LOCAL CONTROL IN FUSION SYSTEMS

OF P -BLOCKS OF FINITE GROUPS

Radha Kessar, Markus Linckelmann, Geoffrey R. Robinson

August 2001

Abstract. If p is an odd prime, b a p-block of a finite group G such that SL(2, p) is not involved in N_{G}(Q, e)/C_{G}(Q) for any b-subpair (Q, e), then N_{G}(Z(J(P))) controls b-fusion, where P is a defect group of b. This is a block theoretic analogue of Glauberman’s ZJ-Theorem [8].

1 Introduction

Glauberman’s ZJ-Theorem [8, Theorem B] states that if p is an odd prime and G is a finite group such that Qd(p) is not involved in G, then N_{G}(Z(J(P))) controls p-fusion in G, for P a Sylow p-subgroup of G. Here, J(P) denotes the Thompson subgroup of P (that is, the subgroup generated by all abelian subgroups of P of maximal order) and Qd(p) denotes the semi-direct product of C_{p} × C_{p} with SL(2, p) (with the natural action). This has proved to be an extremely powerful tool in local group-theoretic analysis, as it gives a general condition which ensures that p-fusion is controlled by a single p-local subgroup.

In this paper, we establish block-theoretic analogues of this and other similar results. Along the way, we will obtain results which seem to be new even in the group-theoretic case. A key ingredient, allowing us to exploit the existing group-theoretic methods, is a result of Külshammer and Puig [13] on extensions of nilpotent blocks. We also show (both in a group-theoretic and in a block-theoretic context) that if a normal subgroup of a given group G has a single local subgroup which controls fusion, then G itself has a single local subgroup with the same property. We discuss some consequences of such control of fusion to other problems in block theory.

Throughout the paper, k will denote an algebraically closed field of prime characteristic p. A block of a finite group G is a primitive idempotent b in Z(kG); following Alperin-Broué [1], a (G, b)-subpair is a pair (Q, e) consisting of a p-subgroup Q of G and a block e of C_{G}(Q) such that Br_{Q}(b)e = e, where Br_{Q} : (kG)^{Q} → C_{G}(Q) is the Brauer homomorphism [5]. The set of (G, b)-subpairs is a partially ordered set on which G acts by conjugation, and the maximal (G, b)-subpairs with respect to this partial order are all G-conjugate. If (P, e) is a maximal (G, b)-subpair, then P is called a defect group of the block b (this

Typeset by AMS-TEX

1980 Mathematics Subject Classification (1985 Revision). 20C20.

notion is due to Brauer [2]); moreover, for any subgroup Q of P there is a unique block e_{Q} of C_{G}(Q) such that (Q, e_{Q}) ⊆ (P, e) (cf. [1]). A detailed account of subpairs and their properties may be found in [16] (where subpairs are referred to as Brauer pairs). The local structure of b is the G-set of (G, b)-subpairs viewed as category; the following definition makes this precise.

Definition 1.1. Let G be a finite group, let b be a block of G and let (P, e) be a maximal (G, b)-subpair. For any subgroup Q of P denote by e_{Q} the unique block of C_{G}(Q) such that (Q, e_{Q}) ⊆ (P, e). We denote by F_{(P,e)}(G, b) the category whose objects are the subgroups of P and whose sets of morphisms Hom_{F_{(P,e)}(G,b)}(Q, R) are the sets of group homomorphisms φ : Q → R for which there exists an element x ∈ G satisfying x(Q, e_{Q}) ⊆ (R, e_{R}) and φ(u) = xux^{−1} for all u ∈ Q, where Q, R run over the set of subgroups of P.

Since all maximal (G, b)-subpairs are G-conjugate, the category F_{(P,e)}(G, b) does not depend on the choice of (P, e) up to isomorphism of categories. If b is the principal block of G then P is a Sylow-p-subgroup of G and e_{Q} is the principal block of C_{G}(Q) for any subgroup Q of P; in this case we write F_{P} (G) = F_{(P,e)}(G, b). Glauberman’s ZJ-Theorem reads then F_{P} (G) = F_{P} (N_{G}(Z(J(P)))), provided that p is odd and Qd(p) is not involved in G.

We need a block-theoretic replacement for the hypothesis on Qd(p). Recall that if G is a finite group and b is a block of G, then a (G, b)-subpair (Q, f) is called centric if Z(Q) is a defect group of f and (Q, f) is called radical if O_{p}(N_{G}(Q, f)/QC_{G}(Q)) = 1. The notion of centric subpairs - frequently called self-centralising pairs in the literature - goes back to Brauer [3]).

Definition 1.2. Let G be a finite group. A block b of G is called SL(2, p)-free if SL(2, p) is not isomorphic to a subquotient of any of the groups N_{G}(Q, f)/C_{G}(Q), where (Q, f) is a centric and radical (G, b)-subpair.

The definition of an SL(2, p)-free block is really a local condition on the block, in that it can be formulated purely in terms of the category F_{(P,e)}(G, b), where (P, e) is a maximal subpair of a block b of G. Indeed, b is SL(2, p)-free if and only if SL(2, p) is not involved in the automorphism group in F_{(P,e)}(G, b) of any subgroup Q of P such that (Q, e_{Q}) is centric and radical for the unique e_{Q} such that (Q, e_{Q}) ≤ (P, e). It may well happen that a non principal block b of G is SL(2, p)-free even though SL(2, p) is involved in G. If, however, the principal block of G is SL(2, p)-free, then Qd(p) is not involved in G (cf. Proposition 5.1 and [9, Lemma 10.6]). In this case, our hypothesis “SL(2, p)-free” is in fact slightly more restrictive, since ( in the principal block case) it effectively excludes faithful action of SL(2, p) on any p-subgroup of G, not just the natural action of SL(2, p) on C_{p} × C_{p}.

Examples of SL(2, p)-free blocks include all blocks with abelian defect groups and, for p ≥ 5, all blocks of finite p-solvable groups, or more generally, all blocks for which the groups N_{G}(Q, f)/C_{G}(Q) occurring in 1.2 are p-solvable.

Since Glauberman’s control of fusion theorems also apply to some characteristic subgroups of p-groups other than the center of the Thompson subgroup, we make the following definitions, the first of which is given in [11, §5].

Definition 1.3 A positive characteristic p-functor is a map W sending any finite pgroup P to a subgroup W (P) of P, with the property that W (P) ≠ 1 if P ≠ 1 and that any isomorphism of finite p-groups P ≅ Q maps W (P) onto W (Q). A Glauberman functor is a positive characteristic p-functor W with the following additional property: whenever P is a Sylow-p-subgroup of a finite group L which satisfies C_{L}(O_{p}(L)) = Z(O_{p}(L)) and which does not have a subquotient isomorphic to Qd(p), then W (P) is normal in L.

Of course, by Glauberman’s ZJ-Theorem the map sending a finite p-group P to Z(J(P)) is a Glauberman functor; in fact showing that this map is a Glauberman functor is the essential ingredient of the ZJ-Theorem. By [9, Theorem 14.8] any of the maps sending a finite p-group P to K_{∞}(P) or K^{∞}(P) are Glauberman functors, where K_{∞}(P), K^{∞}(P) are defined in [9, Section 12].

If W is a positive characteristic p-functor, then W (P) is characteristic in P, for any finite p-group P; in particular, if P is a p-subgroup of a finite group G, then N_{G}(W (P)) contains N_{G}(P). If H is any subgroup of G containing N_{G}(P), there is a unique block c of H such that Br_{P} (b) = Br_{P} (c), the Brauer correspondent of b (cf. [1] or [16]). Then P is again a defect group of c, and since C_{G}(P) ⊆ H, every maximal (G, b)-subpair (P, e) is also a maximal (H, c)-subpair.

We are now ready to state our results. In what follows, refer to 2.1 and 2.3 for the exact definition of control of fusion that we are using.

Theorem 1.4. Let G be a finite group, let b be a block of G and let (P, e) be a maximal (G, b)-subpair. Let W be a Glauberman functor, set N = N_{G}(W (P)) and denote by c the unique block of N such that Br_{P} (b) = Br_{P} (c). If p is odd and b is SL(2, p)-free, then F_{(P,e)}(G, b) = F_{(P,e)}(N, c). In other words, the group N controls fusion in F_{(P,e)}(G, b).

The proof of 1.4 is given in section 6. If we specialise Theorem 1.4 to the case of principal blocks and W (P) = Z(J(P)), we obtain the conclusion of Glauberman’s ZJ-Theorem (but, as mentioned above, our hypothesis “SL(2, p)-free” is slightly more restrictive).

