ON SATURATED FUSION SYSTEMS AND BRAUER INDECOMPOSABILITY OF SCOTT MODULES

RADHA KESSAR, NAOKO KUNUGI, NAOFUMI MITSUHASHI

Abstract. Let p be a prime number, G a finite group, P a p-subgroup of G and k an algebraically closed field of characteristic p. We study the relationship between the category F_{P} (G) and the behavior of p-permutation kG-modules with vertex P under the Brauer construction. We give a sufficient condition for F_{P} (G) to be a saturated fusion system. We prove that for Scott modules with abelian vertex, our condition is also necessary. In order to obtain our results, we prove a criterion for the categories arising from the data of (b, G)- Brauer pairs in the sense of Alperin-Broué and Broué-Puig to be saturated fusion systems on the underlying p-group.

1. Introduction

Let p be a prime number and k an algebraically closed field of characteristic p. For a finite group G, a p-subgroup Q of G, and a finite dimensional kG-module M, the Brauer quotient M(Q) of M with respect to Q, is naturally a kN_{G}(Q)/Qmodule and hence by restriction is a kQC_{G}(Q)/Q module (see [4], [5], [17, Section 11]). We will say that M is Brauer indecomposable for any p-subgroup Q of G, M(Q) is indecomposable (or zero) as kQC_{G}(Q)/Q-module.

For subgroups Q, R of G, let Hom_{G}(Q, R) denote the set of all group homomorphisms from Q to R which are induced by conjugation by some element of G. For a p-subgroup P of G, let F_{P} (G) denote the category whose objects are the subgroups of P; whose morphism set from an object Q to an object R is the set Hom_{G}(Q, R), and where composition of morphisms is the usual composition of functions. We prove the following result (for background on fusion systems and saturated fusion systems, we refer the reader to the articles [3] and [13]; we note that we will follow the notational conventions in [3] rather than those of [13] in that all fusion systems will not be assumed to be saturated).

Theorem 1.1. Let G be a finite group, P a p-subgroup of G and M an indecomposable p-permutation kG-module with vertex P. If M is Brauer indecomposable, then F_{P} (G) is a saturated fusion system.

The question of Brauer indecomposability of p-permutation modules (or rather bimodules) plays a role in the “glueing processes” used for proving categorical equivalences between p-blocks of finite groups as predicted by Broué’s abelian defect group conjecture (see [10], [11]). Since splendid equivalences between blocks preserve local structure, it is not unexpected that there is a connection between saturation and the Brauer indecomposability condition. Theorem 1.1 provides a neat formulation of the connection.

The converse of Theorem 1.1 does not hold in general (see remarks after the proof of Theorem 1.1). However, in the special case that M is a Scott module, there seems to be some control in the reverse direction. For the definition and properties of Scott modules we refer the reader to [4]. For a finite group G and a p -subgroup P of G, we denote by S_{P} (G, k) the kG-Scott module with vertex P.

Theorem 1.2. Let P be an abelian p-subgroup of a finite group G. If F_{P} (G) is a saturated fusion system then the Scott module S_{P} (G, k) is Brauer indecomposable.

As a corollary, we obtain the following.

Corollary 1.3. Suppose that the finite group G has cyclic Sylow p-subgroups and let P be a p-subgroup of G. Then the Scott module S_{P} (G, k) is Brauer indecomposable.

Another consequence is the following result, of use for proving categorical equivalences between principal blocks of finite groups.

Corollary 1.4. Let G_{1} and G_{2} be finite groups with common abelian p-subgroup P and let Δ(P) be the diagonal subgroup {(x, x) : x ∈ P } of G_{1} × G_{2}. If F_{P} (G_{1}) = F_{P} (G_{2}), then S_{Δ(P)}(G_{1} × G_{2}, k) is Brauer indecomposable.

We do not know whether Theorem 1.2 holds without the assumption that P is abelian. Using D. Craven’s construction in [6] of the Scott modules for the symmetric groups S_{n}, n ≤ 6, we prove the following.

Proposition 1.5. Let G = S_{n}, n ≤ 6 and P a p-subgroup of S_{n}. If F_{P} (G) is a saturated fusion system, then S_{P} (G, k) is Brauer indecomposable.

Let A be a p-permutation G-algebra, finite dimensional over k, and b a primitive idempotent in the subalgebra of G-fixed points of A. To each triple (A, b, G), there is associated a G-poset of Brauer pairs. These were introduced in [2] for the case A = kG, considered as a G-algebra via the conjugation action of G on itself; the general case was treated in [5]. Roughly speaking, an (A, b, G)-Brauer pair is a pair of the form (P, e), where P is a p-subgroup of G and e is a block of the Brauer quotient A(P) of A in a prescribed relationship with b. For a maximal object (P, e) of the poset of (A, b, G)-Brauer pairs, we let F_{(P,e)}(A, b, G) denote the category whose objects are the subgroups of P and whose morphisms are group homomorphisms induced by the action of G on the underlying poset (for exact definitions we refer the reader to section 2). In case A = kG, the results of [2] imply that F_{(P,e)}(A, b, G) is a saturated fusion system (see [12]). In the general case, it is a consequence of [5] that F_{(P,e)}(A, b, G) is a fusion system in the sense of [3, Definition 1.1] (see Proposition 2.4). However, it is not the case that F_{(P,e)}(A, b, G) is in general saturated (see remarks after the proof of Theorem 1.1 in section 4). Theorem 1.1 is a special case of the following result, due to the first author, which gives a sufficiency criterion for saturation. For an (A, b, G)-brauer pair, (P, e), let C_{G}(P, e) denote the subgroup of C_{G}(P) which stabilizes the block e of A(Q) under the natural action of C_{G}(Q) on A(Q).

Theorem 1.6. Let G be a finite group, A a p-permutation G-algebra, and b a primitive idempotent of A^{G}. Suppose that

(i) b is a central idempotent of A; and

(ii) For each (A, b, G)-Brauer pair (Q, f) the idempotent e is primitive in A(Q)^{C_{G}(Q,e)}. Then for any maximal (A, b, G)-Brauer pair (P, e), F_{(P,e)}(A, b, G) is a saturated fusion system on P.

We will say that a triple (A, b, G) satisfying conditions (i) and (ii) of Theorem 1.6 is a saturated triple or that (A, b, G) is os saturated type. In this case, if G and b are clear from the context, we may also simply say that A is of saturated type. If A = kG, then the primitive idempotents of A^{G} are prcisely the blocks of kG, and it is easy to see that (A, b, G) is a saturated triple (see Remark at the end of Section 3), hence Theorem 1.6 may be viewed as a generalization of the fact that block fusion systems are saturated. But the class of p-permutation G-algebras is very large. One motivation, besides the relevance to Brauer indecomposability, for introducing the notion of saturated type triples is that they provide a new source of saturated fusion systems and hence may contribute to our understanding of these categories.

The paper is divided into four sections. In section 2, we recall the results and definitions of [2] and [5]. Section 3 contains the proof of Theorem 1.6. Section 4 deals with p-permutation modules, and contains the proofs of Theorem 1.1, Theorem 1.2, Corollary 1.3, Corolllary 1.4 and Proposition 1.5.

2. Background and Quoted results

In this section, we set up notation and recall definitions and background results on Brauer pairs from the papers [2] and [5]. For notation and terminology regarding fusion systems and saturated fusion systems, we refer the reader to [13],[3].

Let G be a finite group, and let A be a p-permutation G-algebra, finite dimensional over k. Recall that A is p-permutation if for any p-subgroup Q of G there is a k-basis of A stabilized by Q.

