On finite groups of p-local rank one and a conjecture of Robinson

by

Charles Eaton PhD thesis University of Leicester, 1999

Acknowledgments

I would first and foremost like to thank Professor Geoffrey Robinson, both for his patient supervision and for setting an example in mathematical rigour and imagination well worth following.

Thanks also should go to Dr. Robert Baddeley, Dr. John Hunton and Professor Rick Thomas for acting as temporary supervisors at various times and for their support in general.

My parents, my sister Alice and my partner Anne deserve special thanks not only for their support but for letting me get on with it.

I thank the Department of Mathematics and Computer Science at the University of Leicester for providing a pleasant atmosphere in which to work and teaching work with which to help support myself. I also thank the Mathematics Institute at the University of Warwick for allowing me to use their facilities when I have needed them.

I acknowledge the Engineering and Physical Sciences Research Council for funding the first two years of my research (Award no. 95700575).

Declaration

I declare that to the best of my knowledge the material contained in this thesis is original unless explicitly stated otherwise

Contents

1 Introduction and notation 1 1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Background 7 2.1 Choice of p-modular system . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Radical p-chains and the p-local rank of a finite group . . . . . . . . . . 7 2.3 Properties of finite groups of p-local rank one . . . . . . . . . . . . . . 11 2.4 The Schur Multiplier . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Clifford-theoretic reductions 16 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Character correspondences . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 Characters afforded by relatively projective modules . . . . . . . . . . . 24

4 Conjectures 28 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2 Alperin’s weight conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.3 A reformulation of the weight conjecture by Knörr and Robinson …. 29 4.4 The conjectures of Dade . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.5 Robinson’s reformulation . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.6 Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5 Finite groups with normal p-subgroups 36 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.2 Reduction for Robinson’s conjecture . . . . . . . . . . . . . . . . . . . . 36 5.3 Groups of p-rank one . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6 Characterization and classification of finite groups with almost TI

Sylow p-subgroups 48 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 6.2 Finite groups with normal p′-subgroups . . . . . . . . . . . . . . . . . . 48 6.3 Characterization of finite groups with almost TI Sylow p-subgroups . . 49

7 Checking the conjecture 57 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7.2 Some results of Blau and Michler . . . . . . . . . . . . . . . . . . . . . 57 7.3 Sporadic simple groups of p-local rank one . . . . . . . . . . . . . . . . 59 7.4 Groups of Lie type of p-local rank one . . . . . . . . . . . . . . . . . . 60 7.5 Ree groups of type G_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7.6 The unitary groups of degree 3 in the defining characteristic . . . . . . 68

8 Proof of Theorem 4.7 73

9 Dade’s inductive conjecture for the Ree groups of type G_{2} 75

10 Appendix A 77 10.1 Generators for P SL_{3}(4).2^{2} . . . . . . . . . . . . . . . . . . . . . . . . . 77 10.2 Generators for 2. ^{2}B_{2}(8) . . . . . . . . . . . . . . . . . . . . . . . . . . 77

1 Introduction and notation

We present the beginnings of a possible inductive approach to Alperin’s weight conjecture (see Alperin [2]) suggested by Robinson in [47]. There a conjecture equivalent to Alperin’s is presented along with an inductive invariant, the p-local rank of a finite group. The hope is that one may reduce to the p-local rank one case (which includes those groups with split BN-pairs of rank one in characteristic p). Here we verify Robinson’s refined conjecture (Conjecture 4.4) in just this situation by applying the classification of finite simple groups (CFSG).

Finite groups G of p-local rank one are those in which G/O_{p}(G) has trivial intersection (TI) Sylow p-subgroups (i.e., any two distinct Sylow p-subgroups of G intersect in O_{p}(G), the unique maximal normal p-subgroup of G). Finite groups with TI Sylow p-subgroups have been studied to an extent by Blau and Michler in [7], in which the CFSG is used to prove an earlier conjecture of Alperin, and for a large part of our work here we use similar techniques to generalize their results. All groups considered in this thesis are finite.

Our main result is the following. The description of Conjecture 4.4 is delayed until Chapter 4, since the material of Chapter 3 is necessary to understand it fully.

Theorem 4.7 Conjecture 4.4 holds for every finite group of p-local rank one.

The first five chapters are general in that the results largely apply to all finite groups.

In Chapter 2 there is a brief review of the material necessary for the later results. This in particular includes a brief introduction to the theory of p-chains, an overview of the properties of groups of p-local rank one and a brief discussion on Schur multipliers. We do not include however the basic results of Clifford theory or block theory. When we refer to ‘Clifford theory’ we refer to any results found in, say Chapter 6 of [29]. Our omission of any block theoretic background may be a little more controversial, but we feel that what we do use (such as Brauer’s main theorems) is familiar enough not to warrant explicit reference. Excellent sources for this material however are [15] and [22].

Chapter 3 contains the Clifford-theoretic reductions we use later in eliminating complicated normal p-subgroups or p′-subgroups. The reductions themselves are wellknown, but their compatibility with block theory has only been discussed fully in the setting of twisted group algebras (see [18]). We use notation set up in the first part of the chapter to give a review of some results of Külshammer and Robinson and to sketch their proofs.

Chapter 5 largely consists of a proof that (for finite groups of any p-local rank) in verifying Robinson’s conjecture it suffices to consider only those groups in which O_{p}(G) is both central and cyclic.

The remainder of the thesis concentrates solely on finite groups of p-local rank one, and consists of the proof of Theorem 4.7.

In Chapter 6 there is a reworking and extension of the reduction used in [7]. Having proved in Chapter 5 that we need only consider finite groups with almost TI Sylow psubgroups (‘almost TI’ is defined in Section 2.3), and after some more Clifford theoretic reductions, it is then possible to show that every such group is a central extension of an automorphism group of a nonabelian simple group. By examining the relationship between the Schur multiplier of a simple group and those of its automorphism groups, we are able to reduce to a consideration of automorphism groups of quasisimple groups. Such groups are well-covered in the literature. The efficiency of this reduction owes much to the results of [6] and to the Schur multiplier package in MAGMA. Classification results achieved by Gorenstein and Lyons (p ≠ 2) and Suzuki (p = 2) give a list of all nonabelian simple groups of p-local rank one, allowing us to give a full list of the remaining possibilities for G.

Chapter 7 consists of the verification of Conjecture 4.4 for the automorphism groups of the quasisimple groups listed in Chapter 6.

In Chapter 8 there is a review of the proof of Theorem 4.7. In Chapter 9 we verify that Dade’s final (inductive) conjecture (see [20]) holds for

the finite groups ^{2}G_{2}(3^{2m+1}) for the prime 3, completing the verification of Dade’s conjecture for this class of groups.

Appendix A consists of the information necessary to replicate the calculations performed on MAGMA.

1.1 Notation

G  a finite group
H < G  H is a proper subgroup of G
H ≤ G  H is a subgroup of G
H ◁ G  H is a normal subgroup of G
N_{G}(P)  the normalizer of P in G
C_{G}(P)  the centralizer of P in G
Z(G)  the centre of G
Irr(G)  the set of irreducible characters of G
(K, R, k)  a p-modular system
J(R)  the Jacobson radical of a ring R
IBr(G)  the set of irreducible Brauer characters of G

with respect to (K, R, k)

k(G)  |Irr(G)|
l(G)  |IBr(G)|
f_{0}(G)  the number of p-blocks of defect zero of G
w(G, Q)  the number of Q-projective irreducible characters of G
(χ, θ)  for characters χ, θ of G, (χ, θ) = \frac{1}{|G|} ∑_{x∈G} χ(x)θ(x^{−1})
σ  a p-chain of G
|σ|  the length of σ
G_{σ} = N_{G}(σ) the stabilizer of σ under conjugation in G
V_{σ}  the initial subgroup of σ
V^{σ}  the final subgroup of σ
C(G)  the set of all p-chains of G
X (G)  a subset of C(G)
X (G)/H  a set of orbit representatives for the action of H on X (G)
X (G|Q)  σ ∈ X (G) with V_{σ} = Q

X_{Q}(G)  σ ∈ X (G) with Q ≤ V_{σ}
N (G)  σ ∈ X (G) with V^{σ} ≤ N_{G}(σ)
R(G)  the radical p-chains of G
plr(G)  the p-local rank of G
O_{π}(G)  π a set of primes, the unique maximal normal π-subgroup of G
G_{π}  π a set of primes, the set of elements of G of order

divisible only by primes in π. Note that 1 ∈ G_{π}

H^{2}(G, F^{×}) the second cohomology group of G with respect to the field F
H_{2}(G, ℤ)  the second integral homology group of G
[G, G] = G′ the derived subgroup of G
M(G)  the Schur multiplier of G
Aut(G)  the automorphism group of G
Out(G)  the outer automorphism group of G
s(g)  the G-conjugacy class sum of g ∈ G
w_{χ}  the central character Z(G) → R associated to χ ∈ Irr(G)
Bl(χ)  the p-block of G containing χ ∈ Irr(G)
ζ^{G}  the character of G induced from the character ζ of H ≤ G
χ|_{H}  the restriction of the character χ of G to H ≤ G
M^{⋆}  the dual of a module M
R_{0}(G)  the set of radical p-subgroups of G
x_{π}  for π a set of primes, the π-part of a group element/integer x
b^{G}  for a p-block b of H ≤ G, the Brauer correspondent in G
I_{G}(μ)  inertial subgroup in G of the character μ of N ◁ G
𝔽_{t}  the field with t elements
𝔽_{t}^{×}  multiplicative group of 𝔽_{t}
f_{S}(G)  the number of fixed points of the G-set S

irr(G) the multiset of irreducible character degrees of G,
e.g., irr(G) = {…, m × t, …}, m characters of degree t
Φ(G) Frattini subgroup of G
CFSG Classification of Finite Simple Groups

Define the defect of a character χ ∈ Irr(G) to be the integer d such that |G|_{p} = p^{d}χ(1)_{p}. Write Irr_{d}(G) for the subset of Irr(G) of elements of defect d, and k_{d}(G) for their number. If χ lies in a p-block of defect d′ then the height of χ is d′ − d. This definition of k_{d}(G) is a modern convention - care should be taken when reading references that d does indeed refer to the defect rather than the height.

Denote the number of p-blocks of defect zero of G by f_{0}(G), and if H is a section of G and B is a block of G, then denote by f_{0}^{(B)}(H) the number of blocks of defect zero of H corresponding to B under the Brauer correspondence and the natural correspondence with a quotient group.

In general if we are considering an invariant involving blocks of a section H of G that correspond to B, then we write, for example, k(H, B). Further, if we are considering only characters covering a given irreducible character μ of a normal subgroup then write, for example, k(H, B, μ) (if T, H ≤ G, then we say that χ ∈ Irr(T) covers, or lies over μ ∈ Irr(H) if (χ|_{T ∩H}, μ|_{T ∩H}) ≠ 0).

Notation for specific finite groups is in general as given in [12].

2 Background

2.1 Choice of p-modular system

When working with blocks it is important to choose the ring and fields of definition carefully. We use a p-modular system, as defined for example in [14], but since we make use of the results of [35] (in particular those given in Proposition 3.13) we also take the residue field to be algebraically closed.

Let R be a complete discrete valuation ring with unique maximal ideal J(R) and field of fractions K of characteristic zero. Suppose that the residue field k = R/J(R) is algebraically closed of characteristic p, p a prime. We assume further that R contains all |G|^{3} − th roots of unity for whichever finite group G we are considering, i.e., K is a splitting field for G and any of its covers we will be considering (see [14, 15.16]). (K, R, k) form a p-modular system.

2.2 Radical p-chains and the p-local rank of a finite group

Following the reformulation of Alperin’s weight conjecture by Knörr and Robinson in [34], most work on the verification of the conjecture has centred around the study of chains of p-groups. Definitions of these p-chains vary a little from author to author, mostly in the matter of restrictions of the choice of initial subgroup. The stabilizers of these p-chains may be regarded as generalized p-local subgroups.

Definition A p-chain σ for G of length |σ| = n is a non-empty strictly increasing chain

Q_{0} < ⋯ < Q_{n}

of p-subgroups Q_{i} of G. We call the p-chain

σ_{i} : Q_{0} < ⋯ < Q_{i}

an initial subchain of σ, V^{σ} = Q_{n} the final subgroup of σ and V_{σ} = Q_{0} the initial subgroup of σ. Let C = C(G) denote the set of p-chains for G.

Suppose that A is a group acting on G. For α ∈ A define

σ^{α} : Q_{0}^{α} < ⋯ < Q_{n}^{α},

and define the chain stabilizer by

A_{σ} = N_{A}(σ) = N_{A}(Q_{0}) ∩ ⋯ ∩ N_{A}(Q_{n}).

It is clear that length of a p-chain is invariant under this action.

If X (G) ⊂ C(G) and Q ≤ G, then we write

X (G|Q) = {σ ∈ X (G) : V_{σ} = Q},

the set of those p-chains starting with Q. Write

X_{Q}(G) = {σ ∈ X (G) : V_{σ} ≥ Q}

for the set of those chains whose initial subgroup contains Q.

Denote by N = N (G) the set of p-chains whose stabilizer in G contains the final subgroup of the chain, i.e., all terms of the chain are normal in the final subgroup.

An important class of p-chains are the radical p-chains - a generalization of the radical p-subgroups, of which the p-subgroups under consideration in the statement of Alperin’s weight conjecture are examples:

Definition Let G be a finite group. We say that a p-subgroup Q of G is radical if Q = O_{p}(N_{G}(Q)). The following lemma lists some important properties of radical p-subgroups:

Lemma 2.1 (a) Each defect group of any p-block is radical, (b) O_{p}(G) and the Sylow p-subgroups of G are radical, (c) each radical p-subgroup of G contains O_{p}(G),

(d) each radical p-subgroup may be regarded as the intersection of a collection of Sylow p-subgroups of G. Proof For (a)-(c), see, for example [17]. For (d) see [34].

Definition A p-chain σ : Q_{0} < ⋯ < Q_{n} is radical if Q_{i} = O_{p}(N_{G}(σ_{i})) for each i. In other words, Q_{0} is a radical p-subgroup of G and each Q_{i+1} is a radical p-subgroup of N_{G}(σ_{i}).

We denote the set of radical p-chains for G by R = R(G), and note that when G is a finite group of Lie type (of characteristic p) the normalizers of the radical p-subgroups are nested so that the radical p-chains are just the chains of radical p-subgroups. In this case the radical p-subgroups are the unipotent radicals of the parabolic subgroups.

A treatment of some other families of p-chains may be found in Knörr and Robinson [34]. For example, one may consider the p-chains consisting of elementary abelian p-groups, radical p-groups or indeed intersections of Sylow p-subgroups. However, we shall not be using these here.

The next result is an exercise from Robinson [50], and shows that the three families of p-chains are fairly interchangeable: Lemma 2.2 Let G be a finite group and let f : C(G) → ℤ be a function stable under the conjugation action of G and dependent only on N_{G}(σ) and on the initial subgroup V_{σ}, for σ ∈ C(G). Let Q be a p-subgroup of G. Then

∑_{σ∈C(G|Q)/G}(−1)^{|σ|}f(σ) = ∑_{σ∈N (G|Q)/G}(−1)^{|σ|}f(σ) = ∑_{σ∈R(G|Q)/G}(−1)^{|σ|}f(σ).

Proof It follows from the definititions that R(G|Q) ⊂ N (G|Q). The first equality is demonstrated in [50]. The second is proved in [17], although the result stated is weaker than ours. We include a proof here.

Let σ ∈ N (G|Q) − R(G|Q), σ : Q_{0} < ⋯ < Q_{n}. Then Q_{i} ≠ O_{p}(N_{G}(σ_{i})) for some i. Choose i maximal with this property, and observe that Q_{i} < O_{p}(N_{G}(σ_{i})). Observe that if i < n, then Q_{i+1} = O_{p}(N_{G}(σ_{i+1})) = O_{p}(N_{N_{G}(σ_{i})}(Q_{i+1})) ≥ O_{p}(N_{G}(σ_{i})) by maximality and by Lemma 2.1. There are two cases to consider:

(i) Suppose that i < n and Q_{i+1} = O_{p}(N_{G}(σ_{i})). Then form the chain

\overline{σ} : Q_{0} < ⋯ < Q_{i} < Q_{i+2} < ⋯ < Q_{n}

from σ by deleting the term Q_{i+1}. Clearly \overline{σ} ∈ N (G|Q) − R(G|Q). We have N_{G}(σ) = N_{G}(\overline{σ}) ∩ N_{G}(Q_{i+1}) and N_{G}(\overline{σ}) ≤ N_{G}(Q_{i+1}), so N_{G}(\overline{σ}) = N_{G}(σ).

(ii) Suppose that i = n or Q_{i+1} > O_{p}(N_{G}(σ_{i})). Then form the chain

\overline{σ} : Q_{0} < ⋯ < Q_{i} < O_{p}(N_{G}(σ_{i})) < Q_{i+1} < ⋯ < Q_{n}

from σ by inserting the term O_{p}(N_{G}(σ_{i})) between Q_{i} and Q_{i+1} if i < n or at the end if i = n. Again \overline{σ} ∈ N (G|Q) − R(G|Q). We have N_{G}(\overline{σ}) = N_{G}(σ) ∩ N_{G}(O_{p}(N_{G}(σ_{i}))) and N_{G}(σ) ≤ N_{G}(O_{p}(N_{G}(σ_{i}))), so N_{G}(\overline{σ}) = N_{G}(σ).

In each case V_{σ} = V\overline{σ} and \overline{(σ^{g})} = (\overline{σ})^{g} for g ∈ G, and \overline{\overline{σ}} = σ, so the contributions from orbits of chains containing σ and \overline{σ} cancel as required, since (−1)^{|\overline{σ}|} = (−1)^{|σ|+1}.

The analogous result holds for the other families of p-chains in [34], except for the family of chains of elementary abelian p-subgroups (see [50]). Note that the result does hold for the chains of intersections of Sylow p-subgroups.

We include here a result which will be crucial in defining the variants of Alperin’s conjecture which involve alternating sums of the type found in Lemma 2.2 Lemma 2.3 (Lemma 3.2 of [34]) Let σ ∈ C(G) and suppose that b is a p-block of G_{σ}. Then b has a Brauer correspondent in G. Definition The p-local rank of a finite group G, plr(G) is the length of the longest radical p-chain of G. Robinson proves some strong results concerning this invariant (see [47]).

A trivial observation however is that plr(G) = 0 if and only if G has a normal Sylow p-subgroup. An important and non-trivial property of the p-local rank is that it respects the partial ordering of subgroups:

Lemma 2.4 (Lemma 7.2 of [47]) Let H ≤ G. Then plr(H) ≤ plr(G).

2.3 Properties of finite groups of p-local rank one

Definition A subset X of a finite group G is said to be a trivial intersection (TI) set if X^{g} ∩ X ⊂ 1 for each g ∈ G with g ∉ N_{G}(X). If P ∈ Syl_{p}(G) is a TI set and P ≠ O_{p}(G), then we say that G has TI Sylow p-subgroups. If O_{p}(G) ≤ Z(G) is cyclic and G/O_{p}(G) has TI Sylow p-subgroups, then we say that G has almost TI Sylow p-subgroups.

Lemma 2.5 (Lemma 7.1 of [47]) Let G be a finite group with plr(G) > 0. Then plr(G) = 1 if and only if G/O_{p}(G) has TI Sylow p-subgroups.

We review a few of the properties of finite groups with TI Sylow p-subgroups.

Lemma 2.6 Suppose that P ∈ Syl_{p}(G) is a TI set and 1 ≠ Q ≤ P. Then N_{G}(Q) ≤ N_{G}(P). Further if g ∈ G and Q^{g} ≤ P, then g ∈ N_{G}(P).

Lemma 2.7 Let P, G be as in the previous lemma and suppose that M ◁ G has order divisible by p. Then Q = P ∩ M ∈ Syl_{p}(M) and MN_{G}(P) = MN_{G}(Q) = G.

Proof This follows by using the Frattini argument and the previous lemma.

Lemma 2.8 Suppose that P ∈ Syl_{p}(G) is a TI set of p-rank strictly greater than one. Then O_{p′}(G) ≤ C_{G}(P).

