{1}^{K} = _{1≤j≤r}^{∪}C_{j}, and it follows that {x}^{KL∪LK} = _{1≤i≤s}^{∪}B_{i} ∪ _{1≤j≤r}^{∪}C_{j} = _{1≤t≤5}^{∪}X_{t}. Proposition 2.2 gives G ≠ KL ∪ LK, however, and so X = {x}^{G} ≠ {x}^{KL∪LK} = _{1≤t≤5}^{∪}X_{t}. Thus X_{6} is non-empty, and the rest is easy. □

3 The rank 6 case

We now prove Theorem 1.4, using the notation and preliminary results of the previous Section.

Suppose G_{x} has exactly six orbits on X. By what we have seen above, these must be X_{1}, X_{2}, X_{3}, X_{6} and two other orbits whose union is X_{4} ∪ X_{5}. Moreover, since |X_{4}| = |X_{5}| (= (r − 1)(s − 1)) but X_{4} ≠ X_{5}, we see that each of X_{4}\X_{5} and X_{5}\X_{4} is non-empty, while X_{4} ∩ X_{5} = ∅ (for otherwise X_{4} ∪ X_{5} would contain three different orbits of G_{x}), and hence the six orbits of G_{x} have to be X_{1}, X_{2}, X_{3}, X_{4}, X_{5} and X_{6}. Note here also that X_{5} ∪ X_{6} must be the union of all the blocks B_{i} containing a point of X_{5}, and similarly X_{4} ∪ X_{6} must be the union of all the blocks C_{j} containing a point of X_{4}.

Furthermore, the subgroup K acts transitively on the block B_{1} = {x, y_{2}, y_{3}, …, y_{r}}, and therefore contains elements that take points of X_{3} = C_{1}\X_{1} = {z_{2}, z_{3}, …, z_{s}} to points of X_{5} = {v_{ij} : 2 ≤ i ≤ r, 2 ≤ j ≤ s}, and as these elements take the blocks B_{2}, B_{3}, …, B_{s} to blocks B_{i} for s < i ≤ k, it follows that the orbits of K are X_{1} ∪ X_{2}, X_{3} ∪ X_{5} and X_{4} ∪ X_{6}. Similarly, the orbits of L on X are X_{1} ∪ X_{3}, X_{2} ∪ X_{4} and X_{5} ∪ X_{6}.

In particular, if g is any element of K that takes one of the blocks B_{2}, B_{3}, …, B_{s} to some block B_{i} for s < i ≤ k, then B_{i} contains exactly one point of X_{5} (the image under g of a point of X_{3}) and r − 1 points of X_{6} (the images under g of r − 1 points of X_{4}). From this it follows that the (r − 1)(s − 1) points of X_{5} all lie in different blocks B_{i}, and therefore |X_{6}| = |X_{5}|(r − 1) = (r − 1)(s − 1)(r − 1), and the number of blocks B_{i} is k = s + |X_{5}| = s + (r − 1)(s − 1). But similarly, the (r − 1)(s − 1) points of X_{4} must all lie in different blocks C_{j}, so that |X_{6}| = |X_{4}|(s − 1) = (r − 1)(s − 1)(s − 1), and the number of blocks C_{j} is l = r + |X_{4}| = r + (r − 1)(s − 1). Comparison of the expressions for |X_{6}| now gives r = s, and also k = l = s + (s − 1)(s − 1) = s^{2} − s + 1.

Now consider the blocks B_{i} as ‘points’ and the blocks C_{j} as ‘lines’ of an incidence structure in which the point-line incidence relation is non-empty intersection (that is, so that B and C are incident if and only if B ∩ C is non-empty). Then what we have is a set of k points and a set of k lines, such that every point is incident with s lines and every line is incident with s points. We claim also that every two points are together incident with a unique line, and that every two lines are together incident with a unique point.

