Dade’s conjecture for the McLaughlin Simple Group John C. Murray

Abstract

In [Da92] E.C. Dade announced the first of what was to be a series of conjectures concerned with counting characters in the blocks of finite groups. Specifically, Dade’s so-called Ordinary Conjecture asserts that if a finite group G has trivial p-core, and B is a p-block of G of positive defect, then the number k(B, d) of complex irreducible characters with fixed defect d belonging to B can be expressed as an alternating sum over the corresponding numbers for the normalizers of the non-trivial p-chains of G. Refined versions of this conjecture have been given in subsequent articles. See [Da94] and [DaPr] for further details. Dade claims that the strongest form of his conjectures is true for all finite groups if it is true for all covering groups of finite simple groups. Thus, in order to prove his conjectures, one merely has to go through the complete list of finite simple groups, carefully verifying that the strongest form holds for each one. In this Thesis we prove the most general version of Dade’s Conjectures for all covering groups of the McLaughlin group M_{c}. This is one of the 26 sporadic simple groups. So our Thesis forms a portion of the as yet incomplete proof of Dade’s Conjectures for all finite groups.

Dedication

To my wife Deirdre and our children Clara and Seóirse.

Acknowledgement

I thank my advisor Everett C. Dade for his patience and humour. He was extremely generous with his time, particularly during the painstaking task of removing the numerous typographical and conceptual errors that crept into my manuscript. John Walter and Nigel Boston gave me valuable advice on ways to simplify the proofs of many of my results. I would also like to thank the other members of my Thesis Committee, Derek Robinson and Michio Suzuki, for their time. My fellow graduate student Gary Schwartz suggested using the isomorphism 𝔄_{8} ≅ GL(4, 2) to simplify the tabulation of the radical 2-subgroups of 𝔄_{8}. Finally, I express my gratitude to Nigel Buttimore of Trinity College Dublin for inspiring me to begin my graduate career in mathematics, and Thomas Laffey and Roderick Gow of University College Dublin for introducing me to Group Theory and Representation Theory.

Contents

List of Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2. Representation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3. Radical p-Chains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4. The Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Chapter 2. The Prime p = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.1. The Radical 5-chains of M_{c} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2. Determining the structure of N_{M_{c}.2}(P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3. The Character Degrees of P ⋊ D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4. The Ordinary Conjecture for the prime p = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5. The Invariant Characters of P ⋊ D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.6. The Invariant Conjecture for the prime p = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.7. The Projective Conjecture for the prime p = 5 . . . . . . . . . . . . . . . . . . . . . . . . . 35

Chapter 3. The Prime p=3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.1. The Groups E ⋊ M and F ⋊ L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2. The Radical 3-chains of M_{c} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3. The Character Degrees of F ⋊ L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4. The Character Degrees of W ⋊ Q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.5. The Character Degrees of E ⋊ M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.6. The Ordinary Conjecture for the prime p = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.7. The Groups E ⋊ (M × ⟨c⟩) and F ⋊ (⟨d⟩L) . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.8. The Invariant Characters of E ⋊ M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.9. The Invariant Characters of F ⋊ L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.10. The Invariant Characters of W ⋊ Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.11. The Invariant Conjecture for the prime p = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.12. The Schur Multiplier of M_{c} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.13. The Character Degrees of A.(E ⋊ M). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.14. The Character Degrees of A.(F ⋊ L). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.15. The Character Degrees of A.(W ⋊ Q). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.16. The Projective Conjecture for the prime p = 3 . . . . . . . . . . . . . . . . . . . . . . . . . 82

Chapter 4. The Prime p=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.1. The 2-local structure of M_{c} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2. The Radical p-subgroups of GL(n, q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.3. The Radical 2-subgroups of H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.4. The Radical 2-chains of M_{c} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.5. The Character Degrees of H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.6. The Character Degrees of N_{H}(R_{3,1}) and N_{M_{c}}(R_{3,1}) . . . . . . . . . . . . . . . . . . . . 112 4.7. The Character Degrees of N_{H}(R_{1,2,1}) and N_{M_{c}}(R_{1,2,1}). . . . . . . . . . . . . . . . . . 116 4.8. The Ordinary Conjecture for the prime p = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.9. The Invariant conjecture for the prime p = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.10. The Projective conjecture for the prime p = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Curriculum Vitae. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

List of Tables

2.1 The Radical 5-chains of M_{c} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 The Ordinary Conjecture for p = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3 The Invariant Conjecture for p = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.4 The Projective Conjecture for p = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1 The Radical 3-chains of M_{c} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 The Ordinary Conjecture for p = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3 The Invariant Conjecture for p = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.4 The Projective Conjecture for p = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.1 Notation for Elements and Subgroups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.2 The Radical 2-subgroups of M_{c} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.3 The Remaining Radical 2-chains of M_{c} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.4 The Ordinary Conjecture for p = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.5 Notation for the remainder of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.6 The Irreducible characters of ℤ_{2} ≀ 𝔖_{4} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.7 The Invariant Conjecture for p = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.8 The Projective Conjecture for p = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

x

Chapter 1 Introduction

In [Da92] E.C. Dade announced his “Ordinary Conjecture” which asserts that if a finite group G has trivial p-core, and B is a p-block of G of positive defect, then the number k(B, d) of complex irreducible characters with fixed defect d belonging to B can be expressed as an alternating sum, as in 1.4.2, over the corresponding numbers for the normalizers of the non-trivial p-chains of G. In subsequent articles [Da94] and [DaPr] he gave stronger versions of this conjecture. He has announced that the strongest form of his conjectures, the so-called “Inductive Conjecture”, will have the property that it is true for all finite groups if it is true for all covering groups of finite simple groups.

The aim of this thesis is to verify that the strongest form of Dade’s conjectures holds for all covering groups of the McLaughlin Simple Group M_{c}, for all primes p. We use some basic facts about M_{c} obtained from the Atlas [Con85] to simplify our task. The group M_{c} has exactly two covering groups: M_{c} and 3.M_{c}. The outer automorphism group Out(M_{c}) of M_{c} is cyclic of order 2. In this situation Dade asserts in [DaPr] that the Inductive Conjecture for M_{c} is equivalent to the weaker “Invariant Conjecture” as outlined in 1.4.4. Furthermore, Dade has proved in [Da96] that this Invariant Conjecture is true for all blocks with cyclic defect groups. The group Out(3.M_{c} | A) of outer automorphisms of 3.M_{c} centralizing A = Z(3.M_{c}) is trivial.

In this situation Dade asserts in [DaPr] that the Inductive Conjecture is equivalent to the “Projective Conjecture”, as outlined in 1.4.6. Moreover, this weak form of the Inductive Conjecture holds for blocks with cyclic defect groups, as shown in [Da96]. From the Atlas |M_{c}| = 2^{7} · 3^{6} · 5^{3} · 7 · 11. For both covering groups we need therefore only worry about the primes p = 5, 3, 2.

In the Introduction we outline the theory needed to understand the conjectures and the methods we will use. The rest of the Thesis consists of three chapters; one for each of the primes p = 5, 3, 2. We deal with p = 5 first, as it is the easiest case. It turns out that the 3-local structure is useful for determining the 2-local structure, so we deal with p = 3 before p = 2. In all cases our first task is to obtain enough information on the p-local structure to be able to determine the conjugacy classes of radical p subgroups of M_{c} and their normalizers in M_{c} and M_{c}.2. This in turn enables us to determine the conjugacy classes of radical p-chains and to describe their normalizers. Finally we use Clifford Theory to obtain the character degrees of the the chain normalizers. The Invariant Conjecture for M_{c} and p = 5 follows from Theorems 2.4.2 and 2.6.2, while the Projective Conjecture for 3.M_{c} and p = 5 follows from Theorem 2.7.1; for p = 3 the corresponding Theorems are 3.6.2, 3.11.2 and 3.16.2; while for p = 2 the relevant Theorems are 4.8.3, 4.9.19 and 4.10.12. These results are enough to prove that the Inductive Conjecture holds for all covering groups of M_{c}.

A final word. The philosophy behind Dade’s conjectures is that one can obtain information on the global block structure of a finite group G from knowledge of the local block structure. The group 3.M_{c} is best viewed as a certain maximal subgroup of Conway’s Group Co. However, it turned out that in order to obtain the information needed to prove the results in our thesis, it was enough to consult the Atlas as well as obtain information on the local structure of M_{c} from [JW72] and [Fk73]. Thus the methods of proof found here are in accord with the local-to-global approach promoted by these conjectures.

1.1. Notation Due to the limited number of symbols available, our notation may vary between

chapters. We will try to warn the reader whenever this occurs.

Some of our notation is nonstandard. In particular if G is a finite group and X is a G-set, then Orb(G, X) will denote the multiset of orbit lengths of G on X, rather than the set of orbits of G on X. We will often deal with the situation where G has normal subgroups X and Y, with Y ≤ X. Then Irr(G) will denote the set of ordinary irreducible characters of G, while Irr(G | X) will denote those elements of Irr(G) which have non-trivial restriction to X. The set Irr(G | G) will be denoted by the more usual Irr(G)^{×}. When we wish to refer to the multiset of the degrees of those irreducible characters of G which are nontrivial on X but trivial on Y, we use Deg(G | X/Y). If now G is embedded as a normal subgroup of some finite group E, while X and Y are normal subgroups of E, and not just G, then Inv(G | X/Y) will stand for the multiset of degrees of those irreducible characters of G which are invariant in E, and non-trivial on X but trivial on Y. The group Y is omitted if it is {1}. As would be expected, Deg(G/X) or Inv(G/X) refer to the characters of the factor group G/X. We use Deg(G mod X) and Inv(G mod X) to refer to the characters of G inflated from G/X.

The p-defect of an irreducible character χ ∈ Irr(G) is the largest power of p which divides |G|/χ(1). We use Def_{p}(G) to refer to the multiset of p-defects of the irreducible characters of G. If the p is omitted, the context should make clear what prime we are using. We use InvDef_{p}(G) to refer to the multiset of p-defects of the irreducible characters of G which are invariant in the extension E of G. The group E will be clear from the context. We can extend these notations by using Def_{p}(G | X/Y) and InvDef_{p}(G | X/Y), when referring to characters of G nontrivial on X but trivial on Y. We use Def_{p}(G mod X) and InvDef_{p}(G mod X) to refer to all the characters of G which are inflated from G/X. In particular, the defects are measured using G rather than G/X.

In all cases, if we replace the normal subgroup X of G by an irreducible character ξ of X, then we will be referring to characters of G lying over ξ. For instance Deg(G | ξ) is the multiset of the degrees of those irreducible characters of G whose restriction to X contains ξ as a constituent. Here the normal subgroup X is determined by the character ξ.

There is some notation specific to Dade’s Conjecture which will be explained in Section 1.4. In addition, the following notation will be used throughout:

M_{c}, M_{c}.2  McLaughlin Simple group, extension of M_{c} by ℤ_{2}.
Q_{n}  Generalized quaternion group of order n.
D_{n}  Dihedral group of order n.

M_{10}  Mathieu group on 10 letters.
p  Almost invariably a prime number.
GF(q)  Finite field with q elements, q a power of p.
GL(n, q) = GL_{n}(q) General Linear Group of dimension n over GF(q).
SL(n, q) = SL_{n}(q) Special Linear subgroup of GL(n, q),

consisting of all matrices of determinant 1.

𝔖(X)  Group of permutations of the set X.
𝔖_{n}  Symmetric group on n letters.
𝔄_{n}  Alternating subgroup of 𝔖_{n},

consisting of all permutations of sign +1.

ℤ_{m}, ℤ_{m}^{n}  Cyclic group of order m, direct product of n copies of ℤ_{m}.
G, G^{#}  Arbitrary finite group, non-trivial elements of that group.
n.G  Central extension of a cyclic group of order n by G.
G/N  Quotient group of G by a normal subgroup N.
N : G = N ⋊ G  Semidirect product of a group N with G.
N.G  Group with normal subgroup N and quotient G ≅ (N.G)/N.
B ≀ G  Wreath product of a group B with a permutation group

G of degree n, giving a group of the form B^{n} ⋊ G.

Z(G)  Center of G.
Φ(G)  Frattini subgroup of G.
O_{p}(G)  Largest normal p-subgroup of G, also called the p-core of G.
⟨x_{1}, ⋯, x_{n}⟩  Group generated by x_{1}, x_{2},⋯,x_{n}.

Ω(P)  Subgroup ⟨x ∈ P | x^{p} = 1⟩ of P, where P is a p-group.
ΩZ(P)  Ω(Z(P)), largest exponent p-subgroup of Z(P).
[x_{1}, x_{2}]  x_{1}^{−1}x_{2}^{−1}x_{1}x_{2}, commutator of x_{1} with x_{2}.
H, K  Arbitrary subgroups of G.
[H, K]  ⟨[h, k] | h ∈ H, k ∈ K⟩, commutator subgroup of H with K.
G′ = [G, G]  Derived subgroup of G.
G_{ab} = G/G′  Largest abelian quotient of G.
H^{x} = x^{−1}Hx  Conjugate subgroup of H by x.
x_{1}^{x_{2}} = x_{2}^{−1}x_{1}x_{2}  Conjugate of x_{1} by x_{2}.
N_{G}(X)  Normalizer in G of a subset X of some G-set,

where X will often be a subgroup of G.

C_{G}(X)  Centralizer in G of a subset X of some G-set.
ψ, ψ^{G}  Complex character of H, character of G induced by ψ.
I_{G}(ψ)  Stabilizer or inertial subgroup of ψ in G.
\overline{ψ}  Complex conjugate of ψ.
B(ψ)  p-block of H containing ψ.
b^{G}  If defined, the p-block of G induced,

in the sense of Brauer, from the p-block b of H.

V^{∗} = Hom_{𝔉}(V, 𝔉) Dual space of V, where V is a vector space over a field 𝔉.
(nA), (nB), ⋯  Atlas notation for conjugacy classes.

Notation used throughout the thesis

1.2. Representation Theory Block Theory. Let G be a finite group. We let ℜ be a local principal ideal

domain with field of fractions 𝔉 of characteristic zero, unique maximal ideal 𝔓 equal to its Jacobson Radical J(ℜ) and residue field 𝔎 = ℜ/ J(ℜ) of prime characteristic p.

We have an ℜ-order ℜG which is naturally embedded in the group algebra 𝔉G = 𝔉 ⨂_{ℜ} ℜG of G over 𝔉. We assume that 𝔉G is a split 𝔉-algebra. In these circumstances the center of the ℜ-order ℜG, denoted by ℨ, has the property that ℨ/ J(ℨ) ≅ 𝔎 ⨁ ⋯ ⨁ 𝔎 as vector spaces over 𝔎 by [Da92, §4].

We denote by Z(𝔉G) the center of the group algebra 𝔉G. For our purposes a p-block of G corresponds to a maximal ideal of ℨ. Thus each p-block B is associated to an epimorphism ω_{B} : ℨ → 𝔎 of commutative ℜ-algebras. We denote by M_{B} the maximal ideal of ℨ which is the kernel of ω_{B}. The blocks B arise in the following way. Let χ be an irreducible complex character of G. Then

e_{χ} = \frac{χ(1_{G})}{|G|} ∑_{g∈G} χ(g^{−1})g

is the centrally primitive idempotent of 𝔉G associated to χ. This means that e_{χ} lies in Z(𝔉G), it is an idempotent, and the ideal 𝔉Ge_{χ} of 𝔉G is a simple ideal of 𝔉G. From Wedderburn’s Theorem and the fact that 𝔉G is split, 𝔉Ge_{χ} is a complete matrix algebra over 𝔉. Thus if ζ ∈ Z(𝔉G), we can find f ∈ 𝔉 such that e_{χ}ζ = fe_{χ}. The map sending ζ to f is χ/χ(1).

Now χ/χ(1) restricts to a map ℨ → 𝔉, where the image of ℨ is an ℜ-suborder of 𝔉. Thus the image coincides with ℜ. We let K_{χ} denote the kernel in ℨ of this map. So K_{χ} is a prime ideal of ℨ.

Lemma 1.2.1. ∩_{χ∈Irr(G)}K_{χ} = 0

Proof. This comes from looking at the corresponding intersection of kernels in Z(𝔉G).

Since ∩_{χ∈Irr(G)}K_{χ} ⊂ M_{B} and M_{B} is a prime ideal, there exists χ ∈ Irr(G) such that K_{χ} ⊂ M_{B}. But then M_{B} corresponds to a maximal ideal of ℜ = ℨ/K_{χ}. Hence M_{B} equals the inverse image 𝔓ℨ + K_{χ} of 𝔓 in ℨ. We say χ lies in B if K_{χ} is contained in M_{B}, or alternatively if the epimorphism ω_{B} factors through the map χ/χ(1). The set of irreducible characters of G lying in the block B will be denoted by Irr(B).

Heller’s Theorem, blocks and block idempotents.

Theorem 1.2.2 (Heller’s Theorem). Suppose ℜ is a local PID with maximal ideal 𝔓 = πℜ, with field of fractions 𝔉 of characteristic zero, and residue field ℜ/𝔓 = 𝔎 of prime characteristic p. If A is a split semisimple 𝔉-algebra and 𝔒 is a full ℜ-order inside A, then given an idempotent \overline{e} of 𝔒/ J(𝔒), there exists an idempotent e of 𝔒 such that \overline{e} = e + J(𝔒).

Proof. A proof of this well known theorem can be found in [CR81, I, 30.18].

Lemma 1.2.3. If R is any ring with identity, and e and f are idempotents of R such that e ≡ f modulo J(R) then there exists j ∈ J(R) such that e^{1+j} = f.

Proof. The proof of this lemma is standard and has been omitted.

In our situation, since ℨ/ J(ℨ) ≅ 𝔎 ⨁ ⋯ ⨁ 𝔎, we have that each block B of G corresponds to a primitive (central) idempotent \overline{e_{B}} of ℨ/ J(ℨ). By Heller’s Theorem 1.2.2, to each such idempotent \overline{e_{B}} there corresponds an idempotent e_{B} of ℨ, such that e_{B}+J(ℨ) = \overline{e_{B}}. Since ℨ is commutative, Lemma 1.2.3 shows that such an idempotent is uniquely determined. It is also clearly primitive. Hence the blocks B of G correspond one-to-one to the primitive (central) idempotents of ℨ, which in turn correspond oneto-one with the indecomposable ℜ-order summands of ℜG.

Lemma 1.2.4. The block B of G corresponds to the idempotent e of ℨ if and only if ω_{B′}(e) = δ_{B′,B} for all blocks B′ of G, where δ_{i,j} is zero or one depending on whether i and j are equal or not.

Proof. This is immediate from our definition of a block B.

Lemma 1.2.5. e_{B} = ∑_{χ∈Irr(B)} e_{χ}. If I ⊂ Irr(G) and ∑_{χ∈I} e_{χ} is in ℜG, then I is the union of Irr(B(χ)), where χ runs over the elements of I and B(χ) is the p-block of G containing χ.

Proof. The first statement is proved in [CR81, II, 56.25]. The second statement follows from the discussion above.

The Defect of a Block. There are many ways to define the family of defect groups of a p-block B of G. If C is a conjugacy class of G then a defect group of C is a Sylow p-subgroup of C_{G}(σ) for any σ ∈ C. The family of defect groups of C clearly forms a single G-orbit of p-subgroups of G. We will denote by C̃ the class sum ∑_{σ∈C} σ of C. So C̃ ∈ Z(𝔉G). Define

Υ(B) = {P |P is a defect group of some conjugacy
class C of G satisfying ω_{B}(C̃) ≠ 0}.

(1.2.6)

A defect group D of B is any minimal element of Υ(B). It is not immediately obvious, but the family of all defect groups of B forms a single G-orbit, denoted Def(B), of p-subgroups of G. See [NT89, 5.1.11(ii)].

Definition 1.2.7. The defect d(B) of a p-block B of G is a non-negative integer determined by

p^{d(B)} = |D|, where D ∈ Def(B).

Proposition 1.2.8. Let B be a p-block of G of defect d. Then d is the maximum integer such that

p^{d} | \frac{|G|}{χ(1)} for some χ ∈ Irr(B).

Proof. This follows from [CR81, II, 56.33, 56.41].

We mention that the p-block B_{0} of G containing the trivial character is called the principal block of G. Its defect groups are the Sylow p-subgroups of G. At the other extreme, a p-block B of G of defect 0 is known to contain a single ordinary irreducible character χ_{B}, which satisfies p | |̸G|/χ_{B}(1). Moreover, any irreducible character satisfying this equation lies in a p-block of defect 0.

The following lemma will be used repeatedly in this thesis, usually without being explicitly referenced.

Lemma 1.2.9. If G has a normal p-subgroup P such that C_{G}(P) ≤ P, then the principal p-block B_{0} is the unique p-block of G.

Proof. See [AL93, Q2, p112] or [Ft82, V.3.11].

Some Clifford Theory. Clifford theory connects the representations of a finite group to those of a normal subgroup. In the following G will denote a finite group, N a normal subgroup of G and χ an irreducible character of N.

Theorem 1.2.10. Suppose G/N is cyclic and χ is invariant in G. Then χ extends to G.

Proof. A proof can be found in [Is76, 11.22].

Next we have a very useful result which says that the extendibility of a character depends only on the Sylow subgroups of the quotient group.

Theorem 1.2.11. Suppose χ is invariant in G. Then χ extends to G if and only if it extends to every subgroup P ≥ N of G for which P/N is a Sylow subgroup of G/N.

Proof. A proof can be found in [Is76, 11.31].

We can in fact focus attention on those Sylow p-subgroups for which p | |N|. Let V be an irreducible right G-module, and let χ ∈ Irr(G) be the irreducible character of the corresponding representation τ : G → End(V) of G on V. We give the following two definitions, which can be found in [Is76, p 88].

Definition 1.2.12. The determinant character λ = det(χ), of χ is defined by setting λ(g) = det(T (g)), where T (g) is the matrix of τ(g) with respect to some choice of basis for V, and g is any element of G.

Definition 1.2.13. The determinantal order o(χ) = o(λ) of χ is the order of the determinant character λ as an element of the group of linear characters of G.

Thus o(χ) = |G : ker(λ)|.

Theorem 1.2.14. Suppose χ is invariant in G and

(|G : N|, o(χ)χ(1)) = 1.

Then there is a unique extension χ̃ of χ to G with the property that o(χ̃) = o(χ). In particular, χ has such an extension if (|G : N|, |N|) = 1.

Proof. This is a theorem due to Gallagher, and a proof can be found in [Is76, 8.16].

Generally we will be interested in whether or not χ extends to its stabilizer I_{G}(χ) in G. The following provides the solution when N is abelian and complemented in G.

Theorem 1.2.15. Suppose G = N ⋊ K and N is abelian. Then every χ ∈ Irr(N) can be extended to χ̃ ∈ Irr(I_{G}(χ)). Moreover I_{G}(χ) = N ⋊ I_{K}(χ).

Proof. A proof can be found in [CR81, I, 11.8].

The next theorem gives a complete description of what happens when a character of a normal subgroup extends to its stabilizer.

Theorem 1.2.16. Suppose χ extends to a character χ̃ of its stabilizer S = I_{G}(χ) in G. Then

(1) χ̃ ∈ Irr(S);

(2) (ωχ̃)^{G} ∈ Irr(G) for each ω ∈ Irr(S mod N);

(3) χ^{G} = ∑_{ω∈Irr(S mod N)} ω(1)(ωχ̃)^{G} gives χ^{G} as a linear combination of the distinct irreducible characters (ωχ̃)^{G} of G lying over the G-orbit of χ.

Proof. A proof can be found in [CR81, I, 11.5].

The following is in fact a special case of this last theorem.

Theorem 1.2.17. Suppose C_{G}(x) ≤ N for all x ∈ N^{×}. Then

(1) For ω ∈ Irr(N)^{×} we have I_{G}(ω) = N and ω^{G} ∈ Irr(G);

(2) For χ ∈ Irr(G) with N ⊈ ker(χ), we have χ = ω^{G}, for some ω ∈ Irr(N)^{×};

(3) If ω_{1} and ω_{2} ∈ Irr(N) are conjugate by an element of G, then ω_{1}^{G} = ω_{2}^{G}. Hence there is a 1-1 correspondence between the elements of Irr(G | N) and the G-conjugacy classes of non-trivial characters of N.

Proof. This theorem is due to Frobenius. A proof can be found in [CR81, I, 11.11].

A group G possessing a normal subgroup N satisfying the hypothesis of the above theorem is called a Frobenius Group. The normal subgroup N is called a Frobenius Kernel. It is a fact that in these circumstances G = N ⋊ K, where the complement K is referred to as a Frobenius Complement to N.

If P is an extra-special group of order p^{1+2n} for p prime and n ≥ 0, it can be shown easily, using Clifford Theory and results on alternating bilinear forms, that P has characters of the following degrees.

Lemma 1.2.18. Suppose P is an extra-special group of order p^{1+2n} with center Z(P) of order p. Then,

Deg(P) = {1^{p^{2n}}, (p^{n})^{(p−1)}}.

Moreover the p − 1 irreducible characters of P of degree p^{n} are zero outside Z(P) and each lies over exactly one irreducible character of Z(P).

Proof. A proof of this result can be found in [Hu67, 16.14].

In many cases we shall be dealing with a group G acting as automorphisms of an abelian group Y. Since Clifford theory is concerned with orbits of characters, the following lemma is useful when we are trying to compute the characters of Y ⋊ G induced from Y.

Lemma 1.2.19. If G is a group acting as automorphisms of an abelian group Y, then the number of G-orbits on Y^{#} equals the number of G-orbits on Irr(Y)^{#}.

Proof. The permutation characters of G afforded by the G-sets Y^{#} and Irr(Y)^{#} are the same. The result now follows from [CR81, I, Ex. 10.2(ii)].

We also have the following:

Lemma 1.2.20. Let G be a group acting as linear transformations of a finitedimensional vector space V over a field K, and V^{∗} be the dual KG-module Hom_{K}(V, K). Then

1. Dim_{K}(C_{V^{∗}}(G)) = Dim_{K}(V/[V, G]);

2. If K has finite prime order p and G is a finite p′-group, then Dim_{K}(C_{V} (G)) = Dim_{K}(C_{V^{∗}}(G)).

Proof. 1. Let ψ ∈ V^{∗}. Then

ψ ∈ C_{V^{∗}}(G) ⟺ ψ(v^{g}) = ψ(v), ∀v ∈ V, g ∈ G
⟺ ψ(−v + v^{g}) = 0, ∀v ∈ V, g ∈ G
⟺ ψ([v, g]) = 0, ∀v ∈ V, g ∈ G
⟺ [V, G] ≤ ker(ψ).

So C_{V^{∗}}(G) is isomorphic to the dual space of V/[V, G]. Hence Dim_{K}(C_{V^{∗}}(G)) = Dim_{K}(V/[V, G]).

2. Since V is abelian, V = C_{V} (G) × [V, G] by [As86, 24.6]. So Dim_{K}(V/[V, G]) = Dim_{K} C_{V} (G). The result follows using part 1.

Finally, we present a result which we will use frequently to compute the character degrees of “small” groups. We say N ≤ G is subnormal in G if there is a sequence of subgroups of G of the form

N ⊴ N_{1} ⊴ N_{2} ⊴ ⋯ ⊴ N_{k} ⊴ G.

Theorem 1.2.21. Let N ≤ G be subnormal and abelian. Then χ(1) | |G : N| for every χ ∈ Irr(G).

Proof. This theorem is due to Ito. A proof may be found in [Is76, 11.30].

1.3. Radical p-Chains The objects of our investigation will be the radical p-chains of the McLaughlin

Group.

Let G be a finite group and p any prime. A p-chain C of G is any non-empty, strictly increasing chain

C: P_{0} < P_{1} < P_{2} < ⋯ < P_{n}

of p-subgroups P_{i} of G. The length |C| of the chain is the number n of inclusions. For each i = 0,1,⋯,n, the chain C_{i} is the initial subchain

C_{i} : P_{0} < P_{1} < P_{2} < ⋯ < P_{i}

of C. The family of all p-chains of G will be denoted by P. This is a G-set, with G acting by conjugation. If C is the chain above and g ∈ G, then the conjugate chain C^{g} is given by

C^{g} : P_{0}^{g} < P_{1}^{g} < P_{2}^{g} < ⋯ < P_{n}^{g}.

The stabilizer of C in G is the set of all elements of G which conjugate C to itself. Hence it is the intersection

N_{G}(C) = N_{G}(P_{0}) ∩ N_{G}(P_{1}) ∩ ⋯ ∩ N_{G}(P_{n}).

We let P/G denote an arbitrary set of representatives of the G-orbits in P.

Lemma 1.3.1. If C is a p-chain and b is a p-block of N_{G}(C), then the induced block b^{G} is defined.

Proof. See [KR89, 3.2].

We recall that a radical p-subgroup of G is a p-subgroup R satisfying

R = O_{p}(N_{G}(R)).

A radical p-chain C of G is one satisfying

P_{0} = O_{p}(G), P_{i} = O_{p}(N_{G}(C_{i})), for i = 1, 2, ⋯, n.

Thus P_{0} is the p-core of G, and P_{i} is a radical p-subgroup of the normalizer of the initial subchain C_{i−1} for i = 1, 2 ⋯, n. The fact that P_{0} is a normal subgroup of G implies that P_{1} is a radical p-subgroup of G. In general P_{i}, for i > 1, need not be a radical p-subgoup of G.

We let R denote the family of all radical p-chains of G. It is clear that R is closed under conjugation by elements of G. We let R/G denote an arbitrary set of representatives for the orbits of G on R.

There are other natural families of p-chains of G. The normal p-chains are those satisfying P_{i} ⊴ P_{n}, for i = 0, 1, …, n, while the elementary abelian p-chains are composed of elementary abelian p-groups. These two families are denoted by N and E respectively. We let P′, N′ and E′ denote the subfamilies of all p-chains C in P, N and E, respectively, such that the first p-subgroup in C is P_{0} = {1}.

The following is very useful when dealing with radical p-subgroups.

Theorem 1.3.2. If R is a radical p-subgroup of G, and P is any p-subgroup of G satisfying N_{G}(R) ≤ N_{G}(P), then P ≤ R.

Proof. This proof is taken from [Da92, 1.3]. Since R normalizes P, the product P R is a p-subgroup of G. Let x ∈ N_{G}(R). Then x also normalizes P. So x normalizes RP. Hence N_{G}(R) ≤ N_{G}(P R). If y ∈ N_{P R}(R), then y^{x} ∈ N_{P R}(R) also. Hence N_{G}(R) ≤ N_{G}(N_{P R}(R)). In particular, N_{P R}(R) is a normal p-subgroup of N_{G}(R) containing R. This forces R = N_{P R}(R). So P R = R since P R is nilpotent. Hence P ≤ R.

Interesting consequences of Theorem 1.3.2 include the following.

Corollary 1.3.3. O_{p}(G) is the unique minimal radical p-subgroup of G under inclusion.

Proof. If R is a radical p-subgroup of G, then N_{G}(R) normalizes O_{p}(G).

Corollary 1.3.4. Every radical p-subgroup R of G is the intersection of the family of all Sylow p-subgroups of G containing it.

Proof. It is enough to note that the intersection mentioned above contains R and is normalized by N_{G}(R).

Radical p-subgroups behave extremely well with respect to normal subgroups and factor groups.

Proposition 1.3.5. Suppose η : G^{∗} → G is an epimorphism of finite groups whose kernel K is the direct product K = Q × Z of a normal p-subgroup Q of G with a central p′-subgroup Z of G. Then if P is a p-subgroup of G, there is a unique psubgroup P^{∗} = η̂(P) of G^{∗} containing Q such that η(P^{∗}) = P. The resulting map η̂ is an inclusion-preserving bijection of all p-subgroups of G onto all p-subgroups of G^{∗} containing Q. Moreover η̂ sends the radical p-subgroups of G one-to-one onto the radical p-subgroups of G^{∗}. We can define η̂(C), for any p-chain C of G^{∗}, as the pchain of G formed by the images of the constituent p-subgroups of C under η̂. Then η̂ sends the radical p-chains of G one-to-one onto the radical p-chains of G^{∗}.

Proof. This follows from Propositions 2.2 and 2.7 of [Da94].

Proposition 1.3.6. Say G is a finite group and N ⊴ G. Then

(1) If R is a radical p-subgroup of G then P = R ∩ N is a radical p-subgroup of N with N_{G}(P) ≥ N_{G}(R) and O_{p}(N_{G}(P)) ≤ R;

(2) If P is a radical p-subgroup of N then O_{p}(N_{G}(P)) is a radical p-subgroup of G and P = O_{p}(N_{G}(P)) ∩ N.

Proof. (1) Suppose R is radical in G and P = R ∩ N. Then N_{G}(P) ≥ N_{G}(R) since N ⊴ G. Set Q = O_{p}(N_{N}(P)). Then Q is characteristic in N_{N}(P). So N_{G}(Q) ≥ N_{G}(N_{N}(P)) ≥ N_{G}(R). By Theorem 1.3.2 we have Q ≤ R. But then Q ≤ R ∩ N = P. So Q = P and P is radical in N. Since N_{G}(P) ≥ N_{G}(R), it follows from Theorem 1.3.2 that O_{p}(N_{G}(P)) ≤ R.

(2) Suppose P is radical in N. Then O_{p}(N_{G}(P)) ∩ N is a normal p-subgroup of N_{N}(P) containing P. So P = O_{p}(N_{G}(P)) ∩ N. Since N ⊴ G, it follows that N_{G}(O_{p}(N_{G}(P))) ≤ N_{G}(P) ≤ N_{G}(O_{p}(N_{G}(P))). Hence O_{p}(N_{G}(P)) is radical in G.

Let P be a p-subgroup of G. Then P ◁ N_{G}(P). So P ≤ O_{p}(N_{G}(P)). Moreover N_{G}(P) ≤ N_{G} ( O_{p}(N_{G}(P)) ). Let P_{1} = O_{p}(N_{G}(P)), and in general let P_{n+1} = O_{p}(N_{G}(P_{n})), for n = 1, 2, · · ·. Then the P_{n}’s form a non-decreasing chain of subgroups of G. So for some N ≥ 1 we have P_{N+1} = P_{N}. Then P_{N} is a radical p-subgroup of G, with P ≤ P_{N} and N_{G}(P) ≤ N_{G}(P_{N}). This radical p-subgroup is called the radical closure of P in G. This argument shows that {N_{G}(R) | R is a radical p-subgroup of G} contains the set of maximal p-local subgroups of G.

Remark 1.3.7. If R is a p-subgroup of G with center Z(R), then R is a radical p-subgroup of G if and only if it is a radical p-subgroup of N_{G}(ΩZ(R)). Thus we may limit our search for radical p-subgroups of G to radical p-subgroups of the normalizers of elementary abelian p-subgroups of G.

1.4. The Conjectures The Ordinary Conjecture. Let G be an arbitrary finite group, B a p-block of

G and d an integer ≥ 0.

Definition 1.4.1. For any p-chain C of G we denote by k(C, B, d) the number of characters ψ ∈ Irr(N_{G}(C)) satisfying

d(ψ) = d and B(ψ)^{G} = B.

The integer k(C, B, d) depends only on the G-conjugacy class of the p-chain C.

Conjecture 1.4.2. If O_{p}(G) = {1}, and d(B) > 0 in the above situation, then

∑_{C∈R/G}(−1)^{|C|} k(C, B, d) = 0.

We say that this conjecture “holds for G” if it holds for all choices of blocks B and non-negative integers d.

Knörr and Robinson have shown in [KR89] that the R/G in the above sum can be replaced by P′/G, N′/G or E′/G, without changing the value of that sum.

