Maximal 2-local subgroups of the Monster and Baby Monster

U. Meierfrankenfeld and S. Shpectorov

September 6, 2002

Abstract

The lists of the maximal 2-local subgroups of the Monster and Baby Monster simple groups in the Atlas are complete.

1 Introduction

The Monster and the Baby Monster are the two largest groups among the 26 sporadic finite simple groups. After the classification of the finite simple groups was announced in 1981 the focus of research in the area of finite simple groups moved toward the study of the properties of the known groups. One of the most important pieces of information about a simple group G is its list of maximal subgroups, taken up to conjugation in G.

Methods used to classify maximal subgroups H of G differ significantly depending on whether or not H is p-local. A subgroup H of G is p-local, where p is a prime number dividing the order of G, if H is the normalizer of a nontrivial p-subgroup. We say that H is a maximal p-local subgroup if H is maximal by inclusion among p-local subgroups of G. Notice that a maximal p-local subgroup H may or may not be maximal in G. Nevertheless, the classification of all maximal p-local subgroups of G for each p is an important step toward a complete determination of all maximal subgroups of G.

In the case where G is one of the 26 sporadic finite simple groups the lists of maximal p-locals subgroups–as well as lists of non p-local maximal subgroups–have been compiled and proven complete for almost all G by work of many people, but most notably, R.A. Wilson. One significant omission to date has been the lists of maximal 2-local subgroups of M and BM. The Atlas of Finite Groups [ATLAS] provides lists of the known maximal 2-local subgroups of M and BM. What was missing was a proof that these lists are in fact complete. In this paper and its sequel [M] we bridge this gap by supplying necessary proofs.

Let us now review the lists from [ATLAS]. Seven conjugacy classes of maximal 2-local subgroups were known for G = M. The corresponding structures are as follows (see [ATLAS] for the exact meaning of these structures; however, notice that [ATLAS] uses the notation B for the Baby Monster group; also we use the good old “Ω” and “Sp” where [ATLAS] uses “O” and “S”):

(1) 2 · BM;

(2) 2^{2} · (^{2}E_{6}(2)) : S_{3};

(3) 2_{+}^{1+24}.Co_{1};

(4) 2^{2}.2^{11}.2^{22}.(S_{3} × M_{24});

(5) 2^{3}.2^{6}.2^{12}.2^{18}.(L_{3}(2) × 3 · S_{6});

(6) 2^{5}.2^{10}.2^{20}.(S_{3} × L_{5}(2));and

(7) 2^{10+16} · Ω_{10}^{+}(2).

Eight classes of maximal 2-local subgroups were known for G = BM. Their structures are shown in [ATLAS] as follows.

(1) 2 · (^{2}E_{6}(2)) : S_{3};

(2) (2^{2} × F_{4}(2)) : 2;

(3) S_{4} ×^{2}F_{4}(2);

(4) 2_{+}^{1+22} · Co_{2};

(5) 2^{2}.2^{10}.2^{20}.(M_{22} : 2 × S_{3});

(6) 2^{3}.[2^{32}].(S_{5} × L_{3}(2));

(7) 2^{5}.[2^{25}].L_{5}(2); and

(8) 2^{9}.2^{16}.Sp_{8}(2).

Recall that O_{2}(H) denotes the largest normal 2-subgroup of H. We say that H is of characteristic 2 if C_{H}(Q) ≤ Q, where Q = O_{2}(H). We split the work as follows. In this papers we determine all maximal 2-local subgroups of M and BM that are of characteristic 2. The sequel [M] deals with the remaining classes. In the above lists the partition into the characteristic 2 type and non characteristic 2 type is as follows: For M, classes (1) and (2) are not of characteristic 2, while classes (3)–(7) are of characteristic 2. For BM, classes (1)–(3) are not of characteristic 2, while classes (4)–(8) are of characteristic 2. Notice that classifying the maximal 2-local subgroups that are not of characteristic 2 is rather more simple and some may even say that this part of the lists has been known to be complete. However, as we are unaware of any published proof, we include this subcase in our work. Needless to say, we believe that our result on the maximal 2-local subgroups of characteristic 2 is entirely new.

Our approach was in part motivated by the work on the geometries of the groups M and BM, by A.A. Ivanov and the second author (see [IS]). We noticed that the known maximal 2-local subgroups of characteristic 2 are either the normalizers of certain very special elementary abelian subgroups that we call singular subgroups, or the normalizers of yet another type of elementary abelian subgroups (of order 2^{10}) that we call arks. The abovementioned geometries of M and BM consist of singular subgroups, while arks also have a geometrical meaning, namely, they correspond to certain natural subgeometries.

We introduce singular subgroups in Section 4, in which we also classify them up to conjugation. Arks are introduced in Section 5. They form a single conjugacy class of subgroups. Our choice of the word “ark” was motivated by the fact that an ark contains representatives of all “species” (i.e., conjugacy classes) of singular subgroups.

We will now formally state the principal results of this paper. We start with the following definitions of the groups M and BM: The Monster M is a finite simple group with a large extraspecial 2-subgroup Q whose normalizer C has the following structure: C ∼ 2^{1+24}.Co_{1}. (Here and in what follows we use ∼ as shorthand for “has structure”, while ≅ as usual stands for “is isomorphic”.) Recall that a group G is said to have a large extraspecial subgroup if for some involution z ∈ G its centralizer C = C_{G}(z) contains a normal extraspecial 2-subgroup Q such that C_{G}(Q) ≤ Q. It easily follows from this that z is 2-central in G, that is, the centralizer of Z contains a Sylow 2-subgroup of G. It also follows that G has a unique conjugacy class of 2-central involutions. Returning to the Monster group M, in addition to the above structure of C, we have that the Co_{1}-module arising in the action of C on Q/⟨z⟩ is isomorphic to the module on Λ/2Λ, where Λ is the Leech lattice. This due to R. Griess who showed in [Gr2] that Λ/2Λ is the only faithful Co_{1}-module in dimension 24.

For the purposes of this paper, the Baby Monster BM is simply the group H/⟨t⟩, where t is a non 2-central involution in the Monster M and H = C_{M}(t). All the work in this papers takes place in M; singular subgroups, arks and the Baby Monster BM “live” in M.

Under these definitions, we prove the following.

Theorem 1 The Monster M contains exactly 5 conjugacy classes of maximal 2-local subgroups of characteristic 2. They are: (a) the normalizers of singular subgroups of types 2^{1}, 2^{2}, 2^{3} and 2_{2}^{5}; and (b) the normalizers of arks.

Theorem 2 The Baby Monster BM contains exactly 5 conjugacy classes of maximal 2-local subgroups of characteristic 2. Their preimages in H are:

(a) the normalizers in H of special singular subgroups U of types 2^{1}, 2^{2}, 2^{3} and 2_{1}^{5}; and (b) the normalizers of arks containing t.

The exact meaning of the word “special” in this last theorem is as follows: With each singular subgroup U we associate in Section 4 a second subgroup Q_{U}. The special singular subgroups U are those for which t ∈ Q_{U}.

In simple words, Theorems 1 and 2 state that the lists of maximal 2local subgroups of M and BM given in [ATLAS] are complete in their characteristic 2 part.

Finally, we need to explain our policy with respect to citing vs. proving. A lot of information is available about the two monsters. However, much of it exists as a sort of finite group theory lore, that is, with no proper published proof known. In particular, at least some of the information given in [ATLAS] should be considered as semi-lore because there is no proofs there and very little by way of citation. Of course, we cannot prove everything in a single paper, and so we decided to take an “inductive” approach. We use some “lore” information about the smaller simple groups involved in M (mostly, Co_{1} and Ω_{10}^{+}(2)). At the same time, we prove everything we need as far as the properties of M and BM themselves are concerned. Likely, some of the facts that we prove can be found in the available sources such as [As1], [AsSe], [Gr1], [Se] and many others. However, we believe that the bulk of the detailed information that we need cannot be covered by citation. Notice also that M. Aschbacher in [As1] determines all maximal subgroups of M containing a Sylow 2-subgroup.

2 Classes of involutions, I

In this section we classify conjugacy classes of involutions of a group H satisfying the following conditions:

(H1) O_{2}(H) ∼ 2^{24};

(H2) H/O_{2}(H) ≅ Co_{1}; and

(H3) the action of H/O_{2}(H) on O_{2}(H) is equivalent to the action of Co_{1} on Λ̂ = Λ/2Λ, the Leech lattice taken modulo 2.

If C is the centralizer of a 2-central involution z in the Monster group M then the group H = C/Z, where Z = ⟨z⟩, satisfies (H1)–(H3) and so the results of this section give us some insight into the structure of C.

We refer to [ATLAS], page 180, for a description of the Leech lattice Λ, terminology and notation related to Λ, and a summary of properties of Λ. A whole wealth of information about the Leech lattice can be found in [CS]. Let (x, y) = \frac{1}{8} ∑_{i=1}^{24} x_{i}y_{i} be the integral inner product that exists on Λ. Let Λ_{n} = {x ∈ Λ|(x, x) = 2n}. One useful fact that is the following

Lemma 2.1 The orbits of Co_{1} on W = Λ̂^{#} are the sets Λ̂_{2}, Λ̂_{3} and Λ̂_{4}.

For i = 2, 3 and 4, let w_{i} ∈ Λ̂_{i}. The structure of the stabilizer of w_{i} in Co_{1} is also well-known.

Lemma 2.2 The following hold.

(1) C_{Co_{1}}(w_{2}) ≅ Co_{2};

(2) C_{Co_{1}}(w_{3}) ≅ Co_{3};

(3) C_{Co_{1}}(w_{4}) ≅ 2^{11} : M_{24}.

Recall that in its action on W = Λ̂ the group Co_{1} preserves a nondegenerate quadratic form q defined as follows: if u = x̂ for some x ∈ Λ then q(u) = \frac{1}{2}(x, x) (mod 2). Let Φ denote the symmetric bilinear form that corresponds to q: for u = x̂ and v = ŷ, we have Φ(u, v) = q(u + v) + q(u) + q(v) = (x, y) (mod 2). It follows from the definition of q that w_{2} and w_{4} are singular, while w_{3} is nonsingular.

Before we go on, let us record the following property that can be verified, say, using the description of Λ from [ATLAS].

Lemma 2.3 There is no Λ̂_{2}-pure subgroups 2^{3} in W = Λ̂.

According to [ATLAS], Co_{1} contains three conjugacy classes of involutions. We will need to know how the involutions and their centralizers act on W.

Let t be an involution in Co_{1} and let C_{t} = C_{Co_{1}}(t). Define U_{t} = C_{W} (t) and V_{t} = [W, t]. Since t is an involution, V_{t} ⊆ U_{t}. Furthermore, dim W/U_{t} = dim V_{t}. In fact, U_{t} is the orthogonal complement (with respect to Φ) of V_{t}, and so the C_{t}-modules W/U_{t} and V_{t} are dual to each other.

First, let t be an involution of type 2A. Then C_{t} is an extension of an extraspecial group 2_{+}^{1+8} by Ω_{8}^{+}(2). The action on the 8-dimensional quotient of O_{2}(C_{t}) provides an irreducible module for C_{t}/O_{2}(C_{t}) ≅ Ω_{8}^{+}(2). We will refer to this module as to the natural module. Notice that Ω_{8}^{+}(2) has two more irreducible 8-dimensional modules, and we will refer to those as to the two halfspin modules. Notice also that the natural module and the halfspin modules are all self-dual.

Lemma 2.4 If t is of type 2A then

(1) V_{t} has dimension 8 and U_{t} has dimension 16;

(2) C_{t} acts irreducibly on each of W/U_{t}, U_{t}/V_{t}, and V_{t}; furthermore, V_{t} and U_{t}/V_{t} are two non-isomorphic halfspin modules and W/U_{t} ≅ V_{t}.

Proof: Notice that t (or rather, its preimage in Co_{0}) can be chosen to act on the standard frame, inverting signs in an octad. This allows to establish (1) by direct computation. Let x ∈ C_{t} be an element of order three, such that x has a 6-dimensional centralizer in the natural module. Then 2^{13} divides the order of the centralizer of x in C_{t}, and hence x is of type 3A (a Suzuki 3-element). According to [ATLAS], x acts fixed-point-freely on W, which implies that W/U_{t}, U_{t}/V_{t}, and V_{t} are halfspin modules for C_{t}/O_{2}(C_{t}) ≅ O_{8}^{+}(2). Since the halfspin modules are self-dual, we have W/U_{t} ≅ V_{t}. Finally, if U_{t}/V_{t} ≅ V_{t} then C_{t} contains a 3-element with an 18-dimensional centralizer in W, which contradicts the information from [ATLAS].

Let t be an involution of type 2B. Then C_{t} ∼ (2^{2} × G_{2}(4)).2.

Lemma 2.5 If t is of type 2B then

(1) U_{t} = V_{t} and so they are of dimension 12;

(2) C_{t} acts transitively on V_{t}^{#}; furthermore, V_{t}^{#} ⊆ Λ̂_{4}.

Proof: Notice that 13 divides the order of G_{2}(4), and it does not divide the orders of Co_{2}, Co_{3} and 2^{11} : M_{24}. Therefore, G_{2}(4) fixes no non-zero vector in W. Again, since 13 divides the order of G_{2}(4), the latter group has no nontrivial GF (2)-modules in dimensions less than 12. It follows that V_{t} has dimension at least 12. Hence, U_{t} = V_{t} and they are both of dimension exactly 12. According to [MOD], V_{t} must be the natural module for G_{2}(4). In particular, G_{2}(4) is transitive on V_{t}^{#}. Since Λ̂_{4} has odd length, t fixes a vector in Λ̂_{4}. Now the transitivity implies that V_{t}^{#} ⊆ Λ̂_{4}.

Finally, let t be of type 2C. Then C_{t} ∼ 2^{11} : Aut M_{12}.

Lemma 2.6 If t is of type 2C then

(1) U_{t} = V_{t}, and so they are of dimension 12;

(2) as a C_{t}-module, U_{t} is uniserial with submodules of dimension 1 and 11; furthermore, the non-zero vector fixed by C_{t} is from Λ̂_{4}, and Λ̂_{3}∩V_{t} coincides with the setwise complement of the 11-dimensional submodule.

Proof: In this case again t can be chosen inside the diagonal subgroup stabilizing the standard frame. Namely, t inverts signs in a dodecad. This allows to compute all vectors in W fixed by t and thus establish (1) and also that V_{t} contains some elements from Λ̂_{3}. Since U_{t} = V_{t}, we have that V_{t} is totally isotropic (with regard to Φ). Since the vectors in Λ̂_{3} are non-singular, we obtain that V_{t} ∩Λ̂_{3} coincides with the complement of a hyperplane. So V_{t} contains an 11-dimensional subspace V_{0} left invariant by C_{t}. Observe that C_{t} is fully contained in the stabilizer of the standard frame. So C_{t} stabilizes a vector v ∈ Λ̂_{4}, the image of the standard frame. Clearly, v ∈ V_{0}. Since 11 divides the order of M_{12}, C_{t} acts irreducibly on V_{0}/⟨v⟩. It remains to notice that v is the only vector in W fixed by C_{t} (indeed, already the diagonal group 2^{11} fixes no other vector in W^{#}). The uniseriality now follows.

We can now determine the conjugacy classes of involutions in a group H satisfying conditions (H1)–(H3). Let E = O_{2}(H) and H̄ = H/E. First of all, Lemma 2.1 implies the following.

Lemma 2.7 The group H has exactly three classes of involutions contained in E. If e_{2}, e_{3} and e_{4} are representatives of those classes then C_{H}(e_{2}) ∼ 2^{24}.Co_{2}, C_{H}(e_{3}) ∼ 2^{24}.Co_{3}, and C_{H}(e_{4}) ∼ 2^{24}.(2^{11} : M_{24}).

We will classify the classes of involutions outside E case by case, depending on whether x̄ is of type 2A, 2B or 2C. We start with a general lemma. Let U = C_{E}(x̄) and V = [E, x̄]. Let X = ⟨x, U⟩ and X̃ = X/V. Let C be the full preimage in H of CH̄(x̄).

Lemma 2.8 Suppose x ∈ H is an involution, and x̄ ≠ 1. Then the following hold:

(1) An element y ∈ xE is an involution if and only if y ∈ xU.

(2) The subgroups X and V are invariant under C; furthermore, E acts trivially on X̃.

(3) If y, z ∈ xU then y and z are conjugate in H if and only if ỹ and z̃ are in the same C̄-orbit.

Remark. Part (2) contends that the action of C̄ = CH̄(x̄) on X̃ is well defined, which allows us to view X̃ as a C̄-module. Since U = X ∩ E, Ũ is invariant under C̄, and so C̄ permutes the vectors in X̃ \ Ũ. The meaning of part (3) is that the orbits of C̄ on X̃ \ Ũ bijectively correspond to those conjugacy classes of involutions in H that map onto x̄H̄.

Proof: If e ∈ E then (xe)^{2} = [x, e] since both x and e are involutions. Part (1) follows. Clearly, V is invariant under C. Since X is generated by all the involutions from the coset x̄ = xU, X is C-invariant, too. Clearly, E acts trivially on Ũ. Furthermore, E fixes x̃, because V = [E, x]. This proves (2). For (3), let y, z ∈ xU. If z = y^{h} for some h ∈ H then h̄ ∈ C̄, since ȳ = x̄ = z̄. So ỹ and z̃ are in the same C̄-orbit. Reversely, suppose that ỹc̄ = z̃ for some c ∈ C. Then y^{c} = zv for some v ∈ V. Since V = [E, x] there exists an element e ∈ E such that v = [e, x]. However, [e, x] = [e, z], since z ∈ xU. Therefore, z^{e} = vz = zv, implying that y and z are conjugate.

We will first classify those involutions x for which x̄ is of type 2A. We will need the following fact proved in [Po].

Lemma 2.9 Suppose Y is a GF (2)-module for Ω_{8}^{+}(2) that is an extension of an irreducible 8-dimensional submodule Y_{0} by a 1-dimensional module. Then Y splits.

