The maximal 2-local subgroups of the Monster and Baby Monster, II

U. Meierfrankenfeld

April 17, 2003

Abstract In this paper the maximal 2-local subgroups in the Monster and Baby Monster simple groups which are not of characteristic 2 are determined.

1 Introduction

Let p be a prime and G a finite group. We say G is of characteristic p if C_{G}(O_{p}(G)) ≤ O_{p}(G). A subgroup of G is called a p-local subgroup if its the normalizer of a non-trivial p-subgroup of G. In [MS] all maximal 2-local subgroups of the Monster M and the Baby Monster BM which are of characteristic 2 have been classified. As a follow up in this paper we determine whose maximal 2-local subgroups of the Monster and the Baby Monster which are not of characteristic 2.

Theorem A The Monster group M contains exactly 2 conjugacy classes of maximal 2-local subgroups which are not of characteristic 2 with the structures as follows:

(1) 2.BM;

(2) 2^{2}.^{2}E_{6}(2).S_{3}

Theorem B The Baby Monster group BM contains exactly 3 conjugacy classes of maximal 2-local subgroups which are not of characteristic 2 with the structures as follows:

(1) 2.^{2}E_{6}(2).S_{3};

(2) 2^{2}.F_{4}(2).2; (3) S_{4} ×^{2}F_{4}(2)

Let G be a finite group. For g ∈ G let C_{g} = C_{G}(g). An element z in G is called singular that if there exists a normal subgroup Q_{z} of C_{z} such Q_{z} is an extraspecial p-group for some prime p and C_{G}(Q_{z}) = ⟨z⟩. So Q_{z} is a large extra special subgroups of G. Note that this implies that z has order p and that C_{z} is of characteristic p. If x, y ∈ G and x is singular we say y is perpendicular to x provided that y ∈ Q_{x}. A subgroup E of G is called singular if all the elements in E^{#} are singular and pairwise perpendicular. If E ≤ G, put C_{E} = C_{G}(E) and Q_{E} = ⋂{Q_{z} | z ∈ E, z singular }. If X, Y are subgroups of G with Y singular, we say that X is perpendicular to Y if X ≤ Q_{Y}.

With the Monster we mean a finite group M such that M has a singular involution z with C_{z}/Q_{z} ≅ Co_{1} and |Q_{z}| = 2^{25} Then M has two classes of involutions, see for example [MS, 7.6]. Let t be a non-singular involution in M and put T = ⟨t⟩ and B = C_{t}. With the Baby Monster we mean the group BM = \overline{B} = B/T.

2 Subgroups of p-type

In this section G is a finite group and p a prime.

Lemma 2.1 Let A be p-subgroup of G. Then the following two statements are equivalent.

(a) C_{G}(O_{p}(C_{G}(A))A) is a p-group.

(b) N_{G}(A) is of characteristic p.

Proof: Let Q^{∗} = O_{p}(N_{G}(A)) and Q = O_{p}(C_{G}(A)).

Suppose (a) holds, i.e. that C_{G}(QA) is a p-group. Since AQ is a normal p-subgroup of N_{G})A), AQ ≤ Q^{∗}. Thus C_{G}(Q^{∗}) ≤ C_{G}(AQ). So C_{G}(Q^{∗}) is a normal p-subgroup of N_{G}(A) and C_{G}(Q^{∗}) ≤ Q^{∗}. Thus by definition of ”characteristic p”, (b) holds.

Suppose (b) holds. Let E = O^{p}(C_{G}(QA)). Then (a) is equivalent to E = 1. Note that Q^{∗} normalizes E and E normalizes Q^{∗}. Thus [Q^{∗}, E] ≤ Q^{∗} ∩E ≤ O_{p}(E). Since O_{p}(E) is a normal p-subgroup of C_{G}(A), O_{p}(E) ≤ Q and so [O_{p}(E), E] = 1 and [Q^{∗}, E, E] = 1. Since E = O^{p}(E) and Q^{∗} is a p-group, we get E ≤ C_{G}(Q^{∗}). Since N_{G}(A) is of characteristic p, C_{G}(Q^{∗})) ≤ Q^{∗}. Hence E ≤ Q^{∗} and E = 1.

A p-subgroup A of G is called of p-type in G provided that it fulfills one of the equivalent conditions of the previous lemma.

Lemma 2.2 Let A be p-subgroup of G and D ≤ A. If D is of p-type then A is of p-type.

Proof: Put Q = O_{p}(C_{G}(D)) and T = O_{p}(C_{G}(A)). By induction on |A/D| we may assume that A normalizes D. Let x be a p′-element in G centralizing AT. By 2.1a it suffices to show that x = 1. Since C_{G}(A) ≤ C_{G}(D), C_{G}(A) normalizes Q. So C_{Q}(A) is a normal p-subgroup of C_{G}(A) and hence C_{Q}(A) ≤ T. Thus x centralizes C_{Q}(A). Since A and x normalize Q, the P × Q-lemma implies, [Q, x] = 1. So x ∈ C_{G}(QD). By 2.1a C_{G}(QD) is a p-group. Since x is a p′ element we get x = 1.

Lemma 2.3 Let L be a finite group, X ≤ L and suppose:

(a) C_{L}(t) ≤ X for all involutions t ∈ X.

(b) There exists an involution s ∈ X with s^{L} ∩ X = s^{X}.

(c) L has at least two classes of involutions.

Then X = L.

Proof: Let t be an involution in X and r an involution in L not conjugate to t. Since ⟨r, t⟩ is a dihedral group and r and t are not conjugate, |rt| is even and there exists a unique involution u in ⟨rt⟩. Then by (a) u ∈ C_{L}(t) ≤ X and r ∈ C_{L}(u) ≤ X.

Applying this to t = s we see that X contains all the involutions in L not conjugate to X. So by (c) there exists a involution t ∈ X with t ∉ s^{L}.

