A REDUCTION THEOREM FOR FUSION SYSTEMS OF BLOCKS

RADHA KESSAR AND RADU STANCU^{∗}

Abstract. Let k be an algebraically closed field of characteristic p and G a finite group. An interesting question for fusion systems is whether they can be obtained from the local structure of a block of the group algebra kG. In this paper we develop some methods to reduce this question to the case when G is a central p′-extension of a simple group. As an application of our result, we obtain that the ’exotic’ examples of fusion systems discovered by Ruiz and Viruel [RV] do not occur as fusion systems of p-blocks of finite groups.

1. Introduction

Let p be a prime number. Fusion systems (referred to as full Frobenius systems in [Pu], and as saturated fusion systems in [BLO]) on finite p-groups were introduced by L. Puig and provide an axiomatic framework for studying p-fusion in finite groups. This axiomatic point of view has been very useful in determining many properties of finite groups and of the p-completion of their classifying spaces as well as in modular representation theory. It also underlies the theory of p-local finite groups developed by C. Broto, R. Levi and R. Oliver in [BLO].

To each pair (G, P), where G is a finite group and P is a Sylow p-subgroup of G, is associated a fusion system F_{P} (G) on P called a p-fusion system of G. However, there exist fusion systems which do not arise in this way [RV, Lemma 7.14], [BLO, Example 9.3]; such systems are called exotic. These exotic examples are interesting from two different, albeit related points of view. First, the exoticity of a given fusion has ramifications for classification problems in finite groups. For instance, R. Solomon’s theorems [So, Theorems 3.1 and 3.2] characterizing the sporadic group Co_{3} by the isomorphism type of its Sylow 2-subgroup may be restated, somewhat ahistorically, as asserting that certain fusion systems are exotic. Secondly, each known exotic fusion system has associated to it a unique p-local finite group. In general, [BLO, Theorem E] provides an obstruction theory for the existence and uniqueness of a p-local finite group associated to a given fusion system, and it is not known if the obstructions classes always vanish.

Fusion systems arise as well in block theory. To each quadruple (H, b, Q, e) where H is a finite group, b is a p-block of kH, and (Q, e) is a maximal b-Brauer pair (that is Q is a defect group of H and e is a p-block of C_{H}(Q) in correspondence with b) is associated a fusion system F_{(Q,e)}(H, b) on Q, called a fusion system of b. If b is the principal block of H, then Q is a Sylow p-subgroup of H and by Brauer’s third main theorem, it follows that F_{(Q,e)}(H, b) = F_{Q}(H). However, if b is not the principal block, then Q may be a proper subgroup of a Sylow p-subgroup of H of arbitrarily large index. On the other hand, there are many examples of quadruples (H, b, Q, e) such that for some group L associated to H, Q is a Sylow p-subgroup of L and F_{(Q,e)}(H, b) = F_{Q}(L). This is always the case, for instance, if H is a p-solvable group or a symmetric group.

Date: Mars 26, 2006.

∗ The second author was supported by the Swiss National Science Foundation’s Grant No PBEL2-100846.

The aim of this paper is to shed some light on the relationship between fusion systems of blocks and fusion systems of finite groups. We use as our starting point simple fusion systems as introduced by Linckelmann [Li, Definition 4.1]. If a simple fusion system occurs as F_{P} (G) for some finite group G, then it occurs as F_{P} (L) for some simple finite group L. It would be desirable to obtain an analogous result for fusion systems of blocks with L being possibly a quasi-simple group (see [Li, Proposition 4.2]). There is however one complication which arises when one tries “descent to a normal subgroup” in the context of block theory: Let H be a finite group and L a normal subgroup of H. If P is a Sylow p-subgroup of H, then P ∩ L is a Sylow p-subgroup of L and F_{P ∩L}(L) is a normal subsystem of F_{P} (H) (see Definition 2.9). Now suppose that (H, b, Q, e) is a quadruple as above and c is a block of L covered by b. Then, Q ∩ L is a defect group of c, however it is not the case in general that for some p-block f of C_{L}(Q ∩ L) in correspondence with c, the system F_{(Q∩L,f)}(L, c) is a subsystem of F_{(Q,e)}(H, b). In other words, there may be fusion in the covered block c, which is not seen in b. Our main result, Theorem 4.2 shows that under certain extra hypotheses this difficulty may be circumvented. In order to prove this theorem, we were led to consider categories that arise through the conjugation action of a finite group G on the block algebra kNd of a G-stable block d of a normal subgroup N of G. We show that these categories, which we call generalized Brauer categories are fusion systems (Theorem 3.4).

As an application of Theorem 4.2, we show in Theorem 6.4 that the examples of exotic fusion systems discovered by Ruiz and Viruel [RV] do not occur in block algebras.

The paper has 6 sections and an appendix. In section 2, we recall the relevant definitions and facts on fusion systems and block theory. In section 3 we study the generalized Brauer category and show that it is a fusion system. Section 4 contains the main reduction theorem. In section 5, we recall the properties of exotic fusion systems on extra-special p-groups of order 7^{3} and exponent 7. Section 6 contains the proof of the fact that these exotic systems do not occur as fusion systems of blocks. The defining properties of fusion systems as stated by us are slightly different from those in [BLO]-in the Appendix we show the equivalence of our approach with that in [BLO]. We should reiterate that for us, “fusion systems” are what are called “saturated fusion systems” in [BLO].

2. Fusion systems. Definitions and basic properties

Definition 2.1. A category F on a finite p-group P is a category whose objects are the subgroups of P and whose set of morphisms between the subgroups Q and R of P, is the set Hom_{F}(Q, R) of injective group homomorphisms from Q to R, with the following properties:

(a) if Q ≤ R then the inclusion of Q in R is a morphism in Hom_{F}(Q, R).

(b) for any ϕ ∈ Hom_{F}(Q, R) the induced isomorphism Q ≃ ϕ(Q) and its inverse are morphisms in F.

(c) composition of morphisms in F is the usual composition of group homomorphisms.

Note that the above definition of a category on P differs from what Puig calls divisible Frobenius system and what, equivalently, Broto, Levi and Oliver call fusion system by the fact that we do not ask for the inner automorphisms of P to be in the category.

If there exists an isomorphism ϕ ∈ Hom_{F}(Q, R) we say that Q and R are Fconjugate.

Definition 2.2. Let F be a category on P. A subgroup Q of P is fully Fcentralized, respectively fully F-normalized if |C_{P} (Q)| ≥ |C_{P} (Q′)|, respectively |N_{P} (Q)| ≥ |N_{P} (Q′)|, for all Q′ ≤ P which are F-conjugate to Q.

For Q, R, T ≤ P we denote Hom_{T} (Q, R) := {u ∈ T |^{u}Q ≤ R}/C_{T} (Q) and Aut_{T} (Q) := Hom_{T} (Q, Q). Other useful notation is Aut_{F}(Q) := Hom_{F}(Q, Q) and Out_{F}(Q) := Aut_{F}(Q)/Aut_{Q}(Q).

Definition 2.3. We say that a subgroup Q of P is F-centric if Z(Q′) = C_{P} (Q′) for any Q′ in the F-isomorphism class of Q. We say that Q is F-radical if O_{p}(Out_{F}(Q)) = Out_{P} (Q).

Definition 2.4. A fusion system F on a finite p-group P is a category on P satisfying the following properties:

(FS-1) Hom_{P} (Q, R) ⊂ Hom_{F}(Q, R) for all Q, R ≤ P.

(FS-2) Aut_{P} (P) is a Sylow p-subgroup of Aut_{F}(P).

(FS-3) Every ϕ : Q → P such that ϕ(Q) is fully F-normalized extends to a morphism ϕ̄: N_{ϕ} → P where

N_{ϕ} := {x ∈ N_{P} (Q) | ∃ y ∈ N_{P} (ϕ(Q)), ϕ(^{x}u) = ^{y}ϕ(u)∀u ∈ Q} .

In the rest of the section we give some properties of fusion systems. Let us start with a characterization of being fully F-normalized.

Proposition 2.5 ([Pu, Proposition 2.7]). Let F be a fusion system on P and let Q be a subgroup of P. Then Q is fully F-normalized if and only if Q is fully F-centralized and Aut_{P} (Q) is a Sylow p-subgroup of Aut_{F}(Q).