Lemma 2.10 The group H has exactly three classes of involutions whose images in H̄ are of type 2A. If a_{1}, a_{2} and a_{3} are representatives of these classes then C_{H}(a_{i}) has the structure 2^{16}.2^{1+8}.Ω_{8}^{+}(2), 2^{16}.2^{1+8}.Sp_{6}(2) and 2^{16}.2^{1+8}.(2^{6} : L_{4}(2)), for i = 1, 2 and 3, respectively.

Proof: Let x be an element of H such that x̄ is of type 2A. We will first show that the coset x̄ contains an involution, and so x can be chosen to be an involution. Let R = ⟨x, E⟩. Then R is a normal subgroup of C, where C is defined, as above, as the full preimage in H of CH̄(x̄). Let U = C_{E}(x) and V = [E, x]. Clearly, U = Z(R). Consider X = R/U. According to Lemma 2.4, C has two chief factors within X, of dimensions 8 and 1. This implies that X is an elementary abelian group, which we can view as a module for C̄. Furthermore, by the same Lemma 2.4, the 8-dimension chief factor in X is not a natural module for C̄/O_{2}(C̄) ≅ Ω_{8}^{+}(2). Therefore, O_{2}(C̄) acts trivially on X. Now Lemma 2.9 implies that X contains a 1dimensional subspace T invariant under C. Let R_{0} be the full preimage in R of T. Clearly, R_{0} is normal in C. Next, define X_{0} = R_{0}/V. Again X_{0} is an extension of an 8-dimensional chief factor U/V by a 1-dimensional R_{0}/U. We conclude again that X_{0} is elementary abelian. Clearly, R_{0} acts trivially on X_{0}. Furthermore, by Lemma 2.4 the 8-dimensional chief factor in X_{0} differs, as a module, from the chief factors in O_{2}(C)/R_{0}, which means that O_{2}(C) acts trivially on X_{0}. Applying again Lemma 2.9 we obtain a C-invariant 1-dimensional subspace T_{0} in X_{0}. Let R_{1} be the full preimage of T_{0} in R. Setting X_{1} = R_{1}, we observe for the third time that X_{1} is an extension of an 8-dimensional chief factor V by a 1-dimensional one, R_{1}/V. Hence R_{1} is elementary abelian. Since R_{1} ≰ E, we finally conclude that the coset x̄ contains some involutions. Without loss of generality we can now assume that x is itself an involution and so Lemma 2.8 applies. In the notation introduced before Lemma 2.8, X̃ (which has already appeared above as X_{0} = R_{0}/V) is the direct sum of a halfspin module and a 1dimensional module. Therefore, C has three orbits on X̃ \ Ũ, of sizes 1, 120 and 135, and this immediately leads to the conclusion as in the lemma.

The classification of involutions x with x̄ of type 2B or 2C is an easy corollary of Lemmas 2.5 and 2.6.

Lemma 2.11 For L = B or C, H has a unique conjugacy class of involutions whose images in H̄ are of type 2L. If b and c are representatives of those two classes then C_{H}(b) ∼ 2^{12}.(2^{2} × G_{2}(4)).2 and C_{H}(c) ∼ 2^{12}.(2^{11} : Aut M_{12}).

Proof: Let x be an element of H such that x̄ is of type 2L. Then according to Lemmas 2.5 and 2.6, we have that C_{E}(x) = [E, x]. In particular, x^{2} = [e, x] for some e ∈ E. It follows that (xe)^{2} = x^{2}[x, e] = 1, that is, the coset xE contains involutions. Furthermore, in the notation of Lemma 2.8 we have that X̃ is 1-dimensional, and so the claim follows.

This completes the classification of conjugacy classes of involutions in H. According to Lemmas 2.7, 2.10 and 2.11, the group H contains eight classes of involutions. We will refer to these classes as to the classes 2e_{i}, 2 ≤ i ≤ 4, 2a_{i}, 1 ≤ i ≤ 3, 2b and 2c.

3 A fusion lemma and an application

In the first part of this section G is an arbitrary group having a large extraspecial subgroup. This means that for some involution z ∈ G the centralizer C = C_{G}(z) contains a normal extraspecial 2-subgroup Q and, furthermore, C_{G}(Q) ≤ Q. This implies that G contains a unique class of 2-central involutions (recall that a 2-central involution is an involution in the center of some Sylow 2-subgroup of G) and that z is itself 2-central. We let S denote the class of 2-central involutions in G. For x = z^{g} ∈ S we denote C_{x} = C_{G}(x) = C^{g} and Q_{x} = Q^{g}. Thus, C = C_{z} and Q = Q_{z}.

We will assume throughout this section that

(∗) S ∩ C ≠ {z}.

Indeed the principal case of interest for us is where G is simple. However in that case the Z^{∗}-theorem of Glauberman makes S ∩ C = {z} impossible. In this section we prove that, modulo some small configurations, (∗) implies the following stronger condition:

(∗∗) S ∩ Q ≠ {z}.

Let, as above, Q = Q_{z} and let Q̄ denote Q/Z where Z = ⟨z⟩.

Lemma 3.1 Suppose S ∩ Q = {z} and let x ∈ S ∩ C, x ≠ z. Denote E = Q ∩ Q_{x}. Then one of the following holds:

(1) E = 1; or

(2) |E| = 2 and either

(a) Ē ≰ [Q̄, x], or

(b) z ≠ [x, y] for all y ∈ Q; or

(3) |E| = 4 and furthermore, for W = ⟨E, z, x⟩,

(a) N_{G}(W) induces on W ≅ 2^{4} either O_{4}^{−}(2), or Ω_{4}^{−}(2) acting as on the natural module;

(b) |W ∩ S| = 5 and, under the identification of W with the orthogonal module, the involutions in W ∩ S are the singular vertors; moreover, for each w ∈ W ∩ S, W ∩ Q_{w} is the perp of w.

Proof: Suppose that E ≠ 1. (Otherwise, (1) holds.) If e ∈ E then e^{2} ∈ Z ∩ ⟨x⟩ = 1, since E is contained in both Q and Q_{x}. Hence E is elementary abelian. Let U and V be defined as the full preimages in Q of CQ̄(x) and [Q̄, x], respectively. Observe that since Q is extraspecial we have that V = C_{Q}(U) = Z(U) and that C_{Q}(x) is either equal to U or [U : C_{Q}(x)] = 2. In the latter case, [x, y] = z for all y ∈ U \ C_{Q}(x).

Notice that [C_{Q}(x), E] ≤ [Q, Q] ∩ [C_{Q}(x), Q_{x}] ≤ Z ∩ Q_{x}. By assumption, S ∩ Q = {z} and hence S ∩ Q_{x} = {x}. We conclude that z ∉ Q_{x} and hence [C_{Q}(x), E] = 1.

Let e ∈ E^{#} and suppose e = [x, y] for some y ∈ Q. Then ex = x^{y} is a conjugate of x contained in Q_{x}, i.e., S ∩Q_{x} ≠ {x}. This contradiction shows that no nontrivial element from E is an elementary commutator [x, y], for y ∈ Q.

If C_{Q}(x) ≠ U then [U : C_{Q}(x)] = 2, which implies that [C_{Q}(C_{Q}(x)) : V] = 2. Hence [E : E ∩ V] ≤ 2. If E ∩ V = 1 then |E| = 2, implying (2a). So let us assume that E ∩ V ≠ 1 and let e ∈ (E ∩ V)^{#}. Since e ∈ V, we have that either e = [x, y] or ez = [x, y] for some y ∈ Q. However, by the preceding paragraph e ≠ [x, y]. So ez = [x, y]. Furthermore, z = [x, t] for t ∈ U \ C_{Q}(x). Thus, [x, ty] = [x, t]^{y}[x, y] = zez = e; a contradiction.

Now assume that C_{Q}(x) = U and so z ≠ [x, y] for all y ∈ Q. Also, E ≤ V, since [C_{Q}(x), E] = 1. If |E| = 2 then we obtain (2b). Hence we may assume that |E| ≥ 4. Set W = ⟨E, z, x⟩. Our next step is to determine which involutions from W are in S. First of all, the involutions in W fall into the following types: z, x, e, ze, xe, zx, and zxe, where e denotes an arbitrary involution from E^{#}. Clearly, z, x ∈ S. By assumption, no other involution in Q∪Q_{x} is in S. Hence the involutions e, ze, and xe are not in S. Since E ≤ V, we have that ē = [ȳ, x] for some y ∈ Q. We have shown above that e ≠ [x, y]. Hence ze = [x, y]. It follows that zxe = xze = x[x, y] = x^{y}. Thus, all elements zxe are in S. The element zx may or may not be in S.

Observe also that the element y above normalizes W. Indeed, ⟨E, z⟩ is normal in Q, so y leaves it invariant. Also x^{y} = zxe ∈ W. Thus, W^{y} = W. We conclude that x and all the elements zxe are conjugate under N_{G}(W). Symmetrically, z is conjugate under N_{G}(W) to the elements zxe and so also to x. Notice that W_{x} = ⟨E, x⟩ is an index two subgroup of W such that |W_{x} ∩ S| = {x}. Pick an element e ∈ E^{#}. By transitivity, there exists a subgroup W_{zxe} of index 2 in E such that W_{zxe} ∩ S = {zxe}. Since z and x are not in W_{zxe}, we have that zx ∈ W_{zxe}. Thus, zx ∉ S, which completes the enumeration of the elements in W ∩ S.

If |E| > 4 then for every e′ ∈ E^{#} there exist elements e_{1} and e_{2} = e_{1}e′ such that e_{1} ≠ e ≠ e_{2}. Since both zxe_{1} and zxe_{2} are not in W_{zxe}, we conclude that e′ = (zxe_{1})(zxe_{2}) is in W_{zxe}, i.e., E ≤ W_{zxe}. However, in that case all elements zxe′, e′ ∈ E^{#}, are in W_{zxe} ∩ S, a contradiction. This establishes that |E| = 4 and, consequently, |W ∩ S| = 5. Observe that the elements y from the preceeding paragraph stabilize z. This proves that N_{G}(W) induces a 2-transitive group on W ∩ S. Also y^{2} ∈ Z for all those elements y, which rules out the Frobenius group F_{5}^{4}. Hence N_{G}(W) induces on W ∩ S one of the groups S_{5} ≅ O_{4}^{−}(2) or A_{5} ≅ Ω_{4}^{−}(2) and the claim (2) follows.

In the remainder of this section G = M, the Monster, z is a 2-central involution in G, Z = ⟨z⟩, C = C_{z} = C_{G}(z) and Q = Q_{z} = O_{2}(C_{z}). Since by assumption M is simple, Glauberman’s Z^{∗} theorem [Gl] shows that S ∩C ≠ {z}. Recall that S denotes the conjugacy class of 2-central involutions, z^{M}. Recall alsoy that for x = z^{g} we set C_{x} = C^{g} and Q_{x} = Q^{g}. Since M has a large extraspecial subgroup, Lemma 3.1 applies to it. We use that lemma to prove the following.

Proposition 3.2 Q ∩ S ≠ {z}.

Proof: Suppose Q∩S = {z}. Since C ∩S ≠ {z}, we can choose x ∈ C ∩S, x ≠ z. According to Lemma 3.1, one of the exceptional cases (2a), (2b), or (3) must hold. In particular, E = Q ∩ Q_{x} has size at most four.

Observe now that the group H = C̄ = C/Z satisfies the conditions (H1)–(H3) from Section 2. In particular, we can use the classification of conjugacy classes of involutions obtained in that section. Let D = C ∩ C_{x} and R = Q_{x} ∩ C. Clearly, R is normal in D, and D̄ is of index two or one in CC̄(x̄) depending on whether or not x and xz are conjugate in C.

Suppose first that x̄ is in the class 2a_{i} for some i. Then also z⟨x⟩ is in 2a_{i} in C_{x}/⟨x⟩. In particular, R is of order at least 2^{16} (cf., Lemma 2.4). Consider C̃ = C/Q. Since E = Q ∩ R is of order at most four, we obtain that D̃ contains a normal 2-subgroup R̃ of order at least 2^{14}. Comparing with Lemma 2.10, we see that i = 3 must hold. However, i = 3 also leads to a contradiction. Indeed, let Ỹ be the normal extraspecial subgroup 2^{1+8} of CC̃(x̃) ∼ 2^{1+8}.Ω_{8}^{+}(2). We have that [Ỹ : Ỹ ∩ R̃] ≤ 2 and [R̃, R̃] ≤ ⟨x̃⟩ = Z(Ỹ). Therefore, all elements of R̃ centralize a hyperplane in the 8-dimensional quotient of Ỹ, which is impossible.

Suppose next that x̄ is in 2b. Then also z⟨x⟩ is in the class 2b in C_{x}/⟨x⟩. Hence |R| ≥ 2^{12}. Considering again C̃ = C/Q and taking into account that |E| ≤ 4, we see that D̃ contains a normal 2-group of size at least 2^{10}, clearly contradicting Lemma 2.11.

Finally, suppose x̄ is in the class 2c. In this case our argument must be slightly more subtle. Let U be the full preimage in Q of Ū = CQ̄(x̄). Then U is a subgroup of order 2^{13}. We claim that C_{Q}(x) is a proper subgroup of U. Indeed, according to Lemma 2.4, Ū contains elements from the class 2e_{3}. Observe that the mapping ē ↦ e^{2} defines a nondegenerate quadratic form g on Q̄. Since, as a module for C/Q ≅ Co_{1}, Q̄ is absolutely irreducible, this quadratic form is unique, and hence g is equivalent to the form q (cf. Section 2). In particular, if ē is in 2e_{3} then e is of order four. Since Ū = [Q̄, x̄], we have that ē = [q̄, x̄] for some q ∈ Q. Therefore, [q, x] = e or e^{3}. Since, clearly, q can be chosen to be an involution, we obtain that x inverts e, i.e., e ∉ C_{Q}(x).

This has two consequences. First, C_{Q}(x) is of size 2^{12}, and symmetrically, also |R| = 2^{12}. (Clearly, z⟨x⟩ must also be in the class 2c in C_{x}/⟨x⟩.) Secondly, we record for further use that z = [q, x] for some q ∈ Q.

Since |E| ≤ 4, we have that R̃ (where, as above, C̃ = C/Q) has size 2^{10}, 2^{11}, or 2^{12}. Comparing with Lemma 2.11 and using that R̃ is normal in D̃, we obtain that |R̃| = 2^{11}, and hence |E| = 2. This means that either (2a) or (2b) of Lemma 3.1 must hold. Above we recorded that z = [q, x] for some q ∈ Q. Hence, in fact, it must be the case (2a). To obtain a contradiction in this last case, it remains to see that Ē ≤ [Q̄, x]. However, this is clear because Ē ≤ CQ̄(x̄) = [Q̄, x̄] = [Q̄, x].

4 Singular subgroups

First, let G be again a group with a large extraspecial subgroup, that is, let there be an involution z ∈ G and an extraspecial 2-subgroup Q normal in C = C_{G}(z) such that C_{G}(Q) ≤ Q. Adopt the notation from Section 3, that is, let S = {z^{G}} be the class of 2-central involutions in G and, for x = z^{g} ∈ S, let C_{x} = C_{G}(x) = C^{g} and Q_{x} = Q^{g}.

Let x and y be two 2-central involutions. We will say that x perpendicular to y if and only if y ∈ Q_{x}. The following important lemma is a slight improvement on [As0], Lemma 8.7 (3).

Lemma 4.1 The perpendicularity relation is symmetric.

Proof: If |Q| > 2^{3} then this is proven in [As0], Lemma 8.7 (3). So suppose Q ∼ 2^{1+2}. Suppose that the relation is not symmetric so that for some x ∈ S we have that x ∈ Q, but z ∉ Q_{x}. In particular, there is no g ∈ G such that z^{g} = x and x^{g} = z. Since x ∈ Q, Q must be isomorphic to D_{8}. Since C_{G}(Q_{x}) = ⟨x⟩ and since Out D_{8} is of order two, we have that C_{x} = Q_{x}⟨z⟩ ≅ D_{16}. It remains to notice that an element from N_{C_{x}}(U), where U = ⟨z, x⟩, permutes z and zx and, likewise, an element from N_{C}(U) permutes x and zx. Hence the normalizer of U induces on it the full group S_{3}. Thus, there exists an element g ∈ G such that z^{g} = x and x^{g} = z; a contradiction.

Let U be a purely 2-central (i.e., all involutions in U are in S) elementary abelian 2-subgroup of G. We will say that U is singular if U ≤ Q_{u} for every u ∈ U^{#}. If U is singular define Q_{U} = ∩_{u∈U^{#}} Q_{u} and L_{U} = ⟨Q_{u} | u ∈ U^{#}⟩. Clearly, U ≤ Q_{U} ≤ L_{U}.

Lemma 4.2 Let U be singular. Then the following hold:

(1) U and Q_{U} are normal in L_{U} and L_{U} acts trivially on Q_{U}/U;

(2) if W ≤ Q_{U} and W ∩ U = 1 then C_{L_{U}} (W) induces on U the full group L_{n}(2) (where n is the rank of U); in particular (for W = 1), N_{G}(U) = L_{U}C_{G}(U);

(3) if W_{1}, W_{2} ≤ Q_{U}, W_{1} ∩ U = W_{2} ∩ U = 1 and W_{1}U = W_{2}U then there is an element x ∈ C_{L_{U}} (U) such that W_{1}^{x} = W_{2}; in particular, C_{L_{U}} (U) acts transitively on every coset qU, q ∈ Q \ U;

(4) if |U| > 2 then Q_{U} is elementary abelian.

Proof: Let U ≤ U′ ≤ Q_{U}. If u ∈ U^{#} then U′ ≤ Q_{u}. Since Q_{u} is extraspecial, U′ is normal in Q_{u} and hence U′ is normal in L_{U}. This proves (1).

For W as in (2), take U′ = W U. Clearly, U′ is elementary abelian and so we can view it as a GF (2)-vector space. Notice that Q_{u} induces on U′ all transvections with center ⟨u⟩. Since W ∩ U = 1, C_{Q_{u}}(W) induces on U all transvections with center ⟨u⟩. This implies (2), since the group generated by all transvections of U is L_{n}(2).

