ON A HOMOTOPY EQUIVALENCE BETWEEN THE 2-LOCAL GEOMETRY AND THE BOUC COMPLEX FOR THE SPORADIC GROUP McL

JOHN MAGINNIS AND SILVIA ONOFREI

Abstract. We study the homotopy relation between the standard 2-local geometry Δ and the Bouc complex for the sporadic finite simple group McL.

1. Introduction

The homotopy equivalence between the pure 2-local geometry Δ for the sporadic simple group McL and the Quillen collection was proved in [SY]. The Quillen collection is homotopy equivalent to the Bouc collection of all non-trivial p-radical subgroups. It follows that for G = McL the 2-local geometry Δ and the Bouc collection of non-trivial 2-radical subgroups are also homotopy equivalent.

It is the purpose of this paper to give a direct relationship bewteen the 2-local geometry Δ and the Bouc complex B_{2}(McL). We prove that a certain subcomplex B_{2}^{I} of the Bouc complex, which is isomorphic to a subdivision of the geometry Δ, is homotopy equivalent to the full Bouc complex B_{2} (in fact, this is a retract under an equivariant deformation retraction). Our approach is similar to the one used in our previous work [MgO1], which describes such a relationship for the sporadic group Co_{3}, although for Co_{3} the standard 2-local geometry is homotopy equivalent to the “distinguished” Bouc collection; see [MgO2].

In Section 2 details on the 2-local geometry Δ are given. In Section 3 the Bouc collection B_{2}(McL) is described using geometric interpretations for each of its subgroups. In Section 4, the main result, Theorem 4.1, is proved.

Acknowledgement: We would like to thank Stephen Smith for suggesting this problem to us. We also would like to express our gratitude to Professor Yoshiara who provided us with a copy of [Y03].

2. The 2-local geometry Δ of McL

Let G denote the sporadic simple group of McLaughlin. This is a group with one class of involutions, of 2-rank four and whose Sylow 2-subgroup is of order 2^{7}. The maximal subgroups of G have been determined in [Fin73]. Some details on the subgroup structure as well the computation of the mod-2 cohomology of G can be found in [AM97].

Date: Friday, October 21, 2005.

1991 Mathematics Subject Classification. 20G05, 20C20, 51D20.

A sporadic geometry Δ̃ for McL was initially given in [RStr84]. A very nice description of this geometry can be found in [Str89]. This is a rank 4 geometry with the following diagram:

∘ 3 ∘ 1  ∘ 4 ∘ 2

In [RSY] it was shown that, for any group P of order 2 in G, the fixed point set Δ̃^{P} is contractible. It follows that the complex Δ̃ is ample; that is, it provides a homology decomposition for the classifying space of McL. In [SY] a subcomplex Δ of Δ̃ was introduced. This is a truncation of Δ̃ obtained by removing all objects of type 1 and all the flags containing them. The new complex Δ is also ample but it has the advantage of being pure 2-local, in the sense of [SY], which means that all the simplex stabilizers are 2-local subgroups of G. The stabilizers of the objects are maximal 2-local subgroups of G. Benson and Smith [BS04, Section 6.11] used this complex to obtain a homology decomposition of the classifying space of G.

In what follows the 2-local geometry Δ will be used. It has diagram:

∘^{A} □  ∘^{P} ∘_{B}

The objects are denoted points P, A-spaces and B-spaces. These objects correspond to the elementary abelian 2-subgroups of G as follows: the points correspond to subgroups A_{1} ≃ 2, the A-spaces to A_{4} ≃ 2^{4} and the B-spaces to A_{4′} ≃ 2^{4}. Note that A_{4} and A_{4′} correspond to the two conjugacy classes of maximal elementary abelian 2-subgroups. Incidence between elements of P and elements of A (B respectively) is given by containment. Two spaces A ∈ A and B ∈ B are said to be incident if the corresponding subgroups have a 2^{2} in common, that is A_{4} ∩ A_{4′} = A_{2}. We will refer to the conjugacy class of subgroups of type 2^{2} as lines L. Given a subgroup A_{2} ≃ 2^{2} there exists a unique pair of incident spaces A and B, with A ∩ B = L; see [BS04, Section 6.11].

