A characterization of 3-local geometry of M(24)

A.A. Ivanov and G. Stroth

Dedicated to Professor B. Fischer on the occasion of his sixtieth birthday

Abstract The largest Fischer 3-transposition group M(24) acts flag-transitively on a 3-local incidence geometry G(M(24)) which is a c-extension of the dual polar space associated with the group O_{7}(3). The action of the simple commutator subgroup M(24)′ is still flag-transitive. We show that G(M(24)) is characterized by its diagram under the flag-transitivity assumption. The result implies in particular that G(M(24)) is simply connected. The geometry G(M(24)) appears as a subgeometry in the Buekenhout – Fischer 3-local geometry G(F_{1}) of the Monster group. The simple connectedness of G(M(24)) has played a crucial role in the characterization of G(F_{1}), achieved recently. When determining the possible structure of the parabolic subgroups we have used an unpublished pushing-up result by U. Meierfrankenfeld.

1 Introduction

Let δ be the following diagram:

δ : _{1}∘ c _{3}∘  _{3}∘  _{3}∘.

An (incidence) geometry G with diagram δ has rank 4; the types of its elements are 1, 2, 3 and 4 correspond to the nodes from the left to the right on the diagram. The residue of a flag of type {1, 2}, {1, 4} or {3, 4} is, respectively, a projective plane of order 3, a generalized quadrangle of order (3, 3) or the geometry of vertices and edges of the complete graph on 5 vertices.

Suppose that G is a geometry whose diagram is δ and that it admits a flagtransitive automorphism group G. Let {α_{1}, α_{2}, α_{3}, α_{4}} be a flag in G where α_{i} is of type i. For a subset J of I = {1, 2, 3, 4} let G_{J} denote the stabilizer in G of the flag F_{J} = {α_{i} | i ∈ J}. To simplify the notation we will write for instance 12 for the subset {1, 2}. Let K_{J} denote the kernel of the action of G_{J} on the residue G_{J} of F_{J} in G and let Ḡ_{J} ≅ G_{J}/K_{J} be the action induced on the residue. For an element β ∈ G by G_{β} we denote its stabilizer in G, so that G_{α_{i}} is another name for G_{i}.

Let M(24) be the largest sporadic 3-transposition group discovered by B. Fischer. It is known [RSt] that M(24) acts flag-transitively on a geometry G(M(24)) whose diagram is δ and in which the residue of an element of type 1 is the dual polar space associated with the group O_{7}(3). For the action of G ≅ M(24) on G(M(24)) we have Ḡ_{1} ≅ O_{7}(3), K_{1} ≅ 3^{7}. Moreover, K_{1} is the natural module for Ḡ_{1} and G_{1} does not split over K_{1}. The parabolic G_{4} is a split extension of K_{4} ≅ 3_{+}^{1+10} : 2 by Ḡ_{4} ≅ U_{5}(2) : 2. Every element α in G(M(24)) can be identified with certain elementary abelian 3-subgroup V (α) in M(24) in such a way that the order of V (α) is 3, 3^{2}, 3^{3} and 3^{7} for α being of type 1, 2, 3 and 4, respectively; the stabilizer of α in G coincides with N_{G}(V (α)) and α is incident to β if and only if either V (α) ≤ V (β) or V (β) ≤ V (α). The commutator subgroup M(24)′ also acts flag-transitively on G(M(24)). With respect to the action of G ≅ M(24)′ on the geometry we have Ḡ_{1} ≅ Ω_{7}(3), K_{1} ≅ 3^{7}, Ḡ_{4} ≅ U_{5}(2) : 2, K_{4} ≅ 3_{+}^{1+10}.

A triple cover 3 · M(24) of M(24) is the normalizer of a subgroup of order 3 in the Monster group F_{1} and G(M(24)) appears as a subgeometry in the Buekenhout – Fischer geometry G(F_{1}) of the Monster [BF] which has the following diagram:

1∘ c _{9}∘  _{3}∘  _{3}∘.

Both G(M(24)) and G(F_{1}) belong to the family of c-extended classical dual polar spaces and their characterization in terms of diagrams is very important for classification of all flag-transitive extended dual polar spaces [Yosh]. In the present paper we prove the following.

Theorem 1 Let G be a geometry with the diagram δ admitting a flag-transitive automorphism group G. Assume that the stabilizer of a maximal flag in G is finite. Then G ≅ G(M(24)) and G is isomorphic either to M(24) or to M(24)′.

The geometry G(F_{1}) has been characterized in [IM] and Theorem 1 was crucial for this characterization.

2 The structure of residues

For the rest of the paper let G be a geometry with the diagram δ and G be a flagtransitive automorphism group of G with finite stabilizers of maximal flags. We adopt the notation introduced in the previous section.

Consider the residue G_{1} of α_{1} in G. Then G_{1} is a flag-transitive C_{3}-geometry; its residual projective planes have order 3 and hence they are Desarguesian. The available results on flag-transitive C_{3}-geometries imply the following (cf. Section 14 in [Pas]).

Lemma 2.1 G_{1} is the (dual) polar space associated with a 7-dimensional GF (3)-space equipped with a non-singular orthogonal form. Ḡ_{1} is either the special orthogonal group O_{7}(3) or its index 2 simple commutator subgroup Ω_{7}(3).

Let us recall some well known properties of the dual polar space G_{1}. Let V = V (α_{1}) be a 7-dimensional GF (3)-space with a non-singular quadratic form ϕ on it. Then the elements of type 2, 3 and 4 in G_{1} are the 3-, 2- and 1-dimensional subspaces in V which are totally singular with respect to ϕ. Two subspaces are incident if one of them contains the other one. It is convenient to associate with G_{1} a graph Δ called the dual polar space graph. The vertices of Δ are maximal (which means 3-dimensional) totally isotropic subspaces of V with respect to ϕ with two such subspaces adjacent if their intersection is of dimension 2. In other terms the vertices of Δ are the elements of type 2 in G_{1} and two of them are adjacent if they are incident to a common element of type 3. For a vertex β of Δ and an integer i let Δ_{i}(β) denote the set of vertices in Δ which are at distance i from β in the natural metric of Δ. It is known [BCN] that Δ is distance-transitive on 1120 vertices with the intersection array {39, 36, 27; 1, 4, 13}. In particular the valency of Δ is 39 and the diameter is 3. Both O_{7}(3) and Ω_{7}(3) act primitively on the vertex set of Δ.

The following two lemmas come from the standard information on the actions of O_{7}(3) and Ω_{7}(3) on the associated dual polar space (compare [Atlas]).

Lemma 2.2 Ḡ_{14} ≅ O_{5}(3) ≅ Ω_{5}(3) : 2 and O_{3}(K_{14}/K_{1}) ≅ 3^{5} is the irreducible orthogonal module for Ḡ_{14}. If Ḡ_{1} ≅ Ω_{7}(3) then K_{14}/K_{1} = O_{3}(K_{14}/K_{1}) and if Ḡ_{1} ≅ O_{7}(3) then K_{14}/K_{1} is an extension of O_{3}(K_{14}/K_{1}) by a fixed-point free involution.

Lemma 2.3 Ḡ_{12} ≅ L_{3}(3) and O_{3}(K_{12}/K_{1}) ≅ 3^{3+3}. If Ḡ_{1} ≅ Ω_{7}(3) then K_{12}/K_{1} = O_{3}(K_{12}/K_{1}) and if Ḡ_{1} ≅ O_{7}(3) then K_{12}/K_{1} is an extension of O_{3}(K_{12}/K_{1}) by an involution commuting with a subgroup of order 3^{3} inside O_{3}(K_{12}/K_{1}). In any case the elementwise stabilizer of Δ_{1}(α_{2}) in K_{12}/K_{1} is elementary abelian of order 3^{3}, it acts faithfully on Δ_{2}(α_{2}) and on Δ_{3}(α_{2}) and it is acted on irreducibly by Ḡ_{12}.

The residue G_{4} of the element α_{4} in G is a flag-transitive c-extension of a classical generalized quadrangle. All such geometries are known (cf. [Yosh] and references therein). There is only one flag-transitive c-extension of the generalized quadrangle G_{14} of the orthogonal group O_{5}(3). This extension has 176 elements of type 1 and it possesses two flag-transitive automorphism groups: U_{5}(2) and U_{5}(2) : 2. In the former group the stabilizer of an element of type 1 induces on the residual generalized quadrangle the simple group Ω_{5}(3) and this is in a contradiction with Lemma 2.2. So we have the following :

Lemma 2.4 Ḡ_{4} ≅ U_{5}(2) : 2.