Definition 2.6. Let F be a fusion system on P and let Q be a subgroup of P. The normalizer N_{F}(Q) is the category on N_{P} (Q) having as morphisms, those morphisms ψ ∈ Hom_{F}(R, T) such that there exists a morphism ϕ ∈ Hom_{F}(QR, QT) with ϕ|_{Q} ∈ Aut_{F}(Q) and ϕ|_{R} = ψ. The centralizer C_{F}(Q) is the category on C_{P} (Q) having as morphisms those morphisms ψ ∈ Hom_{F}(R, T) such that there exists a morphism ϕ ∈ Hom_{F}(QR, QT) with ϕ|_{Q} = id_{Q} and ϕ|_{R} = ψ.

N_{F}(Q) is not, in general, a fusion system on N_{P} (Q) (for instance, the property (FS-2) in Definition 2.4 may fail to hold) but it is one if Q is fully F-normalized. It is the same for C_{F}(Q) when Q is fully F-centralized.

Proposition 2.7 ([Pu, Proposition 2.8]). Let F be a fusion system on P. If Q ≤ P is fully F-normalized then N_{F}(Q) is a fusion system on N_{P} (Q).

A special role in our study is played by strongly F-closed subgroups.

Definition 2.8. Let F be a fusion system on a finite p-group P and Q a subgroup of P. We say that Q is strongly F-closed if for any subgroup R of Q and any morphism ϕ ∈ Hom_{F}(R, P) we have ϕ(R) ≤ Q.

Linckelmann has introduced the notion of normal fusion subsystem, [Li, Definition 3.1].

Definition 2.9. Let F be a fusion system on a finite p-group P and F′ a fusion subsystem of F on a subgroup P′ of P. We say that F′ is normal in F if P′ is strongly F-closed and if for every isomorphism ϕ : Q → Q′ in F and any two subgroups R, R′ of Q ∩ P′ we have

ϕ ∘ Hom_{F′}(R, R′) ∘ ϕ^{−1} ⊆ Hom_{F′}(ϕ(R), ϕ(R′)) .

If F is a fusion system on a finite p-group P, a subsystem of F is a category on a subgroup of P that is contained in F and is itself a fusion system. We say that a fusion system is simple if it has no non-trivial normal fusion subsystem.

Finally, we record how fusion systems arise in finite groups.

Definition 2.10. Let G be a finite group, and P a Sylow p-subgroup of G. We denote by F_{P} (G) the category on P with morphisms Hom_{F_{P} (G)}(Q, R) := Hom_{G}(Q, R).

It is easy to check that F_{P} (G) is a fusion system. Also, F_{P} (G) and F_{P′}(G) are isomorphic for different Sylow p-subgroups P and P′ of G.

3. The generalized Brauer category

Let k be an algebraically closed field of characteristic p, G a finite group, N a normal subgroup of G and c a G-stable block of kN, that is c is a primitive idempotent of Z(kN), fixed by the conjugation action of G. Thus kNc is a primitive G-algebra. For any p-subgroup Q of G the canonical projection from kN to kC_{N}(Q) induces an algebra morphism Br_{Q}^{N} from the subalgebra of fixed points of Q, (kN)^{Q} onto kC_{N}(Q) (see [AB]). This morphism is known in the literature as the Brauer morphism. We adopt the approach of Broué and Puig [BP] for generalized Brauer pairs.

Definition 3.1. A (c, G)-Brauer pair is a pair (Q, e_{Q}) where Q is a p-subgroup of G such that Br_{Q}^{N}(c) ≠ 0 and e_{Q} is a block of kC_{N}(Q) such that Br_{Q}^{N}(c)e_{Q} ≠ 0. When G = N, a (c, G)-Brauer pair is also known as a c-Brauer pair.

Let (Q, e_{Q}) and (R, e_{R}) be two (c, G)-Brauer pairs; we say that (Q, e_{Q}) is contained in (R, e_{R}), and we write (Q, e_{Q}) ≤ (R, e_{R}), if Q ≤ R and for any primitive idempotent i ∈ (kN)^{R} such that Br_{R}^{N}(i)e_{R} ≠ 0, we have Br_{Q}^{N}(i)e_{Q} ≠ 0. This defines an order relation on the set of (c, G)-Brauer pairs compatible with the conjugation action of G. We also have that given a (c, G)-Brauer pair (R, e_{R}) and Q ≤ R there exists a unique (c, G)-Brauer pair (Q, e_{Q}) contained in (R, e_{R}) [BP, Theorem 1.8 (i)]. By [BP, Theorem 1.14 (2)] all maximal (c, G)-Brauer pairs are G-conjugate. If (P, e_{P}) is a maximal (c, G)-Brauer pair then the group P is called a (c, G)-defect group. In the case that G = N, the group P is a defect group of c in the usual sense.

Before proceeding, we record a property characterizing the inclusion of generalized Brauer pairs in the case of normal p-subgroups, which is just a reformulation of [BP, Theorem 1.8]. For a (c, G)-Brauer pair (Q, e_{Q}), denote by N_{G}(Q, e_{Q}) the stabilizer in N_{G}(Q) of e_{Q}.

Proposition 3.2. Let G be a finite group, N a normal subgroup of G, c a Gstable block of N and (Q, e_{Q}) a (c, G)-Brauer pair. Let H be a group such that QC_{N}(Q) ≤ H ≤ N_{G}(Q, e_{Q}). Let S be a p-group such that Q ≤ S ≤ H and let f be a block of kC_{N}(S). The following are equivalent.

(i) (S, f) is a (c, G)-Brauer pair such that (Q, e_{Q}) ≤ (S, f).

(ii) (S, f) is an (e_{Q}, H)-Brauer pair.

Proof. First of all, note that the statement makes sense, since C_{N}(Q) is a normal subgroup of H and e_{Q} is an H-stable block of kC_{N}(Q). Also, since Q ≤ S, C_{C_{N} (Q)}(S) = C_{N}(S). Suppose first that (S, f) is a (c, G)-Brauer pair such that (Q, e_{Q}) ≤ (S, f). Since S normalizes Q, it follows from [BP, Theorem 1.8 (iii)] Br_{S}^{C_{N} (Q)}(e_{Q})f = f, hence (S, f) is an (e_{Q}, H)-Brauer pair. Conversely, suppose that (S, f) is a (e_{Q}, H)- Brauer pair. Then, Br_{S}^{C_{N} (Q)}(e_{Q})f = f, which means that

Br_{S}^{N}(c)f = Br_{S}^{N}(c)Br_{S}^{C_{N} (Q)}(e_{Q})f
= Br_{S}^{C_{N} (Q)}(Br_{Q}^{N}(c))Br_{S}^{C_{N} (Q)}(e_{Q})f
= Br_{S}^{C_{N} (Q)}(Br_{Q}^{N}(c)e_{Q})f
= Br_{S}^{C_{N} (Q)}(e_{Q})f
= f

This shows that (S, f) is a (c, G)-Brauer pair. Thus, by [BP, Theorem 1.8 (i)], there is a unique (c, G)-Brauer pair (Q, e′_{Q}) with (Q, e′_{Q}) ≤ (S, f) and by (iii) of the same theorem, this e′_{Q} is S-stable and Br_{S}^{C_{N} (Q)}(e′_{Q})f = f. But by definition Br_{S}^{C_{N} (Q)}(e_{Q})f = f. We claim that e′_{Q} = e_{Q}. Indeed, suppose not. Then, since e and e′ are blocks of kC_{N}(Q), e_{Q}e′_{Q} = 0. Since Br_{S}^{C_{N} (Q)} is an algebra homomorphism from (kC_{N}(Q))^{S} onto kC_{N}(S), this would imply that Br_{S}^{C_{N} (Q)}(e_{Q})Br_{S}^{C_{N} (Q)}(e′_{Q}) = 0. Hence, f being a central idempotent of kC_{N}(S),

f = fBr_{S}^{C_{N} (Q)}(e_{Q})fBr_{S}^{C_{N} (Q)}(e′_{Q}) = 0,

a contradiction. Thus, e_{Q} = e′_{Q}, showing that (Q, e_{Q}) ≤ (S, f) as required.

□

Definition 3.3. Let G be a finite group, N a normal subgroup of G, c a G-stable block of kN and (P, e_{P}) a maximal (c, G)-Brauer pair. For a subgroup Q of P, we let e_{Q} be the unique block of kC_{N}(Q) such that (Q, e_{Q}) ≤ (P, e_{P}). Denote by F_{(P,e_{P})}(G, N, c) the category on P with morphisms:

Hom_{F_{(P,e_{P} )}(G,N,c)}(Q, R) := {conj_{g} : Q → R|g ∈ G, ^{g}(Q, e_{Q}) ≤ (R, e_{R})}

If G = N, then F_{(P,e_{P})}(G, N, c) is the usual fusion system of the block c, and we denote it by F_{(P,e_{P})}(G, c)

We now show that the category F_{(P,e_{P})}(G, N, c) is a fusion system on P. The details of the proof for the case G = N are given in [Li, Theorem 2.4].