Proof Consider E_{2} ≤ P elementary abelian of rank 2. Since this is a noncyclic abelian p-group of automorphisms of O_{p′}(G), by Gorenstein [24, 6.2.4] we have

O_{p′}(G) = ⟨C_{O_{p′}(G)}(α) | α ∈ E_{2} − {1}⟩ ≤ N_{G}(P).

Hence O_{p′}(G) ≤ N_{G}(P), and C_{P} (O_{p′}(G)) = P.

Finally we prove a special case of a result of Robinson [48].

Lemma 2.9 Suppose that O_{p}(G) ≤ Z(G) and that G/O_{p}(G) has TI Sylow p-subgroups. Let B be a block of G with defect group P ∈ Syl_{p}(G). Then

k(G, B) − k(N_{G}(P), B) = |O_{p}(G)| (l(G, B) − l(N_{G}(P), B)) .

Now let \overline{B} be the unique p-block of \overline{G} = G/O_{p}(G) corresponding to B, and let λ ∈ Irr(O_{p}(G)). Suppose further that O_{p}(G) is cyclic. Then

k(G, B, λ) − k(N_{G}(P), B, λ) = k(\overline{G}, \overline{B}) − k(N\overline{G}(\overline{P}), \overline{B}).

Proof \overline{G} possesses an unique p-block \overline{B} corresponding to B by [22, V.4.5]. (6D) of Brauer [8] gives us

k(G, B) = ∑_{u∈G_{p}/G} l(C_{G}(u), B).

Choose P ∈ Syl_{p}(G). By Lemma 2.6 we may choose G_{p}/G so that C_{G}(u) ≤ N_{G}(P) for each u ∈ G_{p}/G − O_{p}(G). Hence we may take G_{p}/G − O_{p}(G) = N_{G}(P)_{p}/N_{G}(P) − O_{p}(G). For each u ∈ O_{p}(G) we have C_{G}(u) = G, and so

k(G, B) − k(N_{G}(P), B) = ∑_{u∈G_{p}/G} l(C_{G}(u), B) − ∑_{u∈N_{G}(P)_{p}/N_{G}(P)} l(C_{N_{G}(P)}(u), B)

= |O_{p}(G)| (l(G, B) − l(N_{G}(P), B)) .

Noting that l(G, B) − l(N_{G}(P), B) = l(\overline{G}, \overline{B}) − l(N\overline{G}(\overline{P}), \overline{B}), we see that

k(G, B) − k(N_{G}(P), B) = |O_{p}(G)| (k(\overline{G}, \overline{B}) − k(N\overline{G}(\overline{P}), \overline{B})) .

To count the irreducible characters lying over λ we use an inductive argument on |O_{p}(G)|.

Suppose first that |O_{p}(G)| = p. Then k(G, B, 1) − k(N_{G}(P), B, 1) = k(\overline{G}, \overline{B}) − k(N\overline{G}(\overline{P}), \overline{B}) trivially and k(G, B, λ) − k(N_{G}(P), B, λ) takes identical values on each of the remaining λ ∈ Irr(O_{p}(G)), so the result holds in this case.

Suppose that the result holds for |O_{p}(G)| = p, p^{2}, …, p^{n−1} and suppose that |O_{p}(G)| = p^{n}. Then k(G, B, λ) − k(N_{G}(P), B, λ) takes the same value for each λ of order p^{n}, and the result follows by the inductive hypothesis.

2.4 The Schur Multiplier

In applying the results of the next chapter we will make extensive use of the Schur multiplier of a group. Although this arises from the study of projective representations (as will be explained in the next chapter), it can also be explained purely in terms of group theory and cohomology/homology. We give a definition of the Schur multiplier and some results which will be of use later on. We direct the reader to [14], [29] or [32] for further detail, although this is by no means a definitive list.

Definition We define the Schur multiplier M(G) of a group G to be H^{2}(G, ℂ^{×}), the second cohomology group of G over ℂ^{×}. Equivalently, M(G) is maximal such that there is a central extension H of G with A ≤ Z(H), H/A ≅ G, A ≤ [H, H] and A ≅ M(G). It is important to note that H^{2}(G, K^{×}) ≅ H^{2}(G, ℂ^{×}) (see, for example the proof of [35, 2.1]).

The following result is well known, and highlights the local nature of the Schur multiplier.

Lemma 2.10 Let G be a finite group and let P ∈ Syl_{p}(G). Then a Sylow p-subgroup of M(G) is isomorphic to a subgroup of M(P).

Proof This is [28, V.25.1]. The method of proof is similar to that used in the next result.

The following is a special case of [25, 4.229].

Lemma 2.11 Let G be a finite group and let N ≤ G with p |̸[G : N] = n. Suppose also that p |̸|M(N)|. Then p |̸|M(G)|.

Proof Suppose that H is a central extension of G with H/A ≅ A and A ≤ Z(H)∩H′. Write L for the preimage of N in H, and let V : H → L/L′ be the transfer map. Let z ∈ O_{p}(A). Then V (z) = z^{n}L′ (see for example [28, IV.2.1]). But H′ ≤ ker(V), so V (z) = L′. Since (p, n) = 1, there is an integer x such that z = z^{nx} ∈ O_{p}(A) ∩ L′ ≤ O_{p}(Z(L))∩L′. So O_{p}(A) is isomorphic to a subgroup of O_{p}(M(N)) = 1, and the result follows.

The next result allows us (after some more work) to avoid the difficulties posed by the presence of an outer automorphism group which is a direct product of two cyclic groups.

Lemma 2.12 ( [6]) Let G be a group and let N ◁ G, where N is perfect and G/N is cyclic. Then M(G) is a homomorphic image of M(N).

Proof This follows immediately from Theorem 4.2 of [6], observing that H^{2}(G, ℂ^{×}) ≅ H_{2}(G, ℤ) (see [58, p.199]).

The next result, although not strictly necessary for this thesis (less sophisticated methods would suffice), is worth including given our concentration on finite groups with TI Sylow p-subgroups.

Lemma 2.13 (9.6.3 of [32]) Let G be a finite group such that P ∈ Syl_{p}(G) is a TI set. Then O_{p}(M(G)) ≅ O_{p}(M(N_{G}(P))).

It is convenient to include here some results concerning the Schur multipliers of various p-groups we will encounter.

Lemma 2.14 If P is a finite p-group which is either (a) cyclic, (b) generalized quaternion, or (c) extraspecial of order p^{3} and exponent p^{2}, then M(P) = 1. If P is elementary abelian then M(P) is elementary abelian.

Proof If P is of the form (a) or (b), then this is [28, V.25.3]. If P is of the form (c), then by [32, 2.4.9]

P = ⟨x, y | x^{p^{2}} = y^{p}, y^{−1}xy = x^{p+1}⟩,

and the result follows by [28, V.25.2] since P may be defined using two generators and two relations.

The last part is exercise 11.16 of [29].

3 Clifford-theoretic reductions

3.1 Introduction

In this chapter we detail block-friendly character correspondences, which we will later apply to demonstrate that a counterexample to Conjecture 4.4 with [G : Z(G)] minimal has only cyclic, central p-subgroups or p′-subgroups.

Paul Fong in the second part of his famous paper [23] introduced two character correspondences, obtained by observing the action of a finite group G on the irreducible characters of a normal p′-subgroup and applying Schur’s theory of projective representations. Dubbed the Fong correspondences, the first shows that to some extent we may replace G by the inertial subgroup I_{G}(μ) of an irreducible character μ of N. The second is considerably more involved and shows that we can further replace N by a central, cyclic p′-subgroup. Refinements and modifications of Fong’s work were studied by Reynolds [46], Cliff [11] and Okuyama and Tsushima [39].

Whereas Fong used elementary character theoretic methods to verify that the correspondences may be refined to correspondences between p-blocks of characters, the subsequent results in [39] use the theory of twisted group algebras to establish that the correspondences are compatible with the Brauer correspondence. Dade in [18] uses similar methods to establish similar correspondences with no restriction on N, but in the (equivalent) setting of blocks of twisted group algebras (of which he gives a comprehensive account).

We work with non-twisted group algebras and use elementary methods to demonstrate correspondences similar to Fong’s second correspondence, where we have no restriction on the order of N but a restriction on the degree of μ, and check that the correspondence may be refined to p-blocks and is compatible with the Brauer correspondence (where it exists). Analogies to Fong’s first correspondence for p-singular normal subgroups are easy to obtain (blockwise refinements following directly from Lemma 3.1) and are constructed where needed.

The following lemma will be the key to the Brauer correspondence. Let G be a finite group.

Lemma 3.1 Let H ≤ G and b a p-block of H. Let ζ ∈ Irr(H, b) and C be a conjugacy class sum of G. If π_{H} : RG → RH is the projection map of G onto H, then

ω_{ζ} ∘ π_{H}(C) = \frac{ζ^{G}(C)}{ζ^{G}(1)}.

Proof This is [37, 5.3.1].

To use this we need the following observation:

Lemma 3.2 Let N ◁ G and write \overline{G} = G/N, indicating objects associated to \overline{G} with a ’ ¯ ’. Let H ≤ G with N ◁ H. Suppose that θ ∈ Irr(G), and that ζ ∈ Irr(H) and χ ∈ Irr(\overline{H}) are characters such that for each x ∈ \overline{x} ∈ \overline{H} we have ζ(x) = χ(\overline{x})θ(x). Then ζ^{G}(x) = χ\overline{G}(\overline{x})θ(x) for each x ∈ G.

Proof

ζ^{G}(x) = \frac{1}{|H|} ∑_{g∈G,x^{g}∈H} ζ(x^{g}) = \frac{1}{|H|} ∑_{g∈G,x^{g}∈H} χ(\overline{x}\overline{g})θ(x^{g}).

But θ(x^{g}) = θ(x), and x^{g} ∈ H ⇔ \overline{x}\overline{g} ∈ \overline{H}, so

ζ^{G}(x) = \frac{θ(x)}{|H|} ∑_{g∈G,\overline{x}\overline{g}∈\overline{H}} χ(\overline{x}\overline{g}) = \frac{|N|θ(x)}{|H|} ∑_{\overline{g}∈\overline{G},\overline{x}\overline{g}∈\overline{H}} χ(\overline{x}\overline{g}) = χ\overline{G}(\overline{x})θ(x).

3.2 Character correspondences

Let G be a finite group and N ◁ G. Suppose that μ ∈ Irr(N) is G-stable. Denote by π the set of primes dividing |N|, and write π′ for its complement in the set of primes. We construct a finite group Ĝ with cyclic π-group M̂ ≤ Z(Ĝ) such that Ĝ/M̂ ≅ G/N and there exists a 1-1 correspondence between irreducible characters of G lying over μ and irreducible characters of Ĝ lying over a given μ̂ ∈ Irr(M̂). This character correspondence is well known, but we include details here in order to establish the notation necessary to check its compatibility with block theory.

We establish the character correspondences by way of a third group G̃ of which both G and Ĝ are factor groups.

Fix a representation τ affording μ.

Lemma 3.3 We may choose a projective representation

ρ : G → GL_{μ(1)}(R)

extending τ such that (i) ρ(n)ρ(g) = ρ(ng) and ρ(g)ρ(n) = ρ(gn) ∀ g ∈ G, n ∈ N, (ii) det(ρ(g)) is a π-power root of unity for each g ∈ G and (iii) ρ has factor set ϵ : G × G → R^{×} of order s dividing |G|_{p}μ(1) in H^{2}(G, K^{×}).

Proof This is [35, 2.1] (that there is a complex projective representation extending τ is [29, 11.2]).

Let α ∈ R be a primitive s-th root of unity, and define the group

G̃ = {(α^{i}, g)|0 ≤ i < s, g ∈ G},

with group operation given by

(α^{i_{1}}, g_{1})(α^{i_{2}}, g_{2}) = (ϵ(g_{1}, g_{2})α^{i_{1}+i_{2}}, g_{1}g_{2}).

Write M̃ = {(α^{i}, 1) : 0 ≤ i < s} ≤ Z(G̃), so that we have a short exact sequence

1 → M̃ → G̃ → G → 1.

Let Ñ = \{(α^{i}, n) : 0 ≤ i < s, n ∈ N\} ◁ G̃, and \overline{N} = \{(1, n) : n ∈ N\} ◁ G̃, so that \overline{N} ≅ N.

Lemma 3.4 ρ̃: G̃ → GL_{μ(1)}(R) defined by ρ̃(α^{i}, g) = α^{i}ρ(g) and τ̃ : Ñ → GL_{μ(1)}(R) defined by τ̃(α^{i}, n) = α^{i}τ(n) are irreducible representations, and τ̃ = ρ̃|Ñ, τ̃|\overline{N} = τ.

Proof That ρ̃ and τ̃ are representations is clear from the definition of a projective representation. It is also clear that τ̃ = ρ̃|Ñ, τ̃|\overline{N} = τ. Now suppose that ρ̃ = ρ̃_{1} + ρ̃_{2}, then τ = ρ̃|\overline{N} = ρ̃_{1}|\overline{N} + ρ̃_{2}|\overline{N}, so that ρ̃_{1} = 0 or ρ̃_{2} = 0. Similarly τ̃ is irreducible.

Denote all objects associated to G̃ with a ‘˜ ’ symbol.

ϵ(g, n) = ϵ(n, g) = 1 for each g ∈ G, n ∈ N, so we may regard ϵ as a factor set for G/N, i.e., ϵ ∈ H^{2}(G/N, K^{×}). Define

Ĝ = \{(α^{i}, \overline{g}) : 0 ≤ i < s, \overline{g} ∈ G/N\} ≅ G̃/\overline{N},

with group operation given by

(α^{i_{1}}, \overline{g}_{1})(α^{i_{2}}, \overline{g}_{2}) = (ϵ(\overline{g}_{1}, \overline{g}_{2})α^{i_{1}+i_{2}}, \overline{g}_{1}\overline{g}_{2}).

Setting M̂ = \{(α^{i}, \overline{1}) : 0 ≤ i < s\} ≤ Z(Ĝ), we have a short exact sequence

1 → M̂ → Ĝ → G/N → 1.

Let μ̂ : M̂ → R be the linear representation given by μ̂(α^{i}, \overline{1}) = α^{−i}. Denote all elements, characters and subgroups of Ĝ with a ‘ˆ ’ symbol.

Remark 3.5 M̂ ≤ [Ĝ, Ĝ] by [29, 11.19], so M̂ is isomorphic to a subgroup of the Schur multiplier of G/N.

Write θ̃ for the irreducible character of G̃ afforded by ρ̃, so that θ̃|\overline{N} = μ. Let μ̃ be the irreducible character of Ñ defined by μ̃(α^{i}, n) = μ(n). As a consequence of Clifford’s theorem, the irreducible characters of G̃ covering μ̃ are precisely the characters χ̃ = θ̃χ̂ where χ̂ ∈ Irr(Ĝ, μ̂). In particular there is a canonical bijection

Irr(G̃, μ̃) ↔ Irr(Ĝ, μ̂),

since for each (α^{i}, n) ∈ Ñ,

χ̃(α^{i}, n) = θ̃(α^{i}, n)χ̂(α^{i}, \overline{1}) = α^{i}μ(n)α^{−i}χ̂(1, \overline{1}) = μ(n)χ̂(1, \overline{1}).

Summarizing, we have:

Lemma 3.6 There is a 1-1 correspondence Irr(G, μ) ↔ Irr(G̃, μ̃) given by χ̃ = 1G̃χ, where χ̃ ∈ Irr(G̃, μ̃) and χ ∈ Irr(G, μ). Note that this correspondence is the natural one obtained by regarding χ as a character of G̃.

There is a 1-1 correspondence Irr(G̃, μ̃) ↔ Irr(Ĝ, μ̂) given by χ̃ = θ̃χ̂, where χ̃ ∈ Irr(G̃, μ̃) and χ̂ ∈ Irr(Ĝ, μ̂).

We examine how the character correspondences of Lemma 3.6 fit with the p-block correspondences between G̃, G and Ĝ.

If g ∈ G, then denote the conjugacy class sum of the class of G containing g by s(g), with similar notation s(g̃) and s(ĝ) applied to g̃ ∈ G̃ and ĝ ∈ Ĝ. We denote the coset of \overline{N} in G̃ containing g̃ by ĝ, and make the appropriate identification with the corresponding element of Ĝ. We use a similar convention for the other factor groups studied here.

Recall that two irreducible characters belong to the same p-block if and only if their central characters agree on the p-block idempotents for the group, and that this occurs precisely when they agree modulo J(R) on the p-regular conjugacy classes (noting that p-block idempotents have p-regular support). Since all the results of the remainder of this section are concerned with values of central characters, we may assume throughout that g̃ ∈ G̃_{p′}.

Let ω_{χ} : Z(RG) → R be the central character corresponding to χ, so that ω_{χ}(s(g)) = \frac{[G:C_{G}(g)]χ(g)}{χ(1)}. Define ωχ̃, ωχ̂ similarly.

Suppose that N = U is a normal p-subgroup of G. Then we may assume that g̃ ∈ CG̃(Ũ)_{p′}:

Lemma 3.7 Suppose that g̃ ∈ G̃ and that g̃ ∉ CG̃(Ũ). Then

ωχ̃(s(g̃)) ≡ 0 mod J(R)

for all χ̃ ∈ Irr(G̃).

Proof By Isaacs [29, 15.38] s(g̃) is nilpotent and so lies in J(Z(kG̃)), which is contained in the kernel of every central character of kG̃.

The first correspondence of the lemma behaves well with respect to blocks:

Lemma 3.8 Let χ_{i} correspond to χ̃_{i}, i = 1, 2, via the natural correspondence. Then χ_{1}, χ_{2} both lie in the same p-block of G if and only if χ̃_{1}, χ̃_{2} both lie in the same p-block of G̃.

Proof M̃ = O_{p}(M̃) × O_{p′}(M̃). By [22, V.4.3] there is a 1-1 correspondence between p-blocks of G̃ covering the trivial character of O_{p′}(M̃) (here all irreducible characters belonging to such a block cover the trivial character) and the p-blocks of G̃/O_{p′}(M̃). By [22, V.4.5] there is a 1-1 correspondence between the p-blocks of G̃/O_{p′}(M̃) and the p-blocks of G̃/M̃ = G. It suffices to show that if two irreducible characters χ_{1}, χ_{2} ∈ Irr(G, μ) lie in the same p-block of G, then χ̃_{1} and χ̃_{2} lie in the same p-block of G̃. Let g̃ ∈ G̃_{p′}.

ωχ̃_{i}(s(g̃)) = [G̃ : CG̃(g̃)]\frac{χ̃_{i}(g̃)}{χ̃_{i}(˜1)} = \frac{[G̃ : CG̃(g̃)]}{[G : C_{G}(g)]}[G : C_{G}(g)]\frac{χ_{i}(g)}{χ_{i}(1)} = \frac{[G̃ : CG̃(g̃)]}{[G : C_{G}(g)]}ω_{χ_{i}}(s(g)).

But [G : C_{G}(g)]|[G̃ : CG̃(g̃)], so if χ_{1}, χ lie in the same p-block of G, i.e., ω_{χ_{i}}(s(g)) − ω_{χ_{i}}(s(g)) ∈ J(R), then ωχ̃_{i}(s(g̃)) − ωχ̃_{i}(s(g̃)) ∈ J(R) and we are done.

We now examine how the second character correspondence of Lemma 3.6 fits with the p-block correspondence between G̃ and Ĝ.

Lemma 3.9 Let χ̃_{i} ∈ Irr(G̃, μ̃) correspond to χ̂_{i} ∈ Irr(Ĝ, μ̂) under the correspondence given in Lemma 3.6, i = 1, 2. Suppose that either N is a p-group or p |̸μ(1). If χ̂_{1}, χ̂_{2} lie in the same p-block of Ĝ then χ̃_{1}, χ̃_{2} lie in the same p-block of G̃.

Proof Let g̃ ∈ G̃.

ωχ̃_{i}(s(g̃)) = [G̃ : CG̃(g̃)]\frac{χ̃_{i}(g̃)}{χ̃_{i}(˜1)} = \frac{[G̃ : CG̃(g̃)]}{[Ĝ : CĜ(ĝ)]}[Ĝ : CĜ(ĝ)]\frac{χ̂_{i}(ĝ)θ̃(g̃)}{χ̂_{i}(ˆ1)θ̃(˜1)} = kg̃ωχ̂_{i}(s(ĝ)),

where

kg̃ := \frac{[G̃ : CG̃(g̃)]θ̃(g̃)}{[Ĝ : CĜ(ĝ)]θ̃(˜1)}.