Our next result shows that the property of being locally controlled by the normaliser of a single non-trivial subgroup of a defect group carries through normal extensions of blocks.

Theorem 1.5. Let G be a finite group, H a normal subgroup of G, c a G-stable block of H and b a block of G such that bc = b. Let (P, e) be a maximal (G, b)-subpair. There is a P -stable maximal (H, c)-subpair (Q, f) such that Q = P ∩ H and fe_{Q} ≠ 0, where (Q, e_{Q}) is the unique (G, b)-subpair contained in (P, e).

Furthermore, if there is a normal subgroup V of Q such that N_{H}(V) controls fusion in F_{(Q,f)}(H, c), then N_{G}(W) controls fusion in F_{(P,e)}(G, b) where W is the subgroup of P generated by the set of N_{G}(Q, f)-conjugates of V.

An interesting consequence of Theorem 1.5 is that it allows us to prove that any block b of a finite group G lying over an SL(2, p)-free block of a normal subgroup N of G with non-trivial defect groups has again a local structure which is controlled by the normaliser of a single non-trivial p-subgroup of G, even though b itself need not be SL(2, p)-free:

Corollary 1.6. Let G be a finite group, let b be a block of G and let (P, e) be a maximal (G, b)-subpair. If there is a normal subgroup H of G such that H ∩ P ≠ 1 and such that b covers an SL(2, p)-free block c of H, then there is a non-trivial normal subgroup W in P such that N_{G}(W) controls fusion in F_{(P,e)}(G, b).

In [9, Section 12] Glauberman showed that for W = K_{∞} or W = K^{∞}, the subgroup W (P) of P is self-centralising; that is, C_{P} (W (P)) = Z(W (P)). Thus, in the situation of Theorem 1.4, the (G, b)-subpair (W (P), e_{W (P)}) is centric; in other words, the normaliser in G of some centric (G, b)-subpair controls b-fusion. The next Theorem shows that there is a canonical choice for such a centric subpair. By results of Külshammer and Puig in [13, Theorem 1.8], associated with any centric (G, b)-subpair (Q, f) and any choice of a maximal (N_{G}(Q, f), f)-subpair (R, g), there is a canonical group extension

1 ⟶ Q ⟶ L ⟶ N_{G}(Q, f)/QC_{G}(Q) ⟶ 1

having the property that R is a Sylow-p-subgroup of L and F_{(R,g)}(N_{G}(Q, f), f) = F_{R}(L) (we explain this in some more detail in 2.4 below); moreover, O_{p′}(L) = 1 and C_{L}(Q) = Z(Q). Thus, if b is SL(2, p)-free, then Qd(p) is not involved in L, and hence W (R) is normal in L for any Glauberman functor W.

Theorem 1.7. Let G be a finite group, let b be a block of G and let (P, e) be a maximal (G, b)-subpair. Assume that p is odd and that b is SL(2, p)-free. There is a unique minimal subgroup Q of P such that (Q, f) is centric and radical, where f is the unique block of C_{G}(Q) such that (Q, f) ⊆ (P, e). Moreover, Q is normal in P and we have F_{(P,e)}(G, b) = F_{P} (L), where L is the middle term of the Külshammer-Puig extension associated with (Q, f).

Remark 1.8. Theorems 1.5 and 1.7 seem to add some new information even in the principal block case. Theorem 1.5 implies that if N is a normal subgroup of a finite group G such that N_{N}(V) controls strong p-fusion in P ∩ N with respect to N for some normal subgroup V of P ∩N then the subgroup, W, of P generated by all N_{G}(P ∩N)-conjugates of V has the property that N_{G}(W) controls strong p-fusion in P with respect to G. Theorem 1.7 translates to the following statement: given a finite group G with a Sylow-p-subgroup P such that SL(2, p) is not involved in N_{G}(Q)/C_{G}(Q) for any p-subgroup Q of G, there is a unique minimal subgroup Q of P such that Z(Q) is a Sylow-p-subgroup of C_{G}(Q) and such that O_{p}(N_{G}(Q)/QC_{G}(Q)) = 1; moreover, N_{G}(Q) controls strong p-fusion in P with respect to G.

A classifying space of b is a p-complete space B(G, b) having the homotopy type of the p-completion of an L-system associated with F_{(P,e)}(G, b) in the sense of Broto, Levi and Oliver [4]. Note that in the situation of Theorem 1.7, the local structure of b is the same as the local structure of the principal block of L. Thus, if we take for B(G, b) the p-completion BL_{p}^{∧} of the classifying space BL of L we obtain the following immediate consequence.

Corollary 1.9. If p is odd, any SL(2, p)-free block has a classifying space, which is unique up to homotopy.

Theorems 1.4 and 1.5 provide many examples of blocks whose fusion pattern is determined by the normaliser of a single non-trivial p-subgroup. The existence of such controlling subgroups has ramifications for the Dade Projective Conjectures (DPC).

Theorem 1.10. Let G be a finite group, let b be a block of G and let (P, e) be a maximal (G, b)-subpair. Assume that there is a normal subgroup R in P such that N_{G}(P, e) ⊆ N_{G}(R) and such that N_{G}(R) controls fusion in F_{(P,e)}(G, b). Let c be the block of N_{G}(R) which satisfies Br_{P} (c)e = e; that is, c is the Brauer correspondent in N_{G}(Q) of b.

(i) If every section of G satisfies DPC, then there is a defect preserving bijection between the sets of irreducible characters of b and irreducible characters of c.

(ii) If every proper section of G satisfies DPC, then DPC holds for b if and only if there is a defect preserving bijection between the sets of irreducible characters of b and irreducible characters of c.

2 On local categories of blocks

We collect in this Section some standard terminology and properties of local categories of blocks.We fix a finite group G, a block b of G and a maximal (G, b)-subpair (P, e). For any subgroup Q of P, denote by e_{Q} the unique block of C_{G}(Q) such that (Q, e_{Q}) ⊆ (P, e) (in particular, e = e_{P}).

By the uniqueness of the inclusion of subpairs (cf. [1]) we have F_{P} (P) ⊆ F_{(P,e)}(G, b). If we choose a Sylow-p-subgroup S of G containing P, we have also F_{(P,e)}(G, b) ⊆ F_{S}(G).

Two subgroups Q, R of P are isomorphic as objects in F_{(P,e)}(G, b) if there is x ∈ G such that ^{x}(Q, e_{Q}) = (R, e_{R}). Any subgroup Q of P is isomorphic in F_{(P,e}(G, b) to a subgroup R of P such that N_{P} (R) is a defect group of e_{R} viewed as block of N_{G}(R, e_{R}) (cf. [1] or [16]). We say that (Q, e_{Q}) is an Alperin-Goldschmidt- pair (for F_{(P,e)}(G, b)), if (Q, e_{Q}) is centric, radical and N_{P} (Q) is a defect group of kN_{G}(Q, e_{Q})e_{Q}. If Q is normal in P, then P is a defect group of e_{Q} as block of N_{G}(Q, e_{Q}), and hence (P, e_{P}) is also a maximal (N_{G}(Q, e_{Q}), e_{Q})-subpair. It has been shown by Puig, that (Q, e_{Q}) is centric if and only if C_{P} (R) = Z(R) for any subgroup R of P which is isomorphic to Q in F_{(P,e)}(G, b). Thus the property of being centric can be read off the category F_{(P,e)}(G, b). Furthermore, the automorphism group of Q in F_{(P,e)}(G, b) is canonically isomorphic to N_{G}(Q, e_{Q})/C_{G}(Q).

A conjugation family for F_{(P,e)}(G, b) is a set C of subgroups of P with the following property: every isomorphism in F_{(P,e)}(G, b) is the composition of isomorphisms of the form φ : Q → R, where Q, R are subgroups of P, such that there exists a subgroups S in C containing both Q, R and an element x ∈ N_{G}(S, e_{S}) satisfying φ(u) = xux^{−1} for all u ∈ Q.

It is well-known and easy to check that if C is a conjugation family for F_{(P,e)}(G, b), then any subset C′ of C such that any object in C is isomorphic to an object of C′ in F_{(P,e)}(G, b) is again a conjugation family.

By Alperin’s fusion theorem (in its refined version by Goldschmidt and adapted to blocks, cf. [1, §4]), the set of subgroups Q of P for which (Q, e_{Q}) is an Alperin-Goldschmidt pair is a conjugation family for F_{(P,e)}(G, b), called the Alperin-Goldschmidt conjugation family for F_{(P,e)}(G, b).