Considering the action of Ḡ_{4} on the residual geometry G_{4} (compare [Atlas]) we obtain :

Lemma 2.5 G_{14}/K_{4} ≅ (3 × Ω_{5}(3)) : 2 and K_{14}/K_{4} is of order 3.

Now we are in a position to prove the following :

Lemma 2.6 K_{1}/(K_{1} ∩ K_{4}) has order 3, in particular K_{1} ≠ 1.

Proof: By definition K_{1}/(K_{1} ∩ K_{4}) is the action induced by K_{1} on the residue G_{4} of α_{4} and it is contained in the action K_{14}/K_{4} induced by K_{14} on this residue. By Lemma 2.5 the latter action has order 3. Suppose that K_{1}/(K_{1} ∩ K_{4}) is trivial. Then O_{3}(K_{14}/K_{1}) must induce an order 3 action on G_{4} but since O_{3}(K_{14}/K_{1}) has order 3^{5} and it is irreducible under Ḡ_{14} ≅ O_{5}(3) (cf. Lemma 2.2) this is impossible.

Let Γ be the collinearity graph of G which means that the vertices of Γ are the elements of type 1 in G and two of them are adjacent if and only if they are incident to a common element of type 2.

It follows directly from the diagram δ of G that every element of type 2 is incident to exactly two elements of type 1. Hence an element of type 2 determines an edge of Γ. Since G_{1} acts primitively on the set of elements of type 2 in G_{1}, different elements of type 2 determine different edges of Γ. In particular there is a bijective mapping ψ of the vertex set of the dual polar space graph Δ associated with α_{1} onto the set Γ(α_{1}) of vertices in Γ adjacent to α_{1} and ψ commutes with the action of G_{1}.

Lemma 2.7 Let β and γ be adjacent vertices of Δ. Then the images of β and γ under ψ are adjacent in Γ.

Proof: Since β and γ are adjacent in Δ, they are incident to a common element of type 3 in G_{1}, say to α_{3}. But the residue of the flag {α_{3}, α_{4}} is formed by vertices and edges of a complete graph on 5 vertices so the result follows.

In what follows we will identify the vertices and the edges of Γ with the elements in G of type 1 and 2, respectively.

Since the automorphism group of the residual geometry G_{1} acts faithfully on the set of elements of type 2 in the residue, we observe that K_{1} coincides with the elementwise stabilizer in G_{1} of the vertices from Γ(α_{1}).

Lemma 2.8 K_{1} is a non-trivial 3-group.

Proof: K_{1} ≠ 1 by Lemma 2.6. As we observed in the above paragraph, K_{1} coincides with the elementwise stabilizer in G of α_{1} and of all the vertices adjacent to α_{1} in Γ. On the other hand Γ is connected and G acts on it vertex-transitively. So to prove the lemma it is sufficient to show that K_{1} induces a 3-group on Γ(β) for every β ∈ Γ(α_{1}). In view of the obvious symmetry between α_{1} and β, by Lemmas 2.3 and 2.7 the action induced by K_{1} on Γ(β) has order 3^{3}, so the result follows.

In order to identify K_{1} we apply the following result proved by U. Meierfrankenfeld using the pushing-up methods developed in his Ph D. Thesis [Meie1] and the classification [Meix] of the failure factorization modules for Lie type groups in odd characteristic.

Proposition 2.9 [Meie2] Let L be a finite group and suppose that

(i) for Q = O_{3}(L) we have L̄ := L/Q ≅ O_{7}(3) or Ω_{7}(3);

(ii) for V being the natural module for L̄ and P being the full preimage of the stabilizer in L̄ of a maximal totally isotropic subspace in V; if N ≤ Q is normal in L and characteristic in P, then N = 1.

Then Q is either trivial or isomorphic to V, that is to the natural module of L̄.

Lemma 2.10 K_{1} ≅ V ≅ 3^{7} is the natural module for Ḡ_{1}.

Proof: We are going to apply Proposition 2.9 for L = G_{1}, P = G_{12} and Q = K_{1}. Let N be a subgroup in K_{1} which is normal in G_{1} and characteristic in G_{12}. Since G_{2} contains G_{12} as a normal subgroup (of index 2), N must be normal in G_{2}. Now, G_{1} and G_{2} generate G since both G and Γ are connected. So N fixes a vertex of Γ and it is normal in the group G which acts on Γ vertex-transitively. This means that N fixes every vertex of Γ. Since the action of G on Γ is faithful, we have N = 1 and the result follows.

Lemma 2.11 The following assertions hold:

(i) either G_{1} ≅ 3^{7}.Ω_{7}(3) or G_{1} ≅ 3^{7}.O_{7}(3);

(ii) if G_{1} ≅ 3^{7}.Ω_{7}(3) then G_{1} possesses an automorphism τ such that ⟨G_{1}, τ⟩ ≅ 3^{7}.O_{7}(3).

Proof: The assertion (i) follows from Lemmas 2.1 and 2.10.

The assertion (ii) is trivial if G_{1} splits over K_{1}. By [Küs] there is a unique isomorphism type of non-split extensions of Ω_{7}(3) by the natural module. This extension is contained in M(24)′ and the automorphism τ is realized in M(24).

Thus we have established that K_{1} is canonically isomorphic to the 7-dimensional GF (3)-module V equipped with a non-singular quadratic form ϕ and the residual geometry G_{1} can be naturally defined in terms of the polar space (V, ϕ). In what follows we will identify V with K_{1}. For an element β ∈ G_{1} let V (β) denote the subspace of V which corresponds to β. The dimension of V (β) is 3, 2 and 1 for β being of type 2, 3 and 4, respectively. Then G_{1j} is the full normalizer of V (α_{j}) in G_{1}. For an arbitrary subspace U in V let U^{⊥} denote the orthogonal complement of U with respect to ϕ.

Let us now specify K_{4}. We have that G_{14} is the normalizer in G_{1} of the subspace V (α_{4}) which is 1-dimensional. It is a standard fact about orthogonal groups that under the action of G_{14}/K_{1} the module V is uniserial, V (α_{4}) is its unique 1dimensional submodule and V (α_{4})^{⊥} is the unique submodule of codimension 1. This and Lemma 2.6 give :

Lemma 2.12 K_{1} ∩ K_{4} = V (α_{4})^{⊥}.

The action induced by G_{14} on G_{4} is isomorphic to (3 × Ω_{5}(3)) : 2 (cf. Lemma 2.5) and K_{1} induces an action of order 3. Hence G_{14}/⟨K_{1}, K_{4}⟩ ≅ O_{5}(3) ≅ Ω_{5}(3) : 2. This and Lemma 2.2 imply :

Lemma 2.13 ⟨K_{1}, K_{4}⟩ = K_{14}.

Lemma 2.14 O_{3}(K_{4}) ≅ 3_{+}^{1+10}. If G_{1} ≅ 3^{7}.Ω_{7}(3) then K_{4} = O_{3}(K_{4}) and if G_{1} ≅ 3^{7}.O_{7}(3) then K_{4} is an extension of O_{3}(K_{4}) by an involution acting fixed-point freely on O_{3}(K_{4})/Z(O_{3}(K_{4})).

Proof: By Lemmas 2.12 and 2.13 O_{3}(K_{4}) has index 3 in O_{3}(K_{14}) and the latter is of order 3^{12}. Hence |O_{3}(K_{4})| = 3^{11}. Clearly V (α_{4}) is in the center of K_{4}. We have that O_{3}(K_{14}/K_{1}) acts faithfully on V (α_{4})^{⊥} and centralizes V (α_{4})^{⊥}/V (α_{4}). Moreover, both O_{3}(K_{14}/K_{1}) and V (α_{4})^{⊥}/V (α_{4}) are elementary abelian of order 3^{5} and are acted on irreducibly by Ḡ_{14} ≅ O_{3}(5). Hence O_{3}(K_{4}) is extraspecial of plus type. Suppose that G_{1} ≅ 3^{7}.O_{7}(3). Then there exists an involution τ in K_{14} \ O_{3}(K_{14}) which centralizes V (α_{4}) and acts fixed-point freely both on O_{3}(K_{14}/K_{4}) and on V (α_{4})^{⊥}/V (α_{4}) so the result follows.