Suppose that χ̂_{1}, χ̂_{2} ∈ Irr(Ĝ, μ̂) lie in the same p-block of Ĝ. Then ωχ̂_{1}(s(ĝ)) − ωχ̂_{2}(s(ĝ)) ∈ J(R) for each ĝ ∈ Ĝ_{p′}. So if kg̃ ∈ R for each g̃ ∈ G̃_{p′}, then χ̃_{1}, χ̃_{2} must lie in the same p-block of G̃. We claim that this is the case.

Since [Ĝ : CĜ(ĝ)] | [G̃ : CG̃(g̃)] and θ̃(g̃) ∈ R for each g̃ ∈ G̃, this does indeed occur when (p, θ̃(˜1)) = 1. Hence we may assume that N = U is a p-group. By Lemma 3.7 we may assume that g̃ ∈ CG̃(Ũ)_{p′}.

Now ρ̃|Ũ is irreducible and for each g̃ ∈ CG̃(Ũ)_{p′}, ρ̃(g̃) commutes with all ρ̃(ũ), ũ ∈ Ũ, so by Schur’s lemma ρ̃(g̃) must be a scalar matrix, and θ̃(g̃) = κθ̃(˜1), where κ is a p′ root of unity. But we have chosen ρ̃ so that (det(θ̃))(g̃) is a p-power root of unity, so κ = 1, and θ̃(g̃) = θ̃(˜1) for each g̃ ∈ CG̃(Ũ)_{p′}.

Fix g̃ ∈ CG̃(Ũ)_{p′}. We have

[G̃ : CG̃(g̃)] = [Ĝ : CĜ(ĝ)]

since \overline{U} ◁ CG̃(g̃) and |\overline{U}| is coprime to the order of g̃ (see for example Dornhoff [21, 64.1]). So kg̃ = 1 and if χ̂_{1}, χ̂_{2} lie in the same p-block of G̃ then χ̃_{1}, χ̃_{2} lie in the same p-block of G̃.

Proposition 3.10 Suppose that one of the following occurs:

(i) p |̸μ(1),

(ii) N is a p-group.

Then for each p-block B of G, we have, in the notation of Lemma 3.6, a 1-1 correspondence

Irr(G, B, μ) ↔ ⋃_{i=1}^{r} Irr(Ĝ, B̂_{i}, μ̂),

where B̂_{1}, …, B̂_{r} are p-blocks of Ĝ.

Suppose that N ≤ H ≤ G and that b is a p-block of H. Then we may apply the same methods to the irreducible characters ζ of H lying over μ. Suppose that ζ ∈ Irr(H, b, μ). Let b̂ be the p-block of Ĥ containing ζ̂. Suppose that the Brauer correspondents of b and b̂ in G and Ĝ are defined, and that b̂Ĝ is one of the B̂_{i} defined above. Then b^{G} = B.

Proof It remains to verify that the correspondence is compatible with the Brauer correspondence.

Let π_{H} denote the projection G → H, where H is a subgroup of G containing N. If b is a block of H, then assume further that ω_{b} ∘ π_{H} is a central character of B. This does indeed occur when H is a p-chain stabilizer, by Lemma 2.3.

Note that ρ̃ restricted to H̃ is irreducible, so we may use θ̃|H̃ to repeat the above process to obtain ζ̃ ∈ Irr(H̃, μ̃) and ζ̂ ∈ Irr(Ĥ, μ̂) corresponding to a given ζ ∈ Irr(H, μ), where H̃ and Ĥ have the obvious meanings. Denote by πH̃ and πĤ the respective projections.

Suppose first that Bl(ζ)^{G} = Bl(χ) = B, with χ, χ̃ and χ̂ defined as before. Hence for each g̃ ∈ G̃_{p′} we have ω_{χ}(s(g)) − (ω_{ζ} ∘ π_{H})(s(g)) ∈ J(R). By Lemma 3.1, this is ω_{χ}(s(g)) − \frac{ζ^{G}(s(g))}{ζ^{G}(1)}. So by Lemma 3.1 and Lemma 3.2

\begin{aligned}(ωζ̃ ∘ πH̃)(s(g̃)) & = & \frac{ζ̃G̃(s(g̃))}{ζ̃G̃(˜1)} & = & [G̃ : CG̃(g̃)]\frac{ζ^{G}(g)}{ζ^{G}(1)} & = & \frac{[G̃:CG̃(g̃)]}{[G:C_{G}(g)]}[G : C_{G}(g)]\frac{ζ^{G}(g)}{ζ^{G}(1)} \\ & & & & & ≡ & \frac{[G̃:CG̃(g̃)]}{[G:C_{G}(g)]}ω_{χ}(s(g)) mod J(R) \end{aligned}

But

\frac{[G̃ : CG̃(g̃)]}{[G : C_{G}(g)]}ω_{χ}(s(g)) = ωχ̃(s(g̃))

by the proof of Lemma 3.8. Hence the first correspondence of Lemma 3.6 is compatible with the Brauer correspondence.

Now suppose that Bl(ζ̂)Ĝ = Bl(χ̂) = B̂. Then for each g̃ ∈ G̃_{p′} we have ωχ̂(s(ĝ)) − \frac{ζ̂Ĝ(s(ĝ))}{ζ̂Ĝ(ˆ1)} ∈ J(R).

So

(ωζ̃ ∘ πH̃)(s(g̃)) = \frac{ζ̃G̃(s(g̃))}{ζ̃G̃(˜1)} = kg̃ \frac{ζ̂Ĝ(s(ĝ))}{ζ̂Ĝ(ˆ1)} ≡ kg̃ωχ̂(s(ĝ)) ≡ ωχ̃(s(g̃)) mod J(R)

by Lemma 3.1, Lemma 3.2 and the proof of Lemma 3.9, where kg̃ ∈ R is defined as before. Hence the correspondence is compatible with the Brauer correspondence.

3.3 Characters afforded by relatively projective modules

Now that we have set up the appropriate notation we outline some results of Külshammer and Robinson [35], and of Robinson [47], who use a similar set-up to determine when an irreducible character of G is afforded by an N-projective RG-module, where N ◁ G. We focus on the situation where N is a p-group, although the results of [35] are considerably more general.

The results of the remainder of this chapter are entirely expository.

Definition Let H ≤ G. We say that χ ∈ Irr(G) is H-projective with respect to R if there is some H-projective RG-module M affording χ. If R is given then we just say that χ is H-projective. We say that a p-subgroup of G is a vertex for χ if it is a vertex for some RG-module affording χ.

Write w(G, H) for the number of H-projective irreducible characters of G.

Of course we may assume that H is a p-group since R/J(R) has characteristic p. Strong results in this direction have been achieved by Knörr in [33], where he proves

the following. Note that he does not assume that the subgroup is normal:

Proposition 3.11 Let χ ∈ Irr(G) lie in a p-block B of G with defect group D. If V is a vertex of χ (which we may suppose to be contained in D), then

C_{D}(V) ≤ V.

Proof This is [33, 3.7].

For the remainder of this chapter suppose that N ◁ G is a p-group and that μ ∈ Irr(N). We examine when χ ∈ Irr(G, μ) is N-projective. It is clear from the definitions that there is a bijection

Irr(G, μ) ↔ Irr(I_{G}(μ), μ)

which respects N-projectivity. This bijection also respects p-blocks and the Brauer correspondence by Lemma 3.1, so we may assume that G = I_{G}(μ).

Lemma 3.12 ( [35]) Apply all the constructions from earlier in the chapter. Then χ is N-projective if and only if χ̂ is M̂-projective.

Sketch proof Suppose that χ is afforded by the N-projective RG-module T and suppose that θ̃ is afforded by the RG̃-module X. Regard T as an Ñ-projective RG̃- module. Then χ̂ is afforded by (T ⊗ X^{⋆})μ̂, the sum of those components of T ⊗ X^{⋆} affording characters covering μ̂. (T ⊗ X^{⋆})μ̂ is an Ñ-projective RG̃-module, and is M̂-projective when regarded as an RĜ-module (identifying M̂ with M̃). The reverse direction is clear.

Proposition 3.13 ( [35]) The N-projective irreducible characters covering μ are in 1-1 correspondence with the p-blocks of defect zero of I_{G}(μ)/N. Hence the number of N-projective irreducible characters lying in a given p-block B of G is

∑_{μ∈Irr(N)/G} f_{0}^{(B)} (\frac{I_{G}(μ)}{N}) .

Sketch proof For the first part fix μ ∈ Irr(N). By Proposition 3.10 and the remarks preceding Lemma 3.12, we may assume that G = I_{G}(μ) and that N is central in G and cyclic. Let χ ∈ Irr(G, μ) be N-projective. If g ∈ G, then write g_{p} for the p-part of g.

Define

μ^{−1} ⋆ χ = \{ \begin{matrix}μ^{−1}(g_{p})χ(g) & if g_{p} ∈ N \\ 0 & otherwise \end{matrix}

Then by [35, 2.3], μ^{−1} ⋆ χ is N-projective. μ^{−1} ⋆ χ ∈ Irr(G, 1_{N}) and each Nprojective element of Irr(G, 1_{N}) may be obtained from an unique χ ∈ Irr(G, μ) in this way.

μ^{−1} ⋆ χ may be regarded as a character for G/N, afforded by a projective R(G/N)- module as required.

Notice that when g ∈ G is p-regular we have μ^{−1} ⋆ χ(g) = χ(g), so that μ^{−1} ⋆ χ and χ lie in the same p-block of G. Hence the correspondence established above respects p-blocks, and the last part of the proposition holds.

Dade in [18] is concerned with χ ∈ Irr(G, μ) satisfying χ(1)_{p} = [G : N]_{p}μ(1)_{p}, and calls them weights with respect to N. Robinson in [47] demonstrates that (within our choice of p-modular system) this is just the same as saying that χ is N-projective:

Proposition 3.14 ( [47]) χ ∈ Irr(G, μ) is N-projective if and only if

χ(1)_{p} = [G : N]_{p}μ(1).

Sketch proof As usual we may assume that μ is stable under the action of G. By Lemma 3.12 χ is N-projective if and only if χ̂ is M̂-projective. Observe that χ(1)_{p} = [G : N]_{p}μ(1)_{p} if and only if χ̂(ˆ1)_{p} = [Ĝ : M̂]_{p}μ̂(ˆ1)_{p} (= [Ĝ : M̂]_{p}). Hence we may assume WLOG that N is central in G and cyclic.

Suppose that χ is N-projective. Then by the proof of Proposition 3.13 χ(1) = μ^{−1} ⋆ χ(1) and μ^{−1} ⋆ χ may be regarded as lying in a p-block of defect zero of G/N, so χ(1)_{p} = [G : N]_{p}, and χ(1)_{p} = [G : N]_{p}μ(1)_{p} as required.

Now suppose that χ(1)_{p} = [G : N]_{p}μ(1)_{p}.

Define χ^{⋆} by

χ^{⋆}(g) = \{ \begin{matrix}|N|χ(g) & if g ∈ G_{p′} \\ 0 & otherwise \end{matrix}

Now for each g ∈ G,

∑_{λ∈Irr(N)}(λ ⋆ χ)(g) = \{ \begin{matrix}(∑_{λ∈Irr(N)} λ(g_{p})) χ(g) & if g_{p} ∈ N \\ 0 & otherwise \end{matrix} .

But

∑_{λ∈Irr(N)} λ(g_{p}) = \{ \begin{matrix}|N| & if g_{p} = 1 \\ 0 & otherwise \end{matrix}

since N is abelian. So χ^{⋆} = ∑_{λ∈Irr(N)}(λ ⋆ χ).

Recall that each λ ⋆ χ is an irreducible character, and that μ^{−1} ⋆ χ and χ lie in the same p-block of G. But μ^{−1} ⋆ χ has kernel containing N and defect log_{p}(|N|), so may be regarded as a p-block of defect zero for G/N. Hence μ^{−1} ⋆ χ, and so χ, lies in a block of G with defect group N. The result follows.

A trivial but useful observation following from the results of this chapter will be needed when dealing with chains of p-subgroups:

Corollary 3.15 Let G be a finite group and N ◁ G be a p-subgroup of G. Let μ ∈ Irr(N). If w(G, μ, N) ≠ 0, then N = O_{p}(I_{G}(μ)).

4 Conjectures

4.1 Introduction

We outline the various conjectures of local representation theory to which we will refer, highlight the relations between them and give a brief review of the progress made at the time of writing.

Alperin’s weight conjecture, announced in the mid eighties, makes sweeping predictions based on the theory of finite groups of Lie type. Almost immediately it was reformulated by Knörr and Robinson based on topological methods. Part of this reformulation put the conjecture in a form compatible with character theory, and Dade used this and the Alperin-McKay conjecture to produce his own refined form of the conjecture. Dade has produced a series of progressively stronger conjectures, with the intention of giving a reduction to simple groups for the last. We are interested in a separate reformulation of Robinson which uses the Knörr-Robinson reformulation in conjunction with his work with Külshammer outlined in the previous chapter. Included in this chapter is a simple demonstration of the equivalence between Robinson’s and Dade’s projective conjecture organized with use of the p-local rank.

4.2 Alperin’s weight conjecture

Alperin’s ‘weight conjecture’ first appeared in his article [2], the proceedings of a symposium held on finite groups in Arcata. It relates the number of simple kG-modules to the number of ‘weights’ (which are locally defined objects).

Alperin defines a weight to be (the isomorphism class of) a projective simple k(N_{G}(Q)/Q)-module, where Q is a representative of a G-conjugacy class of p-subgroups of G, and allocates the weights to p-blocks in the obvious way. If such a (kN_{G}(Q)/Q)- module exists then Q must necessarily be a radical p-subgroup of G, for otherwise N_{G}(Q)/Q has a non-trivial normal p-subgroup and f_{0}(kN_{G}(Q)/Q) = 0. The weight conjecture is then that the number of these weights should be l(G), with a block-wise version saying that the number of weights associated to a p-block B of G should be l(G, B). This is

l(G, B) = ∑_{Q} f_{0}^{(B)} (\frac{N_{G}(Q)}{Q}) ,

where the sum runs over the conjugacy classes of radical p-subgroups of G. We may rewrite this in the language of p-chains using Corollary 3.15 to highlight the similarity to the reformulation due to Robinson which we give later:

Conjecture 4.1 (Alperin) Let B be a p-block of a finite group G, then

l(G, B) = ∑_{σ∈R/G}(−1)^{|σ|}f_{0}^{(B)} (\frac{I_{N_{G}(σ)}(1_{V_{σ}})}{V_{σ}}) .

Alperin’s weight conjecture is known to hold in the following cases (amongst others):

• Finite groups with split BN-pairs in the defining characteristic (non-blockwise version only). See [9] or [53].

• p-solvable groups (see [38] or [30]).

• Symmetric and general linear groups (see [3]).

• The reduction of Alperin’s conjecture to the case O_{p}(G) = 1 is almost trivial. Hence it holds for finite groups of p-local rank one by [7].

4.3 A reformulation of the weight conjecture by Knörr and Robinson

Soon after the publication of [2], Knörr and Robinson presented a reformulation of the weight conjecture in [34], relating it to the work of Quillen (see [43]) and others on simplicial complexes consisting of p-chains. This new conjecture is presented in several (equivalent) forms, varying in the family of p-chains used and in the use of simple kH-modules or simple KH-modules (or indeed the p-rank of the Cartan matrix of a p-block).

The reformulated conjecture states that for each p-block B of positive defect of each finite group G we have

∑_{σ∈C(G|1)/G}(−1)^{|σ|}l(G_{σ}, B) = 0.

This conjecture is equivalent to the weight conjecture in the sense that if one holds for all finite groups then so does the other (the situation is not quite as bad as this, but the important thing to note is that we do not know that if one conjecture holds for a particular block, then that block must also satisfy the other conjecture).

Knörr and Robinson show in [34] that (amongst other things)

∑_{σ∈C(G|1)/G}(−1)^{|σ|}l(G_{σ}, B) = ∑_{σ∈C(G|1)/G}(−1)^{|σ|}k(G_{σ}, B).

It is proved in [34] that if A is the sum of the p-blocks of positive defect of G, then

∑_{σ∈C(G|1)/G}(−1)^{|σ|}l(G_{σ}, A) = 0

for finite groups with split BN-pairs in the defining characteristic (the proof uses the fact that Alperin’s conjecture holds in this situation). Since the principal p-block is generally the only p-block of positive defect for simple groups of Lie type in the defining characteristic, it should be noted that this suffices to demonstrate the fully blockwise conjecture in many cases.

4.4 The conjectures of Dade

Dade announced the first of his conjectures in [17]. This ‘ordinary conjecture’ states that for each p-block B of positive defect of each finite group G satisfying O_{p}(G) = 1 and for each integer d we have

∑_{σ∈R(G|1)/G}(−1)^{|σ|}k_{d}(G_{σ}, B) = 0.

R(G|1) may be replaced by C(G|1) or any other of the families of chains discussed in [34]. Also, being a refinement of the Knörr-Robinson reformulation, if this conjecture is true then so is the weight conjecture.

If one counts only characters of defect equal to that of B, then this conjecture is identical to the Alperin-McKay conjecture for groups with O_{p}(G) = 1. In fact, in this situation the projective form of the conjecture given below is equivalent to the Alperin-McKay conjecture.

Dade has announced a series of progressively stronger conjectures in [20], the strongest of which, the ‘inductive conjecture’, he claims admits a reduction to finite simple groups. Although the reduction step has yet to be published, work has already begun on verifying the final form of the conjecture for the finite simple groups, with a good portion of the sporadic groups already checked.

We give the ‘projective conjecture’, which may be found in [18]. It is convenient to give the form in which G is allowed to have any normal p-subgroup, although Dade demonstrates that it suffices to assume that O_{p}(G) is central. We note that Dade defines w(G, U) to be the number of irreducible characters χ of G such that χ(1)_{p} = [G : U]_{p}μ(1)_{p}, where μ ∈ Irr(U) is covered by χ, and U is a normal psubgroup of G, but as shown in the previous chapter this is equivalent to our definition of w(G, U).

Conjecture 4.2 (Dade’s projective) Let B be a p-block of a finite group G. Let d be an integer and λ ∈ Irr(O_{p}(Z(G))). Then

∑_{σ∈R(G|O_{p}(G))/G}(−1)^{|σ|}k_{d}(G_{σ}, B, λ) = ∑_{σ∈R(G|O_{p}(G))/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, O_{p}(G)).

Dade also gives a ‘weight’ form of this conjecture, which, as may be seen later, is essentially Robinson’s reformulation.

4.5 Robinson’s reformulation

Robinson in [47] uses the results of [35] to present a conjecture equivalent to Alperin’s giving the number of irreducible (K−)characters in a p-block B of G in terms of the number of p-blocks of defect zero of certain locally-determined sections of G. We give the conjecture as presented in [50].

Conjecture 4.3 (Robinson [47]) Let B be a p-block of a finite group G. Then

k(G, B) = ∑_{σ∈R/G}(−1)^{|σ|} ∑_{μ∈Irr(V_{σ})/G_{σ}} f_{0}^{(B)} (\frac{I_{G_{σ}}(μ)}{V_{σ}}) .

Just as with the conjectures of Dade, this may be refined with the Alperin-McKay conjecture in mind:

Conjecture 4.4 (Robinson [47]) Let G be a finite group and λ ∈ Irr(O_{p}(Z(G))). Then for each p-block B of G and each non-negative integer d, we have

k_{d}(G, B, λ) = ∑_{σ∈R/G}(−1)^{|σ|} ∑_{μ∈Irr_{d}(V_{σ},λ)/G_{σ}} f_{0}^{(B)} (\frac{I_{G_{σ}}(μ)}{V_{σ}}) .

This refined form is the conjecture which we will be concerned with in this thesis.

The following is effectively proved in [18], and shows how Robinson’s conjecture and Dade’s projective conjecture are related. Our argument however is organized using the p-local rank.

Proposition 4.5 Suppose that plr(G) = n and that Robinson’s conjecture holds for finite groups of p-local rank strictly less than n. Write U = O_{p}(G). Then

∑_{σ∈R(G|U)/G}(−1)^{|σ|}k_{d}(G_{σ}, B, λ) = ∑_{σ∈R(G|U)/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, U)

if and only if

k_{d}(G, B, λ) = ∑_{σ∈R/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, V_{σ}).