Theorem 3.4. Let N be a normal subgroup of G, let c be a G-stable block of kN and let (P, e_{P}) be a maximal (c, G)-Brauer pair.

(i) The category F_{(P,e_{P})}(G, N, c) is a fusion system on P.

(ii) If (P′, e_{P′}) is another maximal (c, G)-Brauer pair, then F_{(P′,e_{P′})}(G, N, c) is isomorphic to F_{(P,e_{P})}(G, N, c).

Proof. Denote F := F_{(P,e_{P})}(G, N, c). Let u ∈ P and let (Q, e_{Q}) ≤ (P, e_{P}). Then, ( ^{u}Q, ^{u}e_{Q}) ≤ ( ^{u}P, ^{u}e_{P}) = (P, e_{P}) which implies that ^{u}e_{Q} = e _{^{u}Q}. This shows that property (FS-1) of Definition 2.4 holds.

For the second property we check that Aut_{P} (P) ∈ Syl_{p}(Aut_{F}(P)). Denoting by N_{G}(P, e_{P}) the normalizer of (P, e_{P}) and by C_{G}(P, e_{P}) the intersection of C_{G}(P) with N_{G}(P, e_{P}), we have Aut_{F}(P) ≃ N_{G}(P, e_{P})/C_{G}(P, e_{P}) hence we must show that the index [N_{G}(P, e_{P}) : P C_{G}(P, e_{P})] is not divisible by p.

By [BP, Theorem 1.14(b)], (P, e_{P}) being a maximal (c, G)-Brauer pair means that c ∈ Tr_{P}^{G}((kN)^{P}) and hence that Br_{P}^{N}(c) = Tr_{1}^{N_{G}(P)/P} (a′) for some a′ ∈ kC_{N}(P).

Since the maximal (c, G)-Brauer pairs are all conjugate [BP, Theorem 1.14 (2)], the map a → Tr_{N_{G}(P,e_{P})}^{N_{G}(P)}a is an algebra isomorphism from (kC_{N}(P)e_{P})^{N_{G}(P,e_{P})} to (kC_{N}(P)Br_{P}^{N}(c))^{N_{G}(P)} . The reverse map is given by a → ae_{P}.

Let g ∈ N_{G}(P), and let j ∈ kC_{N}(P). Then

Tr_{P}^{N_{G}(P)}(j ^{g}e_{P}) = Tr_{N_{G}(P, ^{g}e_{P})}^{N_{G}(P)}(Tr_{P}^{N_{G}(P, ^{g}e_{P})}(j ^{g}e_{P}))
= Tr_{N_{G}(P, ^{g}e_{P})}^{N_{G}(P)}(Tr_{P}^{N_{G}(P, ^{g}e_{P})}(j) ^{g}e_{P}) .

Thus,

\begin{aligned}Tr_{P}^{N_{G}(P)}(j ^{g}e_{P})e_{P} & = ^{g^{−1}}(Tr_{P}^{N_{G}(P, ^{g}e_{P})}(j ^{g}e_{P})) \\ & = Tr_{P}^{N_{G}(P,e_{P})}(( ^{g^{−1}}j)e_{P}) . \end{aligned}

The above calculation shows that the image of the ideal Tr_{P}^{N_{G}(P)}(kC_{N}(P)Br_{P} (b)) under the map a → ae_{P} is Tr_{P}^{N_{G}(P,e_{P})}(kC_{N}(P)e_{P}). In particular, since Br_{P}^{N}(c) = Tr_{P}^{N_{G}(P)}(a′) for some a′ ∈ kC_{N}(P), it follows that e_{P} = Tr_{P}^{N_{G}(P,e_{P})}(a) for some a ∈ kC_{N}(P)e_{P}.

As N_{G}(P, e_{P}) ≥ P C_{G}(P, e_{P}) ≥ P we have furthermore

e_{P} = Tr_{P (N_{G}(P,e_{P})∩C_{G}(P))}^{N_{G}(P,e_{P})}Tr_{P}^{P C_{G}(P,e_{P})}(a) .

Now Tr_{P}^{P C_{G}(P,e_{P})}(a) ∈ Z(kC_{N}(P)e_{P}) and Z(kC_{N}(P)e_{P}) is a local ring, hence

Tr_{P}^{P C_{G}(P,e_{P})}(a) = αe_{P} + x , x ∈ J(kP C_{N}(P)) , α ∈ k .

If [N_{G}(P, e_{P}) : P C_{G}(P, e_{P})] is not prime to p then

e_{P} = tr_{P (C_{G}(P,e_{P}))}^{N_{G}(P,e_{P})}(x) ∈ J(kP C_{N}(P)) ,

which is impossible since e_{P} is an idempotent.

For the third property, let (Q, e_{Q}) ≤ (P, e_{P}) be fully F-normalized. This means that R := N_{P} (Q) is of maximal order among the normalizers in P of subgroups F-isomorphic to Q. Let H = RC_{G}(Q, e_{Q}). We claim that (R, e_{R}) is a maximal (e_{Q}, H)-Brauer pair. Indeed, it follows from Proposition 3.2 that (R, e_{R}) is an (e_{Q}, H)-Brauer pair. Now, suppose that (S, f) is a (e_{Q}, H)-Brauer pair with (R, e_{R}) ≤ (S, f). Then by Proposition 3.2, (S, f) is a (c, G)-Brauer pair with (Q, e_{Q}) ≤ (S, f). Since all maximal (c, G)-Brauer pairs are G-conjugate, there exists g ∈ G such that ^{g}(S, f) ≤ (P, e_{P}). Now, (Q, e_{Q}) ≤ (S, f) implies g(Q, e_{Q}) ≤ ^{g}(S, f) ≤ (P, e_{P}), so that in particular, ^{g}Q is F-isomorphic to Q. On the other hand, ^{g}S ≤ N_{P} (^{g}Q). The maximality of R forces S = R, proving the claim.

Now let h ∈ G such that (^{h}Q, ^{h}e_{Q}) ≤ (P, e_{P}) and denote by ϕ := conj_{h^{−1}} : hQ → Q. We have to prove that ϕ extends to ϕ̃ : N_{ϕ} → N_{P} (Q). Now N_{ϕ} consists of those elements x of N_{P} (Q) such that conj(h^{−1}xh) : Q → P is equal to conj(y) : Q → P for some y ∈ N_{P} (Q), that is

N_{ϕ} = {x ∈ N_{P} (^{h}Q)|h^{−1}xh ∈ H} .

So we have ^{h^{−1}}N_{ϕ} ≤ H. Thus it suffices to find a z ∈ C_{G}(Q, e_{Q}) such that zh^{−1}N_{ϕ} ≤ N_{P} (Q). Since ^{h}(Q, e_{Q}) ≤ (P, e_{P}) and N_{ϕ} ≤ N_{P} ( ^{h}Q), there is a containment of (c, G)-Brauer pairs, ^{h}(Q, e_{Q}) ≤ (N_{ϕ}, e_{N_{ϕ}}), giving (Q, e_{Q}) ≤ ^{h^{−1}}(N_{ϕ}, e_{N_{ϕ}}). By Proposition 3.2 applied with S = ^{h^{−1}}N_{ϕ} and f = ^{h^{−1}}e_{N_{ϕ}}, it follows that ( ^{h^{−1}}N_{ϕ}, ^{h^{−1}}e_{N_{ϕ}}) is a (e_{Q}, H)-Brauer pair. But we have shown above that (R, e_{R}) is a maximal (e_{Q}, H)-Brauer pair, hence there exists a y ∈ H such that ^{yh^{−1}}N_{ϕ} ≤ N_{P} (Q), proving the existence of z as desired.

This proves (i) of the theorem. Part (ii) is immediate since all maximal (c, G)- Brauer pairs are G-conjugate.

□

Theorem 3.5. Let G be a finite group and let N be a normal subgroup of G. Let c be a G-stable block of N and b a block of kG such that bc = b. Let (P, e_{P}) be a maximal b-Brauer pair. Then there exists a maximal (c, G)-Brauer pair (S, e′_{S}) such that P ≤ S and such that F_{(P,e_{P})}(G, b) ≤ F_{(S,e′_{S})}(G, N, c). Furthermore, P ∩ N = S ∩ N, (S ∩ N, e′_{S∩N}) is a maximal (c, N)-Brauer pair and F_{(S∩N,e′_{S∩N})}(N, N, c) = F_{(S∩N,e′_{S∩N})}(N, c) is a normal subsystem of F_{(S,e′_{S})}(G, N, c).