Let W_{1} and W_{2} be as in (3). We will use induction on the rank of W_{1}. If W_{1} = 1 then there is nothing to prove. Otherwise, choose W′_{1} ≤ W_{1} such that |W_{1}/W′_{1}| = 2, and let W′_{2} = W_{2} ∩ W′_{1}U. By induction, there is an element x′ ∈ C_{L_{U}} (U) such that (W′_{1})^{x} = W′_{2}. Let w_{1} ∈ W_{1} \ W′_{1} and let {w_{2}} = W_{2} ∩ w_{1}U. Notice that, by (1), w_{1}^{x} ∈ w_{1}U = w_{2}U and hence w_{1}^{x} = w_{2}u for some u ∈ U. If u = 1 then take x = x′. Otherwise, let y be an element of Q_{u} that induces on W_{1}U the transvection with center ⟨u⟩ and axis W′_{2}U. Clearly, y ∈ C_{L_{U}} (U) and W_{1}^{x} = W_{2}, where x = x′y. This proves (3).

Finally, if q ∈ Q_{U} then q^{2} ∈ Φ(Q_{u}) = ⟨u⟩ for every u ∈ U^{#}. This proves (4).

Lemma 4.3 A subgroup U is singular if and only if it is generated by a set of pairwise perpendicular 2-central involutions. Furthermore, if U = ⟨u_{1}, …, u_{k}⟩ (u_{i} ≠ 1 for all i) is singular then Q_{U} = ∩_{i=1}^{k} Q_{u_{i}}.

Proof: We only need to prove the ‘if’ part of the first claim. Suppose U = ⟨u_{1}, …, u_{k}⟩, where u_{1}, …, u_{k} are 2-central and pairwise perpendicular. By induction, U′ = ⟨u_{2}, …, u_{k}⟩ is singular. Since u_{1} is perpendicular to u_{2}, …, u_{k}, we have that U′ ≤ Q_{u_{1}} which by Lemma 4.1 implies that u_{1} ∈ Q_{U′}. By Lemma 4.2 (2), all involutions in U \ U′ = u_{1}U′ are 2-central, since u_{1} is 2-central. Finally, let u ∈ U^{#}. Then u ∈ U ≤ Q_{u_{i}} for every i, since the involutions u_{i} are pairwise perpendicular. By Lemma 4.1, U = ⟨u_{1}, …, u_{k}⟩ ≤ Q_{u}.

Suppose now that U = ⟨u_{1}, …, u_{k}⟩ is singular. Clearly, Q_{0} = ∩_{i=1}^{k} Q_{u_{i}} contains Q_{U}. So it remains to see that Q_{0} ≤ Q_{U}. Let us use induction on k. The claim is obviously true if k = 1. Consider now the case k > 1 and set U′ = ⟨u_{2}, …, u_{k}⟩. By induction, Q_{U′} = ∩_{i=2}^{k} Q_{u_{i}} and hence Q_{0} ≤ Q_{u} for all u ∈ U^{#}. However, this means that Q_{0} is normal in Q_{u}. In particular, Q_{0} is invariant under an element x ∈ Q_{u} which induces on U a transvection taking u_{1} to u_{1}u. Hence Q_{0} = Q_{0}^{x} ≤ Q_{u_{1}}^{x} = Q_{u_{1}u}. Thus Q_{0} ≤ Q_{u} for all u ∈ U^{#}.

We now switch back to the case G = M. Our goal is to classify all singular subgroups in M up to conjugation. Notice that Proposition 3.2 means that the perpendicularity relation on 2-central involutions in M is nontrivial, that is, there exist singular subgroups of size more than two. We start by getting the details of the perpendicularity relation in M. For that we need to know the fusion of involutions in Q. Let C̄ = C/Z, where Z = ⟨z⟩. Recall that the classes of involutions in C̄ were determined in Section 2.

Lemma 4.4 The group C has exactly two classes of involutions x ≠ z, contained in Q. If q_{2} and q_{4} are representatives of those classes then C_{C}(q_{2}) ∼ 2^{1+23}.Co_{2} and C_{C}(q_{4}) ∼ 2^{1+23}.(2^{11} : M_{24}). Furthermore, q_{4} is 2-central and q_{2} is not.

Proof: For x ∈ Q \ Z, the mapping x̄ ↦ x^{2} defines a nondegenerate quadratic form g on Q̄. Since the action of C/Q ≅ Co_{1} on Q̄ is absolutely irreducible, g is unique and hence g is equivalent to the form q existing on Λ̂ = Λ/2Λ (cf. Section 2). The form q is zero on Λ̂_{2} and Λ̂_{4}, and it is nonzero on Λ̂_{3}. This means that x is an involution if and only if x̄ belongs to the class 2e_{2} or 2e_{4}. The involutions x and xz are conjugate in Q, because Q is extraspecial. Combined with Lemma 2.7, this establishes the first two claims of the lemma.

According to Proposition 3.2, at least one of q_{2} and q_{4} is conjugate to z in G. So, to complete the proof of the lemma, it suffices to show that x = q_{2} is not 2-central. Suppose that x ∈ S. Let D = C_{C}(x) = C ∩ C_{x} and R = Q ∩ Q_{x}. From the structure of D (see above), it is clear that R = O_{2}(D) ∼ 2^{1+23}. Since R ≤ Q, we have that [R, R] = Z. Symmetrically, since R ≤ Q_{x} we have that [R, R] = ⟨x⟩, implying that z = x, a contradiction.

In particular, if a 2-central involution y ≠ z is perpendicular to z then y is conjugate in C to x = q_{4}. This lemma implies that every singular subgroup U ∼ 2^{2} in M is conjugate to ⟨z, q_{4}⟩. So there is only one conjugacy class of such subgroups.

Lemma 4.5 Let U ∼ 2^{2} be singular. Then W = Q_{U}/U ∼ 2^{11} and C_{M}(U) induces on W a group M_{24} acting as on the Todd module. Under the identification of W with the Todd module, the images of 2-central involutions from Q_{U} \ U correspond to sextets, while the images of non 2-central involutions correspond to pairs.

Proof: Without loss of generality, U = ⟨z, x⟩, where x = q_{4}. Let D = C ∩ C_{x} = C_{M}(U) and R = Q ∩ Q_{x}. Notice that by Lemma 4.3 we have R = Q_{U}. Recall that Q̄ = Q/Z affords a quadratic form g defined by ȳ ↦ y^{2}. By Lemma 4.2 (4), Q_{U} is elementary abelian. In particular, R̄ is a totally singular subspace with respect to g. This implies that |R̄| ≤ 2^{12}, and hence |R| ≤ 2^{13}. On the other hand, both C ∩ Q_{x} and C_{x} ∩ Q have order 2^{24} and they are normal in D. Since (C ∩ Q_{x}) ∩ (C_{x} ∩ Q) = R, the order of (C ∩ Q_{x})(C_{x} ∩ Q) is at least 2^{24+24−13} = 2^{35} = |O_{2}(D)| (see Lemma 4.4 for the structure of D). Hence |R| = 2^{13}.

It follows that R̄ is a 12-dimensional subspace in Q̄ invariant under the monomial group D/Q ∼ 2^{11} : M_{24}. Such a subspace is known to be unique. Identifying Q̄ with Λ̂ and assuming that x̄ is the image of the standard frame, we get that R̄^{#} consists of the images of the vectors of the shape ±8^{1}0^{23} (x̄), ±4^{2}0^{22} (Λ̂_{2}, non 2-central), and ±4^{4}0^{20} (Λ̂_{4}, 2-central). Each pair of coordinates gives four vectors of the second kind, mapping onto two elements in R̄ ∩ Λ̂_{2}. These two elements of R̄ sum up to x̄. Similarly, every sextet produces 96 vectors of the third kind (two frames), mapping onto two elements in R̄ ∩ Λ̂_{4}. These two elements of R̄ again sum up to x̄. Thus, in R/U ≅ R̄/⟨x̄⟩, the nonidentity elements correspond simply to pairs and sextets. By Lemma 4.4, the elements from R/U^{#} corresponding to pairs (respectively, sextets) are the images of non 2-central (respectively, 2-central) involutions from R \ U.

For the record, the normalizer of a singular subgroup U ∼ 2^{2} is now known to be an extension of a normal 2-subgroup of order 2^{35} by S_{3} × M_{24}. (The latter being the action of N_{M}(U) on U × Q_{U}/U.)

In the above proof, if we do not assume that x̄ is the image of the standard frame then the condition for ȳ to be in R̄ looks as follows: Let {v_{i}} be the frame corresponding to x̄ (i.e., v̂_{i} = x̄ and v_{i} ∈ Λ_{4} for all i) and let u be a short vector in Λ (i.e., a vector from Λ_{2} ∪ Λ_{3} ∪ Λ_{4}) such that v̄ = ŷ. Then ȳ ∈ R̄ if and only if (v_{i}, y) ∈ {0, ±4, ±8} for all i. When {v_{i}} is the standard frame, this corresponds to the statement in the above proof about the shapes of the vectors mapping into R̄.

Combining Lemma 4.2 (3) with the fact that M_{24} acts transitively on pairs and on sextets, we obtain the following.

Corollary 4.6 If U ≅ 2^{2} is singular then C_{M}(U) has exactly two conjugacy classes in Q_{U} \U, one consisting of non 2-central involutions, and one other consisting of 2-central involutions.

Since Q_{U} \ U contains a unique class of 2-central involutions, M has exactly one conjugacy class of singular subgroups 2^{3}.

Before we proceed further we need to understand better the perpendicularity relation among the elements in Q_{U} \U, where U is a singular subgroup 2^{2}. For a non 2-central (respectively, 2-central) involution y ∈ Q_{U} \ U, let P (y) (respectively, S(y)) be the pair (respectively, sextet) corresponding to yU ∈ Q_{U}/U.

We say that two sextets S_{1} and S_{2} intersect evenly if |T_{1} ∩ T_{2}| is even for all tetrads T_{1} ∈ S_{1} and T_{2} ∈ S_{2}. Suppose T_{1} and T_{2} are two tetrads and suppose |T_{1} ∩ T_{2}| = 2. Then the sextets defined by T_{1} and T_{2} intersect evenly if and only if T_{1} ∪ T_{2} is contained in an octad. This allows us to compute that every sextet evenly intersects exactly 90 other sextets.

Lemma 4.7 Let U ∼ 2^{2} be singular. Suppose y, t ∈ Q_{U} \ U, and suppose y is 2-central. Then

(1) if t is non 2-central then t ∈ Q_{y} if and only if P (t) is contained in one of the tetrads from S(y); and (2) if t is 2-central then t ∈ Q_{y} if and only if S(t) and S(y) intersect

evenly.

Proof: Assume again that U = ⟨z, x⟩, where x = q_{4}. Let {v_{i}} be the frame in Λ that corresponds to ȳ and let u be a short vector in Λ such that û corresponds to t̄. Then the vectors v_{i} are of the shape ±4^{4}0^{20}, where the nonzero coordinates appear in a tetrad from the sextet S(y). Similarly, u is of shape ±4^{2}0^{22} (respectively, ±4^{4}0^{20}) with the nonzero coordinates appearing in the pair P (t) (respectively, sextet S(t)) if t is non 2-central (respectively, 2-central). According to the remark after the proof of Lemma 4.5 we have t ∈ Q_{y} if and only if (v_{i}, u) ∈ {0, ±4, ±8} for all i. The claim of the lemma follows.

One implication of Corollary 4.6 is that M contains exactly one conjugacy class of singular subgroups 2^{3}. Indeed, pick a 2-central involution y ∈ Q_{⟨z,x⟩} \ ⟨z, x⟩. Then every singular subgroup 2^{3} is conjugate to ⟨z, x, y⟩.

Lemma 4.8 Let U ∼ 2^{3} be singular. Then W = Q_{U}/U ∼ 2^{6} and C_{M}(U) induces on W a group 3 · S_{6} that acts on W irreducibly. Furthermore, N_{M}(U) has two orbits on W^{#}: an orbit of length 18 (images of non 2central involutions from Q_{U} \ U) and an orbit of length 45 (images of 2central involutions).

Proof: Without loss of generality, U = ⟨z, x, y⟩. We set U_{0} = ⟨z, x⟩ and V = Q̃_{U_{0}}/U_{0}. According to Lemma 4.5, C_{M}(U_{0}) induces on V a group M_{24} acting on V as on the Todd module. Let S = S(y) be the sextet corresponding to ỹ under the identification of V with the Todd module. According to Lemma 4.7, Q̃_{U}^{#} consists of elements corresponding to pairs contained in the tetrads of S and to sextets evenly intersecting S. By counting, Q̃_{U}^{#} consists of 91 sextets (including S) and 36 pairs. Hence |Q̃_{U}| = 2^{7}. This means that |Q_{U}/U| = 2^{6}. Furthermore, (Q_{U}/U)^{#} contains 45 (respectively, 18) elements that are images of 2-central (respectively, non 2-central) involutions.

Recall that C_{M}(U_{0}) induces on V a group M_{24}. The stabilizer of S in the latter group is a subgroup 2^{6} : 3 · S_{6}. Let D = N_{M}(U) ∩ C_{M}(U_{0}) be the full preimage in C_{M}(U_{0}) of the stabilizer of S. According to Lemma 4.2 (2), N_{M}(U) ∩ N_{M}(U_{0}) induces on U \ U_{0} a group S_{4}. Since D is normal in N_{M}(U) ∩ N_{M}(U_{0}) and since C_{M}(U) is the kernel of the action of D on U \U_{0}, we conclude that C_{M}(U) induces the whole sextet stabilizer 2^{6} : 3·S_{6} in its action on V. Thus, it induces a quotient of 2^{6} : 3 · S_{6} on Q_{U}/U. Let a ∈ C_{M}(U) be a 3-element mapping into the normal 3-subgroup of the quotient 3 · S_{6}. Consider the action of a on V. Clearly, a stabilizes every tetrad in the sextet S. Let S′ be a sextet evenly intersecting S. Observe that every tetrad from S meets exactly two tetrads from S′. Being a 3element, if a stabilizes S′ then it must stabilize it tetradwise. However, in that case a stabilizes every part of a partition of {1, …, 24} into 12 pairs (intersections of tetrads from S with tetrads from S′), which makes a to act on {1, …, 24} trivially. This contradiction shows that a cannot stabilize S′ and hence a acts nontrivially on Q_{U}/U. Therefore, C_{M}(U) induces on Q_{U}/U either 2^{6} : 3 · S_{6} or 3 · S_{6}.

It is easy to see that the stabilizer of S in M_{24} acts transitively on pairs contained in tetrads from S and on sextets evenly intersecting S. Consequently, C_{M}(U) has orbits of size 18 and 45 on Q_{U}/U. This makes the action on Q_{U}/U irreducible, implying that the group induced by C_{M}(U) is in fact 3 · S_{6}.

For the record, this lemma and Lemma 4.2 (2) imply that N_{M}(U), where U is a singular subgroup 2^{3}, is an extension of a normal subgroup of order 2^{39} by L_{3}(2) × 3 · S_{6}.

Also, let us record what we proved about the classes of 2-central and non 2-central involutions in Q_{U}.

Corollary 4.9 If U ≅ 2^{3} is singular then C_{M}(U) has exactly two conjugacy classes in Q_{U} \U, one consisting of non 2-central involutions, and one other consisting of 2-central involutions.

Proof: Follows from Lemma 4.2 (3).

In particular, M contains a unique conjugacy class of singular subgroups 2^{4}.

Lemma 4.10 Let U ∼ 2^{4} be singular. Then W = Q_{U}/U ∼ 2^{3}. Furthermore, W^{#} contains exactly three elements that are images of non 2-central involutions, and these three elements generate W. The group C_{M}(U) induces on W a group S_{3}.

Proof: Without loss of generality, U ≥ U_{0} = ⟨z, x⟩, say, U = ⟨z, x, y, t⟩. We will work with the Todd module V = Q̃_{U_{0}} = Q_{U_{0}}/U_{0}. Let S = S(y) and S′ = S(t). If s ∈ Q_{U} \ U is a non 2-central involution then P (s) is contained in a tetrad from S and in a tetrad from S′. Hence T (s) must be one the twelve pairs P_{1}, …, P_{12} (partitioning {1, …, 24}) that are intersections of tetrads from S with tetrads from S′. This proves that Q_{U}/U contains exactly three involutions that are images of non 2-central involutions. (Indeed, the twelve involutions in Q̃_{U} merge into three involutions in Q_{U}/U.)

Let us be more specific. Since S and S′ intersect evenly, there is a unique trio T := {O_{1}, O_{2}, O_{3}⟩ of which both S and S′ are refinements (we view trios and sextets as partitions of {1, …, 24}). Then every O_{i} is a union of some four pairs P_{j}. It is easy to see that pairs P_{j} and P_{k} produce the same element in Q_{U}/U if and only if they are contained in the same octad O_{i}. Thus, the octads O_{i} correspond to the “non 2-central” elements a_{i} ∈ Q_{U}/U. Clearly, the stabilizer of S and S′ in M_{24} induces an S_{3} on the trio T. Hence also N_{M}(U) induces an S_{3} on the three involutions a_{i}. Furthermore, since N_{M}(U) induces a simple group L_{4}(2) on U, we also have that C_{M}(U) induces an S_{3} on the a_{i}’s. It remains to see that they are linearly independent and generate Q_{U}/U.