2.1. Let P be a subgroup of G. We denote by A^{P} the subalgebra consisting of the fixed points of A under P; if Q is a subgroup of P, the map Tr_{Q}^{P} : A^{Q} → A^{P} is the k-linear map defined by the formula Tr_{Q}^{P}(a) = ∑_{x∈P/Q} ^{x}a. The image of Tr_{Q}^{P}, denoted by A_{Q}^{P} is a two-sided ideal of A^{P} and we denote by A_{<P}^{P} the sum ∑_{Q} A_{Q}^{P}, where Q ranges over the proper subgroups of P. We denote by A(P) the quotient A^{P} /A_{<P}^{P}, and we denote by Br_{P}^{A} the canonical morphism from A^{P} onto A(P). Recall from [5, Proposition 1.5] that A(P) is a p-permutation N_{G}(P) algebra. For g ∈ G, the map which sends an element Br_{P}^{A}(a), where a ∈ A^{P} to the element ^{g}(Br_{P}^{A}(a)) := Br_{^{g}P}^{A} (^{g}a) is an algebra isomorphism from A(P) to A(^{g}P).

If Q ≤ P are p-groups, then there exists an algebra morphism, Br_{P,Q}^{A} : Br_{Q}^{A}(A^{P}) → A(P) such that Br_{P,Q}^{A}(Br_{Q}^{A}(a)) = Br_{P}^{A}(a) for a ∈ A^{P}. Clearly, ^{g}Br_{P,Q}^{A}(x) = Br_{^{g}P,^{g}Q}^{A}(^{g}x) for any g ∈ G, x ∈ Br_{Q}^{A}(A^{P}).

If, in addition, Q is normal in P, then Br_{Q}^{A}(A^{P}) = A(Q)^{P} and Ker(Br_{P,Q}^{A}) = Ker(Br_{P}^{A(Q)}). Thus, Br_{P,Q}^{A} induces an isomorphism b_{P,Q}^{A} : A(Q)(P) → A(P). Note that b_{P,Q}^{A} satisfies and is completely determined by the condition

b_{P,Q}^{A}(Br_{P}^{A(Q)}(Br_{Q}^{A}(x))) = Br_{P,Q}^{A}(Br_{Q}^{A}(x)))) = Br_{P}^{A}(x) for all x ∈ A^{P} .

Further, ^{g}b_{P,Q}^{A}(w) = b_{^{g}P,Q}^{A}(^{g}w) for all g ∈ N_{G}(Q) and w ∈ A(Q)(P).

2.2. Let b be a primitive idempotent of A^{G}. Recall from [5, Definition 1.6] that an (b, G)-Brauer pair is a pair (P, e) where P is a p-subgroup of G such that Br_{P} (b) ≠ 0 and e is a block of A(P) such that Br_{P} (b)e ≠ 0. Here we recall that a block of a finite dimensional algebra is a primitive idempotent of the center of the algebra. As we will consider Brauer pairs for different algebras simultaneously, we will adopt the more cumbersome notation (A, b, G)-Brauer pair for (b, G)-Brauer pair.

Recall from [5, Definition 1.6] the notion of inclusion of (A, b, G)-Brauer pairs: If (Q, f) and (P, e) are (A, b, G)-Brauer pairs, then (Q, f) ≤ (P, e) if Q ≤ P and whenever i is a primitive idempotent of A^{P} such that Br_{P}^{A}(i)e ≠ 0, then Br_{Q}^{A}(i)f ≠ 0.

Let (P, e) be an (A, b, G)-Brauer pair and let x ∈ G. The conjugate of (P, e) by x is the (A, b, G)-Brauer pair ^{x}(P, e) := (^{x}P, ^{x}e). Clearly, conjugation by x preserves inclusion.

Recall the following fundamental property of inclusion of Brauer pairs [2, Theorem 3.4],[5, Theorem 1.8].

Theorem 2.1. Let (P, e) be an (A, b, G)-Brauer pair, and let Q ≤ P.

(i) There exists a unique block f of A(Q) such that (Q, f) is an (A, b, G)-Brauer pair and (Q, f) ≤ (P, e).

(ii) If (Q, f) is an (A, b, G)-Brauer pair and P normalizes Q, then (Q, f) ≤ (P, e) if and only if P fixes f and Br_{P,Q}^{A}(f)e = e.

(iii) The set of (A, b, G)-Brauer pairs is a G-poset under the action of G defined above.

Recall also [2, Theorem 3.10] and [5, Theorem 1.14]).

Theorem 2.2. Let A be a p-permutation G-algebra and let b be a primitive idempotent of A^{G}. Then,

(i) The group G acts transitively on the set of maximal (A, b, G)-Brauer pairs. (ii) Let (P, e) be an (A, b, G)-Brauer pair. The following are equivalent.

(a) (P, e) is a maximal (A, b, G)-Brauer pair.

(b) Br_{P}^{A}(b) ≠ 0 and P is maximal amongst subgroups p-subgroups Q of G with the property that Br_{Q}^{A}(b) ≠ 0.

(c) b ∈ Tr_{P}^{G}(A^{P}) and P is minimal amongst subgroups H of G such that b ∈ Tr_{H}^{G}(A^{H}).

The equivalence of ii(b) above with ii(a) is not explictly stated in [5, Theorem 1.14], but is an immediate consequence of (i). For clearly, if P satisfies ii(b), then (P, e)-is a maximal (A, b, G)-Brauer pair. Conversely, if (P, e) is a maximal (A, b, G)- Brauer pair and P ≤ R is such that Br_{R}^{A}(b) ≠ 0, then there exists some block t of A(Q) such that (R, t) is an (A, b, G)-Brauer pair. Let (S, u) be a maximal (A, b, G)- Brauer pair with (R, t) ≤ (S, u). Then by (i), (P, e) and (S, u) are G-conjugate. In particular, |P | = |S| ≥ |R| ≥ |P |, hence P = R.

If Q, R are subgroups of G and g ∈ G is such that ^{g}Q ≤ R, then c_{g} : Q → R denotes the map which sends an element x of Q to the element ^{g}x := gxg^{−1} of R.

Definition 2.3. Let (P, e_{P}) be a maximal (A, b, G)-Brauer pair. For each subgroup Q of P, let (Q, e_{Q}) be the unique (A, b, G)-Brauer pair such that (Q, e_{Q}) ≤ (P, e_{P}). The category F_{(P,e_{P})}(A, b, G) is the category whose objects are the subgroups of P, whose morphisms are given by

Hom_{F_{(P,e_{P} )}(A,b,G)}(Q, R) := {c_{g} : Q → R|g ∈ G, ^{g}(Q, e_{Q}) ≤ (R, e_{R})}

for Q, R ≤ P, and where composition of morphisms is the usual composition of functions.

For any Q ≤ R, the inclusion map from Q to R is a morphism in F_{(P,e_{P})}(A, b, G). In particular, the identity map Q → Q is a morphism in F_{(P,e_{P})}(A, b, G) and if R, S ≤ P and g, h ∈ G are such that ^{g}(Q, e_{Q}) ≤ (R, e_{R}) and ^{h}(R, e_{R}) ≤ (S, e_{S}), then

hg(Q, e_{Q}) ≤ ^{h}(R, e_{R}) ≤ (S, e_{S}),

so F_{(P,e_{P})}(A, b, G) is a category. By the uniqueness of inclusion of Brauer pairs for Q, R ≤ P and g ∈ G, ^{g}(Q, e_{Q}) ≤ (R, e_{R}) if and only if ^{g}Q ≤ R and ^{g}e_{Q} = e_{^{g}Q} and this in turn holds if and only if ^{g}Q ≤ R and ^{g}(Q, e_{Q}) ≤ (P, e_{P}). Thus if x ∈ P, then since e_{P} is fixed by P, ^{x}e_{P} = e_{P}. Hence, for Q ≤ P,

x(Q, e_{Q}) ≤ ^{x}(P, e_{P}) = (P, e_{P}).

