On finite groups of p-local rank one and conjectures of Dade and Robinson

Charles W. Eaton ^{1} School of Mathematics and Statistics University of Birmingham Birmingham B15 2TT England email: eatonc@for.mat.bham.ac.uk

Abstract

We use the classification of finite simple groups to demonstrate that Conjecture 4.1 of [22] and Dade’s projective conjecture hold for finite groups of p-local rank one.

1 Introduction

There are several reformulations and strengthenings of Alperin’s weight conjecture (see [1]). Robinson’s (see [22, 4.1]) uses the notion of relative projectivity (albeit for ordinary characters rather than modules over a non-semisimple group ring) to conserve information about any normal p-subgroups present. Dade’s conjectures are formulated with a reduction to finite simple groups in mind. Considering Robinson’s conjecture along with an inductive invariant known as the p-local rank, it may be that an inductive proof of this, and so too of Alperin’s conjecture could be achieved. This paper may be regarded as a necessary step in this inductive procedure, the proof of the p-local rank one case.

We use the classification of finite simple groups to do this, in much the same way as Blau and Michler in their proof of Alperin’s conjecture for finite groups with trivial intersection (TI) Sylow p-subgroups (see [4]). There are two fundamental differences: the need to take account of the defect of each irreducible character and the need to deal with the presence of a nontrivial normal p-subgroup. We also feel our approach to be simpler than that of [4], the use of the Lyndon-Hoschild-Serre spectral sequence to calculate the Schur multiplier M(G) of a group G being an example.

Let G be a finite group and p a prime. The conjectures we are considering are expressed in terms of chains of p-subgroups, as described in [9] and [18]. Briefly, we say that a p-subgroup Q of G is radical if Q = O_{p}(N_{G}(Q)), where O_{p}(H) is the unique maximal normal p-subgroup of H. We say that a chain σ : Q_{0} < ⋯ < Q_{n} of p-subgroups of G (where inclusions are strict) has length |σ| = n, and write G_{σ} = N_{G}(Q_{0}) ∩ ⋯ ∩ N_{G}(Q_{n}) for its stabilizer under the conjugation action of G. Write σ_{i} for the p-chain Q_{0} < ⋯ < Q_{i} and V_{σ} = Q_{0}. We say that the p-chain σ is radical if Q_{i} = O_{p}(N_{G}(σ_{i})) for each i, i.e., if Q_{0} is a radical p-subgroup of G and Q_{i} is a radical p-subgroup of N_{G}(σ_{i−1}) for each i ≠ 0. Write R = R(G) for the set of radical p-chains of G and write R(G|Q) = {σ ∈ R(G) : V_{σ} = Q}. Write R(G|Q)/G for a set of orbit representatives under the action of G. Following [22], define the p-local rank, plr(G) of G to be the length of a longest chain in R(G). By [22, 7.1], if plr(G) > 0, then plr(G) = 1 if and only if G/O_{p}(G) has TI Sylow p-subgroups (recall that P ≤ H is a TI set if P ∩ P^{x} = 1 whenever x ∈ H \ N_{H}(P)).

1Partially supported by EPSRC award no. 95700575

If N ◁ G and μ ∈ Irr(N) (the set of irreducible characters of N), then denote by Irr(G, μ) the set of irreducible characters of G covering μ. Write irr(G) for the multiset of irreducible character degrees for G. If Q◁G is a p-group and μ ∈ Irr(Q), then denote by w(G, μ, Q) the number of Q-projective irreducible characters of G covering μ, i.e., those χ such that χ(1)_{p} = [G : Q]_{p}μ(1). Dade defines the defect of a character χ to be the integer d such that |G|_{p} = χ(1)_{p}p^{d}, and write Irr_{d}(G) for the set of irreducible characters of G with defect d.

Let R be a local complete discrete valuation ring containing a primitive |G|^{3}-th root of unity, and whose residue field k = R/J(R) is algebraically closed and has characteristic p. Denote by K the field of fractions of R, a splitting field for every central extension of every section of G that we shall be using. If B is a block with respect to this system, then define Irr(G, B) to be the set of those irreducible characters which belong to B. We combine this and the above notation freely.

