Simple connectedness of the 3-local geometry of the Monster

A.A. Ivanov

U. Meierfrankenfeld

Abstract We consider the 3-local geometry M of the Monster group M introduced in [BF] as a locally dual polar space of the group Ω_{8}^{−}(3) and independently in [RS] in the context of minimal plocal parabolic geometries for sporadic simple groups. More recently the geometry appeared implicitly in [DM] within the Z_{3}-orbifold construction of the Moonshine module V^{♮}. In this paper we prove the simple connectedness of M. This result makes unnecessary the refereeing to the classification of finite simple groups in the Z_{3}-orbifold construction of V^{♮} and realizes an important step in the classification of the flag-transitive c-extensions of the classical dual polar spaces (cf. [Yo]). We make use of the simple connectedness results for the 2-local geometry of M [Iv1] and for a subgeometry in M which is the 3-local geometry of the Fischer group M(24) [IS].

1 Introduction

The Monster group M acts flag-transitively on a diagram geometry M which is described by the following diagram:

3∘  _{3}∘  _{9}∘ c^{∗} _{1}∘.

The elements of M corresponding to the nodes from the left to the right on the diagram are called points, lines, planes and quadrics, respectively. The residue of a quadric is the classical polar space associated with the group Ω_{8}^{−}(3). The quadrics and planes incident to a line form the geometry of vertices and edges of a complete graph on 11 vertices. The existence of M was independently established in [BF] and [RS]. We follow [BF] to review briefly the construction of M and to formulate its basic properties. The starting point is the description of conjugacy classes of the subgroups of order 3 in the Monster [At].

Lemma 1.1 In the Monster group M every element of order 3 is conjugate to its inverse and there are exactly three conjugacy classes of subgroups of order 3 with representatives σ, μ and τ, so that

(a) N_{M}(σ) ∼ 3^{1+12}.2.Suz.2, where Suz is the Suzuki sporadic simple group;

(b) N_{M}(μ) ∼ 3.M(24), where M(24) is the largest sporadic Fischer 3-transposition group;

(c) N_{M}(τ) ∼ Sym(3) × F_{3}, where F_{3} is the sporadic simple group discovered by Thompson.

We define a subgroup of order 3 in M to be of Suzuki, Fischer or Thompson type if it is conjugate to σ, μ or τ from 1.1, respectively.

A crucial role in the construction of M is played by a subgroup M_{8} ∼ 3^{8}.Ω_{8}^{−}(3).2 in M. If Q_{8} = O_{3}(M_{8}) then M_{8}/Q_{8} is an extension of the simple orthogonal group Ω_{8}^{−}(3) by an automorphism of order 2, Q_{8} is the natural orthogonal module for M_{8}/Q_{8} and N_{M}(Q_{8}) = M_{8}.

Lemma 1.2 Let φ be the orthogonal form of minus type on Q_{8} preserved by M_{8}/Q_{8}. Then M_{8}/Q_{8} acting on the subgroups of order 3 in Q_{8} has two orbits I and N such that

(a) |I| = 1 066, the subgroups in I are isotropic with respect to φ and of Suzuki type in M; for σ ∈ I we have N_{M_{8}/Q_{8}}(σ) ∼ 3^{6}.2.U_{4}(3).2^{2};

(b) |N| = 2 214, the subgroups in N are non-isotropic with respect to φ and of Fischer type in M; for μ ∈ N we have N_{M_{8}/Q_{8}}(μ) ∼ Ω_{7}(3).2.

Proof: Under the action of O^{2}(M_{8}/Q_{8}) ∼ Ω_{8}^{−}(3) the set of order 3 subgroups in Q_{8} splits into three orbits I, N_{1}, N_{2} with lengths 1 066, 1 107, 1 107 and stabilizers isomorphic to 3^{6}.2.U_{4}(3).2, Ω_{7}(3).2, Ω_{7}(3).2, respectively (cf. [At]). As 3^{17} divides the order of each of the stabilizers there are no Thompson type subgroups in Q_{8} and as the elements of I are 3-central they are of Suzuki type. By Lagrange 3^{7}Ω_{7}(3) is not involved in Suz and since 3^{1+12} has no elementary abelian subgroup or order 3^{8}, N_{1} and N_{2} consist of Fischer type subgroups. Finally, in M(24) all subgroups of order 3^{7} whose normalizer involve Ω_{7}(3) are conjugated. Hence N_{1} and N_{2} fuse into a single M_{8}/Q_{8}-orbit.

By 1.2 the polar space acted on flag-transitively by M_{8}/Q_{8} can be identified with the Suzuki-pure subgroups in Q_{8} with two subgroups being incident if one of them contains the other one. Let Q_{1}, Q_{2}, Q_{3} be Suzuki-pure subgroups in Q_{8} with Q_{1} < Q_{2} < Q_{3}, so that |Q_{i}| = 3^{i} for 1 ≤ i ≤ 3. Then the points, lines, planes and quadrics in M are defined to be the subgroups in M conjugate to Q_{1}, Q_{2}, Q_{3} and Q_{8}, respectively, with F = {Q_{1}, Q_{2}, Q_{3}, Q_{8}} being a maximal flag. Let M_{i} = N_{M}(Q_{i}) be the maximal parabolic subgroup corresponding to the flag-transitive action of M on M. Then M_{8} is as above while M_{1} is the normalizer of a Suzuki type subgroup Q_{1} (which we will also denote by σ) and M_{1} ∼ 3^{1+12}.2.Suz.2 by 1.1a. The stabilizer of F in M contains a Sylow 3-subgroup of M. Hence for two elements of M to be incident it is necessary for their common stabilizer in M to contain a Sylow 3-subgroup. Let P_{i} = O_{3}(M_{i}), P_{i}^{∗} be the kernel of the action of M_{i} on the residue of Q_{i} in M and M̄_{i} = M_{i}/P_{i}^{∗} for i = 1, 2, 3 and 8. It is clear that Q_{i} ≤ P_{i} and that Q_{8} = P_{8} = P_{8}^{∗}. For i = 1, 2, 3 and 8 we denote by M_{i} the set of points, lines, planes and quadrics in M, respectively. For an element α in M we denote by M_{i}(α) the set of elements in M_{i} incident to α.

Let Σ be the graph on the Suzuki type subgroups in Q_{8} in which two subgroups are adjacent if they are orthogonal with respect to φ. Then Σ is strongly regular with parameters

v = 1 066, k = 336, l = 729, λ = 92, μ = 112.

(that is, Σ has v = 1 066 vertices, every vertex has k = 336 neighbors and l = 729 vertices in distance two, two adjacent vertices have λ = 92 common neighbors and two vertices of distance two have μ = 112 common neighbors.).

The quotient M_{8}/Q_{8} induces a rank 3 action on Σ, so that if σ ∈ Σ then N_{M_{8}/Q_{8}}(σ) acts transitively on the set Σ_{1}(σ) of points adjacent to σ in Σ and on the set Σ_{2}(σ) of points at distance 2 from σ.

The next statement follows from standard properties of classical groups.

Lemma 1.3 Let L = N_{M_{8}/Q_{8}}(σ) and z be an involution from O_{3,2}(L). Then

(a) L ∼ 3^{6}.2.U_{4}(3).2^{2} and Q_{8}, as a module for L, has a unique composition series:

1 < σ < ⟨Σ_{1}(σ)⟩ = σ^{⊥} < Q_{8};

(b) both σ^{⊥}/σ and O_{3}(L) are isomorphic to the natural orthogonal module for O^{2}(L)/O_{3}(L) ∼ 2.U_{4}(3) ∼ 2.P Ω_{6}^{−}(3); (c) O_{3}(L) acts regularly on Σ_{2}(σ);

(d) z acts fixed point-freely on σ^{⊥}/σ and on O_{3}(L); it centralizes a unique subgroup ε ∈ Σ_{2}(σ) and C_{Q_{8}}(z) = ⟨σ, ε⟩ is 2-dimensional containing two subgroups of Suzuki and two subgroups of Fischer type.

Since N_{M_{8}}(σ) contains a Sylow 3-subgroup of M, it contains P_{1} = O_{3}(N_{M}(σ)). By 1.1a and 1.3, we have P_{1} ∩ Q_{8} = σ^{⊥}, P_{1}Q_{8} = O_{3}(N_{M_{8}}(σ)) and z acts fixed-point freely on P_{1}/Q_{1}. This shows that all the points collinear to σ are contained in P_{1} and that P_{1}^{∗} = P_{1}⟨z⟩. Let ε be as in 1.3d. Then Q_{8} = ⟨ε, C_{P_{1}}(ε)⟩ is uniquely determined by ε and σ. So if Q_{8}^{∗} is another quadric containing σ then Q_{8} ∩ Q_{8}^{∗} is a point, a line or a plane. Furthermore, the image δ of Q_{8} in M̄_{1} = M_{1}/P_{1}^{∗} ∼ Suz.2 is a subgroup of order 3. Moreover, N_{M_{8}}(σ)/P_{1}^{∗} = NM̄_{1}(δ) ∼ 3.U_{4}(3).2^{2} and by [At] is a maximal subgroup in M̄_{1}. Thus the quadrics from M_{8}(σ) correspond to 3-central subgroups of order 3 in M̄_{1}. The next lemma (cf. [BCN], Section 13.7) describes the action of M̄_{1} on its 3-central subgroups of order 3.