Observe that if P_{j} and P_{k} belong to distinct octads O_{i} then the sum (we switch to the additive notation in V and Q_{U}/U) of the elements from V corresponding to P_{j} and P_{k} is of sextet type and, furthermore, that sextet is not a refinement of T. This means that the sum of two distinct involutions a_{i} is nontrivial and “2-central”. This implies the linear independence. Let b be an arbitrary “2-central” element from (Q_{U}/U)^{#}, say, it is the image of an element of V that corresponds to a sextet S′′ = {R_{1}, …, R_{6}}. Observe that S′′ evenly intersects both S and S′. In particular, |O_{i} ∩ R_{j}| is even for all i and j. Suppose for some i and j we have |O_{i} ∩ R_{j}| = 2. (We will say that such an S′′ is of the first kind.) Observe that O_{i} is a union of some four pairs P_{k}. If R_{j} meets two of these pairs then R_{j} meets a tetrad from S or from S′ in just one point, a contradiction. Hence, O_{i} ∩ R_{j} coincides with some P_{k}. Similarly, considering a nontrivial intersection of R_{j} with some other O_{i′} we obtain that R_{j} contains a second pair P_{k′} and hence b is the sum of two of the a_{i}’s. It remains to consider the case where |O_{i} ∩ R_{j}| ∈ {0, 4} for all i and j, that is, every R_{j} is fully contained in some O_{i}. (Then we will say that S′′ is of the second kind.) Fix O_{i} and R_{j} with R_{j} ⊂ O_{j}. If P_{k} ⊂ R_{j} then b + a_{i} is “non 2-central”, which means that R_{j} \ P_{k} = P_{k′}. However, since P_{k} and P_{k′} are both in O_{i}, we get b = a_{i} + a_{i} = 0, a contradiction. Therefore, R_{i} meets each of the four pairs P_{k} partitioning O_{i} in one point. Fix P_{k} ⊂ O_{i} and consider c = b + a_{i}. Then one of the preimages of c in V will correspond to the sextet S′′′ containing the tetrad R_{j}△P_{k} (△ denotes symmetric difference of sets). If S′′′ is of the second kind then S′′′ contains a tetrad contained in O_{i′} ≠ O_{i}. That tetrad of S′′′ will meet some tetrad of S′′ in at least two points. This gives us two octads meeting in five points, a contradiction. Therefore, S′′′ is of first kind. By the above, c is in the span of a_{i}’s and hence so is also b.

We will continue using the notation a_{i} for the three “non 2-central” elements from Q_{U}/U. According to Lemma 4.10, C_{M}(U) has three orbits on (Q_{U}/U)^{#}: {a_{1}, a_{2}, a_{3}} (“non 2-central”), {a_{1} + a_{2}, a_{1} + a_{3}, a_{2} + a_{3}} (“2central”, sextets of the first kind), and {a_{1} + a_{2} + a_{3}} (“2-central”, sextets of the second kind).

We record this as the following

Corollary 4.11 If U ≅ 2^{4} is singular then C_{M}(U) has exactly three conjugacy classes in Q_{U} \ U, two consisting of 2-central involutions, and one other consisting of non 2-central involutions.

For the record, the normalizer of a singular subgroup U ∼ 2^{4} is an extension of a normal subgroup of order 2^{39} by L_{4}(2) × S_{3}.

It follows from Corollary 4.11 that M contains two conjugacy classes of singular subgroups 2^{5}. One of these two classes is represented by ⟨z, x, y, t, s⟩ with the image of s in Q_{⟨z,x,y,t⟩}/⟨z, x, y, t⟩ being a_{1} + a_{2}, while for the other the image of s can be chosen as a_{1} + a_{2} + a_{3}. We will write “a singular subgroup 2_{1}^{5}” (respectively, 2_{2}^{5}) for the two types of singular subgroups 2^{5}.

We will need the following corollary of Lemma 4.11.

Corollary 4.12 Every singular subgroup 2^{4} is contained in exactly three singular subgroups 2_{1}^{5} and a unique singular subgroup 2_{2}^{5}.

To complete the classification of singular subgroups of M we need to discuss perpendicularity between the elements of Q_{U} \ U.

Lemma 4.13 Suppose U ∼ 2^{4} is singular. Let s and r be two elements from Q_{U} \ U, whose images in Q_{U}/U are distinct. If s is 2-central and r ∈ Q_{s} then the image of s is a_{i} + a_{j} for some i and j. Furthermore, the image of r is either a_{i} or a_{j}.

Proof: Without loss of generality, U = ⟨z, x, y, t⟩ as in lemma 4.10. Since r ∈ Q_{s}, we have that r and rs are both 2-central or both non 2-central. This implies that the image of s cannot be a_{1} + a_{2} + a_{3}. Hence the image of s coincides with some a_{i} + a_{j}. Next, it is easy to see that if r′ maps onto a_{i} or a_{j} then r′ ∈ Q_{s}. Since no element mapping onto a_{1} + a_{2} + a_{3} can be in Q_{s}, we conclude that the image of Q_{s} ∩ Q_{U} in Q_{U}/U coincides with ⟨a_{i}, a_{j}⟩.

The information in Lemma 4.13 allows us to determine Q_{U} for singular subgroups U ∼ 2^{5}.

Lemma 4.14 The following hold.

(1) If U is singular 2_{1}^{5} then Q_{U}/U is of order two. Furthermore, all involutions in Q_{U} \ U are non 2-central.

(2) If U is singular 2_{2}^{5} then Q_{U} = U.

Proof: Follows from Lemma 4.13.

For the record, the normalizer of a singular 2_{1}^{5} is an extension of a subgroup of order 2^{36} by L_{5}(2), while the normalizer of a singular 2_{2}^{5} is an extension of a subgroup of order 2^{36}3 by L_{5}(2).

Lemma 4.14 means that M contains no singular subgroups of order more than 2^{5} and so we have completed the classification of the singular subgroups in M.

Proposition 4.15 The Monster group M contains exactly 6 classes of nontrivial singular subgroups. The corresponding orders are 2, 2^{2}, 2^{3}, 2^{4}, 2^{5} and 2^{5}.

Let S_{i}, 1 ≤ i ≤ 4, denote the conjugacy class of all singular subgroups 2^{i} of M. For i = 5, we will use the notation S_{5,1} and S_{5,2} for the conjugacy classes of singular subgroups 2_{1}^{5} and 2_{2}^{5}, respectively.

Notice that in this section we only indicated the order of the normalizers of singular subgroups and their action on U × Q_{U}/U. A more detailed information about the structure of these 2-local subgroups can be found in the appendix.

5 Arks

From this section on, G = M, the Monster simple group. In this section we construct and study a class of subgroups 2^{10} of M, associated with singular subgroups.

Let U be a singular subgroup 2_{1}^{5}. According to Lemma 4.12, every index two subgroup of U is contained in a unique singular 2_{2}^{5}. Let A = {U′ ∈ S_{5,2}|[U : U ∩ U′]| = 2} and let A(U), the ark defined by U, be the subgroup of M generated by all U′ ∈ A. Clearly, A(U) is invariant under N_{M}(U).

Lemma 5.1 The ark A(U) is elementary abelian of order 2^{10}. Furthermore, U and A(U)/U are dual to each other as modules for N_{M}(U).

Proof: Suppose U′, U′′ ∈ A with U′ ≠ U′′. Then W = U′ ∩ U′′ ∩ U is a singular subgroup 2^{3}. Since U′, U′′ ≤ Q_{W} and since Q_{W} is elementary abelian by Lemma 4.2 (4), we have that U′ and U′′ commute elementwise and, therefore, A = A(U) is elementary abelian.

Consider Ā = A/U. If U′ ∈ A then Ū′ is of order two. This yields a mapping V ↦ ā_{V} from the set of index 2 subgroups V < U to Ā^{#}. Namely, ⟨ā_{V} ⟩ = Ū′, where U′ ∈ A is the only singular 2_{1}^{5} containing V. Clearly, the elements ā_{V} generate Ā. Furthermore, the subgroups V correspond to the elements in (U^{∗})^{#}, where U^{∗} is the dual of U. Therefore, in order to complete the proof of this lemma it suffices to establish the three-term relations: ā_{V_{1}} ā_{V_{2}} ā_{V_{3}} = 1 whenever V_{1}, V_{2} and V_{3} are three index two subgroups of U, containing a given index four subgroup W < U.

Consider Q̂_{W} = Q_{W} /W. According to Lemma 4.8, Q̂_{W} is 6-dimensional (as a vector space over GF (2)) and C_{M}(W) induces on Q̂_{W} a group 3 · S_{6}. Let x ∈ C_{M}(W) be a 3-element that maps onto a nontrivial element in the center of that action. Let V be a singular subgroup 2^{4} containing W. Then V̂ is of order two, and we claim that if U′ is the unique singular 2_{2}^{5} containing V then Û′ = V̂ V̂^{x}. Indeed, on the one hand, each of the 45 (cf. Lemma 4.8) subgroups V is contained in a unique U′. On the other hand, each U′ contains three subgroups V. Therefore, W is contained in exactly 15 singular subgroups 2_{2}^{5}. It follows that each of them is invariant under x, since S_{6} cannot nontrivially act on 15/3 = 5 points. This proves our claim.

We can now finish the proof of the lemma. Suppose V_{1}, V_{2} and V_{3} are the three index two subgroups of U, containing W. Let U′_{i}, i = 1, 2, 3, be the unique singular 2_{2}^{5} containing V_{i}. Working again in Q̂_{W} = Q_{W} /W, we obtain that the image of ⟨U′_{1}, U′_{2}, U′_{3}⟩ in Q̂_{W} coincides with ÛÛ^{x}, since ⟨V_{1}, V_{2}, V_{3}⟩ = U. Thus, ⟨ā_{V_{1}}, ā_{V_{2}}, ā_{V_{3}}⟩ = ⟨U′_{1}, U′_{2}, U′_{3}⟩/U is of order four and hence ā_{V_{1}} ā_{V_{2}} ā_{V_{3}} = 1 holds.

Let U ∈ S_{5,1} and let A, A = A(U) and Ā = A/U be as above.

Lemma 5.2 The following hold.

(1) If a ∈ A \ U then ⟨U, a⟩ = ⟨U, U′⟩ for some U′ ∈ A. In particular, every coset of U in A contains a 2-central involution.

(2) If a ∈ A\U is 2-central then U ∩Q_{a} is of index two in U. Furthermore, au (where u ∈ U) is 2-central if and only if u ∈ U ∩ Q_{a}.

Proof: Part (1) follows directly from Lemma 5.1. Let U′ ∈ A be such that ⟨U, a⟩ = ⟨U, U′⟩. Then ⟨U, a⟩ ≤ Q_{W}, where W = U ∩ U′. Since perpendicularity is symmetric, we have that W ≤ Q_{a}. On the other hand, U ≰ Q_{a}, because otherwise ⟨U, a⟩ must be singular in view of Lemma 4.3.

Thus, U ∩ Q_{a} = W is of index two in U. Clearly, ⟨W, a⟩ is a singular subgroup; in particular, au is singular if u ∈ W. Comparing now with Lemma 4.10, we see that all elements in ⟨U, a⟩ \ (U ∪ U′) are non 2-central. Consequently, a ∈ U′ (and hence U′ = ⟨W, a⟩) and au is non 2-central for all u ∈ U \ W.

It follows from this lemma that A contains exactly 31 · 16 = 496 non 2-central and 31 + 31 ∗ 16 = 527 2-central involutions. Moreover, all non 2-central involutions in A are conjugate to q_{2}. Also notice that both non 2-central and 2-central involutions generate A.

Next, we analize the embedding of the ark A = A(U) in C_{u} for u ∈ U^{#}. First of all, we claim that Lemma 5.2 implies that A ∩ Q_{u} has index two in A. Indeed, u is contained in 15 subgroups W = U ∩ U′, U′ ∈ A, and hence \overline{A ∩ Q_{u}} is of order 16. Let a ∈ A \ (A ∩ Q_{u}). Since A is generated by 2-central involutions, we can choose a to be 2-central.

Lemma 5.3 We have A∩Q_{u} = [Q_{u}, a] and the image of a in C_{u}/Q_{u} ≅ Co_{1} is a 2A-involution.

Proof: Notice that Q_{u} normalizes U and hence it also normalizes A. Therefore, [Q_{u}, A] ≤ A ∩ Q_{u}. If the image of a in Co_{1} is of type 2B or 2C then Lemmas 2.5 and 2.6 imply that |[Q_{u}, a]| ≥ 2^{12}, a contradiction. Hence, the image of a is of type 2A. Furthermore, it follows from Lemma 2.4 that [Q_{u}, a]⟨u⟩/⟨u⟩ is of order 2^{8}, implying that [Q_{u}, a]⟨u⟩ = A ∩ Q_{u}. Since [Q_{u}, a] is normal in Q_{u}, it contains u and hence [Q_{u}, a] = A ∩ Q_{u}.

Let D = N_{C_{u}}(Q_{u}⟨a⟩) ∼ 2^{1+24}.2^{1+8}.Ω_{8}^{+}(2). Let C̄_{u} = C_{u}/⟨u⟩.

Lemma 5.4 A is normal in D.

Proof: Let R = ⟨Q_{u}, a⟩. Then R̄ is normal in D̄. Observe that R̄ has exactly two maximal elementary abelian subgroups: Q̄_{u} and R̄_{0} = ⟨CQ̄_{u}(ā), ā⟩. Since Q_{u} is normal in D, we conclude that R_{0} (defined as the full preimage of R̄_{0} in D) is also normal in D. We claim that A = Z(R_{0}). Indeed, clearly, A ∩ Q_{u} = [Q_{u}, a] is the center of R_{0} ∩ Q_{u}, because Q_{u} is extraspecial and because \overline{R_{0} ∩ Q_{u}} = CQ̄_{u}(ā). Hence, it remains to see that [R_{0} ∩ Q_{u}, a] = 1. However, this is clear: since R̄_{0} is abelian, we have that [R_{0} ∩ Q_{u}, a] ≤ ⟨u⟩; on the other hand, by Lemma 5.2 (2), the involution au is not 2-central. Hence u cannot be written as a commutator [r, a] for r ∈ R. Since R_{0} is normal in D and A = Z(R_{0}), we finally obtain that A is normal in D.

Let N = N_{M}(A).

Corollary 5.5 The action of N on A is irreducible. In particular, A = ⟨u^{N}⟩.

Proof: According to Lemma 5.1, A has two 5-dimensional composition factors as a module for N_{M}(U) ≤ N. On the other hand, it follows from Lemmas 5.4, 5.3 and 2.4 that A has composition factors of dimensions 1, 8 and 1 as a module for D ≤ N.

In view of this lemma we can assume that a is conjugate to u in N. Let A_{0} = A ∩ Q_{a} ∩ Q_{u}. Notice that A_{0} ∼ 2^{8}.

Lemma 5.6 We have C_{D}(⟨a, u⟩) ∼ 2^{10}.2^{16}.Ω_{8}^{+}(2). In particular, ā is of type 2a_{1} in C̄_{u} (cf. Section 2) and C_{D}(⟨a, u⟩) = C_{a} ∩ C_{u}. Furthermore, C_{a} ∩ C_{u} induces on A_{0} a group Ω_{8}^{+}(2) acting as on a halfspin module.

Proof: Since A ∩ Q_{u} = [Q_{u}, a], the orbit of a under Q_{u} consists of at least 2^{8} elements. On the other hand, if a′ is a 2-central involution in A \ Q_{u} then a′u is non 2-central by Lemma 5.2. Therefore, A \ (A ∩ Q_{u}) consists of exactly 2^{8} 2-central and 2^{8} non 2-central involutions. Furthermore, all 2central (respectively, non 2-central) involutions in A\(A∩Q_{u}) are conjugate by Q_{u}.

This shows that Q_{u}C_{D}(⟨a, u⟩) = D. Comparing with Lemma 2.4 we obtain that C_{D}(⟨a, u⟩) ∼ 2^{10}.2^{8}.2^{8}.Ω_{8}^{+}(2). Notice that the two 8-dimensional chief factors (again, see Lemma 2.4) provide nonisomorphic modules for the quotient Ω_{8}^{+}(2). Therefore, O_{2}(C_{D}(⟨a, u⟩))/A is elementary abelian and so we can record the structure of C_{D}(⟨a, u⟩) as 2^{10}.2^{16}.Ω_{8}^{+}(2).

Comparing with Lemmas 2.10 and 2.11, we see that ā must be of type 2a_{1} and that C_{D}(⟨a, u⟩) = C_{a} ∩ C_{u}. Clearly, A_{0} is invariant under C_{a} ∩ C_{u}. Since x ↦ x̄ establishes an isomorphism between A_{0} and [Q̄_{u}, a], the last claim follows from Lemma 2.4 (2).

Define a mapping f : A ⟶ GF (2) as follows: for x ∈ A, f(x) = 0 if and only if x is the identity or a 2-central involution.

Lemma 5.7 The mapping f is a nondegenerate quadratic form of plus type.

Proof: We will switch to the additive notation in A. Decompose A as A = ⟨a, u⟩ ⊕ A_{0}. Then the restriction of f on ⟨a, u⟩ is a plus type form, because au is non 2-central. It was shown in the preceding lemma that C_{a} ∩ C_{u} induces on A_{0} a group Ω_{8}^{+}(2) acting as on a halfspin module (which is a triality conjugate of the natural module). In particular, C_{a} ∩ C_{u} has two orbits on A_{0}^{#}, of length 120 and 135. Thus, in order to show that the restriction of f to A_{0} is a quadratic form of plus type it suffices to show that A_{0} contains exactly 120 non 2-central involutions. However, this is clear. Indeed, by Lemma 5.2, each of the 15 cosets a′ + (U ∩ A_{0}) = a′ + (U ∩ Q_{u}), with a′ ∈ A_{0} \ (U ∩ Q_{u}), contains exactly eight 2-central and eight non 2central involutions. We have shown that the restriction of f on A_{0} is also a plus type form.

It remains to verify the values of f on the elements x + y, x ∈ ⟨a, u⟩^{#} and y ∈ A_{0}. If x = a or u then f(x + y) = f(y) because y and x + y are conjugate in Q_{x}. In view of Lemma 5.6, C_{a} ∩ C_{u} has orbits of length 120 and 135 on the set a + u + A_{0}^{#}. Since the total number of non 2-central involutions in A is known to be 496, we compute that among the elements in a + u + A_{0}^{#} there are exactly 135 non 2-central involutions and 120 2-central involutions. Hence f(a + u + y) = 1 + f(y) for all y ∈ A_{0}^{#}.

We can now pin down the structure of N = N_{M}(A). Let P_{A} = O_{2}(N).