Robinson’s conjecture states that:

Conjecture 1.1 Let G be a finite group and λ ∈ Irr(O_{p}(Z(G))). Then for each pblock B of G and each non-negative integer d, we have

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

Dade’s projective conjecture (see [10]) states that:

Conjecture 1.2 Let G be a finite group with cyclic O_{p}(G) ≤ Z(G) ∩ G′. If B is a p-block of G with defect groups not equal to O_{p}(G), then

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

The purpose of this article is to prove the following:

Theorem 1.3 Every finite group G with plr(G) = 1 satisfies Conjecture 1.1.

By the results of [12], Robinson’s conjecture holds for all finite groups of p-local rank one if and only if Dade’s projective conjecture does. So in proving Theorem 1.3 we also demonstrate

Theorem 1.4 Every finite group G with plr(G) = 1 and cyclic O_{p}(G) ≤ Z(G) ∩ G′ satisfies Conjecture 1.2.

Note that by [12] we have:

Proposition 1.5 If Conjecture 1.1 holds for all finite groups G of p-local rank n or smaller and with cyclic O_{p}(G) ≤ Z(G) ∩ G′, then the conjecture holds for all finite groups of p-local rank n.

Hence in proving the conjecture for groups of p-local rank one it suffices to verify it for groups with p-local rank one and cyclic O_{p}(G) ≤ Z(G) ∩ G′. Recall that every defect group is an intersection of Sylow p-subgroups, so in this case defect groups are either central or are Sylow p-subgroups. If B has a central defect group, then Conjecture 1.1 states that k_{d}(G, B, λ) = w_{d}(G, B, λ, O_{p}(G)), which holds trivially since every irreducible character of B must be O_{p}(G)-projective. Hence we may assume that B has maximal defect. In this case the conjecture 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), since a set of representatives of the orbits of radical p-chains {O_{p}(G), O_{p}(G) < P, P }. Since O_{p}(G) is central in G, it follows from [22, 3.2] that w_{d}(G, B, λ, O_{p}(G)) = w_{d}(N_{G}(P), B, λ, O_{p}(G)) = 0. Further, w_{d}(N_{G}(P), B, λ, P) = k_{d}(N_{G}(P), B, λ) since P is a defect group for B, and so in order to establish Theorem 1.3 it suffices to check that k_{d}(G, B, λ) = k_{d}(N_{G}(P), B, λ).

This paper is structured as follows: In Sections 2 and 3 we give a reduction to a short list of cases to be checked (Section 2 consisting of the reduction and 3 dealing with the determination of the covering groups and with the classification). In Section 4 Conjecture 1.1 is checked for each of the cases listed at the end of Section 3. We give an overview of the proof and some consequences in Section 5.

2 Reduction to covers of automorphism groups of finite simple groups

In this section we show that we may assume that G is a covering group of an automorphism group of a finite non-abelian simple group of p-local rank one. Such simple groups have been classified in [15] and [28].

Let G be a finite group with plr(G) = 1. By Proposition 1.5 we may assume that O_{p}(G) ≤ Z(G) ∩ G′ and is cyclic. Let P ∈ Syl_{p}(G). Write \overline{G} = G/O_{p}(G), and similarly for subgroups of G. We first need to reduce to the case that the maximal normal p′-subgroup O_{p′}(G) is central. This is standard, but we reproduce it here. The first step is to ensure that O_{p′}(G) ≤ N_{G}(P). To do this we need to be able to find a non-cyclic elementary abelian subgroup of \overline{P}, so we treat the case that the p-rank of \overline{P}, m_{p}(\overline{P}) = 1 separately.

Proposition 2.1 Suppose that G is as above and that m_{p}(\overline{P}) = 1. Then every p-block of G satisfies Conjecture 1.1.

Proof: By [13, 5.4.10] \overline{P} is either cyclic or generalised quaternion. In either case P/O_{p}(G) has trivial Schur multiplier (see, for example [16, V.25.3]), and so we may take O_{p}(G) = 1 (see [16, V.25.1]). The result then follows directly from [8] and [29]. □

We assume from now on that m_{p}(\overline{P}) > 1. The following lemma is well-known:

Lemma 2.2 Under the above assumptions, we have O_{p′}(G) ≤ N_{G}(P).