Lemma 1.4 The group S ∼ Suz.2 acting on the set Δ of its subgroups of order 3 with normalizer U ∼ 3.U_{4}(3).2^{2} has rank 5 with subdegrees 1, 280, 486, 8 505 and 13 608. If Δ denotes also the graph of valency 280 invariant under this action, then:

(a) two distinct vertices of Δ commute (as subgroups in S) if and only if they are adjacent;

(b) Δ is distance-transitive with distribution diagram given on Figure 1 and S is the full automorphism group of Δ;

(c) if K is a maximal clique in Δ then |K| = 11, the setwise stabilizer T of K is a maximal subgroup in S and T ∼ 3^{5}.(2 × Mat_{11}), so that O_{3}(T) is generated by the subgroups from K and T/O_{3,2}(T) ∼ Mat_{11} acts 5-transitively on the vertices of K while O_{3}(T) fixes none of the vertices outside K;

(d) let δ be the vertex of Δ stabilized by U, then the geometry of cliques and edges containing δ with the incidence relation via inclusion is isomorphic to the geometry of 1- and 2-dimensional totally isotropic subspaces in 6-dimensional orthogonal GF (3)-space of minus type and it is acted on flag-transitively by O^{2}(U)/O_{3}(U) ∼ U_{4}(3) ∼ P Ω_{6}^{−}(3);

(e) if ϱ is a vertex at distance 2 from δ in Δ then the subgraph induced on the vertices adjacent to both δ and ϱ is the complete bipartite graph K_{4,4}.

(f) If ϱ is a vertex of distance 2, 3 or 4 from δ then ⟨δ, ϱ⟩ is isomorphic to SL_{2}(3), Alt(5) and Alt(4), respectively.

Next we make use of the following information about the action of M_{1} on the set of subgroups of order 9 in P_{1} containing σ (cf. [Wi]).

Lemma 1.5 M̄_{1} has two orbits L and K on the set of subgroups of order 9 in P_{1} containing σ, moreover

(a) if l ∈ L then NM̄_{1}(l) ∼ 3^{5}.(2 × Mat_{11}) is the stabilizer of a maximal clique in the graph Δ as in 1.4c and all subgroups of order 3 in l are of Suzuki type;

(b) if k ∈ K then NM̄_{1}(k) ∼ U_{5}(2).2 and all subgroups of order 3 in k except σ are of Fischer type.

[figure: 1; 280; 8505; 13608; 486; 1; 36; 128; 180; 280; 243; 8; 144; 90; 10; 280]

Figure 1: Distribution diagram of Δ.

Since P_{1} is extraspecial, it follows from the above lemma that the subgroups of order 3 in P_{1} other than σ form exactly two conjugacy classes L̃ and K̃ of M_{1} with normalizers

(3 × 3^{1+10}).2.3^{5}.(2 × Mat_{11}) and (3 × 3^{1+10}) : (2 × U_{5}(2).2),

respectively.

It is clear that the subgroups from L in 1.5 are exactly the lines from M_{2}(σ). Comparing 1.5a and 1.4c we can identify M_{2}(σ) with the set of cliques in the graph Δ on M_{8}(σ). Since a flag of M is stabilized by a Sylow 3-subgroup of M, it follows from 1.4c that a line l ∈ M_{2}(σ) and a quadric δ ∈ M_{8}(σ) are incident if and only if l, as a clique of Δ, contains δ. By 1.4d two cliques l_{1} and l_{2} of Δ of maximal possible intersection have exactly two vertices, say δ_{1} and δ_{2} in common. Then the lines l_{1} and l_{2} are in two different quadrics and hence they generate an element of M which has to be the plane p which is the intersection of δ_{1} and δ_{2}. This enables us to identify p with the edge {δ_{1}, δ_{2}} of Δ.

Thus the elements from M_{i}(σ) for i = 2, 3 and 8 can be considered as cliques, edges and vertices of the graph Δ with the natural incidence relation. In particular the planes and quadrics incident to a given line are edges and vertices of the corresponding clique of size 11 in Δ. Hence we have that the diagram of M is as given above and also (compare [Wi]) that

M_{2} ∼ 3^{2+5+10} : (GL_{2}(3) × Mat_{11}), M̄_{2} ∼ Sym(4) × Mat_{11};
M_{3} ∼ 3^{3+6+8} : (L_{3}(3) × D_{8} : 2), M̄_{3} ∼ L_{3}(3) × 2.

The main result of the paper is the following.

Theorem 1.6 The 3-local geometry M of the Monster is simply connected, equivalently, M is the universal completion of the amalgam of maximal parabolic subgroups M_{1}, M_{2}, M_{3} and M_{8} corresponding to the action of M on M.

Here and elsewhere a tuple {H_{i} |∈ I} of subgroups in a group will also be viewed as the amalgam obtained by considering the intersection of the H_{i} and the inclusion maps.

To prove the theorem we define G to be the universal completion of the amalgam {M_{i} | i = 1, 2, 3, 8}. Identify M_{i} with its image in G. Then there is a unique homomorphism χ of G onto M with χ_{M_{i}} = id_{M_{i}} for all i. We will show eventually that χ is an isomorphism.

The second author would like to thank the Imperial College, the Universität Bielefeld and the Martin Luther-Universität Halle-Wittenberg for their hospitality.

2 M(24)-subgeometry

In this section we discuss a subgeometry M(μ) in M stabilized by a subgroup F := N_{M}(μ) ∼ 3.M(24), where μ is a subgroup of order 3 of Fischer type in M. The elements of M(μ) are some (not all) elements of M centralized by μ and the incidence relation is induced by that in M.

As above, let F = {Q_{1}, Q_{2}, Q_{3}, Q_{8}} be a maximal flag in M with Q_{1} = σ and let μ be a Fischer type subgroup of Q_{8} contained in Q_{3}^{⊥}. Define M(μ) to be the subgeometry in M induced by the images under F = N_{M}(μ) of the elements in F. We discuss the diagram of M(μ) and the structure of the parabolic subgroups F_{i} := N_{F} (Q_{i}) = N_{M_{i}}(μ) corresponding to the action of F on M(μ).

Since μ is non-isotropic with respect to φ we have,

Q_{8} = μ ⊕ μ^{⊥},

where μ^{⊥} is the natural orthogonal module for F_{8}/Q_{8} ∼ Ω_{7}(3).2. Moreover, Q_{1}, Q_{2} and Q_{3} are contained in μ^{⊥} and form a maximal flag in the polar space defined on μ^{⊥}. Thus F_{8}/μ ∼ 3^{7}.Ω_{7}(3).2 and the residue of Q_{8} in M(μ) is the non-degenerate orthogonal polar space in dimension 7 over GF (3).

By 1.5 we have F_{1}/μ ∼ 3^{1+10}.(2×U_{5}(2).2) and one can see that the image of Q_{8} in F̄_{1} = F_{1}/(F_{1}∩ P_{1}^{∗}) ∼ U_{5}(2).2 is a 3-central subgroup of order 3 with the normalizer isomorphic to (3 × U_{4}(2)).2. Let Θ be the graph on all these subgroups of order 3 in F̄_{1} in which two subgroups are adjacent if they commute. Then Θ is strongly regular with parameters

v = 176, k = 40, l = 135, λ = 12, μ = 8

and clearly it is a subgraph in the graph Δ as in 1.4. In these terms the quadrics, planes and lines in M(μ) incident to σ are the vertices, edges and cliques (of size 5) in Θ with the natural incidence relation. This shows that the diagram of M(μ) is the following

3∘  _{3}∘  _{3}∘ c^{∗} _{1}∘.

It is easy to deduce the structure of two other parabolic subgroups (compare [RS]):

F_{2}/μ ∼ 3^{2+4+8}.(GL_{2}(3) × Sym(5)), F_{3}/μ ∼ 3^{3+7+3}.2.(L_{3}(3) × 2).

In [IS] the geometry M(μ) was proved to be simply connected.

Lemma 2.1 The geometry M(μ) is simply connected and hence 3.M(24) is the unique faithful completion of the amalgam consisting of the subgroups F_{1}, F_{2}, F_{3} and F_{8}.

This immediately implies the following.

Lemma 2.2 Let X be a faithful completion of the amalgam consisting of the Monster subgroups M_{i}, i = 1, 2, 3, 8. Let μ be a non-isotropic subgroup of order 3 in Q_{8} contained in Q_{3}^{⊥}. Then X contains a subgroup M_{μ} ∼ 3.M(24), which normalizes μ, such that M_{μ} ∩ M_{i} = N_{M_{i}}(μ) = N_{M_{μ}}(Q_{i}) for i = 1, 2, 3 and 8. If X = M then M_{μ} = N_{M}(μ).

A subgroup μ as in the above lemma will be said to be of Fischer type. We remark that the subgroup M_{μ} of X does not only dependent on μ but a priori also on the flag (M_{1}, M_{2}, M_{3}, M_{8}). But as the reader might check M_{μ} is already determined by μ together with any one of the M_{i}’s, i = 1, 2, 3 or 8

3 The 2-local geometry of the Monster

There are exactly two classes of involutions in M, called the Conway type and Baby Monster type involutions with representatives z and t, such that

C_{M}(z) ∼ 2_{+}^{1+24}.Co_{1} and C_{M}(t) ∼ 2.F_{2},

where Co_{1} is the first Conway sporadic simple group and F_{2} is the Fischer Baby Monster group [At].

Let C = C_{M}(z). Then for i = 4 and 8 up to conjugation in C there is a unique Conway-pure subgroup E_{i} of order i in O_{2}(C) containing z, whose normalizer in M contains a Sylow 2-subgroup of M. Moreover these two subgroups can be chosen so that E_{4} < E_{8} and we will assume that the inclusion holds. Let N = N_{M}(E_{4}) and L = N_{M}(E_{8}). Then

C ∼ 2_{+}^{1+24}.Co_{1}, N ∼ 2^{2+11+22}.(Sym(3) × Mat_{24}), L ∼ 2^{3+6+12+18}.(L_{3}(2) × 3.Sym(6)).

Furthermore C, N and L are the stabilizers of a point, a line and a plane from a maximal flag in the 2-local minimal parabolic geometry of the Monster group [RS] having the following diagram:

2∘  _{2}∘  _{2}∘  _{2}∘ ∼ _{2}∘.

This geometry was proved to be 2-simply connected in [Iv1] and by standard principles this result is equivalent to the following.

Lemma 3.1 The Monster group M is the universal completion of the amalgam of its subgroups C, N and L defined as above.

