c - extensions of the Petersen geometry for M_{22}

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

In this paper we consider a group G acting flag-transitively on a geometry with diagram ∘ ∘ ∘ ∘ c ^{P^{∗}}^{1} ^{2} ^{3} ^{4} such that the residue of an element of type 1 is the Petersen geometry for the group M_{22}. Explicitely, G is generated by finite subgroups P_{1}, P_{2}, P_{3}, P_{4} such that

(a) G ≠ G_{i} := ⟨P_{j}|j ≠ i⟩ for 1 ≤ i ≤ 4.

(b) For B := P_{1} ∩ P_{2} ∩ P_{3} ∩ P_{4} we have B_{G} = 1, where B_{G} := ⋂_{g∈G} B^{g}.

(c) If K_{i} = B_{G_{i}}, G_{ij} = ⟨P_{k}|k ≠ i, j⟩ and K_{ij} = B_{G_{ij}}, then G_{1}/K_{1} ≅ M_{22} or Aut(M_{22}) and the structures of the G_{i}/K_{i}, i ≠ 1, and the G_{ij}/K_{ij} are as indicated by the diagram.^{1}

We are going to prove the following

Theorem. Let G be a group with a parabolic system belonging to the diagram ∘ ∘ ∘ ∘ c ^{P^{∗}}^{1} ^{2} ^{3} ^{4} where G_{1}/K_{1} ≅ M_{22} or Aut(M_{22}). Then G is isomorphic to one of the following groups:

a) 2^{10}M_{22}, 2^{11}M_{22}, 2^{10}Aut(M_{22}), 2^{11}Aut(M_{22})

b) U_{6}(2), 2·U_{6}(2), U_{6}(2)·:, 2·U_{6}(2):2

c) M_{24}.

Remark. It follows from the diagram that

– |P_{1} : B_{P_{1}}| = |P_{4} : B_{P_{4}}| = 2 and P_{2}/B_{P_{2}} ≅ P_{3}/B_{P_{3}} ≅ Σ_{3},

– G_{2}/K_{2} ≅ Z_{2} × Σ_{5}, G_{3}/K_{3} ≅ Σ_{4} × Z_{2} and G_{4}/K_{4} ≅ 2^{3}L_{3}(2),

– K_{14} = K_{4}.

This implies that K_{1}K_{4}/K_{1} and K_{1}K_{4}/K_{4} are 2-groups. Hence, by [6] (1.13)- (1.15) K_{1} and K_{4} are 2-groups. Since B/K_{1} is also a 2-group, B and all K_{i} are 2-groups.

The following lemma is due to A. Pasini.

Lemma 1. We have K_{1} = 1.

1The reader unfamiliar with diagram geometries is referred to [2] and [4].

Proof. Let Γ = Γ(G; G_{i}) be the coset geometry belonging to our parabolic system. Let’s 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}, where we identify the coset G_{1} with α_{1}. Clearly, this equivalence relation is preserved by G_{1}. So the action of G_{1}/K_{1} on res(α_{1}) implies that 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 in res(p) which are coplanar with l. The action G_{p}/K_{p} ≅ M_{22} resp. Aut(M_{22}) on res(p) shows that K_{1} ≤ K_{p}, hence K_{1} = K_{p} = 1. □

So from now on we may assume G_{1} ≅ M_{22} or Aut(M_{22}). We are going to prove Theorem 1 by determining generators and relations for G and then applying the Todd-Coxeter-Algorithm in MAGMA. We start with the following generators for G_{1}:

x_{1}, x_{2}, x_{3}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}, x_{10}, y.

All these elements are involutions, ⟨x_{1}, …, x_{10}⟩ ≅ M_{22}, y is the outer automorphism and the following relations are satisfied:

[x_{1}, x_{2}] = 1, [x_{1}, x_{3}] = 1, x_{1}^{−1}x_{5}^{−1}x_{1}x_{5}x_{2} = 1, x_{1}^{−1}x_{6}^{−1}x_{1}x_{6}x_{3} = 1,
[x_{1}, x_{7}] = 1, [x_{1}, x_{8}] = 1, [x_{1}, x_{9}] = 1, [x_{2}, x_{3}] = 1, [x_{2}, x_{5}] = 1,
[x_{2}, x_{6}] = 1, x_{2}^{−1}x_{7}^{−1}x_{2}x_{7}x_{1} = 1, x_{2}^{−1}x_{8}^{−1}x_{2}x_{8}x_{3} = 1, [x_{2}, x_{9}] = 1,
[x_{3}, x_{5}] = 1, [x_{3}, x_{6}] = 1, [x_{3}, x_{7}] = 1, [x_{3}, x_{8}] = 1,
x_{3}^{−1}x_{9}^{−1}x_{3}x_{9}x_{2} = 1, [x_{5}, x_{6}] = 1, (x_{5}x_{7})^{3} = 1, x_{5}^{−1}x_{8}^{−1}x_{5}x_{8}x_{6} = 1,
[x_{5}, x_{9}] = 1, x_{6}^{−1}x_{7}^{−1}x_{6}x_{7}x_{8} = 1, [x_{6}, x_{8}] = 1, x_{6}^{−1}x_{9}^{−1}x_{6}x_{9}x_{5} = 1,  (1)
[x_{7}, x_{8}] = 1, (x_{7}x_{9})^{4} = 1, (x_{8}x_{9})^{3} = 1, [x_{10}, x_{3}] = 1, [x_{10}, x_{6}] = 1,
[x_{10}, x_{8}] = 1, x_{10}^{−1}x_{1}^{−1}x_{10}x_{1}x_{8} = 1, x_{10}^{−1}x_{2}^{−1}x_{10}x_{2}x_{6} = 1,
x_{10}^{−1}x_{5}^{−1}x_{10}x_{5}x_{6}x_{3} = 1, x_{10}^{−1}x_{7}^{−1}x_{10}x_{7}x_{8}x_{3} = 1, (x_{9}x_{10})^{5} = 1, [x_{1}, y] = 1,
[x_{2}, y] = 1, [x_{3}, y] = 1, [x_{5}, y] = 1, [x_{6}, y] = 1, [x_{7}, y] = 1,
[x_{8}, y] = 1, [x_{9}, y] = 1, y^{−1}x_{10}^{−1}yx_{10}x_{3} = 1, (x_{1}x_{10}x_{7}x_{5}x_{9})^{11} = 1.

Moreover, we have

P_{2} = ⟨x_{1}, x_{2}, x_{3}, x_{5}, x_{6}, x_{7}, x_{8}, y^{i}⟩
P_{3} = ⟨x_{1}, x_{2}, x_{3}, x_{5}, x_{6}, x_{8}, x_{9}, y^{i}⟩  (2)
P_{4} = ⟨x_{1}, x_{2}, x_{3}, x_{5}, x_{6}, x_{8}, x_{10}, y^{i}⟩

and

B = ⟨x_{1}, x_{2}, x_{3}, x_{5}, x_{6}, x_{8}, y^{i}⟩  (3)

where i = 1 if G_{1} ≅ Aut(M_{22}) and i = 0 otherwise. In the following, if not stated explicitely anything different, we always assume G_{1} ≅ Aut(M_{22}).

As remarked above, we have G_{2}/K_{2} ≅ Z_{2} × Σ_{5}. Therefore, we can choose an element x ∈ P_{1} such that o(x) = 2, G = ⟨G_{1}, x⟩ and [x, G_{12}] ≤ K_{2}.

Lemma 2. We have [K_{2}, x] = 1.

Proof. Since [x, G_{12}] ≤ K_{2} = C_{G_{12}}(K_{2}), G_{12} acts on [K_{2}, x]. But |[K_{2}, x]| ≤ 2^{3} and K_{2} ∩ G′_{1}, which is of order 2^{4}, is the only nontrivial G_{12}/K_{2}-submodule of K_{2}. So the assertion follows. □

Let Q_{2} := ⟨K_{2}, x⟩ and H := ⟨x_{1}, x_{8}, x_{9}, x_{10}⟩ ≅ Σ_{5}. By Lemma 2. Q_{2} is elementary abelian of order 2^{6} and [H, Q_{2}] ≤ K_{2}. With respect to the basis x_{2}, x_{3}, x_{5}, x_{6}, y, x of Q_{2} the elements of H belong to 6 × 6-matrices over GF (2) of the shape

( \begin{matrix}& & & 0 \\ & A & & ⋮ \\ & & & 0 \\ ∗ & ⋯ & ∗ & 1 \end{matrix} ) ,

where the 5 × 5-submatrix A is determined by the relations in (1). Since x is centralized by an element of order 3 in H, we may assume [x, x_{8}x_{9}] = 1. Using this and the above relations, it is not difficult to calculate that either [x, H] = 1 or

x_{1} = ( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & a_{5} & 1 \end{matrix} ) , x_{8} = ( \begin{matrix}1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ) ,

x_{9} = ( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ) , x_{10} = ( \begin{matrix}1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & d_{2} & 0 & 1 & 0 & 1 \end{matrix} ) ,

with a_{5}, d_{2} ∈ {0, 1}. In particular, [x, x_{8}] = [x, x_{9}] = 1 in any case.