Proof: Consider E_{2} ≤ \overline{P} elementary abelian of rank 2. Since this is a noncyclic abelian p-group of automorphisms of \overline{O_{p′}(G)}, by [13, 6.2.4] we have

\overline{O_{p′}(G)} = ⟨C\overline{O_{p′}(G)}(α) | α ∈ E_{2} − \{1\}⟩ ≤ N\overline{G}(\overline{P}) = \overline{N_{G}(P)}

and the result follows. □

Proposition 2.3 Suppose that Conjecture 1.1 holds for every finite group H with plr(H) = 1 and cyclic O_{p}(H)O_{p′}(H) = Z(H) ≤ H′. Then Conjecture 1.1 holds for all blocks of finite groups of p-local rank one.

Proof: Consider a p-block B of G, where plr(G) = 1. We have already seen that we may assume that O_{p}(G) ≤ Z(G) ∩ G′ and is cyclic, and that m_{p}(G) > 1, so we do so. Let P ∈ Syl_{p}(G). We may suppose that P is a defect group for B. Choose λ ∈ Irr(O_{p}(G)) and an integer d. We must show that k_{d}(G, B, λ) = k_{d}(N_{G}(P), B, λ).

By Lemma 2.2, O_{p′}(G) ≤ N_{G}(P). The p-blocks of O_{p′}(G) (in this case all are blocks of defect zero, so we identify with the unique irreducible character in each block) covered by B form a single G-orbit. Choose a representative μ of this orbit covered by the Brauer correspondent of B in N_{G}(P), then Irr_{d}(G, B, λ) = Irr_{d}(G, B, μ, λ) and Irr_{d}(N_{G}(P), B, λ) = Irr_{d}(N_{G}(P), B, μ, λ). Clifford theory gives a 1-1 correspondence Irr(G, B, μ, λ) ↔ Irr(I_{G}(μ), B, μ, λ) = Irr(I_{G}(μ), B, λ) given by induction, which preserves character defects. We also have a 1-1 correspondence Irr(N_{G}(P), B, μ, λ) ↔ Irr(I_{N_{G}(P)}(μ), B, μ, λ) = Irr(I_{N_{G}(P)}(μ), B, λ), and all correspondences are compatible with the Brauer correspondence. Hence it suffices to assume that μ is G-stable (note that by [22, 7.2] I_{G}(μ) also has p-local rank one).

We now apply the second Fong correspondence (see for example [12]). There is a finite group Ĝ with O_{p′}(Ĝ) ≤ Z(Ĝ)∩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).

Similarly

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. Since O_{p′}(Ĝ) is central, it follows from [22, 7.3] that plr(Ĝ) = 1, and if Conjecture 1.1 holds for every p-block of Ĝ (and it does by our assumptions), then it holds for B. □

We may now assume also that O_{p′}(G) ≤ Z(G) ∩ G′ and is cyclic. The structure of G/Z(G) is well known (see [4]), but we give a self-contained proof here.

Lemma 2.4 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.

Let N be a (nontrivial) minimal normal subgroup of H. Then |N|_{p} ≠ 1, since otherwise N ≤ O_{p′}(H) = 1. Now P ∩ N is a nontrivial Sylow p-subgroup of N so, by a Frattini argument 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, N_{H}(N_{2}∩P) ≤ N_{H}(P), and so H = N_{H}(P), a contradiction. Hence N is the unique nontrivial minimal normal subgroup of H.

By [13, 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 |N|_{p} ≠ 1, contradicting O_{p}(H) = 1. So we may assume the former. We show that r = 1:

Suppose that r > 1. Clearly |S_{1}|_{p} ≠ 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)

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. By [22, 7.2] 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. □

So far we have shown that we may, and do throughout the rest of this paper, assume that G satisfies the following:

Hypotheses 2.5 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. It is important to note that Z(N) 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 [6, p.xxii]).

3 Determination of the structure of G

The purpose of this section is to apply the results of [15] and [28] to produce a list of possibilities for \overline{G}, and to show further that Z(G) < N′ (i.e., that L = N). This is important in that information in [6] is given only for automorphism groups of covers of simple groups, rather than covers of automorphism groups of simple groups.

For a p-subgroup Q of a group H, write Γ_{Q,1}(H) = ⟨N_{H}(X) | X ≤ Q⟩. Recall that a proper subgroup M ≤ H strongly p-embedded if Γ_{P,1}(H) ≤ M for some P ∈ Syl_{p}(H).