Our strategy to prove Theorem 1.6 is to show that the universal completion G of the amalgam of the 3-local parabolics M_{i} is also a completion of the amalgam consisting of the subgroups C, N and L as in 3.1.

Lemma 3.2 Let μ be a subgroup of Fischer type in M. Then M_{μ} has exactly four classes of involutions and for an involution t ∈ M_{μ} exactly one of the following holds:

(a) t inverts μ, t is of Baby Monster type and C_{M_{μ}}(t) ≅ M(23) × C_{2}.

(b) t centralizes μ, t is of Baby Monster type and C_{M_{μ}}(t) ∼ 3.2^{2}M(22).2.

(c) t inverts μ, t is of Conway type and C_{M_{μ}}(t) ∼ 2^{3}.U_{6}(2).Sym(3).

(d) t centralizes μ, t is of Conway type and C_{M_{μ}}(t) ∼ 3.2_{+}^{1+12}.3.U_{4}(3).2^{2}.

Proof: By [At] M_{μ} has four classes of involutions with centralizers as given. By Lagrange, Co_{1} involves neither M(23) nor 3.M(22).2 and so the first two classes are of Baby Monster type. Since Conway type involutions both invert and centralize groups of Fischer type the remaining two classes must be of Conway type.

Lemma 3.3 Let z be an involution from P_{1}^{∗} = O_{3,2}(M_{1}). Then every involution in M_{1} is conjugated to an involution s ∈ C_{M_{1}}(z) and one of the following holds:

(a) s = z, C_{M_{1}}(s) ∼ 6.Suz.2 and s is of Conway type in M;

(b) s inverts σ, C_{M_{1}}(s)P_{1}^{∗}/P_{1}^{∗} ∼ 2.Mat_{12}; s and sz are conjugated in M_{1} and the centralizer of s in P_{1}/σ has order 3^{6};

(c) s centralizes σ and C_{M_{1}}(s)P_{1}^{∗}/P_{1}^{∗} ∼ 2_{−}^{1+6}.O_{6}^{−}(2) (two conjugacy classes).

Proof: The conjugacy classes of involutions in M_{1}/P_{1} ∼ 2.Suz.2 can be read from [At]. Since z centralizes σ and acts fixed point-freely on P_{1}/σ, the structure of C_{M_{1}}(s) in (a) follows. Since the Baby Monster has no elements of order 3 with normalizer of the shape 3 · Suz.2, z is of Conway type. In (b) we have C_{P_{1}/σ}(s) = [P_{1}/σ, sz] and since s and sz are conjugated, both subspaces have dimension 6.

4 The 3-local geometry for Co_{1}

In [Iv2], a relationship between the 3- and 2-local geometries of the Monster via a 2^{24}-cover of the 3-local geometry of the Conway group [BF] was noticed.

Let X be an arbitrary faithful completion of the amalgam (M_{1}, M_{2}, M_{3}, M_{8}) of the 3-local parabolics in M which has M has a quotient and let X be the geometry whose elements are the cosets in X of M_{i} for i = 1, 2, 3, 8 and where two cosets are incident if their intersection is non-empty. If X = M or X = G where as above G is the universal completion of the amalgam, then X is M or the universal cover G of M, respectively. For an element x of X let M_{x} denote the stabilizer of x in X which is a conjugate of M_{i} for i = 1, 2, 3 or 8 depending on the type of x. If x = M_{i}g put Q_{x} = Q_{i}^{g}, P_{x} = P_{i}^{g} and P_{x}^{∗} = P_{i}^{∗g}. When working in the residue of an element we can and will identify x with Q_{x}. If μ is a subgroup of order 3 of Fischer type in Q_{8}^{g}, then M_{μ} denotes the subgroup as in 2.2, i.e, if μ ∈ Q_{3}^{g⊥} then M_{μ} = ⟨N_{M_{i}^{g}}(μ) | i = 1, 2, 3, 8⟩.

Let us pick an involution z from P_{1}^{∗} = O_{3,2}(M_{1}). Then by 3.3a C_{M_{1}}(z) ∼ 6.Suz.2. Let Ξ = Ξ^{X} be the set of points of X such that x ∈ Ξ if and only z ∈ O_{2,3}(M_{x})). Let Ξ denote also the graph on Ξ in which two points are adjacent if they are incident to a common quadric. It is clear that C_{X}(z) preserves Ξ as a whole as well as the adjacency relation on Ξ.

Lemma 4.1 Locally Ξ is the commuting graph Δ of 3-central subgroups of order 3 in M̄_{1} ∼ Suz.2 as in 1.4. Let Ω be a maximal clique in Ξ containing σ and H be the setwise stabilizer of Ω in C_{X}(z). Then |Ω| = 12 and there is a unique point α collinear to σ such that H = C_{M_{α}}(z). Moreover, H ∼ 2.3^{6}.(2.Mat_{12}), O_{3}(H) = P_{α} ∩ H, H induces the natural action of Mat_{12} on the vertices of Ω and O_{3}(H) is an irreducible GF (3)-module for H/O_{3}(H)⟨z⟩ ∼ 2.Mat_{12}.

Proof: Abusing the notation we denote by σ the point stabilized by M_{1} so that σ ∈ Ξ. By 1.3d every quadric incident to σ contains besides σ exactly one point ε centralized by z and ε is not collinear to σ. This means that the set Ξ(σ) of points adjacent to σ in Ξ is in a natural bijection with the set of quadrics incident to σ, i.e. with the vertices of the graph Δ as in 1.4. Moreover, if δ ∈ Δ then there is a unique point centralized by z which maps onto δ under the homomorphism of M_{1} onto M̄_{1}. We will identify δ with this unique point. By definition if x and y are adjacent points in Ξ then [Q_{x}, Q_{y}] = 1. Hence if δ_{1}, δ_{2} ∈ Ξ(σ) are adjacent in Ξ, then the corresponding vertices of Δ are adjacent. In particular a maximal clique in Ξ contains at most 12 vertices. We are going to show that this bound is attained.

Let l be a line incident to σ and let σ, α, β and γ be all the points incident to l. Since z acts fixed-point freely on P_{1}/σ ∼ 3^{12}, we can choose our notation so that z inverts α and permutes β and γ. So on every line incident to σ there is exactly one point which is inverted by z. Since C_{M_{1}}(z)P_{1} = M_{1}, C_{M_{1}}(z) permutes transitively the lines incident to σ and hence also the points collinear to σ and inverted by z. This implies that C_{M_{α}}(z) permutes transitively the points collinear to α and centralized by z.

Let Q_{8} denote a quadric incident to l and let ε be the point in Q_{8} other than σ centralized by z. Then ε is collinear to exactly one point on l. We know that σ and ε are not collinear and since β and γ are permuted by z, ε is collinear to α. Thus in every quadric incident to l besides σ there is exactly one point collinear to α and centralized by z. By the diagram of X there are exactly 11 such quadrics which correspond to a clique K of Δ. Let Ω = {σ} ∪ K and H be the setwise stabilizer of Ω in C_{X}(z). Since locally Ξ is Δ, K is a maximal clique in Δ, C_{M_{1}}(z) acts transitively on the set of cliques in Δ and since C_{X}(z) is vertex-transitive on Ξ, we see that H acts transitively on Ω. Since α is the only point which is collinear to every point in Ω, it is clear that H ≤ C_{M_{α}}(z). Since z acts fixed-point freely on P_{1}/σ, C_{P_{1}^{∗}}(z) = σ × ⟨z⟩. By 1.4c and the Frattini argument (H ∩ M_{1})P_{1}^{∗}/P_{1}^{∗} ∼ (M_{l} ∩ M_{1})/P_{1}^{∗} ∼ 3^{5}.(2 × Mat_{11}). Since H ∩ M_{1} induces the natural action of Mat_{11} on the points in K, H induces on the points in Ω the natural action of Mat_{12}. Thus O_{3}(H) is elementary abelian of order 3^{6} generated by the 12 points in Ω and H/O_{3}(H)⟨z⟩ ∼ 2.Mat_{12} induces a non-trivial action on O_{3}(H). By [MoAt] Mat_{12} does not have a faithful GF (3)-representation of dimension less than or equal to 6 and the smallest faithful GF (3)-representation of 2.Mat_{12} has dimension exactly 6. Thus we have shown that H ∼ 2.3^{6}.(2.Mat_{12}) and by 3.3b H = C_{M_{α}}(z).

In what follows we will need the detailed information on the structure of 6-dimensional GF (3)- modules of 2.Mat_{12} contained in the following lemma.

Lemma 4.2 Let H̄ ∼ 2.Mat_{12} and A be a faithful irreducible 6-dimensional GF (3)H̄-module. Then the following assertions hold:

(a) H̄ has a unique orbit A of length 12 on the 1-spaces of A.

(b) Any five elements from A are linearly independent.

(c) H̄ has a unique orbit L of length less or equal to 12 on the hyperplanes of A. Moreover, |L| = 12 and if L ∈ L then L contains no element from A.

(d) Let B the set of 1-spaces of A of the form ⟨a_{1} + a_{2}⟩, where ⟨a_{1}⟩ and ⟨a_{2}⟩ ∈ A are different elements of A. Then |B| = 132 and H acts transitively on B.

(e) If F ∈ B then there exist unique elements D_{1} and D_{2} in A with F ≤ D_{1} + D_{2}. If L ∈ L and F̃ is the 1-space in D_{1} + D_{2} different from D_{1}, D_{2} and F, then F ≤ L if and only if F̃ ≰ L;

(f) Define L ∈ L and B ∈ B to be incident if B ≤ L. Then (L, B) is a Steiner system of type (5, 6, 12).

(g) Let T ⊂ L with |T | = 4 and put F = ⋂ T . Then F is a 2-subspace of A, all 1-spaces of F are in B and N_{H}(F)/C_{H}(F) ≅ GL_{2}(3).