So by the first paragraph s^{L} ⊆ X. Thus by (b) s^{L} = s^{X} and so by the Frattini argument, L = XC_{L}(s) = X.

3 Purely non-singular subgroups

Define arks as in [MS]. In this section we determine the purely non-singular elementary abelian subgroups of M and their normalizers.

Lemma 3.1 Let z be a singular involution and t a non-singular involution. Suppose that t ∈ C_{z} \ Q_{z}. Then (a) tz is singular.

(b) Let V = (Q_{z}∩Q_{tz})⟨t, z⟩. Then V is ark and so V is elementary abelian of order 2^{10}, |C_{M}(V)/V | = 2^{16} and N_{M}(V)/C_{M}(V) = Ω_{10}^{+}(2).

(c) There exists a N_{M}(V) invariant non-degenerate quadratic form f on V. Let s be the bilinear form associate to f and let a, b ∈ V^{#}. Then a is singular if and only if f(a) = 0. If a is singular, then b is perpendicular to a if and only if s(a, b) = 0,

(d) V Q_{z} = ⟨t⟩Q_{z} and tQ_{z} is in Class 2A of C_{z}/Q_{z} ≅ Co_{1}.

(e) V is the unique conjugate of V in M containing ⟨t, z⟩.

Proof: By [MS, 7.7](and using the notations wherein) t̄ is of type 2a_{1}, 2a_{3} or 2c. Since t is non-singular, [MS, 7.1] shows that t is of type 2a_{1} and that tz is singular. Hence (a) holds. Moreover, [MS, 7.7(2)] gives that V is the unique ark containing t and z. So (e) holds. (b) follows from [MS, 5.1,5.8] and (c) from [MS, 5.7, 5.9]. Note that z is singular, t is non-singular and t ∉ Q_{z}. From (c), V = ⟨t⟩(V ∩ Q_{z}) and so (d) follows from [MS, 5.6].

Lemma 3.2 (a) For each 1 ≤ i ≤ 3, M has a unique orbit on pairs (E_{i}, D_{i}) such that E_{i} is a singular 2^{i}, D_{i} is a purely non-singular fours group and D_{i} is perpendicular to E_{i}.

(b) M has a unique orbits on pairs (D_{4}, V) such that D_{4} is a purely nonsingular fours group, V is an ark and D_{4} ≤ V.

(c) No purely non-singular fours group is perpendicular to a singular 2^{4} or 2^{5}.

(d) Representatives (E_{i}, D_{i}), 1 ≤ i ≤ 3 and (D_{4}, V) for the above orbits can be chosen such that E_{1} ≤ E_{2} ≤ E_{3} ≤ V and D := D_{1} = D_{2} = D_{3} = D_{4} ≤ V. The stabilizers in L̄ := N_{M}(D)/D are as follows:

(a) NL̄(E_{1}) ∼ 2^{1+[20]}.U_{6}(2).

(b) NL̄(E_{2}) ∼ 2^{2+[27]}.(S_{3} × L_{3}(4)).

(c) NL̄(E_{3}) ∼ 2^{3+[28]}.(L_{3}(2) × Alt(5)).

(d) NL̄(V) ∼ 2^{8+[16]}.Ω_{8}^{−}(2).

Proof: Let E be a singular 2^{i} in M and put R = N_{M}(E). We first determine the orbits of R on the purely non-singular fours groups F in Q_{E} and also C_{R}(F).

By [MS, 4.10], a singular 2^{4} is not perpendicular to a purely non-singular fours group. So (c) holds.

So suppose i ≤ 3. Let V = Q_{E}/E, F̃ = F E/E, T = C_{R}(Q_{E}/E), S = C_{T} (E) and K = R/T. By [MS, 4.2(3)], S acts transitively on the 2^{2i} complements to E in EF, and T/S ≅ SL_{i}(2). Note that [N_{T} (F), F] ≤ E ∩ F = 1. Thus N_{T} (F) = C_{T} (F) and so T/C_{T} (F)| = |S/C_{S}(F)| = 2^{2i} = |E|^{2}. In particular, T = C_{T} (F)S. In particular C_{T} (F)/C_{S}(F) ≅ C_{T} (F)S/S = T/S ≅ SL_{i}(2). We proved:

C_{T} (F)/C_{S}(F) ≅ SL_{i}(2)  (1)

Since |S/C_{S}(F)| = 2^{2i} and |EF | = 2^{i+2}

|C_{S}(F)| = \frac{S}{2^{3i+2}} .  (2)

From [MS] the order of |S| is as follows:

\begin{matrix}i & 1 & 2 & 3 \\ |S| & 2^{25} & 2^{35} & 2^{39} \end{matrix}  (3)

From (2) and (3) we obtain the order of C_{S}(F)/F E:

\begin{matrix}i & 1 & 2 & 3 \\ |C_{S}(F)/F E| & 2^{20} & 2^{25} & 2^{28} \end{matrix}  (4)

Since C_{T} (F) induces Aut(E) on E, C_{R}(F) = C_{T} (F)C_{R}(EF). Since C_{T} (F) ∩ C_{R}(EF) = C_{S}(F) we conclude

C_{R}(F)/C_{S}(F) ≅ C_{T} (F)/C_{S}(F) × C_{R}(F)/C_{T} (F)  (5)

Also C_{R}(F) ≤ C_{R}(F̃), where F̃ = F E/E, and by the Frattiargument, C_{R}(F̃) = T C_{R}(F) so

C_{R}(F)/C_{T} (F) ≅ C_{R}(F̃)/T = C_{K}(F̃).  (6)

In view of (1),(4),(5) and (6), the structure of C_{R}(F)/F will be determined once we know C_{K}(F̃). Also the orbits of R on the possible F are in one to one correspondence with the orbits of K on the possible F̃.

