Nilpotent blocks of quasisimple groups for odd primes ^{1} Jianbei An and Charles W. Eaton

Abstract

We investigate the nilpotent blocks of positive defect of the quasisimple groups for odd primes. In particular, it is shown that every nilpotent block of a quasisimple group has abelian defect groups. A conjecture of Puig concerning the recognition of nilpotent blocks is also shown for these groups.

1 Introduction

Let G be a finite group and k an algebraically closed field of odd characteristic p. A block B of kG with defect group D is said to be nilpotent if for each Q ≤ D and each block b_{Q} of C_{G}(Q) with Brauer correspondent B we have that N_{G}(Q, b_{Q})/C_{G}(Q) is a p-group, where N_{G}(Q, b_{Q}) is the stabilizer of b_{Q} under conjugation in N_{G}(Q). In the case of the principal block B_{0}, D is a Sylow p-subgroup of G and N_{G}(Q, b_{Q}) = N_{G}(Q) for each Q ≤ D, so that B_{0} is nilpotent if and only if G is p-nilpotent (i.e., G has a normal p-complement). Note that every block of defect zero must be nilpotent, and the classification of blocks of defect zero for finite simple groups has been the subject of a separate program of research, culminating in [21]. Hence we give attention here only to blocks with non-central defect groups.

Explicit characterizations of nilpotent blocks are obtained for classical groups, and these are used to prove:

Theorem 1.1 Let G be a finite quasisimple group and let B be a nilpotent p-block of G with p odd. Then B has abelian defect groups.

The second main result concerns the conjecture of Puig which states that a block B of G is nilpotent if and only if l(b_{Q}) = 1 for each p-subgroup Q and each block b_{Q} of C_{G}(Q) with Brauer correspondent B (where l(b_{Q}) is the number of irreducible Brauer characters in b_{Q}). The necessary condition for nilpotency is well-known. The converse is known for blocks with abelian defect groups (see [30]), and is also known to be a consequence of Alperin’s weight conjecture (see [33]). We prove that:

Theorem 1.2 Let G be a finite quasisimple group and let B be a p-block of G with p odd. Then B is nilpotent if and only if l(b_{Q}) = 1 for each p-subgroup Q and each block b_{Q} of C_{G}(Q) with (b_{Q})^{G} = B.

The main part of the paper concerns the representation theory of finite groups of Lie type in non-defining characteristic, and makes use of the examination of subpairs of blocks of classical groups given in [17]. The exceptional groups of Lie type are then treated by examination of the centralizer of an element of the centre of a defect group, and the results for the classical groups applied.

1The first author is supported by the Marsden Fund (of New Zealand), via award number UOA 0721 and the second author is supported by a Royal Society University Research Fellowship

In Section 2 we review the basic notation regarding blocks, give some general results concerning nilpotent blocks, particularly with regard to block domination, and also prove some technical lemmas which will be useful later on. In Section 3 we consider the alternating groups and their covering groups. Here we have able to give a rather complete description of the nilpotent blocks. The covering groups of the sporadic simple groups are treated in Section 4. We give some basic notation used for the classical groups in Section 5. In Section 6 we give a treatment of the general linear and unitary groups, where again we are able to give a full characterization of the nilpotent blocks. In Section 7 we state the set of properties which are central to the study of the nilpotent blocks of the groups of Lie type. These are rather technical conditions, none of which can be satisfied by a nilpotent block with non-abelian defect groups, which amongst other things allow us to use inductive argument when studying the exceptional groups. That these conditions hold for the classical groups is the content of Section 8, and for the exceptional groups is the content of Section 9.

2 Notation and general results

Let G be a finite group and p a prime. Although the classification concerns only blocks with respect to a field of characteristic p, we use methods from ordinary character theory, for example canonical characters, and so must use a p-modular system. Let O be a local discrete valuation ring, complete with respect to the p-adic valuation, with field of fractions K of characteristic zero and algebraically closed residue field k = O/J(O) of characteristic p. We assume that O contains a primitive |G|th root of unity. Write Blk(G) for the set of blocks of OG and denote by B_{0}(G) the principal block of G.

Let N be a normal subgroup of G and write Irr(G) the set of irreducible Kcharacters of G. For θ ∈ Irr(N), we denote by Irr(G | θ) the subset of Irr(G) consisting of characters covering θ. We denote by Irr(B) the set of irreducible characters belonging to B, k(B) = |Irr(B)|, and combine with the above notation freely.

Let B be a p-block of a finite group G. A B-subgroup is a subpair (Q, b_{Q}), where Q is a p-subgroup of G and b_{Q} is a block of QC_{G}(Q) with Brauer correspondent (b_{Q})^{G} = B. The B-subgroups with |Q| maximized are called the Sylow B-subgroups, and they are the B-subgroups for which Q is a defect group for B. Recall that the canonical character of b_{Q} is the unique irreducible character in b_{Q} with Q in its kernel. This will be a valuable tool when comparing subpairs of a group with those of a normal subgroup.

A useful, and well-known, result is the following:

Proposition 2.1 Let B be a block of a finite group G. Suppose a defect group D of B is abelian. Then B is nilpotent if and only if N_{G}(D, b_{D}) = C_{G}(D), where (D, b_{D}) is a Sylow B-subgroup.

In general, we cannot say very much about the relationship between nilpotency of blocks and nilpotency of covered blocks, and this is a main reason behind the difficulty of the classification of nilpotent blocks of groups of Lie type.

However, we do have the following lemma by [24, Proposition 6.5].

Lemma 2.2 Let N be a normal subgroup of a finite group G such that G/N is a pgroup. Suppose that B is a block of G and that b ∈ Blk(N) is covered by B. Then B is nilpotent if and only if b is nilpotent.

We note that the analogous result does not hold if G/N is not a p-group. There are many examples of non-nilpotent blocks covering nilpotent blocks, but there are also examples of nilpotent blocks covering non-nilpotent blocks, such as the following (which came to light during a conversation with Radha Kessar):

Example 2.3 Let G = PGL(3, 7), N = P SL(3, 7) and p = 2, so that [G:N] = 3. Then N has a unique block b with defect group D ≅ ℤ_{2} × ℤ_{2} and b is not nilpotent. Moreover, b is covered by a nilpotent block B of G.

Note that C_{N}(D) = ℤ_{6} × ℤ_{2}. Let Irr^{0}(C_{N}(D)) be the subset of Irr(C_{N}(D)) consisting of characters of C_{N}(D) whose kernel contains D. Then |Irr^{0}(C_{N}(D))| = 3. In addition, C_{N}(D) has a unique character (the trivial character) ξ ∈ Irr^{0}(C_{N}(D)) such that N_{N_{N} (D)}(ξ) = N_{N}(D), and two characters ξ ∈ Irr^{0}(C_{N}(D)) such that N_{N_{N} (D)}(ξ) = C_{N}(D).3. It follows that N has exactly one block b with a defect group D and b is non-nilpotent, as N_{N}(D, b_{D}) = C_{N}(D).3 for a Sylow b-subgroup (D, b_{D}).

Moreover, C_{G}(D) = ℤ_{6} × ℤ_{6}, N_{G}(D)/C_{G}(D) ≅ S_{3} and |Irr^{0}(C_{G}(D))| = 9. In addition, C_{G}(D) has a unique character (the trivial character) ξ ∈ Irr^{0}(C_{G}(D)) such that N_{N_{G}(D)}(ξ) = N_{G}(D), and eight characters ξ ∈ Irr^{0}(C_{G}(D)) such that N_{N_{G}(D)}(ξ) = C_{G}(D). It follows that G has exactly one block B with a defect group D and B is nilpotent, as N_{G}(D, b_{D}) = C_{G}(D) for a Sylow b-subgroup (D, b_{D}). Since b is covered by a block of G with a defect group D, it follows that b is covered by B.

Recall that for N ⊲ G, a block B of G is said to dominate the block \overline{B} of G/N if the inflation to G of an irreducible character in \overline{B} lies in B.

The following lemma follows by [33, Lemma 2].

Lemma 2.4 Let Z be a central p-subgroup of a finite group G, B ∈ Blk(G) and \overline{B} the block of \overline{G} := G/Z dominated by B. Then B is nilpotent if and only if \overline{B} is nilpotent.

Let Z be a central p′-subgroup of a finite group G, and write \overline{H} = HZ/Z, where H ≤ G. Let \overline{B} ∈ Blk(\overline{G}). There is a unique block B ∈ Blk(G) dominating \overline{B}. By [25, Theorem 5.8.8], Irr(B) = Irr(\overline{B}) and if D is a defect group of B, then DZ/Z ≅ D is a defect group of \overline{B}.

If Q is a p-subgroup of G, then C\overline{G}(\overline{Q}) = C_{G}(Q)/Z (since Z is a central p′-subgroup). Let (\overline{Q}, b\overline{Q}) be a \overline{B}-subgroup. Then \overline{Q} = QZ/Z for a unique p-subgroup Q of G. Since C\overline{G}(\overline{Q}) = C_{G}(Q)/Z, we may consider the unique subpair (Q, b_{Q}) with b_{Q} dominating b\overline{Q}, which we call the Brauer pair dominating (\overline{Q}, b\overline{Q}).

We show that (Q, b_{Q}) must be a B-subgroup, and that dominance of subpairs respects the usual partial order on B-subgroups:

Lemma 2.5 Let Z be a central p′-subgroup of a finite group G, and let (\overline{Q}, b\overline{Q}) and (\overline{P} , b\overline{P}) be \overline{B}-subgroups, where \overline{B} is the block of \overline{G} dominated by B. Suppose (Q, b_{Q}) and (P, b_{P}) are subpairs of G dominating (\overline{Q}, b\overline{Q}) and (\overline{P} , b\overline{P}), respectively. Then (\overline{Q}, b\overline{Q}) ≤ (\overline{P} , b\overline{P}) if and only if (Q, b_{Q}) ≤ (P, b_{P}). In particular, (Q, b_{Q}) is a B-subgroup.

Proof: Let F be a splitting field of characteristic p of G, and μ_{Z} the map from F G to F \overline{G} defined by μ_{Z}(∑_{x∈G} α_{x}x) = ∑_{x∈G} α_{x}x̄, where x̄ = μ_{Z}(x). For H ≤ G, let Cℓ(G | H) be the H-orbits of G under conjugation. Then {[C] : C ∈ Cℓ(G | H)} forms a F -basis of the fixed point set (F G)^{H}, where [C] := ∑_{x∈C} x.

If H is a p-subgroup, then (F G)^{H} = F C_{G}(H) ⊕ I^{H}(F G) as vector spaces, where I^{H}(F G) = ∑_{W <H}(F G)_{W}^{H} is an ideal of (F G)^{H} and {[C] : C ∈ Cℓ(G | H), |C| ≠ 1} forms an F -basis of I^{H}(F G). Thus Br_{H}([C]) := [C ∩ C_{G}(H)] gives the natural algebra homomorphism from (F G)^{H} onto F C_{G}(H) with kernel I^{H}(F G). Similarly,

(F \overline{G})^{H} = F (C_{G}(H)/Z) ⊕ I^{H}(F \overline{G})

and μ_{Z} : I^{H}(F G) → I^{H}(F \overline{G}) is an isomorphism of algebras. Now

Br\overline{H} : (F \overline{G})^{H} → F (C_{G}(H)/Z)

and μ_{Z} : F C_{G}(H) → F (C_{G}(H)/Z), so μ_{Z} ∘ Br_{H} = Br\overline{H} ∘ μ_{Z}.

Suppose (\overline{Q}, b\overline{Q}) ⊲ (\overline{P} , b\overline{P}), so that \overline{Q} ⊲ \overline{P}. Since \overline{Q} = QZ/Z and \overline{P} = P Z/Z for p-subgroups Q, P of G and since Q is the only Sylow p-subgroup of QZ, it follows that Q ⊲ P. Since P Z = P × Z and b\overline{Q} is \overline{P} -invariant, it follows that for any y ∈ P b_{Q}^{y} is a block of C_{G}(Q) dominating b\overline{Q}, so that by the uniqueness, b_{Q}^{y} = b_{Q} and b_{Q} is P -invariant. Now

μ_{Z}(Br_{P} (b_{Q})b_{P}) = μ_{Z}(Br_{P} (b_{Q}))b\overline{P} = Br\overline{P} (μ_{Z}(b_{Q}))b\overline{P} = Br\overline{P} (b\overline{Q})b\overline{P} = b\overline{P} ≠ 0,

so that Br_{P} (b_{Q})b_{P} ≠ 0 and Br_{P} (b_{Q})b_{P} = b_{P}. It follows that (Q, b_{Q}) ⊲ (P, b_{P}). Using induction we have that (Q, b_{Q}) ≤ (P, b_{P}) if (\overline{Q}, b\overline{Q}) ≤ (\overline{P} , b\overline{P}).

Suppose (Q, b_{Q}) ⊲ (P, b_{P}), so that Q ⊲ P and \overline{Q} ⊲ \overline{P}. Since b_{Q} is P -invariant, it follows that b\overline{Q} is \overline{P} -invariant. Since Br_{P} (b_{Q})b_{P} = b_{P}, it follows that

Br\overline{P} (b\overline{Q})b\overline{P} = μ_{Z}(Br_{P} (b_{Q})b_{P}) = μ_{Z}(b_{P}) = b\overline{P} ,

so that (\overline{Q}, b\overline{Q}) ⊲ (\overline{P} , b\overline{P}). Similarly, if (Q, b_{Q}) ≤ (P, b_{P}), then (\overline{Q}, b\overline{Q}) ≤ (\overline{P} , b\overline{P}). □

We obtain as a consequence:

Proposition 2.6 Let G be a finite group, Z ≤ Z(G) and \overline{G} = G/Z. Suppose \overline{B} ∈ Blk(\overline{G}) and B ∈ Blk(G) dominating \overline{B}. Then \overline{B} is nilpotent if and only if B is nilpotent.

Proof: Write Z_{p} = O_{p}(Z), Z_{p′} = O_{p′}(Z), G_{1} = G/Z_{p′} and let B_{1} ∈ Blk(G_{1}) be the unique block of G_{1} dominated by B. Then Irr(B_{1}) = Irr(B), and B_{1} dominates \overline{B}.

By Lemma 2.4, \overline{B} is nilpotent if and only if B_{1} is nilpotent. Hence we suppose \overline{B} = B_{1} and Z = Z_{p′}.

Let (\overline{D}, b\overline{D}) be a Sylow \overline{B}-subgroup and (D, b_{D}) the unique B-subgroup dominating (\overline{D}, b\overline{D}). Note that (D, b_{D}) is a Sylow B-subgroup.

Suppose (\overline{Q}, b\overline{Q}) is a B-subgroup and (Q, b_{Q}) is the B-subgroup dominating (\overline{Q}, b\overline{Q}). If x̄ ∈ N\overline{G}(\overline{Q}, b\overline{Q}), then x̄ = xZ for some x ∈ G, and xZ ⊆ N_{G}(Q). Since μ_{Z}(b_{Q}) = b\overline{Q} and μ_{Z}(b_{Q}^{x}) = bx̄ ; \overline{Q} = b\overline{Q}, it follows that b_{Q} and b_{Q}^{x} both are blocks of C_{G}(Q) dominating b\overline{Q} and b_{Q} = b_{Q}^{x} by uniqueness. Thus x ∈ N_{G}(Q, b_{Q}) and N\overline{G}(\overline{Q}, b\overline{Q}) = N_{G}(Q, b_{Q})/Z. Since Z ≤ C_{G}(Q) and C\overline{G}(\overline{Q}) = C_{G}(Q)/Z, it follows that

N\overline{G}(\overline{Q}, b\overline{Q})/C\overline{G}(\overline{Q})\overline{Q} ≅ N_{G}(Q, b_{Q})/C_{G}(Q)Q.

Suppose \overline{B} is not nilpotent, so that there is some \overline{B}-subgroup (\overline{Q}, b\overline{Q}) such that N\overline{G}(\overline{Q}, b\overline{Q})/C\overline{G}(\overline{Q})\overline{Q} is not a p-group. Thus N_{G}(Q, b_{Q})/C_{G}(Q)Q is not a p-group and B is not nilpotent.

Suppose B is not nilpotent, so that N_{G}(Q, b_{Q})/C_{G}(Q)Q is not a p-group for some B-subgroup (Q, b_{Q}). We may suppose (Q, b_{Q}) ≤ (D, b_{D}). Thus \overline{Q} = QZ/Z ≤ \overline{D}, and (\overline{Q}, b\overline{Q}) ≤ (\overline{D}, b\overline{D}) for a unique \overline{B}-subgroup (\overline{Q}, b\overline{Q}). Let (Q, b′_{Q}) be a B-subgroup dominating (\overline{Q}, b\overline{Q}). By Lemma 2.5 (Q, b′_{Q}) ≤ (D, b_{D}), so that by the uniqueness (Q, b′_{Q}) = (Q, b_{Q}). Thus N\overline{G}(\overline{Q}, b\overline{Q})/C\overline{G}(\overline{Q})\overline{Q} ≅ N_{G}(Q, b_{Q})/C_{G}(Q)Q, and N\overline{G}(\overline{Q}, b\overline{Q})/C\overline{G}(\overline{Q})\overline{Q} is not a p-group. It follows that \overline{B} is not nilpotent. □

When considering groups of Lie type, we will often examine the centralisers of pelements, which may be written as central products of groups. By a central product G_{1} ∘_{Z} G_{2} of G_{1} and G_{2} over Z ≤ Z(G_{1})∩Z(G_{2}), we mean that G_{1} ∘_{Z} G_{2} = G_{1}G_{2}, where G_{1} and G_{2} are subgroups of G_{1} ∘_{Z} G_{2} with G_{1} ∩ G_{2} = Z and [G_{1}, G_{2}] = 1. When it is clear what Z is, we write G_{1} ∘G_{2} = G_{1} ∘_{Z} G_{2}. Note that G_{1} ∘G_{2} ≅ (G_{1} ×G_{2})/{(z, z^{−1}) : z ∈ Z}. For χ_{i} ∈ Irr(G_{i}) such that χ_{1} and χ_{2} both cover the same irreducible character of Z, we may define χ_{1} ∘ χ_{2} ∈ Irr(G_{1} ∘ G_{2}) so that χ_{1}χ_{2} ∈ Irr(G_{1} × G_{2}) is the inflation of χ_{1} ∘ χ_{2}. We refer to χ_{1} ∘ χ_{2} as the central product of χ_{1} and χ_{2}.

We will need the following technical lemma in certain cases in relation to Property 7.1 (a) in Section 7 holds.

Lemma 2.7 For i = 1, 2, let G_{i} be a finite group, G_{1} ∘ G_{2} a central product of G_{1} and G_{2} over Z ≤ Z(G_{1}) ∩ Z(G_{2}) and N_{i} a normal subgroup of G_{i} such that G_{i}/N_{i} is abelian, and let N := N_{1} × N_{2} ≤ G ≤ G_{1} ∘ G_{2} such that π_{i}(G) = G_{i}/Z, where π_{i} : (G_{1} ∘ G_{2}) → G_{i}/Z is the canonical projection. Let θ_{i} ∈ Irr(N_{i}) such that θ_{2} has an extension θ̃_{2} to G_{2}, and let θ = θ_{1} × θ_{2} and φ ∈ Irr(G | θ).

(i) There exist ψ_{1} ∈ Irr(G_{1}) and λ ∈ Irr(G_{2}/N_{2}) covering the same irreducible character of Z, such that the restriction (ψ_{1} ∘ (θ̃_{2}λ))|_{G} of ψ_{1} ∘ (θ̃_{2}λ) is equal to φ. Moreover, if ψ ∈ Irr(G_{1} ∘ G_{2} | φ), then ψ|_{G} = φ.

(ii) If further Z ∩ N_{2} = 1, then λ in (i) may be chosen with ZN_{2}/N_{2} in its kernel, so that it may be regarded as a character of G_{2}/N_{2}Z.

(iii) Suppose that Z ∩ N_{2} = 1. If we have y ∈ Aut(G_{1} ∘ G_{2}) such that y centralizes G_{1}, stabilizes G, G_{2} and θ̃_{2}, and g_{2}^{y} ∈ g_{2}N_{2}Z for any g_{2} ∈ G_{2}, then y stabilizes φ.

Proof: (i) We first claim that we may suppose Z ≤ G. For since Z ≤ Z(G_{1} ∘ G_{2}), we have that GZ is a central product over G ∩ Z. Now φ|_{G∩Z} = φ(1)α for some α ∈ Irr(G ∩ Z). Since Z is abelian, there exists α̃ ∈ Irr(Z) extending α. Then φ̃ = φα̃ is an extension of φ covering θ. If ψ_{1} ∈ Irr(G_{1}) and λ ∈ Irr(G_{2}/N_{2}) such that ψ_{1} and λ cover the same irreducible character of Z and (ψ_{1} ∘ θ̃_{2}λ)|_{GZ} = φ̃, then (ψ_{1} ∘ θ̃_{2}λ)|_{G} = φ, and similarly for the final statement, proving the claim.

Similarly, (N_{1} × N_{2})Z = (N_{1}Z) ∘ (N_{2}Z) and φ covers an irreducible character ξ ∈ Irr((N_{1}Z) ∘ (N_{2}Z) | θ) with ξ = ξ_{1} ∘ ξ_{2} for some ξ_{i} ∈ Irr(N_{i}Z | θ_{i}) covering the same irreducible character of Z. Note that θ̃_{2}|_{N_{2}Z} is also an extension of θ_{2} to N_{2}Z. By Gallagher’s theorem, (θ̃_{2}|_{N_{2}Z})β_{2} = ξ_{2} for some β_{2} ∈ Irr(N_{2}Z/N_{2}). Since G_{2}/N_{2} is abelian, it follows that β_{2} can be viewed as the restriction of a character β ∈ Irr(G_{2}/N_{2}), so ξ_{2} = (θ̃_{2}β)|_{N_{2}Z}. Write ξ̃_{2} = θ̃_{2}β, so ξ̃_{2} is an extension of ξ_{2} to G_{2}

Let M_{1} be a subgroup of G_{1} such that ξ_{1} has an extension ξ̃_{1} to M_{1} and M_{1} is maximal with this property, that is, either M_{1} = G_{1} or ξ_{1} has no extension to H_{1} for any M_{1} < H_{1} ≤ G_{1}. Since G_{1}/(N_{1}Z) is abelian, it follows that the inertia subgroup I_{G_{1}}(ξ̃_{1}) equals M_{1}.

Let M = (M_{1} ∘ G_{2}) ∩ G ≤ G, M_{2} := G_{2} and γ ∈ Irr(M_{1} ∘ M_{2} | ξ). Then G/M ≅ G_{1}/M_{1} and γ = γ_{1} ∘ γ_{2} for some γ_{i} ∈ Irr(M_{i} | ξ_{i}). Since ξ_{i} has an extension ξ̃_{i} to M_{i}, it follows that γ_{i} = ξ̃_{i}λ_{i} for some λ_{i} ∈ Irr(M_{i}/N_{i}Z), so that γ|_{M} is an extension of ξ to M. Note that M ≤ I_{G}(ξ) and φ also covers an extension of ξ to M. Replacing γ_{i} by γ_{i}α_{i} for some α_{i} ∈ Irr(M_{i}/N_{i}Z) if necessary, we may suppose φ ∈ Irr(G | γ|_{M}).

Since γ_{1}|_{N_{1}Z} = ξ_{1} and G_{2} stabilizes ξ̃_{2} (and G_{1}/N_{1}Z is abelian), it follows that I_{G_{1}}(γ_{1}) = M_{1}, I_{G}(γ) = M and I_{G_{1}∘G_{2}}(γ) = M_{1} ∘ G_{2}. Let ψ_{1} = Ind_{M_{1}}^{G_{1}}(γ_{1}), so that

ζ := ψ_{1} ∘ γ_{2} = Ind_{M_{1}∘G_{2}}^{G_{1}∘G_{2}}(γ_{1} ∘ γ_{2}).

But φ = Ind_{M}^{G}(γ|_{M}), so

(ζ|_{G}, φ)_{G} = (ζ|_{M}, γ|_{M})_{M}.