Δ can be regarded as a simplicial complex of dimension two with three types of vertices. The group G acts (faithfully) flag-transitively on the geometry; see [BS04] for a proof. The stabilizers of the three types of objects are:

G_{p} = 2^{.}A_{8} = N_{G}(A_{1})  for a point p ∈ P;
G_{A} = 2^{4} : A_{7} = N_{G}(A_{4})  for A ∈ A;
G_{B} = 2^{4} : A_{7} = N_{G}(A_{4′}) for B ∈ B.

The flag stabilizers can be easily determined and are as follows:

THE 2-LOCAL GEOMETRY AND THE BOUC COMPLEX FOR McL 3

\begin{matrix}G_{pA} = 2^{4} : L_{3}(2) & G_{AB} = 2^{4} : (3 × A_{4}) : 2 \\ G_{pB} = 2^{4} : L_{3}(2) & G_{pAB} = 2^{4} : S_{4}. \end{matrix}

Some of the properties of Δ which can be read from the diagram:

1). Given p ∈ P, there are 15 A-spaces and 15 B-spaces incident with p; they correspond to the “points” and the “planes” of a projective space of rank 3 over the field with two elements P G(3, 2).

2). Let A ∈ A. Then there are 15 points in A and 35 B-spaces incident with A. The geometry Res_{Δ}A = (P_{A}, B_{A}), of all points and all B-spaces incident with A, has the structure of the point-truncation of a Neumaier (sporadic A_{7}) geometry (similarly for B ∈ B).

Remark 2.1. The Neumaier geometry is defined on a set X of seven points, and has 35 lines and 15 planes. The set of lines consists of all 3 element subsets of X. There exist 30 different collections of 7 lines yielding the structure of a projective plane. Under the action of the alternating group A_{7}, these projective planes fall into two orbits, each of size 15. Choose one of these orbits as the planes of the Neumaier geometry; see [Cam91].

3. A geometric description of the Bouc complex of McL

Recall that a p-subgroup R of G is called p-radical if R = O_{p}(N_{G}(R)). Let B_{p}(G) denote the collection of all non-trivial p-radical subgroups of G; this collection is known in the literature as the Bouc collection.

In this section G = McL and Δ represents the 2-local geometry described in the previous section; further let B_{2} = B_{2}(McL). The Bouc complex of McL was first determined in [M98] and later given by Yoshiara^{1} in [Y03]. There are eight conjugacy classes of 2-radical subgroups.

Simplex  Name  R  Z(R)  N_{G}(R)  Generators
p  R_{p}  2  A^{1}  2^{.}A_{8}  ⟨ab⟩
A  R_{A}  2^{4}  A^{15}  2^{4} : A_{7}  ⟨a, b, c, d⟩
B  R_{B}  2^{4}  A^{15}  2^{4} : A_{7}  ⟨a, b, e, f⟩
AB  R_{AB}  2^{2}.2^{4}  A^{3}  2^{2+4}(3 × 3) : 2 ⟨a, b, c, d, e, f⟩
p□  R_{p□}  2_{+}^{1+4}  A^{1}  2_{+}^{1+4}(S_{3} × S_{3}) ⟨a, b, c, e, u⟩
pA□  R_{pA□}  2.[2^{5}]  A^{1}  2.[2^{5}] : S_{3}  ⟨a, b, c, d, e, u⟩
pB□  R_{pB□}  2.[2^{5}]  A^{1}  2.[2^{5}] : S_{3}  ⟨a, b, c, e, f, u⟩
pAB□  R_{pAB□}  2^{4} : D_{8} A^{1}  2^{4} : D_{8}  ⟨a, b, c, d, e, f, u⟩

The notation □ stands for a typical vertex of the truncated type in Res_{Δ}v, where v ∈ {P, A, B}. It corresponds to the square node in the diagram of Δ. Note that □ does not represent a class of objects in the geometry Δ. In Res_{Δ}p, □ stands for a line in the P G(3, 2) whose “points” and “planes” are the

1This is from Table 5 in [Y03]. Notice that the table has a typographical error; for the third group R = E_{1}^{(2)}, N(R)/R should read (3 × 3) : 2 and not S_{3} × S_{3}.