Proof. Let (P, e_{P}) be a maximal (b, G)-Brauer pair. As bc = c and Br_{P}^{G}(b)e_{P} = e_{P} there exists a central primitive idempotent e′_{P} ∈ kC_{N}(P) such that Br_{P}^{N}(c)e′_{P} = e′_{P} and e_{P} covers e′_{P}, i.e. e_{P} e′_{P} ≠ 0. Let (S, e′_{S}) be a maximal (c, G)-Brauer pair containing (P, e′_{P}). Let (Q, e_{Q}) be a (b, G)-Brauer pair contained in (P, e_{P}) and (Q, e′_{Q}) be a (c, G)-Brauer pair contained in (S, e′_{S}). We prove that e_{Q}e′_{Q} ≠ 0.

Consider a primitive idempotent decomposition of 1 in (kN)^{P}:

1 = j_{1} + j_{2} + ⋯ + j_{n} .

We have

1 = Br_{P}^{N}(1) = Br_{P}^{N}(j_{1}) + Br_{P}^{N}(j_{2}) + ⋯ + Br_{P}^{N}(j_{n}) ⊂ kC_{G}(P)

and multiplying by e_{P} e′_{P} we obtain

e_{P} e′_{P} = Br_{P}^{N}(j_{1})e_{P} e′_{P} + Br_{P}^{N}(j_{2})e_{P} e′_{P} + ⋯ + Br_{P}^{N}(j_{n})e_{P} e′_{P} .

Thus, given that e_{P} e′_{P} ≠ 0, there exists a primitive idempotent j in (kN)^{P} such that Br_{P}^{N}(j)e_{P} e′_{P} ≠ 0. Moreover, as Br_{P}^{N} is surjective we have that Br_{P}^{N}(j) is also primitive in kC_{N}(P) so Br_{P}^{N}(j)e′_{P} = Br_{P}^{N}(j). Consider now a primitive idempotent decomposition of j in (kG)^{P}:

j = i_{1} + i_{2} + ⋯ + i_{m} .

As before we have

Br_{P}^{N}(j) = Br_{P}^{N}(i_{1}) + Br_{P}^{N}(i_{2}) + ⋯ + Br_{P}^{N}(i_{m})

giving that

0≠ Br_{P}^{N}(j)e_{P} e′_{P} = Br_{P}^{G}(j)e_{P} e′_{P} = Br_{P}^{G}(i_{1})e_{P} e′_{P} +Br_{P}^{G}(i_{2})e_{P} e′_{P} +⋯+Br_{P}^{G}(i_{m})e_{P} e′_{P} .

Thus there exists a primitive idempotent i in (kG)^{P} satisfying ij = i and such that Br_{P}^{G}(i)e_{P} e′_{P} ≠ 0. Again Br_{P}^{G}(i) is primitive in kC_{G}(P) so Br_{P}^{G}(i)e_{P} = Br_{P}^{G}(i).

By definition, every primitive idempotent i ∈ (kG)^{P} such that Br_{P}^{G}(i)e_{P} ≠ 0 satisfies Br_{Q}^{G}(i)e_{Q} ≠ 0 and every primitive idempotent j ∈ (kN)^{P} such that Br_{P}^{N}(j)e′_{P} ≠ 0 satisfies Br_{Q}^{N}(j)e′_{Q} ≠ 0. More precisely, we have

Br_{Q}^{G}(i)Br_{Q}^{N}(j)e_{Q}e′_{Q} = Br_{Q}^{G}(i)e_{Q}Br_{Q}^{N}(j)e′_{Q} = Br_{Q}^{G}(i)Br_{Q}^{N}(j)
= Br_{Q}^{G}(i)Br_{Q}^{G}(j) = Br_{Q}^{G}(i)

This proves the claim.

Consider the orbit O = {^{g}e′_{Q}|g ∈ N_{G}(Q, e_{Q})} of e′_{Q} by conjugation with elements of N_{G}(Q, e_{Q}). As e_{Q}e′_{Q} ≠ 0 for any g ∈ N_{G}(Q, e_{Q}) we have e_{Q} ^{g}e′_{Q} = g(e_{Q}e′_{Q}) ≠ 0. Since C_{G}(Q) acts transitively on the set of blocks f of kC_{N}(Q) satisfying e_{Q}f ≠ 0 we have that C_{G}(Q) acts transitively on O. We apply the Frattini argument to the transitive actions of N_{G}(Q, e_{Q}) and C_{G}(Q) on O and we have N_{G}(Q, e_{Q}) = C_{G}(Q)(N_{G}(Q, e_{Q}) ∩ N_{G}(Q, e′_{Q})). So N_{G}(Q, e_{Q})/C_{G}(Q) ≃ (N_{G}(Q, e_{Q}) ∩ N_{G}(Q, e′_{Q}))/(C_{G}(Q) ∩ N_{G}(Q, e′_{Q})) and thus Aut_{F_{(P,e_{P} )}(G,b)}(Q) is included into Aut_{F_{(S,e′_{S})}(G,N,c)}(Q). By Alperin’s fusion theorem F_{(P,e_{P})}(G, b) is isomorphic to a subsystem of F_{(S,e′_{S})}(G, N, c). This proves the first assertion.

Now, since P is a defect group of the block b of kG, P ∩ N is a defect group of the block c of kN [NT, Chapter 5, Theorem 5.16 (iii)]. On the other hand, clearly Br_{S∩N}^{N}(c) ≠ 0, hence N ∩ S is contained in a defect group of the block c of kN. Thus S ∩N = P ∩N . It follows that (S ∩N, e′_{S∩N}) is a maximal (c, N)-Brauer pair and that F_{(S∩N,e′_{S∩N})}(N, c) is the subcategory of F_{(S,e′_{S})}(G, N, c) on S ∩ N with morphisms induced from conjugation by elements of N (note that for a subgroup R of S ∩N a (c, G)-Brauer pair with first component R is a (c, N)-Brauer pair with first component R and that for (c, G)-Brauer pairs (R′, f′) and (R, f) such that R ≤ S ∩ N, (R′, f′) ≤ (R, f) as (c, G)-Brauer pairs if and only if (R′, f′) ≤ (R, f) as (c, N)-Brauer pairs). Since N is a normal subgroup of G, it is easy to check that F_{(S∩N,e′_{S∩N})}(N, c) is a normal subsystem of F_{(S,e′_{S})}(G, N, c).

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4. Main Result

Definition 4.1. Let G be a finite group, k an algebraically closed field of characteristic p, b a block of kG and F a fusion system on a finite p-group. We say that b is a F-block if F_{P,e_{P}} (G, b) is isomorphic to F for some (and hence any) maximal b-Brauer pair (P, e_{P}).

We say that a finite group is a p′-group if its order is not divisible by p.

Theorem 4.2. Let F_{1} and F_{2} be two fusion systems on P, F_{1} containing F_{2}. Suppose that:

a) P has no non-trivial proper strongly F_{2}-closed subgroup (and therefore no nontrivial strongly F_{1}-closed subgroup),

b) if F is a fusion system on P containing F_{2}, then F = F_{1} or F = F_{2}.

c) if F is a non-trivial fusion system normal in F_{1} or F_{2} then F = F_{1} or F = F_{2}. If there exists a finite group G having an F_{1} or an F_{2}-block then there exists a

quasi-simple group L with Z(L) a p′-group having an F_{1} or an F_{2}-block.

Proof. Let G be a minimal order group having an F_{1} or an F_{2}-block b. By a standard reduction (see for example [Ke, Proposition 2.11]), if N is a normal subgroup of G and c is a block of N with bc ≠ 0, then c is G-stable.

By abuse of notation, P is a (b, G)-defect group. Let H :=< ^{g}P | g ∈ G > be the normal subgroup of G generated by all G-conjugates of P. Let d be the unique block of kH covered by b. Given that d is G-stable, G acts by conjugation on kHd. Let N be the kernel of the homomorphism G → Out(kHd) = Aut(kHd)/Inn(kHd). Then by ([Kü2, §6 Theorem]) G/N is a p′-group. We prove, using the minimality of G and the hypothesis on F_{1} and F_{2}, that G = N.

Let c be the block of N covered by b, i.e. bc ≠ 0; (in this case in fact we have c = b). Let (P, e_{P}) be a maximal b-Brauer pair and let (S, e′_{S}) be a maximal (G, c) Brauer pair as in Theorem 3.5. Since G/N is a p′-group, it follows that S = P. Hence, we have F_{(P,e_{P})}(G, b) is a subsystem of F_{(P,e′_{P})}(G, N, c) and that F_{(P,e′_{P})}(N, c) is a normal subsystem of F_{(P,e′_{P})}(G, N, c).