Suppose that G_{1} ≅ 3^{7}.O_{7}(3) and let τ be an involution from K_{4} \ O_{3}(K_{4}). Then C_{G_{4}}(τ) is an extension of ⟨τ⟩×V (α_{4}) by Ḡ_{4} ≅ U_{5}(2) : 2. Since the Schur multiplier of U_{5}(2) is trivial we have C_{G_{4}}(τ) ≅ ⟨τ⟩ × V (α_{4}) × U_{5}(2) : 2 and G_{4} splits over O_{3}(K_{4}).

Proposition 2.15 Let G be a flag-transitive geometry with diagram δ and let G be the full automorphism group of G. Then G_{1} ≅ 3^{7}.O_{7}(3) and G_{4} ≅ 3_{+}^{1+10} : (2 × U_{5}(2) : 2).

Proof: By Lemma 2.11 and the paragraph before the proposition all we have to show is that if G_{1} ≅ 3^{7}.Ω_{7}(3) then G is not the full automorphism group of G. By a standard argument the elements of G can be identified with the right cosets of the subgroups G_{i} in G so that two elements are incident if and only if the intersection of the cosets is non-empty. So to prove the claim it is sufficient to show that G possesses a suitable automorphism τ which normalizes each G_{i}. By Lemma 2.11 (ii) there is an automorphism τ of G_{1} such that ⟨G_{1}, τ⟩ ≅ 3^{7}.O_{7}(3). We can choose τ in such a way that it normalizes G_{14} and centralizes a complement F to K_{14} = O_{3}(K_{14}) in G_{14} (cf. Lemma 2.2). In this case τ acts fixed-points freely on O_{3}(K_{14}/K_{1}) and on (K_{1}∩K_{4})/Z(K_{4}). Hence τ acts fixed-points freely on K_{4}/Z(K_{4}). Then C_{G_{14}}(τ)/Z(K_{4}) is a complement to O_{3}(K_{4})/Z(K_{4}) in G_{14}/Z(K_{4}). Since G_{14} contains a Sylow 3subgroup of G_{4}, by Gaschütz’s theorem and since the Schur multiplier of U_{5}(2) is trivial, there is a complement H to O_{3}(K_{4}) in G_{4} which contains F. Then we can extend the action of τ on G_{14} to an automorphism of G_{4} by declaring that τ commutes with H. The geometry G is connected, hence G_{1} and G_{4} generate G and we can extend τ to an automorphism of G. Since τ fixes the residue G_{4} elementwise, it clearly normalizes G_{2} and G_{3}.

For the rest of the paper G is assumed to be the full automorphism group of G. By the above proposition in this case G_{1} ≅ 3^{7}.O_{7}(3) and G_{4} ≅ 3_{+}^{1+10} : (2 × U_{5}(2) : 2).

The group U_{5}(2) has only one faithful 10-dimensional GF (3)-representation (compare [JLPW]). This representation is irreducible which means that there is a unique invariant quadratic form on the corresponding module. This observation and Proposition 2.15 specify G_{4} up to isomorphism. By Proposition 2.15 and [Küs] the isomorphism type of G_{1} depends only on whether or not it splits over K_{1}. On the other hand there is a unique way to specify G_{14} as a subgroup in G_{4}. Namely, let U ≅ U_{5}(2) be a complement to O_{3}(G_{4}) in O^{2}(G_{4}) (one can easily see that there is one class of such complements) and let λ be an element of order 3 in U such that C_{U}(λ) ≅ ⟨λ⟩×Ω_{5}(3). Then up to conjugacy in G_{4} we have K_{1} = ⟨λ, C_{O_{3}(G_{4})}(λ)⟩ and G_{14} = N_{G_{4}}(K_{1}). Since G_{14} contains a Sylow 3-subgroup of G_{1}, by Gaschütz’s theorem G_{1} splits over K_{1} if and only if G_{14} does so. By the above argument the isomorphism type of G_{14} and the location of K_{1} in G_{14} are uniquely specified and hence the isomorphism type of G_{1} is also uniquely determined. Clearly, G_{1} must be isomorphic to the parabolic subgroup associated with the action of M(24) on G(M(24)), that is to the non-split extension. So the amalgam consisting of the subgroups G_{1} and G_{4} is isomorphic to the one associated with the action of M(24) on G(M(24)).

Let β ∈ Γ(α_{1}) and K(β) be the elementwise stabilizer in G of the vertices from Γ(β) (which is a conjugate of K_{1}). By Lemmas 2.3 and 2.8 K(β) induces on the dual polar space graph Δ an action of order 3^{3} which is faithful on Δ_{2}(β) and on Δ_{3}(β). This implies the following two lemmas :

Lemma 2.16 The intersection of K(β) and K_{1} has order 3^{4} and it coincides with V (β)^{⊥}.

Lemma 2.17 Let ψ be the mapping of Δ onto Γ(α_{1}) which commutes with the action of G_{1}. Then β, γ ∈ Δ are adjacent in Δ if and only if their images under ψ are adjacent in Γ.

By the above lemma locally Γ is the dual polar space graph Δ. This implies that the subgraph Ξ = Ξ(α_{4}) in Γ induced by the vertices incident to α_{4} is strongly regular with parameters v = 176, k = 40, λ = 12, μ = 8 and Ξ ∩ Γ(α_{1}) are all the maximal totally isotropic subspaces in V which contain V (α_{4}).

3 Reconstruction of Σ_{3} × D_{4}(3).Σ_{3}

As above let G be the full automorphism group of a geometry G with diagram δ. To prove Theorem 1 we have to show that G ≅ M(24). Recall that M(24) is generated by a conjugacy class of so-called 3-transpositions. If m is a 3-transposition then it is of order 2 and C_{M(24)}(m) ≅ ⟨m⟩ × M(23) where M(23) is another 3-transposition group discovered by B. Fischer. Two distinct 3-transpositions either commute or generate an Σ_{3}-subgroup whose normalizer T in M(24) is isomorphic to Σ_{3} × D_{4}(3).Σ_{3}. The aim of the present section is to reconstruct T inside G.

Under the action of Ḡ_{1} ≅ O_{7}(3) the set of 1-dimensional subspaces of V (the projective points) split into 3 orbits: S^{0} (isotropic points), S^{+} (plus points) and S^{−} (minus points). The orbit S^{0} naturally corresponds to the elements of type 4 in G_{1}. Let W ∈ S^{+}. Then W is non-singular, V = W ⊕ W^{⊥} and W^{⊥} contains a maximal totally isotropic subspace of V. Let M = C_{G_{1}}(W) and M̄ = M/K_{1}. Then then the action of M̄ ≅ O_{6}^{+}(3) ≅ L_{4}(3).2_{2} on W^{⊥} is faithful and coincides with the automorphism group of the restriction of ϕ to W^{⊥}. Further N_{G_{1}}(W) is an extension of M by an involution m which centralizes W^{⊥} and inverts W.

There are 80 maximal totally isotropic subspaces in W^{⊥} which form a single Morbit and under the action of O^{2}(M) ≅ 3^{7}.L_{4}(3) they split into two orbits of length 40 each. If the elements from different orbits are assumed to be of different types, then together with the set of 1-dimensional singular subspaces contained in W^{⊥} they form the parabolic geometry of O^{2}(M)/K_{1} ≅ L_{4}(3) with respect to the natural incidence relation. This is the geometry of type A_{3} (or D_{3}) over GF (3) and we will denote it by H(α_{1}, W).

We assume that V (α_{2}) is contained in W^{⊥}. Then V (α_{3}) is also in W^{⊥} and there is exactly one maximal totally isotropic subspace Z in W^{⊥} such that V (α_{2})∩Z = V (α_{3}). Moreover, V (α_{2}) and Z are in different O^{2}(M)-orbits and {V (α_{2}), V (α_{4}), Z} is a maximal flag in the geometry H(α_{1}, W).

Let β and γ be the vertices from Γ(α_{1}) such that {α_{1}, β} = α_{2}, V ({α_{1}, γ}) = Z (here {α_{1}, β} and {α_{1}, γ} are edges of Γ also considered as elements of type 2 in G). Since dim (V (α_{2}) ∩ Z) = 2, α_{1}, β and γ form a triangle in Γ by Lemma 2.17.

Lemma 3.1 The setwise stabilizer in C_{G_{4}}(W) of the triangle {α_{1}, β, γ} induces the symmetric group Σ_{3} on the vertices of the triangle.

