Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 40 (1999), No. 2, 503-531.

Flag-transitive Extensions of Dual Affine Spaces

Cécile Huybrechts^{∗}^{†} Antonio Pasini

Université Libre de Bruxelles, Département de Mathématique Campus Plaine, CP 216, Bd du Triomphe, B-1050 Bruxelles, Belgique

Universitá di Siena, Dipartimento di Matematica Via del Capitano 15, 53100 Siena, Italia

Abstract. We construct a simply connected flag-transitive circular extension of the dual affine space AG^{∗}(3, 4), with 2^{13}:3·M_{22}.2 as full automorphism group and we show that this geometry as well as its unique flag-transitive proper quotient are the unique flag-transitive circular extensions of finite thick dual affine spaces whose full automorphism group does not induce a 1-dimensional affine group on their point- or plane-residues. Moreover, we observe that our theorem has some nice repercussions for a problem in projective spaces.

1. Introduction

1.1. Definitions and notation

Let c and AG^{∗} denote respectively the class of circular spaces (namely, linear spaces with all lines of size 2) and the class of dual affine spaces (namely, line-point truncations of affine geometries). A c·AG^{∗}-geometry is a geometry Γ over the type set {0, 1, 2} belonging to the following diagram:

(c · AG^{∗})  •  •  • c  AG^{∗} 0  1  2

Scientifique”.

∗Chargé de Recherches du Fonds National Belge de la Recherche Scientifique.

†This work was made possible thanks to support of the “Fonds National Belge de la Recherche 0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1999 Heldermann Verlag

The elements of Γ of type 0, 1, 2 are also called points, lines and planes respectively. If there are integers n and q, with n > 1 and q > 1, such that the dual of every point residue of Γ is the point-line truncation of an n-dimensional affine space of order q, then we say that Γ has order q and residual dimension n.

Let G ≤ Aut(Γ). Given an element x of Γ, the elementwise stabilizer in G_{x} of the residue Γ_{x} is denoted by K_{x} and we set \overline{G}_{x} = G_{x}/K_{x}. If G is flag-transitive on Γ, then the stabilizers of any two elements of the same type are isomorphic. In that case, the isomorphism types of G_{x}, K_{x} and \overline{G}_{x}, with x of type i, will be denoted by G_{i}, K_{i} and \overline{G}_{i}.

We recall that a permutation group X of degree t is said to be 1-dimensional affine if t is a prime power and X ≤ AΓL_{1}(t).

1.2. A survey of known results

c · AG^{∗}-geometries of order q and residual dimension n have been studied in the two particular cases where q = 2 or n = 2.

When q = 2, they are c · c^{∗}-geometries with 2^{n} points in every plane. A complete classification of these c · c^{∗}-geometries seems difficult to obtain, even assuming flag-transitivity. The reader is referred to Baumeister and Pasechnik [1] and Baumeister and Pasini [2] for some partial results on this problem.

c · AG^{∗}-geometries of residual dimension 2 are also called extended dual affine planes. They have been classified by Baumeister, Del Fra, Meixner and Pasini [3] in the finite and flag-transitive case. The next table summarizes that classification.

order G = Aut(Γ)  G_{0} = \overline{G}_{0}  \overline{G}_{2}  K_{2}
(1)  2 L_{3}(2)  2^{2}.Sym(3)  2^{2}.Sym(3) 1
(2)  2 (2^{3} × 2^{2}).Sym(3) 2^{2}.Sym(3)  2^{2}.Sym(3) 1
(2′) 2 2^{4}.Sym(3)  2^{2}.Sym(3)  2^{2}.Sym(3) 1
(3)  4 2^{6}.3.Sym(6)  AΓL_{2}(4)  Sym(6) 3
(3′) 4 2^{5}.Sym(5)  2^{4}.5.4 < AΓL_{1}(16) Sym(5) 1
(3′′) 4 2^{4}.3.Sym(5)  2^{4}.15.4 = AΓL_{1}(16) Sym(5) 3
(4)  4 L_{3}(4).2  AΓL_{2}(4)  Sym(6) 1
(5)  4 2^{4}.Sym(6)  2^{4}.Sym(5)  Sym(6) 1
(6) 16 2^{9}.L_{2}(17)  2^{8}.17.8 < AΓL_{1}(2^{8}) L_{2}(17)  1
(6′) 16 2^{8}.L_{2}(17)  2^{8}.17.8 < AΓL_{1}(2^{8}) L_{2}(17)  1

Note. (2′), (3′) and (6′) are 2-fold quotients of (2), (3) and (6), respectively; (3′′) is a 4-fold quotient of (3).

1.3. The main result of this paper

In view of the results quoted in Subsection 1.2, we focus on flag-transitive c · AG^{∗}-geometries of order q > 2 and residual dimension n > 2 (whence q is a prime power). We prove the following:

Theorem 1. Let Γ be a flag-transitive c · AG^{∗}-geometry of order q > 2 and residual dimension n > 2 and let G = Aut(Γ). Then one of the following occurs:

(i) n = 3, q = 4 and either (i.a) G = 2^{13}:3·Aut(M_{22}) and Γ ≃ Γ(S), where Γ(S) is described in Section 2, or (i.b) G = 2^{12}:3·Aut(M_{22}) and Γ ≃ Γ(S)/2, where Γ(S)/2 is a uniquely determined quotient of Γ(S) (see Section 2);

(ii) q is odd, q^{n−1} + q^{n−2} + ⋯ + q + 2 is a prime power and both G_{0} = \overline{G}_{0} and \overline{G}_{2} are 1-dimensional affine groups.

Theorem 1 will be proved in Section 3.

The following arithmetical question arises from case (ii) of Theorem 1.

Problem. What are the solutions (n, q, q′) of the following equation, where q and q′ are prime powers, q is odd and n is an integer ≥ 3 ?

(E1)  q^{n−1} + … + q + 2 = q′

We do not know if the equation (E1) admits a solution for q′ odd. If q′ is even, then the equation (E1) is a particular case of the following equation, which seems to be unsolved and for which only the solution (n, q, m, q_{0}) = (3, 5, 5, 2) is known (see Makowski and Schinzel [23] for more information).

(E2)  q^{n−1} + … + q = q_{0}^{m−1} + … + q_{0}

where q and q_{0} are distinct prime powers and n and m are positive integers. Therefore, for q′ even, the equation (E1) has only one known solution, namely (n, q, q′) = (3, 5, 32).

As we will show in Section 4, for every solution (n, q, q′) of the equation (E1), with q′ even, there is at least one flag-transitive c · AG^{∗}-geometry of order q and residual dimension n, obtained as a gluing of a circular space of q′ points with the point-line system of AG(n, q). For (n, q) = (3, 5), we obtain in this way exactly two non-isomorphic flag-transitive c · AG^{∗}- geometries of order 5 and residual dimension 3 since, in that case, it can be shown (see Section 4) that there are exactly two non-isomorphic flag-transitive gluings.

The material of the present paper is part of an attempt to classify flag-transitive (c · L^{∗})- geometries with finite point residues. We refer to Huybrechts-Pasini [16, 17] for some other works in that direction and to [18] for a survey of the situation (see also Huybrechts [14]).

1.4. A consequence of Theorem 1

In Section 5, we exploit Theorem 1 to study some families of subspaces in projective spaces. More precisely, we prove the following:

Theorem 2. Let S be a family of d-dimensional subspaces of P G(m, q) (1 ≤ d < m) such that

(1) every point of P G(m, q) belongs to either 0 or 2 members of S;

(2) any two members of S have precisely one point in common; (3) the set of the points of P G(m, q) belonging to some member of S spans P G(m, q). Furthermore, let G_{S} be the stabilizer of S in G = P ΓL_{m+1}(q) and assume that G_{S} acts 2-transitively on S. Then one of the following holds:

(i) q = 2; (ii) q = 4, m = 2, d = 1, G_{S} = Sym(6) and S is the dual of a regular hyperoval of P G(2, 4); (iii) q = 4, m = 5, d = 2 and G_{S} = Aut(M_{22}); (iv) q is odd, q^{d} + q^{d−1} + ⋯ + q + 2 is a prime power and G_{S} is 1-dimensional affine.

A family S of d-dimensional subspaces of P G(m, q) satisfying the above properties (1)–(3) is called a d-dimensional dual hyperoval. If furthermore G_{S} acts 2-transitively on S, then we say that S is 2-transitive.

According to this terminology, Theorem 2 reads as follows: 2-transitive d-dimensional dual hyperovals only exist in cases (i)–(iv).

J. Thas and B. Cooperstein (Thas [28]) made a construction providing a lot of ddimensional dual hyperovals of P G(2d, 2), for any positive integer d. Some of them are 2-transitive. The family S of 22 planes of P G(5, 4) we consider in Section 2.1 in order to define the geometry Γ(S) of Theorem 1 as an example for (iii).

It seems unlikely that case (iv) of Theorem 2 may occur. No example is known for that case and, more generaly, no d-dimensional dual hyperoval is known in P G(m, q) with q odd.

We refer to Huybrechts-Pasini [19] for further comments on Theorem 2 and to Huybrechts [15] for some geometric results on d-dimensional dual hyperovals.

1.5. More definitions and notation

We end this introduction recalling some basic definitions and stating some notation, to be used in the sequel of the paper.

The properties (LL) and (T). The following properties hold in some c · AG^{∗}-geometries: (LL) no two distinct lines are incident to the same pair of points; (T) every triple of pairwise collinear points is contained in some plane.

For instance, the geometries (1), (2), (3) and (4) of the list of Subsection 1.2 satisfy (LL), whereas (T) holds in all geometries of that list but (1), (4) and (6).

Notice that, in the context of c · AG^{∗}-geometries, (LL) is equivalent to the Intersection Property [26, Lemma 7.25].

Flatness. Following [26], we say that a c · AG^{∗}-geometry Γ is flat if all points of Γ are incident to all planes of Γ. (For instance, the geometries (2′), (3′′), (5) and (6′) of the list of Subsecion 1.2 are flat.) Clearly, the property (LL) is false in flat geometries, but (T) trivially holds in them.

Collinearity, distance and diameter. Given two distinct points a and b of a c · AG^{∗}- geometry Γ, we write a ∼ b to mean that a and b are collinear. Accordingly, the set of points collinear with (or equal to) a given point a will be denoted by a^{∼}.

We recall that the symbol ⊥ is often used in the literature instead of ∼ to denote the collinearity relation. We prefer the notation ∼ in this paper, keeping ⊥ to denote the orthogonality relation in a projective space with respect to a given projective sesquilinear form (Subsection 2.2).

The distance between two points of Γ is their distance in the collinearity graph of Γ. The distance of a point a from a plane u is the minimal distance between a and points of u. The diameter of Γ is the diameter of its collinearity graph.

2. A geometry for 2^{13}:3·Aut(M_{22})

2.1. A remarkable family of 22 planes of P G(5, 4)

Let Δ_{0} be the rank 3 polar space for U_{6}(2). It is known ([21]; see also [11, page 39]) that Δ_{0} contains a family S of 22 maximal singular subspaces such that any two of them meet in precisely one point, every point of S_{0} = ⋃_{X∈S} X is contained in precisely two members of S, and the stabilizer of S in ΣU_{6}(2^{2}) is isomorphic to 3·Aut(M_{22}) (the unique non-trivial central non-split extension of Aut(M_{22})), acting 2-transitively as Aut(M_{22}) on S. (We warn the reader that a misformulation occurs in [11, page 39], in the paragraph where S is described: the authors claim that every point of Δ_{0} is in exactly two members of S, but they meant ‘in either 0 or exactly 2 members’; compare [12].)

