c - extensions of P - and T -geometries

Gernot Stroth, Corinna Wiedorn Martin-Luther-Universität Halle-Wittenberg Institut für Algebra und Geometrie 06099 Halle, Germany

We consider affine extensions of geometries of Petersen and of tilde type, i.e., of flag-transitive geometries Γ belonging to a diagram

∘  ∘  ∘  ∘  ∘ ⋯ c  ^{X}^{1}  ^{2}  ^{n-2}  ^{n-1}  ^{n} ,
where either ∘  ∘^{X}  = ∘  ∘^{P^{∗}} , i.e., the geometry of edges and vertices of
the Petersen graph, or ∘  ∘^{X}  = ∘  ∘^{∼}  , i.e., the 3-fold cover of the ge-

neralized Sp_{4}(2)-quadrangle. We call such geometries c-P - resp. c-T -geometries because the residue of an element of type 1 is a P - resp. T -geometry. Such geometries have been classified by different authors and an overview over the results is given in [7]. In particular, there exist only two P -geometries of rank 3 with automorphism groups Aut(M_{22}) resp. 3Aut(M_{22}) and three T -geometries of rank 3 with automorphism groups M_{24}, He, and 3^{7}Sp_{6}(2). In this paper we will assume:

(∗) If F is a flag of Γ of type {1, …, n − 3} then the residue res(F) of F in Γ is the P -geometry for M_{22} or the T -geometry for M_{24}.

Since there are no P -geometries of rank ≥ 5 and no T -geometries of rank ≥ 5 satisfying (∗), this forces n ≤ 6. On the other hand, it is known that for n = 3 the universal cover of Γ is infinite, and the case that Γ is a c-P -geometry of rank 4 satisfying (∗) has been considered in [4]. There it is shown that Aut(Γ) is isomorphic to a factorgroup of one of the groups 2^{11}n : Aut(M_{22}), M_{24} or 2·U_{6}(2):2 (where by H:K and H·K we denote as usually a split- resp. non-split extension of a group K by a group H). In the first case, the normal 2-subgroup is the universal representation group (see below) of the P -geometry for M_{22}. It is isomorphic to a submodule of index two (the even half) of the Golay co-code. Models for the other two geometries are described in [4].

Definition 1 (see [6]) Let Γ be any geometry which contains some sets of objects P and L (“points” and “lines”) such that any line l ∈ L is incident to exactly three points of P. Let U(Γ) be the abstract group defined by

U(Γ) = ⟨u_{x}|x ∈ P, u_{x}^{2} = 1 = u_{x}u_{y}u_{z} whenever x, y, z are the

three points incident to a line l ∈ L⟩.

Then U(Γ) is called the universal representation group of the point-line system (P, L).

If Γ is a P - or T -geometry with diagram ∘  ∘  ∘  ∘  ∘ ⋯  ^{X}^{1}  ^{2}  ^{n-2}  ^{n-1}  ^{n} we

define the group U(Γ) to be the universal representation group of the point-line system (P, L) where P and L are the objects of Γ of type 1 and 2.

Of course, the group U(Γ) might become trivial, as it does e.g. in case of the rank 4 P -geometry for M_{23}. But if it is nontrivial, then Aut(Γ) acts on U(Γ) in a natural way and the semidirect product of U(Γ) with Aut(Γ) acts flag-transitively on a c-extension of Γ.

The universal representation groups of the P - and T -geometries occurring as residues in this paper are all known (see [6]). For the P -geometry for Co_{2} it is an elementary abelian 2-group of order 2^{23}, and it is isomorphic to the submodule of index 2 stabilized by Co_{2} in the Leech lattice mod 2. This module contains a one-dimensional subspace invariant under the action of Co_{2} and the quotient over this subspace is also a representation group ([6, (5.7)]). The representation groups of the T -geometries for M_{24} and Co_{1} are elementary abelian of order 2^{11} resp. 2^{24} and isomorphic to the 11-dimensional irreducible submodule of the Golay co-code resp. to the Leech lattice mod 2 ([6, (3.2),(5.8)]). Also by [6] the universal representation group of the P -geometry for the Babymonster sporadic simple group BM is a non-split extension 2·BM and the universal representation group of the T -geometry for the Monster M is isomorphic to M. By an unpublished result of A. Ivanov and S. Shpectorov there exist examples of c-extensions with automorphism groups (2·BM ∗2·BM) : 2 and (M ×M) : 2, which are related to the representation group in these cases as well.

The aim of the present paper is to show that under the assumption of (∗) all c-T -geometries arise in the context of the representation group and that there are only three more examples of c-P -geometries , which we will describe in Section 1. More precisely we prove

Theorem 1 Let G ≤ Aut(Γ) be a flag-transitive automorphism-group of a c-P -geometry Γ of rank n ≥ 5. Assume that the residue of each flag of type {1, …, n − 3} is the P -geometry for M_{22}.

(a) If n = 5 and the residue of an element of type 1 is the P -geometry for M_{23} then G ≅ M_{24}.

(b) If n = 5 and the residue of an element of type 1 is the P -geometry for Co_{2} then either G ≅ 2^{22} : Co_{2} resp. 2^{23} : Co_{2} and O_{2}(G) considered as module for G/O_{2}(G) is isomorphic to a section of the Leech lattice mod 2; or G ≅ Co_{1}.

(c) If n = 6 then either G is isomorphic to one of the groups BM ×BM, BM ≀ Z_{2}, (2·BM ∗2·BM), (2·BM) ≀ Z_{2}, or G ≅ M.

Theorem 2 Let G = Aut(Γ) be the automorphism-group of a c-T -geometry Γ of rank n ≥ 4. Assume that the residue of each flag of type {1, …, n − 3} is the tilde geometry for M_{24}.

(a) If n = 4 then G is the semidirect product of M_{24} with the (11-dimensional) Golay code module.

(b) If n = 5 then G is the semidirect product of Co_{1} with the Leech lattice mod 2.

(c) If n = 6 then G ≅ M × M, or M ≀ Z_{2}.

We prove Theorems 1 and 2 in a series of lemmas in Section 4. In Section 2 we derive some general results about subgeometries of Γ, which apart from being useful for the determination of Aut(Γ) also might be of their own interest. Section 3 contains some further preliminary results.

For the rest of the paper we fix the following notation: Γ is a c-P - or a c-T -geometry, Γ^{(i)} is the set of objects of Γ of type i, {α_{1}, …, α_{n}} is a maximal flag in Γ with α_{i} ∈ Γ^{(i)}, G ≤ Aut(Γ) is a flag-transitive automorphism group of Γ, G_{i} = G_{α_{i}} is the stabilizer of α_{i} in G, K_{i} is the kernel of the action of G_{i} on the residue res(α_{i}) of α_{i} in Γ, P_{i} = ⋂_{j≠i} G_{j} are the so-called “minimal parabolics” and B = P_{1}∩…∩P_{n} is the “Borel subgroup”. Elements of type 1 will sometimes be called ”points”, elements of type 2 ”lines”, and elements of type 3 ”planes”. Further, for x ∈ Γ^{(i)}, by res(x)^{−} we denote the geometry induced on the objects of types 1, …, i − 1 in res(x) and by res(x)^{+} the geometry induced on the objects of types i + 1, …, n.

