Source Algebras of Blocks, Sources of Simple Modules, and a Conjecture of Feit

Susanne Danz and Jürgen Müller June 15, 2011

Abstract

We verify a finiteness conjecture of Feit on sources of simple modules over group algebras for various classes of finite groups related to the symmetric groups.

1 Introduction and Results

1.1. Amongst the long-standing conjectures in modular representation theory of finite groups is a finiteness conjecture concerning the sources of simple modules over group algebras, due to Feit [11] and first announced at the Santa Cruz Conference on Finite Groups in 1979:

By Green’s Theorem [14], given a finite group G and an algebraically closed field F of some prime characteristic p, one can assign to each indecomposable F G-module M a G-conjugacy class of p-subgroups of G, the vertices of M. Given a vertex Q of M, there is, moreover, an indecomposable F Q-module L such that M is isomorphic to a direct summand of the induced module Ind_{Q}^{G}(L); such a module L is called a source of M. Any Q-source of M has vertex Q as well, and is determined up to isomorphism and conjugation with elements in N_{G}(Q). Vertices of simple F G-modules have a number of special features not shared by vertices of arbitrary indecomposable F G-modules; see, for instance, [10] and [21]. Feit’s Conjecture in turn predicts also a very restrictive structure of sources of simple modules, and can be formulated as follows:

Conjecture (Feit). Given a finite p-group Q, there are only finitely many isomorphism classes of indecomposable F Q-modules occurring as sources of simple F G-modules with vertices isomorphic to Q. Here G varies over all finite groups containing a subgroup isomorphic to Q.

While the conjecture remains open in this generality, weaker versions of it are known to be true: by work of Dade [4], Feit’s Conjecture holds when demanding the sources in question have dimension at most d, for a given integer d. Furthermore, Puig [28], [30] has shown that Feit’s Conjecture holds when allowing the group G to vary over p-soluble groups only, and Puig [29] has also shown that Feit’s Conjecture holds for the symmetric groups. The aim of this paper now is to pursue the idea of restricting to suitable classes of groups further, and to prove the following:

Theorem. Feit’s Conjecture holds when letting the group G vary over the following classes of groups only:

{𝔖_{n}}_{n⩾1}, {𝔄_{n}}_{n⩾1}, {𝔖̃_{n}}_{n⩾1}, {𝔖̂_{n}}_{n⩾1}, {𝔄̃_{n}}_{n⩾1}, {𝔅_{n}}_{n⩾2}, {𝔇_{n}}_{n⩾4}.

Here, 𝔖_{n} and 𝔄_{n} denote the symmetric and alternating groups on n letters, respectively. Moreover, 𝔖̃_{n} and 𝔖̂_{n} denote the Schur double covers of the symmetric groups, and 𝔄̃_{n} those of the alternating groups; these groups are described in more detail in 5.1. Finally, 𝔅_{n} and 𝔇_{n} denote the Weyl groups of type B_{n} and D_{n}, respectively; these groups are described in more detail in 6.1.

1.2. We will prove the above-mentioned main result by exploiting the connection between Feit’s Conjecture and two other ingredients: Puig’s Conjecture regarding source algebras of blocks of group algebras, and a question raised by Puig (see [33]) relating vertices to defect groups:

Conjecture (Puig). Given a finite p-group P, there are only finitely many isomorphism classes of interior P -algebras that are source algebras of a block of F G. Here G varies over all finite groups.

Question. Suppose that p > 2, that G is a finite group, and that D is a simple F G-module with vertex Q. Is the order of the defect groups of the block containing D bounded in terms of the group order |Q|?

The corresponding question for p = 2 is known to have a negative answer, and we will elaborate on this more detailedly in Remark 3.3. For p > 2, to the authors’ knowledge, there seem to be no examples known where Puig’s Question admits a negative answer, and Zhang [33] has proved a reduction to quasi-simple groups. The Reduction Theorem 3.8 now shows that Puig’s Conjecture together with a positive answer to the previous question imply Feit’s Conjecture for p > 2. A proof of Puig’s Conjecture alone, however, might not suffice to prove Feit’s Conjecture; see also the remarks following [32, Thm. 38.6].

In order to formulate our Reduction Theorem 3.8, we have been trying to make the conjectures involved precise, in the sense of specifying in detail the sets that are conjectured to have finite cardinality. Despite the fact that the conjectures have been around for quite a while, and belong to folklore in modular representation theory of finite groups, we have not been able to find a reference where they have been stated formally and precisely. Thus we first develop a category-theoretic language to do so.

Our strategy for proving our main Theorem 3.10 is as follows: by work of Kessar [18], [19], [20], Puig [29], and Scopes [31], Puig’s Conjecture is known to be true for the classes of groups considered here. We will, therefore, show that Puig’s Question has an affirmative answer when allowing the groups to vary over the classes of groups in the theorem only, by determining in Theorem 3.9 explicit upper bounds on the respective defect group orders, and regardless of whether p is even or odd. It should be pointed out that such bounds can also be derived from work of Zhang [33], which are, however, much weaker than the ones we get, and are thus hardly useful when actually trying to compute vertices of simple modules in practice.

The strategy to prove the bounds on defect group orders in terms of vertices, in turn, is for part of the cases based on Knörr’s Theorem, see Remark 3.12, ensuring the existence of a selfcentralizing Brauer pair for any subgroup of a group G being a vertex of a simple F G-module. This reduces Puig’s Question to asking more strongly whether defect group orders can even be bounded in terms self-centralizing Brauer pairs. This idea turns out to be successful for the alternating groups and the Schur covers of the symmetric and alternating groups. In particular, in Theorem 4.5 and the subsequent Remark 4.6 we derive a detailed picture of the self-centralizing Brauer pairs for the alternating groups in characteristic p = 2, which might be of independent interest.

Moreover, we would like to point out that, in particular, for the case of the alternating groups, we have been examining various examples explicitly, where the computer algebra systems GAP [13] and MAGMA [2] have been of great help; we will specify later on where precisely these have been invoked.

1.3. The paper is organized as follows: in Section 2 we introduce our notational set-up, define our notions of the category of interior algebras and vertex-source pairs of indecomposable modules over group algebras, and recall the notion of source algebras. Then, in Section 3, we formulate Feit’s and Puig’s Conjectures in our category-theoretic language, prove the Reduction Theorem 3.8, and state Theorem 3.9 in order to prove the main Theorem 3.10. Sections 4–6 are then devoted to proving Theorem 3.9 for the alternating groups, the Schur covers of the symmetric and alternating groups, and the Weyl groups appearing in our main theorem, respectively, where in the former two cases we pursue the idea of using self-centralizing Brauer pairs, while for the Weyl groups appearing in our main theorem we are content with looking at vertices directly.

Throughout this article, let p be a prime number, and let F be an algebraically closed field of characteristic p. All groups appearing will be finite and, whenever G is a group, any F G-module is understood to be a finitely generated left module. We assume the reader to be familiar with modular representation theory of finite groups in general, and the standard notation commonly used, as exposed for example in [24] and [32]. For background concerning the representation theory of the symmetric groups and their covering groups, we refer the reader to [17] and [16], respectively.

Acknowledgements. The authors’ research was supported through a Marie Curie Intra-European Fellowship (grant PIEF-GA-2008-219543). In particular the second-named author is grateful for the hospitality of the University of Oxford, where part of the paper has been written. It is a pleasure to thank Burkhard Külshammer for various helpful discussions on the topic.

2 Interior Algebras, Source Algebras, and Vertex-Source Pairs

In this section we introduce a category-theoretic language, to be able to state a precise formulation of Feit’s and Puig’s Conjectures later in this article. The language we use is that of interior algebras, see for example [32, Ch. 10], where we additionally have to allow the group acting to vary.

2.1. A category of interior algebras. We define a category A whose objects are the triples (G, α, A), where G is a finite group, A is a finite-dimensional, associative, unitary F -algebra, and α : G ⟶ A^{×} is a group homomorphism into the group of multiplicative units A^{×} in A. Given objects (G, α, A) and (H, β, B) in A, the morphisms (G, α, A) ⟶ (H, β, B) are the pairs (φ, Φ) where φ : G ⟶ H is a group homomorphism and Φ : A ⟶ B is a homomorphism of F -algebras satisfying

Φ(α(g)a) = β(φ(g))Φ(a) and Φ(aα(g)) = Φ(a)β(φ(g)),  (1)

for all g ∈ G and all a ∈ A. We emphasize that the algebra homomorphism Φ need not be unitary, in general. If it additionally is, that is, if we have Φ(1_{A}) = 1_{B} then the above compatibility condition (1) simplifies to

Φ(α(g)) = β(φ(g)), for all g ∈ G.

Anyway, whenever (φ, Φ) : (G, α, A) ⟶ (H, β, B) and (ψ, Ψ) : (H, β, B) ⟶ (K, γ, C) are morphisms in A, their composition is defined to be (ψ, Ψ) ∘ (φ, Φ) := (ψ ∘ φ, Ψ ∘ Φ), where the compositions of the respective components are the usual compositions of group homomorphisms and algebra homomorphisms, respectively. Hence A is indeed a category, from now on called the category of interior algebras; an object (G, α, A) in A is called an interior G-algebra, and (φ, Φ) : (G, α, A) ⟶ (H, β, B) is called a morphism of interior algebras.

We just remark that for the conjugation automorphisms κ_{a} ∈ Aut(A) and λ_{b} ∈ Aut(B) induced by some a ∈ (A^{α(G)})^{×} and b ∈ (B^{β(H)})^{×}, respectively, we also have the morphism (φ, λ_{b}∘Φ∘κ_{a}) : (G, α, A) ⟶ (H, β, B). This defines an equivalence relation on the morphisms of interior algebras (G, α, A) ⟶ (H, β, B), and the equivalence class

(φ, Φ̂) := {(φ, λ_{b} ∘ Φ ∘ κ_{a}) | a ∈ (A^{α(G)})^{×}, b ∈ (B^{β(H)})^{×}}

is called the associated exomorphism of interior algebras.

By the above definition, (G, α, A) and (H, β, B) are isomorphic in A if and only if there exists a morphism (φ, Φ) : (G, α, A) ⟶ (H, β, B) such that φ is an isomorphism of groups and Φ is an (automatically unitary) isomorphism of algebras. So, in particular, if (G, α, A) is an interior algebra and φ : H ⟶ G is an isomorphism of groups then also (H, α ∘ φ, A) is an interior algebra, and (H, α ∘ φ, A) and (G, α, A) are isomorphic via (φ, id_{A}). Analogously, if (G, α, A) is an interior algebra and if Φ : A ⟶ B is an isomorphism of algebras then (G, Φ ∘ α, B) is also an interior algebra, and (G, α, A) and (G, Φ ∘ α, B) are isomorphic via (id_{G}, Φ).

2.2. An equivalence relation. Let G and H be groups, let M be an F G-module, and let N be an F H-module. Let further α : G ⟶ E_{M}^{×} and β : H ⟶ E_{N}^{×} be the corresponding representations, where E_{M} := End_{F} (M) and E_{N} := End_{F} (N). Then (G, α, E_{M}) and (H, β, E_{N}) are interior algebras.

(a) We say that the pairs (G, M) and (H, N) are equivalent if there are a group isomorphism φ : G ⟶ H and a vector space isomorphism ψ : M ⟶ N such that, for all g ∈ G and all m ∈ M, we have

ψ(α(g) · m) = β(φ(g)) · ψ(m).

This clearly is an equivalence relation on the set of all such pairs.

(b) The case G = H deserves particular attention: pairs (G, M) and (G, N) are equivalent, via (φ, ψ) say, if and only if we have

β(φ(g)) · n = ψ(α(g) · ψ^{−1}(n))

for all g ∈ G and n ∈ N, that is, if and only if M and N are in the same Aut(G)-orbit on the set of isomorphism classes of F G-modules. Moreover, M and N are isomorphic as F G-modules if and only if φ can be chosen to be the identity id_{G}. In particular, there are at most | Aut(G)| isomorphism classes of F G-modules in the equivalence class of (G, M).

Lemma 2.3. We keep the notation of 2.2. Then the pairs (G, M) and (H, N) are equivalent if and only if the associated interior algebras (G, α, E_{M}) and (H, β, E_{N}) are isomorphic in A. Moreover, if G = H then M and N are isomorphic as F G-modules if and only if (G, α, E_{M}) and (G, β, E_{N}) are isomorphic in A via an isomorphism of the form (id_{G},?).

Proof. If (G, M) and (H, N) are equivalent via φ : G ⟶ H and ψ : M ⟶ N then

Ψ : E_{M} ⟶ E_{N}, γ ⟼ ψ ∘ γ ∘ ψ^{−1}

is an isomorphism of algebras, and we have Ψ(α(g)) = ψ ∘ α(g) ∘ ψ^{−1} = β(φ(g)), for all g ∈ G, thus the interior algebras (G, α, E_{M}) and (H, β, E_{N}) are isomorphic in A via (φ, Ψ).

Let, conversely, (G, α, E_{M}) and (H, β, E_{N}) be isomorphic via (φ, Ψ), where φ : G ⟶ H is a group isomorphism and Ψ : E_{M} ⟶ E_{N} is an isomorphism of algebras. Then, letting i ∈ E_{M} be a primitive idempotent, we may assume that M = E_{M}i and, letting j := Ψ(i) ∈ E_{N}, we may similarly assume that N = E_{N}j. Moreover, Ψ(E_{M}i) = Ψ(E_{M})Ψ(i) = E_{N}j shows that ψ := Ψ|_{E_{M} i} : E_{M}i ⟶ E_{N}j is a vector space isomorphism, where for all g ∈ G and γ ∈ E_{M} we have

ψ(α(g) · γi) = Ψ(α(g) · γi) = β(φ(g)) · Ψ(γ)j = β(φ(g)) · ψ(γi),

implying that (G, M) and (H, N) are equivalent via (φ, ψ). This proves the first statement.

The second statement can be found in [32, L. 10.7]. It also follows from the above observations, by recalling that M and N are isomorphic F G-modules if and only if the group isomorphism φ : G ⟶ G inducing an equivalence of pairs can be chosen to be the identity id_{G}.

Lemma 2.4. We keep the notation of 2.2, and let (G, M) and (H, N) be equivalent. Then the equivalence classes of pairs (G, M′) where M′ is an indecomposable direct summand of the F Gmodule M coincide with the equivalence classes of pairs (H, N′) where N′ is an indecomposable direct summand of the F H-module N. In particular, the F G-module M is indecomposable if and only if the F H-module N is.

Proof. Let (G, α, E_{M}) and (H, β, E_{N}) be isomorphic via (φ, Ψ); such an isomorphism exists, by Lemma 2.3. Given an indecomposable direct summand M′ of M, let i ∈ (E_{M})^{α(G)} be the associated primitive idempotent, so that M′ = iM, with associated representation

α′ : G ⟶ E_{iM} = iE_{M}i, g ⟼ iα(g)i.

Hence, for j := Ψ(i) ∈ E_{N} we have

β(φ(g))j = β(φ(g))Ψ(i) = Ψ(α(g)i) = Ψ(iα(g)) = Ψ(i)β(φ(g)) = jβ(φ(g)),

for all g ∈ G. Thus j ∈ (E_{N})^{β(H)} is a primitive idempotent, giving rise to the indecomposable direct summand jN of N, with associated representation

β′ : H ⟶ E_{jN} = jE_{N}j, g ⟼ jβ(g)j.

Moreover, we have an isomorphism of algebras

Ψ′ := Ψ|_{iE_{M} i} : iE_{M}i ⟶ jE_{N}j, ixi ⟼ Ψ(ixi) = jΨ(x)j.

Then we have

Ψ′(α′(g)) = Ψ′(iα(g)i) = jΨ(α(g))j = jβ(φ(g))j = β′(φ(g)),

for all g ∈ G. Thus the interior algebras (G, α′, iE_{M}i) and (H, β′, jE_{N}j) are isomorphic in A via (φ, Ψ′), that is, the pairs (G, iM) and (H, jN) are equivalent.

Remark 2.5. (a) Let (G, α, A) be an interior algebra. For any A-module M with associated representation δ : A ⟶ E_{M} := End_{F} (M) we obtain an F G-module Res_{α}(M) by restriction along α, that is, the associated representation is given as δ ∘ α : G ⟶ E_{M}^{×}. Thus we get a functor

Res_{α} : A-mod ⟶ F G-mod, M ⟼ Res_{α}(M).

Let (H, β, B) be an interior algebra, and let (φ, Φ) : (G, α, A) ⟶ (H, β, B) be a morphism in A. Hence, by restriction along β and φ, respectively, we similarly get functors

Res_{β} : B-mod ⟶ F H-mod and Res_{φ} : F H-mod ⟶ F G-mod.

Moreover, for any B-module N with associated representation γ : B ⟶ E_{N} := End_{F} (N) we obtain an A-module Res_{Φ}(N) := Φ(1_{A})N whose associated representation is given as

A ⟶ End_{F} (Φ(1_{A})N) = Φ(1_{A})E_{N}Φ(1_{A}), x ⟼ Φ(1_{A})γ(Φ(x))Φ(1_{A}).

This gives rise to a functor Res_{Φ} : B-mod ⟶ A-mod.

(b) If additionally Φ is unitary, that is, Φ(1_{A}) = 1_{B} then the representation associated with Res_{Φ}(N) is obtained by restriction along Φ. Moreover, from Φ(α(g)) = β(φ(g)), for all g ∈ G, we infer that we have the following equality of functors

Res_{α} ∘ Res_{Φ} = Res_{φ} ∘ Res_{β} : B-mod ⟶ F G-mod.

In other words, for any B-module N with associated representation γ : B ⟶ E_{N}, we have

(γ ∘ Φ ∘ α)(g) · n = (γ ∘ β ∘ φ)(g) · n,

for all g ∈ G and all n ∈ N. In particular, if φ is an isomorphism then (G, Res_{α}(Res_{Φ}(N))) and (H, Res_{β}(N)) are equivalent via (φ, id_{N}).

Definition 2.6. Let G be a group, and let M be an indecomposable F G-module. Assume that V ⩽ G is a vertex of M and that S is a V -source of M. Then the elements of the equivalence class of the pair (V, S) are called the vertex-source pairs of (G, M).

Proposition 2.7. If G is a group and M is an indecomposable F G-module then the vertexsource pairs of (G, M) are pairwise equivalent. Moreover, if H is a group and N is an indecomposable F H-module such that (G, M) is equivalent to (H, N) then the vertex-source pairs of (G, M) and (H, N) are pairwise equivalent.

Proof. Let V ⩽ G be a vertex, and let S be a V -source of M. Then the set of all vertices of M is given as {^{g}V | g ∈ G} and, for a given g ∈ G, the set of ^{g}V -sources of M is {^{hg}S | h ∈ N_{G}(^{g}V)}. For g ∈ G and h ∈ N_{G}(^{g}V), let

κ : V ⟶ ^{g}V = ^{hg}V, x ⟼ ^{hg}x = hgxg^{−1}h^{−1}

be the associated conjugation homomorphism, and let ψ : S ⟶ ^{hg}S, m ⟼ hg ⊗ m. Then for all x ∈ V and m ∈ S, we have

κ(x) · ψ(m) = (^{hg}x) · (hg ⊗ m) = hg ⊗ (x · m) = ψ(x · m),

hence the pairs (V, S) and (^{g}V,^{hg}S) are equivalent via (κ, ψ). Since every vertex-source pair of (G, M) is equivalent to one of the pairs (^{g}V,^{hg}S), this shows that all vertex-source pairs of (G, M) belong to the same equivalence class.

Moreover, if (G, M) and (H, N) are equivalent via (φ, ψ) then ψ(α(g) · m) = β(φ(g)) · ψ(m) for all g ∈ G and m ∈ M, where α and β are the representations associated with M and N, respectively. From this we infer that φ(V) is a vertex of the F H-module N having ψ(S) as a φ(V)-source, and that (V, S) is equivalent to (φ(V), ψ(S)) via (φ|_{V} , ψ|_{S}).

Remark 2.8. We remark that, given G and an indecomposable F G-module M, specifying a vertex V as a subgroup of G amounts to restricting to those vertex-source pairs of shape (V,?), henceforth only allowing for isomorphisms of the form (id_{V} ,?). The above argument now shows that these vertex-source pairs are given by the F V -modules {^{h}S | h ∈ N_{G}(V)/V C_{G}(V)}, where S is one of the V -sources. Thus we possibly do not obtain the full Aut(V)-orbit of S, but only see its orbit under N_{G}(V)/V C_{G}(V) ⩽ Aut(V), as the following example shows:

Example 2.9. Let p := 2, let G := 𝔖_{6}, and let M := D^{(5,1)} be the natural simple F 𝔖_{6}- module of F -dimension 4. Then, by [23], the vertices of D^{(5,1)} are the Sylow 2-subgroups of 𝔖_{6}. Let P_{6} := P_{4} × P_{2} ≅ D_{8} × C_{2}, where P_{4} = ⟨(1, 2), (1, 3)(2, 4)⟩ and P_{2} := ⟨(5, 6)⟩. Then P_{6} is a Sylow 2-subgroup of 𝔖_{6} and, by [23], the restriction S := Res_{P_{6}}^{𝔖_{6}}(D^{(5,1)}) is indecomposable, thus is the unique P_{6}-source of D^{(5,1)}. Since N_{𝔖_{6}}(P_{6}) = P_{6}, in view of Proposition 2.7 we have to show that the Aut(P_{6})-orbit of S consists of more than a single isomorphism class of F P_{6}-modules.

Let φ ∈ Aut(P_{6}) be the involutory automorphism given by fixing P_{4} = ⟨(1, 2), (1, 3)(2, 4)⟩ and mapping (5, 6) to (1, 2)(3, 4)(5, 6). Since Res_{𝔖_{4}}^{𝔖_{6}}(D^{(5,1)}) is the natural permutation F 𝔖_{4}- module, there is an F -basis of S with respect to which the elements of P_{4} are mapped to the associated permutation matrices, while

(5, 6) ⟼ [\begin{matrix}. & 1 & 1 & 1 \\ 1 & . & 1 & 1 \\ 1 & 1 & . & 1 \\ 1 & 1 & 1 & . \end{matrix}] and φ((5, 6)) = (1, 2)(3, 4)(5, 6) ⟼ [\begin{matrix}1 & 1 & . & 1 \\ 1 & 1 & 1 & . \\ . & 1 & 1 & 1 \\ 1 & . & 1 & 1 \end{matrix}] .

It can be checked, for example with the help of the computer algebra system MAGMA [2], that the F P_{6}-modules S and ^{φ}S are not isomorphic.

2.10. Source algebras. Let G be a group, and let B be a block of F G. Let further P be a p-group such that the defect groups of B are isomorphic to P. Then we have an embedding of groups f : P ⟶ G such that f(P) is a defect group of B. The block B is an indecomposable F [G×G]-module with vertex Δf(P) and trivial source. Moreover, there is an indecomposable direct summand M of Res_{f(P)×G}^{G×G}(B) with vertex Δf(P), where M is unique up to isomorphism and conjugation in N_{G}(f(P)).

So there is a primitive idempotent i ∈ B^{f(P)} such that M = iB, where i is unique up to taking associates in B^{f(P)} and conjugates under the action of N_{G}(f(P)). We call i and M, respectively, a source idempotent and a source module of B, respectively; as a general reference see [32, Ch. 38]. The embedding f gives rise to the group homomorphism

α_{f,i} : P ⟶ (iBi)^{×}, g ⟼ if(g)i,

which turns (P, α_{f,i}, iBi) into an interior P -algebra. Note that α_{f,i} is injective, by [32, Exc. 38.2]. We call an interior algebra that is isomorphic to (P, α_{f,i}, iBi) in A a source algebra of B.

Proposition 2.11. Let G be a group, and let B be a block of F G. Then the source algebras of B are pairwise isomorphic in A. Moreover, if ψ : G ⟶ G′ is a group isomorphism and if B′ := ψ(B) is the block of F G′ obtained by extending ψ to F G then the source algebras of B and B′ are pairwise isomorphic in A.

Proof. Let P be a p-group isomorphic to the defect groups of B, let f : P ⟶ G be an embedding such that f(P) ⩽ G is a defect group of B, and let α_{f,i} : P ⟶ (iBi)^{×}, g ⟼ if(g)i be the associated group homomorphism, where i ∈ B^{f(P)} is a source idempotent. Moreover, let φ : Q ⟶ P be a group isomorphism, and let f′ : Q ⟶ G be an embedding such that f′(Q) ⩽ G is a defect group of B, with associated group homomorphism α_{f′,j} : Q ⟶ (jBj)^{×}, where j ∈ B^{f′(Q)} is a source idempotent. Note that, hence, there is some h ∈ G such that f′(Q) = ^{h}f(P), and the idempotents ^{h}i = hih^{−1} and j are associate in B^{f′(Q)}. To show that (P, α_{f,i}, iBi) is isomorphic to (Q, α_{f′,j}, jBj) in A we proceed in three steps:

(i) We first consider the particular case where Q = P, φ = id, and f′ = f, and let j ∈ B^{f(P)} be a source idempotent that is associate to i. Then there is some a ∈ (B^{f(P)})^{×} such that j = ^{a}i = aia^{−1}, and κ : iBi ⟶ jBj, x = ixi ⟼ ^{a}(ixi) = j(^{a}x)j is an isomorphism of algebras. Hence, we obtain the group homomorphism

κ ∘ α_{f,i} : P ⟶ (jBj)^{×}, g ⟼ ^{a}(if(g)i) = j(af(g)a^{−1})j = jf(g)j,

that is, κ ∘ α_{f,i} = α_{f′,j}. Moreover, κ(α_{f,i}(g)) = ^{a}(if(g)i) = jf(g)j = α_{f′,j}(g), for all g ∈ P, shows that the interior P -algebras (P, α_{f,i}, iBi) and (P, α_{f′,j}, jBj) are isomorphic via (id_{P} , κ).

(ii) Next, let still φ = id, let h ∈ G be arbitrary with associated conjugation automorphism G ⟶ G, g ⟼ ^{h}g = hgh^{−1}, and let f′ : P ⟶ G, g ⟼ hf(g)h^{−1} be the associated conjugated embedding. Since j ∈ B^{f′(P)} is a source idempotent, by (i) we may assume that j = ^{h}i. This yields the isomorphism of algebras γ : iBi ⟶ jBj, x = ixi ⟼ ^{h}(ixi) = j(^{h}x)j and, associated to f′, the group homomorphism

α_{f′,j} = γ ∘ α_{f,i} : P ⟶ (jBj)^{×}, g ⟼ ^{h}(if(g)i) = j(hf(g)h^{−1})j.

Moreover, γ(α_{f,i}(g)) = ^{h}(if(g)i) = j(hf(g)h^{−1})j = α_{f′,j}(g), for all g ∈ P, shows that the interior P -algebras (P, α_{f,i}, iBi) and (P, α_{f′,j}, jBj) are isomorphic via (id_{P} , γ).

(iii) We finally consider the general case of a group isomorphism φ : Q ⟶ P and an embedding f′ : Q ⟶ G as above. By (ii) we may assume that f′(Q) = f(P). Hence, there is a group automorphism ρ : f(P) ⟶ f(P) such that f ∘ φ = ρ ∘ f′. Thus, replacing φ by φ′ := (f^{−1} ∘ ρ^{−1} ∘ f) ∘ φ we get f ∘ φ′ = f′. So we may assume that f ∘ φ = f′. Moreover, by (ii) we may assume that j = i ∈ B^{f(P)}. Hence we have the associated group homomorphisms

α_{f,i} : P ⟶ (iBi)^{×}, g ⟼ if(g)i and α_{f′,i} : Q ⟶ (iBi)^{×}, h ⟼ if′(h)i = if(φ(h))i.

Moreover, α_{f′,i}(h) = if(φ(h))i = α_{f,i}(φ(h)), for all h ∈ Q, which shows that the interior algebras (Q, α_{f′,i}, iBi) and (P, α_{f,i}, iBi) are isomorphic via (φ, id_{iBi}). This proves the first statement.

Let ψ : G ⟶ G′ be a group isomorphism, which extends to an F -algebra isomorphism ψ : F G ⟶ F G′, and let B′ := ψ(B). Then, letting f′ := ψ ∘ f : P ⟶ G′, we conclude that f′(P) is a defect group of B′, and i′ := ψ(i) ∈ (B′)^{f′(P)} is a source idempotent of B′. Thus we get (P, α_{f′,i′}, i′B′i′) as a source algebra associated with B′, where

α_{f′,i′} = ψ ∘ α_{f,i} : P ⟶ (i′B′i′)^{×} = ψ(iBi)^{×}, g ⟼ i′f′(g)i′ = ψ(if(g)i).

Then we have ψ(α_{f,i}(g)) = α_{f′,i′}(g), for all g ∈ P, that is, (P, α_{f,i}, iBi) and (P, α_{f′,i′}, i′B′i′) are isomorphic in A via (id_{P} , ψ|_{iBi}). Since, by what we have shown above, every source algebra of B is A-isomorphic to (P, α_{f,i}, iBi) and every source algebra of B′ is A-isomorphic to (P, α_{f′,i′}, i′B′i′), this completes the proof of the proposition.

Remark 2.12. We remark that specifying a defect group P of the block B as a subgroup of G amounts to keeping the embedding f : P ⟶ G fixed, and thus to restricting to the source algebras of shape (P, α_{f,i}, iBi), for some i ∈ B^{f(P)}, and to isomorphisms of the form (id_{P} , Φ). The above argument now shows that the isomorphisms Φ realized in G are precisely those of the form Φ = γ ∘κ, where κ : iBi ⟶ iBi, x ⟼ ^{a}x is the inner automorphism of iBi induced by some a ∈ (iBi)^{×}, and where γ : iBi ⟶ jBj, x ⟼ ^{h}x is induced by the conjugation automorphism G ⟶ G afforded by some h ∈ N_{G}(f(P)), where j := ^{h}i. Hence possibly not all elements of the isomorphism class of (P, α_{f,i}, iBi) are realized as source algebras in this strict sense, as the following example shows:

Example 2.13. Let p := 3 and let G = P = ⟨z⟩ ≅ C_{3} be the cyclic group of order 3; hence F P is a local F -algebra. Letting f = id_{P} : P ⟶ P, the source algebra of B = F P (by necessarily taking i := 1_{F P}) is given as (P, α_{id_{P}} , F P). Thus (P, α_{id_{P}} , F P) is the only interior algebra in its isomorphism class that is actually realized in the above strict sense.

We describe all interior algebras (P,?, F P) isomorphic to (P, α_{id_{P}} , F P) in A, that is, all source algebras of F P in the sense of 2.10: note first that in this particular case any group automorphism of P can be extended uniquely to an algebra automorphism of F P, so that any isomorphism (P, α_{id_{P}} , F P) ⟶ (P,?, F P) is of the form (id_{P} , Φ), where Φ ∈ Aut(F P) is an algebra automorphism of F P.

Letting y := 1−z ∈ F P, hence y^{2} = 1+z +z^{2}, the F -basis {1, y, y^{2}} is adjusted to the radical series F P = J^{0}(F P) > J^{1}(F P) > J^{2}(F P) > J^{3}(F P) = {0} of F P, and it can be checked, for example with the help of the computer algebra system GAP [13], that, with respect to this basis, we have

Aut(F P) ≅ \{Φ_{a,b} := [\begin{matrix}1 & . & . \\ . & a & . \\ . & b & a^{2} \end{matrix}] ∈ GL_{3}(F) | a ∈ F^{×}, b ∈ F \} .

Hence we have Φ_{a,b}(z) = (1 − a − b) + (a − b)z − bz^{2}; in particular, we have id_{F P} = Φ_{1,0}, and the non-trivial automorphism of P, mapping z ⟼ z^{2}, extends to Φ_{−1,−1}. Thus the interior algebras looked for are given as (P, Φ_{a,b}∘α_{id_{P}} , F P), where (P, α_{id_{P}} , F P) = (P, Φ_{1,0}∘α_{id_{P}} , F P).

Finally note that this does not encompass all possible embeddings P ⟶ (F P)^{×}: since −z−z^{2} ∈ (F P)^{×} has order 3, there is an embedding of groups β : P ⟶ (F P)^{×}, z ⟼ −z−z^{2}, which extends to the unitary algebra endomorphism Φ_{0,1} of F P, which is not an automorphism. Anyway, this gives rise to the interior algebra (P, β, F P), which is not isomorphic to (P, α_{id_{P}} , F P) in A, hence is not a source algebra of F P.

3 Reducing Feit’s Conjecture to Puig’s Conjecture

We have now prepared the language to state Feit’s Conjecture on sources of simple modules over group algebras as well as Puig’s Conjecture on source algebras of blocks precisely. We will then prove the reduction theorem relating these conjectures, which we will use extensively throughout this paper.

3.1. Source algebras vs vertex-source pairs. (a) The relation between source algebras, in the sense of 2.10, and vertex-source pairs, in the sense of Definition 2.6, is given as follows: let G be a group, let B be a block of F G, let f : P ⟶ G be an embedding such that f(P) ⩽ G is a defect group of B, and let (P, α_{f,i}, iBi) be a source algebra of B. Then, by [32, Prop. 38.2], we have a Morita equivalence between the algebras B and iBi, in the language of Remark 2.5 given by the restriction functor

Res_{Ψ} : B-mod ⟶ iBi-mod

with respect to the natural embedding of algebras Ψ : iBi ⟶ B.

Suppose that M is an indecomposable F G-module belonging to the block B. Then the Morita correspondent of M in iBi is Res_{Ψ}(M) = iM. Moreover, restricting iM along α_{f,i}, we get an F P -module Res_{α_{f,i}}(iM), which is, in general, decomposable. By [32, Prop. 38.3], the vertexsource pairs of (G, M) are precisely the vertex-source pairs (Q,?) of the indecomposable direct summands of the F P -module Res_{α_{f,i}}(iM) such that |Q| is maximal.

(b) We show that proceeding like this to determine the vertex-source pairs of (G, M) is independent of the particular choice of a source algebra: let (D, α, A) be any source algebra of B. Hence, by Proposition 2.11, there is an isomorphism (φ, Φ) : (D, α, A) ⟶ (P, α_{f,i}, iBi) in A. By Remark 2.5, we have an equivalence

Res_{Φ} : iBi-mod ⟶ A-mod.

Letting N := Res_{Φ}(iM), we infer that the pairs (P, Res_{α_{f,i}}(iM)) and (D, Res_{α}(N)) are equivalent. Hence, by Lemma 2.4, the equivalence classes of pairs (P, M′) where M′ is a direct summand of the F P -module iM coincide with the equivalence classes of pairs (D, N′) where N′ is a direct summand of the F D-module N. Moreover, if the pairs (P, M′) and (D, N′) are equivalent then, by Proposition 2.7, their vertex-source pairs are pairwise equivalent.

In conclusion, to find the vertex-source pairs of (G, M), we may go over from (P, α_{f,i}, iBi) to an arbitrary source algebra (D, α, A) by considering the module N instead, and check the above maximality condition by varying over the pairs (D, N′).

Definition 3.2. Let G be a class of groups, that is, a set of isomorphism classes of (finite) groups, and let Q be a p-group.

(a) We define V_{G}(Q) to be the set of all equivalence classes of pairs (Q, L) where L is an indecomposable F Q-module such that (Q, L) is a vertex-source pair of some pair (G, M), where G is a group in G and M is a simple F G-module. In the case that G is the class of all (finite) groups, we also write V(Q) rather than V_{G}(Q).

(b) We say that G has the vertex-bounded-defect property with respect to Q if there is an integer c_{G}(Q) such that, for every pair (Q, L) in V_{G}(Q) and for every pair (G, M) consisting of a group G in G and a simple F G-Module M having (Q, L) as a vertex-source pair, M belongs to a block of F G having defect groups of order at most c_{G}(Q).

Remark 3.3. The vertex-bounded-defect property, by [10], holds in the case where Q is cyclic, with c(Q) = |Q|, including the case Q = {1}, covering all blocks of finite representation type. But it does indeed not hold in general, where, in particular, in the realm of blocks of tame representation type there are prominent counterexamples:

Let p = 2. For the class {PSL_{2}(q) | q ≡ 1 (mod 4)}, the Sylow 2-subgroups are isomorphic to the dihedral group D_{(q−1)_{2}}, where (q − 1)_{2} denotes the 2-part of q − 1. Also, there is a simple F [PSL_{2}(q)]-module in the principal block having dimension (q − 1)/2 and whose vertices, by [9], are isomorphic to the Klein four-group V_{4} ≅ C_{2} × C_{2}. Moreover, for the class {SL_{2}(q) | q ≡ 1 (mod 4)}, consisting of the Schur covering groups of groups above, the Sylow 2-subgroups are isomorphic to the generalized quaternion group 𝔔_{2(q−1)_{2}}, and the inflations of the above simple F [PSL_{2}(q)]-modules to F [SL_{2}(q)] have vertices isomorphic to the quaternion group 𝔔_{8}. Finally, for the class {GU_{2}(q) | q ≡ 1 (mod 4)} the Sylow 2-subgroups are isomorphic to the semi-dihedral group SD_{4(q−1)_{2}}, and the identification SL_{2}(q) ≅ SU_{2}(q) shows that there is a simple F [GU_{2}(q)]-module in the principal block having dimension q − 1 whose vertices are isomorphic to V_{4}. (Alternatively, for the class {PSL_{3}(q) | q ≡ 3 (mod 4)}, the Sylow 2-subgroups are isomorphic to the semi-dihedral group SD_{2(q+1)_{2}}, and there is a simple F [PSL_{3}(q)]-module in the principal block having dimension q(q + 1) whose vertices, by [8], are isomorphic to V_{4}.)

From these cases we also obtain blocks of wild representation type violating the vertexbounded-defect property, for example by forming direct products. Hence the question arises for which defect groups P or classes of groups G one might expect the vertex-bounded-defect property to hold. In particular, the following is in [33] attributed to Puig:

Question 3.4. If p is odd, does then G always have the vertex-bounded-defect property with respect to Q?

We can now state Feit’s and Puig’s Conjectures, and prove the reduction theorem.

Conjecture 3.5 (Feit [11]). Let G be a class of groups (which might, in particular, be the class of all groups), let Q be a p-group, and let V_{G}(Q) denote the set of equivalence classes of vertex-source pairs introduced in Definition 3.2. Then V_{G}(Q) is finite.

In consequence of Lemma 2.3, we can reformulate Feit’s Conjecture equivalently also in the following way:

Conjecture 3.6. Let G be a class of groups, and let Q be a p-group. Then there are, up to isomorphism in A, only finitely many interior algebras (Q, α, E_{L}), where E_{L} = End_{F} (L) for an indecomposable F Q-module L with corresponding representation α : Q ⟶ E_{L}^{×}, such that (Q, L) is a vertex-source pair of some pair (G, M), where G is a group in G and M is a simple F G-module.

Conjecture 3.7 (Puig). Let G be a class of groups (which might, in particular, be the class of all groups), and let P be a p-group. Then there are only finitely many A-isomorphism classes of interior algebras of p-blocks of groups in G whose defect groups are isomorphic to P.

As for the origin of this conjecture, see [32, Conj. 38.5], and the comment on [32, p. 340].

Theorem 3.8. Let G be a class of groups satisfying the vertex-bounded-defect property with respect to any p-group. Suppose that Puig’s Conjecture 3.7 holds true for G. Then Feit’s Conjecture 3.5 is true for G as well.

Proof. Let Q be a p-group, and let c_{G}(Q) be the integer appearing in Definition 3.2. Then there are finitely many (mutually non-isomorphic) p-groups R_{1}, …, R_{n} such that, whenever G ∈ G and M is a simple F G-module with vertex isomorphic to Q, the defect groups of the block containing M are isomorphic to one of the groups in {R_{1}, …, R_{n}}.

Let k ∈ {1, …, n}. Then, by Puig’s Conjecture, there are, up to isomorphism in A, only finitely many interior R_{k}-algebras occurring as source algebras of p-blocks for groups in G with defect groups isomorphic to R_{k}. Denote by {(R_{k}, α_{k,1}, A_{k,1}), …, (R_{k}, α_{k,l_{k}}, A_{k,l_{k}})} a transversal for these isomorphism classes.

Let further r ∈ {1, …, l_{k}}, and choose representatives {M_{k,r,1}, …, M_{k,r,d_{k,r}}} for the isomorphism classes of simple A_{k,r}-modules. Via restriction along α_{k,r} we get F R_{k}-modules Res_{α_{k,r}}(M_{k,r,1}), …, Res_{α_{k,r}}(M_{k,r,d_{k,r}}). For each i ∈ {1, …, d_{k,r}} we determine a vertex-source pair (Q_{k,r,i}, S_{k,r,i}) of an indecomposable direct summand of Res_{α_{k,r}}(M_{k,r,i}) such that |Q_{k,r,i}| is maximal. So this gives rise to the finite set of pairs

V := ⋃_{k=1}^{n} ⋃_{r=1}^{l_{k}} ⋃_{i=1}^{d_{k,r}}{(Q_{k,r,i}, S_{k,r,i})}.

Consequently, by [32, Prop. 38.3], any vertex-source pair of some pair (G, M), with G ∈ G and M a simple F G-module, is equivalent to one of the pairs in the finite set V. Hence V_{G}(Q) is finite, proving Feit’s Conjecture.

To prove Feit’s Conjecture for the classes of groups listed in the main theorem, we are going to apply Theorem 3.8. In order to do so, we will show that each of these classes satisfies the vertex-bounded-defect property with respect to any p-group; this will be done by giving explicit bounds as in the next theorem, whose proof will be broken up into several steps in subsequent sections.

Theorem 3.9. Let Q be a p-group, let G be a finite group possessing a simple F G-module M belonging to a block with defect group isomorphic to P, and having vertices isomorphic to Q. Then the following hold:

(a) If G = 𝔖_{n} then |P | ⩽ |Q|!.

(b) If G = 𝔄_{n} and p = 2 then |P | ⩽ (|Q| + 2)!/2.

(c) If G ∈ {𝔖̃_{n}, 𝔖̂_{n}} and p ⩾ 3 then |P | ⩽ |Q|!.

(d) If G = 𝔅_{n} and p ⩾ 3 then |P | ⩽ |Q|!.

(e) If G = 𝔅_{n} and p = 2 then |P | ⩽ |Q| · log_{2}(|Q|)!.

(f) If G = 𝔇_{n} and p = 2 then |P | ⩽ |Q| · (log_{2}(|Q|) + 1)!.

Proof. (a) follows from [5, Thm. 5.1].

(b) is proved in Proposition 4.7.

(c) is proved in Proposition 5.2.

(d) is proved in Proposition 6.4.

(e) and (f) are proved in 6.2.

Using Theorems 3.8 and 3.9, we are now in a position to prove our main result:

Theorem 3.10. Feit’s Conjecture holds for the following classes of groups:

{𝔖_{n}}_{n⩾1}, {𝔄_{n}}_{n⩾1}, {𝔖̃_{n}}_{n⩾1}, {𝔖̂_{n}}_{n⩾1}, {𝔄̃_{n}}_{n⩾1}, {𝔅_{n}}_{n⩾2}, {𝔇_{n}}_{n⩾4}.

Proof. (i) Let G = {𝔖_{n}}_{n⩾1}. Then Puig’s Conjecture holds for G, by work of Puig [29] and Scopes [31]. Moreover, G has the vertex-bounded-defect property with respect to any p-group, by Theorem 3.9(a). Hence Feit’s Conjecture holds, by Theorem 3.8.

(ii) Let G = {𝔄_{n}}_{n⩾1}. Suppose first that p ⩾ 3, and let E be a simple F 𝔄_{n}-module with vertex-source pair (Q, L). Then there is a simple F 𝔖_{n}-module D such that E | Res_{𝔄_{n}}^{𝔖_{n}}(D). Furthermore, (Q, L) is also a vertex-source pair of D. Hence we have V_{G}(Q) ⊆ V_{{𝔖_{n}}}(Q), and we are done using (i).

Let now p = 2. Then Puig’s Conjecture holds for G, by work of Kessar [20]. Moreover, G has the vertex-bounded-defect property with respect to any p-group, by Theorem 3.9(b). Hence Feit’s Conjecture holds, by Theorem 3.8.

(iii) Let G = {𝔖̃_{n}}_{n⩾1}, where we may argue identically for G = {𝔖̂_{n}}_{n⩾1}. Suppose first that p ⩾ 3. Then Puig’s Conjecture holds for G, by work of Kessar [18]. Moreover, G has the vertex-bounded-defect property with respect to any p-group, by Theorem 3.9(c). Hence Feit’s Conjecture holds, by Theorem 3.8.

Let now p = 2, and let D be a simple F 𝔖̃_{n}-module. Since Z := ⟨z⟩ ⩽ Z(𝔖̃_{n}), in the notation of 5.1, is a normal 2-subgroup of 𝔖̃_{n}, it acts trivially on D. Thus there is a simple F 𝔖_{n}- module \overline{D} such that D = Inf𝔖̃_{n} ; Z(\overline{D}), where Inf denotes the inflation from F 𝔖_{n}-modules to F 𝔖̃_{n}-modules via the normal subgroup Z ⊴ 𝔖̃_{n}. If (Q, L) is a vertex-source pair of D then Z ⩽ Q and \overline{Q} := Q/Z is a vertex of \overline{D}. Moreover, there is an indecomposable F \overline{Q}-module \overline{L} such that L ≅ Inf_{Z}^{Q}(\overline{L}) and such that (\overline{Q}, \overline{L}) is a vertex-source pair of \overline{D}, see [22, Prop. 2.1] and [15, Prop. 2]. Hence we have V_{G}(Q) ⊆ V_{\{𝔖_{n}\}}(\overline{Q}), and we are done by (i).

(iv) Let G = {𝔄̃_{n}}_{n⩾1}. Letting again first p ⩾ 3, we may argue as in (ii) to show that V_{G}(Q) ⊆ V_{{𝔖̃_{n}}}(Q), and we are done using (iii). Moreover, letting p = 2, since Z ⩽ 𝔄̃_{n}, again using the notation of 5.1, we may argue as in (iii) to show that V_{G}(Q) ⊆ V_{\{𝔄_{n}\}}(\overline{Q}), and we are done using (ii).

(v) Let G = {𝔅_{n}}_{n⩾2}. Then Puig’s Conjecture holds for G, by work of Kessar [19]. Moreover, G has the vertex-bounded-defect property with respect to any p-group, by Theorem 3.9(d) and (e). Hence Feit’s Conjecture holds, by Theorem 3.8.

(vi) Let G = {𝔇_{n}}_{n⩾4}. Again suppose first that p ⩾ 3. Then we may argue as in (ii) to show that V_{G}(Q) ⊆ V_{{𝔅_{n}}}(Q), and we are done using (v). Moreover, letting p = 2, Puig’s Conjecture holds for G, by work of Kessar [19], and G has the vertex-bounded-defect property by Theorem 3.9(f). Hence Feit’s Conjecture holds, by Theorem 3.8.

Remark 3.11. We remark that the list of groups in Theorem 3.10 in particular encompasses all infinite series of real reflection groups, except the groups of type I_{2}(m), that is, the dihedral groups D_{2m}, where m ⩾ 3. We give a direct proof that Feit’s Conjecture holds for G = {D_{2m}}_{m⩾3} as well; note that, since D_{2m} is soluble, this also follows from much more general work of Puig [28], [30]:

Let first p be odd. Then D_{2m} ≅ C_{m} : C_{2} has a normal cyclic Sylow p-subgroup C_{m_{p}}, where m_{p} denotes the p-part of m. Hence, by [10], any simple F D_{2m}-module has the normal subgroup C_{m_{p}} as its vertex, and thus is a trivial-source module. Hence Feit’s Conjecture holds for G. Note that, by [32, Thm. 45.12], the source algebras of the blocks in question are isomorphic to F C_{m_{p}} or F [C_{m_{p}} : C_{2}] ≅ F D_{2m_{p}} as interior C_{m_{p}}-algebras, thus Puig’s Conjecture holds for G as well.

Let now p = 2, and let D be a simple F D_{2m}-module. Then there are two cases: if D is relatively C_{m}-projective then D has a normal subgroup C_{m_{2}} as its vertex, and thus is a trivial-source module. If D is not relatively C_{m}-projective then its restriction to C_{m} ⊴ D_{2m} is simple, hence one-dimensional, implying again that D is a trivial-source module. Hence Feit’s Conjecture holds for G. Note that, since D_{2m} is 2-nilpotent, by [32, Prop. 49.13] the blocks in question are nilpotent, hence, by Puig’s Theorem [32, Thm. 50.6], their source algebras are isomorphic to End_{F} (iD) ⊗_{F} F P, where i denotes a source idempotent, and the defect groups in the two cases are P = C_{m_{2}} and P = D_{2m_{2}}, respectively; thus, since D is a trivial-source module, we, moreover, conclude that iD is the trivial F P -module, hence the source algebras are isomorphic to F P as interior P -algebras, so that Puig’s Conjecture holds for G as well.

Thus it remains to prove Theorem 3.9. To do so, we will often argue along the lines of [5], where Theorem 3.9(a) has already been established. The key to this line of reasoning is the following:

Remark 3.12. By a Brauer pair of a group G we understand a pair (P, b) where P is a p-subgroup of G and b is a block of F [P C_{G}(P)]. Recall that the Brauer correspondent b^{G}, a block of F G, is defined, and if B = b^{G} then we call (P, b) a Brauer B-pair. Moreover, in the case that P is a defect group of the block b we call (P, b) a self-centralizing Brauer pair.

Let now G be a class of groups, and let Q be a p-group. We say that G has the stronglybounded-defect property with respect to Q if there is an integer d_{G}(Q) such that, for every group G in G, the Brauer correspondent (b_{Q})^{G} of any self-centralizing Brauer pair (Q, b_{Q}) of G has defect groups of order at most d_{G}(Q).

By Knörr’s Theorem [21], given a block B of G, a self-centralizing Brauer B-pair exists, in particular, in the case where Q is a vertex of some simple F G-module M belonging to the block B. Hence to prove the vertex-bounded-defect property of G with respect to a pgroup Q, it suffices to show the strongly-bounded-defect property of G with respect to Q, and we infer c_{G}(Q) ⩽ d_{G}(Q). We remark that the converse of Knörr’s Theorem does not hold, see for example Example 4.8, but, to the authors’ knowledge, there are no general results known towards a characterization of those self-centralizing Brauer pairs whose first component actually occurs as a vertex of some simple module.

Actually, we prove the strongly-bounded-defect property in the Cases (a)–(c) of Theorem 3.9, while for the Cases (d)–(f) we are content with the weaker vertex-bounded-defect property.

Before we proceed, we give a lemma needed later, relating Brauer correspondence to covering of blocks. It should be well known, but we have not been able to find a suitable reference.

Lemma 3.13. Let G be a finite group, and let H ⊴ G. Moreover, let Q ⩽ H be a p-subgroup, let (Q, b) be a Brauer pair of H, that is, b is a block of F [QC_{H}(Q)], and let b̃ be a block of F [QC_{G}(Q)] covering b. Then the Brauer correspondent b̃^{G} of b̃ in G covers the Brauer correspondent b^{H} of b in H.

Proof. By Passman’s Theorem [24, Thm. 5.5.5], we have to show that

ωb̃^{G}(h^{G+}) = ω_{b^{H}} (h^{G+}) for all h ∈ H,

where the ω’s are the associated central characters, h^{G} is the G-conjugacy class of h ∈ H, and M^{+} denotes the sum over any subset M ⊆ G. By definition of the Brauer correspondence, and by Passman’s Theorem again, for all h ∈ H, we have

ω_{b^{H}} (h^{G+})=ω_{b}((h^{G} ∩ QC_{H}(Q))^{+})=ω_{b}((h^{G} ∩ QC_{G}(Q))^{+})=ωb̃((h^{G} ∩ QC_{G}(Q))^{+})=ωb̃^{G}(h^{G+}),

proving the lemma.

4 The Alternating Groups 𝔄_{n}

We proceed to prove the bound given in Theorem 3.9 for the alternating groups. We begin by fixing our notation for the Sylow p-subgroups of the symmetric and alternating groups, respectively; for later use we do this for arbitrary p. Then we focus on the case p = 2, collect the necessary facts about the self-centralizing Brauer pairs of the alternating groups, and use this to finally prove the desired bound.

4.1. Sylow p-subgroups. (a) We will use the following convention for denoting the Sylow p-subgroups of 𝔖_{n} and 𝔄_{n}, respectively. Let 𝔖_{n} act on the set {1, …, n}. Suppose first that n = p^{m}, for some m ∈ ℕ. Moreover, let C_{p} := ⟨(1, 2, …, p)⟩, and set P_{1} := 1, P_{p} := C_{p} and P_{p^{i+1}} := P_{p^{i}} ≀ C_{p} = {(x_{1}, x_{2}, …, x_{p}; σ) | x_{1}, …, x_{p} ∈ P_{p^{i}}, σ ∈ C_{p}} for i ⩾ 1. As usual, for any i ∈ ℕ_{0}, we view P_{p^{i}} as a subgroup of 𝔖_{p^{i}} in the obvious way. Then, by [17, 4.1.22, 4.1.24], P_{p^{m}} is a Sylow p-subgroup of 𝔖_{p^{m}}, and is generated by the following elements, where j = 1, …, m:

g_{j} := ∏_{k=1}^{p^{j−1}}(k, k + p^{j−1}, k + 2p^{j−1}, …, k + (p − 1)p^{j−1}).  (2)

For instance, if p = 2 then P_{8} is generated by g_{1} = (1, 2), g_{2} = (1, 3)(2, 4), and g_{3} = (1, 5)(2, 6)(3, 7)(4, 8).

(b) Next let n ∈ ℕ be divisible by p, with p-adic expansion n = ∑_{j=1}^{s} α_{j}p^{i_{j}}, for some s ⩾ 1, i_{1} > … > i_{s} ⩾ 1, and 1 ⩽ α_{j} ⩽ p − 1 for j = 1, …, s. By [17, 4.1.22, 4.1.24], P_{n} := ∏_{j=1}^{s} ∏_{l_{j}=1}^{α_{j}} P_{p^{i_{j}} ,l_{j}} is then a Sylow p-subgroup of 𝔖_{n}. Here, the direct factor P_{p^{i_{1}},1} is acting on {1, …, p^{i_{1}}}, P_{p^{i_{1}},2} is acting on {p^{i_{1}} + 1, …, 2p^{i_{1}}}, and so on. If, finally, n ⩾ p + 1 is not divisible by p then we set P_{n} := P_{r} where r < n is maximal with p | r. So, in any case, P_{n} is a Sylow p-subgroup of 𝔖_{n}.

(c) We will examine the case p = 2 in more detail, as this will be of particular importance for our subsequent arguments. As above, suppose that n is even, with 2-adic expansion n = ∑_{j=1}^{s} 2^{i_{j}}, for some s ⩾ 2 and i_{1} > i_{2} > … > i_{s} ⩾ 1. Letting n_{j} := 2^{i_{j}}, we get P_{n} = ∏_{j=1}^{s} P_{n_{j}}, where P_{n_{j}} is understood to be acting on the set

Ω_{j} := {(∑_{l=1}^{j−1} n_{l}) + 1, …, ∑_{l=1}^{j} n_{l}} ,

for j = 1, …, s. The corresponding generating set for P_{n_{j}} given by (2) will be denoted by {g_{1,j}, …, g_{i_{j},j}}, for j = 1, …, s. So if, for instance, n = 14 = 8 + 4 + 2 then P_{n} = P_{14} is generated by g_{1,1} = (1, 2), g_{2,1} = (1, 3)(2, 4), g_{3,1} = (1, 5)(2, 6)(3, 7)(4, 8), g_{1,2} = (9, 10), g_{2,2} = (9, 11)(10, 12), and g_{1,3} = (13, 14).

(d) We now set Q_{n} := P_{n} ∩ 𝔄_{n}, so that Q_{n} is a Sylow p-subgroup of the alternating group 𝔄_{n}. If p > 2 then clearly Q_{n} = P_{n}. Thus, suppose again that p = 2. If n = 2 then Q_{n} = Q_{2} = 1. If n = 2^{m}, for some m ⩾ 2, then, by (2), we obtain the following generators for Q_{n}:

h_{1} := (1, 2)(2^{m−1} + 1, 2^{m−1} + 2); h_{j} := g_{j}, for j = 2, …, m.  (3)

For clearly Q := ⟨h_{1}, …, h_{m}⟩ ⩽ Q_{n}, and Q⟨(1, 2)⟩ = ⟨(1, 2)⟩Q = P_{n}. Thus Q = Q_{n}.

If n > 4 is even but not a power of 2 then we again consider the 2-adic expansion n = ∑_{j=1}^{s} 2^{i_{j}}, for some s ⩾ 2 and some i_{1} > … > i_{s} ⩾ 1. Then the following elements generate Q_{n}:

h_{1,j} := g_{1,s}g_{1,j}, for j = 1, …, s − 1; h_{k,j} := g_{k,j}, for j = 1, …, s and k = 2, …, i_{j}. (4)

Namely, these elements generate a subgroup Q of Q_{n} such that Q⟨g_{1,s}⟩ = ⟨g_{1,s}⟩Q = P_{n}. For instance, Q_{14} = P_{14} ∩ 𝔄_{14} = (P_{8} × P_{4} × P_{2}) ∩ 𝔄_{14} is generated by the elements h_{1,1} = (1, 2)(13, 14), h_{1,2} = (9, 10)(13, 14), h_{2,1} = (1, 3)(2, 4), h_{3,1} = (1, 5)(2, 6)(3, 7)(4, 8), and h_{2,2} = (9, 11)(10, 12).

For the remainder of this section, let p = 2.

Proposition 4.2. Let n = ∑_{j=1}^{s} 2^{i_{j}} ⩾ 2 be the 2-adic expansion of n, where s ∈ ℕ and i_{1} > … > i_{s} ⩾ 1, and let again n_{j} := 2^{i_{j}} for i = 1, …, s.

(a) If n ≡ 0 (mod 4) then

C_{𝔖_{n}}(Q_{n}) = C_{𝔄_{n}}(Q_{n}) = Z(Q_{n}) = \begin{cases}Q_{4}, & if n = 4 \\ Z(P_{n}) = Z(P_{n_{1}}) × ⋯ × Z(P_{n_{s}}), & if n > 4. \end{cases}

(b) If n ≡ 2 (mod 4) then i_{s} = 1, and

C_{𝔖_{n}}(Q_{n}) = Z(P_{n}) = Z(P_{n_{1}}) × ⋯ × Z(P_{n_{s}}) = Z(Q_{n}) × P_{2} = C_{𝔄_{n}}(Q_{n}) × P_{2}.

Proof. We may assume that n ⩾ 4. Then Ω_{1}, …, Ω_{s} are the orbits of P_{n} on {1, …, n}, as well as the orbits of Q_{n} on {1, …, n}, where Ω_{j} is as above in 4.1(c). Since |Ω_{1}| > … > |Ω_{s}|, the Q_{n}-sets Ω_{1}, …, Ω_{s} are pairwise non-isomorphic. For j = 1, …, s, let ω_{j} ∈ Ω_{j}, and set R_{j} := Stab_{Q_{n}}(ω_{j}). Then Ω_{j} is as Q_{n}-set isomorphic to Q_{n}/R_{j}, and we have the following group isomorphism, see [5, La. 4.3]:

∏_{j=1}^{s} N_{Q_{n}}(R_{j})/R_{j} ⟶ C_{𝔖_{n}}(Q_{n}).

In particular, C_{𝔖_{n}}(Q_{n}) is a 2-group and, hence, so is Q_{n}C_{𝔖_{n}}(Q_{n}). Thus there is some g ∈ 𝔖_{n} such that ^{g}(Q_{n}C_{𝔖_{n}}(Q_{n})) ⩽ P_{n}. In particular, we have ^{g}Q_{n} ⩽ P_{n} ∩ 𝔄_{n} = Q_{n}, that is, g ∈ N_{𝔖_{n}}(Q_{n}). Hence we have g ∈ N_{𝔖_{n}}(C_{𝔖_{n}}(Q_{n})) as well, implying C_{𝔖_{n}}(Q_{n}) ⩽ P_{n}, and thus C_{𝔖_{n}}(Q_{n}) = C_{P_{n}}(Q_{n}). So it suffices to show that

C_{P_{n}}(Q_{n}) = \begin{cases}Q_{4}, & if n = 4 \\ Z(P_{n}), & if n ≠ 4, \end{cases}

since then we also get

C_{𝔄_{n}}(Q_{n}) = Z(Q_{n}) = \begin{cases}Q_{4}, & if n = 4 \\ Z(P_{n}), & if 4 < n ≡ 0 (mod 4) \\ Z(P_{n_{1}}) × ⋯ × Z(P_{n_{s−1}}), & if n ≡ 2 (mod 4). \end{cases}

The statement for n = 4 is clear. Next suppose that n = 2^{m}, for some m ⩾ 3. We argue with induction on m, and show that Z(P_{n}) = C_{P_{n}}(Q_{n}). For m = 3 this is immediately checked to be true, so that we may now suppose that m > 3. We consider P_{n} again as the wreath product P_{2^{m−1}} ≀ C_{2} = {(x_{1}, x_{2}; σ) | x_{1}, x_{2} ∈ P_{2^{m−1}}, σ ∈ C_{2}}. Let x := (x_{1}, x_{2}; σ) ∈ C_{P_{n}}(Q_{n}), so that, for each y := (y_{1}, y_{2}; π) ∈ Q_{n}, we have

(x_{1}y_{σ(1)}, x_{2}y_{σ(2)}; σπ) = (x_{1}, x_{2}; σ)(y_{1}, y_{2}; π) = (y_{1}, y_{2}; π)(x_{1}, x_{2}; σ) = (y_{1}x_{π(1)}, y_{2}x_{π(2)}; πσ).

(5)

Setting y_{1} := y_{2} := 1 and π := (1, 2), Equation (5) yields x_{1} = x_{2}. Next we set π := 1, y_{1} := 1, and 1 ≠ y_{2} ∈ Q_{2^{m−1}}. Then (5) this time implies x_{1}y_{σ(1)} = x_{1} and x_{1}y_{σ(2)} = y_{2}x_{1}. Therefore σ = 1 and x_{1} ∈ C_{P_{2^{m−1}}} (Q_{2^{m−1}}). Thus, by induction, x_{1} ∈ Z(P_{2^{m−1}}). Consequently, x = (x_{1}, x_{1}; 1) ∈ Z(P_{n}), and we have C_{P_{n}}(Q_{n}) = Z(P_{n}) ⩽ Q_{n}.

Now let n > 4 with s ⩾ 2. We show that also in this case C_{P_{n}}(Q_{n}) = Z(P_{n}) = Z(P_{n_{1}}) × ⋯ × Z(P_{n_{s}}). For this, let x ∈ C_{P_{n}}(Q_{n}), and write x = x_{1} ⋯ x_{s} for appropriate x_{j} ∈ P_{n_{j}} and j = 1, …, s. Since Q_{n_{1}} × ⋯ × Q_{n_{j}} ⩽ Q_{n}, we deduce that x_{j} ∈ C_{P_{n_{j}}} (Q_{n_{j}}), for i = 1, …, j. Hence, by what we have just proved above, x_{j} ∈ Z(P_{n_{j}}) if i_{j} > 2. Moreover, x_{j} ∈ Z(Q_{4}) = Q_{4} if i_{j} = 2, and clearly x_{j} ∈ Z(P_{2}) = P_{2} if i_{j} = 1. Suppose that there is some j ∈ {1, …, s} with i_{j} = 2. Then j ∈ {s − 1, s}. We need to show that x_{j} ∈ Z(P_{4}). Assume that this is not the case. In the notation of 4.1, we may then suppose that x_{j} = h_{i_{j},j} = g_{2,j}. But this leads to the contradiction x(g_{1,j}g_{1,s})x^{−1} = x_{j}g_{1,j}x_{j} · g_{1,s} ≠ g_{1,j}g_{1,s} if j = s − 1, and to the contradiction x(g_{1,1}g_{1,j})x^{−1} = g_{1,1} · x_{j}g_{1,j}x_{j} ≠ g_{1,1}g_{1,j} if j = s. Thus also x_{j} ∈ Z(P_{4}), and we have shown that Z(P_{n}) ⩽ C_{P_{n}}(Q_{n}) ⩽ Z(P_{n_{1}}) × ⋯ × Z(P_{n_{s}}) = Z(P_{n}).

4.3. The 2-Blocks of 𝔄_{n}. (a) Recall from [17, 6.1.21] that each block B of F 𝔖_{n} can be labelled combinatorially by some integer w ⩾ 0 and a 2-regular partition κ of n − 2w. We call w the 2-weight of B, and κ the 2-core of B. Moreover, by [17, Thm. 6.2.39], the defect groups of B are in 𝔖_{n} conjugate to P_{2w} ⩽ 𝔖_{2w} ⩽ 𝔖_{n}.

(b) The following relationships between blocks of F 𝔖_{n} and F 𝔄_{n} are well known; see for instance [26]: for any partition λ of n, we denote its conjugate partition by λ′. That is, the Young diagram [λ′] of λ′ is obtained by transposing the Young diagram [λ].

Suppose now that B is a block of F 𝔖_{n} of weight w and with 2-core κ. Denote the corresponding block idempotent of F 𝔖_{n} by e_{B}. The 2-core κ is a triangular partition, so that κ = κ′. If w ⩾ 1 then e_{B} is a block idempotent of F 𝔄_{n}, and if w = 0 then e_{B} = e_{B′} + e_{B′′} for 𝔖_{n}- conjugate blocks B′ ≠ B′′ of F 𝔄_{n} of defect 0; note that for w = 1 this also yields a (single) block of F 𝔄_{n} of defect 0. Hence, the weight of any block of F 𝔄_{n} is understood to be the weight w of the covering block of F 𝔖_{n}, and the associated defect groups are in 𝔄_{n} conjugate to Q_{2w} ⩽ 𝔄_{2w} ⩽ 𝔄_{n}.

(c) Let B be a block of F 𝔄_{n} of weight w ⩾ 0. Let further (Q, b_{Q}) be a self-centralizing Brauer B-pair. Then, by [24, Thm. 5.5.21], there is a defect group P of B such that Z(P) ⩽ C_{P} (Q) ⩽ Q ⩽ P. Replacing (Q, b_{Q}) by a suitable 𝔄_{n}-conjugate, we may assume that P = Q_{2w}. If w is even then, by Proposition 4.2, Z(Q_{2w}) has no fixed points on {1, …, 2w}, hence, in this case, Q acts fixed point freely on precisely 2w points; note that this also holds for w = 0. If w is odd then, by Proposition 4.2 again, Z(Q_{2w}) has exactly the two fixed points {2w − 1, 2w} on {1, …, 2w}, hence, in this case Q acts fixed point freely precisely on either {1, …, 2w − 2} or {1, …, 2w}, that is, on 2x points, where w − 1 ⩽ x ⩽ w; note that for w = 1 we have x = 0.

The following example shows that even if we restrict ourselves to Brauer pairs arising from vertices of simple modules we have to deal with both cases w − 1 ⩽ x ⩽ w:

Example 4.4. Suppose that n = 2m, for some odd integer m ⩾ 3, and consider the simple F 𝔖_{n}-module D^{(m+1,m−1)} labelled by the partition (m + 1, m − 1) of n. This is the basic spin F 𝔖_{n}-module, belonging to the principal block of F 𝔖_{n}, which has weight m. Since n ≡ 2 (mod 4), the restriction Res_{𝔄_{n}}^{𝔖_{n}}(D^{(m+1,m−1)}) =: E^{(m+1,m−1)} is simple, by [1], and thus belongs to the principal block B_{0} of F 𝔄_{n}. By [6, Thm. 7.2], E^{(m+1,m−1)} has common vertices with the basic spin F 𝔖_{n−1}-module D^{(m,m−1)}. Therefore, the vertices of E^{(m+1,m−1)} are conjugate to subgroups of Q_{n−2} and have, in particular, fixed points on {1, …, n}, while Q_{n} acts of course fixed point freely. This shows that there is indeed a self-centralizing Brauer B_{0}-pair (Q, b_{Q}) of 𝔄_{n}, where Q arises as a vertex of a simple F 𝔄_{n}-module and such that Q has strictly more fixed points on {1, …, n} than the associated defect group Q_{n} of its Brauer correspondent b_{Q}^{𝔄_{n}} = B_{0}.

The next theorem is motivated by the results of [5], where the self-centralizing Brauer pairs of the symmetric groups are examined, for which, using the above notation, we necessarily have x = w. We pursue the analogy to the case of the symmetric groups as far as possible, the treatment being reminiscent of the exposition in [25, Sect. 1].

Theorem 4.5. Let (Q, b_{Q}) be a self-centralizing Brauer pair of 𝔄_{n}.

(a) Let Ω ⊆ {1, …, n} be such that Q acts fixed point freely on Ω and fixes {1, …, n} ∖ Ω pointwise. Then we have C_{𝔄(Ω)}(Q) = Z(Q).

(b) Let P be a defect group of the Brauer correspondent B := b_{Q}^{𝔄_{n}} of b_{Q} in 𝔄_{n} such that C_{P} (Q) ⩽ Q ⩽ P, and let Ω̂ ⊆ {1, …, n} be such that P acts fixed point freely on Ω̂ and fixes {1, …, n} ∖ Ω̂ pointwise. Then we even have C_{𝔄(Ω̂)}(Q) = Z(Q).

Proof. Since (Q, b_{Q}) is self-centralizing, the block b_{Q} of F [QC_{𝔄_{n}}(Q)] has defect group Q. Let w ⩾ 0 be the weight of B = b_{Q}^{𝔄_{n}}. By the observations made in 4.3, we have 2w − 2 ⩽ 2x = |Ω| ⩽ 2w = |Ω̂|, and we may suppose that Ω = {1, …, 2x} and Ω̂ = {1, …, 2w}, that is, Q ⩽ 𝔄_{2x} ⩽ 𝔄_{2w} and P = Q_{2w} ⩽ 𝔄_{2w}. We have C_{𝔖_{n}}(Q) = C_{𝔖_{2x}}(Q) × 𝔖_{n−2x}, and thus also QC_{𝔖_{n}}(Q) = QC_{𝔖_{2x}}(Q) × 𝔖_{n−2x}. Consider the following chain of normal subgroups

QC_{𝔄_{2x}}(Q) × 𝔄_{n−2x} ⊴ QC_{𝔄_{n}}(Q) ⊴ QC_{𝔖_{n}}(Q) = QC_{𝔖_{2x}}(Q) × 𝔖_{n−2x}.  (6)

Since |QC_{𝔖_{n}}(Q) : QC_{𝔄_{n}}(Q)| ⩽ 2, by [24, Cor. 5.5.6] there is a unique block b̃_{Q} of F [QC_{𝔖_{n}}(Q)] covering b_{Q}. In particular, (Q, b̃_{Q}) is a (not necessarily self-centralizing) Brauer pair of 𝔖_{n}. We may write b̃_{Q} = b̃_{0} ⊗ b̃_{1}, for some block b̃_{0} of F [QC_{𝔖_{2x}}(Q)] and some block b̃_{1} of F 𝔖_{n−2x}. Since Q has no fixed points on {1, …, 2x}, by [25, Prop. 1.2, Prop. 1.3] we conclude that F [QC_{𝔖_{2x}}(Q)] has only one block, that is, the principal one. Therefore, each block of F [QC_{𝔄_{2x}}(Q)] is covered by the principal block b̃_{0} of F [QC_{𝔖_{2x}}(Q)]. Hence all blocks of F [QC_{𝔄_{2x}}(Q)] are conjugate in QC_{𝔖_{2x}}(Q). But then also F [QC_{𝔄_{2x}}(Q)] has only one block, that is, the principal block b_{0}.

Moreover, b_{Q} covers some block b_{0}⊗b_{1} of F [QC_{𝔄_{2x}}(Q)×𝔄_{n−2x}], where b_{1} is a block of F 𝔄_{n−2x}, and, by [24, Cor. 5.5.6] again, b_{Q} is the unique block of F [QC_{𝔄_{n}}(Q)] covering b_{0} ⊗ b_{1}. The defect groups of b_{0} ⊗ b_{1} are in QC_{𝔄_{n}}(Q) conjugate to subgroups of Q, by Fong’s Theorem [24, Cor. 5.5.16]. Hence b_{1} has to be a block of defect 0. Thus, since b̃_{1} covers b_{1} we infer from 4.3 that b̃_{1} is a block of weight w̃ = 0 or w̃ = 1, that is, of defect 0 or 1, respectively. Moreover, since b_{0} is the principal block of F [QC_{𝔄_{2x}}(Q)], we infer that Q ∈ Syl_{2}(QC_{𝔄_{2x}}(Q)).

Thus we may summarize the properties of the relevant blocks of the subgroups in (6) as follows:

b_{0} ⊗ b_{1}  b_{Q}  b̃_{Q} =  b̃_{0} ⊗ b̃_{1}
defect Q  defect 0 defect Q  weight w̃ ∈ {0, 1}
principal  principal

To show (a), assume that QC_{𝔄_{2x}}(Q) is not a 2-group, so that there is some 1 ≠ g ∈ QC_{𝔄_{2x}}(Q) of odd order. Thus g ∈ C_{𝔄_{2x}}(Q), and we denote the conjugacy class of g in QC_{𝔄_{2x}}(Q) by C. Since Q ∈ Syl_{2}(QC_{𝔄_{2x}}(Q)), we also have Q ∈ Syl_{2}(C_{QC_{𝔄_{2x}}(Q)}(g)). In particular, Q is a defect group of the conjugacy class C and of the conjugacy class {1} ≠ C. Hence, from [12, Cor. IV.4.17] we infer that F [QC_{𝔄_{2x}}(Q)] has two blocks of maximal defect, contradicting the fact that the principal block is the only block of F [QC_{𝔄_{2x}}(Q)]. Hence QC_{𝔄_{2x}}(Q) is a 2-group, and from this we finally deduce that QC_{𝔄_{2x}}(Q) = Q, that is, C_{𝔄_{2x}}(Q) = Z(Q), proving (a).

To show (b) we may suppose that x = w − 1, hence we have n ⩾ 2x + 2. We show that in this case we have

QC_{𝔄_{n}}(Q) = QC_{𝔄_{2x}}(Q) × 𝔄_{n−2x},

from which we get C_{𝔄_{2w}}(Q) = QC_{𝔄_{2x}}(Q) = Z(Q). So assume, for a contradiction, that QC_{𝔄_{2x}}(Q) × 𝔄_{n−2x} < QC_{𝔄_{n}}(Q). We consider again the chain of subgroups (6), where [QC_{𝔖_{n}}(Q) : (QC_{𝔄_{2x}}(Q) × 𝔄_{n−2x})] ⩽ 4. Since QC_{𝔄_{n}}(Q) ⩽ 𝔄_{n} but 𝔖_{n−2x} ⩽̸ 𝔄_{n}, we get [QC_{𝔖_{n}}(Q) : QC_{𝔄_{n}}(Q)] = 2, and hence [QC_{𝔄_{n}}(Q) : (QC_{𝔄_{2x}}(Q) × 𝔄_{n−2x})] = 2.

Since b_{Q} has defect group Q, from Fong’s Theorem [24, Thm. 5.5.16] we conclude that the inertial group of b_{0} ⊗b_{1} is given as T_{QC_{𝔄n}(Q)}(b_{0} ⊗b_{1}) = QC_{𝔄_{2x}}(Q)×𝔄_{n−2x}. Thus b_{0} ⊗b_{1} is not QC_{𝔄_{n}}(Q)-invariant, hence is not QC_{𝔖_{n}}(Q)-invariant either. Since b_{0} is the principal block of F [QC_{𝔄_{2x}}(Q)], this implies that b_{1} is not 𝔖_{n−2x}-invariant, from which, by 4.3, we infer that b̃_{1} has weight w̃ = 0, that is, b̃_{1} has defect 0.

Let B̃ := b̃_{Q}^{𝔖_{n}} be the Brauer correspondent of b̃_{Q} = b̃_{0} ⊗ b̃_{1} in 𝔖_{n}. Since b̃_{1} has weight w̃ = 0 and Q acts fixed point freely on the set Ω of cardinality 2x, we conclude from [25, Thm. 1.7] that B̃ is a block of weight x. But, by 4.3, the block of F 𝔖_{n} covering B, has weight w, contradicting Lemma 3.13. Thus we have QC_{𝔄_{2x}}(Q) × 𝔄_{n−2x} = QC_{𝔄_{n}}(Q), proving (b).

Remark 4.6. A closer analysis of the arguments in the proof of Theorem 4.5 yields a somewhat more precise description of the self-centralizing Brauer pairs of 𝔄_{n}, which at least helps to exclude certain subgroups from being the first component of a self-centralizing Brauer pair.

It turns out that there are only the cases listed below, all of which, by Example 4.8, actually occur. We give the results without proofs, since we do not use these facts later on. Note that, by Brauer’s Third Main Theorem [24, Thm. 5.6.1], b_{Q} is the principal block if and only if its Brauer correspondent b_{Q}^{𝔄_{n}} is, which in turn is equivalent to n ∈ {2w, 2w + 1}.

(a) Let first n ∈ {2x, 2x + 1}, thus we have x = w, and hence QC_{𝔄_{n}}(Q) = QC_{𝔄_{2x}}(Q) anyway. Moreover, b_{Q} is principal, hence, in particular, is QC_{𝔖_{n}}(Q)-invariant. It turns out that always w̃ = 0, and that both cases d := [C_{𝔖_{2x}}(Q) : C_{𝔄_{2x}}(Q)] ∈ {1, 2} occur.

(b) Now let n ⩾ 2x + 2. Then it turns out that w = x + w̃ and that

d := [C_{𝔖_{2x}}(Q) : C_{𝔄_{2x}}(Q)] = [QC_{𝔄_{n}}(Q) : (QC_{𝔄_{2x}}(Q) × 𝔄_{n−2x})] ∈ {1, 2}.

Moreover, only the following cases occur:

w odd d = 1 w̃ = 1 b_{Q} is QC_{𝔖_{n}}(Q)-invariant
d = 2 w̃ = 0 b_{Q} is QC_{𝔖_{n}}(Q)-invariant
w even d = 1 w̃ = 0 b_{Q} is not QC_{𝔖_{n}}(Q)-invariant
d = 2 w̃ = 0 b_{Q} is QC_{𝔖_{n}}(Q)-invariant

Note that b_{Q} is principal if and only if n ∈ {2x + 2, 2x + 3} where x is even and d = 1.

We are now prepared to prove the bound given in Theorem 3.9.

Proposition 4.7. Let B be a block of F 𝔄_{n} with defect group P. Let further (Q, b_{Q}) be a self-centralizing Brauer B-pair. Then we have

|P | ⩽ \frac{(|Q| + 2)!}{2}.

Proof. Let w be the weight of B. By Theorem 4.5, there is a subset Ω of {1, …, n} with 2w − 2 ⩽ 2x = |Ω| ⩽ 2w, where x is as in 4.3, such that Q acts fixed point freely on Ω and fixes {1, …, n} ∖ Ω pointwise. Moreover, C_{𝔄(Ω)}(Q) = Z(Q).

Assume there are Q-orbits Ω′ = {ω′_{1}, …, ω′_{m}} and Ω′′ = {ω′′_{1}, …, ω′′_{m}} on Ω that are isomorphic as Q-sets. Then we may suppose that there is an isomorphism of Q-sets mapping ω′_{i} to ω′′_{i}, for i = 1, …, m. Since, by our assumption, Q acts fixed point freely on Ω, we deduce that m ⩾ 2 is even, and therefore the permutation (ω′_{1}, ω′′_{1}) ⋯ (ω′_{m}, ω′′_{m}) is contained in C_{𝔄(Ω)}(Q). But, on the other hand, the elements in Z(Q) have to fix every Q-orbit, so that (ω_{1}, ω′_{1}) ⋯ (ω_{m}, ω′_{m}) ∉ Z(Q), a contradiction.

Hence we deduce that Ω = ⨄_{i=1}^{k} Ω_{i} consists of pairwise non-isomorphic Q-orbits. For j = 1, …, k, let ω_{j} ∈ Ω_{j}, and set R_{j} := Stab_{Q}(ω_{j}), where we may choose notation such that |R_{1}| ⩽ … ⩽ |R_{k}|. Then, by [5, La. 4.3], we have the group isomorphism

φ : ∏_{j=1}^{k} N_{Q}(R_{j})/R_{j} ⟶ C_{𝔖(Ω)}(Q),

which can be described as follows: for i = 1, …, k, let π_{i} : Q ⟶ 𝔖(Ω_{i}) ∩ Q be the canonical projection, and let Z_{i} := C_{𝔖(Ω_{i})}(π_{i}(Q)). Then φ(N_{Q}(R_{i})/R_{i}) = Z_{i}, for i = 1, …, k. Hence C_{𝔖(Ω)}(Q) = Z_{1} × ⋯ × Z_{k}, and Z(Q) = C_{𝔄(Ω)}(Q) = (Z_{1} × ⋯ × Z_{k}) ∩ 𝔄(Ω). Since Q acts fixed point freely on Ω, we have R_{i} < Q, thus N_{Q}(R_{i}) > R_{i}, in particular |Z_{i}| ⩾ 2. Since (Z_{2} × ⋯ × Z_{k}) ∩ 𝔄(Ω) ⩽ Z(Q) ⩽ Q acts trivially on Ω_{1}, we have (Z_{2} × ⋯ × Z_{k}) ∩ 𝔄(Ω) ⩽ R_{1}.

Moreover, as we have just mentioned, |Z_{2} × ⋯ × Z_{k}| ⩾ 2^{k−1}, so that for k ⩾ 2 we have |R_{1}| ⩾ |(Z_{2} × ⋯ × Z_{k}) ∩ 𝔄(Ω)| ⩾ 2^{k−2}. Therefore, for k ⩾ 2 we get

|Ω| = ∑_{i=1}^{k} |Q : R_{i}| ⩽ \frac{k|Q|}{|R_{1}|} ⩽ \frac{k|Q|}{2^{k−2}} .

Thus, for k ⩾ 4 we infer |Ω| ⩽ |Q|. It remains to consider the cases k ⩽ 3: if k = 1 then we have |Ω| ⩽ |Q| anyway. If k = 2 then we have |Ω| ⩽ |Q|, except if Ω_{1} is the regular Q-orbit, that is, R_{1} = {1}, and hence Z_{1} ≅ Q. Now Z_{1} ∩ 𝔄(Ω) ⩽ R_{2} entails

\frac{|Q|}{2} ⩽ |Z_{1} ∩ 𝔄(Ω)| ⩽ |R_{2}| ⩽ \frac{|Q|}{2},

thus |R_{2}| = |Q|/2, hence |Z_{2}| = 2, implying |Ω| = |Q| + 2. Moreover, we have |Z_{1} ∩ 𝔄(Ω)| = |Q|/2, hence Z_{1} ⩽̸ 𝔄(Ω). Since Z_{1} acts regularly on Ω_{1} and fixes Ω_{2} pointwise, we infer that Z_{1} contains an |Ω_{1}|-cycle, that is, Z_{1} ≅ Q is cyclic. Note that we have |Q| ⩾ 4, implying that 2x = |Ω| = |Q| + 2 ≡ 2 (mod 4), that is, x is odd, hence, by 4.3, we infer that w = x.

Next we observe that if R_{1} = {1} then k ⩾ 2 forces k = 2, since 1 = |R_{1}| ⩾ |(Z_{2} × ⋯ × Z_{k}) ∩ 𝔄(Ω)| ⩾ 2^{k−2}. Hence, if k = 3 then we have |Ω| ⩽ |Q|, except if |R_{1}| = |R_{2}| = 2. Thus from |(Z_{2} × Z_{3}) ∩ 𝔄(Ω)| ⩾ 2 and |(Z_{1} × Z_{3}) ∩ 𝔄(Ω)| ⩾ 2 we deduce R_{1} = (Z_{2} × Z_{3}) ∩ 𝔄(Ω) ⩽ Z(Q) and R_{2} = (Z_{1} × Z_{3}) ∩ 𝔄(Ω) ⩽ Z(Q), showing |Z_{1}| = |Q|/2 = |Z_{2}|. This yields

|Q| ⩾ |(Z_{1} × Z_{2} × Z_{3}) ∩ 𝔄(Ω)| ⩾ \frac{|Q|}{2} · \frac{|Q|}{2} · 2 · \frac{1}{2} = \frac{|Q|^{2}}{4} ,

hence |Q| ⩽ 4. Therefore we have Q ≅ V_{4}, and |Ω_{1}| = |Ω_{2}| = |Ω_{3}| = 2, thus 2x = |Ω| = 6 = |Q| + 2. Note that x = 3, by 4.3, implies that w = x = 3 as well.

Consequently, in any case we get |Ω| ⩽ |Q| or, in the two exceptions, |Ω| = |Q| + 2 and x = w. This implies 2w ⩽ |Q| + 2 and, since P ⩽ 𝔄_{2w}, we get |P | ⩽ \frac{(|Q|+2)!}{2}.

Example 4.8. We give a few examples, found with the help of the computer algebra system GAP [13], showing that all the cases listed in Remark 4.6 actually occur. In particular, the exceptional cases detected in the proof of Proposition 4.7, namely Q cyclic with two orbits of lengths |Q| and 2, as well as Q ≅ V_{4} with three orbits of length 2 each, occur for the principal block of 𝔄_{6}.

(a) The principal blocks of 𝔄_{4} and of 𝔄_{5} both have weight w = 2, their defect groups are abelian and conjugate to Q_{4} ⩽ 𝔄_{4} and, in each case, up to conjugacy, there is a unique self-centralizing Brauer pair:

Q (n ∈ {4, 5}, w = 2)  |Q| |Z(Q)| x d
⟨(1, 2)(3, 4), (1, 3)(2, 4)⟩ ≅ V_{4} 4  4 2 1

The non-principal blocks of 𝔄_{7} and of 𝔄_{10} both have weight w = 2, their defect groups are abelian and conjugate to Q_{4} ⩽ 𝔄_{4} and, in each case, up to conjugacy, there are two selfcentralizing Brauer pairs:

Q (n ∈ {7, 10}, w = 2)  |Q| |Z(Q)| x d
⟨(1, 2)(3, 4), (1, 3)(2, 4)⟩ ≅ V_{4} 4  4 2 1
⟨(1, 2)(3, 4), (1, 3)(2, 4)⟩ ≅ V_{4} 4  4 2 1

(b) The principal blocks of 𝔄_{6} and of 𝔄_{7}, and the non-principal block of 𝔄_{9} all have weight w = 3, their defect groups are conjugate to Q_{6} ⩽ 𝔄_{6} and, in each case, up to conjugacy, there are four self-centralizing Brauer pairs:

Q (n ∈ {6, 7, 9}, w = 3)  |Q| |Z(Q)| x d
⟨(1, 2)(3, 4), (1, 3)(2, 4)⟩  ≅ V_{4}  4  4 2 1
⟨(1, 2)(3, 4), (3, 4)(5, 6)⟩  ≅ V_{4}  4  4 3 2
⟨(1, 2)(3, 4), (1, 3, 2, 4)(5, 6)⟩ ≅ C_{4} 4  4 3 2
⟨(1, 2)(5, 6), (1, 3)(2, 4)⟩  ≅ D_{8} 8  2 3 2

(c) The non-principal block of 𝔄_{11} has weight w = 4, its defect groups are conjugate to Q_{8} ⩽ 𝔄_{8} and, up to conjugacy, there are thirty-three self-centralizing Brauer pairs. We do not mention all of them, but just one concluding the list of cases in Remark 4.6:

Q (n = 11, w = 4)  |Q| |Z(Q)| x d
...
⟨(1, 2)(3, 4), (1, 2)(5, 6), (5, 6)(7, 8)⟩ ≅ C_{2}^{3} 8  8 4 2
...

Remark 4.9. With the help of the computer algebra system MAGMA [2], and using the techniques described in [7], it can be shown that all the 2-subgroups listed above actually occur as vertices of suitable simple modules, with the exception of the cyclic group in (b), of course, and the group given in (c). We also point out a mistake in [5, Cor. 6.3(iii)], where the 2-groups Q ⩽ 𝔖_{n} of order 4 occurring as vertices of simple F 𝔖_{n}-modules were classified (up to 𝔖_{n}-conjugation). In fact, the case where Q = P_{2} × P_{2} and w = 2 cannot occur.

5 The Schur Double Covers of 𝔖_{n} and 𝔄_{n}

We begin by recalling the group presentations of the Schur double covers of the symmetric and alternating groups, as well as the necessary facts about their blocks. Then we immediately proceed to prove the bound given in Theorem 3.9.

5.1. Notation. (a) Let n ⩾ 1, and consider the group 𝔖̃_{n} := ⟨z, t_{1}, …, t_{n−1}⟩ with relations

z^{2} = 1,
zt_{i} = t_{i}z, for i = 1, …, n − 1,
t_{i}^{2} = z, for i = 1, …, n − 1, (∗)
t_{i}t_{j} = zt_{j}t_{i}, for |i − j| > 1,
(t_{i}t_{i+1})^{3} = z, for i = 1, …, n − 2; (∗∗)

in particular, we have 𝔖̃_{1} := ⟨z⟩ ≅ C_{2}. Note that also 𝔖̃_{n} ⩽ 𝔖̃_{n+1}, for n ⩾ 1. Via θ : 𝔖̃_{n} ⟶ 𝔖_{n}, t_{i} ⟼ (i, i + 1), we obtain a group epimorphism with central kernel ⟨z⟩.

Replacing the relations (∗) by t_{i}^{2} = 1, for i = 1, …, n−1, and the relations (∗∗) by (t_{i}t_{i+1})^{3} = 1, for i = 1, …, n − 2, we get an isoclinic group 𝔖̂_{n}, which also is a central extension of 𝔖_{n}; we have 𝔖̃_{n} ≅ ˆ𝔖̸_{n} if and only if 1 ≠ n ≠ 6. In the case where n ⩾ 4, the groups 𝔖̃_{n} and 𝔖̂_{n} are the Schur double covers of the symmetric group 𝔖_{n}. Whenever we have a subgroup H of 𝔖_{n}, we denote its preimage under θ by H̃, and similarly its preimage in 𝔖̂_{n} is denoted by Ĥ.

In particular, for H = 𝔄_{n}, we get 𝔄̃_{n} ⊴ 𝔖̃_{n} and 𝔄̂_{n} ⊴ 𝔖̂_{n}, where we actually always have 𝔄̃_{n} ≅ 𝔄̂_{n}, and |𝔖̃_{n} : 𝔄̃_{n}| = |𝔖̂_{n} : 𝔄̂_{n}| = 2, for n ⩾ 2. If n ⩾ 4 and 6 ≠ n ≠ 7 then 𝔄̃_{n} is the unique Schur double cover of the alternating group 𝔄_{n}; recall that precisely for 6 ⩽ n ⩽ 7 there are further Schur covers of 𝔄_{n}, namely threefold and sixfold ones. Since we have no distinction between 𝔄̃_{n} and 𝔄̂_{n} anyway, and since it will turn out that all observations for 𝔖̃_{n} immediately translate to 𝔖̂_{n}, we from now on confine ourselves to investigating 𝔄̃_{n} ⊴ 𝔖̃_{n}.

(b) We list the known facts concerning the block theory of 𝔄̃_{n} and 𝔖̃_{n} we will need, where we from now on suppose that p ⩾ 3, for the remainder of this section. Each faithful block B of F 𝔖̃_{n} can be labelled combinatorially by a some integer w ⩾ 0, called the p-bar weight of B, and a 2-regular partition κ of n − pw, called the p-bar core of B; for details we refer to [16, Appendix 10]. Given the p-bar weight w of the block B, by [26, Thm. (1.3)], the defect groups of B are the 𝔖̃_{n}-conjugates of the Sylow p-subgroups of 𝔖̃_{pw}. The latter in turn are via θ mapped to Sylow p-subgroups of 𝔖_{pw}.

Arguing along the lines of [5], we now have:

Proposition 5.2. Let p ⩾ 3, and let B be a faithful block of F 𝔖̃_{n} with defect group P. Let further (Q, b_{Q}) be a self-centralizing Brauer B-pair. Then we have

|P | ⩽ |Q|!.

Proof. Let w be the p-bar weight of B. Then P is conjugate to a Sylow p-subgroup of 𝔖̃_{pw}, and θ(P) is in 𝔖_{n} conjugate to a Sylow p-subgroup of 𝔖_{pw}. Thus we have θ(P) =_{𝔖_{n}} P_{pw}, so that there is a subset Ω of {1, …, n} with |Ω| = pw and such that θ(P) acts fixed point freely on Ω and fixes {1, …, n} ∖ Ω pointwise. Moreover, by [24, Thm. 5.5.21], we may assume that

Z(P) ⩽ C_{P} (Q) ⩽ Q ⩽ P.  (7)

In particular, since θ|_{P} is injective, we infer that Z(θ(P)) = θ(Z(P)) ≅ Z(P) acts fixed point freely on Ω as well. By (7), we have Z(θ(P)) ⩽ θ(Q) ⩽ θ(P), hence θ(Q) acts fixed point freely on Ω and fixes {1, …, n} ∖ Ω pointwise. That is, θ(Q) ⩽ 𝔖(Ω) and Q ⩽ 𝔖̃(Ω). By [3, Prop. 3.8 e], we further have

C_{𝔖̃(Ω)}(Q) = Z(Q) × Z(𝔖̃(Ω)) = Z(Q) × ⟨z⟩.

Since Q is a p-group, this implies

C_{𝔖(Ω)}(θ(Q)) = θ(C_{𝔖̃(Ω)}(Q)) = θ(Z(Q)) = Z(θ(Q)),

thus, applying [5, Thm. 5.1], we get |P | = |θ(P)| ⩽ |𝔖(Ω)| = |Ω|! ⩽ |θ(Q)|! = |Q|!.

6 The Weyl Groups 𝔅_{n} and 𝔇_{n}

We begin by recalling the description of the Weyl groups 𝔅_{n} and 𝔇_{n} of type B_{n} and D_{n}, respectively. Then we recall the necessary facts about their blocks, distinguishing the cases p even and p odd, in order to immediately proceed to prove the bound given in Theorem 3.9.

6.1. Notation. Let C_{2} := ⟨(1, 2)⟩ be the group of order 2, and for n ∈ ℕ set

𝔅_{n} := C_{2} ≀ 𝔖_{n} = {(x_{1}, …, x_{n}; π) | x_{1}, …, x_{n} ∈ C_{2}, π ∈ 𝔖_{n}}.

We define 𝔖_{n}^{∗} := {(1, …, 1; π) | π ∈ 𝔖_{n}} ⩽ 𝔅_{n}, and denote the usual group isomorphism 𝔖_{n}^{∗} ⟶ 𝔖_{n} by φ. Moreover, whenever U is a subgroup of 𝔖_{n}, we denote by U^{∗} ⩽ 𝔖_{n}^{∗} its image under φ^{−1}. Let also H := {(x_{1}, …, x_{n}; 1) | x_{1}, …, x_{n} ∈ C_{2}} ⊴ 𝔅_{n} be the base group of 𝔅_{n}; so H is isomorphic to a direct product of n copies of C_{2}, and we have 𝔅_{n} = H𝔖_{n}^{∗}. We will identify 𝔅_{n} with a subgroup of 𝔖_{2n}, in the usual way by the primitive action. Furthermore, let 𝔇_{n} := 𝔅_{n} ∩ 𝔄_{2n}. For n ⩾ 2, the group 𝔅_{n} is isomorphic to the Weyl group of type B_{n}, and for n ⩾ 4, the group 𝔇_{n} is isomorphic to the Weyl group of type D_{n}; see [17, 4.1.33].

6.2. The case p = 2. (a) Suppose first that p = 2. Since H is a normal 2-subgroup of 𝔅_{n} such that C_{𝔅_{n}}(H) = H is a 2-group, it follows from [24, Thm. 5.2.8] (see also [24, Exc. 5.2.10]) that F 𝔅_{n} has only the principal block. Moreover, H acts trivially on every simple F 𝔅_{n}-module. Thus if Q is a vertex of a simple F 𝔅_{n}-module then H ⩽ Q. In particular, we have |Q| ⩾ 2^{n}, hence n ⩽ log_{2}(|Q|). Therefore, if P is a Sylow 2-subgroup of 𝔅_{n} then

|P | ⩽ |𝔅_{n}| = 2^{n} · n! ⩽ |Q| · log_{2}(|Q|)!.

(b) To deal with 𝔇_{n}, note that 𝔇_{n} = 𝔅_{n} ∩𝔄_{2n} = (H ∩𝔄_{2n})𝔖_{n}^{∗}. An argument analogous to the one used in Part (a) above shows that also F 𝔇_{n} has only the principal block and, whenever Q is a vertex of a simple F 𝔇_{n}-module, H ∩ 𝔄_{2n} ⩽ Q. In particular, we have |Q| ⩾ 2^{n−1}, hence n ⩽ log_{2}(|Q|) + 1. Therefore, if P is a Sylow 2-subgroup of 𝔇_{n} then we deduce

|P | ⩽ |𝔇_{n}| = 2^{n−1} · n! ⩽ |Q| · (log_{2}(|Q|) + 1)!.

6.3. The case p ⩾ 3. Let now p ⩾ 3, for the remainder of this section. We briefly recall the well-known structure of the defect groups of the blocks of F 𝔅_{n}. By the Theorem of Fong–Reynolds [24, Thm. 5.5.10] applied to the base group C_{2}^{n} ≅ H ⊴ 𝔅_{n}, see also [27] or [17, Ch. 4], the blocks of F 𝔅_{n} are parametrized by pairs (κ, w), with κ = (κ_{0}, κ_{1}) and w = (w_{0}, w_{1}), and where, for i = 0, 1, the partition κ_{i} is the p-core of some partition of n_{i} := |κ_{i}| + pw_{i} such that n = n_{0} + n_{1}. Moreover, the inertial group of the block B(κ, w) is given as T_{𝔅_{n}}(B(κ, w)) := (C_{2} ≀ 𝔖_{n_{0}}) × (C_{2} ≀ 𝔖_{n_{1}}) = C_{2} ≀ (𝔖_{n_{0}} × 𝔖_{n_{1}}) ⩽ 𝔅_{n}.

Note that every p-subgroup of 𝔅_{n} is conjugate to a subgroup of 𝔖_{n}^{∗}. If P is a defect group of the block B(κ, w) then P is conjugate to a Sylow p-subgroup P_{pw_{0}}^{∗} × P_{pw_{1}}^{∗} of 𝔖_{pw_{0}}^{∗} × 𝔖_{pw_{1}}^{∗} ⩽ 𝔖_{n_{0}}^{∗} × 𝔖_{n_{1}}^{∗} ⩽ 𝔖_{n}^{∗}. Note that φ(P_{pw_{i}}^{∗}) = P_{pw_{i}} ⩽ 𝔖_{pw_{i}} ⩽ 𝔖_{n_{i}}, for i = 0, 1, is a defect group of the block of 𝔖_{n_{i}} parametrized by κ_{i}.

Proposition 6.4. Let p ⩾ 3, let D be a simple F 𝔅_{n}-module belonging to a block with defect group P, and let Q be a vertex of D. Then we have

|P | ⩽ |Q|!.

Proof. Let B(κ, w) be the block in question, and n_{i} := |κ_{i}| + pw_{i}, for i = 0, 1. Hence we may assume that Q ⩽ P = P_{pw_{0}}^{∗} × P_{pw_{1}}^{∗} ⩽ 𝔖_{n_{0}}^{∗} × 𝔖_{n_{1}}^{∗} ⩽ 𝔖_{n}^{∗}. Let T := T_{𝔅_{n}}(B(κ, w)) = (C_{2} ≀ 𝔖_{n_{0}}) × (C_{2} ≀ 𝔖_{n_{1}}) ⩽ 𝔅_{n} be the inertial group associated with B(κ, w). Then the simple F 𝔅_{n}-modules belonging to B(κ, w) can be described as follows (see [17, Sec. 4.3], and [27] ):

Let F be the trivial F C_{2}-module, and let E be the non-trivial simple F C_{2}-module. Then the outer tensor product F^{⊗n_{0}} ⊗_{F} E^{⊗n_{1}} naturally becomes an F T -module. Letting D_{i}, for i = 0, 1, be a simple F 𝔖_{n_{i}}-module in the block parametrized by κ_{i}, the tensor product D_{0} ⊗_{F} D_{1} becomes an F [𝔖_{n_{0}} × 𝔖_{n_{1}}]-module with respect to the outer-tensor-product action. Inflating with respect to the base group C_{2}^{n_{0}} ×C_{2}^{n_{1}} ⊴T yields the simple F T -module Inf_{C_{2}^{n_{0}}×C_{2}^{n_{1}}}^{T} (D_{0} ⊗_{F} D_{1}). Then inducing the ordinary tensor product

M := (F^{⊗n_{0}} ⊗_{F} E^{⊗n_{1}}) ⊗_{F} Inf_{C_{2}^{n_{0}}×C_{2}^{n_{1}}}^{T} (D_{0} ⊗_{F} D_{1}),

which is a simple F T -module, to 𝔅_{n} we get a simple F 𝔅_{n}-module Ind_{T}^{𝔅_{n}}(M) belonging to B(κ, w). Conversely, every simple F 𝔅_{n}-module belonging to the block B(κ, w) arises in such a way.

Hence, D ≅ Ind_{T}^{𝔅_{n}}(M), for suitably chosen D_{0} and D_{1}. Let Q_{i} ⩽ 𝔖_{n_{i}} be a vertex of D_{i}, for i = 0, 1, where we may assume that Q_{i} ⩽ P_{pw_{i}}. Hence Q_{0} × Q_{1} ⩽ 𝔖_{n_{0}} × 𝔖_{n_{1}} is a vertex of the outer tensor product D_{0} ⊗_{F} D_{1}. Since F and E are projective F C_{2}-modules, letting Q_{i}^{∗} := φ^{−1}(Q_{i}), it follows from [22] that Q_{0}^{∗} × Q_{1}^{∗} ⩽ 𝔖_{n_{0}}^{∗} × 𝔖_{n_{1}}^{∗} ⩽ 𝔖_{n}^{∗} ⩽ 𝔅_{n} is a vertex of Ind_{T}^{𝔅_{n}}(M) ≅ D.

By [5] we have pw_{i} ⩽ |Q_{i}|, and hence

|P | = |P_{pw_{0}}^{∗}| · |P_{pw_{1}}^{∗}| ⩽ (pw_{0})! · (pw_{1})! ⩽ |Q_{0}|! · |Q_{1}|! ⩽ (|Q_{0}| · |Q_{1}|)! = |Q|!.

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S.D.: Mathematical Institute, University of Oxford 24-29 St Giles’, Oxford, OX1 3LB, UK danz@maths.ox.ac.uk

J.M.: Lehrstuhl D für Mathematik, RWTH Aachen Templergraben 64, D-52062 Aachen, Germany Juergen.Mueller@math.rwth-aachen.de