Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 38 (1997), No. 2, 357-375.

On some flag-transitive non-classical c.C_{2}-geometries II

Satoshi Yoshiara

Division of Mathematical Sciences Osaka Kyoiku University Kashiwara, Osaka 582 JAPAN

Abstract. The flag-transitive automorphism groups of the Tits generalized quadrangle T_{2}^{∗}(O_{4}) are classified up to conjugacy in Aut(T_{2}^{∗}(O_{4})). Basing on the classification, it is shown that there is no flag-transitive circular extension of T_{2}^{∗}(O_{4}) or of its dual other than those constructed in [6]. This result completes the classification of flag-transitive extensions of T_{2}^{∗}(O_{4}) and its dual.

1. Introduction

In this paper, the complete classification is given for the flag-transitive circular extensions of the Tits generalized quadrangle T_{2}^{∗}(O_{4}) as well as its dual for a hyperoval O_{4} in P G(2, 4). See corollary 4.5 for the precise statement. Such geometries are classified in [6] under two assumptions; a restriction on the structure of automorphism groups and the so called (LL) property. The latter turns out to be superfluous. In this paper, we remove the former assumption by giving a complete classification of flag-transitive automorphism groups of T_{2}^{∗}(O_{4}). It will be shown that no new example arises.

Given a generalized quadrangle Q, (circular) extension of Q is a residually connected geometry of rank 3 belonging to the following diagram with point-residues isomorphic to Q:

(c.C_{2})  ^{•}^{points}^{1}  ^{•}^{lines}  ^{•}^{planes} c

Extensions of generalized quadrangles are interesting objects, especially when they admit flag-transitive automorphism groups, because some of them arise from sporadic groups. See

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1997 Heldermann Verlag

358 S. Yoshiara: On some flag-transitive non-classical c.C_{2}-geometries II

[2, 1.1] for the known examples. Note that two infinite families of extensions of T_{2}^{∗}(O) and its dual for regular hyperovals O in P G(2, 2^{e}) for any e ≥ 1 have recently been obtained in [2] and [7]. They are flag-transitive only when e = 1, 2.

The structure of a generalized quadrangle Q is rather restricted, if an extension of Q exists. For example, if Q admits a flag-transitive extension, Q is also flag-transitive. There are only four known examples of such quadrangles besides the classical generalized quadrangles: the Tits quadrangle T_{2}^{∗}(O) and its dual for certain hyperoval O in P G(2, q) (q = 4, 16). For each of these quadrangles, their flag-transitive extensions were investigated: Those with Q classical are classified ([1],[5]). Those with Q = T_{2}^{∗}(O_{4}) or its dual for a hyperoval O_{4} in P G(2, 4) are classified under some assumptions ([6]), which will be removed in this paper. When O is a hyperoval in P G(2, 16), it is easy to see there is no flag-transitive extension of the dual of T_{2}^{∗}(O), whereas flag-transitive extensions of T_{2}^{∗}(O) have been studied by Nakagawa. He obtained the possible amalgams of stabilizers [4], though their completions are not yet determined.

It should also be mentioned that under geometric assumptions of (LL) and the triangular properties, Makhnev obtained the distribution diagram of the collinearity graph of an extension of T_{2}^{∗}(O_{4}) (but not that of the dual of T_{2}^{∗}(O_{4})) [3] without assuming any conditions on automorphisms.

2. The quadrangle T_{2}^{∗}(O_{4}) and its automorphisms

In this section, we review some properties of the Tits quadrangle T_{2}^{∗}(O_{4}).

For any prime power q and a non-negative integer n, P G(n, q) denotes a desarguesian n-dimensional projective space associated with the row vector space P G(q)^{n+1}. Each point of P G(n, q) is a 1-dimensional subspace of P G(q)^{n+1} generated by a vector (x_{0}, …, x_{n}), which we denote by [x_{0}, …, x_{n}] throughout the paper.

Recall that a hyperoval O in a projective plane P G(2, q) (q = 2^{e}) is a set of q+2 projective points such that there is no line through three distinct points of O. For example, the set O_{4} consisting of the following six points P_{i} (i = 1, …, 6) is a hyperoval in P G(2, 4), where ω is a generator of the multiplicative group of GF (4)^{#} and ω̄ = ω^{2}:

P_{1} := [1, 0, 0], P_{2} := [0, 1, 0], P_{3} := [0, 0, 1], P_{4} := [1, 1, 1], P_{5} := [1, ω, ω̄], P_{6} := [1, ω̄, ω].

The set O_{4} consists of the zeros [x_{1}, x_{2}, x_{3}] of a quadratic form X_{2}^{2} + X_{1}X_{3} together with its nucleus [1, 0, 0], the unique point lying on all its tangents. It is known that any hyperoval O in P G(2, 4) can be transformed to O_{4} by an element of Aut(P G(2, 4)). Throughout this paper we use the symbol O_{4} to denote the above hyperoval in P G(2, 4).

The group Aut(O_{4}) of automorphisms of P G(2, 4) preserving O_{4} is a permutation group on the six points of O_{4}. We can verify that Aut(O_{4}) ≅ S_{6}, in which the field automorphism corresponds to the transposition (P_{5}, P_{6}) and the matrices of GL_{3}(4) preserving O_{4} induces A_{6} with the kernel of order 3 (scalar multiplication).

For any hyperoval O in P G(2, q) with q even, a generalized quadrangle T_{2}^{∗}(O), called the Tits quadrangle, can be constructed as follows. Take a hyperplane H of the projective space P G(3, q) given by

H := {[0, x_{1}, x_{2}, x_{3}]|x_{i} ∈ GF (q)(i = 1, 2, 3)}

and identify H with P G(2, q) via the map sending [0, x_{1}, x_{2}, x_{3}] to [x_{1}, x_{2}, x_{3}]. Define P to be the set of projective points of P G(3, q) not contained in H, and define L to be the set of projective lines l of P G(3, q) with the property that l intersects H at a point on the hyperoval O. The geometry (P, L; ∗) with incidence ∗ given by inclusion turns out to be a generalized quadrangle, which is denoted T_{2}^{∗}(O). The elements of P and L are called POINTs and LINEs of T_{2}^{∗}(O) respectively.

Now we consider T_{2}^{∗}(O_{4}) for the above hyperoval O_{4} in P G(2, 4). It follows from the construction that every automorphism of P G(3, 4) stabilizing O_{4} (and so H) and every translation by a vector in H (recognized as a subspace of GF (4)^{4}) are automorphisms of T_{2}^{∗}(O_{4}). We use the following conventions to denote these automorphisms:

σ : the field automorphism sending [x_{0}, x_{1}, x_{2}, x_{3}] to [x_{0}^{2}, x_{1}^{2}, x_{2}^{2}, x_{3}^{2}]. L(X) for each matrix X of GL_{3}(4) preserving O_{4}:

the automorphism sending [x_{0}, x_{1}, x_{2}, x_{3}] to [x_{0}, (x_{1}, x_{2}, x_{3})X]. t(v) for each vector v = (0, v_{1}, v_{2}, v_{3}) in H ≅ GF (4)^{3}:

the translation by v sending [x_{0}, x_{1}, x_{2}, x_{3}] to [x_{0}, x_{1} + v_{1}, x_{2} + v_{2}, x_{3} + v_{3}]. S : the subgroup of Aut(T_{2}^{∗}(O_{4})) generated by L(X) for all X ∈ GL(3, 4) preserving O_{4}

together with the filed automorphism σ. N : the subgroup of Aut(T_{2}^{∗}(O_{4})) generated by t(v) for all v ∈ H.

By the remark on Aut(O_{4}) above, the group S induces S_{6} on the six LINEs through a POINT α := [1, 0, 0, 0] with kernel ⟨L(ωI_{3})⟩ of order 3, because they correspond to the six points P_{i} (i = 1, …, 6) on O_{4} in H = P G(2, 4). We set

d := L(ωI_{3}), a generator of the kernel of S on the points on O_{4}.

The group N = {t(v)|v ∈ H} of translations is isomorphic to the additive group of H ≅ GF (4)^{3}, and hence N is an elementary abelian group of order 2^{6}. Clearly the group S acts on N.

The actions of elements of S on N can be easily calculated, since we have t(v)^{L(X)} = t(vX) and t(v)^{L(X)σ} = t(\overline{vX}) for v ∈ H = GF (4)^{3} and X ∈ GL(3, 4), where vX is the matrix product of a vector v with X and \overline{x} means the vector obtained by replacing each entry of v by its square. Throughout this paper, we write a vector (0, v_{1}, v_{2}, v_{3}) of H by (v_{1}, v_{2}, v_{3}) in short. Moreover in §3, for a subset W of H we slightly abuse the notation by using the same symbol W to denote the subset {t(w)|w ∈ W } of N ≅ 2^{6} for short. In particular, we write t(w) by w in short. Moreover, in some proofs of §3 we understand the extended matrix multiplication of S with H (written as wL(X)σ) to be the vector of H corresponding to the conjugation t(w)^{L(X)σ}.

Lemma 2.1. [6, 2.2] The automorphism group Aut(T_{2}^{∗}(O_{4})) is generated by N and S. Thus Aut(T_{2}^{∗}(O_{4})) = N : S ≅ 2^{6} : (3S_{6}). Moreover, Aut(T_{2}^{∗}(O_{4})) acts flag-transitively on T_{2}^{∗}(O_{4}) with the following stabilizers:

The stabilizer of a POINT α = [1, 0, 0, 0] is S, which induces S_{6} on the residue at α with kernel ⟨d⟩; The stabilizer of a LINE β = ⟨[1, 0, 0, 0], [0, 1, 0, 0]⟩ is a group isomorphic to (A_{4} × A_{5})2 generated by t(1, 0, 0), d, L(X) for X ∈ GL(3, 4) fixing O_{4} and the point P_{1} = [1, 0, 0], and σ, in which ⟨t(1, 0, 0), d, σ⟩ induces S_{4} on the four POINTs of β and the kernel on the residue of β is a group isomorphic to A_{5} generated by L(X) for X ∈ GL(3, 4) fixing O_{4} and P_{1} = [1, 0, 0].

2.1. Transitive subgroups of S

For later purposes, we need to know all subgroups of S acting transitively on the six LINEs through the POINT α = [1, 0, 0, 0]. Note that through the natural projection S to S/⟨d⟩ ≅ S_{6} an element L(X) of S (with X ∈ GL(3, 4) preserving O_{4}) corresponds to the element Xmod(ωI) of P GL(3, 4) preserving O_{4}. Thus the images of such subgroups of S under the natural projection correspond to the subgroups of Aut(O_{4}) ≅ S_{6} acting transitively on the 6 points P_{i} (i = 1, …, 6) on the hyperoval O_{4}. The latter subgroups of S_{6} can be easily classified up to conjugacy under S_{6}. The result is, for example, given in [6, Lemma 2.4, Figure 1].