The Invariant Conjecture. We now embed G as a normal subgroup of another finite group E. We fix an epimorphism ϵ : E ↠ \overline{E} of finite groups with kernel G. So we have an exact sequence

1 → G ⟶^{⊴} E ⟶^{ϵ} \overline{E} → 1

of finite groups and their homomorphisms. For any p-chain C of G we define N\overline{E}(C) to be ϵ(N_{E}(C)). Since the kernel of the epimorphism ϵ : N_{E}(C) → N\overline{E}(C) is the normal subgroup N_{G}(C) of N_{E}(C), we have an exact sequence

1 → N_{G}(C) ⟶^{⊴} N_{E}(C) ⟶^{ϵ} N\overline{E}(C) → 1

of finite groups associated with C.

The above group N_{E}(C) acts by conjugation on the set Irr(N_{G}(C)) of all ordinary irreducible characters ψ of its normal subgroup N_{G}(C). We denote by N_{E}(C, ψ) the stabilizer in N_{E}(C) of any such ψ, and by

N\overline{E}(C, ψ) = ϵ(N_{E}(C, ψ))

the image of that stabilizer in \overline{E}. Since N_{G}(C) is contained in N_{E}(C, ψ), we have an exact sequence

1 → N_{G}(C) ⟶^{⊴} N_{E}(C, ψ) ⟶^{ϵ} N\overline{E}(C, ψ) → 1

associated with any p-chain C of G and any character ψ ∈ Irr( N_{G}(C)).

From now on we fix a p-block B of G, an integer d ≥ 0, and a subgroup \overline{F} of \overline{E}.

Definition 1.4.3. For any p-chain C of G, we let k(C, B, d, \overline{F}) denote the number of characters ψ ∈ Irr(N_{G}(C)) satisfying

d(ψ) = d, b(ψ)^{G} = B and N\overline{E}(C, ψ) = \overline{F} .

The integer k(C, B, d, \overline{F}) depends only on the G-conjugacy class of the p-chain C. So the sum in the following conjecture is well defined.

Conjecture 1.4.4. If O_{p}(G) = {1}, and d(B) > 0 in the above situation, then

∑_{C∈R/G}(−1)^{|C|} k(C, B, d, \overline{F}) = 0.

We say that this conjecture “holds for G” if it holds for all choices of groups \overline{F}, blocks B and non-negative integers d.

Note that Conjecture 1.4.2 follows by summing the above equation over all the subgroups \overline{F} of \overline{E}.

The Projective Conjecture. Next we go back to an arbitrary finite group G. We pick some central extension Z.G of a cyclic group Z by G. So we fix an exact sequence

1 → Z ⟶^{⊴} Z.G ⟶^{η} G → 1

of groups such that Z is a cyclic central subgroup of Z.G. Then any p-chain C of G has a normalizer N_{Z.G}(C) in Z.G, the inverse image η^{−1}(N_{G}(C)) of its normalizer in G.

We fix a faithful linear character ρ of Z in addition to the non-negative integer d. The earlier p-block B of G is now replaced by a p-block B^{∗} of Z.G lying over the block B(ρ) of Z containing ρ.

Definition 1.4.5. For any p-chain C of G we define k(C, B^{∗}, d | ρ) to be the number of characters ψ ∈ Irr(N_{Z.G}(C) | ρ) such that d(ψ) = d and B(ψ) induces the p-block B^{∗} of Z.G.

Note that k(C, B^{∗}, d | ρ) depends only on the G-conjugacy class of the p-chain C. So the sum in the following conjecture is well defined.

Conjecture 1.4.6. If in the above situation, O_{p}(G) = {1}, and the Sylow psubgroup Z_{p} of Z is not a defect group of B^{∗}, then

∑_{C∈R/G}(−1)^{|C|} k(C, B^{∗}, d | ρ) = 0.

We say that this conjecture “holds for Z.G” if it holds for all choices of the above characters ρ, blocks B^{∗}, and non-negative integers d.

Chapter 2 The Prime p = 5

2.1. The Radical 5-chains of M_{c} From [Con85] a Sylow 5-subgroup P of M_{c} is of type 5_{+}^{1+2}, which indicates that it

is extra-special of order 5^{3} and exponent 5. In particular Z(P) is cyclic of order 5. Also from the Atlas N_{M_{c}}(P) = P⋊D where D ≅ ℤ_{3}⋊ℤ_{8}. In particular |N_{M_{c}}(P)| = 4×750.

Proposition 2.1.1. P is a trivial intersection subgroup of M_{c}. The elements of P\ Z(P) form a single N_{M_{c}}(P)-orbit.

Proof. Let γ be an element of Z(P)^{#}. So the order |C_{M_{c}}(γ)| of its centralizer is divisible by |P| = 125. From the Atlas the centralizers of elements in (5A) and (5B) have order 750 and 25, respectively. Hence γ must be a member of the class (5A). Since Z(P) = ⟨γ⟩ is normal in N_{M_{c}}(P), the subgroup C_{M_{c}}(γ) ∩ N_{M_{c}}(P) has index dividing |Aut(⟨γ⟩)| = 4 in N_{M_{c}}(P). So this subgroup has order divisible by |N_{M_{c}}(P)|/4 = 750. But |C_{M_{c}}(γ)| = 750. Thus C_{M_{c}}(γ) equals its intersection with N_{M_{c}}(P), and therefore is a subgroup of index 4 in that group. Hence N_{M_{c}}(P)/ C_{M_{c}}(γ) ≅ Aut(⟨γ⟩). We conclude that N_{M_{c}}(P) is all of N_{M_{c}}(⟨γ⟩).

Choose δ ∈ P from the class (5B) of M_{c}. From the Atlas |C_{M_{c}}(δ)| is of order 25. Thus C_{M_{c}}(δ) is the maximal elementary abelian subgroup of P containing δ. As |N_{M_{c}}(P)| = 25×120, there are 120 elements in the N_{M_{c}}(P)-conjugacy class of δ. These conjugates must be all the elements of P\ Z(P). This proves the second statement.

Suppose δ ∈ P^{η} for some η ∈ M_{c}. Then C_{M_{c}}(δ) is a maximal elementary abelian subgroup of both P and P^{η}. Hence Z(P^{η}) ≤ P. This implies that Z(P^{η}) = Z(P), since each element of Z(P^{η})^{#} lies in (5A), and all elements of P\ Z(P) lie in (5B) by the previous paragraph. But P is the unique Sylow p-subgroup of N_{M_{c}}(Z(P)). So P^{η} = P. Hence P is a trivial intersection subgroup of M_{c}.

Corollary 2.1.2. D acts faithfully on P.

Proof. If δ ∈ P\ Z(P), then C_{M_{c}}(δ) ≤ P. The result follows.

From Corollary 1.3.4 any radical p-subgroup of M_{c} is the intersection of the Sylow p-subgroups containing it. But P is a trivial intersection subgroup of M_{c}. Hence a radical 5-subgroup of M_{c} is either trivial or a conjugate of P. It follows that there are exactly two conjugacy classes of radical 5-chains in M_{c}.

Chain C Chain Description N_{M_{c}}(C) N_{M_{c}.2}(C) Parity
C_{1}  {1}  M_{c}  M_{c}.2  +
C_{2}  {1} < P  P ⋊ D P ⋊ N  −

Table 2.1. The Radical 5-chains of M_{c}

The description of N_{M_{c}.2}(C_{2}) comes from the following section.

2.2. Determining the structure of N_{M_{c}.2}(P) From the Atlas N_{M_{c}.2}(P) = P⋊N, where we may assume D ≤ N, and |N : D| = 2.

We let \overline{P} denote the factor group P/ Z(P).

Suppose η ∈ N\D centralizes \overline{P} = P/ Φ(P). From [Go80, 5.3.2] this implies that η centralizes P. Then by Corollary 2.1.2 we have η^{2} = 1. Hence η lies in the conjugacy class (2B) of M_{c}.2. But from the Atlas |C_{M_{c}}(η)| = 7920, which is not divisible by 5^{3}. This contradicts the fact that η centralizes P. Hence N acts faithfully on \overline{P}. So we may regard N as a subgroup of GL(2, 5) ≅ Aut(\overline{P}).

For the rest of this chapter we fix a generator x for the subgroup X = O_{3}(D) ≅ ℤ_{3}. Then N ≤ N_{GL(2,5)}(X). Since 3 | |̸GF(5)^{#}| = 4, the centralizer C_{GL(2,5)}(x) of x in GL(2, 5) is isomorphic to GF(25)^{#}. Moreover the normalizer N_{GL(2,5)}(X) of X in GL(2, 5) is isomorphic to the semi-direct product GF(25)^{#} ⋊ Gal(GF(25)) of GF(25)^{#} with the group Gal(GF(25)) ≅ ℤ_{2} of all automorphisms of the field GF(25). Since both N_{GL(2,5)}(X) and N have order 48, it follows that the two groups are identical.

Let

υ = [\begin{matrix}−1 & 1 \\ −1 & 3 \end{matrix}] ∈ GL(2, 5),

and Υ = ⟨υ⟩. Then υ has order 24. We may suppose x = υ^{8}. Thus

x = [\begin{matrix}−1 & 1 \\ −1 & 3 \end{matrix}]^{8} = [\begin{matrix}0 & 1 \\ −1 & −1 \end{matrix}] ∈ GL(2, 5).

It follows from the discussion above that Υ = C_{GL(2,5)}(x).

If we set

ι = [\begin{matrix}1 & 0 \\ −1 & −1 \end{matrix}] ∈ GL(2, 5),

and T = ⟨ι⟩, then ι is an involution, and

υ^{ι} = [\begin{matrix}1 & 0 \\ −1 & −1 \end{matrix}] [\begin{matrix}−1 & 1 \\ −1 & 3 \end{matrix}] [\begin{matrix}1 & 0 \\ −1 & −1 \end{matrix}] = [ \begin{matrix}3 & −1 \\ 1 & −1 \end{matrix}] = υ^{5}.

Thus Υ ⋊ T ≅ GF(25)^{#} ⋊ Gal(GF(25)). Moreover T normalizes X, so N = Υ ⋊ T. Let y = υ^{3} and Y = ⟨y⟩. So Y ≅ ℤ_{8} and

y = [\begin{matrix}3 & 1 \\ −1 & 2 \end{matrix}] ,

Also Υ = X × Y. Now D = X ⋊ Z for some cyclic group Z of order 8. The 2-group Z is X-conjugate to a subgroup of the Sylow 2-subgroup Y ⋊ T of N. Thus we may assume that Z ≤ Y ⋊ T. As Z is maximal in Y ⋊ T, it contains (Y ⋊ T)′ = ⟨y^{4}⟩. The group D is non-abelian, as M_{c} has no elements of order 24. Thus Z corresponds to a maximal subgroup \overline{Z} of (Y ⋊ T)_{ab} ≅ ℤ_{4} × ℤ_{2}, such that \overline{Z} is cyclic, but not contained in C_{N}(X)/⟨y^{4}⟩. Let \overline{y} and \overline{ι} denote the images of y and ι, respectively, in (Y ⋊ T)_{ab}. Then ⟨\overline{y} \overline{ι}⟩ is the only possibility for \overline{Z}. Thus Z = ⟨yι⟩. For the rest of this chapter we let z denote the generator yι of Z. So

z = yι = [\begin{matrix}3 & 1 \\ −1 & 2 \end{matrix}] [\begin{matrix}1 & 0 \\ −1 & −1 \end{matrix}] = [ \begin{matrix}2 & −1 \\ 2 & 3 \end{matrix}] .

2.3. The Character Degrees of P ⋊ D Since D is non-abelian but D/X is abelian, D′ = X. So D has 8 linear characters.

The subgroup ⟨xz^{2}⟩ of D is abelian and of index 2 in D. Hence by Theorem 1.2.21, all character degrees of D divide 2. We conclude that

(2.3.1)  Deg(D) = {1^{8}, 2^{4}},  Def_{5}(P ⋊ D mod P) = {3^{12}}.

From Proposition 2.1.1, the elements of P\ Z(P) form a single P ⋊ D conjugacy class. Since \overline{P} = P/ Z(P) is abelian, \overline{P}^{#} forms a single orbit of D. As |\overline{P}^{#}| = 24 = |D|, the group \overline{P} ⋊ D is Frobenius. By Theorem 1.2.17 it has a unique character of degree 24 induced from any non-trivial linear character of \overline{P}. Thus

(2.3.2)  Deg(P ⋊ D | P/ Z(P)) = {24},  Def_{5}(P ⋊ D | P/ Z(P)) = {3^{1}}.

We describe how GL(2, 5) acts on Z(P). Let ν_{1}, ν_{2} be elements of P whose images form a basis of \overline{P}. If

M = [\begin{matrix}m_{1,1} & m_{1,2} \\ m_{2,1} & m_{2,2} \end{matrix}] ∈ GL(2, 5),

then ν_{1}^{M} ≡ ν_{1}^{m_{1,1}}ν_{2}^{m_{1,2}} and ν_{2}^{M} ≡ ν_{1}^{m_{2,1}}ν_{2}^{m_{2,2}} modulo Z(P). Since P is extra-special, commutation in P gives a non-zero alternating bilinear form \overline{P} × \overline{P} → Z(P). Hence [ν_{1}, ν_{2}] generates Z(P), and we compute

[ν_{1}, ν_{2}]^{M} = [ν_{1}^{M}, ν_{2}^{M}] = [ν_{1}^{m_{1,1}}ν_{2}^{m_{1,2}}, ν_{1}^{m_{2,1}}ν_{2}^{m_{2,2}}]
= [ν_{1}, ν_{2}]^{m_{1,1}m_{2,2}}[ν_{2}, ν_{1}]^{m_{1,2}m_{2,2}}
= [ν_{1}, ν_{2}]^{m_{1,1}m_{2,2}−m_{1,2}m_{2,2}}
= [ν_{1}, ν_{2}]^{det(M)}.

So any element M of GL(2, 5) raises a generator of Z(P) to its det(M)^{th} power.

Since det(x) = 1 and det(z) = 3, the element x centralizes Z(P), while z sends every element of Z(P) to its third power. In particular D is transitive on Z(P)^{#}. Thus C_{D}(Z(P)) has index 4 in D. But ⟨xz^{4}⟩ ≅ ℤ_{6} centralizes Z(P), and is of index 4 in D. We conclude that

⟨xz^{4}⟩ = C_{D}(Z(P)). (2.3.3)

By Theorem 1.2.18 the extra-special group P has 4 irreducible characters of degree 5. Let ψ be one such. Then ψ|_{P \ Z(P)} ≡ 0. Hence I_{D}(ψ) = C_{D}(Z(P)).

By Theorem 1.2.10 the character ψ extends to ψ̃ ∈ Irr(P ⋊ ⟨xz^{4}⟩). Then by Theorem 1.2.16 the characters of P ⋊ D lying over the orbit of ψ are the 6 elements of the set {(ωψ̃)^{P⋊D} | ω ∈ Irr(⟨xz^{4}⟩)}. Each such character is of degree 20 = 5 · 4. Thus

(2.3.4)  Deg(P ⋊ D | Z(P)) = {20^{6}},  Def_{5}(P ⋊ D | Z(P)) = {2^{6}}.

From (2.3.1), (2.3.2), and (2.3.4) we have

(2.3.5) Deg(P ⋊ D) = {1^{8}, 2^{4}, 24, 20^{6}},  and  Def_{5}(P ⋊ D) = {3^{13}, 2^{6}}.

We can now prove the following

Proposition 2.3.6. The group N_{M_{c}}(C_{2}) = N_{M_{c}}(P) has a unique 5-block, which necessarily induces the principal 5-block, B_{0}, of M_{c}. Hence

k(C_{2}, B_{0}, 3) = 13,  k(C_{2}, B_{0}, 2) = 6,  and
k(C_{2}, B_{0}, d) = 0,  for all other values of d.

(2.3.7)

Proof. Since C_{P⋊D}(P) = Z(P), Lemma 1.2.9 implies that P ⋊ D has a unique 5-block. We obtain (2.3.7) from (2.3.5).

2.4. The Ordinary Conjecture for the prime p = 5

From [Con85] the group M_{c} has 19 characters in its principal block B_{0}, and 5 blocks of defect 0. We list the characters in the principal block and their defects:

Character χ_{1} χ_{2} χ_{3} χ_{4} χ_{5} χ_{6} χ_{7}  χ_{8}  χ_{10} χ_{11}
Degree  1  22 231 252 770 770 896  896 3520 3520
5-Defect  3  3  3  3  2  2  3  3  2  2
Character χ_{13} χ_{14} χ_{15} χ_{16} χ_{17} χ_{21} χ_{22}  χ_{23}  χ_{24}
Degree  4752 5103 5544 8019 8019 9856 9856 10395 10395
5-Defect 3 3 3 3 3 3 3 2 2

Thus

k(C_{1}, B_{0}, 3) = 13, k(C_{1}, B_{0}, 2) = 6, and
k(C_{1}, B_{0}, d) = 0,  for all other values of d.

(2.4.1)

Theorem 2.4.2. The Ordinary Conjecture holds for the McLaughlin simple group for the prime p = 5.

Proof. From Conjecture (1.4.2) and Table 2.1 on page 25, we need to prove

k(C_{1}, B_{0}, d) = k(C_{2}, B_{0}, d) (2.4.3)

for all values of d.

We obtain the following terms for the equation above for various values of d from (2.3.7) and (2.4.1):

5-Defect C_{1}  C_{2}
3  13 = 13
2  6 = 6

Table 2.2. The Ordinary Conjecture for p = 5

The terms in the above equation are zero for all other values of d. This proves the theorem.

2.5. The Invariant Characters of P ⋊ D Recall that N_{M_{c}.2}(P) = P ⋊ N, where N = Υ ⋊ T. Since Υ has index 2 in N,

we conclude from Theorem 1.2.21 that every character degree of N_{GL(2,5)}(X) divides 2. Furthermore [υ, ι] = υ^{4}, so N′ = ⟨υ^{4}⟩ is of index 8 in N. Thus

Deg(N) = {1^{8}, 2^{10}} (2.5.1)

We have Deg(D) = {1^{8}, 2^{4}}, by (2.3.1). So four of the linear characters of D are invariant in N, while the remaining four fuse to give two characters of degree 2. The four characters of D of degree 2 must all be invariant in N. Hence

(2.5.2)  Inv(D) = {1^{4}, 2^{4}}  and  InvDef_{5}(P ⋊ D mod P) = {3^{8}}.

The unique character of degree 24 of P ⋊ D is necessarily invariant in N. Thus

(2.5.3) Inv(P ⋊ D | P/ Z(P)) = {24}  and  InvDef_{5}(P ⋊ D | P/ Z(P)) = {3}.

Finally we deal with the six characters of Irr(P ⋊ D | Z(P)). As N normalises P, it necessarily normalises Z(P). Then since N acts transitively on Z(P)^{#}, it also acts transitively on Irr(Z(P))^{#}. Moreover, the stabilizer of any non-trivial linear character of Z(P) is of index four in N.

Recall from (2.3.3) that C_{D}(Z(P)) = ⟨xz^{4}⟩ is cyclic of order six. Since det(z^{2}) = −1 and det(ι) = −1, the elements z^{2} and ι invert Z(P). So ιz^{2} centralizes Z(P). Hence C_{N}(Z(P)) = ⟨xz^{4}, ιz^{2}⟩.

Now

(2.5.4)  (xz^{4})^{ιz^{2}} = (x^{−1}z^{4})^{z^{2}} = x^{−1}z^{4} = (xz^{4})^{−1},

while

(2.5.5)  (ιz^{2})^{2} = z^{10}z^{2} = z^{12} = (xz^{4})^{3}.

Thus T_{12} = ⟨xz^{4}, ιz^{2}⟩ is the unique nonabelian group ℤ_{3} ⋊ℤ_{4} of order 12 distinct from D_{12} and 𝔄_{4}. It is readily established that Deg(T_{12}) = {1^{4}, 2^{2}}.

Let ψ ∈ Irr(P | Z(P)). Then I_{N}(ψ) = T_{12}. By Theorem 1.2.14 the character ψ extends to a character ψ̃ of P⋊T_{12}. Moreover, by Theorem 1.2.16 the set {(ωψ̃)^{P⋊N} | ω ∈ Irr(T_{12})} gives the irreducible characters of P ⋊ N lying over the P ⋊ N-orbit of ψ, and hence of ζ. So

Deg(P ⋊ N | Z(P)) = {20^{4}, 40^{2}}. (2.5.6)

From (2.3.4) and (2.5.6) we have

(2.5.7) Inv(P ⋊ D | Z(P)) = {20^{2}}  and  InvDef_{5}(P ⋊ D | Z(P)) = {2^{2}}

From (2.5.2), (2.5.3) and (2.5.7) we have

(2.5.8) Inv(P ⋊ D) = {1^{4}, 2^{4}, 20^{2}, 24}, and InvDef_{5}(P ⋊ D) = {3^{9}, 2^{2}}

We can now prove the following

Proposition 2.5.9. The group N_{M_{c}}(C_{2}) = P ⋊ D has a unique 5-block, which necessarily induces the principal 5-block, B_{0}, of M_{c}. Hence

k( C_{2}, B_{0}, 3, \overline{M_{c}.2} ) = 9,  k( C_{2}, B_{0}, 2, \overline{M_{c}.2} ) = 2,
k( C_{2}, B_{0}, d, \overline{M_{c}.2} ) = 0  for all other values of d.

(2.5.10)

Proof. This follows immediately from (2.5.8) and Proposition 2.3.6.

2.6. The Invariant Conjecture for the prime p = 5 From [Con85] the block B_{0} of M_{c} has eleven characters which are invariant in

M_{c}.2. We list these characters and their defects:

Character χ_{1} χ_{2} χ_{3} χ_{4} χ_{7} χ_{8} χ_{10} χ_{11} χ_{13} χ_{14} χ_{15}
Degree  1 22 231 252 896 896 3520 3520 4752 5103 5544
5-Defect 3 3 3 3 3 3 2 2 3 3 3

Thus

k( C_{1}, B_{0}, 3, \overline{M_{c}.2} ) = 9,  k( C_{1}, B_{0}, 2, \overline{M_{c}.2} ) = 2,  and
k( C_{1}, B_{0}, d, \overline{M_{c}.2} ) = 0  for all other values of d.

(2.6.1)

Theorem 2.6.2. The Invariant Conjecture holds for the McLaughlin simple group and the prime p = 5.

Proof. From Conjecture (1.4.4) and Table 2.1 on page 25, we need to prove

(2.6.3) k( C_{1}, B_{0}, d, \overline{M_{c}.2} ) = k( C_{2}, B_{0}, d, \overline{M_{c}.2} ),

for all values of d. We obtain the following terms for the equation above for various values of d from (2.5.10) and (2.6.1):

5-Defect C_{1}  C_{2}
3  9 = 9
2  2 = 2

Table 2.3. The Invariant Conjecture for p = 5

The terms in the above equation are zero for all other values of d. This proves the theorem.

Remark 2.6.4. G. Schneider has also computed the multiset Deg(P ⋊ N), using the CAYLEY computer program. He used it to prove the weaker Alperin Conjecture for the McLaughlin Group and p = 5. For further information see [BM90, P465].

2.7. The Projective Conjecture for the prime p = 5 We prove in Theorem 3.12.8 that the Schur multiplier A of M_{c} is cyclic of order

3. Hence the only covering groups of M_{c} are M_{c} and M̂_{c} ≅ 3.M_{c}. By the Atlas there is no central extension of M_{c}.2. So we need only show that the Projective Conjecture holds for M̂_{c}.

Theorem 2.7.1. The Projective Conjecture holds for McLaughlin’s simple group and the prime p = 5.

Proof. Let ρ ∈ Irr(A)^{#}. Then from the Atlas the group M̂_{c} has two 5-blocks lying over the 5-block of A containing ρ. One of these blocks has a trivial defect group and contains the single irreducible character χ_{38}; the other block B^{∗} has P as a defect group and contains the remaining 19 characters of Irr(M̂_{c} | ρ). We list here the elements of Irr(B^{∗} | ρ):

Character χ_{25} χ_{26} χ_{27} χ_{28} χ_{29} χ_{30}  χ_{31}  χ_{32}  χ_{33} χ_{34}
Degree  126 126 792 1980 2376 2376 2520 2520 2772 4752
5-Defect  3  3  3  2  3  3  2  2  3  3
Character χ_{35} χ_{36} χ_{37} χ_{39} χ_{40} χ_{41}  χ_{42}  χ_{43}  χ_{44}
Degree  5103 6336 6336 8019 8019 8064 10395 10395 10395
5-Defect  3  3  3  3  3  3  2  2  2

Thus

k(C_{1}, B^{∗}, 3 | ρ) = 13, k(C_{1}, B^{∗}, 2 | ρ) = 6 (2.7.2)

and k(C_{1}, B^{∗}, d | ρ) = 0 for all other values of d.

The group A. N_{M_{c}}(P) has a Sylow 3-subgroup of order 9. From the Atlas character table of M_{c}, the group M̂_{c} has no elements of order 9. So a Sylow 3-subgroup of A. N_{M_{c}}(P) splits over A. Hence the group A. N_{M_{c}}(P) itself splits over A ≅ ℤ_{3}, using Theorem [As86, 10.4]. By (2.3.7) a cohort of characters of A. N_{M_{c}}(P) = NM̂_{c}(C_{2}) lying over ρ contains 19 members. Thirteen of these characters are of defect 3 and six are of defect 2. All of these characters lie in a 5-block of A. N_{M_{c}}(P) inducing the 5-block B^{∗} of M̂_{c}. Hence

k(C_{2}, B^{∗}, 3 | ρ) = 13, k(C_{2}, B^{∗}, 2 | ρ) = 6 (2.7.3)

and k(C_{2}, B^{∗}, d | ρ) = 0, for all other values of d.

From Conjecture (1.4.6) and Table 2.1 on page 25, we need to prove

k(C_{1}, B^{∗}, d | ρ) = k(C_{2}, B^{∗}, d | ρ), (2.7.4)

for all values of d.

From (2.7.2) and (2.7.3), we obtain the following terms for the equation above for various values of d:

5-Defect C_{1}  C_{2}
3  13 = 13
2  6 = 6

Table 2.4. The Projective Conjecture for p = 5

The terms in (2.7.4) are zero for all other values of d. This proves the theorem.

Chapter 3 The Prime p=3

3.1. The Groups E ⋊ M and F ⋊ L For the remainder of the thesis α will denote a fixed element of the (3A) conjugacy

class of M_{c}. From the Atlas N_{M_{c}}(⟨α⟩) = F ⋊ L, where F ≅ 3_{+}^{1+4} and L ≅ 2.𝔖_{5}. By [GL83, p55] a Sylow 2-subgroup of L is isomorphic to a generalized quaternion group Q_{16} of order 2^{4}. Thus L is an isoclinic variant of the group SL(2, 5) ⋊ ℤ_{2}. For the remainder of the thesis C will denote the derived group L′ ≅ SL(2, 5) of L. Since O_{3}(L) = {1}, the group F is the 3-core of F ⋊ L. Clearly Z(F) = ⟨α⟩. Hence

F ⋊ L = N_{M_{c}}(F). (3.1.1)

Lemma 3.1.2. Let S_{L} be a Sylow 3-subgroup of L and let N_{L} denote the normalizer of S_{L} in L. Then S_{L} ≅ ℤ_{3} and N_{L} ≅ ℤ_{3} ⋊ Q_{8}. Let Q be a complement to S_{L} in N_{L}. We may choose generators a and b for Q such that a ∈ L\C, b ∈ C and both a and b invert S_{L}.

Proof. This is clear from the structure of L ≅ SL(2, 5).2.

We use W to denote the Sylow 3-subgroup F ⋊ S_{L} of F ⋊ L. From the Atlas it is also a Sylow 3-subgroup of M_{c}.

Let τ be a generator of Z(Q) = Z(L). So a^{2} = b^{2} = τ. From the Atlas the centralizer H = C_{M_{c}}(τ) of τ is isomorphic to the universal covering group 2.𝔄_{8} of 𝔄_{8}.

Lemma 3.1.3. F ⋊ C = C_{M_{c}}(α). Hence S_{L} ⋊ ⟨b⟩ centralizes α, while a inverts α.

Proof. The group L acts on the group ⟨α⟩ = Z(F) ≅ ℤ_{3}. This action induces a homomorphism of L into the abelian group Aut(Z(F)) ≅ ℤ_{2}. Hence C = L′ is contained in the kernel of this homomorphism i.e. it centralizes Z(F).

Since τ ∈ C centralizes α, the group Z(F) ⋊ L is a subgroup of H = C_{M_{c}}(τ). It follows from the structure of H ≅ 2.𝔄_{8} that Z(F) ⋊ L = N_{H}(Z(F)) and Z(F) × C = C_{H}(α). In particular L does not centralize α. We conclude that F ⋊ C = C_{M_{c}}(α).

Now S_{L} ⋊ ⟨b⟩ centralizes α because it is a subgroup of C, while a ∈ L\C inverts α because it acts non-trivially on Z(F). This concludes the lemma.

We let \overline{F} denote the quotient group F/ Z(F). So \overline{F} is elementary abelian of order 3^{4}. Clearly L acts on \overline{F}.

Proposition 3.1.4. C_{W}(τ) = Z(F)×S_{L}. Hence τ inverts \overline{F}. Also C_{W}(Q) = \{1\}. A Sylow 2-subgroup of F ⋊ L acts fixed-point-free on \overline{F}^{#}.

Proof. The group C_{W}(τ) = W ∩ H is a 3-subgroup of H = C_{M_{c}}(τ) containing Z(F) × S_{L}. From the Atlas a Sylow 3-subgroup of H ≅ 2.𝔄_{8} is elementary abelian of order 3^{2}. So C_{W}(τ) = Z(F) × S_{L}. Hence C_{F}(τ) = Z(F). As τ has order relatively prime to 3, this implies that C\overline{F}(τ) = \{1\}. So τ inverts \overline{F}.

Since a ∈ Q inverts C_{W}(τ) = Z(F) × S_{L}, it follows that C_{W}(Q) = {1}.

Let Q_{16} be a Sylow 2-subgroup of F⋊L. We may suppose that τ ∈ Q_{16}. Since Q_{16} is generalized quaternion, τ is its unique involution. Hence Q_{16} acts fixed-point-free on \overline{F}.

Corollary 3.1.5. The group L acts faithfully on \overline{F}.

Proof. This follows immediately from Proposition 3.1.4 and the fact that ⟨τ⟩ is the unique minimal normal subgroup of L ≅ SL(2, 5).2.

Lemma 3.1.6. Z(W) = Z(F).

Proof. As N_{M_{c}}(F) = F ⋊ L, we have C_{M_{c}}(F) ≤ F ⋊ L. But L acts faithfully on \overline{F}, by Corollary 3.1.5. So C_{M_{c}}(F) = Z(F). Thus Z(W) ≤ Z(F). This implies the lemma since Z(F) has prime order 3.

Lemma 3.1.7. Suppose Q acts on an elementary abelian group Ẽ of order 3^{4}, in such a manner that τ ∈ Q inverts Ẽ. Then any proper non-trivial Q-invariant subgroup F̃ of Ẽ is a simple Q-group of order 3^{2}. Moreover Q acts regularly and transitively on F̃^{#}.

Proof. Let F̃ be a proper non-trivial Q-invariant subgroup of Ẽ. Then Q acts faithfully on both F̃ and Ẽ/F̃. Hence both subgroups have order ≥ 3^{2}. The lemma follows.

Proposition 3.1.8. W′/ Z(W) = [S_{L}, \overline{F}] = C\overline{F}(S_{L}), and each of these groups has index 3^{2} in \overline{F} and order 3^{2}. Let E = W′S_{L}. Then W′ = E ∩ F, and E = C_{W}(W′) is elementary abelian of order 3^{4}. Moreover [W, W′] = Z(W). So the 3-group W is of class three.

Proof. We have W′ < F, as F is a maximal subgroup of W. Moreover Z(W) < W′, as W′ contains the inverse image of the non-trivial subgroup [S_{L}, \overline{F}] of \overline{F}. Hence W′/ Z(W) is a simple Q-subgroup of \overline{F} of order 3^{2}, by Lemma 3.1.7. But [S_{L}, \overline{F}] is a non-trivial Q-subgroup of W′/ Z(W). So [S_{L}, \overline{F}] = W′/ Z(W).

Now [W, W′] ≠ {1}, as Z(W) < W′. Since Z(W) is simple, this and [Rt95, 5.41] imply that Z(W) ≤ [W, W′]. Hence [W, W′]/ Z(W) is a proper Q-subgroup of the simple Q-space W′/ Z(W). So [W, W′] = Z(W). We conclude that W is a 3-group of class three. In particular W′ is abelian.

It follows from the previous paragraph that commutation in W gives a ⟨τ⟩- invariant bilinear map

\frac{W′}{Z(W)} × S_{L} → Z(W).

But τ inverts W′/ Z(W), while centralizing S_{L} and Z(W). So this map is trivial. Hence W′S_{L} is abelian. Thus W′/ Z(W) ≤ C\overline{F}(S_{L}). But C\overline{F}(S_{L}) is a non-trivial proper Q-subgroup of \overline{F}. So W′/ Z(W) = C\overline{F}(S_{L}).

Let E = W′S_{L}. Then W′ is a maximal subgroup of E contained in F. Since S_{L} ∩ F = {1}, we conclude that E ∩ F = W′. As W′S_{L} is abelian we have

E = C_{W′S_{L}}(τ) × [W′S_{L}, τ] = Z(W) × S_{L} × [W′S_{L}, τ],

using Proposition 3.1.4 and [As86, 24.6]. Since F has exponent 3, this implies in particular that E is elementary abelian of order 3^{4}.

Now F is a maximal subgroup of W and W′ = C_{F}(W′) is a maximal abelian subgroup of the extra-special group F. So C_{W}(W′) = E.

Corollary 3.1.9. F is the unique extra-special subgroup of W of order 3^{5}.

Proof. Suppose F^{1} ≠ F is an extra-special subgroup of W of order 3^{5}. We claim Z(W) ≤ F^{1}. Otherwise Z(W) ∩ F^{1} = {1}, and so W = Z(W) × F^{1}, which is impossible. Thus Z(W) coincides with Z(F^{1}), since the former is a central subgroup of F^{1} of order 3.

Now F^{1}F = W, as F is a maximal subgroup of W. Hence F^{1} ∩ F has order 3^{5+5−6} = 3^{4}. Moreover Z(W) ≤ F^{1} ∩ F. So \overline{F}^{1} = F^{1}/ Z(W) is an elementary abelian subgroup of \overline{W} = W/ Z(W) of order 3^{4}, distinct from \overline{F}.

Let \overline{S_{L}} denote the subgroup S_{L} Z(W)/ Z(W) of \overline{W}. Then \overline{W} = \overline{F} ⋊ \overline{S_{L}}. Pick x ∈ \overline{F}^{1}\\overline{F}. So we also have \overline{W} = \overline{F} ⋊ ⟨x⟩. Since \overline{F} is abelian, the actions of \overline{S_{L}} and ⟨x⟩ on \overline{F} coincide. But C\overline{F}(\overline{S_{L}}) = C\overline{F}(S_{L}) = W′/ Z(W) has order 3^{2}, while C\overline{F}(⟨x⟩) ≥ \overline{F} ∩ \overline{F}^{1} has order ≥ 3^{3}. This contradiction means that no such group F^{1} exists.

Corollary 3.1.10. E is the unique subgroup of W isomorphic to ℤ_{3}^{4}.

Proof. Let E^{1} be a subgroup of W isomorphic to ℤ_{3}^{4}. Then E^{1} ≰ F. So E^{1} ∩ F is an abelian subgroup of F of order 3^{3} = 3^{5+4−6}. Since F ≅ 3^{1+4}, this implies that Z(F) ≤ E^{1} ∩ F.