Lemma 5.8 We have N ∼ 2^{10+16}.Ω_{10}^{+}(2). In particular, P_{A} = C_{M}(A) ∼ 2^{10+16}.

Proof: First of all, Lemma 5.6 yields that C_{M}(A) is an extension of A by a group 2^{16}, i.e., C_{M}(A) ∼ 2^{10+16}. Consider now the action of N on A. Clearly, N leaves the form f invariant. So N/C_{M}(A) is isomorphic to a subgroup of O_{10}^{+}(2). We claim that it is isomorphic to Ω_{10}^{+}(2). Indeed, observe that D and N_{M}(U) share a Sylow 2-subgroup T (indeed, the 2-parts of the orders of D and N_{M}(U) coincide and hence as T we can take a Sylow 2-subgroup of N_{M}(U) centralizing u). Consider an index two subgroup in U invariant under T and the unique singular 2_{2}^{5}, say U′, containing that subgroup. Both U and U′ are maximal totally singular with respect to f and T leaves invariant both U and U′. This yields that the image of T lies in Ω_{10}^{+}(2) and, moreover, the images of D and N_{M}(U) lie in Ω_{10}^{+}(2), too. Comparing the orders we obtain that they are two maximal parabolics in Ω_{10}^{+}(2). Therefore, N/C_{M}(A) is either Ω_{10}^{+}(2) or O_{10}^{+}(2). It remains to notice that N/C_{M}(A) ≅ O_{10}^{+}(2) is impossible, because U and U′ are not conjugate.

For x ∈ A^{#}, let x^{⊥} be the orthogonal complement of ⟨x⟩ with respect to the symplectic form (x_{1}, x_{2}) = f(x_{1} + x_{2}) − f(x_{1}) − f(x_{2}) on A. (We continue using the additive notation in A.)

Corollary 5.9 If x ∈ A^{#} is 2-central then x^{⊥} = A ∩ Q_{x}.

Proof: First of all, by the preceding lemma, N is transitive on 2-central involutions in A. Hence A ∩ Q_{x} has index two in A. Furthermore, since y and x+y have the same type whenever y ∈ Q_{x}, we have that f(y) = f(x+y) for all y ∈ A ∩ Q_{x}. This proves that A ∩ Q_{x} ≤ x^{⊥}.

This shows that a subgroup of A is singular if and only if it is totally singular with respect to f. Notice that A contains both a singular 2_{1}^{5} and a singular 2_{2}^{5} and so, indeed, an ark contains all species of singular subgroups. Furthermore, all singular subgroups of A of the same kind are conjugate in N. This implies, in particular, the following

Lemma 5.10 If U ∈ S_{5,1} then A(U) is the only ark containing U. If U ∈ S_{5,2} then U is contained in exactly three arks. Furthermore, those three arks are conjugate under N_{M}(U).

Proof: The first claim follows since N_{M}(U) ≤ N if U ∈ S_{5,1}. If U ∈ S_{5,2} and U ≤ A, we compute that N_{N}(U) has index three in N_{M}(U).

6 Elementary abelian subgroups in P_{A}

Let A be an ark and N = N_{M}(A). We first produce an inventory of the elements from P_{A} \ A. Since A ≤ Z(P_{A}), every coset xA with x ∈ P_{A} \ A consists entirely of involutions or entirely of elements of order four.

Let Ñ = N/P_{A} ≅ Ω_{10}^{+}(2).

Lemma 6.1 If u is a 2-central involution from A then R = P_{A} ∩ Q_{u} is of order 2^{17}. In particular, P_{A} is nonabelian.

Proof: Notice that Q_{u} normalizes any singular 2_{1}^{5} subgroup U such that u ∈ U ≤ A, and hence Q_{u} normalizes A = A(U). Thus, Q_{u} ≤ N. Notice further that Q_{u} cannot be fully contained in P_{A}. Indeed, if Q_{u} ≤ P_{A} then Q_{u} has index two in P_{A}, which implies that Q_{u} must have a center of size at least 2^{5}; clearly a contradiction. Thus, Q_{u} ≰ P_{A}, which means that Q̃_{u} is a nontrivial normal subgroup of D̃, where D = N ∩ C_{u}. Since D̃ is a maximal parabolic (the stabilizer of a singular vector from the natural module), we get that Q̃_{u} ∼ 2^{8}. Hence |R| = 2^{25−8} = 2^{17}. Being a subgroup of an extraspecial group 2^{1+24}, R must be nonabelian. Hence, P_{A} is also nonabelian.

We will now classify the cosets xA with x ∈ P_{A} \A. It turns out that the cosets consisting of involutions correspond to singular subgroups 2_{1}^{5} from A.

Lemma 6.2 Suppose U ≤ A, U ∈ S_{5,1}. Then

(1) Q_{U} ≤ P_{A} and Q_{U} ≰ A; hence, X = Q_{U}A \ A is a coset from P_{A} \ A, consisting of involutions; if x ∈ X then U = [P_{A}, x]; and

(2) K = N_{M}(U) has exactly two orbits on X; one of the orbits is Q_{U} \ U, and it consists of non 2-central involutions (conjugate to q_{2}); the other orbit is X \ Q_{U}, and it consists of 2-central involutions.

Proof: Let K = N_{M}(U). Notice that K̃ is a maximal parabolic in Ñ ≅ Ω_{10}^{+}(2); namely, it is the stabilizer of a maximal totally singular subspace U from the natural module A. Clearly, Q_{U} is invariant under K. Since K has two 5-dimensional chief factors in A and since Q_{U}/U has order two, we conclude that Q_{U} ≰ A. If y ∈ A \ U is 2-central then W = U ∩ y^{⊥} is a singular 2^{4}. Since ⟨y, Q_{U}⟩ ≤ Q_{W} (which is abelian), the subgroup Q_{U} centralizes every y and hence Q_{U} ≤ P_{A}.

Recall that K = N_{M}(U) is contained in N, because A = A(U). Clearly, K acts on X = Q_{U}A \ A. Notice that [P_{A}, Q_{U}] ≤ Q_{U} ∩ A = U and hence [P_{A}, Q_{U}] = U, because K acts on U irreducibly. This implies that for x ∈ Q_{U} \ U we have [P_{A}, x] = U. Since A = Z(P_{A}), the same must be true for all x ∈ X. In particular, for all x ∈ X, all elements in xU are conjugate under P_{A}. Let T = Q_{U}A and let T̂ = T/U. Clearly, T̂ is the product of Â ∼ 2^{5} and Q̂_{U} ∼ 2. Furthermore, K stabilizes both Â and Q̂_{U}, and it acts transitively on Â^{#}. Thus, K indeed has exactly two orbits on xA = T \ A.

We already know from Lemma 4.14 (1) that the involutions from Q_{U} \U are non 2-central. Since those involutions are contained in Q_{u} for u ∈ U^{#}, they are conjugate to q_{2}. To see that the involutions in X \ Q_{U} are 2central, consider W ≤ U, W ∼ 2^{4}. Let U′ be the unique singular subgroup 2_{2}^{5} containing W. Then, by definition of A = A(U), we have U′ ≤ A. Let x ∈ Q_{U} \ U and s ∈ U′ \ W. Comparing with Corollary 4.11 and with the definition of singular subgroups 2_{2}^{5} (following Corollary 4.11), we see that xs is 2-central. Clearly, xs ∈ X \ Q_{U}, and so the claim follows.

In particular, this lemma shows that Q_{U} and hence also U can be recognized from the coset X = Q_{U}A \ A. Therefore, such cosets are in a natural bijection with the set of all singular subgroups 2_{1}^{5} from A. The latter set has size 2295, which means that at least 2295 cosets of A in P_{A} \ A consist of involutions. We claim that the remaining (2^{16} − 1) − 2295 cosets of A in P_{A} \ A consist of elements of order four.

Suppose x ∈ P_{A} is of order four. Then s = x^{2} is an element of A and furthermore s = y^{2} for each y ∈ xA.

Lemma 6.3 If u is a 2-central involution from A then (P_{A} ∩ Q_{u})A \ A contains exactly 120 cosets xA such that u = x^{2}. The group K = N ∩ C_{u} transitively permutes those cosets.

Proof: Consider R = P_{A} ∩ Q_{u}. By Lemma 6.1, |R| = 2^{17}. Let a be a 2central involution from A\u^{⊥}. Then R ≤ C_{a}, the image of a in C_{u}/Q_{u} ≅ Co_{1} is of type 2A, and, comparing with Lemma 2.4, we see that R = Q_{u} ∩ C_{a}. Since Q_{u} ≤ N, we have that [Q_{u}, a] ≤ Q_{u} ∩ A = u^{⊥}. Since F = N ∩ C_{a} ∩ C_{u} involves Ω_{8}^{+}(2) (indeed, if we view A as the natural module for Ñ = N/P_{A} ≅ Ω_{10}^{+}(2) then a and u span in A a nondegenerate subspace of plus type), Lemma 2.4 gives us that R̄ ∼ 2^{8} and F induces on R̄ a group Ω_{8}^{+}(2) acting as on a halfspin module. Here the bar indicates the image in P̄_{A} = P_{A}/A.

Define a mapping q : R̄ ⟶ GF (2) by q(xA) = 0 if x^{2} = 1, and q(xA) = 1 if x^{2} = u. Then q is a quadratic form on R̄ and this form is invariant under K. Since the halfspin module for Ω_{8}^{∗}(2) is triality conjugate to the natural module and since the latter admits a unique invariant quadratic form, the claims of the lemma follow.

Since A contains 527 2-central involutions u, Lemma 6.3 accounts for 527 · 120 cosets xA consisting of elements of order four. Since 527 · 120 = (2^{16} −1)−2295, all the cosets of A in P_{A} \A have been accounted for. Thus, we obtain the following.

Lemma 6.4 The group N has exactly two orbits on the nonidentity elements of P̄_{A} = P_{A}/A. The smaller orbit has length 2295 and it consists of cosets containing involutions. The longer orbit has length 527 · 120 and it consists of cosets containing elements of order four.

In particular, P̄_{A} is irreducible as a module for Ñ = N/P_{A} ≅ Ω_{10}^{+}(2). We remark that this module is isomorphic to the halfspin module. Indeed, this follows from the fact that the stabilizer of maximal totally singular subspace U ≤ A, U ∈ S_{5,1}, fixes a vector in P̄_{A}.

Additionally, Lemma 6.2 gives us the following.

Corollary 6.5 The group N has exactly two conjugacy classes of involutions in P_{A}\A. One class has length 2295·32, and it consists of non 2-central involutions conjugate to q_{2}. The other class has length 2295 · (1024 − 32) and it consists of 2-central involutions.

We will not classify the classes of elements of order four in P_{A}. However, we will need the following fact.

Lemma 6.6 If x ∈ P_{A} is of order four then x ∈ (P_{A} ∩ Q_{u})A where u = x^{2}.

Proof: We have seen above that P_{A} ∩ Q_{u} is nonabelian and hence it contains an element y of order four. Clearly, y^{2} = u. Since N has just one orbit on cosets from P_{A}/A that consist of elements of order four, there is a conjugate y^{n}, n ∈ N, of y which lies in the coset xA. Then (y^{n})^{2} = x^{2} = u and hence, without loss of generality, we may assume that y = y^{n} lies in xA. Now since y ∈ (P_{A} ∩ Q_{u})A, we have that x ∈ yA ≤ (P_{A} ∩ Q_{u})A.

Next, we need to know when two involutions from P_{A} \ A commute. For an involution x ∈ P_{A} \ A let U(x, A) (or simply U(x), if A is clear from the context) be the singular subgroup 2_{1}^{5} from A, that corresponds to the coset xA. Recall that U(x) = [x, P_{A}] (cf. Lemma 6.2 (1)).

Lemma 6.7 Involutions x and y from P_{A}\A commute if and only if U(x)∩ U(y) has size at least eight.

Proof: Notice first of all that [x, y] = [x′, y′] for arbitrary x′ ∈ xA and y′ ∈ yA. Hence, commutation of x and y depends solely on U′ = U(x) and U′′ = U(y). In particular, we may assume that x ∈ Q_{U′} and y ∈ Q_{U′′}. Secondly, observe that U′ and U′′ meet in a subgroup of order 2, 2^{3}, or 2^{5}. Since N = N_{M}(A) acts transitively on pairs (U′, U′′) with U′ ∩U′′ of a given size and since P_{A} is nonabelian, it suffices to show that x and y commute if W = U′ ∩ U′′ has size eight. However, this is clear: both x and y are contained in Q_{W}, which is abelian.

This lemma allows us to determine now all maximal elementary abelian subgroups of P_{A}.

Lemma 6.8 With respect to conjugation by N = N_{M}(A), the group P_{A} has exactly two classes of maximal elementary abelian subgroups Y:

(1) for W ≤ A, W ∈ S_{2}, Y consists of A and all involutions y ∈ P_{A} \ A such that W < U(y); and

(2) for V ≤ A, V ∈ S_{5,2}, Y consists of A and all involutions y ∈ P_{A} \ A such that V ∩ U(y) is of order 2^{4}.

Proof: First of all, it follows from Lemma 6.7 that the subgroups Y from (1) and (2) are elementary abelian. (Indeed, in (1) if U(y_{1}) and U(y_{2}) both contain W then U(y_{1}) ∩ U(y_{2}) is of order at least 2^{3}; the other case is even easier.)

Let E now be an elementary abelian subgroup of P_{A}. Let E = {U(x)|x ∈ E \ A}. It follows from Lemma 6.7 that if U and U′ are distinct elements of E then U ∩ U′ has order 2^{3}. Let us now show that if U, U′, U′′ ∈ E are pairwise distinct then U ∩ U′ ∩ U′′ is of order at least 2^{2}. Suppose not. Then U ∩ U′ ∩ U′′ has order two. Observe that by Lemma 4.3 the subgroup W = ⟨U ∩ U′, U ∩ U′′, U′ ∩ U′′⟩ is singular. Furthermore, since (U ∩ U′) ∩ (U ∩ U′′) has order two, we have U = ⟨U ∩ U′, U ∩ U′′⟩, which means that U ≤ W. Similarly, U′, U′′ ≤ W; clearly, a contradiction, since U, U′ and U′′ are maximal singular. Thus, indeed, U ∩ U′ ∩ U′′ has order at least 2^{2}.

Fix U ∈ E and let T = {U ∩ U′|U′ ∈ E, U′ ≠ U}. This is a set of subgroups 2^{3} from U, such that any two of them meet in a subgroup 2^{2}. We claim that one of the following two possibilities holds: (a) there is a subgroup W ≤ U of order four such that every T ∈ T contains W; or (b) there is a subgroup W ≤ U of order 16, such that every T ∈ T is contained in W. Let T_{1}, T_{2} be distinct elements from T , and let W_{1} = T_{1} ∩ T_{2} and W_{2} = ⟨T_{1}, T_{2}⟩. Then clearly W_{1} is of order four and W_{2} is of order 16. If every T ∈ T is contained in W_{2} then we have case (b) with W = W_{2}. So suppose T_{3} ∈ T and T_{3} ≰ W_{2}. Observe that T_{3}∩W_{2} has order four and hence T_{3} ∩ W_{2} = T_{3} ∩ T_{1} = T_{3} ∩ T_{2} = W_{1}. Finally, consider an arbitrary T ∈ T . If T ≱ W_{1} then T ∩ T_{1} ≠ T ∩ T_{2} and hence T = ⟨T ∩ T_{1}, T ∩ T_{2}⟩ ≤ W_{2}. However, this means that T ∩ T_{3} = T_{3} ∩ W_{2} = W_{1}, that is, T ≥ W_{1}, a contradiction. We proved that case (a) holds with W = W_{1}.

We can now complete the proof of the lemma. If T satisfies the condition in (a) then E is contained in the subgroup from (1) defined by W. If, on the other hand, T satisfies the condition from (b) then E is contained in the subgroup from (2), where V is defined as the unique singular 2_{2}^{5} containing W. Indeed, every U′ from E meets V in a subgroup of size at least eight. Since U′ and V are nonconjugate maximal totally singular subgroups from A, we must have that V ∩ U′ has size 16.

We will use the following notation. For an ark A and a singular W ≤ A, W ∈ S_{2}, let Ab_{2}(A, W) (or simply, Ab_{2}(W)) be the maximal elementary abelian subgroup of P_{A} defined by W as in (1). Similarly, if V ≤ A and V ∈ S_{5,2} then let Ab_{5}(A, V) (or just Ab_{5}(V)) denote the maximal elementary abelian subgroup defined by V as in (2). Notice that |Ab_{2}(W)| = 2^{14} and |Ab_{5}(V)| = 2^{15}.

Recall that N = N_{M}(A) and let P̄_{A} = P_{A}/A.

Lemma 6.9 Suppose W, V ≤ A with W ∈ S_{2} and V ∈ S_{5,2}. Let Y = Ab_{2}(W) and Y′ = Ab_{5}(V). The following hold:

(1) N_{N}(W) induces on Ȳ ∼ 2^{4} the group L_{4}(2); and

(2) N_{N}(V) induces on Ȳ′ ∼ 2^{5} the group L_{5}(2).

Proof: The involutions ȳ ∈ Ȳ bijectively correspond to the 2_{1}^{5} subgroups U in A, that contain W. Since W^{⊥}/W is a 6-dimensional orthogonal space of plus type, N_{N}(W) induces on Ȳ the group Ω_{6}^{+}(2) ≅ L_{4}(2). Similarly, the involutions in ȳ′ ∈ Ȳ′ bijectively correspond to index two subgroups in V. So N_{N}(V) induces on Ȳ′ the group L_{5}(2).

Lemma 6.10 Suppose B ≤ P_{A} is elementary abelian and B̄ ∼ 2^{3}. Then there exist unique W, V ≤ A with W ∈ S_{2} and V ∈ S_{5,2}, such that Ab_{2}(W) and Ab_{5}(V) contain B.

Proof: Let b_{1}, b_{2}, b_{3} ∈ B and B̄ = ⟨b̄_{1}, b̄_{2}, b̄_{3}⟩. Then W = U(b_{1}) ∩ U(b_{2}) ∩ U(b_{3}) and V is the unique singular 2_{2}^{5} in A containing (U(b_{1})∩U(b_{2}))(U(b_{1})∩ U(b_{3})) ∼ 2^{4}.