So, whenever ^{x}Q ≤ R, then c_{x} : Q → R is a morphism in F_{(P,e_{P})}(A, b, G).

Also, note that if Q, R ≤ P and g ∈ G are such that ^{g}(Q, e_{Q}) ≤ (R, e_{R}), then c_{g} : Q → R factors as c_{g} : Q → ^{g}Q followed by the inclusion of ^{g}Q into R. Summarizing the above discussion gives the following proposition, the last statement of which is immediate from the fact that any two maximal (A, b, G)- Brauer pairs are G-conjugate.

Proposition 2.4. Let A be a p-permutation G-algebra, b a primitive idempotent of A^{G} and (P, e_{P}) a maximal (A, b, G)-Brauer pair. Then F := F_{(P,e_{P})}(A, b, G) satisfies the following.

(i) Hom_{P} (Q, R) ⊆ Hom_{F}(Q, R) ⊆ Inj(Q, R) for all Q, R ≤ P.

(ii) For any ϕ ∈ Hom_{F}(Q, R), the induced isomorphism Q ≅ ϕ(Q) and its inverse are morphisms in F and its inverse are morphisms in F. In particular, every morphism in F factors as an isomorphism in F followed by an inclusion in F.

Thus, F is a fusion system in the sense of [3, Definiton 1.1]. If (P′, e_{P′}) is another maximal (A, b, G)-Brauer pair, then F_{(P′,e_{P′})}(A, b, G) is isomorphic to F_{(P,e_{P})}(A, b, G).

3. Proof of Theorem 1.6

Throughout this section, G will denote a finite group, A a p-permutation Galgebra, and b a primitive idempotent of A^{G}. Recall from the introduction that (A, b, G) is a saturated triple if conditions (i) and (ii) of Theorem 1.6 hold. Thus, we will prove that if (A, b, G) is a saturated triple, then F_{(P,e_{P})}(A, b, G) is saturated for any maximal (A, b, G)-Brauer pair (P, e_{P}). We need some preliminary results.

Lemma 3.1. Let H be a finite group and let B be an H-algebra. Let R be a subgroup of H and let C be a normal subgroup of H. Suppose that 1_{B} ∈ Tr_{R}^{H}(B^{R}) and 1_{B} is primitive in B^{C}. Then, RC/C contains a Sylow p-subgroup of H/C.

Proof. Let b ∈ B^{R} be such that

1_{B} = Tr_{R}^{H}(b) = Tr_{RC}^{H}(Tr_{R}^{RC}(b)),

and set u := Tr_{R}^{RC}(b). Then, u ∈ B^{RC} ⊆ B^{C}. By hypothesis, the identity 1_{B} = 1_{B^{C}} of B^{C} is the only idempotent of B^{C}. In other words, B^{C} is a local algebra which means that J(B^{C}) has co-dimension 1 in B^{C}. Thus, we may write u = λ1_{B} + v for some λ ∈ k and v ∈ J(B^{C}). Thus,

1_{B} = Tr_{RC}^{H}(λ1_{B} + v) = [H : RC]λ1_{B} + Tr_{RC}^{H}(v).

Now, since C is normal in H, H acts on B^{C} and hence on J(B^{C}). In particular, Tr_{RC}^{H}(v) ∈ J(B^{C}). But 1_{B} ∉ J(B^{C}). Hence, it follows from the above displayed equation that [H : RC] is not divisible by p, proving the lemma.

For (A, b, G)-Brauer pairs (Q, f) ≤ (P, e), set

N_{G}(P, e) := N_{(A,b,G)}((P, e)) := {x ∈ G : such that ^{x}(P, e) = (P, e)},

and

C_{G}(P, e) := N_{G}(P, e) ∩ C_{G}(Q).

Lemma 3.2. Let H be a finite group, B a p-permutation H-algebra and e a primitive idempotent of B^{H}. If e ∈ Z(B), then for a p-subgroup Q of H and a block f of B(Q), (Q, f) is an (B, e, H)-Brauer pair if and only if Br_{Q}^{B}(e)f = f.

Proof. Suppose that e ∈ Z(B) and let Q be a p-subgroup of H. Since

Z(B) ∩ B^{H} ⊆ Z(B) ∩ B^{Q} ⊆ Z(B^{Q}),

e is a central idempotet of B^{Q}. Hence, either Br_{Q}^{B}(e) = 0 or Br_{Q}^{B}(e) is a central idempotent of B(Q) and for any block f of B(R), either Br_{Q}^{B}(e)f = f, or Br_{Q}^{B}(e)f =

0. The result follows.

For the next result, we note the following. For an (A, b, G)-Brauer pair (Q, e), A(Q) is a N_{G}(Q)-algebra and e is an idempotent of A(Q)^{N_{G}(Q,e)}. Thus, if e is primitive in A(Q)^{C_{G}(Q,e)}, then e is a primitive idempotent of A(Q)^{H} for any H such that C_{G}(Q, e_{Q}) ≤ H ≤ N_{G}(Q, e_{Q}) and it makes sense to speak of (A(Q), e, H)- Brauer pairs.

Lemma 3.3. Suppose that (Q, e) is an (A, b, G)-Brauer pair such that e is primitive in A(Q)^{C_{G}(Q,e)} and let H be a subgroup of G with C_{G}(Q, e) ≤ H ≤ N_{G}(Q, e).

(i) The H-poset of (A(Q), e, H)-Brauer pairs is the H-subposet of (A(Q), e, N_{G}(Q, e))- Brauer pairs consisting of those pairs whose first component is contained in H.

(ii) The map

(R, α) → (QR, α)

is an H-poset homomorphism from the set of (A(Q), e, H)-Brauer pairs to the set of (AQ), e, QH)-Brauer pairs and induces a bijection between the set of (A(Q), e, H)- Brauer pairs whose first component contains Q ∩ H and the set of (A(Q), e, QH)- Brauer pairs whose first component contains Q.

(iii) If Q ≤ H, then (Q, e) is the unique (A(Q), e, H)-Brauer pair with first component Q and (Q, e) is contained every maximal (A(Q), e, H)-Brauer pair.

Proof. (i) This is immediate from the definitions.

(ii) Since Q acts trivially on A(Q), for any p-subgroup R of H, A(Q)^{R} = A(Q)^{QR} and Br_{R}^{A(Q)} = Br_{QR}^{A(Q)}. The first assertion is immediate from this observation. The second assertion follows from the first and the fact that R → QR is a bijection between subgroups of H containing Q ∩ H and subgroups of QH containing Q.

(iii) By hypothesis, A(Q)^{Q} = A(Q). Hence, A(Q)_{<Q}^{Q} = 0 and Br_{Q}^{A(Q)} is the identity map on A(Q). Thus, the set of (A(Q), e, H)-Brauer pairs with first component Q consists precisely of the pairs (Q, α), where α is a block of A(Q) such that eα ≠ 0. Since e itself is a block of A(Q) and any two distinct blocks of A(Q) are orthogonal, it follows that (Q, e) is an (A(Q), e, H)-Brauer pair and that it is the unique one with first component Q. Since ^{h}(Q, e) = (Q, e) for all h ∈ H and by Theorem 2.2(a) H acts transitively on the set of maximal (A(Q), e, H)-Brauer pairs, (Q, e) is contained in every maximal (A(Q), e, H)-Brauer pair.