Given that b is an F_{1}- or F_{2}-block and that F_{1} and F_{2} are the only fusion systems on P that contain F_{2} we obtain that F_{(P,e_{P})}(G, N, c) is either F_{1} or F_{2}. Again, since the only normal proper fusion subsystem on P contained in F_{1} is F_{2} and F_{2} has no normal fusion subsystem it follows that F_{(P,e′_{P})}(N, c) is either F_{1} or F_{2}. By the minimality of G we deduce that G = N.

As b and d have the same defect group P and G acts on kHd by inner automorphisms, using another result of Külshammer ([Kü1, Theorem 7]), we have that kGb and kHd have isomorphic source algebras, so c is also an F_{1}- or F_{2}-block. Thus, once again by the minimality of G we have G = H.

Let M be a proper normal subgroup of G. Then P ∩ M is a strongly F_{1} (or F_{2})-closed subgroup of P, hence P ∩ M = 1 or P ∩ M = P. Suppose first that P ∩ M = P. Then P and all its G-conjugates lie in M. Thus G = M, which is a contradiction. Thus we are in the case P ∩ M = 1. A variation of Fong reduction allows us to deduce that there is a central p′-extension G′ of G/M having an F_{1} or F_{2}-block (see for example [Ke, Section 3;3.3 and below]).

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5. The Ruiz-Viruel Exotic Fusion Systems

In their paper [RV], Ruiz and Viruel classified all possible fusion systems on extra-special p-groups of order p^{3}. They showed that there are three exotic fusion systems on the extraspecial 7-group of order 7^{3} and exponent 7. Let P be such a 7-group. A fusion system F on P is completely determined by Out_{F}(P) and the set of F-automorphisms of F-centric, F-radical proper subgroups of P. The three exotic systems of Ruiz and Viruel correspond to the following data. As in Ruiz and Viruel’s tables we denote by #F^{ec} the number of F-centric, F-radical proper subgroups of P. An entry of the form a + b in the #F^{ec} column indicates that there are two F-conjugacy classes of F-centric, F-radical subgroups of cardinality a and b respectively.

name Out_{F}(P) #F^{ec} Aut_{F}(V)
RV_{1} 6^{2} : 2  6 + 2 SL_{2}(7) : 2, GL_{2}(7)
RV_{2} D_{16} × 3 4 + 4 SL_{2}(7) : 2, SL_{2}(7) : 2
RV_{3} SD_{32} × 3 8  SL_{2}(7) : 2

The categories on P generated by the above sets of morphisms satisfy the properties of fusion systems [RV, Lemma 7.14]. For the convenience of the reader, we give here a proof of this fact. Let F be one of the categories on P described above. The properties (FS-1) and (FS-2) are trivially satisfied. For the property (FS-3), we have to study two types of F-morphisms: those between the elementary abelian subgroups of rank 2 and those between the cyclic subgroups of order 7.

1) Take ϕ : Q → R be an isomorphism in F where Q and R are elementary abelian subgroups of rank 2. Remark that R and Q are both fully F-normalized. Suppose that N_{ϕ} = P. By the construction of homomorphism in F, the morphism ϕ decomposes into αψβ where α ∈ N_{F}(R), β ∈ N_{F}(Q) and ψ is the restriction of a morphism in Aut_{F}(P). In fact we can suppose that α = id as αψβ = ψψ^{−1}αψβ and ψ^{−1}αψβ ∈ N_{F}(Q). So without loss of generality we suppose that ϕ = ψβ. Now as N_{ϕ} = P we have that N_{β} = P. Indeed for any x ∈ N_{ϕ}, by the definition there exists a y ∈ N_{P} (R) such that ϕ(^{x}u) =^{y} ϕ(u). Take z = ψ̃^{−1}(y) where ψ̃ is the extension of ψ to P. Then ψβ(^{x}u) =^{y} ψβ(u) implies that β(^{x}u) =^{z} β(u). By construction, all the morphisms in N_{Aut_{F} (Q)}(Aut_{P} (Q)) can be extended to Aut_{F}(P). So there exists β̃ ∈ Aut_{F}(P) extending β. Now ψ̃β̃ ∈ Aut_{F}(P) extends ψβ and we are done.

2) Take ϕ : Q → R be a isomorphism in F where Q and R are cyclic subgroups of order 7 with R fully F-normalized. As the cyclic subgroups of order 7 are all Fconjugated we have that necessary R = Z(P) as Z(P) is the only cyclic subgroup of order 7 having its normalizer equal to P. Now if Q = Z(P) we are done as any F-automorphism of Z(P) lifts to P. If Q ≠ Z(P), by construction ϕ lifts to ϕ̃ : T → U where T and U are elementary abelian subgroups of rank 2 containing Q, respectively R. But then T = N_{P} (Q) so ϕ lifts to N_{P} (Q) and we are done.

Proposition 5.1. Let P = 7_{2}^{1+2} be the extra-special group of order 7^{3} and exponent 7 and let F be an exotic fusion system on P. If F = RV_{2}, let F_{2} = F and let F_{1} = RV_{3}. If F = RV_{3}, let F_{1} = F and let F_{2} = RV_{2}. If F = RV_{1}, set F_{1} = F_{2} = F. Then F_{1} contains F_{2}. Furthermore, F_{1} and F_{2} satisfy conditions (a), (b) and (c) of Theorem 4.2.

Proof. Let F be one of the above three fusion systems on P. The proper nontrivial subgroups of P are either elementary abelian of rank 2 or cyclic of order 7. There are eight elementary abelian subgroups of rank 2 of P and they are all Fcentric, F-radical. Moreover they are not unique in their F-conjugacy class so they are not strongly F-closed. Another fact is that each of the automorphism groups of the elementary abelian subgroups of rank 2 of P contains SL_{2}(7) so the cyclic subgroups in any F-centric, F-radical subgroup of P are transitively permuted by these automorphisms. Thus none of the cyclic subgroups of P are strongly F-closed. This proves that the condition (a) of Theorem 4.2 is satisfied.

If F_{1} = RV_{2} and F_{2} = RV_{3} then F_{1} contains F_{2} by construction as Out_{F_{2}}(P) is a subgroup of Out_{F_{1}}(P) and the F-automorphisms of F-centric, F-radical proper subgroups of P are the same for F = F_{1} and F = F_{2}. From the classification of Ruiz and Viruel there is no fusion system on P containing RV_{1} or RV_{3}. So the condition (b) is satisfied.

Again let F be one of the above three exotic fusion systems on P. Suppose that F has a normal non-trivial subsystem N on a subgroup R of P. We have that R is strongly F-closed, thus, given that F satisfies property (a), we have that R = P so N is a fusion system on P. Our aim is to prove that N is one of the three exotic fusion systems on P. For this it is sufficient to show that N has also eight N -centric N -radical proper subgroups of P since this characterizes the exotic fusion systems by the classification of Ruiz and Viruel. Take Q to be an elementary abelian subgroup of rank 2 of P. As C_{p} ≃ Aut_{P} (Q) ≤ Aut_{N} (Q) ⊲ Aut_{F}(Q) and Aut_{F}(Q) contains SL_{2}(7), we have that Aut_{N} (Q) also contains SL_{2}(7) so Q is an N -centric N -radical subgroup. Now if we take F_{1} and F_{2} as in the proposition we see that F_{2} ⊲ F_{1} and no other exotic fusion system on P is contained in F_{1} or in F_{2}. Thus F_{1} and F_{2} also satisfy the condition (c) of Theorem 4.2.

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With the notations as in the above proposition we remark that Aut_{RV_{3}}(P) = Aut_{RV_{2}}(P).2 and the extra-automorphism of order 2 in Aut_{RV_{3}}(P) preserve the fusion in RV_{2} in the sense of [BCGLO, Definition 4.5]. Using an analogue notation as in [BCGLO, Theorem 5.7 (a)], along the lines of the proof we obtain that RV_{3} = RV_{2}.2.

6. An application

As in sections 3 and 4, in this section k will denote an algebraically closed field of characteristic p. We will be using the following two well known results.

Lemma 6.1. Let G be a finite group and N a normal subgroup of G. Let b be a block of kG and let D be a defect group of b. Then there exists a block c of kN such that c is D-stable, cb ≠ 0, D ∩ N is a defect group of c and Br_{D}^{H}(c) ≠ 0 for any subgroup H of G containing ND.