To prove these claims, we can argue as follows. First, the point B_{1} is incident with the s lines C_{1}, C_{2}, …, C_{s}, while each point B_{i} for 2 ≤ i ≤ s is incident with C_{1} and s − 1 of the lines C_{s+1}, C_{s+2}, …, C_{l}, and each point B_{i} for i > s is incident with just one of the lines C_{2}, C_{3}, …, C_{s} and s − 1 of the lines C_{s+1}, C_{s+2}, …, C_{l}. Thus B_{1} and any given B_{i} for 2 ≤ i ≤ s are together incident with only the line C_{1}, and B_{1} and any given B_{i} for i > s are together incident with only one of the lines C_{2}, C_{3}, …, C_{s}, and therefore the first claim holds when one of the points is B_{1}. But the group G acts transitively on the set {B_{1}, B_{2}, …, B_{k}} of blocks for K, and as the definition of incidence is independent of the choice of x, it follows that the first claim holds for every point B_{i}. The second claim holds by exactly the same argument, with points and lines interchanged.

It follows (by a standard definition) that this incidence structure is a finite projective plane, of order q = s − 1, with k = q^{2} + q + 1.

If s = 2, then q = 1 and this plane is just a triangle, and so we have case (a) of our theorem. From here on we will suppose that s > 2.

Now the group G acts on this plane Π as a group of incidence-preserving automorphisms (or collineations); indeed G acts primitively on both the set of q^{2} + q + 1 points (the blocks B_{i} for the maximal subgroup K) and the set of q^{2} + q + 1 lines (the blocks C_{j} for the maximal subgroup L), and transitively but imprimitively on the set of incident point-line pairs (flags), which correspond to the original elements of X.

By a theorem of Kantor on flag-transitive projective planes [16], it follows that either Π is Desarguesian (with q a prime-power, and with G involving PSL(3, q)), or otherwise G is a Frobenius group, q^{2} + q + 1 is prime, and |G| divides (q^{2} + q + 1)(q + 1) or (q^{2} + q + 1)q. In our case, the second of these is impossible, because it would imply that K and L have order q + 1 or q, yet we know that K acts 2-transitively on B_{1} (since G_{x} is transitive on X_{2} = B_{1}\{x}) and so |K| is divisible by s(s − 1) = (q + 1)q. Hence, by Kantor’s theorem, our incidence structure is a Desarguesian projective plane, of prime-power order q, and with PSL(3, q) involved in its collineation group.

This completes the proof.

4 The rank 7 case

A similar approach can be taken to prove Theorem 1.5. Again using the notation of Section 2, suppose that G_{x} has exactly seven orbits on X, two of which have size 1. By our previous observations, each of the sets X_{1}, X_{2}, X_{3}, X_{4}, X_{5} and X_{6} must be a non-empty union of orbits of G_{x}, with |X_{4}| = |X_{5}| = (r − 1)(s − 1) but X_{4} ≠ X_{5}, and X_{4} ∪ X_{5} must be the union of at least two such orbits. Hence there are just two possibilities to consider:

4.1 Suppose X_{4} ∩ X_{5} ≠ ∅.

In this case the seven orbits of G_{x} on X must be X_{1}, X_{2}, X_{3}, X_{4}\X_{5}, X_{4} ∩ X_{5}, X_{5}\X_{4} and X_{6}, and one of these apart from X_{1} has size 1.

At this stage we make the observation that each of |X_{4}\X_{5}|, |X_{4} ∩ X_{5}| and |X_{5}\X_{4}| must be divisible by both r − 1 and s − 1. For suppose the block C_{i} containing the point y_{i} (of X_{2}) contains also the points p_{1}, p_{2}, …, p_{t} of X_{4} ∩ X_{5}, and y_{j} is any other point of X_{2}. Then there exists an element g ∈ G_{x} taking y_{i} to y_{j}, and as this element must take C_{i} to C_{j} it follows that g takes the t points p_{1}, p_{2}, …, p_{t} (of X_{4} ∩ X_{5}) lying in C_{i} to t points of X_{4} ∩ X_{5} lying in C_{j}. Hence all of the r − 1 blocks C_{2}, C_{3}, …, C_{r} (containing a point of X_{2}) contain the same numbers of points of X_{4} ∩ X_{5}, and so |X_{4} ∩ X_{5}| is divisible by r − 1. The analogous argument for the blocks B_{2}, B_{3}, …, B_{s} containing points of X_{3} shows that |X_{4} ∩ X_{5}| is divisible by s − 1, and since |X_{4}| = |X_{5}| = (r − 1)(s − 1), it follows that also both |X_{4}\X_{5}| and |X_{5}\X_{4}| are divisible by r − 1 and s − 1 as well.