4 JOHN MAGINNIS AND SILVIA ONOFREI

A-spaces and B-spaces incident with p. In Res_{Δ}A, □ can be identified with a point in the Neumaier geometry. The same is true when considering Res_{Δ}B.

In the last column of the table we give generators for each of the 2-radical subgroups. Following Stroth [Str89], a presentation for the 2-Sylow subgroup of G can be written as:

S = ⟨a, b, c, d, e, f, u | R⟩

with a set of relations:

R ={a^{2} = b^{2} = c^{2} = d^{2} = e^{2} = f^{2} = u^{2} = 1,

[a, b] = [a, c] = [a, d] = [b, c] = [b, d] = [c, d] = [a, e] = [b, e] = [a, f] = [b, f] = [e, f] = 1,

c^{e} = abc, d^{e} = bd, c^{f} = bc, d^{f} = ad, a^{u} = b, c^{u} = c, d^{u} = cd, e^{u} = e, f^{u} = ef}

In what follows we shall describe each of the subgroups of B_{2} in geometric terms.

R_{pAB□} = 2^{4} : D_{8}

The group R_{pAB□} of order 2^{7} is the 2-Sylow subgroup of G. Its structure was analyzed in detail by Adem and Milgram in [AM97, Lemma 1.1].

Lemma 3.1. a) The group R_{pAB□} contains exactly one copy (a conjugate in McL) of each of the

subgroups R_{A}, R_{B}, R_{AB}.

b) The group R_{pAB□} contains exactly one copy of R_{p□}.

Proof. a) See [J68] for a proof.

b) We consider the composition of maps 2_{+}^{1+4} → 2^{4} : D_{8} ↠ UT_{4}(2), where UT_{4}(2) denotes the group of upper triangular 4 × 4 matrices over F_{2}. If this map is not injective, the central involution of 2_{+}^{1+4} lies in the center of R_{pAB□}. The quotient UT_{4}(2) contains a unique subgroup of the form 2^{4} and its inverse image in R_{pAB□} is indeed a copy of 2_{+}^{1+4}. If the above composition is injective, note that UT_{4}(2) contains a unique copy of 2_{+}^{1+4} and the inverse image of this subgroup in R_{pAB□} is of the form 2^{2+4}. We reached a contradiction since 2^{2+4} does not contain a subgroup of the form 2_{+}^{1+4}. □

There are 35 involutions in R_{pAB□} and they can be partitioned as: 1 + 2 + 4_{A} + 4_{B} + 8_{A} + 8_{B} + 8_{□}, where 1 stands for the central involution, 1 + 2 represent the involutions contained in the second term of the upper central series of R_{pAB□} and 1+2+4_{A} +4_{B} the involutions contained in the third term of the same series. The involutions 1 + 2 + 4_{A} + 8_{A} lie in the unique R_{A} contained in R_{pAB□}. Similarily 1 + 2 + 4_{B} + 8_{B} lie in R_{B}. Note that 1 + 2 correspond to the subgroup 2^{2} = R_{A} ∩ R_{B}. Then 8_{□} stands for the remaining 35 − 27 involutions in R_{pAB□}.

THE 2-LOCAL GEOMETRY AND THE BOUC COMPLEX FOR McL 5

R_{p} = 2

This corresponds to a single a central involution and thus a point of the geometry.

R_{A} = 2^{4} and R_{B} = 2^{4}

This is an elementary abelian 2-subgroup of rank 4. Its 15 involutions are in one-to-one correspondence with the planes of the Neumaier geometry.