Proof Note that by the inductive hypotheses both equalities hold for all groups of p-local rank less than n.

Suppose that the former holds.

Let R_{0}(G) be the set of all radical p-subgroups of G (i.e., terms of radical p-chains of length zero). We have

∑_{σ∈R/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, V_{σ})
= ∑_{Q∈R_{0}(G)/G} (∑_{σ∈R(N_{G}(Q)|Q)/N_{G}(Q)}(−1)^{|σ|}w_{d}(N_{G}(Q)_{σ}, B, λ, Q))
= ∑_{Q∈R_{0}(G)/G} (∑_{σ∈R(N_{G}(Q)|Q)/N_{G}(Q)}(−1)^{|σ|}k_{d}(N_{G}(Q)_{σ}, B, λ))
=  ∑_{σ∈R/G}(−1)^{|σ|}k_{d}(G_{σ}, B, λ)

We may pair every chain Q_{0} < ⋯ < Q_{r} in R(G) in which Q_{0} ≠ U with the chain U < Q_{0} ⋯ < Q_{r} in R(G) of length r + 1. These two chains have the same stabilizer, so their contributions cancel. We may cancel all chains in R(G) in this way except for the chain U. Hence

∑_{σ∈R/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, V_{σ}) = k_{d}(G, B, λ)

and Robinson’s conjecture is satisfied in that case.

Now suppose that B satisfies the second equality. Then

k_{d}(G, B, λ)

=  ∑_{σ∈R/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, V_{σ})
= ∑_{Q∈R_{0}(G)/G} (∑_{σ∈R(N_{G}(Q)|Q)/N_{G}(Q)}(−1)^{|σ|}w_{d}(N_{G}(Q)_{σ}, B, λ, Q))
=  ∑_{σ∈R(G|U)/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, U)
+ ∑_{U≠Q∈R_{0}(G)/G} (∑_{σ∈R(N_{G}(Q)|Q)/N_{G}(Q)}(−1)^{|σ|}k_{d}(N_{G}(Q)_{σ}, B, λ))

Replacing each chain Q_{0} < ⋯ < Q_{r} in R(G) in which U ≠ Q_{0} by U < Q_{0} ⋯ < Q_{r}, we have

∑_{U≠Q∈R_{0}(G)/G} (  ∑_{σ∈R(N_{G}(Q)|Q)/N_{G}(Q)}(−1)^{|σ|}k_{d}(N_{G}(Q)_{σ}, B, λ))

= ∑_{σ∈R(G|U)/G}(−1)^{|σ|+1}k_{d}(G_{σ}, B, λ) − k_{d}(G, B, λ),

and so

∑_{σ∈R(G|U)/G}(−1)^{|σ|}k_{d}(G_{σ}, B, λ) = ∑_{σ∈R(G|U)/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, U)

as required.

Remark Although the conjectures of Dade and Robinson are similar, one should note that they represent very different approaches to Alperin’s conjecture. The driving force behind Dade’s conjectures is the proof using the CFSG, whereas Robinson’s approach is to conserve the information implicit in the action on a normal p-subgroup and to keep the idea of relative projectivity that lies behind Alperin’s conjecture.

4.6 Statement of results

Proposition 4.6 If Conjecture 4.4 holds for all finite groups G of p-local rank n or smaller and with O_{p}(G) ≤ Z(G) cyclic, then the conjecture holds for all finite groups of p-local rank n. In particular in proving the conjecture it suffices to check it just for those groups with O_{p}(G) ≤ Z(G) cyclic.

Theorem 4.7 Conjecture 4.4 holds for all finite groups of p-local rank one.

Corollary 4.8 Dade’s projective conjecture holds for all finite groups with almost TI Sylow p-subgroups. In particular, Dade’s inductive conjecture holds for all simple groups with TI Sylow p-subgroups, cyclic Schur multiplier and trivial outer automorphism group.

Proof By [20] the inductive conjecture is identical to the projective conjecture in the situation given above.

Proposition 4.9 Dade’s inductive conjecture is satisfied for the Ree groups of type G_{2} for the prime p = 3.

Remark Since the notion of relative projectivity is well understood one may identify group theoretic conditions for a p-chain to contribute nothing to the alternating sum of Conjecture 4.4. For example, suppose that σ : Q_{0} < ⋯ < Q_{n} is a p-chain of G and B a p-block of G_{σ} with defect group D; if

log_{p}(|Q_{0}|) < (log_{p}(|Q_{n}|) + log_{p}(|Z(Q_{n})|)) /2

or C_{D}(Q_{0}) ≰ Q_{0}, then w(G_{σ}, B, Q_{0}) = 0 (see [49], the proof is an application of the results at the end of Chapter 3). Stronger results of this kind may be obtained if we count only characters covering a given θ ∈ Irr(Q_{0}), although the result as it stands is optimal (see for example GL_{3}(q) in the defining characteristic).

Clearly results of this kind are not directly applicable to Dade’s conjecture (at least in its non-weight form). However, by structuring via the p-local rank we may in some circumstances eliminate certain chains from the calculation of the alternating sums of Dade’s projective conjecture (assuming that the conjecture holds for finite groups of smaller p-local rank). As an example, when verifying Dade’s projective conjecture, we may eliminate the need for the calculation of p-chains totally when a group has Sylow p-subgroups of order p^{2} (as is the case when p = 3 and G = M_{22}) by making use of Theorem 4.7. Such reductions would be best done on an ad hoc basis. It should be noted that in Dade’s papers [17], [18], [20], etc., the residue field considered is not assumed to be algebrically closed, so one must use the fact that both Dade’s and our set-up stipulate that K be a splitting field for G in order to apply these kinds of reduction.

5 Finite groups with normal p-subgroups

5.1 Introduction

In this chapter we consider what happens when a finite group G has a normal psubgroup. Whereas Alperin’s weight conjecture reduces quickly to the case where O_{p}(G) = 1, the reduction regarding Robinson’s reformulation of the conjecture is somewhat more involved, and not as complete (for reasons which will become obvious). It is worth noting however that a complete reduction of the weaker form of the conjecture (Conjecture 4.3) to the case O_{p}(G) = 1 follows from the results of [48].

We demonstrate that in order to verify the conjecture for finite groups of p-local rank n it suffices to check the conjecture for p-blocks of finite groups of p-local rank at most n and having only cyclic, central normal p-subgroups. This will be applied in later chapters to the p-local rank one case in order to use the classification of finite simple groups.

The reduction involves using methods developed by Robinson in [47] to ‘prepare’ the formula predicted by Robinson’s conjecture, so that we may apply the results of Chapter 3.

5.2 Reduction for Robinson’s conjecture

Let λ ∈ Irr(O_{p}(Z(G))). By Clifford’s theorem we may divide Irr_{d}(G, B, λ) into sets of characters lying over a given G-orbit of Irr(O_{p}(G)), so that

k_{d}(G, B, λ) = ∑_{μ∈Irr(O_{p}(G),λ)/G} k_{d}(G, B, μ).

We aim to prove the following result, which then allows us to apply the results of the last chapter:

Proposition 5.1 Let B be a p-block of a finite group G. Write U = O_{p}(G). Let λ ∈ Irr(O_{p}(Z(G)) and d be an integer. Then

∑_{σ∈R(G)/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, V_{σ})

= ∑_{μ∈Irr(U,λ)/G} ( ∑_{σ∈R(I_{G}(μ))/I_{G}(μ)}(−1)^{|σ|}w_{d}(I_{G}(μ)_{σ}, B, μ, V_{σ})) .

Recall that C_{U}(G) is the set of all p-chains of G whose initial subgroup contains U (not necessarily properly). By Lemma 2.2 we may replace R in the first alternating sum of Proposition 5.1 with C_{U}, since the initial subgroup of every radical p-chain of G is a radical p-subgroup of G and so contains U by Lemma 2.1. We obtain

∑_{σ∈R(G)/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, V_{σ}) = ∑_{σ∈C_{U} (G)/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, V_{σ}).

For each σ ∈ C_{U},

w_{d}(G_{σ}, B, λ, V_{σ}) = ∑_{μ∈Irr(U,λ)/G_{σ}} w_{d}(G_{σ}, B, μ, V_{σ}),

so

∑_{σ∈C_{U} (G)/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, V_{σ}) = ∑_{σ∈C_{U} (G)/G}(−1)^{|σ|} ( ∑_{μ∈Irr(U,λ)/G_{σ}} w_{d}(G_{σ}, B, μ, V_{σ})) .

Lemma 5.2

∑_{σ∈C_{U} (G)/G}(−1)^{|σ|} (∑_{μ∈Irr(U,λ)/G_{σ}} w_{d}(G_{σ}, B, μ, V_{σ}))
= ∑_{μ∈Irr(U,λ)/G} (∑_{σ∈C_{U} (G)/I_{G}(μ)}(−1)^{|σ|}w_{d}(G_{σ}, B, μ, V_{σ}))

Proof Observe that the pair (σ, μ) lies in an orbit of length [G : G_{σ}][G_{σ} : I_{G_{σ}}(μ)] = [G : I_{G_{σ}}(μ)] in the first instance and [G : I_{G}(μ)][I_{G}(μ) : I_{G_{σ}}(μ)] = [G : I_{G_{σ}}(μ)] in the second, and that w_{d}(G_{σ}, B, μ, V_{σ}) is constant under conjugation of (σ, μ) in G.

We conclude that if

k_{d}(G, B, μ) = ∑_{σ∈C_{U} (G)/I_{G}(μ)}(−1)^{|σ|}w_{d}(G_{σ}, B, μ, V_{σ})

for each μ ∈ Irr(U, λ), then the conjecture holds for that choice of B, d and λ.

We fix μ ∈ Irr(U, λ) and write H = I_{G}(μ). Clifford theory then allows us to move from counting characters of G and G_{σ} to counting characters of H and H_{σ}:

Lemma 5.3 k_{d}(G, B, μ) = k_{d}(H, B, μ) and

∑_{σ∈C_{U} (G)/H}(−1)^{|σ|}w_{d}(G_{σ}, B, μ, V_{σ}) = ∑_{σ∈C_{U} (G)/H}(−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ} ∩ H).

Proof For each σ ∈ C_{U}(G), I_{G_{σ}}(μ) = H_{σ}, and so Clifford’s theorem gives a 1-1 correspondence

Irr(H_{σ}, μ) ↔ Irr(G_{σ}, μ)

given by induction of characters (see, for example [29, 6.11]), and note that G = G_{σ} for the chain σ = U of length 0. This correspondence is clearly defect-preserving. Suppose that χ ∈ Irr(G_{σ}, μ) corresponds to η ∈ Irr(H_{σ}, μ). We claim that χ is V_{σ}-projective if and only if η is V_{σ} ∩ H-projective.

Proof of claim Let φ ∈ Irr(V_{σ} ∩ H, μ) be a character covered by η. Now I_{V_{σ}}(μ) = H ∩ V_{σ}, so by Clifford’s theorem φ^{V_{σ}} = θ for some θ ∈ Irr(V_{σ}, μ). But

(χ|_{V_{σ}}, θ) = (χ|_{V_{σ}}, φ^{V_{σ}}) = (χ|_{V_{σ}∩H}, φ) = (η^{G_{σ}}|_{V_{σ}∩H}, φ) = (η^{G_{σ}}|_{H_{σ}}, φ^{H_{σ}}).

But (η, η^{G_{σ}}|_{H_{σ}}) ≠ 0 and (η, φ^{H_{σ}}) ≠ 0, so χ covers θ. We have χ(1)_{p} = [G_{σ} : H_{σ}]_{p}η(1)_{p} and

[G_{σ} : V_{σ}]_{p}θ(1)_{p} = [G_{σ} : V_{σ}]_{p}[V_{σ} : V_{σ} ∩ H]_{p}φ(1)_{p} = [G_{σ} : H_{σ}]_{p}[H_{σ} : V_{σ} ∩ H]_{p}φ(1)_{p}.

So χ(1)_{p} = [G_{σ} : V_{σ}]_{p}θ(1)_{p} if and only if η(1)_{p} = [H_{σ} : V_{σ} ∩ H]_{p}φ(1)_{p}, and the claim follows by Proposition 3.14.

It follows immediately from Lemma 3.1 that every p-block of H_{σ} containing an irreducible character lying over μ has a Brauer correspondent in G_{σ} and the character correspondence given by induction respects the Brauer correspondence, i.e., if η ∈ Irr(H_{σ}, μ), then η^{G_{σ}} lies in the p-block of G_{σ} which is a Brauer correspondent of the p-block of H_{σ} containing η.

Hence

w_{d}(G_{σ}, B, μ, V_{σ}) = w_{d}(H_{σ}, B, μ, V_{σ} ∩ H)

since the Brauer correspondence is transitive, and the result follows.

We use the theory of deficient p-chains introduced in [47] in order to overcome the problem of summing over I_{G}(μ)-orbits of p-chains of G (note in particular that a given σ ∈ C_{U}(G) need not lie in the stabilizer of μ). This idea originated in [51], but was defined formally in [47].

Definition Given a subgroup T of G, we say that a p-chain

σ : Q_{0} < ⋯ < Q_{n} ∈ C(G)

is T -deficient if Q_{n}∩T ≤ O_{p}(G). Given a p-chain σ ∈ C(G) we call the longest deficient initial subchain the T -deficient part, and denote it by d_{T} (σ). Note that d_{T} (σ) may be empty. As an abuse of notation denote the empty chain by ∅.

Returning to our original hypotheses, for brevity we write D(G) = D_{H}(G) for the set of non-empty H-deficient chains in C_{U}(G). Write D^{♯}(G) = D(G) − {U} and d(σ) = d_{H}(σ).

Observe that we may write

∑_{σ∈C_{U} (G)/H}(−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ} ∩ H)

= ∑_{τ∈D(G)/H} ( ∑_{σ∈C(G|V^{τ})/H_{τ} ,d(σ)=σ_{0}}(−1)^{|τ|+|σ|}w_{d}((H_{τ})_{σ}, B, μ, U))

+ ∑_{σ∈C_{U} (G)/H,d(σ)=∅}(−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ} ∩ H),

since V_{τ} ∩ H = U when τ ∈ D(G). We show that it suffices to consider only those chains contained in the stabilizer of their deficient part. Lemma 5.4

∑_{τ∈D(G)/H} (  ∑_{σ∈C(G|V^{τ})/H_{τ} ,d(σ)=σ_{0}}(−1)^{|τ|+|σ|}w_{d}((H_{τ})_{σ}, B, μ, U))

= ∑_{τ∈D(G)/H} ( ∑_{σ∈C(G_{τ} |V^{τ})/H_{τ} ,d(σ)=σ_{0}}(−1)^{|τ|+|σ|}w_{d}((H_{τ})_{σ}, B, μ, U))

Proof We wish to cancel the contributions to the alternating sum of those chains σ ∈ C_{U}(G) with σ ∉ C_{U}(N_{G}(d(σ))). We do this by pairing such chains with identical stabilizers but with lengths of differing parity. Note that the pairs of chains need not necessarily have the same, or even conjugate deficient part.

Let σ : Q_{0} < ⋯ < Q_{n} < ⋯ < Q_{m} ∈ C_{U}(G) with d(σ) = σ_{n} non-empty. Suppose that σ ∉ C_{U}(N_{G}(d(σ))), i.e., Q_{m} ≰ N_{G}(Q_{i}) for some i ≤ n. Choose i minimal with this property. Given i, choose j maximal such that Q_{i} ◁ Q_{j}. So i ≤ j < m and Q_{j} ≤ N_{Q_{j+1}}(Q_{i}) ≠ Q_{j+1}. There are two cases to consider: (i) Q_{j} = N_{Q_{j+1}}(Q_{i}) and (ii) Q_{j} ≠ N_{Q_{j+1}}(Q_{i}).

(i) Suppose that Q_{j} = N_{Q_{j+1}}(Q_{i}). Then form the chain

\overline{σ} : Q_{0} < ⋯ < Q_{j−1} < Q_{j+1} < ⋯ < Q_{m}

from σ by deleting the term Q_{j}. If i = j, then Q_{j} ≠ N_{Q_{j+1}}(Q_{i}) since Q_{j+1} is a p-group, so in particular we must have j ≠ 0 for case (i) to occur, and V\overline{σ} = V_{σ}. Again using the fact that case (i) cannot occur when i = j, N_{H}(\overline{σ}) ≤ N_{H}(Q_{j}) and so H_{σ} = H\overline{σ}. Also Q_{i} is still a term of \overline{σ} and j < m, so \overline{σ} ∉ C_{U}(N_{G}(d(\overline{σ}))).

(ii) Suppose that Q_{j} ≠ N_{Q_{j+1}}(Q_{i}). Then form the chain

\overline{σ} : Q_{0} < ⋯ < Q_{j} < N_{Q_{j+1}}(Q_{i}) < Q_{j+1} < ⋯ < Q_{m}

from σ. Of course V\overline{σ} = V_{σ}, and it is easy to see that N_{H}(σ) ≤ N_{Q_{j+1}}(Q_{i}) and so H_{σ} = H\overline{σ}. As above Q_{i} is still a term of the deficient part of \overline{σ} and Q_{m} is still the final subgroup of \overline{σ}, so \overline{σ} ∉ C_{U}(N_{G}(d(\overline{σ}))).

In both cases (\overline{σ})^{g} = \overline{(σ^{g})} and \overline{\overline{σ}} = σ, so we may cancel the contributions from orbits of chains containing σ and \overline{σ}, since (−1)^{|\overline{σ}|} = (−1)^{|σ|+1}.

The next lemma may seem somewhat technical but it will later allow us to reduce the summation to a consideration of chains of H rather than G, the key to proving Proposition 5.1.

Lemma 5.5 Given τ ∈ D(G), denote by C^{⋆}(G_{τ}|V^{τ}) the set of those chains

σ : Q_{0} < ⋯ < Q_{n}

in C(G_{τ}|V^{τ}) satisfying Q_{i} = V^{τ}(H ∩ Q_{i}) for each i = 0, …, n. We also allow τ to be the empty chain, in which case we set G_{τ} = G and set C_{U}^{⋆}(G) = C_{U}(H). Then

∑_{σ∈C_{U} (G)/H,d(σ)=∅}(−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ}∩H) = ∑_{σ∈C_{U}^{⋆} (G)/H,d(σ)=∅}(−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ}∩H)

(representing the case τ is empty), and when τ ∈ D(G) we have

∑_{σ∈C(G_{τ} |V^{τ})/H_{τ} ,d(σ)=σ_{0}}(−1)^{|τ|+|σ|}w_{d}((H_{τ})_{σ}, B, μ, U)

= ∑_{σ∈C^{⋆}(G_{τ} |V^{τ})/H_{τ} ,d(σ)=σ_{0}}(−1)^{|τ|+|σ|}w_{d}((H_{τ})_{σ}, B, μ, U).

Proof Let τ ∈ D(G), and let σ : Q_{0} < ⋯ < Q_{n} ∈ C(G_{τ}|V^{τ}) with d(σ) = σ_{0} and suppose that σ ∉ C^{⋆}(G_{τ}|V^{τ}), i.e., Q_{i} ≠ V^{τ}(H ∩ Q_{i}) for some i. Choose i minimal with this property, noting that i > 0. Now

Q_{i} > V^{τ}(H ∩ Q_{i}) ≥ V^{τ}(H ∩ Q_{i−1}) = Q_{i−1}

by the minimality of i. There are two cases to consider: (i) V^{τ}(H ∩ Q_{i}) = Q_{i−1} and (ii) V^{τ}(H ∩ Q_{i}) ≠ Q_{i−1}.

(i) Suppose that V^{τ}(H ∩ Q_{i}) = Q_{i−1}. Observe that i > 1, since if i = 1 and case (i) occurs, then V^{τ}(H ∩ Q_{1}) = V^{τ}, and so H ∩ Q_{1} = H ∩ V^{τ} = U, contradicting our assumption that d(σ) = σ_{0}. Form the chain

\overline{σ} : Q_{0} < ⋯ < Q_{i−2} < Q_{i} < ⋯ < Q_{n}

from σ by deleting the term Q_{i−1}. Since i > 1, we must have V\overline{σ} = V_{σ}, and it is easy to see that N_{H}(\overline{σ}) ≤ N_{H}(Q_{i−1}), so N_{H}(\overline{σ}) = N_{H}(σ). Also \overline{σ} ∈ C(G_{τ}|V^{τ}), with d(\overline{σ}) = \overline{σ}_{0} and \overline{σ} ∉ C^{⋆}(G_{τ}|V^{τ}).