Now look at G_{4} = ⟨x_{1}, x_{2}, x_{3}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}, y, x⟩. We already know that all generators of G_{4} except x_{7} and (possibly) x_{1} commute with x. Moreover, we have K_{4} = ⟨x_{1}, x_{2}, x_{3}, y⟩, H_{4} := ⟨x_{5}, x_{6}, x_{7}, x_{8}, x_{9}⟩ ≅ L_{3}(2) and Q_{4} = ⟨K_{4}, x^{H_{4}}⟩, where Q_{4} = O_{2}(G_{4}). Using the relations in (1), we calculate that, with respect to the basis {x, x^{x_{7}}, x^{x_{7}x_{9}}}, we get the following matrices for the action of H_{4} on Q_{4}/K_{4}:

x_{5} = ( \begin{matrix}1 & 0 & 0 \\ ¯1 & ¯1 & ¯0 \\ ¯1 & ¯0 & ¯1 \end{matrix} ) , x_{6} = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ ¯1 & ¯0 & ¯1 \end{matrix} ) , x_{7} = ( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ ¯1 & ¯1 & ¯1 \end{matrix} ) ,

x_{8} = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ ¯1 & ¯1 & ¯1 \end{matrix} ) , x_{9} = ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ) .

Here we write ¯1 resp. ¯0 if we only know that the corresponding relations hold in the factorgroup Q_{4}/K_{4} and we write 1, 0 if we know that they hold indeed.

Since there exists the involution x in Q_{4} \ K_{4} and an element of order 7 in L_{3}(2) acts fixedpointfreely on Q_{4}/K_{4}, it is easy to see that Q_{4} is elementary abelian of order 2^{7} if [x, H] = 1 or a_{5} = 0, and that Q_{4} ≅ D_{8}^{3} is extraspecial with Z(Q_{4}) = ⟨y⟩ if a_{5} = 1.

Let’s look at x_{5}. We have (x^{x_{7}})^{x_{5}} = axx^{x_{7}} for some a ∈ ⟨x_{1}, x_{2}, x_{3}, y⟩. Now x^{x_{7}x_{5}x_{7}} = x^{x_{5}x_{7}x_{5}} = x^{x_{7}x_{5}}, hence [a, x_{7}] ∈ ⟨y⟩ and a ∈ C_{K_{4}}(x_{7}) = ⟨x_{1}, x_{3}, y⟩. Furthermore, x^{x_{7}x_{5}x_{6}} = x^{x_{6}x_{7}x_{8}x_{5}} = x^{x_{7}x_{5}}, so a ∈ C_{K_{4}}(x_{6}) = ⟨x_{3}, y⟩.

Suppose a = by with b ∈ ⟨x_{3}⟩. Let x̃ = xy and substitute our basis by {x̃, x̃^{x_{7}}, x̃^{x_{7}x_{9}}}. Then we get the same matrices for x_{5}, x_{6}, x_{7}, x_{8}, x_{9} and x̃^{x_{7}x_{5}} = bx̃x̃^{x_{7}}. This means that w.l.o.g. we may assume a ∈ ⟨x_{3}⟩.

It follows from the relations (1) that the action of x_{5} on x^{x_{7}} already determines the action of H_{4} on Q_{4}. Now we are going to use the Todd-Coxeter-Algorithm in MAGMA to calculate |G : G_{1}| for the different cases. We have to distinguish wether a = 1 or a = x_{3}, and wether [x, H] = 1 or [x, H] ≠ 1, where in the last case we have the open parameters a_{5}, d_{2}. We always apply MAGMA for both possibilities G_{1} ≅ M_{22} and G_{1} ≅ Aut(M_{22}) (except in the case a_{5} = 1 which already implies G_{1} ≅ Aut(M_{22}). We get the following results:

1. If [x, H] = 1 and a = 1 then |G : G_{1}| = 2^{11}.

2. If [x, H] = 1 and a = x_{3} then |G : G_{1}| = 2^{9} · 3^{4}.

3. If [x, H] ≠ 1, a_{5} = 1, d_{2} = 0 and a = 1 then |G : G_{1}| = 23 · 11.

In all the other cases G is trivial.

In the first case G ≅ 2^{11}M_{22} resp. 2^{11}Aut(M_{22}) and O_{2}(G) is the universal representation group of the P -geometry for M_{22} (see [3, section 3]). We have |Z(G)| = 2 and G/Z(G) as well has a parabolic system of the desired type. So we get case a) of the Theorem.