R_{AB} = 2^{2+4}

There are exactly 27 involutions in this group. This structure corresponds to a pair {A, B} of incident spaces from different classes. Furthermore R_{AB} = ⟨R_{A}, R_{B}⟩, see [Fin73]. Clearly R_{A} ⊆ R_{AB} and R_{B} ⊆ R_{AB} are flags in B_{2}. The involutions of R_{AB} can be written as: 1 + 2 + 4_{A} + 8_{A} + 4_{B} + 8_{B}.

R_{p□} = 2_{+}^{1+4}

This group contains 19 involutions which can be partitioned as: 1 + 2 + 4_{A} + 4_{B} + 8_{□}. Notice that R_{p□} and R_{AB} are not incident in the Bouc complex. It is easy to see from the presentation given in Table I that R_{p□} is incident with both R_{pA□} and R_{pB□}.

R_{pA□} = 2.[2^{5}] and R_{pB□} = 2.[2^{5}]

These groups contain 27 involutions each. It is easy to see, from the presentations given in Table I that R_{A} ⊆ R_{pA□} and R_{B} ⊆ R_{pB□} are chains in B_{2}. From this observation we deduce that the involutions of R_{pA□} can be written as: 1+2+4_{A}+8_{A}+4_{B}+8_{□} and similarily for R_{pB□}: 1+2+4_{A}+4_{B}+8_{B}+8_{□}.

4. The homotopy relation

In this Section we will prove:

Theorem 4.1. The Bouc complex B_{2} is homotopy equivalent to the subcomplex B_{2}^{I} obtained by removing from B_{2} the vertices of type R_{p□}, R_{pA□}, R_{pB□} and R_{pAB□} together with all the simplices containing them.

Flags in B_{2}. Let F : x_{1} ⊆ x_{2} … ⊆ x_{n} be a flag in B_{2}. Then the stabilizer of the object x_{n} acts on the collection of objects of type x_{1} incident with x_{n}. In most of the cases this action is not transitive. In the first column of the table we give the type of the flag F, in the second column the order of the stabilizer of the object x_{n} in F and in the third column we give the number of elements in the orbits of Stab_{G}(x_{n}) on the collection of objects of type 1 incident with x_{n}.