Lemma 3.1 Suppose that O_{p}(H) = 1. Then H has a TI Sylow p-subgroup P if and only if N_{H}(P) is a strongly p-embedded subgroup of H.

Proof: The ‘only if’ part is clear. Suppose that 1 < Q < P is a radical psubgroup 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 [22, 7.2]. □

Lemma 3.2 (p, \overline{N}) is 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 [28]. Suppose that p is odd. Then \overline{N} has a strongly p-embedded subgroup. So by [15, 24.9] (p, \overline{N}) is one of the pairs listed or (5, M_{22}) or (p, A_{2p}). By [15, 24.2], Γ_{Q,1}(\overline{N}) ≠ N\overline{N}(Q) (for Q ∈ Syl_{p}(\overline{N})) when (p, \overline{N}) is (5, M_{22}) or (p, A_{2p}), so by Lemma 3.1 in these two cases \overline{N} cannot have TI Sylow p-subgroups. Also by [15, 24.2], Γ_{Q,1}(\overline{N}) = N\overline{N}(Q) for each of the remaining cases, so the result holds by Lemma 3.1. □

We show that it is nearly always the case that the outer automorphisms in G are p-regular. The argument used is essentially that used in [30].

Lemma 3.3 (p, [G : N]) = 1, except when (p, \overline{N}) = (3, ^{2}G_{2}(3)′) or (5, ^{2}B_{2}(32)).

Proof: Suppose first that p = 2. Then the result follows from Theorem 2 of [28]. Assume now that p ≠ 2 and that (p, \overline{N}) ≠ (3, ^{2}G_{2}(3)′) or (5, ^{2}B_{2}(32)). By [6], \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 [6] we have [G : N] = p. □

We now aim to show that L = N. We first treat the p-part of Z(G), then the p′-part. To do this we need to recall some results concerning Schur multipliers (we thank Burkhard Külshammer for suggesting the use of the Lyndon-Hochschild-Serre spectral sequence, giving a more general result than our original reference [3]).

\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}4 & 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}

Table 1: Schur multipliers of groups of automorphisms of P SL_{2}(9) and P SL_{3}(4)

Lemma 3.4 Let X and Y be finite groups with X ≤ Y.

(i) Suppose that [Y : X] is not divisible by p and the Schur multiplier M(X) is a p′-group. Then M(Y) is also a p′-group.

(ii) Suppose that X ◁ Y, X is perfect, and that M(Y/X) = 1. Then M(Y) is a homomorphic image of a subgroup of M(X).

Proof: (i) This is a special case of [14, 4.229].

(ii) Taking ℂ to be the trivial ℂG-module, the Lyndon-Hochschild-Serre spectral sequence gives a short exact sequence

1 → H^{2}(Y, ℂ) → H^{2}(X, ℂ)^{Y/X}

since H^{1}(X, ℂ) = 1 (see [2, p.192]). The duality between H^{n}(−, ℂ) and H_{n}(−, ℤ) then gives us

H_{2}(X, ℤ)^{Y/X} → H_{2}(Y, ℤ) → 0.
□

Lemma 3.5 (i) If p|[G : N], then Z(G) = 1.

(ii) If p |̸[G : N] and O_{p}(Z(G)) ≠ 1, then L = N.

Proof: (i) By [6] 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 we may apply Lemma 3.5, and M(\overline{G}) = 1 as required.

(ii) By Lemma 3.5 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 [6] 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 3.4 M(\overline{G}) is a homomorphic image of a subgroup 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)). In this case, the MAGMA package may be used to compute the Schur multipliers of the relevant automorphism groups of P SL_{2}(9) and P SL_{3}(4), and we give these in Table 1. From this it is clear that when G/N is noncyclic, we have O_{p}(Z(G)) = 1.

□

The following is essentially Lemma 3.5 of [4].

Lemma 3.6 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 1.1 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)) since V/Z(X) is a normal p′-subgroup. 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 (taking N_{X}(σ) as our normal subgroup) we have a 1-1 correspondence

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

This takes V_{σ}-projective characters to V_{σ}-projective characters, and is also compatible with p-blocks and with the Brauer correspondence (this time by considering V/Z(X) as our normal subgroup and applying the second Fong 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. □

Finally we complete the proof that we may assume that L = N. Suppose that G satisfies Hypotheses 2.5 and that (p, [G : N]) = 1, O_{p}(Z(G)) = 1. By [6] 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). The proof of the following is based on [4, 3.6].