To prove that a fusion system of a finite group G on a Sylow p-subgroup S of the group is saturated one applies Sylow’s theorem to the local subgroups N_{G}(Q) and N_{S}(Q)C_{G}(Q) of G, for Q a p-subgroup of G. The proof of Theorem 1.6 is based on the same idea with triples of the form (A(Q), e, N_{G}(Q, e_{Q})), (A(Q), e, N_{P} (Q)C_{G}(Q, e_{Q})) playing the role of local subgroups and Theorem 2.2 and Lemma 3.1 playing the role of Sylow’s theorem. The next result allows us to pass back and forth between (A, b, G)-Brauer pairs and (A(Q), e, H)-Brauer pairs. Recall the isomorphisms b_{R,Q}^{A} : A(Q)(R) → A(R) for p-subgroups Q ⊴ R of G introduced at the end of subsection 2.1.

Lemma 3.4. Suppose that (Q, e) is an (A, b, G)-Brauer pair such that e is primitive in (A(Q))^{C_{G}(Q,e)} and let H be a subgroup of G with QC_{G}(Q, e) ≤ H ≤ N_{G}(Q, e).

The map

(R, α) → (R, b_{R,Q}^{A}(α))

is an H-poset isomorphism between the subset of (A(Q), e, H)-Brauer pairs consisting of those pairs whose first component contains Q, and the subset of (A, b, G)- Brauer pairs containing (Q, e) and whose first component is contained in H.

In particular, H acts transitively on the subset of (A, b, G)-Brauer pairs which are maximal with respect to containing (Q, e) and having first component contained in H.

Proof. Let P_{1} be the subset of (A(Q), e, H)-Brauer pairs consisting of those pairs whose first component contains Q, and let P_{2} be the subset of (A, b, G)- Brauer pairs containing (Q, e) and whose first component is contained in H. Since H ≤ N_{G}(Q, e) ≤ N_{G}(Q), P_{1} and P_{2} are H-posets. Now let Q ≤ R ≤ H, and let α be a block of A(Q)(R). By Lemma 3.2, e = Br_{Q}^{A}(b)e, hence

Br_{R,Q}^{A}(e) = b_{R,Q}^{A}(Br_{R}^{A(Q)}(e)) = b_{R,Q}^{A}(Br_{R}^{A(Q)}(Br_{Q}^{A}(b)e)) = Br_{R}^{A}(b)Br_{R,Q}^{A}(e).

Suppose first that (R, α) is an (A(Q), e, H)-Brauer pair. By Lemma 3.2, α = Br_{R}^{A(Q)}(e)α. Applying b_{R,Q}^{A} to both sides of this equation, and using the displayed equation above, we get that

b_{R,Q}^{A}(α) = Br_{R,Q}^{A}(e)b_{R,Q}^{A}(α) = Br_{R}^{A}(b)Br_{R,Q}^{A}(e)b_{R,Q}^{A}(α).

In particular, Br_{R}^{A}(b)b_{R,Q}^{A}(α) ≠ 0, whence (R, b_{R,Q}^{A}(α)) is an (A, b, G)-Brauer pair. By Theorem 2.1 and the first equality above, (Q, e) ≤ (R, b_{R,Q}^{A}(α)) as (A, b, G)- Brauer pairs.

Conversely, if (Q, e) ≤ (R, b_{R,Q}^{A}(α)), then again by Theorem 2.1, b_{R,Q}^{A}(α) = Br_{R,Q}^{A}(e)b_{R,Q}^{A}(α). Applying the inverse of b_{R,Q}^{A} yields that α = Br_{R}^{A(Q)}(e)α, hence that (R, α) is an (A(Q), e, H)-Brauer pair. This shows that (R, α) → (R, b_{R,Q}^{A}(α)) is a bijection between P_{1} and P_{2}.

We show that the bijection is inclusion preserving. Let (R, α) and (S, β) be (A(Q), e, H)-Brauer pairs with Q ⊲ R ≤ S. By Theorem 2.1, it suffices to consider the case that R⊴S. Clearly, α is S-stable if and only if b_{R,Q}^{A}(α) is S-stable. Further, the restrictions of the maps b_{S,Q}^{A} ∘ Br_{S,R}^{A(Q)} ∘ Br_{R}^{A(Q)} ∘ Br_{Q}^{R} and Br_{S,R}^{A} ∘ b_{R,Q}^{A} ∘ Br_{R}^{A(Q)} ∘

Br_{Q}^{A} to A^{S} both equal Br_{S}^{A}. Since Br_{R}^{A(Q)} ∘ Br_{Q}^{R}(A^{S}) = A(Q)(R)^{S}, it follows that b_{S,Q}^{A} ∘ Br_{S,R}^{A(Q)} is equal to the restriction of Br_{S,R}^{A} ∘ b_{R,Q}^{A} to A(Q)(R)^{S}. In particular,

Br_{S,R}^{A(Q)}(α)β = β if and only if Br_{S,R}^{A}(b_{R,Q}^{A}(α))b_{S,Q}^{A}(β) = b_{S,Q}^{A}(β). Thus, by Theorem 2.1 (R, α) ≤ (S, β) if and only if (R, b_{R,Q}^{A}(α)) ≤ (S, b_{S,Q}^{A}(β)), and the bijection is inclusion preserving. Since Q is normal in H,

b_{^{h}R,Q}^{A}(^{h}α) = b_{^{h}R,^{h}Q}^{A}(^{h}α) = ^{h}b_{R,Q}^{A}(α)

for all h ∈ H, all p-subgroups R of G containing Q as normal subgroup and all α ∈ A(Q)(R), and hence the above bijection is compatible with the H-action on P_{1} and P_{2}. This proves that the given map is an isomorphism of H-posets. In particular, the map induces a bijection between the set of maximal elements of P_{1} and P_{2}. But by Lemma 3.3 (c), the set of maximal elements in P_{1} is precisely the set of maximal (A(Q), e, H)-Brauer pairs. The final assertion follows from this and from the fact that H acts transitively on the set of maximal (A(Q), e, H)-pairs (see 2.2 (a)).

We will prove Theorem 1.6 by using the the saturation axioms given by Robertson and Schpectorov in [16] . For this we recall the following terminology: If F is a fusion system on a finite p-group P, then a subgroup Q of P is fully automized if Aut_{P} (Q) is a Sylow p-subgroup of Aut_{F}(Q) and Q is receptive if for any isomorphism φ : R → Q in F, there exists a morphism φ̂ : N_{φ} → P in F such that Res|_{R}φ̂ = φ, where N_{φ} is the subgroup of N_{P} (R) consisting of those elements z ∈ N_{P} (R) such that φ ∘ c_{z} = c_{x} ∘ φ for some x ∈ N_{P} (Q).

Lemma 3.5. Suppose that (A, b, G) is a saturated triple and let (P, e_{P}) be a maximal (A, b, G)-Brauer pair. For each Q ≤ P let e_{Q} be the unique block of A(Q) such that (Q, e_{Q}) ≤ (P, e_{P}) and let F = F_{(P,e_{P})}(A, b, G). If Q ≤ P is such that (N_{P} (Q), e_{N_{P} (Q)}) is maximal amongst (A, b, G)-Brauer pairs (R, f) with (Q, e_{Q}) ≤ (R, f) and R ≤ N_{G}(Q, e_{Q}), then Q is fully F-automised and F-receptive.