Suppose that i = 1. Let Λ be the Leech lattice and \overline{Λ} = Λ/2Λ. Let (x, y) = \frac{1}{8} ∑_{i=1}^{24} x_{i}y_{i} be the unimodular inner product on Λ. \frac{1}{2}(x, x) is called the type of x. The type of \overline{x} = x + 2Λ is the minimum type of a vector in x + 2Λ. By our definition of the Monster, V is as a C_{z}/Q_{z} ≅ Co_{1}-module isomorphic to \overline{Λ}. By [MS, 4.4] the non-singular elements in Q_{z} corresponds to the vectors of type 2 in \overline{Λ}. By [ATLAS], Co_{1} acts transitively on the fours groups in \overline{Λ} all of whose non-trivial elements have type 2. Moreover, the centralizer of such a fours group is U_{6}(2). So (a) and (d:a) hold for i = 1. holds in this case.

Suppose next that i = 2. By [MS, 4.5], V = Q_{E}/E is the Todd-module for K ≅ M_{24} and if t is a non-singular involutions in Q_{E}, tE corresponds to a pair in Ω := {1, 2, …, 24}. Hence F̃^{#} corresponds to three pairs in a subset of size three of Ω. Since M_{24} is five transitive on Ω we conclude that C_{K}(F̃) ≅ M_{21} ≅ L_{3}(4) and so (a) and (d:b) holds for i = 2.

Suppose now that i = 3. Then by [MS, 4.8], K ≅ 3.Sym(6), V is irreducible of order 2^{6} and the non-singular involutions in Q_{E} lie in the orbit of length for 18 for K on V^{#}. Note that Z(K′) has order three and acts fixed-point freely on V. For v ∈ V let v^{∗} = Z(K′)v ∪ {1}. It follows that v^{∗} is a fours group in V. Let a, b ∈ F̃^{#}. Suppose that a^{∗} ≠ b^{∗} and observe that (ab)^{∗} ≤ a^{∗}b^{∗} and a^{∗} ≠ (ab)^{∗} ≠ b^{∗}. Let I = {(a^{∗})^{k} | k ∈ K}. Since |a^{K}| = 18, |I| = 6. Hence K induces Sym(I) on I and N_{K}(a^{∗})∩N_{K}(b^{∗}) acts transitively on I \ {a^{∗}, b^{∗}} and so d^{∗} ≤ a^{∗}b^{∗} for all d^{∗} ∈ I. But then ⟨I⟩ ≤ a^{∗}b^{∗}, contradicting the irreducible action of K on V ( and |a^{∗}b^{∗}| = 2^{4} < |V |). Hence a^{∗} = b^{∗} and F̃ = a^{∗}. Thus N_{K}(F̃) = 3.Sym(5). Since [Z(K′), K] ≠ 1, N_{K}(F̃) induced Sym(3) on F̃ = a^{∗} and so C_{K}(F̃) = Alt(5). Thus (a) and (d:c) holds for i = 3.

Now let V be an ark with D ≤ V. Then by 3.1, D is a non-degenerate 2-space in V. Thus by 3.1, D is unique up to conjugacy in N_{M}(V), N_{M}(V)∩ N_{M}(D) induces Sym(3) on D and N_{M}(V) ∩ C_{M}(D) ∼ 2^{10+16}Ω_{8}^{+}(2). So (b) and (dd) hold. Finally that exists a chain E_{1} < E_{2} < E_{3} of subgroups of order 2, 4 and 8 in V which are (with respect to the quadratic form on V) singular and perpendicular to D. Thus by 3.1 E_{i} is ( in M) singular and perpendicular to D.

Lemma 3.3 Let z be a singular involution in M. Then C_{z} is transitive on the purely non-singular fours group in Q_{z}. Moreover, there does not exist any purely non-singular subgroups of order larger than four in Q_{z}.

Proof: The first statement follows from 3.2(a). For the second we use the use the same notations for Q_{z}/ < z >≅ \overline{Λ} as in the the previous lemma.

Suppose that exist vectors a, b, c of type 2 in Λ such that ⟨\overline{a}, \overline{b}, \overline{c}⟩ has order eight and only contains vectors of type 2. Since \overline{a + b} has type two it is easy to see that (a, b) = ±2. Moreover, if (a, b) = 2, then a − b has type 2, and if (a, b) = −2 then a + b has type 2. Replacing b and c by their negatives, if necessary, we may assume (a, b) = (a, c) = 2. Thus (a, b + c) = 4 and (a, b − c) = 0. Since either b + c or b − c has type 2, we get that \overline{a + b + c} = \overline{a + b − c} is not of type 2, a contradiction.

Lemma 3.4 There exists a unique class of purely non-singular fours groups in M.

Proof: Let F = {1, a, b, c} be a purely non-singular fours group in M. By 3.3 it suffices to show that F ≤ Q_{z} for some singular involution. For this let z be a singular involution with a ∈ Q_{z}. Then C_{a} ∩ C_{z} contains a Sylow 2-subgroup of C_{a} and so we may choose z such that F ≤ C_{a} ∩ C_{z}. If F ≤ Q_{z} we are done. So we may assume that F ≰ Q_{z}. Then b, z ∈ C_{z} \ Q_{z} and by 3.1 bz and cz are singular and there exists a unique ark V containing b and z. Since a ∈ Q_{z}, az is conjugate to a and so az is non-singular. Also (az)(bz) = c is non-singular and thus 3.1a applied with to bz in place of z shows that az ∈ Q_{bz}. Thus az ∈ Q_{z} ∩ Q_{bz} ≤ V and so F ≤ V. By 3.1c there exists a singular involution in V perpendicular to F.

Lemma 3.5 Let D be purely non-singular fours group. Then there exist eights groups D_{1}, D_{2} and D_{3} containing F such that

(a) D_{1} contains a unique singular involution z and D_{1} ≤ Q_{z}.