(ii) Suppose that V^{τ}(H ∩ Q_{i}) ≠ Q_{i−1}. Then form the chain

\overline{σ} : Q_{0} < ⋯ < Q_{i−1} < V^{τ}(H ∩ Q_{i}) < Q_{i} < ⋯ < Q_{n}

from σ. V\overline{σ} = V_{σ} since i > 0. Clearly N_{H}(σ) ≤ N_{H}(V^{τ}(H ∩ Q_{i})), so that N_{H}(\overline{σ}) = N_{H}(σ). Since V^{τ}(H ∩ Q_{i}) ∩ H = H ∩ Q_{i} > U, we have d(\overline{σ}) = \overline{σ}_{0}, and \overline{σ} ∉ C^{⋆}(G_{τ}|V^{τ}).

In both cases (\overline{σ})^{g} = \overline{(σ^{g})} and \overline{\overline{σ}} = σ, so we may cancel the contributions from orbits of chains containing σ and \overline{σ}, since (−1)^{|\overline{σ}|} = (−1)^{|σ|+1}. Hence the second part of the Lemma holds.

To consider those chains with deficient part the empty chain we may use a similar argument to cancel all chains in which Q_{i} ≠ H ∩ Q_{i} for some i. We leave this to the reader. Lemma 5.6

∑_{σ∈C_{U}^{⋆} (G)/H,d(σ)=∅}(−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ} ∩ H) = ∑_{σ∈C_{U} (H)/H,d(σ)=∅}(−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ}),

and if τ ∈ D(G), then

∑_{σ∈C^{⋆}(G_{τ} |V^{τ})/H_{τ} ,d(σ)=σ_{0}}(−1)^{|τ|+|σ|}w_{d}((H_{τ})_{σ}, B, μ, U)

= ∑_{σ∈C(H_{τ} |U)/H_{τ}}(−1)^{|τ|+|σ|}w_{d}((H_{τ})_{σ}, B, μ, U).

Proof Suppose that τ ∈ D(G) and σ ∈ {α ∈ C^{⋆}(G_{τ}|V^{τ}) : d(α) = α_{0}}, where σ : Q_{0} < ⋯ < Q_{n}. Then by definition Q_{i} = V^{τ}(H ∩ Q_{i}) for each i ≥ 0, and so

|Q_{i}| = \frac{|V^{τ}||H ∩ Q_{i}|}{|V^{τ} ∩ H ∩ Q_{i}|} = \frac{|V^{τ}||H ∩ Q_{i}|}{|U|}

(recall that H ∩ V^{τ} = U). Since the Q_{i}’s are distinct, the chain φ(σ) defined to be

φ(σ) : H ∩ V^{τ} < H ∩ Q_{1} < ⋯ < H ∩ Q_{n}

has distinct terms. Since H ∩ V^{τ} = U, we have φ(σ) ∈ C(H_{τ}|U).

Conversely, if θ ∈ C(H_{τ}|U), where θ : P_{0} < ⋯ < P_{n}, then define ψ(θ) to be the chain

ψ(θ) : V^{τ}P_{0} < V^{τ}P_{1} < ⋯ < V^{τ}P_{n}.

Observe that V^{τ}P_{i} = V^{τ}(H ∩ V^{τ}P_{i}) for each i and that ψ(θ) ∈ {C^{⋆}(G_{τ}|V^{τ}) : d(α) = α_{0}}. Hence φ and ψ are inverse maps between {α ∈ C^{⋆}(G_{τ}|V^{τ}) : d(α) = α_{0}} and C(H_{τ}|U) which preserve length. It is clear that chains in {α ∈ C^{⋆}(G_{τ}|V^{τ}) : d(α) = α_{0}} are conjugate in H_{τ} if and only if their images in C(H_{τ}|U) are conjugate in H_{τ}. Hence we have a bijection {α ∈ C^{⋆}(G_{τ}|V^{τ})/H_{τ} : d(α) = α_{0}} ↔ C(H_{τ}|U)/H_{τ}. It remains to check that N_{H_{τ}} (φ(σ)) = N_{H_{τ}} (σ). But this is clear since N_{H_{τ}} (σ) ≤ N_{H_{τ}} (φ(σ)) and (since τ is stabiized by H_{τ}) N_{H_{τ}} (φ(σ)) ≤ N_{H_{τ}} (ψ(φ(σ))) = N_{H_{τ}} (σ).

The first equality of the lemma is immediate since C_{U}^{⋆}(G) is by definition C_{U}(H).

Proof of Proposition 5.1 We have seen that

∑_{σ∈R(G)/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, V_{σ}) = ∑_{μ∈Irr(U,λ)/G} ( ∑_{σ∈C_{U} (G)/I_{G}(μ)}(−1)^{|σ|}w_{d}(G_{σ}, B, μ, V_{σ}))

(see Lemma 5.2 and the discussion preceding it). We fix μ ∈ Irr(U, λ) and write H = I_{G}(μ). By Lemmas 5.3, 5.4, 5.5 and 5.6,

∑_{σ∈C_{U} (G)/H}(−1)^{|σ|}w_{d}(G_{σ}, B, μ, V_{σ})

= ∑_{τ∈D(G)/H} ( ∑_{σ∈C(H_{τ} |U)/H_{τ}}(−1)^{|τ|+|σ|}w_{d}((H_{τ})_{σ}, B, μ, U))

+ ∑_{σ∈C_{U} (H)/H,d(σ)=∅}(−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ}),

where the final alternating sum represents those chains in C_{U}(G) with empty deficient part.

Now if we set α ∈ D(G) to be the chain α = U (of length zero), then

∑_{σ∈C(H_{α}|U)/H_{α}}(−1)^{|α|+|σ|}w_{d}((H_{α})_{σ}, B, μ, U) + ∑_{σ∈C_{U} (H)/H,d(σ)=∅}(−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ})

= ∑_{σ∈C_{U} (H)/H}(−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ}),

as we are considering on the one hand chains whose initial subgroup is U and on the other chains whose initial subgroup strictly contains U. We claim that

∑_{τ∈D^{♯}(G)/H} ( ∑_{σ∈C(H_{τ} |U)/H_{τ}}(−1)^{|τ|+|σ|}w_{d}((H_{τ})_{σ}, B, μ, U)) = 0.

Proof of claim Suppose that τ ∈ D^{♯}(G), and consider a chain σ ∈ C(H_{τ}|U), σ : Q_{0} < ⋯ < Q_{n}, of length |σ| > 0. Then U < Q_{1} ◁ N_{H_{τ}} (σ) = I_{N_{Hτ} (σ)}(μ), i.e., Q_{1} is a normal p-subgroup of (H_{τ})_{σ} strictly containing U and stabilizing μ, and so w_{d}((H_{τ})_{σ}, B, μ, U) = 0 by Corollary 3.15. Hence

∑_{τ∈D^{♯}(G)/H} ( ∑_{σ∈C(H_{τ} |U)/H_{τ}}(−1)^{|τ|+|σ|}w_{d}((H_{τ})_{σ}, B, μ, U)) = ∑_{τ∈D^{♯}(G)/H}(−1)^{|τ|}w_{d}(H_{τ}, B, μ, U).

But notice that we may pair each deficient chain τ ∈ D^{♯}(G) satisfying V_{τ} = U with another chain in D^{♯}(G) with initial term strictly containing U, since we have defined D^{♯}(G) to exclude the chain τ = U. Clearly paired chains lie in H-orbits of the same size, and the lengths of the chains in each pair differ in parity, so we may cancel their contributions to this last alternating sum. This gives ∑_{τ∈D^{♯}(G)/H}(−1)^{|τ|}w_{d}(H_{τ}, B, μ, U) = 0, and the claim follows.

Hence we have

∑_{σ∈C_{U} (G)/H}(−1)^{|σ|}w_{d}(G_{σ}, B, μ, V_{σ}) = ∑_{σ∈C_{U} (H)/H}(−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ})

+ ∑_{τ∈D^{♯}(G)/H} ( ∑_{σ∈C(H_{τ} |U)/H_{τ}}(−1)^{|τ|+|σ|}w_{d}((H_{τ})_{σ}, B, μ, U))

= ∑_{σ∈C_{U} (H)/H}(−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ}) =^{2.2} ∑_{σ∈R(H)/H}(−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ}),

completing the proof of the Proposition.

Proposition 4.6 If Conjecture 4.4 holds for all finite groups G of p-local rank n or smaller and with O_{p}(G) ≤ Z(G) cyclic, then the conjecture holds for all finite groups of p-local rank n. In particular in proving the conjecture it suffices to check it just for those groups with O_{p}(G) ≤ Z(G) cyclic.

Proof Let B be a p-block of a finite group G with plr(G) = n. We have seen (by Proposition 5.1, its proof and the accompanying discussion) that in order to verify Conjecture 4.4 for B (for a given λ ∈ Irr(O_{p}(Z(G))) and integer d) it suffices to show that

k_{d}(I_{G}(μ), B, μ) = ∑_{σ∈R(I_{G}(μ))/I_{G}(μ)}(−1)^{|σ|}w_{d}(I_{G}(μ)_{σ}, B, μ, V_{σ})  (1)

for each μ ∈ Irr(U, λ). Hence we may assume that I_{G}(μ) = G.

Now by the results of Chapter 3 we may choose an irreducible extension θ̃ of μ to a covering group G̃ of G, and we may use restrictions of this extension to examine the characters of each G_{σ} lying over μ via covering groups of G_{σ}/U. We use the notation of Proposition 3.10, with N = U.

Let B̂ be the sum of the p-blocks of Ĝ corresponding to B in Proposition 3.10 (i.e., the sum of the p-blocks Bl(χ̂), where χ̂ corresponds to some χ ∈ Irr(G, B, μ)). It is clear that R(Ĝ/M̂)/Ĝ and R(G/U)/G may be identified, so Ĝ and G have ‘similar’ p-local stucture. For each σ ∈ R(G), define σ̂ in the obvious way. We may make the identification

Irr(G_{σ}, B, μ, V_{σ}) ↔ Irr(Ĝσ̂, B̂, μ̂, Vσ̂).

Let a = log_{p}(|G|_{p}) and a′ = log_{p}(|Ĝ|_{p}). In order to count characters of defect d in G, we must count characters of defect d′ = d + a′ − a + log_{p}(μ(1)) in Ĝ. Hence the correspondence above may be refined to

Irr_{d}(G_{σ}, B, μ, V_{σ}) ↔ Irr_{d′}(Ĝσ̂, B̂, μ̂, Vσ̂).

Of course we also have a 1-1 correspondence

Irr_{d}(G, B, μ) ↔ Irr_{d′}(Ĝ, B̂, μ̂).

Hence (1) if and only if

k_{d′}(Ĝ, B̂, μ̂) = ∑_{σ̂∈R(Ĝ)/Ĝ}(−1)^{|σ̂|}w_{d′}(Ĝσ̂, B̂, μ̂, Vσ̂),

since by Proposition 3.10 the correspondences respect the Brauer correspondence. It is clear that M̂ = O_{p}(Ĝ) ≤ Z(Ĝ), and by definition M̂ is cyclic. Hence the proposition holds, since by Lemma 2.4 we have plr(I_{G}(μ)) ≤ plr(G).

5.3 Groups of p-rank one

We are now in a position to verify Theorem 4.7 for those finite groups G (of p-local rank one) in which G/O_{p}(G) has p-rank one (i.e., G/O_{p}(G) has no elementary abelian subgroups of order p^{2}). It is well-known that a p-group has p-rank one if and only if it is cyclic or generalized quaternion (see for example [24, 5.4.10]).

Proposition 5.7 Let G be a counterexample to Theorem 4.7. Then G/O_{p}(G) contains an elementary abelian subgroup of order p^{2}.

Proof By Proposition 5.1 and its proof, it suffices to verify the conjecture for central extensions, with centre a cyclic p-group, of each subgroup of G/O_{p}(G). Since each subgroup of G/O_{p}(G) is either a p′-group or has p-rank one, it then suffices to assume that G has p-local rank one, O_{p}(G) = O_{p}(Z(G)) ≤ G′ and G/O_{p}(G) has cyclic or generalized quaternion Sylow p-subgroups. Let B be a p-block of G. We may assume that B has maximal defect.

Suppose first that G/O_{p}(G) has a cyclic Sylow p-subgroup P. Then by Lemma 2.14 P has trivial Schur multiplier, and so by Lemma 2.10 O_{p}(M(G/O_{p}(G))) = 1, so that O_{p}(G) = 1. By the results of Dade [16] k(B) = k(b), where b is the unique block of N_{G}(P) corresponding to B, and all irreducible characters in B and b have height zero. Hence Conjecture 4.4 holds for B. (Note that Dade considers his conjectures for blocks of cyclic defect in [19]).

Now suppose that p = 2 and that G/O_{p}(G) has a generalized quaternion Sylow 2subgroup P. By Lemma 2.14 M(P) = 1, and so by Lemma 2.10 O_{p}(M(G/O_{p}(G))) = 1, so that O_{p}(G) = 1. It then follows from [56], which in turn relies upon the description of the irreducible characters of such a group given by Olsson in [40], that the conjecture holds for B, and the result follows.

6 Characterization and classification of finite groups

with almost TI Sylow p-subgroups

6.1 Introduction

We have already seen that it suffices to assume that O_{p}(G) is central in G and cyclic, and that G has p-rank at least two, so from now on we will assume that this is so. As in [7], we demonstrate that it also suffices to assume that O_{p′}(G) is cyclic and central in G. These reductions are crucial in order to make use of the CFSG.

Having first established such groups as automorphism groups of quasi-simple groups, we use the classifications of Gorenstein and Lyons [26] (which assumes that p is odd, O_{p′}(G) = 1 and lists simple groups with strongly p-embedded subgroups - a condition similar to TI-ness) and of Suzuki (which assumes p = 2 and looks directly at the TI case) to list all such quasisimple groups.

6.2 Finite groups with normal p′-subgroups

Lemma 6.1 In order to verify that Conjecture 4.4 holds for finite groups with almost TI Sylow p-subgroups of p-rank greater than one, it suffices to verify it just for those with only cyclic, central normal p′-subgroups.

Proof Let B be a p-block of a finite group G with almost TI Sylow p-subgroups. Let P ∈ Syl_{p}(G). In this case, by Corollary 3.15 Conjecture 4.4 states that k_{d}(G, B, λ) = k_{d}(N_{G}(P), B, λ) for each integer d and each λ ∈ Irr(O_{p}(Z(G))). Let K = O_{p′}(G). By Lemma 2.8 K centralizes P and so K ≤ N_{G}(P). The p-blocks of K (in this case all are blocks of defect zero) covered by B form a single G-orbit (see [22, V.2.3]), and Irr_{d}(G, B, λ) = Irr_{d}(G, B, μ, λ) for some μ ∈ Irr(K). Clifford theory gives a 1-1 correspondence

Irr(G, B, λ) = Irr(G, B, μ, λ) ↔ Irr(I_{G}(μ), B, μ, λ)

given by induction, which preserves character defects. It is clear that we also have a 1-1 correspondence Irr(N_{G}(P), B, μ, λ) ↔ Irr(I_{N_{G}(P)}(μ), B, μ, λ). Hence we assume that μ is G-stable.

By Proposition 3.10 there is a finite group Ĝ with O_{p′}(Ĝ) ≤ Z(Ĝ) cyclic, Ĝ/O_{p′}(Ĝ) ≅ G/O_{p′}(G), μ̂ ∈ Irr(O_{p′}(Ĝ)) and a sum of p-blocks B̂ of Ĝ such that

k_{d}(G, B, λ) = k_{d}(G, B, μ, λ) = k_{d}(Ĝ, B̂, μ̂, λ) = k_{d}(Ĝ, B̂, λ).

Clearly R(Ĝ/O_{p′}(Ĝ))/Ĝ = R(G/O_{p′}(G))/G. By Proposition 3.10 we also have

k_{d}(N_{G}(P), B, λ) = k_{d}(N_{G}(P), B, μ, λ) = k_{d}(NĜ(P̂), B̂, μ̂, λ) = k_{d}(NĜ(P̂), B̂, λ),

where P̂ is defined in the obvious way. Hence if Conjecture 4.4 holds for Ĝ then it holds for G, and we are done.

6.3 Characterization of finite groups with almost TI Sylow p-subgroups

Lemma 6.2 Let H be a finite group with plr(H) = 1 and O_{p}(H) = O_{p′}(H) = 1. Then there is an unique nontrivial minimal normal subgroup N of H. Further, N is nonabelian simple, plr(N) = 1 and H is isomorphic to a subgroup of Aut(N), the automorphism group of N.

Proof Let P ∈ Syl_{p}(H). We note that P is not normal in H, since otherwise plr(H) = 0 by the remarks preceding Lemma 2.4.

Let N be a (nontrivial) minimal normal subgroup of H. Then p | |N|, since otherwise N ≤ O_{p′}(H) = 1.

Now P ∩N is a nontrivial Sylow p-subgroup of N so, by Lemma 2.7 H = N_{H}(P)N. Suppose that N_{1} and N_{2} are distinct nontrivial minimal normal subgroups of H. By minimality we have N_{1} ∩ N_{2} = 1, and N_{1} ≤ C_{H}(N_{2}) ≤ C_{H}(N_{2} ∩ P) ≤ N_{H}(N_{2} ∩ P). Since N_{2}∩P is nontrivial, by Lemma 2.6 N_{H}(N_{2}∩P) ≤ N_{H}(P). But then H = N_{H}(P), a contradiction. So N is the unique nontrivial minimal normal subgroup of H.

Now by Gorenstein [24, 2.1.5] either N is isomorphic to a direct product S_{1}×⋯×S_{r} of isomorphic nonabelian simple groups or N is elementary abelian. If the latter occurs, then N is a normal p-subgroup of H since p||N|, contradicting O_{p}(H) = 1. So we may assume the former. We show that r = 1:

Suppose that r > 1. Clearly p | |S_{1}|, so we have

S_{2} × ⋯ × S_{r} ≤ C_{H}(S_{1}) ≤ C_{H}(S_{1} ∩ P) ≤ N_{H}(S_{1} ∩ P) ≤ N_{H}(P)

by Lemma 2.6 since S_{1} ∩ P is nontrivial. By the same argument each S_{i} is contained in N_{H}(P) and N ≤ N_{H}(P), again a contradiction. So r = 1 and N is nonabelian simple. It is clear from Lemma 2.4 that plr(N) = 1. It remains to show that C_{H}(N) = 1, so that H is isomorphic to a subgroup of Aut(N).

Since C_{H}(N) ◁ H and N is the unique minimal normal subgroup of H, either C_{H}(N) ≥ N or C_{H}(N) = 1. Since N is nonabelian we must have the latter, and the result follows.

Now by Proposition 4.6, Proposition 5.7 and Lemma 6.1, in proving Theorem 4.7 we may assume that G has O_{p}(G)O_{p′}(G) = Z(G) cyclic and is a central extension of G/Z(G) with Z(G) ≤ [G, G], and that G/Z(G) has p-rank greater than one. Of course plr(G) = 1. By Lemma 6.2, \overline{G} = G/Z(G) is an automorphism group of a finite simple group \overline{N} = N/Z(G) of p-local rank one, where Z(G) < N. We use the classification of finite groups with TI Sylow 2-subgroups by Suzuki, and the classification of finite groups with a proper strongly p-embedded subgroup, non-cyclic Sylow p-subgroups and no nontrivial normal p′-subgroup, for p an odd prime, by Gorenstein and Lyons to give a list of possibilities for N.

The following hypotheses hold throughout the remainder of the thesis.

Hypotheses 6.3 G is a finite group of p-local rank one and p-rank greater than one. Z(G) ≤ G′, Z(G) < N ◁ G and Z(G) is cyclic. \overline{N} = N/Z(G) is a nonabelian simple group of p-local rank one and \overline{N} ≤ \overline{G} ≤ Aut(\overline{N}). Write L = N′.

Remark 6.4 It is important to note that Z(L) is central in G. This excludes some automorphism groups of quasisimple groups which do not act trivially on the centre of that quasisimple group (see [12, p.xxii]).