Definition 2.1 A subgroup H of G controls fusion in F_{(P,e)}(G, b) if H contains P and if F_{(P,e)}(G, b) ⊆ F_{S}(H) for some Sylow-p-subgroup S of H which contains P.

By Alperin’s fusion theorem, a subgroup H of G containing P controls fusion in F_{(P,e)}(G, b) if and only if N_{G}(Q, e_{Q}) = N_{H}(Q, e_{Q})C_{G}(Q) for any subgroup Q of P.

Lemma 2.2. Let W be a normal subgroup in P, and let H be a subgroup of G such that P ⊆ H ⊆ N_{G}(W). Assume that H controls fusion in F_{(P,e)}(G, b). Then W is contained in any subgroup Q of P such that (Q, e_{Q}) is centric and radical.

Proof. Let Q be a subgroup of P such that (Q, e_{Q}) is centric and radical. Since N_{G}(Q, e_{Q}) = N_{H}(Q, e_{Q})C_{G}(Q) and W is normal in H, the image of N_{W} (Q) is normal in N_{G}(Q, e_{Q})/QC_{G}(Q), hence N_{W} (Q) ⊆ QC_{G}(Q) as (Q, e_{Q}) is radical. Thus N_{W} (Q) ⊆ Q because (Q, e_{Q}) is centric, and therefore W ⊆ Q. □

The first statement of the following Proposition is a variation of [12, Statement 1]. The second statement makes precise what it means, in certain circumstances, for a subgroup to control fusion.

Proposition 2.3. Let Q be a subgroup of P, let H be a subgroup of N_{G}(Q) containing QC_{G}(Q), and let c be the unique block of H such that Br_{Q}(c)e_{Q} = e_{Q}. Assume that c has a defect group R contained in P. Then (R, e_{R}) is a maximal (H, c)-subpair, and we have F_{(R,e_{R})}(H, c) ⊆ F_{(P,e)}(G, b); moreover, this inclusion is an equality if and only if H controls fusion in F_{(P,e)}(G, b).

Proof. Since Q is normal in H, Q is contained in any defect group of H. If R is a defect group of c contained in P, then C_{G}(R) ⊆ C_{G}(Q) ⊆ H, and thus (R, e_{R}) is a - necessarily maximal - (H, c)-subpair. Let (S, f) be a centric radical (H, c)-subpair contained in (R, e_{R}). Again, since Q is normal in H, we have Q ⊆ S by 2.2. Then C_{G}(S) = C_{H}(S), and so f = e_{S}. Thus N_{H}(S, f) = N_{H}(S, e_{S}) ⊆ N_{G}(S, e_{S}). The inclusion F_{(R,e_{R})}(H, c) ⊆ F_{(P,e)}(G, b) follows, using Alperin’s fusion theorem.

Assume that H controls fusion in F_{(P,e)}(G, b). Then in particular R = P is a defect group of c, as Q is normal in H and P is contained in H. Thus (P, e) is also a maximal (H, c)-subpair. Let now S be a subgroup of P such that (S, e_{S}) is a radical centric (G, b)- subpair. Thus Q ⊆ S by 2.2. But then C_{G}(S) ⊆ H, and so (S, e_{S}) is also a centric (H, c)- subpair. Thus the inclusion N_{G}(S, e_{S}) ⊆ N_{H}(S, e_{S})C_{G}(S) translates to F_{(P,e)}(G, b) ⊆ F_{(P,e)}(H, c), hence equality by the first statement. The rest is clear. □

Proposition 2.3 applies in the following two situations. If H contains N_{G}(P) and if c is the unique block of H such that Br_{P} (c) = Br_{P} (b), then (P, e) is also a maximal (H, c)-subpair. Thus if P has a subgroup Q such that C_{G}(Q) ⊆ H ⊆ N_{G}(Q), we have F_{(P,e)}(H, c) ⊆ F_{(P,e)}(G, b). The second situation, in which we are going to apply 2.3 arises if H = N_{G}(Q, e_{Q}) for some subgroup Q of P and if c = e_{Q} such that N_{P} (Q) is a defect group of c (viewed as block of H).

The next Proposition is a particular case of Külshammer-Puig [13, Theorem 1.8], translated to our terminology (see also [12, Statement 8]).

Proposition 2.4. Assume that G = N_{G}(Q, e_{Q}) for some subgroup Q of P such that (Q, e_{Q}) is centric. Then b = e_{Q}, and there is a short exact sequence of finite groups

1 ⟶ Q ⟶ L ⟶ G/QC_{G}(Q) ⟶ 1

such that P is a Sylow-p-subgroup of L and such that F_{(P,e)}(G, b) = F_{P} (L). Moreover, we have O_{p′}(L) = 1 and C_{L}(Q) = Z(Q); in particular, L is p-constrained.

Proof. As Q is normal in G, the block idempotent b is contained in kC_{G}(Q), and as G stabilises e_{Q}, we have b = e_{Q} (this is a standard argument; see [1]). To establish the link with the terminology in [13, 1.8], note first that P is also a defect group of {b} viewed as point of G on kC_{G}(Q), because P is maximal with respect to the property Br_{P} (b) ≠ 0. The existence of a canonical exact sequence as stated such that P is a Sylow-p-subgroup of L is a particular case of [13, 1.8]. This extension has the property, that for any y ∈ L, the outer automorphisms of Q induced by conjugation with y and by conjugation with some element x ∈ G such that xQC_{G}(Q) is the image of y in G/QC_{G}(Q) coincide. In particular, if y ∈ C_{L}(Q) then x ∈ QC_{G}(Q), and hence y ∈ Q. This shows that C_{L}(Q) = Z(Q), and since Q is normal in L, we have O_{p′}(L) = O_{p′}(C_{L}(Q)) = 1. The equality F_{(P,e)}(G, b) = F_{P} (L) is essentially a reformulation of [13, 1.8.2]; we reproduce the argument from [12, Statement 8]. Since Q is normal in L and in G, it suffices to show that the images in Aut(R) of N_{G}(R, e_{R}) and N_{L}(R) are equal, where R is a subgroup of P containing Q. As (Q, e_{Q}) is centric, so is (R, e_{R}). Similarly, as C_{L}(Q) = Z(Q), we have C_{L}(R) = Z(R). Setting Ḡ = G/QC_{G}(Q), with the notation of [13, 1.8] (which is defined in [13, 2.8]) we have E_{G,Ḡ}(R, e_{R}) = E_{L,Ḡ}(R). By [13, (2.8.1)], the canonical maps E_{G,Ḡ}(R, e_{R}) → E_{G}(R, e_{R}) and E_{L,Ḡ}(R) → E_{L}(R) are surjective. Thus E_{G}(R, e_{R}) = E_{L}(R). This implies the equality F_{(P,e)}(G, b) = F_{P} (L). □

We need the following generalisation of [12, Statement 9].

Proposition 2.5. Let Q be a normal subgroup of P, set H = N_{G}(Q) and denote by c the unique block of H such that e_{Q}c = e_{Q}. Suppose there is a finite group L having P as Sylow-p-subgroup such that F_{(P,e)}(G, b) = F_{P} (L). Then (P, e) is a maximal (H, c)-subpair, P is a Sylow-p-subgroup of N_{L}(Q), and we have F_{(P,e)}(H, c) = F_{P} (N_{L}(Q)).

Proof. Since Q is normal in P, the pair (P, e_{P}) is also a maximal (H, c)-subpair, and clearly P is a Sylow-p-subgroup of N_{L}(Q). By 2.3, we have F_{(P,e)}(H, c) ⊆ F_{(P,e)}(G, b). In order to show the equality F_{(P,e)}(H, c) = F_{P} (N_{L}(Q)), it suffices to show that N_{H}(S, f) and N_{L}(S) ∩ N_{L}(Q) have the same images in Aut(S), where (S, f) is an (H, c)-Brauer pair contained in (P, e). Since Q is normal in H and N_{L}(Q), we may assume that Q ⊆ S, by 2.2. Then C_{G}(S) ⊆ H and f = e_{S}. The assumption F_{(P,e)}(G, b) = F_{P} (L) implies that given any x ∈ N_{G}(S, e_{S}), there is y ∈ N_{L}(S) such that ^{x}u = ^{y}u for all u ∈ S. Since Q ⊆ S, clearly x ∈ N_{H}(S, e_{S}) if and only if y ∈ N_{L}(S) ∩ N_{L}(Q). The equality F_{(P,e)}(H, c) = F_{P} (N_{L}(Q)) follows. □

The following Lemma appears in a slightly more general version in Puig [14].

Lemma 2.6. Let G be a finite group, let b be a block of G and let (Q, e), (R, f) be centric (G, b)-subpairs such that (Q, e) ⊆ (R, f). We have

N_{G}(R, f) ∩ C_{G}(Q) = Z(Q)C_{G}(R) .