(b) D_{2} \ D contains a unique non-singular involution and D_{2} lies in an unique ark.

(c) All elements of D_{3}\D are singular, D_{3} is not contained in a ark and the singular involutions in M perpendicular to D_{3} generate a non-trivial singular subgroup of M.

Proof: By 3.1c groups D_{1} and D_{2} as in (a) and (b) can be found in any ark. It remains find a group D_{3} as in (c). Let E be a singular fours group with D ≤ Q_{E}. By [MS, 4.5] Q_{E}/E is the Todd-module for C_{E}/O_{2}(C_{E}) ≅ M_{24}. Moreover, the pairs represent the non-singular involutions, while the sextetts represent singular involutions. Let S = (T_{i} | 1 ≤ i ≤ 6) be a sextett and k_{i} ∈ T_{i}. Let a and b be non-singular involutions in Q_{E} such that Ea and Eb correspond to {k_{1}, k_{2}} and {k_{1}, k_{3}} respectively. Then Eab correspond to {k_{2}, k_{3}} and so D := ⟨a, b⟩ is a purely non-singular fours group. Let Ez correspond to S and put D_{3} = ⟨a, b, z⟩. Then az corresponds to the sextett determined by the T_{1}∪{k_{2}}\{k_{1}} and so az is singular. Similarly bz and abz are singular. In particular, no element in D^{#} is perpendicular to z and so D_{3} cannot be contained contained in an arc. Also E is perpendicular to D_{3}. So it remains to show that if d and e are singular involutions perpendicular to D_{3}, then d is perpendicular to e. Note that d and e commute. [MS, 7.7] lists the orbits of M on pairs of commuting singular involutions and their common perp. We apply this to (c, d). Since D_{3} ≤ Q_{c}∩Q_{d}, Q_{c}∩Q_{d} contains a purely non-singular fours group ( namely D). This rules out Cases (3) and (4) of the list in [MS, 7.7]. In Case (1) we would conclude that D_{3} lies in the ark (Q_{c} ∩ Q_{d})⟨c, d⟩. So Case (2) holds, d and e are perpendicular and (c) is established.

Lemma 3.6 Let D be a purely non-singular fours group and put L = C_{M}(D) and L̄ = L/D.

(a) L̄ ≅ ^{2}E_{6}(2) and N_{M}(A)/A ≅ Aut(^{2}E_{6}(2)) ∼ ^{2}E_{6}(2).Sym(3)

(b) M has exactly three classes of eights groups containing a pure nonsingular fours group. Representatives are as given in 3.5. In particular, any purely non-singular subgroup has order at most four.

(c) If x ∈ L with x^{2} ∈ D then x^{2} = 1.

Proof: (a) Let E_{1}, E_{2}, E_{3}, D and V be as in 3.2(d). Define X = ⟨N_{L}(E_{i}), N_{L}(V) | 1 ≤ i ≤ 3⟩. It is straightforward from 3.2(d) to show that \overline{X} acts faithfully and flag transitively on a geometry B with F_{4}-diagram. By [Ti2, Theorem 1] B is covered by a building. By the classifications of spherical buildings [Ti1] B is the building associate to ^{2}E_{6}(2). Thus X ≤ Aut(B ≅ Aut(^{2}E_{6}(2)) ∼ ^{2}E_{6}(2).Sym(3). It is now easy to see that NL̄(E_{1}) ≤ ^{2}E_{6}(2) and is a maximal parabolic subgroup of ^{2}E_{6}(2). Hence X̄ ≅ ^{2}E_{6}(2).

We may choose the D_{i} such that E_{1} contains the unique singular involution in D_{1}, V is the unique ark containing D_{2} and for some i, E_{i} is the subgroup generated by the singular involutions perpendicular to D_{3}. It follows that CL̄(D_{1}/D) ≤ NL̄(E_{1}) ≤ X̄, CL̄(D_{2}/D) ≤ NL̄(V) ≤ X̄ and CL̄(D_{3}/D) ≤ NL̄(E_{i}) ≤ X̄..

By [AS] X̄ has three classes of involutions. Thus D_{1}/D, D_{2}/D and D_{3}/D are representatives for the classes of subgroups of order two in X̄. Since for i ≠ j, D_{i} is not conjugate to D_{j} in M, two involutions in X̄ are conjugated in X̄ if and only if they are conjugate in \overline{L}. Hence all assumptions of 2.3 are fulfilled and so \overline{X} = \overline{L}.

In N_{M}(V) we see that N_{M}(A)/C_{M}(A) ≅ Sym(3) and C_{M}(\overline{L}) = 1. Thus N_{M}(A)/A ≅ Aut(^{2}E_{6}(2)). So (a) holds.

(b) As L = X this was proved in (a).

(c) By the prove of (a), D⟨x⟩/D is conjugate in L̄ to some D_{i}/D. Since D_{i} is elementary abelian, (c) holds.

4 Proof of Theorem A

In this section we prove Theorem A. So let P be a maximal p-local subgroup of M which is not of characteristic p. Let A = Ω_{1}Z(O_{p}(P)). Then N_{M}(A) = P and as P is not of characteristic p, A is not of p-type. Thus by 2.2 none of the involutions are of 2-type and so all the involutions in A are non-singular. By 3.6 |A| ≤ 4. If |A| = 2 then by [MS, 7.6], A is conjugate to T and so Case (1) of Theorem A holds. If |A| = 4 then by 3.4 A is unique up to conjugacy and by 3.6(a), Case (2) of Theorem A holds.

5 Involutions in the Baby Monster

In this section we determine the conjugacy classes of involution in the Baby Monster. The conjugacy classes of involution in the Baby Monster which lift to involution in the monster already have been determined. So we need to investigate the set F of elements of order four in M which square to a non-singular involution. We start with a technical lemma used later to show that elements in F normalize a purely non-singular fours group.