Now if |X_{2}| = 1, then r = 2, but then the above observation gives |X_{4} ∩ X_{5}| = s − 1 = |X_{4}|, and so X_{4} = X_{5}, a contradiction. Thus s ≥ r > 2. In particular, this rules out the possibilities that |X_{3}| = 1 or |X_{4}\X_{5}| = 1 or |X_{4} ∩ X_{5}| = 1 and |X_{5}\X_{4}| = 1, and so we conclude that |X_{6}| = 1.

By definition of X_{6}, it follows that the number of blocks B_{i} for K is k = s + 1 and the number of blocks C_{j} for L is l = r + 1. In particular, d = kr = r(s + 1) = rs + r while also d = ls = (r + 1)s = rs + s, and therefore r = s and k = l = s + 1 and d = s(s + 1).

Now consider the action of G on the set L = {C_{1}, C_{2}, …, C_{l}} of blocks for L. First the subgroup K permutes transitively the s blocks C_{1}, C_{2}, …, C_{s} containing a point of B_{1}, so fixes C_{l} = C_{s+1} (setwise), and therefore G is 2-transitive on L. Furthermore, the subgroup G_{x} fixes each of C_{1} and C_{s+1} (setwise) and permutes transitively the s − 1 blocks C_{2}, C_{3}, …, C_{s} containing a point of X_{2}, so G is 3-transitive on L.

Next, clearly the subgroup G_{x} = K ∩ L is the stabilizer of the ordered pair (C_{1}, C_{s+1}) in this action of G on L, and it follows that the original action of G on X is equivalent to its action on ordered pairs of distinct members of L. Letting α = C_{1} and ω = C_{s+1}, we now see that K and L are the stabilizers in G of ω and α respectively, and that the seven orbits of G_{x} on X must be equivalent to the following:

O_{1} = {(α, ω)},
O_{2} = {(λ, ω) : λ ∈ L\{α, ω}},
O_{3} = {(α, λ) : λ ∈ L\{α, ω}},
O_{4} = {(ω, λ) : λ ∈ L\{α, ω}},
O_{5} = {(λ, α) : λ ∈ L\{α, ω}},
O_{6} = {(λ, μ) : λ, μ ∈ L\{α, ω}, λ ≠ μ}, and
O_{7} = {(ω, α)}.

(In fact O_{1}, O_{2}, O_{3}, O_{4}, O_{5}, O_{6} and O_{7} are equivalent to X_{1}, X_{2}, X_{3}, X_{4}\X_{5}, X_{5}\X_{4}, X_{4} ∩X_{5} and X_{6} respectively; and the orbits of K are O_{1} ∪O_{2} (≈ X_{1} ∪X_{2}), O_{3} ∪O_{5} ∪O_{6} (≈ X_{3} ∪X_{5}) and O_{4} ∪O_{7} (≈ (X_{4}\X_{5})∪X_{6}), while the orbits of L are O_{1} ∪O_{3} (≈ X_{1} ∪X_{3}), O_{2} ∪O_{4} ∪O_{6} (≈ X_{2} ∪ X_{4}) and O_{5} ∪ O_{7} (≈ (X_{5}\X_{4}) ∪ X_{6}).)

In particular, as G_{x} = K ∩ L = G_{αω} has to be transitive on the set O_{6}, it follows that G is 4-transitive on L. By the classification of 2-transitive finite groups, all finite 4-transitive permutation groups are known; see [6] or [7]. Accordingly, the action of G on L is equivalent to the natural action of either one of the Mathieu groups M_{11}, M_{12}, M_{23} or M_{24}, or the alternating group A_{k} (for k ≥ 5) or the symmetric group S_{k} (for k ≥ 4). As each of these groups is ‘almost simple’, the permutation group induced by the action of G on ordered pairs of members of L is isomorphic to the permutation group induced by the action of G on L, and therefore G itself is isomorphic to one of them, as required.