Proof. Suppose that (N_{P} (Q), e_{N_{P} (Q)}) is maximal amongst (A, b, G)-Brauer pairs (R, f) such that (Q, e_{Q}) ≤ (R, f) and R ≤ N_{G}(Q, e_{Q}). Let

α = b_{N_{P} (Q),Q}^{A}(e_{N_{P} (Q)}).

By Lemma 3.4, (N_{P} (Q), α) is a maximal (A(Q), e_{Q}, N_{G}(Q, e_{Q}))-Brauer pair. Thus, by Theorem 2.2 (b), e_{Q} ∈ Tr_{N_{P} (Q)}^{N_{G}(Q,e_{Q})}(A(Q)^{N_{P} (Q)}). Since e_{Q} is central in A(Q), idempotent and an element of A^{N_{G}(Q,e_{Q})} multiplying on both sides by e_{Q} gives that

e_{Q} ∈ Tr_{N_{P} (Q)}^{N_{G}(Q,e_{Q})}((e_{Q}A(Q)e_{Q})^{N_{P} (Q)}).

Now, C_{G}(Q, e_{Q}) is a normal subgroup of N_{G}(Q, e_{Q}) and since (A, b, G) is a saturated triple e_{Q} is a primitive idempotent of (A(Q))^{C_{G}(Q,e_{Q})} and hence also of (e_{Q}A(Q)e_{Q})^{C_{G}(Q,e_{Q})}. Thus, by Lemma 3.1 applied with B = e_{Q}A(Q)e_{Q}, H = N_{G}(Q, e_{Q}), C = C_{G}(Q, e_{Q}) and R = N_{P} (Q), we have that N_{P} (Q)C_{G}(Q, e_{Q})/C_{G}(Q, e_{Q}) is a Sylow p-subgroup of N_{G}(Q, e_{Q})/C_{G}(Q, e_{Q}). Since N_{P} (Q)C_{G}(Q, e_{Q})/C_{G}(Q, e_{Q}) ≅ N_{P} (Q)/C_{P} (Q) ≅ Aut_{P} (Q) and N_{G}(Q, e_{Q})/C_{G}(Q, e_{Q}) ≅ Aut_{F}(Q), it follows that Q is fully F-automised.

It remains to show that Q is F-receptive. For this, we first observe that the hypothesis on Q implies that (N_{P} (Q), e_{N_{P} (Q)}) is also maximal amongst (A, b, G)- Brauer pairs (R, f) such that (Q, e_{Q}) ≤ (R, f) and R ≤ N_{P} (Q)C_{G}(Q, e_{Q}). Hence, by Lemma 3.4, now applied with H = N_{P} (Q)(C_{G}(Q, e_{Q}), (N_{P} (Q), e_{N_{P} (Q)}) contains an N_{P} (Q)C_{G}(Q, e_{Q}) conjugate of any (A, b, G)-Brauer pair which contains (Q, e_{Q}) and whose first component is contained in N_{P} (Q)C_{G}(Q, e_{Q}). Now let φ : R → Q be an isomorphism in F, and let g ∈ G induce φ, that is, ^{g}(R, e_{R}) = (Q, e_{Q}) and φ(x) = gxg^{−1} for all x ∈ R. Then, it is an easy check that N_{φ} = N_{P} (R)∩^{g^{−1}}N_{P} (Q)C_{G}(Q, e_{Q}). Set N′ = ^{g}N_{φ} = ^{g}N_{P} (R)∩N_{P} (Q)C_{G}(Q, e_{Q}), e′_{N′} = ge_{N_{φ}} and consider the (A, b, G)-Brauer pair (N′, e_{N}). Since (R, e_{R}) ≤ (N_{φ}, e_{N_{φ}}), (Q, e_{Q}) ≤ ^{g}(N_{φ}, e_{N_{φ}}) = (N′, e′_{N′}). Also, N′ ≤ N_{P} (Q)C_{G}(Q, e_{Q}). Thus, as pointed out above ^{h}(N′, e′_{N′}) ≤ (N_{P} (Q), e_{N_{P} (Q)}) for some h ∈ N_{P} (Q)C_{G}(Q, e_{Q}). Multiplying by some element of N_{P} (Q) if necessary, we may assume that h ∈ C_{G}(Q, e_{Q}). Since ^{hg}(N_{φ}, e_{N_{φ}}) ≤ (P, e_{P}) and hence φ̄ := c_{hg} : N_{φ} → P is a morphism in F. and since h ∈ C_{G}(Q, e_{Q}), φ̄ extends φ. Thus Q is F-receptive.

We now give the proof of Theorem 1.6.

Proof. Keep the notation of the theorem, set F = F_{(P,e_{P})}(A, b, G) and for each Q ≤ P, let e_{Q} be the unique block of A(Q) such that (Q, e_{Q}) ≤ (P, e_{P}). We have shown in Proposition 2.4 that F is a fusion system on P. Thus, by Lemma 3.5 nad by the saturation axioms of [16] it suffices to show that each subgroup of P is F-conjugate to a subgroup Q of P such that (N_{P} (Q), e_{N_{P} (Q)}) is maximal amongst (A, b, G)-Brauer pairs (R, f) with (Q, e_{Q}) ≤ (R, f) and R ≤ N_{G}(Q, e_{Q}). So, let Q′ ≤ P, and let (T, α) be a maximal (A(Q′), e_{Q′}, N_{G}(Q′, e_{Q′}))- Brauer pair. By Lemma 3.3 (c), Q′ ≤ T. Let f = b_{R,Q}^{A}^{−1}(α). By Lemma 3.4, (T, f) is an (A, b, G)- Brauer pair with (Q′, e_{Q′}) ≤ (T, f). Since (P, e_{P}) is a maximal (A, b, G)-Brauer pair, we have

g(Q′, e_{Q′}) ≤ ^{g}(T, f) ≤ (P, e_{P})

for some g ∈ G. Set Q = ^{g}Q′. By the above, c_{g} : Q′ → Q is a morphism in F, so Q is F-conjugate to Q′. We will show that (N_{P} (Q), e_{N_{P} (Q)}) has the required maximality property. Note that by Lemma 3.4, (T, f) is maximal amongst (A, b, G)-Brauer pairs which contain (Q′, e_{Q′}) and whose first component is contained in N_{G}(Q′, e_{Q′}). Thus, by transport of structure ^{g}(T, f) is maximal amongst (A, b, G)-Brauer pairs which contain (Q, e_{Q}) and whose first component is contained in N_{G}(Q, e_{Q}). Since ^{g}(T, f) ≤ (P, e_{P}), ^{g}T ≤ N_{P} (Q) and ^{g}f = e_{^{g}T}. Consequently, ^{g}(T, f) ≤ (N_{P} (Q), e_{N_{P} (Q)}). Since (N_{P} (Q), e_{N_{P} (Q)}) contains (Q, e_{Q}) and N_{P} (Q) is contained in N_{G}(Q, e_{Q}), the maximality of ^{g}(T, f) forces ^{g}(T, f) = (N_{P} (Q), e_{N_{P} (Q)}), and completes the proof of the theorem.

4. p-permutation modules and saturation

Let G be a finite group, M an indecomposable p-permutation kG-module, and P a vertex of M and set A = End_{k}(M). Then A is a G-algebra via the map

G × A → A,

sending the pair (g, ϕ) to the element ^{g}ϕ of A defined by

gϕ(m) = gϕ(g^{−1}m), m ∈ M.

Since M is a p-permutation module, M is a p-permutation G-algebra and since M is indecomposable, 1_{A} = id_{M} is primitive in End_{k}(M)^{G}.