Let x ∈ E^{1}\F. Then ⟨x⟩ ≡ S_{L} modulo F. Since \overline{F} is abelian, this implies that the actions of ⟨x⟩ and S_{L} on \overline{F} are identical. So

(E^{1} ∩ F)/ Z(F) ≤ C\overline{F}(x) = C\overline{F}(S_{L}) = W′/ Z(F).

Since the first and last terms have order 3^{2}, we conclude that E^{1} ∩ F = W′. Then E^{1} ≤ C_{W}(W′) = E. So E^{1} = E.

Lemma 3.1.11. N_{M_{c}}(E) = E ⋊ M, where M ≅ M_{10} can be chosen to contain Q. Let S_{M} = M ∩ W and N_{M} = N_{M}(S_{M}). Then S_{M} is a Sylow 3-subgroup of M while N_{M} = S_{M} ⋊ Q and

N_{M_{c}}(W) = W ⋊ Q = F ⋊ N_{L} = E ⋊ N_{M} = (F ⋊ L) ∩ (E ⋊ M).

Proof. It follows from Lemma 3.1.6 that N_{M_{c}}(W) ≤ N_{M_{c}}(Z(F)) = F ⋊ L. So N_{M_{c}}(W) = F ⋊ N_{L}(W) = F ⋊ (S_{L} ⋊ Q) = F ⋊ N_{L}.

From the Atlas N_{M_{c}}(E) = E ⋊ M, where M ≅ M_{10}. Clearly W ≤ N_{M_{c}}(E). Let S_{M} = M ∩ W. Then S_{M} is a Sylow 3-subgroup of M. Let N_{M} = N_{M}(S_{M}). Then N_{M} = S_{M} ⋊ Q^{1}, where Q^{1} ≅ Q_{8}. Clearly Q^{1} ≤ N_{M_{c}}(W).

From the first paragraph it follows that Q and Q^{1} are Sylow 2-subgroups of N_{M_{c}}(W). Hence (Q^{1})^{x} = Q, for some x ∈ W. We replace M by M^{x} and S_{M} by (S_{M})^{x}. Then we still have N_{M_{c}}(E) = E ⋊ M. Also N_{M} = S_{M} ⋊ Q and N_{M_{c}}(W) = E ⋊ N_{M}.

Finally (E ⋊ M) ∩ (F ⋊ L) ≤ N_{M_{c}}(W), since W = EF. But the opposite inclusion follows from the first and third paragraphs of this proof. Hence N_{M_{c}}(W) = (F ⋊ L) ∩ (E ⋊ M).

Note that the Sylow 3-subgroup S_{M} of M ≅ M_{10} is isomorphic to ℤ_{3}^{2}.

Proposition 3.1.12. E ⋊ M has two orbits on E^{#}. One consists of 20 elements from the class (3A) of M_{c} containing α, while the other consists of 60 elements from the class (3B) of M_{c}. Let β ∈ E come from the class (3B) of M_{c}. Then

C_{M}(α) = S_{M} ⋊ Z_{4}^{(1)} < S_{M} ⋊ Q = N_{M}(⟨α⟩), where Z_{4}^{(1)} ≅ ℤ_{4},
C_{M}(β) = A_{4} < S_{4} = N_{M}(⟨β⟩) < M′ ≅ 𝔄_{6}, where A_{4} ≅ 𝔄_{4} and S_{4} ≅ 𝔖_{4}.

(3.1.13)

Furthermore

C_{M_{c}}(β) = E ⋊ A_{4} < N_{M_{c}}(⟨β⟩) = E ⋊ S_{4}.

Proof. Since α ∈ E, it follows from Lemmas 3.1.3 and 3.1.11 that N_{E⋊M}(⟨α⟩) = (F ⋊ L) ∩ (E ⋊ M) = E ⋊ (S_{M} ⋊ Q), and C_{E⋊M}(α) = (F ⋊ C) ∩ (E ⋊ M) = E ⋊ (S_{M} ⋊ Z_{4}^{(1)}), where Z_{4}^{(1)} < Q < M is isomorphic to ℤ_{4}. So α lies in an M-orbit of length |M : S_{M} ⋊ Z_{4}^{(1)}| = 20.

Let β be an element of the (3B) class of M_{c}. Then N_{M_{c}}(⟨β⟩) ≅ ℤ_{3}^{4} ⋊ 𝔖_{4}, by [GL83, p55]. Hence by Corollary 3.1.10, we may assume that N_{M_{c}}(⟨β⟩) = E ⋊ S_{4}, where S_{4} ≅ 𝔖_{4}. But E = O_{3}(E⋊S_{4}). So β ∈ E. Moreover E⋊S_{4} ≤ E⋊M = N_{M_{c}}(E). So, replacing S_{4} by (E ⋊ S_{4}) ∩ M if necessary, we may also assume that S_{4} ≤ M. From the maximal subgroups of M ≅ M_{10} listed on page 4 of the Atlas it follows that any subgroup of M isomorphic to 𝔖_{4} is contained in M′ ≅ 𝔄_{6}. Choose an involution ι ∈ C_{M}(β). Such an involution exists because C_{M}(β) is a subgroup of index at most 2 in N_{M}(⟨β⟩) ≅ 𝔖_{4}. All 3-elements of C_{M_{c}}(ι) ≅ 2.𝔄_{8} are real. So β is real in M_{c}.

Hence C_{M_{c}}(β) = E ⋊ A_{4}, where A_{4} = S_{4}′ ≅ 𝔄_{4} is the unique subgroup of index 2 in S_{4}. We conclude that β lies in an M-orbit of length |M : A_{4}| = 60.

The M-orbits of α and β account for all elements of E^{#}.

For the rest of the thesis β will denote an element of E which is a member of the class (3B) of M_{c}.

Proposition 3.1.14. L has a single orbit on F\ Z(F), consisting of 240 elements from the class (3B) of M_{c}. Hence L acts transitively on \overline{F}^{#}.

Proof. By Proposition 3.1.12 the group E contains only 20 elements of the class (3A) of M_{c}. But |F ∩ E| = 3^{3}. So F ∩ E intersects the class (3B) of M_{c}. We suppose that the element β in Proposition 3.1.12 lies in (F ∩ E)\ Z(F).

From the Atlas |C_{M_{c}}(β)| = 972 = 3^{5} · 2^{2}, and by Proposition 3.1.4 a Sylow 2subgroup of C_{F⋊L}(β) is trivial. Thus |C_{F⋊L}(β)| ≤ 3^{5}. So β has at least |F ⋊ L|/3^{5} = 240 distinct F ⋊ L-conjugates. But no element of Z(F) is conjugate to β. Hence the F ⋊ L-orbit of β is contained in F\ Z(F). Since this latter set is of cardinality 240, we conclude that F\ Z(F) is the F ⋊ L-orbit of β.

Now \overline{F} is abelian. So L acts transitively on \overline{F}^{#}.

Corollary 3.1.15. Let A_{6} = M′. Then C_{E}(A_{6}) = {1}. Hence M acts faithfully on E. Also E is an irreducible M-group.

Proof. The group A_{6} ≅ 𝔄_{6} is the unique non-trivial proper normal subgroup of M ≅ M_{10}. But it follows from (3.1.13) that C_{E}(A_{6}) = {1}. We conclude that M acts faithfully on E.

From Proposition 3.1.12 the group M has orbits of length 1, 20, 60 on E. Hence M acts irreducibly on E.

3.2. The Radical 3-chains of M_{c}

We find the radical 3-subgroups of M_{c} using the following lemma.

Lemma 3.2.1. If X is a ℤ_{3}^{n} subgroup of M_{c}, for n ≥ 1, then N_{M_{c}}(X) is conjugate to a subgroup of F ⋊ L or E ⋊ M.

Proof. The result follows from [Fk73, 5.6, p70].

We can now give the radical 3-subgroups of M_{c}.

Proposition 3.2.2. Let X be a radical 3-subgroup of M_{c}. Then X is conjugate in M_{c} to exactly one of

{1}, E, F or W.

Furthermore any radical 3-subgroup of N_{M_{c}}(E) is N_{M_{c}}(E)-conjugate to E or W, and any radical 3-subgroup of N_{M_{c}}(F) is N_{M_{c}}(F)-conjugate to F or W.

Proof. Assume X ≠ {1}. We have N_{M_{c}}(X) ≤ N_{M_{c}}(ΩZ(X)). So the normalizer N_{M_{c}}(X) is either contained in a conjugate of N_{M_{c}}(E) = E⋊M or of N_{M_{c}}(F) = F⋊L, by Lemma 3.2.1. Hence we may assume that X is a radical 3-subgroup of one of these groups.

Suppose X is a radical 3-subgroup of F ⋊ L. Then X ≥ F, since F = O_{3}(F ⋊ L). Now X is contained in some Sylow 3-subgroup W^{1} of F ⋊ L. Since F is maximal in W^{1}, either X = F or X = W^{1} is conjugate to W in F ⋊ L.

Suppose on the other hand that X is a radical 3-subgroup of E ⋊ M. Then X ≥ E, since E = O_{3}(E⋊M). If X ≠ E, then X = E⋊X_{R}, where X_{R} = X∩M is a non-trivial radical 3-subgroup of M. It follows that N_{E⋊M}(X) = E⋊N_{M}(X) = E⋊N_{M}(X_{R}). From the Atlas, C_{M}(X_{R}) is a Sylow 3-subgroup of M. Hence X_{R} = O_{3}(N_{M}(X_{R})) = C_{M}(X_{R}) is an M-conjugate of S_{M}. Thus X = E ⋊ X_{R} is conjugate to W in M.

Corollary 3.2.3. Every radical 3-chain of M_{c} is conjugate to exactly one of

{1}, {1} < E, {1} < E < W, {1} < F, {1} < F < W or {1} < W.

Furthermore, each of these 3-chains is a radical 3-chain of M_{c}.

Proof. This follows immediately from the previous proposition.

Three of the radical 3-chains of M_{c} have the same stabilizers, as we now show.

Lemma 3.2.4. The normalizer of each of the radical 3-chains

{1} < W, {1} < F < W, and {1} < E < W

in either M_{c} or M_{c}.2 is the same as the normalizer of W in that group.

Proof. The groups E and F are characteristic subgroups of W, by Corollaries 3.1.9 and 3.1.10. Thus the normalizer of W in M_{c} or M_{c}.2 also normalizes each of the three given radical 3-chains. But since W occurs in each of the chains, the normalizer of any of the chains is contained in the normalizer of W. This proves the lemma.

Using (3.1.1), Lemmas 3.1.11 and 3.2.4 above, and Lemmas 3.7.1 and 3.7.2 below, we obtain the following descriptions of the normalizers of the radical 3-chains in M_{c} and M_{c}.2:

Chain C Chain Description N_{M_{c}}(C) N_{M_{c}.2}(C)  Parity
C_{1}  {1}  M_{c}  M_{c}.2  +
C_{2}  {1} < E  E ⋊ M E ⋊ (M × ⟨c⟩) −
C_{3}  {1} < E < W W ⋊ Q W ⋊ (⟨d⟩Q)  +
C_{4}  {1} < F  F ⋊ L F ⋊ (⟨d⟩L)  −
C_{5}  {1} < F < W W ⋊ Q W ⋊ (⟨d⟩Q)  +
C_{6}  {1} < W  W ⋊ Q W ⋊ (⟨d⟩Q)  −

Table 3.1. The Radical 3-chains of M_{c}

3.3. The Character Degrees of F ⋊ L

Since L ≅ 2.𝔖_{5}, we have from the Atlas

Deg((F ⋊ L)/F) = {1^{2}, 4^{5}, 5^{2}, 6^{3}},
Def_{3}(F ⋊ L mod F) = {6^{9}, 5^{3}}.

(3.3.1)

The group L acts transitively on \overline{F}^{#} by Proposition 3.1.14. So by Lemma 1.2.19, it also acts transitively on Irr(\overline{F})^{#}. Hence any non-trivial linear character ψ of \overline{F} has 80 distinct L-conjugates and I_{L}(ψ) is cyclic of order 3. It follows from Theorem 1.2.10 that ψ extends to the cyclic group I_{L}(ψ). Let ψ̃ be one such extension. Then from Theorem 1.2.16 we have Irr(\overline{F} ⋊ L | ψ) = \{(ωψ̃)^{\overline{F}⋊L} | ω ∈ Irr(ℤ_{3})\}. Each (ωψ̃)^{\overline{F}⋊L} is of degree deg(ω) · deg(ψ̃) · |L : I_{L}(ψ)| = 80. Thus

(3.3.2)  Deg(F ⋊ L | F/ Z(F)) = {80^{3}},  Def_{3}(F ⋊ L | F/ Z(F)) = {6^{3}}.

The group F is extra-special of order 3^{5}. So by Theorem 1.2.18 it has two characters of degree 9 which are non-trivial on Z(F). Moreover these characters are determined by their restrictions to Z(F). By Lemma 3.1.3 this implies that they form a single F ⋊ Lorbit. Let χ be one of these characters. Then I_{L}(χ) = C_{L}(Z(F)) = C. Since a Sylow 3-subgroup of C is cyclic, χ extends to I_{L}(χ) = F ⋊ C. Let χ̃ be one such extension. Then Irr(F ⋊ L | χ) = {(ωχ̃)^{F⋊L} | ω ∈ Irr(C)}. Since Deg(C) = Deg(2.𝔄_{5}) = {1, 2^{2}, 3^{2}, 4^{2}, 5, 6}, and Deg( (ωχ̃)^{F⋊L} ) = deg(ω) · deg(χ̃) · 2 = 18 · deg(ω), we have

Deg(F ⋊ L | Z(F)) = {18, 36^{2}, 54^{2}, 72^{2}, 90, 108},
Def_{3}(F ⋊ L | Z(F)) = {4^{6}, 3^{3}}.

(3.3.3)

From equations (3.3.1), (3.3.2) and (3.3.3) we obtain

Deg(F ⋊ L) = {1^{2}, 4^{5}, 5^{2}, 6^{3}, 18, 36^{2}, 54^{2}, 72^{2}, 80^{3}, 90, 108},
Def_{3}(F ⋊ L) = {6^{12}, 5^{3}, 4^{6}, 3^{3}}.

(3.3.4)

This allows us to prove the following

Proposition 3.3.5. The group N_{M_{c}}(C_{4}) = F ⋊ L has a unique 3-block, which necessarily induces the principal 3-block, B_{0}, of M_{c}. Thus

k(C_{4}, B_{0}, 6) = 12,  k(C_{4}, B_{0}, 5) = 3,
k(C_{4}, B_{0}, 4) = 6,  k(C_{4}, B_{0}, 3) = 3,
k(C_{4}, B_{0}, d) = 0,  for all other values of d.

(3.3.6)

Proof. Since C_{F⋊L}(F) = Z(F) ≤ F, the group F ⋊ L has only one 3-block, b_{0}, which must be its principal 3-block. So b_{0} induces the principal 3-block, B_{0}, of M_{c}. We now obtain (3.3.6) from (3.3.4).

3.4. The Character Degrees of W ⋊ Q Recall from Lemma 3.1.11 that W⋊Q = F⋊N_{L}. Since Q acts transitively on S_{L}^{#},

by Lemma 1.2.19 it also acts transitively on Irr(S_{L})^{#}. Let ϕ be an element of Irr(S_{L})^{#}. Then I_{N_{L}}(ϕ) ≅ ℤ_{3}×ℤ_{4}. By Theorems 1.2.10 and 1.2.16 we have Deg(N_{L} | S_{L}) = {2^{4}}. Also Deg(N_{L}/S_{L}) = Deg(Q) = {1^{4}, 2}. Hence

(3.4.1)  Deg(N_{L}) = {1^{4}, 2^{5}},  Def_{3}(W ⋊ Q mod F) = {6^{9}}.

Next we consider the group \overline{F} ⋊ N_{L}. We identify Irr(\overline{F}) with the dual group \overline{F}^{∗} of \overline{F}. The action of N_{L} on \overline{F} induces an action of N_{L} on \overline{F}^{∗}. Lemma 1.2.20 and Proposition 3.1.8 imply that C_{\overline{F}}^{∗}(S_{L}) has order 3^{2}. It then follows from Lemma 3.1.7 that Q acts regularly and transitively on C_{\overline{F}}^{∗}(S_{L})^{#}. Hence C_{\overline{F}}^{∗}(S_{L})^{#} is a N_{L}-orbit of length 8. If ψ ∈ \overline{F}^{∗}, then by Proposition 3.1.4, its centralizer C_{N_{L}}(ψ) is a 3-group. But S_{L} is the unique non-trivial 3-subgroup of N_{L}. Thus, if ψ is not centralized by S_{L}, it must have a trivial centralizer in N_{L}, in which case it lies in an N_{L}-orbit of length 24. We conclude that

Orb(N_{L}, (\overline{F}^{∗})^{#}) = \{8, 24^{3}\}. (3.4.2)

By Theorem 1.2.15 each irreducible character of \overline{F} extends to its stabilizer in N_{L}. Using Theorem 1.2.16 we find

Deg(W ⋊ Q | F/ Z(F)) = {8^{3}, 24^{3}},
Def_{3}(W ⋊ Q | F/ Z(F)) = {6^{3}, 5^{3}}.

(3.4.3)

As noted in Section 3.3, the group F has two characters of degree 9 which are nontrivial on Z(F). Moreover these characters are determined by their restrictions to Z(F). By Lemma 3.1.3 this implies that they form a single W ⋊ Q-orbit. Let χ be one of these characters. Then I_{N_{L}}(χ) = C_{N_{L}}(Z(F)) = S_{L} ⋊ ⟨b⟩. The character χ extends to χ̃ ∈ Irr(F ⋊ I_{N_{L}}(χ)), as S_{L} ⋊ ⟨b⟩ has cyclic Sylow subgroups. So Irr(W ⋊ Q | Z(F)) = { (ωχ̃)^{W⋊Q} | ω ∈ Irr(S_{L} ⋊ ⟨b⟩) }. Hence Deg(F ⋊ I_{N_{L}}(χ) | χ) = 9 · Deg(S_{L} ⋊ ⟨b⟩). Now b inverts S_{L} by Lemma 3.1.2. So Deg(F ⋊ I_{N_{L}}(χ) | χ) = {9^{4}, 18^{2}}. Then by Clifford theory

Deg(W ⋊ Q | Z(F)) = {18^{4}, 36^{2}},
Def_{3}(W ⋊ Q | Z(F)) = {4^{6}}.

(3.4.4)

From equations (3.4.1), (3.4.3) and (3.4.4) we have

Deg(W ⋊ Q) = {1^{4}, 2^{5}, 8^{3}, 18^{4}, 24^{3}, 36^{2}},
Def_{3}(W ⋊ Q) = {6^{12}, 5^{3}, 4^{6}}.

(3.4.5)

This allows us to prove the following

Proposition 3.4.6. The group N_{M_{c}}(C_{3}) = W ⋊ Q has a unique 3-block, which necessarily induces the principal 3-block, B_{0}, of M_{c}. Hence

k(C_{3}, B_{0}, 6) = 12,  k(C_{3}, B_{0}, 5) = 3,
k(C_{3}, B_{0}, 4) = 6,  k(C_{3}, B_{0}, 3) = 0,
k(C_{3}, B_{0}, d) = 0,  for all other values of d.

(3.4.7)

Proof. Since C_{W⋊Q}(W) = Z(W) ≤ W, the group W ⋊ Q has only one 3-block. This block necessarily induces the principal block B_{0} of M_{c}. We now obtain (3.4.7) from (3.4.5).

3.5. The Character Degrees of E ⋊ M Let E^{∗} = Hom(E, GF(3)) be the dual group to E. We identify E^{∗} and Irr(E).

Both E and E^{∗} can be considered as GF(3)M-modules. By Proposition 3.1.12 the group M has two orbits on E^{#}, of lengths 20 and 60. So M has two orbits on (E^{∗})^{#} also.

Lemma 3.5.1. There exists an increasing chain

K < A_{4} < S_{4} < A_{6} < M (3.5.2)

of subgroups of M, with A_{6} = M′ ≅ 𝔄_{6}, S_{4} ≅ 𝔖_{4}, A_{4} = S_{4}′ ≅ 𝔄_{4} and K = O_{2}(A_{4}) ≅ ℤ_{2}^{2}. The GF(3)-vector space E^{∗} is isomorphic to the regular GF(3)K-module. Hence M acts absolutely irreducibly on E.

Proof. From the Atlas the derived group A_{6} ≅ 𝔄_{6} of M contains two conjugacy classes of maximal subgroups isomorphic to 𝔖_{4}, and these two classes fuse in M. We choose a member S_{4} of one of these conjugacy classes. We let A_{4} denote the derived group S_{4}′ ≅ 𝔄_{4} of S_{4}, and let K denote O_{2}(A_{4}) ≅ ℤ_{2}^{2}. The existence of (3.5.2) now follows.

We claim that E^{∗} is the regular GF(3)K-module. Since K is an elementary abelian 2-group, E^{∗} ≅ E_{1} ⊕ E_{2} ⊕ E_{3} ⊕ E_{4} as GF(3)K-modules, where each E_{i} is a onedimensional GF(3)K-submodule of E^{∗}. Let ι be an involution in K. Since A_{6} is simple and acts faithfully on E, we can regard K as a subgroup of SL(4, 3). So ι inverts an even, non-zero number of the E_{i}. But K ∩ Z(SL(4, 3)) = {1}, as Z(A_{6}) = {1}. So ι inverts exactly two of the A_{i}. The claim now follows easily.

By Corollary 3.1.15 the group M acts irreducibly on E. So End_{GF(3)M}(E) is a subfield of the ring End_{GF(3)K}(E). But from the regularity of the GF(3)K-module E, the ring E ≅ GF(3)K is isomorphic to a direct sum of four copies of GF(3). Any subfield of such a direct sum is isomorphic to GF(3). The result now follows.

We can now describe the orbits of M on E^{∗}.

Lemma 3.5.3. There exist ψ_{1} and ψ_{2} in E^{∗} such that

I_{M}(ψ_{1}) = S_{M} ⋊ Z_{4}^{(2)} < S_{M} ⋊ Q = N_{M}(⟨ψ_{1}⟩), where Q > Z_{4}^{(2)} ≅ ℤ_{4},
I_{M}(ψ_{2}) = A_{4} < S_{4} = N_{M}(⟨ψ_{2}⟩), where A_{4} ≅ 𝔄_{4} and S_{4} ≅ 𝔖_{4}.

(3.5.4)

Hence

Orb(M, E^{∗}) = {1, 20, 60}. (3.5.5)

Proof. It follows from Proposition 3.1.8 that the GF(3)M-module W′ = E ∩ F has codimension 1 in E. Since W = E ⋊ S_{M}, and E and S_{M} are abelian, it follows that W′ = [E, S_{M}]. So C_{E^{∗}}(S_{M}) has dimension 1, by Lemma 1.2.20.

Let ψ_{1} be a generator of C_{E^{∗}}(S_{M}). Since Q normalizes S_{M}, it must stabilize ⟨ψ_{1}⟩. But it follows easily from Proposition 3.1.12 that C_{E}(Q) = {1}. So C_{E^{∗}}(Q) = {1}, using Lemma 1.2.20. Hence I_{N_{M}}(ψ_{1}) = S_{M} ⋊ Z_{4}^{(2)}, where Z_{4}^{(2)} ≅ ℤ_{4}. But from the Atlas N_{M} = S_{M} ⋊ Q is a maximal subgroup of M ≅ M_{10}. It follows from this and the irreducibility of E^{∗} as a GF(3)M-module (see Corollary 3.1.15) that N_{M} is the stabilizer of ⟨ψ_{1}⟩ in M. Hence I_{M}(ψ_{1}) = S_{M} ⋊ Z_{4}^{(2)}. We conclude that the M-orbit of ψ_{1} contains 720/36 = 20 elements.

Lemma 3.5.1 shows that E^{∗} is isomorphic to the regular GF(3)K-module. In particular dim_{GF(3)}(C_{E^{∗}}(K)) = 1. Let ψ_{2} be a generator of C_{E^{∗}}(K). Then ⟨ψ_{2}⟩ is stabilized by S_{4}, since K ◁ S_{4}. Let D_{8} be a Sylow 2-subgroup of S_{4}. Then we see from (3.1.13) that C_{E}(D_{8}) = {1}. So C_{E^{∗}}(D_{8}) = {1} by Lemma 1.2.20. Hence C_{E^{∗}}(S_{4}) = {1}. This implies that I_{S_{4}}(ψ_{2}) = A_{4}. But S_{4}, A_{6} and M are the only subgroups of M containing S_{4}. If A_{6} stabilizes ⟨ψ_{2}⟩, then it has an orbit of length 2 on (E^{∗})^{#}. This is impossible since A_{6} is simple. So S_{4} is precisely the stabilizer of ⟨ψ_{2}⟩ in M. Hence A_{4} = I_{M}(ψ_{2}). We conclude that the E ⋊ M-orbit of ψ_{2} contains 60 elements.

The 20 M-conjugates of ψ_{1} and the 60 M-conjugates of ψ_{2} account for all nontrivial elements of E^{∗}.

We consider the two non-trivial orbits of M on E^{∗} in turn.

Let μ be a non-trivial linear character of S_{M}. All involutions in I_{M}(ψ_{1}) invert S_{M}. So I_{S_{M}⋊Z_{4}^{(2)}}(μ) = S_{M}. Hence Deg(I_{M}(ψ_{1}) | μ) = {4}. Since I_{M}(ψ_{1}) has two orbits on S_{M}^{#} we get Deg(I_{M}(ψ_{1}) | S_{M}) = {4^{2}}. Also Deg(I_{M}(ψ_{1})/S_{M}) = {1^{4}}. Thus Deg(I_{M}(ψ_{1})) = {1^{4}, 4^{2}}.

Now ψ_{1} extends to E ⋊ I_{M}(ψ_{1}) by Theorem 1.2.15. So Deg(E ⋊ I_{M}(ψ_{1}) | ψ_{1}) = {1^{4}, 4^{2}}. The M-orbit of ψ_{1} has length 20. So from Clifford theory we obtain

(3.5.6)  Deg(E ⋊ M | ψ_{1}) = {20^{4}, 80^{2}},  Def_{3}(E ⋊ M | ψ_{1}) = {6^{6}}.

Since I_{M}(ψ_{2}) = A_{4}, we obtain Deg(E ⋊ I_{M}(ψ_{2}) | ψ_{2}) = Deg(A_{4}) = {1^{3}, 3}. The M-orbit of ψ_{2} has length 60. It follows from Clifford theory that

(3.5.7)  Deg(E ⋊ M | ψ_{2}) = {60^{3}, 180},  Def_{3}(E ⋊ M | ψ_{2}) = {5^{3}, 4}.

Since M ≅ M_{10}, we obtain from the Atlas

(3.5.8) Deg((E ⋊ M)/E) = {1^{2}, 9^{2}, 10^{3}, 16}, Def_{3}(E ⋊ M mod E) = {6^{6}, 4^{2}}.

From (3.5.6), (3.5.7) and (3.5.8) we have

Deg(E ⋊ M) = {1^{2}, 9^{2}, 10^{3}, 16, 20^{4}, 60^{3}, 80^{2}, 180},
Def_{3}(E ⋊ M) = {6^{12}, 5^{3}, 4^{3}}.

(3.5.9)

We now have enough information to prove

Proposition 3.5.10. The group N_{M_{c}}(C_{2}) = E ⋊ M has a unique 3-block, namely the principal 3-block. This block necessarily induces the principal 3-block, B_{0}, of M_{c}.

k(C_{2}, B_{0}, 6) = 12,  k(C_{2}, B_{0}, 5) = 3,
k(C_{2}, B_{0}, 4) = 3,  k(C_{2}, B_{0}, 3) = 0,
k(C_{2}, B_{0}, d) = 0,  for all other values of d.

(3.5.11)

Proof. The first statement follows from the fact that C_{E⋊M}(E) = E. Then (3.5.11) comes from (3.5.9).

3.6. The Ordinary Conjecture for the prime p = 3 From [Con85, p101], the group N_{M_{c}}(C_{1}) = M_{c} has three 3-blocks of defect 0, and

the principal block B_{0} of defect 6, which contains the remaining 21 characters. We list the characters of the principal block and their defects:

Character χ_{1} χ_{2} χ_{3} χ_{4} χ_{5} χ_{6} χ_{7} χ_{8}  χ_{9}  χ_{10} χ_{11}
Degree  1  22 231 252 770 770 896 896 1750 3520 3520
3-Defect  6  6  5  4  6  6  6  6  6  6  6
Character χ_{12} χ_{13} χ_{15} χ_{18} χ_{19} χ_{20} χ_{21} χ_{22}  χ_{23}  χ_{24}
Degree  4500 4752 5544 8250 8250 9625 9856 9856 10395 10395
3-Defect  4  3  4  5  5  6  6  6  3  3

Thus

k(C_{1}, B_{0}, 6) = 12,  k(C_{1}, B_{0}, 5) = 3,
k(C_{1}, B_{0}, 4) = 3,  k(C_{1}, B_{0}, 3) = 3,
k(C_{1}, B_{0}, d) = 0,  for all other values of d.

(3.6.1)

We now have enough information for the following theorem.

Theorem 3.6.2. The Ordinary Conjecture holds for McLaughlin’s simple group and the prime p = 3.

Proof. From Conjecture 1.4.2 and Table 3.1 on page 48, the Ordinary Conjecture for the prime p = 3 asserts that

(3.6.3)  k(C_{1}, B_{0}, d) + k(C_{3}, B_{0}, d) = k(C_{2}, B_{0}, d) + k(C_{4}, B_{0}, d)

for all values of d ∈ ℤ.

From (3.3.6), (3.4.7), (3.5.11) and (3.6.1) we obtain the following sums for the equation above for various values of d:

3-Defect C_{1}  C_{3}  C_{2}  C_{4}
6  12 + 12 = 12 + 12
5  3 + 3 = 3 + 3
4  3 + 6 = 3 + 6
3  3 + 0 = 0 + 3

Table 3.2. The Ordinary Conjecture for p = 3

The summands in (3.6.3) are zero for all other values of d. This completes the proof of the theorem.

3.7. The Groups E ⋊ (M × ⟨c⟩) and F ⋊ (⟨d⟩L)

We begin with descriptions of the normalizers of E and F in M_{c}.2.

Lemma 3.7.1. N_{M_{c}.2}(E) = E ⋊ (M × ⟨c⟩), where c is an involution inverting E. Hence N_{M_{c}.2}(W) = W ⋊ (Q × ⟨c⟩). The group M × ⟨c⟩ acts faithfully on E.

Proof. From the Atlas, N_{M_{c}.2}(E) has the form E⋊(M×⟨c⟩) for some involution c in M_{c}.2\M_{c}. centralizing M ≅ M_{10}. From the order of C_{M_{c}}(c) in the Atlas, it is clear that c acts as a non-trivial automorphism of the GF(3)M-module E. But End_{GF(3)M}(E) ≅ GF(3) by Corollary 3.5.1. So c inverts E.

It is clear that c normalizes W = E ⋊ S_{M}. So N_{M_{c}.2}(W) = W ⋊ (Q × ⟨c⟩). Since Z(M) = {1}, the embedding of M in GL(4, 3) = GL(E) does not contain

− I. If x ∈ (M × ⟨c⟩)\M centralizes E, then xc ∈ M inverts E, contradicting the previous sentence. Hence M × ⟨c⟩ acts faithfully on E.

Lemma 3.7.2. N_{M_{c}.2}(F) = F ⋊ (⟨d⟩L), where d is an element of M_{c}.2 of order 4. The element d satisfies the equations d^{2} = τ, [d, C] = {1} and [d, a] = τ. Hence N_{M_{c}.2}(W) = W ⋊ (⟨d⟩Q). The group ⟨d⟩L acts faithfully on \overline{F}.

Proof. From [Con85] we have N_{M_{c}.2}(F) = F ⋊ (L.2), where L.2 ≅ 4.𝔖_{5}. We denote by d a generator of the normal subgroup ℤ_{4} of L.2. Then d^{2} is the only generator τ of the unique normal subgroup ⟨τ⟩ of order 2 in L ≅ 2.𝔖_{5}.

Since L normalizes the cyclic group ⟨d⟩, its derived group C centralizes d. In particular d normalizes W = F ⋊ S_{L}, since S_{L} is contained in C. So N_{M_{c}.2}(W) = W ⋊ (⟨d⟩Q).

The Sylow 2-subgroups Q × ⟨c⟩ and ⟨d⟩Q of N_{M_{c}.2}(W) are isomorphic. But C_{Q×⟨c⟩}(Q) = ⟨τ⟩ × ⟨c⟩ has exponent 2. So the 4-element ⟨d⟩ is not centralized by Q. Hence a does not centralize d. We conclude that d^{a} = d^{−1}.

Let x be an element of C_{⟨d⟩L}(\overline{F}). Then x^{2} ∈ L centralizes \overline{F}. So x^{2} = 1 and x lies in the class (2B) of M_{c}.2. But then x centralizes F, since x is a 3′-element and \overline{F} = F/Φ(F). This contradicts the fact that |F| = 3^{5} ||̸C_{M_{c}}(x)|.

Corollary 3.7.3. Q × ⟨c⟩ = ⟨d⟩Q. So ⟨d⟩ = ⟨cb⟩, and d inverts Z(F).

Proof. The Sylow 2-subgroups Q × ⟨c⟩ and ⟨d⟩Q of N_{M_{c}.2}(W) are conjugate by some w ∈ W. Then Q = ⟨d⟩Q ∩ M_{c} = Q^{w}. But Q does not centralize any element of W, by Proposition 3.1.4. So w = 1. Hence Q × ⟨c⟩ = ⟨d⟩Q.

Now d centralizes b ∈ C. So d ∈ C_{Q×⟨c⟩}(b) = ⟨b⟩ × ⟨c⟩. The latter group has exactly two cyclic subgroups ⟨b⟩ and ⟨cb⟩ of order 4. Since ⟨d⟩ is such a subgroup, and is not equal to ⟨b⟩, it must equal ⟨bc⟩. The involution c inverts Z(F), since by Lemma 3.7.1 it inverts E and Z(F) = Z(W) is a subgroup of E. Also b centralizes Z(F), by Lemma 3.1.3. We conclude that d inverts Z(F).

3.8. The Invariant Characters of E ⋊ M Let ψ_{1} and ψ_{2} be representatives of the two non-trivial orbits of M on E^{∗}, as in

Lemma 3.5.3. Then ψ_{1} and ψ_{2} are not conjugate in M × ⟨c⟩ since their stabilizers in M are not isomorphic. Hence they are representatives of the two non-trivial orbits of M × ⟨c⟩ on E^{∗}. Since c inverts E, it also inverts E^{∗}.

From (3.5.4) there exists x ∈ Q which inverts ψ_{1}. Hence xc centralizes ψ_{1}. So I_{M×⟨c⟩}(ψ_{1}) = ⟨I_{M}(ψ_{1}), xc⟩. But c centralizes M. So I_{M×⟨c⟩}(ψ_{1}) ≅ I_{M}(ψ_{1})⟨x⟩ = S_{M} ⋊ Q. This group is Frobenius. It follows easily that Deg(I_{M×⟨c⟩}(ψ_{1})) = {1^{4}, 2, 8}.