Lemma 3.7 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 1.1 holds for G if and only if it holds for H.

Proof: 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 3.5 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. The result now follows from Lemma 3.6. □

To summarize, in order to prove Theorem 1.3, it suffices to assume Hypotheses 2.5 and L = N, i.e., G is an automorphism group of the quasisimple group N:

Proposition 3.8 Suppose that G is as in Hypotheses 2.5 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})).

4 Checking the conjecture

In this section we verify Conjecture 1.1 for the groups listed in Proposition 3.8. Part of this has already been done for us in [4], since Conjecture 1.1 is equivalent to that handled in that paper for p-blocks of finite groups with abelian TI Sylow p-subgroups:

Proposition 4.1 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 1.1.

Proof: It suffices to consider p-blocks B of defect a = log_{p}(|G|_{p}). Let P ∈ Syl_{p}(G). By [4] we have k(G, B) = k(N_{G}(P), B) and k_{a}(G, B) = k_{a}(N_{G}(P), B). Ito’s theorem then gives k_{a}(G, B) = k_{a}(N_{G}(P), B) = k(N_{G}(P), B) = k(G, B), so all the irreducible characters of B have height zero. The result follows. □

The following result will prove useful:

Proposition 4.2 (Robinson) Let B be any p-block of any finite group H with O_{p}(H) ≤ Z(H). If Z_{1} ≤ Z(H), then denote by \overline{G} the image under the natural homomorphism G → G/Z_{1}. Denote by \overline{B} the unique p-block of \overline{G} in the image of B under the extension of this map to RG. There is a natural 1-1 correspondence R(G|Z)/G ↔ R(\overline{G}, \overline{Z})/\overline{G}.

Suppose that for every such Z_{1}

∑_{\overline{σ}∈R(\overline{G},\overline{Z})/\overline{G}}(−1)^{|\overline{σ}|}k\overline{d}(N\overline{G}(\overline{σ}), \overline{B}) = 0

for every integer \overline{d}. Then

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

for every integer d and every λ ∈ Irr(Z).

Proof: See the comments at the end of [25, Section 1]], noting that we may replace N by R (see [24, 1.1]). □

The following lemma is a simple extension of [4, 2.1], and can be proved using the same well-known consequence of Brauer’s second main theorem:

Lemma 4.3 Let G be a finite group with plr(G) = 1 and O_{p}(G) ≤ Z(G). Let B be a block of G with defect group P ∈ Syl_{p}(G). Denote by \overline{B} the unique p-block of \overline{G} = G/O_{p}(G) corresponding to B. Then

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

Proposition 4.4 Conjecture 1.1 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 1.1 holds in this case by Proposition 4.1.

If (p, \overline{N}) = (11, J_{4}), then again we have G = \overline{N}. By [21] 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 [21, 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 [6, pp.188] k_{3}(G, B_{0}) = 42, k_{2}(G, B_{0}) = 7 and k(G, B_{0}) = 49. So Conjecture 1.1 holds in this case.

Now suppose that (p, \overline{N}) = (5, McL). The conjecture has been verified independently in this case by Murray (see [20]). By [6] the outer automorphisms of McL do not act trivially on the centre of 3.McL, so by the remark following Hypotheses 2.5 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 [21] McL has just one p-block B_{0} of positive defect. It follows from the character tables given in [21] and [6] 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 5. 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 [6, 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. It follows from [17, 2.4.9] and [16, V.25.2,V.25.3] that 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 1.1 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 [4] (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 [6] gives the required result. □

Proposition 4.5 G satisfies Conjecture 1.1 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. Since ^{2}B_{2}(2^{5}) has cyclic Sylow 5-subgroups we need only to consider G = ^{2}B_{2}(2^{5}).5. Let P ∈ Syl_{5}(G). By [6, 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 (see for example [7, 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 [6] we get k_{3}(G, B_{0}) = 25, k_{2}(G, B_{0}) = 1 and k(G, B_{0}) = 26, so Conjecture 1.1 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). Since ^{2}G_{2}(3)′ has cyclic Sylow 3-subgroups we need only to consider G =^{2}G_{2}(3). Let P ∈ Syl_{3}(G). By [6, 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 [6] we get k_{3}(G, B_{0}) = 9, k_{2}(G, B_{0}) = 1 and k(G, B_{0}) = 10, so Conjecture 1.1 holds for G in this case. □