Proposition 4.1. With the notation above, the (A, 1_{A}, G)-Brauer pairs are the pairs (Q, 1_{A(Q)}) such that M(Q) ≠ 0 and (P, 1_{A(P)}) is a maximal (End_{k}(M), 1_{End_{k}(M)}, G)- Brauer pair. Further,

(i) F_{(P,1_{A(P)})}(A, 1_{A}, G) = F_{P} (G).

(ii) The triple (A, 1_{A}, G) is of saturated type if and only if M is Brauer indecomposable.

Proof. Let Q be a p-subgroup of G. There is a natural action of A(Q) on M(Q) which induces an isomorphism of kN_{G}(Q)/Q-algebras. between A(Q) and End_{k}(M(Q)) (see for instance [17, Proposition 27.6]). Since the identity element is the only central idempotent of a matrix algebra, it follows that the (A, 1_{A}, G)- Brauer pairs are the pairs (Q, 1_{A(Q)}) such that M(Q) ≠ 0. The maximality of (P, 1_{A(P)}) is immediate from the fact that P is a vertex of P and that M(Q) ≠ 0 if and only if Q is contained in a vertex of M (see [17, Corollary 27.6]). Clearly, g1_{A(Q)} = 1_{A(^{g}Q)}, for any g ∈ G and (i) is immediate from this. Under the natural identification of A(Q) and End_{k}(M(Q)) 1_{A(Q)} = id_{M(Q)}. Hence 1_{A(Q)} is primitive in (A(Q))^{C_{G}(Q)} if and only if M(Q) is an indecomposable kQC_{G}(Q)/Q-module. The equivalence of (ii) is immediate from this and the fact that 1_{A} is a central idempotent of A and hence of A^{G}.

Proof of Theorem 1.1. In light of Proposition 4.1, this is a special case of Theorem 1.6.

Remarks 1. Let P be a p-subgroup of G. Since there exist indecomposable p-permutation kG-modules with vertex P, the analysis before the statement of Theorem 1.1 shows that given any p-subgroup P of a finite group G, there exists a p-permutation G-algebra A, and a primitive idempotent b of A^{G} such that there is a maximal (A, b, G)-Brauer pair, say (P, e_{P}) with first component P and such that F_{(P,e_{P})}(A, b, G) = F_{P} (G). On the other hand, there exist pairs P, G where G is a finite group and P is a p-subgroup of G such that F_{P} (G) is not a saturated system-for instance if P is a non-Sylow p-subgroup of G such that N_{S}(P) strictly contains P C_{S}(P) for some Sylow p-subgroup S of G containing P. Thus, the fusion system F_{(P,e_{P})}(A, b, G) is not always saturated.

2. Suppose that b is a (non-principal) block of kG such that a defect group P of kGb is a Sylow p-subgroups of G, but Br_{P}^{kG}(b) is a sum of more than one block of kC_{G}(P). Let M be an indecomposable p-permutation module kG-module in the block b and with vertex P. Then, since N_{G}(P) acts transitively on the set E of blocks e of kC_{G}(P) such that Br_{P}^{kG}(b)e = e and M(P) ≠ 0, M(P)e ≠ 0 for any e ∈ E, and in particular, M(P) is not indecomposable as kC_{G}(P)-module. However, since P is a Sylow p-subgroup of G, F_{P} (G) is a saturated fusion system on P (see [3]). Thus, the converse of Theorem 1.1 does not hold in general. Since Theorem 1.1 is a special case of Theorem 1.6, it follows also that the converse of Theorem 1.6 does not hold. It might be that the methods of proof of Theorem 1.6 can be refined to yield a condition on (A, b, G) which in certain situations (as in the one just discussed) is weaker than the condition of (A, b, G) being a saturated triple, and which in all cases is necessary and sufficient for the saturation of the corresponding fusion systems.

We now prove Theorem 1.2. We need some lemmas. The following is well known.

Lemma 4.2. Let H be a finite group and N a normal subgroup of H such that H/N is a p′-group. Then, the restriction of the projective cover of the trivial kH-module to kN is indecomposable.

Proof. Under the hypothesis, J(kG) = J(kN)kH. Let V be a projective kHmodule. Then,

Res_{N}^{H}Rad(V) = Res_{N}^{H}J(kH)V = Res_{N}^{H}J(kN)kHV = Res_{N}^{H}J(kN)V = Rad(Res_{N}^{H}V).

Consequently,

Res_{N}^{H}(V/Rad(V)) = Res_{N}^{H}V/Rad(Res_{N}^{H}V).

The result is immediate.

Remark. The above indecomposability result holds for the projective cover of any simple kH-module whose restriction to N remains simple.

Lemma 4.3. Let G be a finite group, P a p-subgroup of G and M := S_{P} (G, k) the Scott module of kG relative to P.

(i) M(P) is indecomposable as P C_{G}(P)/P -module if and only if N_{G}(P)/P C_{G}(P) is a p′-group.

(ii) If F_{P} (G) is a saturated fusion system, then M(P) is indecomposable as P C_{G}(P)/P -module.

Proof. (i) M(P) is the projective cover of the trivial N_{G}(P)/P -module and in particular is indecomposable as kN_{G}(P)/P -module. The forward implication follows from Lemma 3.1, applied with B = End_{k}(M(P)), H = N_{G}(P), R = P and C = C_{G}(P). The backward implication is clear from Lemma 4.2.

(ii) Suppose that F_{P} (G) is a saturated fusion system. Then, Aut_{P} (P) is a Sylow p-subgroup of Aut_{F}(P). On the other hand, the image of Aut_{P} (P) under the natural isomorphism from Aut_{F}(P) to N_{G}(P) is P C_{G}(P)/P. Thus, N_{G}(P)/P C_{G}(P) is a p′-group. The result is immediate from (i).

Lemma 4.4. Let G be a finite group, P a p-subgroup of G, M = S_{P} (G, k) the Scott module of kG relative to P. Suppose that F_{P} (G) is a saturated fusion system and let Q ≤ Z(P). If M(Q) is indecomposable as kN_{G}(Q)/Q-module, then M(Q) is indecomposable as kC_{G}(Q)/Q-module.

Proof. Suppose that M(Q) is indecomposable as N_{G}(Q)/Q-module and set L = N_{G}(Q) and C = C_{G}(Q). Since Q ≤ Z(P) the extension axiom for saturated fusion systems implies that L = C[N_{G}(P) ∩ L]. We consider M(Q) as kL-module via inflation. Since M(Q) has vertex P and P ≤ C, there exists an indecomposable p-permutation kC-module V with vertex P such that M(Q) is a direct summand of Ind_{C}^{L}V. Let W be an indecomposable summand of Res_{C}^{L}Ind_{C}^{L}V. By the Mackey formula, W ≅ ^{x}V for some x ∈ L. In particular, ^{x}P is a vertex of ^{x}V. By the decomposition of L given above, x = uv for some u ∈ C_{G}(Q), v ∈ N_{G}(P). Thus, xP = ^{u}P is C-conjugate to P, and it follows that P is a vertex of W. In particular, W (P) ≠ 0. Let

Res_{C}^{L}M(Q) = W_{1} ⊕ ⋯ ⊕ W_{s}

be a decomposition of M(P) as a direct sum of indecomposable kC-modules and suppose if possible that s > 1. By the above argument, W_{i}(P) ≠ 0 for i, 1 ≤ i ≤ s, hence

Res_{C∩N_{G}(P)}^{N_{G}(P)}M(P) ≅ (Res_{C}^{L}M(Q))(P) = W_{1}(P) ⊕ ⋯ ⊕ W_{s}(P)

is not indecomposable. Since C_{G}(P) ≤ C ∩ N_{G}(P), it follows that Res_{C_{G}(P)}^{N_{G}(P)}M(P) is not indecomposable. But by the Sylow axiom for saturated fusion systems, N_{G}(P)/P C_{G}(P) = Aut_{F}(P) is a p′-group. This contradicts Lemma 4.3.