As E is abelian, Theorem 1.2.15 shows that ψ_{1} extends to E ⋊ I_{M⋊⟨c⟩}(ψ_{1}). Then by Clifford theory

Deg(E ⋊ (M × ⟨c⟩) | ψ_{1}) = {20^{4}, 40, 160}. (3.8.1)

We deduce from (3.5.6) and (3.8.1) that

(3.8.2)  Inv(E ⋊ M | ψ_{1}) = {20^{2}},  InvDef_{3}(E ⋊ M | ψ_{1}) = {6^{2}}.

From (3.5.4) there exists y ∈ S_{4}\A_{4} which inverts ψ_{2}. Then, by an argument analogous to that used for ψ_{1}, we find that I_{M×⟨c⟩}(ψ_{2}) ≅ S_{4}. Hence Deg(I_{M×⟨c⟩}(ψ_{2}))= {1^{2}, 2, 3^{2}}.

As ψ_{2} also extends to its stabilizer in E ⋊ (M × ⟨c⟩), we have

Deg(E ⋊ (M × ⟨c⟩) | ψ_{2}) = {60^{2}, 120, 180^{2}}. (3.8.3)

Using (3.5.7) and (3.8.3) we obtain

(3.8.4)  Inv(E ⋊ M | ψ_{2}) = {60, 180},  InvDef_{3}(E ⋊ M | ψ_{2}) = {5, 4}.

All characters of M are invariant in M × ⟨c⟩. This and (3.5.8) imply that

(3.8.5)  Inv(M) = {1^{2}, 9^{2}, 10^{3}, 16},  InvDef_{3}(E ⋊ M mod E) = {6^{6}, 4^{2}}.

From (3.8.1), (3.8.3) and (3.8.5) we have

(3.8.6) Inv(E ⋊ M) = {1^{2}, 9^{2}, 10^{3}, 16, 20^{2}, 60, 180}, InvDef_{3}(E ⋊ M) = {6^{8}, 5, 4^{3}}.

We can now prove the following

Proposition 3.8.7. The group N_{M_{c}}(C_{2}) = E ⋊ M has a unique 3-block. This block induces of the principal 3-block, B_{0}, of M_{c}. Moreover

k( C_{2}, B_{0}, 6, \overline{M_{c}.2} ) = 8,  k( C_{2}, B_{0}, 5, \overline{M_{c}.2} ) = 1,
k( C_{2}, B_{0}, 4, \overline{M_{c}.2} ) = 3,  k( C_{2}, B_{0}, 3, \overline{M_{c}.2} ) = 0,
k( C_{2}, B_{0}, d, \overline{M_{c}.2} ) = 0,  for all other values of d.

(3.8.8)

Proof. This follows immediately from (3.8.6) and Proposition 3.5.10.

3.9. The Invariant Characters of F ⋊ L

Lemma 3.9.1. We have

(3.9.2)  Inv(L) = {1^{2}, 4^{3}, 5^{2}, 6},  InvDef_{3}(F ⋊ L mod F) = {6^{7}, 5}.

Proof. Let μ ∈ Irr(L). Recall that [d, C] = 1 and [d, a] = τ. Hence μ^{d}(x) = μ(x) for x ∈ C, and μ^{d}(x) = μ(τx) for x ∈ L\C. Since τ is central in L, there exists ν ∈ Irr(⟨τ⟩) such that μ(τx) = ν(τ)μ(x) for all x ∈ L\C. But ν(τ) = ±1, since ⟨τ⟩ ≅ ℤ_{2}. So μ^{d}(x) = ±μ(x), depending on whether μ(τ) = ±μ(1).

From the Atlas, we see that d stabilizes all characters of L ≅ 2.𝔄_{5}.2, except the four extensions of the characters χ_{8}, χ_{9} in the Atlas character table for 2.𝔄_{5}. The lemma now follows easily from that table.

Let χ be one of the two characters of the extra-special 3-group F of degree 9. Since these two characters form a single L-orbit, they also form a single L.2-orbit. Hence I_{L.2}(χ) has index 2 in L.2. Both a and d invert Z(F). So I_{L.2}(χ) = C.⟨ad⟩. But d centralizes C. Also (ad)^{2} = τ = a^{2}, since a^{2} = d^{2} = [a, d] = τ. Hence I_{L.2}(χ) = C.⟨ad⟩ ≅ L ≅ 2.𝔖_{5}. A Sylow 3-Subgroup of 2.𝔖_{5} is cyclic. So by Theorem 1.2.10, the character χ extends to F ⋊ I_{L.2}(χ). Since Deg(2.𝔖_{5}) = {1^{2}, 4^{5}, 5^{2}, 6^{3}}, it follows from Theorem 1.2.16 that

Deg(F ⋊ (L.2) | Z(F)) = {18^{2}, 72^{5}, 90^{2}, 108^{3}} (3.9.3)

Comparing (3.3.3) and (3.9.3) we have

Inv(F ⋊ L | Z(F)) = {18, 72^{2}, 90, 108},
InvDef_{3}(F ⋊ L | Z(F)) = {4^{4}, 3}.

(3.9.4)

Let ψ ∈ (\overline{F}^{∗})^{#}. By Proposition 3.1.14, the group L.2 acts transitively on \overline{F}^{#}, and hence also on (\overline{F}^{∗})^{#}. So I_{L.2}(ψ) has order 6. We may assume without loss of generality that S_{L} ≤ I_{L.2}(ψ).

It follows from Corollary 3.7.3 that c ∈ ⟨d⟩Q normalizes F (this could also be deduced from Corollary 3.1.9). So c acts on \overline{F} = F/ Z(F) as well as on S_{L} ≤ E. Hence c acts on C\overline{F}(S_{L}). By Proposition 3.1.8 the group W′ = E ∩ F is the inverse image of C\overline{F}(S_{L}) in F. So c inverts C\overline{F}(S_{L}). Commutation in F induces a non-singular c-invariant bilinear form

\frac{\overline{F}}{[\overline{F}, S_{L}]} × C\overline{F}(S_{L}) → Z(F).

As c inverts C\overline{F}(S_{L}) and Z(F), it necessarily centralizes \overline{F}/[\overline{F}, S_{L}]. But C_{\overline{F}}^{∗}(S_{L}) ≅ (\overline{F}/[\overline{F}, S_{L}])^{∗}, by Lemma 1.2.20. So c centralizes C_{\overline{F}}^{∗}(S_{L}). We conclude that I_{L.2}(ψ) = S_{L} ⋊ ⟨c⟩ ≅ 𝔖_{3}.

As \overline{F} is abelian, ψ extends to I_{F⋊L.2}(ψ). Then by Clifford theory (3.9.5) Deg(F ⋊ L | F/ Z(F)) = {80^{2}, 160}

Comparing (3.3.2) and (3.9.5) we see that (3.9.6) Inv(F ⋊ L | F/ Z(F)) = {80}, InvDef_{3}(F ⋊ L | F mod Z(F)) = {6}.

From (3.9.2), (3.9.4), and (3.9.6), we have

Inv(F ⋊ L) = {1^{2}, 4^{3}, 5^{2}, 6, 18, 72^{2}, 80, 90, 108},
InvDef_{3}(F ⋊ L) = {6^{8}, 5, 4^{4}, 3}.

(3.9.7)

We can now prove

Proposition 3.9.8. The group N_{M_{c}}(C_{4}) = F⋊L has a unique 3-block. This block induces the principal 3-block, B_{0}, of M_{c}. Moreover

k( C_{4}, B_{0}, 6, \overline{M_{c}.2} ) = 8,  k( C_{4}, B_{0}, 5, \overline{M_{c}.2} ) = 1,
k( C_{4}, B_{0}, 4, \overline{M_{c}.2} ) = 4,  k( C_{4}, B_{0}, 3, \overline{M_{c}.2} ) = 1,
k( C_{4}, B_{0}, d, \overline{M_{c}.2} ) = 0,  for all other values of d.

(3.9.9)

Proof. This follows immediately from Proposition 3.3.6 and (3.9.7).

3.10. The Invariant Characters of W ⋊ Q We have N_{M_{c}.2}(W) = W ⋊ (Q × ⟨c⟩) = F ⋊ (N_{L} ⋊ ⟨c⟩), where N_{L} = N_{L}(S_{L}) =

S_{L} ⋊ Q. Choose a non-trivial character μ ∈ Irr(S_{L}). Now C_{Q}(S_{L}) = ⟨ab⟩, and both b and c invert S_{L}. So I_{Q×⟨c⟩}(μ) = ⟨ab, bc⟩ ≅ Q_{8}. As S_{L} is abelian, μ extends to I_{N_{L}⋊⟨c⟩}(μ) = S_{L} ⋊ ⟨ab, bc⟩. Thus Deg(N_{L} ⋊ ⟨c⟩ | S_{L}) = {2^{4}, 4}.

Now Deg((N_{L} ⋊ ⟨c⟩)/S_{L}) = Deg(Q × ⟨c⟩) = {1^{8}, 2^{2}}. So

Deg(N_{L} ⋊ ⟨c⟩) = {1^{8}, 2^{6}, 4}. (3.10.1)

Comparing (3.10.1) with (3.4.1), we see that

(3.10.2)  Inv(S_{L} ⋊ Q) = {1^{4}, 2^{3}},  InvDef_{3}(W ⋊ Q mod F) = {6^{7}}.

Let χ be one of the two irreducible characters of F of degree 9. Both a and c invert Z(F). So ac centralizes Z(F). Hence I_{N_{L}⋊⟨c⟩}(χ) = S_{L} ⋊ ⟨b, ac⟩ ≅ S_{L} ⋊ Q = N_{L}. Using (3.4.1), we see that Deg(I_{N_{L}⋊⟨c⟩}(χ)) = Deg(N_{L}) = {1^{4}, 2^{5}}.

Since χ extends to W, it also extends to I_{F⋊(N_{L}⋊⟨c⟩)}(χ). Then from Clifford theory

Deg(W ⋊ (Q × ⟨c⟩) | Z(F)) = {18^{4}, 36^{5}}. (3.10.3)

Comparing (3.10.3) with (3.4.4), we see that

(3.10.4) Inv(W ⋊ Q | Z(F)) = {18^{2}, 36^{2}}, InvDef_{3}(W ⋊ Q | Z(F)) = {4^{4}}.

If x ∈ Q × ⟨c⟩ has order 2, then Q centralizes ⟨x⟩ and hence stabilizes C_{\overline{F}}^{∗}(x). By Proposition 3.1.4 and Lemma 3.1.7, the group Q acts regularly on each of its nontrivial orbits in \overline{F}^{∗}. So |Q| | |C_{\overline{F}}^{∗}(x)| − 1. Hence |C_{\overline{F}}^{∗}(x)| = 1 or 9. We conclude that the nontrivial elements of C_{\overline{F}}^{∗}(x) form either a single Q-orbit or the empty set.

From (3.4.2) we know that Orb(N_{L}, (\overline{F}^{∗})^{#}) = \{8, 24^{3}\}. The N_{L}-orbit of length 8 must be an N_{L} × ⟨c⟩-orbit, while one of the three N_{L}-orbits of length 24 must be an N_{L} × ⟨c⟩-orbit, and the other two either join to form a single N_{L} × ⟨c⟩-orbit of length 48, or are both N_{L} ×⟨c⟩-orbits. If ν ∈ \overline{F}^{∗} lies in an N_{L} ⋊⟨c⟩-orbit of length 24, then I_{N_{L}⋊⟨c⟩}(ν) is isomorphic to ℤ_{2}. Thus some N_{L} ⋊ ⟨c⟩-conjugate of I_{N_{L}⋊⟨c⟩}(ν) is contained in the Sylow 2-subgroup Q × ⟨c⟩ of N_{L} ⋊ ⟨c⟩. Since τ inverts \overline{F}^{∗}, it follows that I_{N_{L}⋊⟨c⟩}(ν) is conjugate to one of the remaining two subgroups of Q × ⟨c⟩ of order 2. By the previous paragraph, each of these ℤ_{2}’s centralizes elements from at most one N_{L} ⋊ ⟨c⟩-orbit of (\overline{F}^{∗})^{#}. Hence there are at most two N_{L} ⋊ ⟨c⟩-orbits of length 24. But then there is exactly one such orbit, coinciding with one of the N_{L}-orbits of length 24. The remaining two N_{L}-orbits of length 24 must fuse in N_{L} ⋊ ⟨c⟩. Thus

Orb(N_{L} ⋊ ⟨c⟩, (\overline{F}^{∗})^{#}) = \{8, 24, 48\}.

Let ν_{1} lie in the N_{L} ⋊ ⟨c⟩-orbit of length 8. Without loss of generality S_{L} ≤ I_{N_{L}⋊⟨c⟩}(ν_{1}). We know there exists an involution x ∈ {τ, c, τc} such that I_{N_{L}⋊⟨c⟩}(ν_{1}) = S_{L} ⋊ ⟨x⟩. We cannot have x ≠ τ, since τ inverts \overline{F}^{∗}. So x is either c or τc. In both cases x inverts S_{L}, because τ centralizes S_{L} and c inverts S_{L}. Hence I_{N_{L}⋊⟨c⟩}(ν_{1}) = S_{L} ⋊ ⟨x⟩ ≅ 𝔖_{3}. Invoking Theorems 1.2.15 and 1.2.16 we see that

Deg(\overline{F} ⋊ (N_{L} ⋊ ⟨c⟩) | ν_{1}) = \{8^{2}, 16\}. (3.10.5)

Let ν_{2} ∈ (\overline{F}^{∗})^{#} have N_{L} ⋊ ⟨c⟩-orbit length 24. Then I_{N_{L}⋊⟨c⟩}(ν_{2}) ≅ ℤ_{2}. So

Deg(\overline{F} ⋊ (N_{L} ⋊ ⟨c⟩) | ν_{2}) = \{24^{2}\}. (3.10.6)

Let ν_{3} ∈ (\overline{F}^{∗})^{#} have N_{L} ⋊ ⟨c⟩-orbit length 48. Then ν_{3} induces an irreducible character of \overline{F} ⋊ (N_{L} ⋊ ⟨c⟩) of degree 48. So Deg(\overline{F} ⋊ (N_{L} ⋊ ⟨c⟩) | ν_{3}) = \{48\}. (3.10.7)

From (3.10.5), (3.10.6) and (3.10.7) we obtain (3.10.8) Deg(W ⋊ (Q × ⟨c⟩) | F/ Z(F)) = {8^{2}, 16, 24^{2}, 48}.

Comparing (3.10.8) with (3.4.3) we conclude that (3.10.9) Inv(W ⋊ Q | F/ Z(F)) = {8, 24}, InvDef_{3}(W ⋊ Q | F/ Z(F)) = {6, 5}.

From (3.10.2), (3.10.4) and (3.10.8) we have

Inv(W ⋊ Q) = {1^{4}, 2^{3}, 8, 18^{2}, 24, 36^{2}},
InvDef_{3}(W ⋊ Q) = {6^{8}, 5, 4^{4}}.

(3.10.10)

We now can prove the following proposition.

Proposition 3.10.11. The group N_{M_{c}}(C_{3}) = W ⋊ Q has unique 3-block, which necessarily induces the principal 3-block, B_{0}, of M_{c}. Hence

k( C_{3}, B_{0}, 6, \overline{M_{c}.2} ) = 8,  k( C_{3}, B_{0}, 5, \overline{M_{c}.2} ) = 1,
k( C_{3}, B_{0}, 4, \overline{M_{c}.2} ) = 4,  k( C_{3}, B_{0}, 3, \overline{M_{c}.2} ) = 0,
k( C_{3}, B_{0}, d, \overline{M_{c}.2} ) = 0,  for all other values of d.

(3.10.12)

Proof. This follows immediately from Proposition 3.4.6 and (3.10.10). From Table 3.1 on page 48 and Lemma 3.7.1, the normalizer in M_{c}.2 of each

of the chains C_{3}, C_{5} and C_{6} coincides with N_{M_{c}.2}(W) = W ⋊ (⟨dQ⟩). Hence k( C_{3}, B_{0}, d, \overline{M_{c}.2} ) = k( C_{5}, B_{0}, d, \overline{M_{c}.2} ) = k( C_{6}, B_{0}, d, \overline{M_{c}.2} ) for all values of d.

3.11. The Invariant Conjecture for the prime p = 3

We obtain the invariant characters of the principal block B_{0} of M_{c} from [Con85, p100]. We list here these characters and their defects:

Character χ_{1} χ_{2} χ_{3} χ_{4} χ_{7} χ_{8} χ_{9} χ_{10} χ_{11} χ_{12} χ_{13} χ_{15} χ_{20}
Degree  1 22 231 252 896 896 1750 3520 3520 4500 4752 5544 9625
3-Defect 6 6 5 4 6 6 6 6 6 4 3 4 6

Thus

k( C_{1}, B_{0}, 6, \overline{M_{c}.2} ) = 8,  k( C_{1}, B_{0}, 5, \overline{M_{c}.2} ) = 1,
k( C_{1}, B_{0}, 4, \overline{M_{c}.2} ) = 3,  k( C_{1}, B_{0}, 3, \overline{M_{c}.2} ) = 1,
k( C_{1}, B_{0}, d, \overline{M_{c}.2} ) = 0,  for all other values of d.

(3.11.1)

We can now prove the following Theorem 3.11.2. The Invariant Conjecture holds for the McLaughlin simple group

and the prime p = 3.

Proof. From Conjecture 1.4.4 and Table 3.1 on page 48, we must prove

k( C_{1}, B_{0}, d, \overline{M_{c}.2} )+ k( C_{3}, B_{0}, d, \overline{M_{c}.2} ) =
k( C_{2}, B_{0}, d, \overline{M_{c}.2} ) + k( C_{4}, B_{0}, d, \overline{M_{c}.2} ),

(3.11.3)

for all values of d ∈ ℤ.

From (3.8.8), (3.9.9), (3.10.12) and (3.11.1) we obtain the following sums for the equation above for d ∈ {3, 4, 5, 6}:

3-Defect C_{1}  C_{3}  C_{2}  C_{4}
6  8 + 8 = 8 + 8
5  1 + 1 = 1 + 1
4  3 + 4 = 3 + 4
3  1 + 0 = 0 + 1

Table 3.3. The Invariant Conjecture for p = 3

The summands in (3.11.3) are zero for all other values of d. This completes the proof of the theorem.

3.12. The Schur Multiplier of M_{c} We let A.M_{c} denote the universal covering group of M_{c} by its Schur Multiplier

A. In [Gr87], R. Greiss shows that A is isomorphic to ℤ_{3}. The arguments he uses are somewhat terse. So we expound them in more detail here. We will assume only that A has order a non-trivial power of 3.

In this section A_{1} will denote a fixed subgroup of A, properly contained in A. We let M̂_{c} = A^{1}.M_{c} denote the factor extension A.M_{c}/A_{1} of M_{c} by A^{1} = A/A_{1}. So M̂_{c} is a perfect central extension of M_{c} by a non-trivial 3-group, and any factor extension M̂_{c}/A_{1}^{1} of M_{c} by A^{1}/A_{1}^{1} is non-split, for every subgroup A_{1}^{1} < A^{1}.

If X is any subgroup of M_{c} then we will use A^{1}.X or X̂ to denote its inverse image in M̂_{c}. We note that there is a natural conjugation action of M_{c} on M̂_{c}, since M̂_{c} is a central extension of M_{c}.

We will use the notation of Section 3.1. In particular, F is an extra-special 3-group of M_{c} of type 3_{+}^{1+4}, and N_{M_{c}}(F) = F ⋊ L, where L ≅ SL(2, 5).2. We let C = L′. So C ≅ SL(2, 5). Also W will denote a fixed Sylow 3-subgroup of M_{c} containing F, and S_{L} will denote the intersection of W with L. Since F = O_{3}(F ⋊ L), it follows that F̂ = O_{3}(A^{1}.(F ⋊ L)).

Lemma 3.12.1. L̂ splits over A^{1}.

Proof. From the Atlas, the group C ≅ 2.𝔄_{5} is isomorphic to the universal covering group of 𝔄_{5}. Hence Ĉ splits over A^{1}. But C contains a Sylow 3-subgroup S_{L} of L. So Ŝ_{L} splits over A^{1}. We conclude that L̂ splits over A^{1}.

We will denote by L^{1} a fixed complement to A^{1} in L̂. So L^{1} is a complement to F̂ in A^{1}.(F ⋊ L).

Lemma 3.12.2. F̂ does not split over A^{1}.

Proof. Suppose there exists a complement F^{1} to A^{1} in F̂. Since A^{1} is central, there is a one-to-one correspondence between the complements to A^{1} in F̂ and the elements of the set H^{1}(F^{1}, A^{1}) = Hom(F^{1}, A^{1}) = Hom(\overline{F}, A^{1}). The complement corresponding to f ∈ Hom(F^{1}, A^{1}) is the set {f(x) × x | x ∈ F^{1}}.

Now |Hom(\overline{F}, A^{1})| = |Ω(A^{1})|^{4} is a power of 3, as \overline{F} is an elementary abelian 3-group. Consider the central involution τ^{1} of L^{1}. This element normalizes F̂ and A^{1}. Thus it permutes the complements to A^{1} in F̂. Since there are an odd number of complements, there is at least one which is fixed by τ^{1}. We assume that τ^{1} fixes F^{1}. As τ^{1} inverts \overline{F} and centralizes A^{1}, it inverts all homomorphisms \overline{F} → A^{1}. So τ^{1} fixes no non-trivial elements of the 3-group Hom(\overline{F}, A^{1}). Hence F^{1} is the unique complement fixed by τ^{1}.

Let x be an arbitrary element of L^{1} ≤ CM̂_{c}(τ^{1}). Then (F^{1})^{x} is a complement to A^{1} in F̂ and ((F^{1})^{x})^{τ^{1}} = (F^{1})^{xτ^{1}} = (F^{1})^{τ^{1}x} = (F^{1})^{x}. As F^{1} is the unique complement fixed by τ^{1}, this implies that (F^{1})^{x} = F^{1}. So L^{1} normalizes F^{1}. Hence F^{1} ⋊ L^{1} is a complement to A^{1} in A^{1}.(F ⋊ L). Since Ŵ is a subgroup of A^{1}.(F ⋊ L), it splits over A^{1}. This implies that M̂_{c} itself splits over A^{1}, as Ŵ is a Sylow 3-subgroup of M̂_{c}. This is impossible. The lemma now follows.

We now prove

Lemma 3.12.3. F̂′ = F̂′ = { \atop Z(F)}.

Proof. Since Z(F) is cyclic, { \atop Z(F)} is abelian. The derived group F̂′ of F̂ is contained in { \atop Z(F)}, since F′ = Z(F). Furthermore, F̂′ is stabilized by ⟨a⟩ ∈ L. Let A_{1}^{1} = CF̂′(a) and Z^{1} = [F̂′, a]. Then by [As86, 24.6] we have F̂′ = A_{1}^{1} × Z^{1}. Also A_{1}^{1} is contained in A^{1} and { \atop Z(F)} = A^{1} × Z^{1}.

We claim that A_{1}^{1} = A^{1}. If not, A^{1}/A_{1}^{1} is a non-trivial quotient of A. Since A^{1} is an arbitrary non-trivial quotient of A, we can replace it by A^{1}/A_{1}^{1}. So for the rest of the proof we assume we have done so. This is equivalent to assuming that A_{1}^{1} = {1}.

By [As86, 24.6], the abelian group F̂/F̂′ can be written as F̂/F̂′ = C_{F̂/F̂′}(τ) × [F̂/F̂′, τ]. Since τ centralizes A^{1}F̂′/F̂′ but inverts F̂/(A^{1}F̂′) = F̂/{ \atop Z(F)} ≅ \overline{F}, we have

F̂/F̂′ = (A^{1}F̂′/F̂′) × [F̂/F̂′, τ].

Let F^{1} be the inverse image of [F̂/F̂′, τ] in F̂. Then A^{1}F̂′ ∩ F^{1} = F̂′, since (A^{1}F̂′/F̂′) ∩ (F^{1}/F̂′) = {1}. But A^{1} ∩ F̂′ = A_{1}^{1} = {1}. So A^{1} ∩ F^{1} = {1}. It follows that F^{1} is a complement to A^{1} in F̂, contradicting Lemma 3.12.2. This proves our claim. The lemma follows.

Corollary 3.12.4. The 3-group F̂ is of class two. Hence F̂′ = Z(F̂), while both F̂′ and A^{1} are elementary abelian.

Proof. Since [F, F′] = {1}, we have [F̂, F̂′] ≤ A^{1}. So commutation in F̂ induces a ⟨τ⟩-invariant bilinear map F̂/F̂′ ×F̂′/A^{1} → A^{1}. Since F̂′/A^{1} and A^{1} are centralized by τ, while F̂/F̂′ is inverted by τ, the above map is trivial. Hence [F̂, F̂′] = {1}.

As F̂/F̂′ ≅ \overline{F} is elementary abelian, [R93, 5.2.5] implies that F̂′/[F̂, F̂′] = F̂′ is also elementary abelian. In particular A^{1} is elementary abelian.

Corollary 3.12.5. C = L′ centralizes F̂′. The elementary abelian 3-group F̂′ = Z(F̂) can be written as

F̂′ = A^{1} × Z^{1},

where Z^{1} ≅ ℤ_{3} is an L-invariant subgroup of F̂′.

Proof. The action of M_{c} on M̂_{c} restricts to an action of C and L on the 3-group F̂′ = Z(F̂) = { \atop Z(F)}. By Lemma 3.1.3, the action of C on each section for the normal series F̂′ > A^{1} > {1} of F̂′ is trivial. Since C ≅ SL(2, 5) has no non-trivial quotients which are 3-groups, this implies that it centralizes F̂′.

Recall there exists an element a of order 4 in L\C which inverts Z(F). By [As86, 24.6] we may write F̂′ = Z(F̂) = CF̂′(a) × [F̂′, a]. But a centralizes A^{1} and inverts F̂′/A^{1}. So F̂′ = A^{1} × Z^{1}, where Z^{1} = [F̂′, a]. This decomposition is L-invariant as L = ⟨C, a⟩, and C centralizes F̂′.

For the remainder of this chapter we let S_{L^{1}} denote the Sylow 3-subgroup of L^{1} contained in Ŵ. So S_{L^{1}} is cyclic of order 3.

Corollary 3.12.6. Z(Ŵ) = Z(F̂).

Proof. Since Z(W) = Z(F), we have Z(Ŵ) ≤ { \atop Z(F)} = Z(F̂). By the previous corollary [C, Z(F̂)] = 1. So the Sylow 3-subgroup S_{L^{1}} = Ŵ ∩ L^{1} of L^{1} centralizes Z(F̂). But S_{L^{1}} F̂ = Ŵ. The result now follows immediately.

The following lemma could have been proved earlier. We use it here to show that A has order 3.

Lemma 3.12.7. C acts irreducibly, but not absolutely irreducibly on \overline{F}. Also L acts absolutely irreducibly on \overline{F}.

Proof. From Proposition 3.1.14 the group L acts transitively on \overline{F}^{#}, and the stabilizer of a point is isomorphic to S_{L}. Since S_{L} ≤ C has index 40 in C it follows that Orb(C | \overline{F}^{#}) = \{40^{2}\}. Hence C acts irreducibly on \overline{F}. Consider \overline{F} as an additive GF(3)C-module. Then 𝔉 = End_{GF(3)C}(\overline{F}) is a finite field of characteristic three.

Recall from Lemma 3.7.2 that N_{M_{c}.2}(F) = F ⋊ (⟨d⟩L), where d is an element of M_{c}.2 of order four, [d, C] = 1, and [d, a] = τ.

We identify GL(4, 3) with End_{GF(3)}(\overline{F}). Then we can regard ⟨d⟩L as a subgroup of GL(4, 3), as ⟨d⟩L acts faithfully on \overline{F}. Thus d ∈ 𝔉, and hence 4 | |𝔉^{#}|. Since 𝔉^{#} ≤ GL(4, 3), this implies that 𝔉 ≅ GF(9) or GF(81). However GF(81)^{#} is a self centralizing subgroup of GL(4, 3), and thus does not contain C. So 𝔉 ≅ GF(9). This proves the first statement.

Now C ≤ L. So End_{GF(3)L}(\overline{F}) ⊆ 𝔉. But [L, d] ≠ 1. So End_{GF(3)L}(\overline{F}) < 𝔉. We conclude that End_{GF(3)L}(\overline{F}) = GF(3). This proves the second statement.

We now specialize the previous results to A^{1} = A. Then M̂_{c} becomes the universal covering group A.M_{c} of M_{c} by its Schur multiplier A, and X̂ becomes A.X, for any X ≤ M_{c}.

Theorem 3.12.8. A ≅ ℤ_{3}.

Proof. By Corollary 3.12.4, the group A is an elementary abelian 3-group. Suppose A ≅ ℤ_{3}^{n}, for some n ∈ ℤ. We can find x_{1}, …, x_{n}, y_{1}, …, y_{n} ∈ F̂ such that {[x_{i}, y_{i}] | i = 1, …, n} is a basis for A.

Commutation in F̂ gives rise to the L-invariant alternating bilinear forms ρ_{i} : \overline{F} × \overline{F} → GF(3), for i = 1, …, n, where ρ_{i}(\overline{x}, \overline{y}) is defined by the equation

[x, y] = ∏_{i=1}^{n}[x_{i}, y_{i}]^{ρ_{i}(\overline{x},\overline{y})},

for any x, y ∈ F̂ with respective images \overline{x}, \overline{y} ∈ \overline{F}. Each ρ_{i} induces a map f_{i} ∈ Hom_{GF(3)L}(\overline{F}, \overline{F}^{∗}), given by f_{i}(\overline{x}) : \overline{y} → ρ_{i}(\overline{x}, \overline{y}), for all \overline{x}, \overline{y} ∈ \overline{F}. The map f_{i} is non-trivial, as ρ_{i}(\overline{x}_{i}, \overline{y}_{i}) = 1_{GF(3)}.

Since \overline{F} and \overline{F}^{∗} are irreducible L-modules, Schur’s Lemma implies that f_{i} is in fact an L-isomorphism of \overline{F} onto \overline{F}^{∗}, for i = 1, …, n. Thus each f_{1}^{−1}f_{i} is an element of End_{GF(3)L}(\overline{F}). But L acts absolutely irreducibly on \overline{F} by Lemma 3.12.7. So f_{1}^{−1}f_{i} = λ_{i}, for some λ_{i} ∈ GF(3)^{#}.

Suppose n > 1. Then

0 = ρ_{2}(\overline{x}_{1}, \overline{y}_{1}) = (f_{2}(\overline{x}_{1}))(\overline{y}_{1}) = (λ_{2}f_{1}(\overline{x}_{1}))(\overline{y}_{1}) = λ_{2} ρ_{1}(\overline{x}_{1}, \overline{y}_{1}) = λ_{2} ≠ 0.

This contradiction forces n = 1. We conclude that A ≅ ℤ_{3}.

3.13. The Character Degrees of A.(E ⋊ M) The action of M_{c} on M̂_{c} restricts to an action of M ≅ M̂/A on Ê. If x ∈ M,

then ỹ^{x} = ỹx̃, for any ỹ ∈ Ê, where x̃ is any element of M̂ lying over x. Notice that y^{x} is the image of ỹ^{x} in M_{c}, where y is the image of ỹ in E.

Recall from (3.5.2) the chain K < A_{4} < S_{4} < A_{6} < M of subgroups of M, where A_{6} = M′ ≅ 𝔄_{6}, S_{4} ≅ 𝔖_{4}, A_{4} = S_{4}′ ≅ 𝔄_{4} and K = O_{2}(A_{4}) ≅ ℤ_{2}^{2}.

Proposition 3.13.1. Ê ≅ ℤ_{3}^{5}.

Proof. From Lemma 3.5.1 the elementary abelian 3-group E is isomorphic to the regular GF(3)K-module. In particular E ≅ E_{1} ⊕ E_{2} ⊕ E_{3} ⊕ E_{4} as GF(3)K-modules, where each E_{i} is cyclic of order 3.

Suppose i ≠ j. Commutation in Ê gives rise to an K-invariant bilinear map Ê_{i}/A × Ê_{j}/A → A. By the regularity of E, we can find x ∈ K which centralizes one of E_{i}, E_{j} and inverts the other. So the map must be trivial. Hence Ê_{i} commutes with Ê_{i}. Moreover each Ê_{i} is abelian, since each E_{i} is cyclic. We conclude that Ê is abelian.

By Proposition 3.1.4 the centralizer of Q in E ≤ W is trivial. Hence

Ê = A × [Q, Ê] ≅ A × E,

using [As86, 24.6]. So Ê is elementary abelian of order 3^{5}.

We will use the following lemma to show that the (3B) conjugacy class of M_{c} fuses in M̂_{c}.

Lemma 3.13.2. Let X be a special p-group of type p^{2+4}, i.e. X′ = Φ(X) = Z(X) is elementary abelian of order p^{2}, while \overline{X} = X/X′ is elementary abelian of order p^{4}. Then there is some element x ∈ X such that X′ = [x, X].

Proof. Let \overline{X} = X/X′. We can regard \overline{X} and X′ as vector spaces over GF(p) with dimensions 4 and 2, respectively. Commutation in X gives a bilinear map [ , ] : \overline{X} × \overline{X} → X′.

Let \overline{x}_{1} ∈ \overline{X}. If [\overline{x}_{1}, \overline{X}] = X′, we are done. Otherwise, there exists \overline{x}_{2} ∈ \overline{X} such that \overline{X} = ⟨\overline{x}_{2}⟩ ⊕ ⟨\overline{x}_{1}⟩^{⊥}, where ⟨\overline{x}_{1}⟩^{⊥} = \{\overline{x} ∈ \overline{X} | [\overline{x}_{1}, \overline{x}] = 1\}. If [\overline{x}_{2}, \overline{X}] = X′, we are done. Otherwise \overline{X} = ⟨\overline{x}_{1}⟩ ⊕ ⟨\overline{x}_{2}⟩^{⊥} and ⟨\overline{x}_{1}⟩^{⊥} ≠ ⟨\overline{x}_{2}⟩^{⊥}, since \overline{x}_{1} ∈ ⟨\overline{x}_{1}⟩^{⊥}\⟨\overline{x}_{2}⟩^{⊥}. Thus ⟨\overline{x}_{1}⟩^{⊥} ∩ ⟨\overline{x}_{2}⟩^{⊥} is a 2-dimensional subspace of \overline{X}. Let \overline{x}_{3}, \overline{x}_{4} be generators for ⟨\overline{x}_{1}⟩^{⊥} ∩ ⟨\overline{x}_{2}⟩^{⊥}. By the bilinearity of the commutator, it is clear that [⟨\overline{x}_{1}, \overline{x}_{2}⟩, ⟨\overline{x}_{3}, \overline{x}_{4}⟩] = \{1\}.

Since \overline{X} = ⟨\overline{x}_{1}, \overline{x}_{2}, \overline{x}_{3}, \overline{x}_{4}⟩, the vector space X′ is spanned by \{[\overline{x}_{i}, \overline{x}_{j}]\}, for 1 ≤ i < j ≤ 4. But then X′ = ⟨[\overline{x}_{1}, \overline{x}_{2}], [\overline{x}_{3}, \overline{x}_{4}]⟩, since all the other terms in the spanning set are trivial.

Let \overline{x} = \overline{x}_{1} + \overline{x}_{3}. Then [\overline{x}, \overline{x}_{2}] = [\overline{x}_{1}, \overline{x}_{2}] and [\overline{x}, \overline{x}_{4}] = [\overline{x}_{3}, \overline{x}_{4}]. So [\overline{x}, \overline{X}] = X′. The lemma follows.

We need the next result in order to investigate the action of A.(E ⋊ M) on Ê.

Lemma 3.13.3. The (3A) conjugacy class of M_{c} splits in M̂_{c}, while the (3B) conjugacy class of M_{c} fuses in M̂_{c}.