Rank 1 flags  Rank 3 flags
p  2^{7} · 3^{2} · 5 · 7 1  p ⊆ A ⊆ pAB□  2^{7}  1, 2, 4, 8
A  2^{7} · 3^{2} · 5 · 7 1  p ⊆ B ⊆ pAB□  2^{7}  1, 2, 4, 8
B  2^{7} · 3^{2} · 5 · 7 1  p ⊆ p□ ⊆ pAB□  2^{7}  1, 2, 4, 4, 8
pA□  2^{7} · 3  1  p ⊆ pA□ ⊆ pAB□  2^{7}  1, 2, 4, 4, 8, 8
pB□  2^{7} · 3  1  p ⊆ pB□ ⊆ pAB□  2^{7}  1, 2, 4, 4, 8, 8
AB  2^{7} · 3^{2}  1  p ⊆ AB ⊆ pAB□  2^{7}  1, 2, 4, 4, 8, 8
p□  2^{7} · 3^{2}  1  p ⊆ A ⊆ AB  2^{7} · 3^{2} 3, 12
pAB□  2^{7}  1  p ⊆ B ⊆ AB  2^{7} · 3^{2} 3, 12
Rank 2 flags  p ⊆ p□ ⊆ pA□  2^{7} · 3 1, 6, 12
p ⊆ A  2^{7} · 3^{2} · 5 · 7 15  p ⊆ p□ ⊆ pB□  2^{7} · 3 1, 6, 12
p ⊆ B  2^{7} · 3^{2} · 5 · 7 15  p ⊆ A ⊆ pA□  2^{7} · 3 1, 6, 8
p ⊆ p□  2^{7} · 3^{2}  1, 18  p ⊆ B ⊆ pB□  2^{7} · 3 1, 6, 8
p ⊆ AB  2^{7} · 3^{2}  3, 12, 12  A ⊆ AB ⊆ pAB□  2^{7}  1
p ⊆ pA□  2^{7} · 3  1, 6, 8, 12  B ⊆ AB ⊆ pAB□  2^{7}  1
p ⊆ pB□  2^{7} · 3  1, 6, 8, 12  A ⊆ pA□ ⊆ pAB□  2^{7}  1
p ⊆ pAB□  2^{7}  1, 2, 4, 4, 8, 8, 8 B ⊆ pB□ ⊆ pAB□  2^{7}  1
A ⊆ pA□  2^{7} · 3  1  p□ ⊆ pA□ ⊆ pAB□  2^{7}  1
A ⊆ AB  2^{7} · 3^{2}  1  p□ ⊆ pB□ ⊆ pAB□  2^{7}  1
A ⊆ pAB□  2^{7}  1
B ⊆ pB□  2^{7} · 3  1  Rank 4 flags
B ⊆ AB  2^{7} · 3^{2}  1  p ⊆ A ⊆ AB ⊆ pAB□  2^{7}  1, 2, 4, 8
B ⊆ pAB□  2^{7}  1  p ⊆ B ⊆ AB ⊆ pAB□  2^{7}  1, 2, 4, 8
p□ ⊆ pA□  2^{7} · 3  1  p ⊆ A ⊆ pA□ ⊆ pAB□ 2^{7}  1, 2, 4, 8
p□ ⊆ pB□  2^{7} · 3  1  p ⊆ B ⊆ pB□ ⊆ pAB□ 2^{7}  1, 2, 4, 8
p□ ⊆ pAB□  2^{7}  1  p ⊆ p□ ⊆ pA□ ⊆ pAB□ 2^{7}  1, 2, 4, 4, 8
AB ⊆ pAB□ 2^{7}  1  p ⊆ p□ ⊆ pB□ ⊆ pAB□ 2^{7}  1, 2, 4, 4, 8
pA□ ⊆ pAB□ 2^{7}  1
pB□ ⊆ pAB□ 2^{7}  1

Table I

Proof of the Theorem 4.1. The proof of Theorem is straightforward, involving a sequence of steps of homotopy retractions which use the collapsibility property of certain flags. We will make repeated use of the following:

Lemma 4.2 ([RSY], Section 1). Let Σ ∈ Δ be a simplex of maximal dimension with σ as a face. Assume that Σ is the only simplex of maximal dimension with σ as a face. Then the process of removing Σ from Δ, by collapsing Σ down onto its faces other than σ, is a homotopy equivalence.

In what follows we will homotopically retract B_{2} down to B_{2}^{I}. Recall that a maximal simplex Σ is free over some maximal face σ, if Σ is the only maximal simplex with σ as a face. In this case we can use Lemma 3.1 to collapse Σ down onto its faces other than σ. This means that we can remove Σ and σ from the complex to obtain a homotopy equivalent subcomplex. Note that since this method depends on the maximality of Σ and since the removal of some simplices might make other simplices maximal,

THE 2-LOCAL GEOMETRY AND THE BOUC COMPLEX FOR McL 7

the sequence of homotopy retractions we are performing has to be done in the order indicated here.

In the first column we label the step. In the second column we give the maximal simplex Σ and in the third column we give a face σ of Σ such that Σ is free over σ. In the fourth column we specify the collection of points in σ.