Proof: Let Θ be the set of vertices of Γ such that for λ ∈ Θ, W is contained in O_{3}(G_{λ}) and is of plus type with respect to the unique quadratic form on O_{3}(G_{λ}) preserved by G_{λ}. Since G_{1} acts transitively on S^{+}, C_{G}(W) acts transitively on Θ. Since O^{2}(M) acts flag-transitively on H(α_{1}, W) and M/O^{2}(M) induces a diagram automorphism of the geometry, the result follows.

Let a_{1} = β, a_{2} = α_{4}, a_{3} = γ, a_{4} = α_{1}. Let P_{i} be the intersection of the centralizers C_{G_{a_{j}}} (W) for all 1 ≤ j ≤ 4, j ≠ i, in other words P_{i} is the largest subgroup in G which centralizes W and stabilizes a_{j} for every 1 ≤ j ≤ 4, j ≠ i. Put P̄_{i} = P_{i}/O_{3}(P_{i}) and P̄_{ij} = ⟨P_{i}, P_{j}⟩/O_{3}(⟨P_{i}, P_{j}⟩), 1 ≤ i, j ≤ 4.

Lemma 3.2 P̄_{i} ≅ GL_{2}(3) for 1 ≤ i ≤ 4. If i ≠ 2 and j ≠ 2 then P̄_{ij} ≅ (SL_{2}(3) ∗ SL_{2}(3)).2 and P̄_{2i} ≅ L_{3}(3) for i = 1, 3 and 4.

Proof: Let 1 ≤ i < j ≤ 4. By Lemma 3.1 we can assume that i ≠ 4 and j ≠ 4. In this case P_{i}, P_{j} ≤ O^{2}(M) where M = C_{G_{1}}(W) as above. Then P_{i} and P_{j} are minimal parabolic subgroups associated with the (unfaithful) action of O^{2}(M) on H(α_{1}, W) and the result follows from the standard facts about the action of L_{4}(3) on its parabolic geometry (that is on the projective space P G(3, 3)).

Notice that for every 1 ≤ i ≤ 4, P_{i} is contained either in C_{G_{1}}(W) or in C_{G_{4}}(W) or in both.

Lemma 3.3 Let T be the subgroup in G generated by N_{G_{1}}(W) and N_{G_{4}}(W). Then T ≅ Σ_{3} × D_{4}(3).Σ_{3}.

Proof: Let P = ⟨P_{i} | 1 ≤ i ≤ 4⟩. It is easy to see that W is the largest subgroup which is contained and normal in every P_{i} for 1 ≤ i ≤ 4. This and Lemma 3.2 imply that {P_{i} | 1 ≤ i ≤ 4} is a parabolic system of type D_{4} over GF (3) and by [Tit] P/W ≅ D_{4}(3). Since the 3-part of the Schur multiplier of D_{4}(3) is trivial, P ≅ 3 × D_{4}(3). There is an element τ_{13} ∈ M \ O^{2}(M) which normalizes P_{2} and P_{4} and permutes P_{1} and P_{3}. By Lemma 3.1 there are similar elements τ_{14} and τ_{34} and these three elements generate Σ_{3} modulo P. This shows that the centralizers of W in G_{1} and G_{4} generate in G a subgroup T_{1} ≅ 3 × D_{4}(3).Σ_{3}. Finally T contains T_{1} with index 2 and since the outer automorphism group of D_{4}(3) (isomorphic to Σ_{4}) does not contain Σ_{3} × 2, the result follows.

4 Reconstruction of 2 × M(23)

In this section we reconstruct in G a subgroup Q isomorphic to the centralizer 2 × M(23) of a 3-transposition in M(24). The reconstruction is based on enumeration by a computer of the cosets of a subgroup in a group given in terms of generators and relations.

We start with the Steinberg presentation for D ≅ D_{4}(3) to obtain generators and relations for T ≅ Σ_{3} × D_{4}(3).Σ_{3}.

We use the numbering for the fundamental system Π = {1, 2, 3, 4} in the root system of type D_{4} as in the previous section, so that 2 is the central node in the Dynkin diagram. Then the positive roots correspond to multisets from the following family (here, for instance, 12234 stays for the multiset {1, 2, 2, 3, 4}):

J = {1, 2, 3, 4, 12, 23, 24, 123, 124, 234, 1234, 12234}.

With every I ∈ J a subgroup X_{I} of order 3 in D ≅ D_{4}(3) (a root subgroup) is associated so that such subgroups taken for all I ∈ J generate a Sylow 3-subgroup of D. The root subgroups satisfy the following commutator relations:

[X_{I}, X_{J}] = \{ \begin{matrix}X_{I∪J}, & if I ∪ J ∈ J; \\ 1, & otherwise. \end{matrix}

Here ∪ is understood as the union operation on multisets.

For every fundamental root i ∈ Π we have a Weyl element w_{i} in D, which is of order 4 and together with X_{i} generate in D a subgroup isomorphic to SL_{2}(3). In addition for I ∈ J and I ≠ {i} we have

X_{I}^{w_{i}} = \{ \begin{matrix}X_{I∪\{i\}}, & if I ∪ \{i\} ∈ J; \\ X_{I\\{i\}}, & if I \ \{i\} ∈ J; \\ X_{I}, & otherwise. \end{matrix}

The product of two Weyl elements w_{i} and w_{j} is an element of order 3 if i and j are adjacent in the Dynkin diagram of D_{4} and they commute otherwise.

To obtain a concrete realization of the Steinberg presentation for D_{4}(3), in each root subgroup X_{I} for I ∈ J we have to choose a generator x_{I} (a root element) and the concrete form of the relations depends on this choice. We have chosen the generators so that the non-trivial commutator relations among the root elements are determined by the following ones:

[x_{1}, x_{2}]x_{12}, [x_{3}, x_{2}]x_{23}, [x_{1}, x_{23}]x_{123}, [x_{1}, x_{24}]x_{124}, [x_{1}, x_{234}]x_{1234}, [x_{2}, x_{4}]x_{24},

[x_{2}, x_{1234}]x_{12234}, [x_{3}, x_{24}]x_{234}, [x_{3}, x_{124}]x_{1234}, [x_{4}, x_{23}]x_{234}^{−1}, [x_{4}, x_{123}]x_{1234}^{−1},

[x_{234}, x_{12}]x_{12234}, [x_{124}, x_{23}]x_{12234}, [x_{123}, x_{23}]x_{12234}, [x_{12}, x_{4}]x_{124} , [x_{12}, x_{3}]x_{123}^{−1}.

For 1 ≤ i ≤ 4 the SL_{2}(3)-subgroup generated by x_{i} and w_{i} is given by the following relations:

(x_{i}^{w_{i}}x_{i}^{−1})^{4}, [(x_{i}^{w_{i}}x_{i}^{−1})^{2}, x_{i}], (x_{i}^{w_{i}}x_{i}^{−1})^{2}w_{i}^{2}, w_{i}^{−1}x_{i}x_{i}^{w_{i}}x_{i}.

The center of ⟨x_{i}, w_{i}⟩ is generated by w_{i}^{2}.

For 1 ≤ i ≤ 4 and I ∈ J the Weyl element w_{i} commutes with the root element x_{I} if I ∪{i} ∉ J and I \{i} ∉ J and the non-trivial conjugation relations are determined by the following:

x_{2}^{w_{1}}x_{12}^{−1}, x_{23}^{w_{1}}x_{123}^{−1}, x_{24}^{w_{1}}x_{124}^{−1}, x_{234}^{w_{1}}x_{1234}^{−1}, x_{1}^{w_{2}}x_{12}, x_{3}^{w_{2}}x_{23}, x_{1234}^{w_{2}}x_{12234}^{−1}, x_{4}^{w_{2}}x_{24}^{−1},

x_{12}^{w_{3}}x_{123}^{−1}, x_{2}^{w_{3}}x_{23}^{−1}, x_{24}^{w_{3}}x_{234}^{−1}, x_{124}^{w_{3}}x_{1234}^{−1}, x_{2}^{w_{4}}x_{24}, x_{23}^{w_{4}}x_{234}, x_{12}^{w_{4}}x_{124}, x_{123}^{w_{4}}x_{1234}.

The above relations (where it is also assumed that x_{I}^{3} = w_{i}^{4} = 1 for I ∈ J and 1 ≤ i ≤ 4) constitute the Steinberg presentation for the universal Lie group of type D_{4} over GF (3), that is of an extension of D_{4}(3) by a center of order 4. In order to obtain a presentation for the simple group we put the additional relations

w_{1}^{2} = w_{3}^{2} = w_{4}^{2}.