Proof. Let α̃ be an element of M̂_{c} whose image in M_{c} is α. Then α̃ is central in Ŵ, by Corollary 3.12.6. The derived group C^{1} of L^{1} centralizes { \atop Z(F)}, by Corollary 3.12.5. Hence

CM̂_{c}(α̃) = F̂ ⋊ C^{1} = A.(C_{M_{c}}(α)).

In particular, the conjugacy class (3A) of M_{c} containing α splits in M̂_{c}.

By Lemma 3.13.2, there exist x, y ∈ F̂\F̂′ such that A = ⟨[x, y]⟩. This implies that the conjugacy class of xA in M_{c} = (M̂_{c})/A fuses in M̂_{c}. But by Proposition 3.1.14, the element xA ∈ F\F′ comes from the (3B) conjugacy class of M_{c}. So this class fuses in M̂_{c}.

We fix ρ ∈ Irr(A)^{#} for the remainder of this chapter. In the next lemma we compute the degrees and 3-defects of the irreducible characters of A.(E ⋊ M) which lie over ρ.

Lemma 3.13.4. Deg(A.(E ⋊ M) | ρ) = {36^{4}, 45^{4}, 90^{3}, 144}. Hence

Def_{3}(A.(E ⋊ M) | ρ) = {5^{12}} (3.13.5)

Proof. As before α̃ is an element of M̂_{c} whose image in M_{c} is α. We deduce from Proposition 3.1.12 and Lemma 3.13.3 that α̃ lies in an M-orbit of size 20, and M has three such orbits in Ê.

Recall that after Proposition 3.1.12 we set β as an element of E belonging to the (3B) conjugacy class of M_{c}. We let β̃ be an element of M̂_{c} whose image in M_{c} is β. By (3.1.13), we have C_{M}(β) = A_{4}, where A_{4} ≅ 𝔄_{4}. It follows from Lemma 3.13.3 that CM̂_{c}(β̃) is a subgroup of (Ê).A_{4} of index 3. In particular β̃ lies in an M-orbit of length 180.

Since [A, M] = 1, we have Orb(M, A^{#}) = {1^{2}}. Thus

Orb(M, Ê^{#}) = {1^{2}, 20^{3}, 180}.

As M has 6 orbits on Ê^{#}, it also has 6 orbits on Irr(Ê)^{#}. By Proposition 3.1.12, the set Irr(Ê mod A)^{#} accounts for 2 of these orbits. There exists a Galois automorphism γ of the characters of subgroups of Ê such that ρ^{γ} = ρ^{2}. Hence γ sends Irr(Ê | ρ) bijectively onto the set Irr(Ê | ρ^{2}). In particular these two sets have the same cardinality. Since Irr(E)^{#} is the disjoint union of these two sets and Irr(E)^{#}, all of which are M-invariant, we have

6 = |Orb(M, Irr(Ê))^{#}|
= |Orb(M, Irr(E)^{#})| + |Orb(M, Irr(Ê | ρ))| + |Orb(M, Irr(Ê | ρ^{2}))|
= 2 + 2|Orb(M, Irr(Ê | ρ))|.

We conclude that |Orb(M, Irr(Ê | ρ))| = 2.

Let T_{2} be a Sylow 2-subgroup of M. Then from the Atlas T_{2} is a maximal subgroup of M. Moreover T_{2} is semi-dihedral of order 16.

Now C_{E}(T_{2}) = {1} by (3.1.13). So CÊ(T_{2}) = A. Since Ê is an abelian 3-group and T_{2} is a 3′-group, by [As86, 24.6] we have Ê = A × [Ê, T_{2}]. Let ψ̂_{1} be the unique character in Irr(Ê) which is trivial on [Ê, T_{2}] and restricts to ρ on A. So ψ̂_{1} ∈ CÊ^{∗}(T_{2}). Moreover I_{M}(ψ̂_{1}) = T_{2}, since T_{2} is maximal in M. So ψ̂_{1} lies in an M-orbit of length 720/16 = 45.

Let N_{5} be the normalizer of a Sylow 5-subgroup of M. Then N_{5} is a maximal subgroup of M, isomorphic to the Frobenius group 5 : 4. As with T_{2}, we can find ψ̂_{2} ∈ Irr(Ê | ρ) such that I_{M}(ψ̂_{2}) = N_{5}. So ψ̂_{2} lies in an M-orbit of length 720/20 = 36.

Since I_{M}(ψ̂_{i}) is a 3′-group, ψ̂_{i} extends to I_{A.(E⋊M)}(ψ̂_{i}), for i = 1, 2. It is routine to show that Deg(T_{2}) = {1^{4}, 2^{3}} and Deg(N_{5}) = {1^{4}, 4}. Hence by Clifford theory

Deg(A.(E ⋊ M) | ψ̂_{1}) = {45^{4}, 90^{3}},  Def_{3}(A.(E ⋊ M) | ψ̂_{1}) = {5^{7}},
Deg(A.(E ⋊ M) | ψ̂_{2}) = {36^{4}, 144},  Def_{3}(A.(E ⋊ M) | ψ̂_{2}) = {5^{5}}.

(3.13.6)

Also Irr(A.(E⋊M) | ρ) is the disjoint union of Irr(A.(E⋊M) | ψ̂_{1}) and Irr(A.(E⋊M) | ψ̂_{2}). The results of the lemma now follow.

Finally we have

Proposition 3.13.7. The group NM̂_{c}(C_{2}) = A.(E ⋊ M) has a unique 3-block. This block necessarily induces the principal 3-block, B_{0}^{∗}, of M̂_{c}. Hence

k(C_{2}, B_{0}^{∗}, 5 | ρ) = 12,
k(C_{2}, B_{0}^{∗}, d | ρ) = 0,  for all other values of d.

(3.13.8)

Proof. The first statement follows from the fact that Ê is self centralizing in M̂_{c}. We obtain (3.13.8) from (3.13.6).

3.14. The Character Degrees of A.(F ⋊ L)

Recall from Corollary 3.12.5 that Z(F̂) = F̂′ = A × Z^{1} is an L-invariant decomposition of the elementary abelian 3-group Z(F̂) ≅ ℤ_{3}^{2}.

Lemma 3.14.1. If X is a subgroup of Z(F̂) of order 3, then F̂/X is an extra-special group of type 3_{+}^{1+4}.

Proof. Let X ≤ Z(F̂) have order 3. Let Z denote the inverse image of Z(F̂/X) in F̂. Then Z(F̂) = F̂′ ≤ Z < F̂ because X < F̂′.

Recall that the subgroup C of L acts trivially on Z(F̂). Hence C normalizes Z, and Z/ Z(F̂) is a C-invariant subspace of F̂/ Z(F̂). But by Lemma 3.12.7 the group C acts irreducibly on F̂/ Z(F̂). So Z = Z(F̂). It follows that Z(F̂/X) = Z(F̂)/X = F̂′/X. But (F̂/X)′ = F̂′/X, as X ≤ F̂′. So Z(F̂/X) = (F̂/X)′.

Finally, F̂/X has exponent 3, since by the Atlas M̂_{c} has no elements of order 9.

We can now prove the following

Lemma 3.14.2. There exist characters χ̂_{1}, χ̂_{2} ∈ Irr(F̂ | ρ) such that I_{A.(F⋊L)}(χ̂_{1}) = A.(F ⋊ L), and I_{A.(F⋊L)}(χ̂_{2}) = A.(F ⋊ C). Each χ̂_{i} extends to its stabilizer in A.(F ⋊ L). Hence

Deg(A.(F ⋊ L) | χ̂_{1}) = {9^{2}, 36^{5}, 45^{2}, 54^{3}},  Def_{3}(A.(F ⋊ L) | χ̂_{1}) = {5^{9}, 4^{3}},
Deg(A.(F ⋊ L) | χ̂_{2}) = {18, 36^{2}, 54^{2}, 72^{2}, 90, 108}, Def_{3}(A.(F ⋊ L) | χ̂_{2}) = {5^{6}, 4^{3}}.

(3.14.3)

Also Irr(A.(F ⋊ L) | ρ) is the disjoint union of Irr(A.(F ⋊ L) | χ̂_{1}) and Irr(A.(F ⋊ L) | χ̂_{2}).

Proof. Let λ ∈ Irr(Z(F̂))^{#}. We can regard λ as an element of Irr(Z(F̂)/ Ker(λ))^{#}. Then by Theorem 1.2.18, there is unique irreducible character λ̂ of F̂/ Ker(λ) lying over λ, and λ̂ has degree 9 and vanishes outside Z(F̂)/ Ker(λ). By inflation, we may regard λ̂ as an irreducible character of F̂. We see in this way that F̂ has eight characters of degree 9 which vanish outside Z(F̂). Since Irr(F̂ | ρ) and Irr(F̂ | ρ^{2}) are of the same size, and account for all irreducible characters of Irr(F̂ | Z(F̂)) non-trivial on A, both of these sets contain (8 − 2)/2 = 3 characters.

We pick χ̂_{1}, χ̂_{2} ∈ Irr(F̂ | ρ), with χ̂_{1} trivial on Z^{1} and χ̂_{2} non-trivial on Z^{1}. Then I_{A.(F⋊L)}(χ̂_{1}) = A.(F ⋊ L), and I_{A.(F⋊L)}(χ̂_{2}) = A.(F ⋊ C). Since A.(F ⋊ L)/F̂ ≅ L has cyclic Sylow 3-subgroups, both χ̂_{1} and χ̂_{2} extend to their stabilizers in A.(F ⋊ L). We obtain (3.14.3) immediately using Clifford theory.

The A.(F ⋊ L)-invariant character χ̂_{1} and the two A.(F ⋊ L)-conjugates of χ̂_{2} account for all three elements of Irr(F̂ | ρ).

To conclude our analysis of A.(F ⋊ L) we now prove

Proposition 3.14.4. The group NM̂_{c}(C_{4}) = A.(F ⋊ L) has a unique 3-block, which necessarily induces the principal 3-block, B_{0}^{∗}, of M̂_{c}. Hence

k(C_{4}, B_{0}^{∗}, 5 | ρ) = 15,

k(C_{4}, B_{0}^{∗}, 4 | ρ) = 6,
k(C_{4}, B_{0}^{∗}, d | ρ) = 0,  for all other values of d.

(3.14.5)

Proof. Since C_{A.(F⋊L)}(F̂) = Z(F̂), the group A.(F ⋊ L) has a unique 3-block. Then (3.14.5) follows from (3.14.3).

3.15. The Character Degrees of A.(W ⋊ Q) Most of the work of this section has already been done in Section 3.14. We have

Lemma 3.15.1. There exist χ̂_{1}, χ̂_{2} ∈ Irr(F̂) such that I_{A.(W⋊Q)}(χ̂_{1}) = A.(W⋊Q) and I_{A.(W⋊Q)}(χ̂_{2}) = A.(W ⋊ ⟨b⟩), where b is an element of Q ∩ C of order 4. Hence

Deg(A.(W ⋊ Q) | χ̂_{1}) = {9^{4}, 18^{5}},  Def_{3}(A.(W ⋊ Q) | χ̂_{1}) = {5^{9}},
Deg(A.(W ⋊ Q) | χ̂_{2}) = {18^{4}, 36^{2}},  Def_{3}(A.(W ⋊ Q) | χ̂_{2}) = {5^{6}}.

(3.15.2)

Also Irr(A.(W⋊Q) | ρ) is the disjoint union of Irr(A.(W⋊Q) | χ̂_{1}) and Irr(A.(W⋊Q) | χ̂_{2}).

Proof. The existence of χ̂_{1} and χ̂_{2} comes from Lemma 3.14.2. The character χ̂_{1} is invariant in A.(F⋊L), and hence in A.(W⋊Q). The stabilizer of χ̂_{2} in A.(W⋊Q) is A.(W ⋊ Q) ∩ A.(F ⋊ C) = A.(F ⋊ (S_{L} ⋊ ⟨b⟩)) It also follows immediately from Lemma 3.14.2 that both χ̂_{1} and χ̂_{2} extend to their stabilizers in A.(W ⋊ Q). We obtain (3.15.2) using (3.4.1), (3.4.4) and Clifford Theory.

We can now prove the following

Proposition 3.15.3. The group NM̂_{c}(C_{3}) = A.(W ⋊ Q) has a unique 3-block. This block necessarily induces the principal 3-block, B_{0}^{∗}, of M̂_{c}. Hence

k(C_{3}, B_{0}^{∗}, 5 | ρ) = 15,
k(C_{3}, B_{0}^{∗}, d | ρ) = 0,  for all other values of d.

(3.15.4)

Proof. The group A.(W ⋊ Q) has a unique 3-block since C_{A.(W⋊Q)}(W) = Z(Ŵ). Then (3.15.4) follows from (3.15.2).

3.16. The Projective Conjecture for the prime p = 3 The universal covering group M̂_{c} of M_{c} has a single 3-block of defect 7, namely

the principal block B_{0}^{∗}. All other 3-blocks are of defect 1. We list here the characters of B_{0}^{∗} lying over some fixed ρ ∈ Irr(A)^{#} and their 3-defects:

Character χ_{25} χ_{26} χ_{27} χ_{28} χ_{29}  χ_{30}  χ_{31}  χ_{32}  χ_{33}
Degree  126 126 792 1980 2376 2376 2520 2520 2772
3-Defect  5  5  5  5  4  4  5  5  5
Character χ_{34} χ_{36} χ_{37} χ_{38} χ_{41}  χ_{42}  χ_{43}  χ_{44}  χ_{45}
Degree  4752 6336 6336 7875 8064 10395 10395 10395 12375
3-Defect  4  5  5  5  5  4  4  4  5

Thus

k(C_{1}, B_{0}^{∗}, 5 | ρ) = 12,
k(C_{1}, B_{0}^{∗}, 4 | ρ) = 6,
k(C_{1}, B_{0}^{∗}, d | ρ) = 0,  for all other values of d.

(3.16.1)

Theorem 3.16.2. The Projective Conjecture holds for the McLaughlin Simple Group and the prime p = 3.

Proof. From Conjecture 1.4.6 and Table 3.1 on page 48 we need to prove

k(C_{1}, B_{0}^{∗}, d | ρ) + k(C_{3}, B_{0}^{∗}, d | ρ) = k(C_{2}, B_{0}^{∗}, d | ρ) + k(C_{4}, B_{0}^{∗}, d | ρ),

for all values of d ∈ ℤ_{+}. From (3.13.8), (3.14.5), (3.15.4) and (3.16.1) we obtain the following sums for the equation above for various values of d:

3-Defect C_{1} C_{3} C_{2} C_{4}
5  12 + 15 = 12 + 15
4  6 + 0 = 0 + 6

Table 3.4. The Projective Conjecture for p = 3

The summands in the above equation are zero for all other values of d. This proves the Projective Conjecture for McLaughlin’s simple group for the prime p = 3.

Chapter 4 The Prime p=2

4.1. The 2-local structure of M_{c}

We begin with a fundamental property of M_{c}.

Proposition 4.1.1. M_{c} has exactly one class of involutions and the centralizer of an involution is isomorphic to 2.𝔄_{8}.

Proof. In [JW72] Janko and Wong characterised the McLaughlin Group as the unique finite simple group possessing a single class of involutions, with the centralizer of an involution isomorphic to the covering group 2.𝔄_{8} of 𝔄_{8}.

For the rest of the thesis τ will denote a fixed involution of M_{c} and H will denote the centralizer C_{M_{c}}(τ) of τ in M_{c}. The group ⟨τ⟩ will be denoted by T.

Lemma 4.1.2. The group H/T ≅ 𝔄_{8} has exactly two conjugacy classes of involutions: the class (2A) consisting of all involutions of cycle structure (..)(..)(..)(..), and the class (2B) consisting of all involutions of cycle structure (..)(..). The inverse image in H of the class (2A) is the unique conjugacy class of non-central involutions of H, while that of (2B) is a single conjugacy class of elements of order 4 in H.

Proof. This comes from the character table of 𝔄_{8} found in the Atlas.

We now prove

Proposition 4.1.3. M_{c} has exactly one conjugacy class of elementary abelian 2subgroups of order 2^{2}. If K is a representative of one of these classes then N_{M_{c}}(K) acts doubly transitively on K^{#}.

Proof. It is clear from Lemma 4.1.2 that H, and hence M_{c}, possesses elementary abelian 2-groups of order 2^{2}.

Suppose K_{1}, K_{2} are elementary abelian 2-subgroups of M_{c} of order 2^{2}. Choose arbitrary pairs of distinct elements (τ_{1}, ε̃_{1}) and (τ_{2}, ε̃_{2}) from K_{1}^{#} and K_{2}^{#} respectively. Note that K_{1} = ⟨τ_{1}, ε̃_{1}⟩ and K_{2} = ⟨τ_{2}, ε̃_{2}⟩.

In view of Proposition 4.1.1, the elements τ_{1}, τ_{2} and τ are all conjugate in M_{c}. Since we are interested in K_{1} and K_{2} only up to conjugacy in M_{c}, we may assume, and we do, that τ_{1} = τ_{2} = τ. Hence both K_{1} and K_{2} are subgroups of H = C_{M_{c}}(τ).

By Lemma 4.1.2, there exists x ∈ H such that ε̃_{1}^{x} = ε̃_{2}. Obviously τ^{x} = τ. So K_{1}^{x} = K_{2}. This shows that M_{c} has a single conjugacy class of elementary abelian 2subgroups of order 2^{2}. Taking K_{1} = K_{2} to begin with, the above argument also shows that N_{M_{c}}(K_{1}) acts doubly transitively on K_{1}^{#}.

We collect information on certain classes of 2-subgroups of 𝔄_{8}, which we will use to classify the maximal elementary abelian 2-subgroups of H.

Lemma 4.1.4. Let A_{0} be a regular transitive embedding of ℤ_{2}^{3} in 𝔄_{8}. Then A_{0} is a self-centralizing trivial intersection subgroup of 𝔄_{8}, and all elements of A_{0}^{#} come from the (2A) conjugacy class of 𝔄_{8}. The normalizer N_{𝔄_{8}}(A_{0}) of A_{0} in 𝔄_{8} is isomorphic to the holomorph A_{0} ⋊ Aut(A_{0}) of A_{0}. There are exactly two 𝔄_{8}-conjugacy classes of such subgroups A_{0}. These two classes fuse in 𝔖_{8}.

Proof. Since A_{0} is an abelian regular transitive subgroup of 𝔖_{8}, it is self-centralizing in 𝔖_{8}, and all its non-trivial elements come from the (2A) conjugacy class of 𝔄_{8} consisting of involutions of cycle structure (..)(..)(..)(..). Furthermore N_{𝔖_{8}}(A_{0}) is the holomorph of A_{0} i.e., the semi-direct product of A_{0} with its automorphism group Aut(A_{0}) ≅ GL(3, 2). It is clear that N_{𝔖_{8}}(A_{0}) is contained in 𝔄_{8} (for instance because the simple group GL(3, 2) has no subgroups of index 2). Hence there are two 𝔄_{8}-conjugacy classes of subgroups of this type. However the full symmetric group 𝔖_{8} contains only one such class.

From the Atlas, the conjugacy class (2A) of 𝔄_{8} contains 105 elements. Every element in this class is contained in some 𝔄_{8}-conjugate of A_{0}. From the structure of N_{𝔄_{8}}(A_{0}), the group A_{0} has (8!/2)/(2^{3} · 168) = 15 conjugates in 𝔄_{8}. So the 𝔄_{8}- conjugates of A_{0} contain at most |A_{0}^{#}| · |𝔄_{8} : A_{0}| = 7 · 15 = 105 non-trivial elements. Hence every element of the class (2A) is contained in exactly one conjugate of A_{0}. It follows that distinct 𝔄_{8}-conjugates of A_{0} intersect trivially.

We now discuss the covering group of A_{0} in H.

Lemma 4.1.5. Let A_{0} be a regular transitive embedding of ℤ_{2}^{3} in 𝔄_{8}. Then the covering group R_{0} of A_{0} in H is elementary abelian of order 2^{4}, and is a selfcentralizing subgroup of M_{c}. Distinct H-conjugates of R_{0} intersect in T. Also N_{H}(R_{0}) = R_{0} ⋊ G_{0}, where G_{0} ≅ GL(3, 2) acts as the full automorphism group on A_{0} = R_{0}/T.

Proof. By Lemmas 4.1.2 and 4.1.4, all elements of R_{0}\T come from the unique conjugacy class of non-central involutions of H. In particular R_{0} has exponent 2. So R_{0} is an elementary abelian 2-group of order 2^{4}.

Since A_{0} is a self-centralizing subgroup of 𝔄_{8}, it follows that the abelian group R_{0} is self-centralizing in H. But C_{M_{c}}(R_{0}) ≤ H = C_{M_{c}}(τ). So R_{0} is also self-centralizing in M_{c}.

Distinct H-conjugates of R_{0} intersect in T, since A_{0} is a trivial intersection subgroup of 𝔄_{8} ≅ H/T, and each H-conjugate contains T.

As N_{𝔄_{8}}(A_{0}) ≅ A_{0} ⋊ GL(3, 2), it follows that N_{H}(R_{0}) ≅ R_{0}. GL(3, 2), where the factor N_{H}(R_{0})/R_{0} ≅ GL(3, 2) acts as the full automorphism group on A_{0} = R_{0}/T.

To show that N_{H}(R_{0}) splits over R_{0}, we need only show that one of its Sylow 2-subgroups splits over R_{0}. To this end we fix

\begin{aligned}A_{0} & = ⟨(1, 2)(3, 4)(5, 6)(7, 8), (1, 3)(2, 4)(5, 7)(6, 8), (1, 5)(2, 6)(3, 7)(4, 8)⟩, \\ x & = (1, 3)(2, 4)(5, 8)(6, 7), y = (1, 5)(2, 6)(3, 8)(4, 7), z = (1, 2)(3, 4)(5, 7)(6, 8), \end{aligned}

and set K = ⟨x, y⟩ and D_{8} = K ⋊ ⟨z⟩. Then K ≅ ℤ_{2}^{2} while D_{8} ≅ D_{8}, and the group S_{2} = A_{0} ⋊ D_{8} is a Sylow 2-subgroup of N_{𝔄_{8}}(A_{0}).

Let x̃, ỹ and z̃ be elements of H whose images in H/T are x, y and z respectively. Then x̃, ỹ and z̃ are involutions and x̃ commutes with both ỹ and z̃. If ỹz̃ = x̃ỹ then ⟨x̃, ỹ⟩ ⋊ ⟨z̃⟩ ≅ D_{8} is a complement to T in T.D_{8}. Otherwise ỹz̃ = τx̃ỹ and ⟨τx̃, ỹ⟩ ⋊ ⟨z̃⟩ ≅ D_{8} is a complement to T in T.D_{8}. In any event T.D_{8} splits over T. But this shows that the Sylow 2-subgroup T.S_{2} = R_{0}.D_{8} of N_{H}(R_{0}) splits over R_{0}. The rest of the lemma now follows.

Corollary 4.1.6. N_{H}(R_{0}) acts transitively on R_{0}\T.

Proof. Given x, y ∈ R_{0}\T, there exists z ∈ H such that x^{z} = y, by Lemma 4.1.2. Then R ∩ R^{z} ≥ ⟨τ, y⟩ > T. It follows from Lemma 4.1.5 that R_{0} = R_{0}^{z}. Hence in fact z ∈ N_{H}(R_{0}). This proves the corollary.

The next lemma shows why we are interested in the two conjugacy classes of subgroups of H of type R_{0}.

Lemma 4.1.7. Every elementary abelian 2-subgroup of H is contained in the inverse image R_{0} ≅ ℤ_{2}^{4} of some regular transitive embedding A_{0} of ℤ_{2}^{3} in H/T ≅ 𝔄_{8}. Hence H has exactly two conjugacy classes of maximal elementary abelian 2-subgroups. These two classes fuse in H.2 = C_{M_{c}.2}(τ).

Proof. Let X be an elementary abelian 2-subgroup of H, and let \overline{X} denote the group XT/T. Recall from Lemma 4.1.2 that the single conjugacy class of non-central involutions of H lies over the conjugacy class (2A) consisting of involutions of H/T ≅ 𝔄_{8} of cycle structure (..)(..)(..)(..). Hence no non-trivial element of \overline{X} fixes a point. It follows that \overline{X} acts regularly on each of its orbits. In particular each orbit of \overline{X} has length |\overline{X}|.

There is a single conjugacy class of elementary abelian 2-subgroups of 𝔖_{8} of order |\overline{X}| which act regularly on their orbits. Moreover a regular transitive embedding of ℤ_{2}^{3} in 𝔖_{8} contains representatives of each such class. We conclude that \overline{X} ≤ A_{0}, where A_{0} is a member of one of the two classes of regular transitive embeddings of ℤ_{2}^{3} in 𝔄_{8} given by Lemma 4.1.4.

The previous paragraph shows that X ≤ R_{0}, where R_{0} is the inverse image of A_{0} in H. But by Lemma 4.1.5, the group R_{0} is itself elementary abelian of order 2^{4}. Hence H has two conjugacy classes of maximal elementary abelian 2-subgroups, and a member of either class is the inverse image of a regular transitive embedding of ℤ_{2}^{3} in 𝔄_{8}.

From the Atlas C_{M_{c}.2}(τ) = H.2 ≅ 2.𝔖_{8}. It then follows from Lemmas 4.1.4 and 4.1.5 that the two classes of maximal elementary abelian 2-subgroups of H fuse in H.2.

For the rest of this section we let R_{1} and R_{2} be fixed representatives of the two classes of maximal elementary abelian 2-subgroups of H of order 2^{4}. If we prove any result for R_{1}, then an analogous result will hold for R_{2}. In view of Lemma 4.1.7, the factor group A_{1} = R_{1}/T is a regular transitive embedding of ℤ_{2}^{3} in H/T ≅ 𝔄_{8}.

Lemma 4.1.8. N_{M_{c}}(R_{1}) = R_{1} ⋊ Λ_{1}, where Λ_{1} ≅ 𝔄_{7} acts flag transitively on R_{1}.

Proof. Let ε̃ be any element of R_{1}\T. By Proposition 4.1.1 there exists x ∈ M_{c} such that τ^{x} = ε̃. Hence R_{1} ≤ C_{M_{c}}(ε̃) = H^{x} is in one of the two classes of maximal elementary abelian 2-subgroups of H^{x} given by Lemma 4.1.7. Then by Corollary 4.1.6, all elements of R_{1}\⟨ε̃⟩ are conjugate in N_{H^{x}}(R_{1}), and all elements of R_{1}\T are conjugate in N_{H}(R_{1}). It follows that N_{M_{c}}(R_{1}) acts transitively on R_{1}^{#}. In parlicular N_{H}(R_{1}) is a subgroup of index |R_{1}^{#}| = 15 in N_{M_{c}}(R_{1}).

From Lemma 4.1.5 we have N_{H}(R_{1}) = R_{1} ⋊ G_{1}, where G_{1} ≅ GL(3, 2) acts as the full automorphism group on R_{1}/T. But N_{H}(R_{1}) is the centralizer of the element τ ∈ R_{1}^{#} in N_{M_{c}}(R_{1}). We deduce from this, and the transitivity of N_{M_{c}}(R_{1}) on R_{1}^{#}, that N_{M_{c}}(R_{1}) acts flag transitively on R_{1}.

Now R_{1} is a self-centralizing subgroup of M_{c}, by Lemma 4.1.5. Hence N_{M_{c}}(R_{1})/R_{1} is isomorphic to a subgroup of index |𝔄_{8}|/(|N_{M_{c}}(R_{1}) : N_{H}(R_{1})| · |N_{H}(R_{1})|) = (8!/2)/(15 · 168) = 8 in Aut(R_{1}) ≅ GL(4, 2) ≅ 𝔄_{8}. From the list of maximal subgroups of 𝔄_{8} found in the Atlas, it follows that N_{M_{c}}(R_{1})/R_{1} ≅ 𝔄_{7}.

By Lemma 4.1.5, the group N_{H}(R_{1}) splits over R_{1}. It also contains a Sylow 2subgroup of N_{M_{c}}(R_{1}). Hence the latter group also splits over R_{1}. This completes the proof of the lemma.

Corollary 4.1.9. M_{c} has exactly two conjugacy classes of maximal elementary abelian 2-groups, corresponding to the two classes of subgroups of H isomorphic to ℤ_{2}^{4}. The former two classes fuse in M_{c}.2.

Proof. It suffices to show that R_{1} is conjugate to R_{2} in M_{c}.2 but not in M_{c}, since any maximal elementary abelian 2-subgroup of M_{c} is M_{c}-conjugate to a maximal elementary abelian 2-subgroup of H, and hence to either R_{1} or R_{2}.

That R_{1} and R_{2} are conjugate in M_{c}.2 is a consequence of Lemma 4.1.7.

Suppose R_{1} = R_{2}^{x}, for some x ∈ M_{c}. Then τ^{x} ∈ R_{1}, as τ ∈ R_{2}. Since both τ and τ^{x} lie in R_{1}, it follows from Lemma 4.1.8 that there exists y ∈ N_{M_{c}}(R_{1}) such that τ^{xy} = τ. Hence R_{1} = R_{2}^{xy}, where xy ∈ H = C_{M_{c}}(τ). This contradicts our choice of R_{1} and R_{2}. We conclude that R_{1} and R_{2} are not conjugate in M_{c}.

Corollary 4.1.10. Let S be a Sylow 2-subgroup of M_{c}. Then S contains exactly one M_{c}-conjugate of R_{1}. In particular R_{1} is weakly closed in S with respect to M_{c}.

Proof. We may assume that S is a Sylow 2-subgroup of H containing R_{1}. Let R_{1}^{(1)} be an M_{c}-conjugate of R_{1} contained in S, and set A_{1}^{(1)} = R_{1}^{(1)}/T. We need to show that R_{1} = R_{1}^{(1)}.

The two conjugacy classes of elementary abelian 2-subgroups of H of order 2^{4} do not fuse in M_{c}, by Corollary 4.1.9. So R_{1} and R_{1}^{(1)} are conjugate in H. Also Z(S/T) is contained in A_{1} ∩ A_{1}^{(1)}, since by Lemma 4.1.4 both A_{1} and A_{1}^{(1)} are self-centralizing in H/T. But A_{1} is a trivial intersection subgroup of H/T, by the same lemma. So A_{1} = A_{1}^{(1)}. We conclude that R_{1} = R_{1}^{(1)}.

The next result is due to Larry Finkelstein [Fk73, Lemma 5.3].

Corollary 4.1.11. If X is a subgroup of R_{1} properly containing T then N_{M_{c}}(X) ≤ N_{M_{c}}(R_{1}).

Proof. Suppose x ∈ N_{M_{c}}(X). Then T^{x} ≤ X^{x} = X ≤ R_{1}. By Lemma 4.1.8 we can find y ∈ N_{M_{c}}(R_{1}) such that T^{xy} = T and X^{xy} = X. So xy ∈ N_{H}(X) and T < X ≤ R_{1} ∩ R_{1}^{xy}. We conclude from Lemma 4.1.5 that xy normalizes R_{1}. Hence so too does x. This proves the corollary.

We now prove the most important result of this section.

Proposition 4.1.12. If R is a non-trivial radical 2-subgroup of M_{c}, then N_{M_{c}}(R) is conjugate in M_{c} to a subgroup of H, N_{M_{c}}(R_{1}) or N_{M_{c}}(R_{2}).

Proof. Since R is non-trivial, there is some involution τ^{1} in Z(R). After a M_{c}- conjugation, we may assume that τ^{1} = τ. Then T = ⟨τ⟩ ≤ Z(R) ≤ R implies that R ≤ H = C_{M_{c}}(τ). Let X = ΩZ(R). Then T ≤ X, and by Corollary 4.1.9 the group X is contained in a H-conjugate of either R_{1} or R_{2}.

If T = X, then N_{M_{c}}(X) = H. If T < X, then N_{M_{c}}(X) is contained in a Hconjugate of either N_{M_{c}}(R_{1}) or N_{M_{c}}(R_{2}), by Corollary 4.1.11. The proposition now follows from the fact that N_{M_{c}}(R) ≤ N_{M_{c}}(X).

The easiest way to describe the radical 2-subgroups of H/T is to use the GL(4, 2) representation of this group, and this we do. We then give a description of the radical 2subgroups of 𝔄_{7}. Using these descriptions, and Propositions 1.3.5 and 4.1.12, we obtain a list of the conjugacy classes of radical 2-subgroups of M_{c} and their normalizers.

4.2. The Radical p-subgroups of GL(n, q) Let V be a vector space of finite dimension n over a finite field of characteristic p

and order q, and let A be a radical p-subgroup of GL(V). We denote by [V, A^{i}] the subspace of V spanned by all commutators [v, x_{1}, x_{2}, …, x_{i}], with i ≥ 1, v ∈ V and x_{j} ∈ A, for j = 1, …, i. For convenience we identify V and [V, A^{0}]. So we have a descending chain of subspaces:

C : V = [V, A^{0}] ⊃ [V, A] ⊃ [V, A^{2}] ⊃ ⋯ ⊃ [V, A^{t−1}] ⊃ [V, A^{t}] = 0,

where t is some integer between 1 and n. The normalizer N_{GL(V)}(A) of A in GL(V) acts on each factor [V, A^{i}]/[V, A^{i+1}] of C.

We let \overline{A} = \overline{A}(C) denote the largest subgroup of N_{GL(V)}(A) which acts trivially on each of the factors of C. Then \overline{A} is a p-group by [Go80, 5.3.3]. Also \overline{A} is a normal subgroup of N_{GL(V)}(A). It follows from Theorem 1.3.2 that \overline{A} ≤ A. But A ≤ \overline{A}, by definition of \overline{A}. So \overline{A} = A.

If [V, A^{i}]/[V, A^{i+1}] has dimension n_{i+1}, for i = 0,1,…,t − 1, then we say A is a subgroup of GL(V) of type (n_{1}, n_{2}, …, n_{t}). Notice that n_{1} + n_{2} + ⋯ + n_{t} = n. Clearly the above tuple of numbers uniquely determines A up to conjugacy in GL(V).

Conversely, suppose A is any p-subgroup of GL(V) of type (n_{1}, n_{2},…, n_{t}). Then

A = [\begin{matrix}I_{n_{1}} & 0 & ⋯ & 0 \\ ∗ & I_{n_{2}} & ⋯ & 0 \\ ∗ & ∗ & ... & ⋮ \\ ∗ & ∗ & ⋯ & I_{n_{t}} \end{matrix} ] , N_{GL(V)}(A) = [\begin{matrix}GL_{n_{1}}(q) & 0 & ⋯ & 0 \\ ∗ & GL_{n_{2}}(q) & ⋯ & 0 \\ ∗ & ∗ & ... & ⋮ \\ ∗ & ∗ & ⋯ & GL_{n_{t}}(q) \end{matrix}] .

Hence N_{GL(V)}(A) ≅ A⋊(GL_{n_{1}}(q)×GL_{n_{2}}(q)×⋯×GL_{n_{t}}(q)). Since GL_{m}(q) has trivial p-core for all positive integers m, it follows that A is a radical p-subgroup of GL(V).

So we have determined the conjugacy classes of radical p-subgroups of GL(V). In fact we have shown a little more. A Sylow p-subgroup S of GL(n, q) is of type (1, 1, …, 1). The chain

C_{S} : V = [V, S^{0}] ⊃ [V, S^{1}] ⊃ [V, S^{2}] ⊃ ⋯ ⊃ [V, S^{n−1}] ⊃ [V, S^{n}] = {0}

of subspaces of V is called a flag, since each factor is of dimension 1. Then the radical 2-subgroups of GL(V) contained in S correspond 1−1 to the subchains of C_{S} containing both V and {0}. The one of type (n_{1}, n_{2}, …, n_{t}) corresponds to the subchain:

V ⊃ [V, S^{n_{1}}] ⊃ [V, S^{n_{1}+n_{2}}] ⊃ ⋯ ⊃ [V, S^{n_{1}+n_{2}+···+n_{t}}] = {0}.