Remark. Before determining the subgroups of S in question, let me correct several typographic errors found in [6]: First, a line joining F_{7} and F_{16} is missing in [6, Figure 1]. The permutation induced by f should be (2536)(14), not (2356)(14) (see the Table in [6, Lemma 2.4] and [6, p.64, line 7]). The entry ω̄ in the matrix q in [6, p.64] should be ω, and the upper F(n) should be F(m) in the relations collected in [6, p.64]. A sloppy argument found in [6, 3.6], which does not affect the conclusions of that paragraph, is also mentioned at the end of this paper. Finally, the geometric construction given in [6, 4.4] should be modified, though it is totally independent of the rest of [6] and of the present paper (See [7] for the detail).

To describe the result, we set several explicit automorphisms of T_{2}^{∗}(O_{4}): the same notation as in [6, 2.5] will be used. We give both u and d^{2}u explicitly, and give d^{2}l, instead of l.

u := L ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & ω \\ ω̄ & 0 & 0 \end{matrix} ) , d^{2}u := L ( \begin{matrix}0 & ω̄ & 0 \\ 0 & 0 & 1 \\ ω & 0 & 0 \end{matrix} ) , p := L ( \begin{matrix}1 & 0 & 0 \\ 1 & ω & ω̄ \\ 1 & ω̄ & ω \end{matrix} );

m := L ( \begin{matrix}ω & ω & ω \\ ω & ω̄ & 1 \\ ω & 1 & ω̄ \end{matrix} ) · σ, d^{2}l := L ( \begin{matrix}ω & ω & ω \\ ω & 1 & ω̄ \\ ω & ω̄ & 1 \end{matrix} ) · σ,

From now on throughout this paper, the letters u, p, m, l as well as d and σ denote the above automorphisms, unless otherwise stated. The other automorphims in [6, 2.5] can be obtained as follows: a := (duu^{σ})^{2}, b := a^{−1}du, n := md^{2}l, q := anb^{a} and f := d^{2}mσ. These symbols are used only in Lemma 2.2 and in the proof of Lemma 3.3.

Lemma 2.2. Assume that a subgroup G of S is transitive on the six LINEs of T_{2}^{∗}(O_{4}) through the POINT α = [1, 0, 0, 0]. Then G is conjugate under S to one of the following groups under the above notation:

i = 1  H_{1} := S ≅ 3S_{6}
i = 2  H_{2} := S′ ≅ 3A_{6}
i = 3 G_{3} := ⟨u, p, q, m⟩  H_{3} := ⟨d, u, p, q, m⟩ ≅ 3S_{5}
i = 4 G_{4} := ⟨u, p, q⟩  H_{4} := ⟨d, u, p, q⟩ ≅ (3 × A_{5})
i = 5 G_{5} := ⟨d^{2}u, p, m, d^{2}l⟩  H_{5} := ⟨d, u, p, m, l⟩ ≅ 3(S_{4} × 2)
i = 6 G_{6} := ⟨d^{2}u, p, n⟩  H_{6} := ⟨d, u, p, n⟩ ≅ 3S_{4}
i = 7 G_{7} := ⟨d^{2}u, p, d^{2}l⟩  H_{7} := ⟨d, u, p, l⟩ ≅ 3(A_{4} × 2)
i = 8 G_{8−1} := ⟨u, p, m⟩  G_{8−2} := ⟨d^{2}u, p, m⟩ H_{8} := ⟨d, u, p, m⟩ ≅ 3S_{4}
i = 9 G_{9−1} := ⟨u, p⟩ G_{9−2} := ⟨d^{2}u, p⟩ H_{9} := ⟨d, u, p⟩ ≅ 3A_{4}
i = 10  H_{10} := ⟨a, b, f, ana^{−1}σ⟩ ≅ 3_{+}^{1+2}D_{8}
i = 11  H_{11} := ⟨a, b, f⟩ ≅ 3_{+}^{1+2}4
i = 12  H_{12} := ⟨a, b, m, l⟩ ≅ 3_{+}^{1+2}2^{2}
i = 13  H_{13} := ⟨a, b, l⟩ ≅ 3_{+}^{1+2}2
i = 14 G_{14} := ⟨d^{2}u, m, d^{2}l⟩  H_{14} := ⟨d, u, m, l⟩ ≅ 3(S_{3} × 2)
i = 15 G_{15} := ⟨d^{2}u, m⟩  H_{15} := ⟨d, u, m⟩ ≅ 3S_{3}
i = 16 G_{16} := ⟨d^{2}u, d^{2}l⟩  H_{16} := ⟨d, u, l⟩ ≅ 3.6

Proof. As G⟨d⟩/⟨d⟩ is transitive on the six points on O_{4} (via the correspondence at the beginning of 2.1), it follows from [6, Lemma 2.4] that it is conjugate under S/⟨d⟩ to exactly one of the subgroups F_{i} = H_{i}/⟨d⟩ of S/⟨d⟩ ≅ S_{6} (i = 1, …, 16). (Note that every H_{i} above contains the kernel ⟨d⟩ on the residue at α.) Thus we may assume that G⟨d⟩ = H_{i}. Then we have either G = H_{i} or G ∩ ⟨d⟩ = 1. We may assume that the latter case holds. In particular, the extension H_{i}/⟨d⟩ splits.

If F_{i} contained a Sylow 3-subgroup of S_{6}, G with G⟨d⟩/⟨d⟩ = F_{i} would contain d, as a Sylow 3-subgroup of S is an extraspecial group 3_{+}^{1+2}, and hence it does not split over its center. This eliminates the possibilities for i = 1, 2, 10, 11, 12, 13.

If F_{i} does not contain a Sylow 3-subgroup of S_{6}, a Sylow 3-subgroup of G is a subgroup of 3^{2}, and hence the extension H_{i}/⟨d⟩ in fact splits. Explicit complements are given as G_{i} (or G_{i−1} and G_{i−2} for i = 8, 9) in the statement of the Lemma. We can verify this fact using the relations in [6, 2.5]. Since G is also a complement for H_{i}/⟨d⟩, it follows from Sylow’s theorem that we may assume G contains a Sylow 2-subgroup of G_{i} (or G_{i−1} for i = 8, 9).

If i = 3 or 4, H′′_{i} is the unique subgroup of H_{i} isomorphic to A_{5}, as H_{i}/⟨d⟩ splits. Since G is generated by H′′_{i} and a Sylow 2-subgroup of G_{i}, it coincides with G_{i}.

For i = 5, 7, 14, 16, G contains an involution d^{2}l in its center. Since ⟨d, u⟩ is a d^{2}l-invariant Sylow 3-subgroup of H_{i} and C(d^{2}l) ∩ ⟨d, u⟩ = ⟨d^{2}u⟩, G is generated by a Sylow 2-subgroup of G_{i} and d^{2}u, and hence G = G_{i}.

In the remaining cases, G is generated by a Sylow 2-subgroup of G_{i} (or G_{i−1} for i = 8, 9) and an element of order 3 of shape d^{s}u for some s = 0, 1, 2. For i = 15, G is generated by ⟨m⟩ and d^{s}u. Since an element g := d^{−1}ab^{−1} of order 3 satisfies u^{g} = du, (du)^{g} = d^{2}u and [m, g] = 1, G is conjugate to G_{15} under ⟨g⟩. For i = 6, G contains (d^{s}u)(d^{s}u)^{n} = d^{2s}uu^{n} = d^{−s+2}, and hence s = 2 and G = G_{6}.

Since u^{d^{2}l} = du, [p, d^{2}l] = [m, d^{2}l] = [n, d^{2}l] = 1, we may take G = G_{i−1} or G_{i−2} in each of the remaining cases i = 8, 9. Since the centralizer in S/⟨d⟩ ≅ S_{6} of the image of G_{i−j} (j = 1, 2) is a group of order 2 generated by the image of d^{2}l, C_{S}(G_{i−j}) = 1 or ⟨d^{2}l⟩. Thus C_{S}(G_{i−1}) = 1 and C_{S}(G_{i−2}) = ⟨d^{2}l⟩ for each i = 8, 9. In particular, G_{i−1} is not conjugate to G_{i−2} under S. □

3. Flag-transitive automorphism groups on T_{2}^{∗}(O_{4})

In this section, we will classify all flag-transitive subgroups of Aut(T_{2}^{∗}(O_{4})) up to conjugacy. We use the notation in §2, and ρ denotes the homomorphism of the semidirect product Aut(T_{2}^{∗}(O_{4})) = N : S onto S sending an element ns (n ∈ N, s ∈ S) to s. Let F be a subgroup of Aut(T_{2}^{∗}(O_{4})) = N : S acting flag-transitively on T_{2}^{∗}(O_{4}), and let ρ(F) be the image of F by ρ. In other words,

ρ(F) := {s ∈ S|ns ∈ F for some n ∈ N}.

The group ρ(F) is a subgroup of S containing F ∩ S and normalizing F ∩ N, as N is abelian. Moreover, ρ induces an isomorphism of F/(F ∩ N) with ρ(F).

Since F is flag-transitive on the generalized quadrangle T_{2}^{∗}(O_{4}) of order (3, 5), the stabilizer F ∩ S in F of a POINT α := [1, 0, 0, 0] is of index 2^{6} = |N| in F and it is transitive on the six LINEs through α. Then [ρ(F) : S ∩ F] = |F |/|F ∩ N||S ∩ F | = |N|/|F ∩ N|.

Summarizing, we observed the following, which is a starting point of the classification.

Lemma 3.1. Let F be a flag-transitive automorphism group on T_{2}^{∗}(O_{4}). Then ρ(F) is a subgroup of S containing S ∩ F of index [N : F ∩ N] and it normalizes F ∩ N. Moreover, S ∩ F is transitive on the six LINEs through α.

Note that ρ(F)⟨d⟩/⟨d⟩ is also a subgroup of S/⟨d⟩ ≅ S_{6} transitive on the six points of O_{4} corresponding to the six LINEs through α (see the remark in 2.1). Thus we may assume that under the conjugacy by S the group ρ(F) is one of the groups listed in Lemma 2.2. It has a subgroup S ∩ F with index [N : N ∩ F], which is also a conjugate of one of the groups given in Lemma 2.2.

If [N : N ∩ F] = 1, or equivalently F contains N, we have ρ(F) = S ∩ F, and the transitivity of S ∩ F on the six LINEs through α is equivalent for F to be flag-transitive on T_{2}^{∗}(O_{4}). Thus, as we saw in [6],

Proposition 3.2. For F a subgroup of Aut(T_{2}^{∗}(O_{4})) containing N, F is flag-transitive group on T_{2}^{∗}(O_{4}) if and only if F is conjugate under S to NG_{i}, NG_{i−j} or NH_{i} (i = 1, …, 16; j = 1, 2), the semi-direct products of N with the subgroups of S in Lemma 2.2.

Hence from now on we may assume that a flag-transitive group F does not contain N. Examining the inclusion relations among subgroups in Lemma 2.2 up to conjugacy (see also [6, Figure 1]), we conclude that the all possible 2-power indices between these groups are 2, 2^{2} and 2^{3}. Moreover, we can list all the pairs of such subgroups with given 2-power index.