It is clear from the above that the pair (S, S_{0}), with the incidence relation inherited from Δ_{0}, is a circular space with S as the set of points and S_{0} as the set of lines. Furthermore,

Lemma 2.1. For every point a of Δ_{0}, at least one line of Δ_{0} through a contains at least two points of S_{0}.

Proof. The statement is obvious when a ∈ S_{0} (take a line through a contained in a member of S). Assume a ∉ S_{0}. Then |X ∩a^{⊥}| = 5 for every member X of S. Hence |S_{0} ∩a^{⊥}| = 55, as every point of S_{0} belongs to precisely two members of S. On the other hand, 45 is the number of lines of Δ_{0} through a. Therefore some of those lines meet S_{0} in at least two points. □

Corollary 2.2. The set S_{0} spans Δ_{0}.

Proof. Obvious, by Lemma 2.1. □

Remark. According to the above corollary, S satisfies (3) of Theorem 2. Hence S is a 2-dimensional dual hyperoval of P G(5, 4).

2.2. The geometry Γ(S)

Let Δ be the polar space for U_{8}(2) and, given a point p of Δ, let H = p^{⊥} and let Δ\H be the affine polar space obtained by removing the hyperplane H from Δ. The residue Δ_{p} = H/p of p in Δ is isomorphic to Δ_{0} and we may assume that the above mentioned family S is a set of planes of Δ_{p}.

Let Γ^{∗}(S) be the substructure of Δ \ H defined as follows. There are three types of elements in Γ(S), which we call points, lines and planes. The points of Γ^{∗}(S) are the ones of Δ \ H. The lines of Γ^{∗}(S) are the lines of Δ \ H that, viewed as lines of Δ, meet H in a point belonging to a member of S. The planes of Γ^{∗}(S) are the 3-spaces of Δ \ H that, viewed as maximal singular subspaces of Δ, meet H in a plane contained in a member of S. The incidence relation is the natural one, inherited from Δ.

Lemma 2.3. The residues of the planes of Γ^{∗}(S) are isomorphic to the point-line system of AG(3, 4). The residues of the points of Γ^{∗}(S) are isomorphic to the dual of the circular space (S, S_{0}).

Proof. The first claim is obvious. Let us prove the second one. Given a point a of Δ \ H, the set a^{⊥} meets each member of S (resp. S_{0}) in a plane (a point). For every X ∈ S the 3-space X is the unique member of S containing the plane X ∩ a^{⊥} and, for every member Y of S_{0}, the point Y ∩ a^{⊥} is contained in just two members of S. The conclusion follows. □

Lemma 2.4. The structure Γ^{∗}(S) is connected.

Proof. In view of Lemma 2.3, we only need to prove that the point-line system of Γ^{∗}(S) is connected.

Let a_{1}, a_{2} be collinear points of Δ \ H and let L be the line joining them in Δ \ H. The line L, viewed as a line of Δ, meets H in a point b ≠ p. If b belongs to a member of S, then L is a line of Γ^{∗}(S) and a_{1} and a_{2} are collinear in Γ^{∗}(S). Otherwise, it follows from Lemma 2.1 that there is a plane X of Δ containing the points p and b and at least two lines L_{1}, L_{2} of S_{0}. For i = 1, 2, let b_{i} be the intersection of L_{i} with the line M = L^{⊥} ∩ X and let M_{i} be the line of Δ joining a_{i} with b_{i}. The lines M_{1}, M_{2} belong to Γ^{∗}(S). Furthermore, the lines M and L meet in b. Hence they span a plane Y of P G(7, 4). Both M_{1} and M_{2} belong to Y. Whence they meet in a point, say c. Clearly, c is collinear with both a_{1} and a_{2} in Γ^{∗}(S).

The connectedness of the point-line system of Γ^{∗}(S) now follows from the connectedness of the collinearity graph of Δ \ H. □

We denote the dual of Γ^{∗}(S) by Γ(S). Namely, Γ(S) and Γ^{∗}(S) are the same structure, but for calling the points and the planes of Γ^{∗}(S) planes and points of Γ(S), respectively.

Proposition 2.5. The structure Γ(S) is a c · AG^{∗}-geometry of order 4 and residual dimension 3.

Proof. Obvious, by Lemmas 2.3 and 2.4. □

2.3. The group G = Aut(Γ(S))

Let G be the stabilizer of p and S in Aut(Δ). Clearly, G = N:G_{2} where N = 2^{13} acts regularly on the set of points of Δ \ H and G_{2} = 3·Aut(M_{22}) is the stabilizer in G of a point of Δ \ H (namely, a plane of Γ(S)). Furthermore, G stabilizes Γ(S) and acts faithfully and flag-transitively on it, with point-stabilizer G_{0} = AΣL_{3}(4). It will turn out from a theorem of Section 3 (Theorem 3.5) that G = Aut(Γ(S)). Thus,

Aut(Γ(S)) = G = 2^{13}:3·Aut(M_{22}).

Furthermore (compare Section 3, Theorem 3.5(i) and proof of Theorem 3.6),

G_{0} = \overline{G}_{0} = AΣL_{3}(4)
G_{1} = ASL_{2}(4):(2 × S_{4}) with K_{1} = ASL_{2}(4) (hence \overline{G}_{1} = 2 × S_{4})
G_{2} = 3·Aut(M_{22}) with K_{2} = 3 (hence \overline{G}_{2} = Aut(M_{22}))

(notation as in Subsection 1.1). The commutator subgroup G′ = 2^{13}:3·M_{22} of G also acts flagtransitively in Γ(S). According to Theorem 3.5 of Section 3, G′ is the unique flag-transitive proper subgroup of G.

Writing G′_{i} for G′ ∩ G_{i}, we have K_{i} ≤ G′_{i} for i = 1, 2 and

G′_{0} = ASL_{3}(4), G′_{1} = ASL_{2}(4):(2 × A_{4}), G′_{2} = 3·M_{22}.

2.4. The quotient Γ(S)/2

The hyperplane H of Δ spans a hyperplane ⟨H⟩ of P G(7, 4). Let Z_{0} be the intersection of Aut(Δ) with the group of elations of P G(7, 4) with axis ⟨H⟩ and center p. It is well-known that Z_{0} has order 2 and it is the center of the stabilizer of p in Aut(Δ).

Proposition 2.6. We have Z(G′) = Z(G) = Z_{0}.

Proof. Clearly, Z_{0} ≤ Z(G′). Assume that Z_{0} < Z(G′). Then G′/Z_{0} has a non-trivial center. However, G′/Z_{0} is a subgroup of V:SU_{6}(2), with V the additive group of the natural GF (4)-module V (6, 4) of SU_{6}(2) (see [11]). Furthermore, G′/Z_{0} = V:3·M_{22}, with 3·M_{22} acting faithfully on V. Hence Z(G′/Z_{0}) ≤ V. Consequently, some non-trivial element of V is central in G′/Z_{0}. However, the centralizer in SU_{6}(2) of a non-trivial element of V does not involve M_{22} (see [11]); contradiction.

Therefore, Z(G′) = Z_{0}. As |Z_{0}| = 2, Z(G′) is central in G, too. Hence Z(G) = Z(G′). □

It is well-known [26, 8.4.7] that Z_{0} defines a quotient of the affine polar space Δ\H. Accordingly, Z_{0} (= Z(G)) defines a quotient of Γ(S). We denote that quotient by Γ(S)/2. Clearly, Γ(S)/2 is flag-transitive with

Aut(Γ(S)/2) = G/Z_{0} = 2^{12}:3·Aut(M_{22})

(but Z(G) ≤ G′ and the factor group G′/Z(G) = 2^{12}:3·M_{22} is also flag-transitive on Γ(S/2)).

2.5. Another way to construct Γ(S)/2

Writing P for P G(7, 4), let P/p (resp. ⟨H⟩/p) be the star of p in P (in ⟨H⟩). Clearly, ⟨H⟩/p ≅ P G(5, 4) and, as H/p contains a copy of S, we may assume that S is a family of planes of the projective space ⟨H⟩/p. Let A be the complement of ⟨H⟩/p in P/p. Clearly, A ≅ AG(6, 4). Furthermore, every line of P through p not contained in ⟨H⟩ meets Δ in two points different from p, which form an orbit of Z_{0}. Thus, the orbits of Z_{0} on the set of points of Δ \ H bijectively correspond to the points of A. It is now clear that the dual Γ^{∗}(S)/2 of Γ(S)/2 can be described as follows:

points: the points of A; lines: the lines of A with point at infinity belonging to a member of S; planes: the 3-spaces of A with a member of S as the plane at infinity; incidence relation: inherited from A.

In short, Γ^{∗}(S)/2 is the affine expansion of the dual circular space (S_{0}, S) to A (see Buekenhout, Huybrechts and Pasini [7, Section 4] for the definition of affine expansions; also [26, Section 2.3]).

The stabilizer of S in Aut(A) (= AΓL_{6}(4)) is indeed 2^{12}:3·Aut(M_{22}). As the latter is just Aut(Γ(S)/2), all automorphisms of Γ(S)/2 are inherited from Aut(A).

2.6. Some properties of Γ(S) and Γ(S)/2

As the points and the lines of Γ^{∗}(S) are points and lines of a projective geometry, no two distinct lines of Γ^{∗}(S) are incident to the same pair of points. The same property holds in Γ^{∗}(S)/2. Consequently, Γ^{∗}(S) and Γ^{∗}(S)/2 also satisfy the Intersection Property ([26, Lemma 7.25]). Needless to say, the same holds in Γ(S) and Γ(S)/2. In particular, the property (LL) (Subsection 1.5) holds in Γ(S) and Γ(S)/2.

Both Γ(S) and Γ(S)/2 satisfy (T). (This follows from Proposition 3.11 of Section 3, but it can also be proved straightforwardly.)

The geometry Γ(S)/2 has 1408 points. The number of points of Γ(S) is twice that, namely 2816. As it will be proved in Section 3 (Proposition 3.36), the collinearity graph of Γ(S)/2 is strongly regular with parameters k = 336 and λ = μ = 80. The collinearity graph of Γ(S) is regular of diameter 4, with valency k = 336 and μ = 40 (see Section 3, 3.7).

As both (LL) and (T) hold in Γ(S) and in Γ(S)/2, both Γ(S) and Γ(S)/2 can be erected to geometries of rank 4 with diagram and orders as follows (Huybrechts [15]):

•  •  • • 1  4  3 4 c  AG^{∗}

3. Proof of Theorem 1

3.1. Outline of the proof

We first determine the structure of point- and plane-stabilizers and show in Theorem 3.5 (Subsection 3.2) that we can focus on the case of order q = 4 and residual dimension n = 3.

In Subsection 3.3 (Theorem 3.6), we prove that there is just one simply connected flagtransitive c · AG^{∗}-geometry with n = 3 and q = 4, say Γ̃. Our next step is to find a geometric characterization of Γ̃ by which we can eventually show that Γ̃ is indeed the geometry Γ(S) of Section 2. In view of that, we first investigate properties of a generic flag-transitive c · AG^{∗}- geometry Γ of order 4 and residual dimension 3. After having proved that Γ satisfies (LL) and (T) (Subsection 3.4), we study the common neighbourhood of two points at distance two (Subsection 3.5), eventually proving that Γ has diameter at most 4 (Subsection 3.6). Then we assume that Γ also admits a flag-transitive proper quotient. As we prove in Subsection 3.7, that additional hypothesis forces Γ = Γ̃. At this stage, the conclusion Γ̃ = Γ(S) follows quite easily.