Proof. Clearly the right side is contained in the left side. For the converse, assume first that Q is normal in R. Let x ∈ N_{G}(R, f) ∩ C_{G}(Q). It is easy to check that [R, x] ⊆ C_{R}(Q) = Z(Q). Thus [R, x, x] = 1. If x is a p′-element, this forces x ∈ C_{G}(R) by standard properties of coprime group actions (cf. [10]). Note that the image of a defect group of f as block of N_{G}(R, f) is a Sylow-p-subgroup of N_{G}(R, f)/C_{G}(R). Thus if x is a p-element, we may assume that x belongs to a defect group of f as block of N_{G}(R, f), which implies x ∈ Z(Q), as (Q, e) is centric. The general case follows by induction. □

Proposition 2.7. Assume that there is a unique minimal subgroup R of P such that (R, e_{R}) is centric and radical. Then R is normal in P, the pair (P, e) is a maximal (N_{G}(R, e_{R}), e_{R})-subpair, and we have F_{(P,e)}(G, b) = F_{(P,e)}(N_{G}(R, e_{R}), e_{R}).

Proof. The uniqueness of R implies that R is normal in P, and hence (P, e) is also a maximal (N_{G}(R, e_{R}), e_{R})-subpair. Let S be a subgroup of P such that (S, e_{S}) is centric and radical. Then R ⊆ S by the uniqueness of (R, e_{R}). If x ∈ N_{G}(S, e_{S}), then ^{x}(R, e_{R}) ⊆ (S, e_{S}), and again, by the uniqueness of (R, e_{R}), we deduce that ^{x}(R, e_{R}) = (R, e_{R}). In other words, N_{G}(S, e_{S}) ⊆ N_{G}(R, e_{R}), which implies F_{(P,e)}(G, b) ⊆ F_{(P,e)}(N_{G}(R, e_{R}), e_{R}), hence the equality by 2.3. □

We provide a criterion for when the Alperin-Goldschmidt conjugation family has a unique minimal element.

Proposition 2.8. Assume that F_{(P,e)}(G, b) = F_{(P,e)}(N_{G}(Q, e_{Q}), e_{Q}) for some normal subgroup Q of P such that (Q, e_{Q}) is centric. Then there is a unique subgroup R of P containing Q such that O_{p}(N_{G}(Q, e_{Q})/QC_{G}(Q)) = RC_{G}(Q)/QC_{G}(Q). The group R is then the unique minimal subgroup of P such that (R, e_{R}) is centric and radical. In particular, R is normal in P and F_{(P,e)}(G, b) = F_{(P,e)}(N_{G}(R, e_{R}), e_{R}).

Proof. We may assume that G = N_{G}(Q, e_{Q}) and hence that b = e_{Q}. The image of P in G/QC_{G}(Q) is a Sylow-p- subgroup; since (Q, e_{Q}) is centric, this image is isomorphic to P/Q. Therefore, there is a unique subgroup R of P containing Q such that the image of R in G/QC_{G}(Q) is O_{p}(G/QC_{G}(Q)). The uniqueness of R implies that R is normal in P. Note that b is still a block of RC_{G}(Q), and then (R, e_{R}) is a maximal (RC_{G}(Q), b)- subpair. By our choice of R, the group RC_{G}(Q) is normal in G, and since RC_{G}(Q) acts transitively on the set of maximal (RC_{G}(Q), b)-subpairs, the Frattini argument shows that G = N_{G}(R, e_{R})C_{G}(Q).

Let S be a subgroup of P such that (S, e_{S}) is centric and radical. By Lemma 2.6, we have N_{G}(S, e_{S}) ∩ QC_{G}(Q) = QC_{G}(S). Thus the inclusion N_{G}(S, e_{S}) ⊂ G induces an injective group homomorphism N_{G}(S, e_{S})/QC_{G}(S) → G/QC_{G}(Q). The image of R in G/QC_{G}(Q) is O_{p}(G/QC_{G}(Q)); thus the image of N_{R}(S) in N_{G}(S, e_{S})/QC_{G}(S)) is contained in O_{p}(N_{G}(S, e_{S})/QC_{G}(S)), and hence the image of N_{R}(S) in N_{G}(S, e_{S})/SC_{G}(S) is contained in O_{p}(N_{G}(S, e_{S})/SC_{G}(S)) = 1. This forces N_{R}(S) ⊆ SC_{G}(S). As (S, e_{S}) is centric, we get N_{R}(S) ⊆ S, hence R ⊆ S.

By Lemma 2.6 again, we have N_{G}(R, e_{R}) ∩ RC_{G}(Q) = RC_{G}(R). As G = N_{G}(R, e_{R})C_{G}(Q), it follows that N_{G}(R, e_{R})/RC_{G}(R) ≅ G/RC_{G}(Q), and hence O_{p}(N_{G}(R, e_{R})/RC_{G}(R)) = 1 by our choice of R. This shows that R is indeed the unique minimal subgroup of P such that (R, e_{R}) is centric and radical. The rest is clear by 2.7. □

3 Local control of characteristic p-functors

Let G be a finite group, let b be a block of G, let (P, e) be a maximal (G, b)-subpair, and for any subgroup Q of P, denote by e_{Q} the unique block of C_{G}(Q) such that (Q, e_{Q}) ⊆ (P, e).

Given a positive characteristic p-functor W and a subgroup Q of P, we set W_{1}(Q) = Q and P_{1}(Q) = N_{P} (Q). For any positive integer i, we define inductively W_{i+1}(Q) = W (P_{i}(Q)) and P_{i+1}(Q) = N_{P} (W_{i+1}(Q)). For all positive integers i we have W_{i}(Q) ⊆ P_{i}(Q), and if P_{i}(Q) is a proper subgroup of P, in fact P_{i}(Q) is a proper subgroup of P_{i+1}(Q). In particular, P_{i}(Q) = P for all large enough i. We will say that Q is well-placed in P (with respect to W and F_{(P,e)}(G, b)) if P_{i}(Q) is a defect group of the block e_{W_{i}(Q)} as block of N_{G}(W_{i}(Q), e_{W_{i}(Q)}) for all positive integer i. Clearly P is always well-placed in P.

The next Lemma states essentially that every subgroup of P is isomorphic to a wellplaced subgroup with respect to F_{(P,e)}(G, b) and a positive characteristic p-functor.

Lemma 3.1. Let W be a positive characteristic p-functor. For any subgroup Q of P, there is an element x ∈ G such that ^{x}(Q, e_{Q}) ⊂ (P, e), ^{x}N_{P} (Q) ⊆ P and such that ^{x}Q is well-placed in P.

Proof. Define sequences of subgroups and blocks as follows. Let V_{1} := Q, v_{1} := e_{Q}. Let (R_{1}, r_{1}) be a b-subpair which is maximal with respect to normalising (V_{1}, v_{1}) and such that (N_{P} (Q), e_{N_{P} (Q)}) ≤ (R_{1}, r_{1}). For i ≥ 1 let V_{i+1} = W (R_{i+1}) and let (V_{i+1}, v_{i+1}) be the b-subpair contained in (R_{i+1}, r_{i+1}). Let (R_{i+1}, r_{i+1}) be a b-subpair which is maximal with respect to normalising (V_{i+1}, v_{i+1}) and such that (R_{i}, r_{i}) ≤ (R_{i+1}, r_{i+1}). Note that if (S, f) is a maximal b-subpair containing (R_{i+1}, r_{i+1}), then R_{i+1} = N_{S}(V_{i+1}). On the other hand, N_{S}(R_{i}) ⊂ N_{S}(V_{i+1}). Thus, either R_{i} = S or R_{i+1} properly contains R_{i}. In other words, there exists an integer t such that for all i ≥ t, (R_{i}, r_{i}) = (R_{t}, r_{t}) is a maximal b-Brauer pair, (V_{i}, v_{i}) = (W (R_{t}), v_{t}). Let x ∈ G be such that ^{x}(R_{t}, r_{t}) = (P, e). Then x(Q, e_{Q}) ≤ (P, e), and since for every i ≥ 1, ^{x}R_{i} ⊂ ^{g}R_{t} = P, it is clear that ^{x}(Q, e_{Q}) is well placed in (P, e). The second assertion is clear since N_{P} (Q) ⊂ R_{1} ⊂ ^{x^{−1}}P. □

The next results states roughly speaking, that “if a positive characteristic p-functor controls fusion locally, it controls fusion globally”. This generalises a result by Alperin and Gorenstein (cf. [11, Ch. X, Theorem 9.3])

Proposition 3.2. Let W be a positive characteristic p-functor. Assume that for any non-trivial subgroup Q of P and any maximal (N_{G}(Q, e_{Q}), e_{Q})-subpair (R, f), the group N_{N_{G}(Q,e_{Q})}(W (R)) controls fusion in F_{(R,f)}(N_{G}(Q, e_{Q}), e_{Q}). Then N_{G}(W (P)) controls fusion in F_{(P,e)}(G, b).