Now to obtain a presentation for T ≅ Σ_{3}×D_{4}(3).Σ_{3} we have to add automorphisms of D inducing Σ_{3} and elements generating another copy of Σ_{3} as a direct factor. It is clear that D = ⟨x_{i}, w_{j} | 1 ≤ i ≤ 4, 1 ≤ j ≤ 4⟩ so it is sufficient to define the automorphisms on the x_{i} and w_{j}. The automorphisms generating Σ_{3} correspond to symmetries of the Dynkin diagram and can be generated by elements t and ν satisfying the relations:

t^{2}, w_{1}^{t}w_{3}^{−1}, [t, w_{2}], [t, w_{4}], x_{1}^{t}x_{3}^{−1}, [t, x_{2}], [t, x_{4}]

and

ν^{3}, (tν)^{2}, w_{1}^{ν}w_{3}^{−1}, w_{3}^{ν}w_{4}, x_{1}^{ν}x_{3}^{−1}, x_{3}^{ν}x_{4}, x_{2}^{ν}x_{2}^{−1}, w_{2}^{ν}w_{2}^{−1}.

The Σ_{3} direct factor comes by adjoining the generators m and ϱ satisfying the following obvious relations, where 1 ≤ i ≤ 4 and I ∈ J:

m^{2}, [m, w_{i}], [m, x_{I}], [m, t], [m, ν];
ϱ^{3}, (ϱm)^{2}, [ϱ, w_{i}], [ϱ, x_{I}], [ρ, t], [ϱ, ν].

If P_{i} is the minimal parabolic subgroup of D ≅ D_{4}(3) as in the previous section, then O^{2}(P_{i}) is generated by the root elements x_{I} for all I ∈ J together with the Weyl element w_{i}, and the root elements other than x_{i} generate O_{3}(P_{i}), 1 ≤ i ≤ 4. The elements t and ν normalize P_{2} and conjugate the parabolics P_{1}, P_{3} and P_{4} in the way they conjugate the corresponding Weyl elements.

We are going to produce Q ≅ 2 × M(23) as the subgroup in G generated by C_{T} (m) and C_{G_{4}}(m). The former of the centralizers is isomorphic to 2 × D_{4}(3).Σ_{3}; it is generated by all the above generators of T except for ϱ subject to all the defining relations of T, except for the relations which involve ϱ.

Let us discuss C_{G_{4}}(m). First of all, it follows directly from the construction of T in the previous section that T ∩ G_{4} contains the normalizer T_{2} in T of the maximal parabolic subgroup of D_{4}(3) generated by P_{1}, P_{3} and P_{4}. Since T_{2} is maximal in T we have T_{2} = T ∩G_{4}. In order to modify the above presentation for T into a presentation for T_{2} we have to drop the generator w_{2} together with all the relations which involve w_{2} and to introduce a new generator h (which corresponds to the square of w_{2} in T) and the following set of relations:

h^{2}; (w_{i}h)^{2} for i = 1, 3, 4; [h, t], [h, ν], [h, m], [h, ρ];
(x_{I}h)^{2}, if I ∪ {2} ∈ J or I \ {2} ∈ J and [x_{I}, h], otherwise.

Similarly by dropping the generator ϱ together with the relations which involve ϱ we obtain a presentation for C_{T_{2}}(m).

One can see in [Atlas] that C_{T_{2}}(m) ≅ ⟨m⟩ × 3_{+}^{1+8}2_{−}^{1+6}3_{+}^{1+2}(2 × Σ_{3}), in particular O_{3}(C_{T_{2}}(m)) is extraspecial of order 3^{9}. Let us locate m and C_{T_{2}}(m) in G_{4}. Comparing the 3-parts in the orders of C_{T_{2}}(m) and U_{5}(2) : 2 × 2 ≅ G_{4}/O_{3}(G_{4}) we see that O_{3}(C_{T_{2}}(m)) is contained in O_{3}(G_{4}) which means that m has 8-dimensional centralizer in O_{3}(G_{4})/Z(O_{3}(G_{4})) ≅ 3^{10}. Comparing this with [JLPW], we conclude that m is the product of an involution from O_{3,2}(G_{4}) (acting fixed-point freely on O_{3}(G_{4})/Z(O_{3}(G_{4})) and a central involution from O^{2}(G_{4}) ≅ 3_{+}^{1+10}.U_{5}(2). This shows (compare [Atlas]) that C_{G_{4}}(m) ≅ ⟨m⟩.3_{+}^{1+8}2_{−}^{1+6}3_{+}^{1+2}GL_{2}(3), which means that C_{G_{4}}(m) contains C_{T_{2}}(m) with index 4. In more details the situation is the following. Let R = O_{3,2}(C_{G_{4}}(m)) ≅ ⟨m⟩ × 3_{+}^{1+8}2_{−}^{1+6}. Then C_{G_{4}}(m)/R and C_{T_{2}}(m)/R are, respectively, a maximal 3-local parabolic subgroup and the corresponding Borel subgroup in Out(R/O_{3}(R)) ≅ Sp_{4}(3).

Thus [C_{G_{4}}(m) : C_{T_{2}}(m)] = 4 and C_{G_{4}}(m) ≅ ⟨m⟩.3_{+}^{1+8}2_{−}^{1+6}3_{+}^{1+2}GL_{2}(3) while C_{T_{2}}(m) ≅ ⟨m⟩ × 3_{+}^{1+8}2_{−}^{1+6}3_{+}^{1+2}(2 × Σ_{3}). Let us introduce some subgroups in C_{T_{2}}(m):

X_{1} = ⟨x_{12234}, x_{1234}, x_{123}, x_{124}, x_{234}, x_{23}, x_{12}, x_{24}, x_{2}⟩,
X_{2} = ⟨w_{1}, w_{3}, w_{4}, w_{1}^{x_{1}}, w_{3}^{x_{3}}, w_{4}^{x_{4}}⟩,
X_{3} = ⟨x_{1}x_{3}x_{4}^{−1}, ν, x_{1}x_{4}⟩, X_{4} = ⟨t, h, x_{4}⟩.

Then X_{1} = O_{3}(C_{T_{2}}(m)) ≅ 3_{+}^{1+8}, X_{2} is a complement to X_{1} × ⟨m⟩ in O_{3,2}(C_{T_{2}}(m)), which is isomorphic to the central product of three copies of the quaternion group, that is to 2_{−}^{1+6}, X_{3} ≅ 3_{+}^{1+2} and X_{4} ≅ (2 × Σ_{3}). So that C_{T_{2}}(m) = ⟨m⟩ × X_{1}X_{2}X_{3}X_{4}. Our intention is to obtain a presentation for C_{G_{4}}(m) by adjoining to C_{T_{2}}(m) an element t_{1} so that ⟨t_{1}, x_{4}, h⟩(⟨m⟩X_{1}X_{2}X_{3})/(⟨m⟩X_{1}X_{2}X_{3}) ≅ GL_{2}(3). To do so we have to determine the action of t_{1} on X_{3}, X_{2} and X_{1}. After that, we will find a relation between t_{1} and w_{2}. Eventually we obtain generators and relations for the subgroup Q in G generated by C_{T} (m) and C_{G_{4}}(m) and identify Q with 2 × M(23) using coset enumeration.

The additional generator t_{1} can be taken so that ⟨t_{1}, x_{4}, h⟩ ≅ GL_{2}(3) and the following relations hold:

(t_{1}^{x_{4}})^{2}t_{1}^{2}, [t_{1}, t_{1}^{x_{4}}]t_{1}^{2}, t_{1}^{x_{4}^{−1}}t_{1}t_{1}^{x_{4}}, [t_{1}, t], t_{1}^{h}t_{1}.

Then t_{1} is of order 4 and t_{1}^{2} must be equal to t, tm, w_{1}^{2}t, or w_{1}^{2}tm. Since w_{1}^{2} is in the center of X_{2}X_{3}, the action of t_{1} on X_{2}X_{3} is independent on the exact form of t_{1}^{2}. Since ⟨t_{1}, x_{4}, h⟩ ≅ GL_{3}(2) is the full outer automorphism group of X_{3}, the action of t_{1} on X_{3} is uniquely determined and we get the following relations :

[x_{1}x_{3}x_{4}^{−1}, t_{1}], (x_{1}x_{4})^{t_{1}}ν^{−1}x_{1}x_{3}x_{4}^{−1}.