It follows that each radical p-subgroup of GL(V) contained in S is weakly closed in S with respect to GL(V).

We can now list representatives of the seven conjugacy classes of non-trivial radical 2-subgroups of GL(4, 2). We let A_{i,j,...,k}, Z_{i,j,...,k}, and N_{i,j,...,k} denote a radical 2-group of type (i, j, …, k), its center, and its normalizer in GL(4, 2), respectively.

A_{1,1,1,1} = [\begin{matrix}\begin{matrix}1 & 0 \\ ∗ & 1 \end{matrix} & 0 0 0 0 \\ \begin{matrix}∗ & ∗ \\ ∗ & ∗ \end{matrix} & \begin{matrix}1 & 0 \\ ∗ & 1 \end{matrix} \end{matrix}] , Z_{1,1,1,1} = [\begin{matrix}1 0 0 1 & 0 0 0 0 \\ \begin{matrix}0 & 0 \\ ∗ & 0 \end{matrix} & 1 0 0 1 \end{matrix}] , N_{1,1,1,1} = [\begin{matrix}\begin{matrix}1 & 0 \\ ∗ & 1 \end{matrix} & 0 0 0 0 \\ \begin{matrix}∗ & ∗ \\ ∗ & ∗ \end{matrix} & \begin{matrix}1 & 0 \\ ∗ & 1 \end{matrix} \end{matrix}]

A_{2,1,1} = [\begin{matrix}1 0 0 1 & 0 0 0 0 \\ \begin{matrix}∗ & ∗ \\ ∗ & ∗ \end{matrix} & \begin{matrix}1 & 0 \\ ∗ & 1 \end{matrix} \end{matrix}] , Z_{2,1,1} = [\begin{matrix}1 0 0 1 & 0 0 0 0 \\ \begin{matrix}0 & 0 \\ ∗ & ∗ \end{matrix} & 1 0 0 1 \end{matrix}] , N_{2,1,1} = [\begin{matrix}GL_{2}(2) & 0 0 0 0 \\ \begin{matrix}∗ & ∗ \\ ∗ & ∗ \end{matrix} & \begin{matrix}1 & 0 \\ ∗ & 1 \end{matrix} \end{matrix}]

A_{1,2,1} = [\begin{matrix}1 & 0 0 & 0 \\ \begin{matrix}∗ \\ ∗ \end{matrix} & 1 0 0 1 & {0 \atop 0} \\ ∗ & ∗ ∗ & 1 \end{matrix}] , Z_{1,2,1} = [\begin{matrix}1 & 0 0 & 0 \\ {0 \atop 0} & 1 0 0 1 & {0 \atop 0} \\ ∗ & 0 0 & 1 \end{matrix}] , N_{1,2,1} = [\begin{matrix}1 & 0 0 & 0 \\ \begin{matrix}∗ \\ ∗ \end{matrix} & GL_{2}(2) & {0 \atop 0} \\ ∗ & ∗ ∗ & 1 \end{matrix}]

(4.2.1)

A_{1,1,2} = [\begin{matrix}\begin{matrix}1 & 0 \\ ∗ & 1 \end{matrix} & 0 0 0 0 \\ \begin{matrix}∗ & ∗ \\ ∗ & ∗ \end{matrix} & 1 0 0 1 \end{matrix}] , Z_{1,1,2} = [\begin{matrix}1 0 0 1 & 0 0 0 0 \\ \begin{matrix}∗ & 0 \\ ∗ & 0 \end{matrix} & 1 0 0 1 \end{matrix}] , N_{1,1,2} = [\begin{matrix}\begin{matrix}1 & 0 \\ ∗ & 1 \end{matrix} & 0 0 0 0 \\ \begin{matrix}∗ & ∗ \\ ∗ & ∗ \end{matrix} & GL_{2}(2) \end{matrix}]

A_{1,3} = [\begin{matrix}1 & 0 0 0 \\ \begin{matrix}∗ \\ ∗ \\ ∗ \end{matrix} & 1 0 0 0 1 0 0 0 1 \end{matrix}] , Z_{1,3} = [\begin{matrix}1 & 0 0 0 \\ \begin{matrix}∗ \\ ∗ \\ ∗ \end{matrix} & 1 0 0 0 1 0 0 0 1 \end{matrix}] , N_{1,3} = [\begin{matrix}1 & 0 0 0 \\ \begin{matrix}∗ \\ ∗ \\ ∗ \end{matrix} & GL_{3}(2) \end{matrix} ]

A_{3,1} = [\begin{matrix}1 0 0 0 1 0 0 0 1 & 0 0 0 \\ ∗ ∗ ∗ & 1 \end{matrix}] , Z_{3,1} = [\begin{matrix}1 0 0 0 1 0 0 0 1 & 0 0 0 \\ ∗ ∗ ∗ & 1 \end{matrix}] , N_{3,1} = [\begin{matrix}GL_{3}(2) & 0 0 0 \\ ∗ ∗ ∗ & 1 \end{matrix}]

A_{2,2} = [\begin{matrix}1 0 0 1 & 0 0 0 0 \\ \begin{matrix}∗ & ∗ \\ ∗ & ∗ \end{matrix} & 1 0 0 1 \end{matrix}] , Z_{2,2} = [\begin{matrix}1 0 0 1 & 0 0 0 0 \\ \begin{matrix}∗ & ∗ \\ ∗ & ∗ \end{matrix} & 1 0 0 1 \end{matrix}] , N_{2,2} = [\begin{matrix}GL_{2}(2) & 0 0 0 0 \\ \begin{matrix}∗ & ∗ \\ ∗ & ∗ \end{matrix} & GL_{2}(2) \end{matrix}]

We immediately obtain the following inclusions among the radical 2-subgroups of GL(4, 2) and their normalizers:

[figure: GL(4, 2); N_{1,3}; N_{3,1}; N_{2,2}; N_{1,2,1}; N_{1,1,2}; N_{2,1,1}; A_{1,1,1,1}; A_{2,1,1}; A_{1,1,2}; A_{1,2,1}; A_{2,2}; A_{3,1}; A_{1,3}; {1}]

Figure 1. The radical 2-subgroups of GL(4, 2) and their normalizers in GL(4, 2)

τ  a fixed involution of M_{c}
H ≅ 2.𝔄_{8} the centralizer of τ in M_{c}
H.2 ≅ 2.𝔖_{8} the centralizer of τ in M_{c}.2
T  the group ⟨τ⟩ = Z(H)
R_{1,1,1,1} a fixed Sylow 2-subgroup of H
A_{i,j,...,k} the unique radical 2-subgroup of H/T of type

(i, j, …, k) contained in R_{1,1,1,1}/T

Z_{i,j,...,k}  the center of A_{i,j,...,k}
N_{i,j,...,k} the normalizer of A_{i,j,...,k} in H/T
R_{i,j,...,k} the covering group of A_{i,j,...,k} in H
ε  the generator of the center of A_{1,1,1,1}
ε̃  an element of H whose image in H/T is ε
Λ_{3,1}  a fixed complement to R_{3,1} in N_{M_{c}}(R_{3,1}) ≅ R_{3,1} ⋊ 𝔄_{7}
Λ_{1,3}  a fixed complement to R_{1,3} in N_{M_{c}}(R_{1,3}) ≅ R_{1,3} ⋊ 𝔄_{7}
G_{3,1}  a fixed complement to R_{3,1} in N_{H}(R_{3,1}) ≅ R_{3,1} ⋊ GL(3, 2)
G_{1,3}  a fixed complement to R_{1,3} in N_{H}(R_{1,3}) ≅ R_{1,3} ⋊ GL(3, 2)

Table 4.1. Notation for Elements and Subgroups

4.3. The Radical 2-subgroups of H

Table 4.1 summarises the notation we shall use for the rest of Chapter 4. We remark that from (4.2.1) the center Z_{1,1,1,1} of A_{1,1,1,1} = R_{1,1,1,1}/T is cyclic of

order 2. Let ε denote a generator of this group, as specified in Table 4.1. Then the order of the centralizer of ε in H/T ≅ 𝔄_{8} is divisible by |A_{1,1,1,1}| = 2^{6}. From the orders of the centralizers of elements of 𝔄_{8}, it is clear that ε comes from the class (2A) of 𝔄_{8}.

We now investigate the radical 2-subgroups of H.

Lemma 4.3.1. Z(R_{1,1,1,1}) = T.

Proof. Let ε̃denote an element of H whose image in H/T is ε. Then Z(R_{1,1,1,1}) ≤ ⟨τ, ε̃⟩. But from the Atlas the class (2A) does not split in H. So ε̃ ∉ Z(R_{1,1,1,1}), and the latter group must coincide with T.

Lemma 4.3.2. Z(R_{2,1,1}) = T and Z(R_{1,1,2}) = T.

Proof. From (4.2.1) the center Z_{2,1,1} of A_{2,1,1} is an elementary abelian group of order 2^{2} which properly contains ⟨ε⟩. Moreover the normalizer N_{2,1,1} acts on Z_{2,1,1} as the full automorphism group Aut(Z_{2,1,1}) ≅ GL(2, 2). It follows that Z(R_{2,1,1}) is either T or the full inverse image Z̃_{2,1,1} of Z_{2,1,1} in H. Similarly Z(R_{1,1,2}) is either T or the full inverse image Z̃_{1,1,2} of Z_{1,1,2} in H.

Suppose Z(R_{2,1,1}) = Z̃_{2,1,1}. Then Z(R_{1,1,2}) = Z̃_{1,1,2}, since R_{2,1,1} and R_{1,1,2} are conjugate in H.2, and in particular their centers are isomorphic. Hence ε̃ ∈ Z̃_{2,1,1}∩Z̃_{1,1,2} centralizes R_{2,1,1}R_{1,1,2} = R_{1,1,1,1}. This contradicts the fact that Z(R_{1,1,1,1}) = T. We conclude that Z(R_{2,1,1}) = T and Z(R_{1,1,2}) = T.

Corollary 4.3.3. The normalizer in M_{c} of each of the groups R_{1,1,1,1}, R_{2,1,1} and R_{1,1,2} is contained in H, and the normalizer in M_{c}.2 is contained in H.2. In particular, each of these groups is a radical 2-subgroup of M_{c}.

Proof. Each of the groups R_{1,1,1,1}, R_{2,1,1} and R_{1,1,2} is radical in H. So the result follows at once from Lemmas 4.3.1 and 4.3.2, and the fact that H = C_{M_{c}}(T).

It turns out to be useful to represent A_{2,2} as a subgroup of 𝔄_{8} in order to describe its covering group R_{2,2} in H. So we first need some results on the elementary abelian 2-subgroups of 𝔄_{8}.

Lemma 4.3.4. Suppose V is a finite set and X is a maximal elementary abelian p-subgroup of 𝔖(V). Then there exist decompositions:

V = V_{0} ∪ V_{1} ∪ V_{2} ∪ ⋯ ∪ V_{t},
X = X_{1} × X_{2} × ⋯ × X_{t},

(4.3.5)

of V and X, where X fixes each element of V_{0}, the V_{i} are pairwise disjoint subsets of V\V_{0}, and X_{i} is a non-trivial elementary abelian p-subgroup of 𝔖(V_{i}) acting regularly and transitively on V_{i}, for i = 1, 2, …, t.

Proof. Let V_{0} be the set of fixed points, and V_{1}, V_{2}, …, V_{t} be the distinct Xorbits of length > 1 on V. We identify 𝔖(V_{i}), for i = 1, 2, …, t, with the subgroup of all permutations in 𝔖(V) fixing each element in V\V_{i}. Hence 𝔖(V_{1})×𝔖(V_{2})×⋯×𝔖(V_{t}) is a subgroup of 𝔖(V). Let X_{i} be the image of X under restriction to V_{i}, for each i = 1, 2, …, t. Each X_{i} is a regular transitive elementary abelian p-subgroup of 𝔖(V_{i}). Also X ≤ X_{1} × X_{2} × ⋯ × X_{t}. By the maximality of X, this inequality is actually an equality.

We list the maximal elementary abelian 2-subgroups of 𝔖_{8} to within conjugacy in 𝔖_{8}. Since 8 is even, the number of trivial orbits of any 2-subgroup is even. So no maximal elementary abelian 2-subgroup of 𝔖_{8} has any trivial orbits.

Orbit Lengths Elementary Abelian 2-group Intersection with 𝔄_{8}
(8)  ℤ_{2}^{3}  ℤ_{2}^{3}
(4, 4)  ℤ_{2}^{2} × ℤ_{2}^{2}  ℤ_{2}^{2} × ℤ_{2}^{2}
(4, 2, 2)  ℤ_{2}^{2} × ℤ_{2} × ℤ_{2}  ℤ_{2}^{2} × ℤ_{2}
(2, 2, 2, 2)  ℤ_{2} × ℤ_{2} × ℤ_{2} × ℤ_{2}  ℤ_{2} × ℤ_{2} × ℤ_{2}

Using the above table, it is clear that 𝔖_{8} has a unique conjugacy class ℤ_{2}^{2} × ℤ_{2}^{2} of elementary abelian 2-subgroups of order ≥ 2^{4} contained in 𝔄_{8}. Since the normalizer in 𝔖_{8} of such a subgroup is not contained in 𝔄_{8}, we see that 𝔄_{8} also has a unique conjugacy class of such subgroups. Clearly A_{2,2} is of this form. So we may choose an identification of GL(4, 2) with 𝔄_{8} so that

A_{2,2} = ⟨(1, 2)(3, 4), (1, 3)(2, 4)⟩ × ⟨(5, 6)(7, 8), (5, 7)(6, 8)⟩. (4.3.6)

Lemma 4.3.7. Let K be any four subgroup of 𝔄_{8} whose nontrivial elements come from the class (2B) of involutions of type (..)(..). Then the covering group of K in H = 2.𝔄_{8} is isomorphic to a quaternion group Q_{8} of order 8.

Proof. From the Atlas, the inverse images of elements of type (i, j)(k, l) are of order 4 in H. Thus the covering group of K has a unique involution, namely τ. Since the covering group is non-cyclic of order 8, it must be isomorphic to Q_{8}.

Lemma 4.3.8. Let σ ∈ 𝔖_{8}, and suppose x_{1} and x_{2} are elements of H whose images in H/T ≅ 𝔄_{8} are (σ(1), σ(2))(σ(3), σ(4)) and (σ(5), σ(6))(σ(7), σ(8)), respectively. Then [x_{1}, x_{2}] = 1.

Proof. Let x_{3} and x_{4} be elements of H having images (σ(1), σ(2))(σ(5), σ(6)) and (σ(1), σ(2))(σ(7), σ(8)), respectively in 𝔄_{8}. By Lemma 4.3.7 the groups ⟨x_{1}, x_{3}⟩ and ⟨x_{1}, x_{4}⟩ are quaternion. So [x_{1}, x_{3}] = [x_{1}, x_{4}] = τ. Then [x_{1}, x_{3}x_{4}] = [x_{1}, x_{4}][x_{1}, x_{3}]^{x_{4}} = τ^{2} = 1. But x_{2} ≡ x_{3}x_{4} modulo ⟨τ⟩. The result now follows.

We can now prove the following

Lemma 4.3.9. The covering group R_{2,2} of A_{2,2} in H is extra-special of order 2^{5}. In particular, its center is generated by τ. So N_{M_{c}}(R_{2,2}) ≤ H and N_{M_{c}.2}(R_{2,2}) ≤ H.2. Hence R_{2,2} is a radical 2-subgroup of M_{c}.

Proof. By (4.3.6) the group A_{2,2} is the direct product of two regular elementary abelian 2-groups of type ℤ_{2}^{2}. By Lemma 4.3.7 the covering group of each is quaternion, and by Lemma 4.3.8 the two covering groups commute. Hence R_{2,2} is the central product of two Q_{8}’s and Z(R_{2,2}) = T.

Finally N_{M_{c}}(R_{2,2}) ≤ H, since H = N_{M_{c}}(T), and N_{M_{c}.2}(R_{2,2}) ≤ H.2, since H.2 = N_{M_{c}.2}(T).

Next we investigate the covering groups R_{3,1} and R_{1,3} of A_{3,1} and A_{1,3} in H.

Lemma 4.3.10. The groups R_{3,1} and R_{1,3} are representatives of the two classes of maximal elementary abelian 2-subgroups of M_{c} of order 2^{4}. In particular they are conjugate in M_{c}.2.

Proof. From (4.2.1) the normalizer N_{3,1} of A_{3,1} is isomorphic to the holomorph A_{3,1} ⋊ GL(3, 2) of A_{3,1}. Hence all elements of A_{3,1}^{#} are conjugate in N_{3,1}. As remarked before Lemma 4.3.1, the element ε ∈ A_{3,1} comes from the (2A) conjugacy class of 𝔄_{8}. So all elements of A_{3,1}^{#} come from this class also. Then from Lemma 4.1.2, all elements of R_{3,1}\T come from the single class of non-central involutions of H. Hence R_{3,1} has exponent 2. We conclude that it is in one of the two classes of maximal elementary abelian 2-subgroups of H of order 2^{4} given by Lemma 4.1.7.

A similar statement can be made for R_{1,3}.

Since A_{3,1} and A_{1,3} are not conjugate in H/T, their inverse images R_{3,1} and R_{1,3} cannot be conjugate in H. The lemma now follows from Corollary 4.1.9.

Corollary 4.3.11. N_{M_{c}}(R_{3,1}) = R_{3,1} ⋊ Λ_{3,1}, where Λ_{3,1} ≅ 𝔄_{7} acts flag transitively on R_{3,1}. Hence R_{3,1} is a radical 2-subgroup of M_{c}.

Proof. The first statement follows from Lemmas 4.1.8 and 4.3.10. Since 𝔄_{7} is simple, R_{3,1} is the 2-core of N_{M_{c}}(R_{3,1}). This proves the second statement.

Similar statements can be made about R_{1,3} and its normalizer N_{M_{c}}(R_{1,3}) = R_{1,3}⋊ Λ_{1,3}.

Corollary 4.3.12. The groups R_{3,1} and R_{1,3} are the unique subgroups of R_{1,1,1,1} isomorphic to ℤ_{2}^{4}. In particular both are weakly closed in R_{1,1,1,1} with respect to M_{c}.

Proof. This is an immediate consequence of the lemma above and Corollary 4.1.10.

Next we take a look at the covering group R_{1,2,1} of A_{1,2,1} in H. The first result is easy.

Lemma 4.3.13. R_{1,2,1} = R_{3,1}R_{1,3}.

Proof. This follows at once from (4.2.1).

Lemma 4.3.14. Z(R_{1,2,1}) = C_{M_{c}}(R_{1,2,1}) = ⟨τ, ε̃⟩ ≅ ℤ_{2}^{2}. Hence N_{M_{c}}(R_{1,2,1}) = N_{M_{c}}(Z(R_{1,2,1})) = N_{M_{c}}(R_{3,1}) ∩ N_{M_{c}}(R_{1,3}). Moreover N_{M_{c}}(R_{1,2,1}) acts as the full automorphism group on Z(R_{1,2,1}).

Proof. By Lemmas 4.1.5 and 4.3.10 the groups R_{3,1} and R_{1,3} are self-centralizing in M_{c}. Hence C_{M_{c}}(R_{1,2,1}) = C_{M_{c}}(R_{3,1}R_{1,3}) = R_{3,1} ∩ R_{1,3} = ⟨τ, ε̃⟩. So Z(R_{1,2,1}) = ⟨τ, ε̃⟩.

The group Z(R_{1,2,1}) = R_{3,1} ∩ R_{1,3} strictly contains T. It then follows from Corollary 4.1.11 that N_{M_{c}}(Z(R_{1,2,1})) ≤ N_{M_{c}}(R_{3,1}) ∩ N_{M_{c}}(R_{1,3}). But R_{1,2,1} = R_{3,1}R_{1,3}. So N_{M_{c}}(R_{1,2,1}) ≥ N_{M_{c}}(R_{3,1}) ∩ N_{M_{c}}(R_{1,3}). Hence N_{M_{c}}(R_{1,2,1}) ≤ N_{M_{c}}(Z(R_{1,2,1})) ≤ N_{M_{c}}(R_{3,1}) ∩ N_{M_{c}}(R_{1,3}) ≤ N_{M_{c}}(R_{1,2,1}). So these inequalities are all equalities.

From Corollary 4.3.11 the group N_{M_{c}}(R_{3,1}) acts flag transitively on R_{3,1}. Hence the normalizer of Z(R_{1,2,1}) = ⟨τ, ε̃⟩ in N_{M_{c}}(R_{3,1}) acts as the full automorphism group on Z(R_{1,2,1}) ≅ ℤ_{2}^{2}. But this normalizer is N_{M_{c}}(R_{1,2,1}). This completes the lemma.

Corollary 4.3.15. [R_{1,2,1}, R_{1,2,1}] = Φ(R_{1,2,1}) = Z(R_{1,2,1}). If T^{(1)} is any subgroup of Z(R_{1,2,1}) of order 2, then R_{1,2,1}/T^{(1)} is extra-special of order 2^{5}.

Proof. Since R_{1,2,1}/ Z(R_{1,2,1}) ≅ A_{1,2,1}/ Z(A_{1,2,1}) is an elementary abelian 2group, it follows that [R_{1,2,1}, R_{1,2,1}] and Φ(R_{1,2,1}) are non-trivial N_{M_{c}}(R_{1,2,1})-invariant subgroups of Z(R_{1,2,1}). But N_{M_{c}}(R_{1,2,1}) acts as the full automorphism group on Z(R_{1,2,1}), by Lemma 4.3.14. Hence [R_{1,2,1}, R_{1,2,1}] = Φ(R_{1,2,1}) = Z(R_{1,2,1}).

It also follows from the action of N_{M_{c}}(R_{1,2,1}) on Z(R_{1,2,1}) that R_{1,2,1}/T^{(1)} ≅ R_{1,2,1}/T. Since this latter group is extra-special of order 2^{5}, this completes the proof of the lemma.

4.4. The Radical 2-chains of M_{c} Lemma 4.4.1. A Sylow 2-subgroup B_{1} of 𝔄_{7} is isomorphic to D_{8}. So it has two

non-cyclic subgroups B_{2} and B_{3} of order 4, each isomorphic to ℤ_{2}^{2}. They can be ordered so that N_{𝔄_{7}}(B_{2}) is isomorphic to 𝔖_{4}, while N_{𝔄_{7}}(B_{3}) has the form (𝔄_{4} × 3): 2. Both B_{2} and B_{3} are radical 2-subgroups of 𝔄_{7}. So too are B_{1}, which is self-normalizing in 𝔄_{7}, and B_{4} = {1}, whose normalizer is 𝔄_{7}. Any radical 2-subgroup of 𝔄_{7} is conjugate to exactly one of B_{1}, B_{2}, B_{3}, B_{4}.

Proof. Let B_{1} = ⟨(1, 3, 2, 4)(5, 6), (1, 2)(5, 6)⟩ be a fixed Sylow 2-subgroup of 𝔄_{7}. Then B_{1} is dihedral of order 8. Also B_{1} is self-normalizing in 𝔄_{7}. Hence it is a radical 2-subgroup of 𝔄_{7}.

The trivial group B_{4} = {1} is a radical 2-subgroup of 𝔄_{7}, since it is the 2-core of its normalizer 𝔄_{7}.

The centralizer of any involution of 𝔄_{7} is isomorphic to (ℤ_{2}^{2} × ℤ_{3}) : 2. Hence there are no radical 2-subgroups of 𝔄_{7} of order 2.

Any radical 2-subgroup of 𝔄_{7} of order 4 is conjugate to one of the three subgroups of order 4 in B_{1} ≅ D_{8}.

The group ⟨(1, 3, 2, 4)(5, 6)⟩ is self-centralizing in 𝔄_{7}. Hence its normalizer is B_{1}. So this group is not radical in 𝔄_{7}.

Let B_{2} = ⟨(1, 2)(3, 4), (1, 2)(5, 6)⟩. Then N_{𝔄_{7}}(B_{2}) ≅ 𝔖_{4}. So B_{2} is a radical 2subgroup of 𝔄_{7}.

We set B_{3} = ⟨(1, 2)(3, 4), (1, 3)(2, 4)⟩. Its normalizer in 𝔖_{7} is isomorphic to 𝔖_{4} × 𝔖_{3}. Thus N_{3} = (⟨B_{3}, (1, 2, 3)⟩ × ⟨(5, 6, 7)⟩) ⋊ ⟨(1, 2)(5, 6)⟩ ≅ (𝔄_{4} × 3) : 2 is the normalizer of B_{3} in 𝔄_{7}. So B_{3} is also radical in 𝔄_{7}.

Recall from Corollary 4.3.11 and Table 4.1 on page 97 that N_{M_{c}}(R_{3,1}) = R_{3,1} ⋊ Λ_{3,1}, where Λ_{3,1} ≅ 𝔄_{7}. We identify Λ_{3,1} with 𝔄_{7} in such a way that the Sylow 2subgroup B_{1} of Λ_{3,1} = 𝔄_{7} is R_{1,1,1,1} ∩ Λ_{3,1}. So B_{1}, B_{2} and B_{3} are radical 2-subgroups of Λ_{3,1} contained in R_{1,1,1,1}.

Lemma 4.4.2. R_{3,1} ⋊ B_{1} = R_{1,1,1,1}, while R_{3,1} ⋊ B_{2} = R_{2,1,1} and R_{3,1} ⋊ B_{3} = R_{1,2,1}. Hence N_{M_{c}}(R_{1,2,1}) = N_{R_{3,1}⋊Λ_{3,1}}(R_{1,2,1}) = R_{3,1}⋊N_{Λ_{3,1}}(B_{3}) ≅ R_{3,1}⋊((𝔄_{4}×3) :

2) ≅ R_{1,2,1} ⋊ (3^{2} : 2). In particular R_{1,2,1} is a radical 2-subgroup of M_{c}.

Proof. From the group orders, R_{3,1} ⋊ B_{1} = R_{1,1,1,1}.

Since R_{3,1} ≤ R_{2,1,1} ≤ R_{3,1} ⋊ Λ_{3,1}, the group R_{2,1,1} ∩ Λ_{3,1} is a maximal subgroup of B_{1} isomorphic to ℤ_{2}^{2}. Thus R_{2,1,1} ∩ Λ_{3,1} is a radical 2-subgroup of Λ_{3,1}. Also N_{R_{3,1}⋊Λ_{3,1}}(R_{2,1,1}) = R_{3,1} ⋊ (N_{Λ_{3,1}}(R_{2,1,1} ∩ Λ_{3,1})). But by Equations (4.2.1) and Corollary 4.3.1 the group N_{M_{c}}(R_{2,1,1}) = N_{H}(R_{2,1,1}) = 2.N_{2,1,1} has order 2^{7} · 3. Hence |N_{Λ_{3,1}}(R_{2,1,1} ∩ Λ_{3,1})| has order ≤ 2^{3} · 3. We conclude from this and Lemma 4.4.1 that R_{2,1,1} ∩ Λ_{3,1} = B_{2}.

Similarly R_{1,2,1} ∩ Λ_{3,1} is a maximal non-cyclic subgroup of B_{1} which is different from R_{2,1,1} ∩ Λ_{3,1} = B_{2}. Hence R_{1,2,1} ∩ Λ_{3,1} = B_{3}.

From Lemma 4.3.14, the normalizer N_{M_{c}}(R_{1,2,1}) of R_{1,2,1} in M_{c} is the intersection N_{M_{c}}(R_{3,1}) ∩ N_{M_{c}}(R_{1,3}) of the normalizers of R_{3,1} and R_{1,3}. In particular N_{M_{c}}(R_{1,2,1}) = N_{R_{3,1}⋊Λ_{3,1}}(R_{1,2,1}) = R_{3,1} ⋊ N_{Λ_{3,1}}(B_{3}). The description of this last group comes from Lemma 4.4.1.

We can now prove the following

Proposition 4.4.3. M_{c} has nine conjugacy classes of radical 2-subgroups, with representatives given in Table 4.2 on the following page.

Proof. This follows from Proposition 4.1.12, Equations (4.2.1), Corollaries 4.3.3 and 4.3.11, and Lemmas 4.3.9 and 4.4.2. The radical 2-subgroups R_{3,1} and R_{1,3} are conjugate in H.2, and hence in M_{c}.2 by Lemma 4.3.10. Then from Figure 1 on page 96, so too are R_{2,1,1} and R_{1,1,2}. No other pairs of radical 2-subgroups in Table 4.2 on the following page are isomorphic as groups. So N_{M_{c}.2}(R) is some extension N_{M_{c}}(R).2 of N_{M_{c}}(R), when R is one of R_{1,1,1,1}, R_{1,2,1}, R_{2,2}, T or {1}. For reasons outlined in Proposition 4.4.9 below, we do not need detailed information about the normalizers of R_{1,1,1,1}, R_{2,1,1}, R_{1,1,2} or R_{2,2} in either M_{c} or M_{c}.2. We have already seen that N_{M_{c}}(T) = H.2 ≅ 2.𝔖_{8}. Clearly N_{M_{c}.2}({1}) = M_{c}.2. The structure of N_{M_{c}.2}(R_{1,2,1}) is investigated in Section 4.9 below.

We will also need a list of the conjugacy classes of radical 2-subgroups of GL(3, 2). The notation and methods are as in Section 4.2.

Radical 2-subgroup R Structure N_{M_{c}}(R)  N_{M_{c}.2}(R)
R_{1,1,1,1}  ℤ_{2}^{4} ⋊ D_{8} R_{1,1,1,1}  R_{1,1,1,1}.2
R_{2,1,1}  ℤ_{2}^{4} ⋊ ℤ_{2}^{2} R_{2,1,1} ⋊ 𝔖_{3}  R_{2,1,1} ⋊ 𝔖_{3}
R_{1,1,2}  ℤ_{2}^{4} ⋊ ℤ_{2}^{2} R_{1,1,2} ⋊ 𝔖_{3}  R_{1,1,2} ⋊ 𝔖_{3}
R_{3,1}  ℤ_{2}^{4}  R_{3,1} ⋊ Λ_{3,1}  R_{3,1} ⋊ Λ_{3,1}
R_{1,3}  ℤ_{2}^{4}  R_{1,3} ⋊ Λ_{1,3}  R_{1,3} ⋊ Λ_{1,3}
R_{1,2,1}  ℤ_{2}^{2}.ℤ_{2}^{4}  R_{1,2,1} ⋊ (3^{2} : 2) R_{1,2,1} ⋊ (𝔖_{3} × 𝔖_{3})
R_{2,2}  2^{1+4}  R_{2,2}.(𝔖_{3} × 𝔖_{3}) R_{2,2}.(𝔖_{3} × 𝔖_{3}).2
T  ℤ_{2}  H ≅ 2.𝔄_{8}  H.2 ≅ 2.𝔖_{8}
{1}  {1}  M_{c}  M_{c}.2

Table 4.2. The Radical 2-subgroups of M_{c}

Lemma 4.4.4. There are four conjugacy classes of radical 2-subgroups of GL(3, 2), represented by:

A_{3} = [\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] , A_{1,2} = [\begin{matrix}1 & 0 & 0 \\ ∗ & 1 & 0 \\ ∗ & 0 & 1 \end{matrix}] , A_{2,1} = [\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ ∗ & ∗ & 1 \end{matrix}] , A_{1,1,1} = [\begin{matrix}1 & 0 & 0 \\ ∗ & 1 & 0 \\ ∗ & ∗ & 1 \end{matrix}] .

Their normalizers in GL(3, 2) are N_{3} = GL(3, 2), N_{1,2} ≅ A_{1,2} ⋊ GL(2, 2) ≅ 𝔖_{4}, N_{2,1} ≅ A_{2,1} ⋊ GL(2, 2) ≅ 𝔖_{4}, and N_{1,1,1} = A_{1,1,1} ≅ D_{8} respectively.

Proof. This follows directly from the results of Section 4.2.

Recall from Lemma 4.1.5 and Table 4.1 on page 97 that N_{H}(R_{3,1}) = R_{3,1} ⋊ G_{3,1}, where G_{3,1} is isomorphic to GL(3, 2). We identify G_{3,1} and GL(3, 2) in such a way that the Sylow 2-subgroup A_{1,1,1} of G_{3,1} = GL(3, 2) corresponds to R_{1,1,1,1} ∩ G_{3,1}.

Lemma 4.4.5. Every radical 2-subgroup of N_{H}(R_{3,1}) = R_{3,1}⋊G_{3,1} is conjugate in N_{H}(R_{3,1}) to one of the groups R_{3,1}, R_{2,1,1}, R_{1,2,1} or R_{1,1,1,1}. Similarly every radical 2-subgroup of N_{H}(R_{1,3}) = R_{1,3} ⋊ G_{1,3} is conjugate in N_{H}(R_{1,3}) to one of the groups R_{1,3}, R_{1,1,2}, R_{1,2,1} or R_{1,1,1,1}.

Proof. There are 4 conjugacy classes of radical 2-subgroups of N_{H}(R_{3,1}) = R_{3,1}⋊ G_{3,1}. These classes are represented by R_{3,1} ⋊ A_{3}, R_{3,1} ⋊ A_{1,2}, R_{3,1} ⋊ A_{2,1}, and R_{3,1} ⋊ A_{1,1,1}, using the notation of Lemma 4.4.4. It is easy to see that R_{3,1} ⋊ A_{3} = R_{3,1} and R_{3,1} ⋊ A_{1,1,1} = R_{1,1,1,1}, while R_{3,1} ⋊ A_{1,2} and R_{3,1} ⋊ A_{2,1} are R_{2,1,1} and R_{1,2,1} in some order.

We now deal with the radical 2-subgroups of of N_{M_{c}}(R_{2,2}).

Lemma 4.4.6. Every radical 2-subgroup of N_{M_{c}}(R_{2,2}) is conjugate in N_{M_{c}}(R_{2,2}) to one of the groups R_{2,2}, R_{2,1,1}, R_{1,1,2} or R_{1,1,1,1}.

Proof. The factor group N_{M_{c}}(R_{2,2})/R_{2,2} is isomorphic to 𝔖_{3} × 𝔖_{3}. There are four conjugacy classes of radical 2-subgroups in this factor group, represented by 1 × 1, 𝔖_{2} × 1, 1 × 𝔖_{2} and 𝔖_{2} × 𝔖_{2}. It follows from Lemma 4.3.9 and Figure 1 on page 96, that the inverse images in N_{M_{c}}(R_{2,2}) of these classes are represented by R_{2,2}, R_{2,1,1}, R_{1,1,2} and R_{1,1,1,1} in some order.

The previous results allow us to prove the following.

Lemma 4.4.7. Any radical 2-chain of M_{c} is M_{c}-conjugate to a unique chain of the form

C: P_{0} = 1 < P_{1} < ⋯ < P_{n}, (4.4.8)

where P_{i}, for i = 0, 1, …, n, is one of the radical 2-subgroups of M_{c}, as given in Table 4.2 on page 107. Any chain of this form is a radical 2-chain of M_{c}.

Proof. This is clear from Lemmas 4.4.1, 4.4.2, 4.4.5 and 4.4.6, Proposition 4.4.3, and the fact that R_{2,1,1}, R_{1,2,1} and R_{1,1,2} are maximal in the Sylow 2-subgroup R_{1,1,1,1} of M_{c}.

The following proposition halves the number of radical 2-chains we must consider.