Proof. Set H = N_{G}(W (P)). Suppose, if possible that the result is not true. Then by 3.1 above, there exists a non-trivial subgroup Q of P such that (Q, e_{Q}) is well placed in (P, e) such that N_{G}(Q, e_{Q}) is not contained in C_{G}(Q)N_{H}(Q, e_{Q}).

We introduce the following notation. For i ≥ 1, let W_{i} = W_{i}(Q), P_{i} = P_{i}(Q), e_{i} = e_{W_{i}}, N_{i} = N_{G}(W_{i}, e_{i}), M_{i} = N_{G}(W_{i}) and L_{i} = N_{i} ∩ N_{G}(W_{i+1}). Let f_{i} be the block of M_{i} satisfying e_{i}f_{i} = e_{i}. Let s_{i} be the block of L_{i} such that Br_{P_{i}}(s_{i}) = Br_{P_{i}}(e_{i}).

Set F_{i} = F_{(P_{i},e_{P_{i}})}(N_{i}, e_{i}), set G_{i} = F_{(P_{i},e_{P_{i}})}(L_{i}, s_{i}), and set H_{i} = F_{(P_{i},e_{P_{i}})}(M_{i}, f_{i}).

It is clear from 2.3 that G_{i} ⊂ F_{i}. On the other hand, P_{i}C_{M_{i+1}}(W_{i}) ⊂ L_{i} ⊂ N_{M_{i+1}}(W_{i}). Since (W_{i+1}, e_{i+1}) ≤ (P_{i}, e_{P_{i}}), Br_{P_{i}}(f_{i})e_{P_{i}} = e_{P_{i}} and hence by 2.3 it follows that G_{i} ⊂ H_{i+1}. Since, clearly H_{i+1} = F_{i+1}, we get that G_{i} ⊂ F_{i+1}.

By the hypothesis of proposition, we have that G_{i} = F_{i}, hence, we get that for all i ≥ 1, F_{1} ⊂ F_{i} ⊂ F_{i+1}.

Let i be such that P_{i} = P, so that F_{i+1} = F_{H,c}. Let g be an element of N_{G}(Q, e_{Q}). Then conjugation by g determines an element, say ϕ of of End_{F_{1}}(Q). Then ϕ is induced by conjugation with an element x ∈ H. Then g = cx for some c ∈ C_{G}(Q). Thus, N_{G}(Q, e_{Q}) ⊂ C_{G}(Q)(H ∩ N_{G}(Q, e_{Q})), contradicting our choice of (Q, e_{Q}). □

4 On the local structure of central p-extensions

Let G be a finite group, let b be a block of G, and let (P, e) be a maximal (G, b)-subpair. We assume in this section that P contains a subgroup Z of Z(G). We set Ḡ = G/Z and P̄ = P/Z; for any element or subset a of kG, we denote by ā its canonical image in kḠ. It is well-known that the image b̄ of b in kḠ is a block of Ḡ having P̄ as defect group. The following (equally well-known) Lemma relates the local structures of b and b̄.

Lemma 4.1. For every (G, b)-subpair (Q, f) there is a unique (Ḡ, b̄)-subpair of the form (Q̄, g) such that f̄g = f̄, and then the canonical map G → Ḡ induces a surjective group homomorphism N_{G}(Q, f)/C_{G}(Q) → NḠ(Q̄, g)/CḠ(Q̄) whose kernel is an abelian p-group. In particular, if O_{p}(N_{G}(Q, f)/QC_{G}(Q)) = 1, this map induces an isomorphism N_{G}(Q, f)/QC_{G}(Q) ≅ NḠ(Q̄, g)/Q̄CḠ(Q̄).

Proof. It is well-known (and easy to check) that the group \overline{C_{G}(Q)} is a normal subgroup of CḠ(Q̄) and that CḠ(Q̄)/\overline{C_{G}(Q)} is an abelian p-group. Thus any block of CḠ(Q̄) is contained in k\overline{C_{G}(Q)}. Hence the sum of the different CḠ(Q̄)- conjugates of f̄ is the unique block g of CḠ(Q̄) fulfilling f̄g = f̄, and we have NḠ(Q̄, g) = \overline{N_{G}(Q, f)}CḠ(Q̄). The Lemma follows. □

The above Lemma implies in particular, that the maximal (G, b)-subpair (P, e) determines a unique maximal (Ḡ, b̄)-subpair (P̄, f) by the condition ēf = ē. With this choice of maximal subpairs, 4.1 translates to the following statement.

Proposition 4.2. The canonical map G → Ḡ induces a surjective functor F_{(P,e)}(G, b) → F_{(P̄,f)}(Ḡ, b̄). In particular, b is SL(2, p)-free, if and only if b̄ is SL(2, p)-free.

Proof. Clear by 4.1. □

Proposition 4.3. Let H be a subgroup of G containing N_{G}(P) and denote by c the unique block of H such that Br_{P} (c) = Br_{P} (b). Assume that there is a subgroup Q of P containing Z such that Q is normal in H and such that CḠ(Q̄) ⊆ H̄. Then BrP̄(c̄) = BrP̄(b̄), and we have F_{(P,e)}(G, b) = F_{(P,e)}(H, c) if and only if F_{(P̄,f)}(Ḡ, b̄) = F_{(P̄,f)}(H̄, c̄).

Proof. The equality BrP̄(c̄) = BrP̄(b̄) is clear by [12, Statement 5]. Suppose that F_{Ḡ,b̄} = F_{H̄,c̄}. Let (R, t) be a centric radical (G, b)-subpair. Let s be the unique block of CḠ(R̄) such that t̄s = t̄. By Lemma 4.1, we have N_{G}(R, t)/RC_{G}(R) ≅ NḠ(R̄, s)/R̄CḠ(R̄) = NH̄(R̄, s)/R̄CḠ(R̄)∩H̄. Now Q is normal in H, and thus the canonical image of N_{Q}(R) is normal in N_{G}(R, t)/RC_{G}(R). Therefore we have N_{Q}(R) ⊆ RC_{G}(R). As the subpair (R, t) is centric, we have N_{Q}(R) ⊆ R, which forces Q ⊆ R. Thus CḠ(R̄) ⊆ H̄ by the assumptions, and so (R̄, s) is also an (H̄, c̄)-subpair and (R, t) is an (H, c)-subpair. Therefore N_{H}(R, t)/RC_{G}(R) is a subgroup of N_{G}(R, t)/RC_{G}(R) ≅ NḠ(R̄, s)/R̄CḠ(R̄) = NH̄(R̄, s)/R̄CH̄(R̄). But then Lemma 4.1, applied to H and c instead of G and b, respectively, shows that N_{H}(R, t) = N_{G}(R, t), which implies the equality F_{(P,e)}(G, b) = F_{(P,e)}(H, c). The converse is trivial. □

5 On SL(2, p)-free blocks

Proposition 5.1. Let G be a finite group and let b be a block of G. Suppose that SL(2, p) is involved in N_{G}(Q, f)/C_{G}(Q) for some non-trivial (G, b)-subpair (Q, f). Then SL(2, p) is involved in N_{G}(Q, e)/C_{G}(Q) for some centric and radical (G, b)-subpair (Q, e).

Proof. Fix a maximal (G, b)-subpair (P, e), and for any subgroup Q of P, denote by (Q, e_{Q}) the unique (G, b)-subpair contained in (P, e). Let Q be a subgroup of P with |Q| maximum such that SL(2, p) is involved in N_{G}(Q, e_{Q})/C_{G}(Q). Replacing (Q, e_{Q}) with a G-conjugate if necessary, we may assume that N_{P} (Q) is a defect group of kN_{G}(Q, e_{Q})e_{Q}, so that in particular, R = QC_{P} (Q) is a defect group of kQC_{G}(Q)e_{Q} and (R, e_{R}) is a maximal (QC_{G}(Q), e_{Q})-pair. Since QC_{G}(Q) is normal in N_{G}(Q, e_{Q}), the Frattini argument gives N_{G}(Q, e_{Q}) = C_{G}(Q)[N_{G}(R, e_{R})∩N_{G}(Q, e_{Q})]. But then, N_{G}(Q, e_{Q})/C_{G}(Q) ≅ N_{G}(R, e_{R})∩ N_{G}(Q, e_{Q})/N_{G}(R, e_{R}) ∩ C_{G}(Q). On the other hand, since C_{G}(R) ⊆ C_{G}(Q), N_{G}(R, e_{R}) ∩ N_{G}(Q, e_{Q})/N_{G}(R, e_{R})∩C_{G}(Q) is isomorphic to a subquotient of N_{G}(R, e_{R})/C_{G}(R). Hence SL(2, p) is involved in N_{G}(R, e_{R})/C_{G}(R). The choice of Q now implies that R = Q whence (Q, e_{Q}) is a centric (G, b)-pair.