By 2^{n} we usually denote an elementary abelian 2-group of order 2^{n}, by [2^{n}] any 2-group of order 2^{n}, and by 2^{n_{1}+n_{2}+...+n_{k}}H we mean the extension of a 2-group P by a group H such that P has an H-invariant composition series 1 = P_{0} ≤ P_{1} ≤ … ≤ P_{k} = P with P_{i}/P_{i−1} ≅ 2^{n_{i}}. The rest of the notation is standard.

1 Examples of c-P -geometries

1.1 A geometry for M_{24} with diagram

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

This example is probably well known but we include it for completeness. Let S = S(24, 8, 5) be the Steiner system for G = M_{24} with underlying set Ω (for some properties of S see e.g. [2, chapters 6,7]). Let O denote the set of octads and T the set of trios. We set

– Γ^{(1)} = Ω,

– Γ^{(2)} = {{a_{1}, a_{2}}|a_{i} ∈ Ω, a_{1} ≠ a_{2}} – Γ^{(3)} = {{a_{1}, a_{2}, a_{3}, a_{4}}|a_{i} ∈ Ω, a_{i} ≠ a_{j} for i ≠ j}

– Γ^{(4)} = {(O_{1}, {O_{2}, O_{3}})|{O_{1}, O_{2}, O_{3}} ∈ T }

– Γ^{(5)} = O.

Incidences between objects of types 1,2,3 are defined by inclusion. An element a ∈ Γ^{(1)} is incident to (O_{1}, {O_{2}, O_{3}}) ∈ Γ^{(4)} if a ∈ O_{1}, and to O ∈ Γ^{(4)} if a ∉ O. Elements x ∈ Γ^{(2)} ∪ Γ^{(3)} and y ∈ Γ^{(4)} ∪ Γ^{(5)} are incident if all elements of x are incident to y. The elements of type 5 in res(x) for x = (O_{1}, {O_{2}, O_{3}}) ∈ Γ^{(4)} are O_{2} and O_{3}.

1.2 A geometry for Co_{1} with diagram

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

In order to define the geometry for Co_{1} first we briefly recall the definition of the P-geometry for Co_{2} as it is given in [9].

Let Λ denote the Leech lattice with inner product ( , ) and let Λ_{2} = {v ∈ Λ, (v, v) = 32}. Consider the group H = Co_{2} as the stabilizer in Aut(Λ) of a fixed vector v_{0} ∈ Λ_{2}. Let Σ = {{v, −v}|v ∈ Λ_{2}, (v_{0}, v) = 0} and define a graph on Σ in the way that σ and τ ∈ Σ are adjacent if H_{σ} ∩ H_{τ} ≅ 2^{9}.2^{5}Σ_{5}. A clique C in this graph is called closed if for each {v, −v}, {w, −w} ∈ C there exists {u, −u} ∈ C such that v_{0} + v + w + u ∈ 2Λ. It is shown in [9] that each closed clique is of size 1,3,7 or 15, and that, if we take as objects of type i the closed cliques of size 2^{i} − 1 and define incidence by inclusion, then we get the P-geometry for Co_{2}.

Now we can define the c-P -geometry for Co_{1}: Let

– Γ^{(1)} = {{v, −v}|v ∈ Λ_{2}}.

For x = {v, −v} ∈ Γ^{(1)} let Σ_{x} = {{w, −w} ∈ Γ^{(1)}|(v, w) = 0} and consider Σ_{x} as the graph just described. Let

– Γ^{(i)} = {{x} ∪ C|x ∈ Γ^{(1)}, C ⊆ Σ_{x}, |C| = 2^{i−1} − 1, C is a closed clique in the graph on Σ_{x}}, for 2 ≤ i ≤ 5,

and define incidence by inclusion. Then it is straightforward to show that Γ = ⋃_{i=1}^{5} Γ^{(i)} is a geometry with the desired diagram. Since Co_{1} is transitive on {{v, −v}|v ∈ Λ_{2}}, it is clear that Aut(Γ) ≅ Co_{1}.

1.3 A geometry for the Monster with diagram

∘  ∘  ∘  ∘  ∘  ∘ c  ^{P^{∗}}^{1}  ^{2}  ^{3}  ^{4}  ^{5}  ^{6}

The idea of the construction is quite similar to the previous example. Recall that 2 · BM = C_{M}(z) for an involution z ∈ M of type 2A in the notation of [3] and that the objects of the P-geometry for BM can be identified with certain conjugacy classes of elementary abelian 2-groups of orders 2, 4, 8, 16 resp. 32 of BM (see e.g. [6]). Therefore, we define Γ in the following way:

– Γ^{(1)} = 2A,

– Γ^{(i)} = {zU|z ∈ 2A, U ≤ C_{M}(z), U is elementary abelian of order 2^{i−1} and corresponds to an object of the P-geometry for C_{M}(z)/⟨z⟩ ≅ BM}, for 2 ≤ i ≤ 6.

Incidence is again defined by inclusion.

If zU ∈ Γ^{(i)} for some i > 1 and u ∈ U, then u is of type 2B in M, i.e. C_{M}(u) ≅ 2^{1+24}Co_{1}, and ⟨z, U⟩ ≤ O_{2}(C_{M}(u)). Since O_{2}(C_{M}(u)) is extraspecial and N_{M}(U) ∩ C_{M}(z) is transitive on U, N_{M}(⟨z, U⟩) is transitive on zU. Now it is straightforward to see that Γ is a geometry of c-P -type.

2 Subgeometries

In this section Γ always denotes a c-P -geometry of rank n ≥ 4. The reader might take it as an exercise to generalize the results to any flag-transitive geometry

with diagram ∘  ∘  ∘  ∘  ∘ ⋯ c  ^{X}^{1}  ^{2}  ^{n-2}  ^{n-1}  ^{n} , where ∘  ∘^{X}  may be an arbi-

trary rank-2-geometry in which every object of the rightmost type is incident to exactly three objects of the other type. Further one has to assume that the stabilizer of an element x ∈ Γ of type i ≥ 2 induces the full affine linear group 2^{i−1}L_{i−1}(2) on res(x)^{−}. In particular, we need the fact that G_{x} acts 3-fold transitively on the set of points and 2-fold transitively on the set of partitions of res(x)^{−} into classes of parallel lines. (This is automatically true if X = P^{∗}.)

2.1 Shrinking

In this subsection we define a new incidence structure S related to Γ. In general, the structure S might not be connected (in fact, for most of the c-P -geometries it is disconnected), but it is not dificult to see that its connected components are geometries with the same diagram as Γ but of rank n − 1. (Therefore, we call this construction “shrinking”.)

The structure S has the following sets of objects:

– S^{(1)} = Γ^{(2)},

– S^{(i)} = {(x, P_{x})|x ∈ Γ^{(i+1)}, P_{x} is a partition of res(x)^{−} into parallel lines},

for 2 ≤ i ≤ n − 1. The elements (x, P_{x}) ∈ S^{(i)}, (y, P_{y}) ∈ S^{(j)} with i, j > 1, are incident if x and y are incident in Γ and P_{x} is compatible with P_{y}; x ∈ S^{(1)} and (y, P_{y}) ∈ S^{(i)}, i > 1, are incident if x is incident to y in Γ and is part of the partition P_{y} (recall that x is a line).