We also observe the following facts: The group ρ(F) contains d iff S ∩F contains d, since [ρ(F) : F ∩ S] = [N : N ∩ F] is a power of 2. If ρ(F) contains d, the dimension of N ∩ F over GF (2) is even, since d acts fixed point freely on N. A subgroup of G_{5} conjugate to G_{9−j} (j = 1, 2) should be conjugate to G_{9−2}, as it is centralized by an involution d^{2}l. By the same reason, a subgroup of G_{8−i} conjugate to G_{9−j} for some j = 1, 2 is conjugate to G_{9−i}.

The above considerations yield the following conclusion:

Lemma 3.3. If a flag-transitive subgroup F of Aut(T_{2}^{∗}(O_{4})) does not contain F, the triple (F ∩ N, ρ(F), S ∩ F) of subgroups of N and S is one of the following table, up to conjugacy in Aut(T_{2}^{∗}(O_{4})) = NS. Here, the column for S ∩ F is determined up to conjugacy: that is, if (ρ(F), S ∩ F) = (X_{i}, X_{j}^{g}) for some g ∈ S with X_{j}^{g} ≤ X_{i}, we just put X_{i} and X_{j} in the columns for ρ(F) and S ∩ F respectively.

|F ∩ N| ρ(F)  S ∩ F
2^{5}  G_{3} (≅ S_{5})  G_{4}  (≅ A_{5})
G_{5} (≅ S_{4} × 2)  G_{6}  (≅ S_{4})
G_{5}  G_{7}  (≅ A_{4} × 2)
G_{5}  G_{8}  (≅ S_{4})
G_{6} (≅ S_{4})  G_{9−i} i = 1, 2 (≅ A_{4})
G_{7} (≅ A_{4} × 2)  G_{9−i} i = 1, 2
G_{8−i} i = 1, 2 (≅ S_{4}) G_{9−i} i = 1, 2
G_{14} (≅ S_{3} × 2)  G_{15}  (≅ S_{3})
G_{14}  G_{16}  (≅ Z_{6})
2^{4}  G_{5} (≅ S_{4} × 2)  G_{9−2} (≅ A_{4})
H_{5} (≅ 3.(S_{4} × 2)) H_{9}  (≅ 3.A_{4})
G_{7} (≅ A_{4} × 2) G_{16} (≅ Z_{6})
H_{7} (≅ 3(A_{4} × 2)) H_{16}  (≅ 3.Z_{6})
G_{8−i} i = 1, 2 (≅ S_{4}) G_{15}  (≅ S_{3})
H_{8} (≅ 3.S_{4})  H_{15}  (≅ 3.S_{3})
2^{3}  G_{5} (≅ S_{4} × 2)  G_{16}  (≅ Z_{6})

To each candidate for ρ(F) (that is, G_{i} (or G_{i−j}) of H_{i} in the table of Lemma 3.3), we now examine the subspaces of N invariant under ρ(F) in order to determine N ∩ F. The actions of elements of S on N can be easily computed as we observed in the paragraphs previous to Lemma 2.1. Note also that for a subset W of H we also denote the subset {t(w)|w ∈ W } of N by the same symbol W for short. In particular, t(w) is also denoted by w in short, if there is no risk of confusion. Furthermore, a vector (0, v_{1}, v_{2}, v_{3}) of H is denoted simply by (v_{1}, v_{2}, v_{3}).

We first treat the subgroup ⟨d^{2}u, p⟩ of S, because it is contained in many candidates for ρ(F) in Lemma 3.3. The ⟨d^{2}u, p⟩-orbits on N^{#} can be easily determined as follows. (Note that the commutativity of ⟨d^{2}u, p⟩ with d implies that the multiples by ω and ω̄ of each ⟨d^{2}u, p⟩-orbit are also such orbits.)

Lemma 3.4. The group ⟨d^{2}u, p⟩ ≅ A_{4} has the following 11 orbits O_{i} (i = 1, …, 11) on N^{#} of the indicated lengths: In particular, there are exactly 3 (resp. 6 and 2) orbits of length 1 (resp. 6 and 12).

O_{1} 1  {(1, ω̄, ω̄)}
O_{2} = ωO_{1} 1  {(ω, 1, 1)}
O_{3} = ω̄O_{1} 1  {(ω̄, ω, ω)}
O_{4} 6 {(1, 1, ω̄), (1, ω̄, 1), (ω, ω̄, ω̄), (0, ω, ω), (ω̄, 0, ω), (ω̄, ω, 0)}
O_{5} = ωO_{4} 6 {(ω, ω, 1), (ω, 1, ω), (ω̄, 1, 1), (0, ω̄, ω̄), (1, 0, ω̄), (1, ω̄, 0)}
O_{6} = ω̄O_{4} 6 {(ω̄, ω̄, ω), (ω̄, ω, ω̄), (1, ω, ω), (0, 1, 1), (ω, 0, 1), (ω, 1, 0)}

O_{7} 6 {(1, 1, 1), (ω, ω̄, 1), (ω, 1, ω̄), (0, 0, ω), (ω̄, 0, 0), (0, ω, 0)}
O_{8} = ωO_{7} 6 {(ω, ω, ω), (ω̄, 1, ω), (ω̄, ω, 1), (0, 0, ω̄), (1, 0, 0), (0, ω̄, 0)}
O_{9} = ω̄O_{7} 6 {(ω̄, ω̄, ω̄), (1, ω, ω̄), (1, ω̄, ω), (0, 0, 1), (ω, 0, 0), (0, 1, 0)}
O_{10} 12 {(1, ω, 1), (ω, ω̄, ω), (ω̄, 1, ω̄), (0, 1, ω), (ω̄, 0, 1), (ω, ω, 0),
(0, ω, ω̄), (1, 0, ω), (ω̄, ω̄, 0), (0, ω̄, 1), (ω, 0, ω̄), (1, 1, 0)}
O_{11} 12 {(1, 1, ω), (ω, ω, ω̄), (ω̄, ω̄, 1), (0, 1, ω̄), (1, 0, 1), (ω, ω̄, 0),
(0, ω, 1), (ω, 0, ω), (ω̄, 1, 0), (0, ω̄, ω), (ω̄, 0, ω̄), (1, ω, 0)}

The above lemma eliminates many candidates for ρ(F) when |N ∩ F | = 2^{5}.

Lemma 3.5. There is no subgroup of N of order 2^{5} invariant under ⟨d^{2}u, p⟩. In particular, if N ∩ F ≅ 2^{5}, then ρ(F) is conjugate under S to G_{8−1} or G_{14}.

Proof. Let M be a subgroup of N of order 2^{5} invariant under ⟨d^{2}u, p⟩. Since M^{#} is a disjoint union of some ⟨d^{2}u, p⟩-orbits O_{i} (i = 1, …, 11) in Lemma 3.4, there are integers a, b, c satisfying

a + 6b + 12c = 31, 0 ≤ a ≤ 3, 0 ≤ b ≤ 6, 0 ≤ c ≤ 2.

In particular, a is an odd integer prime to 3. Thus a = 1 and 5 = b + 2c.

We can easily verify that O_{1} ∪ O_{2} ∪ O_{3} ⊆ ⟨O_{j}⟩ for j = 7, 10, 11 (see also Lemma 3.6(1)). This implies that M ∩ O_{j} ≠ ∅ for every j = 7, …, 11, for otherwise we would have a = 3. Thus c = 0 and b ≤ 3, but this contradicts the equation 5 = b + 2c above. Hence there is no such subgroup M of N.

The candidates ρ(F) with |N ∩ F | = 2^{5} in the table in Lemma 3.3 contain ⟨d^{2}u, p⟩ up to conjugacy in S, except G_{8−1} and G_{14}. (Note that G_{3} ≅ S_{5} contains a subgroup conjugate to G_{6} ≅ S_{4}.) Hence the latter claim follows from the non-existence of a ⟨d^{2}u, p⟩-invariant subgroup of order 2^{5} of N. □

By the lemma above and Lemma 3.3, ρ(F) is conjugate to one of the following subgroups of S: G_{14}, G_{8−i} (i = 1, 2), G_{5}, G_{7}, H_{5}, H_{7} and H_{8}. In order to examine the subgroups of N invariant under one of these subgroups of S, we need the lemmas 3.6, 3.7, 3.8 below. Using the remarks on the actions of elements of S on N in the paragraphs previous to Lemma 2.1, these results can be verified by taking multiplications of matrices and vectors as well as applying the field automorphism.

Lemma 3.6. (1) Under the notation in Lemma 3.4, the group G_{8−2} = ⟨d^{2}u, p, m⟩ (≅ S_{4}) has the following 8 orbits on N^{#}:

one orbit O_{2} of length 1, one orbit O_{1} ∪ O_{3} of length 2, two orbits O_{4} and O_{7} of length 6; four orbits O_{10}, O_{11}, O_{5} ∪ O_{6}, and O_{8} ∪ O_{9} of length 12.

Moreover they generate the following subspaces of N:

⟨O_{2}⟩ ≅ 2, ⟨O_{1} ∪ O_{3}⟩ = {0} ∪ O_{2} ∪ (O_{1} ∪ O_{3}) ≅ 2^{2}; ⟨O_{4}⟩ = {0} ∪ O_{2} ∪ O_{4} ≅ 2^{3}; ⟨O_{7}⟩ = {0} ∪ O_{2} ∪ (O_{1} ∪ O_{3}) ∪ O_{4} ∪ O_{7} ≅ 2^{4}; ⟨O_{10}⟩ = {0} ∪ O_{2} ∪ (O_{1} ∪ O_{3}) ∪ O_{10} ≅ 2^{4}, ⟨O_{11}⟩ = {0} ∪ O_{2} ∪ (O_{1} ∪ O_{3}) ∪ O_{11} ≅ 2^{4}; ⟨O_{5} ∪ O_{6}⟩ = ⟨O_{8} ∪ O_{9}⟩ = N.

(2) The involution d^{2}l preserves each of the above G_{8−2}-orbits, except O_{10} and O_{11} which are fused by d^{2}l. Then the following 7 subsets are orbits on N^{#} under G_{7} = ⟨d^{2}u, p, d^{2}l⟩ (≅ A_{4} × 2) as well as G_{5} = ⟨G_{7}, m⟩ = ⟨G_{8−2}, d^{2}l⟩ ≅ S_{4} × 2:

O_{2}, O_{1} ∪ O_{3}, O_{4}, O_{7}, O_{5} ∪ O_{6}, O_{8} ∪ O_{9}; and O_{10} ∪ O_{11} of length 24.

Lemma 3.7. (1) The group G_{14} = ⟨d^{2}u, m, d^{2}l⟩ ≅ S_{3} × 2 has the following 12 orbits A_{i} (i = 1, …, 12) on N^{#} of the indicated lengths.