Now we determine the action of t_{1} on X_{2}. One can see that C_{X_{2}}(⟨t, h⟩)/⟨w_{1}^{2}⟩ = ⟨w_{1}w_{3}, w_{1}^{2}⟩/⟨w_{1}^{2}⟩, which shows that ⟨t_{1}, t_{1}^{x_{4}}⟩ has to centralize w_{1}w_{3}. In addition [t_{1}, x_{1}x_{3}x_{4}^{−1}] = 1, t_{1}^{h} = t_{1}^{−1} and t_{1}^{2} ∈ {t, tm, w_{1}^{2}t, w_{1}^{2}tm} which imply the following relations :

[w_{1}w_{3}, t_{1}], [w_{1}^{x_{1}}w_{3}^{x_{3}}, t_{1}], [w_{1}^{2}, t_{1}], [t_{1}^{x_{1}x_{3}}, w_{1}w_{3}], [t_{1}^{x_{1}x_{3}}, w_{1}^{x_{1}}w_{3}^{x_{3}}],
[t_{1}^{x_{1}x_{3}}, w_{1}^{2}], (w_{3}^{t_{1}})^{−1}(w_{1}w_{3}w_{3}^{x_{3}}w_{4}^{−1}(w_{4}^{−1})^{x_{4}})^{x_{1}^{−1}x_{3}^{−1}x_{4}}w_{1}^{2}.

Now it is the right time to decide whether t_{1}^{2} = t, tm, w_{1}^{2}t, or w_{1}^{2}tm. We can calculate the action of t and w_{1}^{2}t on Y = X_{1}/⟨x_{12234}⟩. We get that |C_{Y} (w_{1}^{2}t)| = 3^{2} while |C_{Y} (t)| = 3^{6}. This shows that the images of w_{1}^{2} and t in U := O^{2}(G_{4})/O_{3}(G_{4}) ≅ U_{5}(2) are not conjugated. On the other hand we have the following :

Lemma 4.1 In the above notation the images of t_{1}^{2} and w_{1}^{2} in U are conjugated.

Proof: We identify t_{1}^{2} and w_{1}^{2} with their images in U and follow notation from [Atlas]. Then w := w_{1}^{2} is a central involution (that is of type 2a) in U and t_{1} is an element of order 4 in C_{U}(w) which commutes with the element x := x_{1}x_{3}x_{4}^{−1} which is also contained in C_{U}(w) and acts fixed-point freely on O_{2}(C_{U}(w))/⟨w⟩ ≅ 2^{6}. Then C_{U}(wx) has order 1 296 and hence wx is of type 6cd. This means that x is of type 3cd. The square of t_{1}^{2}x is an element of type 3cd, hence t_{1}^{2}x is of type 6cd, 6ef or 6jk. In the former two cases the cube is of type 2a and in the latter case it is of type 2b. But the elements of type 6jk are not squares in U. Hence t_{1}^{2} is of type 2a and the result follows.

Since |C_{Y} (w_{1}^{2})| = 1, we see that w_{1}^{2} must be conjugate to w_{1}^{2}t or w_{1}^{2}tm. As |C_{O_{3}(G_{4})/Z(O_{3}(G_{4}))}(w_{1}^{2})| = 3^{2} and [ρ, w_{1}^{2}t] = 1, we see that |C_{O_{3}(G_{4})/Z(O_{3}(G_{4}))}(w_{1}^{2}t)| ≥ 3^{3}. This gives that w_{1}^{2} is conjugate to w_{1}^{2}tm and then

t_{1}^{2} = w_{1}^{2}tm.

Next we determine the action of t_{1} on X_{1}. Set Y_{1} = [X_{1}, w_{1}^{2}t]. Then

Y_{1} = ⟨x_{1234}, x_{123}, x_{124}x_{234}, x_{12}x_{23}, x_{24}, x_{2}⟩.

Since t_{1}^{2} = w_{1}^{2}tm, h^{t_{1}} = hw_{1}^{2}tm, [x_{1}x_{3}x_{4}^{−1}, t_{1}] = 1, and t_{1}^{x_{4}^{−1}}t_{1}t_{1}^{x_{4}} = 1 we get the relations:

x_{12234}^{t_{1}}x_{12234}^{−1}, x_{1234}^{t_{1}}x_{123}x_{234}x_{124}, (x_{124}x_{234})^{t_{1}}(x_{1234}(x_{12}x_{23})^{−1}x_{24}^{−1})^{−1},
(x_{12}x_{23}x_{24})^{t_{1}}x_{123}, x_{24}^{t_{1}}(x_{124}x_{234}x_{2}^{−1})^{−1}, x_{2}^{t_{1}}(x_{1234}x_{23}^{−1}x_{12}^{−1})^{−1}.

Now using the fact that O_{2}(⟨t_{1}, x_{4}, h⟩) acts trivially on C_{X_{1}}(w_{1}^{2}tm), we see that

[t_{1}, x_{124}x_{234}^{−1}] = 1 = [t_{1}, x_{12}x_{23}^{−1}].

This completes the set of relations presenting C_{G_{4}}(m).

To generate the subgroup Q = ⟨C_{T} (m), C_{G_{4}}(m)⟩ we need all the generators of T except for ϱ together with the element t_{1}. Nevertheless, the relations given above still define an infinite group, namely the free amalgamated product of C_{T} (m) and C_{G_{4}}(m). In fact, for each of the above relations, all the generators involved in the relation are contained either in T or in G_{4} and none of the relations involve both t_{1} (which is not in T) and w_{2} (which is not in G_{4}). To find the missing relation we are going to analyse the parabolic subgroup G_{3} associated with the action of G on G.

First we are going to identify the intersections G_{i} ∩ T for 1 ≤ i ≤ 4. It follows directly from the construction of T in Section 3, that G_{1} ∩ T is the normalizer in T of the subgroup generated by the minimal parabolics P_{1}, P_{2} and P_{3}. This means that G_{1} ∩ T is generated by all the generators of T except for w_{4} and ν. Moreover, O_{3}(G_{1}) is contained in G_{1} ∩T and it is generated by the root elements x_{I} such that I contains 4, and ϱ. Similarly, directly from the construction of T we see that G_{2} ∩ T is the normalizer in T of the subgroup generated by P_{2} and P_{3}; O_{3}(G_{2}) ∩ T is generated by the root elements x_{I} such that I contains either 1 or 4 or both of them. This means that O_{3}(G_{2}) ∩ T is of order 3^{9} and hence its index in O_{3}(G_{2}) is 3^{4}. The element α_{2} is identified with the edge {α_{1}, β} of the collinearity graph Γ of G. Clearly G_{β} ∩ T is the normalizer in T of the subgroup generated by the minimal parabolics P_{2}, P_{3} and P_{4}, so that O_{3}(G_{β}) is generated by the root elements x_{I} for I containing 1. This and Lemma 2.16 imply that E := ⟨x_{124}, x_{1234}, x_{12234}, ϱ⟩ is the intersection of O_{3}(G_{1}) and O_{3}(G_{β}), hence N_{G_{i}}(E) = G_{i} ∩ G_{2} for i = 1, 3 and 4. Moreover it is easy to see, that in terms introduced in Section 2 we have V (α_{2}) = ⟨x_{124}, x_{1234}, x_{12234}⟩.

We have already observed that G_{4} ∩ T is the normalizer in T of the subgroup generated by P_{1}, P_{3} and P_{4}, in particular Z(O_{3}(G_{4})) = ⟨x_{12234}⟩.