Lemma 5.1 Let H be a finite group, Q a normal subgroup of H and f ∈ H. Suppose that Q is an extra-special 2-group, f^{2} ∈ Q \ Z(Q) and that there exists an involution in Qf. Then one of the following holds:

(a) There exists an involution q ∈ Q with [q, f] = f^{2}.

(b) N_{H}(Qf) normalizes f^{2}Z(Q).

Proof: Recall the commutator formulas:

[a, bc] = [a, c][a, b]^{c}  (1)

[ab, c] = [a, c]^{b}[b, c]  (2)

Let i be an involution in Qf. Set s = fi and z = s^{2}. Then s ∈ Q, f = si and since Q is extra special, z ∈ Z(Q). By (1), [s, f] = [s, si] = [s, i] and so f^{2} = sisi = s^{2}[s, i] = z[s, f]. We record

f^{2} = z[s, f]  (3)

If z = 1, (a) holds with q = s. So suppose that z ≠ 1. Then Z(Q) = {1, z}.

Assume first that there exist an involutions r ∈ [Q, i] with [r, s] ≠ 1. Since [Q, i] = {[u, i], z[u, i] | u ∈ Q} we may assume that r = [u, i] = i^{u}i for some u ∈ Q. Then⟨i^{u}, i⟩ is a Dihedral group of order four and so

[r, i] = 1  (4)

Since Q is extraspecial and r, s ∈ Q, [r, s] = z. Hence r inverts s and q := rs is an involution. Moreover, by (2)

[q, s] = [rs, s] = [r, s]^{s}[s, s] = z^{s} = z  (5)

by (2) and (4)

[q, i] = [rs, i] = [r, i]^{s}[s, i] = [s, i]  (6)

and so by (1),(5),(6) and (3)

[q, f] = [q, si] = [q, i][q, s]^{i} = [s, i]z = f^{2}.  (7)

Thus (a) holds in this case.

Assume finally that s centralizes E := Ω_{1}([Q, i]). Note that [Q, i] = [Q, Qf] and so N_{H}(Qf) normalizes E. Moreover, [Q, i] is abelian and so E has index at most two in [Q, i]. Let D := C_{Q}([Q, i]) and R := C_{Q}(E). Then s ∈ R, |R/D| = |[Q, i]/E| ≤ 2 and D/Z(Q) = C_{Q/Z(Q)}(i). If s ∈ D we get [s, i] ∈ Z(Q) and f^{2} = z[s, i] ∈ Z(Q), a contradiction to the assumptions of the lemma. Thus s ∉ D, E has index two in [Q, i] and ⟨s⟩D = R. Thus

⟨f^{2}⟩Z(Q) = ⟨[s, i]⟩Z(Q) = [R, i]Z(Q) = [R, Qi]Z(Q) = [R, Qf]Z(Q).

As N_{H}(Qf) normalizes Q, E, R and Z(Q) we conclude that (b) holds.

Lemma 5.2 Let f ∈ F. Then there exists a dihedral group H of order eight in M with f ∈ H and such that H contains exactly two singular involutions.

Proof: Let z be a singular involution with f^{2} ∈ Q_{z}. Then z is 2-central in C_{f^{2}} and so we may assume f ∈ C_{z}. Since f^{2} is non-singular, f̃^{2} is of 2-type in Q̃_{z} = Q_{z}/⟨Z⟩ ≅ Λ/2Λ. By [MS, 2.5] involutions of type 2B in Co_{1} do not centralizes elements of type 2 in Λ/2Λ. Hence Q_{z}f is of 2A or 2C. By [MS, 7.1] in both case there exists an involution i ∈ Q_{z}f. Moreover by [MS, 2.4,2.5] N_{C_{z}}(Q_{z}f) normalizes no elements type 2 in the Leech lattice Q̃_{z}. Thus by 5.1 there exists an involution q ∈ Q_{z} with [q, f] = f^{2}.

Suppose that all such q’s are non-singular. Let X = Q_{z} ∩ C_{f} ∩ C_{q}. \overline{X} has index at most 4 in C_{\overline{Q}_{x}}(f) and C_{\overline{Q}_{x}}(f) has order at least 2^{12}. Thus |\overline{X}| ≥ 2^{10}. Hence X contains elementary abelian subgroup of order 2^{5} and so also 2^{4} E with z ∉ E. Since M contains no pure non-singular 2^{4}, there exists a singular involution s in E. Then s ≠ z, s ∈ Q_{z} and s centralizes q and f. In particular, sq is an involution in Q_{z} with [sq, f] = f^{2} and so by assumption sq is non-singular. Then as q and sq are non-singular, q ∈ Q_{s}. If sqf is non-singular, then qf ∈ Q_{s} and so also f = q^{−1}qf ∈ Q_{s} and f^{2} ∈ ⟨s⟩, a contradiction since s is singular and f^{2} is not. Thus sqf is singular and we can choose H = ⟨sq, sqf⟩.

So we may assume now that q is singular. If qf is non-singular we can choose H = ⟨q, qf⟩. So we may assume that qf is singular.

Put X = {1, f^{2}, q, q^{f}}. Then X is a fours group normalized by f and [X, f] = ⟨f^{2}⟩. Put Y = Q_{X}. Since q and q^{f} are singular and f^{2} is not, 3.1 implies that X lies in a unique ark V, and that Y is a non-degenerate subspace of order 2^{8} in V. Namely Y is the orthogonal complement to X in V. Suppose that C_{Y} (f) is singular. Since C_{Y} (f) has order at least 2^{4} we get |C_{Y} (f)| = 2^{4} and |C_{V} (f)| = 2^{5}. But this contradicts the fact that any involution in Ω(V) centralizes an even dimensional subspace of V.