In the following we derive some properties of S. The first lemma is due to A. Pasini.

Lemma 1 Let S be disconnected. Then, for every x ∈ Γ^{(i)}, i ≥ 3, and two different partitions P_{1},P_{2} of res(x)^{−} into parallel lines, the objects (x, P_{1}) and (x, P_{2}) belong to different connected components of S.

Proof. The relation “giving rise to objects in the same connected component” is clearly an equivalence relation on the set of partitions of res(x)^{−} into parallel lines and it is preserved by the action of G_{x} on that set. But as mentioned above that action is 2-transitive and therefore primitive. So either we have the statement or all partitions give rise to objects in the same connected component of S. However, in the latter case the connectedness of Γ forces S to be connected. □

We can also describe the shrinking in terms of the corresponding parabolic subgroups: If P_{1}, …, P_{n} are the minimal parabolics corresponding to a fixed maximal flag of Γ and R_{1}, …, R_{n−1} the minimal parabolics belonging to a certain maximal flag in a suitable connected component Σ of S, then R_{i} = ⟨P_{1}, P_{i+1}⟩ for i ≥ 2 and R_{1} = N_{⟨P_{1},P_{2}⟩}(P_{1}). If G_{Σ} denotes the stabilizer of Σ in G, then clearly G_{Σ} = ⟨R_{1}, …, R_{n−1}⟩ and S is connected iff G = G_{Σ}. For the Borel-subgroup B_{Σ} we have B_{Σ} = R_{1} ∩ … ∩ R_{n−1} = P_{1}, hence |B_{Σ} : B| = 2.

Lemma 2 Let G_{Σ} be the stabilizer of a connected component Σ of S, N ≤ G_{Σ} the kernel of the action of G_{Σ} on Σ and H = G_{Σ}/N. Suppose

(a) the kernel of the action of H_{p} on the residue of p in Σ is trivial for p ∈ Σ of type 1,

(b) G_{2}/K_{2} ≅ Z_{2} × G_{12}/K_{2}.

Let Q_{2} ≤ G_{2} such that |Q_{2} : K_{2}| = 2 and G_{2}/Q_{2} ≅ G_{12}/K_{2}. Then N = Q_{2}.

Proof. We identify p ∈ Σ with α_{2} ∈ Γ and H_{p} with G_{2}N/N. Now Q_{2} is trivial on the elements of types 3, …, n of res(α_{2}) in Γ, hence it is trivial on the residue of p in Σ. By assumption (a) Q_{2} ≤ N.

Let F_{Σ} be a maximal flag in Σ. Then F_{Σ} is stabilized by N. By construction, F_{Σ} comes from a unique flag F of cotype 1 in Γ. So N stabilizes F. Since F is contained in exactly two maximal flags, we have |N : N ∩ B| ≤ 2. On the other hand, our construction implies that N ∩ B ≤ K_{2}. So the assertion follows from Q_{2} ≤ N and |Q_{2} : K_{2}| = 2. □

Corollary 1 Suppose the assumptions of Lemma 2. Then S is disconnected.

Proof. By Lemma 2 Q_{2} = N ≤ G_{Σ}. Since Q_{2} ≤ G_{2}, therefore Q_{2} must stabilize each element of {α_{2}^{G_{Σ}}}. Hence G_{Σ} ≠ G. □

If n ≥ 5, we can repeat the shrinking construction applying it to S. Then we get a new incidence structure, say T , whose connected components are contained in the connected components of S and constitute c-P -geometries of rank n − 2. For the set T^{(1)} of objects of T of type 1 we have by construction

T^{(1)} = S^{(2)} = {(x, P_{x})|x ∈ Γ^{(3)}, P_{x} is a partition of res(x)^{−} into parallel lines}.

Now, for each x ∈ Γ^{(3)}, there are exactly three different partitions P_{1}, P_{2}, P_{3} of res(x)^{−} into two parallel lines, and by Lemma 1 either S is connected or each pair (x, P_{i}) belongs to a different connected component of S, hence also to a different connected component of T . Define a relation ≡ on the set of connected components of T by Θ_{1} :≡ Θ_{2} if there exist x ∈ Γ^{(3)} and partitions P_{1}, P_{2} of res(x)^{−} into parallel lines such that (x, P_{i}) belongs to Σ_{i} for i = 1, 2. Then the following holds.

Lemma 3 Suppose S is disconnected and define the relation ≡ as just described. If Θ_{1} ≡ Θ_{2} then the elements of Θ_{1}^{(1)} and Θ_{2}^{(1)} arise from the same set of elements from Γ^{(3)}. Moreover, ≡ is an equivalence relation on the set of connected components of T , each equivalence class is of size three and, if {Θ_{1}, Θ_{2}, Θ_{3}} is such an equivalence class, then the Θ_{i} are contained in different connected components of S.

Proof. Obviously ≡ is reflexive and symmetric. Moreover, if the first statement of the lemma is shown, then it is clear that ≡ is also transitive, hence an equivalence relation, and Lemma 1 together with the fact that, for any x ∈ Γ^{(3)}, there are exactly three partitions of res(x)^{−} into parallel lines implies the rest of the assertions.

Let Θ_{1} ≡ Θ_{2} and x ∈ Γ^{(3)} such that (x, P_{i}) ∈ Θ_{i} for certain partitions P_{1}, P_{2} of res(x)^{−}. By res(x, P_{i})^{(2)} denote the set of elements of type 2 in the residue of (x, P_{i}) in Θ_{i}. It follows from the construction of T that

res (x, P_{i})^{(2)} = {(l, P_{S,l})|l ∈ S^{(3)}, l is incident to (x, P_{i}) in S, and (x, P_{i}) is part of the partition P_{S,l} of the residue of (l, P_{S,l}) into parallel lines of S}.

As l ∈ S^{(3)} and l is incident to (x, P_{i}), by construction of S there is l_{0} ∈ Γ^{(4)} ∩ res(x) and a partition P_{0} of res(l_{0})^{−} into parallel lines of Γ such that l = (l_{0}, P_{0}) and P_{0} is compatible with the partition P_{i}. In terms of Γ then x is a plane of l_{0} and P_{S,l} induces a partition of res(l_{0})^{−} into two parallel planes. If (x′, P′_{i}) ∈ Θ_{i}^{(1)} is the unique element incident to (l, P_{S,l}) in Θ_{i} and different from (x, P_{i}), then x′ is also a plane of res(l_{0})^{−} and (x′, P′_{i}) must be part of the partition P_{S,l}. So x′ must be the plane in res(l_{0})^{−} parallel to x. This means that there is a bijection between the set of points collinear to (x, P_{i}) in Θ_{i} and the planes parallel to x in Γ. Since Θ_{i} is connected, this implies the assertion. □

2.2 From c-P - to T -geometries

In this subsection we show how, under certain conditions, starting from a c-P -geometry Γ we can construct a tilde geometry with the same automorphism group.