A_{1} 1  {(ω, 1, 1)}
A_{2} 2  {(1, ω̄, ω̄), (ω̄, ω, ω)}
A_{3} 3  {(1, 1, ω̄), (1, ω̄, 1), (ω, ω̄, ω̄)}
A_{4} 3  {(0, ω, ω), (ω̄, 0, ω), (ω̄, ω, 0)}
A_{5} 6 {(1, ω, 1), (ω, ω̄, ω), (ω̄, 1, ω̄), (1, 1, ω), (ω, om, ω̄), (ω̄, ω̄, 1)}
A_{6} 6 {(ω, ω, 1), (ω, 1, ω), (ω̄, 1, 1), (ω̄, ω̄, ω), (ω̄, ω, ω̄), (1, ω, ω)}
A_{7} 6 {(1, 1, 1), (ω, ω̄, 1), (ω, 1, ω̄), (0, 0, ω), (ω̄, 0, 0), (0, ω, 0)}
A_{8} 6 {(ω, ω, ω), (ω̄, 1, ω), (ω̄, ω, 1), (0, 0, 1), (ω, 0, 0), (0, 1, 0)}
A_{9} 6 {(ω̄, ω̄, ω̄), (1, ω, ω̄), (1, ω̄, ω), (0, 0, ω̄), (1, 0, 0), (0, ω̄, 0)}
A_{10} 6 {(0, 1, 1), (ω, 0, 1), (ω, 1, 0), (0, ω̄, ω̄), (1, 0, ω̄), (1, ω̄, 0)}
A_{11} 6 {(0, ω, ω̄), (1, 0, ω), (ω̄, ω̄, 0), (0, ω̄, ω), (ω̄, 0, ω̄), (1, ω, 0)}
A_{12} 12 {(0, 1, ω), (ω̄, 0, 1), (ω, ω, 0), (0, ω̄, 1), (ω, 0, ω̄), (1, 1, 0),
(0, 1, ω̄), (1, 0, 1), (ω, ω̄, 0), (0, ω, 1), (ω, 0, ω), (ω̄, 1, 0)}

(2) The subspaces of N generated by A_{i} (i = 1, …, 12) are as follows:

⟨A_{1}⟩ ≅ 2, ⟨A_{2}⟩ = {0} ∪ A_{1} ∪ A_{2} ≅ 2^{2}; ⟨A_{3}⟩ = ⟨A_{4}⟩ = {0} ∪ A_{1} ∪ A_{3} ∪ A_{4} ≅ 2^{3}; ⟨A_{7}⟩ = {0} ∪ A_{1} ∪ A_{2} ∪ A_{3} ∪ A_{4} ∪ A_{7} ≅ 2^{4}; ⟨A_{i}⟩ = {0} ∪ A_{1} ∪ A_{3} ∪ A_{4} ∪ A_{5} ∪ A_{8} ∪ A_{10} ∪ A_{11} ≅ 2^{5} for i = 5, 8, 10, 11; ⟨A_{6}⟩ = ⟨A_{9}⟩ = ⟨A_{12}⟩ = N.

Lemma 3.8. (1) The group G_{8−1} = ⟨u, p, m⟩ ≅ S_{4} has the following 8 orbits B_{i} (i = 1, …, 8) on N^{#} of the indicated lengths.

B_{1} 3  {(ω, 1, 1), (1, ω̄, ω̄), (ω̄, ω, ω)}
B_{2} 4  {(0, 1, ω̄), (ω, 0, ω), (1, ω, 0), (ω̄, ω̄, 1)}
B_{3} 6  {(0, ω, ω), (1, 0, ω̄), (ω, 1, 0), (ω, ω, 1), (ω̄, ω, ω̄), (ω, ω̄, ω̄)}
B_{4} 6  {(0, 0, 1), (ω̄, 0, 0), (1, ω̄, ω), (ω̄, 1, ω), (0, ω̄, 0), (1, 1, 1)}
B_{5} 8 {(0, ω, 1), (ω̄, 0, ω̄), (1, 0, 1), (ω, ω̄, 0), (0, ω̄, ω), (ω̄, 1, 0), (1, 1, ω̄), (ω, ω, ω̄)}
B_{6} 12  {(0, ω, ω̄), (ω, 0, ω̄), (1, 1, 0), (ω̄, ω̄, 0), (ω, ω, 0), (1, ω, 1),
(ω̄, 0, 1), (0, 1, ω), (0, ω̄, 1), (ω̄, 1, ω̄), (ω, ω̄, ω), (1, 0, ω)}

B_{7} 12  {(0, 0, ω), (1, 0, 0), (ω, 1, ω̄), (ω, ω̄, 1), (0, 1, 0), (ω̄, ω̄, ω̄),
(ω, ω, ω), (0, ω, 0), (ω̄, ω, 1), (1, ω, ω̄), (ω, 0, 0), (0, 0, ω̄)}
B_{8} 12  {(0, 1, 1), (ω̄, 0, ω), (0, ω̄, ω̄), (1, ω̄, 0), (1, 1, ω̄), (ω̄, ω, 0),
(ω, 0, 1), (ω, 1, ω), (1, ω̄, 1), (ω̄, ω̄, ω), (1, ω, ω), (ω̄, 1, 1)}

(2) The subspaces of N generated by B_{i} (i = 1, …, 8) are as follows:

⟨B_{1}⟩ ≅ {0} ∪ B_{1} ≅ 2^{2}; ⟨B_{2}⟩ = {0} ∪ B_{1} ∪ B_{2} ≅ 2^{3};

⟨B_{6}⟩ = {0} ∪ B_{1} ∪ B_{6} ≅ 2^{4}, ⟨B_{5}⟩ = {0} ∪ B_{1} ∪ B_{2} ∪ B_{5} ≅ 2^{4}; ⟨B_{3}⟩ = ⟨B_{4}⟩ = {0} ∪ B_{1} ∪ B_{2} ∪ B_{3} ∪ B_{4} ∪ B_{6} ≅ 2^{5}; ⟨B_{7}⟩ = ⟨B_{8}⟩ = N.

Basing on the informations in Lemma 3.4, 3.6, 3.7 and 3.8 above, we can obtain a list of the possible triples (N ∩ F, S ∩ F, ρ(F)).

Proposition 3.9. Let F be a flag-transitive automorphism group of T_{2}^{∗}(O_{4}) not containing N = O_{2}(Aut(T_{2}^{∗}(O_{4}))). Then, up to conjugacy by S, the triple (F ∩ N, F ∩ S, ρ(F)) is one of the following.

(1) If F ∩ N ≅ 2^{3}, (⟨O_{4}⟩, G_{16}, G_{5}).

(2) If F ∩ N ≅ 2^{4}, (⟨O_{7}⟩, G_{9−2}, G_{5}), (⟨O_{7}⟩, G_{16}, G_{7}); (⟨B_{5}⟩, ⟨u, m⟩, G_{8−1}), (⟨B_{6}⟩, ⟨u, m⟩, G_{8−1}); (⟨O_{7}⟩, G_{15}, G_{8−2}), (⟨O_{10}⟩, G_{15}, G_{8−2}); or (⟨O_{10}⟩, H_{15}, H_{8}).

(3) If F ∩ N ≅ 2^{5}, (⟨A_{5}⟩, G_{16}, G_{14}), (⟨A_{5}⟩, G_{15}, G_{14}), (⟨A_{5}⟩, ⟨d^{2}u, md^{2}l⟩, G_{14}) or (⟨B_{3}⟩, G_{9−1}, G_{8−1}).

Proof. By Lemma 3.3, if N ∩ F ≅ 2^{3}, we may assume that ρ(F) = G_{5} and S ∩ F is a conjugate of G_{16}. As the subgroups of G_{5} ≅ S_{4} × 2 isomorphic to G_{16} ≅ Z_{6} are conjugate to each other, we may also take G_{16} = S ∩ F. By Lemma 3.6(1)(2), the unique G_{5}-invariant subspace of N of order 2^{3} is ⟨O_{4}⟩. Thus N ∩ F = ⟨O_{4}⟩ by Lemma 3.1, which establishes the claim (1).

If N ∩ F ≅ 2^{5}, it follows from Lemma 3.5 that ρ(F) = G_{8−1} (or G_{14}) and S ∩ F is a conjugate of G_{9−1} (resp. G_{15} or G_{16}). If ρ(F) = G_{8−1} ≅ S_{4}, the subgroup S ∩ F ≅ A_{4} of ρ(F) is uniquely determined as the derived subgroup of ρ(F), and hence S ∩ F = G_{9−1}. Moreover, ⟨B_{3}⟩ is the unique G_{8−1}-invariant subspace of N of order 2^{5} by Lemma 3.8, and so N ∩ F = ⟨B_{3}⟩. If ρ(F) = G_{14} = ⟨d^{2}u, m, d^{2}l⟩ ≅ S_{3} × 2, ⟨A_{5}⟩ is the unique G_{14}-invariant subspace of order 2^{5} by Lemma 3.7. The group G_{16} is the unique subgroup of G_{14} isomorphic to Z_{6}, and so S ∩ F = G_{16} if S ∩ F is conjugate to G_{16}. If S ∩ F is conjugate to G_{15} ≅ S_{3}, we have S ∩ F = ⟨d^{2}u, m⟩ or ⟨d^{2}u, md^{2}l⟩. Since G_{14} is self-normalizing in S and the above two subgroups are not conjugate in ρ(F) = G_{14}, there is no element g ∈ NS such that F^{g} = F_{1} satisfies (N ∩ F_{1}, S ∩ F_{1}, ρ(F_{1})) = (⟨A_{5}⟩, ⟨d^{2}u, md^{2}l⟩, G_{14}) for F with (N ∩ F, S ∩ F, ρ(F)) = (⟨A_{5}⟩, G_{15}, G_{14}). The claim (3) is verified.

Now assume that N ∩ F ≅ 2^{4}. If ρ(F) contains d, ρ(F) = H_{5}, H_{7} or H_{8}. It follows from Lemma 3.6 that there is no subgroup of N of order 2^{4} invariant under H_{5} or H_{7}, while ⟨O_{10}⟩ and ⟨O_{11}⟩ are the set of all H_{8}-invariant subspaces of N of order 2^{4}. Since d^{2}l interchanges O_{10} and O_{11} and normalizes H_{8}, we may assume N ∩ F = ⟨O_{10}⟩. Moreover, as S ∩ F is a subgroup of H_{8} ≅ 3.S_{4} isomorphic to H_{15} ≅ 3.S_{3}, it is a Sylow 2-normalizer, and hence we may take S ∩ F = H_{15}. By the same reasoning, if ρ(F) = G_{8−2}, we may take S ∩ F = G_{15}. There are three subgroups ⟨O_{7}⟩, ⟨O_{10}⟩ and ⟨O_{11}⟩ of N of order 2^{4} invariant under G_{8−2}, among which the latter two are interchanged by the involution d^{2}l normalizing G_{8−2}. The claim (2) is verified if ρ(F) = H_{i} (i = 5, 7, 8) or G_{8−2}.