Hence C_{Y} (f) is not singular. Let b be any non-singular involution in C_{Y} (f). Since b ∈ Y ≤ Q_{q} also bq is non-singular. If bqf is singular, we can choose H = ⟨bq, bqf⟩. So we may assume that bqf is non-singular for all such b. Since b and bqf are non-singular we conclude that b ∈ Q_{qf}. Let E be the group of generated by the non-singular involution in C_{Y} (f). Then E ≤ Q_{qf}. If E contains a singular involution s we get ⟨q, qf⟩ ∈ Q_{s}. But then f^{2} = [q, f] ∈ ⟨s⟩, a contradiction, as f^{2} is non-singular. We conclude that E is purely non-singular and so |E| ≤ 4. But then also |C_{Y} (f)| ≤ 4, a contradiction.

Lemma 5.3 Let V be an ark and t, z ∈ V^{#} with z singular, t non-singular and t ∈ Q_{z}. Then Ṽ is in class 2A of C̃_{z} ∩ C_{t} ≅ Co_{2}.

Proof: V is the natural module for N_{M}(V)/C_{M}(V) ≅ Ω_{10}(2) we get C_{z} ∩ C_{t} ∩N_{M}(V) has factor group isomorphic to Sp_{6}(2). Thus the lemma follows from Lagrange’s Theorem and [ATLAS].

Lemma 5.4 There exist subgroups D and D^{∘} of M such that with X = N_{M}(D):

(a) D ≅ D_{8} and all involutions in D are non-singular.

(b) X interchanges the two fours groups in D.

(c) Let f be the element of order four in D and J a Sylow 2-subgroup of N_{G}(⟨f⟩) containing D. Then D is the unique conjugate of D under N_{G}(⟨f⟩) contained in J.

(d) Let e be an involution in C_{M}(D) \ D. Then e or ef^{2} is singular.

(e) X/D ≅ Aut(F_{4}(2)).

(f) f ∈ D^{∘}, D^{∘} ≅ D_{8} and D^{∘} contains exactly two singular involutions.

(g) D and D^{∘} is a complete set of representatives for the conjugacy classes of subgroups isomorphic to D_{8} in M which contain a purely nonsingular fours group. In particular, any two D_{8}’s in M containing only non-singular involutions are conjugate.

(h) X/D has a unique class of subgroup of order two not contained in (X/D)′. Moreover if S is the inverse image in X of such a group, then X ≅ SD_{16} and N_{X}(S)/S ≅ ^{2}F_{4}(2).

Proof: Let t = f^{2}.

(a) Let A be a purely non-singular fours group in G. Then by 3.6 N_{M}(A)/A ≅ Aut(^{2}E_{6}(2)) and so by [AS] there exists a unique class of subgroup D ≤ N_{M}(A) with A ≤ D, |D/A| = 2, (N_{M}(A) ∩ N_{M}(D))/D ≅ F_{4}(2) and D ≰ C_{G}(A). Then D ≅ D_{8}. Let t be an involution in D. As 17 divides the order of F_{4}(2) and so also of C_{M}(t), t is non-singular. Thus (a) holds.

(b) By (a) the fours group B in D distinct from A is also pure nonsingular. Hence B = A^{s} for some s ∈ M. But then D and D^{s} are conjugate in N_{M}(B) and we may and do choose s such that D^{s} = D. Let Y = N_{M}(A) ∩ N_{M}(D). Then X = Y ⟨s⟩.

(c) Suppose that is D is the unique conjugate of D under N_{G}(⟨f⟩) normalizing D. Then N_{N_{J} (D)}(N_{J}(D)) ≤ N_{J}(D), D is normal in J and (c) holds in this case.

So to prove (c) we may assume (for a contradiction) that there exists a conjugate D^{∗} of D under N_{G}(⟨f⟩) normalizing D. Put U = ⟨D, D^{∗}⟩. As f ∈ D ∩ D^{∗}, |U| = 16.

Assume that D^{∗} induces only inner automorphism on D. Then U = DC_{U}(D), |C_{U}(D)| = 4, Z(U) = C_{D}(U) and either Z(U) is cyclic or Z(U) is a fours group. Note that D∪D^{∗} contains eight involutions from U \Z(U). If Z(U) ≅ C_{4}, every non -trivial coset of Z(U) contains exactly two involutions and U \Z(U) contains six involutions, a contradiction. Thus Z(U) is a fours group and there are exactly eight involution in U \ Z(U). We conclude that all the involutions of U \ Z(U) are in D ∪ D^{∗} and so are non-singular. Thus U contains at most two singular involutions. Let E be an eights group in U. Then D ∩ E is a purely non-singular fours group in E and E contains at most two singular involutions. By 3.6b and 3.5, E contains a unique singular involution z. But then z is the unique singular involution in U, all involutions in U are in Q_{z}, U ≤ Q_{z} and f^{2} = z a contradiction, a contradiction as f^{2} = t is non-singular.

Hence D^{∗} induces an outer automorphism on D and so interchanges the two fours groups in D. Let a and a^{∗} be non central involutions in D and D^{∗} respectively. Put b = a^{a^{∗}}. Then D = ⟨a, b⟩ and U = ⟨a, a^{∗}⟩ ≅ D_{16}. Put t = f^{2} and let z be a singular involution perpendicular with t ∈ Q_{z}. As z is 2-central in C_{t} we may assume that U ≤ C_{z}. Since f^{2} ∉ ⟨z⟩, f ∉ Q_{z}. Let N be a normal subgroup of U with N ≰ Z(U) = ⟨t⟩. Then |N| ≥ 4, |U/N| ≤ 4 and so ⟨f⟩ = U′ ≤ N. Thus U ∩ Q_{z} = ⟨t⟩. Let C̃_{z} = C_{z}/Q_{z} ≅ Co_{1}. As U centralizes t, Ũ is contained in C̃_{z} ∩ C_{t} ≅ Co_{2}. In particular, f̃ is not of G_{2}(4)-type.