Let us first assume the rank of Γ is 4 and G = Aut(Γ) ≅ M_{24}. We will identify Γ with the geometry from [4, Example 2]. So Γ has the following sets of objects (where Ω, T , and O are as in Example 1.1 the points, trios and octads of the Steiner system):

– Γ^{(1)} = {{p, q}|p, q ∈ Ω, p ≠ q}

– Γ^{(2)} = {{a, b}|a, b ∈ Γ_{1}, a ∩ b = ∅}

– Γ^{(3)} = {(O_{1}, P, {O_{2}, O_{3}})|O_{i} ∈ O, {O_{1}, O_{2}, O_{3}} ∈ T , P is a partition of O_{1} into four parallel lines in the affine geometry on Ω \ O_{2}}

– Γ^{(4)} = {(O, P)|O ∈ O, P is a partition of Ω \ O into parallel lines }.

(Recall that for O ∈ O the complement Ω \ O bears the structure of an affine space over GF (2) and G_{O} ≅ 2^{4}A_{8} acts as affine group AGL(4, 2) on Ω \ O.)

Let Σ be a connected component of the structure S obtained from Γ by the shrinking construction described in the previous section. Then Σ has the

diagram ∘  ∘  ∘ c  ^{P^{∗}}^{1}  ^{2}  ^{3} . We have G_{2} ≤ G_{Σ} and G_{2} ≅ 2^{4+1}(Z_{2} × Σ_{5}), and

Lemma 2 resp. Corollary 1 yield that Σ is disconnected and that O_{2}(G_{2}) is the kernel of the action of G_{Σ} on Σ. In particular, O_{2}(G_{2}) ≤ G_{Σ}, and the list of maximal subgroups of M_{24} in [3] shows that G_{Σ} ≤ G_{S}, where G_{S} ≅ 2^{6}3Σ_{6} and G_{S} is the stabilizer of a sextet S in the Steiner system. From the structure of G_{S} we see that either G_{Σ} = G_{S} or G_{Σ} ≅ 2^{6}(Z_{3} × Σ_{5}). But in the latter |G_{Σ} : G_{2}| = 3, so Σ would just contain 3 points, which is not true (consider the residue of an element of type 3). Hence G_{Σ} = G_{S} ≅ 2^{6}3Σ_{6}.

Let T^{(1)} be the set of connected components of S, T^{(2)} the set of triples of elements from Γ^{(3)} of the shape

{(O_{1}, P_{1}, {O_{2}, O_{3}}), (O_{2}, P_{2}, {O_{1}, O_{3}}), (O_{3}, P_{3}, {O_{1}, O_{2}})}

such that P_{i} ∪P_{j} is a partition into parallel lines of the affine space Ω\O_{k} for all choices of {i, j, k} = {1, 2, 3}, and let T^{(3)} = Γ^{(4)}. Set T = T^{(1)} ∪ T^{(2)} ∪ T^{(3)} and define an incidence relation on T in the way that Σ ∈ T^{(1)} and x ∈ T^{(2)} ∪ T^{(3)} are incident if (x, P_{x}) (resp. (y, P_{y}) for all y ∈ x) belong to Σ, where P_{x}, P_{y} are suitable partitions of res(x)^{−} resp. res(y)^{−} into parallel lines of Γ. The elements in T^{(3)} incident to a typical element

{(O_{1}, P_{1}, {O_{2}, O_{3}}), (O_{2}, P_{2}, {O_{1}, O_{3}}), (O_{3}, P_{3}, {O_{1}, O_{2}})} ∈ T^{(2)}

are the three elements (O_{i}, P_{j} ∪ P_{k}), {i, j, k} = {1, 2, 3}.

If x = (O, P) ∈ T^{(3)} then G_{x} ≤ G_{O} ≅ 2^{4}A_{8}. Since G_{x} stabilizes the partition P we see G_{x} ≅ 2^{1+6}L_{3}(2). If x ∈ T^{(2)}, x as above, then G_{x} ≤ G_{t} where G_{t} is the stabilizer of the trio t = {O_{1}, O_{2}, O_{3}} and G_{t} ≅ [2^{6}](Σ_{3} × L_{3}(2)). The Σ_{3}-factor permutes the three octads in t, so it will stabilize x as a set. The L_{3}(2)-factor can be viewed as the stabilizer of a partition of Ω \ O_{i} into two parallel planes for each of the O_{i}. Since G_{x} also stabilizes the partitions P_{j} ∪ P_{k} we deduce G_{x} ≅ [2^{8}](Σ_{3} × Σ_{3}). So we have shown that G_{x} is of the right shape for all x ∈ T. We leave it to the reader to convince himself that the incidence relation is defined in the right way to get a model of the tilde geometry for M_{24}.

Now let Γ be of arbitrary rank n ≥ 5. Apply the shrinking construction n−3 times and call the obtained incidence structures S_{1}, S_{2}, …,S_{n−3}. Suppose that S_{1} is disconnected and that the connected components of S_{i+1} are not equal to those of S_{i}, i = 1, …, n − 4. Then, similarly as in section 2.1, for each i ≥ 2, we can define an equivalence relation on the set of connected components of S_{i}, whose equivalence classes correspond to the set of partitions into two parallel hyperplanes of an i-dimensional affine space over GF (2). Hence they are of size 2^{i} − 1. Assume the connected components of S_{n−4} are the c-P -geometries for M_{24}. Let

– T^{(1)} = connected components of S_{1},

– T^{(i)} = equivalence classes of connected components of S_{i}, for 2 ≤ i ≤ n−3,

– T^{(n−2)} = triples of elements of Γ^{(n−1)} which, for suitable partitions, are contained in a common connected component of S_{n−2} and there form a triple which corresponds to an element of type 2 of the T -geometry for M_{24},

– T^{(n−1)} = Γ^{(n)}

and T be the geometry on T^{(1)}∪…∪T^{(n−1)} with the following incidene relation:

Two equivalence classes x ∈ T^{(i)}, y ∈ T^{(j)}, 1 < i < j < n − 2, resp. Σ ∈ T^{(1)} and y ∈ T^{(i)} are incident if a representative of y is contained in a representative of x resp. in Σ. Elements x ∈ T^{(i)}, i < n − 2, and y ∈ T^{(n−2)} ∪ T^{(n−1)} are incident if y resp. all members of y belong to (a representative of) x. Incidence between elements in T^{(n−2)} and T^{(n−1)} is defined by inclusion.

By construction the diagram of T is a string, the residue of an element of type n−1 is an n−1-dimensional projective space over GF (2), and the geometry induced on the elements of types n − 2 and n − 1 in the residue of an element of type n − 3 is the rank 2 tilde geometry for 3Σ_{6}. So T is a tilde geometry of rank n − 1. Since the stabilizer of a suitable connected component Σ ∈ T^{(1)} contains G_{2}, since T^{(n−1)} = Γ^{(n)} and since G = ⟨G_{2}, G_{n}⟩, it is straightforward to see that G is a flag-transitive automorphism group of it. We formulate this result in the following

Proposition 1 Suppose G acts flag-transitively on a c-P -geometry Γ of rank n ≥ 5. Apply the shrinking construction n − 3 times on Γ and call the obtained incidence structures S_{1}, S_{2}, …,S_{n−3}. Suppose that S_{1} is disconnected, that the connected components of S_{i+1} are not equal to those of S_{i}, i = 1, …, n − 4, and that the connected components of S_{n−4} are the c-P -geometries for M_{24}. Then G acts flag-transitively on a tilde geometry of rank n − 1.