In the second case, we have |G : G_{1}| = 2 · |U_{6}(2) : M_{22}| and we have shown below (Example 1) that there exists a parabolic system for the group U_{6}(2) satisfying these relations. This means that, if G_{1} ≅ M_{22}, then there exists a subgroup N ≤ G such that G/N ≅ U_{6}(2). Now |N| = 2, and since G = ⟨G_{1}, G_{4}⟩ = ⟨G′_{1}, G′_{4}⟩ ≤ G′, we get a non-split extension G ≅ 2 · U_{6}(2). If G_{1} ≅ Aut(M_{22}) then H := ⟨G′_{1}, G′_{4}⟩ ≅ 2·U_{6}(2) and |G : H| = 2. Hence G ≅ 2·U_{6}(2) : 2. Again in both cases G/Z(G) also has such a parabolic system and we get b).

In the third case we can use similar arguments to deduce from Example 2 that G ≅ M_{24}, i.e., c). □

Example 1. A parabolic system with diagram ∘  ∘  ∘  ∘ c  ^{P^{∗}}^{1}  ^{2}  ^{3}  ^{4} for the

group U_{6}(2):

We will give two different descriptions for the parabolic system: one by just giving matrices which satisfy the relations deduced above, and a second one which is more geometric and uses the acton of U_{6}(2) : 2 ≤ Co_{2} on the Leech lattice.

Let V be a 6-dimensional unitary space over GF (4) = {0, 1, ω, ω^{2}}, G = SU_{6}(2) and choose a basis {v_{1}, v_{2}, v_{3}, w_{3}, w_{2}, w_{1}} for V such that ⟨v_{i}, w_{i}⟩ are pairwise orthogonal hyperbolic planes. Then the following matrices belong to G:

x_{1} = ( \begin{matrix}1 \\ & & & 1 & 0 \\ & & & 0 & 1 \\ & 1 & 0 \\ & 0 & 1 \\ & & & & & 1 \end{matrix} ) , x_{2} = ( \begin{matrix}1 \\ 1 & 1 \\ 1 & & 1 \\ 1 & & & 1 \\ 1 & & & & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \end{matrix} ) ,

x_{3} = ( \begin{matrix}1 \\ 0 & 1 \\ 1 & & 1 \\ 0 & & & 1 \\ 1 & & & & 1 \\ 0 & 1 & 0 & 1 & 0 & 1 \end{matrix} ) , x_{5} = ( \begin{matrix}1 \\ ω & 1 \\ ω & & 1 \\ ω^{2} & & & 1 \\ ω^{2} & & & & 1 \\ 1 & ω & ω & ω^{2} & ω^{2} & 1 \end{matrix} ) ,

x_{6} = ( \begin{matrix}1 \\ 0 & 1 \\ ω & & 1 \\ 0 & & & 1 \\ ω^{2} & & & & 1 \\ 1 & ω & 0 & ω^{2} & 0 & 1 \end{matrix} ) , x_{7} = ( \begin{matrix}1 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ) ,

x_{8} = ( \begin{matrix}1 \\ 0 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & & & 1 & 0 \\ 1 & & & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 & 1 \end{matrix} ) , x_{9} = ( \begin{matrix}1 \\ & 0 & 1 \\ & 1 & 0 \\ & & & 0 & 1 \\ & & & 1 & 0 \\ & & & & & 1 \end{matrix} ) ,

x_{10} = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & ω & 1 \\ & & & 1 & 0 & 0 \\ & & & ω^{2} & 1 & 0 \\ & & & 1 & 0 & 1 \end{matrix} ) , x = ( \begin{matrix}1 \\ & 1 \\ & & 1 \\ & & & 1 \\ & & & & 1 \\ 1 & & & & & 1 \end{matrix} ) .

Moreover, one can calculate that there images in Ḡ := G/Z(G) ≅ U_{6}(2) satisfy the relations in (1) resp. the relations we have deduced for x in the proof of Theorem 1. In particular, if we set

P_{1} = ⟨x_{1}, x_{2}, x_{3}, x_{5}, x_{6}, x_{8}, x⟩, P_{2} = ⟨x_{1}, x_{2}, x_{3}, x_{5}, x_{6}, x_{7}, x_{8}⟩,
P_{3} = ⟨x_{1}, x_{2}, x_{3}, x_{5}, x_{6}, x_{8}, x_{9}⟩, P_{4} = ⟨x_{1}, x_{2}, x_{3}, x_{5}, x_{6}, x_{8}, x_{10}⟩,

then {P_{1}, P_{2}, P_{3}, P_{4}} is a parabolic system of the desired type. So it remains to show that G = ⟨P_{1}, P_{2}, P_{3}, P_{4}⟩.