Since M and M_{1} ∘ G_{2} are both normal in G_{1} ∘ G_{2} and M ≤ M_{1} ∘ G_{2}, it follows that (M_{1} ∘ G_{2})\(G_{1} ∘ G_{2})/M = (G_{1} ∘ G_{2})/(M_{1} ∘ G_{2}) ≅ G_{1}/M_{1} ≅ G/M. Note also (M_{1} ∘ G_{2})^{t} ∩ M = M for any t ∈ G/M. Hence the Mackey decomposition gives us

ζ|_{M} = ∑_{t∈G/M}((γ_{1} ∘ γ_{2})^{t}|_{M}) = ∑_{t∈G/M}((γ|_{M})^{t})

and so (ζ|_{M}, γ|_{M})_{M} = 1. Since ζ(1) = φ(1) = [G:M]γ(1), it follows that ζ|_{G} = φ.

Note that I_{G_{1}∘G_{2}}(φ) = G_{1} ∘ G_{2} and φ has an extension ζ to G_{1} ∘ G_{2}. If ψ ∈ Irr(G_{1} ∘ G_{2} | φ), then by Gallagher’s theorem, ψ = ζη for some η ∈ Irr((G_{1} ∘ G_{2})/G) and so ψ|_{G} = φ.

(ii) First note that ZN_{2} = Z × N_{2} and (N_{1} × N_{2})Z = (N_{1} ∘ Z) × N_{2}. If ξ ∈ Irr((N_{1} × N_{2})Z | θ), then ξ = (θ_{1} ∘ η) × θ_{2} for some η ∈ Irr(Z). Thus we may suppose ξ_{1} = (θ_{1} ∘ η) ∈ Irr(N_{1}Z) and ξ_{2} = (θ_{2} × 1_{Z}) ∈ Irr(N_{2}Z), and take ξ̃_{2} = θ̃_{2} as an extension of ξ_{2} to G_{2}. As shown in the proof of part (i) φ = (ψ_{1} ∘ (θ̃_{2}λ))|_{G} for some ψ_{1} ∈ Irr(G_{1}) and λ ∈ Irr(G_{2}/N_{2}Z).

(iii) Since y centralizes the factor group G_{2}/ZN_{2}, it follows that y stabilizes λ, so does θ̃_{2}λ. But y centralizes G_{1}, so y stabilizes ψ_{1} ∘ (θ̃_{2}λ) and hence y stabilizes φ. □

3 The alternating groups

To handle the case p = 3 we will need the following. The first lemma will be used in determining non-faithful nilpotent blocks of the double covers of alternating groups. Recall that a partition is self-associate if its Young diagram is symmetric.

Lemma 3.1 Let n be a positive integer. There is a self-associate 3-core partition λ ⊢ n if and only if there is a positive integer m such that n = 3m^{2} + 2m or n = 3m^{2} − 2m.

Proof: We claim that the self-associate 3-cores, i.e., those Young diagrams which are symmetric about the leading diagonal and have no 3-hooks, are those which arise from partitions of the form

(3m, 3m − 2, 3m − 4, …, 3m − 2(m − 1), m^{2}, (m − 1)^{2}, …, 2^{2}, 1^{2})

and

(3m − 2, 3m − 4, 3m − 6, …, 3m − 2m, (m − 1)^{2}, (m − 2)^{2}, …, 2^{2}, 1^{2})

for integers m ≥ 1.

This may be seen directly from methodical construction of the possible Young diagrams. However, we give a formal proof here using [19]. For a fixed t, Garson, Kim and Stanton give a bijection ϕ between the set of t-cores and {n_{0}, n_{1}, …, n_{t−1} ∈ ℤ^{t} : n_{0} + ⋯ + n_{t−1} = 0}, defined as follows. We of course only need to consider the case t = 3. Let λ be a 3-core. We take the 3-residue diagram, i.e., in the (i, j)th cell of the Young diagram we put the residue of j−i modulo 3 (see [23, p.84]). We also include the 0th column (with infinitely many entries), calling this the extended 3-residue diagram. Divide this into regions labelled by the integers as follows: the (i, j)th cell lies in the region r if 3(r − 1) ≤ j − i < 3r. Say that a cell is exposed if it lies at the end of a row. Define n_{i} to be the maximal r such that an exposed cell with value i lies in the region r (the inclusion of the 0th row ensures the existence of such an r).

It is verified in [19] that ϕ does indeed give a bijection. It is also shown that λ is selfassociate if and only if ϕ(λ) = (n_{0}, n_{1}, n_{2}) = (−n_{2}, −n_{1}, −n_{0}), i.e., if ϕ(λ) = (m, 0, −m) for some m ∈ ℤ.

Suppose first that m > 0. Then the end cell on the first row is labelled 0, so the first row has length λ_{1} = 3(m − 1) + 1 = 3m − 2. Since regions 0 and −m lie below the leading diagonal, the end cells lying above the diagonal are all labelled 0. Since λ is a 3-core, the difference between adjacent row lengths is at most 2, hence the row lengths decrease in steps of two until the mth row (which has end cell on the leading diagonal). Since λ, is self adjoint, this determines the whole Young diagram and we are done in this case.

Suppose that m ≤ 0. Then the end cell of the first row is labelled 2, so the first row has length λ_{1} = 3m, and by a similar argument to the above the difference between adjacent row lengths is 2 until the (m + 1)th row (which has end cell below the leading diagonal). Again this determines λ, and we are done. □

We now consider the analogue of the above lemma which will be used for faithful blocks. We write λ ≻ n for a bar partition of n (i.e., a partition with distinct parts). Recall that a bar partition λ ≻ n is odd or even according as n − r is odd or even, where r is the number of parts in the partition. We refer to [26] for definitions of bars and p̄-cores.

Lemma 3.2 Let n be a positive integer. There is an even ¯3-core bar partition λ ≻ n if and only if there is a positive integer m_{1} with m_{1} ≡ 0, 1 mod 4 and n = (3m_{1}^{2} − m_{1})/2 or a positive integer m_{2} with m_{2} ≡ 0, 3 mod 4 and n = (3m_{2}^{2} + m_{2})/2.

Proof: Determining the ¯3-core partitions is a little more straight-forward than determining 3-core partitions, and the reader can easily verify that the ¯3-core bar partitions are precisely those of the form

λ_{m}^{−} := (3m − 2, 3m − 5, …, 3m − 2 − 3i, …, 4, 1)

or

λ_{m}^{+} := (3m − 1, 3m − 4, …, 3m − 1 − 3i, …, 5, 2).

Note that λ_{m}^{−} ≻ (3r^{2} − m)/2 and λ_{m}^{+} ≻ (3m^{2} + m)/2. Also note that λ_{m}^{−} is even if and only if m ≡ 0, 1 mod 4; λ_{m}^{+} is even if and only if m ≡ 0, 3 mod 4. □

Theorem 3.3 Let n be an integer with n ≥ 5 and G = Â_{n}, the double cover of A_{n}. Let p be an odd prime. If p ≠ 3, then G does not possess a nilpotent p-block of positive defect. If p = 3, then G possesses a non-faithful nilpotent block of positive defect if and only if n = 3m^{2} + 2m + 3 or n = 3m^{2} − 2m + 3 for some positive integer m. Also if p = 3, then G possesses a faithful nilpotent block of positive defect if and only if n = (3m_{1}^{2} − m_{1} + 6)/2 for a positive integer m_{1} with m_{1} ≡ 0, 1 mod 4 or n = (3m_{2}^{2} + m_{2} + 6)/2 for a positive integer m_{2} with m_{2} ≡ 0, 3 mod 4. In each case the nilpotent blocks have defect groups of order 3 generated by (the preimage of) a 3-cycle.

Proof: The properties of Â_{n} used here are described in [20, 5.2]. We consider Â_{n} ≤ Ŝ_{n}, the double cover of the symmetric group. Write Z = Z(Ŝ_{n}) and \overline{X} = XZ/Z whenever X ≤ Ŝ_{n}. For convenience of notation we write \overline{Ŝ_{n}} = S_{n} and \overline{Â_{n}} = A_{n}. Since we are taking p odd, for every p-subgroup Q ≤ Ŝ_{n} we have C_{S_{n}}(\overline{Q}) = \overline{CŜ_{n}(Q)} and N_{S_{n}}(\overline{Q}) = \overline{NŜ_{n}(Q)}. Suppose that B is a nilpotent p-block of Â_{n} with non-trivial defect group D. Choose y ∈ D of order p. Then yZ is a product of say t disjoint p-cycles, fixing the other n−pt points. Then C_{S_{n}}(yZ) ≅ (ℤ_{p}≀S_{t})×S_{n−pt}, and so CÂ_{n}(y) contains a normal elementary abelian p-group R such that \overline{R} is generated by t disjoint p-cycles. Now R is contained in a conjugate of D, and so in particular D contains an element x for which xZ is a p-cycle. Write Q = ⟨x⟩. We have C_{A_{n}}(\overline{Q}) ≅ \overline{Q} × A_{n−p}. By [20, 5.2.6] we have CÂ_{n}(Q) ≅ Q × Â_{n−p} (the point here being that the central extension of A_{n−p} does not split). We have N_{S_{n}}(\overline{Q}) ≅ N_{S_{p}}(\overline{Q}) × S_{n−p} and N_{A_{n}}(\overline{Q}) ≅ (N_{A_{p}}(\overline{Q}) × A_{n−p})⟨\overline{a}⟩ where \overline{a}^{2} = 1. Note that [NÂ_{n}(Q) : CÂ_{n}(Q)] = p − 1.

The p-blocks of CÂ_{n}(Q) are in 1-1 correspondence with the p-blocks of Â_{n−p} and the action of NÂ_{n}(Q) on these blocks is determined by the action of Ŝ_{n−p} on the blocks of Â_{n−p}. Hence for each block b_{Q} of CÂ_{n}(Q) we have [NÂ_{n}(Q, b_{Q}) : CÂ_{n}(Q)] = (p−1)/2 or p − 1.

If p > 3, then this shows that NÂ_{n}(Q, b_{Q})/CÂ_{n}(Q) is not a p-group, contradicting our choice of B nilpotent.

Now suppose that p = 3. We first show that \overline{D} is generated by a 3-cycle. Suppose that D is not cyclic. Then D contains an elementary abelian subgroup of order 9, and in particular contains distinct elements x and y for which xZ and yZ is the product of s and t disjoint 3-cycles respectively (briefly, consider the centralizer of gh, for which ghZ is the product of all the disjoint 3-cycles in gZ and hZ. This has an elementary abelian subgroup contained in a conjugate of D and containing elements whose images in S_{n} are all the 3-cycles making up ghZ). Then C_{S_{n}}(xZ) ≅ (ℤ_{p} ≀ S_{s}) × S_{n−ps} and C_{S_{n}}(yZ) ≅ (ℤ_{p} ≀ S_{t}) × S_{n−pt}. Hence D contains elements g and h for which gZ and hZ are each a 3-cycle and these 3-cycles are disjoint. Write R = ⟨g, h⟩ ≤ D. We have C_{S_{n}}(\overline{R}) ≅ \overline{R}×S_{n−6} and C_{A_{n}}(\overline{R}) ≅ \overline{R}×A_{n−6}. Now [NŜ_{n}(R) : CŜ_{n}(R)] = 8, and arguing as above we see that [NÂ_{n}(R, b_{R}) : CÂ_{n}(R)] is even for every block b_{R} of CÂ_{n}(R), a contradiction.

Hence D is cyclic. Suppose that |D| > 3. Then D possesses an element y of order 9. By an argument similar to above we may assume yZ is a 9-cycle. But then y^{3}Z is a product of three distinct 3-cycles, which as we have seen cannot happen.

Hence D has order three and is generated by an element x where xZ is a single 3-cycle. We have CÂ_{n}(D) ≅ D × Â_{n−3}. The blocks of CÂ_{n}(D) with defect group D are in 1-1 correspondence with the blocks of defect zero of Â_{n−3}, and the action of NÂ_{n}(D) on these blocks is given by the action of Ŝ_{n−3} on the blocks of defect zero of Â_{n−3}. Hence the nilpotent blocks of Â_{n} with defect group D are in 1-1 correspondence with orbits of length two of blocks of defect zero of Â_{n−3} under the action of Ŝ_{n−3}.

Now blocks of defect zero of Â_{n−3} are covered by blocks of defect zero of Ŝ_{n−3}. We consider faithful and non-faithful blocks separately. Note that B is faithful if and only if the B-subpairs have kernel intersecting trivially with Z (i.e., if and only if they correspond to faithful blocks of Â_{n−3}).

Suppose that B is non-faithful. Blocks of defect zero correspond to 3-core partitions of n − 3. By [23, 2.5.7] irreducible characters of S_{n−3} remain irreducible when restricted to A_{n−3} if and only if the corresponding partition is not self-associate. Hence [NÂ_{n}(D, b_{D}) : CÂ_{n}(D)] = 1 if and only if the block of defect zero of A_{n−3} corresponding to b_{D} is labelled by a self-associate partition, and so the result follows in this case from Lemma 3.1.

Suppose that B is faithful. Blocks of defect zero correspond to ¯3-core bar partitions of n − 3. By [27, p.212] faithful irreducible characters of Ŝ_{n−3} remain irreducible when restricted to Â_{n−3} if and only if the corresponding bar partition is odd. Hence [NÂ_{n}(D, b_{D}) : CÂ_{n}(D)] = 1 if and only if the block of defect zero of Â_{n−3} corresponding to b_{D} is labelled by an even ¯3-core bar partition, and so the result follows in this case from Lemma 3.2. □

We have not yet considered all the perfect central extensions of A_{6} and A_{7}. However, by the above theorem, neither yields a nilpotent 3-block with non-central defect group, and further it is easy to check that there are no nilpotent blocks of positive defect for the other odd primes.

It is appropriate here to extend our study to the double covers of the symmetric groups.

Proposition 3.4 Let G = Ŝ_{n} be the double cover of the symmetric group S_{n} for n ≥ 5 and let p be an odd prime. If p ≥ 5, then G does not possess a nilpotent p-block of positive defect. If p = 3, then every nilpotent block of positive defect is faithful. These have defect one, and occur if and only if there is a positive integer m_{1} with m_{1} ≡ 2, 3 mod 4 and n = (3m_{1}^{2} − m_{1} + 6)/2 or a positive integer m_{2} such that m_{2} ≡ 1, 2 mod 4 and n = (3m_{2}^{2} + m_{2} + 6)/2.

Proof: Suppose first that B is a non-faithful block of positive defect. By, for example, [23, 6.2.2] l(B) = ∑ p(w_{1}) ⋯ p(w_{p−1}), where the sum runs over improper partitions (w_{1}, …, w_{p−1}) of the weight w of B and p(x) is the number of partitions of x. But p − 1 ≥ 2 and (w, 0, …, 0) and (0, w, 0, …, 0) are improper partitions of w, so l(B) > 1 and B cannot be nilpotent.

Now suppose that B is faithful of positive defect. Then by [28] l(B) is at least k((p−1)/2, w), the number of (p−1)/2-tuples of (possibly empty) partitions with sum w, where again w is the weight of B. If w ≤ 2, then ((w), ∅, …, ∅) and ((1^{w}), ∅, …, ∅) are such (p − 1)/2-tuples of partitions, so l(B) > 1 and B cannot be nilpotent. Now suppose that w = 1. If p ≥ 5, then (1, ∅, …, ∅) and (∅, 1, ∅, …, ∅) are such (p − 1)/2tuples, and again B cannot be nilpotent. We are left with the case w = 1 and p = 3. By [28, 13.17] l(B) = 2 if the \overline{3}-core μ of B is even (in the sense that n − 3 − r is even, where r is the number of parts in μ), and l(B) = 1 if μ is odd. Note that since B has cyclic defect groups, B is nilpotent if and only if l(B) = 1, and so the result follows from Lemma 3.2. □

We now turn our attention to Puig’s conjecture.

Lemma 3.5 Let G = Ŝ_{n} be the double cover of S_{n}, and let B be a block of G with defect group D. If |D| > p^{2}, then l(B) ≥ 3.

Proof: Suppose first that B is a non-faithful block. As above,

l(B) = ∑ p(w_{1}) ⋯ p(w_{p−1}),

where the sum runs over improper partitions (w_{1}, …, w_{p−1}) of the weight w of D. If |D| > p^{2}, then w ≥ 2. But (w, 0, …, 0), (0, w, 0, …, 0) and (w−1, 1, 0, …, 0) are three such improper partitions, so l(B) ≥ 3.

If B is faithful, then by [28] l(B) is at least k((p−1)/2, w), the number of (p−1)/2tuples of (possibly empty) partitions with sum w, where again w is the weight of B. We have w ≥ 2. Here ((w), ∅, …, ∅), ((1^{w}), ∅, …, ∅) and ((w − 1, 1), 1, ∅, …, ∅) are three such tuples, so l(B) ≥ 3. □

Corollary 3.6 Let B a p-block of G for p odd, where G is quasisimple with G/Z(G) ≅ A_{n} for some n. Then B is nilpotent if and only if l(b_{Q}) = 1 for every B-subgroup (Q, b_{Q}).

Proof: If B has abelian defect group D, then this is [30]. So we may assume |D| > p^{2}. Suppose G ⊲ H, where H ≅ Ŝ_{n}, and let B_{H} ∈ Blk(H) covering B. Then B_{H} has defect group D, and by Lemma 3.5 l(B_{H}) ≥ 3. But l(B) ≥ l(B_{H})/2 > 1, so B is not nilpotent, and of course we can take the B-subgroup (1, B) to show the proposed equivalent condition is also not satisfied.

It remains to consider the exceptional covers, but in these cases it is easy to check that every block with non-central defect groups has more than one irreducible Brauer character. □

4 Sporadic groups

In this section we determine the nilpotent blocks with non-central defect groups of quasisimple groups G where G/Z(G) is one of the 26 sporadic simple groups. Note that due to Lemma 2.4 it suffices to consider the case Z(G) is a p′-group.

In order to provide a reasonably unified treatment of the classification of nilpotent blocks of the sporadic groups, we work from [20, Table 5.3]. However, in all cases the number of irreducible Brauer characters in the blocks are known, which would lead to a shorter but less illuminating proof. To avoid an overly long proof we do use these results in showing that Puig’s conjecture holds.

We use [20, Table 5.3] and apply the following simple results to demonstrate the non-existence of such blocks in many cases:

Lemma 4.1 Suppose that D is a defect group for a nilpotent block of a finite group G. Let x ∈ D have order p and write Q = ⟨x⟩ and R = O_{p}(C_{G}(Q)). Then

(i) there is no p-regular g ∈ N_{G}(Q) − C_{G}(Q) which fixes every block of C_{G}(Q); (ii) there is no p-regular g ∈ N_{G}(R) − C_{G}(R) which fixes every block of C_{G}(R).

Proof: Note that R is contained in every defect group of every block of C_{G}(Q). Hence R ≤ D. The result then follows from the definition of a nilpotent block. □

Lemma 4.2 Let Q be a p-subgroup of G. If |N_{G}(Q)/C_{G}(Q)| is prime to p and, for every n, is strictly greater than the number of p-blocks of C_{G}(Q) of dimension n (or is greater than or equal to n in the case n is the dimension of the principal block of C_{G}(Q)), then Q cannot be a subgroup of a defect group of a nilpotent block of G.

Proof: In this case every p-block of C_{G}(Q) must be fixed by a p-regular element of N_{G}(Q) − C_{G}(Q), and we apply Lemma 4.1. □

Lemma 4.3 Let B be a nilpotent block with defect group D, and let 1 ≠ Q ≤ Z(D). Then C_{G}(Q)/Q possesses a nilpotent block with defect group D/Q.

Proof: Let b_{D} ∈ Blk(DC_{G}(D)) with b_{D}^{G} = B. Now DC_{G}(D) ≤ C_{G}(Q), and b = b_{D}^{C_{G}(Q)} is nilpotent. D is the unique defect group of b_{D} and b^{G} = B, so D is a defect group of b. There is a one-to-one correspondence between the blocks of C_{G}(Q) with defect group D and the blocks of C_{G}(Q)/Q with defect group D/Q. Let \overline{b} be the correspondent of b. By Lemma 2.4 \overline{b} is nilpotent. □

Write Z = Z(G) and \overline{G} = G/Z. Note that when Z is a p′-group, for every psubgroup Q of G we have \overline{C_{G}(Q)} = C\overline{G}(\overline{Q}) and \overline{N_{G}(Q)} = N\overline{G}(\overline{Q}).

Throughout our notation for the conjugacy classes of \overline{G} follows that of [20].

Proposition 4.4 Let G be a quasisimple group such that \overline{G} is a sporadic simple group, with |\overline{G}|_{p} = p. Let B be a p-block of maximal defect of G. Then B is not nilpotent.

Proof: Let D ∈ Syl_{p}(G). Note that D is abelian. If C\overline{G}(\overline{D}) ≤ \overline{D}, then C_{G}(D) ≤ DZ and every p-block of C_{G}(D) is N_{G}(D)-stable. But p does not divide [N_{G}(D) : DC_{G}(D)], whilst by Burnside’s transfer theorem we cannot have N_{G}(D) = C_{G}(D), so a block with defect group D cannot be nilpotent. Hence, using [20, Table 5.3], we may rule out all but the following cases: p = 3 and \overline{G} = J_{1}; p = 5 and \overline{G} = M_{24},J_{1},J_{3},J_{4}; p = 7 and \overline{G} = M_{24},J_{4},Co_{3}, Co_{2}, Suz, Ly, Ru, F i_{22}, F i_{23}, HN; p = 11 and \overline{G} = Co_{3}, Co_{1}, Ly, F i_{22}, F i_{23}, F i′_{24}, HN, F_{2}; p = 13 and \overline{G} = Co_{1}, Ru, F i_{23}, F i′_{24}, T h, F_{2}; p = 17 and \overline{G} = F_{2}, F_{1}; p = 19 and G = F_{2}, F_{1}; p = 23 and \overline{G} = F_{2}, F_{1}; p = 29 and \overline{G} = F_{1}; p = 31 and \overline{G} = F_{1}.

Applying Lemma 4.2 with Q = D to these cases eliminates all but the case p = 3 and G = J_{1}. Here C\overline{G}(\overline{D}) ≅ D × D_{10} and N\overline{G}(\overline{D}) ≅ S_{3} × D_{10}, and it is clear that N_{G}(D) fixes every block of C_{G}(D). □

Theorem 4.5 Let B be a nilpotent p-block with non-central defect group D of a quasisimple group G such that \overline{G} is a sporadic simple group. Then |D| = 3 and G is one of M_{23}, J_{4}, Ly. In each of these cases G does indeed possess a nilpotent block with defect group D.

Proof: We need only consider the case p^{2} divides |\overline{G}|.

Suppose that D is a non-central defect group of a nilpotent p-block B. We assume that Z is a p′-group. Choose x ∈ D of order p, and write Q = ⟨x⟩. In each case N\overline{G}(\overline{Q}) is given by [20, Table 5.3], and C\overline{G}(\overline{Q}) may be deduced using [14].

Let P = O_{p}(N_{G}(Q)). Then P ≤ D.

We eliminate each possibility for the conjugacy class containing x in turn using a succession of methods until we are left with the three cases listed. For each of these we then verify the existence of a nilpotent block with defect group Q.

Suppose that N\overline{G}(\overline{Q}) ≅ \overline{H}_{1} × \overline{H}_{2} and C\overline{G}(\overline{Q}) ≅ \overline{Q} × \overline{H}_{2} for some H_{1}, H_{2} such that Q ◁ H_{1}, and \overline{H}_{1}/\overline{Q} not a p-group. Then every p-block of C_{G}(Q) is fixed by N_{G}(Q) and N_{G}(Q)/C_{G}(Q) is not a p-group, so B cannot be nilpotent. In this way we eliminate the following pairs (\overline{G}, C), where C is the conjugacy class in \overline{G} containing xZ: (M_{11}, 3A), (M_{12}, 3B), (M_{24}, 3B), (J_{2}, 3B), (J_{2}, 5), (Co_{3}, 3C), (Co_{3}, 5B), (Co_{2}, 3B), (Co_{2}, 5B), (Co_{1}, 3D), (HS, 3A), (HS, 5B), (He, 3B), (He, 7A), (He, 7B), (Ru, 5B), (F i_{22}, 3A), (F i_{22}, 5A), (F i_{23}, 3A), (F i_{23}, 5A), (F i′_{24}, 7A), (F_{2}, 3A), (F_{2}, 5A), (F_{1}, 3C).