For ρ(F) = G_{5} or G_{7}, ⟨O_{7}⟩ is the unique invariant subgroup of N of order 2^{4} by Lemma 3.6. If ρ(F) = G_{7} ≅ A_{4}×2, S∩F ≅ Z_{6} is a Sylow 3-normalizer, and we may take S∩F = G_{16}. If ρ(F) ≅ G_{5} ≅ S_{4} × 2, S ∩ F is a conjugate of G_{9−2} ≅ A_{4} contained in ρ(F). If S ∩ F contains a product of an involution of G_{9−2} with d^{2}l, it induces a permutation of type 2^{3} on the set of six LINEs through α, while G_{9−2} induces even permutations only. Thus S ∩ F coincides with ⟨d^{2}u, p⟩ = G_{9−2}. This verifies the claim (2) for ρ(G) = G_{5} and G_{7}.

The remaining case ρ(F) = G_{8−1} is easy to treat: we may take S ∩ F = ⟨u, m⟩ as it is a Sylow 3-normalizer of ρ(F). By Lemma 3.8, ⟨B_{5}⟩ and ⟨B_{6}⟩ are all G_{8−1}-invariant subspaces of N of order 2^{4}. □

We will now examine for each possible triple in Proposition 3.9 whether or not there exists a flag-transitive subgroup F of NS = Aut(T_{2}^{∗}(O_{4})) having that triple. The outline is as follows: Take a generator of ρ(F), say s_{1}, …, s_{m}, outside from S ∩ F. By the definition of ρ(F), there are some elements n_{i} ∈ N with n_{i}s_{i} ∈ F (i = 1, …, m). Note that n_{i} is not contained in F ∩ N, so that we may take n_{i} as a representative of (N/N ∩ F)^{#}. Since F is generated by N ∩ F, S ∩ F and n_{i}s_{i} (i = 1, …, m), we will first examine for each non-trivial representative n_{i} of N/(N ∩ F) whether the intersection of ⟨N ∩ F, S ∩ F, n_{i}s_{i}|i = 1, …, m⟩ with S in fact coincides with S ∩ F. For representatives n_{i} satisfying this condition, F is an extension of F ∩ N by ρ(F). Observing the stabilizer of the flag (α, β) (see Lemma 2.1), we can check the flag-transitivity of F. Proposition 3.10. Under the notation in §2, we have the following.

(1) There is no flag-transitive subgroup F of Aut(T_{2}^{∗}(O_{4})) with triple (⟨O_{7}⟩, G_{9−2}, G_{5}), (⟨A_{5}⟩, ⟨d^{2}u, md^{2}l⟩, G_{14}), (⟨B_{6}⟩, ⟨u, m⟩, G_{8−1}), (⟨O_{7}⟩, G_{15}, G_{8−2}), (⟨O_{10}⟩, G_{15}, G_{8−2}), or (⟨O_{10}⟩, H_{15}, H_{8}).

(2) The group F_{1} = ⟨⟨O_{4}⟩, G_{16}, t(1, ω, ω)p, t(1, ω̄, ω̄)m⟩ (≅ 2^{3}(S_{4} × 2)) is a unique flagtransitive group with triple (⟨O_{4}⟩, G_{16}, G_{5}). The stabilizers in F_{1} of a POINT α and a LINE β are G_{16} ≅ Z_{6} and ⟨t(0, ω, ω) · t(1, ω, ω)p, t(ω, ω̄, ω̄) · t(1, ω̄, ω̄)m · d^{2}l⟩ ≅ 2^{2} respectively.

(3) The group F_{2} = ⟨⟨B_{5}⟩, ⟨u, m⟩, t(0, 1, 0)p⟩ (≅ 2^{4}S_{4}) is a unique flag-transitive group with triple (⟨B_{5}⟩, ⟨u, m⟩, G_{8−1}). The stabilizers in F_{2} of a POINT α and a LINE β are ⟨u, m⟩ ≅ S_{3} and ⟨t(ω, 0, ω) · (t(0, 1, 0)p)^{u} · m⟩ ≅ Z_{4} respectively.

(4) There are exactly two flag-transitive groups with triple (⟨O_{7}⟩, G_{16}, G_{7}). They are F_{3} := ⟨⟨O_{7}⟩, G_{16}, t(1, 0, 0)p⟩ (≅ 2^{4}(A_{4} × 2)) and F_{4} := ⟨⟨O_{7}⟩, G_{16}, t(0, 1, 0)p⟩ (≅ 2^{4}(A_{4} × 2)). The stabilizers in F_{3} (resp. F_{4}) of a POINT α and a LINE β are G_{16} ≅ Z_{6} and ⟨t(ω̄, 0, 0), t(0, 1, 0)p⟩ ≅ 2^{2} (resp. ⟨t(ω, 1, ω̄) · (t(0, 1, 0)p)^{d^{2}u} · d^{2}l · t(1, 0, 0)p⟩ ≅ Z_{4}) respectively.

(5) The group F_{5} = ⟨⟨A_{5}⟩, G_{16}, t(1, 0, 0)m⟩ (≅ 2^{5}(S_{3} × 2)) is a unique flag-transitive group with triple (⟨A_{5}⟩, G_{16}, G_{14}). The stabilizers in F_{5} of a POINT α and a LINE β are G_{16} ≅ Z_{6} and ⟨t(ω, 0, 0), t(1, 0, 0)m · d^{2}l⟩ ≅ 2^{2} respectively.

(6) The group F_{6} = ⟨⟨A_{5}⟩, G_{15}, t(1, 0, 0)d^{2}l⟩ (≅ 2^{5}(S_{3} × 2)) is a unique flag-transitive group with triple (⟨A_{5}⟩, G_{15}, G_{14}). The stabilizers in F_{6} of a POINT α and a LINE β are G_{15} ≅ S_{3} and ⟨t(ω, 0, 0), t(ω, ω̄, ω̄) · m · t(1, 0, 0)d^{2}l⟩ ≅ 2^{2} respectively.

(7) The group F_{7} = ⟨⟨B_{3}⟩, G_{9−1}, t(1, 0, 0)m⟩ (≅ 2^{5}S_{4}) is a unique flag-transitive group with triple (⟨B_{3}⟩, G_{9−1}, G_{8−1}). The stabilizers in F_{7} of a POINT α and a LINE β are G_{9−1} ≅ A_{4} and ⟨t(ω̄, 0, 0), t(1, 0, 0)m · p^{u}⟩ ≅ D_{8} respectively.

Proof. We first describe the above procedure in detail for the triple (N ∩ F, S ∩ F, ρ(F)) = (⟨O_{4}⟩, G_{16}, G_{5}), and establish the claim (2). Note that the set ⟨O_{6}⟩^{#} = O_{1} ∪ O_{6} consisting of (ω, 1, 1), (ω̄, ω̄, ω), (ω̄, ω, ω̄), (1, ω, ω), (0, 1, 1), (ω, 0, 1) and (ω, 1, 0) intersects trivially with ⟨O_{4}⟩ (see Lemma 3.6), and hence ⟨O_{6}⟩ ≅ 2^{3} is a set of representatives for N/⟨O_{4}⟩. As ρ(F) = G_{5} = ⟨G_{16}, p, m⟩, we may assume t(x)p and t(y)m are elements of F for some x, y ∈ ⟨O_{6}⟩^{#} (identified with a subset of H ≅ GF (4)^{3}).

Now the condition F/(F ∩ N) ≅ ρ(F) = G_{5} ≅ S_{4} × Z_{2} is equivalent to saying that the following elements are contained in F ∩ N: (t(x)p)^{2}, (t(y)m)^{2}, (t(x)p · d^{2}u)^{3}, (d^{2}u · t(y)m)^{2}, (d^{2}l · t(x)p)^{2}, and (d^{2}l · t(y)m)^{2}. As p^{2} = m^{2} = (d^{2}u)^{3} = 1, [d^{2}u, p] = 1 and (d^{2}u)^{m} = (d^{2}u)^{−1}, we have (t(x)p)^{2} = t(x)·t(x)^{p}, (t(y)m)^{2} = t(y)·t(y)^{m}, (t(x)p·d^{2}u)^{3} = t(x)·t(x)^{(d^{2}u)^{2}} ·t(x)^{(d^{2}u)}, and (d^{2}u · t(y)m)^{2} = t(y)^{(d^{2}u)^{2}}t(y)^{m}.

As we saw in the paragraphs previous to Lemma 2.1, we can easily compute these elements by matrix multiplication and the field automorphism for each x, y ∈ ⟨O_{6}⟩^{#}. For example, for y = (ω̄, ω̄, ω), we have t(y) · t(y)^{m} = t(z), where under the notation for the extended matrix multiplication in the paragraphs previous to Lemma 2.1

z = y + ym = (ω̄, ω̄, ω) + (ω̄, ω̄, ω) ( \begin{matrix}ω & ω & ω \\ ω & ω̄ & 1 \\ ω & 1 & ω̄ \end{matrix} ) σ =

(ω̄, ω̄, ω̄) + \overline{(ω̄, 1, ω̄)} = (ω̄, ω̄, ω) + (ω, 1, ω) = (1, ω, 0)

As (1, ω, 0) ∈ O_{11} ⊈ ⟨O_{4}⟩, we conclude that y ≠ (ω̄, ω̄, ω). Similarly, we can verify that the conditions x + xp ∈ ⟨O_{4}⟩ and y + ym ∈ ⟨O_{4}⟩ for x, y ∈ ⟨O_{6}⟩^{#} imply x, y ∈ {(1, ω̄, ω̄), (1, ω, ω), (0, 1, 1)}. Then the conditions x + x(d^{2}u) + x(d^{2}u)^{2} ∈ ⟨O_{4}⟩ (equivalent to t(x) · t(x)^{(d^{2}u)^{2}} · t(x)^{(d^{2}u)} ∈ N ∩ F) and y(d^{2}u)^{2} + ym ∈ ⟨O_{4}⟩ (equivalent to t(y)^{(d^{2}u)^{2}} · t(y)^{m} ∈ N ∩ F) implies x = (1, ω, ω) and y = (1, ω̄, ω̄). These x, y satisfy the remaining conditions for F to be an extension of N ∩ F by ρ(F).

Thus F is uniquely determined to be F_{1} in this case. In particular, |F_{1}| = 2^{3}|S_{4}|2 = 2^{7}3. As the stablizer of a POINT α in F_{1} is F_{1} ∩ S = G_{16} ≅ Z_{6}, the stabilizer of a flag (α, β) is trivial. (Recall that the description of the stabilizer of β in Lemma 2.1.) Thus F_{1} acts flag-transitively on the 2^{7}3 = |F_{1}|/1 flags of T_{2}^{∗}(O_{4}) with the stabilizer of β of order 2^{7}3/2^{5}3 = 4. As t(0, ω, ω), t(ω, ω̄, ω̄) ∈ ⟨O_{4}⟩, we can see that t(0, ω, ω) · t(1, ω, ω)p and t(ω, ω̄, ω̄) · t(1, ω̄, ω̄)m · d^{2}l are mutually commuting elements of the stabilizer of β in F_{1}. Then the stabilizer is the four group. The claim (2) is verified.