Remark. One can also start from the rank 3 case and first construct the T -geometry for 3Σ_{6} from the c-P -geometry. This can be done, e.g., considering a model of the latter which exists inside AGL(3, 4) or getting a model via the shrinking process from the c-P -geometry for M_{24}. We did not choose this approach here because there was no obvious need and it would have made the whole construction even more technical.

Notice that it is essential for the construction of a tilde geometry that the connected components of S_{n−3} are the c-P -geometries for 3Σ_{6}. Otherwise their stabilizers might not act on a tilde geometry at all and one will not be able to construct even a rank-2-tilde geometry.

3 Some preliminary lemmas

The first two lemmas are maybe well known but as we had no reference we provide the proofs.

Lemma 4 Let G = L_{4}(2) act on an 8-dimensional GF (2)-vectorspace V which contains a G-invariant subspace V_{1} such that V_{1} and V/V_{1} are nontrivial 4dimensional modules for G. Suppose there exists x ∈ V \V_{1} such that |x^{G}| = 15. Then V ≠ ⟨x^{G}⟩.

Proof. Since |x^{G}| = 15, C_{G}(x) ≅ 2^{3}L_{3}(2) is a maximal parabolic subgroup of G and each nontrivial coset of V/V_{1} contains exactly one conjugate of x. Since L_{4}(2) acts doubly transitively on the 4-dimensional module and hence also on the set of conjugates of x, it suffices to show that for some conjugate y of x the element xy is also conjugate to x. Then ⟨x^{G}⟩ = {x^{g}|g ∈ G} ∪ {1}, the assertion.

There exists an element g ∈ G of order 3 acting fixed point freely on both 4-dimensional modules, hence g acts fixed point freely on V. Since xx^{g}x^{g^{2}} is centralized by g, we get xx^{g}x^{g^{2}} = 1 and xx^{g} = x^{g^{2}} has the desired property. □

Lemma 5 Let G = M_{24} and V be a 12-dimensional GF (2)G-module. If V contains a G-submodule V_{1} with |V_{1}| = 2^{11} and V_{1} isomorphic to the (irreducible) Golay code module then V splits over V_{1}.

Proof. Suppose not and consider the orbits of G on V \ V_{1}. Since 2048 is not divisible by 23, each element of order 23 in G must have a fixed point on V \ V_{1}. Now the list of maximal subgroups of M_{24} in [3] shows that G has an orbit of length 24. Since any two subgroups of G isomorphic to M_{23} intersect in an M_{22} then the sum of any two vectors in this orbit is fixed by a subgroup of G isomorphic to M_{22}. But M_{22} does not stabilize a nontrivial vector in V_{1}, a contradiction. □

In the next three lemmas G is a flag-transitive automorphism group of some c-P - or c-T -geometry Γ satisfying (∗) and all other notations are as in the introduction.

Lemma 6 Suppose there exists an involution x ∈ G_{2}\G_{1} such that [G_{12}, x] = 1. Let X_{1} = ⟨x^{G_{1}}⟩ and X = ⟨x^{G_{1n}}⟩. If X is elementary abelian and 2^{n−1} ≤ |X| ≤ 2^{n} then |X| = 2^{n−1} and X_{1} is isomorphic to a factor group of the universal representation group of res(α_{1}).

Proof. If p ∈ res(α_{1}) ∩ Γ^{(2)} then p = α_{2}^{g} for a suitable g ∈ G_{1} and we set u_{p} = x^{g}. Since [G_{12}, x] = 1 this is well-defined and it suffices to show that u_{p}u_{q}u_{r} = 1 whenever p, q, r are incident to a common element l ∈ res(α_{1})∩Γ^{(3)}. (Then X_{1} satisfies the axioms of Definition 1.)

By flag-transitivity we may assume l = α_{3}, p = α_{2}. Then l ∈ res(α_{n}), q and r are conjugate to p in G_{13n} and ⟨u_{p}, u_{q}, u_{r}⟩ ≤ X. As x ∉ K_{n} and G_{n}/K_{n} ≅ 2^{n−1}L_{n−1}(2) we have G_{n} = XG_{1n}. So either |X| = 2^{n−1} or |X ∩ K_{n}| = 2, and since X ∩ K_{n} ≤ G_{n}, in the latter we get X ∩ K_{n} ≤ Z(G_{n}). Furthermore, as C_{G_{1n}}(x) = G_{1n} ∩ G_{2}, the action of G_{1n} on X/X ∩ K_{n} corresponds to the action on the projective GF (2)-space res({α_{1}, α_{n}}). Since α_{3} corresponds to a line in this space the assertion holds in X/X ∩ K_{n}, particularly it holds if |X| = 2^{n−1}. Suppose 1 ≠ z = u_{p}u_{q}u_{r} ∈ X ∩ K_{n} ≤ Z(G_{n}). Let X_{3} = ⟨x^{G_{3}}⟩. Since G_{3} = G_{123}G_{3...n} and [x, G_{123}] = 1 we have X_{3} = ⟨x^{G_{3...n}}⟩ = ⟨u_{p}, u_{q}, u_{r}⟩. Now X_{3} is invariant under P_{n}, z ∈ X_{3}, and as z ∉ Z(G) we have [z, P_{n}] ≠ 1. However, we see in G_{1} that z^{P_{n}} ⊆ B. Since X ∩ B = X ∩ K_{n} this yields a contradiction. □

Lemma 7 We have K_{i} = K_{1i} for i ≥ 2.

Proof. Let g ∈ K_{1i}. Then g fixes all elements in res(α_{1}) ∩ res(α_{i}). Let p ∈ Γ^{(1)} be incident to α_{i}. Then there exists l ∈ Γ^{(2)} incident to α_{i} such that res(l) ∩ Γ^{(1)} = {α_{1}, p}. Now g fixes α_{1} and l, so g must fix p. This implies that g acts trivially on res(α_{i}) and hence g ∈ K_{i}. The other inclusion is trivial. □

Lemma 8 If n ≤ 5 then K_{1} = 1. If n = 6 then |K_{1}| ≤ 2.

Proof. By [7] we know that, if Γ is of c-P -type, then G_{1}/K_{1} ≅ M_{23} or Co_{2}, if n = 5, and G_{1}/K_{1} ≅ BM, if n = 6. If Γ is of c-T -type, then G_{1}/K_{1} ≅ M_{24}, Co_{1}, M for n = 4, 5, 6, respectively. We will make use of the descriptions of the corresponding residual geometries as given in [6].

Let us call the elements of Γ of type 1 “points” and the elements of type 2 “lines”. Then each line consists of 2 points and the relation “having the same points” is an equivalence relation on the set of lines on the point α_{1}, which is preserved by G_{1}. Since in the considered cases G_{12}/K_{1} is a maximal subgroup of G_{1}/K_{1}, the action of G_{1}/K_{1} on that set is primitive. Hence there are no two lines with the same points. In particular, G_{p} ∩ G_{q} ≤ G_{l} for two points p, q on a line l.