Since U ∩ Q_{z} = ⟨t⟩ we have a ∉ Q_{z} and so az is singular. By 3.1 there exists a unique ark V_{a} containing z and a. Consider the eights group E = ⟨t, a, z⟩ and the purely non-singular subgroup B = ⟨a, t⟩. Also tz is non-singular and so Case (b) of 3.5 must hold for E. So E lies in an ark. Since V_{a} is the unique arl containing z and a, we get t ∈ E ≤ V_{a}. Thus 5.3 implies that ã ∈ Ṽ_{a} is a 2A involution in Co_{2}. By symmetry all involution in Ũ except maybe f̃ are in 2A.

Suppose first that f̃ is in one of the classes 2A or 2B of Co_{2}. Then f̃ is 2-central in Co_{1} and there exists an ark V_{f} containing z and such that f̃ ∈ Ṽ_{f}. Since Q_{z}V_{f}/V_{f} is elementary abelian, t = f^{2} ∈ V_{f}. So 5.3 implies that f̃ ∈ Ṽ_{f} is 2A in Co_{2}. Hence {1, f̃, ã, b̃} is a pure 2A-subgroup of Co_{2}. But no such fours groups exists. ( For example by [ATLAS] the 2A involutions have trace −9 on the complex 23-space and 23 − 9 − 9 − 9 is negative, contradicting the orthogonally relations)

Thus f̃ is in the class 2C. Let x be an element of order four in Ũ. Let m be the trace of x on the complex 23-space. By [ATLAS] the 2A elements have trace −9 and the 2C-elements have trace −1. Thus the sum of the traces of the elements in Ũ is 23 + 4 · (−9) + (−1) + 2m = 2m − 14. Hence m ≥ 7. From [ATLAS] we conclude m = 7, x is in the class 4A and x^{2} is in the class 2A. A contradiction, since x^{2} = f̃.

(d) Suppose that both e and et are non-singular. Then by 3.6b and 3.5 applied to A⟨e⟩, there exist a unique singular involution z in A⟨e⟩. As D normalizes A⟨e⟩ we conclude that D centralizes z. But then D centralizes ⟨t, e, z⟩ = A⟨e⟩, a contradiction.

(e) Let L = ⟨t⟩X^{∞}. Then L ∼ 2.F_{4}(2). So by [AS] there exists e ∈ L with e^{2} ∈ ⟨t⟩ such C_{L/⟨t⟩}(e) ∼ [2^{15}]Sp_{6}(2). By [ATLAS] e has order two and e is not conjugate to et in L. Thus C_{L}(e) ∼ [2^{16}].Sp_{6}(2). By (d) (and replacing e be et if necessary) we may assume that e is singular. Note that f ∉ Q_{e}. Let s = f if f^{2} ∈ Q_{e} and s = f^{2} if f^{2} ∉ Q_{e}. Then s̃ is an involution in C̃_{e} = C_{e}/Q_{e}. Also [DC_{L}(e), s] ≤ ⟨s^{2}⟩ ≤ Q_{e}. Since C̃_{L}(e) has factor group Sp_{6}(2) we conclude form [ATLAS] that C_{C_{e}}(s̃) ∼ 2^{1+8}Ω_{8}^{+}(2). Since no fours group in 2^{1+8}Ω_{8}^{+}(2) has a centralizer involving Sp_{6}(2) we conclude that D̃ is neither isomorphic to D_{8} nor a fours group. Since f̃ ≠ 1 we conclude that D ∩ Q_{e} ∈ {A, B}. If X induces only inner automorphisms on L/⟨t⟩. Since L/⟨t⟩ is perfect, we conclude that X = C_{X}(L)L and so there exists x ∈ C_{X}(L) \ DL. Then by the proven part of (b), A^{x} = B, a contradiction as x normalizes D ∩ Q_{z}. Thus (e) holds.

(f) Moreover, a ∈ A \ Z(D) and b ∈ B with ab = f. As a ∈ Q_{e}, ae is non-singular and as b ∉ Q_{e}, be is singular. Put D^{∘} = ⟨ae, be⟩. Then (f) holds.

(g) By [AS]^{2}E_{6}(2).2 has exactly two classes of involutions outside^{2}E_{6}(2). Thus (g) follows from (a) and (f).

(h) The uniqueness part of (h) and the structure of N_{X}(S)/S follows from [AS]. Suppose that there exists an involution d in S \ D and let a be an involution D with a ≠ t. Then D = ⟨a, a^{d}⟩ and so S = ⟨a, d⟩ ≅ D_{16} and D^{∗} := ⟨d, d^{a}⟩ ≅ D_{8}. Moreover, C_{X}(d) involves ^{2}F_{4}(2)′ and so C_{M}(d) is divisible by 13. Thus d is non-singular. By (e) D and D^{∗} are conjugate in M and as ⟨f⟩ is the unique cyclic subgroup of order four in D ( in D^{∗}), D and D^{∗} are conjugate in N_{G}(⟨f⟩). But this contradicts (c). Hence S \ D contains no involutions, S ≅ SD_{16} and all part of 5.4 are proved.

Lemma 5.5 Let F be the set of elements of order four squaring to a nonsingular involution, f ∈ F and F = N_{M}(⟨f⟩). Then (a) M acts transitively on F.

(b) F ∼ D_{8}.Aut(F_{4}(2)).

(c) F/O_{2}(F) has a unique class of subgroup of order two not contained in (F/O_{2}(F))′. Moreover if S is the inverse image in F of such a group, then S ≅ SD_{16} and N_{F} (S)/S ≅ ^{2}F_{4}(2).

Proof: (a) Let f ∈ F and choose H as in 5.2. By 5.4 H is conjugate to D^{∘}. Since the two elements of order four in D^{∘} are conjugate in D^{∘}, (a) is proved.