Proof: Let X and Y be two non-conjugate subgroups in H̄ isomorphic to Mat_{11}. Then every proper subgroup of index at most 12 in H̄/Z(H̄) ≅ Mat_{12} is conjugate to the image of either X or Y. Moreover, H̄ = ⟨X, Y ⟩ and X ∩Y ≅ L_{2}(11). Let Z be one of the subgroups X, Y and X ∩Y. By [MoAt] every faithful irreducible GF (3)Z-module is 5-dimensional. This means that Z normalizes in A at most one 1-subspace and at most one 5-subspace. Suppose that A contains a 1-subspace normalized by X and a 1-subspace normalized by Y. Then both these 1-spaces are normalized by X ∩ Y and hence this is the same 1-space, normalized by the whole H̄ = ⟨X, Y ⟩, a contradiction to the irreducibility of A. Applying the same argument to the module dual to A, we obtain that the subspaces in A normalized by X and Y have different dimensions and we can choose our notation so that X normalizes a 1-space D and Y normalizes a 5-space E. In this case A = D ⊕ E as a module for X ∩ Y. Moreover, A := DH̄ is the only orbit of length 12 of H̄ on 1-spaces in A and L := EH̄ is the only orbit of length 12 of H̄ on hyperplanes in A and (c) holds.

The actions induced by H̄ on A and L are two non-equivalent 5-transitive actions of Mat_{12}. Since A is irreducible, A spans A and so there is a set of six linearly independent elements in A. Since H̄ induces on A a 5-transitive action, every set of five elements in A is linearly independent and thus (b) holds.

Let D_{1} ≠ D_{2} ∈ A and let D_{1}, D_{2}, F, F̃ be the set of all 1-spaces in D_{1} + D_{2}. Then F, F̃ ∈ B. We are going to show that B satisfies the properties stated in (d) - (f). If there are D_{i}, D_{j} ∈ A with {i, j} ≠ {1, 2} such that F is contained in D_{i} + D_{j} then the set {D_{k} | k = 1, 2, i, j} of size at most four in A would be linearly dependent, a contradiction to (b). Hence the pair {D_{1}, D_{2}}is uniquely determined by F. Let L ∈ L. Since L is a hyperplane in A, its intersection with D_{1} + D_{2} is at least 1-dimensional. By (c) neither D_{1} nor D_{2} are in L, hence (e) follows. Moreover, F or F̃ is contained in at least 6 elements of L. Since the action of H̄ on L is 5-transitive, we conclude that the intersection of any five elements of L is in B. Let D be the set of elements in L containing F. Suppose that |D| ≥ 7. Then by 5-transitivity of H̄ on L there exists h ∈ H̄ with |D ∩ D^{h}| ≥ 5 and D ≠ D^{h}. But then the intersection of the elements on D, D ∩ D^{h} and D^{h}, respectively, are all equal to F, a contradiction to D ≠ D^{h}. Hence |D| ≤ 6 and both F and F̃ are contained in exactly six elements of L. Thus (f) holds. As H̄ acts transitively on the blocks of any associate Steiner systems,

(d) follows.

By (f), T is incident to exactly four elements say B_{1}, B_{2}, B_{3}, B_{4} of B. By the dual of (b), F is a 2-space and so B_{1}, B_{2}, B_{3}, B_{4} are exactly the 1-spaces of F. Since NH̄(T ) induces Sym(4) on {B_{1}, B_{2}, B_{3}, B_{4}} we conclude get NH̄(F)/CH̄(F) ≅ GL_{2}(3).

By 4.1 and 1.4d two maximal cliques in Ξ are either disjoint or have intersection of size 1, 2 or 3. Moreover, if C = C^{X} is a geometry whose elements are maximal cliques, triangles, edges and vertices of Ξ^{X} with respect to the incidence relation given by inclusion, then C corresponds to the following diagram:

3∘ _{9}∘ c^{∗} _{1}∘ c^{∗} _{1}∘.

The geometry C is connected precisely when Ξ is connected. Let σ = Ω_{1} ⊂ Ω_{2} ⊂ Ω_{3} ⊂ Ω_{12} = Ω be maximal flag in C. Then Ω_{i} is a complete subgraph of size i in Ξ. Let C_{i} denote the stabilizer in C_{X}(z) of Ω_{i}. Then

C_{1}/⟨z⟩ ∼ 3.Suz.2, C_{2}/⟨z⟩ ∼ 3^{2}.U_{4}(3).D_{8},
C_{3}/⟨z⟩ ∼ 3^{3+4}.[2^{3}].S_{4}.S_{3}, C_{12}/⟨z⟩ ∼ 3^{6}.2.Mat_{12}.

Consider the situation when X = M. By 3.3a z is of Conway type and C_{M}(z) = C ∼ 2_{+}^{1+24}.Co_{1}. Put R = O_{2}(C).

Lemma 4.3 The graph Ξ^{M} is connected.

Figure 2: Distribution diagram of Ψ.

Proof: Let A be the setwise stabilizer in C_{M}(z) of the connected component of Ξ^{M} which contains σ. Then A contains C_{1} ∼ 6.Suz.2. Let ε be a vertex adjacent to σ in Ξ^{M}. Then [σ, ε] = 1 and since σ acts fixed-point freely on R/⟨z⟩, we have σR ≠ εR. Since C_{1}R is maximal in C, this means that AR = C. Finally, C/⟨z⟩ does not split over R/⟨z⟩ and hence A = C and Ξ^{M} is connected.

The homomorphism χ : G → M induces morphisms G → M and C^{G} → C^{M} of geometries which will be denoted by the same letter χ. Our goal is to show that the restriction of χ to the connected component of C^{G} containing σ is an isomorphism onto C^{M}. This will immediately imply that the setwise stabilizer in C_{G}(z) of the connected component of C^{G} maps isomorphicly onto C ∼ 2_{+}^{1+24}.Co_{1}. An important role in the realization of this step will be played by a simply connected subgeometry in G.

Let μ be a subgroup of Fischer type as in in section 2. Then k := σμ is a subgroup of order 9 in P_{1} which is not a line (so that k is as in 1.5b). Since z acts fixed-point freely on P_{1}/Q_{1}, as in the proof of 4.1 we have a unique subgroup of order 3 in k which is normalized and inverted by z. Hence we can and do choose μ so that z inverts μ. By 2.2 there is a subgroup M_{μ} ∼ 3.M(24) in X which normalizes μ such that M_{μ} ∩ M_{i} = N_{M_{i}}(μ) for i = 1, 2, 3 and 8. Let W = C_{M_{μ}}(z) and let Ψ be the orbit of W on Ξ which contains σ.

Lemma 4.4 (a) |Ψ| = 2 688 and W/⟨z⟩ ∼ 2^{2}.U_{6}(2).Sym(3) acts faithfully on Ψ;

(b) locally Ψ is the commuting graph Θ on the 3-central subgroups of order 3 in U_{5}(2).2.

Proof: By 1.5b and since M_{μ} ∩ M_{1} = N_{M_{1}}(μ), C_{M_{μ}}(z) ∩ M_{1} ∼ 2.(3 × U_{5}(2)).2. By 3.2 and since z is of Conway type and inverts μ, W ∼ 2^{3}.U_{6}(2).Sym(3). Thus (a) holds.

For (b) we may by (a) assume that X = M. The subgroups of Fischer type in P_{1} normalized by z, are permuted transitively by C_{M_{1}}(z) and hence Ψ contains a vertex x of Ξ if and only if μ is contained in P_{x}, or equivalently if x is contained in M(μ) and hence (b) follows.

Since Ψ is locally Θ, its maximal cliques have size 6 and two such cliques are either disjoint or have intersection of size 1, 2 or 3. Define U to be a geometry whose elements are maximal cliques, triangle, edges and vertices of Ψ with the natural incidence relation. Since Ψ = Ξ ∩ M(μ), it is easy to see that the diagram of U is the following.

3∘ _{3}∘ c^{∗} _{1}∘ c^{∗} _{1}∘.

As follows from 4.4, the isomorphism type of U is independent on whether X = M or X = G, since U is contained in M(μ) which is simply connected. It worth mentioning that U itself is simply connected as proved in [Me] and that Ψ is distance-transitive with the distribution diagram given on Figure 2.

5 A characterization of C^{M}

It is not know whether the geometry C^{M} is simply connected. In this section we establish a sufficient condition for a covering of C^{M} to be an isomorphism in terms of the subgeometry U and its images under C_{M}(z). Let R = O_{2}(C_{M}(z)) which is extraspecial of order 2^{25}. We start by defining the folding C̄ of C^{M} with respect to the action of R.

The kernel of the action of C = C_{M}(z) on C^{M} is ⟨z⟩ and since O_{2}(C_{i}/⟨z⟩) = 1 for i = 1, 2, 3 and 12, the action of R/⟨z⟩ is fixed-point free. Let C̄ be the folding of C^{M} with respect to the action of R. This means that C̄ is a geometry whose elements are the orbits of R on C^{M} with two such orbits O_{1} and O_{2} incident if and only if an element from O_{1} is incident in C^{M} to an element from O_{2}. Since R/⟨z⟩ acts fixed-point freely on C^{M}, it is easy to see that if O_{1} and O_{2} are incident in C̄ then each element from O_{1} is incident in C^{M} to exactly one element from O_{2}. Let Ξ̄ be the collinearity graph of C̄ which is also the folding with respect to the action of R of the collinearity graph Ξ^{M} of C^{M}.

We put C̄ = C/R and use the bar notation for the images in C̄ of subgroups of C. Then σ̄ is a subgroup of order 3 in C̄ and NC̄(σ̄) ∼ 3.Suz.2 which is a maximal subgroup in C̄. This enables us to identify the vertices of Ξ̄ with the Suzuki-type subgroups of order 3 in C̄ ∼ Co_{1}. We will use the following properties of the action of C̄ on Ξ̄.

