Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 38 (1997), No. 2, 337-342.

A Tilde Geometry for F_{4}(2)

Corinna Wiedorn^{1}

Martin-Luther-Universität Halle-Wittenberg Fachbereich Mathematik und Informatik D-06099 Halle, Germany

1. Introduction

The aim of this paper is to construct a geometry G (resp. its flag-transitive automorphism group) with diagram

(F̃_{4})  ∘ ∘ ∘^{∼}^{1}  ^{2}  ^{3} ∘^{4}

where the rank-3-residues of objects of types 1 and 4 are so-called tilde geometries belonging to the group 3^{7}Sp_{6}(2). The question of the existence (and uniqueness) of such a geometry arises in the classification of semiclassical geometries (resp. semiclassical parabolic systems), i.e. of flag-transitive geometries in which each residue of corank 2 is either a generalized digon, a generalized m-gon, m ≥ 3, coming from some group of Lie type in char 2 or the 3-fold cover of the generalized Sp_{4}(2)-quadrangle. Since some of the sporadic simple groups act on a semiclassical geometry (see [8]), these geometries might be interesting in the context of trying to find a uniform approach to the classification of the finite simple groups via their action on certain “nice” geometries.

For example, apart from the above mentioned group 3^{7}Sp_{6}(2), also the sporadic simple groups M_{24} and He act flag-transitively on a geometry with diagram∘ ∘ ∘^{∼} . But it was shown in [3] that these three are the only possibilities. The group 3^{7}Sp_{6}(2) belongs to an infinite series of non-split extensions 3^{k}Sp_{2n}(2), k = (2^{n} − 1)(2^{n−1} − 1)/3, acting on a tilde geometry, i.e. a geometry with diagram

∘ ^{∼} ∘ ∘^{1}  ^{2}  ^{3} ∘ ∘ ⋯^{n−1} ^{n}

338 C. Wiedorn: A Tilde Geometry for F_{4}(2)

The existence and uniqueness of these groups/geometries was proved in [4], [10]. The M_{24}- geometry extends to tilde geometries of rank 4 resp. 5 for the groups Co_{1} and the Monster (see [6]), but it is shown in [2] that the geometries for M_{24} and He cannot arise as residues in a geometry with diagram (F̃_{4}).

1This work is part of the Ph.D. thesis of the author.

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

In the case that G is a group acting flag-transitively on a geometry with diagram (F̃_{4}) where the rank-3-residues of elements of types 1 and 4 belong to 3^{7}Sp_{6}(2), it is shown in [11] that there exists a homomorphism of G onto the finite simple group of Lie type F_{4}(2). It is also shown that, under the assumption that the kernel of this homomorphism is abelian, G is a non-split extension of F_{4}(2) by an elementary abelian subgroup of order 3^{833}, which can be viewed as an irreducible module over GF (3). In this paper we prove the existence of such a group G ≅ 3^{833}F_{4}(2) with diagram of type (F̃_{4}).

2. A group of type (F̃_{4})

It is well-known that (2^{2} × F_{4}(2))2 is the centralizer of an involution of type 2C in the Babymonster sporadic simple group (see [1]), and the idea is to look at the corresponding subgroup of the non-split extension of the Babymonster by an elementary abelian group of order 3^{4371} (which exists by [5]). First we describe how one can get the maximal parabolics of F_{4}(2) from certain subgroups of the Babymonster.

Let’s denote the Babymonster by H. The group H acts on a rank-5-geometry G(H) of Petersen-type (for short P -geometry) with diagram

∘ ∘ ∘ ∘ ∘^{1}  ^{2}  ^{3}  ^{4}  ^{5}^{P}

(see [7]). If {a_{1}, a_{2}, a_{3}} is a flag in G(H) with a_{i} of type i and H_{i} = H_{a_{i}} the stabilizer in H of a_{i}, then

H_{1} ≅ 2^{1+22}.Co_{2},
H_{2} ≅ [2^{32}].(Σ_{3} × Aut(M_{22})),
H_{3} ≅ [2^{35}].(L_{3}(2) × Σ_{5}).