To prove (b) and (c) let D, X and f be as in 5.4. Put L = N_{M}(⟨f⟩). Then (b) and (c) follow from 5.4a,g once we establish that L = X. By 5.4c and the Z^{∗} theorem ([Gl]) applied to L/⟨f⟩ we have L = O(L)X, where O(L) is the largest normal subgroup of order odd order in L. Let z be an involution in C_{M}(D) with z ≠ f^{2}. By 5.4d z or zf^{2} is singular and so by 2.2 ⟨f, z⟩ is of 2-type. Thus O(C_{M}(⟨f, z⟩)) = 1 and z inverts O(L). Since this is true for any involution in C_{M}(D) distinct from f^{2} and as C_{M}(D) contains elementary abelian groups of order eight we conclude that O(L) = 1 and L = X.

6 Proof of Theorem B

In this section we prove Theorem B. So let \overline{P} be a maximal 2-local of BM = \overline{B} such that \overline{P} is not of characteristic 2. Let P its preimage in B, R = O_{2}(P) and T = ⟨t⟩. Then P = N_{B}(R). We claim that R is not of 2-type in M.

Let U be the preimage of C\overline{B}(\overline{Q}) in B. Since \overline{P} is not of characteristic p, U ≰ R and U is not a 2-group. Since [R, U] ≤ T and [T, U] = 1 we get [R, O^{2}(U)] = 1. Hence C_{M}(R) is not a 2-group. Since t ∈ R,C_{M}(R) = C_{B}(R) = C_{P} (R) and so O_{2}(C_{M}(R)) ≤ R. Thus by 2.1(b) R is not a of 2-type, proving the claim.

So 2.2 none of the involutions in R are of 2-type. That is all the involutions in R are non-singular.

Let A be a normal subgroup of P minimal with respect to A ≰ T. Then P = N_{B}(A). Note that AT/T

is elementary abelian and AT ≤ R. So if x ∈ T A^{#} then x is a nonsingular involution or x has order 4.

Suppose that A contains an involution a with a ≠ t. Let D = ⟨a, t⟩. Then D is a purely non-singular fours group. Note that b^{2} ∈ T ≤ A for all b ∈ C_{AT} (D). Thus by 3.6c, C_{AT} (D) is elementary abelian and by 3.6b, |C_{AT} (D))| ≤ 4. Thus C_{AT} (D) = D. Since D is normal in AT and |D| = 4 we get |AT/D| ≤ 2. Hence AT is a fours group or AT ≅ D_{8}. If AT is a fours group, then by 3.4 AT is unique up to conjugacy in M. By 3.5 we conclude that N_{M}(AT) induces Sym(3) on AT. Thus N_{B}(AT) induces C_{2} on AT, A = AT and A is unique up to conjugacy in B. So by 3.5, Case (1) of Theorem B holds. If AT ≅ D_{8}, then A = AT and P normalizes the unique cyclic group of order four in A, a contradiction to the minimal choice of A.

Suppose next that t is the only involution in A. Then A ≅ C_{4} or Q_{8}. If A ≅ C_{4}, then by 5.5, Case (2) of Theorem B holds.

So suppose A ≅ Q_{8}. Let R be one of the three cyclic subgroups of order four in A. Then T ≤ R. Put D = O_{2}(N_{M}(R)). Then by 5.5, D ≅ D_{8}. Note that A ≤ N_{M}(R) and so A normalizes D. Let F be a fours group in D and suppose that A normalizes F. Then |A/C_{A}(F)| ≤ 2. So there exists H ≤ C_{A}(F) with H ≅ C_{4}. Then Φ(H) = T ≤ A and |HA| = 8, a contradiction two 3.6c.

Thus A does not normalize F and so A ≰ DC_{M}(D). Put S = DA. Then S/D has order two and by 5.5 S is unique up to conjugation in N_{M}(D) and N_{M}(S)/S ≅ ^{2}F_{4}(2). Note that A (D) is the unique subgroup of S isomorphic to Q_{8} (D_{8}). . So A,D and R = A ∩ D are all normal subgroups of N_{M}(S). Since S = DA and N_{M}(D) = N_{M}(R) we get N_{M}(S) = N_{M}(D) ∩ N_{M}(A) = N_{M}(R) ∩ N_{M}(A). Moreover, N_{M}(S) = D(N_{M}(S) ∩ C_{M}(A)) = DC_{M}(A). Since D∩C_{M}(A) = T we conclude C_{M}(A)/T ≅ ^{2}F_{4}(2) and C_{M}(A) ≤ N_{G}(S).

Let L be the subgroup of M generated by the various S as R runs through the three subgroups of order four in A. Since [S, C_{M}(A)] ≤ C_{S}(A) = T we get [L, C_{M}(A)] ≤ T and so C_{L}(A)/T ≤ Z(C_{M}(A)/A) = 1. Thus C_{L}(A) = T. Since [S, A] = R, L induces Sym(3) on A/T and so L/T = L/C_{L}(A) ≅ Aut(A) ≅ Sym(4). Thus N_{M}(A) = LC_{M}(A) and N_{M}(A)/T ≅ Sym(4) ×^{2}F_{4}(2). Thus Case (3) of Theorem B holds.

References

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

[CS] J.H.Conway, N.J.A.Sloane Sphere Packings, Lattices and Groups Springer Verlag, New York, 1988

[Gl] G. Glauberman, Central elements in core-free groups, J. Algebra 4 (1966) 403–420.

[MS] U. Meierfrankenfeld and S. Shpectorov, The maximal 2-local subgroups of the Monster and Baby Monster, I.

[Ti1] J. Tits, Buildings of Spherical Type and finite BN-pairs , Lecture Notes in Mathematics 386, Springer-Verlag (1974).

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