Lemma 5.1 Let C̄ ≅ Co_{1}, Ξ̄ be the set of Suzuki-type subgroups of order 3 in C̄, σ̄ ∈ Ξ̄ and C̄(σ̄) = NC̄(σ̄) ∼ 3.Suz.2. Then C̄ acts primitively on Ξ̄ while C̄(σ̄) has 5 orbits on Ξ̄: {σ̄}, Ξ̄_{1}(σ̄), Ξ̄_{2}(σ̄), Ξ̄_{3}(σ̄) and Ξ̄_{4}(σ̄) with lengths 1, 22 880, 405 405, 1 111 968 and 5 346, respectively. Let Ξ̄ denote also the graph on Ξ̄ invariant under the action of C̄, in which σ̄ is adjacent to the vertices from Ξ̄_{1}(σ̄). Let μ̄_{i} ∈ Ξ̄_{i}(σ̄) and B̄_{i} = C̄(σ̄) ∩ C̄(μ̄_{i}) for i = 1, 2, 3, 4. Then

(a) δ̄ ∈ Ξ̄ \ {σ̄} is adjacent to σ̄ in Ξ̄ if and only if [σ̄, δ̄] = 1, so that Ξ̄ is the folding of Ξ^{M} with respect to the action of R; the distribution diagram of Ξ̄ is given on Figure 3;

(b) B̄_{1} ∼ 3^{2}.U_{4}(3).2^{2}, locally Ξ̄ is the commuting graph Δ of central subgroups of order 3 in C̄(σ̄)/σ̄ ∼ Suz.2 ;

(c) B̄_{2} ∼ 2_{−}^{1+6}.U_{4}(2).2 acts transitively on Ξ̄_{i}(σ̄) ∩ Ξ̄_{1}(μ̄_{2}) for i = 1, 2 and 3, the subgraph induced on Ξ̄_{1}(μ̄_{2}) ∩ Ξ̄_{1}(σ̄) is the disjoint union of 40 copies of the complete 3-partite graph K_{4,4,4}, these copies are permuted primitively by B̄_{2}/O_{2}(B̄_{2}) ∼ U_{4}(2).2, ⟨σ̄, μ̄_{2}⟩ ≅ SL_{2}(3);

(d) B̄_{3} ∼ J_{2} : 2 × 2 acts primitively on Ξ̄_{1}(μ̄_{3}) ∩ Ξ̄_{i}(σ̄) for i = 1, 4 and transitive for i = 2, ⟨σ̄, μ̄_{3}⟩ ≅ Alt(5);

(e) B̄_{4} ∼ G_{2}(4).2 acts primitively on Ξ̄_{1}(μ̄_{4}) ∩ Ξ̄_{i}(σ̄) for i = 1 and 3, ⟨σ̄, μ̄_{4}⟩ ≅ Alt(4) ; (f) the subgraph induced on Ξ̄_{1}(μ̄_{i}) ∩ Ξ̄_{1}(σ̄) is empty for i = 3 and 4;

(g) each vertex from Ξ̄_{1}(μ̄_{3}) ∩ Ξ̄_{3}(σ̄) is adjacent to a vertex from Ξ̄_{1}(μ̄_{3}) ∩ Ξ̄_{1}(σ̄) or to a vertex from Ξ̄_{1}(μ̄_{3}) ∩ Ξ̄_{2}(σ̄)

Proof: The subdegrees, 2-point stabilizers B̄_{i} of the action of C̄ on Ξ̄ and ⟨σ̄, μ̄_{i}⟩ are well known (cf. Lemma 49.8 in [As] or Lemma 2.22.1 (ii) in [ILLSS]). The distribution diagram on Figure 3 is taken from [PS]. This diagram and the structure of B̄_{1} show that the subgraph induced on Ξ̄_{1}(σ̄) is isomorphic to the graph Δ as in 1.4 and that C̄(σ̄) induces its full automorphism group. This means that B̄_{1} acts transitively on Ξ̄_{1}(μ̄_{1}) ∩ Ξ̄_{i}(σ̄) for i = 1, 2, 3, 4 and hence for every vertex γ̄ at distance 2 from σ̄ in Ξ̄, C̄(σ̄)∩C̄(γ̄) acts transitively on Ξ̄_{1}(σ̄)∩Ξ̄_{1}(γ̄). Let χ_{i} be the permutational character of C̄(σ̄) on the cosets of B̄_{i}σ̄ for i = 1, 2 and 4. By Lemma 2.13.1 in [ILLSS] the inner product of χ_{1} and χ_{i} is 5,3 and 2 for i = 1, 2 and 4, respectively. This implies the transitivity statements in (c), (d) and (e). By [At] every action of B̄_{3} of degree 100 or 280 as well as every action of B̄_{4} of degree 2 080 or 20 800 is primitive.

Let δ̄_{i} ∈ Ξ̄_{1}(σ̄) ∩ Ξ̄_{1}(μ̄_{i}) for i = 2, 3, 4, then since locally Ξ̄ is Δ, the distance from σ̄ to μ̄_{j} in the subgraph induced on Ξ̄_{1}(δ̄_{i}) is i. Hence the subgraph induced by Ξ̄_{1}(μ̄_{i}) ∩ Ξ̄_{1}(σ̄) is empty for i = 3 and 4, while for i = 2 it is locally K_{4,4} (compare 1.4e). It is well known and easy to check that K_{4,4,4} is the only connected graph which is locally K_{4,4} and the structure of the subgraph induced on Ξ̄_{1}(μ̄_{2}) ∩ Ξ̄_{1}(σ̄) follows. Finally, every transitive action of B̄_{2} of degree 40 is primitive and has O_{2}(B̄_{2}) in its kernel. Thus all statements except (g) are proved.

We will prove (g) with the roles of σ̄ and μ̄_{3} interchanged. For this we first determine the orbits of B̄_{3} on Ξ̄_{1}(σ). Let A = ⟨σ̄, μ̄_{3}⟩. Then A ≅ Alt(5). Note that there exist exactly two elements ρ ∈ Ξ̄ ∩ A such that ⟨ρ, σ̄⟩ ≅ Alt(4) and ⟨ρ, μ̄_{3}⟩ ≅ Alt(5). Without loss μ̄_{4} is one of these two. Put J = NB̄_{3}(μ̄_{4}) = B̄_{4} ∩ B̄_{3}. Then J is of index two in B̄_{3} and J ∼ J_{2}.2. Put K := B̄_{4}. Then K ∼ G_{2}(4).2.

As the main step in determining the orbits of B̄_{3} on Ξ̄_{1}(σ̄) we compute the orbits of J by decomposing the orbits of K. By (e) K acting on Ξ̄_{1}(σ̄) has two orbits, Γ_{1} = Ξ̄_{1}(σ̄) ∩ Ξ̄_{3}(μ̄_{4}) and Γ_{2} = Ξ̄_{1}(σ̄) ∩ Ξ̄_{1}(μ̄_{4}) with lengths 20 800 and 2 080, respectively, moreover if K_{1} and K_{2} are the respective stabilizers, then K_{1} ∼ U_{3}(3) : 2 × 2 and K_{2} ∼ 3.L_{3}(4).2^{2}. Consider the graph Σ with 416 vertices of valency 100 on which K acts as a rank 3 automorphism group ( see [BvL]). Then the parameters of Σ are the following:

v = 416, k = 100, l = 315, λ = 36, μ = 20.

It follows from the list of maximal subgroups in K, that Γ_{1} can be identified with the set of edges of Σ while J is the stabilizer in K of a vertex x of Σ. By well known properties of the action of K on Σ [BvL] the orbit of J on the edge-set of Σ containing an edge {y_{1}, y_{2}} of Σ is uniquely determined by the pair {d_{1}, d_{2}} where d_{i} is the distance from x to y_{i} in Σ. This and the parameters of Σ given above show that under the action of J the set of edges of Σ (identified with the set Γ_{1}) splits into four orbits Ω_{1}, Ω_{2}, Ω_{3} and Ω_{4} corresponding to the pairs of distances {0, 1}, {1, 1}, {1, 2} and {2, 2} and having lengths 100, 1 800, 6 300 and 12 600, respectively. Let Ω_{5} = Ξ̄_{1}(σ̄) ∩ Ξ̄_{1}(μ̄_{3}) and γ ∈ Ω_{5}. Note that J acts transitively on Ω_{5} and |Ω_{5}| = 280. By (a) γ commutes with μ̄_{3}. Thus γ ≤ J ≤ K′ and so γ ∈ Γ_{2} and Ω_{5} is an orbit for J on Γ_{2}. Let K_{2} be the stabilizer of γ in K. Then γ = O_{3}(K_{2}). By (f) all 280 vertices adjacent to γ in the subgraph induced on Ξ̄_{1}(σ̄) are in Γ_{1} and by (a) these 280 vertices are fixed by γ. Let Σ(γ) be the set of vertices in Σ fixed by γ. Comparing the permutation characters of K on σ with the permutational character of K on Γ_{2}, we see that K_{2} has exactly two orbits on the vertex set of Σ. On one hand this means that under the action of J the set Γ_{2} splits into two orbits namely Ω_{5} and an orbit Ω_{6} of length 1 800. On the other hand K_{2}/γ ∼ L_{3}(4).2^{2} acts

[figure: 1; 22880; 1111968; 5346; 405405; 280; 16200; 5120; 22880; 13608; 8505; 280; 17280; 6300; 100; 2080; 480; 20800; 486; 1]

Figure 3: Distribution diagram of Ξ̄.

transitively on Σ(γ) and so |Σ(γ)| = 280 · \frac{|J|}{|K_{2}|} = 56. Any transitive action of the latter group of degree 56 is the rank 3 action on the vertex set of the Gewirtz graph which is strongly regular with parameters

v = 56, k = 10, l = 45, λ = 0, μ = 2.

Hence we conclude that K_{2} acts transitively on the set of edges in Σ fixed by γ. Again since γ is adjacent in Ξ̄ to exactly 280 vertices from Γ_{1} there are 280 edges in the subgraph of Σ induced on Σ(γ) and hence this subgraph is the Gewirtz graph rather than its complement.