Let us consider the intersection G_{3}∩T. We know that G_{3} is the setwise stabilizer in G of the subgraph Φ in the collinearity graph Γ of G, induced by the vertices (elements of type 1) incident to α_{3}. It follows directly from the diagram of G that Φ is a complete subgraph on 5 vertices and that it contains α_{1} and β. Since G_{13} is the full preimage in G_{1} of the stabilizer in Ḡ_{1} ≅ O_{7}(3) of an isotropic 2-dimensional subspace in V, one can see (compare [Atlas]) that G_{13}/O_{3}(G_{13}) ≅ Σ_{4} × GL_{2}(3) and that G_{13} induces the natural action of Σ_{4} on Φ \ {α_{1}}. This implies that G_{3}/O_{3}(G_{3}) ≅ Σ_{5} × GL_{2}(3) and that G_{3} induces on Ξ the natural action of Σ_{5}. By Lemma 2.17 Φ is a maximal clique in Γ and each triangle in Γ is contained in a unique clique of size 5 which is an image of Φ under G. So without loss of generality we assume that the triangle {α_{1}, β, γ} is contained in Φ (cf. the terms introduced before Lemma 3.1). Moreover this triangle is the intersection of Φ with the subgraph Θ defined in the proof of Lemma 3.1. This means that G_{3} ∩ T is the setwise stabilizer in G_{3} of the triangle {α_{1}, β, γ} and (G_{3} ∩ T)O_{3}(G_{3})/O_{3}(G_{3}) ≅ GL_{2}(3) × Σ_{3} × 2. Similarly G_{3} ∩ T is the setwise stabilizer of {α_{1}, β, γ} in T, which means that G_{3} ∩ T is the normalizer of P_{2} in T. This shows that ⟨x_{2}, w_{2}⟩ is contained in G_{3} ∩ T and it maps isomorphically into the GL_{2}(3)-direct factor of G_{3}/O_{3}(G_{3}). Notice that ν is not in O_{3}(G_{3}) since it rotates the vertices α_{1}, β and γ. Thus O_{3}(G_{3}) ∩ T is generated by the root elements x_{I} for I ∈ J \ {2} together with ϱ. In particular O_{3}(G_{3}) ∩ T is of index 3^{2} in O_{3}(G_{3}).

Let V (α_{3}) be the subspace of dimension 2 in V = O_{3}(G_{1}) as introduced in Section 2, such that G_{1}∩G_{3} = N_{G_{1}}(V (α_{3})). Since O_{3}(G_{3}) is contained in G_{1}, we can calculate in the latter group that V (α_{3}) is the center of O_{3}(G_{3}). Using the relations for T one can see that ⟨x_{12234}, x_{1234}⟩ is the center of O_{3}(G_{3}) ∩ T. This shows that

Z_{3} := Z(O_{3}(G_{3})) = V (α_{3}) = ⟨x_{12234}, x_{1234}⟩

and that ⟨x_{2}, w_{2}⟩ acts faithfully on Z_{3}.

Put S = C_{O_{3}(G_{3})}(⟨x_{2}, w_{2}⟩)Z_{3}. One can calculate in G_{1} that S is of order 3^{6} and that S/Z_{3} = Z(O_{3}(G_{3})/Z_{3}). This and direct calculations with the relations for T, show that S = ⟨x_{12234}, x_{1234}, x_{234}, x_{123}, x_{124}, ϱ⟩ and that [S, O_{3}(G_{3})] = Z_{3}.

We assume that Φ = {α_{1}, β, γ, δ, ϵ}. Notice that every pair of vertices from Φ form an edge of Γ and hence can be considered as an element of type 2 in G incident to α_{3}. Clearly an automorphism from O_{3}(G_{2})\O_{3}(G_{3}) stabilizes α_{1} and β, permuting γ, δ and ϵ. This means that ⟨ν⟩O_{3}(G_{3})/O_{3}(G_{3}) and O_{3}(G_{2})O_{3}(G_{3})/O_{3}(G_{3}) both are Sylow 3-subgroups in the Σ_{5}-direct factor of G_{3}/O_{3}(G_{3}) ≅ Σ_{5} × GL_{2}(3).

Let W := C_{S/Z_{3}}(⟨ν⟩)Z_{3}. Then direct calculations with the relations for T show that

W = C_{S/Z_{3}}(⟨ν⟩)Z_{3} = ⟨x_{12234}, x_{1234}, x_{234}x_{123}x_{124}, x_{124}^{−1}x_{234}⟩.

As above let E = O_{3}(G_{1}) ∩ O_{3}(G_{β}). Then E is contained S, contains Z_{3} and O_{3}(G_{3}) centralizes E/Z_{3}. This and order reason give

E = C_{S/Z_{3}}(O_{3}(G_{2}))Z_{3} = ⟨x_{124}, x_{1234}, x_{12234}, ϱ⟩.

Since ⟨ν⟩O_{3}(G_{3})/O_{3}(G_{3}) and O_{3}(G_{2})O_{3}(G_{3})/O_{3}(G_{3}) are Sylow 3-subgroups in the Σ_{5}-direct factor of G_{3}/O_{3}(G_{3}), there is an element y ∈ G_{3} ∩ G_{4} which conjugates W to E and by Frattini argument we can choose y so that [y, w_{2}] = 1. Since [t_{1}, W] ⊆ W and G_{4} ∩ G_{2} = N_{G_{4}}(E) we see t_{1}^{y} ∈ G_{2} and o(t_{1}^{y}w_{2}) = o(t_{1}w_{2}).

The relations for T and C_{G_{4}}(m) describe the actions of w_{2} and t_{1} on W. Comparing these actions we observe that the product w_{2}t_{1} induces on W an action of order 3 which means that (w_{2}t_{1})^{3} centralizes W. Hence (w_{2}t_{1}^{y})^{3} is in C_{G_{2}}(E). The structure of G_{2} shows that C_{G_{2}}(E) is contained in O_{3}(G_{2}) and has order 3^{10}. On the other hand the relations for T show that the subgroup

U_{1} = ⟨x_{12234}, x_{1234}, x_{234}, ϱ, x_{124}, x_{123}, x_{24}, x_{23}, x_{3}, x_{4}⟩

has order 3^{10} and centralizes W. Hence U_{1} = C_{G_{2}^{y^{−1}}}(W) and so (w_{2}t_{1})^{3} ∈ U_{1}. We

know that ⟨w_{1}, w_{2}⟩ ≅ Σ_{4} and so w_{1}^{2}^{w_{2}} = w_{1}^{2}w_{2}^{2} = w_{1}^{2}h. Further (w_{2}t_{1})^{h} = w_{2}t_{1}w_{1}^{2}tm. As t and m both centralize w_{2} and t_{1}, we see ((w_{2}t_{1})^{3})^{h} = (w_{2}t_{1}w_{1}^{2})^{3}tm. Now

w_{2}t_{1}w_{1}^{2}w_{2}t_{1}w_{1}^{2}w_{2}t_{1}w_{1}^{2} = w_{2}t_{1}w_{2}w_{1}^{2}ht_{1}w_{1}^{2}w_{2}t_{1}w_{1}^{2} =
w_{2}t_{1}w_{2}t_{1}w_{1}^{2}tmhw_{2}t_{1}w_{1}^{2} = (w_{2}t_{1})^{3}tm.

So we have that [h, (w_{2}t_{1})^{3}] = 1. Hence (w_{2}t_{1})^{3} ∈ C_{U_{1}}(h) = ⟨x_{234}, x_{123}, x_{124}, ϱ⟩. As both w_{2} and t_{1} commute with m we see that (w_{2}t_{1})^{3} ∈ ⟨x_{234}, x_{123}, x_{124}⟩. As w_{2}t_{1} centralizes (w_{2}t_{1})^{3}, we have (w_{2}t_{1})^{3} ∈ ⟨x_{124}^{−1}x_{234}⟩. But t centralizes w_{2}t_{1} and inverts x_{124}^{−1}x_{234}, so we finally get

(w_{2}t_{1})^{3} = 1.

Now the subgroup Q = ⟨C_{T} (m), C_{G_{4}}(m)⟩ is generated by t_{1} and all the generators of T except for ϱ. These generators satisfy the relations given above. The coset enumeration with a computer program implementing the Todd-Coxeter algorithm has shown that the presentation defines a group whose order is equal to the order of 2 × M(23) (we enumerated the cosets of C_{T} (m) in Q and have obtained the index 137 632). Since the relations are uniquely determined by the conditions on the geometry and since M(24) acts as the full automorphism group of a geometry of the considered type, we have the following result.

Lemma 4.2 Let Q be the subgroup in G generated by the subgroups C_{T} (m) and C_{G_{4}}(m). Them Q ≅ 2 × M(23).

5 The transposition graph

In this section we complete the proof of Theorem 1 by showing that the full automorphism group G of the geometry G under consideration is isomorphic to the Fischer group M(24). We will achieve this by considering a graph Λ on the set of cosets in G of the subgroup Q ≅ 2 × M(23). We will establish the isomorphism between Λ and the graph on the set of 3-transpositions of M(24) in which two transpositions are adjacent if they do not commute (i.e. if their product is of order 3).