Step Σ  σ  Points in σ
1.A p ⊆ A ⊆ AB ⊆ pAB□  p ⊆ AB ⊆ pAB□ 4_{A} + 8_{A}
1.B p ⊆ B ⊆ AB ⊆ pAB□  p ⊆ AB ⊆ pAB□ 4_{B} + 8_{B}
2.A p ⊆ A ⊆ pA□ ⊆ pAB□  p ⊆ pA□ ⊆ pAB□ 8_{A}
2.B p ⊆ B ⊆ pB□ ⊆ pAB□  p ⊆ pB□ ⊆ pAB□ 8_{B}
3.A p ⊆ p□ ⊆ pA□ ⊆ pAB□  p ⊆ pA□ ⊆ pAB□ 4_{B} + 8_{□}
3.B p ⊆ p□ ⊆ pB□ ⊆ pAB□  p ⊆ pB□ ⊆ pAB□ 4_{A} + 8_{□}
4.A p ⊆ A ⊆ pA□ ⊆ pAB□  p ⊆ A ⊆ pAB□  4_{A}
4.B p ⊆ B ⊆ pB□ ⊆ pAB□  p ⊆ B ⊆ pAB□  4_{B}
5.A p ⊆ p□ ⊆ pA□ ⊆ pAB□  p ⊆ pA□ ⊆ pAB□ 4_{A}
5.B p ⊆ p□ ⊆ pB□ ⊆ pAB□  p ⊆ pB□ ⊆ pAB□ 4_{B}
6.A 3 · 2p ⊆ A ⊆ pA□ ⊆ pAB□ p ⊆ A ⊆ pA□  2 + 4_{A}
6.B 3 · 2p ⊆ B ⊆ pB□ ⊆ pAB□ p ⊆ B ⊆ pB□  2 + 4_{B}
7.A 3 · 2p ⊆ p□ ⊆ pA□ ⊆ pAB□ p ⊆ p□ ⊆ pA□  2 + 4_{A}
7.B 3 · 2p ⊆ p□ ⊆ pB□ ⊆ pAB□ p ⊆ p□ ⊆ pB□  2 + 4_{B}
8.A p ⊆ A ⊆ AB ⊆ pAB□  p ⊆ A ⊆ pAB□  2
8.B p ⊆ B ⊆ AB ⊆ pAB□  p ⊆ B ⊆ pAB□  2
9.A p ⊆ A ⊆ pAB□  p ⊆ pAB□  8_{A}
9.B p ⊆ B ⊆ pAB□  p ⊆ pAB□  8_{B}
10 p ⊆ p□ ⊆ pAB□  p ⊆ pAB□  4_{A} + 4_{B} + 8_{□}
11.A p ⊆ p□ ⊆ pA□  p ⊆ pA□  4_{B} + 8_{□}
11.B p ⊆ p□ ⊆ pB□  p ⊆ pB□  4_{A} + 8_{□}
12 9 · 2p ⊆ p□ ⊆ pAB□  p ⊆ p□  2 + 4_{A} + 4_{B} + 8_{□}
13.A 3 · 2p ⊆ pA□ ⊆ pAB□  p ⊆ pA□  2 + 4_{A}
13.B 3 · 2p ⊆ pB□ ⊆ pAB□  p ⊆ pB□  2 + 4_{B}
14.A p ⊆ A ⊆ pA□  p ⊆ pA□  8_{A}
14.B p ⊆ B ⊆ pB□  p ⊆ pB□  8_{B}
15 p ⊆ AB ⊆ pAB□  p ⊆ pAB□  2

Table II

We give details for Step 6.A which is suggestive for the approach we use. Step 6.A: Consider the simplex σ : p ⊆ A ⊆ pA□. We are concerned here with the points denoted

8 JOHN MAGINNIS AND SILVIA ONOFREI

2+4_{A}. There are 6 such points, 2 points in each of the three flags of the form p ⊆ A ⊆ pA□ ⊆ pAB□ containing σ. The element of order 3 in N_{G}(R_{pA□}) permutes the three Sylow subgroups containing R_{pA□}, stabilizing the collection of involutions of R_{pA□}; in particular, it permutes the three lines on the point p in the plane 1 + 2 + 4_{A}, thus we can write: 2 + 4_{A} = 2 + 2 + 2. This means that for a fixed σ, there are 3 simplexes Σ of the type chosen above. We write this as 3 · 2 p ⊆ A ⊆ pA□ ⊆ pAB□, emphasizing also the 2 points of Σ. Fix one such simplex Σ. This is a simplex of maximal dimension and free over the face σ, but only after the homotopy retractions of steps 1 − 5. Thus we can apply Lemma 3.2 and remove these two simplices. A similar argument can be used for the steps 6.B, 7, 12, 13.