4.2 Suppose X_{4} ∩ X_{5} = ∅.

In this case one of X_{2}, X_{3}, X_{4}, X_{5} and X_{6} is a union of two orbits of G_{x}, while the four others and X_{1} = {x} are all single orbits of G_{x}.

Now if X_{2} is the union of two orbits of G_{x}, say U and V, then also X_{5} will split into two orbits of G_{x} (one consisting of points of X_{5} from blocks C_{j} containing a point of U and the other consisting of points of X_{5} from blocks C_{j} containing a point of V), so X_{2} must be a single orbit. Similarly, X_{3} is a single orbit.

Next, just as in the proof of Theorem 1.4, we can show that X_{1} ∪ X_{2} and X_{3} ∪ X_{5} are orbits of K, and that X_{1} ∪ X_{3} and X_{2} ∪ X_{4} are orbits of L. (Note: the possibility that X_{4} or X_{5} is a union of two orbits of G_{x} does not affect the argument, since X_{2} and X_{3} are orbits of K ∩ L = G_{x}, and only one of X_{4}, X_{5} and X_{6} is not.) Moreover, again we find that the (r − 1)(s − 1) points of X_{5} all lie in different blocks B_{i} (for s < i ≤ k), and that the (r − 1)(s − 1) points of X_{4} must all lie in different blocks C_{j} (for r < j ≤ l).

Now if X_{6} were a single orbit, then it would have to consist of all the points in the blocks B_{s+1}, B_{s+2}, …, B_{k} other than those already lying in X_{5}, and at the same time consist of all points in the blocks C_{r+1}, C_{r+2}, …, C_{l} other than those already lying in X_{4}, and in particular, we would have |X_{6}| = |X_{5}|(r − 1) and |X_{6}| = |X_{4}|(s − 1) in this case. But if p and q are any points of X_{6} lying in different blocks B_{i} and B_{j}, and g is any element of G_{x} taking p to q, then g takes B_{i} to B_{j} and therefore takes the unique point of X_{5} in B_{i} to the unique point of X_{5} lying in B_{j}, and it follows that X_{5} forms a single orbit of G_{x}. The analogous argument for the blocks C_{r+1}, C_{r+2}, …, C_{l} shows that X_{4} is a single orbit of G_{x}. It follows that G_{x} has only six orbits, a contradiction. Thus X_{6} is the union of two orbits of G_{x}, while X_{4} and X_{5} are single orbits of G_{x}.

Next, let U be the union of all blocks B_{i} that contain a point of X_{5}, and let V be the union of all blocks C_{j} containing a point of X_{4}. Then |U| = |X_{5}|r = r(r − 1)(s − 1) while |V | = |X_{4}|s = s(r − 1)(s − 1). Also U and V are preserved by G_{x} and hence each of U \X_{5} and V \X_{4} is a union of orbits of G_{x} contained in X_{6}. It follows that each of (U \X_{5}) \ (V \X_{4}) and (V \X_{4}) \ (U \X_{5}) and (U \X_{5}) ∩ (V \X_{4}) is a union of orbits of G_{x}, but at most two of these can be non-empty.

If r = s then |U| = |V | = s(s−1)^{2} so |U\X_{5}| = |V \X_{4}| = s(s−1)^{2} −(s−1)^{2} = (s−1)^{3}. Now if U \X_{5} ≠ V \X_{4}, then (U \X_{5}) ∩ (V \X_{4}) must be empty, and hence the seven orbits of G_{x} are X_{1}, X_{2}, X_{3}, X_{4}, X_{5}, U \X_{5} and V \X_{4}, of sizes 1, s − 1, s − 1, (s − 1)^{2}, (s − 1)^{2}, (s − 1)^{3} and (s − 1)^{3} respectively, but this makes it impossible for just two of the orbits of G_{x} to have size 1. Thus U \X_{5} = V \X_{4} = T, say. Again none of the orbits X_{2}, X_{3}, X_{4} and X_{5} can have size 1 (for otherwise r = s = 1 and then all of them have size 1), so either T is an orbit of G_{x} and |X_{6}| = |T | + 1, or T = X_{6} is a union of two orbits of G_{x}, one of which has size 1. In the former case, summing the orbit sizes gives d = |X| = 1 + (s − 1) + (s − 1) + (s − 1)^{2} + (s − 1)^{2} + (s − 1)^{3} + 1 ≡ 1 mod s, but since d = ls, this gives a contradiction; in the latter case, the block B_{i} containing both a point of X_{5} and the orbit of G_{x} of size 1 must be fixed by G_{x}, and hence the point of X_{5} forms another single orbit for G_{x} of size 1, another contradiction.