Since the remaining claims are similarly verified, we just sketch the proofs.

Note that {(1, 0, 0), (0, 1, 0), (1, 1, 0)} is a set of representatives for (N/⟨O_{7}⟩)^{#}. For a triple (F ∩ N, S ∩ F, ρ(F)) = (⟨O_{7}⟩, G_{9−2}, G_{5}), there are some x ∈ {(1, 0, 0), (0, 1, 0), (1, 1, 0)} with t(x)d^{2}l ∈ F. As d^{2}l centralizes d^{2}u ∈ S∩F, we have [(t(x)d^{2}l)^{−1}, (d^{2}u)^{−1}] = t(x)·t(x)^{(d^{2}u)^{−1}} ∈ F ∩ N = ⟨O_{7}⟩. However, none of (1, 0, 0) + (1, 0, 0)(u^{2}d) = (1, 0, ω̄), (0, 1, 0) + (0, 1, 0)(u^{2}d) = (ω, 1, 0) and (1, 1, 0) + (1, 1, 0)(u^{2}d) = (ω̄, 1, ω̄) is contained in ⟨O_{7}⟩. Similarly, there is no F with triple (⟨O_{7}⟩, G_{15}, G_{8−2}), since (t((1, 0, 0))p·u)^{3} = t((1, 0, 0)+(1, 0, 0)pu+(1, 0, 0)(pu)^{2}) = t((ω̄, 0, ω̄)) ∉ ⟨O_{7}⟩, (t((0, 1, 0)p · m)^{2} = t((0, 1, 0) + (0, 1, 0)pm) = t((0, 1, ω̄)) ∉ ⟨O_{7}⟩, and (t((1, 1, 0)p · m)^{2} = t((1, 1, 0) + (1, 1, 0)pm) = t((ω, ω, 0)) ∉ ⟨O_{7}⟩. This verifies the claim (1) for F with F ∩ N = ⟨O_{7}⟩.

On the other hand, for a triple (F ∩N, S ∩F, ρ(F)) = (⟨O_{7}⟩, G_{16}, G_{7}), we may verify that any x ∈ {(1, 0, 0), (0, 1, 0), (1, 1, 0)} satisfies the condition for F = ⟨⟨O_{7}⟩, G_{16}, t(x)p⟩ to be an extension of ⟨O_{7}⟩ by G_{7}. Since (t((0, 1, 0))p)^{m} = t((ω̄, ω̄, ω̄))p = t((ω, ω̄, 1)) · t((1, 1, 0))p with (ω, ω̄, 1) ∈ ⟨O_{7}⟩, we have ⟨⟨O_{7}⟩, G_{16}, t((0, 1, 0))p⟩^{m} = ⟨⟨O_{7}⟩, G_{16}, t((1, 1, 0))p⟩. Thus we have two possible F up to conjugacy in this case. As above, we can verify that both of them are flag-transitive and determine the stabilizes. As the stabilizers of β in them are not isomorphic, they are not conjugate. This proves the claim (4).

Note that {(0, ω, 0), (0, 0, ω), (0, ω, ω)} is a set of non-trivial representatives of N/⟨O_{10}⟩. There is no F with triple (⟨O_{10}⟩, G_{15}, G_{8−2}), since for x = (0, ω, 0), (0, 0, ω) and (0, ω, ω), we have x + xpu + x(pu)^{2} = (1, 0, 1), (ω̄, 0, ω̄) and (ω, 0, ω) respectively, none of which lies in ⟨O_{10}⟩, and hence the relation (t(x)p·u)^{3} = 1 does not hold. This also proves the non-existence of F with triple (⟨O_{10}⟩, H_{15}, H_{8}).

There is no F with triple (⟨B_{6}⟩, ⟨u, m⟩, G_{8−2}), since for representatives x = (1, 0, 0), (0, 0, 1) and (1, 0, 1) of (N/⟨B_{6}⟩)^{#}, we have (1, 0, 0)p + (1, 0, 0)m = (ω, ω̄, ω̄), (0, 0, 1) + (0, 0, 1)pu+(0, 0, 1)(pu)^{2} = (ω, 0, ω) and (1, 0, 1)p+(1, 0, 1)m = (0, 1, ω̄) respectively, none of which lies in ⟨O_{10}⟩, and hence one of the relations (m·t(x)p)^{2} ∈ N ∩F and (t(x)p·u)^{3} ∈ N ∩F does not hold.

It is easy to see that there is no F with triple (⟨A_{5}⟩, ⟨d^{2}u, md^{2}l⟩, G_{14}), since md^{2}l fixes a flag (α, β) and so the group F of order 2^{5}|G_{14}| = 2^{7}3 is not flag-transitive. Hence the claim (1) is established.

For the case (3), the group (N/⟨B_{5}⟩)^{#} has representatives x = (1, 0, 0),(0, 1, 0),(1, 1, 0), among which (0, 1, 0) is a unique element satisfying the condition x^{p} + x^{m} ∈ ⟨B_{5}⟩. For the remaining possible triples, we have F ∩ N ≅ 2^{5}, and the non-trivial representative of N/(N ∩ F) is uniquely determined ((1, 0, 0) for the claims (5) and (6); and (0, 1, 0) for the claim (7)). It is now routine to verify that the unique possible F is flag-transitive with the stabilizers in each claim. □

By Proposition 3.9 and 3.10 we complete the classification of flag-transitive subgroups of Aut(T_{2}^{∗}(O_{4})).

Theorem 3.11. A flag-transitive subgroup F of Aut(T_{2}^{∗}(O_{4})) = NS not containing N is conjugate under S to exactly one of the seven groups F_{i} (i = 1, …, 7) in Proposition 3.10.

4. Flag-transitive extensions of T_{2}^{∗}(O_{4}) and its dual

In this section, we complete the classification of flag-transitive c.C_{2}-geometries with point residues isomorphic to T_{2}^{∗}(O_{4}) or its dual. Let G = (G_{0}, G_{1}, G_{2}; ∗) be such a geometry and G a flag-transitive subgroup of Aut(G). Then the stabilizer G_{P} of a point P ∈ G_{0} acts on the residue Res(P) at P, which is a generalized quadrangle T_{2}^{∗}(O_{4}) or its dual. First note that the kernel K_{P}, the normal subgroup of G_{P} acting trivially on Res(P), is trivial by a standard argument given in [6, 3.2]. Thus G_{P} is isomorphic to a flag-transitive subgroup of Aut(T_{2}^{∗}(O_{4})) = NS, and hence, identifying G_{P} with a subgroup of NS, either it contains N or it is conjugate to one of the seven groups F_{i} (i = 1, …, 7) in Proposition 3.10.

In the former case, G is determined in [6] under the assumption of (LL) property. However, we did not essentially use this assumption in [6]. This was used to identify a line l with the set of two points, say P, Q on it, so that G_{l} = G_{{P,Q}}, the setwise stabilizer of {P, Q}. However, this information is not needed in [6]: though in [6, 3.6,4.2] the group G_{P,Q} was examined, every argument there goes through if we replace G_{P,Q} by G_{P,l}. Hence the classification of [6] was done without using the (LL)-property.

Thus we may assume that G_{P} is one of the flag-transitive subgroup F_{i} (i = 1, …, 7) in Proposition 3.10. We will treat this case by the same method adopted in [6], namely generators and relations. For that purpose, we first determine presentations of some F_{i} in terms of their generators of POINT and LINE stabilizers. To describe the result, besides the notation in §3 we fix some more notation.

Notation 4.1. We take the following basis of F_{i} ∩ N for i = 1, …, 5.

For F_{2} ∩ N = ⟨B_{5}⟩ ≅ 2^{4}:

b_{1} := t(ω, 1, ω), b_{2} := t(1, 0, 1), b_{3} := t(0, ω, 1), b_{4} := t(ω, 1, 1);
For F_{3} ∩ N = F_{4} ∩ N = ⟨O_{7}⟩ ≅ 2^{4}:
v_{1} := t(ω̄, 0, 0), v_{2} := t(0, ω, 0), v_{3} := t(0, 0, ω), v_{4} := t(1, 1, 1);
For F_{1} ∩ N = ⟨O_{4}⟩ ≅ 2^{3}: c_{i} := v_{i} + v_{4} (i = 1, 2, 3);
For F_{5} ∩ N = ⟨A_{5}⟩ ≅ 2^{5}:
w_{1} := t(ω̄, 0, 0), w_{2} := t(0, 1, 0), w_{3} := t(0, 0, 1),
w_{4} := t(ω, ω, ω), w_{5} := t(ω̄, ω̄, 0).

Consider also the following elements of the stabilizer of a LINE β in Aut(T_{2}^{∗}(O_{4})):

k := t(ω, 0, ω) · (t(0, 1, 0)p)^{u} · m = t(ω, 0, 0) · ( \begin{matrix}ω̄ & 0 & 0 \\ ω & ω̄ & 1 \\ 0 & ω & 0 \end{matrix} ) · σ,

h := t(ω, 1, ω̄) · (t(0, 1, 0)p)^{d^{2}u} · d^{2}l · t(0, 1, 0)p = t(1, 0, 0) · ( \begin{matrix}ω̄ & 0 & 0 \\ 0 & ω & 0 \\ ω & ω̄ & 1 \end{matrix} ) · σ,

x := t(0, ω, ω) · t(1, ω, ω)p = t(1, 0, 0)p = t(1, 0, 0) · ( \begin{matrix}1 & 0 & 0 \\ 1 & ω & ω̄ \\ 1 & ω̄ & ω \end{matrix} ) · σ,

y := t(ω, ω̄, ω̄) · t(1, ω̄, ω̄) · (m · d^{2}l) = t(ω̄, 0, 0) · ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ) · σ, and z := t(1, 0, 0) · (m · d^{2}l).