Now let g ∈ K_{1}, p a point collinear to α_{1} and l the line on α_{1} and p. Then g fixes l and α_{1}, hence g ∈ G_{p}. Let k be a line on p such that l and k are incident to a common element x of type 3 and q the other point on k. Then x is incident to α_{1} and there is a line through α_{1} and q in res(x). So g fixes q and g ∈ G_{p} ∩ G_{q} ≤ G_{k}. This means that g fixes all lines incident to p which are coplanar with l. In terms of the residual geometry res(p), where l corresponds to a point, this means that g fixes all points collinear to l. Since the points of the P -geometry for M_{23} are just the 23 points of the Steiner system S(23, 7, 4), the lines are all the pairs of different points, and incidence is defined by inclusion, any two points are collinear in this case. So K_{1} fixes all points of the Steiner system and K_{1} ≤ K_{p}. Hence K_{1} = K_{p} = 1. In the M_{24}-geometry l corresponds to a sextet of the Steiner system and g must fix all sextets which refine a common trio with l. So we get K_{1} ≤ K_{p} again. For the geometries of the Conway groups one can deduce the same from the descriptions in terms of the Leech lattice given in Example 2 resp. [6].

Finally, also by [6] the points of the geometries for BM and M can be identified with the conjugacy class of 2-central involutions and the involutions corresponding to the set of points collinear to l generate the subgroup O_{2}(G_{pl}/K_{p}). Since this is an extraspecial group of order 2^{1+22} resp. 2^{1+24} we get |K_{1} : K_{1} ∩ K_{p}| ≤ 2 in those cases. So it remains to show that K_{1} ∩ K_{p} = 1. From Lemma 7 and G_{6}/K_{6} ≅ 2^{5}L_{5}(2) we see that K_{1}K_{6}/K_{6} and K_{1}K_{6}/K_{1} are 2-groups. Hence K_{1} and K_{6} are 2-groups by the Frattini argument. Since B/K_{1} is also a 2-group, we get that B and all K_{i} are 2-groups. Suppose first that C_{G_{1}}(K_{1}) ≤ K_{1}. Let S ∈ Syl_{2}(G_{1} ∩ G_{2}) and Z_{1} = ⟨Ω_{1}(Z(S))^{G_{1}}⟩. Notice that |G_{2} : G_{1}∩G_{2}| = 2, so G_{1}∩G_{2}≤G_{2} and G_{2} = (G_{1}∩G_{2})N_{G_{2}}(S). If Z_{1} = Ω_{1}(Z(S)) then Z_{1} ≤ ⟨G_{1}, N_{G_{2}}(S)⟩ = G, which is impossible. So Z_{1} ≠ Ω_{1}(Z(S)) and Z_{1} involves a non-trivial G_{1}/K_{1}-module. By [1, (11.1)] BM and M do not have F -modules. Hence J(S) ≤ K_{1} (where J(S) is the Thompson subgroup of S). But then J(S) = J(K_{1}) ≤ ⟨G_{1}, N_{G_{2}}(S)⟩ = G, again a contradiction. So we have C_{G_{1}}(K_{1}) ≰ K_{1}, hence G_{1} = C_{G_{1}}(K_{1})K_{1} and, by symmetry, G_{p} = C_{G_{p}}(K_{p})K_{p}. Therefore K_{1} ∩K_{p} ≤ ⟨O^{2}(G_{1}), O^{2}(G_{p}), K_{1}, K_{p}⟩. But as G_{6} = ⟨G_{1} ∩G_{6}, G_{p} ∩G_{6}⟩ we have G = ⟨G_{1}, G_{6}⟩ ≤ ⟨G_{1}, G_{p}⟩ = ⟨O^{2}(G_{1}), O^{2}(G_{p}), K_{1}, K_{p}⟩. So K_{1} ∩ K_{p} ≤ G and K_{1} ∩ K_{p} = 1. This proves the assertion. □

4 Classification of the geometries

Lemma 9 If Γ is a c-P -geometry of rank 5 and G_{1} ≅ M_{23}, then G ≅ M_{24}.

Proof. We show that |G : G_{1}| = |Γ^{(1)}| = 24. Since G_{12} ≅ M_{22}, we have |res(α_{1})^{(2)}| = |G_{1} : G_{12}| = 23. So there are 23 points which are collinear to α_{1} and it suffices to show that, if p and q are collinear to α_{1}, then p is collinear to q. Let l_{p}, l_{q} ∈ Γ^{(2)} be the lines through α_{1} and p resp. q. Since l_{p}, l_{q} correspond to points in the geometry for M_{23} and any two points in this geometry are collinear by the 2-transitive action of M_{23}, we see in res(α_{1}) that there exists x ∈ Γ^{(3)} such that x is incident to l_{p} and l_{q}. But now p and q are incident to x, and since the elements of res(x)^{−} form an affine space, the claim follows. □

For the rest of the section we fix the following situation: From Lemma 7 we know that K_{2} = K_{12}. Further we have |G_{2} : G_{12}| = 2. So G_{12} ≤ G_{2} and there exists x ∈ G_{2} \ G_{12} such that x^{2} ∈ G_{12}. Since K_{2} = O_{2}(G_{12}) we see that x acts on G_{12}/K_{2}. But except in one case, where we will have G_{12}/K_{2} ≅ 3Σ_{6}, the group G_{12}/K_{2} will not possess outer automorphisms of order 2, so G_{2}/K_{2} ≅ Z_{2} × G_{12}/K_{2} and we may assume [x, G_{12}] ≤ K_{2}. If G_{12}/K_{2} ≅ 3Σ_{6} then |Out(G_{12}/K_{2})| = 2 and the outer automorphism of 3Σ_{6} interchanges the two conjugacy classes of subgroups of type Z_{2} × Σ_{4}. But these correspond to stabilizers of elements of different types in res(α_{2})^{+} in Γ, and as x ∈ Aut(Γ), x preserves the types. So x cannot act as outer automorphism and we get [x, G_{12}] ≤ K_{2} in this case as well.

Unless stated otherwise, from now on x will always denote such an element. Then [x, B] ≤ [x, G_{12}] ≤ K_{2} ≤ B and x ∈ N_{G_{2}}(B). If Γ is of c-T -type this yields x ∈ P_{1}, if Γ is of c-P -type x ∈ P_{1}P_{n}. Furthermore, by the Three subgroup lemma G_{12} acts on [K_{2}, x]K′_{2}. In particular, if K_{2} is elementary abelian and irreducible as G_{12}/K_{2}-module, then [K_{2}, x] = 1. Further, since then [G_{12}, x^{2}] = 1 we get that x is of order 2.

Lemma 10 If Γ is a c-T -geometry of rank 4 and G_{1} ≅ M_{24}, then G ≅ 2^{11} :M_{24} and O_{2}(G) is the universal representation group of the T -geometry for M_{24} (so it is the Golay code module).