Thus r < s. Clearly this gives |U| < |V | and so |U\X_{5}| < |V \X_{4}|. It follows that U\X_{5} must be an orbit of G_{x}, and the seventh orbit of G_{x} must be X_{6} \(U \X_{5}), which could be either V \X_{4} or (V \X_{4}) \ (U \X_{5}). The sizes of the seven orbits of G_{x} are therefore 1, r − 1, s − 1, (r − 1)(s − 1), (r − 1)(s − 1), (r − 1)^{2}(s − 1) and either (r − 1)(s − 1)^{2} or (r − 1)(s − 1)(s − r). Since (r − 1)(s − 1) > s − 1 > r − 1, the smallest of these sizes are 1 and r − 1, and so the second orbit of G_{x} of size 1 must be X_{2}, and thus r = 2.

In particular, we now know that the orbits of G_{x} have sizes 1, 1, s − 1, s − 1, s − 1, s − 1 and either (s − 1)^{2} or (s − 1)(s − 2). Note that the sum of first five of these is 3s − 1, so |X_{6}| = d − (3s − 1) = d − 3s + 1. Since d = ls, however, |X_{6}| = d − 3s + 1 ≡ 1 modulo s, so |X_{6}| cannot be the sum of |U \X_{5}| = s − 1 and |V \X_{4}| = (s − 1)^{2}, and it follows that X_{6} = V \X_{4}, and the seventh orbit of G_{x} has to be (V \X_{4}) \ (U\X_{5}), of size (s − 1)(s − 2). Thus d = |X_{6}| + 3s − 1 = (s − 1)^{2} + 3s − 1 = s^{2} + s = s(s + 1).

Also k = d/r = s(s + 1)/2, while l = d/s = s + 1.

Now consider the action of G on the set L = {C_{1}, C_{2}, …, C_{l}} of blocks for L. Just as in case 4.1, we see that G is 3-transitive on L, and that the action of G on X is equivalent to its action on ordered pairs of distinct members of L. In this case, however, if (α, ω) = (C_{1}, C_{l}) = (C_{1}, C_{s+1}) then K is the stabilizer in G of {α, ω} (while L is still the stabilizer of α), and the orbits of G_{x} on X are still equivalent to the sets O_{1}, O_{2}, …, O_{7} defined in case 4.1. (Also O_{1}, O_{2}, O_{3}, O_{4}, O_{5}, O_{6} and O_{7} are equivalent to X_{1}, U \X_{5}, X_{3}, X_{5}, X_{4}, (V \X_{4})\(U\X_{5}) and X_{2} respectively; and the orbits of K are O_{1} ∪O_{7} (≈ X_{1} ∪X_{2}), O_{3} ∪ O_{4} (≈ X_{3} ∪ X_{5}), O_{2} ∪ O_{5} (≈ X_{4} ∪ (U \X_{5})) and O_{6} (≈ (V \X_{4}) \ (U \X_{5})), while the orbits of L are O_{1} ∪ O_{3} (≈ X_{1} ∪ X_{3}), O_{5} ∪ O_{7} (≈ X_{2} ∪ X_{4}) and O_{2} ∪ O_{4} ∪ O_{6} (≈ X_{5} ∪ V).) Again G_{x} = G_{αω} has to be transitive on the set O_{6}, and so G is 4-transitive on L, and the rest follows.

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