Suppose that C\overline{G}(\overline{P}) is a p-group and N_{G}(P)/C_{G}(P) is not a p-group. Then every p-block of C_{G}(P) is N_{G}(P)-stable, and B cannot be nilpotent. In this way we may eliminate the pairs (M_{12}, 3A), (J_{3}, 3B), (J_{4}, 11), (Co_{3}, 3A), (Co_{3}, 3B), (Co_{3}, 5A), (Co_{2}, 3A), (Co_{2}, 5A), (Co_{1}, 3C), (Co_{1}, 5C), (HS, 5A), (HS, 5C), (McL, 3), (McL, 5), (Suz, 3B), (He, 7C), (He, 7D), (He, 7E), (Ly, 3B), (Ly, 5), (Ru, 5A), (O′N, 7), (F i_{22}, 3B), (F i_{22}, 3C), (F i_{22}, 3D), (F i_{23}, 3B), (F i_{23}, 3C), (F i′_{24}, 3B) [since in this case no involution in G centralizes a subgroup of the form 3^{1+10}], (F i′_{24}, 3C) [since in this case no involution in \overline{G} centralizes a subgroup of the form C_{3}^{7}], (F i′_{24}, 7B), (HN, 3B), (HN, 5B), (HN, 5C), (HN, 5D), (HN, 5E), (T h, 3B), (T h, 3C), (T h, 5A), (F_{2}, 3B) [since no involution in \overline{G} centralizes a subgroup of the form 3^{1+8}], (F_{2}, 5B), (F_{1}, 3B), (F_{1}, 5B), (F_{1}, 7B), (F_{1}, 13B).

Suppose that N\overline{G}(\overline{Q}) ≅ (\overline{H}_{1} × \overline{H}_{2})n, where n is an integer, and C\overline{G}(\overline{Q}) ≤ \overline{H}_{1} × \overline{H}_{2} for some H_{1} and H_{2} such that \overline{Q} is a proper normal self-centralizing Sylow p-subgroup of \overline{H}_{1}. Then H_{1} ≤ N_{G}(Q) fixes every p-block of C_{G}(Q) and N_{G}(Q)/C_{G}(Q) is not a p-group. It follows that B cannot be nilpotent. In this way we may eliminate the pairs (Co_{1}, 5A), (Co_{1}, 5B), (Co_{1}, 7), (Suz, 3C), (Suz, 5), (He, 5A), (O′N, 3A), (F i′_{24}, 3E), (F i′_{24}, 5A), (HN, 5A), (T h, 7A), (F_{2}, 7A), (F_{1}, 5A), (F_{1}, 7A), (F_{1}, 11A), (F_{1}, 13A).

Suppose that N_{G}(Q) ≅ (H_{1}×H_{2})n for some H_{1} and H_{2}, where n is an integer which is not a power of p, Q ≤ H_{1}, and H_{i}n (with the appropriate action) fixes every p-block of H_{i} for i = 1, 2. Then N_{G}(Q) fixes every p-block of C_{G}(Q) and N_{G}(Q)/C_{G}(Q) is not a p-group. It follows that B cannot be nilpotent. In this way we may eliminate the pairs (M_{22}, 3A), (M_{24}, 3A), (J_{2}, 3A), (J_{3}, 3A), (Co_{1}, 3A), (Co_{1}, 3B), (Suz, 3A), (He, 3A), (Ru, 3A), (F i′_{24}, 3A), (HN, 3A), (T h, 3A), (F_{1}, 3A).

The only cases left unaccounted for are F i_{23} and F i′_{24}, where in each case xZ belongs to the class labelled 3D. Suppose \overline{G} = F i_{23} or F i′_{24} and xZ ∈ 3D. In this case Z = 1 (since p = 3). We have already seen that a nilpotent 3-block of G cannot contain elements of order three outside of 3D. Note that x is conjugate to x^{−1} (to see this consider the orders of the centralizers). Irreducible characters in such a block must vanish on 3A, 3B and 3C. This happens for only one irreducible character, and this lies in a block of defect zero.

If \overline{G} = M_{23} and p = 3, then Z = 1 and N_{G}(Q) ≅ (ℤ_{3}×A_{5})·2, C_{G}(Q) ≅ ℤ_{3}×A_{5}. Note that M_{23} possesses just one 3-block of maximal defect, which cannot then be nilpotent. Hence we may assume D = Q, and so if b_{Q} is a block of C_{G}(Q) with b_{Q}^{G} = B, then b_{Q} has defect group Q. Now C_{G}(Q) has two blocks with defect group Q. The action of N_{G}(Q) on the blocks of C_{G}(Q) is given by the action of S_{5} on the blocks of A_{5}, so the two blocks with defect group Q are fused by N_{G}(Q). Hence [N_{G}(Q, b_{Q}) : C_{G}(Q)] = 1, and b_{Q}^{G} is nilpotent.

If \overline{G} = J_{4} and p = 3, then Z = 1 and N_{G}(Q) ≅ (6M_{22}) · 2 and C_{G}(Q) ≅ (6M_{22}). By [14] 2M_{22} possesses precisely two 3-blocks of defect zero fused by 2M_{22}2 (the rest are fixed). These correspond to two 3-blocks of C_{G}(Q) with defect group Q fused by N_{G}(Q). Hence G possesses a nilpotent block with defect group Q (the Brauer correspondent of the above blocks of C_{G}(Q)).

If \overline{G} = Ly and x ∈ 3A, then Z = 1 and G possesses a nilpotent block with defect group Q, since N_{G}(Q) ≅ (3McL) · 2, C_{G}(Q) ≅ 3McL, and McL possesses precisely two 3-blocks of defect zero which are fused in McL · 2 (all other 3-blocks of McL are fixed by McL · 2). These correspond to two 3-blocks of C_{G}(Q) with defect group Q fused by N_{G}(Q). Hence G possesses a nilpotent block with defect group Q (the Brauer correspondent of the above blocks of C_{G}(Q)).

Note that we have shown in particular that whenever p divides the Schur multiplier of a sporadic simple group, there is no nilpotent block of positive defect of the quotient group (by the Sylow p-subgroup of the centre). □

We conclude:

Proposition 4.6 Let G be a quasisimple group such that G/Z(G) is a sporadic simple group and let p be an odd prime. If B is a nilpotent block of G, then B has defect groups of order at most three.

We now address Puig’s conjecture.

Proposition 4.7 Let G be a quasisimple group such that G/Z(G) is a sporadic simple group and let p be an odd prime. Let B be a p-block of G. If B has positive defect, then l(B) > 1. In particular, B is nilpotent if and only if l(b_{Q}) = 1 for every B-subgroup (Q, b_{Q}).

Proof: We may assume that Z(G) is a p′-group. Let D be a defect group of B. If D is cyclic, then the result follows from the theory of blocks with cyclic defect groups. In the following table we list all the numbers of irreducible Brauer characters in blocks with non-cyclic defect groups, along with a reference. A ’*’ will be used to denote a faithful block in a group with non-trivial centre. The result then follows from examination of the table.

G/Z(G) |D|  ℓ(B)  reference
M_{11}  3^{2}  7  [18]
M_{12}  3^{3}/3^{3}  8/8^{∗}  [18]
M_{22}  3^{2}/3^{2}/3^{2}  5/5^{∗}/5^{∗}  [18]
M_{23}  3^{2}  7  [18]
M_{24}  3^{3}  7  [18]
J_{2}  3^{3}/3^{3}/5^{2}/5^{2}  8/8^{∗}/6/6^{∗}  [18]
J_{3}  3^{5}  8  [18]
J_{4}  3^{3}/3^{3}/3^{2}/11^{3}  9/9/5/40  [8]/[10]
HS  3^{2}/3^{2}/3^{2}/5^{3}/5^{3}  7/7/5^{∗}/10/10^{∗}  [18]
McL  3^{6}/5^{3}/5^{3}  10/12/12^{∗}  [18]
Suz  3^{7}/3^{2}/3^{7}  13/5/10^{∗}  [18]
Ly  3^{7}/5^{6}  21/35
He  3^{3}/3^{2}/5^{2}/7^{3}  7/7/14/10  [18]
Ru  3^{3}/3^{3}/5^{3}/5^{3}  9/9^{∗}/18/18^{∗}  [18]
O′N  3^{4}/3^{2}/7^{3}/7^{3}  14/5/19/19^{∗}  [18]
Co_{3}  3^{7}/5^{3}  20/18  [18]
Co_{2}  3^{6}/5^{3}  23/16  [18]
Co_{1}  3^{9}/3^{3}/3^{2}/5^{4}/5^{2}/7^{2}  29/7/5/29/12/21  [7]
F i_{22}  3^{9}/3^{9}/5^{2}/5^{2}/5^{2}/5^{2}  22/18^{∗}/16/16^{∗}/16^{∗}/16^{∗}  [18]
F i_{23}  3^{13}/5^{2}/5^{2}  32/16/16  [5]
F i′_{24}  3^{16}/3^{2}/5^{2}/5^{2}/5^{2}/5^{2}/5^{2}  25/4/16/16/14/16^{∗}/16^{∗}  [4]
5^{2}/5^{2}/7^{3}/7^{3}/7^{3}  14^{∗}/14^{∗}/22/22^{∗}/22^{∗}  [4]
T h  3^{10}/5^{3}/7^{2}  10/30/24
HN  3^{6}/3^{2}/5^{6}  20/7/16  [6]
F_{2} = B 3^{13}/3^{2}/3^{2}/3^{2}/3^{13}  71/7/7/5/31^{∗}  [9]
5^{6}/5^{2}/5^{2}/5^{6}/7^{2}/7^{2}/7^{2}/7^{2}  51/16/16/33^{∗}/24/24/21/24^{∗} [9]
F_{1} = M 3^{20}/3^{3}/5^{9}/5^{2}/7^{6}/7^{2}/11^{2}/13^{3} 83/7/91/16/70/24/45/52  [18]

Table 1: Numbers of irreducible Brauer characters in blocks with non-cyclic, noncentral defect groups of sporadic groups

If G = Ly and p = 3, then by [14] G has thirty 3-regular conjugacy classes. By [31], aside from the principal block, G has five 3-blocks of defect zero and two 3-blocks of defect one. Since we have shown that neither of these blocks of defect one is nilpotent, it follows that they each have two irreducible Brauer characters. Hence the principal block has 21 irreducible Brauer characters for p = 3. A similar computation for p = 5 shows that the principal 5-block of Ly has 35 irreducible Brauer characters (and this is the unique 5-block with non-cyclic defect groups).

If G = T h and p = 3, then by [14] G has sixteen 3-regular conjugacy classes. By [32] G has four 3-blocks of defect zero and one 3-block of defect one (which we have seen cannot be nilpotent, so has two irreducible Brauer characters. Hence the principal block 3-possesses ten irreducible Brauer characters. A similar computation for p = 5 shows that the principal 5-block of T h has 30 irreducible Brauer characters (and this is the unique 5-block with non-cyclic defect groups). For p = 7, by [32] G has fourteen blocks of defect zero, a block of defect one (with six irreducible Brauer characters, by consideration of the inertial quotient) and the principal block, which must then have 24 irreducible Brauer characters.

If G/Z(G) = F i′_{24}, then the result follows from [4] when p = 3, and when p = 5 or 7 for non-faithful blocks in the case |Z(G)| = 3. Suppose |Z(G)| = 3, and consider faithful blocks B with a defect group D covering a block, say c of Z(G). Suppose first p = 5. We have D = 5^{2} and from [4, p.141] k(B) = 20. Note that G has only one conjugacy class of elements of order 5. If x ∈ D \{1} and b ∈ Blk(C_{G}(x)) with b^{G} = B, then C_{G}(x) = 3×5×A_{9} and b = c×B_{0}(5)×b′ for some b′ ∈ Blk(A_{9}) with D(b′) = 5. As shown in [4, p.114] A_{9} has three such blocks b′_{0} = B_{0}(A_{9}), b′_{1}, b′_{2} and l(b′_{0}) = l(b′_{1}) = 4, l(b′_{2}) = 2. The canonical characters of the root blocks of b′_{2} and b′_{1} are linear and degree 3 characters of C_{A_{9}}(5) = 5 × A_{4}, respectively. Since N_{G}(D) = 3.(5^{2}:4A_{4} × A_{4}).2 and a Sylow 3-subgroup of N_{G}(D) is isomorphic to 3_{+}^{1+2}, it follows that c × B_{0}(5) × b′_{0} and c × B_{0}(5) × b′_{2} induce the same block B of G and so l(B) = 20 − 4 − 2 = 14. Also c × B_{0}(5) × b′_{1} induces another block B of G and l(B) = 20 − 4 = 16. If p = 7, then by [4, p.141], k(B) = k(B_{0}(F i′_{24})) and C_{G}(x) = 3 × C_{F i′_{24}}(x) for any x ∈ D \ {1}. Thus l(B) = l(B_{0}(F i_{24})) = 22. □

5 Notation for classical groups and their blocks

Let V be a linear, unitary, non-degenerate orthogonal or symplectic space over the field 𝔽_{q}, where q = r^{a} for some prime r ≠ p. We will follow the notation of [3], [11], [16] and [17].

If V is orthogonal (and q is odd), then there is a choice of equivalence classes of quadratic forms. Write η(V) for the type of V as defined in [17], so η(V) = η = + or −. Write η(V) = + if V is linear and η(V) = − if V is unitary. If V is non-degenerate orthogonal or symplectic, then denote by I(V) the group of isometries on V and let I_{0}(V) = I(V) ∩ SL(V).

If V is symplectic, then I(V) = I_{0}(V) = Sp_{2n}(q).

If V is a (2n + 1)-dimensional orthogonal space, then I(V) = ⟨−1_{V} ⟩ × I_{0}(V) with I_{0}(V) = SO_{2n+1}(q).

If V is a 2n-dimensional orthogonal space, then I(V) = O^{η}(V) = O_{2n}^{η}(q) and I_{0}(V) = SO_{2n}^{η}(q).

If V is a 2n-dimensional non-degenerate orthogonal or symplectic space, then denote by J_{0}(V) the conformal isometries of V with square determinant. If V is orthogonal of dimensional at least two, then write D_{0}(V) for the special Clifford group of V (cf. [17]).

Denote by GL^{+}(V) the general linear group GL(V) and GL^{−}(V) the unitary group U(V).

Let G = GL^{η}(V) or I(V). Write F_{q} = F_{q}(G) for the set of polynomials (with coefficients in 𝔽_{q}) serving as elementary divisors for semisimple elements of G (cf. [3, p.6]). For Γ ∈ F_{q}(G), let d_{Γ} be the degree of Γ, and δ_{Γ} be the reduced degree defined as in [3], [16] and [17]. So δ_{Γ} = d_{Γ} or δ_{Γ} = \frac{1}{2}d_{Γ} according as d_{Γ} is even or odd (note that if V is symplectic or orthogonal, Γ must have even degree unless Γ = X ± 1).

If G = GL(V), then let ϵ_{Γ} = 1. Otherwise ϵ_{Γ} is given by [3, p.6]. Let e_{Γ} be the multiplicative order of ϵ_{Γ}q^{δ_{Γ}} modulo p. Thus we may write e_{Γ}δ_{Γ} = ep^{α_{Γ}}δ′_{Γ} for some α_{Γ} and δ′_{Γ} with p ∤ δ′_{Γ}, where e = e_{X−1}.

Given a semisimple element s ∈ G, there is a unique orthogonal decomposition V = ∑_{Γ∈F_{q}} V_{Γ}(s), with s = ∏_{Γ∈F_{q}} s(Γ), where the V_{Γ}(s) are nondegenerate subspaces of V and s(Γ) ∈ GL(V_{Γ}(s)), U(V_{Γ}(s)) or I(V_{Γ}(s)) (depending on G) has minimal polynomial Γ. This is called the primary decomposition of s. Write m_{Γ}(s) for the multiplicity of Γ in s(Γ). We have C_{G}(s) = ∏_{Γ∈F_{q}} C_{Γ}(s), where C_{Γ}(s) = I(V_{Γ}(s)) or GL^{ϵ_{Γ}}(m_{Γ}(s), q^{δ_{Γ}}) as appropriate.

6 Blocks of linear and unitary groups

Suppose G = GL_{n}^{η}(q) = GL^{η}(V) and p is odd and distinct to r, and let B be a p-block of G with a defect group D and label (s, κ). Then we may write

V = V_{0} ⊥ V_{+}, D = D_{0} × D_{+}, s = s_{0} × s_{+},  (6.1)

where V_{0} = C_{V} (D), V_{+} = [D, V], s_{0} ∈ G_{0} = GL^{η}(V_{0}) and s_{+} ∈ G_{+} := GL^{η}(V_{+}). For convenience we denote GL^{η}(V) by G(V) and SL^{η}(V) by S(V).

Theorem 6.1 Let G = GL^{η}(V) = GL^{η}(n, q) and suppose p is odd with p ∤ q. Then the following are equivalent.

(a) B is a nilpotent block of G.

(b) m_{Γ}(s_{+}) = e_{Γ} = 1 for all Γ ∈ F_{q} which are elementary divisors of s_{+}.

(c) κ_{Γ} is a e_{Γ}-core of m_{Γ}(s) whenever e ∤ δ_{Γ}, and m_{Γ}(s) ≤ 1 whenever e | δ_{Γ}, where κ_{Γ} = ∅ is viewed as an e_{Γ}-core of 0 = m_{Γ}(s).

(d) Let (D, b_{D}) be a Sylow B-subgroup and θ the canonical character of b_{D}. Then C_{G}(D) = G_{0} × C_{+} and θ = θ_{0} × θ_{+}, where C_{+} := C_{G_{+}}(D_{+}) is regular in G_{+}, θ_{0} is an irreducible character of defect 0 of G_{0} labelled by (s_{0}, κ) and θ_{+} = ±R_{T_{+}}^{C_{+}}(s_{+}) with T_{+} = C_{C_{+}}(s_{+}) a torus of both G_{+} and C_{+}, and D_{+} = O_{p}(T_{+}). Here R_{T_{+}}^{C_{+}}(s_{+}) is the Deligne-Lusztig generalized character.

In particular, if B is nilpotent, then D is abelian.

Proof: Let s_{+} = ∏_{Γ} s(Γ) be a primary decomposition, so that V_{+} = ⨁_{Γ} V_{Γ} with V_{Γ} the underlying space of s(Γ). Write m_{Γ} for m_{Γ}(s_{+}). Then

C_{G_{+}}(s_{+}) = ∏_{Γ} C_{Γ},  (6.2)

where C_{Γ} ≅ GL^{ϵ_{Γ}}(m_{Γ}, q^{δ_{Γ}}). We may suppose D_{+} ∈ Syl_{p}(C_{G_{+}}(s_{+})), so that

D_{+} = ∏_{Γ} D_{Γ}, D_{Γ} ∈ Syl_{p}(C_{Γ}).  (6.3)

So D is a direct product of wreath product p-groups.

Let Γ be an elementary divisor of s_{+}. Since C_{V_{Γ}}(D_{Γ}) = 0, it follows that p divides q^{δ_{Γ}m_{Γ}} − ϵ_{Γ} and so e_{Γ} | m_{Γ}. Hence we may write m_{Γ} = e_{Γ}w_{Γ} for some w_{Γ}. Let A(D) be the subgroup of D generated by all the abelian normal subgroups of D. By [1, Theorem 2], A(D) is the base subgroup of D. Write R = A(D). Then

R = D_{0} × ∏_{Γ}(R_{Γ})^{w_{Γ}}, C_{G}(R) = G_{0} × ∏_{Γ}(K_{Γ})^{w_{Γ}}, K_{Γ} ≅ GL^{ϵ}(δ′_{Γ}, q^{ep^{α_{Γ}}}) (6.4)

where ϵ = ϵ_{X−1} and R_{Γ} = O_{p}(Z(K_{Γ})) is cyclic and (R_{Γ})^{w_{Γ}} is a diagonal subgroup of GL^{ϵ_{Γ}}(w_{Γ}, q^{δ_{Γ}e_{Γ}}) ≤ C_{Γ}. Thus C_{G_{Γ}}((R_{Γ})^{w_{Γ}}) = (K_{Γ})^{w_{Γ}}, C_{G}(R) is regular in G,

N_{G_{Γ}}((R_{Γ})^{w_{Γ}}) = K_{Γ} ≀ S(w_{Γ}),

and we may suppose s ∈ C_{G}(R), where G_{Γ} := G(V_{Γ}) and S(m) is the symmetric group on m letters.

Suppose w_{Δ} ≥ 2 for some Δ. Then there is P (D_{Δ}) ≤ (R_{Δ})^{w_{Δ}} such that

C_{G_{Δ}}(P (D_{Δ})) = (K_{Δ})^{w_{Δ}−2} × GL^{ϵ}(2δ′_{Δ}, q^{ep^{α_{Δ}}}) and P (D_{Δ}) = O_{p}(C_{G_{Δ}}(P (D_{Δ}))).

Thus C_{C_{Δ}}(P (D_{Δ})) = GL^{ϵ_{Δ}}(1, q^{δ_{Δ}e_{Δ}})^{w_{Δ}−2} × GL^{ϵ_{Δ}}(2, q^{δ_{Δ}e_{Δ}}) and

N_{C_{G_{Δ}}(P (D_{Δ}))}((R_{Δ})^{w_{Δ}}) = (K_{Δ})^{w_{Δ}−2} × K_{Δ} ≀ S(2).

There is an element y_{Δ} of (N_{C_{Δ}}((R_{Δ})^{w_{Δ}}) ∩ C_{G_{Δ}}(P (D_{Δ}))) \ C_{C_{Δ}}((R_{Δ})^{w_{Δ}}) which swaps exactly two factors K_{Δ} in C_{G_{Δ}}((R_{Δ})^{w_{Δ}}), |y_{Δ}| = 4, y_{Δ}^{2} ∈ C_{C_{Δ}}((R_{Δ})^{w_{Δ}}) and det(y_{Δ}) = 1.

Writing y_{Γ} := 1 when Γ ≠ Δ, define

y := 1_{V_{0}} × ∏_{Γ} y_{Γ} and P (D) := D_{0} × ∏_{Γ≠Δ}(R_{Γ})^{w_{Γ}} × P (D_{Δ}).  (6.5)

Then y ∈ (N_{C_{G}(P (D))}(R)∩C_{G}(s))\C_{G}(R) and y^{2} ∈ C_{G}(R). Let (R, b_{R}) be a B-subgroup, so that D(b_{R}) = R and we may suppose b_{R} ⊆ E_{p}(C_{G}(R), (s)). Since y ∈ C_{G}(s), it follows that y ∈ N_{G}(R, b_{R}) \ C_{G}(R), so B is not nilpotent, a contradiction. Thus m_{Γ} = e_{Γ} for all Γ and D is abelian with each D_{Γ} cyclic.

For each Γ,

N_{GL^{ϵ_{Γ}}(e_{Γ},q^{δ_{Γ}})}(D_{Γ}) = ⟨τ_{Γ}, C_{GL^{ϵ_{Γ}}(e_{Γ},q^{δ_{Γ}})}(D_{Γ})⟩, C_{GL^{ϵ_{Γ}}(e_{Γ},q^{δ_{Γ}})}(D_{Γ}) = GL^{ϵ_{Γ}}(1, q^{e_{Γ}δ_{Γ}}), where τ_{Γ} ∈ GL^{ϵ_{Γ}}(e_{Γ}, q^{δ_{Γ}}) has order e_{Γ} modulo GL^{ϵ_{Γ}}(1, q^{e_{Γ}δ_{Γ}}). If τ = 1_{V_{0}} × ∏_{Γ} τ_{Γ}, then τ ∈ N_{G}(D) ∩ C_{G}(s) and so τ ∈ N_{G}(D, b_{D}), where b_{D} is the block of C_{G}(D) labelled by (s, κ).

Since e_{Γ} and p are coprime, it follows that τ is a p′-element, e_{Γ} = 1 and C_{Γ} = GL^{ϵ_{Γ}}(1, q^{δ_{Γ}}). In particular, C_{G_{+}}(s_{+}) is a torus and e | δ_{Γ}.