Note that Ω_{i}, 1 ≤ i ≤ 6 are the orbits for J on Ξ̄_{1}(σ̄). If \overline{B}_{3} normalizes K′ then K′ centralizes ⟨σ̄, μ̄B̄_{3} ; 4⟩ = A and so K′ ≤ B̄_{3}, a contradiction. Since K′ is generated by the elements of Γ_{2} = Ξ̄ ∩ K we conclude that B̄_{3} does not normalize Γ_{2}. Thus some of the orbits of J must be fuzed by B̄_{3}. Since J is normal in B̄_{3}, only orbits with the same lengths can fuse. Thus Ω_{2} ∪ Ω_{6} is a single orbit of B̄_{3}. The distribution diagram of Ξ̄ enables us to identify Ω_{5}, Ω_{3}, Ω_{2} ∪ Ω_{4} ∪ Ω_{6} and Ω_{1} with Ξ̄_{1}(σ̄) ∩ Ξ̄_{i}(μ̄_{3}) for i = 1, 2, 3 and 4, respectively. A vertex from Γ_{1} is adjacent to γ in Ξ̄ if and only if the corresponding edge of Σ is fixed by γ. The parameters of the Gewirtz graph imply that γ is adjacent to 10, 90 and 180 vertices from Ω_{i} for i = 1, 3 and 4, respectively. Since every vertex from Γ_{1} is adjacent to 28 = 280 · |Γ_{2}|/|Γ_{1}| vertices from Γ_{2} and every vertex from Ω_{3} is adjacent to 4 = 90 · |Ω_{5}|/|Ω_{3}| vertices of Ω_{5}, we observe that a vertex v ∈ Ω_{3} is adjacent to 24=28-4 vertices from Ω_{6}. Since Ω_{2} and Ω_{6} are fuzed under B̄_{3} this means that v is also adjacent to 24 vertices from Ω_{2}. Hence every vertex from Ξ̄_{1}(σ̄) ∩ Ξ̄_{3}(μ̄_{3}) = Ω_{2} ∪ Ω_{4} ∪ Ω_{6} is adjacent to a vertex from Ξ̄_{1}(σ̄) ∩ Ξ̄_{1}(μ̄_{3}) = Ω_{5} or a vertex from Ξ̄_{1}(σ̄) ∩ Ξ̄_{2}(μ̄_{3}) = Ω_{3} (or both).

Let Ψ̄ be the image in Ξ̄ of the subgraph Ψ in Ξ as in 4.4. Since none of the 2-point stabilizers of the action of C̄ on Ξ̄ involve U_{5}(2), every vertex from the antipodal block containing σ maps onto σ̄ and we have the following

Lemma 5.2 Let Ψ̄ be the image of Ψ in Ξ̄. Then Ψ̄ is the antipodal folding of Ψ which is a strongly regular graph with parameters

v = 672, k = 176, l = 495, λ = 40, μ = 48.

The image W̄ of W = C_{M_{μ}}(z) in C̄ is isomorphic to U_{6}(2).Sym(3).

Since locally Ψ̄ (as well as Ψ) is the commuting graph Θ of 3-central subgroups of order 3 in U_{5}(2).2 which is strongly regular, it is easy to see that in terms of 5.1 Ψ̄ ⊆ {σ̄} ∪ Ξ̄_{1}(σ̄) ∪ Ξ̄_{2}(σ̄).

Let ϱ : C̃ → C^{M} be a covering of C^{M} such that there is a flag-transitive automorphism group C̃ of C̃ which commutes with ϱ and whose induced action on C^{M} coincides with that of C/⟨z⟩. In particular ϱ can be the restriction to a connected component of C^{G} of the morphism of C^{G} onto C^{M} induced by the homomorphism χ : G → M. In this case C̃ is the setwise stabilizer in C_{G}(z)/⟨z⟩ of that connected component. Let R̃ be the kernel of the natural homomorphism of C̃ onto C̄ ∼ C/R. Let Ξ̃ be the collinearity graph of C̃ so that there is a natural morphism of Ξ̃ onto Ξ̄.

Let Ψ and W be as in 4.4. Let Ψ̄ be the image of Ψ in Ξ̄ and W̄ be the image of W in C̄. Let Ψ̃ be a connected component of the preimage of Ψ under ϱ and let W̃ be the stabilizer of Ψ̃ in the preimage of W/⟨z⟩ in C̃.

Lemma 5.3 In the above notation Ψ̃ is isomorphic to Ψ, W̃ ∼ W/⟨z⟩ ∼ 2^{2}.U_{6}(2).Sym(3) and hence W̃ ∩ R̃ is elementary abelian of order 2^{2}.

Proof: The result follows from 4.4 and the fact that Ψ is the collinearity graph of the geometry U which is simply connected by [Me].

Let T̄(σ̄) be the set of images of Ψ̄ under C̄ which contain σ̄. Equivalently we can define T̄(σ̄) to be the set of images of Ψ̄ under NC̄(σ̄). Let σ̃ be a preimage of σ̄ in C̃. Let T̃(σ̃) be the set of connected subgraphs Φ̃ such that σ̃ ∈ Φ̃ and Φ̃ maps onto some Φ̄ ∈ T̄(σ̄). If Φ̃ ∈ T̃(σ̃) and Ũ := C̃(Φ̃) is the setwise stabilizer of Φ̃ in C̃, then by 5.3 O_{2}(Ũ) = Ũ ∩ R̃ is of order 2^{2}. Let

R̃_{σ} = ⟨O_{2}(C̃(Φ̃)) | Φ̃ ∈ T̃(σ̃)⟩.

Lemma 5.4 R̃_{σ} = R̃.

Proof: Let Ξ̂ be the folding of Ξ̃ with respect to the orbits of R̃_{σ}. This means that the vertices of Ξ̂ are the orbits of R̃_{σ} on the vertex set of Ξ̃ with the induced adjacency relation. Notice that in the way it is defined Ξ̂ is not necessary vertex-transitive although every automorphism from C̃ stabilizing σ̃ can be realized as an automorphism of Ξ̂. Nevertheless eventually we will see that Ξ̂ is equal to Ξ̄ and in particular it is vertex-transitive. Since the vertices of Ξ̄ can be considered as orbits of R̃ on Ξ̃ and R̃_{σ} is contained in R̃, there is a covering ω : Ξ̂ → Ξ̄ and R̃_{σ} = R̃ if and only if ω is an isomorphism. Let σ̂ be the image of σ̃ in Ξ̂. Since R̃_{σ} is normalized by the stabilizer C̃(σ̃) of σ̃ in C̃, there is a subgroup Ĉ(σ̂) in the automorphism group of Ξ̂ which stabilizes σ̂ and maps isomorphically onto C̄(σ̄) ∼ 3.Suz.2. We will identify Ĉ(σ̂) and C̄(σ̄). For δ̂ ∈ Ξ̂ let Ξ̂_{1}(δ̂) be the set of vertices adjacent to δ̂ in Ξ̂. Since ω is a covering, the subgraph induced on Ξ̂_{1}(δ̂) is isomorphic to Δ and if δ̂ = σ̂ then Ĉ(σ̂) induces the full automorphism group of this subgraph. Hence Ĉ(σ̂) has exactly three orbits, on the vertices at distance 2 from σ̂. We denote these orbits by Ξ̂_{i}(σ̂), so that ω(Ξ̂_{i}(σ̂)) = Ξ̄_{i}(σ̄) for 2 ≤ i ≤ 4. Let μ̂_{i} ∈ Ξ̂_{i}(σ̂) and B̂_{i} be the stabilizer of μ̂_{i} in Ĉ(σ̂). We assume that there is a vertex μ̂_{1} ∈ Ξ̂_{1}(σ̂), adjacent to μ̂_{i} for 2 ≤ i ≤ 4 and that μ̂_{3} is adjacent to μ̂_{2} and μ̂_{4}. Assuming also that ω(μ̂_{i}) = μ̄_{i}, we can consider B̂_{i} as a subgroup in B̄_{i}, 1 ≤ i ≤ 4. Notice that B̂_{i} acts transitively on the set Ξ̂_{1}(σ̂) ∩ Ξ̂_{1}(μ̂_{i}). Since ω is a covering, the subgraph induced by Ξ̂_{1}(σ̂) ∩ Ξ̂_{1}(μ̂_{2}) is union of m disjoint copies of K_{4,4,4} where 1 ≤ m ≤ 40. For Φ̃ ∈ T̃(σ̃) the image Φ̂ of Φ̃ in Ξ̂ is isomorphic to Ψ̄ as in 5.2 and is contained in {σ̂} ∪ Ξ̂_{1}(σ̂) ∪ Ξ̂_{2}(σ̂). The parameters of Ψ̄ imply that m ≥ 3. Since B̄_{2} acts primitively on the 40 copies of K_{4,4,4} as in 5.1c we have m = 40 and B̂_{2} = B̄_{2}. By 5.1c B̂_{2} has three orbits on the vertices from Ξ̂_{1}(μ̂_{2}) with lengths 480, 5 120 and 17 280, moreover, these orbits are contained in Ξ̂_{i}(σ̂) for i = 1, 2 and 3 respectively. In particular B̂_{2} ∩ B̂_{3} has order order divisible by 2^{7}. By 5.1d the stabilizer in B̂_{3} of a vertex from Ξ̂_{1}(σ̂) ∩ Ξ̂_{1}(μ̂_{2}) has order not divisible by 2^{7} and so B̂_{3} ∩ B̂_{1} is a maximal subgroup of B̄_{3} not containing B̂_{2} ∩ B̂_{3}. Thus B̂_{3} = B̄_{3}. Arguing similarly B̂_{3} ∩ B̂_{4} and B̂_{1} ∩ B̂_{4} are two different maximal subgroups of B̄_{4} and so B̂_{4} = B̄_{4}. Let ρ̂ be a vertex adjacent to μ̂_{i} for i = 2 or 4. By 5.1c,e ρ̂ is conjugate under Ĉ(σ̂) to μ̂_{j} for some 1 ≤ j ≤ 4, except maybe in the case where ρ̂ is adjacent to μ̂_{2} and ρ̂ maps onto an element of Ξ̄_{2}(σ̄). In the latter case we see from the distribution diagram of Δ that such a ρ̂ can already be found in the residue of μ_{1}. Hence in any case a vertex adjacent to μ̂_{i} for i = 2 or 4 is in Ξ̂_{j}(σ̂) for 1 ≤ j ≤ 4. Suppose that there is a vertex ν̂ which is adjacent to μ̂_{3} and whose distance from σ̂ is 3. By 5.1g there must be a vertex in Ξ̂_{1}(μ̂_{3}) ∩ Ξ̂_{j}(σ̂) for j = 1, or 2 which is adjacent to ν̂. As we have seen above, this is impossible. Hence there are no vertices at distance 3 from σ̂ and ω is an isomorphism.