We finish the proof (Subsection 3.8) by showing that the equality Γ̃ = Γ(S) implies that Γ(S)/2 is the unique flag-transitive quotient of Γ(S).

3.2. Feasible structures of point- and plane-stabilizers

Henceforth Γ is a flag-transitive c · AG^{∗}-geometry of order q > 2 and residual dimension n > 2 and G is a flag-transitive subgroup of Aut(Γ). The prime which q is a power of and its exponent in q will be denoted by p and h, respectively; namely, q = p^{h}.

In the sequel {a, u} is a given point-plane flag. It will be linguistically convenient to distinguish between the residue of a and its dual. We denote the latter by Γ_{a}^{∗}, calling the planes of Γ incident to a the points of Γ_{a}^{∗}.

We have K_{a} = 1 for every point a (this is true in any c.L^{∗}-geometry, as proved in Huybrechts and Pasini [16, Lemma 2.8]). Hence \overline{G}_{a} = G_{a}, namely G_{a} acts faithfully on the residue Γ_{a} of a. Thus, G_{a} is a subgroup of AΓL_{n}(q) = Aut(AG(n, q)). As G_{a} is flag-transitive on the affine space Γ_{a}^{∗}, it contains all translations of Γ_{a}^{∗} (see Buekenhout, Delandtsheer, Doyen [5]). Note also that, for every plane u, K_{u} is the kernel of the action of G_{a,u} = G_{a} ∩ G_{u} on the set of lines incident to a and u, for every point a incident to u.

According to the notation stated in Subsection 1.1, we denote the stabilizer in G of u (resp. a) by G_{2} (resp. G_{0}), K_{2} stands for the kernel of the action of G_{2} on the residue Γ_{u} of u and \overline{G}_{2} = G_{2}/K_{2}.

Lemma 3.1. Let G_{0} be not 1-dimensional affine. Then G_{0} is 2-transitive on the set of points of the affine space Γ_{a}^{∗}. Furthermore, the following are the possibilities for G_{0,2} = G_{0} ∩ G_{2}:

(i) SL_{m}(q_{1})⊴G_{0,2} for some m with 1 < m ≤ n and some power q_{1} = p^{k} of p with mk = nh; (ii) Sp_{2m}(q_{1}) ⊴ G_{0,2} for some m with 2 < 2m ≤ n and some power q_{1} = p^{k} of p with

2mk = nh; (iii) p = 2 and G_{2}(q_{1})′ ⊴ G_{0,2} for some power q_{1} = 2^{k} of 2 with 6k = nh;

Proof. As G_{0} is not 1-dimensional affine, one of the following holds (Buekenhout et al. [6, 3.1]):

(a) G_{0} is 2-transitive on the set of points of the affine space Γ_{a}^{∗};

(b) n = 4, q = 3 and G_{0,2}^{(∞)} = 2.Alt(5). In case (b), u has 41 points and G_{0,2}^{(∞)}/K_{2} = Alt(5). On the other hand, G_{2} acts 2-transitively on the set of points incident to u. Thus, \overline{G}_{2} is a 2-transitive group of degree 41 with point stabilizer (\overline{G}_{2})_{0} such that (\overline{G}_{2})_{0}^{(∞)} ≅ Alt(5). On the one hand, such a permutation group, if any, is of affine type. Indeed, Alt(41) and Sym(41) are the unique almost simple 2-transitive permutation groups of degree 41 (Cameron [8]), but they do not fit with the condition (\overline{G}_{2})_{0}^{(∞)} ≅ Alt(5). On the other hand, the condition (\overline{G}_{2})_{0}^{(∞)} ≅ Alt(5) is not satisfied by any affine 2-transitive permutation group of degree 41, since any such group is solvable.

Thus, we are left with case (a): G_{0} is one of the 2-transitive permutation groups of affine type listed in Liebeck [22, Appendix 1]. In view of the assumption that n, q > 2 and recalling that G_{0} ≤ AΓL_{n}(q), the possibilities that might occur for G_{0,2} are listed below:

(i) SL_{m}(q_{1})⊴G_{0,2} for some m with 1 < m ≤ n and some power q_{1} = p^{k} of p with mk = nh; (ii) Sp_{2m}(q_{1}) ⊴ G_{0,2} for some m with 2 < 2m ≤ n and some power q_{1} = p^{k} of p with

2mk = nh; (iii) p = 2 and G_{2}(q_{1})′ ⊴ G_{0,2} for some power q_{1} = 2^{k} of 2 with 6k = nh; (iv) q = p = 3, n = 4 and G_{0,2} is an extension of a central product of D_{8} and Q_{8} by a

subgroup of Sym(5);

(v) p = 3, nh = 6 and G_{0,2} = SL_{2}(13).

In case (iv) the group \overline{G}_{2} is 2-transitive of degree 41 with point-stabilizer of order divisible by 16. By looking at the list of 2-transitive groups, one can readily see that no such group exists.

Let (v) occur. Then either n = 3 and q = 9 or n = 6 and q = 3. The group \overline{G}_{2} is 2transitive of degree 92 in the first case and 365 in the latter, with point-stabilizer isomorphic to L_{2}(13) in any case. However, no such permutation group exists. Case (v) is also impossible. □

The group \overline{G}_{2}, being flag-transitive in Γ_{u}, is 2-transitive. Its degree (namely, the number of points of Γ_{u}) is

d = q^{n−1} + q^{n−2} + ⋯ + q + 2  (1)

and its point-stabilizer is G_{0,2}/K_{2}. Note that

d ≥ 14  (2)

(by (1), and since n, q ≥ 3 by assumption).

Lemma 3.2. Let G_{0} be not 1-dimensional affine. Then \overline{G}_{2} is not of affine type.

Proof. Assume the contrary. Then

d = p_{1}^{s}  (3)

for some prime p_{1} and some integer s. Comparing (3) with (1) we obtain

p_{1}^{s} = 2 + p^{h} + p^{2h} + ⋯ + p^{(n−1)h}.  (4)

This, together with the assumptions n, q > 2, forces p ≠ 2 and p ≠ p_{1}. It is also clear that the largest power of p_{1} dividing q^{n} − 1 divides p^{h} − 1 and that p is not a divisor of p_{1}^{s} − 1.

As p ≠ 2, case (iii) of Lemma 3.1 is impossible. Thus, we are left with cases (i) and (ii) of Lemma 3.1. In both (i) and (ii) of Lemma 3.1, the center of G_{0,2} is the kernel of the action of G_{0,2} on the set of lines of Γ_{a}^{∗} on u, namely K_{2} = Z(G_{0,2}). Furthermore, (G_{0,2}/K_{2})^{(∞)} is non-trivial and simple. Hence \overline{G}_{2} is not 1-dimensional affine. (Indeed, 1-dimensional affine permutation groups have solvable point-stabilizers.) Consequently, by Liebeck [22, Appendix 1] and recalling that (4) holds and that n, q > 2, the following three possibilities are obtained for G_{0,2}/K_{2}:

(j) SL_{r}(p_{1}^{t}) ⊴ G_{0,2}/K_{2} for some integers r, t with rt = s and r > 1; (jj) Sp_{2r}(p_{1}^{t}) ⊴ G_{0,2}/K_{2} for some integers r, t with 2rt = s and r > 1; (jjj) p_{1} = 2 and G_{2}(2^{t})′ ⊴ G_{0,2}/K_{2} for some integer t with 6t = s.

Furthermore, according to Lemma 3.1 and since K_{2} = Z(G_{0,2}), the center of G_{0,2}/K_{2} is trivial. This forces (r, p_{1}^{t} − 1) = 1 in (j) and p_{1} = 2 in (jj). We should now match each of (j), (jj) and (jjj) with (i) and (ii) of Lemma 3.1. The isomorphisms between groups of Lie type are well-known (see [11, 2, 3], for instance). Exploiting that information and recalling that n > 2 by assumption, one can see that no matching as required above is possible. □

Lemma 3.3. Let G_{0} be not 1-dimensional affine. Then n = 3, q = 4, K_{2} = 3 and either G_{0} = ASL_{3}(4) and G_{2} = 3·M_{22} or G_{0} = AΣL_{3}(4) and G_{2} = 3·Aut(M_{22}).

Proof. By Lemma 3.2, the permutation group \overline{G}_{2} is not of affine type. As noticed in the proof of Lemma 3.2, in each of the cases (i), (ii) and (iii) we have K_{2} = Z(G_{0,2}) and (G_{0,2}/K_{2})^{(∞)} is non-trivial and simple. Hence \overline{G}_{2} is not a group of Lie type acting on the set of points of its natural geometry. Indeed, if X is a group of Lie type and X_{0} is its natural point-stabilizer, then X_{0}^{(∞)} is either non-simple or trivial.

Also, \overline{G}_{2} is not isomorphic to either of Sym(d) and Alt(d), as (G_{0,2}/K_{2})^{(∞)} ≅ A̸lt(d − 1). Furthermore, there is no s such that \overline{G}_{2} ≅ O_{2s+1}(2) in its action on one of the two families of non-singular hyperplanes of P G(2s, 2). Indeed, if such an s existed, then either d = 2^{s}(2^{s} +1) or d = 2^{s}(2^{s} − 1) but, comparing those equalities with (1) and recalling that n, q > 2, a contradiction is easily obtained.

Thus, \overline{G}_{2} falls into one of the following exceptional almost simple cases:

\begin{matrix}\overline{G}_{2} & d \\ L_{2}(11) & 11 \\ Alt(7) & 15 \\ M_{11} & 12 \\ M_{11} & 11 \\ M_{12} & 12 \\ M_{22} or Aut(M_{22}) & 22 \\ M_{23} & 23 \\ M_{24} & 24 \\ HS & 176 \\ Co_{3} & 276 \end{matrix}

However, it is easily checked that none of the values listed above for d but 22 survives through the conditions (1) and (2). On the other hand, 22 = 4^{2} + 4 + 2. Thus, d = 22, q = 4, n = 3 and \overline{G}_{2} = M_{22} or Aut(M_{22}). Accordingly, G_{0,2}/K_{2} = L_{3}(4) or L_{3}(4).2 and G_{0} = ASL_{3}(4) or AΣL_{3}(4). In any case, K_{2} = 3. □

Lemma 3.4. Let G_{0} be 1-dimensional affine. Then q is odd, d is a prime power and the permutation group \overline{G}_{2} is of 1-dimensional affine type.

Proof. We recall that a group X is said to be metacyclic if either it is cyclic or it admits a normal subgroup Y such that both Y and X/Y are cyclic. It is well-known (and easy to prove) that, if a group X is metacyclic, then all subgroups and factor groups of X are metacyclic. We also recall that point-stabilizers in 1-dimensional affine groups are metacyclic.