Conversely, if m_{Γ}(s_{+}) = e_{Γ} = 1, then C_{Γ} = GL^{ϵ_{Γ}}(1, q^{δ_{Γ}}) and so N_{C_{Γ}}(D_{Γ}) = C_{C_{Γ}}(D_{Γ}) = C_{Γ}. Thus D is abelian,

N_{C_{G}(s)}(D) = C_{C_{G}(s)}(D) = C_{G}(s)

and N_{G}(D, b_{D}) = C_{G}(D). By Proposition 2.1, B is nilpotent.

Note that e_{Γ} = 1 if and only if e | δ_{Γ}, so since D is a Sylow p-subgroup of C_{G}(s), it follows that m_{Γ}(s_{0}) = 0 whenever m_{Γ}(s_{+}) = 1, so that m_{Γ}(s) = 1. If m_{Γ}(s_{0}) = m_{Γ}(s) ≠ 0, then by definition, κ_{Γ} is an e_{Γ}-core of m_{Γ}(s). The equivalences of (b), (c) and (d) now follow by [11] and [16]. □

For integers c and m, we write p^{c}∥m when p^{c} | m and p^{c+1} ∤ m.

Remark 6.2 In the notation of the proof above, we may suppose the element τ ∈ C_{G_{+}}(s_{+}) has determinant 1 whenever e_{Γ} ≥ 2 for some Γ.

Proof We may suppose q = q^{δ_{Γ}}, so that e_{Γ} = e. Let T be the diagonal maximal torus of G = GL(𝔽 ⊗ V), σ the Frobenius map of G such that C_{G}(σ) = GL^{ϵ}(e, q), where 𝔽 is the algebraic closure of 𝔽_{q}.

Choose matrices P_{ij} (with i ≠ j) of G such that P_{ij} acts on T as the permutation swapping the (i, i) and (j, j) entries of T, the entries of P_{ij} are 1 or −1, det(P_{ij}) = 1 and P_{ij} is fixed by σ. If W is generated by the matrices P_{ij}, then N_{G}(T) = TW and W T/T ≅ S(e).

Note ω^{σ} = ω for each ω ∈ W. Let ω_{0} ∈ W such that

C_{T}(σω_{0}) ≅ GL^{ϵ}(1, q^{e}),

so that ω_{0} acts on T as the cycle (1, 2, ⋯, e). Now C_{T}(σω_{0}) is conjugate in G to the Coxeter torus GL^{ϵ}(1, q^{e}) of C_{G}(σ) and ω_{0} normalizes C_{T}(σω_{0}). Thus there is an element β ∈ SL^{η}(e, q) such that β normalizes the torus GL^{ϵ}(1, q^{e}) and β has order e modulo GL^{ϵ}(1, q^{e}). Since N_{GL^{ϵ}(e,q)}(GL^{ϵ}(1, q^{e}))/GL^{ϵ}(1, q^{e}) ≅ ZZ_{e}, it follows that

N_{GL^{ϵ}(e,q)}(GL^{ϵ}(1, q^{e})) = ⟨β, GL^{ϵ}(1, q^{e})⟩

and we may suppose τ = β.

7 A set of technical conditions

In order to investigate nilpotent blocks of exceptional groups of Lie type it is not sufficient just to find the nilpotent blocks of classical groups. We need in addition some somewhat technical properties which we will identify in classical groups and their extensions by diagonal automorphisms which relate to nilpotency (in particular, they will be used to examine centralizers of elements of defect groups of nilpotent blocks).

These properties also ensure that Puig’s conjecture holds for the groups under consideration.

We state these properties in this section, along with some general results which will be needed in proving that they hold for classical groups.

Let G be a finite group, Q a p-subgroup of G, and B ∈ Blk(G). If p is odd, we denote by A(Q) the subgroup of Q generated by all the abelian normal subgroups of Q. Recall that a B-subgroup (R, b_{R}) is called self-centralizing if Z(R) is a defect group of b_{R} ∈ Blk(C_{G}(R)).

We will prove for some finite groups of Lie type that one of (a)-(d) of the following holds. A feature of these properties is that none can be satisfied by a nilpotent block with non-abelian defect groups.

Property 7.1 Let K be a normal subgroup of a finite group H, and let B ∈ Blk(K) and B_{H} ∈ Blk(H) such that B_{H} covers B.

(a) There exist B-subgroups (P, g) ≤ (R, b), where R is abelian, with abelian defect groups D(g) and D(b) respectively such that D(g) = D(b), and an element y ∈ N_{C_{K}(P)}(R) such that y^{4} = 1, [y, x] ∉ Z(K) for some x ∈ R and y^{2} ∈ C_{K}(R), and such that θ^{y} = θ, where θ is the canonical character of b. There exist B_{H}- subgroups (P, g_{H}) ≤ (R, b_{H}) where g_{H} covers g, b_{H} covers b, such that b_{H}^{y} = b_{H} and D(b_{H}) = D(g_{H}) is abelian for defect groups D(b_{H}) and D(g_{H}) of b_{H} and g_{H} respectively.

(a∗) Property (a) above holds, and there exist subgroups N_{i} ⊲ M_{i} of H, and characters θ_{i} ∈ Irr(N_{i}) for i = 1, 2 such that M_{i}/N_{i} is abelian,

Z(K) ≤ N_{1} × N_{2} ≤ C_{K}(R) ≤ C_{H}(R) ≤ M_{1} ∘ M_{2},

θ covers θ_{1} × θ_{2}, Z_{0} ∩ N_{2} = 1, θ_{2} has a y-stable extension to M_{2} and [y, x] = 1 or [y, x] ∈ Z_{0}N_{2} according as x ∈ M_{1} or M_{2}, where Z_{0} ≤ Z(M_{1}) ∩ Z(M_{2}) such that M_{1} ∘ M_{2} is a central product over Z_{0}

(b) D(B) ≅ 3_{+}^{1+2}, ℓ(B) ≥ 2, and either D(B_{H}) = D(B) or D(B_{H}) ≅ ℤ_{3} ≀ ℤ_{3}.

(c) D(B) = 3^{2}, ℓ(B) ≥ 2, and either D(B_{H}) = D(B) or D(B_{H}) ≅ 3_{+}^{1+2}.

(d) Both D(B) and D(B_{H}) are abelian.

Remark 7.2 (i) Suppose that (P, g) ≤ (R, b) are B-subgroups with abelian defect groups D(g) and D(b), and R is abelian. By [15, Lemma 4.1], there exists a B_{H}- subgroup (P, g_{H}) such that g_{H} covers g. Since R ≤ D(b) is abelian and (P, g) ≤ (R, b), it follows that (R, b) is a g-subgroup and by [15, Lemma 4.1] again, there exists a g_{H}- subgroup (R, b_{H}) such that b_{H} covers b. Thus (R, b_{H}) is a B_{H}-subgroup and (P, g_{H}) ≤ (R, b_{H}).

(ii) Note that in the notation of the proof of Theorem 6.1, (R, b_{R}) is self-centralizing.

We also observe that there is some redundancy in (a) when (a∗) holds:

Remark 7.3 In the notation of Property 7.1 (a) and (a∗) suppose N_{1} × N_{2} ≤ E ≤ M_{1} ∘ M_{2} such that y normalizes E and suppose φ ∈ Irr(E | θ_{1} × θ_{2}). Then φ^{y} = φ.

Proof: Since φ has an extension φ̃ to EZ_{0}, it follows that we may suppose Z_{0} ≤ E. Let π_{i} : (M_{1} ∘ M_{2}) → M_{i}/Z_{0} be the canonical projection and let E_{i}/Z_{0} = π_{i}(E) for some E_{i} ≤ M_{i}. Then

N_{1} × N_{2} ≤ E ≤ E_{1} ∘ E_{2} ≤ M_{1} ∘ M_{2}.

Let ζ be the y-stable extension of θ_{2} to M_{2}, and set θ̃_{2} = ζ|_{E_{2}}. Then θ̃_{2} is an extension of θ_{2} to E_{2} which is stabilized by y. Since [y, x] = 1 or N_{2}Z_{0} according as x ∈ M_{1} or x ∈ M_{2}, it follows that y centralizes E_{1} and [y, x] ∈ N_{2}Z_{0} for any x ∈ E_{2}. By Lemma 2.7, φ^{y} = φ. □

Applying Remark 7.3 to the canonical characters of b and b_{H}, we see that in the notation of (a), θ^{y} = θ and b_{H}^{y} = b_{H} are automatically satisfied if the other parts of (a∗) hold.

Before stating the key consequence of Property 7.1, we need the following:

Lemma 7.4 Let B be a block of a finite group G and suppose there are B-subgroups (P, b_{P}) ≤ (R, b_{R}) such that R is abelian, b_{P} has abelian defect groups and there is pregular element y ∈ N_{C_{G}(P)}(R) \ C_{G}(R) such that b_{R}^{y} = b_{R}. Then there is a B-subgroup (Q, b_{Q}) such that l(b_{Q}) > 1.

Proof: For convenience write L = C_{G}(P). Note first that b_{P} is not nilpotent, since yC_{G}(R) ∈ N_{L}(R, b_{R})/C_{G}(R) is not a p-element and (R, b_{R}) is a b_{P} -subgroup. Hence by Lemma 2.4 the unique block \overline{b_{P}} of \overline{L} := L/P dominated by b_{P} is not nilpotent either. But \overline{b_{P}} has abelian defect groups, so by [30, Theorem 3] there is a \overline{b_{P}} -subgroup (\overline{Q}, b\overline{Q}), where \overline{Q} = Q/P for some Q ≥ P, such that l(b\overline{Q}) > 1.

Note that \overline{C_{L}(Q)} ⊲ C\overline{L}(\overline{Q}) and C\overline{L}(\overline{Q})/\overline{C_{L}(Q)} is a p-group. By [33, Lemma 1] there is a b_{P} -subgroup (Q, b_{Q}) such that b\overline{Q} is the unique block of C\overline{L}(\overline{Q}) covering the block b′ ; \overline{Q} of \overline{C_{L}(Q)} dominated by b_{Q}. Note that l(b_{Q}) = l(b′ ; \overline{Q}) = l(b\overline{Q}) > 1. But P ≤ Q, so (Q, b_{Q}) is also a B-subgroup, and we are done. □

We make the key observation, and see also that the conjecture of Puig is a consequence of any of Properties 7.1 (a)-(d).

Corollary 7.5 Suppose one of Property 7.1 (a) − (d) holds for a block B. If B has non-abelian defect groups, then there is a B-subgroup (Q, b_{Q}) such that l(b_{Q}) > 1, and hence B is not nilpotent. In particular, B is nilpotent if and only if l(b_{Q}) = 1 for every B-subgroup (Q, b_{Q}).

Proof: If (a) holds, then the result follows immediately from Lemma 7.4, since B is not nilpotent and there is a B-subgroup (Q, b_{Q}) such that l(b_{Q}) > 1. If (b) or (c) holds, then l(B) > 1, so B is not nilpotent, and we may take (Q, b_{Q}) = (1, B). If (d) holds, then this is [30, Theorem 3]. □

We prove a lemma which will be useful in establishing the given properties. Let H be a finite group, K ⊲H, Z ≤ Z(H)∩K and \overline{K} := K/Z ≤ \overline{H} := H/Z. Let \overline{B} ∈ Blk(\overline{K}) and B ∈ Blk(K) dominating \overline{B}, and (Q, b_{Q}) a B-subgroup. Let γ : H → \overline{H} be the natural homomorphism, and write \overline{X} = γ(X) for any X ⊆ H,

If Z is a p′-group, then (\overline{Q}, b̄_{Q}) is defined in Section 2 and it is a \overline{B}-subgroup. Suppose Z is a p-group. Then γ^{−1}(C\overline{K}(\overline{Q})) ≤ N_{K}(QZ) and γ^{−1}(C\overline{K}(\overline{Q}))/C_{K}(Q) is a p-group. Thus γ^{−1}(C\overline{K}(\overline{Q})) has a unique block b̂_{Q} covering b_{Q} and we denote b̄_{Q} the block of C\overline{K}(\overline{Q}) corresponding to b̂_{Q}, so that by [33, Lemma 1], (\overline{Q}, b̄_{Q}) is also a \overline{B}-subgroup.

In general, since \overline{K} ≅ (K/O_{p}(Z))/(Z/O_{p}(Z)) and Z/O_{p}(Z) ≤ Z(K/O_{p}(Z)), it follows that (\overline{Q}, b̄_{Q}) is defined and is a \overline{B}-subgroup.

Lemma 7.6 Let H be a finite group, K ⊲ H, Z ≤ Z(H) ∩ K. Define \overline{K} := K/Z and \overline{H} := H/Z. Let \overline{B} ∈ Blk(\overline{K}) and B ∈ Blk(K) dominating \overline{B}. Suppose B-subgroups (P, g) ≤ (R, b) satisfy Property 7.1 (a∗), and suppose Z(\overline{K}) = Z(K)/Z. In addition, if Z = O_{p}(Z), then suppose, moreover that C_{H}(P)/Z = C\overline{H}(\overline{P}) and C_{H}(R)/Z = C\overline{H}(\overline{R}). Then the \overline{B}-subgroups (\overline{P} , ḡ) ≤ (\overline{R}, b̄) satisfies Property 7.1 (a∗).

Proof: Let \overline{B}\overline{H} ∈ Blk(\overline{H}) covering \overline{B}, and B_{H} ∈ Blk(H) dominating \overline{B}\overline{H} and χ ∈ Irr(\overline{B}\overline{H}), so that χ covers some ψ ∈ Irr(\overline{B}). But Irr(\overline{B}\overline{H}) ⊆ Irr(B_{H}) and Irr(\overline{B}) ⊆ Irr(B), so B_{H} covers B.

Let f be the unique block of Z covered by B. Then each character χ in Irr(B) covers a character in Irr(f). Since Irr(\overline{B}) ⊆ Irr(B), it follows that f is the principal block. Since (P, g) is a B-subgroup and Z ≤ Z(K) and since B covers f, it follows that g covers f, and similarly, b covers f. The same applies to B_{H} and to B_{H}-subgroups.

Since C_{K}(P Z) = C_{K}(P), it follows that we may suppose O_{p}(Z) ≤ P. Let y ∈ N_{C_{K}(P)}(R) such that y^{4} = 1, [y, x] ∉ Z(K) for some x ∈ R, y^{2} ∈ C_{K}(R), and suppose N_{i} and M_{i} are subgroups of H, and θ_{i} ∈ Irr(N_{i}) for i = 1, 2, such that M_{i}/N_{i} is abelian,

Z ≤ Z(K) ≤ N_{1} × N_{2} ≤ C_{K}(R) ≤ C_{H}(R) ≤ M_{1} ∘ M_{2},

θ covers θ_{1} × θ_{2}, Z_{0} ∩ N_{2} = 1, θ_{2} has a y-stable extension θ̃_{2} to M_{2} and [y, x] = 1 or in Z_{0}N_{2} according as x ∈ M_{1} or M_{2}, where Z_{0} ≤ Z(M_{1}) ∩ Z(M_{2}) such that M_{1} ∘ M_{2} is the central product over Z_{0} and θ is the canonical character of b.

It suffices to consider the cases p-group and a p′-group separately. Then by [33, Lemma 1 (iii)] and Lemma 2.5, (\overline{P} , ḡ) ≤ (\overline{R}, b̄) are \overline{B}-subgroups. If Z is a p′-group, then C_{K}(R)/Z = C\overline{K}(\overline{R}). If Z is a p-group, then C_{H}(R)/Z = C\overline{H}(\overline{R}), γ^{−1}(C\overline{H}(\overline{R})) = C_{H}(R) and so γ^{−1}(C\overline{K}(\overline{R})) = C_{K}(R). Thus in either case C_{K}(R)/Z = C\overline{K}(\overline{R}) and D(b)Z/Z = D(b̄). Similarly, D(ḡ) = D(g)Z/Z, and D(ḡ) = D(b̄) is abelian as D(g) = D(b) is abelian.

Let ȳ = γ(y), so that ȳ^{4} = 1, ȳ ∈ C\overline{K}(\overline{P})∩N\overline{K}(\overline{R}), ȳ^{2} ∈ C\overline{K}(\overline{R}). Since [y, x] ∉ Z(K) for some x ∈ R and since Z(\overline{K}) = Z(K)/Z, it follows that [ȳ, x̄] ∉ Z(\overline{K}) and in particular, ȳ ∉ C\overline{K}(\overline{R}).

Let \overline{N}_{i} = γ(N_{i}) and \overline{M}_{i} = γ(M_{i}). Then \overline{N}_{i} ⊲ \overline{M}_{i} and \overline{M}_{i}/\overline{N}_{i} is abelian such that

Z(\overline{K}) ≤ \overline{N}_{1} × \overline{N}_{2} ≤ C\overline{K}(\overline{R}) ≤ C\overline{H}(\overline{R}) ≤ \overline{M}_{1} ∘ \overline{M}_{2},

where \overline{M}_{1} ∘ \overline{M}_{2} is the central product over \overline{Z}_{0} = γ(Z_{0}).

Since Z ≤ Z(K) ≤ N_{1} ×N_{2} and θ is the canonical character of b, it follows that θ is the lift of the canonical character θ̄ of b̄. Similarly, since θ covers θ_{1} ×θ_{2}, it follows that Z ≤ ker(θ_{1} × θ_{2}) and θ_{1} × θ_{2} is the lift of θ̄_{1} × θ̄_{2} for some θ̄_{i} ∈ Irr(\overline{N}_{i}). In particular, θ̄ ∈ Irr(C\overline{K}(\overline{R}) | θ̄_{1} ×θ̄_{2}). If φ ∈ Irr(M_{1} ∘M_{2} | θ_{1} ×θ_{2}), then by Lemma 2.7 φ = ψ ∘(θ̃_{2}λ) for some ψ ∈ Irr(M_{1} | θ_{1}) and λ ∈ Irr(M_{2}/Z_{0}N_{2}). Since [y, x] ∈ N_{2}Z_{0} for all x ∈ M_{2}, it follows that λ^{y} = λ and so θ̃_{2}λ is y-invariant. But Z ≤ ker(φ), so Z ∩ M_{2} ≤ ker(θ̃_{2}λ), and θ̃_{2}λ can be viewed as a character of Irr(\overline{M}_{2}), which is a ȳ-invariant extension of θ̄_{2} to \overline{M}_{2}. Thus Property 7.1 (a∗) holds for \overline{B}.

Similarly, if (P, g_{H}) ≤ (R, b_{H}) are B_{H}-blocks such that D(g_{H}) = D(b_{H}) is abelian, and g_{H} and b_{H} cover g and b, respectively, then there exist \overline{B}\overline{H}-subgroups (\overline{P} , ḡ\overline{H}) ≤ (\overline{R}, b̄\overline{H}) such that ḡ\overline{H} is dominated by g_{H} and b̄\overline{H} is dominated by b_{H} and D(ḡ\overline{H}) = D(g_{H})Z/Z and D(b̄\overline{H}) = D(b_{H})Z/Z. Thus D(ḡ\overline{H}) = D(b̄\overline{H}) = D(b_{H})Z/Z is abelian.

Since g_{H} covers g, it follows that the canonical character θ_{H} of g_{H} covers the canonical character θ of g. But θ_{H} is the lift of the canonical character of ḡ\overline{H} and θ is the lift of the canonical character of ḡ, so ḡ\overline{H} covers ḡ. Similarly, b̄\overline{H} covers b̄. This proves that Property 7.1 (a) holds for (\overline{P} , ḡ) ≤ (\overline{R}, b̄). □

8 Classical groups

Suppose p is odd. In this section we demonstrate that every nilpotent block of a classical group of has abelian defect groups.

Proposition 8.1 Let K := SL_{n}^{η}(q) ≤ H ≤ G := GL_{n}^{η}(q) = GL^{η}(V), Z ≤ Z(K), B ∈ Blk(K), B_{H} ∈ Blk(H) covering B. Let B_{G} ∈ Blk(G) be a weakly regular cover of B_{H}. Write R := A(D(B_{G})) ∩ K. Then Property 7.1 (a∗) holds for some B-subgroups (P, g) ≤ (R, b) with C_{H}(P)/Z = C\overline{H}(\overline{P}) and C_{H}(R)/Z = C\overline{H}(\overline{R}), or Property 7.1 (b) or (d) holds, where \overline{X} = XZ/Z for any X ≤ G. Moreover, if Property 7.1 (b) holds, then n = 3d with gcd(6, d) = 1 and 3∥(q − η).

Proof: Suppose B_{G} is labelled by (s, κ). Since B_{H} covers B, it follows that D(B) = D(B_{H}) ∩ K for some defect group D(B_{H}). There exists a defect group D(B_{G}) such that D(B_{H}) = D(B_{G}) ∩ H, so

D(B) = D(B_{H}) ∩ K = D(B_{G}) ∩ K and D(B_{H}) = D(B_{G}) ∩ H.

We may suppose D(B_{G}) ∈ Syl_{p}(C_{G}(s)).

Suppose the decompositions V = V_{0} ⊥ V_{+}, D(B_{G}) = D_{0} × D_{+} and s = s_{0} × s_{+} are given as in (6.1). Set R_{G} = A(D(B_{G})). Then R_{G} and C_{G}(R_{G}) are given by (6.4) with R replaced by R_{G}.

In the notation of the proof of Theorem 6.1, suppose each w_{Γ} ≤ (p − 1). Then D(B_{G}) is abelian, and both D(B) and D(B_{H}) are abelian. Thus Property 7.1 (d) holds. So we suppose that w_{Δ} ≥ p for some Δ. There exists y ∈ C_{G}(s) ∩ K such that y ∈ N_{G}(R_{G}) \ C_{G}(R_{G}), |y| = 4, y|_{V_{0}} = 1_{V_{0}}, y|_{V_{Γ}} = 1_{V_{Γ}} for all Γ ≠ Δ, and y swaps exactly two factors K_{Δ} of C_{G}(R_{G}). Let R_{H} := R_{G} ∩ H and R = R_{G} ∩ K.

Let P_{G} := P (D(B_{G})) be defined by (6.5), so that P_{G} ≤ R_{G} and we may suppose y ∈ C_{G}(P_{G}) ∩ K. Let P := P_{G} ∩ K and P_{H} = P_{G} ∩ H. Since w_{Δ} ≥ p, it follows that |Ω_{1}(P_{G})| ≥ p^{p−1}, and P is cyclic if and only if p = 3, w_{Δ} = 3, w_{Γ} = 0 for all Γ ≠ Δ and P_{G} ≰ K.

We claim that C_{G}(P) ≠ C_{G}(P_{G}) if and only if V_{0} = 0, p = 3 = w_{Δ}, w_{Γ} = 0 when Γ ≠ Δ, η = ϵ, α_{Δ} = 0, e_{Δ} = 1 and 3∥(q − ϵ). In particular, D(B) = 3_{+}^{1+2} in this case.

Indeed, if w_{Δ} > 3, then P is noncyclic and so C_{G}(P) = C_{G}(P_{G}). Thus w_{Δ} = 3, and so p = 3. If w_{Γ} ≠ 0, then P is also noncyclic and hence C_{G}(P) = C_{G}(P_{G}). Suppose p = 3 = w_{Δ} and w_{Γ} = 0, so that |Ω_{1}(P_{G})| = 3^{2}. Define c ≥ 1 by 3^{c}∥(q^{e3^{α_{Δ}}} − ϵ), and choose β ∈ 𝔽_{q^{2e3^{α_{Δ}}}}^{×} with |β| = 3^{c}. Note that x_{β} := 1_{V_{0}} × diag{β^{−2}, β, β} ∈ P and so if c ≥ 2, then C_{G}(P) = C_{G}(P_{G}). Thus 3∥(q^{e3^{α_{Δ}}} − ϵ) and so α_{Δ} = 0. Note that e = 1 or 2 as p = 3. If e = 2, then we may suppose β ∈ SL_{2}(q), since SL_{2}(q) contains a maximal torus ℤ_{q−ϵ}. Thus det(β) = 1, P = P_{G} and C_{G}(P) = C_{G}(P_{G}), a contradiction. So e = 1 and 3∥(q − ϵ). But Z(G(V_{0})) × Z(G(V_{+})) ≤ Z(C_{G}(P)), so η = ϵ. Since 3 | (q − η) and 1 = D_{0} = O_{3}(G(V_{0})), it follows that V_{0} = 0. Similarly, since Z(G) = ℤ_{q−η} ≤ Z(C_{G}(s)) = ℤ_{q^{d_{Δ}}−ϵ_{Δ}}, it follows that e_{Δ} = 1 and the claim holds.