Definition Let Q ≤ H be a p-group, H a finite group, and k ∈ ℕ. Let m(X) be the p-rank of the group X. Define

Γ_{Q,k}(H) = ⟨N_{H}(X) | X ≤ Q, m(X) ≥ k⟩.

For P ∈ Syl_{p}(H) we call Γ_{P,k}(H) the k-generated p-core of H.

We call a proper subgroup M ≤ H strongly p-embedded if Γ_{P,1}(H) ≤ M for some P ∈ Syl_{p}(H).

Lemma 6.5 Suppose that O_{p}(H) = 1. Then H has a TI Sylow p-subgroup P if and only if Γ_{P,1}(H) = N_{H}(P), i.e., N_{H}(P) is a proper strongly p-embedded subgroup of H.

Proof The ‘only if’ part is clear from Lemma 2.6. Suppose that 1 < Q < P is a radical p-subgroup of H and that Γ_{P,1}(H) = N_{H}(P). Then N_{H}(Q) ≤ Γ_{P,1}(H) = N_{H}(P). So Q < P ∩ N_{H}(Q) ≤ O_{p}(N_{H}(Q)), contradicting our assumption that Q is radical. Hence the only radical p-subgroups of H are 1 and the Sylow p-subgroups of H, so that plr(H) = 1. The result then follows by Lemma 2.5.

Lemma 6.6 Consider Hypotheses 6.3. Then (p, \overline{N}) is isomorphic to one of the following:

(a) (2,^{2} B_{2}(2^{2m+1})), m ≥ 1

(b) (3, ^{2}G_{2}(3^{2m+1})), (3, P SL_{3}(4)), (3, ^{2}G_{2}(3)′), (3, M_{11}), m ≥ 1

(c) (5, ^{2}B_{2}(32)), (5, ^{2}F_{4}(2)′), (5, McL)

(d) (11, J_{4})

(e) (p, P SL_{2}(p^{m})), (p, P SU_{3}(p^{m})), m ≥ 1.

Proof For p = 2 this is Theorem 1 of [55]. Suppose that p is odd. Then \overline{N} has a proper strongly p-embedded subgroup. So by [26, 24.9] (p, \overline{N}) is isomorphic to one of the pairs listed or to (5, M_{22}) or (p, A_{2p}). By [26, 24.2], Γ_{Q,1}(\overline{N}) ≠ N\overline{N}(Q) (for Q ∈ Syl_{p}(\overline{N})) when (p, \overline{N}) is isomorphic to (5, M_{22}) or (p, A_{2p}), so by Lemma 6.5 in these two cases \overline{N} cannot have TI Sylow p-subgroups. Also by [26, 24.2], Γ_{Q,1}(\overline{N}) = N\overline{N}(Q) for each of the remaining cases, so the result holds by Lemma 6.5.

The argument used in proving the following result is essentially that used in [59].

Lemma 6.7 Consider Hypotheses 6.3. Then (p, [G : N]) = 1, except when (p, \overline{N}) ≅ (3, 2G_{2}(3)′) or (5, ^{2}B_{2}(32)).

Proof Suppose first that p = 2. Then the result follows from Theorem 2 of [55].

Assume now that p ≠ 2 and that (p, \overline{N}) ≅ (3, ^{2}G_{2}(3)′) or (5, ^{2}B_{2}(32)). By [12], \overline{N} has p-regular outer automorphism group except when possibly (p, N) ≅ (p, P SL_{2}(p^{m})), (p, P SU_{3}(p^{m})) or (3, ^{2}G_{2}(3^{2m+1})). In each of these cases an outer automorphism of p-power order must be a field automorphism, i.e., a power of the Frobenius map τ : x → x^{p}.

Assuming that \overline{N} is one of these classes of groups, let P ∈ Syl_{p}(\overline{G}) and Q = P ∩\overline{N}. We assume that p | [\overline{G} : \overline{N}] and derive a contradiction.

Let g ∈ P be a field automorphism of order p. Then g fixes a nontrivial proper subfield of 𝔽_{q}, say 𝔽_{t}, where 𝔽_{q} is the field of definition of \overline{N} as a group of Lie type. Hence g fixes a subgroup T of \overline{N} isomorphic to P SL_{2}(𝔽_{t}), P SU_{3}(𝔽_{t}) or ^{2}G_{2}(𝔽_{t}) respectively, so that

T ≤ C\overline{G}(g) = C\overline{G}(⟨g⟩) ≤ N\overline{G}(⟨g⟩).

But N\overline{G}(⟨g⟩) ≤ N\overline{G}(P) since \overline{G} has TI Sylow p-subgroups, so T ≤ N\overline{G}(P)∩\overline{N} = N\overline{N}(P). But N\overline{N}(P) is of the form P B, where B is abelian, so N\overline{G}(P) is solvable, a contradiction since N\overline{G}(P) contains the nonabelian simple group T and subgroups of solvable groups must be solvable.

When (p, \overline{N}) ≅ (3, ^{2}G_{2}(3)′) or (5, ^{2}B_{2}(32)), by [12] we have [G : N] = p.

In verifying Conjecture 4.4 for the various possibilities for G we rely upon knowledge of the automorphism groups of covers of simple groups. The remainder of this chapter is devoted to overcoming the problem of deriving information concerning covering groups of the automorphism groups. We do this both by determining the Schur multipliers explicitly and by using reduction techniques developed in [7]. The aim is to show that we may assume L = N.

We first deal with the cases where [G : N] is divisible by p.

Lemma 6.8 Consider Hypotheses 6.3. If (p, \overline{N}) ≅ (3, ^{2}G_{2}(3)′) or (5, ^{2}B_{2}(32)), then Z(G) = 1.

Proof By [12] the Schur multiplier M(\overline{N}) is trivial in these cases, and [\overline{G} : \overline{N}] = p. Then since \overline{G}/\overline{N} is cyclic, by Lemma 2.12 M(\overline{G}) = 1 and the result follows.

We now assume that p |̸[G : N].

Lemma 6.9 Consider Hypotheses 6.3, and suppose also that p |̸[G : N]. If O_{p}(Z(G)) ≠ 1, then L = N.

Proof By Lemma 2.11 if O_{p}(M(\overline{N})) = 1, then O_{p}(M(\overline{G})) = 1, so we need only consider those cases where O_{p}(M(\overline{N})) ≠ 1. By [12] this occurs only when (p, \overline{N}) ≅ (2, P SL_{2}(4)), (2, ^{2}B_{2}(8)), (3, P SL_{2}(9)) or (3, P SL_{3}(4)).

If G/N is cyclic, then by Lemma 2.12 M(\overline{G}) is a homomorphic image of M(\overline{N}). But Z(L) ≤ Z(G), so in this case Z(L) = Z(G), and L = N as required.

Hence we may assume that G/N is noncyclic. This occurs only when (p, \overline{G}) ≅ (3, P SL_{2}(9)) or (3, P SL_{3}(4)). We use the MAGMA package to compute the Schur multipliers of the relevant automorphism groups of P SL_{2}(9) and P SL_{3}(4). I would like to thank Derek Holt for suggesting a set of generators for P SL_{2}(4).2^{2}, which we give in Appendix B. P SL_{2}(9).2^{2} may be obtained as P ΓL_{2}(9).

\begin{matrix}\overline{G} & M(\overline{G}) & \overline{G} & M(\overline{G}) \\ P SL_{2}(9) & C_{6} & P SL_{3}(4) & C_{4} × C_{4} × C_{3} \\ P SL_{2}(9).2_{1} & C_{2} & P SL_{3}(4).2_{1} & C_{2} × C_{2} × C_{3} \\ P SL_{2}(9).2_{2} & C_{2} & P SL_{3}(4).2_{2} & C_{4} \\ P SL_{2}(9).2_{3} & C_{3} & P SL_{3}(4).2_{3} & C_{4} \\ P SL_{2}(9).2^{2} & C_{2} × C_{2} & P SL_{3}(4).2^{2} & C_{2} × C_{2} \end{matrix}

Hence when G/N is noncyclic, we have O_{p}(Z(G)) = 1 and we are done.

The following is essentially Lemma 3.5 of [7]. We leave the precise details of the proof to the reader since most of the methods have been used in Chapter 3.

Lemma 6.10 Let B be a p-block of a finite group X with Z(X) p-regular. Let V be a cyclic p′-group containing Z(X) and let A = X ⋆ V be a central product with V ∩ X = Z(X). Suppose that B_{1} is a p-block of A covering B. Then Conjecture 4.4 holds for B if and only if it holds for B_{1}.

Proof Fix an integer d. The irreducible characters in B_{1} all lie over a single μ ∈ Irr(V/Z(X)) by [22, V.2.3]

Since V ≤ Z(A) and (p, [A : X]) = 1 we may identify R(X)/X and R(A)/A, and N_{A}(σ) = N_{X}(σ) ⋆ V for each σ ∈ R(X), with N_{X}(σ) ∩ V = Z(X).

Fix σ ∈ R(X). Since N_{A}(σ)/N_{X}(σ) is cyclic and each χ ∈ Irr(N_{X}(σ)) is N_{A}(σ)- stable, by Clifford’s Theorem we have a 1-1 correspondence

Irr_{d}(N_{A}(σ), μ) ↔ Irr_{d}(N_{X}(σ)).

By Lemma 3.14 this takes V_{σ}-projective characters to V_{σ}-projective characters. It is clear from the proof of Proposition 3.10 that this correspondence is compatible with p-blocks and with the Brauer correspondence. Hence we have a 1-1 correspondence

Irr_{d}(N_{A}(σ), B_{1}) ↔ Irr_{d}(N_{X}(σ), B)

and w_{d}(N_{A}(σ), B_{1}, V_{σ}) = w_{d}(N_{X}(σ), B, V_{σ}). The result follows.

Suppose that G satisfies Hypotheses 6.3 and that (p, [G : N]) = 1, O_{p}(Z(G)) = 1. By [12] G/N is a product of two cyclic p′-groups. Let N ◁ T ◁ G, where T/N and G/T are cyclic. Write W for the central extension of T/Z(G) by T′ ∩ Z(G) such that W ≤ T. We have Z(L) ≤ Z(W) ≤ Z(G).

Lemma 6.11 Let G be as above. Then there is a finite group H satisfying (i) Z(H) ≤ L ◁ H, (ii) O_{p′}(H) = Z(H) ≤ H′, Z(H) is cyclic and O_{p}(H) = 1, (iii) C_{H}(L) = Z(H) = Z(L), (iv) H/L is a p′-group,

(v) Conjecture 4.4 holds for G if and only if it holds for H.

Proof This is based upon [7, 3.6].

Choose u ∈ G such that uW Z(G) = uT generates the cyclic p′-group G/W Z(G) = G/T of order m. Hence u^{m} ∈ W Z(G), say u^{m} = wz where w ∈ W, z ∈ Z(G).

Let V be a cyclic p′-group such that Z(G) ≤ V, and containing an element v ∈ V such that v^{m} = z. Write A = G ⋆ V, where we identify Z(G) with G ∩ V.

Setting h = uv^{−1} ∈ A, define H = ⟨W, h⟩ ◁ A. Now

\frac{A}{W V} = \frac{GV}{W V} ≅ \frac{G}{G ∩ W V} = \frac{G}{W Z(G)}

and hW V has order m, so hW V generates A/W V. Hence A = H ⋆ V with Z(H) and V ∩ H identified.

Clearly L ◁ W. Since h = uv^{−1} and v ∈ Z(A), conjugation by h induces the same (outer) automorphism on L as does u. Hence

C_{H}(L) = C_{W} (L) = Z(W) = W ∩ Z(G) = W ∩ (G ∩ V) = W ∩ V = H ∩ V = Z(H).

Now by Lemma 2.12 Z(W) = Z(L), so Z(H) = Z(L) as required. It is clear that H/L is a p′-group, O_{p′}(H) = Z(H) = H ∩ V is cyclic and O_{p}(H) = 1.

Hence the result follows from Lemma 6.10. To summarize, in order to prove Theorem 4.7, it suffices to assume Hypotheses 6.3

and L = N, i.e., G is an automorphism group of the quasisimple group N:

Proposition 6.12 Suppose that G is as in Hypotheses 6.3 and L = N. Then one of the following possibilities for (p, N) occurs:

(a) (p, P SL_{2}(p^{m})), (p, SL_{2}(p^{m})), (p, P SU_{3}(p^{m})), (p, SU_{3}(p^{m})), m > 1

(b) (2, ^{2}B_{2}(2^{2m+1}), m ≥ 1, (2, 2. ^{2}B_{2}(2^{3}), (2, 2.P SL_{2}(4)).

(c) (3,^{2}G_{2}(3^{m}), m > 1 odd, (3,^{2}G_{2}(3)′), (3, P SL_{3}(4)), (3, 2.P SL_{3}(4)), (3, 3.P SL_{3}(4)), (3, 4_{1}.P SL_{3}(4)), (3, 4_{2}.P SL_{3}(4)), (3, 6.P SL_{3}(4)), (3, 12_{1}.P SL_{3}(4)), (3, 12_{2}.P SL_{3}(4)), (3, 3.P SL_{2}(9)), (3, 6.P SL_{2}(9)) or (3, M_{11}).

(d) (5, ^{2}B_{2}(2^{5})), (5, ^{2}F_{4}(2)′), (5, McL) or (5, 3.McL).

(e) (11, J_{4}). Further, p|[G : N] only when (p, N) ≅ (3, ^{2}G_{2}(3)′) or (5, ^{2}B_{2}(2^{5})).

7 Checking the conjecture

7.1 Introduction

Over this chapter we verify the conjecture for the groups listed in Proposition 6.12.

Since some of the calculations have been performed in part or in full by Blau and Michler in [7], we give a summary of their results, and list the cases where Robinson’s conjecture follows directly from their work.

We assume Hypotheses 6.3 and assume (as we may) that N is perfect. Let B be a p-block of G with defect groups the Sylow p-subgroups of G (otherwise the conjecture holds trivially).

7.2 Some results of Blau and Michler

Although we endeavour to cite any results of [7] used as fully as possible, we feel that due to the closeness of the two sets of calculations some lapse is inevitable and so make our apologies in advance.

Proposition 7.1 (Blau and Michler [7]) Let G be a finite group with TI Sylow psubgroups. Let a = log_{p}(|G|_{p}) and let B be a p-block of G of defect a. Let P ∈ Syl_{p}(G). Then the following are true:

(a) k(G, B) = k(N_{G}(P), B).

(b) k_{a}(G, B) = k_{a}(N_{G}(P), B).

(c) l(G, B) = l(N_{G}(P), B).

Corollary 7.2 Let B be a p-block of a finite group G with TI Sylow p-subgroups. If B has abelian defect groups then B satisfies Conjecture 4.4.

Proof By Ito’s theorem and Proposition 7.1 k_{a}(G, B) = k_{a}(N_{G}(P), B) = k(N_{G}(P), B) = k(G, B).

Much of [7] is devoted to the calculation of the fixed points of the action of outer automorphisms on conjugacy classes of groups of Lie type. We summarize one of the main results here. Lemma 7.3 ( [7]) Let N = ^{2}G_{2}(3^{2m+1}), ^{2}B_{2}(2^{2m+1}) or P SU_{3}(p^{m}), where m > 0. Then the (outer) field automorphisms of N consist of automorphisms of the form τ^{r}, where τ is the Frobenius automorphism x → x^{p}. Suppose that r|2m + 1 in the first two cases or r|2m if N = P SU_{3}(p^{m}). Then τ^{r} fixes precisely p^{r} p-regular conjugacy classes of N.

In their proof of Proposition 7.1 Blau and Michler use some neat combinatorial results for dealing with the action of the outer automorphisms on the irreducible (ordinary or Brauer) characters of the group. We give these here for convenience, and make the trivial observation that the results can be extended to counting characters of any given height (given that the index is coprime to p). Lemma 7.4 (Lemma 2.2 of [7]) Let A be a finite group acting on two sets S_{1} and S_{2} as a permutation group. For H ≤ A, write f_{S_{i}}(H) = |{s ∈ S_{i}|s^{a} = s ∀ a ∈ H}| and m_{S_{i}}(H) = |{s ∈ S_{i}|C_{A}(s) = H}|.

If f_{S_{1}}(H) = f_{S_{2}}(H) for all H ≤ A then m_{S_{1}}(H) = m_{S_{2}}(H) for all H ≤ A. Proof This follows by an easy inductive argument on [A : H], given in [7]. Lemma 7.5 (Lemma 2.3 of [7]) Let H ≤ G◁E, where H ◁E and E/G is cyclic and p-regular. Suppose further that for a generator y ∈ E of E/G we have [E : G] = [H⟨y⟩ : H]. Let B be a p-block of G and b a p-block of H, with both B and b are stable under the action of y. Let d be an integer. Then if (in the notation of the previous lemma) f_{Irr_{d}(G,B)}(C) = f_{Irr_{d}(H,b)}(C) for each subgroup C of ⟨y⟩, then k_{d}(E, B_{1}) = k_{d}(H⟨y⟩, b_{1}), where B_{1}, b_{1} is the collection of p-blocks of E, H⟨y⟩ covering B, b respectively.

The analagous result holds when we consider Brauer characters instead of ordinary characters.

Proof This is a standard application of Clifford’s theorem using Lemma 7.4, and is given in [7].

7.3 Sporadic simple groups of p-local rank one

Proposition 7.6 Conjecture 4.4 holds for G when (p, \overline{N}) ≅ (3, M_{11}), (5, McL), and (11, J_{4}).

Proof If (p, \overline{N}) ≅ (3, M_{11}), then G = \overline{N} since M(M_{11}) = Out(M_{11}) = 1. M_{11} has abelian Sylow 3-subgroups , so Conjecture 4.4 holds in this case by Corollary 7.2.

If (p, \overline{N}) ≅ (11, J_{4}), then again we have G = \overline{N}. By Ostermann [41] the principal 11-block B_{0} is the only 11-block of J_{4} of positive defect. Let P ∈ Syl_{p}(G), so |P | = 11^{3}. By Ostermann’s character table for N_{G}(P) ( [41, pp69]) k_{3}(N_{G}(P), B_{0}) = 42, k_{2}(N_{G}(P), B_{0}) = 7 and k(N_{G}(P), B_{0}) = 49. By [12, pp.188] k_{3}(G, B_{0}) = 42, k_{2}(G, B_{0}) = 7 and k(G, B_{0}) = 49. So Conjecture 4.4 holds in this case.

Now suppose that (p, \overline{N}) ≅ (5, McL). The conjecture has been verified independently in this case by Murray [36]. By [12] the outer automorphisms of McL do not act trivially on the centre of 3.McL, so by Remark 6.4 we do not have to consider the group 3.McL.2. Hence G ≅ McL, 3.McL or McL.2. First suppose that G ≅ McL. By [41] McL has just one p-block B_{0} of positive defect. It follows from the character tables given in [41] and [12] that for P ∈ Syl_{5}(G) we have k_{3}(G, B_{0}) = 13 = k_{3}(N_{G}(P), B_{0}) and k_{2}(G, B_{0}) = 6 = k_{2}(N_{G}(P), B_{0}) (this accounting for all the irreducible characters of B_{0} and its Brauer correspondent in N_{G}(P)).

Suppose that G ≅ 3.McL. Then N_{G}(P) has the form 3.5_{+}^{1+2} : 3 : 8, where 5_{+}^{1+2} is the non-abelian group of order 5^{3} and exponent 25. Now C_{G}(P) ≅ 3.5_{+}^{1+2}, so G has three 3-blocks B_{0}, B_{1}, B_{2} of positive defect, each lying over a distinct λ ∈ Irr(Z(G)), B_{0} the principal 3-block. By [12, p.101] k_{3}(G, B_{i}) = k_{3}(G, B_{0}) = 13, k_{2}(G, B_{i}) = k_{2}(G, B_{0}) = 6 and k(G, B_{i}) = k(G, B_{0}) = 19 for each i. By Lemma 2.10 and Lemma 2.14 we have M(N_{G}(P)/Z(G)) = 1, so each λ ∈ Irr(Z(G)) extends to an irreducible character of N_{G}(P), and irr(N_{G}(P), B_{i}) is the same for each i. Since irr(N_{G}(P), B_{0}) = irr(N_{G}(P)/Z(G), B_{0}) and irr(G, B_{0}) = irr(G/Z(G), B_{0}), it follows that Conjecture 4.4 holds each each B_{i}.