As G_{0} is 1-dimensional affine, G_{0,2} is metacyclic. Hence (\overline{G}_{2})_{0} (= G_{0,2}/K_{2}) is metacyclic. Assume that \overline{G}_{2} is not of affine type. Then, as (\overline{G}_{2})_{0} is metacyclic, by the the list of 2transitive permutation groups we see that \overline{G}_{2} = L_{2}(p_{1}) for some prime p_{1} with p_{1} + 1 = d. Thus,

p_{1} = \frac{q^{n} − 1}{q − 1}  (5)

(by (1)) and G_{0,2}/K_{2} has order (p_{1} − 1)p_{1}/2 (notice that p_{1} ≠ 2, by (2)). However, as G_{0} is 1-dimensional affine, the order of G_{0,2}/K_{2} divides the order (q^{n}−1)nh of ΓL_{1}(p^{nh}). Therefore,

(p_{1} − 1)p_{1} divides 2(q^{n} − 1)nh.  (6)

Comparing (5) and (6) we see that q^{n−1}+q^{n−2}+⋯+q divides 2(q−1)nh, which is impossible, since n > 2.

Thus, \overline{G}_{2} is of affine type; namely, it is one of the groups of [22, Appendix 1]. Actually, the only groups listed there with metacyclic point-stabilizer and degree d satisfying (1) with n, q ≥ 3 are the 1-dimensional affine ones.

As \overline{G}_{2} is of affine type, d is a prime power. This, together with (1) and the assumption n, q > 2, forces q to be odd. □

The following theorem is an assembling of Lemmas 3.3 and 3.4.

Theorem 3.5. One of the following holds:

(i) n = 3, q = 4 and either (i.a) G_{0} = ASL_{3}(4) and G_{2} = 3·M_{22} with K_{2} = 3, or (i.b) G_{0} = AΣL_{3}(4) and G_{2} = 3·Aut(M_{22}) with K_{2} = 3;

(ii) q is odd, q^{n−1} + q^{n−2} + ⋯ + q + 2 is a prime power and both G_{0} = \overline{G}_{0} and \overline{G}_{2} are 1-dimensional affine.

3.3. Uniqueness of the amalgam when (n, q) = (3, 4)

From now on we focus on the case of q = 4 and n = 3, that is on case (i) of Theorem 3.5 (for indeed, when (n, q) = (3, 4), case (ii) of Theorem 3.5 cannot hold since 4^{2} + 4 + 2 = 22 is not a prime power).

Theorem 3.6. There exists precisely one flag-transitive and simply connected c · AG^{∗}-geom-etry of order 4 and residual dimension 3.

Proof. At least one such geometry exists, namely the universal cover of the geometry Γ(S) described in Section 2. Thus, we only need to prove the uniqueness claim.

As observed above, we are in case (i) of Theorem 3.5. As both cases (i.a) and (i.b) of Theorem 3.5 are realized in Γ(S) for G = Aut(Γ(S)) and G = Aut(Γ(S))′ respectively, we only need to show the uniqueness of the amalgam of the parabolics G_{0}, G_{1}, G_{2} for each of the cases (i.a) and (i.b) of Theorem 3.5. We will consider case (i.a) first. Then we shall shortly discuss (i.b).

Thus, let (i.a) of Theorem 3.5 hold. The intersections G_{0,2} = G_{0} ∩ G_{2}, G_{0,1} = G_{0} ∩ G_{1}, G_{1,2} = G_{1} ∩ G_{2} and G_{0,1,2} = G_{0} ∩ G_{1} ∩ G_{2} are uniquely determined inside G_{0} and G_{2}. In particular, G_{0,1} (resp. G_{0,2}, G_{0,1,2}) is the stabilizer of a line (a point, a point-line flag) of AG(3, 4) in ASL_{3}(4) = G_{0}. Thus,

G_{0,1} ≅ (V × W):L

with V = 2^{2}, W = 2^{4}, L = GL_{2}(4) = Z × L′, where |Z| = 3 and L′ = L_{2}(4). The group L normalizes both V and W and we have

W L = Z × W L′ = 3 × ASL_{2}(4), V L = L′ × V Z = L′ × GL_{1}(4).

Also,

G_{0,1,2} = W L and G_{0,2} = SL_{3}(4) with center Z.

Note that Z is also the center of G_{2} (= Z·M_{22}.) Finally, G_{1,2} is the stabilizer in 3:M_{22} of a duad of S(22, 6, 3) (namely, a point of the subset S_{0} of P G(5, 4) considered in Subsection 2.1). Clearly, G_{1,2} = G_{0,1,2}2 and Z is also the center of G_{1,2}. (Indeed, it is the center of G_{2}). Furthermore,

G_{1,2}/Z = 2^{4}:S_{5} withG_{0,1,2}/Z = 2^{4}:A_{5}

(see [11]). Therefore,

G_{1,2} = ⟨G_{0,1,2}, i⟩

for a suitable involution i normalizing G_{0,1,2} = W L = Z × W L′ and centralizing Z. As W L′ is the derived group (W L)′ of W L, the involution i also normalizes W L′ and its action on W L′ is uniquely determined via the isomorphisms W L′ ≅ 2^{4}:A_{5} and (W L)⟨i⟩/Z ≅ 2^{4}:S_{5}. Thus,

G_{1,2} = Z × W L′⟨i⟩ = 3 × 2^{4}:S_{5}.

To finish the proof in case (i.a) we only need to prove that G_{1} is also uniquely determined by the above. We have G_{1} = ⟨G_{1,2}, v⟩ and ⟨G_{0,1,2}, v⟩ = G_{0,1}, for an involution v ∈ V. Thus, G_{1} = ⟨G_{0,1,2}, v, i⟩. As i permutes the two points of the line stabilized by G_{1} and G_{0,1} stabilizes each of those two points, i normalizes G_{0,1} = (V × W)L. Furthermore, as noticed above, i centralizes Z and normalizes W L′. Therefore, as V Z is the centralizer of W L′ in G_{0,1}, the involution i also normalizes V. As seen above, the action of i on W L is uniquely determined. Thus, we only need to determine the action of i on V. As i is an involution, it fixes an element w of V \ {1}. Thus, w^{i} = w. On the other hand, w, w^{z} and w^{z^{2}} (with z a generator of Z) are the non-trivial elements of V. However, as noticed above, i centralizes Z. Hence w^{zi} = w^{iz} = w^{z} and w^{z^{2}i} = w^{iz^{2}} = w^{z^{2}}. That is, i centralizes V. Thus, G_{1} is uniquely determined:

G_{1} = ((V × W):L)⟨i⟩

with W L⟨i⟩ = Z × W L′⟨i⟩ = 3 × 2^{4}:S_{5} and V L⟨i⟩ = V Z × L′⟨i⟩ = GL_{1}(4) × S_{5}.

The above holds in (i.b), too, but for replacing L = L_{2}(4) with LF = ΣL_{2}(4), where the factor F has order 2 and is contributed by the non-trivial automorphism of GF (4). Accordingly, F permutes the two non-trivial elements of Z and acts non-trivially on both V and W. Thus,

G_{1} = ((V × W):LF)⟨i⟩

where

W LF = (Z × W L′)F with ZF = S_{3} and W L′F = AΣL_{2}(4),
V LF = (V Z × L′)F with V ZF = AΓL_{1}(4) and (Z × L′)F = ΓL_{2}(4).

Furthermore, i centralizes V ZF. (Indeed, i centralizes V and Z, as above, and normalizes ZF ≅ S_{3}.) Thus,

\begin{matrix}G_{0,1} = (V × W):LF = (2^{2} × 2^{4})ΓL_{2}(4), \\ G_{1,2}/Z ≅ W L′F ⟨i⟩ = 2^{5}:S_{5} \end{matrix}

(note also that G_{1,2}/Z can be recognized inside Aut(M_{22}) as the stabilizer of a duad; see [11]). □

Corollary 3.7. Let Γ be a flag-transitive c · AG^{∗}-geometry of order 4 and residual dimension 3. Then (i.a) of Theorem 3.5 holds for some flag-transitive subgroup G of Aut(Γ).

Proof. Let Γ̃ be the unique flag-transitive simply connected c · AG^{∗}-geometry of order 4 and residual dimension 3 (Theorem 3.6) and let Γ(S) be as in Section 2. The group Aut(Γ(S)) satisfies (i.b) of Theorem 3.5 whereas (i.a) is satisfied by the commutator subgroup of Aut(Γ(S)). Furthermore, as Γ̃ is the universal cover of Γ(S), Aut(Γ(S)) lifts to a subgroup G̃ of Aut(Γ̃) and, as Aut(Γ(S)) satisfies (i.b) of Theorem 3.5, that condition holds in G̃, too. Hence, according to Theorem 3.5(i), G̃ = Aut(Γ̃).

On the other hand, the commutator subgroup of Aut(Γ(S)) lifts to a flag-transitive proper normal subgroup G̃′ of G̃ satisfying (i.a). (It is not difficult to see that G̃′ is indeed the commutator subgroup of G̃, but we will not exploit this fact here.) By Theorem 3.5(i), G̃′ is the unique flag-transitive proper subgroup of G̃.

Let Γ be a flag-transitive c · AG^{∗}-geometry of order 4 and residual dimension 3. By Theorem 3.6, we have Γ = Γ̃/Z for some subgroup Z of G̃. As Γ is flag-transitive, the normalizer of Z in G̃ is also flag-transitive. Hence it contains G̃′. Therefore the group G = ZG̃′/Z is flag-transitive on Γ. As G̃′ satisfies (i.a) of Theorem 3.5, that condition also holds for G. □

3.4. The case of (n, q) = (3, 4). Properties (LL) and (T)

In the sequel Γ is a flag-transitive c · AG^{∗}-geometry of order 4 and residual dimension 3. By Corollary 3.7, Aut(Γ) admits a flag-transitive subgroup G with G_{0} = ASL_{3}(4) and G_{2} = 3·M_{22}. Henceforth G will stand for such a subgroup of Aut(Γ).

The reader is referred to Subsection 1.5 for the statement of the properties (LL) and (T). According to the conventions stated in that subsection, the symbol ∼ denotes the collinearity relation of Γ.

Proposition 3.8. The property (LL) holds in Γ.

Proof. The relation ‘having the same points’ is an equivalence relation on the set of lines of Γ. Let us denote that relation by Θ. Clearly, no two lines in the same class of Θ are incident to the same plane. Furthermore, by the flag-transitivity of G, all classes of Θ have the same size, say ν. We have ν = 1 if and only if (LL) holds.

Assume ν > 1. Given a point a of Γ, let Θ_{a} be the relation induced by Θ on the set of lines incident to a. On the one hand, Θ_{a} is not the trivial relation (where any two lines correspond). Indeed no two lines in the same class of Θ are incident to the same plane. On the other hand, Θ_{a} is not even the identity relation, as ν > 1 by assumption. Furthermore, G_{a} (= ASL_{3}(4)) permutes the classes of Θ_{a}. However, the partition in parallel classes is the unique non-trivial partition of the set of lines of AG(3, 4), different from the identity partition and preserved by ASL_{3}(4). Consequently, Θ_{a} is the parallelism of the affine space Γ_{a}^{∗}.

Thus, ν = 16 and the points of a^{∼} \ {a} bijectively correspond to the parallel classes of Γ_{a}^{∗}. Hence |a^{∼} \ {a}| = 21. Furthermore, for any parallel class X of Γ_{a}^{∗} and any line l of Γ_{a}^{∗} not in X, there are lines of X meeting l in a point of Γ_{a}^{∗}, namely coplanar with l in Γ. Therefore, any two points of a^{∼} belong to a common plane of Γ, whence they are collinear. It is now clear that Γ has exactly 22 points. As 22 is the number of points in a plane, Γ is flat. Accordingly, |G : G_{0}| = 22 and |G : G_{2}| = 64.