As we stated in Proposition 3.10, N ∩ F_{i}, the stabilizers of the POINT α = [1, 0, 0, 0] and the LINE β = ⟨[1, 0, 0, 0], [0, 1, 0, 0]⟩ in F_{i} for i = 1, 2, 3, 4 and 5 are given as follows:

N ∩ F_{i}  POINT stabilizer  LINE stabilizer
F_{1} ⟨O_{4}⟩ = ⟨c_{1}, c_{2}, c_{3}⟩ ≅ 2^{3}  G_{16} = ⟨d^{2}u · d^{2}l⟩ ≅ Z_{6} ⟨x, y⟩ ≅ 2^{2}
F_{2} ⟨B_{5}⟩ = ⟨b_{i}|i = 1, …, 4⟩ ≅ 2^{4} ⟨u, m⟩ ≅ S_{3}  ⟨k⟩ ≅ Z_{4}
F_{3} ⟨O_{7}⟩ = ⟨v_{i}|i = 1, …, 4⟩ ≅ 2^{4} G_{16} = ⟨d^{2}u · d^{2}l⟩ ≅ Z_{6} ⟨t(ω̄, 0, 0), x⟩ ≅ 2^{2}
F_{4} ⟨O_{7}⟩ = ⟨v_{i}|i = 1, …, 4⟩ ≅ 2^{4} G_{16} = ⟨d^{2}u · d^{2}l⟩ ≅ Z_{6} ⟨h⟩ ≅ Z_{4}
F_{5} ⟨A_{5}⟩ = ⟨w_{i}|i = 1, …, 5⟩ ≅ 2^{5} G_{16} = ⟨d^{2}u · d^{2}l⟩ ≅ Z_{6} ⟨t(ω, 0, 0), z⟩ ≅ 2^{2}

Since F_{i} is flag-transitive, each F_{i} is generated by the stabilizers of α and β. Thus we do not need basis of N ∩F_{i} as generators for F_{i}. However, in the above we include them to explicitly exhibit the actions of generators of stabilizers on N ∩ F_{i}.

Lemma 4.2. For each i in {1, 2, 3, 4, 5}, the above generators of F_{i} (see 4.1) satisfies the following set R(F_{i}) of relations. Furthermore, as an abstract group, F_{i} can be presented by the set R(F_{i}) of relations.

R(F_{1}) Generators: c_{i} (i = 1, 2, 3), d^{2}u, d^{2}l, x, y:

Relations: c_{i}^{2} = [c_{i}, c_{j}] = 1 (1 ≤ i, j ≤ 3), (d^{2}u)^{3} = (d^{2}l)^{2} = [d^{2}u, d^{2}l] = x^{2} = y^{2} = 1, c_{1}^{d^{2}u} = c_{1}, c_{2}^{d^{2}u} = c_{1}c_{2}c_{3}, c_{3}^{d^{2}u} = c_{2}, c_{1}^{d^{2}l} = c_{1}, c_{2}^{d^{2}l} = c_{2}, c_{3}^{d^{2}l} = c_{3}, c_{1}^{x} = c_{1}, c_{2}^{x} = c_{1}c_{2}, c_{3}^{x} = c_{1}c_{3}, c_{1}^{y} = c_{1}, c_{2}^{y} = c_{3}, c_{3}^{y} = c_{2}, (x · d^{2}u)^{3} = c_{1}, [x, d^{2}l] = c_{1}c_{2}c_{3}, (y · d^{2}u)^{3} = c_{1}c_{2}, [y, d^{2}l] = c_{1}, and [x, y] = 1.

R(F_{2}) Generators: b_{i} (i = 1, …, 4), u, m, k:

Relations: b_{i}^{2} = [b_{i}, b_{j}] = 1 (1 ≤ i, j ≤ 4), u^{3} = m^{2} = (u · m)^{2} = k^{4} = 1, b_{1}^{u} = b_{2}b_{3}, b_{2}^{u} = b_{2}b_{4}, b_{3}^{u} = b_{1}b_{2}, b_{4}^{u} = b_{1}b_{2}b_{3}, b_{1}^{m} = b_{1}b_{4}, b_{2}^{m} = b_{3}, b_{3}^{m} = b_{2}, b_{4}^{m} = b_{4} b_{1}^{k} = b_{2}b_{3}, b_{2}^{k} = b_{3}b_{4}, b_{3}^{k} = b_{1}b_{2}, b_{4}^{k} = b_{4} (ku̇)^{4} = 1, (k · m)^{2} = b_{1}b_{2}b_{3}b_{4}, (k^{2} · u)^{3} = b_{1}, (k^{2}m)^{2} = b_{4}, (k^{−1}u)^{2} = b_{2}b_{3}, and (k · m · u)^{3} = b_{2}b_{4}.

R(F_{3})] Generators: v_{i} (i = 1, …, 4), d^{2}u, d^{2}l, x:

Relations: v_{i}^{2} = [v_{i}, v_{j}] = 1 (1 ≤ i, j ≤ 4), d^{2}u^{3} = d^{2}l^{2} = [d^{2}u, d^{2}l] = x^{2} = 1, v_{1}^{d^{2}u} = v_{2}, v_{2}^{d^{2}u} = v_{3}, v_{3}^{d^{2}u} = v_{1}, v_{4}^{d^{2}u} = v_{1}v_{2}v_{4}, v_{1}^{d^{2}l} = v_{4}, v_{2}^{d^{2}l} = v_{1}v_{2}v_{4}, v_{3}^{d^{2}l} = v_{1}v_{3}v_{4}, v_{4}^{d^{2}l} = v_{1}, v_{1}^{x} = v_{1}, v_{2}^{x} = v_{1}v_{2}v_{4}, v_{3}^{x} = v_{1}v_{3}v_{4}, v_{4}^{x} = v_{4}, (x · d^{2}u)^{3} = v_{1}v_{4} and [x, d^{2}l] = v_{1}v_{2}v_{3}v_{4}.

R(F_{4}) Generators: v_{i} (i = 1, …, 4), d^{2}u, d^{2}l, h:

Relations: v_{i}^{2} = [v_{i}, v_{j}] = 1 (1 ≤ i, j ≤ 4), d^{2}u^{3} = d^{2}l^{2} = [d^{2}u, d^{2}l] = 1, h^{2} = v_{1}, v_{1}^{d^{2}u} = v_{2}, v_{2}^{d^{2}u} = v_{3}, v_{3}^{d^{2}u} = v_{1}, v_{4}^{d^{2}u} = v_{1}v_{2}v_{4}, v_{1}^{d^{2}l} = v_{4}, v_{2}^{d^{2}l} = v_{1}v_{2}v_{4}, v_{3}^{d^{2}l} = v_{1}v_{3}v_{4}, v_{4}^{d^{2}l} = v_{1}, v_{1}^{h} = v_{1}, v_{2}^{h} = v_{2}, v_{3}^{h} = v_{1}v_{3}v_{4}, v_{4}^{h} = v_{4}, (h · d^{2}u · d^{2}l)^{3} = v_{1}v_{4} and [h, d^{2}l] = v_{2}v_{3}.

R(F_{5}) Generators: w_{i} (i = 1, …, 5), d^{2}u, d^{2}l, z:

Relations: w_{i}^{2} = [w_{i}, w_{j}] = 1 (1 ≤ i, j ≤ 4), d^{2}u^{3} = d^{2}l^{2} = [d^{2}u, d^{2}l] = 1, h^{2} = w_{1}, w_{1}^{d^{2}u} = w_{2}, w_{2}^{d^{2}u} = w_{3}, w_{3}^{d^{2}u} = w_{1}, w_{4}^{d^{2}u} = w_{1}w_{4}w_{5}, w_{5}^{d^{2}u} = w_{1}w_{3}w_{4},

w_{1}^{d^{2}l} = w_{4}, w_{2}^{d^{2}l} = w_{1}w_{4}w_{5}, w_{3}^{d^{2}l} = w_{2}w_{3}w_{5}, w_{4}^{d^{2}l} = w_{1}, w_{5}^{d^{2}u} = w_{1}w_{2}w_{4},
w_{1}^{z} = w_{1}, w_{2}^{z} = w_{3}, w_{3}^{z} = w_{2}, w_{4}^{z} = w_{4}, w_{5}^{z} = w_{1}w_{2}w_{3}w_{4}w_{5},
(z · d^{2}u)^{2} = w_{1}w_{5} and [w, d^{2}l] = w_{2}w_{3}w_{4}.

Proof. It is straightforward to verify that the above explicit elements satisfy the relations in R(F_{i}). Thus in order to see that R(F_{i}) presents F_{i}, it suffices to verify that the group X_{i} presented by R(F_{i}) is of order at most |F_{i}| (= 2^{7}3 for all i = 1, …, 5). This can be done as follows.

For example, take i = 1. Then X_{1} normalizes an elemntary abelian group N_{1} = ⟨c_{1}, c_{2}, c_{3}⟩. The factor group X_{1}/N_{1} is generated by the image of d^{2}u, d^{2}l, x and y, which are d^{2}u, d^{2}l, p and m · (d^{2}l), respectively, modulo N_{1}. The relations among the former generators, each c_{i} being reduced to 1, are nothing but the relations among the latter generators, which can be easily observed to present ⟨d^{2}u, p, m⟩ × ⟨d^{2}l⟩ ≅ S_{4} × 2. Hence |X_{1}| ≤ |N_{1}||X_{1}/N_{1}| ≤ 2^{3}|S_{4}×2| = |F_{1}|. By the same argument, one can prove |X_{i}| ≤ |F_{i}| for the remaining cases. □

Theorem 4.3. There is no flag-transitive extension of the dual of T_{2}^{∗}(O_{4}) admitting a flagtransitive automorphism group G in which the stabilizer G_{P} of a point P does not contain N.

Proof. Let (P, l, u) be a maximal flag of G, a flag-transitive extension of T_{2}^{∗}(O_{4}), admitting a flag-transitive automorphism group G such that G_{P} ⊉ N. Thus G_{P} = F_{i} for some i = 1, …, 7, as we remarked above.

Since the dual of Res(P) is T_{2}^{∗}(O_{4}), the stabilizer G_{P,u} is the stabilizer in F_{i} of a POINT u of T_{2}^{∗}(O_{4}), and the six lines incident with (P, u) are identified with six LINEs of T_{2}^{∗}(O_{4}) through a POINT u. By Proposition 3.10, G_{P,u} is isomorphic to Z_{6}, S_{3} or A_{4}. On the other hand, Res(u) is a circle geometry with 5 + 2 = 7 points, on which the stabilizer G_{u} acts doubly transitively by the flag-transitivity of G. Thus (G_{u}/K_{u}, G_{P,u}/K_{u}) is isomorphic to (S_{7}, S_{6}), (A_{7}, A_{6}), (L_{3}(2), S_{4}) or (F_{7}^{6}, Z_{6}). Hence we have G_{P,u} ≅ Z_{6}, K_{u} = 1 and G_{u} ≅ F_{7}^{6}. In particular, G_{P,u} = G_{16} = ⟨d^{2}u, d^{2}l⟩, and G_{P} is F_{1}, F_{3}, F_{4} or F_{5}. Moreover, G_{P,l,u} = 1 and G_{l,u} ≅ Z_{2} is generated by an involution v inducing a permutation of type 2^{3} on the seven points on u.

We will determine the possible relations satisfied by generators of G. Note first that in Lemma 4.2 we have already determined presentations of G_{P} in terms of generators of G_{P} ∩N, G_{P,u} and G_{P,l}.