It remains to demonstrate the conjecture for G ≅ McL.2. This has just one 5block B_{0} of positive defect. The character degrees of N_{G}(P) are given in full in [7] (irr(N_{G}(P), B_{0}) = {8 × 1, 10 × 2, 4 × 20, 2 × 24, 2 × 40}). Comparing these to the character degrees given for G in [12] gives the required result.

7.4 Groups of Lie type of p-local rank one

Proposition 7.7 G satisfies Conjecture 4.4 when (p, \overline{N}) ≅ (3,^{2}G_{2}(3)′) or (5,^{2}B_{2}(2^{5})).

Proof If (p, \overline{N}) ≅ (5, ^{2}B_{2}(2^{5}), then G ≅ ^{2}B_{2}(2^{5}) or ^{2}B_{2}(2^{5}).5. Now ^{2}B_{2}(2^{5}) has cyclic Sylow 5-subgroups, so the conjecture holds when G ≅ ^{2}B_{2}(2^{5}) by Proposition 5.7. Let G ≡ ^{2}B_{2}(2^{5}).5 and P ∈ Syl_{5}(G). By [12, p.77] N_{G}(P) has the form C_{25} : C_{20} and C_{20} acts on C_{25} with orbits of length 1, 4 and 20. Hence by Brauer’s theorem ( [14, 11.9]) C_{20} acts on Irr(C_{25}) with three orbits, which by counting must have lengths 1, 4 and 20. Hence by Clifford’s Theorem irr(N_{G}(P)) = {20 × 1, 5 × 4, 1 × 20}. We have C_{G}(P) ≤ P, so G has only one 5-block B_{0} of positive defect. By the character table in [12] we get k_{3}(G, B_{0}) = 25, k_{2}(G, B_{0}) = 1 and k(G, B_{0}) = 26, so Conjecture 4.4 holds for G in this case.

If (p, \overline{N}) ≅ (3, ^{2}G_{2}(3)′), then G ≅ ^{2}G_{2}(3)′ or ^{2}G_{2}(3). Now ^{2}G_{2}(3)′ has cyclic Sylow 3-subgroups, so the 3-blocks of ^{2}G_{2}(3)′ satisfy Conjecture 4.4. Let G ≡ ^{2}G_{2}(3) and P ∈ Syl_{3}(G). By [12, p.6] N_{G}(P) has the form C_{9} : C_{6} and C_{6} acts on C_{9} with orbits of length 1, 2 and 6. Again by Brauer’s theorem C_{6} acts on Irr(C_{9}) with three orbits, which by counting must have lengths 1, 2 and 6. Hence by Clifford’s Theorem irr(N_{G}(P)) = {6 × 1, 3 × 2, 1 × 6}. We have C_{G}(P) ≤ P, so G has only one 3-block B_{0} with defect group P. By the character table in [12] we get k_{3}(G, B_{0}) = 9, k_{2}(G, B_{0}) = 1 and k(G, B_{0}) = 10, so Conjecture 4.4 holds for G in this case.

Proposition 7.8 Conjecture 4.4 holds for G when (p, \overline{N}) ≅ (2, ^{2}B_{2}(2^{2m+1})), m ≥ 1.

Proof In this case G ≅ ^{2}B_{2}(2^{2m+1}).C_{t}, where t|2m + 1, or G ≅ 2. ^{2}B_{2}(2^{3}) (there is no finite group of the form 2. ^{2}B_{2}(2^{3}).3). Write q = 2^{2m+1}. The first part of the proof is [7, 6.3].

First suppose that G ≅ ^{2}B_{2}(2^{2m+1}).C_{t}, and let P ∈ Syl_{2}(G). By [54, p.107] and since the outer automorphisms are field automorphisms, C_{G}(P) ≤ P and so G has only one 2-block B_{0} of positive defect. By [54, pp.126,142], irr(N_{N}(P)) = {q − 1 × 1, 1 × q − 1, 2 × 2^{m}(q − 1)}, and by [54, p.141] irr(N) = {1 × 1, 1 × q^{2}, q/2 − 1 × q^{2} + 1, (q + 2^{m+1})/4 × (q − 2^{m+1} + 1)(q − 1), (q − 2^{m+1})/4 × (q + 2^{m+1} + 1)(q − 1), 2 × 2^{m}(q − 1)} (the only 2-block of defect zero being that consisting of the Steinberg character).

Write τ : x → x^{2} for the Frobenius map, and let r|2m + 1 so that τ^{r} generates a subgroup of C_{t}. Note that every 2-regular outer automorphism of N can be described in such a way. By Lemma 7.5 it suffices to show that f_{Irr_{d}(N,B_{0})}(⟨τ^{r}⟩) = f_{Irr_{d}(N_{N} (P),B_{0})}(⟨τ^{r}⟩) for each choice of τ^{r}.

By Lemma 7.3 τ^{r} fixes 2^{r} 2-regular conjugacy classes of N. By examination of the character table τ^{r} fixes the 2-singular conjugacy classes of N. Hence τ^{r} fixes 2^{r} + 3 conjugacy classes of N, and so 2^{r} + 3 irreducible characters (including the Steinberg character) by Brauer’s theorem. Hence f_{Irr_{4m+2}(N,B_{0})}(⟨τ^{r}⟩) = 2^{r}, since τ^{r} must fix the three irreducible characters of N of even degree.

Now N_{N}(P) = P W where W ≅ 𝔽_{q}^{×}. Hence τ^{r} fixes 2^{r} − 1 of the q − 1 2-regular conjugacy classes of N_{N}(P), and so 2^{r} − 1 Brauer characters of N_{N}(P). But τ^{r} fixes 2^{r} −1 Brauer characters of N (excluding the Steinberg character), so f_{IBr(N,B_{0})}(⟨τ^{r}⟩) = f_{IBr(N_{N} (P),B_{0})}(⟨τ^{r}⟩) and by Lemma 7.5 l(G, B_{0}) = l(N_{G}(P), B_{0}). Hence by Lemma 2.9 k(G, B_{0}) = k(N_{G}(P), B_{0}). Since all but one of the irreducible characters of height one of N_{N}(P) are linear, it follows that f_{Irr_{4m+2}(N_{N} (P),B_{0})}(⟨τ^{r}⟩) = 2^{r}. So by Lemma 7.5 k_{4m+2}(G, B_{0}) = k_{4m+2}(N_{G}(P), B_{0}). The result follows in this case since all positive height charaters of N, N_{N}(P) have the same defect.

Finally suppose that G ≅ 2. ^{2}B_{2}(2^{3}). Again we have just one 2-block B_{0} of positive defect. Let Irr(Z(G)) = {1, λ} and P ∈ Syl_{2}(G). We have already shown that Conjecture 4.4 holds for B_{0} and the character 1 ∈ Irr(Z(G)). It remains to verify that k_{d}(G, B_{0}, λ) = k_{d}(N_{G}(P), B_{0}, λ) for each d. By [12, p.28] N_{G}(P) has the structure 2.2^{3+3} : 7. Using the MAGMA package we may construct the character tables of P and P/Z(G) (see Appendix B for details), to obtain irr(P, λ) = {8}. By Clifford’s theorem this gives irr(N_{G}(P), B_{0}, λ) = {7 × 8}. By [12] k_{4}(G, B_{0}, λ) = 7 = k(G, B_{0}, λ) and we are done.

Proposition 7.9 G satisfies Conjecture 4.4 when (p, \overline{N}) ≅ (2, P SL_{2}(4)), (2, P SL_{2}(9)) and (3, P SL_{3}(4)).

Proof In each of these cases P ∈ Syl_{p}(G) has order p^{3} and |O_{p}(Z(G))| = p, so that P is extraspecial. Let B be a p-block of G with P as a defect group. Write \overline{G} = G/O_{p}(Z(G)) and let \overline{B} be the block of \overline{G} corresponding to B. Since P/O_{p}(Z(G)) is abelian, Conjecture 4.4 holds for \overline{B} by Corollary 7.2. Hence by Lemma 2.9, k(G, B, λ) = k(N_{G}(P), B, λ) for each λ ∈ Irr(O_{p}(Z(G))). Now

irr(P, λ) = \{ \begin{matrix}\{p^{2} × 1\} & if λ = 1 \\ \{p\} & if λ ≠ 1 \end{matrix}

so by Clifford’s theorem k(N_{G}(P), λ) = k_{3}(N_{G}(P), λ) if λ = 1 and k(N_{G}(P), λ) = k_{2}(N_{G}(P), λ) if λ ≠ 1.

From the character tables at [12, pp.2,4,24] we see that k(G, B, λ) = k_{3}(G, B, λ) if λ = 1 and k(G, B, λ) = k_{2}(G, B, λ) if λ ≠ 1. Hence k_{3}(G, B, λ) = k_{3}(N_{G}(P), B, λ) and k_{2}(G, B, λ) = k_{2}(N_{G}(P), B, λ) for each λ ∈ Irr(O_{p}(Z(G))) as required.

Proposition 7.10 G satisfies Conjecture 4.4 when (p, \overline{N}) ≅ (p, P SL_{2}(p^{m})) or (5, 2F_{4}(2)′), m > 1.

Proof In each case G has abelian Sylow p-subgroups and the result follows by Lemma 7.2.

7.5 Ree groups of type G_{2}

Our aim is to prove the following:

Proposition 7.11 Every 3-block of G satisfies Conjecture 4.4 when (p, \overline{N}) ≅ (3, 2G_{2}(3^{2m+1}), m > 0.

Note that \overline{N} = N. Blau and Michler use Ward [57] in their proof [7] of Proposition 7.1. However, since 3-blocks (of positive defect) of N contain irreducible characters of three distinct heights we need more exact information on the character degrees. To do this we return to Ree’s original description of N as a twisted Chevalley group.

We follow Carter [10] in our desciption of N = ^{2}G_{2}(q), where q = 3^{2m+1}, m > 0.

Lemma 7.12 Let P ∈ Syl_{3}(N). Consider the automorphism of 𝔽_{q} defined by λ ↦ λ^{f}, where f = 3^{m}.

P has order q^{3}, and we may label its elements as x(t, u, v), where t, u, v ∈ 𝔽_{q}, with multiplication given by

x(t_{1}, u_{1}, v_{1})x(t_{2}, u_{2}, v_{2}) = x(t_{1} + t_{2}, u_{1} + u_{2} − t_{1}t_{2}^{3f}, v_{1} + v_{2} − t_{2}u_{1} + t_{1}t_{2}^{3f+1} − t_{1}^{2}t_{2}^{3f}).

Z(P) = {x(0, 0, v) | v ∈ 𝔽_{q}} and |Z(P)| = q. The subgroup

P_{1} = {x(0, u, v) ∈ P | u, v ∈ 𝔽_{q}}

of order q^{2} is normal in P, P/P_{1} is elementary abelian and P_{1} = P′.

Proof The description of the elements and their multiplication follows from [10]. This gives x(t, u, v)^{−1} = x(−t, −u − t^{3f+1}, −v − tu + t^{3f+2}) ∀ t, u, v ∈ 𝔽_{q}, so we see that conjugation in P is given by

x(t_{1}, u_{1}, v_{1})x(t_{2}, u_{2}, v_{2})x(t_{1}, u_{1}, v_{1})^{−1}

= x(t_{2}, u_{2} − t_{1}t_{2}^{3f} + t_{2}t_{1}^{3f}, v_{2} − t_{2}u_{1} + t_{1}u_{2} + t_{1}t_{2}^{3f+1} + t_{1}^{3f+1}t_{2} + t_{1}^{2}t_{2}^{3f} + t_{1}^{3f}t_{2}^{2}).

Suppose that x(t_{2}, u_{2}, v_{2}) ∈ Z(P). Then

−t_{1}t_{2}^{3f} + t_{2}t_{1}^{3f} = 0,
−t_{2}u_{1} + t_{1}u_{2} + t_{1}t_{2}^{3f+1} + t_{1}^{3f+1}t_{2} + t_{1}^{2}t_{2}^{3f} + t_{1}^{3f}t_{2}^{2} = 0

∀ x(t_{1}, u_{1}, v_{1}) ∈ P, so t_{2} = u_{2} = 0. Hence

Z(P) = {x(0, 0, v) | v ∈ 𝔽_{q}}

and |Z(P)| = q as required. The last part is clear.

Lemma 7.13 P has q conjugacy classes of length 1, q − 1 of length q and 3(q − 1) of length \frac{1}{3}q^{2}.

Proof

x(t_{1}, u_{1}, v_{1})x(0, u_{2}, 0)x(t_{1}, u_{1}, v_{1})^{−1} = x(0, u_{2}, t_{1}u_{2}),

so for each u_{2} ∈ 𝔽_{q}^{×}, x(0, u_{2}, 0) is conjugate to all elements x(0, u_{2}, v) ∈ P where v ∈ 𝔽_{q}. In this way we obtain q − 1 conjugacy classes of length q, each corresponding to a distinct value of u_{2} in 𝔽_{q}^{×}. It is clear that we have q conjugacy classes of length one, corresponding to the elements of Z(P). We have thus accounted for every element of P_{1}.

Suppose now that x(t_{2}, u_{2}, v_{2}) ∈ P with t_{2} ≠ 0. If x(t_{1}, u_{1}, v_{1}) ∈ C_{P} (x(t_{2}, u_{2}, v_{2})), then t_{2}t_{1}^{3f} = t_{1}t_{2}^{3f}. Let ϕ : 𝔽_{q} → 𝔽_{q} be the additive endomorphism sending λ to t_{2}λ^{3f} − λt_{2}^{3f}. If 0 ≠ λ ∈ ker(ϕ), then λ^{3f} = λt_{2}^{3f−1}. Without loss of generality, assume t_{2}^{3f−1} = 1, so λ^{3f} = λ. Hence

λ^{3^{m+1}−1} = 1 = λ^{q−1} = λ^{3^{m}(3^{m+1}−1)+3^{m}−1}

and

1 = λ^{q−1} = λ^{3(3^{m}−1)(3^{m}+1)+2} = λ^{2},

so |ker(ϕ)| = 3. This gives |C_{P} (x(t_{2}, u_{2}, v_{2}))| = 3q since we can choose u_{1} to give t_{2}u_{1} any value in 𝔽_{q}. Hence we have found 3(q − 1) conjugacy classes of length \frac{1}{3}q^{2}. This accounts for all the elements of P.

We use the action of P on Irr(P_{1}) to calculate the nature and degrees of the irreducible characters of P.

Lemma 7.14 irr(P) = {q × 1, (q − 1) × q, 3(q − 1) × 3^{m}}.

Proof Considering 𝔽_{q} as an n = (2m + 1)-dimensional vector space over 𝔽_{3}, write ζ ∈ 𝔽_{q} as ζ = (ζ_{1}, …, ζ_{n}). Write the elements of Irr(P_{1}) as χ_{x,y}, x, y ∈ 𝔽_{q}, where, for x(0, u, v) ∈ P_{1},

χ_{x,y} (x(0, u, v)) = ∏_{k=1}^{n} e^{\frac{2u_{k}x_{k}πi}{3}} ∏_{j=1}^{n} e^{\frac{2v_{j} y_{j} πi}{3}} .

Recall that

x(t_{1}, u_{1}, v_{1})x(0, u_{2}, 0)x(t_{1}, u_{1}, v_{1})^{−1} = x(0, u_{2}, t_{1}u_{2}) ∀ t_{1}, u_{1}, v_{1}, u_{2} ∈ 𝔽_{q}.

From this we see that for any x ∈ 𝔽_{q}, we have I_{P} (χ_{x,0}) = P and that for any x ∈ 𝔽_{q}, y ∈ 𝔽_{q}^{×}, we have I_{P} (χ_{x,y}) = P_{1}. This gives us q orbits of length one and q − 1 orbits of length q. Representatives of the long orbits are χ_{0,y}, y ∈ 𝔽_{q}^{×}.

By Clifford theory representatives of the long orbits induce to give distinct irreducible characters of P of degree q. χ_{0,0} = 1_{P_{1}} extends to P in q distinct ways to give q linear characters of q. Since P_{1} is the derived subgroup of P, every linear character of P is an extension of χ_{0,0}.

Now consider the characters χ_{x,0}, where x ∈ 𝔽_{q}^{×}. Although these characters are stable under the action of P, they do not extend to irreducible characters of P. We show that all irreducible characters of P covering such characters of P_{1} have the same degree.

Fix μ = χ_{x,0}, x ∈ 𝔽_{q}^{×}. Since I_{P} (χ_{x,0}) = P, by Lemma 3.6 there is a degree preserving 1-1 correspondence Irr(P, μ) ↔ Irr({ \atop (P/P_{1})}, μ̂), where { \atop (P/P_{1})} is a central extension of

P/P_{1} and μ̂ ∈ Irr(Z({ \atop (P/P_{1})})).

Now by Lemma 3.6 we may take Z({ \atop (P/P_{1})}) to be a cyclic p-group. However, by Lemma 2.14 the Schur multiplier of an elementary abelian group is itself elementary abelian and so we have |Z({ \atop (P/P_{1})})| = 3. Note that μ̂ is non-trivial since we may not extend μ to P. This demonstrates that irr(P, χ_{x,0}) is independent of the choice of x ∈ 𝔽_{q}^{×}.

Since P possesses 5q − 4 conjugacy classes, we must have |irr(P, χ_{x,0})| = 3, and all irreducible characters of P lying over χ_{x,0} have the same degree, which must then be 3^{m}, as required.

We examine the action of N_{N}(P) on the irreducible characters of P to calculate the irreducible character degrees of N_{N}(P). It is interesting to note that by using Ward’s character table for G (see [57]) and the fact that the non-blockwise version of the Knörr-Robinson reformulation of Alperin’s weight conjecture is known to hold for finite groups with split BN-pairs in the defining characteristic, we may already observe that N_{N}(P) possesses q + 7 conjugacy classes.

Lemma 7.15 N_{N}(P) has irreducible characters of degrees 1, q −1, 3^{m}(q −1)/2, 3^{m}(q − 1), q(q − 1) with multiplicity q − 1, 1, 4, 2, 1 respectively.

Proof As noted at [57, p.63], N_{N}(P) = P W, where W is cyclic of order q −1 and acts fixed-point freely on P/P_{1}. Hence W acts transitively on the q − 1 non-trivial linear characters of P, giving q−1 linear characters of N_{N}(P) and one irreducible character of degree q − 1. The irreducible characters of degree q of P are also permuted transitively by W, giving one irreducible character of degree q(q − 1). Since k(N_{N}(P)) = q + 7 we are left with k_{5m+3}(N_{N}(P)) = 6. By Ree [44], W possesses an unique involution, h_{0}, and C_{P} (h_{0}) = {x(0, u, 0)|u ∈ 𝔽_{q}}. W acts transitively on Z(P) and P/P_{1}, and W/ < h_{0} > acts on P_{1}/Z(P) with 2 orbits of length (q − 1)/2 and one of length one. Recall that each χ_{x,0} with x ∈ 𝔽_{q}^{×} is covered by three irreducible characters of P. It follows that the irreducible characters of N_{N}(P) of height 3^{m} consist of two of degree 3^{m}(q − 1) and four of degree 3^{m}(q − 1)/2.

Proof of Proposition 7.11

By Ward [57, p.63], C_{N}(P) ≤ P. Since the outer automorphisms of N are field automorphisms, C_{G}(P) ≤ P. So G possesses only one 3-block B_{0} of positive defect.

Ward [57, p.85] gives k(N) = q + 8 and from the character table at [57, p.87], we have k_{6m+3}(N, B_{0}) = q, k_{5m+3}(N, B_{0}) = 6 and k_{4m+2}(N, B_{0}) = 1

Let τ be the Frobenius map as in Lemma 7.3 and let r be an integer dividing 2m+1 such that τ^{r} generates a subgroup G/N. We calculate the number of characters of N, N_{N}(P) of each height fixed by τ^{r}. Note that we are not assuming that τ^{r} is 3-regular (this will be important in Chapter 9). By Lemma 7.3 τ^{r} fixes 3^{r} 3-regular conjugacy classes of N. Examination of the character table given in [57] reveals that τ^{r} fixes every 3-singular conjugacy class of N, hence τ^{r} fixes 3^{r} + 8 conjugacy classes in total. So f_{Irr(N,B_{0})}(⟨τ^{r}⟩) = 3^{r} + 7 by Brauer’s theorem. Further examination of the character table and using the fact that τ^{r} has odd order reveals that τ fixes every irreducible character of positive height, and so 3^{r} irreducible characters of height zero (excluding the Steinberg character), i.e., f_{Irr_{6m+3}(N,B_{0})}(⟨τ^{r}⟩) = 3^{r}, f_{Irr_{5m+3}(N,B_{0})}(⟨τ^{r}⟩) = 6 and f_{Irr_{4m+2}(N,B_{0})}(⟨τ^{r}⟩) = 1.