We complete this section with a different construction of the maximal elementary abelian subgroups Ab_{5}(A, V). Suppose V is a singular subgroup 2_{2}^{5} and let A_{1}, A_{2} and A_{3} be the three arks containing V (cf. Lemma 5.10). Furthermore, let P_{i} = P_{A_{i}} for all i.

Lemma 6.11 Let {i, j, k} = {1, 2, 3} and R = A_{i}A_{j}. Then the following hold:

(1) the subgroups A_{i} and A_{j} commute elementwise and A_{i} ∩ A_{j} = V; in particular, R ∼ 2^{15}; furthermore, R = Ab_{5}(A_{i}, V) = Ab_{5}(A_{j}, V);

(2) P_{i} ∩ P_{j} = R; in particular, R is maximal abelian;

(3) R contains A_{k}; in particular, R = A_{i}A_{k} = A_{j}A_{k}.

Proof: If a ∈ A_{i} and b ∈ A_{j} then a and b lie in Q_{W} for some subgroup W ≅ 2^{3} of V. Since Q_{W} is abelian, we have that a and b commute. In particular, A_{j} ≤ P_{i} and hence also R ≤ P_{i}. The stabilizer K of V in N_{M}(A_{i}) involves L_{5}(2) acting irreducibly on both V and A_{i}/V (see Lemma 5.1). Since the stabilizer of A_{j} in K is of index at most two, we obtain that A_{i} ∩ A_{j} = V. If y ∈ A_{j} \ V then U(y) contains an index two subgroup from V. Hence A_{j} ≤ Ab_{5}(A_{i}, V), proving (1).

For (2), let N = N_{M}(A_{j}), K = N_{N}(V) and S = P_{i} ∩ N. Since K acts on {A_{i}, A_{k}}, the index of K_{0} = N_{M}(A_{i}) ∩ N in K is at most two. Notice that the image of K in N/P_{j} ≅ Ω_{10}^{+}(2) is a maximal parabolic 2^{10}.L_{5}(2) with an irreducible action of L_{5}(2) on the normal 2^{10}. This means that S, being normal in K_{0}, is either contained in P_{j} and so S = P_{i} ∩ P_{j}, or R has index 2^{10} in S. In its turn, S has index at most two in P_{i}. Thus, in the first case P_{i} ∩ P_{j} has index two in P_{i}, which is clearly impossible. In the second case, P_{i} ∩ P_{j} has order 2^{15} or 2^{16}, implying that R has index one or two in P_{i} ∩ P_{j}. Suppose R ≠ P_{i} ∩ P_{j}. Since R is in the center of P_{i} ∩ P_{j} and since R is a maximal elementary abelian subgroup by (1), all elements in (P_{i} ∩ P_{j}) \ R are of order four and they all square to the same involution in A_{i} ∩ A_{j} = V. Since K_{0} involves L_{5}(2) acting transitively on the involutions from V, we obtain a contradiction, proving that R = P_{i} ∩ P_{j}. This shows that R is self-centralized, which means that R is maximal abelian. So (2) is proven.

Since A_{k} commutes with both A_{i} and A_{j}, it is contained in C_{M}(R) = R.

In particular, this lemma shows that the subgroup Ab_{5}(A, V) depends, in fact, only on V. So we will use the notation Ab_{5}(V).

7 Classes of involutions, II

In this section z is a 2-central involution in M, the Monster, Z = ⟨z⟩, C = C_{z}, and Q = Q_{z}. We classify conjugacy classes of involution in C and determine the fusion of these classes in M.

Let C̄ = C/Z. If x ≠ z is an involution from C then c̄ is again an involution. Notice that the group C̄ satisfies the conditions (H1)–(H3) from Section 2. The results from that section tell us that C̄ has exactly eight classes of involutions. If x̄ is an involution then x is either an involution or an element of order four, having the property that x^{2} = z. Suppose x is an involution. Then y = xz is also an involution, and either x and y are conjugate in C, or they are not. As a result, each conjugacy class of involutions from C̄ leads to zero, one, or two conjugacy classes of involutions in C. We now have to decide which case takes place for each of the eight classes of involutions from C̄.

The three classes contained in Q̄, namely, 2e_{2}, 2e_{3} and 2e_{4}, were discussed in Lemma 4.4. There we proved that 2e_{3} produces a class of elements of order four, while each of 2e_{2} and 2e_{4} leads to a class of involutions. We denoted by q_{2} and q_{4} representatives of those classes and noted that q_{4} is 2-central (i.e., conjugate to z) and q_{2} is non 2-central. Thus, in this section we only need to discuss the classes 2a_{i}, 2b and 2c.

Let q = q_{4}, D = C ∩ C_{q} and R = Q ∩ Q_{q} ∼ 2^{1+23}.(2^{11} : M_{24}). According to Lemma 4.5, Q ∩ Q_{q} has order 2^{13} and hence (C ∩ Q_{q})(C_{q} ∩ Q) = O_{2}(D). That is, the subgroup C ∩ Q_{q} maps in C̃ = C/Q ≅ Co_{1} onto the diagonal subgroup 2^{11}. Since the latter contains representatives of the conjugacy classes 2A and 2C from Co_{1}, we may be able to find elements y with ȳ in the classes 2a_{1}, 2a_{2}, 2a_{3} and 2c by looking at C ∩ Q_{q}.

If x is an involution in C ∩ Q_{q} and x ∉ ⟨z, q⟩ then ⟨z, x⟩ maps onto a size four subgroup of Q_{q}/⟨q⟩. Under the identification of the latter with Λ̂ = Λ/2Λ, the Leech lattice modulo two, the images of z, x, and xz belong to Λ̂_{2} ∪ Λ̂_{4}, because these elements are involutions (cf. Lemma 4.4).

To proceed, we will need some information about the subgroups of order four in Λ̂. According to [ATLAS], Co_{1} has exactly fifteen orbits on such subgroups. Representatives of nine of those orbits contain elements from Λ̂_{3}. The remaining six orbits are shown in Table 1. Let Û ∼ 2^{2} be a representative of one of these orbits. Then the second column contains the types of the three involutions from Û. For example, if the entry there is 244 then one involution lies in Λ̂_{2}, while two involutions lie in Λ̂_{4}. The third column shows the structure of the elementwise stabilizer (centralizer) of Û in Co_{1}. The fourth column shows the group induced on Û by the setwise stabilizer (normalizer) of Û in Co_{1}. Notice that [ATLAS] shows a different structure (namely, [2^{12}].L_{3}(2)) of the elementwise stabilizer of Û for Û from orbit 6. That structure is incorrect.

Orbit Type Stabilizer Induced group
1  222  U_{6}(2)  S_{3}
2  224 2^{10} : Aut M_{22}  2
3  244  2^{1+8}A_{8}  2
4  444 2^{4+12}.3 · S_{6}  S_{3}
5  444  Aut M_{12}  S_{3}
6  444  [2^{11}].L_{3}(2)  S_{3}

Table 1: (Λ̂_{2} ∪ Λ̂_{4})-pure subgroups 2^{2} in Λ̂

Lemma 7.1 Suppose x ∈ C. Then the following hold:

(1) if x̄ is of type 2a_{1} then x and xz are involutions; one of them is non 2-central (fused with q_{2}) and the other one is 2-central;

(2) if x̄ is of type 2a_{3} then x and xz are 2-central involutions and they are conjugate in C; (3) if x̄ is of type 2c then x and xz are 2-central involutions and they are

conjugate in C.

Proof: Let x be an involution from C ∩ Q_{q} \ ⟨z, q⟩. Let U = ⟨z, x⟩ and Û be the image of U in Λ̂. Notice that z maps onto an element from Λ̂_{4} (cf. Lemma 4.4). Thus, Û cannot be in orbit 1 from Table 1. If x ∈ Q and x is 2-central then ⟨z, q, x⟩ is singular. Comparing with with Lemma 4.8, we see that in this case Û is in orbit 4. If x ∈ Q and x is non 2-central, then xz is also non 2-central and hence Û is of type 224, that is, Û is in orbit 2. Thus, orbits 2 and 4 correspond to x’s from Q.

Suppose next that we choose x so that Û is in orbit 3. Then, according to Table 1 and Lemma 4.4, one of the elements x and xz is 2-central and the other one is non 2-central. In particular, x ∉ Q. Let A be an ark containing z and q and let a ∈ A be a 2-central involution contained in q^{⊥}, but not in z^{⊥}. Then az is non 2-central and hence ⟨z, a⟩ is conjugate to U in C_{q}. Furthermore, since the normalizer of Û in Co_{1} permutes the two elements from Λ̂_{4} ∩ Û and since U is contained in the extraspecial group Q_{q}, we get that ⟨z, a⟩ is conjugate to U in C ∩ C_{q}. It follows from Lemma 5.8 that C ∩ C_{a} involves Ω_{8}^{+}(2). Comparing with Lemmas 2.10 and 2.11, we obtain that ā (and hence also x̄) is of type 2a_{1}.

Next choose x so that Û is in orbit 5. Then C ∩ C_{x} involves M_{12}, whose order is divisible by 11. Comparing with Lemmas 2.10 and 2.11, we immediately obtain that x̄ is of type 2c.

Finally, let x be such that Û is in orbit 6. In this case both x and xz are 2-central, which rules out the possibility that x̄ is of type 2a_{1}. It follows from Table 1 that 2^{38} divides |C ∩ C_{x}|. Comparing again with Lemmas 2.10 and 2.11, we see that x̄ can only be of type 2a_{3}.

In fact, we have proved a bit more.

Lemma 7.2 Suppose x is an involution in C. Then there exists a 2-central involution q ∉ ⟨z, x⟩, such that z, x ∈ Q_{q}, if and only if either x ∈ Q or x̄ is of type 2a_{1}, 2a_{3}, or 2c. Furthermore, if such q’s exist then C ∩ C_{x} permutes them transitively.

Proof: Only the transitivity claim requires proof. Let q and q′ be 2-central involutions such that z, x ∈ Q_{q} ∩Q_{q′}. Since q, q′ ∈ Q, there exists c ∈ C such that (q′)^{c} = q. Then \overline{x^{c}} is of the same type as x̄. It follows from the above that ⟨z, x^{c}⟩ and U are conjugate in C_{q} and moreover there is an element s ∈ C ∩ C_{q} such that (x^{c})^{s} = x. Clearly, cs ∈ C ∩ C_{x} and (q′)^{cs} = q.

Our next goal is to show that the classes 2a_{2} and 2b do not lead to involutions. We start with 2a_{2}.

Lemma 7.3 If x ∈ C and x̄ is of type 2a_{2} then x is of order four.

Proof: Let R be the full preimage in Q of CQ̄(x̄) and let F = ⟨R, x⟩. Then |R| = 2^{17} and |F | = 2^{18}. The coset F \R consists entirely of elements y with ȳ of type 2a_{i} for some i. According to Lemma 7.1, y is an involution if i = 1 or 3. Suppose x is also an involution. Then all elements in F \R are involutions. This implies that C_{R}(x) contains no elements of order four, implying that C_{R}(x) is elementary abelian. Since C_{R}(x) ≤ Q and |C_{R}(x)| ≥ 2^{16}, we get a contradiction.

It remains to consider the class 2b.

Lemma 7.4 If x ∈ C and x̄ is of type 2b then x is of order four.

Proof: Suppose by contradiction that x is an involution. We first show that x and xz are not conjugate in C. Indeed, according to Lemma 2.11, C ∩ C_{x} involves G_{2}(4). Let D be the Sylow 13-subgroup from C ∩ C_{x}. Clearly, if x and xz are conjugate in C then they are also conjugate in N_{C}(D). Let C̃ = C/Q ≅ Co_{1}. According to [ATLAS], CC̃(D̃) ≅ 13 × A_{4}. Notice also that C_{Q}(D) = Z. This means that F = C_{C}(D) is an extension of Z by a group 13 × A_{4}. Now the fact that x is an involution yields that F ≅ 2 × 13 × A_{4}. (Here Z is the direct factor of order two.) Consequently, one of x and xz is contained in the commutator subgroup of F and the other is not. Thus, x and xz cannot be conjugate in N_{C}(D), and hence they are not conjugate in C.

Choose an involution a ∈ Q such that ā ∉ CQ̄(x̄). Then for b = [a, x] we have that b̄ ∈ CQ̄(x̄). It follows from Lemma 2.5 that b is a 2-central involution. In particular, x commutes with b and, furthermore, x and xb are conjugate in ⟨a, x⟩ ≤ C_{b}. Shifting now our attention to C_{b} we see that the image of x in C_{b}/⟨b⟩ cannot be of type 2a_{1} or 2b because x and xb are conjugate in C_{b} (cf. Lemma 7.1 for 2a_{1}). Also, by Lemma 7.3, it cannot be of type 2a_{2}. Thus, the image of x in C_{b}/⟨b⟩ must be of type 2a_{3} or 2c. Lemma 7.1 forces now that x is 2-central.

Clearly, the image of z in C_{x}/⟨x⟩ must also be of type 2b. Observe that Q ∩ C_{x} is of index at most four in O_{2}(C ∩ C_{x}). Symmetrically, Q_{x} ∩ C is of index at most four in O_{2}(C ∩ C_{x}). This shows that Q ∩ Q_{x} ≠ 1, that is, there exists a 2-central involution q such that z, x ∈ Q_{q}. Now Lemma 7.2 provides a contradiction.

We summarize all the above as follows.

Proposition 7.5 The group C has exactly seven conjugacy classes of involutions, three in Q and four in C \ Q.

One important corollary of Lemmas 7.1, 7.3 and 7.4 is that the group M contains exactly two classes of involutions. Indeed, we have shown that every involution is either 2-central and hence fused with z, or non 2-central, fused with q_{2}. We record this as the following

Proposition 7.6 Every involution in M is either conjugate with z or with q_{2}.

A second important corollary is that we now know all pairs of commuting 2-central involutions in M.

Lemma 7.7 Let a and b be two commuting 2-central involutions, a ≠ b, and let R = Q_{a} ∩ Q_{b}. Let bar indicate the image in C_{a}/⟨a⟩. Then one of the following is true:

(1) b̄ is of type 2e_{4}, i.e., b ∈ Q_{a} and ⟨a, b⟩ is singular;

(2) b̄ is of type 2a_{1}; moreover, ⟨a, b⟩ is contained in a unique ark A and R = a^{⊥} ∩ b^{⊥}; in particular, A = ⟨a, b⟩R;

(3) b̄ is of type 2a_{3} and R ∼ 2^{5}; R contains a singular 2^{4} subgroup W and every involution in R \ W is non 2-central; subgroups W_{a} = W ⟨a⟩ and W_{b} = W ⟨b⟩ are singular 2_{1}^{5}; we have that R⟨a⟩ = Q_{W_{a}} and R⟨b⟩ = Q_{W_{b}}; finally, Q_{W} = R⟨a, b⟩; or

(4) b̄ is of type 2c, R ≅ 2, and the involution in R is 2-central.

Proof: According to Lemmas 4.4, 7.1, 7.3 and 7.4, b̄ is of type 2e_{4}, 2a_{1}, 2a_{3} or 2c in C̄_{a}. By Lemma 7.2 C_{a} ∩ C_{b} is transitive on the (nonempty) set of 2-central involutions q ∉ ⟨a, b⟩ such that a and b are in Q_{q}. Dividing |C_{a} ∩ C_{b}| by |C_{a} ∩ C_{b} ∩ C_{q}| (the latter can be found using Table 1) we obtain that the number of involutions q is equal to 7084, 135, 15 and 1 depending on whether the type of b̄ is 2e_{4}, 2a_{1}, 2a_{3} or 2c. We now turn to the concrete cases.

If b̄ is of type 2e_{4} then, clearly, (1) holds. Suppose b̄ is of type 2a_{1}. Then a and b are contained in an ark A. Clearly, a^{⊥} ∩ b^{⊥} ∼ 2^{8} lies in R. On the other hand, R is abelian and invariant under the action of C_{a} ∩ C_{b}. It follows from Lemma 2.4 that R cannot have size more than 2^{8} and hence it coincides with a^{⊥} ∩ b^{⊥}. Now all claims of (2) follow.

Consider next the case where b̄ is of type 2a_{3}. Choose a singular 2^{4} subgroup W and let A and B be two singular 2_{1}^{5} containing W. Let a′ ∈ A \ W and b′ ∈ B \ W. If the image of b′ is of type 2a_{1} in C_{a′}/⟨a′⟩ then both A and B are contained in the unique ark containing a′ and b′. This is impossible because in an ark every singular 2^{4} is contained in a unique singular 2_{1}^{5}. Also the image of b′ cannot be of type 2c because all the involutions q ∈ W have the property that a′, b′ ∈ Q_{q}. Hence the image of b′ is of type 2a_{3} and without loss of generality we may assume that a′ = a and b′ = b. Clearly, W ≤ R and W contains all 15 2-central involutions from R. Hence all involutions in R \ W are non 2-central. In particular, if c ∈ R \ W and w ∈ W then cw is non 2-central. According to Lemmas 7.1, 7.3 and 7.4, this means that c ∈ Q_{w}. That is, R ≤ Q_{W}. Since also a, b ∈ Q_{W}, the claim (3) follows from Lemmas 4.10 and 4.13.

Finally, suppose that b̄ is of type 2c. Let q be the only 2-central involution in R. Then it follows from Lemma 2.3 that [R : ⟨q⟩] ≤ 4. Comparing with Lemmas 2.6 and 2.11 we see that R = ⟨q⟩.

8 More on P_{A}

In this section A is an arc. For an involution x ∈ P_{A} \ A let W (x, A) (or simply W (x)) be the subgroup of A generated by all 2-central involutions in a ∈ A such that x ∈ Q_{a}. Recall from Section 6 that U(x, A) (or simply U(x)) is the singular subgroup 2_{1}^{5} in A that corresponds to the coset xA.