Proposition 4.6 G satisfies Conjecture 1.1 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 1.1 holds for \overline{B} by Proposition 4.1. The result now follows by Lemma 4.3 and Proposition 4.2, observing that the irreducible characters (of both G and N_{G}(P)) of defect two are precisely those covering non-trivial linear characters of O_{p}(G). □

For the remaining cases we need to consider the action of the Frobenius automorphism on a group of Lie type. We need the following three results found in [4] (the proof of the third is a trivial extension of the proof given in [4]). We include them here in order to establish the necessary notation:

Lemma 4.7 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.

Lemma 4.8 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.

Lemma 4.9 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.

We continue checking the cases given in Proposition 3.8.

Proposition 4.10 Conjecture 1.1 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 [4, 6.3].

First suppose that G = ^{2}B_{2}(2^{2m+1}).C_{t}, and let P ∈ Syl_{2}(G). By [27, 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 [27, pp.126,142], irr(N_{N}(P)) = {q − 1 × 1, 1 × q − 1, 2 × 2^{m}(q − 1)}, and by [27, 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 4.9 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 4.7 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 4.9 l(G, B_{0}) = l(N_{G}(P), B_{0}). Hence by Lemma 4.3 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 4.9 k_{4m+2}(G, B_{0}) = k_{4m+2}(N_{G}(P), B_{0}). The result follows in this case since all positive height characters of N, N_{N}(P) have the same defect.

Finally suppose that G = 2. ^{2}B_{2}(2^{3}). Then the result follows from the first part of the proof, Lemma 4.3 and Proposition 4.2, observing that the irreducible characters (of both N_{G}(P) and G) of defect 7 are precisely those covering the trivial character of Z(G). □

Proposition 4.11 G satisfies Conjecture 1.1 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 4.1. □

Proposition 4.12 G satisfies Conjecture 1.1 when (p, \overline{N}) = (3, ^{2}G_{2}(3^{2m+1}), where m > 0.

Proof: Note that \overline{N} = N. Let τ be the Frobenius map as in Lemma 4.7 and let r be an integer dividing 2m + 1 such that τ^{r} generates a subgroup G/N. Let P ∈ Syl_{3}(G) and let B_{0} be the unique p-block of N of positive defect. Then by [11] we have f_{Irr_{6m+3}(N,B_{0})}(⟨τ^{r}⟩) = f_{Irr_{6m+3}(N_{N} (P),B_{0})}(⟨τ^{r}⟩), f_{Irr_{5m+3}(N,B_{0})}(⟨τ^{r}⟩) = f_{Irr_{5m+3}(N_{N} (P),B_{0})}(⟨τ^{r}⟩) and f_{Irr_{4m+2}(N,B_{0})}(⟨τ^{r}⟩) = f_{Irr_{4m+2}(N_{N} (P),B_{0})}(⟨τ^{r}⟩), this accounting for all the irreducible characters in B_{0} and its Brauer correspondent in N_{N}(P). The result then follows from Lemmas 4.8 and 4.9, since there is an unique p-block of G covering B_{0}. □

It remains to verify the conjecture when \overline{N} = P SU_{3}(p^{m}). The case where G = N has been treated in [19] (where Dade’s conjecture is verified for all unitary groups in the defining characteristic), and a less detailed conjecture has been verified for G in [4]. It suffices in demonstrating Conjecture 1.1 for these groups to show that all irreducible characters in p-blocks of positive defect of N_{G}(P) and G have one of two defects, since the conjecture will then follow from [4].

We determine the irreducible characters of a Sylow p-subgroup P of N. Let q = p^{m}, m > 1.

Lemma 4.13 Let P ∈ Syl_{p}(N). Consider the automorphism of 𝔽_{q^{2}} defined by λ ↦ λ^{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^{q} = t^{q+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}^{q}t_{2}).

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

Proof: The characterization and multiplication of the elements of P follow from [5], so that

x(t, u)^{−1} = x(−t, −u − t^{q+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}^{q}t_{1} − t_{1}^{q}t_{2}).