Corollary 5.5 C̄ is the universal completion of the amalgam (C̄_{1}, C̄_{2}, C̄_{12}, W̄).

6 Construction of the 2-locals

As above let G denote the universal completion of the amalgam (M_{i} | i = 1, 2, 3, 8) and χ be the homomorphism of G onto M which is identical on this amalgam. We will consider the M_{i}’s as subgroups both in M and G. The group G acts flag-transitively on the universal cover G of M. The points, lines, planes and quadrics in G and M are the cosets of M_{1}, M_{2}, M_{3} and M_{8} in G and M, respectively. We follow notation introduced in the beginning of Section 4, so that X stays for an arbitrary completion of the amalgam which has M as an quotient.

Let σ = M_{1} viewed as a point stabilized by M_{1}, d = M_{8} viewed as a quadric stabilized by M_{8}, z an involution from P_{1}^{∗}, C = C_{M}(z) ∼ 2_{+}^{1+24}.Co_{1} and R = O_{2}(C). Our nearest goal is to construct in C_{G}(z) a subgroup C̃ which maps isomorphically onto C. As above let Ξ be the graph on the set of points τ with z ∈ P_{τ}^{∗} in which two points are adjacent if they are incident to a common quadric. We will obtain C̃ as the stabilizer in C_{G}(z) of the connected component of Ξ containing σ. Let Ω be a maximal clique in Ξ containing σ, H be the setwise stabilizer of Ω in C_{X}(z) and put A = O_{3}(H). Then by 4.1 H ∼ ⟨z⟩ × 3^{6}.2.Mat_{12}, moreover there is a unique point α collinear to σ, and inverted by z, such that H = C_{M_{α}}(z) and O_{3}(H) = P_{α} ∩ H. We use notation introduced in 4.2, so that A and B are orbits of H̄ = H/⟨A, z⟩ on the set of subgroups of order 3 in A with lengths 12 and 132 respectively while L is the unique orbit of length 12 of H̄ on the set of hyperplanes of A. Then it is straightforward to identify A with the vertices in Ω.

Let {σ, δ} be the edge of Ω incident to d. Then ⟨σ, δ⟩ = C_{Q_{d}}(z). Besides σ and δ there are two subgroups, say ρ and ρ′ of order 3 in C_{Q_{d}}(z). These subgroups are of Fischer type, and lie in the orbit B. Since ρ ≤ P_{α} we can define M_{ρ} as in 2.2. Since C_{M_{d}}(z) ∼ 2.3^{2}.U_{4}(3).D_{8}, we have C_{M_{d}}(z) ∩ M_{ρ} ∼ 2.3^{2}.U_{4}(3).2^{2}. Moreover by 3.2 z is a 2-central involution in M_{ρ} and

C_{M_{ρ}}(z) ∼ (3 × 2_{+}^{1+12}).3.U_{4}(3).2^{2}.

Put C_{0} = C_{M_{ρ}}(z) and R_{0} = O_{2}(C_{0}). Recall the choice of μ and the definition of W before 4.4. In particular σ, δ ≤ W and both σ and δ act non-trivially on O_{2}(W). Thus one of ρ and ρ′ centralizes O_{2}(W). We choose notation so that ρ centralizes O_{2}(W). Recall the definition of C_{i}, i = 1, 2, 3, 12 before 4.3, where we choose Ω_{2} = {σ, δ}. So C_{1} = C_{M_{σ}}(z), C_{2} = C_{M_{{σ,δ}}}(z), and C_{12} = H = C_{M_{α}}(z).

Lemma 6.1 (a) R = ∏_{L∈L} C_{R}(L);

(b) R_{0} = ∏_{ρ≤L∈L} C_{R}(L).

Proof: The image in C/R ≅ Co_{1} of H is the full normalizer of the image of A which shows that R_{0} ≤ R and R_{0} = C_{R}(ρ). Note that [R/⟨z⟩, A] is a non-trivial GF (2)-module for H of dimension at most 24. The restriction of this module to A is a direct sum of irreducible 2-dimensional modules and the kernel of such a summand is a hyperplane. The hyperplanes appearing as kernels form a union of orbits under H̄. By 4.2 there are no orbit of length less than 12 and L is the only orbit of length 12. This implies (a). Since ρ acts fixed-point freely on R/R_{0}, we have (b).

Proposition 6.2 C is the universal completion of the amalgam (C_{0}, C_{1}, C_{2}, C_{12}, W) of subgroups of C.

Proof: Let C̃ be the universal completion of the amalgam and as usual view the C_{i} and W has subgroups of C̃. By 4.4b, C_{2} ∩ W ∼ 3^{2}.U_{4}(2).2 and so C_{2} ∩ W normalizes no non-trivial 2-subgroup of O^{2}(C_{0}/R_{0}). Thus O_{2}(W) ≤ R_{0}.

Since H ∩ W ∼ 3^{4+1}.2.Sym(6) we conclude from 4.2 applied to the dual of A that (H ∩ W)A = N_{H}(A ∩ W) and that there exists unique elements L_{1} and L_{2} in L with L_{1} ∩ L_{2} ≤ A ∩ W. Let U = ⟨O_{2}(W)^{A}⟩. Then U/⟨z⟩ is a subspace in R_{0}/⟨z⟩ of dimension at least 4 centralized by C_{A}(O_{2}(W)). Thus by 6.1b, C_{A}(O_{2}(W)) is the intersection of two members of L. Hence C_{A}(O_{2}(W)) = L_{1} ∩ L_{2}, U = C_{R_{0}}(L_{1})C_{R_{0}}(L_{2}), ρ ≤ L_{1} ∩ L_{2} and |U| = 2^{5}.

Put V = C_{R_{0}}(L_{1}). We conclude from 6.1(b) that N_{H∩C_{0}}(V) ∼ 3^{6}.2.Sym(5). On the other hand (H∩W)^{∞} is normal in (H∩W)A and so (H∩W)^{∞} centralizes all conjugates of O_{2}(W) under A. Thus (H ∩W)^{∞} ≤ N_{H}(V). It follows that N_{H}(V) = ⟨N_{H∩C_{0}}(V), (H ∩W)^{∞}⟩ ∼ 3^{6}.2.Mat_{11}. In particular, H acts doubly transitive on the 12 elements of V^{H} and since V V^{h} ≅ 2_{+}^{1+4} for h ∈ H ∩ C_{0} \ N_{H}(V) we conclude that R̃ := ⟨V^{H}⟩ ≅ 2_{+}^{1+24}.

We claim that R̃ is normal in C̃. By definition H normalizes R̃. Moreover, R_{0} = ⟨V^{H∩C_{0}}⟩. Let t ∈ H ∩ C_{2} \ C_{0}. As C_{0} ∩ C_{2} is of index two in C_{2}, t normalizes C_{0} ∩ C_{2}. Also t permutes ρ and ρ′ and we conclude that R̃ = R_{0}R_{0}^{t} is normalised by R_{0}, C_{0} ∩ C_{2} and t. Thus both C_{0} = R_{0}(C_{0} ∩ C_{2}) and C_{2} = (C_{0} ∩ C_{2})⟨t⟩ normalize R̃. Since C_{1} = ⟨C_{1} ∩ C_{2}, C_{1} ∩ H⟩, R̃ is indeed normal in C̃.

Note that C̃/R̃ is a completion of the amalgam (C_{1}R̃/R̃, C_{2}R̃/R̃, C_{12}R̃/R̃, W R̃/R̃). As O_{2}(W) ≤ R̃, we can apply 5.5 and conclude that C̃/R̃ ≅ C̄ ≅ Co_{1}. Thus C̃ ∼ 2_{+}^{1+24}.Co_{1} and since C is a quotient of C̃, we obtain C̃ ≅ C

In view of the preceding proposition our nearest goal is to find such an amalgam inside of G. The first part, namely finding the subgroups, is already accomplished. Indeed the groups C_{0}, C_{1}, C_{2}, , C_{12} = H and W had been defined for X, in particular for G and for M. It remains to show that the pairwise intersections are the same when regarded as subgroups of G and M, respectively. The fact that the pairwise intersections between C_{1}, C_{2}, H and W are correct follows immediately from the definitions of these groups. Also H ≤ M_{α} and C_{2} ≤ M_{d}. Since ρ is perpendicular to Q_{α} in Q_{d} we conclude from 2.2 that C_{0} intersects C_{2} and H correctly. Moreover, N_{C_{1}}(ρ) ≤ N_{M_{σ}}(ρ) ≤ N_{M_{σ}}(⟨σδ⟩) ≤ M_{d} and so C_{0} and C_{1} intersect correctly. It remains to check the intersection C_{0} ∩ W. As C_{0} ≤ M_{ρ} and W ≤ M_{μ} this is accomplished by

Lemma 6.3 N_{M_{ρ}}(μ) = M_{ρ} ∩ M_{μ}.

Proof: Let F = ρμ. Then F is a non-degenerated 2-space of ”plus”-type with respect to the M_{d} invariant quadratic form on Q_{d}. Hence N_{M_{d}}(ρ, μ′) ∼ 3^{8}.Ω_{6}^{−}(3).2 and F/ρ is of type 3C in M_{ρ}/ρ ≅ M(24) (compare [At]). This shows that N_{M_{ρ}}(μ) = N_{M_{d}}(ρ, μ) ≤ N_{M_{d}}(μ) ≤ M_{μ}.