Lemma 8.1 We have W (x) ≤ U(x). If x is non 2-central then W (x) = U(x); otherwise, W (x) is an index two subgroup in U(x).

Proof: We first notice that W (x) is a singular subgroup. Indeed, suppose a, b ∈ A are 2-central involutions such that x ∈ Q_{a} ∩ Q_{b}. If a and b are not perpendicular then (a, b) is as in Lemma 7.7 (2). However, in this case Q_{a} ∩ Q_{b} ≤ A, which means that x ∈ A, a contradiction. Hence W (x) is singular.

If x is non 2-central then x ∈ Q_{U}, where U = U(x). Clearly, this means that U ≤ W (x). Since U is maximal singular, we obtain that W (x) = U. Now suppose x is 2-central. Then for some a ∈ A, we have that t = xa is non 2-central. Let U = U(x) = U(t) and W = U ∩ a^{⊥}. Clearly, the index of W in U is at most two. If w ∈ W then a ∈ Q_{w} and t ∈ Q_{w}. Hence also x ∈ Q_{w}, i.e., W ≤ W (x). On the other hand, W (x)⟨x⟩ is singular, and hence |W (x)| ≤ 2^{4}. Thus, W (x) = W.

The next result adds to Lemma 6.1.

Lemma 8.2 Suppose u is a 2-central involution in A and x is an involution in P_{A} \ A. Then x ∈ (P_{A} ∩ Q_{u})A if and only if u ∈ U(x).

Proof: Suppose first that x ∈ (P_{A} ∩ Q_{u})A, that is, for some a ∈ A the element t = xa is in Q_{u}. By Lemma 8.1, this means that u ∈ U(t) = U(x). Reversely, suppose u ∈ U(x). Let a ∈ A be such that t = xa is non 2central. Since u ∈ U(x) = U(t), Lemma 8.1 implies that t ∈ Q_{u}. Therefore, x ∈ (P_{A} ∩ Q_{u})A.

9 Heart of the proof

Suppose t ∈ M, where either t = 1, or t is a non 2-central involution. We call a 2-central involution u marked if t ∈ Q_{u}. In this section we are going to prove the following result.

Proposition 9.1 Let Q be a 2-subgroup of M such that C_{M}(Q) = Z(Q) and t ∈ Z(Q). Let E = Ω_{1}Z(Q). Let J be the set of all those marked involutions u ∈ E for which |E ∩ Q_{u}| reaches maximum. Then either ⟨J ⟩ is singular, or E is an ark.

The proposition will be proven in a sequence of lemmas. We first show that J is nonempty.

Lemma 9.2 There exists a marked 2-central involution in E. In particular, J is nonempty.

Proof: If t = 1 then E contains a 2-central involution, since if T is a Sylow 2-subgroup of M containing Q then Z(T) ≤ Z(Q). So suppose now that t ≠ 1. Suppose T_{0} is a Sylow 2-subgroup of C_{M}(t) containing Q and T is a Sylow 2-subgroup of M containing T_{0}. Suppose z is the 2-central involution in the center of T. Then z ∈ C_{M}(Q) and hence z ∈ E. Also, since C_{z} contains a full Sylow 2-subgroup of C_{M}(t), it follows that t̄ cannot be of type 2a_{1} in C̄_{z} = C_{z}/⟨z⟩. Hence t ∈ Q_{z}, i.e., z is marked.

Thus, J is nonempty. We need to show that either ⟨J ⟩ is singular, or E is an ark. Notice that in both cases J is fully contained in an ark. Inside an ark, every pair of 2-central involutions (in particular, two involutions from J) is either as in case (1), or as in the case (2) of Lemma 7.7. Therefore, we first show that the other two cases are impossible. Fix a, b ∈ J, a ≠ b. Let U = ⟨a, b⟩ and R = Q_{a} ∩ Q_{b}. Notice that t ∈ R because a and b are marked. Let Q̄_{a} = Q_{a}/⟨a⟩.

Lemma 9.3 The pair (a, b) is not in case (4) of Lemma 7.7.

Proof: Indeed, suppose the pair (a, b) is in case (4). Since the only nontrivial element q from R is 2-central, we have t = 1. In particular, q is marked. According to Lemma 7.7, b̄ is of type 2c, which means that, with respect to the identification of Q_{q}/⟨q⟩ with Λ̂, we have that Û is in orbit 5 from Table 1. In particular, the image of C_{a} ∩ C_{b} in C_{q}/Q_{q} ≅ Co_{1} is isomorphic to M_{12} or Aut M_{12}. We obtain that C_{a} ∩ Q_{b} (which is normal in C_{a} ∩ C_{b}!) is contained in Q_{q}. Similarly, C_{b} ∩ Q_{a} ≤ Q_{q}.

We claim that q ∈ E. Indeed, Q centralizes a and b, hence also q, as it is the only nontrivial element in R = Q_{a} ∩ Q_{b}. Therefore, q ∈ C_{M}(Q) = Z(Q), that is, q ∈ E, as claimed. On the other hand, we have E ∩ Q_{b} ≤ C_{a} ∩ Q_{b} ≤ Q_{q}. This means that E ∩ Q_{b} ≤ E ∩ Q_{q}, hence, by maximality of |E ∩ Q_{b}|, we have q ∈ J and E ∩ Q_{b} = E ∩ Q_{q}. Symmetrically, E ∩ Q_{a} = E ∩ Q_{q}. However, as b ∉ Q_{a}, we have E ∩ Q_{b} ≠ E ∩ Q_{a}; a contradiction.

The second case, where (a, b) is as in Lemma 7.7 (3), is harder and requires several lemmas. Let us start with some additional notation. By Lemma 7.7, R = Q_{a} ∩ Q_{b} ≅ 2^{5} and it contains a singular 2^{4} subgroup W. All elements in R \ W are non 2-central. Furthermore, W_{a} = ⟨a, W ⟩ and W_{b} = ⟨b, W ⟩ are singular subgroups 2_{1}^{5}. By Lemma 5.10, W_{a} is contained in a unique ark A_{a} = A(W_{a}). Similarly, let A_{b} = A(W_{b}) be the only ark containing W_{b}. We have A_{a} ≠ A_{b}, since (a, b) is not as in case (1) or (2) of Lemma 7.7. Let also P_{a} = P_{A_{a}} and P_{b} = P_{A_{b}}.

By Lemma 4.12, W is contained in a unique singular 2_{2}^{5} subgroup T. By the definition of an ark, T ≤ A_{a} and T ≤ A_{b}. It follows from Lemma 6.11 that A_{a} ∩ A_{b} = T and that S = P_{a} ∩ P_{b} coincides with A_{a}A_{b} and is elementary abelian. (It coincides with Ab_{5}(T).) In particular, a ∈ P_{b} and b ∈ P_{a}. Notice that W = T ∩ W_{a} = T ∩ W_{b}. Also W = W (b, A_{a}) < U(b, A_{a}) (cf. Lemma 8.1) and similarly W = W (a, A_{b}) < U(a, A_{b}).

First of all, we note the following.

Lemma 9.4 For x, y ∈ {a, b}, x ≠ y, we have that W_{x} = U(y, A_{x}). Also, 1 ≠ C_{W_{x}}(Q) ≤ E and all involutions in W_{x} are marked.

Proof: By Lemma 8.1, W = W (y, A_{x}) < U(y, A_{x}). Since W is contained in A_{x} in a unique singular 2_{1}^{5}, it follows that W_{x} = U(y, A_{x}). The next claim follows from the fact that E = C_{M}(Q). Since t ∈ R (because a and b are marked) and since R ≤ Q_{W} by Lemma 7.7 (3), we conclude that t ∈ Q_{W}. Taking now in the account that also t ∈ Q_{x} and using Lemma 4.3, we obtain that t ∈ Q_{W_{x}}, because W_{x} = ⟨x, W ⟩. Thus, every involution in W_{x} is marked.

Let E_{a} = E ∩ P_{a} and E_{b} = E ∩ P_{b}. Since a ∈ P_{b} and b ∈ P_{a}, we have that a ∈ E_{b} and b ∈ E_{a}.

Lemma 9.5 We have E ∩Q_{a} ≤ P_{a} and hence E ∩Q_{a} ≤ E_{a}. Symmetrically, E ∩ Q_{b} ≤ E_{b}.

Proof: Since E ≤ C_{b} it suffices to show that C_{b} ∩ Q_{a} ≤ P_{a}. Let c ∈ T \ W. Clearly, A_{a} is the only ark containing a and c. Since b, c ∈ Q_{W} and since Q_{a} ∩ Q_{W} = Q_{W_{a}} has index two in Q_{W} we have that bQ_{a} = cQ_{a}. It follows from Lemma 7.1 that C_{c} ∩ Q_{a} has order 2^{17}, namely, it is the full preimage in Q_{a} of CQ̄_{a}(c). Since CQ̄_{a}(b) = CQ̄_{a}(c), we obtain that C_{b} ∩ Q_{a} ≤ C_{c} ∩ Q_{a}. On the other hand, P_{a} ∩ Q_{a} ≤ C_{c} and according to Lemma 6.1 the size of P_{a} ∩ Q_{a} is exactly 2^{17}. Therefore, Q_{a} ∩ C_{c} = P_{a} ∩ Q_{a} ≤ Q_{a}.

Since E ∩ Q_{a} is fully contained in E_{a}, the maximality property of a ∈ J implies that for every marked 2-central involution s ∈ E_{a} we have |E_{a}∩Q_{s}| ≤ |E_{a} ∩ Q_{a}|. Symmetrically, for every marked 2-central involution s ∈ E_{b} we have that |E_{b} ∩ Q_{s}| ≤ |E_{b} ∩ Q_{b}|. Since b ∉ Q_{a}, it follows that E_{a} ∩ Q_{a} ≠ E_{a} and similarly E_{b} ∩ Q_{b} ≠ E_{b}.

Our argument depends on how E_{a} and E_{b} embed into P_{a} and P_{b}, respectively. Since E_{a} and E_{b} are elementary abelian we can make use of the classification from Section 6.

Lemma 9.6 For x ∈ {a, b}, if E_{x} ∩ Q_{x} has index more than two in E_{x} then E_{x}A_{x} = Ab_{5}(V) for some singular 2_{2}^{5} subgroup V_{x} ≤ A_{x}, and the index of E_{x} ∩ Q_{x} in E_{x} is four.

Proof: We may assume that x = a. Suppose [E_{a} : E_{a} ∩ Q_{a}] > 2. Notice that E_{a}A_{a} is elementary abelian. It follows from Lemma 6.8 that either E_{a}A_{a} = Ab_{5}(V_{a}) for some singular 2_{2}^{5} subgroup V_{a} ≤ A_{a}, or the intersection F of all U(s), s ∈ E_{a}\A_{a}, is nontrivial. Suppose the latter. Since b ∈ E_{a}\A_{a}, we get F ≤ W_{a}. Therefore Q centralizes some 1 ≠ e ∈ F, as Q clearly normalizes F. It follows from Lemma 9.4 that e ∈ E and e is marked. If s ∈ E_{a} then e ∈ U(s), which by Lemma 8.2 means that E_{a} ≤ (P_{a} ∩ Q_{a})A_{a}. The latter group contains P_{a} ∩ Q_{a} as an index two subgroup. It follows that [E_{a} : E_{a} ∩ Q_{e}] ≤ 2. By the maximality property of a we now have that [E_{a} : E_{a} ∩ Q_{a}] = 2, since E_{a} ≰ Q_{a}. This is a contradiction. Thus, if [E_{a} : E_{a} ∩ Q_{a}] > 2 then E_{a}A_{a} = Ab_{5}(V_{a}) for some singular 2_{2}^{5} subgroup V_{a} ≤ A_{a}. Let K = Ab_{5}(V_{a}). Observe that Q acts trivially on V_{a}, since it acts trivially on K/A_{a}. Hence V_{a} ≤ E. Since b ∈ E_{a}, we have that V_{a} ∩ W_{a} ∼ 2^{4}. Let e ∈ V_{a} ∩ W_{a}. Then e is a marked 2-central involution in E. Using Lemma 8.2, we see that [K : K ∩ Q_{e}] = 4 and hence [E_{a} : E_{a} ∩ Q_{e}] ≤ 4. By the maximality of a, we now must conclude that [E_{a} : E_{a} ∩ Q_{a}] ≤ 4.

Out of the two options given by this lemma, we will first dispose of the possibility that E_{x}A_{x} = Ab_{5}(V_{x}) for some singular 2_{2}^{5} subgroup V_{x} ≤ A_{x}.

Lemma 9.7 For x ∈ {a, b}, we have that E_{x}A_{x}/A_{x} ≁ 2^{5}. In particular, [E_{x} : E_{x} ∩ Q_{x}] = 2.

Proof: Suppose by contradiction that K = E_{x}A_{x} = Ab_{5}(V_{x}) for some singular 2_{2}^{5} subgroup V_{x} ≤ A_{x}. Then again Q acts trivially on V_{x} because it acts trivially on K/A_{x}. Suppose x ∉ V_{x}. If s ∈ E_{x} ∩ Q_{x} then U(s) contains x (cf. Lemma 8.2) and also U(s) meets V_{x} in a subgroup of index two (because s ∈ K = Ab_{5}(V_{x})). Since U(s) is singular, we have that U(s) ∩ V_{x} = V_{x} ∩ Q_{x}. This shows that U(s) = ⟨x, V_{x} ∩ Q_{x}⟩ is unique, which means that [E_{x} ∩ Q_{x} : E_{x} ∩ Q_{x} ∩ A_{x}] = 2. However, in that case the index of E_{x} ∩ Q_{x} in E_{x} is at least 16, a contradiction with Lemma 9.6. Thus, x ∈ V_{x}. In particular, V_{x} ≠ T. Now let y ∈ {a, b}, y ≠ x. Observe that V_{x} ∩ Q_{y} ≤ W (y) = W ≤ T. Consequently, V_{x} ∩ Q_{y} ≤ V_{x} ∩ T, which gives us that [V_{x} : V_{x} ∩ Q_{y}] ≥ 4. (Here we use that both V_{x} and T are singular subgroups 2_{2}^{5} and hence |V_{x} ∩ T | ≤ 2^{3}.) As V_{x} ≤ E ∩ P_{y} = E_{y}, it follows that [E_{y} : E_{y} ∩ Q_{y}] ≥ 4. By Lemma 9.6, E_{y}A_{y} = Ab_{5}(V_{y}) for some singular 2_{2}^{5} subgroup V_{y} ≤ A_{y}. Repeating the above argument with y in place of x, we obtain that y ∈ V_{y} and hence V_{y} ≠ T, and also that [E_{x} : E_{x} ∩ Q_{x}] = 4.

Since U(y, A_{x}) meets both V_{x} and T in a subgroup of order 16, we conclude that that |V_{x} ∩ T | ≥ 8. Symmetrically, |V_{y} ∩ T | ≥ 8. As a result, F = V_{x} ∩ V_{y} ∩ T ≠ 1. Clearly, Q normalizes F and so we can choose e ∈ C_{F} (Q). Then e is a marked 2-central involution and e ∈ E. Since e ∈ V_{x}, we have that [K : K ∩ Q_{e}] = 4 and therefore [E_{x} : E_{x} ∩ Q_{e}] ≥ 4. Because of the maximality of x, we must have that E ∩ Q_{e} = E_{x} ∩ Q_{e} and that this subgroup has index four in E_{x}. In particular, E ∩ Q_{x} ≤ E_{x} and, symmetrically, E ∩ Q_{e} ≤ E_{y}. Hence E ∩ Q_{e} ≤ E_{x} ∩ E_{y}] ≤ P_{x} ∩ P_{y} = S = Ab_{5}(T). This shows that E ∩ Q_{e} is contained in K ∩ S. By Lemma 6.10, [E ∩ Q_{e} : E ∩ Q_{e} ∩ A_{x}] ≤ 4. However, this means that [E_{x} : E ∩ Q_{e}] ≥ 8, a contradiction.

Thus, we now know that |E_{x}A_{x}/A_{x}| ≤ 2^{4} and [E_{x} : E_{x} ∩ Q_{x}] = 2 for x = a and b. We will obtain the final contradiction by showing that [E_{b} : E_{b} ∩ Q_{b}] must at the same time be equal to four. However, the proof of this claim will be different for the following two cases: (a) |E_{a}A_{a}/A_{a}| ≤ 4, and (b) |E_{a}A_{a}/A_{a}| ≥ 8. We first consider the case (a).

Lemma 9.8 If |E_{a}A_{a}/A_{a}| ≤ 4 then the index of E_{b} ∩ Q_{b} in E_{b} is at least four.

Proof: By assumption, E_{a} = ⟨E_{a} ∩ A_{a}, b, c⟩ for some c ∈ E_{a}. Notice that since b ∉ Q_{a} and since [E_{a} : E_{a} ∩ Q_{a}] = 2, we can choose c ∈ Q_{a}. Since b and c commute, Lemma 6.7 tells us that U(b) ∩ U(c) has size at least eight. Since b is 2-central, W (b) = W is of index two in U(b). Depending on the type of c, the subgroup W (c) either coincides with U(c) or is an index two subgroup in it. In any case, F = W (b) ∩ W (c) is nontrivial. In particular, we can select 1 ≠ e ∈ C_{F} (Q). This e is a marked 2-central involution and e ∈ E. Since b, c ∈ Q_{e}, we have [⟨A_{a}, b, c⟩ : ⟨A_{a}, b, c⟩ ∩ Q_{e}] ≤ 2, which gives us that [E_{a} : E_{a} ∩ Q_{e}] ≤ 2. By the maximality of a, we have that e ∈ J and, furthermore, that E ∩ Q_{e} is an index two subgroup in E_{x}.