Let K be the kernel of the action of G on the set of points of Γ. Then K is a normal subgroup of G_{a}, stabilizing all parallel classes of Γ_{a}^{∗}. On the other hand, if an element g ∈ G_{a} stabilizes all parallel classes of Γ_{a}^{∗}, then g also stabilizes all points of Γ. Consequently, g ∈ K. Therefore, K = 2^{6}.3. Hence |G : G_{2}| = |K|/3. It is also clear that G_{2} ∩ K = K_{2} = 3. Thus, G = K.\overline{G}_{2} = 2^{6}.3.M_{22}. Also, G = O_{2}(K):G_{2} = 2^{6}:3·M_{22}.

Let X be the elementwise stabilizer in G of the set of planes of Γ. As noticed in Subsection 1.1, we have K_{0} = 1. Hence X acts semi-regularly on the set of points of Γ. Therefore, X is a normal subgroup of \overline{G}_{2} acting semi-regularly on the 22 points of Γ. However, \overline{G}_{2} = M_{22}. This makes it clear that X = 1. Namely, G acts faithfully on the set of planes of Γ. Furthermore, G_{0} acts 2-transitively as ASL_{3}(4) on that set of planes. Hence G is also 2-transitive on the 64 planes of Γ. As a consequence, the stabilizer G_{u} in G of a plane u permutes transitively the 63 planes of Γ other than u, whence it admits a subgroup of index 63. However, G_{u} = 3·M_{22} and the group 3·M_{22} has no subgroups of index 63, since M_{22} has no subgroup of index 63 or 27 (see [11]). We have reached a final contradiction. □

More notation. As (LL) holds in Γ, the lines of Γ are just the pairs of distinct collinear points. Given two distinct collinear points a, b, we denote the line {a, b} by ab.

As Γ satisfies (LL), it also satisfies the Intersection Property. In particular, distinct planes of Γ are incident to distinct sets of points and a line ab is incident to a plane u precisely when both a and b are incident to u. Thus, planes may be viewed as sets of points. Accordingly, given a point a and a plane u, we write a ∈ u if a is incident to u; if a ∉ u, we denote the set of points of u collinear with a by a^{∼} ∩ u; and so on.

Lemma 3.9. The group G is transitive on the set of triples (a, b, u) with a, b distinct points of Γ, u a plane of Γ at distance 1 from a and b ∈ a^{∼} ∩ u.

Proof. Given a point b, its stabilizer G_{b} (≅ ASL_{3}(4)) is transitive on the non-incident pointline pairs of the affine space Γ_{b}^{∗}. The lemma follows from this and from the transitivity of G on the set of points of Γ. □

As a consequence, G is also transitive on the set of point-plane pairs (a, u) of Γ, with a at distance 1 from u. (Note that such pairs exist, otherwise Γ would be flat, contrary to Proposition 3.8.)

According to the above, given a point-plane pair (a, u) with a at distance 1 from u, the number |a^{∼} ∩ u| does not depend on the particular choice of (a, u). We shall denote that number by γ. Clearly, γ ≤ 22.

Lemma 3.10. We have γ < 22.

Proof. Assume the contrary. Given three points a, b, c with a ∼ b ∼ c, let u be a plane on the line bc. As γ = 22, a is collinear with all points of u. In particular, a ∼ c. Consequently, any two points of Γ are collinear. Hence Γ has 1 + 16 · 21 = 337 points. On the other hand, every point is in 64 planes and every plane has 22 points. Thus, the number of planes is 337 · 64/22. However, this number is not an integer; contradiction. □

Proposition 3.11. The property (T) holds in Γ.

Proof. It is easily seen that (T) holds if and only if γ = 5. Thus, we need to prove that γ = 5.

Given a plane u and a point a at distance 1 from u, let b ∈ a^{∼} ∩ u. The plane u, viewed as a point of the affine space Γ_{b}^{∗}, does not belong to the line L = ab of Γ_{b}^{∗}. The points of a^{∼} ∩ (u \ {b}) bijectively correspond to certain lines of Γ_{b}^{∗} through u. Let L(b) be the set of those lines. Then γ = |L(b)| + 1.

The stabilizer G_{a,b,u} of a and u in G_{b} is the stabilizer of the pair (u, L) of Γ_{b}^{∗}. However, G_{b} ≅ ASL_{3}(4) in its natural action on AG(3, 4) ≅ Γ_{b}^{∗}. The stabilizer of (u, L) in ASL_{3}(4) has three orbits on the set of lines of AG(3, 4) through u. One of those orbits, say L_{1}(b), contains the four lines joining u with points of L. Another orbit, say L_{2}(b), contains just one line, namely the line through u parallel to L. Finally, the third orbit, say L_{3}(b), contains all lines through u skew with L. Note that the stabilizer of (u, L) in ASL_{3}(4) acts 2-transitively on L_{1}(b).

Clearly, L_{1}(b) ⊆ L(b). Also, L(b) ≠ L_{1}(b) ∪ L_{2}(b) ∪ L_{3}(b), by Lemma 3.10. Therefore, L(b) is equal to one of L_{1}(b), L_{1}(b) ∪ L_{2}(b) or L_{1}(b) ∪ L_{3}(b). Accordingly, γ = 5, 6 or 21. To finish, we only need to show that each of the hypotheses γ = 6 and γ = 21 yields a contradiction.

Let γ = 6. Given b ∈ a^{∼} ∩ u, let U(b) be the set of points different from b and incident to lines of L_{1}(b) and let u(b) be the point of the line of L_{2}(b) different from b. Clearly, |U(b)| = 4 and {b} ∪ {u(b)} ∪ U(b) = a^{∼} ∩ u. Furthermore, G_{a,b,u} stabilizes u(b) and acts 2-transitively on U(b). Note also that {b, u(b)} is the unique line of u on b contained in a^{∼} but not coplanar with a. Therefore, if b′ = u(b), then u(b′) = b. Consequently, the group G_{a,u}, which transitively permutes the six points of a^{∼} ∩ u, preserves a partition of that set in the three pairs. However, for every point b ∈ a^{∼} ∩ u, G_{a,b,u} is 2-transitive on the four points of a^{∼} ∩ u not in the pair containing b. Hence G_{a,u} cannot preserve a partition of a^{∼} ∩ u in three pairs; contradiction.

Let γ = 21. With the above notation, u(b) is the unique point of u at distance 2 from a. Namely, given three points a, b, c with a ∼ b ∼ c, we have a ≁ c if and only if the lines ba and bc are parallel in Γ_{b}^{∗}. As a consequence, for every point b ∼ a, there are precisely 15 points collinear with b at distance 2 from a. Furthermore, given two points b, c with c ∼ b ∼ a and c ≁ a, let v be a plane on c. There are four planes on bc meeting v in a line. Let u be one of them and let d be the point of u ∩ v different from c. The plane u has distance 1 from a. Hence all its points but one are collinear with a. However, c ≁ a. Hence d ∼ a. Thus, v has distance 1 from a. As a consequence, all planes of Γ have distance at most 1 from a. Therefore, for every plane u of Γ, at most one point of u has distance 2 from a.

Given two collinear points c, d, let u be a plane on the line cd. As at most one point of u has distance 2 from a, at most one of c and d has distance 2 from a. Therefore, given a point c at distance 2 from a, all points collinear with c are also collinear with a. Furthermore, as noticed above, every point collinear with a is collinear with exactly 15 points at distance 2 from u. Hence there are precisely 15 points at distance 2 from a. Each of them belongs to 64 planes and these have distance 1 from a. On the other hand, G_{a} transitively permutes the planes at distance 1 from a and every such plane contains just one point at distance 2 from a. Hence G_{a} transitively permutes the 15 points at distance 2 from a. Consequently, G_{a} contains a subgroup of index 15. However, G_{a} ≅ ASL_{3}(4) and it is easily seen that no subgroup of index 15 exists in ASL_{3}(4). A contradiction has been reached. □

3.5. The case of (n, q) = (3, 4). Local geometries

Let Γ and G be as in the previous subsection.

Definition. Given two points a, b of Γ at distance 2, an (a, b)-local subspace is a connected component of the graph induced by the point-line system of Γ on the set a^{∼} ∩ b^{∼}. If X is an (a, b)-local subspace, then the structure formed by the graph X and by the planes of Γ with at least three points in X will be called an (a, b)-local geometry of a and b, also the local geometry of X.

The second neighbourhood of a point. Given a point a, by P (a) (respectively, S(a)) we mean the set of points at distance 2 from a such that for some point c ∈ a^{∼} ∩ b^{∼} the lines ac and bc are parallel (skew) in the affine space Γ_{c}^{∗}. Given a point b at distance 2 from a, by P (a, b) (respectively, S(a, b)) we mean the set of points c ∈ a^{∼} ∩ b^{∼} such that the lines ac and bc are parallel (skew) in Γ_{c}^{∗}. Given a point c ∼ a, the group G_{a,c} has two orbits on the set of lines on c non-coplanar with ac. Therefore,

Lemma 3.12. Either S(a) = P (a) or S(a) ∩ P (a) = ∅.

In the latter case, |c^{∼} ∩ P (a)| = 15 and |c^{∼} ∩ S(a)| = 16 · 15. In any case, P (a) ≠ ∅. The following is also clear.

Lemma 3.13. Let b ∈ P (a) and c ∈ P (a, b). Let π be the plane of Γ_{c}^{∗} spanned by the two parallel lines ac and ab. Then the lines of c coplanar with both ac and bc in Γ are the 16 lines of π not parallel to ac and ab.

Terminology. We recall that, given a Laguerre plane Ξ, a rank 3 geometry can be built over Ξ by ‘enrichment’ [26, Chapter 2], namely by taking the 2-subsets of the circles as new elements, to be placed between the points and the circles of Ξ. The incidence relation is the natural one (symmetrized inclusion). Henceforth, when speaking of a Laguerre plane, we will always mean its rank 3 enrichment, constructed as above. Accordingly, the circles of a Laguerre plane will be called planes and the 2-subsets of the circles will be called lines.

Lemma 3.14. Let b ∈ P (a). Then P (a, b) is a join of (a, b)-local subspaces. Each of them has size 20 and its local geometry is a Laguerre plane of order 4.

Proof. Let c ∼ d with c ∈ P (a, b) and d ∈ S(a, b), if possible. By (T) (Proposition 3.11), the line cd is coplanar with both ca and cb and, by Lemma 3.13, there are 12 lines in Γ_{c} coplanar with ac, bc and dc. Therefore, and by (T), there are 12 lines on d coplanar with ad, bd and cd. However, the lines ad and bd are skew in the affine space Γ_{d}^{∗} and cd, being a transversal of them in Γ_{d}^{∗}, cannot meet 12 distinct transversals of ad and cd; contradiction. Consequently, P (a, b) is the disjoint union of (a, b)-local subspaces.

Let Ξ be an (a, b)-local geometry contained in P (a, b). By the above, the residues of the points of Ξ are isomorphic to the geometry obtained from AG(2, 4) by removing a bundle of parallel lines. Hence Ξ has 20 points and it is a Laguerre plane. □

Lemma 3.15. The number |P (a)| divides 252.

Proof. For every point c ∈ a^{∼} there are precisely 15 points b ∈ P (a) for which c ∈ P (a, b). On the other hand, |a^{∼}| = 2^{4} · 3 · 7. The conclusion follows from Lemma 3.14. □

Lemma 3.16. We have P (a) ∩ S(a) = ∅.