Suppose V_{0} = 0, p = 3 = w_{Δ}, w_{Γ} = 0 when Γ ≠ Δ, η = ϵ, α_{Δ} = 0, e_{Δ} = 1 and 3∥(q − ϵ). Then

C_{G}(P) = GL^{ϵ}(3d_{Δ}, q) = G, C_{G}(P_{G}) = GL^{ϵ}(d_{Δ}, q) × GL^{ϵ}(2d_{Δ}, q),

so P = O_{3}(Z(G)) = O_{3}(Z(K)) = ℤ_{3} and P_{G} = O_{3}(Z(C_{G}(P_{G}))) = ℤ_{3}^{2}. In addition, C_{G}(s) ≅ GL^{ϵ_{Δ}}(3, q^{δ_{Δ}}) and D(B_{G}) ∈ Syl_{3}(GL^{ϵ_{Δ}}(3, q^{δ_{Δ}})). Since α_{Δ} = 0, it follows that 3∥(q^{δ_{Δ}} − ϵ_{Δ}), D(B_{G}) = ℤ_{3} ≀ ℤ_{3} and |D(B)| = 3^{3}. But 3_{+}^{1+2} ∈ Syl_{3}(SL^{ϵ_{Δ}}(3, q^{δ_{Δ}})), so D(B) ≅ 3_{+}^{1+2}.

Write D_{G} = D(B_{G}), D_{H} = D(B_{H}) and D = D(B). Since [D_{G}:D] = 3, it follows that D_{H} = D or D_{H} = D_{G}. Note that D_{G} = ⟨R_{G}, σ⟩ for some permutation σ of order 3. So D = ⟨R, σ⟩, C_{G}(D_{G}) = C_{G}(D) ≅ GL^{ϵ}(d_{Δ}, q), C_{G}(C_{G}(D)) = GL^{ϵ}(3, q) and ⟨D_{G}, y⟩ ≤ C_{G}(C_{G}(D)). Thus y centralizes C_{G}(D) and so y ∈ N_{K}(D, b_{D}), where (D, b_{D}) is a Sylow B-subgroup. Since Z(D) ≤ Z(K), it follows that B dominates a block \overline{B} ∈ Blk(K/Z(D)), D(\overline{B}) = D/Z(D) ≅ 3^{2} and ȳ stabilizes the \overline{B}-subgroup (D/Z(D), b̄_{D}) with |ȳ| = 4, where ȳ = yZ(D). In particular, \overline{B} is non-nilpotent. But D(\overline{B}) is abelian, so ℓ(\overline{b}\overline{Q}) ≥ 2 for some \overline{B}-subgroup (\overline{Q}, \overline{b}\overline{Q}). If \overline{Q} ≅ ℤ_{3}, then C_{K/Z(D)}(\overline{Q}) = C_{K}(Q)D/Z(D) = (C_{K}(Q)/Z(D)).3, where Q = ⟨Z(K), w⟩ for some w ∈ D\Z(D) of order 3. Let B_{Q} be a block of C_{K}(Q)D dominating \overline{b}\overline{Q} and b_{Q} ∈ Blk(C_{K}(Q)) covered by B_{Q}. Then D(b_{Q}) = Q and the canonical character θ_{Q} of b_{Q} is the only irreducible Brauer character of b_{Q}. If ϕ is any irreducible Brauer character of B_{Q}, then ϕ covers θ_{Q}. But C_{K}(Q) contains a representative set of the conjugacy 3′-classes of C_{K}(Q)D, so ℓ(B_{Q}) = 1. In particular, ℓ(\overline{b}\overline{Q}) = 1. Similarly, if \overline{Q} = D/Z(D), then ℓ(\overline{b}\overline{Q}) = 1. Thus if ℓ(\overline{b}\overline{Q}) ≥ 2, then \overline{Q} = 1 and \overline{b}\overline{Q} = \overline{B}. It follows that ℓ(B) ≥ ℓ(\overline{B}) ≥ 2, and hence Property 7.1 (b) holds.

Since p is odd, it follows that C_{G}(R) = C_{G}(R_{H}) = C_{G}(R_{G}). Let x ∈ G such that for any u ∈ R, there exists z ∈ Z(G) such that x^{−1}ux = uz. Then x^{−1}ux = cu for some c ∈ O_{p}(𝔽_{q^{2}}^{×}) and so x ∈ N_{G}(R) = N_{G}(R_{G}). If λ is an eigenvalue of u in some algebraic closure of 𝔽_{q^{2}} and m_{X−λ}(u) is the multiplicity, then cλ is also an eigenvalue of u with the same multiplicity. In particular, m_{X−1}(u) = m_{X−c}(u). It follows that if we choose u ∈ R such that m_{X−1}(u) ≠ m_{X−c}(u) for any c ∈ 𝔽_{q^{2}}^{×}, then x^{−1}ux = u. Since |Ω_{1}(R)| ≥ p^{p−1}, it follows that x ∈ C_{G}(R) and so C_{G}(R)/Z = C_{G/Z}(R/Z) for any Z ≤ Z(G), except when p = 3 = w_{Δ}, α_{Δ} = 0 and 3∥(q^{e} − ϵ), in which case D(B) = 3_{+}^{1+2}. If 3 ∤ (q − η), then e = 2 and as shown above R = R_{G}, in which case we still have x ∈ C_{G}(R). If 3 | (q − η), then 3∥(q − η), P ≅ ℤ_{3} and C_{G}(P) ≠ C_{G}(P_{G}), which is discussed above.

Similarly, C_{G}(P)/Z = C_{G/Z}(P/Z) for any Z ≤ Z(G) when C_{G}(P) = C_{G}(P_{G}) (note in this case that for any u ∈ P, we have m_{X−λ}(u) ≥ 2 for some eigenvalue λ).

Suppose C_{G}(P) = C_{G}(P_{H}) = C_{G}(P_{G}) and C_{G}(R) = C_{G}(R_{G}) = C_{G}(R_{H}). Thus C_{G}(R) is regular in G and s ∈ C_{G}(R). Let (P, g) and (R, b) be B-subgroups such that (P, g) ≤ (R, b).

Let (P, g_{H}) ≤ (R, b_{H}) be B_{H}-subgroups such that g_{H} covers g and b_{H} covers b (see the remark after the definition of Property 7.1), and (P, g_{G}) ≤ (R, b_{G}) B_{G}-subgroups such that g_{G} covers g_{H} and b_{G} covers b_{H}. By [12, Theorem 3.2], we may suppose b_{G} ⊆ E_{p}(C_{G}(R), (s)) and so b_{G}^{y} = b_{G} as y ∈ C_{G}(s) ∩ K. Now

b_{G} = b(0) × ∏_{Γ} b(Γ)^{w_{Γ}},

where b(0) is a block of G_{0} labelled by (s_{0}, κ) with defect 0, and b(Γ) = E_{p}(K_{Γ}, (t_{Γ})) with t_{Γ} the restriction of s to K_{Γ}. Note that if U_{Γ} is the underlying space of K_{Γ} and view t_{Γ} as an element of G(U_{Γ}), then we have m_{Γ}(t_{Γ}) = e_{Γ}. Thus for any generator x_{Γ} ∈ R_{Γ}

T_{Γ} := C_{K_{Γ}}(t_{Γ}) = C_{G(U_{Γ})}(x_{Γ}t_{Γ}) ≅ GL^{ϵ_{Γ}}(1, q^{e_{Γ}δ_{Γ}})

is a Coxeter torus of both K_{Γ} and G(U_{Γ}), and R_{Γ} = O_{p}(T_{Γ}) is a defect group of b(Γ). In particular, R_{G} is a defect group of b_{G}. We may suppose D(b) = D(b_{G}) ∩ C_{K}(R) and D(b_{H}) = D(b_{G}) ∩ H, so that D(b) = R and D(b_{H}) = R_{H}.

Since (P, g_{G}) ≤ (R, b_{G}) and D(b_{G}) ≤ C_{G}(P), it follows that D(g_{G}) = D(b_{G}) = R_{G} and so D(g) = D(b_{G}) ∩ C_{K}(P) = D(b_{G}) ∩ K = R is abelian. Similarly, D(g_{H}) = R_{H} = D(b_{H}).

Let θ, θ_{H} and θ_{G} be canonical characters of b, b_{H} and b_{G}, respectively. Then θ_{G} covers θ_{H} and θ_{H} covers θ. Now

θ_{G} = θ(0) × θ(+), θ(+) := ∏_{Γ} θ(Γ)^{w_{Γ}}, θ(Γ) := ±R_{T_{Γ}}^{K_{Γ}}(t_{Γ})

where Irr(b(0)) = {θ(0)}. If H_{Γ} := K_{Γ} ∩ S(U_{Γ}), then there exists an element x_{Γ} ∈ K_{Γ} permutes all the irreducible constituents of the restriction θ(Γ)|_{H_{Γ}}. Since w_{Δ} ≥ p, it follows that θ(+)|_{C_{S(V_{+})}(R_{+})} is irreducible, where R_{+} = S(V_{+}) ∩ (∏_{Γ}(R_{Γ})^{w_{Γ}}).

Let K_{0} = S(V_{0}) and K_{+} = S(V_{+}), so that

C_{K}(R) = ⟨K_{0} × C_{K_{+}}(R_{+}), u_{K}⟩, u_{K} = u_{0} × u_{+}

with u_{0} ∈ G_{0} \ K_{0} and u_{+} ∈ C_{G_{+}}(R_{+}) \ C_{K_{+}}(R_{+}). Note that G_{0} = ⟨K_{0}, u_{0}⟩ and C_{G_{+}}(R_{+}) = ⟨C_{K_{+}}(R_{+}), u_{+}⟩.

Let θ_{0} and θ_{+} be irreducible constituents of θ|_{K_{0}} and θ|_{C_{K_{+}}(R_{+})}, respectively. Then θ ∈ Irr(C_{K}(R) | θ_{0} × θ_{+}) and θ_{G} covers θ_{0} × θ_{+}. But θ(+)|_{C_{K_{+}}(R_{+})} is irreducible, so θ_{+} = θ(+)|_{C_{K_{+}}(R_{+})} and θ_{+} has an extension θ(+) to C_{G_{+}}(R_{+}). Applying Lemma 2.7 to

K_{0} × C_{K_{+}}(R_{+}) ≤ C_{K}(R) ≤ C_{G}(R) = G_{0} × C_{G_{+}}(R_{+})

(with Z_{0} = 1), we have that θ_{G}|_{C_{K}(R)} = θ. In particular, θ_{G}|_{C_{K}(R)} is irreducible and so is θ_{G}|_{C_{H}(R)}. But θ_{G} covers θ_{H}, so θ_{G}|_{C_{H}(R)} = θ_{H}. Thus θ^{y} = θ, θ_{H}^{y} = θ_{H}, b^{y} = b and b_{H}^{y} = b_{H}. Note that θ_{H}|_{C_{K}(R)} = θ. Thus Property 7.1 (a∗) holds. □

Let V be a non-degenerate orthogonal or symplectic space, G = I_{0}(V) and let G^{∗} be the dual group of G. Then

Sp_{2n}(q)^{∗} = SO_{2n+1}(q), SO_{2n+1}(q)^{∗} = Sp_{2n}(q), SO_{2n}^{η}(q)^{∗} = SO_{2n}^{η}(q).

If B is a block of I_{0}(V), then there exists a semisimple p′-element s ∈ I_{0}(V)^{∗} such that

B ⊆ E_{p}(I_{0}(V), (s)).

Let (D, b_{D}) be a Sylow B-subgroup of I_{0}(V). Then V and D have corresponding decompositions

V = V_{0} ⊥ V_{+}, D = D_{0} × D_{+}.  (8.1)

We have V_{0} = C_{V} (D), V_{+} = [ V, D ], D_{0} = {1_{V_{0}}} and D_{+} ≤ I_{0}(V_{+}). Let G_{0} := I_{0}(V_{0}), G_{+} := I_{0}(V_{+}), C_{+} := C_{I_{0}(V_{+})}(D_{+}) and let V^{∗} be the underlying space of I_{0}(V)^{∗}.

Let z ∈ D be a primitive element. Then z ∈ Z(D) with |z| = p (cf. [17, p.176]). Thus

z = z_{0} × z_{+}, L := C_{G}(z) = L_{0} × L_{+}, L_{0} = G_{0}, L_{+} := GL^{ϵ}(m, q^{e}),  (8.2)

where z_{0} = 1_{V_{0}}, z_{+} ≤ D_{+} and dim V_{+} = 2em. Then L is a regular subgroup of G and we may suppose s ∈ L^{∗} ≤ G^{∗}. In particular,

V^{∗} = U_{0} ⊥ U_{+} and s = s_{0} × s_{+},  (8.3)

where U_{0} = V_{0}^{∗}, s_{0} ∈ L_{0}^{∗} = I_{0}(U_{0}), U_{+} is the underlying space of L_{+}^{∗} and s_{+} ∈ L_{+}^{∗} ≤ I_{0}(U_{+}).

Let C_{I(U_{+})}(s_{+}) = ∏_{Γ} C_{Γ} and let U_{Γ} be the underlying vector space of C_{Γ}, so that

C_{Γ} = GL^{ϵ_{Γ}}(m_{Γ}(s_{+}), q^{δ_{Γ}}) or I(U_{Γ})  (8.4)

according as Γ ≠ X ± 1 or Γ = X ± 1.

Proposition 8.2 Let K := Ω_{2n}^{η}(q) := Ω^{η}(V) ≤ H ≤ J_{0}(V), B_{K} ∈ Blk(K) and B_{H} ∈ Blk(H) covering B_{K}. Write R := A(D(B_{K})). Then either Property 7.1 (a∗) holds for some B_{K}-subgroups (P, g) ≤ (R, b) or Property 7.1 (d) holds, where P is some subgroup of R.

Proof: Let G := SO_{2n}^{η}(q) := SO(V) and B ∈ Blk(G) covering B_{K}. Then D(B_{K}) = D ∩ K for some defect group D := D(B). Since G is self dual, we have V = V^{∗}, U_{0} = V_{0}, U_{+} = V_{+} in (8.3).

(1). Since |G:K| = 2, it follows that D = D(B_{K}) and D = D(B_{H}) ∩ G for some D(B_{H}). In the notation above we have

C_{K}(z) ≤ L = L_{0} × L_{+}.

Let K_{0} = Ω(V_{0}), K_{+} = Ω(V_{+}) and M_{+} := SL^{ϵ}(m, q^{e}) ≤ L_{+} ∩ K_{+}, so that

M_{+} ≤ C_{K_{+}}(z_{+}) ≤ L_{+}, C_{K}(z) = ⟨K_{0} × C_{K_{+}}(z_{+}), t_{0} × t_{+}⟩  (8.5)

and [L_{+}:C_{K_{+}}(z_{+})] ≤ 2, where t_{0} ∈ L_{0} \ K_{0} and t_{+} ∈ L_{+}. Let (z, B_{z}) be a major subsection of B_{K}. Then B_{z} covers a block B_{0} × B_{+} of K_{0} × C_{K_{+}}(z_{+}) with B_{0} ∈ Blk(K_{0}) and B_{+} ∈ Blk(C_{K_{+}}(z_{+})) such that D(B_{z}) = D(B_{0} × B_{+}) = D. Note since [L_{+}:C_{K_{+}}(z_{+})] ≤ 2, it follows that D(B_{+}) = D_{+} ≅ 3̸_{+}^{1+2}.

By [15, Lemma 4.1], there exists a B-subgroup (z, B_{L}) such that B_{L} covers B_{z}. Thus (z, B_{L}) is a major subsection of B.

Let R := A(D), so that z ∈ R and

R = D_{0} × ∏_{Γ}(R_{Γ})^{w_{Γ}}, C_{G}(R) = SO(V_{0}) × ∏_{Γ}(K_{Γ})^{w_{Γ}},

where R_{Γ} = O_{p}(K_{Γ}), m_{Γ}(s_{+}) = w_{Γ}e_{Γ} or 2w_{Γ}e_{Γ} according as Γ ≠ X ± 1 or Γ = X ± 1, K_{Γ} ≅ GL^{ϵ}(δ′_{Γ}, q^{ep^{α_{Γ}}}). Thus R = O_{p}(C_{G}(R)), C_{G}(R) is a regular subgroup of G and we may suppose s ∈ C_{G}(R). Set R_{+} = ∏_{Γ}(R_{Γ})^{w_{Γ}}, so that R_{+} ≤ C_{K_{+}}(z_{+}) and

K_{0} × C_{K_{+}}(R_{+}) ≤ C_{K}(R) ≤ C_{G}(R) = L_{0} × C_{L_{+}}(R_{+}).

Let (R_{+}, b_{+}) be a B_{+}-subgroup, so that

(B_{0} × b_{+})^{K_{0}×C_{K_{+}}(z_{+})} = B_{0} × B_{+}

and b_{+} ∈ Blk(C_{K_{+}}(R_{+})) as C_{K_{+}}(R_{+}) ≤ C_{K_{+}}(z_{+}). Now K_{0} × C_{K_{+}}(z_{+}) is normal in C_{K}(z), R ≤ K_{0} × C_{K_{+}}(z_{+}) and B_{z} covers B_{0} × B_{+}. It follows by [15, Lemma 4.1] that there exists a B_{z}-subgroup (R, b) such that b covers B_{0} × b_{+}, so that (R, b) is a B_{K}-subgroup.

Similarly, there exists a B_{L}-subgroup (R, b_{G}) such that b_{G} covers b. Thus b_{G} ⊆ E_{p}(C_{G}(R), (s)), R = D(b_{G}), and so R = D(b) = D(B_{0} × b_{+}).

Since L = L_{0} × L_{+} with L_{0} = G_{0} = SO(V_{0}), it follows that B_{L} = B_{L_{0}} × B_{L_{+}} with B_{L_{0}} ∈ Blk(G_{0}) and B_{L_{+}} ∈ Blk(L_{+}). But B_{L} covers B_{z} and B_{z} covers B_{0} × B_{+}, so B_{L_{0}} covers B_{0} and B_{L_{+}} covers B_{+}. In particular, D_{+} = D(B_{+}) = D(B_{L_{+}}). Similarly, b_{G} = b_{G_{0}} × b_{G_{+}} with b_{G_{0}} = B_{L_{0}} ∈ Blk(G_{0}) and b_{G_{+}} ∈ Blk(C_{L_{+}}(R_{+})), and b_{G_{+}} covers b_{+}.

Suppose D is non-abelian, so that w_{Δ} ≥ p for some Δ. Let P := D_{0} × P_{+} ≤ R and (P, g) ≤ (R, b), where P_{+} := P (D_{+}) is given by (6.5). By Proposition 8.1, there exists B_{+}-subgroup (P_{+}, g_{+}) such that (P_{+}, g_{+}) ≤ (R_{+}, b_{+}) and D(g_{+}) = D(b_{+}) = R_{+}. By the remark after Property 7.1, we may choose (P, g) such that g covering B_{0} × g_{+}, so D(g) = D_{0} × D(g_{+}) = D_{0} × D(b_{+}) = D(b) = R.

In addition, there exists y ∈ C_{L_{+}}(s_{+}) ∩ M_{+} such that y ∈ N_{C_{L_{+}}(P_{+})}(R_{+}) \ C_{L_{+}}(R_{+}), |y| = 4, y^{2} ∈ C_{L_{+}}(R_{+}), y|_{V_{0}} = 1_{V_{0}}, y|_{V_{Γ}} = 1_{V_{Γ}} for all Γ ≠ Δ and y swaps exactly two factors K_{Δ} of C_{G}(R). Since y ∈ C_{L_{+}}(s_{+}), it follows that (b_{G_{+}})^{y} = b_{G_{+}}.

Let θ_{G} and θ be canonical characters of b_{G} and b, respectively. Then θ_{G} covers θ,

θ_{G} = θ_{G_{0}} × θ_{G_{+}}

with Irr(b_{G_{0}}) = {θ_{G_{0}}} and θ_{G_{+}} the canonical character of b_{G_{+}}. If Irr(B_{0}) = {θ_{0}} and θ_{+} is the canonical character of b_{+}, then θ_{G_{+}} covers θ_{+}. Since C_{K_{+}}(R_{+}) = C_{C_{K_{+}}(z_{+})}(R_{+}), it follows by the proof of Proposition 8.1 that θ_{+} = θ_{G_{+}}|_{C_{K_{+}}(R_{+})} and θ_{+}^{y} = θ_{+}.

Now

C_{K}(R) = ⟨K_{0} × C_{K_{+}}(R_{+}), u_{K}⟩, u_{K} = u_{0} × u_{+}  (8.6)

with [C_{K}(R):K_{0} × C_{K_{+}}(R_{+})] ≤ 2, where u_{0} ∈ G_{0} and u_{+} ∈ C_{L_{+}}(R_{+}). If C_{K}(R) = K_{0} × C_{K_{+}}(R_{+}), then θ = θ_{0} × θ_{+} and θ^{y} = θ. If [C_{K}(R):K_{0} × C_{K_{+}}(R_{+})] = 2, then L_{0} = ⟨K_{0}, u_{0}⟩ and C_{L_{+}}(R_{+}) = ⟨C_{K_{+}}(R_{+}), u_{+}⟩. Now θ ∈ Irr(C_{K}(R) | θ_{0} × θ_{+}) and θ_{G} ∈ Irr(C_{G}(R) | θ) and θ_{+} has an extension θ_{G_{+}} to C_{L_{+}}(R_{+}), so by Lemma 2.7, θ_{G}|_{C_{K}(R)} = θ and hence θ^{y} = θ and y ∈ N_{K}(R, b).

(2). Suppose K ≤ H ≤ J_{0}(V). Let (P, g_{H}) ≤ (R, b_{H}) be B_{H}-subgroups such that g_{H} covers g and b_{H} covering b. Since J_{0}(V)/Z(J_{0}(V))K is a 2-group and p is odd, it follows that D(g_{H}) = O_{p}(Z(H))D(g) = O_{p}(Z(H))D(b) = D(b_{H}) and both are abelian, and the canonical character θ_{H} of b_{H} covers θ. Now

K_{0} × C_{K_{+}}(R_{+}) ≤ C_{K}(R) ≤ C_{H}(R) ≤ C_{J_{0}(V)}(R) ≤ J_{0}(V_{0}) × C_{J_{0}(V_{+})}(R_{+}).

By [17, (1A)], C_{J_{0}(V_{+})}(z_{+}) = ⟨L_{+}, τ_{+}⟩ with [τ_{+}, L_{+}] = 1 and so C_{J_{0}(V_{+})}(R_{+}) is a central product C_{L_{+}}(R_{+}) ∘ ⟨τ_{+}⟩. In particular, θ_{G_{+}} has an extension θ̃_{G_{+}} to C_{J_{0}(V_{+})}(R_{+}), and θ̃_{G_{+}} is also an extension of θ_{+}. Moreover, y stabilizes θ̃_{G_{+}}, since θ̃_{G_{+}} is a central produce θ_{G_{+}} ∘ β for some β ∈ Irr(⟨τ_{+}⟩) and θ_{G_{+}}^{y} = θ_{G_{+}}. Since y ∈ M_{+} ≤ K_{+} and y normalizes K_{+}, it follows that [y, x] ∈ C_{K_{+}}(R_{+}) = C_{L_{+}}(R_{+}) ∩ K_{+} for any x ∈ C_{L_{+}}(R_{+}). But C_{J_{0}(V_{+})}(R_{+}) = C_{L_{+}}(R_{+}) ∘ ⟨τ_{+}⟩, so [y, x] ∈ C_{K_{+}}(R_{+}) for all x ∈ C_{J_{0}(V_{+})}(R_{+}). It follows by Remark 7.3 that y stabilizes θ_{H} and so b_{H}^{y} = b_{H}. Thus Property 7.1 (a∗) holds.