Proof. Let d be a block of kN such that bd ≠ 0 and let I be the stabilizer in G of d. Then there is a block b_{1} of kI such that db_{1} ≠ 0 and such that any defect group of b_{1} is a defect group of b [NT, Chapter 5, Theorem 5.10]. Let D_{1} ≤ I be a defect group of b_{1} and of b. Then Br_{D_{1}}^{I}b_{1} ≠ 0. Since I stabilizes d, d is the unique block of kN such that db_{1} ≠ 0, hence db_{1} = b_{1}. It follows that Br_{D_{1}}^{I}(d) ≠ 0 and hence that Br_{D_{1}}^{H}(d) ≠ 0 for any subgroup H of G containing ND. Also, D_{1} ∩ N is a defect group of d [NT, Chapter 5, Theorem 5.16 (ii)]. Now, D = ^{g}D_{1} for some g ∈ G. Set c = ^{g}d. It is easy to check that c has all the desired properties.

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Lemma 6.2. Let H = LD be a finite group such that L is normal in H and D is a p-group. Let c be a block of kL, stabilized by D. Suppose that D ∩ L is a defect group of c as a block of kL and that Br_{D}^{H}(c) ≠ 0. Let D′ be a subgroup of D containing D ∩ L. Then,

(i) The idempotent c is a block of LD′ and D′ is a defect group of c as a block of LD′.

(ii) If the elements of D′ induce inner automorphisms of L, then D′ = (D′ ∩ L)C_{D′}(L).

Proof. The fact that c is a block of LD′ is immediate since D′ is a p-group. By hypothesis, Br_{D}^{H}(c) ≠ 0, hence Br_{D′}(c) ≠ 0. Hence there is a p-subgroup, say D′′, of LD′ containing D′ such that D′′ is a defect group of c as a block of LD′. Now D′′ ∩ L is a defect group of c as block of L [NT, Chapter 5, Theorem 5.16 (ii)], hence |D′′ ∩ L| = |D ∩ L| = |D′ ∩ L|. On the other hand, D′′L/L is a a subgroup of D′L/L, proving (i).

Now suppose that the elements of D′ induce inner automorphisms of L. Let x ∈ D′, and let w_{x} ∈ L be a p-element such that ^{w_{x}}u = ^{x}u for all u ∈ L. Then w_{x}^{−1}x is a central p-element of L < x >. In particular, w_{x}^{−1}x is contained in any defect group of any block of L < x >. On the other hand, by (i), c is a block of L < x > with defect group (D ∩ L) < x >. Since (D ∩ L) < x >≤ D′, it follows that w_{x} ∈ D′ ∩ L and w_{x}^{−1}x ∈ C_{D′}(L). The result follows.

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Proposition 6.3. Let p ≥ 7 be prime and let D be an extra-special p-group. Let G be a quasisimple finite group, and let Ḡ be the simple group G/Z(G). Suppose that Ḡ = G(q) is a finite group of Lie type with p ∤ q. If D is a defect group of a block of G, then there exists an integer n, a power q′ of q and a subgroup H of the finite general linear group GL_{n}(q′) (or GU_{n}(q′)) with H ≥ SL_{n}(q′) (or SU_{n}(q′)), a block c of H and a defect group D̃ of c such that D̃/ < ζ > is extra-special of order |D| for some cyclic subgroup < ζ > of D̃ ∩ Z(H). Consequently, G has no blocks with defect groups extra-special of order p^{3}.

Proof. Suppose that G has a block with defect group isomorphic to D. Then Ḡ has non-abelian Sylow p-subgroups which means in particular that the order of the Weyl group of the algebraic group corresponding to Ḡ is divisible by p ([GLS, Theorem 4.10.2]). Since p ≥ 7, this means that exceptional part of the Schur multiplier of Ḡ is trivial ([GLS, Table 6.1.3]). Thus there is a simple simply connected algebraic group K̄ over the algebraic closure of the field of q elements and a Frobenius morphism F : K̄ → K̄ such that K̄^{F} is a central extension of G. If K̄ is of type A, then set H := K̄^{F} and let c be the unique block of H whose image under the algebra homomorphism kH → kG induced by the canonical surjection of H onto G. Then c clearly has the required properties and the first assertion holds.

Thus we may assume that K̄ is not of type A. Since p ≥ 7, the kernel of the surjection K^{F} is an p′-group. In particular, K̄^{F} has a block with defect group isomorphic to D. Thus, we may assume that G = K̄^{F}.

Let Z(D) =< z >. By Brauer’s first main theorem, the group C_{G}(z) has a block, say b with defect group D. Since p ≥ 7, p is good for K̄ . Thus, since K̄ is simply connected, CK̄(z) is a Levi subgroup of K̄.

Let Z̄ denote the connected center of CK̄(z). Then

CK̄(z) = [CK̄(z), CK̄(z)]Z̄ .

Furthermore, [CK̄(z), CK̄(z)] being simply connected ([GLS, Theorem 1.13.2]) is a direct product of its components, each of which is also simply connected and which are permuted by F. That is, we may write

[CK̄(z), CK̄(z)] = ∏_{i=1}^{t} ∏_{j=1}^{r_{i}} L̄_{ij} ,

where each L̄_{ij} is a simply connected simple group, such that for each i, 1 ≤ j ≤ r_{i}, the groups L̄_{ij}, 1 ≤ j ≤ r_{i} are in a single orbit under the action of F.

Set L_{i} := (∏_{j=1}^{r_{i}} L̄_{ij})^{F}. Then L_{i} is the diagonal subgroup consisting of elements ∏_{j=0}^{r_{i}−1} F^{j}(u) where u ∈ L̄_{1i}^{F^{r_{i}}}. In particular, L_{i} ≅ L̄_{i1}^{F^{r_{i}}}. Furthermore,

C_{G}(z) ≅ (L_{1} × ⋯ × L_{t})T ,

where T is an abelian group of order prime to q, inducing inner-diagonal automorphisms ([GLS, Definition 2.5.13]) on each L_{i} (T is the subgroup of F -fixed points of a F -stable maximal torus of CK̄(z)).

Since T is abelian, D∩(L_{1} ×⋯×L_{t}) ≠ 1. On the other hand, D∩(L_{1} ×⋯×L_{t}) is a defect group of a block of L_{1} × ⋯ × L_{t} (see Lemma 6.1). But a defect group of a block of a direct product of groups is the direct product of defect groups of blocks of each factor. Thus, since Z(D) is cyclic of prime order, we may assume that Z(D) ≤ L_{1} and that D ∩ (L_{2} × ⋯ × L_{t}) = 1. Since Z(D) is central in C_{G}(z), it follows that each L̄_{1j} is of type A and of Lie rank at least p, hence that L_{1} is isomorphic to SL_{n}(q′) or SU_{n}(q′) for some power q′ of q.

Let x be a non-central element of D. We claim that x does not centralize L_{1}. Indeed, first note that if L̄ = SL_{n}(𝔽̄_{q}) and σ : L̄ → L̄ is a Frobenius endomorphism, then L̄^{σ} ≅ SL_{n}(q′) or L̄^{σ} ≅ SU_{n}(q′) for some power q′ of q and CL̄(L̄^{σ}) ≤ Z(L̄).

Write

x = (∏_{i=1}^{t} ∏_{j=1}^{r_{i}} x_{ij})t_{1}, x_{ij} ∈ L̄_{ij}, t_{1} ∈ Z̄ .

Let y in D be such that [x, y] = z. Write

y = (∏_{i=1}^{t} ∏_{j=1}^{r_{i}} y_{ij})t_{2}, y_{ij} ∈ L̄_{ij}, t_{2} ∈ Z̄ .

Since [x, y] = z ∈ L_{1}, [x_{11}, y_{11}] ≠ 1. In particular, x_{11} is not in the center of L̄_{11} and by the remark above, x_{11} does not centralize L_{11}^{F^{r_{i}}}. It follows that x does not centralize L_{1}.

Let c be a block of kL_{1} such that bc ≠ 0, c is D-stable, Br_{D}^{L_{1}D}(c) ≠ 0 and such that D ∩ L_{1} is a defect group of c (see Lemma 6.1). Let D_{0} be the kernel of the map D → Out(L_{1}). Then, < z >= Z(D) ≤ (D ∩ L_{1}) ≤ D_{0}. Thus, by Lemma 6.2 applied to the group L_{1}D, the block c of kL_{1} and the subgroup D_{0} of D, we have that D_{0} = (D_{0} ∩ L_{1})C_{D_{0}}(L_{1}). But it was shown above that C_{D}(L_{1}) = Z(D) ≤ D ∩ L_{1}. Hence D_{0} ≤ L_{1}.