Proof. Assume the contrary. Then S(a) = P (a) by Lemma 3.12. However, given c ∼ a, there are precisely 17 · 15 = 255 points b ∼ c at distance 2 from a: too many! (Compare Lemma 3.15.) □

Lemma 3.17. The group G_{a} has two orbits on the set of points at distance 2 from a, namely P (a) and S(a). Furthermore, it is transitive on the set of paths a ∼ c ∼ b with b ∈ P (a) (respectively, b ∈ S(a)).

Proof. Clear, by Lemma 3.16 and because, given c ∈ a^{∼}, G_{a,c} is transitive on P (a) ∩ c^{∼} and S(a) ∩ c^{∼}. □

We recall that an extended grid of order s is a c.C_{2}-geometry satisfying (LL) and where the residues of the points are grids of order s (Cameron, Hughes and Pasini [10]).

Lemma 3.18. Let b ∈ S(a). Then the (a, b)-local geometries are extended grids of order 3 satisfying the property (T).

Proof. The lines of AG(3, 4) meeting two given skew lines form a dual grid of order 3. The conclusion follows from this, from Lemma 3.17 and from the fact that (T) holds in Γ. □

Lemma 3.19. Distinct points of S(a) give rise to distinct local geometries.

Proof. Indeed, given c ∈ a^{∼} and b, b′ ∈ S(a) ∩ c^{∼}, let L_{ab} be the set of lines of Γ_{c}^{∗} meeting both ca and cb. If all those lines also meet cb′, then cb = cb′. □

Definition. Given b at distance 2 from a, let Ξ be an (a, b)-local geometry. We say that Ξ is of Laguerre type (of grid type) if it is a Laguerre plane (an extended grid), namely if b ∈ P (a) (if b ∈ S(a)).

Planes at distance 1 from a point. Given a plane u at distance 1 from a, let c ∈ u ∩ a^{∼}. As (T) holds in Γ, the other points of u ∩ a^{∼} are the four lines on c contained in u and coplanar with ac. Thus,

Lemma 3.20. We have |u ∩ a^{∼}| = 5.

Given u and c as above, in Γ_{c}^{∗} we see that G_{a,u} stabilizes a unique point b of u distinct from c, namely the point b for which cb is parallel to ca, and permutes the remaining 16 points of u\a^{∼} transitively. We denote that point by p_{u}(a) and we call it the projection of a onto u. Clearly, c ∈ P (a, p_{u}(a)). Therefore,

Lemma 3.21. We have u ∩ P (a) = {p_{u}(a)} and u\(a^{∼} ∪ {p_{u}(a)}) ⊂ S(a).

Lemma 3.22. There are 2^{6} · 3^{2} · 7 planes at distance 1 from a.

Proof. There are 16·21 points in a^{∼}, each of them is in 4^{3} −4 planes at distance 1 from a and each such plane is counted 5 times in this way. Therefore, the number of planes at distance 1 from a is 16 · 21 · 12. □

Lemma 3.23. Let u be a plane at distance 1 from a and let c_{1}, c_{2}, c_{3}, c_{4}, c_{5} be the five points of u collinear with a. Then the lines ac_{1}, ac_{2}, ac_{3}, ac_{4}, ac_{5} belong to a plane of the affine space Γ_{a}^{∗} and form a non-degenerate dual conic in that plane.

Proof. Any two of those lines meet in the affine space Γ_{a}^{∗}. On the other hand, no plane of Γ on a contains three of them. Indeed, if v were such a plane, then v = u by (T) and this cannot be. The conclusion follows. □

Local geometries of Laguerre type. Let Ξ be an (a, b)-local geometry of Laguerre type. For every point c of Ξ, the line ac belongs to 4 planes of Ξ and each of these planes contains 5 points of Ξ. Therefore, precisely 16 of the 64 planes of Ξ contain a.

Lemma 3.24. The 16 planes of Ξ on a and the 20 lines of Ξ on a are the 16 points and the 20 lines of a plane π_{Ξ} of the affine space Γ_{a}^{∗}. The remaining planes of Ξ are the 3 · 16 non-degenerate dual conics of π_{Ξ} with the line at infinity of π_{Ξ} as the nuclear line. These conics are shared in three orbits by the stabilizer ASL_{2}(4) of π_{Ξ} in G_{a}. One of these orbits is contributed by the 16 planes of Γ on b at distance 1 from a.

Proof. Let u be a plane of Ξ on a. In the affine space Γ_{a}^{∗}, let L_{1}, L_{2}, …, L_{5} be the five lines of Ξ ∩ Γ_{a}^{∗} through u. The points of ⋃_{i=1}^{5} L_{i} in Γ_{a}^{∗} are the 16 points of Γ_{a}^{∗} ∩ Ξ. Similarly, if v ∈ L_{i}\{u}, the five lines of Γ_{a}^{∗} ∩Ξ on v contribute 15 points of Γ_{a}^{∗} ∩Ξ other than v. However, the 5 lines on u already give us all points of Γ_{a}^{∗} ∩ Ξ. Therefore, the 5 lines of Γ_{a}^{∗} ∩ Ξ on v meet each of L_{1}, L_{2}, …, L_{5}. The conclusion is clear (see also Lemma 3.23). □

Local geometries of grid type. By Lemma 3.17, all local geometries of grid type have the same isomorphism type.

Let b ∈ S(a) and let Ξ be an (a, b)-local geometry. We denote by Π_{a} (by Π_{b}) the set of planes of Ξ containing a (respectively, b) and Ξ_{a} will denote the configuration of points and lines of the affine space Γ_{a}^{∗} formed by the planes of Π_{a} and the lines joining a with points of Ξ.

Lemma 3.25. The pair {Π_{a}, Π_{b}} is a partition of the set of planes of Ξ with the property that no two planes in the same class have a line in common.

Proof. If n is the number of points of Ξ, then |Π_{a}| = |Π_{b}| = 4n/5. However, Ξ has 8n/5 planes in total and no two planes on a (or on b) meet in a line of Ξ. The statement follows.□

By Lemma 3.18, Ξ is an extended grid of order 3 satisfying (T). Furthermore, by Lemma 3.17, Ξ admits a point-transitive automorphism group and, by Lemma 3.25, its planes are partitioned in two classes in such a way that no two planes of the same class meet in a line. By Fisher [13] (also Blokhuis and Brouwer [4]), Ξ is one of the following:

(I) The geometry obtained from the Coxeter complex of type A_{7} by truncating the wings of the diagram but saving three nodes in the middle, then forming the Grassmann geometry [26, Chapter 5] with respect to the central node. This extended grid is flag-transitive, it has 70 points and 112 planes, partitioned in two classes of size 56.

(II) The extended grid called 2.CF^{−}(4) in [25]. This extended grid is flag-transitive, it has 40 points and 64 planes, partitioned in two classes of size 32.

Lemma 3.26. The extended grid Ξ is of type (II).

Proof. Let Ξ be of type (I), if possible. Then Ξ_{a} contains 56 points of the affine space Γ_{a}^{∗}. Thus, 8 out of the 64 points of Γ_{a}^{∗} are missing in Ξ_{a}. G_{a,b} acts transitively on the set of lines of Ξ_{a}. Given a line ac ∈ Ξ_{a}, G_{a,c,b} has order 2^{4} · 3. As there are 70 lines in Ξ_{a}, the stabilizer H of Ξ in G_{a} has order 70·2^{4} ·3 = 2^{5} ·3·5·7. The group H acts on the 8 points of Γ_{a}^{∗} missing in Ξ_{a}. Let u be one of them. Its stabilizer H_{u} in H has order 2^{2} · 3 · 5 · 7 or a multiple of this number. It is a subgroup of SL_{3}(4) (= G_{a,u}). Thus, its index in SL_{3}(4) is a divisor of 9 · 16. However, no maximal subgroup of SL_{3}(4) has index a divisor of 9 · 16 (see [11]). Therefore H_{u} = SL_{3}(4). But this is clearly impossible: if H_{u} = SL_{3}(4) then H cannot have an orbit of size 70 on the set of lines of Γ_{a}^{∗}. □

As Ξ is of type (II), Ξ_{a} contains 32 points and 40 lines of the affine space Γ_{a}^{∗}. Furthermore, G_{a,b} acts transitively on the set of lines of Ξ_{a}. Given a line ac ∈ Ξ_{a}, we have |G_{a,c,b}| = 2^{4} · 3. As there are 40 lines in Ξ_{a}, the stabilizer H of Ξ in G_{a} has order 40 · 2^{4} · 3 = 2^{7} · 3 · 5. (Note that H also stabilizes b, by Lemma 3.19.) The group H acts on the 32 points of Ξ_{a}. Let u be one of them.

Lemma 3.27. The stabilizer H_{u} of u in H is isomorphic to the alternating group Alt(5) of degree 5 and the five lines of u through a belonging to Ξ_{a} form a non-degenerate quadratic cone of Γ_{a}^{∗} with vertex u.

Proof. Given a line ac in Ξ_{a}, G_{a,b,c} is on the four points of Γ_{c}^{∗} belonging to the line ac. Hence H is transitive on the 32 points of Γ_{a}^{∗} belonging to Ξ_{a} and the stabilizer H_{u} of u in H has order 2^{2} · 3 · 5. H_{u} is a subgroup of G_{a,u} = SL_{3}(4) of index 63 · 16. There are only two maximal subgroups of SL_{3}(4) that might fit with this situation (see [11]), namely 3.Alt(6) and 3.(2^{4}:Alt(5)). Assume H_{u} ≤ 3.Alt(6). As H_{u} has index 18 in 3.Alt(6), we have H_{u} = 3.Alt(5) (= 3 × Alt(5)) (see [11]), acting as Alt(5) on the set of lines of Γ_{a}^{∗} on u. On the other hand, if H_{u} is a subgroup of 3.(2^{4}:Alt(5)), it has index 3 · 16 in it. In this case, comparing the information we have on stabilizers of planes of Γ, we get H_{u} = Alt(5). In any case, H_{u} acts as Alt(5) on the star of u in Γ_{a}^{∗}. In the first case, H_{u,c} = 3 × Alt(4). In the second case, H_{u,c} = Alt(4). However, it is easy to check that H_{u,c} = Alt(4). Hence H_{u} = Alt(5).

Each subgroup of L_{3}(4) of type Alt(5) is the stabilizer of a non-degenerate conic (and its nucleus) or of a non-degenerate dual conic (and its nuclear line). Therefore, H_{u} is either the stabilizer of a set of 5 planes of Γ_{a}^{∗} on u forming a non-degenerate dual conic in the star of u or the stabilizer of a non-degenerate quadratic cone of Γ_{a}^{∗} of vertex u.