Proof of Theorem 1.2. Let M = S_{P} (G, k). Suppose that F := F_{P} (G) is saturated and let Q ≤ P. We will show that M(Q) is indecomposable as kC_{G}(Q)-module. We proceed by induction on the index of Q in P. If Q = P, then by Lemma 4.3, M(Q) is indecomposable as kP C_{G}(P)/P -module. Suppose now that Q is proper in P and that M(R) is indecomposable as kRC_{G}(R)/R-module for any p-subgroup R of P properly containing Q. Since P ≤ N_{G}(Q), S_{P} (N_{G}(Q), k) is a direct summand of Res_{N_{G}(Q)}^{G}M (see [14, Chapter 4, Theorem 8.6]). Write

Res_{N_{G}(Q)}^{G}M = S_{P} (N_{G}(Q), k) ⊕ X.

We claim that X(Q) = 0. Indeed, suppose if possible that there exists a direct summand, say N of X such that N(Q) ≠ 0 and let R be a vertex of N. Since Q is normal in N_{G}(Q), we have that Q ≤ R. The group Q is not a vertex of the indecomposable kG-module M. Hence by the Burry-Carlson-Puig theorem (see [14, Chapter 4, Theorem 4.6 (ii)]), Res_{N_{G}(Q)}^{G}M does not have any indecomposable summand with vertex Q. Thus Q is a proper subgroup of R. On the other hand, since M is a summand of Ind_{P}^{G}k, and N is a summand of Res_{N_{G}(Q)}^{G}M, by the Mackey formula, N is relatively ^{x}P ∩ N_{G}(Q)-projective for some x ∈ G. Thus,

Q < R < ^{x}P and Q < P.

In particular, conjugation by x is an F-isomorphism from ^{x^{−1}}Q to Q. Now P is abelian and F is saturated. So, by the extension axiom there exists a g ∈ N_{G}(P) such that gx^{−1} ∈ C_{G}(Q). Setting h = gx^{−1}, and conjugating all terms in the above by h, we get

Q = ^{h}Q < ^{h}R < ^{hx}P = ^{g}P = P.

Since h ∈ N_{G}(Q), replacing R by ^{h}R, we may assume that R ≤ P. Since N is a summand of X and N(R) ≠ 0, we have X(R) ≠ 0. Since S_{P} (N_{G}(Q), k) has vertex P and R ≤ P, we also have that S_{P} (N_{G}(Q), k)(R) ≠ 0. The equation

Res_{N_{G}(Q)}^{G}M = S_{P} (N_{G}(Q), k) ⊕ X,

implies that M(R) is not indecomposable as k[N_{G}(Q) ∩ N_{G}(R)]-module. Since RC_{G}(R) ≤ N_{G}(Q)∩N_{G}(R), it follows that M(R) is not indecomposable as kRC_{G}(R)- module or equivalently as kRC_{G}(R)/R-module, a contradiction. This proves the claim. Thus,

M(Q) = S_{P} (N_{G}(Q), k)(Q) ⊕ X(Q) = S_{P} (N_{G}(Q), k)

as kN_{G}(Q) and hence as kN_{G}(Q)/Q-module. In particular, M(Q) is indecomposable as kN_{G}(Q)/Q-module. By Lemma 4.4, M(Q) is indecomposable as kQC_{G}(Q)/Qmodule, completing the proof.

Proof of Corollary 1.3. If G has cyclic Sylow p-subgroups, then it is easy to see that F_{P} (G) is saturated for any p-subgroup P of G. The result is immediate from the Theorem 1.2.

Proof of Corollary 1.4. With the hypothesis of the statement, it is immediate that F_{Δ(P)}(G_{1} × G_{2}) ≅ F_{P} (G_{1}). Thus, since P is Sylow in G_{1}, F_{P} (G_{1}) and hence F_{Δ(P)}(G_{1} ×G_{2}) is a saturated fusion system on P (see [3]). The result follows from Theorem 1.2.

Finally, we prove Proposition 1.5. For this we set up some more notation and recall a few facts about Scott modules. Let (K, O, k)-be a p-modular system (we assume here that k is an algebraic closure of the field of p-elements). Let G = S_{n}, and let P be a p-subgroup of G. Let M = S_{P} (G, k) be the kG-Scott module with vertex P and let M̃ = S_{P} (G, O) be the OG-Scott module with vertex P, so that M = k ⊗_{O} M̃. Let χ : M̃ → K be the character of the OG-module M̃. Since M̃ is a p-permutation OG-module, for any p- element x of G, dim_{k} M(⟨x⟩) = χ(x). In particular, if Q is a p-subgroup of G, then dim_{k} M(Q) ≤ χ(x) for any element x of Q, with equality if Q = ⟨x⟩.

Proof of Proposition 1.5. Suppose that n ≤ 6 and that F_{P} (G) is saturated. We will show that M(Q) is indecomposable as kC_{G}(Q)/Q-module for every subgroup Q of P. By Theorem 1.2, we may assume that P is not abelian. If P is a Sylow p-subgroup of G, then M = k [4, Theorem 2.5] and the result is immediate. So, we may assume that P is a non-abelian, non-Sylow p-subgroup of G. Consequently, p = 2, n = 6 and P is isomorphic to the dihedral group of order 8.

By the Sylow axiom for saturated fusion systems, P C_{G}(P) is a Sylow 2-subgroup of N_{G}(P). So, up to G-conjugacy P is one of ⟨(1, 2, 3, 4), (1, 3)⟩, ⟨(1, 2, 3, 4)(5, 6), (1, 3)⟩ ⟨(1, 2, 3, 4), (1, 3)(5, 6)⟩ or ⟨(1, 2, 3, 4)(5, 6), (1, 3)(5, 6)⟩.

We will show that in each case above, M is Brauer indecomposable. It can be checked directly that F_{P} (G) is saturated in each case above- the second case corresponds to the nilpotent fusion system, the remaining three correspond to the saturated fusion system on D_{8} in which the automorphism of exactly one Klein-4 subgroup contains an element of order 3. However, we do not prove saturation as by Theorem 1.1 this will follow after the fact of Brauer indecomposability.

Before embarking on our case by case analysis, we recall the 2-decomposition matrix of S_{6} [7, Page 414]:

1  4_{1}  4_{2}  16
(1) (5, 1) (4, 2) (3, 2, 1)
1 (6)  1
5 (5, 1)  1  1
9 (4, 2)  1  1  1
16 (3, 2, 1)  1
10 (4, 1^{2}) 2  1  1
5 (3^{2})  1  1
10 (3, 1^{3}) 2  1  1
5 (2^{3})  1  1
9 (2^{2}, 1^{2}) 1  1  1
5 (2, 1^{4}) 1  1
1 (1^{6})  1

\underline{Case:} P = ⟨(1, 2, 3, 4), (1, 3)⟩. Then P is a Sylow p-subgroup of S_{5}, naturally considered as a subgroup of S_{6} as a one-point stabilizer, whence M̃ is a direct summand of Ind_{S_{5}}^{S_{6}}(O) (see for instance [4, Theorem 2.5]). On the other hand by [6, Page 32], M has dimension 6. So, M̃ = Ind_{S_{5}}^{S_{6}}(O). Now, if u = (1, 3), then χ(u) = 4 and if u = (1, 2)(3, 4) or u = (1, 2, 3, 4) then χ(u) = 2. Hence, it follows that unless Q ≤ P is G-conjugate to ⟨(1, 3)⟩, the dimension of M(Q) ≤ 2 and if Q = ⟨(1, 3)⟩, then M(Q) has dimension 4. On the other hand, since M(P) as kN_{G}(P)/P -module is the projective cover of the trivial module, M(P) has dimension at least 2. So, if Q ≤ P is not G-conjugate to ⟨(1, 3)⟩, then for any R ≤ P containing Q as a normal subgroup, M(Q) ≅ M(R) as k(N_{G}(Q)∩N_{G}(R))-module, hence as kC_{G}(R)- modules. Arguing inductively, it follows that M(Q) ≅ M(P) as kC_{G}(P)-modules. By Lemma 4.3, M(P) is indecomposable as kP C_{G}(P)/P -module, hence as kC_{G}(P)- module. Since C_{G}(P) ≤ QC_{G}(Q), it follows that M(Q) is indecomposable as kQC_{G}(Q)/Q-module.