Proposition 4.4.9. The collection of radical 2-chains (4.4.8) of M_{c} containing any one of the groups R_{1,1,1,1}, R_{2,1,1}, R_{1,1,2} or R_{2,2} contribute zero to the alternating sums of Conjectures 1.4.2, 1.4.4 and 1.4.6.

Proof. Let C be a radical 2-chain of M_{c}, as in (4.4.8). Then T ≤ P_{1} since each non-trivial group in Table 4.2 contains T.

If T < P_{1}, form the 2-chain C′: P_{0} = 1 < T < P_{1} < P_{2} < ⋯ < P_{n} by inserting T between P_{0} and P_{1}. If T = P_{1}, form the 2-chain C′: P_{0} = 1 < P_{2} < ⋯ < P_{n} by deleting P_{1} (if n = 1, we just drop P_{n}). In either case C′ is a radical 2-chain of M_{c}, since it has the same form as (4.4.8). Moreover the chains C and C′ have opposite parity i.e. |C| = |C′| ± 1.

Clearly if we apply the operation ′ twice, we recover the original chain. In other words (C′)′ = C.

If N_{M_{c}}(C) = N_{M_{c}}(C′), then the contribution of C and C′ to any alternating sum is zero. In particular this occurs if, for some value of i = 1, 2, …, n we have T < P_{i} and N_{M_{c}}(P_{i}) ≤ N_{M_{c}}(T) = H. By Corollary 4.3.3 the normalizer of any one of the groups R_{1,1,1,1}, R_{2,1,1} or R_{1,1,2} is contained in H. The same is true of R_{2,2} by Lemma 4.3.9. Hence we can ignore all 2-chains containing any of these groups.

Chain C Chain Description  N_{M_{c}}(C)  N_{M_{c}.2}(C)  Parity
C_{1}  {1}  M_{c}  M_{c}.2  +
C_{2}  {1} < R_{3,1}  N_{M_{c}}(R_{3,1}) N_{M_{c}}(R_{3,1})  -
C_{3}  {1} < R_{3,1} < R_{1,2,1}  N_{M_{c}}(R_{1,2,1}) N_{M_{c}}(R_{1,2,1}) +
C_{4}  {1} < R_{1,3}  N_{M_{c}}(R_{1,3}) N_{M_{c}}(R_{1,3})  -
C_{5}  {1} < R_{1,3} < R_{1,2,1}  N_{M_{c}}(R_{1,2,1}) N_{M_{c}}(R_{1,2,1}) +
C_{6}  {1} < R_{1,2,1}  N_{M_{c}}(R_{1,2,1}) N_{M_{c}.2}(R_{1,2,1}) -
C_{7}  {1} < T  H  H.2  -
C_{8}  {1} < T < R_{3,1}  N_{H}(R_{3,1})  N_{H}(R_{3,1})  +
C_{9}  {1} < T < R_{3,1} < R_{1,2,1} N_{H}(R_{1,2,1}) N_{H}(R_{1,2,1})  -
C_{10}  {1} < T < R_{1,3}  N_{H}(R_{1,3})  N_{H}(R_{1,3})  +
C_{11}  {1} < T < R_{1,3} < R_{1,2,1} N_{H}(R_{1,2,1}) N_{H}(R_{1,2,1})  -
C_{12}  {1} < T < R_{1,2,1}  N_{H}(R_{1,2,1}) N_{H.2}(R_{1,2,1}) +

Table 4.3. The Remaining Radical 2-chains of M_{c}

We can now give a list of radical 2-chains of M_{c} which is sufficient for the purpose of proving the Conjectures.

Theorem 4.4.10. If C is a radical 2-chain of M_{c} then C is redundant from the point of view of the Conjectures, or C is M_{c}-conjugate to exactly one of the radical 2-chains given in Table 4.3 on the preceding page. The only chains in this table which are conjugate in M_{c}.2 are the pairs {C_{2}, C_{4}}, {C_{3}, C_{5}}, {C_{8}, C_{10}} and {C_{9}, C_{11}}.

Proof. This is clear from Section 4.3, Lemma 4.4.7 and Proposition 4.4.9.

Note that the 2-chains C_{1}, …, C_{6} in Table 4.3 are not the same as the 3-chains having the same names in Table 3.1 on page 48.

4.5. The Character Degrees of H From the character table of H ≅ 2.𝔄_{8} given in [Con85, p22], the group N_{M_{c}}(C_{7}) =

H has two 2-blocks. The characters χ_{13} and χ_{23} lie in a block b_{1} of defect 1, while the remaining 21 characters lie in the principal 2-block b_{0}. Clearly T is the unique defect group of b_{1}. Since H = N_{M_{c}}(T), we conclude from Brauer’s First Main Theorem that b_{1}^{M_{c}} is a block of M_{c} of defect 1. The characters of b_{1} are handled by the theory of blocks with cyclic defect groups.

We list here the characters in b_{0} and their defects:

Character χ_{1} χ_{2} χ_{3} χ_{4} χ_{5} χ_{6} χ_{7} χ_{8} χ_{9} χ_{10} χ_{11}
Degree  1 7 14 20 21 21 21 28 35 45 45
Defect  7 7 6 5 7 7 7 5 7 7 7
Character χ_{12} χ_{14} χ_{15} χ_{16} χ_{17} χ_{18} χ_{19} χ_{20} χ_{21} χ_{22}
Degree  56 70 8 24 24 48 56 56 56 56
Defect  4 6 4 4 4 3 4 4 4 4

Since N_{M_{c}}(C_{7}) = H, we have

k(C_{7}, B_{0}, 7) = 8,  k(C_{7}, B_{0}, 6) = 2,
k(C_{7}, B_{0}, 5) = 2,  k(C_{7}, B_{0}, 4) = 8,
k(C_{7}, B_{0}, 3) = 1,  k(C_{7}, B_{0}, d) = 0, for all other values of d.

(4.5.1)

From [Con85, p22] the characters of b_{0} invariant in H.2 are χ_{1}, χ_{2}, χ_{3}, χ_{4}, χ_{5}, χ_{8}, χ_{9}, χ_{12}, χ_{14}, χ_{15} and χ_{18}.

Thus

k(C_{7}, B_{0}, 7, \overline{M_{c}.2}) = 4, k(C_{7}, B_{0}, 6, \overline{M_{c}.2}) = 2,
k(C_{7}, B_{0}, 5, \overline{M_{c}.2}) = 2, k(C_{7}, B_{0}, 4, \overline{M_{c}.2}) = 2,
k(C_{7}, B_{0}, 3, \overline{M_{c}.2}) = 1, k(C_{7}, B_{0}, d, \overline{M_{c}.2}) = 0, for all other values of d.

(4.5.2)

4.6. The Character Degrees of N_{H}(R_{3,1}) and N_{M_{c}}(R_{3,1})

We note that R_{3,1} is a self-centralizing 2-subgroup of N_{M_{c}}(R_{3,1}). Hence the principal 2-block is the unique 2-block of each of the groups N_{H}(R_{3,1}) and N_{M_{c}}(R_{3,1}).

So the characters of both groups lie in 2-blocks inducing the principal 2-block, B_{0}, of M_{c}.

Recall from Lemma 4.1.5 and Table 4.1 on page 97 that N_{H}(R_{3,1}) = R_{3,1} ⋊ G_{3,1}, where G_{3,1} ≅ GL(3, 2) acts as the full automorphsim group on A_{3,1} = R_{3,1}/T.

Lemma 4.6.1. Orb(N_{H}(R_{3,1}), R_{3,1}^{∗}) = {1, 7, 8}.

Proof. The group G_{3,1} acts transitively on the subspaces of codimension 1 of A_{3,1}. Therefore the 7 elements of Irr(R_{3,1} mod T)^{#} form a single G_{3,1}-orbit. The trivial character forms another G_{3,1}-orbit on Irr(R_{3,1} mod T).

Since N_{H}(R_{3,1}) acts transitively on R_{3,1}\T (by Corollary 4.1.6) and fixes both elements of T, there are exactly three G_{3,1}-orbits in R_{3,1}. So there are just three G_{3,1}- orbits in Irr(R_{3,1}). We have just seen that Irr(R_{3,1} mod T) accounts for two of these orbits. So the complement Irr(R_{3,1} | T) to Irr(R_{3,1} mod T) is the third orbit.

We now deal with the chains C_{8} and C_{10}.

Proposition 4.6.2. The group N_{M_{c}}(C_{8}) = N_{H}(R_{3,1}) has a unique 2-block, which necessarily induces the principal 2-block, B_{0}, of M_{c}. Moreover Deg(N_{H}(R_{3,1})) = {1, 3^{2}, 6, 7^{3}, 8^{4}, 14, 21^{2}, 24^{2}} and Def_{2}(N_{H}(R_{3,1})) = {7^{8}, 6^{2}, 4^{6}}.

Thus

k(C_{8}, B_{0}, 7) = 8,  k(C_{8}, B_{0}, 6) = 2,
k(C_{8}, B_{0}, 5) = 0,  k(C_{8}, B_{0}, 4) = 6,
k(C_{8}, B_{0}, 3) = 0,  k(C_{8}, B_{0}, d) = 0, for all other values of d.

(4.6.3)

Also

(4.6.4) k(C_{10}, B_{0}, d) = k(C_{8}, B_{0}, d), for all values of d.

Proof. By Lemma 4.6.1 the group N_{H}(R_{3,1}) has two orbits on Irr(R_{3,1})^{#}, one of length 7 and the other of length 8. We let ψ_{1} be a representative of the former orbit, and ψ_{2} be a representative of the latter orbit.

From the Atlas, the group G_{3,1} ≅ GL(3, 2) has two conjugacy classes of maximal subgroups with index 7. Each consists of subgroups isomorphic to 𝔖_{4}. The stabilizer of ψ_{1} in G_{3,1} must lie in one of these classes of subgroups. By Theorem 1.2.15, the character ψ_{1} extends to its stabilizer in N_{H}(R_{3,1}). It then follows from Theorem 1.2.16 that Deg(I_{N_{H}(R_{3,1})}(ψ_{1}) | ψ_{1}) = Deg(𝔖_{4}) = {1^{2}, 2, 3^{2}}. Using Clifford Theory we conclude

Deg( N_{H}(R_{3,1}) | ψ_{1} ) = {7^{2}, 14, 21^{2}}. (4.6.5)

From the Atlas, the group G_{3,1} ≅ GL(3, 2) has a unique conjugacy class of subgroups with index 8, consisting of subgroups isomorphic to 7: 3. Moreover the group 7: 3 is Frobenius (for instance because GL(3, 2) has no elements of order 21). The stabilizer of ψ_{2} in G_{3,1} must lie in this class of subgroups. By Theorem 1.2.15, the character ψ_{2} extends to its stabilizer in N_{H}(R_{3,1}). It then follows from Theorem 1.2.16 that Deg(I_{N_{H}(R_{3,1})}(ψ_{2}) | ψ_{2}) = Deg(7: 3) = {1^{3}, 3^{2}}. Using Clifford Theory we conclude

Deg( N_{H}(R_{3,1}) | ψ_{2} ) = {8^{3}, 24^{2}}. (4.6.6)

Since N_{H}(R_{3,1}) ≅ R_{3,1} ⋊ GL(3, 2), we also have

(4.6.7) Deg( N_{H}(R_{3,1})/R_{3,1} ) = Deg( GL(3, 2) ) = {1, 3^{2}, 6, 7, 8}.

Equations (4.6.3) now follow from (4.6.5), (4.6.6) and (4.6.7). Equation (4.6.4) holds because the 2-chains C_{8} and C_{10} are conjugate in M_{c}.2.

We now consider the chains C_{2} and C_{4}.

Lemma 4.6.8. Orb(Λ_{3,1}, R_{3,1}^{∗}) = {1, 15}. So the stabilizer in Λ_{3,1} of any nontrivial character of R_{3,1} is isomorphic to GL(3, 2). Hence

(4.6.9) Deg( N_{M_{c}}(R_{3,1}) | R_{3,1} ) = {15, 45^{2}, 90, 105, 120}.

Proof. By Corollary 4.3.11, the group Λ_{3,1} acts transitively on the subspaces of R_{3,1} of codimension 1. Hence it also acts transitively on (R_{3,1}^{∗})^{#}. From the Atlas, there are two conjugacy classes of subgroups of index |(R_{3,1}^{∗})^{#}| = 15 in Λ_{3,1} ≅ 𝔄_{7}. A representative of either of these classes is isomorphic to GL(3, 2). Hence the stabilizer in Λ_{3,1} of any non-trivial character of R_{3,1} is isomorphic to GL(3, 2). Clifford Theory now gives us (4.6.9).

The following completes our analysis of the stabilizers of the radical 2-chains C_{2} and C_{4}.

Proposition 4.6.10. The group N_{M_{c}}(C_{2}) = N_{M_{c}}(R_{3,1}) has a unique 2-block, which necessarily induces the principal 2-block, B_{0}, of M_{c}. Moreover Deg(N_{M_{c}}(R_{3,1}))= {1, 6, 10^{2}, 14^{2}, 15^{2}, 21, 35, 45^{2}, 90, 105, 120} and Def_{2}(N_{M_{c}}(R_{3,1})) = {7^{8}, 6^{6}, 4}. Hence

k(C_{2}, B_{0}, 7) = 8,  k(C_{2}, B_{0}, 6) = 6,
k(C_{2}, B_{0}, 5) = 0,  k(C_{2}, B_{0}, 4) = 1,
k(C_{2}, B_{0}, 3) = 0,  k(C_{2}, B_{0}, d) = 0, for all other values of d.

(4.6.11)

Also (4.6.12) k(C_{4}, B_{0}, d) = k(C_{2}, B_{0}, d), for all values of d.

Proof. Since N_{M_{c}}(R_{3,1})/R_{3,1} ≅ 𝔄_{7}, we have (4.6.13) Deg( N_{M_{c}}(R_{3,1})/R_{3,1} ) = Deg(𝔄_{7}) = {1, 6, 10^{2}, 14^{2}, 15, 21, 35}.

Equations (4.6.11) now follow from (4.6.9) and (4.6.13). Equation (4.6.12) holds because the 2-chains C_{2} and C_{4} are conjugate in M_{c}.2.

4.7. The Character Degrees of N_{H}(R_{1,2,1}) and N_{M_{c}}(R_{1,2,1}) It follows from Lemma 4.3.14 that C_{M_{c}}(R_{1,2,1}) = C_{H}(R_{1,2,1}) = Z(R_{1,2,1}) is con-

tained in R_{1,2,1}. Hence the principal 2-block is the unique 2-block of each of the groups N_{H}(R_{1,2,1}) and N_{M_{c}}(R_{1,2,1}). So the characters of both groups lie in 2-blocks inducing the principal 2-block, B_{0}, of M_{c}.

We deal first with the stabilizer N_{H}(R_{1,2,1}) of the radical 2-chains C_{9}, C_{11} and C_{12}.

Lemma 4.7.1. R_{1,2,1} has exactly three irreducible characters which are non-trivial on Z(R_{1,2,1}). Let χ be one such. Then χ vanishes outside Z(R_{1,2,1}) and χ|_{Z(R_{1,2,1})} = 4λ, where λ is a non-trivial linear character of Z(R_{1,2,1}). The group N_{M_{c}}(R_{1,2,1}) acts doubly transitively on the three elements of Irr(R_{1,2,1} | Z(R_{1,2,1})), while the normalizer N_{H}(R_{1,2,1}) of T ≤ Z(R_{1,2,1}) in N_{M_{c}}(R_{1,2,1}) acts transitively on the two elements of Irr(R_{1,2,1} | T) ⊆ Irr(R_{1,2,1} | Z(R_{1,2,1})).

Proof. Let λ be a non-trivial character of Z(R_{1,2,1}). By Corollary 4.3.15 the group R_{1,2,1}/ Ker(λ) is extra-special of type 2^{1+4}. It follows that Irr(R_{1,2,1} | λ) consists of a single irreducible character χ, which vanishes outside Z(R_{1,2,1}), and equals 4λ on Z(R_{1,2,1}).

Since there are three choices for λ, the group R_{1,2,1} has exactly three irreducible characters lying over non-trivial characters of its center.

By Lemma 4.3.14 the group N_{M_{c}}(R_{1,2,1}) acts doubly transitively on Z(R_{1,2,1})^{#}. It follows that it acts doubly transitively on Irr(Z(R_{1,2,1}))^{#}, and that the normalizer N_{H}(R_{1,2,1}) of T acts transitively on the subset Irr( Z(R_{1,2,1}) | T) of Irr( Z(R_{1,2,1}))^{#}. This implies the rest of the lemma.

We now compute the degrees of the irreducible characters of N_{H}(R_{1,2,1}) lying over the single non-trivial character of Z(R_{1,2,1}) containing T in its kernel.

Lemma 4.7.2. Deg( N_{H}(R_{1,2,1}) | Z(R_{1,2,1})/T ) = {4^{2}, 8}.

Proof. From Lemma 4.7.1, it is clear that R_{1,2,1} has a unique irreducible character χ_{1} lying over the unique non-trivial linear character in Irr(Z(R_{1,2,1}) mod T). Also χ_{1} is invariant in the group N_{H}(R_{1,2,1}). Then χ_{1} extends to this group, since all Sylow subgroups of N_{H}(R_{1,2,1})/R_{1,2,1} ≅ 𝔖_{3} are cyclic. The lemma now follows from Clifford Theory.

Next we compute the degrees of all irreducible characters of N_{H}(R_{1,2,1}) lying over non-trivial characters of R_{3,1}.

Lemma 4.7.3.

Deg( N_{H}(R_{1,2,1}) | R_{3,1} ) = {3^{4}, 4^{2}, 6, 8^{4}}, and
Deg( N_{H}(R_{1,2,1}) | R_{3,1}/T ) = {3^{4}, 4^{2}, 6, 8}.

Proof. By Lemma 4.7.1 the two elements of Irr( Z(R_{1,2,1}) | T ) form a single N_{H}(R_{1,2,1})-orbit. Let χ_{2} be a character in this orbit. Then I_{N_{H}(R_{1,2,1})}(χ_{2}) ≅ R_{1,2,1}.ℤ_{3}. From Clifford Theory we get

Deg(N_{H}(R_{1,2,1}) | T) = {8^{3}}. (4.7.4)

The group N_{1,2,1} acts flag transitively on A_{3,1}/⟨ε⟩, by (4.2.1). Hence there is a single N_{1,2,1}-orbit of non-trivial linear characters of A_{3,1} whose kernels contain ⟨ε⟩. Let χ_{3} be a member of this orbit. Identify χ_{3} with its inflation to R_{3,1}. Then I_{N_{H}(R_{1,2,1})}(χ_{3}) has index 3 in N_{H}(R_{1,2,1}). So I_{N_{H}(R_{1,2,1})}(χ_{3}) ≅ R_{3,1} ⋊ D_{8}. Since R_{3,1} is abelian, it follows immediately from Clifford Theory that

(4.7.5) Deg(N_{H}(R_{1,2,1}) | R_{3,1}/ Z(R_{1,2,1})) = {3^{4}, 6}.

Now the set Deg(N_{H}(R_{1,2,1}) | R_{3,1}) is obtained from Lemma 4.7.2 and Equations (4.7.4) and (4.7.5), while Deg(N_{H}(R_{1,2,1}) | R_{3,1}/T) is obtained from Lemma 4.7.2 and Equation (4.7.5).

We can now prove

Proposition 4.7.6. The group N_{M_{c}}(C_{9}) = N_{H}(R_{1,2,1}) has a unique 2-block, which necessarily induces the principal 2-block, B_{0}, of M_{c}. Moreover

Deg(N_{H}(R_{1,2,1})/T) = {1^{2}, 2, 3^{6}, 4^{2}, 6, 8},
Deg(N_{H}(R_{1,2,1})) = {1^{2}, 2, 3^{6}, 4^{2}, 6, 8^{4}},
Def_{2}(N_{H}(R_{1,2,1})) = {7^{8}, 6^{2}, 5^{2}, 4^{4}}.

(4.7.7)

Hence

k(C_{9}, B_{0}, 7) = 8,  k(C_{9}, B_{0}, 6) = 2,
k(C_{9}, B_{0}, 5) = 2,  k(C_{9}, B_{0}, 4) = 4,
k(C_{9}, B_{0}, 3) = 0,  k(C_{9}, B_{0}, d) = 0, for all other values of d.

(4.7.8)

Also

(4.7.9)  k(C_{9}, B_{0}, d) = k(C_{11}, B_{0}, d) = k(C_{12}, B_{0}, d), for all values of d.

Proof. Lemma 4.7.3 dealt with all characters of N_{H}(R_{1,2,1}) non-trivial on R_{3,1}. Since N_{H}(R_{1,2,1}) = R_{3,1} ⋊ 𝔖_{4}, the remaining characters are just the characters of 𝔖_{4} inflated to N_{H}(R_{1,2,1}). Equations (4.7.7) follow from this and Lemma 4.7.3. Since N_{H}(R_{1,2,1}) has a unique 2-block, we immediately obtain (4.7.8). The equalities in (4.7.9) follow from the fact that N_{H}(R_{1,2,1}) is the stabilizer of each of the radical 2-chains C_{9}, C_{11} and C_{12} in M_{c}.

We now consider the stabilizer Deg(N_{M_{c}}(R_{1,2,1})) of the radical 2-chains C_{3}, C_{5} and C_{6}.

Lemma 4.7.10. Deg( N_{M_{c}}(R_{1,2,1}) | Z(R_{1,2,1}) ) = {12^{2}, 24}.

Proof. By Lemma 4.7.1, the group N_{M_{c}}(R_{1,2,1}) acts transitively on the three elements of Irr(R_{1,2,1} | Z(R_{1,2,1})). Since an element of N_{M_{c}}(R_{1,2,1}) fixes a linear character λ of Z(R_{1,2,1}) if and only if it normalizes Ker(λ), it follows from Lemma 4.7.1 that the unique character χ ∈ Irr(R_{1,2,1} | Z(R_{1,2,1})) with kernel T has N_{H}(R_{1,2,1}) as its stabilizer in N_{M_{c}}(R_{1,2,1}). The result now follows from Lemma 4.7.2 and Clifford Theory.

Corollary 4.7.11. Let χ ∈ Irr(R_{1,2,1} | Z(R_{1,2,1})). Then the stabilizer I_{N_{Mc} (R_{1,2,1})}(χ) of χ in N_{M_{c}}(R_{1,2,1}) is conjugate to N_{H}(R_{1,2,1}). Hence the stabilizer has Sylow 3-subgroups of order 3.

Proof. The proof of the lemma shows that I_{N_{Mc} (R_{1,2,1})}(χ) is conjugate to N_{H}(R_{1,2,1}). This last group has Sylow 3-subgroups of order 3, since R_{1,2,1} is a 2-group, and the quotient group N_{H}(R_{1,2,1})/R_{1,2,1} ≅ N_{1,2,1}/A_{1,2,1} is isomorphic to 𝔖_{3} by (4.2.1).

Lemma 4.7.12. Deg( N_{M_{c}}(R_{1,2,1})/ Z(R_{1,2,1}) ) = {1^{2}, 2^{4}, 3^{4}, 6^{2}, 9^{2}}.

Proof. We give detailed descriptions of the structure of N_{M_{c}}(R_{1,2,1})/ Z(R_{1,2,1}) in the beginning of Section 4.9. It turns out to be expedient to quote these results here, although we could prove the present lemma without this material.

It follows from Corollary 4.9.5 that N_{M_{c}}(R_{1,2,1})/ Z(R_{1,2,1}) = (A_{4}^{(1)} ×A_{4}^{(2)})⋊⟨ι⟩, where A_{4}^{(i)} ≅ 𝔄_{4} and A_{4}^{(i)} ⋊ ⟨ι⟩ ≅ 𝔖_{4} for i = 1, 2. The result of the lemma is now a straightforward exercise in Clifford Theory.

The next result will be needed in Section 4.10.

Corollary 4.7.13. Let χ ∈ Irr(R_{1,2,1})^{#}. Then the stabilizer I_{N_{Mc} (R_{1,2,1})}(χ) has Sylow 3-subgroups of order 1 or 3.

Proof. If χ is an element of Irr(R_{1,2,1} | Z(R_{1,2,1})), then this is a consequence of Corollary 4.7.11.

If χ is trivial on Z(R_{1,2,1}), then it is inflated from a non-trivial character \overline{χ} of R_{1,2,1}/ Z(R_{1,2,1}). In the notation of the proof of the lemma, this last group is the direct product V_{1} × V_{2} of the four-groups V_{1} = O_{2}(A_{4}^{(1)}) and V_{2} = O_{2}(A_{4}^{(2)}). So \overline{χ} is the direct product \overline{χ}_{1}×\overline{χ}_{2} of some linear characters \overline{χ}_{1} and \overline{χ}_{2} of V_{1} and V_{2}, respectively. A Sylow 3-subgroup of its stabilizer in N_{M_{c}}(R_{1,2,1})/ Z(R_{1,2,1}) = (A_{4}^{(1)} ×A_{4}^{(2)})⋊⟨ι⟩ has order strictly bigger than 3 if and only if A_{4}^{(i)} stabilizes \overline{χ}_{i} for i = 1, 2. This happens if and only if \overline{χ}_{1} and \overline{χ}_{2} are both trivial, in which case \overline{χ} is trivial, contradicting our hypothesis.

We can now prove the following

Proposition 4.7.14. The group N_{M_{c}}(C_{3}) = N_{M_{c}}(R_{1,2,1}) has a unique 2-block, which necessarily induces the principal 2-block, B_{0}, of M_{c}. Moreover

Deg(N_{M_{c}}(R_{1,2,1})) = {1^{2}, 2^{4}, 3^{4}, 6^{2}, 9^{2}, 12^{2}, 24}, and
Def_{2}(N_{M_{c}}(R_{1,2,1})) = {7^{8}, 6^{6}, 5^{2}, 4}.

Hence

k(C_{3}, B_{0}, 7) = 8,  k(C_{3}, B_{0}, 6) = 6,
k(C_{3}, B_{0}, 5) = 2,  k(C_{3}, B_{0}, 4) = 1,
k(C_{3}, B_{0}, 3) = 0,  k(C_{3}, B_{0}, d) = 0, for all other values of d.

(4.7.15)

Also

(4.7.16)  k(C_{3}, B_{0}, 7) = k(C_{5}, B_{0}, 7) = k(C_{6}, B_{0}, 7), for all values of d.

Proof. We obtain the degrees of the irreducible characters of N_{M_{c}}(R_{1,2,1}) from Lemmas 4.7.10 and 4.7.12. Then (4.7.15) is an immediate consequence, given the fact that N_{M_{c}}(R_{1,2,1}) has a unique 2-block.

The equalities in (4.7.16) follow from the fact that the radical 2-chains C_{3}, C_{5} and C_{6} have identical stabilizers in M_{c}.

4.8. The Ordinary Conjecture for the prime p = 2 From [Con85, p101] the group M_{c} has four 2-blocks of defect 0, one block of

defect 1 containing two ordinary characters, and the principal block B_{0} containing the remaining 18 characters. The conjecture holds for the block of defect 1 by the theory of blocks with cyclic defect. For details see [Da96]. So we need only worry about the principal block. We list here the characters of the principal block and their defects, with the notation taken from [Con85]:

Character χ_{1} χ_{2} χ_{3} χ_{4} χ_{5} χ_{6} χ_{9}  χ_{12}  χ_{13}
Degree  1  22 231 252 770 770 1750 4500 4752
Defect  7  6  7  5  6  6  6  5  3
Character χ_{14} χ_{15} χ_{16} χ_{17} χ_{18} χ_{19} χ_{20}  χ_{23}  χ_{24}
Degree  5103 5544 8019 8019 8250 8250 9625 10395 10395
Defect 7 4 7 7 6 6 7 7 7

Thus, since N_{M_{c}}(C_{1}) = M_{c}, we have

k(C_{1}, B_{0}, 7) = 8,  k(C_{1}, B_{0}, 6) = 6,
k(C_{1}, B_{0}, 5) = 2,  k(C_{1}, B_{0}, 4) = 1,
k(C_{1}, B_{0}, 3) = 1,  k(C_{1}, B_{0}, d) = 0, for all other values of d.

(4.8.1)

From [Con85, p101] the characters χ_{1}, χ_{2}, χ_{3}, χ_{4}, χ_{9}, χ_{12}, χ_{13}, χ_{14}, χ_{15}, χ_{20} are the characters of B_{0} invariant in M_{c}.2. Thus

k(C_{1}, B_{0}, 7, \overline{M_{c}.2}) = 4, k(C_{1}, B_{0}, 6, \overline{M_{c}.2}) = 2,
k(C_{1}, B_{0}, 5, \overline{M_{c}.2}) = 2, k(C_{1}, B_{0}, 4, \overline{M_{c}.2}) = 1,
k(C_{1}, B_{0}, 3, \overline{M_{c}.2}) = 1, k(C_{1}, B_{0}, d, \overline{M_{c}.2}) = 0, for all other values of d.

(4.8.2)

We now have enough information for the following theorem. Theorem 4.8.3. The Ordinary Conjecture holds for McLaughlin’s simple group

and the prime p = 2.

Proof. From Conjecture 1.4.2 and Table 4.3 on page 110 we need to prove k(C_{1}, B_{0}, d) + k(C_{3}, B_{0}, d) + k(C_{8}, B_{0}, d) + k(C_{10}, B_{0}, d) =

k(C_{2}, B_{0}, d) + k(C_{4}, B_{0}, d) + k(C_{7}, B_{0}, d) + k(C_{9}, B_{0}, d),

(4.8.4)

for all values of d ∈ ℤ.

From (4.8.1), (4.7.15), (4.6.3), (4.6.4), (4.6.11), (4.6.12), (4.5.1) and (4.7.8) we obtain the following sums for the above equation for various values of d:

2-Defect C_{1}  C_{3}  C_{8}  C_{10}  C_{2}  C_{4}  C_{7}  C_{9}
7  8 + 8 + 8 + 8 = 8 + 8 + 8 + 8
6  6 + 6 + 2 + 2 = 6 + 6 + 2 + 2
5  2 + 2 + 0 + 0 = 0 + 0 + 2 + 2
4  1 + 1 + 6 + 6 = 1 + 1 + 8 + 4
3  1 + 0 + 0 + 0 = 0 + 0 + 1 + 0

Table 4.4. The Ordinary Conjecture for p = 2

The summands in Equation (4.8.4) are zero for all other values of d. This completes the proof.

4.9. The Invariant conjecture for the prime p = 2 In view of Table 4.3, and Equations (4.8.2) and (4.5.2), it remains only to compute

the degrees of the characters of N_{M_{c}}(C_{6}) = N_{M_{c}}(R_{1,2,1}) and N_{M_{c}}(C_{12}) = N_{H}(R_{1,2,1}), which are invariant in N_{M_{c}.2}(C_{6}) = N_{M_{c}.2}(R_{1,2,1}) and N_{M_{c}.2}(C_{12}) = N_{H.2}(R_{1,2,1}) respectively.

First we introduce some notation, which we will use for the remainder of this chapter.

Let V = R_{1,2,1}/ Z(R_{1,2,1}), V_{1} = R_{3,1}/ Z(R_{1,2,1}) and V_{2} = R_{1,3}/ Z(R_{1,2,1}). So V is elementary abelian of order 2^{4}, and V = V_{1} × V_{2}.

Recall from Section 3.1 that E is an elementary abelian 3-subgroup of M_{c} of order 3^{4}, and N_{M_{c}}(E) = E ⋊ M, where M ≅ M_{10}. By Proposition 3.1.12 there exists an element β_{1} of E which comes from the (3B) conjugacy class of M_{c}, such that C_{M_{c}}(β_{1}) = E ⋊ A_{4}, where A_{4} ≅ 𝔄_{4} is contained in M. Also A_{4} < N_{M}(⟨β_{1}⟩) = S_{4}, where S_{4} ≅ 𝔖_{4} is contained in M′ ≅ 𝔄_{6}. Clearly each element of S_{4}\A_{4} inverts β_{1}.

Let β_{2} be an element of order 3 in A_{4}, and let ι be an involution in S_{4}\A_{4} inverting β_{2}. Then ι inverts β_{1}. Furthermore S_{4} is the semi-direct product of the four-group K = O_{2}(A_{4}) with ⟨β_{2}, ι⟩ ≅ 𝔖_{3}.

Let S = ⟨β_{1}, β_{2}⟩. Then S is an elementary abelian 3-subgroup of E ⋊ M of order 3^{2} inverted by ι. Also S ⋊ ⟨ι⟩ normalizes K, but intersects it trivially. Hence E ⋊ M contains the group K ⋊ (S ⋊ ⟨ι⟩).

Now N_{M_{c}.2}(E) = E ⋊ (M × ⟨c⟩), where c is an involution in M_{c}.2\M_{c} inverting E, by Lemma 3.7.1. Then c inverts β_{1} ∈ E and centralizes S_{4} = K ⋊ ⟨β_{2}, ι⟩ ≤ M. It follows that K^{(1)} = ⟨ι, c⟩ is a four-subgroup of M_{c}.2 normalizing ⟨β_{1}⟩, ⟨β_{2}⟩ and K. So we can form the subgroup K ⋊ (S ⋊ K^{(1)}) of E ⋊ (M × ⟨c⟩).

By Proposition 4.1.3, the four-groups K and Z(R_{1,2,1}) are conjugate in M_{c}. Conjugating by an element of M_{c}, if necessary, we may assume that K = Z(R_{1,2,1}). The same proposition also shows that N_{M_{c}}(K) acts transitively on K^{#}. Conjugating by an element of N_{M_{c}}(K), if necessary, we may assume that τ ∈ Z(R_{1,2,1}) = K, is centralized by ι ∈ K^{(1)}.

We summarise our notation in the following table:

V ≅ ℤ_{2}^{4} The elementary abelian group R_{1,2,1}/ Z(R_{1,2,1}).
V_{1} ≅ ℤ_{2}^{2} The subgroup R_{3,1}/ Z(R_{1,2,1}) of V.
V_{2} ≅ ℤ_{2}^{2} The subgroup R_{1,3}/ Z(R_{1,2,1}) of V.
E ≅ ℤ_{3}^{4} A subgroup of M_{c} isomorphic to ℤ_{3}^{4}.
M ≅ M_{10} A complement to E in N_{M_{c}}(E).
β_{1}  A fixed element of E from the (3B) class of M_{c}.
A_{4} ≅ 𝔄_{4} The centralizer of β_{1} in M.
S_{4} ≅ 𝔖_{4} The normalizer of ⟨β_{1}⟩ in M.
β_{2}  An element of order 3 in A_{4}.
S ≅ ℤ_{3}^{2} The group ⟨β_{1}, β_{2}⟩.
ι  An involution in S_{4}\A_{4} inverting β_{2} and centralizing τ ∈ Z(R_{1,2,1}).
c  An involution in M_{c}.2\M_{c} inverting E and centralizing M.
K ≅ ℤ_{2}^{2} The 2-core O_{2}(A_{4}) of A_{4}, which coincides with Z(R_{1,2,1}).
K^{(1)} ≅ ℤ_{2}^{2} The four-group ⟨ι, c⟩.

Table 4.5. Notation for the remainder of the chapter

We note the following for future use.

Lemma 4.9.1. The group S ⋊ K^{(1)} is the direct product ⟨β_{1}, c⟩ × ⟨β_{2}, ιc⟩ of two copies of 𝔖_{3}. Hence O_{2}(S ⋊ K^{(1)}) = {1}.

Proof. This is a consequence of the fact that c inverts β_{1} and centralizes β_{2}, while ι inverts S.

Lemma 4.9.2. N_{M_{c}.2}(R_{1,2,1}) = R_{1,2,1} ⋊ (S ⋊ K^{(1)}).