Denote by ϕ : 𝔽_{q^{2}} → 𝔽_{q^{2}} the additive homomorphism given by ϕ(λ) = λ + λ^{q}. |ker(ϕ)| = q, so |P | = q^{3}. Suppose that x(t_{2}, u_{2}) ∈ Z(P). Then t_{2}^{q}t_{1} = t_{1}^{q}t_{2} for all t_{1} ∈ 𝔽_{q^{2}}. So t_{2} = 0 and u_{2} ∈ ker(ϕ). Hence |Z(P)| = q. □

Lemma 4.14 P has q conjugacy classes of length one and q^{2} − 1 of length q.

Proof: In Lemma 4.13 we saw that Z(P) has order q and consists of elements of the form x(0, u), where u^{q} = −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}^{q}t_{1} = t_{1}^{q}t_{2}. Assume that t_{1} ≠ 0, then we want t_{1}^{q−1} = t_{2}^{q−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 4.15 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}^{q}t_{1} − t_{1}^{q}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, there is a degree-preserving 1-1 correspon-

dence 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. P/Z(P) is elementary abelian and so has elementary abelian Schur multiplier. 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. □

Proposition 4.16 Every p-block of G satisfies Conjecture 1.1 when \overline{N} = P SU_{3}(p^{m}), m > 1.

Proof: It follows from Lemma 4.15 that every irreducible character of N_{N}(P) has defect 3m or 2m. By the character table in [26] every irreducible character of N has defect 3m, 2m or zero. It follows from Lemma 3.3 that the same can be said for N_{G}(P) and G. By [4] k_{3m}(G, B) = k_{3m}(N_{G}(P), B) and k(G, B) = k(N_{G}(P), B) for every block B of positive defect. It hence follows that k_{2m}(G, B) = k_{2m}(N_{G}(P), B) and we are done. □

5 Proof of the main theorem

Proof of Theorem 1.3 By Proposition 1.5 it suffices to verify the conjecture for pblocks B of finite groups G with cyclic O_{p}(G) ≤ Z(G) ∩ G′, and plr(G) ≤ 1. Hence we may assume that plr(G) = 1, and so G/O_{p}(G) has TI Sylow p-subgroups. In this case Conjecture 1.1 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 holds trivially. Hence we may assume that B is a p-block of maximal defect, so we have only to check that k_{d}(G, B, λ) = k_{d}(N_{G}(P), B, λ). By Proposition 2.1, Conjecture 1.1 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 2.2. By Proposition 2.3 we may then assume that O_{p′}(G) ≤ Z(G) ∩ G′ and is cyclic. So Z(G) = O_{p}(G)O_{p′}(G) ≤ G′ and is cyclic.

By Lemma 2.4 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 3.2 gives a list of the possibilities for N/Z(G) using the classification of finite simple groups. By Lemma 3.3, Lemma 3.5 and Proposition 4.5 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 3.5 tells us that if O_{p}(Z(G)) > 1, then N is quasisimple; (b) Lemma 3.7 tells us that we may assume that N is quasisimple when O_{p}(Z(G)) = 1.

We are left with the list of cases to be checked given in Proposition 3.8. Section 4 consists of the case-by-case verification of Conjecture 1.1 for each of the possibilities for G where N/Z(G) given in the list, and so completes the proof.

Corollary 5.1 When plr(G) = 2 and B is a p-block of G with abelian defect groups, then the assertion of Dade’s projective conjecture is equivalent to the statement that k_{d}(G, B, λ) = k_{d}(N_{G}(D), B, λ) for each λ ∈ Irr(O_{p}(Z(G))) and each integer d.

Proof: Immediate from Theorem 1.3, the equivalence of Dade’s and Robinson’s conjectures demonstrated in [12], and the discussion of Robinson’s conjecture for blocks with abelian defect groups found in [22]. □

Remark Similar results may be applied to cancel chains σ : Q_{0} < ⋯ < Q_{n} and σ_{n−1} where plr(N_{G}(σ_{n−1})) = 1 (such as occurs when plr(G) = n) and w(N_{G}(σ), B, Q_{0}) = w(N_{G}(σ_{n−1}), B, Q_{0}) = 0 (which occurs for example when B has abelian defect groups - this is discussed in [23].