Since b, c ∈ Q_{e} and E_{a} = ⟨E_{a} ∩ A_{a}, b, c⟩, we can choose s ∈ E_{a} ∩ A_{a} such that s ∉ Q_{e}. Consider the subgroup X = ⟨a, s⟩ = {1, a, s, as}. Clearly, X ≤ E_{b}, because X ≤ E and X ≤ A_{a} ≤ P_{b}. We have a ∉ Q_{b}. Also, both s and as are not contained in Q_{e}, because a ∈ Q_{e} (the latter holds since both a and e are in U(b)). We claim that s and as do not belong to Q_{b}. For that, view A_{a} ∩ Q_{b} as a subspace of the orthogonal space A_{a} (cf. Lemma 5.7). All singular vectors in A_{a} ∩ Q_{b} form the subspace W. This implies that W is in the radical of A_{a} ∩ Q_{b}. So if s or as is in Q_{b} then it must be also perpendicular to e, because e ∈ W.

We have shown that X trivially intersects Q_{b}. Since X ≤ E_{b}, this means that [E_{b} : E_{b} ∩ Q_{b}] ≥ 4.

This lemma and Lemma 9.7 rule out case (a). So it remains to deal with case (b), i.e., we now assume that |E_{a}A_{a}/A_{a}| ≥ 8. We borrow ideas for our argument from the proof of Lemma 9.7.

First a preparatory lemma.

Lemma 9.9 There is a subgroup D ≤ E_{a} such that

(1) |DA_{a}/A_{a}| ≥ 2^{3} and b ∈ D;

(2) D is contained in Ab_{5}(V) for some singular 2_{2}^{5} subgroup V ≤ A_{a};

(3) for every 2-central involution e ∈ A_{a} we have that D ≰ Q_{e}.

Proof: We consider two cases: Either (a) E_{a}A_{a} = Ab_{2}(F) for some singular 2^{2} subgroup F ≤ A_{a}, or (b) E_{a}A_{a} ≠ Ab_{2}(F) for all such F. Suppose we are in case (a), that is, E_{a}A_{a} = Ab_{2}(F) for some F. Let P̄_{a} = P_{a}/A_{a}. Notice that since b ∈ E_{a} we have that F ≤ U(b) = W_{a}. Let F = {1, f_{1}, f_{2}, f_{3}}. Lemma 9.4 yields that each f_{i} is marked. Suppose E_{a} ≤ Q_{f_{i}} for some i. The set of all such f_{i} together with 1 forms a Q-invariant subspace F_{0} of F. Hence Q centralizes some f_{i} with E_{a} ≤ Q_{f_{i}}; a contradiction to the maximality of a. Thus, E_{a} ≰ Q_{f_{i}} for all i. In particular, there exist x_{1}, x_{2} and x_{3} ∈ E_{a} such that x_{i} ∉ Q_{f_{i}}. Moreover, we can choose x_{1} and x_{2} equal, because no group can be fully covered by two proper subgroups. Let D be the full preimage in E_{a} of a subgroup 2^{3} from Ē_{a} that contains the three elements x̄_{1} = x̄_{2}, x̄_{3}, and b̄. Clearly, (1) is satisfied for this D. Also, (2) follows from Lemma 6.10. In case (b) we simply take D = E_{a}. Then (1) is trivially satisfied, while (2) follows from Lemmas 6.8 and 6.10.

It remains to show that (3) holds in both cases. If e ∈ A_{a} is a 2-central involution such that D ≤ Q_{e} then e lies in the intersection X of all W (s), s ∈ D \ A_{a}. Reversely, if e ∈ X then e is 2-central and D ≤ Q_{e}. Suppose X ≠ 1. In case (a) we have that F is the intersection of all U(s), s ∈ D \A_{a}. Therefore, X ≤ F. However, no f_{i} can be in X because the corresponding x_{i} is in D, a contradiction. Suppose now we are in case (b). Since b ∈ D, we have that X ≤ W (b) = W, and Lemma 9.4 implies that all e ∈ X are marked. Finally, since Q centralizes D, it normalizes X, and hence it centralizes some 1 ≠ e ∈ X. That e is a marked 2-central involution in E_{a} with the property that D = E_{a} ≤ Q_{e}. However, we know that no such e exists.

Choose D as in this lemma and let V ≤ A_{a} be the singular subgroup 2_{2}^{5} such that D ≤ Ab_{5}(V). In view of Lemma 6.10 this V is unique. Since V is unique, Q normalizes V.

Lemma 9.10 The group Q centralizes V and, in particular, V ≤ E.

Proof: If s ∈ D \A_{a} then we set B(s) = V ∩U(s). If s ∈ (D ∩A_{a})\V then we set B(s) = V ∩s^{⊥}. In both cases B(s) is a hyperplane of V. Let F be the intersection of subgroups B(s) for all s ∈ D \ V. Since Q normalizes every B(s), it normalizes F and acts trivially on V/F. That is, [V, Q] ≤ F. If F = 1 then there is nothing else to prove, so we assume that F ≠ 1. Notice that since |DA_{a}/A_{a}| ≥ 2^{3} and since F ≤ U(s) for all s ∈ D \ A_{a}, the size of F is at most four. Notice also that every e ∈ F is perpendicular to all of D ∩ A_{a}. Finally, since F ≤ U(s) for all s ∈ D \ A_{a}, we have that D ≤ A_{a}Q_{e} for every e ∈ F, e ≠ 1. Therefore, [D : D ∩ Q_{e}] = 2 for every such e.

Set X = D ∩ Q_{F} and for a hyperplane H of F set Y = Y (H) to be D ∩ Q_{H}. (Here if H = 1 then Q_{H} is the entire group M and Y = D.) Since [D : D ∩ Q_{e}] = 2 for every e ∈ F, we have |D/Y | ≤ 2, |Y/X| ≤ 2, and so |XA_{a}/A_{a}| ≥ 2.

We claim that Y is never equal to X. Indeed, suppose Y ≤ Q_{F}. If |F | = 2, we have Y = D and so D ≤ Q_{F}, a contradiction with the definition of D. Thus |F | = 4. Pick x ∈ D \ Y. Since W (x) is of index at most two in U(x) and since F ≤ U(x), there exists 1 ≠ e ∈ F with e ∈ W (x). For that e, we have D = ⟨Y, x⟩ ≤ Q_{e}, since Y = X ≤ Q_{e}. This is a contradiction, proving that Y ≠ X.

Thus, D ∩ A_{a} < X < Y, and there exist s_{1}, s_{2} ∈ Y \ X with s_{1}A_{a} ≠ s_{2}A_{a}. Then U(s_{1}) ≠ U(s_{2}), and since every singular subgroup 2^{4} in A_{a} lies in a unique singular 2_{2}^{5}, we also get B(s_{1}) ≠ B(s_{2}). Since s_{i} ∉ Q_{F} but F ≤ U(s_{i}), the involution s_{i} is 2-central and U(s_{i}) = F W (s_{i}). Notice that F ∩ W (s_{i}) = H for i = 1, 2. As Q centralies s_{i}, Q normalizes W (s_{i}), F and H. Observe that [F, Q] ≤ H and also [W (s_{i}), Q] ≤ F ∩ W (s_{i}) = H. Thus, [U(s_{i}), Q] ≤ H. Finally, since V = U(s_{1})U(s_{2}), we have [V, Q] ≤ H. As H was an arbitrary hyperplane of F we conclude [V, Q] = 1 and V ≤ E.

The next question is whether V = T or not.

Lemma 9.11 We have V ≠ T. In particular, [E_{b} : E_{b} ∩ Q_{b}] ≥ 4.

Proof: Since a ∉ T, it suffices to show that a ∈ V. Suppose a ∉ V. If s ∈ Ab_{5}(V) ∩ Q_{a} then U(s) = ⟨a, V ∩ U(s)⟩. Since U(s) is singular, V ∩ U(s) must coincide with V ∩ a^{⊥}. Hence U(s) is unique. This shows that Ab_{5}(V) ∩ Q_{a} ∩ A_{a} has index at most two in Ab_{5}(V) ∩ Q_{a}. Therefore, also |(D ∩ Q_{a})A_{a}/A_{a}| ≤ 2. Since DA_{a}/A_{a} has size at least eight, we must have that [D : D ∩ Q_{a}] ≥ 4, yielding [E_{a} : E_{a} ∩ Q_{a}] ≥ 4, which is a contradition with Lemma 9.7. Hence V ≠ T. Finally, notice that V ≤ E_{b} and V ∩ Q_{b} ≤ V ∩ W (b) ≤ V ∩ T. Since [V : V ∩ T] ≥ 4, we conclude that [E_{b} : E_{b} ∩ Q_{b}] ≥ 4.

Manifestly, the conclusion of this lemma contradicts Lemma 9.7, ruling out case (2) and thus showing that b̄ cannot be of type 2a_{3} in C̄_{a} = C_{a}/⟨a⟩.

Corollary 9.12 The pair (a, b) is not in case (3) of Lemma 7.7.

According to Lemma 9.3 and Corollary 9.12, any two involutions a, b ∈ J are either perpendicular, or (a, b) is as in case (2) of Lemma 7.7. If any two involutions in J are perpendicular then the subgroup generated by J is singular. Thus, it order to complete the proof of Proposition 9.1 all we need is to prove the following lemma.

Lemma 9.13 If the pair (a, b) is as in case (2) of Lemma 7.7 then E is an ark and ⟨J ⟩ = E ∩ t^{⊥}.

Proof: In this case b̄ is of type 2a_{1} in C̄_{a} = C_{a}/⟨a⟩. By Lemma 7.7, A = ⟨a, b⟩(Q_{a}∩Q_{b}) is the unique ark containing a and b. Let P = P_{A}. Notice that Q_{a}∩C_{b} has order 2^{17} by Lemma 7.1 and that P ∩Q_{a} also has size 2^{17} by Lemma 6.1. Since P ≤ C_{b}, we conclude that Q_{a}∩C_{b} = P ∩Q_{a}. In particular, E ∩ Q_{a} ≤ P. Symmetrically, E ∩ Q_{b} ≤ P. Let D = (E ∩ Q_{a})(E ∩ Q_{b}). Then D ≤ P and E ∩Q_{x} = D ∩Q_{x} for x ∈ {a, b}. The maximality of x now shows that |D ∩ Q_{e}| ≤ |D ∩ Q_{x}| for all marked 2-central involutions from E. Let y ∈ {a, b}, y ≠ x. Since y ∉ Q_{x}, we have that E ∩ A ∩ Q_{x} is a hyperplane of E ∩ A. This implies that |E ∩ A ∩ Q_{x}| = |E ∩ A ∩ Q_{y}| and hence also |(E ∩ Q_{x})A/A| = |(E ∩ Q_{y})A/A|. We will denote this latter number by r. We intend to prove that r = 1, that is, E ∩ Q_{x} ≤ A.

If s ∈ P \A then U(s) = U(s, A) is singular, and hence it cannot contain both x and y. This means that (E ∩ Q_{x})A/A and (E ∩ Q_{y})A/A meet trivially in P/A. Therefore, |DA/A| = r^{2}. Furthermore, this shows that [D : D ∩ Q_{x}] = [DA/A : (D ∩ Q_{x})A/A] · [D ∩ A : D ∩ Q_{x} ∩ A] = 2r. Since D is abelian we have that |DA/A| ≤ 2^{5} and hence r ≤ 4.

Suppose first that r = 2 and let F be the intersection of all U(s), s ∈ D \ A. Then F is a singular subgroup 2^{3}. Note that t ∈ Q_{a} ∩ Q_{b} ≤ A. This means that t^{⊥} is a hyperplane in A and hence F_{0} = F ∩ t^{⊥} is nontrivial. Clearly, Q normalizes F_{0} so we can select 1 ≠ e ∈ C_{F_{0}}(Q). This e is a marked 2-central involution in E. Since e ∈ F we have that [D : D ∩ Q_{e}] ≤ 2. In view of maximality of x we must have that 2 ≥ 2r = 4, a contradiction.

Suppose now that r = 4. Pick a hyperplane H in D such that D∩A ≤ H. Let F be the intersection of all U(s), s ∈ H \ A. Then F ∼ 2^{2} and hence again F_{0} = F ∩ t^{⊥} ≠ 1. Choosing 1 ≠ e ∈ C_{F_{0}}(Q), we see that e is a marked 2-central involution from E and [D : D ∩ Q_{e}] ≤ 2[H : H ∩ Q_{e}] ≤ 4 < 2r = 8. So again we have a contradiction with the maximality of x.

Thus, r = 1, that is, E ∩ Q_{x} ≤ A for x ∈ {a, b}. Let e be any 2-central involution in E ∩A that is perpendicular to t. Since [E ∩A : E ∩A∩Q_{a}] ≤ 2 the maximality of x implies that e ∈ J, E∩Q_{e} ≤ A and [E∩A : E∩A∩Q_{e}] = 2. Consider F = (E ∩ A)^{⊥}. We would like to show that A ≤ E, that is we need to prove that F = 1. Suppose the contrary. We first remark that t ∈ E ∩ A. Indeed, t ∈ A and t ∈ E = Ω_{1}Z(Q) by assumption. Since t ∈ E ∩ A, every 2-central involution in F is marked. Recall now that the 2-central involutions in A are simply the singular vectors with respect to the quadratic form f on A. Since Q normalizes F, if the number of singular vectors in F is odd then Q centralizes a 2-central involution e ∈ F. Then e is marked and 2-central, e ∈ E and E ∩ A ≤ Q_{e}, a contradiction with the maximality of x. Hence the number of singular vectors in F is even. This means that that either F is 1-dimensional containing a nonsingular vector, or 2-dimensional nondegenerate. Suppose F is 2-dimensional. Then F contains an odd number of nonsingular vectors and hence Q centralizes one of them, say c. Since E = C_{M}(Q), we have that c ∈ E. Hence c is in the radical of the symplectic form on F, a contradiction, since that form is nondegenerate. If F is 1-dimensional then Q centralizes a hyperplane in A. Since N_{M}(A) induces on A the group Ω_{10}^{+}(2), no element of Q can act on A as a transvection. This means that Q centralizes A, that is, A ≤ E and W = 1, a contradiction. Thus, F = 1 and A ≤ E.

Now, A ≤ E implies that Q ≤ C_{M}(A) = P. In particular, E ≤ P. Suppose s ∈ E \ A. Then W (s) ∩ t^{⊥} ≠ 1. By the above every involution e ∈ W (s)∩t^{⊥} is in J and E ∩Q_{e} ≤ A. This contradicts the fact that s ∈ Q_{e}.

The proof of Proposition 9.1 is now complete.

10 Proof of the theorems

In this section we derive Theorems 1 and 2. We start with the Monster group M.

Proof of Theorem 1. Suppose N is a maximal 2-local subgroup in M such that C_{N}(Q) ≤ Q, where Q = O_{2}(N). Let E = Ω_{1}Z(Q). If E is an ark then N is the normalizer of an ark, which agrees with Theorem 1. So now suppose that E is not an ark. Set t = 1 and let J be the set of all 2-central involutions e ∈ E for which |E ∩ Q_{e}| reaches maximum. According to Proposition 9.1, U = ⟨J ⟩ is singular, since E is not an ark. Thus, N coincides with the normalizer of a singular subgroup. The singular subgroups of M have been classified in Section 4 (cf. Proposition 4.15). The normalizer of a singular 2^{4} is not maximal because it is contained in the larger normalizer of a singular subgroup 2_{2}^{5} (cf. Lemma 4.12). Also the normalizer of a singular subgroup 2_{1}^{5} is not maximal, because it is contained in the normalizer of an ark (cf. Lemma 5.10). The remaining possibilities agree with Theorem 1.

We now turn to the case of the Baby Monster BM. Recall that for us BM is defined as the group H̄, where H = C_{M}(t), t is a non 2-central involution in M, and H̄ = H/⟨t⟩.

Proof of Theorem 2. Let N̄ be a maximal 2-local subgroup of H̄ and suppose CH̄(Q̄) ≤ Q̄, where Q̄ = O_{2}(H̄). Let N and Q be the full preimages of N̄ and Q̄ in H. Then C_{H}(Q) ≤ Q and t ∈ Z(Q). Thus Proposition 9.1 applies to Q. Let E = Ω_{1}Z(Q) and let J be the set of all marked 2central involutions e ∈ E, for which |E ∩ Q_{e}| reaches maximum. According to Proposition 9.1, either E is an ark, or U = ⟨J ⟩ is singular. Suppose first that E is an ark. Notice that t ∈ E and that t is a nonsingular vector in the orthogonal space E. The group Ω_{10}^{+}(2) induced on E acts transitively on nonsingular vectors. Thus, H has a unique conjugacy class of arks containing t. This leads to one of the cases from Theorem 2. Next suppose that E is not an ark and hence U is singular. Observe that U is generated by marked 2-central involutions which means that t ∈ Q_{U}. It follows from the results of Section 4 (namely, Lemma 4.4, Corollaries 4.6, 4.9 and 4.11, and Lemma 4.14; see also Lemma 4.2 (3)) that U cannot be a singular 2_{2}^{5} and that in every other case the normalizer of U has a unique conjugacy class of non 2-central involutions in Q_{U}. Thus, H contains exactly five classes of singular subgroups U′ having the property that t ∈ Q_{U′}. For the case U′ ∼ 2^{4} we claim that in fact N_{H}(U′) is not a maximal subgroup (so U = ⟨J ⟩ can never be a singular 2^{4}). Namely, we claim that U′ and t are contained together in a unique ark A and hence N_{H}(U′) ≤ N_{H}(A). First notice that U′ and t are contained in some ark. Indeed, if A is an ark containing U′ then A ∩ Q_{U′} does contain some non 2-central involutions. Since N_{M}(U′) is transitive on non 2-central involutions from Q_{U′}, there must be an ark containing U′ and t. It follows from Lemmas 4.12 that U′ is contained in exactly three singular 2_{1}^{5} subgroups V, each of which is in turn contained in a unique ark A. By Lemma 4.13, t belongs to Q_{V} for two of these V. If t ∈ Q_{V} then by Lemma 6.2 we have for the corresponding ark A that t ∈ P_{A} \ A. Thus, there is at most one ark containing U′ and t. We have shown that U ≁ 2^{4}. The remaining four classes of possible singular subgroups U appear in Theorem 2.

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Department of Mathematics Michigan State University East Lansing, MI 48824 meier@math.msu.edu

Department of Mathematics and Statistics Bowling Green State University Bowling Green, OH 43403 sergey@bgnet.bgsu.edu