Corollary 6.4 Let C̃ be the subgroup of G generated by C_{0}, C_{1}, C_{2}, H and W. Then C̃ ∼ 2_{+}^{1+24}.Co_{1} and C̃ is the normalizer of the connected component of Ξ containing σ.

We now proceed finding the remaining terms E_{4} and E_{8} (cf. Section 3) of the 2-local geometry of M. Of the 3-local subgroups considered so far only the normalizers of Fischer type subgroups contain a conjugate of E_{4}. (This follows from the fact E_{4} centralizes all subgroups of odd order in M which are normalized by E_{4}.) This is not enough to reconstruct N as a subgroup of G and we are forced to first locate a further 3-local subgroup of G containing E_{4}. By 4.2g there exists a 2-space F in A all of whose 1-spaces are in B and so of Fischer type. Moreover N_{H}(F)/C_{H}(F) ≅ GL_{2}(3) and there exists L_{1}, L_{2} in L with F ≤ L_{1} ∩ L_{2}. Choose F so that ρ ≤ F and let δ be a further Fischer type subgroup of F.

We are trying to locate subgroups of N_{G}(F) and for this we will produce a quadric d′ with F ≤ Q_{d′}. Let z′ be an involution in H so that P_{α}^{∗}z = P_{α}^{∗}z′, but P_{α}z ≠ P_{α}z′. Then by 3.3b z′ = z^{r} for some r ∈ M_{α}. Let A′ = A^{r} and Ω′ = Ω^{r}. Since σ_{i} = C_{A′}(L_{i}) has 12-conjugates under H ∩ H^{r} ∼ 2^{2}.Mat_{12}, σ_{i} ∈ Ω′. Thus {σ_{1}, σ_{2}} is an edge in Ω′ and there exists an unique quadric d′ adjacent to α, σ_{1} and σ_{2}. In Q_{d′} we see that Q_{d′} ∩P_{α} = σ_{1}σ_{2}[Q_{d′} ∩P_{α}, z′] and [Q_{d′} ∩P_{α}, z′] has order 3^{5}. As [Q_{α}, z′] = AQ_{α} and C_{AQ_{α}}(σ_{1}σ_{2}) has order 3^{5} we conclude that C_{AQ_{α}}(σ_{1}σ_{2}) = [Q_{d′} ∩ P_{α}, z′]. Hence F ≤ Q_{d′}.

Since all 1-spaces in F are of Fischer type, F is a non-degenerate 2-space of ”minus”-type in Q_{d′} and C_{M_{d′}} (F) ∼ 3^{8}.Ω_{6}^{+}(3). Since C_{M_{d′}} (F) ≤ M_{ρ} we conclude [At] that F/ρ is of type 3A in M_{ρ}/ρ, which means that C_{M_{ρ}}(F) ∼ 3^{2}.P Ω_{8}^{+}(3). Let g be a point incident to d′ such that Q_{g} is perpendicular to F in Q_{d′}. Then Q_{g} is centralized by a Sylow 3-subgroup of C_{M_{d′}} (F). Hence Q_{g}F/F is 3-central in C_{M_{d′}} (F)/F and so also 3-central in C_{M_{ρ}}(F). Thus C_{M_{ρ}}(F) ∩ N_{M_{ρ}}(Q_{g}) is a maximal subgroup of C_{M_{ρ}}(F) different from C_{M_{d′}} (F). Hence

C_{M_{ρ}}(F) = ⟨C_{M_{d′}} (F), C_{M_{ρ}}(F) ∩ N_{M_{ρ}}(Q_{g})⟩ ≤ ⟨N_{M_{d′}} (δ), N_{M_{g}}(δ)⟩ ≤ M_{δ}.

Put T = C_{M_{ρ}}(F). We conclude that T = C_{M_{δ}}(F) and so N_{H}(F) normalizes T. Put M_{F} = T N_{H}(F). Then M_{F} ∼ (3^{2} × D_{4}(3)).GL_{2}(3) and in particular, M_{F} maps isomorphically onto the full normalizer of F in M.

Note that C_{M_{F}} (z) = N_{H}(F)C_{T} (z) ⊆ HC_{M_{ρ}}(z) ⊆ C̃. As z centralizes F, z ∈ O^{3}(T) = T′ ≅ D_{4}(3). As N_{H}(F) induces the full group of outer automorphisms on T′ and by [At] T′ has a unique class of involutions invariant under all automorphisms, z is 2-central in T′. In particular, there exists a pure Conway foursgroup E in T′ with z ∈ E ≤ O_{2}(C_{T′}(z)) ≤ O_{2}(C_{M_{ρ}}(z)) = R_{0} ≤ R̃. Let t be an involution in E distinct from z. Then t = z^{g} for some g ∈ T′ ≤ M_{ρ} ∩ M_{F}. Put C̃_{t} = C̃^{g}. Then by conjugation of the corresponding statements for z we get C_{M_{ρ}}(t) ≤ C̃_{t} and C_{M_{F}} (t) ≤ C̃_{t}.

Lemma 6.5 CC̃(E) ≤ C̃_{t}.

Proof: Put C_{E} = CC̃(E). Then C_{E} ∼ 2^{2+11+22}.Mat_{24}. Moreover C_{E} ∩ M_{ρ} = N_{C_{E}}(ρ) and so modulo O_{2}(C_{E}), C_{E} ∩ M_{ρ} has shape 3.Sym(6). Similarly modulo O_{2}(C_{E}) the intersection C_{E} ∩ M_{F} is of shape 3^{2}.GL_{2}(3). By [At] no proper subgroup of Mat_{24} has two such subgroups and thus C_{E} = ⟨C_{E} ∩ M_{ρ}, C_{E} ∩ M_{F} ⟩O_{2}(C_{E}). Since ρ has fixed points on any composition factor for C_{E} on O_{2}(C_{E}) this implies C_{E} = ⟨C_{E} ∩ M_{ρ}, C_{E} ∩ M_{F} ⟩ ≤ C̃_{t}.

Let E_{8} be a pure Conway type eights subgroup of T′ such that E_{8} ≤ O_{2}(C_{T′}(x)) for all 1 ≠ x ∈ E_{8} and E ≤ E_{8}. Put E_{4} = E and for i = 4, 8 put C_{E_{i}} = ⋂_{1≠x∈E_{i}} C̃_{x}. Then by 6.5 C_{E_{i}} = CC̃(E_{i}). Moreover N_{T} (E_{i}) normalizes C_{E_{i}} and induces on E_{i} its full automorphism group. Put Ñ = C_{E_{4}}N_{T} (E_{4}) and L̃ = C_{E_{8}}N_{T} (E_{8}). Then χ maps the amalgam (C̃, Ñ, L̃) isomorphicly onto the amalgam (C, N, L) as in Section 3. Let M̃ be the group generated by C̃, Ñ and L̃. Then by 3.1 χ maps M̃ isomorphicly onto M. Thus to complete the proof of 1.6 it remains to show that G = M̃. For this note first that M_{ρ} is generated by its intersection with C̃ and Ñ. Moreover, M_{1} and M_{8} are both generated by their intersections with M_{ρ} and C̃. Finally M_{1} and M_{8} generate G and so G = M̃ and 1.6 is proved.

References

[As] M. Aschbacher, Sporadic Groups, Cambridge Tracts in Mathematics 104, Cambridge University Press, 1994.

[At] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Perkel and R.A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985.

[BCN] A.E. Brouwer, A.M. Cohen and A. Neumaier, Distance-Regular Graphs, Springer Verlag, Berlin, 1989.

[BvL] A.E. Brouwer and J.H. van Lint, Strongly regular graphs and partial geometries, In: Enumeration and Design, pp. 85-122, Academic Press, Toronto, 1984.

[BF] F. Buekenhout and B. Fischer, A locally dual polar space for the Monster, unpublished manuscript dated around 1983.

[DM] C. Dong and G. Mason, The construction of the Moonshine module as a Z_{p}-orbifold, Preprint 1993.

[Iv1] A.A. Ivanov, A geometric characterization of the Monster, In: “Groups, Combinatorics and Geometry” (M. Liebeck and J. Saxl eds.), pp. 46–62, London Math. Soc. Lect. Notes 165, Cambridge Univ. Press, Cambridge, 1992.

[Iv2] A.A. Ivanov, On the Buekenhout–Fischer geometry of the Monster, In: Moonshine, the Monster and Related Topics, C.Dong and G.Mason eds., pp. 149–158, Contemp. Math. 193 AMS, 1996.

[ILLSS] A.A. Ivanov, S.A. Linton, K. Lux, J. Saxl and L.H. Soicher, Distance-transitive representations of the sporadic groups, Comm. Algebra, 23(9) (1995), 3379–3427.

[IS] A.A. Ivanov and G. Stroth, A characterization of 3-local geometry of M(24), Geom. Dedic. (to appear).

[MoAt] Ch. Jansen, K. Lux, R.A. Parker and R.A. Wilson, An Atlas of Brauer Characters, Clarendon Press, Oxford, 1995.

[Me] T. Meixner, Some polar towers, Europ. J. Combin. 12 (1991), 397–415.

[PS] C.E. Praeger and L.H. Soicher, Low Rank Representations and Graphs for Sporadic Groups, Australian Math. Soc. Lect. Note Ser., Cambridge Univ. Press (to appear).

[RS] M. Ronan and G. Stroth, Minimal parabolic geometries for the sporadic groups, Europ. J. Combin. 5 (1984), 59-91.

[Wi] R.A. Wilson, The odd-local subgroups of the Monster, J. Austral. Math. Soc. (A) 44 (1988), 1–16.

[Yo] S. Yoshiara, On some extended dual polar spaces, I, Europ. J. Combin. 15 (1994), 73–86.

Department of Mathematics Imperial College London SW7 2BZ, UK a.ivanov@ic.ac.uk Department of Mathematics Michigan State University East Lansing, Michigan 48824 meier@math.msu.edu