Let M be the inverse image of O_{p}(N_{G}(Q, e_{Q})/QC_{G}(Q)) in N_{G}(Q, e_{Q}) and let S = M ∩ N_{P} (Q). Then S is a defect group of kMe_{Q}, (S, e_{S}) is a maximal (M, e_{Q})-pair. Since N_{G}(Q, e_{Q}) normalises M, the Frattini argument again gives that N_{G}(Q, e_{Q}) = M[N_{G}(S, e_{S}) ∩ N_{G}(Q, e_{Q})]. But M = (QC_{G}(Q))S whence N_{G}(Q, e_{Q}) = C_{G}(Q)[N_{G}(S, e_{S}) ∩ N_{G}(Q, e_{Q})]. Arguing as before, we conclude that S = Q and hence that M = QC_{G}(Q). This completes the proof. □

The main application of 5.1 is the following proposition which shows that the property of being SL(2, p)-free passes down to corresponding blocks of normalisers of subpairs.

Proposition 5.2. Let G be a finite group and let b be an SL(2, p)-free block of G. For every (G, b)-subpair (Q, f) the block f of N_{G}(Q, f) is SL(2, p)-free.

Proof. Let (R, g) be a centric radical (N_{G}(Q, f), f)-subpair. Then Q ⊆ R by 2.2, and hence C_{G}(R) ⊆ N_{Q}(Q, f). Thus (R, g) is a (G, b)-Brauer pair, and hence SL(2, p) is not a subquotient of N_{G}(R, g)/C_{G}(R) by 5.1. But then SL(2, p) is obviously not a subquotient of N_{N_{G}(Q,f)}(R, g)/C_{N_{G}(Q,f)}(R). □

6 Proof of Theorem 1.4 and Theorem 1.7

Proof of Theorem 1.4. Let G be a finite group, let b be a block of G, let (P, e) be a maximal (G, b)-subpair, and for any subgroup Q of P, denote by e_{Q} the unique block of C_{G}(Q) such that (Q, e_{Q}) ⊆ (P, e). Let W be a Glauberman functor, set N = N_{G}(W (P)) and denote by c the unique block of N such that Br_{P} (c) = Br_{P} (b). Assume that p is odd.

Suppose that Theorem 1.4 fails for the blocks b and c of G and N, respectively, and assume that |G| has minimal order with this property. We are going to derive a contradiction, proceeding in several steps.

6.1. We have O_{p}(G) ≠ 1.

Proof. If O_{p}(G) = 1, then for any nontrivial (G, b)-Brauer pair (Q, f), the group N_{G}(Q, f) is a proper subgroup of G. Since f is SL(2, p)-free by 5.2, the induction hypothesis implies that Theorem 1.4 holds for the block f of N_{G}(Q, f). But then 3.2 implies, that Theorem 1.4 holds for the block b of G, contradicting our choice of b. □

¿From now on, we set Q = O_{p}(G). Since Q is normal in G, the block b lies in kC_{G}(Q) (cf. [1, (2.9)(1)]). Thus b = Tr_{N_{G}(Q,e_{Q})}^{G}(e_{Q}). But then F_{(P,e)}(G, b) = F_{(P,e)}(N_{G}(Q, e_{Q}), e_{Q}). If N_{G}(Q, e_{Q}) is a proper subgroup of G, the induction hypothesis implies that Theorem 1.4 holds for the block e_{Q} of N_{G}(Q, e_{Q}), and hence for the block b, contradicting again our choice of b. This proves the following.

6.2. We have G = N_{G}(Q, e_{Q}) and b = e_{Q}.

Then b is a block for any subgroup of G containing C_{G}(Q). In particular, b is a block of QC_{G}(Q). Set R = QC_{P} (Q). Then (R, e_{R}) is a maximal (QC_{G}(Q), b)-subpair (cf. [1, (2.9)(6)]). Note that C_{G}(R) ⊆ C_{G}(Q) and that R is normal in P. Since the maximal (QC_{G}(Q), b)-subpairs are QC_{G}(Q)-conjugate, a Frattini argument shows that

6.3. we have G = N_{G}(R, e_{R})C_{G}(Q).

If R = Q, then (R, e_{R}) = (Q, b) is (G, b)-centric. The group L occurring in the Külshammer-Puig-extension [13, Theorem 1.8]

1 ⟶ Q ⟶ L ⟶ G/QC_{G}(Q) ⟶ 1

is p-constrained and does not have SL(2, p) as subquotient (cf. 2.4). Thus W (P) is normal in L. Since F_{P} (L) = F_{(P,e)}(G, b), this implies F_{(P,e)}(G, b) = F_{(P,e)}(N, c), contradicting our choice of b.

Thus Q is a proper subgroup of R. Since Q = O_{p}(G), it follows that N_{G}(R, e_{R}) is a proper subgroup of G. So Theorem 1.4 applies to the block e_{R} of N_{G}(R, e_{R}), which has still P as defect group as R is normal in P.

We assume now that P C_{G}(Q) is a proper subgroup of G, and derive a contradiction; that is, we are going to show that then (G, b) cannot be a counterexample to Theorem 1.4. We do this by showing that N_{G}(S, e_{S}) = N_{N}(S, e_{S})C_{G}(S) for any subgroup S of P containing Q such that (S, e_{S}) is (G, b)-centric. We argue by induction over the order of S. Up to replacing (S, e_{S}) by some G-conjugate, we may assume that N_{P} (S) is a defect group of e_{S} as block of N_{G}(S, e_{S}). The subgroup N_{G}(S, e_{S}) ∩ SC_{G}(Q) is normal in N_{G}(S, e_{S}) and contains C_{G}(S). Thus e_{S} is a block of N_{G}(S, e_{S})∩SC_{G}(Q) having as defect group the group T = N_{P} (S) ∩ SC_{G}(Q) = N_{P} (S) ∩ SC_{P} (Q) = N_{SR}(Q), as R = QC_{P} (Q). Therefore, (T, e_{T}) is a maximal (N_{G}(S, e_{S}) ∩ SC_{G}(Q), e_{S})-subpair. The Frattini argument yields

N_{G}(S, e_{S}) = (N_{G}(S, e_{S}) ∩ N_{G}(T, e_{T})) · (N_{G}(S, e_{S}) ∩ SC_{G}(Q)) .

Now (S, e_{S}) is also a (P C_{G}(Q), b)-subpair contained in (P, e). As P C_{G}(Q) is assumed to be a proper subgroup of G, it follows that N_{G}(S, e_{S})∩SC_{G}(Q) = (N_{N}(S, e_{S})∩SC_{G}(Q))C_{G}(S). If S does not contain R, then S is properly contained in SR, hence properly contained in T = N_{SR}(S). By induction, we get N_{G}(T, e_{T}) = N_{N}(T, e_{T})C_{G}(T). Together we get N_{G}(S, e_{S}) ⊆ NC_{G}(S), hence N_{G}(S, e_{S}) = N_{N}(S, e_{S})C_{G}(S).