We next examine the possible relations betweeen v and generators of G_{P,u}. Let P = P_{1} and Q = P_{2} be the two points on a line l and P_{i} (i = 3, …, 7) be the remaining points on u. Suitably arranging the five points on u not on l, we may assume that a generator d^{2}u · d^{2}l of G_{P,u} = G_{16} ≅ Z_{6} induces a 6-cycle (P_{2}, P_{3}, P_{4}, P_{5}, P_{6}, P_{7}). Now v is an involution of type 2^{3} on the points on u with cycle (P_{1}, P_{2}) such that G_{u} = ⟨G_{P,u}, G_{l,u}⟩ = ⟨d^{2}u · d^{2}l, v⟩ ≅ F_{7}^{6}. We may verify that this is satisfied exactly when v induces either (P_{1}, P_{2})(P_{3}, P_{7})(P_{5}, P_{6}) or (P_{1}, P_{2})(P_{3}, P_{7})(P_{4}, P_{5}). In the former (resp. latter) case we can check that v and d^{2}ud^{2}l satisfy the following set R(4) (resp. R(6)) of relations:

Setting s := d^{2}u · d^{2}l,
R(4): v^{2} = 1, s^{6} = 1, (vs)^{3} = 1, [s, v]^{7} = 1, v^{s} = s^{3} ;
R(6): v^{2} = 1, s^{6} = 1, (vs)^{3} = 1, [s, v]^{7} = 1, v^{s^{2}} = s^{3}.

By coset enumerations (available e.g. in GAP), we can verify that both R(4) and R(6) define a group of order 42, and hence they present the Frobenius group F_{7}^{6}.

Finally we examine the possible relations betweeen v and generators of G_{P,l}. Here we use the notation 4.1 for elements of G_{P,l} freely. The involution v ∈ G_{l} acts on the subgroup G_{P,l} of G_{l} of index 2. Note that G_{P,l} is either Z_{4} or E_{4}, as G_{P} = F_{i} for i = 1, 3, 4, or 5. If i = 4, then G_{P,l} = ⟨h⟩ ≅ Z_{4}, and hence

R_{4}(ε): h^{v} = h^{ε} for ε = ±1 In the remaining cases, G_{P,l} is the four group generated by two involutions, say a and b. Under the notation 4.1, they are explicitly given as

a = x, b = y if G_{P} = F_{1},

a = t(ω̄, 0, 0), b = x if G_{P} = F_{3},
a = t(ω, 0, 0), b = z if G_{P} = F_{5}.

Since v induces on G_{P,l} either the identity element or one of the three involutions of GL(2, 2), we have four possible relations:

R_{a,b}(A): a^{v} = a, b^{v} = b; R_{a,b}(B): a^{v} = b, b^{v} = a R_{a,b}(C): a^{v} = a, b^{v} = ab; R_{a,b}(D): a^{v} = ab, b^{v} = b The flag-transitivity implies that G is generated by G_{P,l}, G_{P,u} and G_{l,u}, and hence G =

⟨G_{P} , v⟩. In the above paragraphs, we have determined possible sets of relations satisfied by some generators of G. Explicitly, G satisfies one of the following sets of relations for some j = 4 or 6 and some X = A, B, C, D (or ε = ±1), where R(F_{i}) means the set of relations for G_{P} = F_{i} in Lemma 4.2.

(i) Generators: c_{i} (i = 1, …, 3), d^{2}u, d^{2}l, x, y and v; Relations: R(F_{1}) ∪ R(j) ∪ R_{x,y}(X)

(ii) Generators: v_{i} (i = 1, …, 4), d^{2}u, d^{2}l, x and v;

Relations: R(F_{3}) ∪ R(j) ∪ R_{t(ω̄,0,0),x}(X) (iii) Generators: v_{i} (i = 1, …, 4), d^{2}u, d^{2}l, h and v;

Relations: R(F_{3}) ∪ R(j) ∪ R_{4}(ε) (iv) Generators: w_{i} (i = 1, …, 5), d^{2}u, d^{2}l, z and v;

Relations: R(F_{3}) ∪ R(j) ∪ R_{t(ω,0,0),z}(X)

Now coset enumeration (performed with the aid of computer) tells us the index of G_{P}, presented by a subset of those relations, in the group G̃ presented by the whole relations. It is always 1 in each of the possible cases above, which implies that v ∈ G_{P}, a contradiction. Thus there is no flag-transitive extension of the dual of T_{2}^{∗}(O_{4}) with G_{P} not containing N. □

Theorem 4.4. If an extension G of T_{2}^{∗}(O_{4}) admits a flag-transitive automorphism group G in which the stabilizer G_{P} of a point P does not contain N, then G_{P} is conjugate to the group F_{2} in Proposition 3.10 and G is the simply connected geometry G with 160 points in [6, Theorem 1.1] or its quotient on 80 points.

Proof. The proof is quite similar to that for Theorem 4.3. In this case, the group G_{P,u}/K_{u} is the stabilizer of a point in a doubly transitive group on 3+2 = 5 points. As G_{P,u} ≅ E_{4}, Z_{4} or D_{8} by Proposition 3.10, we have G_{u} ≅ F_{5}^{4}, G_{P,u} ≅ Z_{4}, K_{u} = 1 and G_{l,u} ≅ Z_{2}. In particular, (G_{P} , G_{P,u}) = (F_{2}, ⟨h⟩) or (F_{4}, ⟨k⟩). We may assume that a generator s := h or k of G_{P,u} induces a 4-cycle (P_{2}, P_{3}, P_{4}, P_{5}) with P = P_{1} and P_{2} = Q lying on l. Then the involution v of G_{l,u} ≅ Z_{2} induces either (P_{1}, P_{2})(P_{3}, P_{4}) or (P_{1}, P_{2})(P_{4}, P_{5}) to generate F_{5}^{4} with s. We can verify that in each of these cases, v and s satisfy the following relations, which present F_{5}^{4}.

R(5) : If v induces (P_{1}, P_{2})(P_{3}, P_{4}),
v^{2} = s^{4} = 1, (vs)^{4} = 1, [s, v]^{5} = 1, (s^{2})^{[s,v]^{2}} = v.
R(3) : If v induces (P_{1}, P_{2})(P_{4}, P_{5}),
v^{2} = s^{4} = 1, (vs)^{4} = 1, [s, v]^{5} = 1, (s^{2})^{[s,v]} = v.

Since v ∈ G_{l} acts on G_{P,l}, which is ⟨u, m⟩ ≅ S_{3} if F_{2} and ⟨d^{2}u · d^{2}l⟩ ≅ Z_{6} if F_{4}, we have the following relations.

R_{2}(i, j) : If G_{P} = F_{2}, u^{v} = u^{i}, m^{v} = mu^{j} for some i = 1, 2 and j = 0, 1, 2.
R_{4}(i) : If G_{P} = F_{4}, (d^{2}u)^{v} = (d^{2}u)^{i}, (d^{2}l)^{v} = d^{2}l for some i = 1, 2.

The relation v^{2} = 1 restricts (i, j) in R_{2}(i, j), because it implies m = m^{v^{2}} = mu^{j(1+i)}, and so (i, j) = (1, 0), (2, 0), (2, 1), (2, 2). Now G = ⟨G_{P} , v⟩ satisfies one of the following sets of relations:

Generators: b_{i} (i = 1, 2, 3, 4), u, m, k, v:

Relations: R(F_{2}) ∪ R_{2}(i, j) ∪ R(q) for some (i, j) = (1, 0), (2, 0), (2, 1), (2, 2) and

q = 5, 3.
Generators: v_{i} (i = 1, 2, 3, 4), d^{2}u, d^{2}l, h, v:
Relations: R(F_{4}) ∪ R_{4}(i) ∪ R(q) for some i = 1, 2 and q = 5, 3.

The coset enumeration shows that the index of G_{P} in the group G̃ presented by one of these sets of relations is 1 except when G̃ is presented by R(F_{2}) ∪ R(2, 2) ∪ R(5). In this case, the index is 160. Since the group F_{2} is contained in the group N⟨d, p, q, m⟩ of type F_{3} in [6, 3.7], the resulting geometry coincides with that for the group of type F_{3} on 160 points, denoted by G^{(0)} in [6]. □

Remark. Our F_{2} is also contained in the group N⟨d, p, m⟩ of type F_{15} in [6, 3.6]. Thus a geometry on 160 points should arise from the consideration of group of type F_{15} in [6, 3.6]. This possibility was overlooked there, because in [6] (Page 68, line 9 from the bottom), I carelessly concluded that m^{v} = md^{i} for some i, but in fact m^{v} = md^{i}u^{j} for some i, j, from which the overlooked case arises. Fortunately, the overlooked group is conjugate under ⟨b⟩ into the group arising from the flag-transitive group of type F_{3}, and hence there is no mistake in conclusion.

Together with [6], Theorem 4.4 completes the classification of flag-transitive extensions of T_{2}^{∗}(O_{4}) and its dual. For completeness, we summarize the results.

Corollary 4.5.

(1) A flag-transitive extension of T_{2}^{∗}(O_{4}) is isomorphic to one of the following two geometries: (their geometric constructions are given in [2]).

(i) the simply connected geometry G on 160 points and 3072 planes with Aut(G) ≅ 2_{+}^{1+8} · (A_{5} × A_{5})2 found in [6, Theorem 1.1]

(ii) the half quotient Ḡ of G on 80 points and 1536 planes with Aut(Ḡ) ≅ 2^{8}·(A_{5}×A_{5})2 (2) A flag-transitive extension of the dual of T_{2}^{∗}(O_{4}) is isomorphic to one of the following

three geometries:

(i) the simply connected geometry G^{(0)} on 896 points and 8192 planes with Aut(G^{(0)}) ≅ 2_{+}^{1+12} : 3S_{7} found in [6, Theorem 1.2]

(ii) the half quotient Ḡ^{(0)} of G^{(0)} on 448 points and 4096 planes with Aut(Ḡ^{(0)}) ≅ 2^{12} : 3S_{7} (its geometric construction is given in [7])

(iii) the simply connected geometry G^{(1)} on 448 points and 4096 lines with Aut(Ḡ^{(1)}) ≅ 2^{6+6} : L_{3}(2)

References

[1] Del Fra, A.; Ghinelli, D.; Meixner, T.; Pasini, A.: Flag-transitive extensions of C_{n} geometries. Geom. Dedicata 37 (1991), 253–273.

[2] Del Fra, A.; Pasechnik, D.; Pasini, A.: A new family of extended generalized quadrangles. European J. Combin. (to appear).

[3] Makhnev, A. A.: On strongly regular graphs with parameters (64, 18, 2, 6). Discrete Math. (Russian) 1995, 121–128.

[4] Nakagawa, N.: Private communication (see also the report on the conference on algebraic combinatorics, Kyoto RIMS, 1994).

[5] Weiss, R.; Yoshiara, S.: A geometric characterization of the groups Suz and HS. J. Algebra 133 (1990), 182–196.

[6] Yoshiara, S.: On some flag-transitive non-classical c.C_{2}-geometries. European J. Combin. 14 (1993), 59–77.

[7] Yoshiara, S.: A construction of extended generalized quadrangles using the Veronesean. European. J. Combin. (to appear).

Received October 30, 1996; revised version January 21, 1997