(3). Suppose w_{Γ} < p for any Γ with m_{Γ}(s_{+}) ≠ 0. Then D = D(B_{K}) is abelian, and so D(B_{H}) = DO_{p}(Z(H)) is abelian. Thus Property 7.1 (d) holds. □

Proposition 8.3 Let K := Ω_{2n+1}(q) = Ω(V) or K := Sp_{2n}(q) = Sp(V), and

K ≤ H ≤ J_{0}(V),

B_{K} ∈ Blk(K) and B_{H} ∈ Blk(H) covering B_{K}, where H = SO(V) when K = Ω(V). Write R := A(D(B_{K})). Then either Property 7.1 (a∗) holds for some B_{K}-subgroups (P, g) ≤ (R, b) or Property 7.1 (d) holds, where P is some subgroup of R.

Proof: Suppose V is orthogonal. Replacing G by H in the proof (1) of Proposition 8.2 with some obvious modifications, we have that Property 7.1 (a∗) holds for (P, g) ≤ (R, b). Suppose V is symplectic, so that H/K is cyclic. Applying the proofs (1) and (2) of Proposition 8.2 with some obvious modifications, we have that Property 7.1 (a∗) holds for (P, g) ≤ (R, b).

If D(B_{K}) is abelian, then D(B_{K}) = D(B_{H}) ∩ K for some D(B_{H}). Since the outerdiagonal group of K is order 2, it follows that D(B_{H}) ≤ KZ(H) and so D(B_{H}) = D(B_{K})O_{p}(Z(H)) is abelian. □

Theorem 8.4 Let G = I_{0}(V), B ∈ Blk(G), and (D, b_{D}) a Sylow B-subgroup. Follow the notation in (8.1), (8.2) and (8.3). Then the following are equivalent:

(a) B is nilpotent.

(b) C_{I_{0}(V)}(D) = G_{0}×C_{I_{0}(V_{+})}(D_{+}) is a regular subgroup of I_{0}(V) and s ∈ C_{I_{0}(V)}(D)^{∗} ≤ L^{∗} satisfies the following conditions.

(i) Suppose I_{0}(V) = Sp_{2n}(q) or SO_{2n+1}(q). Then

m_{Γ}(s_{+}) = \{ \begin{matrix}0 or 1 & if Γ ≠ X ± 1 and e | δ_{Γ}, \\ 0 & otherwise. \end{matrix}

(ii) Suppose I_{0}(V) = SO_{2n}^{η}(q). Then

m_{Γ}(s_{+}) = 0 or 1  if Γ ≠ X ± 1 and e | δ_{Γ},
m_{X−1}(s_{+}) + m_{X+1}(s_{+}) = 0 or 2 if p | (q − ϵ),
m_{Γ}(s_{+}) = 0  otherwise,

where ϵ is the type of the underlying space of (s_{+})_{X±1} when m_{X−1}(s_{+}) + m_{X+1}(s_{+}) = 2.

(c) C_{I_{0}(V)}(D) = G_{0}×C_{I_{0}(V_{+})}(D_{+}) is a regular subgroup of I_{0}(V) and s ∈ C_{I_{0}(V)}(D)^{∗} ≤ L^{∗} such that T_{+}^{∗} := C_{I_{0}(U_{+})}(s_{+}) is a maximal torus of I_{0}(U_{+}). In particular, if θ = θ_{0} × θ_{+} is the canonical character of b_{D} with θ_{0} ∈ Irr(G_{0}) and θ_{+} ∈ Irr(C_{+}), then θ_{0} has defect zero and θ_{+} = ±R_{T_{+}}^{C_{+}}(s_{+}), where C_{+} = C_{I_{0}(V_{+})}(D_{+}) and T_{+} ≤ C_{+} is a dual of T_{+}^{∗}.

Proof: Suppose B is nilpotent. By Propositions 8.2 and 8.3, D is abelian, so m_{Γ}(s_{+}) = e_{Γ} or 0 when Γ ≠ X ± 1 and m_{Γ}(s_{+}) = 2e or 0 when Γ = X ± 1.

Suppose m_{Γ}(s_{+}) = e_{Γ} ≥ 2 with Γ ≠ X ± 1. As shown in the proof of Theorem 6.1 there exists a p′-element τ_{Γ} ∈ C_{Γ}^{∗} of order e_{Γ} normalizing D_{Γ} := D ∩ C_{Γ}^{∗}, so that there exists a p′-element of order e_{Γ} normalizing the Sylow B-subgroup (D, b_{D}), a contradiction, so e_{Γ} ≤ 1 and e | δ_{Γ}.

Suppose Γ = X ± 1, so that m_{Γ}(s_{+}) = 2e or 0 and C_{Γ} = I(U_{Γ}).

Suppose, moreover that G = SO_{2n}^{η}(q), so that G = G^{∗}. Let Δ = X ± 1 and suppose m_{Δ}(s_{+}) = 2e. By [17, (1.14)],

|N_{I(U_{Δ})}(⟨z_{Δ}⟩)/C_{I(U_{Δ})}(z_{Δ})| = 2e.

If e ≥ 2, then there exists y_{Δ} ∈ N_{I_{0}(U_{Δ})}(⟨z_{Δ}⟩) \ C_{I_{0}(U_{Δ})}(z_{Δ}) of order e, so that y_{Δ} normalizes the Sylow subgroup D_{Δ} of C_{I_{0}(U_{Δ})}(z_{Δ}). Let y_{Γ} = 1 ∈ I_{0}(U_{Γ}) and y = 1_{V_{0}} × ∏_{Γ} y_{Γ}. Then

y ∈ (N_{G}(D) ∩ C_{G}(s)) \ C_{G}(D)

and y normalizes (D, b_{D}). Since |y| = e ≠ 1, it follows that B is not nilpotent, which is impossible. Thus e = 1, I_{0}(U_{Δ}) = SO^{ϵ}(2, q) with ϵ = η(U_{Δ}), so p | (q − ϵ).

Similarly, suppose m_{X−1}(s_{+}) = m_{X+1}(s_{+}) = 2. Since Ω_{4}^{−}(q) = P SL_{2}(q^{2}), Ω_{4}^{+}(q) = SL_{2}(q) ∘ SL_{2}(q) and SO_{4}^{±}(q) = Ω_{4}^{±}(q).2, it follows that there exists an element w ∈ (N_{G}(D) ∩ C_{G}(s)) \ C_{G}(D) of order 2 such that w ∈ N_{G}(D, b_{D}), a contradiction.

If G = Sp_{2n}(q) or SO_{2n+1}(q), then by [17, (1.14)] again, there exists an element w ∈ N_{G}(D) \ C_{G}(D) of order 2e such that w normalizes (D, b_{D}), which is impossible. Thus m_{X±1}(s_{+}) = 0 and (b) holds.

Suppose (b) holds. Then T_{+}^{∗} := C_{C_{+}^{∗}}(s_{+}) is a maximal torus of I_{0}(U_{+}) and so D is abelian. Since C_{G}(D) is regular in G, we may suppose s ∈ C_{G}(D)^{∗} and so

b_{D} ⊆ E_{p}(C_{G}(D), (s)).

Thus θ_{+} = ±R_{T_{+}}^{C_{+}}(s_{+}) and θ = θ_{0} × θ_{+} is the canonical character of b_{D}, where θ_{0} ∈ Irr(G_{0}) has defect 0. In particular, N_{G}(D, θ) = C_{G}(D) and B is nilpotent. □

Proposition 8.5 Let K := Spin^{η}(V) ⊲ H such that H/K is abelian, C_{H}(K) ≤ Z(H) and H/Z(H) ≤ SO(V) or J_{0}(V)/Z(J_{0}(V)) according as dim V is odd or even. Let B ∈ Blk(K), B_{H} ∈ Blk(H) covering B, and Z ≤ Z(K) such that K_{c} := K/Z = Ω^{η}(V), so that |Z| = gcd(2, q − η). Write R := A(D(B)). Then either Property 7.1 (a∗) holds for some B_{K}-subgroups (P, g) ≤ (R, b) or Property 7.1 (d) holds, where P is some subgroup of R.

Proof: Let D := D(B), G := SO^{η}(V) and Z_{+} ≤ Z(D_{0}(V)) such that G = D_{0}(V)/Z_{+}, so that Z = Z_{+} ∩ K and Z_{+} ≅ ℤ_{q−1}.

We may suppose D = DZ/Z ≤ K_{c}. Thus D is of defect type in K_{c}, where a p-subgroup Q of K_{c} is of defect type if Q is a Sylow p-subgroup of a centralizer C_{K_{c}}(t) of a semisimple p′-element t. So D is of defect type of G, and D has a primary element z ∈ Z(D) (see [17, Section 5]). Thus we have the corresponding decompositions V = V_{0} ⊥ V_{+}, D = D_{0} × D_{+}, z = z_{0} × z_{+}, C_{G}(z) = L_{0} × L_{+} given by (8.1) and (8.2). Since K is universal, it follows by [20, Theorem 4.2.2] and [17, (2E)] that

C := C_{K}(z) = L_{C}T_{C}, L_{C} = L_{1} × SL^{ϵ}(m, q^{e}), L_{1} = Spin(V_{0}),

where T_{C} is an abelian r′-group inducing inner-diagonal automorphisms on SL^{ϵ}(m, q^{e}) and L_{1}. Here for simplicity, we identify z with its preimage (with the same order) in K. Since p is odd and D ≤ C, it follows that D_{+} ∩ L_{1} = 1 and so L_{1}L_{2} = L_{1} × L_{2}, where L_{2} = SL^{ϵ}(m, q^{e})D_{+} ≤ L_{+}. Let L = L_{1} × L_{2} ≤ C, so that C = LT_{C}. Let (z, B_{z}) be a major subsection of B, and B_{L} = B_{1} × B_{2} a block of L covered by B_{z}, where B_{1} ∈ Blk(L_{1}) and B_{2} ∈ Blk(L_{2}). We may suppose D(B_{2}) = D_{+} ∩ L_{2}, so that D(B_{2}) = D_{+}.

Suppose B_{2} satisfies Property 7.1 (a∗). Let R_{2} := A(D_{+}) ≤ D(B_{2}), R := R_{1} × R_{2}, P_{2} := P (D_{+}) ≤ R_{2}, P = P_{1} × P_{2} ≤ R and let (P, g_{L}) ≤ (R, b_{L}) be B_{L}-subgroups, where P_{1} = R_{1} = D_{0}. So g_{L} = g_{1} × g_{2}, b_{L} = b_{1} × b_{2} with g_{1} = b_{1} = B_{1}, g_{2}^{L_{2}} = b_{2}^{L_{2}} = B_{2} and D(b_{2}) = R_{2} = D(g_{2}). In addition, there exists y ∈ (N_{L_{2}}(R_{2}) ∩ C_{L_{2}}(P_{2})) \ C_{L_{2}}(R_{2}) such that y^{4} = 1, y^{2} ∈ C_{L_{2}}(R_{2}), and b_{2}^{y} = b_{2}. Thus b_{L}^{y} = b_{L} and D(b_{L}) = R = D(g_{L}).

(1). For t ∈ T_{C} write t = t_{1}t_{2} such that [t_{1}, t_{2}] = 1 and t_{2} induces inner-diagonal automorphism on L_{2}. Let J_{i} = ⟨L_{i}, t_{i} : t = t_{1}t_{2} ∈ T_{C}⟩, so that C ⊲ J := J_{1} × J_{2} and L_{2} ≤ J_{2} ≤ L_{+}. Let B_{J} ∈ Blk(J) be a weakly regular cover of B_{z}, (P, g) ≤ (R, b) B_{z}- subgroups such that g covers g_{L} and b covers b_{L}, and (P, g_{J}) ≤ (R, b_{J}) B_{J}-subgroups such that g_{J} covers g and b_{J} covers b, so that g_{J} covers g_{L} and b_{J} covers b_{L}.

If g_{J} = g_{J_{1}} × g_{J_{2}} and b_{J} = b_{J_{1}} × b_{J_{2}} for some g_{J_{i}} ∈ Blk(C_{J_{i}}(P_{i})) and b_{J_{i}} ∈ Blk(C_{J_{i}}(R_{i})), then g_{J_{2}} covers g_{2} and b_{J_{2}} covers b_{2} and by Proposition 8.1 and its proof (1), D(g_{J_{2}}) = D(b_{J_{2}}) is abelian θ_{J_{2}}|_{C_{L_{2}}(R_{2})} = θ_{2} and θ_{J_{2}}^{y} = θ_{J_{2}}, where θ_{J_{i}} and θ_{i} are canonical characters of b_{J_{i}} and b_{i}, respectively. Thus D(g_{J}) = D(b_{J}) is abelian. But P ≤ R ≤ D(g_{J}) and D(g_{J}) is abelian, so

D(g) = D(g_{J}) ∩ C_{K}(P) = D(g_{J}) ∩ K = D(b_{J}) ∩ K = D(b_{J}) ∩ C_{K}(R) = D(b),

which is also abelian. In addition, as shown in the proof (1) of Proposition 8.1 θ_{J_{2}} has an extension to C_{L_{+}}(R_{2}).

Let θ_{J} and θ be canonical characters of b_{J} and b, respectively, so that θ_{J} = θ_{J_{1}} ×θ_{J_{2}} covers θ, θ_{J}^{y} = θ_{J} and θ ∈ Irr(C_{K}(R) | θ_{1} × θ_{2}). Applying Lemma 2.7 to

L_{1} × C_{L_{2}}(R_{2}) ≤ C_{K}(R) ≤ J_{1} × C_{J_{2}}(R_{2})

we have that θ_{J}|_{C_{K}(R)} = θ, so θ^{y} = θ and b^{y} = b.

(2). If q is even, then the outer-diagonal group of K is trivial, so we may suppose q is odd. Let (P, g_{H}) ≤ (R, b_{H}) be B_{H}-subgroups such that g_{H} covers g and b_{H} covers b. Then b_{H} is a block of C_{H}(R) and R = D(b_{H}) ∩ C_{K}(R) for some D(b_{H}). Since H induces inner-diagonal automorphisms and since the outer-diagonal group of K is a 2group, it follows that D(b_{H}) ≤ RZ(H) and D(b_{H}) = RO_{p}(Z(H)) is abelian. Similarly, R = D(b) = D(g) = D(g_{H}) ∩ C_{K}(P) and D(g_{H}) = RO_{p}(Z(H)) = D(b_{H}).

Now z ∈ K ≤ D_{0}(V), so by [17, (2E)], C_{D_{0}(V)}(z) = D_{0}(V_{0}) ∘ L̃_{+}, C_{D_{0}(V)}(z)/Z_{+} = C_{G}(z) and C = C_{D_{0}(V)}(z) ∩ K, where L̃_{+} is a central extension of L_{+} by Z_{+}.

To show that b_{H}^{y} = b_{H} we may suppose H/KZ(H) = J_{0}(V)/K_{c}Z(J_{0}(V)) = Outdiag(K), so that H/KZ(H) is a 2-group. Let t ∈ C_{H}(z) \ C_{K}(z), so that t^{4} ∈ C_{K}(z)Z(H). In the notation of [20, Table 4.5.2], t induces an element of C^{∗} := C_{Inndiag(K)}(zZ(K)) (note here C^{∗} is not the dual group of C). But C^{∗}/C^{∘∗} is a pgroup, so t induce an element of C^{∘∗} and hence t ∈ C_{K}(z)Z(H). Thus

C_{H}(z) = ⟨C_{K}(z), x_{H}, t_{H}⟩, x_{H} = x_{1}x_{2}, t_{H} = t_{1}t_{2}

where x_{1} ∈ D_{0}(V_{0}) and x_{2} ∈ L̃_{+}, and t_{H} centralizes L̃_{+}. So

⟨C_{K}(z), x_{H}⟩ ≤ ⟨J_{1}, x_{1}⟩ ∘ ⟨J_{2}, x_{2}⟩ ≤ D_{0}(V_{0}) ∘ L̃_{+}.

Let H_{1} = ⟨J_{1}, x_{1}, t_{H}⟩ and H_{2} = ⟨J_{2}, x_{2}⟩, so that

L_{1} × L_{2} ≤ C_{K}(z) ≤ C_{H}(z) ≤ H_{1} ∘ H_{2}.

It follows that

L_{1} × C_{L_{2}}(R_{2}) ≤ C_{K}(R) ≤ C_{H}(R) ≤ H_{1} ∘ C_{H_{2}}(R_{2}).  (8.26)

By [20, Table 4.5.2],

L̃_{+} = (Z × L_{2} ∘ ℤ_{q−ϵ})⟨x_{+}⟩

where x_{+} induces outer-diagonal automorphism of order dividing gcd(m, q − ϵ) on L_{2}. Thus C_{H_{2}}(R_{2}) = (Z × C_{L_{2}}(R_{2}) ∘ ℤ_{q−ϵ})⟨y_{2}⟩ for some y_{2} ∈ L̃_{+} inducing outer-diagonal automorphism on L_{2}. View θ_{2} as character of Z × C_{L_{2}}(R_{2}) with Z ≤ ker(θ_{2}). Now C_{L_{2}}(R_{2}) ≤ C_{H_{2}}(R_{2})/Z ≤ C_{L_{+}}(R_{2}). As shown in the proof (1) of Proposition 8.1 θ_{2} has an extension θ̃_{2} to C_{H_{2}}(R_{2})/Z such that θ̃_{2}^{y} = θ̃_{2}. By Remark 7.3 θ_{H}^{y} = θ_{H} and b_{H}^{y} = b_{H}. Thus the Property 7.1 (a∗) holds for (R, b).

Suppose B_{2} satisfies Property 7.1 (d), so that D(B_{2}) is abelian. Thus D_{+} and so D(B) are abelian. Since D(B) = D(B_{H}) ∩ K for some D(B_{H}), it follows that D(B_{H}) ≤ KZ(H) and D(B_{H}) = D(B)O_{p}(Z(H)) which is abelian. □

Theorem 8.6 Let K be a finite quasi-simple group of classical type over a field 𝔽_{q} and B ∈ Blk(K), and let K ⊲ H such that H/K is abelian, C_{H}(K) ≤ Z(H), H induces inner-diagonal automorphisms on K and B_{H} ∈ Blk(H) covering B. If p | q, then either D(B) = D(B_{H}) is cyclic or l(B) ≥ 2. Suppose p ∤ q and p is odd. Then one of Properties 7.1 (a∗), (b), (c) and (d) holds. In addition, if Property 7.1 (b) or (c) holds, then p = 3, K = SL^{η}(3d, q)/Z for some Z ≤ Z(SL^{η}(3d, q)) with gcd(6, d) = 1 and 3∥(q − η).

Proof: We will follow the notation of [20]. In particular, K_{u} denotes the universal group with the same type as K. If p|q and D(B) is noncyclic, then D(B) is a Sylow subgroup of K and l(B) = l(B_{0}) with principal B_{0} := B_{0}(K) ∈ Blk(K). But B_{0} dominates the principal block \overline{B} of K/Z(K) = K_{a} and l(\overline{B}) + 1 is the number of p′-conjugacy classes of K_{a}, so l(B_{0}) ≥ l(\overline{B}) ≥ 2. Suppose p ∤ q.

If K = A_{n}^{η}(q), then set K̂ = K_{u} = SL_{n+1}^{η}(q) ≤ GL_{n+1}^{η}(q), so that K = K̂/Z for some Z ≤ Z(GL_{n+1}^{η}(q)) ∩ K̂. We may take Ĥ ≤ GL_{n+1}^{η}(q) such that H = Ĥ/Z.

If K = B_{n}(q) = K_{a} = Ω_{2n+1}(q), then set K̂ = Ω_{2n+1}(q) ≤ Ĥ ≤ SO_{2n+1}(q) such that H = Ĥ/Z. If K = B_{n}(q) = K_{u} = Spin_{2n+1}(q) = Spin(V), then take K = K̂ ⊲ Ĥ = H such that H/Z(K) ≤ SO(V).

If K = C_{n}(q), then we may take K̂ = Sp_{2n}(q) = Sp(V) ≤ Ĥ ≤ J_{0}(V) such that H = Ĥ/Z.

Suppose K = D_{n}^{η}(q) with (n, η) = (2k + 1, ±) or (2k, −). If K = Ω_{2n}^{η}(q) = Ω(V), then K = K̂ ⊲ Ĥ = H ≤ J_{0}(V). If K = P Ω_{2n}^{η}(q) = P Ω(V), then take K̂ = Ω_{2n}^{ϵ}(q) ≤ Ĥ ≤ J_{0}(V) such that H = Ĥ/Z. If K = Spin_{2n}^{η}(q) = Spin(V), then take K = K̂ ⊲ Ĥ = H such that H/Z(K) ≤ J_{0}(V).

Suppose K = D_{2k}^{+}(q) with q even. Then K = H and we may take K̂ = Ĥ = H. Suppose K = D_{2k}^{+}(q) with q odd, so that Z(K_{u}) = {1, z, z_{s}, z_{c}} and K_{u}/Z = Ω_{4k}^{+}(q), where Z = ⟨z⟩. If K = Ω_{4k}^{+}(q) = Ω(V), then take K̂ = K ≤ Ĥ ≤ J_{0}(V). If K = P Ω_{4k}^{+}(q) = P Ω(V), then take K̂ = Ω(V) ≤ Ĥ ≤ J_{0}(V) such that Ĥ/Z = H. If K = Spin_{4k}^{+}(q)/Z′ for Z′ = ⟨z_{s}⟩ or ⟨z_{c}⟩, then we may take K̂ = Spin_{4k}^{+}(q) = Spin(V) ≤ Ĥ ≤ D_{0}(V) such that H = Ĥ/Z′. If K = Spin_{4k}^{+}(q) = Spin(V), then take K̂ = K and Ĥ = H.

Let B̂ ∈ Blk(K̂) dominating B and B̂_{H} ∈ Blk(Ĥ) dominating B_{H}, so that B̂_{H} covers B̂. By Propositions 8.1, 8.2, 8.3 and 8.5, one of Properties 7.1 (a∗), (b) of (d) holds for B̂.

If Property 7.1 (a∗) holds for B̂, then there exist B̂-subgroups (P̂, ĝ) ≤ (R̂, b̂) satisfying Property 7.1 (a∗). By Lemma 7.6, Property 7.1 (a∗) holds for some Bsubgroups (P, g) ≤ (R, b).

Suppose Property 7.1 (b) holds for B̂. By Proposition 8.1, K̂ = SL_{n+1}^{η}(q), D(B̂) = 3_{+}^{1+2} and n + 1 = 3d with gcd(6, d) = 1 and 3∥(q − η). In particular, Z(D(B̂)) = O_{3}(Z(K̂)) and we may suppose D(B) ≤ D(B̂)Z/Z. If O_{3}(Z(K̂)) ≤ Z, then D(B) = ℤ_{3}^{2}, D(B_{H}) = 3^{2} or ℤ_{3} ≀ ℤ_{3}/O_{3}(Z(K̂)) ≅ 3_{+}^{1+2} and (c) holds. If Z is a 3′-group, then D(B) = D(B̂) and (b) holds.

If Property 7.1 (d) holds for B̂, then D(B̂) and D(B̂_{H}) are both abelian. Since Z ≤ Z(Ĥ) ∩ K̂, it follows that D(B) = D(B̂)Z/Z and D(B_{H}) = D(B̂_{H})Z/Z, and so D(B) and D(B_{H}) are both abelian. □

9 Exceptional groups

Suppose p is odd. We will follow the notation of [20]. In this section we demonstrate that every nilpotent block of an exceptional group of Lie type has abelian defect groups. We first prove a simple lemma.

Lemma 9.1 Let J_{i} be a finite group and P_{i} a p-subgroup of J_{i} such that C_{J_{i}/Z_{i}}(P_{i}/Z_{i}) = C_{J_{i}}(P_{i})/Z_{i} for i = 1, 2, where Z_{i} = O_{p}(Z(J_{i})). Let J = J_{1} × J_{2}, P = P_{1} × P_{2} and Z ≤ O_{p}(Z(J)). Then

C_{J}(P)/Z = C_{J/Z}(P/Z).

Proof: Let Z_{+} = Z_{1} × Z_{2} = O_{p}(Z(J)). Then

C_{J/Z_{+}}(P/Z_{+}) = C_{J_{1}/Z_{1}}(P_{1}/Z_{1})×C_{J_{1}/Z_{1}}(P_{1}/Z_{1}) = C_{J_{1}}(P_{1})/Z_{1}×C_{J_{2}}(P_{2})/Z_{2} = C_{J}(P)/Z_{+}.