Now consider the action of τ^{r} on Irr(N_{N}(P)). The 3-regular conjugacy classes of N_{N}(P) = P W are represented by the elements of W, and so τ^{r} fixes 3^{r} − 1 3-regular conjugacy classes of N_{N}(P). So f_{IBr(N_{N} (P),B_{0})}(⟨τ^{r}⟩) = 3^{r} − 1 and τ^{r} fixes 3^{r} − 1 linear characters of N_{G}(P). Clearly τ^{r} fixes the unique irreducible character of degree q − 1, so f_{Irr_{6m+3}(N_{N} (P),B_{0})}(⟨τ^{r}⟩) = 3^{r}.

It remains to examine the action on the characters of N_{N}(P) of positive height. Clearly the unique irreducible character of degree q(q − 1) is fixed by τ^{r}, as are the two of degree 3^{m}(q − 1) since τ^{r} has odd order. Finally it is clear from the construction of the remaining characters that τ^{r} cannot act on the four irreducible characters of degree 3^{m}(q − 1)/2 with an orbit of length three, and so τ^{r} fixes these also. The result then follows from Lemma 7.5.

7.6 The unitary groups of degree 3 in the defining characteristic

Our aim is to prove the following: Proposition 7.16 Every p-block of G satisfies Conjecture 4.4 when \overline{N} ≅ P SU_{3}(p^{m}), m > 1.

Our approach is similar to that used in verifying the conjecture for the Ree groups in defining characteristic. Again we use Carter [10] to view N as a twisted Chevalley group and so calculate the irreducible character degrees for the normalizer for a Sylow p-subgroup of N, and compare the numbers obtained with those from the character tables for N found in Simpson and Frame [52].

Let q = p^{m}, m > 1. Lemma 7.17 Let P ∈ Syl_{p}(N). Consider the automorphism of 𝔽_{q^{2}} defined by λ ↦ λ^{f}, where f = q.

P has order q^{3}, and we may label its elements as x(t, u), where t, u ∈ 𝔽_{q^{2}} and u + u^{f} = t^{f+1}, with multiplication given by

x(t_{1}, u_{1})x(t_{2}, u_{2}) = x(t_{1} + t_{2}, u_{1} + u_{2} − t_{1}^{f}t_{2}).

Z(P) = {x(0, u) | u^{f} = −u} and |Z(P)| = q.

Proof Note that N ≅^{2}A_{2}(q^{2}), and construct G as a twisted A_{2}(q^{2}) following Carter [10].

The characterization and multiplication of the elements of P follow from [10], so that

x(t, u)^{−1} = x(−t, −u − t^{f+1}) ∀ x(t, u) ∈ P,

and conjugation in P is given by

x(t_{1}, u_{1})x(t_{2}, u_{2})x(t_{1}, u_{1})^{−1} = x(t_{2}, u_{2} + t_{2}^{f}t_{1} − t_{1}^{f}t_{2}).

Denote by ϕ : 𝔽_{q^{2}} → 𝔽_{q^{2}} the additive homomorphism given by

ϕ(λ) = λ + λ^{f}.

|ker(ϕ)| = q, so |P | = q^{3}.

Suppose that x(t_{2}, u_{2}) ∈ Z(P). Then t_{2}^{f}t_{1} = t_{1}^{f}t_{2} for all t_{1} ∈ 𝔽_{q^{2}}. So t_{2} = 0 and u_{2} ∈ ker(ϕ). Hence |Z(P)| = q.

Lemma 7.18 P has q conjugacy classes of length one and q^{2} − 1 of length q.

Proof In Lemma 7.17 we saw that Z(P) has order q and consists of elements of the form x(0, u), where u^{f} = −u. This gives us the q conjugacy classes of length one.

Now suppose that x(t_{2}, u_{2}) ∈ P, t_{2} ≠ 0. Suppose that x(t_{1}, u_{1}) ∈ C_{P} (x(t_{2}, u_{2})). Then t_{2}^{f}t_{1} = t_{1}^{f}t_{2}. Assume that t_{1} ≠ 0, then we want t_{1}^{f−1} = t_{2}^{f−1}. This has q − 1 solutions for each t_{2}, and so t_{1} can take q values in 𝔽_{q^{2}}. So |C_{P} (x(t_{2}, u_{2}))| = q^{2}, and we have found the remaining q^{2} − 1 conjugacy classes.

We use the action of P on Irr(Z(P)) to calculate the degrees of the irreducible characters of P.

Lemma 7.19 irr(P) = {q^{2} × 1, (q − 1) × q} Proof Consider 𝔽_{q^{2}} as an n = 2m-dimensional vector space over 𝔽_{p}, and write ζ ∈ 𝔽_{q^{2}} as ζ = (ζ_{1}, …, ζ_{n}). Denote the elements of Irr(Z(P)) as χ_{y}, y ∈ 𝔽_{q^{2}}, where, for x(0, u) ∈ Z(P) we have

χ_{y}(x(0, u)) = ∏_{k=1}^{n} e^{\frac{2u_{k}y_{k}πi}{p}} .

Each χ_{y} is stable under the action of P.

Note that for all x(t_{1}, u_{1}), x(t_{2}, u_{2}) ∈ P, we have

[x(t_{1}, u_{1}), x(t_{2}, u_{2})] = x(0, t_{2}^{f}t_{1} − t_{1}^{f}t_{2}),

so P′ = Z(P). Hence P has q^{2} linear characters, namely those which are extensions of χ_{0}.

Fix μ = χ_{y}, y ∈ 𝔽_{q^{2}}^{×}. Since I_{P} (χ_{y}) = P, by Lemma 3.6 there is a degree-preserving 1-1 correspondence Irr(P, μ) ↔ Irr({ \atop (P/Z(P))}, μ̂), where { \atop (P/Z(P))} is a central extension of P/Z(P), μ̂ ∈ Irr(Z({ \atop (P/Z(P))})) and Z({ \atop (P/Z(P))}) is a cyclic p-group. As in Lemma 7.14, P/Z(P) is elementary abelian and so has elementary abelian Schur multiplier by Lemma 2.14. Hence |Z({ \atop (P/Z(P))})| = p. Again since we cannot extend μ to an irreducible character of P, we have μ̂ ≠ 1 and so irr(P, χ_{y}) is independent of the choice of y ∈ 𝔽_{q^{2}}^{×}. Hence all the non-linear irreducible characters of P have the same degree and the result follows.

Let d = (3, q + 1). If N = P SU_{3}(q) then by [52, p.487] C_{N}(P) ≤ P, and N has just one p-block of positive defect. Out(N) consists of diagonal and field automorphisms. By [7, p.436] and since a nontrivial field automorphism cannot centralize P, C_{G}(P) ≤ P and G has just one p-block of positive defect. If N = SU_{3}(q), then G has d p-blocks of positive defect since |Z(G)| = d, which is coprime to p. Each p-block of positive defect consists only of irreducible characters lying over a distinct irreducible character of Z(G). So the principal p-block of G/Z(G) possesses precisely the same irreducible characters as the principal p-block of G (after factoring out the kernel of course), and similarly for N_{G/Z(G)}(Z(G)P/Z(G)) and N_{G}(P). Hence it suffices in proving Proposition 7.16 to assume that N = SU_{3}(q), which we do.

Let B_{λ} be the p-block of G containing only irreducible characters lying over λ ∈ Irr(Z(G)).

Lemma 7.20 Each p-block of N_{N}(P) possesses irreducible characters of degrees 1, (q^{2}− 1)/d, q(q − 1) with multiplicities (q^{2} − 1)/d, d, (q + 1)/d respectively.

Proof Write \overline{N} = N/Z(N). By [28, II.10.12], N\overline{N}(\overline{P}) = \overline{N_{N}(P)} = W \overline{P}, where W is cyclic of order (q^{2} − 1)/d. Also by [28, II.10.12], W acts on the linear characters of \overline{P} with one orbit of length one and d orbits of length (q^{2} − 1)/d, and transitively on the remaining q − 1 irreducible characters. Hence by Clifford theory irr(\overline{N_{N}(P)}) = {(q^{2} − 1)/d × 1, d × (q^{2} − 1)/d, (q + 1)/d × q(q − 1)}. By Lemmas 2.13, 2.10 and 2.14 M(N_{N}(P)) = 1, so each irreducible character of Z(N) extends uniquely to an irreducible character of N_{N}(P) and the result follows.

Blau and Michler in [7] give a full account of the action of the diagonal and field automorphisms on the conjugacy classes of N and N_{N}(P). Let δ be an outer diagonal automorphism such that ⟨δN⟩ has order d. Write E =< δ > N and note that δ stabilizes P.

Using Lemma 7.20, [7, 4.5] and the character tables given in [52], we see that all irreducible characters have defects 3m, 2m or zero, and

k_{3m}(N_{N}(P), B_{λ}) = k_{3m}(N, B_{λ}) = (q^{2} − 1)/d + d,
k_{2m}(N_{N}(P), B_{λ}) = k_{2m}(N, B_{λ}) = (q + 1)/d,
k_{3m}(N_{E}(P), B_{λ}) = k_{3m}(E, B_{λ}) = q^{2},
k_{2m}(N_{E}(P), B_{λ}) = k_{2m}(E, B_{λ}) = q + 1

for each λ ∈ Irr(Z(G)).

Let τ^{r} be a field automorphism as in Lemma 7.3 generating a subgroup of G/E, so r|2m. By [7, 4.5],

\begin{aligned}f_{Irr_{3m}(N_{N} (P),B_{λ})}(⟨τ^{r}⟩) & = & f_{Irr_{3m}(N,B_{λ})}(⟨τ^{r}⟩) \\ & = & \{ \begin{matrix}p^{m} & if r|m \\ p^{m} + 2 & if r |̸m and 3|(2m/r) \\ (p^{m} − 1)/3 + 3 & if r |̸m and 3 |̸(2m/r) \end{matrix} \end{aligned} ,

\begin{aligned}f_{Irr_{2m}(N_{N} (P),B_{λ})}(⟨τ^{r}⟩) & = & f_{Irr_{2m}(N,B_{λ})}(⟨τ^{r}⟩) \\ & = & \{ \begin{matrix}(2, p − 1) & if r|m \\ p^{r/2} + 1 & if r |̸m and 3|(2m/r) \\ (p^{r/2} + 1)/3 & if r |̸m and 3 |̸(2m/r) \end{matrix} \end{aligned} ,

f_{Irr_{3m}(N_{E}(P),B_{λ})}(⟨τ^{r}⟩) = f_{Irr_{3m}(E,B_{λ})}(⟨τ^{r}⟩) = p^{m}

and

\begin{aligned}f_{Irr_{2m}(N_{E}(P),B_{λ})}(⟨τ^{r}⟩) & = & f_{Irr_{2m}(E,B_{λ})}(⟨τ^{r}⟩) \\ & = & \{ \begin{matrix}(2, p − 1) & if r|m \\ p^{r/2} + 1 & if r |̸m \end{matrix} \end{aligned} .

The result follows by Lemma 7.5.

8 Proof of Theorem 4.7

Theorem 4.7 Conjecture 4.4 holds for every finite group of p-local rank one.

Proof By Proposition 4.6 it suffices to verify the conjecture for p-blocks B of finite groups G with O_{p}(G) ≤ Z(G), O_{p}(G) ≤ G′, O_{p}(G) cyclic and plr(G) ≤ 1. If plr(G) = 0, then Conjecture 4.4 states that k_{d}(G, B, λ) = w_{d}(G, B, λ, O_{p}(G)) for each integer d and each λ ∈ Irr(O_{p}(Z(G))). But in this case O_{p}(G) ∈ Syl_{p}(G), so the conjecture holds trivially. Hence we may assume that G has p-local rank one, and by Lemma 2.5 G has almost TI Sylow p-subgroups. In this case Conjecture 4.4 states that k_{d}(G, B, λ) = w_{d}(G, B, λ, O_{p}(G))−w_{d}(N_{G}(P), B, λ, O_{p}(G))+w_{d}(N_{G}(P), B, λ, P), where P ∈ Syl_{p}(G). If B has defect group O_{p}(G), then the conjecture predicts that k_{d}(G, B, λ) = w_{d}(G, B, λ, O_{p}(G)), which holds trivially. Hence we may assume that B is a p-block of maximal defect, and so it has P ∈ Syl_{p}(G) as a defect group (since defect groups are radical p-subgroups and the only radical p-subgroups of G are O_{p}(G) and the Sylow p-subgroups). Now w_{d}(N_{G}(P), B, λ, P) = k_{d}(N_{G}(P), B, λ) and by Corollary 3.15 w_{d}(G, B, λ, O_{p}(G)) = w_{d}(N_{G}(P), B, λ, O_{p}(G)) = 0, so we have only to check that k_{d}(G, B, λ) = k_{d}(N_{G}(P), B, λ).

By Proposition 5.7, Conjecture 4.4 holds for every such p-block when P/O_{p}(G) has p-rank one. Hence we may assume that P/O_{p}(G) has p-rank strictly greater than one, and so O_{p′}(G) ≤ N_{G}(P) by Lemma 2.8. By Lemma 6.1 we may then assume that O_{p′}(G) ≤ Z(G) is cyclic and Z(G) ≤ G′. So Z(G) = O_{p}(G)O_{p′}(G) is cyclic.

By Lemma 6.2 there is a unique normal subgroup N of G minimal such that it strictly contains Z(G). The factor group N/Z(G) has TI Sylow p-subgroups and is nonabelian simple. Further N/Z(G) ≤ G/Z(G) ≤ Aut(N/Z(G)). Lemma 6.12 gives a list of the possibilities for N/Z(G) using the classification of finite simple groups. By Lemma 6.7, Lemma 6.8 and Proposition 7.7 we may assume that (p, [G : N]) = 1.

Next we show that we may assume that N is quasisimple, i.e., that G is an automorphism group of a quasisimple group: (a) Lemma 6.9 tells us that if O_{p}(Z(G)) > 1, then N is quasisimple; (b) Lemma 6.11 tells us that we may assume that N is quasisimple when O_{p}(Z(G)) = 1.

The results of Chapter 7 give a case-by-case verification of Conjecture 4.4 for each of the possibilities for G where N/Z(G) is one of the nonabelian simple groups listed in Proposition 6.12, and the result follows.

9 Dade’s inductive conjecture for the Ree groups

of type G_{2}

As an aside and whilst we have introduced the notation necessary to study the Ree groups of type G_{2}, we go on to demonstrate the inductive form of Dade’s conjecture for these groups for the prime p = 3. Jianbei An has achieved a verification for all primes but p = 3 in [4], and so the results of this section will complete the verification for this class of groups.

Since Out(^{2}G_{2}(q)) is cyclic and the Schur multiplier of ^{2}G_{2}(q) is trivial, it follows from the remarks at the end of Chapter 3 of [20] that in order to verify the inductive conjecture for this class of groups it suffices to check only that they satisfy the invariant conjecture. We state the invariant conjecture here and direct the reader to [20] for a statement of the inductive conjecture. Conjecture 9.1 (Dade’s invariant) Let B be a p-block of a finite group G satisfying O_{p}(G) = 1, and suppose that G ◁ E. If G ◁ H ≤ E and σ ∈ R(G), then denote by Irr_{d}(G_{σ}, B, H) the set of those characters in Irr_{d}(G_{σ}, B) with inertial subgroup N_{H}(σ) in N_{E}(σ). Write k_{d}(G_{σ}, B, H) = |Irr_{d}(G_{σ}, B, H)|. Then

∑_{σ∈R(G)/G}(−1)^{|σ|}k_{d}(G_{σ}, B, H) = 0.

Proposition 4.9 Dade’s inductive conjecture is satisfied for the Ree groups of type G_{2} for the prime p = 3.

Proof Let G = ^{2}G_{2}(q), where q = 3^{2m+1}. Let G ◁ E and let B_{0} denote the unique 3-block of G of positive defect. Choose t|2m + 1 such that τ^{t} generates E/G, where τ is as in Lemma 7.3. Suppose that G ◁ H ≤ E, and choose r|2m + 1 such that τ^{r} generates H/G. Note that we may now have 3|[E : G]. Let P ∈ Syl_{p}(G).

From the proof of Proposition 7.11, f_{Irr_{6m+3}(G,B_{0})}(⟨τ^{r}⟩) = 3^{r}, f_{Irr_{5m+3}(G,B_{0})}(⟨τ^{r}⟩) = 6 and f_{Irr_{4m+2}(G,B_{0})}(⟨τ^{r}⟩) = 1, this accounting for all the irreducible characters of B_{0}. Also from the proof of Proposition 7.11, we have f_{Irr_{6m+3}(N_{G}(P),B_{0})}(⟨τ^{r}⟩) = 3^{r}, f_{Irr_{5m+3}(N_{G}(P),B_{0})}(⟨τ^{r}⟩) = 6 and f_{Irr_{4m+2}(N_{G}(P),B_{0})}(⟨τ^{r}⟩) = 1, this accounting for all the irreducible characters of N_{G}(P). The result follows by Lemma 7.4.

10 Appendix A

10.1 Generators for P SL_{3}(4).2^{2}

Regarded as a permutation group acting on a set of 56 elements, a system of generators for P SL_{3}(4).2^{2} is ⟨a, b, c, d⟩, where

a = (2, 6, 22, 31, 8)(3, 9, 32, 39, 12)(4, 14, 42, 46, 16)(5, 18, 38, 51, 21)(7, 26, 49, 52, 27) (10, 19, 23, 54, 35)(11, 36, 50, 33, 37)(13, 40, 20, 45, 41)(15, 30, 55, 44, 28) (17, 47, 29, 24, 48)(25, 43, 56, 53, 34),

b = (1, 2, 5)(3, 10, 13)(4, 15, 17)(6, 23, 25)(7, 20, 19)(8, 28, 32)(9, 33, 21) (11, 30, 38)(12, 18, 14)(16, 45, 40)(22, 24, 53)(26, 48, 37)(27, 39, 43)(29, 47, 46) (31, 44, 41)(34, 49, 55)(35, 42, 51)(50, 52, 54)

c = (1, 3)(4, 11)(5, 19)(6, 24)(8, 29)(9, 32)(10, 34)(12, 39)(14, 43)(15, 23)(16, 44) (20, 49)(21, 50)(25, 42)(27, 45)(30, 36)(33, 56)(38, 53)(40, 41)(46, 54)(47, 48)

d = (1, 4)(2, 7)(3, 11)(5, 20)(6, 24)(8, 30)(9, 34)(10, 32)(12, 39)(13, 35)(14, 33) (15, 44)(16, 23)(17, 18)(19, 49)(21, 46)(22, 52)(25, 53)(26, 55)(27, 41)(28, 51)(29, 36) (31, 37)(38, 42)(40, 45)(43, 56)(47, 48)(50, 54)

It can easily be checked using MAGMA that P SL_{3}(4).2^{2} ≅ ⟨a, b, c, d⟩ by computing the centre of P SL_{3}(4).2^{2}, the centralizer of P SL_{3}(4) and making use of the CompositionFactors command, or simply by using the CharacterTable command and comparing with that given in [12].

10.2 Generators for 2. ^{2}B_{2}(8)

Following [12, pp.28] 2. ^{2}B_{2}(8) may be regarded as the subgroup of GL_{8}(5) generated by the matrices a,b and c given below:

a = ( \begin{matrix}0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \end{matrix} ) , b = ( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ) c = ( \begin{matrix}2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 4 & 3 & 3 & 4 & 1 & 0 & 3 \\ 1 & 3 & 3 & 4 & 1 & 0 & 3 & 4 \\ 1 & 3 & 4 & 1 & 0 & 3 & 4 & 3 \\ 1 & 4 & 1 & 0 & 3 & 4 & 3 & 3 \\ 1 & 1 & 0 & 3 & 4 & 3 & 3 & 4 \\ 1 & 0 & 3 & 4 & 3 & 3 & 4 & 1 \\ 1 & 3 & 4 & 3 & 3 & 4 & 1 & 0 \end{matrix} )

78

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