Assume the first case and let C be the set of five planes on u stabilized by H_{u}. Then C is a non-degenerate conic in the projective plane dual of the star of u in Γ_{a}^{∗}. Let π be its nucleus. The group H_{u} has three orbits on the set of lines of Γ_{a}^{∗} on u, of size 5, 6 and 10 and the plane π gives us the orbit of size 5. Comparing the information we have given on stabilizers of planes of Γ, the 5 lines of Ξ_{a} on u are the lines of π on u, say them L_{1}, L_{2}, L_{3}, L_{4}, L_{5}. Let v ∈ L_{1} \{u}. If the five lines of Ξ_{a} on v still belong to π, then, by the transitivity of H on the set of point-line pairs of Ξ_{a} and by the connectivity of Ξ_{a}, all lines of Ξ_{a} belong to π. This is impossible (there are 40 lines in Ξ_{a}, but only 20 of them can stay in π). Therefore, the plane of Γ_{a}^{∗} containing the five lines on v is different from π. Thus we get at least 16 + 3 · 12 points for Ξ_{a}, namely 16 points in π, and 12 more points in the plane of Γ_{a}^{∗} containing the five lines of Ξ_{a} on v, for each of the three points v ∈ L_{1} \ {u}. However, 16 + 3 · 12 > 32; contradiction. Therefore, H_{u} stabilizes a non-degenerate quadratic cone of Γ_{a}^{∗} with vertex u. □

Let C_{0} be the cone of Γ_{a}^{∗} formed by the five lines of u belonging to Ξ_{a} and let L_{0} be the nuclear line of C_{0} in Γ_{a}^{∗}. Let u_{0}, u_{1}, u_{2}, u_{3} be the four points of L_{0}, with u_{0} = u, and for i = 1, 2, 3, let C_{i} be the image of C_{0} by the translation of Γ_{a}^{∗} sending u_{0} to u_{i} and let \overline{C}_{i} be the set of points of Γ_{a}^{∗} belonging to lines of C_{i} but different from u_{i}. The following is straightforward:

Lemma 3.28. The sets \overline{C}_{0}, \overline{C}_{1}, \overline{C}_{2}, \overline{C}_{3}, \{u_{0}\}, \{u_{1}\}, \{u_{2}\}, \{u_{3}\} are the orbits of H_{u} on the point-set of the affine space Γ_{a}^{∗}.

Clearly, the set of points of Γ_{a}^{∗} belonging to Ξ_{a} is a union of orbits of H on Ξ_{a}. Hence, it is a union of orbits of H_{u}. As Ξ_{a} has 32 points, it is the union of C_{0} with one of the cones C_{1}, C_{2} or C_{3}. Needless to say, we may assume to have chosen indices in such a way that \overline{C}_{0} ∪ \overline{C}_{1} ∪ \{u_{0}, u_{1}\} is the point-set of Ξ_{a}.

Lemma 3.29. Every line of Γ_{a}^{∗} not contained in Ξ_{a} meets Ξ_{a} in either 0 or 2 points.

Proof. Let L be a line of Γ_{a}^{∗} not contained in Ξ_{a} but meeting Ξ_{a} in a point u. By Lemma 3.28, Ξ_{a} is the union of two parallel cones C_{0} and C_{1} with vertices u_{0} = u and u_{1}, respectively, the latter lying on a line L_{0}, which is the nuclear line of both C_{0} and C_{1}. As L is not contained in Ξ_{a}, it is not a line of C_{0}. If L = L_{0}, then L ∩ Ξ_{a} = {u_{0}, u_{1}}. In this case we are done.

Otherwise, let π be the plane of Γ_{a}^{∗} spanned by L and L_{0}. As L_{0} is the nuclear line of both C_{0} and C_{1}, the plane π meets C_{i} (for i = 0, 1) in a line M_{i} through u_{i}, and the lines M_{0}, M_{1} are parallel, the translation sending u_{0} to u_{1} also maps C_{0} onto C_{1}. The points of L in Ξ_{a} are the meet-points of L with M_{0} and M_{1}. □

3.6. The case of (n, q) = (3, 4). The diameter of Γ

With Γ as in the previous subsection, we have the following:

Proposition 3.30. The geometry Γ has diameter δ ≤ 4.

Proof. Let b, b′ be points of Γ at distance 5, if possible, and let b ∼ c ∼ a ∼ a′ ∼ c′ ∼ b′ be a path from b to b′. Let u be a plane of Γ on the line aa′. As b, b′ have distance 5, the plane u contains no points of either a^{∼} ∩ b^{∼} or a^{′∼} ∩ b^{′∼}.

Let Ξ be an (a, b)-local geometry. Assume that b ∈ P (a). By Lemma 3.24, 16 out of the 21 lines of Γ_{a}^{∗} through u are incident to a plane of Ξ. Hence u contains 16 points of Γ_{a}^{∗} different from a and at distance 2 from b.

On the other hand, let b ∈ S(a). By Lemma 3.29, every line of Γ_{a}^{∗} through u meets Ξ_{a} in either 0 or 2 points. Furthermore, Ξ_{a} has 32 points. Hence u contains precisely 16 points of Γ at distance 2 from b and different from a.

Thus, in any case, u contains 17 points at distance 2 from b, including a among them. Similarly, u contains 17 points at distance 2 from b′. However, 17 + 17 > 22, which is the total number of points of u. Hence u contains some points at distance 2 from both b and b′, contrary to the hypothesis that b and b′ have distance 5. □

3.7. The case of (n, q) = (3, 4). Quotients

Let Γ and G be as in Subsection 3.4 and 3.5, but now we also assume that Γ admits a flag-transitive proper quotient \overline{Γ}. Let f : Γ → \overline{Γ} be a covering from Γ to \overline{Γ}.

Lemma 3.31. The geometry Γ has diameter δ = 4 and any two distinct points in the same fiber of f have mutual distance 4.

Proof. We have δ ≤ 4 by Proposition 3.30. On the other hand, both properties (LL) and (T) also hold in \overline{Γ}, by Propositions 3.8 and 3.11. If f(a) = f(a′) for distinct points a, a′ of Γ at distance ≤ 3, then either (T) or (LL) would fail in \overline{Γ}. The conclusion follows. □

Lemma 3.32. Let a, b be distinct points of Γ at distance 2. Then f(a), f(b) have distance 2 in \overline{Γ} and we have f(b) ∈ S(f(a)) (or f(b) ∈ P (f(a))) if and only if b ∈ S(a) (resp. b ∈ P (a)).

Proof. The points f(a), f(b) are distinct by Lemma 3.31. Clearly, they have distance 2. Furthermore, f isomorphically maps every (a, b)-local geometry onto an (f(a), f(b))-local geometry. The conclusion is obvious. □

Lemma 3.33. Let a, a′ be distinct points of Γ in the same fiber of f. Then there is a point b at distance 2 from both a and a′ and such that b ∈ S(a) ∩ S(a′).

Proof. The points a, a′ have distance 4 by Lemma 3.31. Let b be a point at distance 2 from both a and a′. By Lemma 3.32, a ∈ S(b) if and only if f(a) ∈ S(f(b)). Similarly, a′ ∈ S(b) if and only if f(a′) ∈ S(f(b)). Hence a ∈ S(b) if and only if a′ ∈ S(b).

If b ∈ S(a), then we are done. Otherwise, let b ∈ P (a). Hence b ∈ P (a′), by the above. Let c ∈ a^{∼} ∩ b^{∼} and c′ ∈ a^{′∼} ∩ b^{∼}. Let L = bb′ be a line of Γ_{b}^{∗} meeting each of the lines bc and bc′ in the affine space Γ_{b}^{∗}. That is, bb′ is coplanar with each of bc and bc′ in Γ. As the points b, b′, c belong to the same plane of Γ, the lines cb and cb′ meet in a point of the affine space Γ_{c}^{∗}. Hence cb′ is not parallel to ca, as ca is parallel to cb. Therefore, b′ ∈ S(a). Similarly, b′ ∈ S(a′). □

Lemma 3.34. Let a, b be points of Γ at distance 2 with b ∈ S(a). Then we have |a^{∼} ∩ b^{∼}| = 40 and |f(a)^{∼} ∩ f(b)^{∼}| = 80.

Proof. Let a, a′ be distinct points of Γ in the same fiber of f. Then a, a′ have distance 4 by Lemma 3.31 and, by Lemma 3.33, there is a point b at distance 2 from both a and a′ and such that b ∈ S(a) ∩ S(a′). By Lemma 3.26, the connected components of a^{∼} ∩ b^{∼} have size 40. More precisely, either a^{∼} ∩ b^{∼} has size 40 and is connected, or it has two connected components of size 40.

On the other hand, as every point of a^{∼} ∩ b^{∼} has distance 2 from every point of a^{′∼} ∩ b^{∼}, the restriction of f to (a^{∼} ∩ b^{∼}) ∪ (a^{′∼} ∩ b^{∼}) is injective. Furthermore, each of the sets a^{∼} ∩ b^{∼} and a^{′∼} ∩ b^{∼} is mapped by f onto a connected component of f(a)^{∼} ∩ f(b)^{∼} (= f(a′)^{∼} ∩ f(b)^{∼}, as f(a) = f(a′)). Therefore, f(a)^{∼} ∩ f(b)^{∼} has two connected components, which are the images by f of a^{∼} ∩ b^{∼} and a^{′∼} ∩ b^{∼}, and we have |a^{∼} ∩ b^{∼}| = |a^{′∼} ∩ b^{∼}| = 40. The conclusion follows from the transitivity of G on the set of ordered pairs (a, b) of points at distance 2 with b ∈ S(a) (Lemma 3.17) and from the analogous property of Aut(\overline{Γ}). □

Proposition 3.35. The geometry Γ is simply connected.

Proof. Let Γ̃ be a flag-transitive proper cover of Γ. By replacing the pair (Γ, \overline{Γ}) with (Γ̃, Γ) in Lemmas 3.32 and 3.34, we obtain that |a^{∼} ∩ b^{∼}| = 80 for any two points a, b of Γ at distance 2 with b ∈ S(a), whereas |a^{∼} ∩ b^{∼}| = 40 by the statement of Lemma 3.34 for the pair (Γ, \overline{Γ}); contradiction. Therefore, Γ does not admit any flag-transitive proper cover. However, the universal cover of a flag-transitive geometry is flag-transitive [26, Theorem 12.13]. Hence Γ is its own universal cover. □

Proposition 3.36. The geometry \overline{Γ} has at least 1408 points. It has precisely that number of points if and only if its collinearity graph is strongly regular with parameters (k, λ, μ) = (336, 80, 80).

Proof. The number of points of \overline{Γ} is minimal when \overline{Γ} has diameter 2 and, for any two points ā, b̄ of Γ at distance 2, the number |ā^{∼} ∩ b̄^{∼}| is maximal, namely equal to 80. If that is the case, then the collinearity graph of \overline{Γ} is strongly regular with parameters (336, 80, 80). A strongly regular graph with these parameters has 1408 vertices. □

3.8. End of the proof of Theorem 1

The geometry Γ(S) of Section 2 admits a flag-transitive proper quotient, namely Γ(S)/2. Hence Γ(S) is simply connected, by Proposition 3.35. According to Theorem 3.6, Γ(S) is the unique simply connected flag-transitive c · AG^{∗}-geometry of order 4 and residual dimension 3.

In order to finish the proof of Theorem 1, we only need to show that Γ(S)/2 is the unique flag-transitive proper quotient of Γ(S). Let Γ be such a quotient. As Γ(S) is simply connected, we have Γ = Γ(S)/X for some subgroup X of Aut(Γ(S)) acting semi-regularly on the set of elements of Γ(S) and normalized by a flag-transitive subgroup G of Aut(Γ(S)) ([26, Theorem 12.13]). As X is semi-regular on the set of points of Γ(S), the order of X divides 2816, which is the number of points of Γ(S), and 2816/|X| is the number of points of Γ. However, Γ has at least 1408 points, by Proposition 3.36, and 1408 is a half of 2816. Hence |X| = 2 and X, being normal in G, is also central in G. The commutator subgroup of Aut(Γ(S)) is the minimal flag-transitive automorphism group of Γ(S) (compare Theorem 3.5(i)). Hence G contains the commutator subgroup of Aut(Γ(S)). This forces X to be the center of Aut(Γ(S)). That is, Γ = Γ(S)/2.