Proof. We have G_{12} ≅ 2^{6}3Σ_{6} where K_{2} is an irreducible G_{12}/K_{2}-module. So as just remarked, Q_{2} = ⟨K_{2}, x⟩ must be an elementary abelian group of order 2^{7}. Let ρ ∈ O_{23}(G_{12}), o(ρ) = 3. Then ρ acts fixed point freely on K_{2}. So C_{Q_{2}}(ρ) must be of order 2. Since C_{Q_{2}}(ρ) is G_{12}-invariant we may assume [x, G_{12}] = 1.

Now consider G_{4}. We have G_{14} ≅ 2^{1+6}L_{3}(2), G_{4}/K_{4} ≅ 2^{3}L_{3}(2), and G_{4} = G_{14}X where X = ⟨x^{G_{14}}⟩. Since K_{4} ≤ G_{12} we get [X, K_{4}] = [x, K_{4}]^{G_{14}} = 1. As K_{4} is extraspecial this yields |X ∩ K_{4}| ≤ 2 and |X| ≤ 16. So X_{1} = ⟨x^{G_{1}}⟩ is a representation group for the tilde geometry for M_{24} by Lemma 6 and |X_{1}| = 2^{11} by [6, (3.2)] resp. [8]. Since X_{1} is normalized by G_{1} and G = ⟨x, G_{1}⟩, we get G = X_{1}G_{1} and the lemma is proved. □

In the rank 5 and rank 6 cases we will consider c-P - and c-T -geometries at the same time. This has the following reason: Many of our calculations mainly depend on the 2-structures of some of the maximal parabolic subgroups. Since the P -geometries for Co_{2} and BM can be defined as certain subgeometries in the T -geometries for Co_{1} resp. M, these structures are quite similar for both classes of geometries. In particular, it turned out that often we just can use the same arguments without distinguishing between T - and P -geometries.

Lemma 11 Let rank(Γ) = 5.

(a) If Γ is of c-P -type and G_{1} ≅ Co_{2}, then either G ≅ 2^{22} :Co_{2} resp. 2^{23} :Co_{2} and O_{2}(G) is a representation group of the P -geometry for Co_{2} (so it is isomorphic to a section of the Leech lattice mod 2); or G ≅ Co_{1}.

(b) If Γ is of c-T -type and G_{1} ≅ Co_{1}, then G ≅ 2^{24} : Co_{1} and O_{2}(G) is the universal representation group of the tilde geometry for Co_{1} (so it is isomorphic to the Leech lattice mod 2).

Proof. From the geometries of Co_{2} and Co_{1} we know that G_{12} ≅ 2^{10}Aut(M_{22}) resp. 2^{11}M_{24} and that K_{2} is the irreducible Golay code module for G_{12}/K_{2}. So [x, K_{2}] = 1 and O_{2}(G_{2}) = ⟨x, K_{2}⟩ is an elementary abelian group of order 2^{11} resp. 2^{12} by the above remark. Now if G_{12}/K_{12} ≅ Aut(M_{22}), then we have two general possibilities for the structure of G_{2}:

(a.1) G_{2} ≅ Z_{2} × 2^{10}Aut(M_{22}),

(a.2) G_{2} ≅ 2^{10+1}Aut(M_{22}), where the action of G_{2}/O_{2}(G_{2}) on O_{2}(G_{2}) is nonsplit,

whereas in (b) by Lemma 5 we must have G_{2} ≅ Z_{2} × 2^{11}M_{24}.

Next look at G_{5}. We have G_{5}/K_{5} ≅ 2^{4}L_{4}(2). From the structure of G_{1} we know that in (a) G_{15} ≅ (2^{1+6} × 2^{4})L_{4}(2), |Z(K_{5})| = 2^{5} and |K_{5} ∩ K_{2}| = 2^{7}, and in (b) G_{15} ≅ 2^{1+8+6}L_{4}(2), |Z(K_{5})| = 2, and |K_{5} ∩ K_{2}| = 2^{8}.

Let us first consider (a.1) and (b). In these cases [x, K_{5}] = 1 and x ∈ N_{G_{2}}(K_{5}) = G_{2} ∩G_{5}. Hence if we set X = ⟨x^{G_{15}}⟩, then G_{5} = XG_{15}, [X, K_{5}] = 1, and X ≤ Z(O_{2}(G_{5})). In (a.1) we apply Lemma 4 to see that X splits over the 4-dimensional module in Z(K_{5}). So in both cases we may assume |X| ≤ 32. Now Lemma 6 yields that X_{1} = ⟨x^{g}|g ∈ G_{1}⟩ is a representation group for the residual geometry of G_{1}/K_{1}. So X_{1} is elementary abelian of order 2^{22} or 2^{23} in (a.1) and of order 2^{24} in (b) by [6, (5.7),(5.8)]. Again because X_{1} is normalized by G_{1} and G = ⟨x, G_{1}⟩ we get that G = X_{1}G_{1} is as stated.

Now consider (a.2). Suppose [y, K_{5}] ≤ Z(G_{15}) for all y ∈ O_{2}(G_{2}) \ K_{2}. Because O_{2}(G_{2}) = ⟨y|y ∈ O_{2}(G_{2})\K_{2}⟩ then we would get [O_{2}(G_{2}), K_{5}] ≤ Z(G_{15}). But K_{5}K_{2}/K_{2} does not induce transvections on K_{2}. So we get a contradiction. Hence we may choose x with [x, K_{5}] ≰ Z(G_{15}). Then, particularly, [X, K_{5}] = [⟨x^{G_{5}}⟩, K_{5}] = [x, K_{5}]^{G_{5}} is not contained in K_{5} ∩ O_{2}(G_{2}).

Now let Σ be a connected component of the structure S obtained from Γ by shrinking as described in section 2. We can choose Σ such that the flag {α_{2}, …, α_{n}} maps onto a maximal flag in Σ. Let H = G_{Σ} and let N be the kernel of the action of H on Σ. By Lemma 2 we have N = O_{2}(G_{2}) and by construction of S O_{2}(G_{5})N/N = O_{2}(H_{4}/N). Since [X, K_{5}] ≰ N, we see that O_{2}(H_{4}/N) is non-abelian. So [4] implies that H/N ≅ M_{24}. Now G acts flagtransitively on a rank-4-tilde geometry by Proposition 1, and since H is the stabilizer of an object of this geometry, by [7] we get G ≅ Co_{1}. □

Lemma 12 Let rank(Γ) = 6.

(a) If Γ is of c-P -type and G_{1}/K_{1} ≅ BM, then either G is isomorphic to a section of (2·BM ∗2·BM):Z_{2} or G ≅ M.

(b) If Γ is of c-T -type and G_{1}/K_{1} ≅ M, then G ≅ M × M or M ≀ Z_{2}.

Proof. By Lemma 8 we have |K_{1}| ≤ 2. Hence we get the following possibilities for G_{1}:

(a.1) K_{1} = 1, G_{1} ≅ BM,

(a.2) |K_{1}| = 2, G_{1} ≅ Z_{2} × BM,

(a.3) |K_{1}| = 2, G_{1} ≅ 2·BM, a non-split extension,

respectively

(b.1) K_{1} = 1, G_{1} ≅ M,

(b.2) |K_{1}| = 2, G_{1} ≅ Z_{2} × M, (as the Schurmultiplier of M is trivial by [3]).