As above let W = ⟨ρ⟩ be the subgroup of order 3 in V = O_{3}(G_{1}) which is of plus type with respect to the orthogonal form on V preserved by G_{1}/V and T = ⟨N_{G_{1}}(W), N_{G_{4}}(W)⟩. Then T ≅ Σ_{3} × D_{4}(3).Σ_{3} by Lemma 3.3. Let m be an involution in T which commutes with the subgroup D ≅ D_{4}(3) in T and let Q = ⟨C_{T} (m), C_{G_{4}}(m)⟩. Then Q ≅ 2 × M(23) by Lemma 4.2 and it is straightforward that Q ∩ T = C_{T} (m) ≅ 2 × D_{4}(3).Σ_{3}, so that [T : T ∩ Q] = 3.

Let R = O_{3}(G_{4})C_{G_{4}}(m). Then directly from the construction of Q in the previous section we have the following :

Lemma 5.1 The following assertions hold.

(i) R ∩ T ≅ Σ_{3} × 3_{+}^{1+8}2_{−}^{1+6}3_{+}^{1+2}(2 × Σ_{3}) and [R : R ∩ T] = 12;

(ii) R ∩ Q ≅ 2 × 3_{+}^{1+8}2_{−}^{1+6}3_{+}^{1+2}GL_{2}(3) and [R : R ∩ Q] = 9;

(iii) W ≤ O_{3}(G_{4}) and |W ∩ Q| = 1.

Define Λ to be a graph of the set of (right) cosets of Q in G in which two cosets are adjacent if and only if they intersect a common coset of T in G. Let λ ∈ Λ be the coset containing the identity element so that G_{λ} = Q. Let Θ be the orbit of T on the vertex set of Λ which contains λ. Since [T : T ∩ Q] = 3, Θ contains besides λ exactly two vertices, say μ and ν. Then G_{λ} ∩ G_{μ} contains the subgroup D ≅ D_{4}(3) from T. As one can check with [Atlas], every proper subgroup in Q containing D is contained in T ∩ Q. This shows that G_{λ} ∩ G_{μ} ≤ T and we have the following.

Lemma 5.2 (i) The valency of Λ is 275 264 = 2 · [Q : Q ∩ T];

(ii) Θ induces a triangle in Λ and T is its full stabilizer in G;

(iii) ⟨W, m⟩ ≅ Σ_{3} acts faithfully on Θ.

Let Ω be the orbit of R on the vertex set of Λ which contains λ. By Lemma 5.1 (iii) and Lemma 5.2 (iii), Ω contains Θ and by Lemma 5.1 (ii) |Ω| = 9. Let L be the kernel of the action of R on Ω. By Lemma 5.1 (ii) we see that O_{3}(R)/(L ∩ O_{3}(R)) is elementary abelian of order 9 and acts regularly in Ω. Hence R/O_{3}(R)L is a subgroup in GL_{2}(3) ≅ Aut(O_{3}(R)/(L ∩ O_{3}(R))). On the other hand L is contained and normal in both R ∩ T and R ∩ Q, by Lemma 5.1 this gives the following :

[figure: 1; 218 400; 1; 28 431; 28 431; 1; 1; 22 400; 22 400; 173 200; 22 599; 173 600; 22 599; 173 600; 22 400; 22 400]

Figure 1: Distribution diagram of M(23) acting of D_{4}(3).3

Lemma 5.3 Let N = R/O_{3}(R)L. Then N ≅ GL_{2}(3) and ⟨m⟩ maps onto the center of N.

Let Q′ be the commutator subgroup of Q, so that Q′ ≅ M(23) and Q = ⟨m⟩ × Q′. As a direct consequence of the above lemma we have the following :

Lemma 5.4 (i) The subgraph in Λ induced by Ω is complete and R induces a doubly transitive action on Ω;

(ii) both Q ∩ R and Q′ ∩ R induce GL_{2}(3) on Ω \ {λ};

(iii) Q′ acts transitively on the set Λ(λ) of vertices adjacent to λ and the stabilizer in Q′ of μ ∈ Λ(λ) is isomorphic to D_{4}(3).3;

(iv) if σ ∈ Ω \ Θ, then the order of Q′_{μ} ∩ Q′_{σ} is divisible by 3^{12} · 2^{7}.

Lemma 5.5 G_{λ} ∩ G_{μ} acting on Λ(λ) \ {μ} has exactly 4 orbits: Λ^{1}(λ, μ) = {ν}, Λ^{2}(λ, μ), Λ^{3}(λ, μ) and Λ^{4}(λ, μ) with lengths 1, 218 400, 28 431 and 28 431, respectively. The intersection diagram of the graph on Λ(λ) associated with Λ^{2}(λ, μ) is given on Figure 1 and Ω \ Θ ⊆ Λ^{2}(λ, μ).

Proof: By Lemma 5.2 (i) G_{λ} = Q acts on Λ(λ) as it acts on the cosets of an index 2 subgroup in its maximal subgroup isomorphic to 2 × D_{4}(3).Σ_{3} and by Lemma 5.4 (ii) Q′ acts transitively on Λ(λ). This specifies the action under consideration up to similarity. The subdegrees of this action and the intersection diagrams can be found in [PS]. It is clear that ν is stabilized by G_{λ} ∩ G_{μ}. If σ ∈ Ω \ Θ then Lemma 5.4 (iv) and straightforward congruences show that σ ∈ Λ^{2}(λ, μ).

Lemma 5.6 Let Ξ be the graph on the set of 3-transpositions of M(24) in which two transpositions are adjacent if they are different and commute. Then Λ is isomorphic to the complement of Ξ and G ≅ M(24).

Proof: For α ∈ Λ let m_{α} be the unique non-trivial element in the center of G_{α} ≅ 2 × M(23), so that m = m_{λ}. If β ∈ Λ(α) then the product of m_{α} and m_{β} has order 3. On the other hand it is clear that Q contains elements conjugated to m and different from m. Hence m_{α} and m_{β} commute for some vertices α and β of Λ and the diameter of Λ is at least 2. Let α be a vertex adjacent to μ and at distance 2 from λ and let Φ the orbit of Q containing α. Then by Lemma 5.5 Φ contains Λ^{3}(λ, μ) or Λ^{4}(λ, μ) or both. The intersection diagram of the graph on Λ(λ) associated with Λ^{2}(λ, μ) shows that α is adjacent to 173 600 vertices from Λ^{2}(μ, λ) ⊆ Λ(λ). Hence Φ contains at most 275 264 · (28 431 + 28 431)/173 600 < 90 162. It is clear that Q induces a non-trivial action on Φ. Comparing this with the indices of maximal subgroups in Q′ ≅ M(23) [Atlas] we conclude that Q′ acting on Φ has one or two orbits of length 31 671 and the action on each of the orbits is similar to the action of M(23) on the cosets of its subgroup 2 · M(22). Suppose that |Φ| = 2 · 31 671. Since α is adjacent to at least 173 600 vertices from Λ(λ), in this case Λ^{3}(μ, λ) ∪ Λ^{4}(μ, λ) ⊆ Φ and α is adjacent to 247 104 vertices from Λ(λ). We may assume that α ∈ Λ^{3}(μ, λ). Then the intersection diagram shows that α is adjacent to 22 400 vertices from Λ^{3}(μ, λ) and to 22 400 vertices from Λ^{4}(μ, λ). Since 247 104 + 2 · 22 400 = 291 904 is larger than the valency of Λ, which is 275 264, this is impossible. Hence |Φ| = 31 671 and Q′ acts transitively on Φ, Λ(μ) ∩ Φ = Λ^{3}(μ, λ) and α is adjacent to 247 104 vertices from Λ(λ). Comparing this with the valency of Λ we see that α is adjacent to 28 160 vertices outside Λ(λ). The subdegrees of the action of M(23) on the cosets of 2·M(22) are well known to be 1, 3 510 and 28 160. Since α is adjacent to 22 400 vertices from Λ^{3}(μ, λ) we see that Λ = {λ} ∪ Λ(λ) ∪ Φ. Now the complement of Λ has valency 31 671 and locally it is the commuting graph of transpositions in M(23). Also for any two vertices α and β in Λ the product m_{α}m_{β} has order at most 3, which means that {m_{α} | α ∈ Λ} is a class of 3-transpositions in G and to complete the proof we can apply one of the available characterizations of M(24), for instance [BW].

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A.A. Ivanov, Department of Mathematics, Imperial College, 180 Queen’s Gate, London, SW7 2BZ, UK

G. Stroth, Institut für Algebra und Geometrie, Fachbereich Mathematik u. Informatik, Martin-Luther Universität Halle-Wittenberg, 06099 Halle, Germany