After the 15 steps from Table II we are left with the complex B_{2}^{I}, with a cone attached at each point p. This is of course homotopy equivalent to B_{2}^{I}.

□

Remark 4.3. Note that the subcomplex B_{2}^{I} is essentially |A_{2}^{−}| from [BS04, Section 6.11]. Further Δ corresponds to the semi-order complex Δ_{H_{I}}. Note, also that |A_{2}^{−}| is a subdivision of Δ_{H_{I}} where we subdivided by adding a vertex AB (denoted 44′ in [BS04]) and a new edge p ⊆ AB. It was proved in [SY] that Δ, |A_{2}^{−}| and the Quillen complex are homotopy equivalent.

Remark 4.4. Using the information from Table I, the reduced Euler characteristic of the Bouc complex can be easily computed^{2} and it is:

χ̃(B_{2}) = 2^{7} · 7483

A p-subgroup P of G is called p-centric if Z(P) is a Sylow p-subgroup of C_{G}(P). Then B_{p}^{cen} will denote the collection of p-radical and p-centric subgroups of G. Among the 2-radical subgroups of McL all but the first conjugacy class of groups are 2-centric. The subcomplex B_{2}^{cen} of the centric subgroups of the group McL is the complex obtained by removing from B_{2} the vertex R_{p} and all the flags containing it. This subcomplex has reduced Euler characteristic:

χ̃(B_{2}^{cen}) = 2^{6} · 37241

Therefore, in this case Δ and B_{2}^{cen} are not homotopy equivalent.

References

[AM97] A. Adem, R.J. Milgram, The cohomology of the McLaughlin group and some associated groups, Math. Z. 224, 495-517 (1997).

[BS04] D. Benson, S.D. Smith, Homotopy decompositions and 2-completed classifying spaces for sporadic simple groups, preliminary version (2004).

[Cam91] P.J. Cameron, Projective and polar spaces, QMW Math. Notes 13 (1991).

[Fin73] L. Finkelstein, The maximal subgroups of Conway’s group C_{3} and McLaughlin’s group, J. Algebra 25, 58-89 (1973).

[J68] Z. Janko, Characterization of the Mathieu simple groups, J. Algebra 9, 20-41 (1968).

2This is given also in Table I of [SY].

[MgO1] J.S. Maginnis, S.E. Onofrei, On a homotopy relation between the 2-local geometry and the Bouc complex for the sporadic group Co_{3} (preprint 2004).

[MgO2] J.S. Maginnis, S.E. Onofrei, New collections of p-subgroups and homology decompositions for classifying spaces of finite groups (preprint 2005).

[M98] J.C. Murray, Dade’s conjecture for the McLaughlin simple group, PhD Thesis, University of Illinois at Urbana-Champaign (1998).

[RStr84] M.A. Ronan, G. Stroth, Minimal parabolic geometries for the sporadic groups, Europ. J. Combin. 5, 59-91 (1984).

[RSY] A.J.E. Ryba, S.D. Smith, S. Yoshiara, Some projective modules determined by sporadic geometries, J. Algebra 129, 279-311 (1990).

[SY] S.D. Smith, S. Yoshiara, Some homotopy equivalences for sporadic geometries, J. Algebra 192, 326-379 (1997).

[Str89] G. Stroth, Some geometry for McL, Comm. Algebra 17(11), 2825-2833 (1989).

[Y03] S. Yoshiara, Radical 2-subgroups of the monster and baby monster, preliminary version, May 2003.