4. Two 1-dimensional examples

4.1. Constructing c · AG^{∗}-geometries as gluings

Given a prime power q and an integer n > 1, let K be a complete graph on v = v(n, q) vertices, with

(i)  v(n, q) = 1 + \frac{q^{n} − 1}{q − 1}

(namely, v(n, q) − 1 is the number of points of P G(n − 1, q)) and let A be the n-dimensional affine space of order q. If v is even (which, in view of (i), means that either q is even or both q and n are odd), then K admits a 1-factorization. Hence, given a 1-factorization χ of K and a bijection α from the set χ^{∞} of classes of χ to the set A^{∞} of points at infinity of A, we can glue K with A via α (Buekenhout, Huybrechts and Pasini [7]). Let Gl_{α}(K, A) be that glued geometry. We recall that its elements are the points of A (which we call planes), the vertices of K (called points) and the pairs (e, L) with e and L an edge of K and a line of A such that α(χ(e)) = L^{∞}, where L^{∞} stands for the point at infinity of L. These pairs are called lines. A line (e, L) of Gl_{α}(K, A) is incident with precisely the vertices of e and the points of L, whereas all vertices of K are declared to be incident to all points of A (namely, Gl_{α}(K, A) is flat).

Gl_{α}(K, A) is c · AG^{∗}-geometry of order q and residual dimension n, but it need not be flag-transitive. According to Theorem 1, if Gl_{α}(K, A) is flag-transitive, then v = 2^{m} for some positive integer m and χ is the natural parallelism of AG(m, 2) (compare Cameron and Korchmaros [9]). Consequently, and by (i), the prime power q and the integers n, m satisfy the following equation

(ii)  2^{m} − 1 = \frac{q^{n} − 1}{q − 1}

(which is a special case of equation (E1) of Subsection 1.3, with q′ = 2^{m}). For any solution (q, n, m) of (ii), a flag-transitive gluing exists. Indeed, given Singer cycles θ and ζ in L_{m}(2) and P GL_{n}(q) respectively and chosen a class c^{∞} ∈ χ^{∞} and a point p^{∞} ∈ A^{∞}, let α be the bijection from χ^{∞} to A^{∞} sending c ∈ χ^{∞} to ζ^{k}(p^{∞}), with k the unique positive integer less than v and such that θ^{k}(c^{∞}) = c. Then, according to Buekenhout, Huybrechts and Pasini [7], the geometry Gl_{α}(K, A) admits a flag-transitive automorphism group G as follows:

(iii)  G = (2^{m} × p^{dm}Z).Z_{s}

where p is prime, d is a positive integer such that q = p^{d}, s = (q^{n} − 1)/(q − 1) (which is the period of θ and ζ) and Z = ⟨ζ̂^{s}⟩, where ζ̂ stands for a representative of ζ in GL_{n}(q). (We warn that G need not be the full automorphism group of Gl_{α}(K, A), even if the latter is never much larger than G.)

Clearly, once θ and ζ are given, the choice of c and p has no influence on the isomorphism type of Gl_{α}(K, A). Accordingly, we write Gl(θ, ζ) for Gl_{α}(K, A).

4.2. The examples

As remarked in Subsection 1.3, the triple (q, n, m) = (5, 3, 5) is the only known solution of the equation (ii) where n, q > 2. Chosen Singer cycles θ and ζ in L_{5}(2) and SL_{3}(5) respectively, for any pair (r, k) with r, k = 1, 2, …, 30 we can consider the gluing Gl(θ^{r}, ζ^{k}). Clearly, Gl(θ^{rh}, ζ^{kh}) = Gl(θ^{r}, ζ^{k}) for all h = 1, 2, …, 30. Thus, we may assume that r = 1. Consequently, at most 30 geometries can be obtained by the above gluing construction.

Lemma 4.1. We have Gl(θ, ζ^{k}) ≅ Gl(θ, ζ^{h}) if and only if there is some r ∈ Z_{31} such that fθf^{−1} = θ^{r} and gζ^{k}g^{−1} = ζ^{hr} for some f ∈ L_{5}(2) normalizing ⟨θ⟩ and some g ∈ GL_{3}(5) normalizing ⟨ζ⟩.

Proof. We have Gl(θ, ζ^{k}) ≅ Gl(θ^{r}, ζ^{t}) if and only if θ^{r} = fθf^{−1} and ζ^{t} = gζ^{k}g^{−1} for suitable elements f ∈ L_{5}(2) and g ∈ GL_{3}(5) normalizing ⟨ζ⟩ and ⟨θ⟩, respectively (Buekenhout, Huybrechts and Pasini [7, §3.4]). On the other hand, Gl(θ, ζ^{k}) = Gl(θ^{r}, ζ^{kr}). The conclusion follows. □

The following are the orbits on ⟨θ⟩ \ {1} for the normalizer of ⟨θ⟩ in L_{5}(2). (In the rightmost column we record the minimal polynomial of the elements of each row, modulo a choice of one out of the six possibilities for the first row.)

TABLE 1 ; \begin{matrix}θ & θ^{2} & θ^{4} & θ^{8} & θ^{16} & x^{5} + x^{2} + 1 \\ θ^{3} & θ^{6} & θ^{24} & θ^{12} & θ^{17} & x^{5} + x^{4} + x^{3} + x^{2} + 1 \\ θ^{5} & θ^{10} & θ^{20} & θ^{9} & θ^{18} & x^{5} + x^{4} + x^{2} + x + 1 \\ θ^{7} & θ^{14} & θ^{28} & θ^{25} & θ^{19} & x^{5} + x^{3} + x^{2} + x + 1 \\ θ^{11} & θ^{21} & θ^{22} & θ^{13} & θ^{26} & x^{5} + x^{4} + x^{3} + x + 1 \\ θ^{15} & θ^{30} & θ^{27} & θ^{29} & θ^{23} & x^{5} + x^{3} + 1 \end{matrix}

These are the orbits on ⟨ζ⟩ \ {1} for the normalizer of ⟨ζ⟩ in GL_{3}(5).

TABLE 2 ; \begin{matrix}ζ & ζ^{5} & ζ^{25} & x^{3} + x − 1 \\ ζ^{2} & ζ^{10} & ζ^{19} & x^{3} + 2x^{2} + x − 1 \\ ζ^{3} & ζ^{15} & ζ^{13} & x^{3} + 2x^{2} − x − 1 \\ ζ^{4} & ζ^{20} & ζ^{7} & x^{3} − 2x^{2} − 1 \\ ζ^{6} & ζ^{30} & ζ^{26} & x^{3} − x^{2} − 1 \\ ζ^{8} & ζ^{9} & ζ^{14} & x^{3} + x^{2} + x − 1 \\ ζ^{11} & ζ^{24} & ζ^{27} & x^{3} + 2x − 1 \\ ζ^{12} & ζ^{29} & ζ^{21} & x^{3} − x^{2} − 2x − 1 \\ ζ^{16} & ζ^{18} & ζ^{28} & x^{3} + x^{2} − 2x − 1 \\ ζ^{17} & ζ^{23} & ζ^{22} & x^{3} − x^{2} − x − 1 \end{matrix}

By comparing Tables 1 and 2 and by Lemma 4.1 we easily obtain the following:

Proposition 4.2. Up to isomorphism, Gl(θ, ζ) and Gl(θ, ζ^{3}) are the unique flag-transitive gluings of the complete graph on 32 vertices with the affine space AG(3, 5).

Writing Γ for Gl(θ, ζ) or Gl(θ, ζ^{3}), let G be the (flag-transitive) subgroup of Aut(Γ) of this form: G = (2^{5} × 5^{3}).Z_{31} (as Subsection 4.1, (iii)). Let us consider the following set of relations

\begin{matrix}(1) & a^{2} = 1 \\ (2) & b^{5} = 1 \\ (3) & c^{31} = 1 \\ (4) & ab = ba \\ (5) & [a, a^{c^{i}}] = 1 (for i = 1, 2, 3, 4) \\ (6) & [b, b^{c}] = [b, b^{c^{2}}] = 1 \\ (7) & a^{c^{5}+c^{2}+1} = 1 \\ (8) & b^{p(c)} = 1 \end{matrix}

with p(c) = c^{3} + c − 1 or c^{3} + 2c^{2} − c − 1 in (8), according to whether Γ is Gl(θ, ζ) or Gl(θ, ζ^{3}). The relations (1), (3), (5) and (7) give a presentation of G_{2}, the relations (2), (3),

(6) and (8) form a presentation of G_{0} and the relations (1)–(4) present G_{1}. By a computer aided calculation (we have done it by GAP [27]) one can see that the group presented by the relations (1)–(8) has size 2^{5} · 5^{3} · 31. Hence, that group is G. Therefore [26, Corollary 12.30]:

Proposition 4.3. Both Gl(θ, ζ) and Gl(θ, ζ^{3}) are simply connected.

5. Proof of Theorem 2

Before to turn to the proof of Theorem 2, we make some remarks on d-dimensional dual hyperovals and c · AG^{∗}-geometries, to be exploited for that proof.

Dimensional dual hyperovals and c · AG^{∗}-geometries. Let S be a d-dimensional dual hyperoval of P G(m, q) and S_{0} be the set of points of P G(m, q) belonging to some member of S. Clearly, (S, S_{0}), with the incidence relation inherited from P G(m, q), is a circular space where every ‘point’ (namely, every member of S) is incident to precisely q^{d} +q^{d−1} +⋯+q +1 ‘lines’ (points of S_{0}). Therefore,

|S| = q^{d} + q^{d−1} + ⋯ + q + 2.

Let Af(S) be the affine expansion of the dual circular space (S_{0}, S) to AG(m+1, q) (Buekenhout, Huybrechts and Pasini [7, Section 4]). Clearly, the dual of Af(S) is a c · AG^{∗}-geometry of order q and residual dimension n = d + 1.

Let d = 1. Then, by definition of S, the span of any two members of S contains all remaining members of S. Therefore m = 2 and S is just a dual hyperoval of a projective plane (whence, q is even). Moreover, the dual of Af(S) is a Laguerre structure.

Let d = 2. The family of 22 planes of P G(5, 4) considered in Subsection 2.1 is a 2dimensional dual hyperoval in P G(5, 4). If S is this family, then Af(S) = Γ^{∗}(S)/2 (see Subsection 2.5).

If q = 2, then Af(S) is a semi-biplane. A lot of semi-biplanes can be obtained in this way since, as said in Subsection 1.4, a lot of d-dimensional dual hyperovals exist in P G(2d, 2) for any d ≥ 1.

Proof of Theorem 2. Let S be a d-dimensional dual hyperoval of P G(m, q). The automorphism group of Af(S) contains the extension V:Z.G_{S} of the translation group V of AG(m + 1, q) by Z.G_{S}, with Z the center of GL_{m+1}(q) and G_{S} the stabilizer of S in P ΓL_{m+1}(q).

Let G_{S} be 2-transitive. Then Af(S) is flag-transitive and the conclusion of Theorem 2 follows from Theorem 1 and from the well-known fact that the only 2-transitive hyperovals are the (regular) hyperovals of P G(2, 2) and P G(2, 4). □

Acknowledgment. We are indebted to H. Cuypers for having pointed out a gap in the earliest draft of this paper. His remark forced us to push the investigation further, eventually discovering the geometries Γ(S) and Γ(S)/2. We also thank S. Shpectorov for some helpful informations and the referee for many suggestions and for having noticed a number of errors in the earliest submitted version of this paper.

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Received July 6, 1998