ACKNOWLEDGEMENTS

The material for this paper consists of part of my doctoral research, and I thank my supervisor Geoffrey Robinson without whose guidance this would have been impossible. I also thank Robert Baddeley, Burkhard Külshammer and Wayne Wheeler for their helpful comments and suggestions.

References

[1] J.L.Alperin, Weights for finite groups, Proceedings of Symposia in Pure Mathematics 47(1987), 369-379.

[2] A.Babakhanian, “Cohomological Methods in Group Theory,” Dekker, New York, 1972.

[3] N.Blackburn and L.Evens, Schur multipliers of p-groups, J. Reine Angew. Math. 309(1979), 100-113.

[4] H.I.Blau and G.O.Michler, Modular representation theory of finite groups with T.I. Sylow p-subgroups, Trans. AMS. 319(1990), 417-468.

[5] R.Carter, “Simple Groups of Lie Type,” Wiley-Interscience, New York, 1972.

[6] J.Conway, J.Curtis, S.P.Norton, R.A.Parker and R.A.Wilson, “Atlas of Finite Groups,” Clarendon Press, Oxford, 1985.

[7] C.W.Curtis and I.Reiner, “Methods of Representation Theory I,” Wiley-Interscience, 1987.

[8] E.Dade, Blocks with cyclic defect groups, Ann. of Math. (2) 84(1966), 20-48.

[9] E.Dade, Counting characters in blocks I, Invent. Math. 109(1992), 187-210.

[10] E.Dade, Counting characters in blocks II, J. Reine Angew. Math. 448(1994), 97-190.

[11] C.W.Eaton, Dade’s projective conjecture for the Ree groups of type G_{2} in the defining characteristic, J. Algebra 226(2000), 614-620.

[12] C.W.Eaton, On the equivalence of some conjectures of Dade and Robinson, preprint.

[13] D.Gorenstein, “Finite Groups,” Harper and Row, 1968.

[14] D.Gorenstein, “Finite Simple Groups: An Introduction to Their Classification,” Plenum Press, New York, 1982.

[15] D.Gorenstein and R.Lyons, On finite groups of characteristic 2-type, Mem. AMS. 276(1982).

[16] B.Huppert, “Endliche Gruppen I,” Springer-Verlag, Berlin-Heidelberg, 1967.

[17] G.Karpilovsky, “The Schur Multiplier,” Clarendon Press, Oxford, 1987.

[18] R.Knörr and G.R.Robinson, Some remarks on a conjecture of Alperin, J. London Math. Soc. (2) 39(1989), 48-60.

[19] C. Ku, “Dade’s Ordinary Conjecture for the Finite Unitary Goups in the Defining Characteristic,” PhD thesis, California Institute of Technology, 1999.

[20] J.Murray, “Dade’s Conjecture for the McLaughlin Simple Group,” PhD thesis, University of Illinois, 1997.

[21] T.Ostermann, “Irreduzible Charaktere von Sylownormalisatoren der sporadischen einfachen Gruppen,” Dissertation, Universität Essen, 1986.

[22] G.R.Robinson, Local structure, vertices and Alperin’s conjecture, Proc. London Math. Soc. (3) 72(1996), 312-330.

[23] G.R.Robinson, Some open conjectures on representation theory, in Representation Theory of Finite Groups, Ed. R.Solomon, Walter de Gruyter and Co., Berlin-New York, 1997, 127-131.

[24] G.R.Robinson, Further consequences of conjectures like Alperin’s, J. Group Theory 1(1998), 131-141.

[25] G.R.Robinson, Dade’s projective conjecture for p-solvable groups, J. Algebra 229(2000), 234-248.

[26] W.A.Simpson and J.S.Frame, The character tables for SL(3, q), SU(3, q^{2}), P SL(3, q), P SU(3, q^{2}), Can. J. Math.25 no.3(1973), 486-494.

[27] M.Suzuki, On a class of doubly transitive groups, Ann. of Math. (2) 75(1962), 105-145.

[28] M.Suzuki, Finite groups of even order in which Sylow 2-subgroups are independent, Ann. of Math. (2) 80(1964), 58-77.

[29] K.Uno, Dade’s conjecture for tame blocks, Osaka J. Math. 31(1994), 747-772.

[30] J.P.Zhang, Studies on defect groups, J. Algebra 166(1994), 310-316.