We set ⟨z⟩ = Z(G_{1}) = K_{1}. Then G_{12}/⟨z⟩ ≅ 2^{1+22}Co_{2} resp. 2^{1+24}Co_{1}.

Let us first assume we are not in case (a.3). Then Q_{2} = [K_{2}, G_{12}] is extraspecial and [K_{2}, Q_{2}] = Q′_{2}. Since [x, G_{12}, Q_{2}] ≤ [K_{2}, Q_{2}] = Q′_{2} and [Q_{2}, x, G_{12}] ≤ [K_{2}, G_{12}] = Q_{2}, the Three-Subgroup-Lemma yields [Q_{2}, x] = [G_{12}, Q_{2}, x] ≤ Q_{2}. Hence Q_{2} ≤ G_{2}. Now also C_{G_{2}}(Q_{2}) ≤ G_{2} and [C_{G_{2}}(Q_{2}), O^{2}(G_{12})] = 1. Notice that |C_{G_{2}}(Q_{2})| = 4 resp. 8 (depending whether we have K_{1} = 1 or not) and that this group is not necessarily abelian but it has exponent at most 4.

From now on we assume x ∈ C_{G_{2}}(Q_{2}). Then o(x) ≤ 4 and x centralizes a Sylow-2-subgroup S ≤ G′_{1}. In particular x centralizes O_{2}(H_{16}), where H_{16} = G_{6} ∩ G′_{1}. So x ∈ N_{G_{2}}(O_{2}(H_{16})) = G_{2} ∩ G_{6}. Notice that H_{16}/O_{2}(H_{16}) ≅ L_{5}(2) with |O_{2}(H_{16})| = 2^{30} resp. 2^{36}, and that G_{6} = ⟨z⟩H_{16}X where X = ⟨x^{H_{16}}⟩. Now XK_{6}/K_{6} is an elementary abelian group of order 2^{5} and with x also X centralizes O_{2}(H_{16}). Hence X ≤ Z(O_{2}(H_{16})X). Let Z = Z(O_{2}(H_{16})). Then |Z| = 2^{5} and |Z(O_{2}(H_{16})X)| = 2^{10}. Now if o(x) = 4 then X = Z(O_{2}(H_{16})X) ≅ Z_{4}^{5}, but by [5] GL_{5}(Z_{4}) does not contain a subgroup isomorphic to L_{5}(2). So o(x) = 2.

We claim that XZ splits over Z. We know that Z and XK_{6}/K_{6} ≅ X/(X ∩ K_{6}) = X/(X ∩ Z) ≅ XZ/Z are isomorphic as L_{5}(2)-modules. Consider a subgroup L of L_{5}(2) of order 31 · 5. Since L acts completely reducibly on XZ, we see that L has three orbits of length 31 and 6 orbits of length 155 on (XZ)^{#} where each of the three orbits of length 31 generates a 5-dimensional subspace of XZ. As x has exactly 31 conjugates under H_{6} the claim follows, i.e., X is elementary abelian of order 2^{5}. Replacing G_{1} by G′_{1} in Lemma 6 now we get that X_{1} = ⟨X^{G′_{1}}⟩ is a representation group for the residual geometry for G′_{1} ≅ G_{1}/K_{1}. Hence X_{1} ≅ BM or 2· BM resp. M by [6].

Consider the semidirect product X_{1}G′_{1}. If z = 1 then X_{1}G′_{1} = X_{1}G_{1} = G and since Out(G_{1}) = 1 we get G ≅ G_{1} × G_{1}. If z ≠ 1 then [z, x] ≠ 1 (otherwise we would get z ∈ Z(⟨x, G_{1}⟩) = Z(G). But as z ∈ K_{6} we have [z, X] ≤ K_{6} and [z, X_{1}] ≤ X_{1}G′_{1}. So z normalizes X_{1}G′_{1} ≅ X_{1} ∗X_{1} but not X_{1} and we get the first case of (a) resp. (b). (Notice that G_{1} is the diagonal subgroup of G).

Now consider case (a.3). Let Σ be a connected component of the structure S as described in Section 2. Let H = G_{Σ} and N the kernel of the action of H on Σ. By Lemma 2 we have N = K_{2}⟨x⟩, and by Lemma 11 we have H/N ≅ Co_{1} or 2^{22}Co_{2} (resp. 2^{23}Co_{2}).

If H/N ≅ Co_{1}, then Proposition 1 yields that G acts flag-transitively on a rank-5-tilde geometry. So G ≅ M by [7] in this case.

Suppose H/N ≅ 2^{22}Co_{2} (resp. 2^{23}Co_{2}). Since G_{12} acts on [K_{2}, x]K′_{2}, the structure of K_{2} as G_{12}/K_{2}-module implies that [x, K_{2}] ≤ Z(G_{12}). Hence |K_{2} : C_{K_{2}}(x)| ≤ 4. If [K_{2}, x] ≰ K′_{2}, then |K_{2}/K′_{2} : C_{K_{2}/K′_{2}}(x)| = |[K_{2}/K′_{2}, x]| = 2, but K_{2}/K′_{2} does not contain a G_{12}-submodule of index 2. So [K_{2}, x] ≤ K′_{2}, and since N = ⟨x, K_{2}⟩, N′ = K′_{2} is of order 2. As x^{2} centralizes K_{2} we get o(x) ≤ 4 again. As above [x, z] ≠ 1, so Z(N) = N′ and N is extraspecial of order 2^{1+24}.

Let N_{1} = O_{2}(H). Then N_{1} acts on N/N′ and C_{N/N′}(N_{1}) ≠ 1. Let C be the full preimage in N of C_{N/N′}(N_{1}). Then C ≤ H and the action of Co_{2} on N/N′ implies z ∈ C. Therefore, [z, N_{1}] ≤ N′ and |N_{1} : C_{N_{1}}(z)| ≤ 2. Since N_{1}/N does not contain a G_{12}-submodule of index 2, we get N_{1} = C_{N_{1}}(z)N. Now consider the action of N_{1}/N on Σ. Because N acts trivially on Σ and N_{1}/N acts regularly on the points of Σ, i.e., on Σ ∩ Γ^{(2)}, we see that even C_{N_{1}}(z) acts regularly. We may assume α_{2} ∈ Σ. Let β_{2} ∈ Γ^{(2)} be the line parallel to α_{2} in res(α_{3}). Our construction implies that β_{2} ∈ Σ. Choose n ∈ C_{N_{1}}(z) such that β_{2} = α_{2}^{n} and let β_{1} = α_{1}^{n}. (Notice that β_{1} ≠ α_{1} because β_{1} is incident to β_{2}, and α_{2} and β_{2} are parallel.) Then

K_{β_{1}} = K_{α_{1}^{n}} = K_{1}^{n} = ⟨z⟩^{n} = ⟨z⟩ = K_{1}.

But α_{1}, β_{1} ∈ res(α_{3}), so α_{1} and β_{1} lie on a common line l ∈ Γ^{(2)} ∩ res(α_{3}). By flag-transitivity α_{1} and β_{1} are conjugate in G_{l}, so G_{l} ≤ N_{G}(K_{1}) and K_{1} ≤ ⟨G_{1}, G_{l}⟩ = G, a contradiction. □

Literatur

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