Proof. The group S ⋊ K^{(1)} normalizes N_{M_{c}}(Z(R_{1,2,1})), as it normalizes K = Z(R_{1,2,1}) and M_{c}. By Lemmas 4.3.14 and 4.4.2 it follows that it normalizes R_{1,2,1} = O_{2}(N_{M_{c}}(R_{1,2,1})) = O_{2}(N_{M_{c}}(Z(R_{1,2,1}))). Hence (S ⋊ K^{(1)}) ∩ R_{1,2,1} is contained in O_{2}(S ⋊ K^{(1)}), which is {1} by Lemma 4.9.1.

The group N_{M_{c}.2}(R_{1,2,1})/R_{1,2,1} has order |3^{2} : 2|·2 = 36, by Lemma 4.4.2. This is also the order of S ⋊ K^{(1)}. We conclude that N_{M_{c}.2}(R_{1,2,1}) = R_{1,2,1} ⋊ (S ⋊ K^{(1)}).

Corollary 4.9.3. N_{M_{c}.2}(R_{1,2,1})/R_{1,2,1} acts faithfully on V.

Proof. We have V = R_{1,2,1}/ Φ(R_{1,2,1}), by Corollary 4.3.15. Then [As86, 24.1] implies that C_{N_{Mc.2}(R_{1,2,1})}(V) is a 2-group. So C_{S⋊K^{(1)}}(V) is a normal 2-subgroup of S ⋊ K^{(1)}. But O_{2}(S ⋊ K^{(1)}) = {1}, by Lemma 4.9.1. The result follows.

Corollary 4.9.4. N_{H.2}(R_{1,2,1}) = R_{1,2,1} ⋊ (⟨β_{1}, c⟩ × ⟨ιc⟩) ≅ R_{1,2,1} ⋊ (𝔖_{3} × ℤ_{2}) and C_{M_{c}.2}(Z(R_{1,2,1})) = R_{1,2,1} ⋊ ⟨β_{1}, c⟩ ≅ R_{1,2,1} ⋊ 𝔖_{3}.

Proof. By Lemma 4.3.14, we know that N_{M_{c}}(R_{1,2,1}) acts as the full automorphism group on Z(R_{1,2,1}). In particular S ⋊ K^{(1)} acts doubly transitively on K^{#} = Z(R_{1,2,1})^{#}.

The group ⟨β_{1}, K^{(1)}⟩ is a subgroup of index 3 in S ⋊ K^{(1)} which centralizes τ ∈ K, since K^{(1)} centralizes τ and K ≤ A_{4} = C_{M}(β_{1}). Hence it must coincide with C_{S⋊K^{(1)}}(τ). So N_{H.2}(R_{1,2,1}) = C_{N_{Mc.2}(R_{1,2,1})}(τ) = R_{1,2,1} ⋊ (⟨β_{1}, c⟩ × ⟨ιc⟩).

The group ⟨β_{1}, c⟩ is a subgroup of index 2 in C_{S⋊K^{(1)}}(τ) which centralizes K. Hence it must coincide with C_{S⋊K^{(1)}}(Z(R_{1,2,1})). So C_{M_{c}.2}(Z(R_{1,2,1})) = R_{1,2,1} ⋊ ⟨β_{1}, c⟩.

Corollary 4.9.5. If we identify S and K^{(1)} with their images in N_{M_{c}.2}(R_{1,2,1})/ Z(R_{1,2,1}), then there exists a decomposition S = Y_{1} × Y_{2} of S into cyclic subgroups, such that N_{M_{c}.2}(R_{1,2,1})/ Z(R_{1,2,1}) = (A_{4}^{(1)} × A_{4}^{(2)}) ⋊ K^{(1)}, where A_{4}^{(i)} = V_{i} ⋊ Y_{i} ≅ 𝔄_{4} and A_{4}^{(i)} ⋊ ⟨ι⟩ ≅ 𝔖_{4}, for i = 1, 2. Moreover c transposes A_{4}^{(1)} and A_{4}^{(2)}.

Proof. We identify the group S ⋊ ⟨ι⟩ with its image in N_{M_{c}}(R_{1,2,1})/ Z(R_{1,2,1}). So this group acts faithfully on V. Both R_{3,1} and R_{1,3} are normal subgroups of N_{M_{c}}(R_{1,2,1}), by Lemma 4.3.14. Hence V_{1} and V_{2} are subgroups of V stabilized by S. Since ι inverts all elements of S, the fact that S acts non-trivially on both V_{1} and V_{2} implies that ι acts faithfully on both V_{1} and V_{2}.

Let Y_{i} be the kernel of the action of S ⋊ ⟨ι⟩ on V_{3−i}, for i = 1, 2. Then Y_{1} ∩ Y_{2} = {1}. So each of these kernels has order 3 and S = Y_{1} × Y_{2}.

Consider the involution c in N_{M_{c}.2}(R_{1,2,1})\M_{c}. By Corollary 4.1.9 and Lemma 4.3.10, this element cannot normalize R_{3,1}. So R_{3,1}^{c} is an elementary abelian subgroup of R_{1,2,1} of order 2^{4} distinct from R_{3,1}. It then follows from Corollary 4.3.12 that R_{3,1}^{c} = R_{1,3}. In the same way R_{1,3}^{c} = R_{3,1}. So c exchanges V_{1} and V_{2} in its conjugation action on V. Hence it also exchanges Y_{1} and Y_{2}. Now S has four subgroups of order 3, namely ⟨β_{1}⟩, ⟨β_{2}⟩, ⟨β_{1}β_{2}⟩ and ⟨β_{1}β_{2}^{−1}⟩. Since c inverts β_{1} and centralizes β_{2}, we see that it transposes ⟨β_{1}β_{2}⟩ and ⟨β_{1}β_{2}^{−1}⟩, while fixing ⟨β_{1}⟩ and ⟨β_{2}⟩. Hence Y_{1} is one of the subgroups ⟨β_{1}β_{2}⟩ and ⟨β_{1}β_{2}^{−1}⟩, while Y_{2} is the other. We set A_{4}^{(1)} = V_{1} ⋊ Y_{1} and A_{4}^{(2)} = V_{2} ⋊ Y_{2}.

Since Y_{i} acts faithfully on V_{i}, it follows that A_{4}^{(i)} ≅ 𝔄_{4}, for i = 1, 2. It is also clear that A_{4}^{(i)} ⋊ ⟨ι⟩ ≅ 𝔖_{4}, while c transposes A_{4}^{(1)} and A_{4}^{(2)}.

We can now prove the following

Proposition 4.9.6. Deg(N_{H.2}(R_{1,2,1}) | Z(R_{1,2,1})/T) = {4^{4}, 8^{2}} and Deg(N_{H.2}(R_{1,2,1}) | T) = {8^{2}, 16}. Hence

Inv(N_{H}(R_{1,2,1})) = {1^{2}, 2, 3^{2}, 4^{2}, 6, 8^{2}}, and
InvDef_{2}(N_{H}(R_{1,2,1})) = {7^{4}, 6^{2}, 5^{2}, 4^{2}}.

Thus

k( C_{12}, B_{0}, 7, \overline{M_{c}.2} ) = 4, k( C_{12}, B_{0}, 6, \overline{M_{c}.2} ) = 2,
k( C_{12}, B_{0}, 5, \overline{M_{c}.2} ) = 2, k( C_{12}, B_{0}, 4, \overline{M_{c}.2} ) = 2,
k( C_{12}, B_{0}, d, \overline{M_{c}.2} ) = 0, for all other values of d.

(4.9.7)

Proof. From the list of maximal subgroups of 𝔄_{8} in the Atlas, N_{H.2}(R_{1,2,1}) is isomorphic to 2.(ℤ_{2}≀𝔖_{4}). We compute the character degrees of ℤ_{2}≀𝔖_{4} by using Clifford Theory on the linear characters of the base group ℤ_{2}^{4}. Fixing some ordered basis of the base group ℤ_{2}^{4}, we let (i_{1}, i_{2}, i_{3}, i_{4}) denote the character of ℤ_{2}^{4} sending the j^{th} basis element to (−1)^{i_{j}}, for each j = 1, 2, 3, 4. Table 4.6 on the following page summarizes our results. It follows from this table that Deg(N_{H.2}(R_{1,2,1})/T) = {1^{4}, 2^{2}, 3^{4}, 4^{4}, 6^{4}, 8^{2}}. Comparing this with (4.7.7), we see that

Inv(N_{H}(R_{1,2,1})/T) = {1^{2}, 2, 3^{2}, 4^{2}, 6, 8}. (4.9.8)

Base  Orbit Stabilizer Stabilizer Character Induced Character
Character Length Structure Degrees  Degrees
(1, 1, 1, 1)  1  𝔖_{4}  1^{2}, 2, 3^{2}  1^{2}, 2, 3^{2}
(1, 1, 1, 0)  4  𝔖_{3} × 𝔖_{1}  1^{2}, 2  4^{2}, 8
(1, 1, 0, 0)  6  𝔖_{2} × 𝔖_{2}  1^{4}  6^{4}
(1, 0, 0, 0)  4  𝔖_{1} × 𝔖_{3}  1^{2}, 2  4^{2}, 8
(0, 0, 0, 0)  1  𝔖_{4}  1^{2}, 2, 3^{2}  1^{2}, 2, 3^{2}

Table 4.6. The Irreducible characters of ℤ_{2} ≀ 𝔖_{4}

The group Z(R_{1,2,1})/T is the diagonal embedding of ℤ_{2} in the base subgroup ℤ_{2}^{4} of ℤ_{2} ≀𝔖_{4}. The characters (0, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 1) are all trivial on this subgroup, while (1, 0, 0, 0) and (1, 1, 1, 0) are not. Hence Irr(N_{H.2}(R_{1,2,1}) | Z(R_{1,2,1})/T) consists of those irreducible characters of ℤ_{2} ≀ 𝔖_{4} lying over the latter two characters. So from the table above, Deg(N_{H.2}(R_{1,2,1}) | Z(R_{1,2,1})/T) = {4^{4}, 8^{2}}.

By Lemma 4.7.1 the group N_{H.2}(R_{1,2,1}) acts transitively on the two elements of Irr(R_{1,2,1} | T). Moreover the stabilizer of an element of Irr(R_{1,2,1} | T) in N_{H.2}(R_{1,2,1}) is C_{M_{c}.2}(Z(R_{1,2,1})). From Corollary 4.9.4 the group C_{M_{c}.2}(Z(R_{1,2,1})) is of the form R_{1,2,1} ⋊ 𝔖_{3}. Since all Sylow subgroups of 𝔖_{3} are cyclic, we immediately obtain from Clifford Theory and Lemma 4.7.1 that Deg(N_{H.2}(R_{1,2,1}) | T) = {8^{2}, 16}. Comparing this with (4.7.4) we deduce that

Inv(N_{H.2}(R_{1,2,1}) | T) = {8}. (4.9.9)

We obtain Inv(N_{H}(R_{1,2,1})) from (4.9.8) and (4.9.9). By Proposition 4.7.6, the group N_{H}(R_{1,2,1}) = N_{M_{c}}(C_{12}) has a unique 2-block, which necessarily induces the principal 2-block, B_{0}, of M_{c}. The rest of the proposition now follows.

Lemma 4.9.10. Deg( N_{M_{c}.2}(R_{1,2,1})/ Z(R_{1,2,1}) ) = {1^{4}, 2^{4}, 4, 6^{2}, 9^{4}, 12}. Hence
Inv( N_{M_{c}.2}(R_{1,2,1})/ Z(R_{1,2,1}) ) = {1^{2}, 2^{2}, 9^{2}}, and
InvDef_{2}( N_{M_{c}.2}(R_{1,2,1}) mod Z(R_{1,2,1}) ) = {7^{4}, 6^{2}}.

(4.9.11)

Proof. We use the notation of Corollary 4.9.5. So N_{M_{c}.2}(R_{1,2,1})/ Z(R_{1,2,1}) = (A_{4}^{(1)} × A_{4}^{(2)}) ⋊ K^{(1)}, where A_{4}^{(i)} ≅ 𝔄_{4} for i = 1, 2. Let λ_{1}^{i}, λ_{2}^{i}, λ_{3}^{i} be the distinct linear characters of A_{4}^{(i)}, with λ_{1}^{i} the trivial character. Let μ^{i} denote the single irreducible character of A_{4}^{(i)} of degree 3.

Then

λ_{1}^{1} × μ^{2}, μ^{1} × λ_{1}^{2}, λ_{2}^{1} × μ^{2}, μ^{1} × λ_{2}^{2}, λ_{3}^{1} × μ^{2}, μ^{1} × λ_{3}^{2}, μ^{1} × μ^{2},

is a complete list of those characters of A_{4}^{(1)} × A_{4}^{(2)} which are non-trivial on V.

The determinantal character det(μ^{i}) of μ^{i}, as given in Definition 1.2.12, satisfies \overline{det(μ^{i})} = det(\overline{μ^{i}}) = det(μ^{i}), for i = 1, 2. So det(μ^{i}) is the trivial character of A_{4}^{(i)}. Hence det(μ^{1} × μ^{2}) = det(μ^{1}) × det(μ^{2}) is the trivial character of A_{4}^{(1)} × A_{4}^{(2)}. It follows that the determinantal order o(μ^{1} × μ^{2}), as given in Definition 1.2.13, is 1.

The unique character μ^{1} × μ^{2} of A_{4}^{(1)} × A_{4}^{(2)} of degree 9 is necessarily K^{(1)}- invariant. Then, since

( |(A_{4}^{(1)} × A_{4}^{(2)}) ⋊ K^{(1)} : A_{4}^{(1)} × A_{4}^{(2)}| , o(μ^{1} × μ^{2})(μ^{1} × μ^{2})(1) ) = (4, 9) = 1, Theorem 1.2.14 guarantees that μ^{1} × μ^{2} extends to (A_{4}^{(1)} × A_{4}^{(2)}) ⋊ K^{(1)}. Hence (4.9.12) Deg(N_{M_{c}.2}(R_{1,2,1})/ Z(R_{1,2,1}) | μ^{1} × μ^{2}) = {9^{4}}.

Recall from Corollary 4.9.5 that A_{4}^{(i)} ⋊ ⟨ι⟩ ≅ 𝔖_{4}, for i = 1, 2, and that c interchanges A_{4}^{(1)} and A_{4}^{(2)}. We now examine the action of K^{(1)} = ⟨ι, c⟩ on Irr(A_{4}^{(1)} × A_{4}^{(2)}).

The characters λ_{1}^{1} × μ^{2} and μ^{1} × λ_{1}^{2} are stabilized by ι and transposed by c. So they form a single K^{(1)}-orbit. Thus (4.9.13) Deg(N_{M_{c}.2}(R_{1,2,1})/ Z(R_{1,2,1}) | λ_{1}^{1} × μ^{2}) = {6^{2}}.

The stabilizer of λ_{2}^{1} × μ^{2} in K^{(1)} is trivial. Hence the remaining four characters λ_{2}^{1} × μ^{2}, μ^{1} × λ_{2}^{2}, λ_{3}^{1} × μ^{2}, μ^{1} × λ_{3}^{2} of A_{4}^{(1)} × A_{4}^{(2)} form a single K^{(1)}-orbit. So (4.9.14) Deg(N_{M_{c}.2}(R_{1,2,1})/ Z(R_{1,2,1}) | λ_{2}^{1} × μ^{2}) = {12}.

The quotient ((A_{4}^{(1)} × A_{4}^{(2)}) ⋊ K^{(1)}) /V is isomorphic to 𝔖_{3} × 𝔖_{3}. Hence (4.9.15) Deg(N_{M_{c}.2}(R_{1,2,1})/R_{1,2,1}) = Deg(𝔖_{3} × 𝔖_{3}) = {1^{4}, 2^{4}, 4}.

The multiset Deg(N_{M_{c}.2}(R_{1,2,1})/ Z(R_{1,2,1})) can be obtained from (4.9.12), (4.9.13), (4.9.14) and (4.9.15). The rest of the lemma now follows from a comparison with Lemma 4.7.12.

Lemma 4.9.16.

Inv(N_{M_{c}}(R_{1,2,1}) | Z(R_{1,2,1})) = {12^{2}, 24},
InvDef_{2}(N_{M_{c}}(R_{1,2,1}) | Z(R_{1,2,1})) = {5^{2}, 4}.

Proof. From Lemma 4.9.2, we have N_{M_{c}.2}(R_{1,2,1})/R_{1,2,1} ≅ (𝔖_{3} × 𝔖_{3}). By Lemma 4.7.1 there is a single N_{M_{c}.2}(R_{1,2,1})-orbit of characters of R_{1,2,1} non-trivial on Z(R_{1,2,1}). Moreover, a character χ of this orbit vanishes outside Z(R_{1,2,1}). We may suppose, without loss of generality, that Ker(χ) = T. Then χ is invariant in N_{H.2}(R_{1,2,1}). But N_{H.2}(R_{1,2,1}) is a sugroup of index three in N_{M_{c}.2}(R_{1,2,1}). So it must coincide with I_{N_{Mc.2}(R_{1,2,1})}(χ). Now Deg(N_{H.2}(R_{1,2,1}) | Z(R_{1,2,1})/T) = {4^{4}, 8^{2}}, by Proposition 4.9.6. Hence Deg(N_{M_{c}.2}(R_{1,2,1}) | Z(R_{1,2,1})) = {12^{4}, 24^{2}}.

The lemma now follows from a comparison with the result of Lemma 4.7.10.

From the previous two lemmas we obtain the following

Proposition 4.9.17.

Inv(N_{M_{c}.2}(R_{1,2,1})) = {1^{2}, 2^{2}, 9^{2}, 12^{2}, 24},
InvDef_{2}(N_{M_{c}.2}(R_{1,2,1})) = {7^{4}, 6^{2}, 5^{2}, 4}.

Hence

k( C_{6}, B_{0}, 7, \overline{M_{c}.2} ) = 4, k( C_{6}, B_{0}, 6, \overline{M_{c}.2} ) = 2,
k( C_{6}, B_{0}, 5, \overline{M_{c}.2} ) = 2, k( C_{6}, B_{0}, 4, \overline{M_{c}.2} ) = 1,
k( C_{6}, B_{0}, d, \overline{M_{c}.2} ) = 0, for all other values of d.

(4.9.18)

Proof. This follows at once from Lemmas 4.9.10 and 4.9.16 and the fact that N_{M_{c}.2}(C_{6})= N_{M_{c}.2}(R_{1,2,1}).

We now have enough information for the following theorem.

Theorem 4.9.19. The Invariant Conjecture holds for McLaughlin’s simple group and the prime p = 2.

Proof. By [Da96] we need only consider the unique 2-block, B_{0}, of M_{c} having a non-cyclic defect group. In view of Table 4.3 on page 110, Conjecture 1.4.4 for the block B_{0} is equivalent to the equation

k( C_{1}, B_{0}, d, \overline{M_{c}.2} )+ k( C_{12}, B_{0}, d, \overline{M_{c}.2} ) =
k( C_{6}, B_{0}, d, \overline{M_{c}.2} ) + k( C_{7}, B_{0}, d, \overline{M_{c}.2} ),

(4.9.20)

for all values of d ∈ ℤ.

From (4.8.2), (4.9.7), (4.9.18) and (4.5.2) we obtain the following sums for the equation above for various values of d:

2-Defect C_{1}  C_{12}  C_{6}  C_{7}
7  4 + 4 = 4 + 4
6  2 + 2 = 2 + 2
5  2 + 2 = 2 + 2
4  1 + 2 = 1 + 2
3  1 + 0 = 0 + 1

Table 4.7. The Invariant Conjecture for p = 2

The summands in Equation (4.9.20) are zero for all other values of d. This completes the proof.

4.10. The Projective conjecture for the prime p = 2 By Theorem 3.12.8, the McLaughlin group has a cyclic Schur multiplier of order 3.

As in Section 3.12, we let A denote the center of the universal perfect covering group M̂_{c} of M_{c}. So A is cyclic of order 3. If X is a subgroup of M_{c}, then we will denote its inverse image in M̂_{c} by A.X or X̂.

Let ρ be some fixed non-trivial character of A.

Lemma 4.10.1. M̂_{c} has three 2-blocks, B_{0}^{∗}, B_{1}^{∗} and B_{2}^{∗}, lying over the 2-block of A containing ρ. The block B_{0}^{∗} has defect 7 and contains 18 characters, while B_{1}^{∗} has defect 1 and contains 2 characters, and B_{2}^{∗} has defect 0.

Proof. We use the Atlas notation for the characters in Irr(M̂_{c} | ρ). Two of these characters, χ_{36} and χ_{37}, have defect 1 and lie in a 2-block B_{1}^{∗} of defect 1, while another, χ_{41}, lies in a 2-block B_{2}^{∗} of defect 0. The remaining 18 characters lie in a single 2-block, B_{0}^{∗}, of defect 7.

We list the irreducible characters of B_{0}^{∗} and their 2-defects:

Character χ_{25} χ_{26} χ_{27} χ_{28} χ_{29}  χ_{30}  χ_{31}  χ_{32}  χ_{33}
Degree  126 126 792 1980 2376 2376 2520 2520 2772
Defect  6  6  4  5  4  4  4  4  5
Character χ_{34} χ_{35} χ_{38} χ_{39} χ_{40}  χ_{42}  χ_{43}  χ_{44}  χ_{45}
Degree  4752 5103 7875 8019 8019 10395 10395 10395 12375
Defect  3  7  7  7  7  7  7  7  7

k(C_{1}, B_{0}^{∗}, 7 | ρ) = 8,  k(C_{1}, B_{0}^{∗}, 6 | ρ) = 2,
k(C_{1}, B_{0}^{∗}, 5 | ρ) = 2,  k(C_{1}, B_{0}^{∗}, 4 | ρ) = 5,
k(C_{1}, B_{0}^{∗}, 3 | ρ) = 1,  k(C_{1}, B_{0}^{∗}, d | ρ) = 0, for all other values of d.

(4.10.2)

The covering group Ĥ of H ≅ 2.𝔄_{8} splits over A, because H is the universal covering group of the alternating group 𝔄_{8}. So its subgroups A. N_{H}(R_{3,1}), A. N_{H}(R_{1,3}) and A. N_{H}(R_{1,2,1}) also split over A. Now H has a unique 2-block of defect greater than 1, and by Propositions 4.6.2 and 4.7.6, each of its subgroups R_{3,1}, R_{1,3} and R_{1,2,1} has a unique 2-block. Then in view of Table 4.3 on page 110, it follows that

(4.10.3) k(C_{i}, B_{0}^{∗}, d | ρ) = k(C_{i}, B_{0}, d), for 7 ≤ i ≤ 12, and all values of d.

Next we consider the groups A. N_{M_{c}}(R_{3,1}), A. N_{M_{c}}(R_{1,3}) and A. N_{M_{c}}(R_{1,2,1}). Recall from Table 4.5 on page 126 and from Lemma 4.9.2 that the Sylow 3-subgroup S of N_{M_{c}}(R_{1,2,1}) is elementary abelian of order 9. Using Lemma 4.3.14 and Corollary 4.3.11, we see that S is also a Sylow 3-subgroup of both N_{M_{c}}(R_{3,1}) and N_{M_{c}}(R_{1,3}).

Lemma 4.10.4. The covering group Ŝ of S in M̂_{c} does not split over A. Hence it is extra-special of type 3_{+}^{1+2}.

Proof. By construction S = ⟨β_{1}, β_{2}⟩, where β_{1} ∈ E is an element of the (3B) class of M_{c}, and β_{2} is a 3-element of A_{4} ≤ M. In particular S ≰ E.

Let β̂_{1} be an element of M̂_{c} having β_{1} as its image in M_{c}. From the Atlas, β̂_{1} is a 3-element of the unique conjugacy class of M̂_{c} lying over the conjugacy class (3B) of M_{c}. In particular |CM̂_{c}(β̂_{1})| = |C_{M_{c}}(β_{1})|.

It follows from Proposition 3.1.12 that C_{M_{c}}(β_{1}) = E ⋊ A_{4}. This group normalizes E and has a Sylow 3-subgroup of order 3^{5}. Hence the group CM̂_{c}(β̂_{1}) ≤ A. C_{M_{c}}(β_{1}) normalizes Ê, and has a Sylow 3-subgroup of order 3^{5}. But Ê is isomorphic to ℤ_{3}^{5}, by Proposition 3.13.1. Hence it centralizes β̂_{1}. So Ê is the unique Sylow 3-subgroup of CM̂_{c}(β̂_{1}). Since Ŝ is a 3-group normalizing Ê, but not contained in Ê, we conclude that Ŝ ≰ CM̂_{c}(β̂_{1}). In particular Ŝ is not abelian. So Ŝ does not split over A.

From the Atlas, any element of M̂_{c} whose image in M_{c} has order 3 itself has order 3. Hence Ŝ has exponent 3.

The previous two paragraphs show that Ŝ is a non-abelian 3-group of exponent 3 and order 3^{3}. The lemma follows.

We now study NM̂_{c}(C_{3}) = NM̂_{c}(R_{1,2,1}).

Proposition 4.10.5.

Deg(NM̂_{c}(R_{1,2,1}) | ρ) = {3^{6}, 6^{2}, 9^{2}, 12^{2}, 24},
Def_{2}(NM̂_{c}(R_{1,2,1}) | ρ) = {7^{8}, 6^{2}, 5^{2}, 4}.

Hence

k(C_{3}, B_{0}^{∗}, 7 | ρ) = 8,  k(C_{3}, B_{0}^{∗}, 6 | ρ) = 2,
k(C_{3}, B_{0}^{∗}, 5 | ρ) = 2,  k(C_{3}, B_{0}^{∗}, 4 | ρ) = 1,
k(C_{3}, B_{0}^{∗}, 3 | ρ) = 0,  k(C_{3}, B_{0}^{∗}, d | ρ) = 0, for all other values of d.

(4.10.6)

(4.10.7) k(C_{3}, B_{0}^{∗}, d | ρ) = k(C_{5}, B_{0}^{∗}, d | ρ) = k(C_{6}, B_{0}^{∗}, d | ρ), for all values of d.

Proof. From Lemma 4.4.2 the normalizer of R_{1,2,1} in M_{c} is of the form R_{1,2,1} ⋊ (3^{2} : 2). Then from Lemma 4.10.4, its covering group NM̂_{c}(R_{1,2,1}) in M̂_{c} is of the form R_{1,2,1}^{(1)} ⋊ (3_{+}^{1+2} : 2), where R_{1,2,1}^{(1)} is the unique Sylow 2-subgroup of the inverse image R̂_{1,2,1} of R_{1,2,1} in M̂_{c}.

The Sylow 3-subgroup 3_{+}^{1+2}, of a complement 3_{+}^{1+2} : 2 to R_{1,2,1}^{(1)} in NM̂_{c}(R_{1,2,1}), has exactly one irreducible character μ lying over ρ. Now μ has degree 3 and is invariant in 3_{+}^{1+2} : 2. So there are exactly two characters of 3_{+}^{1+2} : 2 lying over μ, and each has degree 3. We inflate these to characters of NM̂_{c}(R_{1,2,1}). In this way we obtain all irreducible characters of Irr(NM̂_{c}(R_{1,2,1}) | ρ) whose kernels contain R_{1,2,1}^{(1)}.

Let χ be a non-trivial irreducible character of R_{1,2,1}^{(1)}. Then there is one extension χ̃ of χ to R̂_{1,2,1} = A × R_{1,2,1}^{(1)} which lies over ρ. The stabilizer I_{NM̂c (R_{1,2,1})}(χ̃) of this extension is then just A. I_{N_{Mc} (R_{1,2,1})}(χ). By Corollary 4.7.13, we know that A. I_{N_{Mc} (R_{1,2,1})}(χ)/(A × R_{1,2,1}^{(1)}) has a Sylow 3-subgroup of order 1 or 3. Also, a Sylow 2-subgroup of N_{M_{c}}(R_{1,2,1})/(A × R_{1,2,1}^{(1)}) is cyclic of order 2. Hence χ̃ extends to its entire stabilizer, by Theorem 1.2.11. We conclude that there is a degree preserving bijection between Irr(N_{M_{c}}(R_{1,2,1}) | R_{1,2,1}) and the set of those characters of Irr(NM̂_{c}(R_{1,2,1}) | ρ) whose restriction to R_{1,2,1} is non-trivial.

All elements of Irr(NM̂_{c}(R_{1,2,1}) | ρ) belong to a single 2-block b_{0}^{∗} of NM̂_{c}(R_{1,2,1}) lying over the block of A containing ρ. This block b_{0}^{∗} has a defect group containing R_{1,2,1}^{(1)}. In particular, the defect of (b_{0}^{∗})M̂_{c} is at least log_{2}(|R_{1,2,1}|) = 6. It then follows from Lemma 4.10.1 that (b_{0}^{∗})M̂_{c} = B_{0}^{∗}.

The character degrees and defects given in the statement of the proposition now follow from the first paragraph of the proof, Lemmas 4.7.10 and 4.7.12, and the fact that Deg(N_{M_{c}}(R_{1,2,1})/R_{1,2,1}) = Deg(3^{2} : 2) = {1^{2}, 2^{4}}.

Next we deal with NM̂_{c}(C_{2}) = NM̂_{c}(R_{3,1}).

Proposition 4.10.8.

Deg(NM̂_{c}(R_{3,1}) | ρ) = {6, 15^{3}, 21^{2}, 24^{2}, 45^{2}, 90, 105, 120},
Def_{2}(NM̂_{c}(R_{3,1}) | ρ) = {7^{8}, 6^{2}, 4^{3}}.

Hence

k(C_{2}, B_{0}^{∗}, 7 | ρ) = 8,  k(C_{2}, B_{0}^{∗}, 6 | ρ) = 2,
k(C_{2}, B_{0}^{∗}, 5 | ρ) = 0,  k(C_{2}, B_{0}^{∗}, 4 | ρ) = 3,
k(C_{2}, B_{0}^{∗}, 3 | ρ) = 0,  k(C_{2}, B_{0}^{∗}, d | ρ) = 0, for all other values of d.

(4.10.9)

Also

(4.10.10) k(C_{4}, B_{0}^{∗}, d | ρ) = k(C_{2}, B_{0}^{∗}, d | ρ), for all values of d.

Proof. From Corollary 4.3.11 and Lemma 4.10.4, the group NM̂_{c}(R_{3,1}) is of the form R_{3,1}^{(1)} ⋊ Λ̂_{3,1}, where R_{3,1}^{(1)} is the unique Sylow 2-subgroup of the cover of R_{3,1} in M̂_{c}, and the complement Λ̂_{3,1} is a non-split central extension of Λ_{3,1} ≅ 𝔄_{7} by the cyclic group A of order 3.

From the character table of 𝔄_{7} in the Atlas, we obtain

Deg(Λ̂_{3,1} | ρ) = {6, 15^{2}, 21^{2}, 24^{2}},
Def_{2}(Λ̂_{3,1} | ρ) = {7^{4}, 6, 4^{2}}

(4.10.11)

Let χ be a non-trivial linear character of R_{3,1}. By Lemma 4.6.8, the Λ_{3,1}-orbit of χ is Irr(R_{3,1})^{#}, and I_{N_{Mc} (R3,1)}(χ) is of the form R_{3,1} ⋊ GL(3, 2). We let χ̂ be a linear character of R̂_{3,1} = A × R_{3,1}^{(1)} lying over ρ and non-trivial on R_{3,1}^{(1)}. Then the stabilizer of χ̂ in NM̂_{c}(R_{3,1}) is of the form A.(R_{3,1} ⋊ GL(3, 2)). But A.(R_{3,1} ⋊ GL(3, 2)) splits over A, since a Sylow 3-subgroup of R_{3,1} ⋊ GL(3, 2) is cyclic of order 3, and is not central in its normalizer. Hence there is a degree preserving bijection between Irr(I_{N_{Mc} (R_{3,1})}(χ) | χ) and Irr(I_{NM̂c (R_{3,1})}(χ̂) | χ̂). We conclude that there is also a degree preserving bijection between Irr(N_{M_{c}}(R_{3,1}) | R_{3,1}) and the set of those characters of Irr(NM̂_{c}(R_{3,1}) | ρ) non-trivial on R_{3,1}.

We now obtain Deg(NM̂_{c}(R_{3,1}) | ρ) and Def_{2}(NM̂_{c}(R_{3,1}) | ρ) using (4.6.9) and (4.10.11).

All elements of Irr(NM̂_{c}(R_{3,1}) | ρ) belong to a single 2-block b_{0}^{∗} of NM̂_{c}(R_{3,1}) lying over the block of A containing ρ. This block b_{0}^{∗} has a defect group containing R_{3,1}^{(1)}. In particular, the defect of (b_{0}^{∗})M̂_{c} is at least log_{2}(|R_{3,1}|) = 4. It then follows from Lemma 4.10.1 that (b_{0}^{∗})M̂_{c} = B_{0}^{∗}.

We can now compute (4.10.9) using the multiset Def_{2}(NM̂_{c}(R_{3,1}) | ρ) and the fact that NM̂_{c}(C_{2}) = NM̂_{c}(R_{3,1}). Equation (4.10.10) holds because NM̂_{c}(R_{3,1}) ≅ NM̂_{c}(R_{1,3}) = NM̂_{c}(C_{4}).

We can now prove the following

Theorem 4.10.12. The Projective Conjecture holds for McLaughlin’s Simple Group M_{c} and the prime p = 2.

Proof. We need only consider the 2-blocks B_{0}^{∗} and B_{1}^{∗} of positive defect in Lemma 4.10.1. Since B_{1}^{∗} has defect 1, it has a cyclic defect group. So Conjecture 1.4.6 holds for it by [Da96]. In view of Table 4.3 on page 110, Conjecture 1.4.6 for the block B_{0}^{∗} is equivalent to the equation

k(C_{1}, B_{0}^{∗}, d | ρ) + k(C_{3}, B_{0}^{∗}, d | ρ) + k(C_{8}, B_{0}^{∗}, d | ρ) + k(C_{10}, B_{0}^{∗}, d | ρ) =

k(C_{2}, B_{0}^{∗}, d | ρ) + k(C_{4}, B_{0}^{∗}, d | ρ) + k(C_{7}, B_{0}^{∗}, d | ρ) + k(C_{9}, B_{0}^{∗}, d | ρ),

(4.10.13)

for all values of d ∈ ℤ.

We obtain Table 4.8 using Equations (4.10.2), (4.10.6), (4.10.9) and (4.10.10), as well as (4.5.1), (4.6.3), (4.7.8), (4.6.4) and (4.10.3).

2-Defect C_{1}  C_{3}  C_{8}  C_{10}  C_{2}  C_{4}  C_{7}  C_{9}
7  8 + 8 + 8 + 8 = 8 + 8 + 8 + 8
6  2 + 2 + 2 + 2 = 2 + 2 + 2 + 2
5  2 + 2 + 0 + 0 = 0 + 0 + 2 + 2
4  5 + 1 + 6 + 6 = 3 + 3 + 8 + 4
3  1 + 0 + 0 + 0 = 0 + 0 + 1 + 0

Table 4.8. The Projective Conjecture for p = 2

The summands in the above equation are zero for all other values of d. This completes the proof.

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Curriculum Vitae

John Cyril Murray was born in Clonmel, Ireland on February 9, 1968. He received a Certificate in Visual Education from Letterkenny Regional Technical College in 1986, a B. Electronic Engineering from University College Dublin in 1990, a Diploma in Mathematical Science from University College Dublin in 1992 and an M.Sc. in Mathematics from University College Dublin in 1993. He will be starting a Forbairt Postdoctoral Fellowship in September 1997 and is a member of the IMS and AMS.

John and his wife Deirdre McGowan have two children, Clara and Seóirse.