It is clear that C_{J}(P)/Z ≤ C_{J/Z}(P/Z). If xZ ∈ C_{J/Z}(P/Z) for some x ∈ J, then xZ_{+} ∈ C_{J/Z_{+}}(P/Z_{+}) and so xZ_{+} = (x_{1}, x_{2})Z_{+} for some x_{i} ∈ C_{J_{i}}(P_{i}). In particular, x = (x_{1}, x_{2})x_{+} for some x_{+} ∈ Z_{+}. Since Z_{+} ≤ C_{J}(P), it follows that x ∈ C_{J}(P) and hence C_{J/Z}(P/Z) = C_{J}(P)/Z. □

The lemma will be applied to a central product J_{1} ∘ J_{2} as J_{1} ∘ J_{2} = (J_{1} × J_{2})/Z_{0} for some Z_{0} ≤ Z(J_{1}) ∩ Z(J_{2}).

Theorem 9.2 Let K be a finite quasi-simple group of exceptional type over a field 𝔽_{q}, let B ∈ Blk(K), and let K ⊲ H such that C_{H}(K) ≤ Z(H), H/K is cyclic, and H induces inner-diagonal automorphisms on K. Let B_{H} ∈ Blk(H) be a block covering B. Choose (as we may) defect groups D(B) and D(B_{H}) of B and B_{H} respectively such that D(B) = D(B_{H}) ∩ K. If p|q, then either D(B) = D(B_{H}) is cyclic or l(B) ≥ 2. If p ∤ q and p is odd, then one of the Properties 7.1 (a), (b) and (d) holds.

Proof: If p|q, then a proof similar to that of Theorem 8.6 shows that either D(B) = D(B_{H}) is cyclic or l(B) ≥ 2.

Suppose p ∤ q. Let K_{u} be the universal group, so that K = K_{u}/Z for some Z ≤ Z(K_{u}). Since Z(K_{u}) is cyclic of order 1, 2 or 3, it follows that H induces the trivial action on Z(K_{u}).

Before beginning the proof proper we introduce some notation.

Write D := D(B). If Z(K) ≠ Ω_{1}(Z(D)), then take z ∈ Z(D) \ Z(K) with |z| = p. If Z(K) = Ω_{1}(Z(D)) (so in particular p = 3), then take z ∈ D such that |z| = p^{2} and zZ(K) ∈ Z(D/Z(K)). Let (z, B_{z}) be a B-subsection, which we choose to be major in the case z ∈ Z(D) (that such a major subsection exists is [2, 4.15]). In the case that z ∈ Z(D), we may further choose (z, B_{z}) so that B_{z} (a block of C := C_{K}(z)) has defect group D. By [20, Theorem 4.2.2] C = O^{r′}(C)T, where O^{r′}(C) is a central product

O^{r′}(C) = L_{1} ∘ L_{2} ∘ ⋯ ∘ L_{ℓ}

with each L_{i} ∈ Lie(r), and T is an abelian r′-group inducing inner-diagonal automorphisms on each L_{i}. In general, it may be the case that z ∉ O^{r′}(C). We introduce some more notation as follows to allow for this inconvenience: If Z(C) ≤ O^{r′}(C), then define s := ℓ and L := O^{r′}(C). If Z(C) ≰ O^{r′}(C), then define s = ℓ + 1, L_{s} = Z(C) and

L := L_{1} ∘ L_{2} ∘ ⋯ ∘ L_{s}.  (9.1)

In all cases C = LT, z ∈ L and L ⊲ C. Let B_{L} be a block of L covered by B_{z}. There are uniquely defined blocks B_{i} ∈ Blk(L_{i}) such that if χ ∈ Irr(B_{L}) with χ = χ_{1} ∘ ⋯ ∘ χ_{s} for some χ_{i} ∈ Irr(L_{i}), then χ_{i} ∈ Irr(B_{i}). We write

B_{L} = B_{1} ∘ B_{2} ∘ ⋯ ∘ B_{s}.

Each element t ∈ T has the form t_{1}t_{2} ⋯ t_{s}t′, where t′ centralizes L and t_{i} induces an inner-diagonal automorphism on L_{i} and [L_{i}, t_{j}] = 1 for i ≠ j. Let T′ = ⟨t′ : t = t_{1}t_{2} ⋯ t_{s} ∘ t′ ∈ T ⟩, and

J_{i} := ⟨L_{i}, t_{i} : t = t_{1}t_{2} ⋯ t_{s}t′ ∈ T ⟩, and J := J_{1} ∘ J_{2} ∘ ⋯ ∘ J_{s} ∘ T′.  (9.2)

Then LT ⊲ J and T′ is abelian. Let B_{J} be a block of J covering B_{z}, so that B_{J} covers B_{L}. Thus

B_{J} = B_{J_{1}} ∘ B_{J_{2}} ∘ ⋯ ∘ B_{J_{s}} ∘ B_{T′},

where B_{J_{i}} ∈ Blk(J_{i}) covering B_{i} and B_{T′} ∈ Blk(T′). Note that if C_{J_{i}}(L_{i}) ≤ Z(L_{i}) for all i, then the central product J is over a subgroup of Z(L).

Case 1. Suppose each B_{i} satisfies Property 7.1 (d). Then each D(B_{J_{i}}) is abelian and so is D(B_{J}). Thus D(B_{z}) = D(B_{J}) ∩ C is abelian.

Case 2. Suppose L is a direct product of L_{i}’s, C_{J_{i}}(L_{i}) ≤ Z(L_{i}) for all i and some B_{j} satisfies Property 7.1 (a∗). Without loss of generality, take j = 1. In addition, suppose each L_{i} is classical and universal (or L_{s} = Z(C)). Thus

L = L_{1} × ⋯ × L_{s} ≤ C ⊲ J = J_{1} × ⋯ × J_{s} × T′.  (9.3)

We now define R_{i}. If L_{i} = SL^{η}(V_{i}), then denote G_{i} = GL^{η}(V_{i}) and let B_{G_{i}} ∈ Blk(G_{i}) be a weakly regular cover of B_{J_{i}} and R_{i} := A(D(B_{G_{i}})) ∩ L_{i}. If L_{i} is not linear and unitary, then set R_{i} = A(D(B_{i})). In addition, let (R_{i}, b_{i}) be a B_{i}-subgroup, and note that (R_{i}, b_{i}) is a Sylow B_{i}-subgroup when D(B_{i}) is abelian. Let

R := R_{1} × ⋯ × R_{s} ≤ L, b_{L} := b_{1} × ⋯ × b_{s},

so that (R, b_{L}) is a B_{L}-subgroup. Since z ∈ O_{p}(Z(L)) and R is abelian, it follows that z ∈ R and R ≤ C. By Propositions 8.1, 8.2, 8.3 and 8.5, each defect group D(b_{i}) of b_{i} is abelian. Let (R, b_{z}) be a B_{z}-subgroup such that b_{z} covers b_{L}, and (R, b_{J}) be a B_{J}-subgroup such that b_{J} covers b_{z}, so that b_{J} covers b_{L} and

b_{J} = b_{J_{1}} × ⋯ × b_{J_{s}} × B_{T′}  (9.4)

where each b_{J_{i}} covers b_{i}. By Propositions 8.1, 8.2, 8.3 and 8.5 again, each defect group D(b_{J_{i}}) is abelian, so defect groups D(b_{J}) and D(b_{z}) of b_{J} and b_{z} respectively are both abelian, since we may suppose D(b_{z}) = D(b_{J})∩C. Note in the proof above that (R, b_{z}) can be any B_{z}-subgroup such that b_{z} covers b_{L}. Later we will choose a special such B_{z}-subgroup.

Suppose further that B_{1} satisfies Property 7.1 (a∗) in Propositions 8.1, 8.2, 8.3 or 8.5 for B_{1}-subgroups (P_{1}, g_{1}) ≤ (R_{1}, b_{1}). Let

L_{+} = ∏_{i=2}^{s} L_{i}, R_{+} := ∏_{i=2}^{s} R_{i}, P = P_{1} × R_{+}, g_{L} = g_{1} × (∏_{i=2}^{s} b_{i}) , J_{+} = (∏_{i=2}^{s} J_{i}) × T′,

so that C_{L}(R) = C_{L_{1}}(R_{1}) × C_{L_{+}}(R_{+}) ≤ C_{C}(R) ≤ C_{J_{1}}(R_{1}) × C_{J_{+}}(R_{+}), and (P, g_{L}) ≤ (R, b_{L}). Since B_{z} covers B_{L}, it follows that there exist B_{z}-subgroups (P, g_{z}) ≤ (R, b_{z}) such that g_{z} covers g_{L} and b_{z} covers b_{L}. Let (P, g_{J}) ≤ (R, b_{J}) be B_{J}-subgroups such that g_{J} covers g_{z} and b_{J} covers b_{z}. Thus g_{J} covers g_{L}, b_{J} covers b_{L} and g_{J} = g_{J_{1}} × ⋯ × g_{J_{s}} × B_{T′}. In particular, g_{J_{i}} = b_{J_{i}} for i ≥ 2, where b_{J_{i}} are given in (9.4). By Propositions 8.1, 8.2, 8.3 and 8.5 again, each D(b_{J_{i}}) is abelian and D(g_{J_{1}}) = D(b_{J_{1}}), and hence D(g_{J}) = D(b_{J}) is abelian and D(g_{z}) = D(g_{J}) ∩ C = D(b_{J}) ∩ C = D(b_{z}).

Let θ_{i} be the canonical character of b_{i} and θ_{+} = ∏_{i≥2} θ_{i}, so that θ := θ_{1} × θ_{+} is the canonical character of b_{L} and the canonical character θ_{z} of b_{z} covers θ.

Since (P_{1}, g_{1}) ≤ (R_{1}, b_{1}) satisfy Property 7.1 (a∗), it follows that there exists y ∈ N_{C_{L_{1}}(P_{1})}(R_{1}, b_{1}) \ C_{L_{1}}(R_{1}) such that y^{4} = 1, y^{2} ∈ C_{L_{1}}(R_{1}) and [y, x] ∉ Z(L_{1}) for some x ∈ R_{1}. Moreover, there exist subgroups N_{i} ⊲ M_{i} of J_{1}, and character ϕ_{i} ∈ Irr(N_{i}) for i = 1, 2 such that M_{i}/N_{i} is abelian,

Z(L_{1}) ≤ N_{1} × N_{2} ≤ C_{L_{1}}(R_{1}) ≤ C_{J_{1}}(R_{1}) ≤ M_{1} ∘ M_{2},

θ_{1} covers ϕ_{1} × ϕ_{2}, Z ∩ N_{2} = 1, ϕ_{2} has a y-stable extension ϕ̃_{2} to M_{2} and [y, x] = 1 or in ZN_{2} according as x ∈ M_{1} or M_{2}, where Z ≤ Z(M_{1}) ∩ Z(M_{2}) such that M_{1} ∘ M_{2} is the central product over Z.

Let N′_{2} = N_{1} × C_{L_{+}}(R_{+}), M′_{2} := M_{1} × C_{J_{+}}(R_{+}), and ϕ′_{2} = ϕ_{1} × θ_{+}. Then M_{2}/N_{2} and M′_{2}/N′_{2} are abelian,

Z(C) ≤ N′_{2} × N_{2} ≤ C_{C}(R) ≤ C_{J}(R) ≤ M′_{2} ∘ M_{2},

ϕ_{2} has an extension ϕ̃_{2} to M_{2} which is y-invariant, [y, x] = 1 for any x ∈ M′_{2}, [y, x] ∈ ZN_{2} for any x ∈ M_{2}, M_{2}∘M′_{2} is a central product over Z and θ_{z} ∈ Irr(C_{C}(R) | ϕ′_{2}×ϕ_{2}). By Remark 7.3, θ_{z}^{y} = θ_{z}, and so b_{z}^{y} = b_{z}. If [y, x] ∈ Z(C) for all x ∈ R, then [y, x] ∈ Z(C) ∩ L_{1} = Z(L_{1}) for all x ∈ R_{1}, which is impossible. Thus [y, x] ∉ Z(C) for some x ∈ R and Property 7.1 (a∗) holds for (P, g_{z}) ≤ (R, b_{z}) (with H := K).

Case 3. Suppose that K := ^{2}B(2^{2a+1}),^{2}G_{2}(3^{2a+1}),^{2}F_{4}(2^{2a+1}), G_{2}(q), ^{3}D_{4}(q), F_{4}(q) or E_{6}^{−ϵ}(q) with q ≡ ϵ mod 3, and B ∈ Blk(K). Then B satisfies one of Property 7.1 (a∗), (b) or (d).

In each case K = K_{u} and z induces an inner automorphism on K, so it follows that each L_{i} is a classical group (or possibly L_{s} is abelian). Hence by the results of Section 8 each B_{i} satisfies one of Property 7.1 (a∗), (b), (c) or (d).

Case 3.1. Suppose B_{i} satisfies either Property 7.1 (b) or (c) for some i. Without loss of generality, take i = 1. By Theorem 8.6, p = 3, L_{1} = SL^{ϵ_{1}}(3d_{1}, q_{1})/Z for some Z ≤ Z(SL^{ϵ_{1}}(3d_{1}, q_{1})), gcd(6, d_{1}) = 1 and 3∥(q_{1} − ϵ_{1}). By [20, Table 4.7.3A], (q_{1}, ϵ_{1}) = (q, ϵ) or (q^{2}, 1) and (K, C) are given in Table 2, where L_{ϵ} := SL_{3}^{ϵ}(q).

\begin{matrix}K & C & K & C \\ ^{3}D_{4}(q) & (ℤ_{q^{2}+ϵq+1} ∘ L_{ϵ}).3 & G_{2}(q) & L_{ϵ} \\ ^{2}F_{4}(2^{2m+1}) & SU_{3}(2^{2m+1}) & F_{4}(q) & (L_{ϵ} ∘ L_{ϵ}).(3:3) \\ E_{6}^{−ϵ}(q) & (L_{ϵ} ∘ SL_{3}(q^{2})).(3:3) & E_{6}^{ϵ}(q)_{u} & (L_{ϵ} × L_{ϵ} ∘ L_{ϵ}).(3:3:3) \\ E_{7}(q)_{u} & (L_{ϵ} ∘ SL_{6}^{ϵ}(q)).(3:3) & E_{8}(q) & (E_{6}^{ϵ}(q)_{u} ∘ L_{ϵ}).(3:3) \end{matrix}

Table 2: Possible (K, C) with some B_{i} satisfying Property 7.1 (b) or (c)

Case 3.1.1. If K = G_{2}(q) or ^{2}F_{4}(2^{2m+1}), then s = 1 and L = C and B_{z} = B_{L} = B_{0}(L), so B = B_{0}(K) with D(B) = 3_{+}^{1+2}. In particular, l(B) ≥ 2.

Case 3.1.2. Let K = ^{3}D_{4}(q), so that C = ℤ_{\frac{1}{3} (q^{2}+ϵq+1)} × H_{ϵ}, where H_{ϵ} = ⟨L_{ϵ}, x⟩ with x inducing outer-diagonal automorphism of order 3 on L_{ϵ}. So D(B_{z}) = ℤ_{3} ≀ ℤ_{3} ∈ Syl_{3}(C) and we may suppose D(B_{z}) ∈ Syl_{3}(G_{ϵ}), where G_{ϵ} = GL_{3}^{ϵ}(q) contains H_{ϵ}. Let R_{ϵ} = A(D(B_{z})) = (ℤ_{3})^{3} and P_{ϵ} = ⟨Z(L_{ϵ}), diag{1, w, w}⟩ ≤ R_{ϵ} such that |w| = 3 in 𝔽_{q^{2}}^{×}. Then C_{G_{ϵ}}(P_{ϵ}) = ℤ_{q−ϵ} × GL_{2}^{ϵ}(q) and C_{G_{ϵ}}(R_{ϵ}) = (ℤ_{q−ϵ})^{3}. Thus

ℤ_{\frac{1}{3} (q^{2}+ϵq+1)} × C_{H_{ϵ}}(R_{ϵ}) = C_{C}(R_{ϵ}) ≤ C_{C}(P_{ϵ}) = ℤ_{\frac{1}{3} (q^{2}+ϵq+1)} × C_{H_{ϵ}}(P_{ϵ}).

As shown in the proof of Proposition 8.1 the B_{z}-subgroups (P_{ϵ}, g) ≤ (R_{ϵ}, b) satisfy Property 7.1 (a∗).

Note that in the notation above C_{G_{ϵ}}(P_{ϵ})/Z = C_{G_{ϵ}/Z}(P_{ϵ}/Z) and C_{G_{ϵ}}(R_{ϵ})/Z = C_{G_{ϵ}/Z}(R_{ϵ}/Z) for any Z ≤ Z(L_{ϵ}). Let b_{ϵ} ∈ Blk(H_{ϵ}) and B_{ϵ} ∈ Blk(G_{ϵ}) covering b_{ϵ}, so that D(b_{ϵ}) = D(B_{ϵ}) ∩ H_{ϵ} = D(B_{ϵ}). Thus D(b_{ϵ}) ∈ Syl_{3}(C_{G_{ϵ}}(t)) for some semisimple 3′-element t. In particular, D(b_{ϵ}) is either abelian with |D(b_{ϵ})| ≥ 9 and D(b_{ϵ}) ≰ L_{ϵ} or D(b_{ϵ}) = ℤ_{3} ≀ ℤ_{3}. In the former case, C_{G_{ϵ}}(D(b_{ϵ}))/Z = C_{G_{ϵ}/Z}(D(b_{ϵ})/Z) for any Z ≤ Z(L_{ϵ}).

Case 3.1.3. Suppose K = E_{6}^{−ϵ}(q) or F_{4}(q) and L_{1} = L_{ϵ}, so that C = ⟨L_{1} ∘ L_{2}, x⟩, where L_{2} = L_{ϵ} or SL_{3}(q^{2}), and x = x_{1}x_{2} such that each x_{i} induces outer-diagonal automorphism of order 3 on L_{i}. Let J_{i} = ⟨L_{i}, x_{i}⟩ and B_{J_{2}} ∈ Blk(J_{2}) covering B_{2}. Let R_{1} = R_{ϵ} ≤ J_{1}, P_{1} = P_{ϵ} ≤ R_{1} and P_{2} = R_{2} = A(B_{J_{2}}), so that by the remark of Case 3.1.2 above, C_{J_{i}}(P_{i})/Z = C_{J_{i}/Z}(P_{i}/Z) and C_{J_{i}}(R_{i})/Z = C_{J_{i}/Z}(R_{i}/Z) for any Z ≤ Z(L_{i}). By Lemmas 9.1 and 7.6, we may suppose

L = L_{1} × L_{2} ≤ C ⊲ J := J_{1} × J_{2}

Let R = (R_{1} × R_{2}) ∩ C, P = (P_{1} × P_{2}) ∩ C and let (P, g) ≤ (R, b) be B_{z}-subgroups, so that π_{1}(P) = P_{ϵ} and π_{1}(R) = R_{ϵ}, where π_{i} is the natural projection form J to J_{i}. A proof similar to that of Case 2 shows that (P, g) ≤ (R, b) satisfy Property 7.1 (a∗). If K = E_{6}^{−ϵ}(q) and L_{1} = SL_{3}(q^{2}), then L_{2} = L_{ϵ} and a similar proof shows that Property 7.1 (a∗) holds for some B-subgroups (P, g) ≤ (R, b).

Note that C_{J}(P)/Z = C_{J/Z}(P/Z) and C_{J}(R)/Z = C_{J/Z}(R/Z) for any Z ≤ O_{3}(Z(L)).

Case 3.2. Suppose that each B_{i} satisfies either Property 7.1 (a∗) or (d). By Case 1, we may suppose, moreover that B_{1} satisfies Property 7.1 (a∗). In particular, a Sylow p-subgroup of L_{1} is nonabelian.

Case 3.2.1. Suppose p ≥ 5, so that z is of parabolic type. By [20, Theorem 4.2.2 (f)], O^{r′}(C) is a direct product and each L_{i} is universal. In addition, if a Sylow p-subgroup of L_{1} is nonabelian, then ℓ = 1 or 2 and each L_{i} is universal.

Suppose ℓ = 1, so that s = 1 or 2. Since B_{1} satisfies Property 7.1 (a∗) and L_{s} = Z(C) when s = 2, it follows by Lemma 9.1 that we may suppose L = L_{1} × L_{s} and C_{J_{i}}(L_{i}) ≤ Z(L_{i}).

Suppose ℓ = 2, so that s = 2 or 3. Since L_{1} has a nonabelian Sylow p-subgroup, it follows that L_{2} has an abelian Sylow p-subgroup and, moreover L_{2} ∩ O_{p}(Z(C)) = 1. Since the central product L_{1} ∘ L_{2} ∘ L_{s} is over a subgroup of Z(C), it follows that each p-subgroup of L_{2} satisfies the assumption of Lemma 9.1. Since B_{1} satisfies Property 7.1 (a∗) and L_{s} = Z(C) when s = 3, it follows by Lemma 9.1 that we may suppose L = L_{1} × L_{s} × L_{s} and C_{J_{i}}(L_{i}) ≤ Z(L_{i}).

By Case 2, B satisfies Property 7.1 (a∗).

Case 3.2.2. Suppose p = 3, so that C is given by [20, Table 4.7.3a]. Thus either ℓ = 1 or ℓ = 2 with C given by Table 2. In addition, each L_{i} is also universal for 1 ≤ i ≤ ℓ and C_{J_{i}}(L_{i}) ≤ Z(L_{i}).

A proof similar to that of Case 3.2.1 shows that we may suppose (9.3) holds and by Case 2, B satisfies Property 7.1 (a∗).

Case 4. Let 3 | (q − ϵ), K = K_{u} = 3.E_{6}^{ϵ}(q) ≤ E := 3.E_{6}^{ϵ}(q).3, B ∈ Blk(K) and B_{E} ∈ Blk(E) covering B. Either Property 7.1 (a∗) holds for some B-subgroups (P, g) ≤ (R, b) with C_{E}(P)/Z = C_{E/Z}(P/Z) and C_{E}(R)/Z = C_{E/Z}(R/Z) for Z ≤ O_{p}(K), or Property 7.1 (d) holds for B.

Let D := D(B) and m^{∗} := gcd(m, q − ϵ).

If some B_{i} satisfies either Property 7.1 (b) or (c), then p = 3 and C is given by Table 2. A proof similar to that of Case 3.2.2 shows that Property 7.1 (a*) holds for some B-subgroups (P, g) ≤ (R, b).

Suppose some B_{i} satisfies Property 7.1 (a∗). Since z is parabolic or equal-rank type and z induces an inner automorphism on K, it follows that each L_{i} is classical. We first show that there exist B-subgroups (P, g) ≤ (R, b) and y satisfying the Property 7.1 (a∗) with H := K.

If p ≥ 5, then z is parabiloc. A proof similar to that of Case 3.2.1 shows that the Property 7.1 (a∗) holds for B-subgroups (P, g) ≤ (R, g).

Suppose p = 3. By [20, Table 4.7.3A],

C_{K}(z) = ⟨SL_{3}^{ϵ}(q) × SL_{3}^{ϵ}(q) ∘ SL_{3}^{ϵ}(q), 3:3:3⟩, (SL_{6}^{ϵ}(q) ∘_{2^{∗}} (q − ϵ)).2^{∗},

Spin_{8}^{+}(q) ∘_{2^{∗}} ((q − ϵ) × (q − ϵ)).(2^{∗} × 2^{∗}), Spin_{10}^{ϵ}(q) ∘ (q − ϵ).(2^{∗} × 2_{ϵ}^{∗}) (when q ≡ ϵ (mod 9)) with 2_{ϵ}^{∗} = 1 or 2^{∗} according as ϵ = − or +, or (SL_{2}(q) × SL_{5}^{ϵ}(q)) ∘ (q − ϵ).2^{∗} (when q ≡ ϵ (mod 9)).

Thus ℓ = 1, 2 or 3. If ℓ = 1 or 2, then a proof similar to that of Cases 3.2.1 and 3.2.2 shows that the Property 7.1 (a∗) holds for B-subgroups (P, g) ≤ (R, g). If ℓ = 3, then each L_{i} = SL_{3}^{ϵ}(q) and C_{J_{i}}(L_{i}) ≤ Z(L_{i}). A proof similar to that of Cases 3.1.3 shows that we may suppose (9.3) holds and by Case 2, the Property 7.1 (a∗) holds for B-subgroups (P, g) ≤ (R, g).