Thus, we may assume that R ⊆ S. Then C_{G}(S) ⊆ C_{G}(R) ⊆ N_{G}(R, e_{R}) ∩ P C_{G}(Q). Therefore, e_{R} is a block of the group N_{G}(R, e_{R}) ∩ P C_{G}(Q), having still (P, e) as maximal subpair. Let x ∈ N_{G}(S, e_{S}). Since G = N_{G}(R, e_{R})C_{G}(Q), we can write x = nc for some n ∈ N_{G}(R, e_{R}) and some c ∈ C_{G}(Q). Then ^{c}(S, e_{S}) = ^{n^{−1}}(S, e_{S}). This implies that ^{c}S ⊆ N_{G}(R, e_{R}) ∩ P C_{G}(Q) and that (R, e_{R}) ⊆ ^{c}(S, e_{S}). Thus ^{c}(S, e_{S}) is a (N_{G}(R, e_{R}) ∩ P C_{G}(Q), e_{R})-subpair. Therefore, there is y ∈ N_{G}(R, e_{R}) ∩ P C_{G}(Q) such that ^{yc}(S, e_{S}) ⊆ (P, e). We have x = nc = (ny^{−1})(yc). The element yc belongs to the group P C_{G}(Q), and conjugation by yc is a morphism in the category F_{(P,e)}(P C_{G}(Q), b) from S to ^{yc}S. As P C_{G}(Q) is assumed to be a proper subgroup of G, this implies that yc ∈ (N ∩P C_{G}(Q))C_{G}(S). The element ny^{−1} belongs to the group N_{G}(R, e_{R}), and conjugation by ny^{−1} is a morphism from ^{yc}S to ^{x}S = S in the category F_{(P,e)}(N_{G}(R, e_{R}), e_{R}). Since N_{G}(R, e_{R}) is a proper subgroup of G, it follows that ny^{−1} ∈ N_{N}(R, e_{R})C_{G}(^{yc}S). Together, we get x = (ny^{−1})(yc) ∈ NC_{G}(S), hence N_{G}(S, e_{S}) = N_{N}(S, e_{S})C_{G}(S). This contradicts the fact that (G, b) is a counterexample to the Theorem. Therefore,

6.4. we have G = P C_{G}(Q).

Set Z = Z(P) ∩ Q; since Q is normal in G, the group Z is non-trivial. Set Ḡ = G/Z, and denote by b̄ the image of b in kḠ. Thus b̄ is a block of kḠ with defect group P̄ = P/Z. By 4.2, the block b̄ is SL(2, p)-free. Denote by H the inverse image in G of NḠ(W (P̄)). Then H is the normaliser in G of a subgroup of P which contains Q properly, and so H is a proper subgroup of G fulfilling the hypotheses of 4.3. Denote by d the unique block of H such that Br_{P} (d) = Br_{P} (b), and denote by d̄ the image of d in kH̄. By the induction hypothesis, we have F_{(P̄,f)}(Ḡ, b̄) = F_{(P̄,f)}(H̄, d̄), where f is the unique block of CḠ(P̄) such that ēf = f. But then 4.3 implies that we have F_{(P,e)}(G, b) = F_{(P,e)}(H, d). Since H is a proper subgroup of G, by induction again, we have F_{(P,e)}(G, b) = F_{(P,e)}(N, c) by 2.3. This contradicts our choice of b and completes the proof of Theorem 1.4. □

Proof of Theorem 1.7. Since b is SL(2, p)-free, we apply Theorem 1.4 with the Glauberman functor W mapping P to Q = K_{∞}(P). Then the subpair (Q, e_{Q}) is centric, and its normaliser controls fusion in F_{(P,e)}(G, b). The Theorem follows immediately from 2.8 and 2.7. □

7 Proof of Theorem 1.5 and 1.6

Proof of Theorem 1.5. In this section, we keep the notation of Theorem 1.5; that is, we let G be a finite group, and let H be a normal subgroup of G. Let c be a G-stable block of H and let b be a block of G, which covers c; that is, b satisfies bc = b. Let (P, e) be a maximal (G, b)-subpair and set Q = P ∩ H. Then clearly Q is a defect group of the block c of N. Let (Q, e_{Q}) be the unique (G, b)-subpair contained in (P, e) and let f be a block of C_{H}(Q) covered by the block e_{Q} of C_{G}(Q), i.e. such that e_{Q}f ≠ 0 . Then (Q, f) is a maximal (H, c)-subpair. If x ∈ N_{G}(Q, e_{Q}), then ^{x}f is a block of kC_{H}(Q) which is covered by e_{Q}, hence x = yz for some y ∈ C_{G}(Q) and some z ∈ [N_{G}(Q, e_{Q}) ∩ N_{G}(Q, f)]. In other words, we have

N_{G}(Q, e_{Q}) = C_{G}(Q)[N_{G}(Q, e_{Q}) ∩ N_{G}(Q, f)].

The group in square brackets has a block which induces up to the block kN_{G}(Q, e_{Q})e_{Q} and thus contains a defect group of kN_{G}(Q, e_{Q})e_{Q}. For some y ∈ C_{G}(Q), we thus have

yP ≤ N_{G}(Q, e_{Q}) ∩ N_{G}(Q, f),

hence

P ≤ N_{G}(^{y^{−1}}(Q, e_{Q})) ∩ N_{G}(^{y^{−1}}(Q, f)),

Since ^{y^{−1}}(Q) = Q and since e_{Q}^{y^{−1}}f = ^{y^{−1}}(e_{Q}f) ≠ 0, on replacing (Q, f) by ^{y^{−1}}(Q, f), we may assume that P stabilises f, and this proves the first statement of the Theorem.

For any subgroup R of P, we let e_{R} be the unique block of C_{G}(R) such that (R, e_{R}) ≤ (P, e), and for a subgroup S of Q, we let f_{S} be the unique block of C_{H}(S) such that (S, f_{S}) ≤ (Q, f). Note that whenever R is a subgroup of P, the pair (R ∩ H, f_{R∩H}) is stabilised by N_{P} (R ∩ H) because this last group stabilzes R ∩ H and (Q, f).

Let F denote the Brauer category F_{(P,e)}(G, b) and let H denote the Brauer category F_{(Q,f)}(H, c). Let C denote the Alperin-Goldschmidt conjugation family for F.

Let D denote the set of objects R of F such that

(i) N_{P} (R) is a defect group of kN_{G}(R, e_{R})e_{R}.

(ii)N_{P} (R ∩ H) is a defect group of kN_{G}(R ∩ H, e_{R∩H})e_{R∩H}.

(iii) N_{P} (R ∩ H) stabilises f_{R∩H}.

7.1. Every object in F is isomorphic to an object in D and C ∩ D is a conjugation family for F.

Proof. Consider (R, e_{R}) ≤ (P, e). Let (S, u) be a (G, b)-Brauer pair such that S is maximal with respect to normalising (R, e_{R}). Since N_{G}(R, e_{R}) ≤ N_{G}(R ∩ H, e_{R∩H}), we may find a (G, b)- subpair (T, v) such that T is maximal with respect to normalising (R ∩ H, e_{R∩H}) and such that S ≤ T. Note that T is a defect group of kN_{G}(R ∩ H, e_{R∩H})e_{R∩H} and S is a defect group of N_{G}(R, e_{R})e_{R}.

For some element x of G, we have ^{x}(T, v) ≤ (P, e). Thus we have ^{x}(R ∩ H, e_{R∩H}) ≤ x(R, e_{R}) ≤ ^{x}(S, u) ≤ ^{x}(T, v) ≤ (P, e).

Clearly, ^{x}T is a defect group of kN_{G}(^{x}(R ∩ H, e_{R∩H}))^{x}e_{R∩H} , and ^{x}S is a defect group of kN_{G}(^{x}(R, e_{R}))^{x}e_{R}. Also, ^{x}S = N_{P} (^{x}R) and ^{x}T = N_{P} (^{x}(R ∩ H)).

Hence, on replacing (R, e_{R}) by ^{x}(R, e_{R}), we may assume that (R, e_{R}) satisfies (i) and (ii) above. Statement (iii) is immediate from (i) and (ii), since P stabilses (Q, f). This proves the first part of the proposition. Since the set of objects R of F_{(P,e)}(G, b) for which (R, e_{R}) is a centric and radical (G, b)-subpair is invariant under F isomorphism, this proves also the second part of Statement 7.1. □

Let E be the Alperin-Goldschmidt conjugation family for F_{(Q,f)}(H, c).

7.2. If R ∈ C ∩ D, then R ∩ H ∈ E.

Proof. Let R ∈ C ∩D and let ẽ_{R∩H} and f̃_{R∩H} respectively denote the blocks of N_{G}(R ∩H) and N_{H}(R ∩ H) induced from e_{R∩H} and f_{R∩H}. Since N_{P} (R ∩ H) is a defect group of kN_{G}(R ∩ H, e_{R∩H})e_{R∩H}, N_{P} (R ∩ H) is a defect group of kN_{G}(R ∩ H)ẽ_{R∩H}. Since the block ẽ_{R∩H} of kN_{G}(R ∩ H) covers the block f̃_{R∩H} of kN_{H}(R ∩ H), N_{Q}(R ∩ H) is a defect group of kN_{H}(R∩H)f̃_{R∩H}; hence the defect groups of kN_{H}(R∩H, f_{R∩H})f_{R∩H} have order |N_{Q}(R ∩ H)|. On the other hand, since (R, e_{R}) ∈ D, N_{Q}(R ∩ H) ⊆ N_{H}(R ∩ H, f_{R∩H}), thus N_{Q}(R ∩ H) is a defect group of kN_{H}(R ∩ H, f_{R∩H})f_{R∩H}.