Let G_{1} = ⟨P_{2}, P_{3}, P_{4}⟩. Since Ḡ_{1} satisfies the relations in (1), we have Ḡ_{1} ≅ M_{22}. Now by [1] M_{22} is a maximal subgroup of U_{6}(2), and since x ∉ G_{1}, and the assertion follows.

Before starting with the second description we recall the definition of the P -geometry for Co_{2} as it is given in [5] and some more facts about the action of this group on the Leech lattice.

Let Λ denote the Leech lattice with inner product ( , ) and let Λ_{2} = {v ∈ Λ_{2}|(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}. Then H_{σ} ≅ 2^{10}.Aut(M_{22}) for σ ∈ Σ and the orbits of H_{σ} on Σ are described in [5]. Define a graph on Σ in the way that σ and τ ∈ Σ are adjacent iff H_{σ} ∩ H_{τ} ≅ 2^{9}.2^{5}.Σ_{5}. A clique C in this graph is called closed iff for each {v, −v}, {w, −w} ∈ C there exists {u, −u} ∈ C such that v_{0} + v + w + u ∈ 2Λ. It is shown in [5] 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, than we get the P-geometry for Co_{2}.

Let Ω = {{v, v_{0} − v}|v ∈ Λ_{2}, (v, v_{0}) = 16}. Then H acts transitively on Ω and H_{ω} ≅U_{6}(2):2 for ω ∈ Ω (see [5], section 6). By table 2 of [5] H_{σ} has three orbits Ω_{1}, Ω_{2}, Ω_{3} on Ω and if σ = {u, −u}, ω = {w, v_{0} − w} we may assume (u, v) = 16, 0 resp. 8. Moreover, for ω ∈ Ω_{3} we have H_{σ} ∩ H_{ω} ≅ Aut(M_{22}). Since the orbits of H_{σ} on Ω bijectively correspond to the orbits of H_{ω} on Σ, for a fixed ω_{0} = {v, v_{0} − v} ∈ Ω, H_{ω_{0}} has exactly one orbit, say Σ_{3}, on Σ of length |Σ_{3}| = |(U_{6}(2):2) : Aut(M_{22})| = 20736 which consists of the pairs {v, −v} ∈ Σ such that w.l.o.g. (v, w) = 8. Now, for i = 1, 2, 3, 4, let

Γ_{i} = {C ∩ Σ_{3}|C is a closed clique of size 2^{i} − 1 in Σ }

and define an incidence relation between the Γ_{i} by inclusion. Considering the shapes of vectors from Λ_{2} as given in [5] it is easy to see that |x| = 2^{i−1} for x ∈ Γ_{i} and that this really defines a geometry of the desired type with automorphism group H_{ω_{0}} ≅ U_{6}(2):2.

Example 2. A geometry with diagram ∘  ∘  ∘  ∘ c  ^{P^{∗}}^{1}  ^{2}  ^{3}  ^{4} for the group M_{24}:

Let S = S(24, 8, 5) be the Steiner system for G = M_{24} with underlying set Ω. Let O denote the set of octads and T the set of trios. We assume that the reader is familiar with the action of G on S, particularly we need the following property:

Let O ∈ O and G_{O} the stabilizer of O in G. Then there is defined the structure of an affine space on the complement Ω\O and G_{O} ≅ 2^{4}A_{8} acts as affine group AGL(4, 2) on Ω \ O.

Now we define a rank-4-geometry Γ = Γ_{1} ∪ Γ_{2} ∪ Γ_{3} ∪ Γ_{4} by

– 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 partiton of Ω \ O into parallel lines }.

We define incidences between the Γ_{i} as follows:

Let x ∈ Γ_{1}, {a, b} ∈ Γ_{2} and y ∈ Γ_{3} or Γ_{4}. Then x is incident to {a, b} iff x ∈ {a, b}, x is incident to y iff x ∈ P and {a, b} is incident to y iff a and b are incident to y. Finally, (O_{1}, P_{1}, {O_{2}, O_{3}}) ∈ Γ_{3} and (O, P) ∈ Γ_{4} are incident iff O = O_{2} or O_{3} and P_{1} ⊂ P.

Literatur

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[4] A. Pasini, Diagram Geometries, Oxford, 1994.

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