If all B_{i} satisfies Property 7.1 (d), then by Case 1, D(B_{z}) is abelian.

Now we prove the rest of Property 7.1 (a∗) with H = E. Suppose (P, g_{E}) ≤ (R, b_{E}) are B_{E}-subgroups such that g_{E} covers g and b_{E} covering b.

Case 4.1. If p ≥ 5, then D(b) = D(b_{E}), D(g) = D(g_{E}), and so D(g_{E}) = D(b_{E}) is abelian. Now C_{E}(z) = C or ⟨C, x⟩ for some x ∈ E \ K. If C_{E}(z) = C, then C_{E}(R) = C_{C}(R). Apply the proof of Case 2 we have that

Z(C) ≤ N′_{2} × N_{2} ≤ C_{C}(R) ≤ C_{C_{E}(z)}(R) ≤ M′_{2} ∘ M_{2}.  (9.5)

Suppose C_{E}(z) = ⟨LT, x⟩ for some x ∈ E \ K, so that x induces inner-diagonal automorphism on each L_{i}. Thus x = x_{1}x_{2} ⋯ x_{s}x′. Replacing J_{i} by ⟨J_{i}, x_{i}⟩ and T′ by ⟨T′, x′⟩ in the proof Case 2 with some obvious modifications, we have that (9.5) still holds. Thus Property 7.1 (a∗) holds for B-subgroups (P, g) ≤ (R, b) (with H := E).

Case 4.2. Suppose p = 3. By [20, Table 4.7.3A],

C_{E}(z) = ⟨SL_{3}^{ϵ}(q) × SL_{3}^{ϵ}(q) ∘ SL_{3}^{ϵ}(q), 3:3:1, 1:3:3⟩, (SL_{6}^{ϵ}(q) ∘_{2^{∗}} (q − ϵ)).(3 × 2^{∗}),

Spin_{8}^{+}(q) ∘_{2^{∗}} ((q − ϵ) × (q − ϵ)).(2^{∗} × 2^{∗} × 3), Spin_{10}^{ϵ}(q) ∘ (q − ϵ).(3 × 2^{∗} × 2_{ϵ}^{∗}) (when q ≡ ϵ (mod 9)) with 2_{ϵ}^{∗} = 1 or 2^{∗} according as ϵ = − or +, or (SL_{2}(q)×SL_{5}^{ϵ}(q))∘(q−ϵ).(2^{∗}×3) (when q ≡ ϵ (mod 9)).

Suppose C_{E}(z) = ⟨SL_{3}^{ϵ}(q) × SL_{3}^{ϵ}(q) ∘ SL_{3}^{ϵ}(q), t, x⟩, so that L = SL_{3}^{ϵ}(q) × SL_{3}^{ϵ}(q) ∘ SL_{3}^{ϵ}(q), T = ⟨t⟩ ≤ K with t induces 3:3:3 on L, and x ∈ E \ K induces 1:3:3 on L. Let L_{i} = SL_{3}^{ϵ}(q) ≤ G_{i} := GL_{3}^{ϵ}(q), t = t_{1} × t_{2} × t_{3}, x = x_{1} × x_{2} × x_{3} with t_{i}, x_{i} act on L_{i} and centralizes L_{j} when i ≠ j. In addition, let H_{i} = ⟨L_{i}, t_{i}, x_{i}⟩, so that H_{i} ≤ G_{i}. Let S = ℤ_{q−ϵ} ×ℤ_{q−ϵ} ≤ SL_{3}^{ϵ}(q) be a maximal torus, and S ×S ∘_{3} S ≤ L. Since C_{GL_{3}^{ϵ}(q)}(S) = ℤ_{q−ϵ} ×ℤ_{q−ϵ} ×ℤ_{q−ϵ} is a maximal torus, it follows that A := C_{E}(S ×S ∘_{3} S) is abelian such that A ∩ K ≅ ℤ_{q−ϵ}^{6} is a maximal torus of K and A/(A ∩ K) = ℤ_{3}. In particular, we may suppose t, x ∈ A and C_{E}(z) = LA with abelian A and L ⊲ C_{E}(z).

Similarly, if C_{E}(z) = (SL_{6}^{ϵ}(q) × (q − ϵ)).6^{∗}, Spin_{8}^{+}(q) ∘_{2} ((q − ϵ) × (q − ϵ)).(2^{∗} × 6^{∗}), Spin_{10}^{ϵ}(q) ∘ (q − ϵ).(6^{∗} × 2_{ϵ}^{∗}) or (SL_{2}(q) × SL_{5}^{ϵ}(q)) ∘ (q − ϵ).6^{∗}, then A ≤ C_{E}(z) and so C_{E}(z) = LA with abelian A and L ⊲ C_{E}(z), and A induces inner-diagonal automorphisms on each L_{i}.

A proof similar to that of Case 2 with LT replaced by LA and some modifications shows that D(g_{E}) = D(b_{E}) is abelian, and hence Property 7.1 (a∗) holds.

Case 4.3. Suppose p = 3 and Z = Z(K). If C_{E}(z)/Z = C_{E/Z}(zZ), then C_{E}(P)/Z = C_{E/Z}(P/Z) and C_{E}(R)/Z = C_{E/Z}(R/Z). Suppose C_{E}(z)/Z ≠ C_{E/Z}(zZ). By [20, Table 4.7.3A], either L = L_{1} ∘ L_{2} with L_{1} = Spin_{8}^{+}(q) and L_{2} = Z(C) = ℤ_{q−ϵ} × ℤ_{q−ϵ} or L = L_{ϵ} × L_{ϵ} ∘ L_{ϵ} ≤ K. In the former case,

C_{E/Z}(zZ) = ⟨C_{E}(z)/Z, wZ⟩,

where w ∈ E such that w = γ:ω, that is, w acts on L_{1} as a graph automorphism of order 3 and (h_{1}, h_{2})^{ω} = (h_{2}, (h_{1}h_{2})^{−1}) for any (h_{1}, h_{2}) ∈ L_{2}. Now O_{3}(L_{2}) = O_{3}(ℤ_{q−ϵ}) × O_{3}(ℤ_{q−ϵ}) ≤ P and Z(K) = {(x, x) : x ∈ Ω_{1}(O_{3}(ℤ_{q−ϵ}))} ≤ Z(C). Suppose h ∈ E such that for any u ∈ P we have h^{−1}uh = cu for some c ∈ Z(K) and suppose h ∉ C_{E}(z). Then we may suppose h = tw for some t ∈ C_{E}(z), and so (1, h_{2})^{h} = (1, h_{2})^{ω} = (h_{2}, 1) for any h_{2} ∈ O_{3}(ℤ_{q−ϵ}) \ {1}. But (h_{2}, 1) ≠ c(1, h_{2}) for any c ∈ Z(K), which is a contradiction. Thus h ∈ C_{E}(z) and so C_{E}(P)/Z = C_{E/Z}(P/Z). Similarly, C_{E}(R)/Z = C_{E/Z}(R/Z). If L = L_{ϵ} × L_{ϵ} ∘ L_{ϵ}, then L/Z = L_{ϵ} ∘ L_{ϵ} ∘ L_{ϵ} and C_{E/Z}(zZ) = ⟨L/Z, tZ, xZ, wZ⟩, where t, x are given above and w ∈ E \ K permutes transitively the three components L_{ϵ} of L. The proof in this case is similar. Suppose h ∈ E such that for all u ∈ P h^{−1}uh = cu for some c ∈ Z, so that hZ ∈ C_{E/Z}(zZ). Since |Ω_{1}(P)| ≥ 3^{4} and C_{J}(P)/Z = C_{J/Z}(P/Z), it follows that h ∈ C_{E}(z) = ⟨L, x, t⟩ and hence C_{E}(P)/Z = C_{J/Z}(P/Z). Similarly, C_{E}(R)/Z = C_{E/Z}(R/Z).

Case 4.4. Now we prove the rest of Property 7.1 (d). Suppose Property 7.1 (d) holds for each B_{i} and suppose D_{E} ∩ K = D for some D_{E} = D(B_{E}), so that D(B_{J_{i}}) is abelian and so is D(B_{z}) = D(B_{J}) ∩ C. If B_{z} is a major subsection, then D = D(B_{z}) is abelian. If p ≥ 5, then D_{E} = D. Suppose p = 3 and there exists x ∈ Z(D_{E}) \ D. Then x ∈ E \ K, x ∈ C_{E}(D) and D_{E} = ⟨D, x⟩ is abelian. If Z(D_{E}) ≤ D, then take z ∈ Z(D_{E}) with |z| = 3, so that D_{E} ≤ C_{E}(z) = LA. A proof similar to that of Case 1.2 with some obvious modifications shows that D_{E} is abelian.

Suppose z ∈ D with |z| = 9 and zZ(K) ∈ Z(D/Z(K)). By [20, Table 4.7.3A], 9 | (q − ϵ) and C_{E}(z) = Spin_{10}^{ϵ}(q) ∘ (q − ϵ).(6^{∗} × 2_{ϵ}^{∗}) or (SL_{2}(q) × SL_{5}^{ϵ}(q)) ∘ (q − ϵ).6^{∗}. In this case, C_{E/Z(K)}(zZ(K)) is also given by [20, Table 4.7.3A], and we have C_{K/Z(K)}(zZ(K)) = C_{K}(z)/Z(K). Thus D/Z(K) ≤ C_{K}(z)/Z(K) and D ≤ C_{K}(z). In particular, z ∈ Z(D) and we may suppose (z, B_{z}) is major. Hence D = D(B_{z}) is abelian. It follows that Property 7.1 (d) holds for B.

Case 5. Let K := E_{7}(q) with q even and B ∈ Blk(K). Then either L_{i} is classical, or L_{i} is exceptional given in Cases 3 or 4. If L_{i} is classical, then apply Propositions 8.1, 8.2, 8.3 and 8.5. If L_{i} is exceptional, then apply the results given in Cases 3 and 4. Either Property 7.1 (a∗) holds for B or Property 7.1 (d) holds for B (with H := K := E_{7}(q)).

Let q be odd, K = 2.E_{7}(q) ≤ E := 2.E_{7}(q).2, B ∈ Blk(K) and B_{E} ∈ Blk(E) covering B. Either Property 7.1 (a∗) holds for some B-subgroups (P, g) ≤ (R, b) or Property 7.1 (d) holds for B.

Again let D := D(B) and m^{∗} := gcd(m, q − ϵ).

Suppose Property 7.1 (a∗) holds for some B_{i}-subgroups (P_{i}, g_{i}) ≤ (R_{i}, b_{i}). A proof similar to that of Case 3.2 with some obvious modifications shows that there exist B-subgroups (P, g) ≤ (R, b) and y satisfying the first part of Property 7.1 (a∗) (with H = K). Suppose B_{i} satisfies either Property 7.1 (b) or (c) for some i. Then p = 3 and C is given by Table 2. A proof similar to that of Case 3.1.3 shows that Property 7.1 (a*) holds for some B-subgroups (P, g) ≤ (R, b). Suppose (P, g_{E}) ≤ (R, b_{E}) are B_{E}-subgroups such that g_{E} covering g and b_{E} covering b. Then D(g) = D(g_{E}) and D(b) = D(b_{E}) for some D(g_{E}) and D(b_{E}). But D(g) = D(b) is abelian, so D(g_{E}) = D(b_{E}) is abelian. A proof similar to that of Case 4.1 shows that (9.5) holds and so Property 7.1 (a∗) (with H = E) holds for B-subgroups (P, g) ≤ (R, b).

Since E/K = 2 and p is odd, it follows that D(B_{E}) is abelian whenever D(B) is abelian.

Case 6. Suppose K := E_{8}(q). Either Property 7.1 (a) holds for some B-subgroups (P, g) ≤ (R, b) or Property 7.1 (d) holds for B.

In this case (z, B_{z}) is a major subsection of B and either L_{i} is classical, or L_{i} is exceptional given in Cases 3, 4 or 5. If L_{i} is classical, then apply Theorem 8.6. If L_{i} is exceptional, then apply the results in Cases 3, 4 or 5. Thus if each D(B_{i}) is abelian, then D(B_{J_{i}}) is abelian, and so D = D(B_{z}) = D(B_{J}) ∩ C is abelian. Suppose D(B_{i}) is non-abelian for some i, say i = 1.

If p ≥ 7, then B_{1} satisfies Property 7.1 (a∗), z is of parabolic type and the proof is similar to that of Case 3.2.1.

Suppose p = 5, so that B_{1} satisfies Property 7.1 (a∗) and C is given by [20, Table 4.7.3B]. Thus ℓ = 1 or 2. If z is parabolic, then a proof similar to that of Case 3.2.1 shows that we may suppose (9.3) holds.

If z is equal-rank, then

C = ⟨L_{1} ∘ L_{2}, 5:5⟩, L_{1} = L_{2} = SL_{5}^{ϵ}(q),

so that L = L_{1} ∘ L_{2}. Here ϵ = ±1 such that q ≡ ϵ (mod 5). A proof similar to that of Case 3.1.3 shows that we may suppose L = L_{1} × L_{2}. By Case 2, B satisfies Property 7.1 (a∗).

Suppose p = 3, so that C is given by [20, Table 4.7.3A] and ℓ = 1 or 2.

If some B_{i} satisfies either Property 7.1 (b) or (c), then C is given by Table 2. In particular, ℓ = 2 and D(B_{i}) = 3_{+}^{1+2}.

If ℓ = 1, then B_{1} satisfies Property 7.1 (a∗) and we may suppose (9.3) holds. By Case 2, B satisfies Property 7.1 (a∗).

Suppose ℓ = 2, so that L = L_{ϵ} ∘ E_{6}(q)_{u} and C = ⟨L, 3:3⟩, where L_{ϵ} = SL_{3}^{ϵ}(q) with q ≡ ϵ (mod 3).

If L_{1} = E_{6}(q)_{u} and B_{1} satisfies Property 7.1 (a∗) for (P_{1}, g_{1}) ≤ (R_{1}, b_{1}). Let P_{2} = R_{2} = A(D(B_{2})), and let (R_{2}, b_{2}) be a B_{2}-subgroup and set (P_{2}, g_{2}) = (R_{2}, b_{2}). By Case 4 and the ramark of Case 3.1.2, C_{J_{i}}(R_{i})/Z_{i} = C_{J_{i}/Z_{i}}(R_{i}/Z_{i}) for i = 1, 2. By Lemma 9.1, we may suppose L = L_{1} × L_{2} and a similar to that of Case 2 shows that B satisfies Property 7.1 (a∗).

Suppose L_{1} = L_{ϵ} and B_{2} satisfies Property 7.1 (d), so that D(B_{2}) is abelian. In this case B_{1} satisfies Property 7.1 (b) with D(B_{1}) = 3_{+}^{1+2} or B_{1} satisfies Property 7.1 (a*). Note that J = J_{1} ∘ J_{2} over Z(L) = 3 and J_{1} = H_{ϵ} given in Case 3.1.2.

Let (P_{ϵ}, g) ≤ (R_{ϵ}, b) be defined in Case 3.1.2. Then there exists y ∈ C_{H_{ϵ}}(P_{ϵ}) ∩ N_{H_{ϵ}}(R_{ϵ}) \ C_{H_{ϵ}}(R_{ϵ}) satisfies Property 7.1 (a∗).

Let P_{2} = R_{2} = A(D(B_{J_{2}})) = D(B_{2}) and let (P_{2}, b_{2}) = (R_{2}, b_{2}) be the Sylow B_{J_{2}}-subgroup. Since H_{ϵ} = J_{1}, it follows that P_{ϵ} ∘ P_{2} ≤ R_{ϵ} ∘ R_{2} ≤ J. Set

P = (P_{ϵ} ∘ P_{2}) ∩ C, and R = (R_{ϵ} ∘ R_{2}) ∩ C.

Then P ≤ R ≤ D and P, R are abelian. Since Z(L) = 3, it follows that b ∘ b_{2} is a block C_{J}(R_{ϵ} ∘ R_{2}) and g ∘ g_{2} ∈ Blk(C_{J}(P_{ϵ} ∘ P_{2})). Since J/C = 3, it follows that b ∘ b_{2} covers a unique block b_{R} of C_{C}(R) and similarly, g ∘ g_{2} covers a unique block g_{P} of C_{C}(P). Since B_{J} is the unique block covering B_{z}, it follows that (P, b_{P}) ≤ (R, b_{R}) are B_{z}-subgroups. Since H_{ϵ}/L_{ϵ} = 3, it follows that y ∈ L_{ϵ} = L_{1}. Now B satisfies Property 7.1 (a) (not (a∗)) for B-subgroups (P, b_{P}) ≤ (R, b_{R}) (with H = K). □

Lemma 9.3 Let G be a quasisimple group such that G/Z(G) is alternating or of Lie type and G is an exceptional cover. Let p be an odd prime. Then every p-block of G with nonabelian defect groups has a subpair with at least two irreducible Brauer characters.

Proof: We must consider the cases G/Z(G) ≅ P SL_{2}(4), P SL_{2}(9), A_{7}, P SL_{3}(2), P SL_{3}(4), P SU_{4}(2), P SU_{4}(3), P SU_{6}(2), ^{2}B_{2}(8), O_{7}(2), O_{7}(3), O_{8}^{+}(2), G_{2}(3), G_{2}(4), F_{4}(2) and ^{2}E_{6}(2). We may use [18] to confirm all but the cases F_{4}(2) for p = 3, and 2E_{6}(2) for p = 3, 5, 7 (noting that the three double covers of O_{8}^{+}(2) have the same block structure - see [14]), as in each case the block itself has at least two irreducible Brauer characters. The result holds for F_{4}(2) for p = 3 by [22]. Note that ^{2}E_{6}(2) has abelian Sylow 5- and 7-subgroups, so we are left with p = 3 and G/Z(G) ≅ ^{2}E_{6}(2). In this case we do not know the Brauer characters of G, so we are forced into a slightly involved argument to make use of the current literature. Note that it suffices to consider the case |Z(G)| = 4. Our group G has three conjugacy classes of elements of order three, 3A, 3B and 3C. For each such x ∈ G, we have C_{G}(x)/Z(G) ≅ C_{G/Z(G)}(xZ(G)). Consider a block B covering the block c of Z(G) containing the irreducible character λ, say. We may assume that c is faithful. By examination of the character table in [14], only two irreducible characters lying over λ vanish on 3A, 3B but not 3C (χ_{184} and χ_{202} in the notation of [14]). Since a 3-block of positive defect must possess at least three irreducible characters, it follows by a theorem of Green that B must have a defect group D containing elements of 3A or 3B. Suppose x ∈ Z(D) has order three. Write Q = ⟨x⟩. Note that DC_{G}(D) ≤ C_{G}(Q), so there is a B-subgroup (Q, b_{Q}) with defect group D. If x ∈ 3A, then C_{G/Z(G)}(xZ(G)) ≅ Q × P SU_{6}(2). We have seen that every block with nonabelian defect groups of a double cover of P SU_{6}(2) has at least two irreducible Brauer characters, so it follows that l(b_{Q}) = 1 if D is nonabelian. If x ∈ 3B, then C_{G/Z(G)}(xZ(G)) ≅ Q × O_{8}^{+}(2).3, and the same argument applies as for 3A. □

Theorem 9.4 Let G be a quasisimple group and B a nilpotent p-block of G with defect group D, where p is odd. Then D is abelian.

Proof: If G/Z(G) is an alternating group, then the result follows by Theorem 3.3 and the remarks following it. For G/Z(G) sporadic see Theorem 4.5. If G/Z(G) is a classical group and G is a non-exceptional cover, see Propositions 8.1, 8.2, 8.3 and 8.5. For G/Z(G) an exceptional group of Lie type and G is a non-exceptional cover, see Theorem 9.2. For the exceptional covers, see Lemma 9.3. □

10 Puig’s conjecture

We complete the proof of Puig’s conjecture for quasisimple groups for odd primes, present some general results and deduce some corollaries.

Theorem 10.1 Let G be a finite quasisimple group and let B be a p-block of G with p odd. Then B is nilpotent if and only if l(b_{Q}) = 1 for each p-subgroup Q and each block b_{Q} of C_{G}(Q) with (b_{Q})^{G} = B.

Proof: The necessary condition for nilpotency follows from [13, 1.2]. By Corollary 7.5 and Propositions 8.1, 8.2, 8.3 and 8.5 the result holds for the classical groups. By Theorem 9.2 it holds for the exceptional groups of Lie type. The result holds for the double covers of the alternating groups by Theorem 3.3, and when G/Z(G) is sporadic by Theorem 4.5. For the exceptional covers of the alternating groups and of the finite simple groups of Lie type, see Lemma 9.3. □

Lemma 10.2 Let N ⊲ G such that G/N is cyclic and of order prime to p, and let B ∈ Blk(G) cover b ∈ Blk(N). Suppose there are abelian R and P and b-subgroups (P, b_{P}) ≤ (R, b_{R}) such that b_{P} and b_{R} have abelian defect groups and there is x ∈ C_{N}(P) of order prime to [G:N] such that x ∈ N_{N}(R, b_{R}) \ C_{N}(R). Then there are B-subgroups (P, B_{P}) ≤ (R, B_{R}) such that B_{R} and B_{P} have abelian defect groups and x ∈ N_{G}(R, B_{R}) \ C_{G}(R).

Proof: By [15, 4.1] there is B_{R} ∈ Blk(C_{G}(R)) such that (B_{R})^{G} = B and B_{R} covers b_{R}. We claim that the number of such B_{R} divides [G:N]. Now C_{G}(R)/C_{N}(R) is cyclic of order dividing [G:N]. The blocks of C_{N}(R) and of C_{G}(R) are in 1-1 correspondence with their canonical characters. Let θ_{R} be the canonical character for b_{R}. Since C_{G}(R)/C_{N}(R) is cyclic, θ_{R} extends to an irreducible character of C_{G}(R), and since [C_{G}(R) : C_{N}(R)] is not divisible by p, the extensions are precisely the canonical characters of the blocks of C_{G}(R) covering b_{R}. By Clifford theory, the group of irreducible characters of C_{G}(R)/C_{N}(R) acts transitively on the blocks B_{1}, …, B_{n} of C_{G}(R) covering b_{R} by inflation and multiplication, and also transitively on the set {B_{i}^{G} : 1 ≤ i ≤ n}. Consequently the number of blocks of C_{G}(R) covering b_{R} with Brauer correspondent B divides [C_{G}(R) : C_{N}(R)], and the claim follows.

For each i, we have (B_{i}^{x})^{G} = (B_{i}^{G})^{x} = B^{x} = B, and B_{i}^{x} overs b_{i}^{x} = b_{i}. Hence x permutes {B_{i} : B_{i}^{G} = B}. Since this set has order prime to the order of x, it follows that x must fix some such B_{i}. Call it B_{R}. Letting B_{P} = (B_{R})^{C_{G}(P)}, we are done. □

As an almost immediate corollary we have:

Corollary 10.3 Let G be a finite group such that there is N ⊲ G with [G : N] odd and G/N is a p-regular cyclic group, where N is quasisimple and p > 3 is a prime. Let B be a p-block of G. Then B is nilpotent if and only if l(b_{Q}) = 1 for every B-subgroup (Q, b_{Q}).

Proof: Since the alternating and sporadic groups have outer automorphism groups of order at most two, it follows that it suffices to consider the groups of Lie type. Suppose first that N is not an exceptional covering group. Since p > 3, every block of N satisfies one of Property 7.1 (a) or (d), and the result follows by Lemma 10.2 and Corollary 7.5.

Suppose that N is an exceptional cover. Then the outer automorphism group is a 2-group except when N/Z(N) ≅ P SL_{3}(4), P SU_{6}(2) or ^{2}E_{6}(2), in which case it has order three, and consists of diagonal automorphisms. In each case Z(N) is a Klein-four group. However, in each of these cases the non-inner automorphisms transitively permute the blocks whose kernel does not contain Z(N), and the result follows in these cases too, since B is nilpotent if and only if b is, and B-subgroups (Q, B_{Q}) covering b-subgroups (Q, b_{Q}) satisfy l(B_{Q}) = l(b_{Q}). □

Acknowledgement. The second author thanks the University of Auckland for their hospitality during two visits to the University of Auckland, where part of this work was done.

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