If D_{0} = D, then the first assertion of the proposition holds with H = L_{1}, D̃ = D and < ζ >= 1 for the block c. We assume from now on that D ≠ D_{0}. The elements of T and hence of C_{G}(z) induce inner diagonal automorphisms of L_{1}. Since L_{1} is isomorphic to a special linear or special unitary group, Inndiag(L_{1})/Inn(L_{1}) is cyclic ([GLS, Section 2.7]). In particular, D/D_{0} is cyclic. But since Z(D) ≤ D_{0} and D is extra-special, in fact |D/D_{0}| = p. Let y ∈ D be such that D/D_{0} =< yD_{0} > and let η be a p-element in GL_{n}(q′) (or GU_{n}(q′)) such that ^{η}u = ^{y}u for all u ∈ L_{1}. In particular, c is stabilized by < η >. Let H = L_{1} < η >. Then H is a subgroup of GL_{n}(q′) (or GU_{n}(q′)) containing SL_{n}(q′) (or SU_{n}(q′)). Let D̃ be the subgroup of H generated by D_{0} and η. Then H = L_{1}D̃ and c is an H-stable block of L_{1}. Also, since C_{D}(L_{1}) = CD̃(L_{1}), we have that BrH ; D̃(c) ≠ 0. Finally, D_{0} = D̃ ∩ L_{1} is a defect group of the block c of kH. Thus, by Lemma 6.2, applied with H = L_{1}D̃, the block c is of kL_{1} and the subgroup D̃ of D̃, we have that c is a block of kH with D̃ as defect group.

Now y^{p} ∈ Z(D) ≤ Z(L_{1}), hence η^{p} centralizes L_{1}. Thus, η^{p} is a central element of GL_{n}(q′) (or GU_{n}(q′)). It follows that < η^{p} > ∩D_{0} ≤ Z(D) =< z >. If z ∈< η >, then η has order at least p^{2} and we set < ζ > to be the subgroup of < η > of index p^{2}. If z ∉< η >, then we set < ζ > to be the subgroup of < η > of index p. Then it is easy to check that D̃/ < ζ > is extra-special of order |D|. Since < ζ >≤< η^{p} >, < ζ > is a central subgroup of H. This proves the first part of the proposition.

Now suppose that G has a block with a defect group D which is extra-special of order p^{3}. Let H, c and D̃ be as in the the first assertion of the proposition. Suppose first that H ≤ GL_{n}(q′) and let p^{a} be the exact power of p dividing q′ − 1.

Then since SL_{n}(q′) ≤ H, it follows that |D̃| ≤ p^{3+a} and that there is a block of kGL_{n}(q′) covering c, with non-abelian defect groups of order at most p^{2a+3}. The structure of defect groups of finite general linear and unitary groups groups is well known. In particular, non-abelian defect groups of GL_{n}(q′) have order at least p^{pa+1} ([FS, Theorem 3C]). So, p^{pa+1} ≤ p^{2a+3}, which is impossible since p > 3. A similar argument, taking p^{a} to be the exact power of p dividing q′ + 1 handles the case H ≤ GU_{n}(q′).

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We now state and prove the main theorem of this section.

Theorem 6.4. Let F be an exotic fusion system on the extra-special group P of order 7^{3}. Then F is not a fusion system of a 7 block of any finite group.

Proof. Let G be a finite group with an F-block, say b. If F = RV_{2}, let F_{2} = F and let F_{1} = RV_{3}. If F = RV_{3}, let F_{1} = F and let F_{2} = RV_{2}. If F = RV_{1}, set F_{1} = F_{2} = F. Then, by Proposition 5.1 and Theorem 4.2, we may assume that G is quasisimple, 7 ∤ |Z(G)| and that b is an F_{1}− or F_{2}− block. Also, by Proposition 5.1, we may assume that P is not Sylow in G since neither F_{1} nor F_{2} is contained in a non-exotic fusion system on P.

Let Ḡ be the simple quotient of G. By the previous proposition, Ḡ is not a finite field of Lie type in characteristic different from 7. Suppose that Ḡ is a finite field of Lie type in characteristic 7. Then the exceptional part of the Schur multiplier of Ḡ is trivial, ([GLS, Table 6.1.3]). Thus there is a central 7′ extension G̃ of G such that G̃ = K̄^{F} where K̄ is a simply connected simple algebraic group and F is a Frobenius endomorphism of K̄. Then it follows from the theory of finite groups with strongly split BN-pair [CE, Theorem 6.18], (see also [Ke, Lemma 5.1]) that the defect groups of a 7 block of G̃ are either trivial or Sylow 7-subgroups of G̃. Hence, P is a Sylow 7-subgroup of G, a contradiction.

For odd p, a defect group of a p-block of a finite alternating group or a double cover of a finite alternating group is isomorphic to the Sylow p-subgroups of a finite symmetric group, hence Ḡ is not an alternating group.

Now, if Ḡ is a sporadic group then Ḡ must be one of He, O′N, F i′_{24} and the monster F_{1} as these are the only sporadic groups whose order is divisible by 7^{3}. Furthermore, if Ḡ is one of He, O′N, F i′_{24}, then 7^{3} is the exact power of 7 dividing |G|. Hence, P is a Sylow 7-subgroup of G, a contradiction.

Finally, suppose that G = Ḡ = F_{1}. Thus G has two conjugacy classes of elements of order 7 denoted by 7A and 7B(ATLAS notation). As in the ATLAS we denote by 7A^{2} and 7B^{2} the abelian elementary 7-groups of rank 2 generated by elements in 7A, respectively 7B. Then the maximal 7-local subgroups of Ḡ are of the type T_{1} = (7 : 3×He) : 2, normalizer of an element in 7A, S_{1} = (7^{2} : (3×2S_{7})×L_{2}(7)).2 normalizer of a group of type 7A^{2}, T_{2} = 7_{+}^{1+4} : (3 × 2S_{7}), normalizer of an element in 7B and S_{2} = 7^{2}.7.7^{2} : GL_{2}(7) normalizer of a group of type 7B^{2}.

The cyclic subgroups of order 7 of P are all conjugate in F as they are in the conjugacy class of the centre of P given by the automorphisms of the elementary abelian subgroups of rank 2 of P. Thus the elements of order 7 of P are in the same F-conjugacy class. Also Aut_{F}(P) normalizes the centre of P. So we have that Aut_{F}(P) is a section of T_{i}, for i = 1 or 2. Moreover, Aut_{F}(V) is a section of S_{i} for the same index i as above (where V is an F-centric, F-radical subgroup of P). But this is not possible as S_{1} has no section containing SL_{2}(7) : 2 and T_{2} has no section containing D_{16} × 3 or 6^{2} : 2.

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7. Appendix

We prove in the appendix that the definition we give in this paper for fusion systems is equivalent to the definition of Broto, Levi and Oliver [BLO, Definition 1.2] for saturated fusion systems. Given that the notion of ’fusion system’ in Broto, Levi and Oliver’s approach and our notion of ’fusion system’ are not the same, we call the former ’BLO-fusion system’.

Definition 7.1. [BLO, Definition 1.1] A BLO-fusion system on a finite p-group P is a category whose objects are the subgroups in P and whose morphism set Hom_{F}(P, Q) satisfy the following

(1) Hom_{P} (Q, R) ⊂ Hom_{F}(Q, R) ⊂ Inj(Q, R) for all Q, R ≤ P .

(2) Every morphism in F factors as an isomorphism in F followed by an inclusion.

The definition that Broto, Levi and Oliver give for a fully F-centralized subgroup is the same as ours. But the definition for a fully F-normalized subgroup is different.

Definition 7.2. [BLO, Definition 1.2] A subgroup Q of P is fully F-centralized if |C_{P} (Q)| ≥ |C_{P} (Q′)| for all Q′ ≤ P which is F-conjugated to Q.

Definition 7.3. [BLO, Definition 1.2] A subgroup Q of P is BLO-fully F-normalized if Q is fully F-centralized and Aut_{P} (Q) ∈ Syl_{p}(Aut_{F}(Q)).

Now the definition of saturated BLO-fusion systems.

Definition 7.4. [BLO, Definition 1.2] F is a saturated BLO-fusion system if the two following conditions hold:

(BLO-i) Each subgroup Q ≤ P is F-conjugated to at least one BLO-fully Fnormalized subgroup.