Further, the geometry G(H) contains a family of rank-4-subgeometries (including types 1 to 4 of the above diagram), which are classical symplectic spaces over GF (2). If A denotes the stabilizer in H of such a geometry, then A ≅ 2^{9+16}Sp_{8}(2) and A/O_{2}(A) induces Sp_{8}(2) on the subgeometry G(A). Moreover, we may choose A such that G(A) contains the flag {a_{1}, a_{2}, a_{3}}.

Now take an involution σ ∈ O_{2}(H_{1}) of type 2A and let E_{i} = C_{H_{i}}(σ), i = 1, 2, 3, and E_{4} = C_{A}(σ). Then

\begin{matrix}E_{1} ≅ 2^{2+20}.U_{6}(2).2, & E_{2} ≅ [2^{30}].(Σ_{3} × L_{3}(4).2), \\ E_{3} ≅ [2^{32}].(L_{3}(2) × Σ_{5}), & E_{4} ≅ [2^{25}].O_{8}^{−}(2) \end{matrix}

and if E = ⟨E_{1}, E_{2}, E_{3}, E_{4}⟩, then E ≅ 2^{2}E_{6}(2)2 and the groups E_{i}/⟨σ⟩ are just the maximal parabolics of E/⟨σ⟩ ≅ ^{2}E_{6}(2)2 corresponding to the diagram

∘ ∘ ∘^{1}  ^{2}  ^{3}_{2}  _{2}  _{4} ∘^{4}_{4}

Let τ be the outer automorphism in 2^{2}E_{6}(2)2, F_{i} = C_{E_{i}}(τ) and F = ⟨F_{1}, F_{2}, F_{3}, F_{4}⟩ = C_{E}(τ). Then F/⟨σ, τ⟩ ≅ F_{4}(2), and it is not difficult to see that the groups F_{i}/⟨σ, τ⟩ are the maximal parabolics of F_{4}(2). (The construction of the group 2^{2}E_{6}(2)2 was carried out in detail in [7].)

It was shown in [5] that the universal 2-cover G(H̃) of the P -geometry G(H) admits a flagtransitive automorphism group H̃ ≅ 3^{4371}H, and we are now going to apply the above procedure to the group H̃. So let {ã_{1}, ã_{2}, ã_{3}} be a flag in G(H̃) mapping onto {a_{1}, a_{2}, a_{3}} and H̃_{i} = H̃ã_{i}. By [5]

H̃_{1} ≅ (2^{1+22} × 3^{23}).Co_{2},
H̃_{2} ≅ [2^{32}].(Σ_{3} × 3 · Aut(M_{22})),
H̃_{3} ≅ H_{3} ≅ [2^{35}].(L_{3}(2) × Σ_{5}).

Also by [5] G(H̃) contains a family of symplectic type tilde-subgeometries with stabilizer Ã ≅ (2^{25} × 3^{35})Sp_{8}(2).

Lemma 1. Let σ̃ ∈ O_{2}(H̃_{1}) be an involution mapping onto σ under the natural epimorphism from H̃ onto H, Ẽ_{i} = CH̃_{i}(σ̃), i = 1, 2, 3, and Ẽ_{4} = CÃ(σ̃). Then

(1) Ẽ_{1} ≅ (2^{2+20} × 3^{23}).U_{6}(2).2,

(2) Ẽ_{2} ≅ [2^{30}].(Σ_{3} × SL_{3}(4).2),

(3) Ẽ_{3} ≅ [2^{32}].(L_{3}(2) × Σ_{5}),

(4) Ẽ_{4} ≅ (2^{25} × 3^{35}).O_{8}^{−}(2),

and Ẽ/O_{3}(Ẽ) ≅ 2^{2}E_{6}(2)2, where Ẽ = ⟨Ẽ_{1}, Ẽ_{2}, Ẽ_{3}, Ẽ_{4}⟩.

Proof. As σ̃ ∈ O_{2}(H̃_{1}) we have [σ̃, O_{3}(H̃_{1})] = 1. Let Ê_{1} be the preimage of E_{1} in H̃_{1}. Then clearly [σ̃, Ê_{1}] ≤ O_{3}(H̃) ∩ H̃_{1} = O_{3}(H̃_{1}). So [σ̃, Ê_{1}] ≤ O_{3}(H̃_{1}) ∩ O_{2}(H̃_{1}) = 1, i.e. Ê_{1} = Ẽ_{1} and (1) is proved.

Since O_{3}(H̃_{2}) ≤ O_{3}(H̃_{1}), in (2) we get SL_{3}(4) instead of L_{3}(4), and Ẽ_{3} ≅ E_{3} because H̃_{3} ∩ O_{3}(H̃) = 1.

Further, H̃_{1}∩Ã is the stabilizer of the element ã_{1} ∈ G(Ã), which implies (H̃_{1}∩Ã)/O_{2}(H̃_{1}∩ Ã) ≅ 3^{7}Sp_{6}(2) and |O_{3}(H̃_{1}) ∩ O_{3}(Ã)| = 3^{7}. Hence |C_{O_{3}(Ã)}(σ̃)| ≥ 3^{7}. But O_{8}^{−}(2) acts on C_{O_{3}(Ã)}(σ̃) and O_{3}(Ã) possesses the decomposition 1 + 34 with respect to the action of O_{8}^{−}(2). So C_{O_{3}(Ã)}(σ̃) = O_{3}(Ã) and we can argue as above to get (4). □

Lemma 2. Let τ̃ ∈ Ẽ be an involution mapping onto τ under the natural epimorphism from H̃ onto H and F̃_{i} = CẼ_{i}(τ̃), i = 1, …, 4. Then

(1) F̃_{1} ≅ 2^{2} × (2^{15} × 3^{7}).Sp_{6}(2),

(2) F̃_{2} ≅ 2^{2} × [2^{20}]..(Σ_{3} × L_{3}(2)),

(3) F̃_{3} ≅ 2^{2} × [2^{20}].(L_{3}(2) × Σ_{3}),

(4) F̃_{4} ≅ 2^{2} × ([2^{15}] × 3^{7}).Sp_{6}(2),

F̃/O_{3}(F̃) ≅ 2^{2} × F_{4}(2), where F̃ = ⟨F̃_{1}, F̃_{2}, F̃_{3}, F̃_{4}⟩, and the cosets of the groups F̃_{i} form a geometry with diagram of type (F̃_{4}).

Proof. (3) is clear because Ẽ_{3} ≅ E_{3}, and as τ ∈ Aut(M_{22})\M_{22}, τ̃ inverts O_{3}(H̃_{2}), which shows (2). Next we look at Ẽ_{4}. We have τ ∈ A\O_{2}(A) and C_{A}(⟨σ, τ⟩)/O_{2}(C_{A}(⟨σ, τ⟩)) ≅ Sp_{6}(2), i.e. τ corresponds to the involution in the center of a maximal parabolic 2^{1+6}Sp_{6}(2) of Sp_{8}(2). Now by [9, (3.5)] |C_{O_{3}(Ã)}(τ̃)| = 3^{7} and we get (4).

Also F̃_{1}/O_{2}(F̃_{1}) is an extension of Sp_{6}(2) by a 3-group. In order to determine O_{3}(F̃_{1}) we look at the groups F̃_{1i} = F̃_{1}∩F̃_{i} for i ≠ 1. Obviously F̃_{12}/O_{2}(F̃_{12}) ≅ L_{3}(2) and F̃_{13}/O_{2}(F̃_{13}) ≅ Σ_{3} × Σ_{3}. Further F̃_{14}/O_{23}(F̃_{4}) ≅ Σ_{6} and it follows from [9] that |O_{3}(F̃_{14})| = 3 or 3^{7}. Assume the latter possibility holds. Then O_{3}(F̃_{4}) = O_{3}(F̃_{14}) ≤ O_{3}(H̃_{1} ∩ Ã). But as said in the proof of Lemma 1 |O_{3}(H̃_{1} ∩ Ã)| = 3^{7}, so O_{3}(F̃_{4}) = O_{3}(H̃_{1} ∩ Ã). This yields F̃_{4} ≤ NÃ(O_{3}(H̃_{1} ∩ Ã)) = H̃_{1} ∩ Ã which contradicts the fact that F_{4} ≰ H_{1}. So F̃_{14}/O_{2}(F̃_{14}) ≅ 3Σ_{6}, and as F̃_{1} = ⟨F̃_{12}, F̃_{13}, F̃_{14}⟩, (1) and the last assertion follow. □

For the determination of |O_{3}(F̃)| we need a little lemma about the action of dihedral groups on vector spaces over fields of odd characteristic.

Lemma 3. Let P ≅ D_{8}^{n} act on a vectorspace V over a field F such that 2 does not divide charF. If Z(P) acts fixpointfreely on V, then 2^{n} divides dim V.

Proof. We proceed by induction on n. If n = 1, then

P = ⟨x, y|x^{2} = y^{2} = (xy)^{4} = 1⟩.

Let ⟨z⟩ = Z(P) and consider the action of ⟨x, z⟩ ≅ E_{4} on V. Since 2 does not divide charF and C_{V} (z) = 0 we have V = C_{V} (x) ⊕ C_{V} (xz). But xz = x^{y}, so dim V = dim C_{V} (xz) + dim C_{V} (x) = 2 · dim C_{V} (x) and the induction is anchored.

If n > 1, take x ∈ P, o(x) = 2. Then C_{P} (x) ≅ D_{8}^{n−1} × ⟨x⟩ and the pair P_{1} = C_{P} (x)/⟨x⟩, V_{1} = C_{V} (x) satisfies the hypothesis of the lemma, so by induction 2^{n−1} divides dim V_{1}. As before we can show dim V = 2 · dim V_{1}, so the assertion follows. □

Lemma 4. F̃ ≅ 2^{2} × 3^{833}.F_{4}(2).

Proof. We set U = O_{3}(F̃), V = O_{3}(H̃) and regard the images σ, τ ∈ H of σ̃, τ̃ as involutions of H ≅ H̃/V acting on the GF (3)H-module V (which we write additively). Then |U| = 3^{k} for some k ≤ dim C_{V} (⟨σ, τ⟩). It follows from [5] that we can read dim C_{V} (σ) and dim C_{V} (τ) from the usual character table of H in [1] and that χ_{1} is the character corresponding to V. As σ ∈ 2A and τ ∈ 2C we have dim C_{V} (σ) = 1939, dim[V, σ] = 2432, dim C_{V} (τ) = 2159 and dim[V, τ] = 2212. Now ⟨σ, τ⟩ = ⟨σ, στ⟩ acts on W = [V, τ] and

W = C_{W} (σ) ⊕ [W, σ] = C_{W} (στ) ⊕ [W, στ].

From C_{W} (σ) ∩ C_{W} (στ) ≤ C_{W} (τ) = 0 follows W = C_{W} (σ) ⊕ C_{W} (στ). But since the whole centralizer of an involution of type 2C in the Babymonster is isomorphic to (2^{2} × F_{4}(2))2, σ and στ are conjugate in C_{H}(τ). So dim C_{W} (σ) = \frac{1}{2} dim W = 1106. Further C_{W} (σ) = C_{[V,τ]}(σ) = [C_{V} (σ), τ] and dim C_{V} (⟨σ, τ⟩) = dim C_{V} (σ) − dim[C_{V} (σ), τ] = 1939 − 1106 = 833.

So we have k ≤ 833 and it remains to show that also k ≥ 833. Let C = CF̃(U), F = F̃/C ≅ F_{4}(2), F_{i} = F̃_{i}C/C, and consider the action of F on U. Further, let V_{i} = V ∩ F̃_{i}, i = 1, 4, and U_{1} = ⟨(V_{1}V_{4})F̃_{1}⟩ ≤ U. We have |V_{1}| = |V_{4}| = 3^{7}, |V_{1}∩V_{4}| = 3 and [V_{i}, O_{2}(F_{i})] = 1. Let Z_{1} = Z(O_{2}(F_{1})). Then

U_{1} = C_{U_{1}}(Z_{1}) ⊕ [U_{1}, Z_{1}] and [U_{1}, Z_{1}] = ⨁_{Z∈H} C_{U_{1}}(Z),

where H denotes the set of hyperplanes Z ≤ Z_{1} such that C_{U_{1}}(Z) ≠ 0. It follows from the structure of F_{4}(2) that |Z_{1} : Z_{1} ∩ O_{2}(F_{4})| = 2, and by [9, (3.5)] any element t ∈ Z_{1} \ O_{2}(F_{4}) acts fixpointfreely on V_{4}/V_{4} ∩ V_{1}. This implies that dim[V_{4}, Z_{1}] = 6 and that C_{U_{1}}(Z_{1}) = V_{1}. Hence

dim U_{1} = 7 + |F_{1} : F_{1} ∩ F_{4}| · 6 = 7 + 63 · 6 = 385.

Now let ⟨z_{1}⟩ = Z(F_{1}). Then z_{1} ∈ Z(O_{2}(F_{4})) and [V_{4}, z_{1}] = 0. Hence also [z_{1}, U_{1}] = [z_{1}, V_{1}V_{4}]^{F_{1}} = 0. Choose g ∈ F_{4} such that z_{1} ∈ F_{1}^{g}, but z_{1} ∉ O_{2}(F_{1}^{g}) (such a g exists by the structure of F_{4}(2)). Then z_{1} acts nontrivially on V_{1}^{g}. By similar arguments as before we have dim[V_{1}^{g}, z_{1}] = 6 and C_{V_{1}^{g}}(z_{1}) = V_{1}^{g} ∩ V_{4} ≤ U_{1}, i.e. z_{1} acts fixpointfreely on V_{1}^{g}/V_{1}^{g} ∩ U_{1}. Let U_{2} = ⟨(U_{1}V_{1}^{g})^{F_{1}}⟩ and Ū_{2} = U_{2}/U_{1}. Then z_{1} is also fixpointfree on Ū_{2}. As above

Ū_{2} = ⨁_{Z∈H} CŪ_{2}(Z), where H = {Z ≤ Z_{1}||Z_{1} : Z| = 2, CŪ_{2}(Z) ≠ 0}.

Choose Z ∈ H. Then z_{1} ∉ Z, so N_{F_{1}}(Z)/Z ≅ 2^{1+8}O_{6}^{−}(2) or 2^{1+8}O_{6}^{+}(2). Moreover, O_{2}(N_{F_{1}}(Z)/Z) ≅ D_{8}^{4} and Z(O_{2}(N_{F_{1}}(Z)/Z)) = ⟨z_{1}⟩Z/Z acts fixpointfreely on Ū_{2}. Now Lemma 3 yields that 16 must divide dim CŪ_{2}(Z), and we get

dim Ū_{2} ≥ |F_{1} : N_{F_{1}}(Z)| · 16 = \{ \begin{matrix}28 · 16 = 448 resp. \\ 36 · 16 = 576. \end{matrix}

Since dim Ū_{2} ≤ dim U − dim U_{1} ≤ 833 − 385 = 448, we must have U = U_{2}, N_{F_{1}}(Z)/Z ≅ 2^{1+8}O_{6}^{−}(2) and dim U = 833. □

Theorem. There exists a non-split extension G ≅ 3^{833}F_{4}(2) which acts on a geometry of type (F̃_{4}).

Proof. Let G = F̃/⟨σ̃, τ̃⟩ with F̃, σ̃, τ̃ as above, and let M = O_{3}(G). Then G acts on a geometry of type (F̃_{4}) by Lemma 2 and |M| = 3^{833} by Lemma 4. So it remains to show that the extension M ⊴ G is non-split.

Suppose G = MF with F ≅ F_{4}(2). Let φ : G → F be the canonical epimorphism, G_{1} = F̃_{1}/⟨σ̃, τ̃⟩ ≅ (2^{15} × 3^{7})Sp_{6}(2), F_{1} = φ(G_{1}) ≅ 2^{15}Sp_{6}(2) and K_{1} = O_{2}(F_{1}). Since F_{1} and G_{1} both contain a Sylow-2-subgroup of G, we may assume K_{1} = O_{2}(G_{1}). Now N_{M}(K_{1}) = C_{M}(K_{1}) and in the proof of Lemma 4 we have seen that C_{M}(K_{1}) = M_{1}. This implies that G_{1} = N_{G}(K_{1}) = M_{1}F_{1}, a contradiction to the fact that G_{1}/K_{1} ≅ 3^{7}Sp_{6}(2) is a non-split extension. □

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Received January 3, 1997