Next we show that (R ∩ H, f_{R∩H}) is (H, c)-centric. For this, by the above remarks, it suffices to show that C_{Q}(R∩H) = Z(R∩H). Choose p-regular y ∈ C_{H}(R∩H)∩N_{G}(R, e_{R}).

Then [R, y] ⊆ R ∩ H, so that [R, y, y] = 1, and hence [R, y] = 1 as y is p-regular. Hence [C_{H}(R∩H)∩N_{G}(R, e_{R})]/C_{H}(R) is a p-group. On the other hand, C_{H}(R∩H)∩N_{G}(R, e_{R}) is clearly a normal subgroup of N_{G}(R, e_{R}), and O_{p}(N_{G}(R, e_{R})/RC_{G}(R)) = 1. Hence, C_{H}(R∩H)∩N_{G}(R, e_{R}) ⊆ RC_{G}(R). Since C_{P} (R) = Z(R), we get C_{Q}(R∩H)∩N_{Q}(R) ⊆ R. Since R normalises C_{Q}(R ∩ H), this means that R is its own normaliser in the p-group C_{Q}(R ∩ H)R whence C_{Q}(R ∩ H) ⊆ R.

It remains to show that O_{p}(N_{H}(R∩H, f_{R∩H})/(R∩H)C_{H}(R∩H)) = 1. So, let M be the full inverse image of O_{p}(N_{H}(R∩H, f_{R∩H})/(R∩H)C_{H}(R∩H)) in N_{H}(R∩H, f_{R∩H}). Since N_{P} (R∩H)C_{H}(R∩H)/(R∩H)C_{H}(R∩H) is a Sylow-p subgroup of N_{H}(R∩H, f_{R∩H})/(R∩ H)C_{H}(R ∩ H)), we have M = (M ∩ P)C_{H}(R ∩ H). We will show that M ∩ P ⊂ R ∩ H.

We have N_{G}(R ∩ H, e_{R∩H}) = C_{G}(R ∩ H)[N_{G}(R ∩ H, f_{R∩H}) ∩ N_{G}(R ∩ H, e_{R∩H})], and C_{H}(R ∩ H) is normal in N_{G}(R ∩ H, e_{R∩H}); hence C_{H}(R ∩ H)[M ∩ N_{G}(R ∩ H, e_{R∩H})] is normal in N_{G}(R ∩ H, e_{R∩H}). Since C_{H}(R ∩ H) ⊂ [M ∩ N_{G}(R ∩ H, e_{R∩H})], this means that [M∩N_{G}(R∩H, e_{R∩H})] is normal in N_{G}(R∩H, e_{R∩H}) and hence is normal in N_{G}(R, e_{R}). By the definition of M, it follows that M ∩N_{G}(R ∩H, e_{R∩H})/C_{H}(R ∩H) is a p-group. On the other hand, we have shown before that C_{H}(R∩H)∩N_{G}(R, e_{R})/C_{H}(R) is a p-group. Hence, M ∩ N_{G}(R, e_{R})/C_{H}(R) is a normal p subgroup of N_{G}(R, e_{R})/C_{H}(R), and is therefore isomorphic to a normal p-subgroup of N_{G}(R, e_{R})/C_{G}(R). But then by choice of (R, e_{R}) it follows that M ∩ N_{G}(R, e_{R}) ⊂ RC_{G}(R) whence M ∩ N_{P} (R) ⊂ RC_{P} (R) ∩ H ⊂ R ∩ H. Since R normalises M ∩ P, we see that M ∩ P ⊂ R ∩ H. This completes the proof. □

Now let V be a normal subgroup of Q and suppose that N_{H}(V) controls fusion in H and let W be as in the statement of the Theorem.

7.3. N_{H}(W) controls fusion in H. Further, if S is a subgroup of Q containing W then N_{G}(S, e_{S}) ⊂ N_{G}(W).

Proof. Let (S, f_{S}) ≤ (Q, f) and let x ∈ N_{G}(Q, f). Since ^{x^{−1}}(S, f_{S}) ≤ (Q, f), we have that N_{H}( ^{x^{−1}}(S, f_{S})) ⊂ C_{H}( ^{x^{−1}}S)N_{H}(V) whence N_{H}(S, f_{S}) ⊂ C_{H}(S)N_{H}( ^{x}V).Thus N_{H}( ^{x}V) controls fusion in F_{H,c} for all x ∈ N_{G}(Q, f). It follows by Lemma 2.1 that if S ∈ E, then N_{H}(S, e_{S}) ⊆ N_{H}( ^{x}V) for all x ∈ N_{G}(Q, f), so that in particular, N_{H}(S, e_{S}) ⊆ N_{H}(W). Hence N_{H}(W) controls fusion in F_{H,c}.

Let S be a subgroup of Q containing W and let x ∈ N_{G}(S, e_{S}). By the Frattini argument, we may write x = yz, where y ∈ N_{G}(Q, f) and z ∈ H. Then ^{z}(S, f_{S}) = y^{−1}x(S, f_{S}) ≤ (Q, f). Since N_{H}(W) controls fusion in F_{H,c}, we may write z = ct, where c ∈ C_{H}(S) ⊂ N_{H}(W) and t ∈ N_{H}(W). Since by definition of W, y ∈ N_{G}(W), we have x = yct ∈ N_{G}(W). □

Let R ∈ C ∩ D. Then by 7.2, R ∩ H ∈ E. In particular, by Lemma 2.1, we have that W ⊂ R ∩ H and it follows by 7.3 that N_{G}(R ∩ H, f_{R∩H}) ⊂ N_{G}(W). Hence, N_{G}(R, e_{R}) ⊂ N_{G}(R∩H, e_{R∩H}) ⊂ C_{G}(R∩H)[N_{G}(R∩H, e_{R∩H})∩N_{G}(R∩H, f_{R∩H})] ⊂ N_{G}(W). Theorem 1.5 now follows from 7.2 and the fact that P ⊆ N_{G}(W). □

Proof of 1.6. By a standard argument we may assume that G stabilises the block c. Then 1.6 is an immediate consequence of 2.3 and Theorems 1.4 and 1.5. □

Remark 7.4 The advantage of Theorem 1.6 is that if we wish to produce a single local subgroup controlling fusion in F_{(P,e)}(G, b), it is not really necessary to assume that b is SL(2, p) free. This could be useful in some instances; for example, suppose that G = XwrS_{n} for some large integer n and some non-Abelian finite simple group X, while H is the base-group of the wreath product. It is quite possible for automizers of “diagonaltype” (G, b)-subpairs to involve SL(2, p) because of the action of the S_{n}, while automizers (in H) of (H, c) might not involve SL(2, p).

8 Proof of 1.10

Proof of 1.10. It is clear that the pair (P, e) is a maximal (N_{G}(R), c)-subpair. For a subgroup Q of R, we let (Q, f_{Q}) be the unique (N_{G}(R), c) subpair contained in (P, e).

In [15], it is shown that if we are considering a group G such that DPC holds in every section of G, then in calculating the various quantities k_{d}(B, λ), it is only necessary to consider chains of (G, b)-pairs whose initial objects are pairs (Q, e_{Q}) contained in (P, e) which are (G, b)-centric and radical. By Lemma 2.2, we have that for any such subpair (Q, e_{Q}), R ≤ Q, and thus N_{G}(Q, e_{Q}) ⊂ C_{G}(Q)N_{G}(R) ⊂ N_{G}(R). The fact that R ≤ Q also implies that f_{Q} = e_{Q}. It follows that in the subpair version of (W)DPC, the contribution in kGb from chains beginning with (Q, e_{Q}) is the same as the contribution in kN_{G}(R)c from chains beginning with (Q, e_{Q}). Similarly, it follows that if DPC holds in every proper section of G, then checking DPC for G reduces to checking that there is a defect-preserving bijection between irreducible characters of B lying over λ and irreducible characters in c lying over λ. □

Acknowledgement. This work was completed while the third author was a Visiting Fellow at All Souls College, Oxford. The author is grateful to the College for its support and hospitality.

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Radha Kessar University College High Street Oxford OX14BH

U.K.

Markus Linckelmann CNRS, Université Paris 7 UFR Mathématiques 2, place Jussieu 75251 Paris Cedex 05

FRANCE

Geoffrey R. Robinson School of Mathematics and Statistics University of Birmingham Birmingham B15 2TT

U.K.