(BLO-ii) If Q ≤ P and ϕ ∈ Hom_{F}(Q, P) are such that ϕ(Q) is fully Fcentralized and one set N_{ϕ} := {g ∈ N_{P} (Q)|ϕc_{g}ϕ^{−1} ∈ Aut_{P} (ϕ(Q))}, then there is ϕ̄ ∈ Hom_{F}(N_{ϕ}, P) such that ϕ̄|_{P} = ϕ.

Here by c_{g} we mean the automorphism of Q given by conjugation by g. Remark that the definition of N_{ϕ} given here, although apparently different from the one in Definition 2.4, it is equivalent to the latter. This is an easy verification.

It is obvious that a saturated BLO-fusion system satisfies the three properties (FS-1) (FS-2) and (FS-3) in Definition 2.4. So a saturated BLO-fusion system is a fusion system in our approach. We prove now that Definition 2.4 in this paper implies the one in Definition 7.4. A fusion system in our approach clearly satisfies (BLO-i) so we only have to prove that it also satisfies (BLO-ii).

We start by proving that in a fusion system (Definition 2.4) the definition for a BLO-fully F-normalized subgroup (Definition 7.3) is obtained as a property from our setting. This is in fact Proposition 2.5 in this paper and the result was originally proved by Puig but given that the setting is different, we prefer to give a proof for the convenience of the reader.

Proposition 7.5 ([Pu, Proposition 2.7]). Let F be a fusion system (Definition 2.4) on a finite p-group P and let Q be a subgroup of P. Then Q is fully F-normalized (Definition 2.2) if and only if Q is fully F-centralized and Aut_{P} (Q) is a Sylow p-subgroup of Aut_{F}(Q).

Proof. First, a fully F-normalized subgroup Q of P is also fully F-centralized. Indeed, for any F-isomorphic subgroup Q′ of P the morphism ϕ ∈ Hom_{F}(Q′, Q) extends to a morphism ϕ̄ ∈ Hom_{F}(N_{ϕ}, N_{P} (Q)). But then C_{P} (Q′) ⊂ N_{ϕ} and ϕ̄(C_{P} (Q′)) ⊂ C_{P} (Q) giving that |C_{P} (Q′)| ≤ |C_{P} (Q)|. So Q is fully F-centralized.

Second, if Q is fully F-normalized then Aut_{P} (Q) is a Sylow p-subgroup of Aut_{F}(Q). To prove this, we will follow the proof of Proposition 1.5 in [Li]. Let Q be a subgroup of maximal order such that Aut_{P} (Q) is not a Sylow p-subgroup of Aut_{F}(Q). Then Q is a proper subgroup of P by property (FS-2). Choose a p-subgroup S of Aut_{F}(Q) such that Aut_{P} (Q) is a proper normal subgroup of S. Let ϕ ∈ S \ Aut_{P} (Q). Since ϕ normalizes Aut_{P} (Q), for every y ∈ N_{P} (Q) there is z ∈ N_{P} (Q) such that ϕ(^{y}u) = ^{z}ϕ(u) for all u ∈ Q. Thus we have N_{ϕ} = N_{P} (Q). Since Q is fully F-normalized, by property (FS-3) ϕ extends to ϕ̄ ∈ Hom_{F}(N_{ϕ}, N_{P} (Q)) so ϕ̄ ∈ Aut_{F}(N_{P} (Q)). Since ϕ has p-power order, by decomposing ϕ̄ into its p-part and its p′-part we may assume that ϕ̄ has also ppower order. Let ψ ∈ Hom_{F}(N_{P} (Q), P) be such that N′ := ψ(N_{P} (Q)) is fully F-normalized. As the order of N′ is greater then the order of Q, we have that Aut_{P} (N′) is a Sylow p-subgroup of Aut_{F}(N′). Now ψϕ̄ψ^{−1} is a p-element of Aut_{F}(N′), thus conjugated to an element in Aut_{P} (N′). Therefore we may choose ψ in such a way that there is y ∈ N_{P} (N′) satisfying ψϕ̄ψ^{−1}(v) = ^{y}v for all v ∈ N′. Since ϕ̄|_{Q} = ϕ, the automorphism ψϕ̄ψ^{−1} of N′ stabilizes ψ(Q). Thus y ∈ N_{P} (ψ(Q)). Since Q is fully F-normalized and ψ(N_{P} (Q)) ⊂ N_{P} (ψ(Q)) we have that ψ(N_{P} (Q)) = N_{P} (ψ(Q)), hence ϕ̄(u) = ^{ψ^{−1}(y)}u, for all u ∈ N_{P} (Q). And, in particular, ϕ ∈ Aut_{P} (Q), contradicting our first choice of ϕ.

The converse is straight forward as |N_{P} (Q)| = |Aut_{P} (Q)| · |C_{P} (Q)|.

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The following proposition gives the last ingredient for the equivalence of the two approaches. In our approach, property (FS-3) guarantees the extension to N_{ϕ} for the F-isomorphisms ϕ ending in fully F-normalized subgroups. But this is sufficient in order to have the extension to N_{ϕ} for all the F-isomorphisms ending in fully F-centralized subgroups.

Proposition 7.6. Let F be a fusion system (Definition 2.4) on a finite p-group P. Every morphism ϕ ∈ Hom_{F}(Q, P) such that ϕ(Q) is fully F-centralized extends to a morphism ϕ̄ ∈ Hom_{F}(N_{ϕ}, P). Thus F satisfies (BLO-ii).

Proof. We note Q′ := ϕ(Q). Choose θ ∈ Hom(Q′, P) such that θ(Q′) is fully F-normalized and, as Aut_{P} (θ(Q′)) is a Sylow p-subgroup of Aut_{F}(θ(Q′)) we can modify θ by a morphism in Aut_{F}(θ(Q′)) in order to have that N_{θ} = N_{P} (Q′).

By the property (FS-3) we have that θ extends to θ̄ ∈ Hom_{F}(N_{θ}, P). Denote by ψ := θϕ. By the same property (FS-3) ψ extends to ψ̄ ∈ Hom(N_{ψ}, P).

Our aim in what follows is to prove that N_{ϕ} ⊂ N_{ψ} and ψ̄(N_{ϕ}) ⊂ θ̄(N_{θ}) so that (θ̄)^{−1}ψ̄|_{N_{ϕ}} is the extension of ϕ to N_{ϕ}.

Both are simple verifications. Take y ∈ N_{ϕ} then by definition, there exists z ∈ N_{P} (Q′) such that ϕ(^{y}u) = ^{z}ϕ(u) for all u ∈ Q. By composing with θ we obtain θϕ(^{y}u) = θ(^{z}ϕ(u)). But as N_{θ} = N_{P} (Q′) we have that there exists x ∈ N_{P} (θ(Q′)) such that θ(^{z}ϕ(u)) = ^{x}θ(ϕ(u)) = ^{x}ψ(u). Thus, we have ψ(^{y}u) = ^{x}ψ(u) which means that y ∈ N_{ψ}. As this is true for all y ∈ N_{ϕ} we obtain that N_{ϕ} ⊂ N_{ψ}.

Take now x ∈ ψ̄(N_{ϕ}). Suppose that x = ψ̄(y), y ∈ N_{ϕ}. By definition, there exists z ∈ N_{P} (Q′) such that ϕ(^{y}u) = ^{z}ϕ(u) for all u ∈ Q. Now x = ψ̄(y) implies that ψ(^{y}u) = ^{x}ψ(u), so θ(^{z}ϕ(u)) = ψ(^{y}u) = ^{x}ψ(u), which is equivalent to ^{θ̄(z)}ψ(u) = xψ(u) for all u ∈ Q. This gives that x = θ̄(z)c for some c ∈ C_{P} (ψ(Q)). But as C_{P} (Q′) ⊂ N_{θ} and θ̄(C_{P} (Q′)) ⊂ C_{P} (θ(Q′)) and using the fact that Q′ is fully F-centralized we have that θ̄(C_{P} (Q′)) = C_{P} (θ(Q′)). This means that c ∈ θ̄(N_{θ}), so x ∈ θ̄(N_{θ}). Given that this is true for all x ∈ ψ̄(N_{ϕ}) we obtain ψ̄(N_{ϕ}) ⊂ θ̄(N_{θ}).

Thus (θ̄)^{−1}ψ̄|_{N_{ϕ}} extends ϕ to N_{ϕ}.

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Radha Kessar, Department of Mathematical Sciences, University of Aberdeen, Meston Building 318, Aberdeen AB24 3UE, U.K.

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

Radu Stancu, Ohio State University, 100 Mathematics Tower, 231, West 18th Avenue, Columbus, OH 43210, USA

   E-mail address: stancu@math.ohio-state.edu