Now suppose that Q = ⟨(1, 3)⟩. Then M(Q) is a 4-dimensional p-permutation kN_{G}(Q)-module. Let V be an indecomposable kN_{G}(Q)-module summand of M(Q) and let Q ≤ R ≤ N_{G}(Q) be a vertex of M(Q). Then

M(R) = M(Q)(R) ≠ 0,

whence ^{g}Q ≤ ^{g}R ≤ P or ^{g}R ≤ N_{P} (^{g}Q). Since no transposition in P is central in P, R has order at most 4 (and for some summand V exactly 4). Let S be a Sylow p-subgroup of N_{G}(Q) containing R. Since V is a direct summand of Ind_{R}^{N_{G}(Q)}(k), the Mackey formula and the Green indecomposability theorem imply that any direct summand of Res_{S}^{N_{G}(Q)}V is isomorphic to Ind_{^{x}R∩S}^{S}k for some x ∈ N_{G}(Q). In particular, the dimension of V is divisible by the index of R in V. Since the Sylow p-subgroups of C_{G}(Q) = N_{G}(Q) have order 16 and R has order 8, it follows that V has dimension divisible by 4. Thus, V = M(Q). In particular, M(Q) is indecomposable as kN_{G}(Q), and N_{G}(Q) = C_{G}(Q).

\underline{Case:} P = ⟨(1, 2, 3, 4), (1, 3)(5, 6)⟩. By [6] M has composition factors 1_{G}, 4_{1} ⊕ 4_{2}, 1_{G}. An inspection of the decomposition matrix and the character table of S_{6} gives that χ = χ_{(6)} + χ_{(4,2)}. Further, the values of χ on non-trivial 2 elements of G are as follows:

χ((1, 3)) = 4, χ((1, 3)(2, 4)) = 2, χ((1, 2)(3, 4)(5, 6)) = 4,
χ((1, 2, 3, 4)) = 0, χ((1, 2, 3, 4)(5, 6)) = 2.

Since C_{G}(P)/Z(P) contains an element of order 2, it follows as in the previous case that M(Q) is indecomposable as kC_{G}(Q)-module for any p-subgroup Q of G such that M(Q) has dimension 2. From the above character calculations, we may assume that the only non-trivial elements of Q are in the G-conjugacy class of (1, 3) and (1, 2)(3, 4)(5, 6) and in particular are non-central involutions in P. If Q contains two such involutions, then Q = P, so we may assume that either Q = ⟨(1, 3)⟩ or Q = ⟨(1, 2)(3, 4)(5, 6)⟩. But now the result follows as above since both of these involutions are central in some Sylow p-subgroup and in both cases M(Q) has dimension 4.

\underline{Case:} P = ⟨(1, 2, 3, 4), (1, 3)(5, 6)⟩. The image of P under the exceptional noninner automorphism of S_{6} is S_{6}-conjugate to ⟨(1, 2, 3, 4), (1, 3)⟩. The result follows from Case 1 by transport of structure.

\underline{Case:} P = ⟨(1, 2, 3, 4)(5, 6), (1, 3)(5, 6)⟩. By [6] M is two dimensional with composition factors 1_{G}, 1_{G}. Since M(P) has dimension at least 2, M(Q) = M(P) = M for all Q ≤ P. By Lemma 4.3, M = M(P) is indecomposable as kP C_{G}(P)/P - module. Hence, M(Q) = M is indecomposable as kQC_{G}(Q)/Q-module for all Q ≤ P as required. This completes the proof of Proposition 1.5.

Concluding Remarks. Given a saturated fusion system, F on a finite p-group P, Park has shown that there exists a finite group G with P ≤ G and such that F = F_{P} (G) (cf.[15]). We pose the following question:

Given a saturated fusion system F on a finite p-group P, does there exist a saturated triple (A, b, G) such that F = F_{(P,e_{P})}(A, b, G) for some maximal (A, b, G)- Brauer pair (P, e_{P}) ?

Acknowledgments The second author was partially supported by the Ministry of Education, Culture, Sports and Technology (MEXT) Grant-in-Aid for Young Scientists (B)19740018, 2007-2009, and also Grant-in-Aid for Young Scientists (B) 22740025, 2010-2012.

References

[1] J. L. Alperin, Local representation theory, Modular representations as an introduction to the local representation theory of finite groups. Cambridge Studies in Advanced Mathematics, 11. Cambridge University Press, Cambridge, 1986.

[2] J. L. Alperin and M. Broué, Local methods in block theory, Ann. of Math. 110 (1979), 143–157.

[3] C. Broto, R. Levi, R. Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), 779-856.

[4] M. Broué, On Scott modules and p-permutation modules: an approach through the Brauer morphism, Proc. Amer. Math. Soc. 93 (1985), 401–408.

[5] M. Broué and L. Puig, Characters and Local structures in G-Algebras, Journal of Algebra 63 (1980), 51–59.

[6] D. Craven, Scott modules and lower defect groups, Transfer Thesis, Oxford, 2005.

[7] G. James, A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its applications, Addison-Wesley, 1981.

[8] R. Kessar, Introduction to block theory, Group representation theory, EPFL Press, Lausanne (2007), 47–77.

[9] R. Kessar, R. Stancu, A reduction theorem for fusion systems of blocks, J. Algebra 319 (2008), 806–823.

[10] S. Koshitani, N. Kunugi, K.Waki, Broué’s s abelian defect group conjecture for the Held group and the sporadic Suzuki group, J. Algebra 279 (2004), 638–666.

[11] S. Koshitani, M. Linckelmann, The indecomposability of a certain bimodule given by the Brauer construction, Journal of Algebra 285 (2005), 726–729.

[12] M. Linckelmann, Simple fusion systems and the Solomon 2-local groups, J. Algebra 296 (2006), no. 2, 385–401.

[13] M. Linckelmann, An introduction to fusion systems. Group representation theory, 79-113, EPFL Press, Lausanne, 2007.

[14] H. Nagao, Y. Tsushima, “Representations of finite groups”, Academic Press, 1988.

[15] S. Park, Realising a fusion system by a single finite group, preprint.

[16] K. Robertson, S. Shpectorov, On the definition of saturated fusion systems J. Group theory 12 (2009) 679–687.

[17] J. Thévenaz, “G-Algebras and Modular Representation Theory,” Oxford Science Publications, 1995.

Radha Kessar, Institute of Mathematics, University of Aberdeen, Fraser Noble Building, Aberdeen AB24 3UE, U.K.

E-mail address: kessar@maths.abdn.ac.uk

Naoko Kunugi, Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

E-mail address: kunugi@rs.kagu.tus.ac.jp

Naofumi Mitsuhashi, Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan