Cancellation Theorems related to Conjectures of Alperin and Dade Geoffrey R. Robinson School of Mathematics and Statistics University of Birmingham Edgbaston Birmingham B15 2TT England e-mail : g.r.robinson@bham.ac.uk, Fax UK 121 4143389

INTRODUCTION: In the 1990 paper Astérisque paper [14], the present author and R.Staszewski showed that the Knörr-Robinson formulation ( in [11]) of Alperin’s weight conjecture (henceforth denoted by AWC, see [1]) was compatible with Clifford-theoretic reductions, and proved a number of results which could be viewed as simplifying certain alternating sum calculations in the presence of normal subgroups, among other things. We discuss these results at some length here ( necessary notation, if unfamiliar, can be found in [16] and/or the following section).

In this paper we prove some analogues of these results for Dade’s projective conjecture ( henceforth denoted by DPC). We also prove a general condition in terms of covered nilpotent blocks of normal subgroups which ensures that a block of a finite group satisfies the equalities predicted by DPC. In particular, this last condition is sufficient to give a new, relatively short, proof of DPC for p-solvable groups and enables us to show that the only chains of subpairs which contribute to the alternating sum appearing in the subpair formulation of DPC for a block B are those for which the normalizer of the initial subpair (Q, b_{Q}) is what Alperin terms ( in [1]) a B-parabolic.

DPC and some of its variants, are discussed in Dade [3],[4] and [5]. DPC would imply AWC in the sense that it is a refinement of the Knörr-Robinson formulation of AWC. The key new idea in DPC is that it predicts a precise p-locally determined formula for the number of irreducible characters of any given positive defect in a p-block, so simultaneously generalizes AWC and the Alperin-McKay conjecture.

As well as any intrinsic interest, we view our main result here and the consequences discussed as representing progress on three fronts. Firstly, they represent a significant Clifford-theoretic reduction in studying a minimal counterexample to DPC. Secondly, they can simplify the calculations necessary for DPC, even in a group G such that F^{∗}(G)/Z(G) is simple, in the case that [G : F^{∗}(G)] is divisible by p. Thirdly, we consider that they provide further evidence that relatively projective modules provide a useful theoretical tool for studying these and related questions. For example, the use of relatively projective modules enables us to provide a link between DPC and Broué’s conjecture on blocks with Abelian defect groups ( the link concerns blocks which cover stable blocks with Abelian defect groups, though the covering blocks need not themselves have Abelian defect group).

The three results of [14] which we extend here are as follows: we proved there that for G a purported minimal counterexample to AWC conjecture, and N a minimal normal subgroup of G, we have

∑_{σ∈N (G)/G}(−1)^{|σ|}ℓ(B_{σ}) = ∑_{σ∈N (N)/G}(−1)^{|σ|}ℓ(B_{σ})

for any p-block B of G which does not cover blocks of defect 0 of N. The key observation for the proof of this fact is that if B has defect group D, while Q is a non-trivial p-subgroup of G with Q ∩ N = 1, then N_{G}(Q)/Q has no projective simple module in a Brauer correspondent of B ( using a well-known result of Knörr [10]). For if it did, we would have C_{D}(Q) ≤ Q ( possibly after replacing Q by a G-conjugate ). On the other hand, our assumptions imply that D ∩ N ≠ 1, and hence Z(D) ∩ N ≠ 1, as D is a p-group. Since Z(D) ∩ N ≤ C_{D}(Q) ≤ Q, this would contradict the fact that Q ∩ N = 1.

This first result, together with an inductive argument, allowed the observation that AWC ( for a fixed prime p), is equivalent to the fact that whenever N is a normal subgroup of a finite group G and B is a p-block of G, then we have

ℓ(B) = ∑_{Q/G}( the number of simple F N_{G}(Q)/Q − modules in Brauer

correspondents of B which lie over p-blocks of defect 0 of N_{N}(Q)/Q),

where Q runs over p-subgroups of N, up to G-conjugacy, as indicated. The argument is of a straighforward “inversion” type, similar to those of [11], this time considering chains in N (N) which begin with ( conjugates of ) Q for Q a non-trivial p-subgroup of N.

As in [14], it follows (implicitly using Knörr’s result again) that when B covers a block of N with an Abelian defect group D, then if the formula of the second result holds, there is no proper subgroup Q of D such that N_{N}(Q)/Q has a projective simple module in a block covered by a Brauer correspondent of B for N_{G}(Q). Similar conclusions can be drawn when B covers a nilpotent block of N, or a block of N with a TI-defect group. It is unclear to us how to formulate the correct analogue of the TI-defect result for DPC, but the Abelian and the nilpotent block situations do have analogues.

Extending these results to to be compatible with DPC is not entirely straightforward. The first major obstacle is that as DPC is concerned with groups G for which O_{p}(G) = Z is central, but not necessarily trivial, the argument used for the first result above is not available as it stands. This problem is compounded by the fact that we need to deal with characters afforded by relatively projective modules (over a suitable discrete valuation ring R), rather than simple modules in prime characteristic p.

If a group G has a normal subgroup N ≥ Z, and a p-subgroup Q such that Q ∩ N = Z, then a p-block B of G can have a defect group D such that C_{D}(Q) ≤ Q, B contains a Q-projective irreducible character, yet B does not cover a p-block of N with defect group Z. Perhaps the simplest example of such a situation is when G is extra-special of order p^{3} and exponent p ( for p odd), Q is one maximal subgroup of G and N is a different maximal subgroup of G. While this type of example is not relevant to DPC as it stands, it serves to illustrate the type of problems which can arise. The root of the problem highlighted here is that, in general, if χ is a U-projective irreducible character of a finite group G, where U is a normal p-subgroup of G, then it need not be the case that when N is a normal subgroup of G the irreducible constituents of Res_{N}^{G}(χ) are U ∩ N-projective irreducible characters of N.

However, as so often seems to be the case, the use of p-subgroup complexes somewhat miraculously allows us to “smooth out” problems when calculating alternating sums which could not be eliminated at the level of individual subgroups.

We will explain in detail later, but for the analogue of the first result, the idea is that, ( as long as the block B does not cover blocks of N ⊲ G with defect group Z) chains outside N (N) make zero contribution to the relevant alternating sum. The strategy is that ( as in [14]), the poset map X → X ∩ N should cancel most of the unwanted contribution. As mentioned above, unlike the situation in [14], we need to consider chains which start with p-subgroups U not contained in N, but which satisfy U ∩ N = Z. Under appropriate minimality assumptions, and using characters of relatively projective modules, we can also choose a defect group D for B such that C_{D}(U) ≤ U. Now the contribution to the alternating sum from chains beginning with ( conjugates of ) U can be calculated within N_{G}(U), and it turns out we only need to consider Brauer correspondents of B for N_{G}(U) which cover blocks of O^{p}(N_{N}(U)) with defect group contained in Z.

Once again, the paper of Külshammer-Puig [12] is an indispensable tool for our work. The blocks of N_{G}(U) discussed above cover nilpotent blocks of UN_{N}(U) which have defect groups strictly containing U. The paper [12] provides Morita equivalences which allow us, in conjunction with manipulation of chains of p-subgroups, to conclude that chains starting with conjugates of U make zero contribution to the relevant alternating sum.

¿From the point of view of Clifford-theoretic reductions, Theorem 1 appears useful for the following reason. Using the results of [16], we may choose our normal subgroup N to be the central product of a single conjugacy class of components ( quasi-simple subnormal subgroups) of G. If there are t such components, then G has a normal subgroup K ≥ N such that G/K acts faithfully as a transitive permutation group of degree t on these components. To continue the reductions to reach a point where a minimal counterexample G to DPC has E(G) quasi-simple (and F (G) central), it would be necessary to show that C_{G}(N) = Z(G) for this choice of N, (equivalently, G has a single conjugacy class of components), and then that G/K is the trivial group. Using the results of this paper, as well as some of the Clifford Theory developed by E.C Dade, the first step has been accomplished by C.W. Eaton and the present author. This will appear in a forthcoming paper [6].

The analogues of the other two results follow relatively easily, once the first main result of this paper is established. In the second result, we have to deal with another alternating sum, as we are dealing with ordinary characters rather than simple modules in characteristic p. However, the basic idea is the same: in the second quoted result of the Astérisque paper, it was shown that in the local calculation, for a p-subgroup Q of N, we only needed to consider simple modules of N_{G}(Q) which covered Q-projective simple modules of N_{N}(Q). The second result of this paper shows that when we consider a chain σ ∈ N^{#}(N), in the analogous local computations for Dade’s projective conjecture, we only need to consider irreducible characters which lie over V_{σ}-projective irreducible characters of N_{σ}. The third result exploits the fact that if the block B covers a block of N with Abelian defect group D then, unless D = V_{σ}, there are no V_{σ}-projective irreducible characters of N_{σ} in blocks covered by Brauer correspondents of B. In that case, the alternating sum in question reduces to a single G-conjugacy class of chains, containing just one non-central p-subgroup. Even if we were reduced to the case that F^{∗}(G)/Z(G) is simple, this result could be used to simplify the calculations necessary for checking DPC in the case that E(G) has Abelian Sylow p-subgroups and [G : F^{∗}(G)] is divisible by p.

SECTION 1: NOTATION

Let G be a finite group, p be a rational prime number, and let (𝕂, R, F) be a “sufficiently large” p-modular system. This includes the requirement that R is a complete discrete valuation ring of characteristic 0, with field of fractions 𝕂, such that F = R/J(R) is algebraically closed of characteristic p. We also require that 𝕂 contains a primitive |G|_{p}-th root of unity. As usual, for a non-zero integer n and a prime number q, the highest power of q dividing n is denoted by n_{q}. The hypotheses on F and R imply that 𝕂 contains p′-roots of unity of all orders. Hence we may, and do, suppose that 𝕂 contains all p′-roots of unity in ℂ, the complex root of unity e^{\frac{2πi}{|G|p}} , and the rational field ℚ.

Let X be a finite p′-central extension of a section of G. Then the Krull-Schmidt theorem holds for finitely generated RX modules, as R is complete. We may identify complex characters of X with 𝕂-valued characters of X. Whenever we speak of characters from now on, we will be referring to complex characters. As is now customary, we say that an irreducible character, χ, of X with p^{d}χ(1)_{p} = |X|_{p} has defect d.

We will sometimes refer to blocks of the group algebra RX simply as “p-blocks of X”. Let b be a block of RX. We denote the set of irreducible characters in b by Irr(b). We denote the multiplicative identity of b by 1_{b}.

As is now customary, we employ the notation L < M for finite groups L and M to indicate that L is a proper subgroup of M. We denote by N (X) the set of all normal chains of non-trivial p-subgroups of X of the form

σ = (Q_{1} < … < Q_{n}),

where Q_{i} ⊲ Q_{n} for i = 1, …, n ( we also include the empty chain ϕ- if we wish to omit the empty chain, then we will employ the notation N^{#}(X)). For such a chain σ, we let |σ| = n, the number of non-trivial p-subgroups appearing in σ. We let V^{σ} denote the maximal p-subgroup appearing in σ and let X_{σ} denote the stabilizer of σ under the natural conjugation action of X on N (X). We let V_{σ} denote the first non-trivial subgroup of σ ( if σ is empty, we set V_{σ} = V^{σ} = {1_{G}}). We define the sum of blocks b_{σ} of RX_{σ} as in [11] ( as we are dealing with normal chains, this is just the sum of the usual Brauer correspondents of b for X_{σ}). When Y is a non-trivial p-subgroup of X, we denote by N (X, Y) the set of all chains from N (X) with initial subgroup Y and by N^{#}(X, Y) the set of all chains from N (X) whose initial subgroup strictly contains Y. It may happen in what follows that it is difficult to ascertain whether Y is trivial or not, so we need to define N (X, 1). As a matter of technical convenience, we define this to be the set of all strictly increasing chains of p-subgroups of X whose initial term is trivial, and we define the length of a chain σ in N (X, 1) to be the number of subgroups appearing in σ, including the trivial subgroup. Hence N^{#}(X, 1), sometimes abbreviated to N^{#}(X), is just the set of non-empty chains in N (X). For σ ∈ N (X, 1), we define V_{σ} as before, except that when σ = (1_{G}), we set V_{σ} = 1_{G}. We will somtimes have occasion to work with related p-subgroup complexes, in particular P(X), which is analogous to N (X), except that the p-subgroups in the chains are not necessarily normal in the maximal p-subgroup of the chain. Another useful collection of chains, though not in general a simplicial complex, is the collection of radical p-chains R(X), which were exhibited in [11], but so named later by Dade. These are the chains σ ∈ N (X) with the property that for any initial subchain τ of σ, the largest subgroup V^{τ} is equal to O_{p}(X_{τ}). We will use variants of the notation established for N with P and R as necessary. It follows from Lemma 3.2 and Proposition 3.3 of [11] ( after identifying N (X, U) with N (X/U, 1) in the case that U is a normal p-subgroup of X ) that alternating sums of the type which appear in this paper taken over N (X, U)/X can be freely interchanged with the corresponding sums over P(X, U)/X or over R(X, U)/X. Such interchangeability is made more explicit in the case that U > 1 in Dade [3,4]. In any case, we will freely engage in it throughout this paper without further explanation.

For b′ a sum of blocks of RX and d a non-negative integer, we let k_{d}(b′) denote the number of irreducible characters of defect d in b′. More generally, for γ a linear character of Y, we denote the number of irreducible characters of defect d in b′ which lie over γ by k_{d}(b′, γ). We let Irr(b′) denote the set of irreducible characters in b′, Irr_{d}(b′) denote the set of irreducible characters of defect d in b′ and Irr_{d}(b′, γ) denote the the set of irreducible characters of defect d in b′ which lie over γ.

For a p-subgroup, Q, of X, we let Br_{Q} denote the composition of the natural epimorphism from RX to F X followed by the vector space projection from F X onto F C_{X}(Q) with kernel F [X\C_{X}(Q)]. Then Br_{Q} restricts to a homomorphism of R-algebras from RX^{Q} ( the subalgebra of RX consisting of the Q-fixed points under conjugation action) to F C_{X}(Q).

If B is a block of X, and Y is a section of X such that Brauer correspondence is defined, we use a superscript (B) to denote “in Brauer correspondents of B” for invariants of p-blocks of Y. It is occasionally convenient to extend this notation to p-blocks of a p-power finite central extension, say Ỹ, of Y, since there is a canonical bijection between p-blocks of Y and p-blocks of Ỹ . We use the notation f_{0}(A) to denote the number of projective simple modules associated to the reduction modJ(R) of A, when A is a sum of p-blocks of Y ( or Ỹ ) in that situation.

Let us now consider a group finite group G such that O_{p}(G), the unique largest normal psubgroup of G, is contained in the centre, Z(G), of G. We set Z = O_{p}(G) and we fix a block, B, of RG whose defect groups are not central in G.

DPC asserts that

∑_{σ∈N (G,Z)/G}(−1)^{|σ|}k_{d}(B_{σ}, λ) = 0

for each linear character, λ, of Z and every positive integer d. The notation /G indicates that we are taking a full set of representatives for the G-orbits under the given action. REMARKS: We remind the reader that, as we are dealing with a “blockwise” conjecture, it follows that for a linear character, μ, of a central p′-subgroup, W, of G, either all irreducible characters in B (and, in fact, in every B_{σ}) lie over μ, or none do.

We also remind the reader that ( as remarked in [16]), if the formula

∑_{σ̄∈N (Ḡ,Z̄)/Ḡ}(−1)^{|σ|}k_{d′}(B̄σ̄) = 0

( which we henceforth denote by WDPC (for Ḡ in this case)) holds whenever d′ is a positive integer, Ḡ is a homomorphic image of G of the form G/Z_{1} for a subgroup, Z_{1}, of Z, and B̄ is the unique block of RḠ which is the image of B under the natural epimorphism from RG to RḠ, then we have

∑_{σ∈N (G,Z)/G}(−1)^{|σ|}k_{d}(B_{σ}, λ) = 0

for each linear character, λ, of Z, and each positive defect d.

SECTION 2: STATEMENT OF THE THEOREMS

AUXILIARY THEOREM: Let X be a finite group such that U = O_{p}(X) ≤ Z(X). Let B be a block of RX which covers a nilpotent block b, with non-central defect group, of the normal subgroup Y of X. Then B satisfies all the equalities predicted by Dade’s projective conjecture. REMARK: Notice that this is a perfectly general statement, and makes no assumption of the type that X is a minimal counterexample to DPC. THEOREM 1: Suppose that the formula appearing in Dade’s projective conjecture fails to hold ( for some defect d and some linear character, λ, of the central subgroup Z = O_{p}(G)) for the block B of RG, and that first [G : Z(G)], then |G|, have been minimized subject to such a failure occurring. Then whenever N ≥ Z is a non-central normal subgroup of G, the block B covers blocks of N with defect groups strictly containing Z, and we have

∑_{σ∈N (G,Z)/G}(−1)^{|σ|}k_{d}(B_{σ}, λ) = ∑_{σ∈N (N,Z)/G}(−1)^{|σ|}k_{d}(B_{σ}, λ)

for each linear character, λ, of Z, and each positive defect d. REMARK: Since we know from [16] that G as in Theorem 1 has no non-central normal p′- subgroup, while also O_{p}(G) is central in G, it follows that a normal subgroup which is minimal subject to being non-central in G is a central product of a single conjugacy class of components of G, so this would be a natural choice of the normal subgroup N. It may also be helpful to observe that under the hypotheses of Theorem 1, an elementary argument shows that Z ≤ Φ(G), since if G = MZ for a maximal (normal) subgroup M, then each irreducible character of M extends in [Z : Z ∩ M] ways to an irreducible character of G and this affects the character defect in a transparent fashion. Also, it may be useful to remark that Z(G) is cyclic by the minimality of G. First of all, λ is a faithful linear character of Z, otherwise DPC holds in G/kerλ, from which it quickly follows that G is no counterexample after all, a contradiction. In particular, Z is cyclic. Similarly, the linear character of O_{p′}(Z(G)) covered by B must be faithful, and so O_{p′}(Z(G)) is cyclic.

We remark that although Theorem 1 is stated in terms of a minimal counterexample to DPC, the proof will show that it could equally well be re-stated as a Theorem for a group in which DPC ( for the prime p) holds in every section. Similar remarks apply to the following two results. THEOREM 2: Dade’s projective conjecture is equivalent to the assertion that whenever N is a normal subgroup of a finite group G, B is a block of RG which does not lie over blocks of defect 0 of N and d is a non-negative integer, then

k_{d}(B) = ∑_{σ∈N (N)/G}(−1)^{|σ|+1} (the number of irreducible characters of defect d of B_{σ} lying over

V_{σ} − projective irreducible characters of N_{σ}).

THEOREM 3: Dade’s projective conjecture implies that whenever N is a normal subgroup of a finite group G and B is a block of RG which lies over a G-stable block, b, of N which has an Abelian defect group D, then there is a defect-preserving bijection between irreducible characters of B and irreducible characters of the Harris-Knörr correspondent of B ( that is, the unique Brauer correspondent of B for RN_{G}(D) which lies over the Brauer correspondent of b for N_{N}(D)).

REMARK: Though we have not mentioned it explicitly, it follows from slight modifications of our arguments for Theorem 3 that whenever N is a normal subgroup of a minimal counterexample G ( in the appropriate sense) to DPC and B is a block of RG which lies over a block, b, of N which has an Abelian defect group D, then b is G-stable, and there is a non-negative d such that k_{d}(B) ≠ k_{d}(B′), where B′ is the Harris-Knörr correspondent of B.

SECTION 3 : PROOF OF THE THEOREMS

3.1: EXPLOITING KÜLSHAMMER-PUIG TYPE MORITA EQUIVALENCES

We first discuss some of the results of Külshammer-Puig [12]. Suppose that G is a finite group, and that B is a block of RG which lies over a ( not necessarily G-stable) nilpotent block b of RN, where N is a normal subgroup of G. Suppose further that the defect group, D say of B satisfies D ∩ N = D_{0}. Let I = I_{G}(b). Now B is Morita equivalent to the unique block B^{∗} of RI whose irreducible characters induce to the elements of Irr(B). Let b′ denote the unique Brauer correspondent of b for N_{N}(D_{0}). The block B^{∗} of RI has exactly one Brauer correspondent for N_{I}(D_{0}) lying over b′, say B′, by Harris-Knörr [9].

The results of [12] describe ( in particular) the structure of the blocks of G which lie over b in terms of a twisted group algebra RL̂ constructed from a group extension L of I/N by D_{0}. We have I = NN_{I}(D_{0}, b^{∗}), where (D_{0}, b^{∗}) is a maximal b-subpair. A good deal of effort in [12], which we do not repeat here, goes into showing that the group extension L ( and twisted group algebra L̂) are the same whether computed with respect to I ( and B′) or with respect to N_{I}(D_{0}, b^{∗}) (and its unique block lying over b^{∗}). This has the consequence that 1_{b′}RN_{I}(D_{0}) and 1_{b}RI are Morita equivalent (see 1.20.3 of [12]). In turn, this implies that B and B′ are Morita equivalent, which is the particular fact that will be most useful to us.

We discuss a particular special case more carefully, as it is the case which we will require most often, and the necessary results simplify substantially in that situation. Suppose that we have a finite group X and a block B of RX which covers an X-stable block b of defect 0 of RY, where Y is a normal subgroup of X. Suppose that we have a factorization of the form X = Y N_{X}(Q) for some non-trivial p-group Q such that Q ∩ Y = 1.

Then Y Q⊲X, and b extends to a unique nilpotent block of RY Q, say b̃, which has defect group Q. This block has a unique Brauer correspondent, b′ say, for N_{Y Q}(Q) = C_{Y} (Q) × Q, which covers a unique block of defect zero of RC_{Y} (Q), say b^{∗}. Notice that both b′ and b^{∗} are N_{X}(Q)-stable. By the Harris-Knörr result [9], we know that there is a unique Brauer correspondent of B for RN_{X}(Q) which lies over b′, say B′.

This situation naturally affords two 2-cocycles for N_{X}(Q)/QC_{Y} (Q)(≅ X/Y Q) with values in F^{×} ( which, if desired, may be “lifted” to take values in the set of p′-roots of unity in R). Firstly, there is a unique simple F Y Q-module, say S, in the block b̃, which is necessarily X-stable. The associated action of X on End_{F} (S) gives rise to a projective representation ( in Schur’s sense) of X/Y Q.

Also, there is a unique simple F QC_{Y} (Q)-module, say T, in the block b′, which is necessarily N_{X}(Q)-stable. The action of N_{X}(Q) on End_{F} (T) gives rise to a projective representation ( in Schur’s sense) of N_{X}(Q)/QC_{Y} (Q).

One consequence of the Külshammer-Puig results in this particular situation is that when X/Y Q and N_{X}(Q)/QC_{Y} (Q) are canonically identified, the two 2-cocycles arising above are the same.

This has the consequence that we may replace X by a finite p′-central extension X̂ with a central subgroup Ẑ, and a normal subgroup Ŷ Q̂, canonically isomorphic to Y Q and intersecting Ẑ trivially, so that ( with natural identifications) S extends to a simple F X̂-module, say Ŝ, lying over a fixed 1-dimensional F Ẑ-module, U say, and T extends to a simple F NX̂(Q̂)-module, say T̂, which also lies over U.

Now ( as explained in [12]), it is the case that S has a canonical “lift” to an RY Q-module ( which may be realized over a subring of R in which the prime p is unramified) and this lift extends to an RX̂-module which lifts Ŝ. Let α denote the irreducible character afforded by this lift. Let X̃ = X̂/Ŷ . Then there is a block B̃, which is Morita equivalent to B, and has the property that Irr(B) = {α.β : β ∈ Irr(B̃)}.

Similarly, T̂ has a lift to an RNX̂(Q̂)-module. Let γ denote the irreducible character afforded by this lift. Then, identifying X̂/Ŷ with NX̂(Q̂)/NŶ (Q̂), it follows from [12] that B′ is also Morita equivalent to B̃ as above, and that Irr(B′) = {γ.β : β ∈ Irr(B̃)}.

Suppose now that we replace X by a central extension X^{∗} by a finite Abelian p-group Z^{∗}, and Y^{∗} and Q^{∗} by their respective full pre-images in X^{∗}. Then there is a unique block B^{∗} of RX^{∗} which maps onto B under the canonical epimorphism from RX^{∗} onto RX, and similarly, there is a unique block B^{′∗} of RY^{∗} corresponding to B′. The results of [12] tell us that the blocks B^{∗} and B^{′∗} are still Morita equivalent. The particular consequence we wish to draw from this, and one which we will make repeated use of from now on, is that whenever λ is a linear character of Z^{∗} and d is a non-negative integer, we have k_{d}(B^{∗}, λ) = k_{d}(B^{′∗}, λ). We have dealt with the case that Z^{∗} is trivial, so we proceed by induction on |Z^{∗}|. Fix a choice of d and λ. We may suppose that λ is faithful ( otherwise, we pass to X^{∗}/kerλ). Hence Z^{∗} is now cyclic. We may also assume that k_{d}(B^{∗}, μ) = k_{d}(B^{′∗}, μ) whenever μ is a non-faithful linear character of Z^{∗}. The Morita equivalence between B^{∗} and B^{′∗} tells us that k_{d}(B^{∗}) = k_{d}(B^{′∗}). Furthermore, as Irr(B^{∗}) is invariant under Galois automorphisms fixing p′-roots of unity, we know that k_{d}(B^{∗}, λ) = k_{d}(B^{∗}, λ^{σ}) whenever λ^{σ} is an algebraic conjugate of λ. Since all faithful linear characters of Z^{∗} are algebraically conjugate, we obtain the desired conclusion.

Now let θ be the unique irreducible R-valued character associated to the block b of RY, and let ϕ be the unique irreducible character associated to the block b^{∗} of RC_{Y} (Q). Then the irreducible characters of B^{∗} which lie over the trivial character of Z^{∗} are precisely the irreducible characters of B^{∗} which lie over θ, while the irreducible characters of B^{′∗} which lie over the trivial character of Z^{∗} are the irreducible characters in B^{′∗} which lie over ϕ. For any linear character λ of Z^{∗}, there is a unique irreducible character in the block of RY^{∗} corresponding to b which lies over λ, usually denoted by λ ∗ θ, which agrees with θ on p-regular elements of Y^{∗} and vanishes on elements whose p-parts lie outside Z^{∗}. The irreducible characters of B^{∗} which lie over λ are precisely the irreducible characters of B^{∗} which lie over λ ∗ θ, while the irreducible characters of B^{′∗} which lie over λ are the irreducible characters in B^{′∗} which lie over λ ∗ ϕ.

The reason that we have explained this situation at some length is that we wished to emphasise that ( in the special situation considered above) the existence of defect-preserving bijections ( respecting linear characters of Z^{∗}) between irreducible characters of B^{∗} and B^{′∗} does not really depend on the particular isomorphism type of the central extension of X by Z^{∗} ( though individual numbers might change for a non-isomorphic extension X_{1} of X by Z^{∗}). Similarly, the final statements did not really depend on which particular cocycle was used to construct X̂. In Section 3.4, we will make use of this freedom of choice without further reference.

We also take the opportunity here to remind the reader of the particular result of Harris-Knörr [9] which we will frequently make use of. Let B be a block of RY with defect group D_{0}, where Y is a normal subgroup of the finite group X. Let b be the unique Brauer correspondent of B for N_{Y} (D_{0}). Then each block of RX which covers the block B of RY has a unique Brauer correspondent for RN_{X}(D_{0}) ( which necessarily covers b), and this Brauer correspondent has a common defect group with the original block.

3.2: THE AUXILIARY THEOREM

Let us consider: HYPOTHESIS 3.2.1: X is a finite group, B is a block of RX. There is a normal subgroup, Y, of X which contains U = O_{p}(X). The block B has a defect group D such that D ∩ Y strictly contains U and B covers a nilpotent block ( say b), of RY. Without loss of generality, we assume that b has defect group D ∩ Y. THEOREM 3.2.2 : Under hypothesis 3.2.1, we have

∑_{σ∈N (X,U)/X}(−1)^{|σ|}k_{d}(B_{σ}) = 0

for each non-negative integer d. PROOF: Before we commence the proof, it is useful to recall some basic well-known facts. Almost from the very definition,it is clear that if B_{1} is a nilpotent block of RH for a finite group H and b_{1}^{H} = B_{1} for some block b_{1} of N_{H}(Q), some p-subgroup Q of H, then b_{1} is also nilpotent. Also, if M is a normal subgroup of a finite group L and T = O_{p}(L), then every nilpotent block of RM is covered by a unique nilpotent block of RMT. We will frequently make use of these facts in the proof below without further reference.

Suppose then that the theorem is false, and that X is a counterexample of minimal order. Choose a non-negative integer d for which the alternating sum above is non-zero. Let V be a psubgroup of X, contained in D, strictly containing U, and such that D∩Y > V ∩Y. We consider the contribution to the alternating sum from ( conjugates of ) chains in N (X, U) whose second subgroup is V. Up to a change of sign, this is the same as the corresponding sum in N (N_{X}(V), V), with the sum of blocks B_{(V)} in place of B. By Proposition 3.3 of [11], ( after identifying N (N_{X}(V), V) with N (N_{X}(V)/V, 1)), we can work with P(N_{X}(V), V) instead, without affecting the alternating sum. Then, as usual, if V is not radical, the alternating sum vanishes, using Quillen’s contraction W → W O_{p}(N_{X}(V)) → O_{p}(N_{X}(V)). So we may suppose that V is radical, and do so.

We set M = N_{Y} (V)V. Let B′ be a Brauer correspondent of B for N_{X}(V) ( equivalently, a block summand of B_{(V)}). Let D_{1} be a defect group for B′. Since V > U = O_{p}(X), we know that N_{X}(V) is a proper subgroup of X. As B′ was an arbitrary block summand of B_{(V)}, to deduce that the contribution in question vanishes, it suffices to prove that D_{1} ∩ M > V. For in that case, the hypotheses for (X, Y, U, B) are inherited by (N_{X}(V), M, V, B′), and the minimality of X allows the desired conclusion.

If D_{1} ∩M = V, then RM has a block with defect group V lying under B′. But M = V N_{Y} (V) = N_{V Y} (V), so that RV Y has a block with defect group V lying under B, by Brauer’s First Main Theorem. Hence RY has a block with defect group V ∩ Y lying under B. However V ∩ Y is strictly contained in D ∩ Y under the present assumptions, while all blocks of RY lying under B have defect group X-conjugate to D ∩ Y, a contradiction. Consequently, we have the desired conclusion that there is zero contribution to the alternating sum from chains in N (X, U) whose second term is conjugate to V ( in the case that V ∩ Y < D ∩ Y).

On the other hand, if V ∩ Y = D ∩ Y, but V ≰ Y, then the contribution to the alternating sum from ( conjugates of) chains with second term V may be paired with the contribution to the alternating sum from chains whose second term is V ∩ Y and third term is V by inserting V ∩ Y before V in the former chains ( recall also that D ∩ Y > U). The only terms not yet considered in the alternating sum are contributions from the singleton chain U and the chain U < D ∩ Y. Hence the alternating sum is reduced to k_{d}(B) − k_{d}(b′), where b′ is the sum of the Brauer correspondents of B for N_{X}(D ∩ Y). By Harris-Knörr [9], there is a single such Brauer correspondent, and by the results of Külshammer-Puig [12], as explained in the previous section, this is Morita equivalent to B, as B covers the nilpotent block b of RY. Hence k_{d}(B) = k_{d}(b′), a contradiction, as we assumed that the alternating sum did not vanish. PROOF OF THE AUXILIARY THEOREM : Suppose that X, B, and U are as in the statement of Theorem 3.2.2, but in addition that U ≤ Z(X). Let Y be a normal subgroup of X which is maximal subject to having B cover a nilpotent block of Y with a non-central defect group. Then U ≤ Y, otherwise UY would contradict the maximality of Y. Hence Theorem 3.2.2 allows us to conclude that WDPC holds for B. Similarly, WDPC holds for X/W ( and its unique block corresponding to B) for each subgroup W of U. As we have already seen, this suffices to prove that DPC holds for B. REMARK: As mentioned earlier, this auxiliary Theorem is already sufficiently strong to provide an alternative proof of DPC for p-solvable groups. For if X is p-solvable with U = O_{p}(X) ≤ Z(X), and we are given a block B of RX whose defect groups strictly contain U, then we may take T to be a normal subgroup of X which is maximal subject to B covering blocks of RT with defect group U. Then T ≠ X, and it is clear that O_{p′}(X/T) = 1 by the maximality of T. Let Y be the full pre-image of O_{p}(X/T) in X. Then Y > T as X > T and X is p-solvable. Consequently, B does not cover blocks of Y with defect group U. Let b be any block of RT lying under B. Since T ⊲ Y with [Y : T] a power of p, and since b covers blocks of RT with defect group U, we see that b is a nilpotent block of RY. Since b has a non-central defect group, the auxiliary theorem tells us that X satisfies DPC.

Another useful consequence of Theorem 3.2.2 is worth singling out. The Theorem tells us that if B covers a nilpotent block of RY which has defect groups strictly containing U, then the alternating sum which appears in the statement of the Theorem vanishes. Suppose now that X occurs as N_{H}(U) for some larger group H, and B is a Brauer correspondent of the block B̃ of RH. then the Theorem tells us that the contribution from B ( and chains in N (H, U)) to the alternating sum which appears in WDPC for B̃ is 0. Let b be a block of UC_{X}(U) covered by B, chosen so that D is still a defect group for the block B^{∗} of N_{X}(U, b) covering b. If WDPC ( for p) holds in X and all its proper sections, then the equivalence of Dade’s weight conjecture with DPC, and the explanation in Section 4 of [15] of the interpretation of Dade’s weights in terms of characters of relatively projective modules, tell us that there is a radical chain σ ∈ R(X, U) such that X_{σ} has a U-projective irreducible ordinary irreducible character in Irr(B_{σ}). This is afforded by an RX_{σ} lattice with vertex contained in U. By a careful choice of defect groups, we may suppose that this lattice is contained in a block summand of B_{σ} which has defect group D_{σ} = D ∩ G_{σ}. Invoking Proposition 3.2 of [16], which is a special case of a Theorem of R. Knörr, we see that C_{D_{σ}}(U) ≤ U. But this is far from the desired conclusion that C_{D}(U) ≤ U. However V^{σ} ≤ D_{σ} as σ is radical, so that C_{V^{σ}}(U) ≤ U. Let Q = C_{D}(U) ∩ N_{X}(V^{σ}). Now [Q, V^{σ}] ≤ V^{σ} ∩ C_{X}(U) ≤ U. Hence Q ≤ (X_{σ} ∩ D) = D_{σ}. Therefore, Q ≤ C_{D_{σ}}(U) ≤ U. We conclude that V^{σ} is self-normalizing in V^{σ}C_{D}(U) ( and this is indeed a group), so that C_{D}(U) ≤ V^{σ}. Consequently C_{D}(U) ≤ U. Hence b has defect group Z(U). In particular, b is a nilpotent block of C_{X}(U), and B covers nilpotent blocks of UC_{X}(U).

Suppose that O_{p}(N_{X}(U, b)/UC_{X}(U)) > 1. Taking Y to be the full pre-image in N_{X}(U, b) of O_{p}(N_{X}(U, b)/UC_{X}(U)), we see that B^{∗} covers a nilpotent block of RY whose defect group strictly contains U ( for if P ∈ Syl_{p}(N_{X}(U, b)), then b extends to a nilpotent block of P C_{X}(U) with defect group Q satisfying P C_{X}(U) = QC_{X}(U) and Q ∩ UC_{X}(U) = U This block covers a unique block of RY which lies over b, is nilpotent, and has defect group Q ∩ Y, which strictly contains U.

Applying Theorem 3.2.2 with N_{X}(U, b) in place of X and B^{∗} in place of B, ( then returning to the Morita equivalent block B, as explained in [16]), we conclude that

∑_{σ∈N (X,U)/X}(−1)^{|σ|}k_{d}(B_{σ}) = 0

for each non-negative integer d. In terms of the subpair formulation of DPC ( or WDPC) as explained in [16], and to be discussed more fully in the next section, this means that in calculating the alternating sum, we need only consider chains of subpairs such that the normalizer of the initial subpair (Q, b_{Q}) is what Alperin terms a B-parabolic ( see [1]). That is, if DPC is valid ( for p) in every proper section of the finite group H, and B is a block of RH with non-central defect group B, then in checking whether or not the appropriate alternating sum is 0 ( for any given d, λ,) we only need to consider chains of subpairs with an initial subpair (Q, b_{Q}) satisfying C_{D}(Q) = Z(Q) and O_{p}(N_{H}(Q, b_{Q})/QC_{H}(Q)) = 1.

3.3: REDUCING TO QUASI-PRIMITIVE BLOCKS

For parts of the proof of our main Theorem, it is convenient to deal with the conjecture via B-subpairs. As in [16], instead of working with N (G, Z), we work with N_{B}(G, Z), whose chains have the form

(σ, b′) = ((Q_{0}, b_{0}) < … < (Q_{n}, b_{n})),

where the p-subgroups appearing form a chain, σ, in N (G, Z), and, for each Q_{i}, the subpair (Q_{i}, b_{i}) is the unique B-subpair contained in the B-subpair (V^{σ}, b′)(= (Q_{n}, b_{n})). There is a natural conjugation action of G on N_{B}(G, Z). Notice that the notation we adopt is justified by the fact that once the chain σ ∈ N (G, Z), and the block, b′ of RC_{G}(V^{σ}) are specified, the chain of subpairs is uniquely determined. Furthermore, this uniqueness shows that the stabilizer G_{(σ,b′)} is precisely I_{G_{σ}}(b′), the inertial subgroup of b′ in G_{σ}. We observe that 1_{b′} is also a centrally primitive idempotent of RG_{(σ,b′)}, and we let β_{(σ,b′)} denote the block 1_{b′}.RG_{(σ,b′)}.

For any σ ∈ N (G, Z), the blocks which occur as summands of B_{σ} are in bijection with the orbits of G_{σ} on B-subpairs of the form (V^{σ}, b′), as V^{σ}C_{G}(V^{σ})⊲G_{σ}. Furthermore, given a choice of b′, the block of RG_{σ} induced (characterwise) from β_{(σ,b′)} is the block summand of B_{σ} lying over b′. We note that, for any positive integer d, this induced block contributes (−1)^{|σ|}k_{d}(β_{(σ,b′)}) towards (−1)^{|σ|}k_{d}(B_{σ}), as its irreducible characters are in defect-preserving bijection with the irreducible characters of β_{(σ,b′)}.

For any positive defect d′ we may calculate

∑_{σ∈N (G,Z)/G}(−1)^{|σ|}k_{d′}(B_{σ})

by counting ( for all σ ∈ N (G, Z)/G ) the contribution from G_{σ}-orbits of B-subpairs of the form (V^{σ}, b′). By the discussion above, this is the same as taking the appropriate alternating sum over all (σ, b′) ∈ N_{B}(G, Z)/G.

Hence WDPC may be equivalently stated as the requirement that

∑_{(σ,b′)∈N_{B}(G,Z)/G}(−1)^{|σ|}k_{d′}(β_{(σ,b′)}) = 0

for each positive defect d′.

Now suppose that B covers a block b of N ⊲ G, where N is as in Theorem 1. We assume, as we may, that b has defect group D_{0} = D ∩ N, where D is a defect group for B. We claim that b is G-stable under the minimality assumption of Theorem 1. Suppose otherwise.

The inertial subgroup I = I_{G}(b) satisfies [I : Z(I)] < [G : Z(G)], as I contains Z(G) and b is not G-stable. Let B′ be the block of RI which lies over b and whose irreducible characters induce to those of B. There is a bijection between N_{B}(G, Z)/G and N_{B′}(I, Z)/I which we now describe. The block idempotent for b remains primitive as a central idempotent of RDN. Hence its image under Br_{Q} is well-defined for each subgroup, Q, of D. We choose ( as we may) a maximal B-subpair (D, b_{D}) such that a root b^{∗} of the Brauer correspondent of b for DN_{N}(D) lies under b_{D} ( implicit here is the assumption that D ≤ I. Then D is also a defect group for B′). Notice that b^{∗} is a block of RC_{N}(D), uniquely specified up to N_{N}(D)-conjugacy. For each subgroup, Q, of D, we let (Q, b_{Q}) denote the unique B-subpair contained in (D, b_{D}). Such subpairs include a full set of representatives for the G-conjugacy classes of B-subpairs.

Regarding b as a block of ND, we may regard (D, b^{∗}) as a b-subpair. For each subgroup, Q of D, we have a unique b-subpair (Q, b_{Q}^{∗}) ≤ (D, b^{∗}). Then b_{Q}^{∗} is a block of C_{N}(Q) which lies under the block b_{Q} of RC_{G}(Q).

Now (b_{Q}^{∗})^{NQ} = b for each such Q. If x ∈ C_{G}(Q) stabilizes b_{Q}^{∗}, then we have

b^{x} = [(b_{Q}^{∗})^{NQ}]^{x} = [(b_{Q}^{∗})^{x}]^{x^{−1}NQx} = b,

so that x ∈ C_{I}(Q). The block b_{Q} is induced ( characterwise) from a block of the inertial subgroup in C_{G}(Q) of b_{Q}^{∗}. Since this last inertial subgroup is contained in C_{I}(Q), there is an intermediate block b′_{Q} of C_{I}(Q) which is Morita equivalent to b_{Q}, and whose irreducible characters induce to those of b_{Q}.

Hence we have defined a a B′-subpair (Q, b′_{Q}) corresponding to the B-subpair (Q, b_{Q}), and we have a canonical bijection between B-subpairs contained in (D, b_{D}) and B′-subpairs contained in (D, b′_{D}). This bijection extends to a canonical bijection between chains (σ, b) with (V^{σ}, b) ≤ (D, b_{D}) and chains (σ′, b′) with (V^{σ′}, b′) ≤ (D, b′_{D}).

We note that we have N_{G}(Q, b_{Q}) ∩ I = N_{I}(Q, b′_{Q}), and that T r_{C_{I}(Q)}^{C_{G}(Q)}(1_{b′_{Q}}) = 1_{b_{Q}}. From this, we see that N_{G}(Q, b_{Q}) = C_{G}(Q)N_{I}(Q, b′_{Q}).

It follows from the Alperin-Broué subpair version of Alperin’s fusion theorem [2] that given a subpair (Q, b_{Q}) ≤ (D, b_{D}) and g ∈ G such that (Q, b_{Q})^{g} ≤ (D, b_{D}), then we may write g = cx for some c ∈ C_{G}(Q) and some x ∈ I. We deduce that the canonical correspondence between chains (σ, b) with (V^{σ}, b) ≤ (D, b_{D}) and chains (σ′, b′) with (V^{σ′}, b′) ≤ (D, b′_{D}) induces a bijection between N_{B}(G, Z)/G and N_{B′}(I, Z)/I. Hence, using the Morita equivalences discussed earlier, we see that

∑_{(σ,b′)∈N_{B}(G,Z)/G}(−1)^{|σ|}k_{d′}(β_{(σ,b′)}) =

∑_{(σ′,b′)∈N_{B′} (I,Z)/I}(−1)^{|σ|}k_{d′}(β_{(σ′,b′)})

for each positive defect d′.

If O_{p}(I) > Z, then the right hand expression may be re-written as

∑_{σ′∈N (I,Z)/I}(−1)^{|σ|}k_{d′}(B′_{σ′}),

and then as

∑_{σ′∈P(I,Z)/I}(−1)^{|σ|}k_{d′}(B′_{σ′}).

This last alternating sum vanishes by the usual argument using Quillen’s contraction T → T O_{p}(I)/Z → O_{p}(I)/Z on P(I/Z, 1) in the present case O_{p}(I) > Z.

If O_{p}(I) = Z, then the right hand expression is the quantity occurring in the subpair formulation of WDPC ( for I and B′), so is 0, by the minimality of [G : Z(G)], as D > Z. This contradiction shows that the block B is quasi-primitive, that is to say, B covers a G-stable block of M whenever M ⊲ G.

3.4: REMOVING COVERED BLOCKS WITH CENTRAL DEFECT GROUPS

We are still concerned with our minimal counterexample G to WDPC. Suppose now that our chosen block B covers a ( now known to be G-stable) block b of a non-central normal subgroup N ≥ Z such that b has defect group Z. We assume that N has been chosen maximal with respect to this property. Then, arguing as in the remarks at the end of Section 3.2, we may conclude that O_{p′}(G/N) = 1 and O_{p}(G/N) = 1. Certainly, then, Z(G) ≤ N.

We will show that the Külshammer-Puig results allow us to replace G by a central extension G̃ of G/N and exhibit a block B̃ of RG̃ for which WDPC still fails. We certainly have [G̃ : Z(G̃)] ≤ [G : N] < [G : Z(G)], so this will contradict the minimal choice of G.

First, some general remarks which we have already implicitly made use of is some cases. If X is a finite group having a central p-subgroup W, and B is a block of RX having defect group D, B̄ is the unique corresponding block of RX̄ ( obtained via the natural projection), then there is a canonical fusion respecting bijection between B-subpairs whose p-subgroup contains W and B̄-subpairs. It is not generally the case that if Q is a p-subgroup of X, containing W, then we have \overline{C_{X}(Q)} = CX̄(Q̄), but we do always have \overline{O^{p}(C_{X}(Q))} = O^{p}(CX̄(Q̄)), which is sufficient to set up the bijection of subpairs, as block idempotents of C_{X}(Q) have support in O^{p}(C_{X}(Q)).

Returning to G, let ϕ be the unique irreducible character in b which has Z in its kernel. This irreducible character of N is G-stable, as b is, and provides us with a 2-cocycle of G/N which may be ”normalized” to take values in the set of p′-roots of unity of R ( strictly speaking, it is necessary to realize ϕ over a subring of R in which the prime p is unramified, but this can always be achieved, a fact which was originally observed in essence by P. Fong). Notice that Z has unimodular action on the source of any RN-module affording ϕ, as this source is trivial.

We may inflate the 2-cocycle of G/N associated to ϕ to a 2-cocycle of G, which yields a p′- central extension Ĝ of G having a normal subgroup N̂ naturally isomorphic to N and central p′-subgroup Ŵ such that Ĝ/Ŵ is isomorphic to G and such that Ĝ/Ŵ N̂ is isomorphic to G/N.

Let ϕ̂ be the extension to Ĝ of ϕ ( viewed as a character of N̂) associated to the above 2-cocycle.

The block B may now be identified with a block B̂ of RĜ lying over a certain linear character of O_{p′}(Z(Ĝ)). It still covers a block of N̂ with central defect group. We wish to construct a block B̃ of RG̃ for which WDPC fails, where G̃ = Ĝ/N̂. This will still violate our minimal choice of G.

Hence, in what follows, it is no real loss of generality to consider the case that ϕ extends to an irreducible character ϕ̂ of G and that L is a central extension of G/N by Z ( otherwise, we could work with Ĝ rather than G). In that case, we may assume that the Külshammer-Puig cocycle will be trivial and take L̂ = L. Set L̄ = G/N. Let B̃ be the unique block of L corresponding to B. Let B̄ be the block of L̄ corresponding to B̃ under the natural epimorphism. Then the irreducible characters of in B which have Z in their kernels are precisely those of the form ϕ̂.θ, where θ runs through the ( inflations to G of ) the irreducible characters of B̃.

We claim that there is a canonical bijection between N_{B}(G, Z)/G and NB̃(L, Z)/L, where Z is identified with O_{p}(L). By the earlier general remarks, for this claim it suffices to show that there is a canonical bijection between N_{B}(G, Z)/G and NB̄(L̄, 1)/L̄.

It suffices, then ( for ease of notation) to explain this bijection of subpairs in the case that Z = 1, so we assume for the moment that Z = 1 ( so that L = L̄).

Notice that if (Q, b_{Q}) is a B-subpair, then Q ∩ N = 1 and Q is a defect group for the unique block of RNQ covering b. It follows that N_{L}(Q̄) = \overline{N_{G}(Q)}, using the uniqueness up to conjugacy of defect groups. We may argue similarly that C_{L}(Q̄) = \overline{C_{G}(Q)}. Now there is a unique Harris-Knörr correspondent of b_{Q} ( viewed as a block of RQC_{G}(Q)) for NQC_{G}(Q) which covers b and has defect group containing Q, say b_{Q}^{∗}. Then we may use the fact that ϕ extends to an irreducible character of NQC_{G}(Q) to obtain a block b̄Q̄ of Q̄C_{L}(Q̄)(= NQC_{G}(Q)/N) and this gives a subpair (Q̄, b̄Q̄) for L. Letting B̄ denote the block so obtained using the subpair (1, B), it follows as in [16], pages 241-243, that (Q̄, b̄Q̄) is a B̄-subpair for each such Q.We have explained in less than complete detail, as there is no essential difference between the situation here and that of [16], page 241-.

The fact that ϕ here lies in a block of defect 0 of N while μ considered on pages 241- of [16] is of p′-degree makes no substantial difference to the necessary arguments.

Since

∑_{n∈N} |ϕ̂(xn)|^{2}/ϕ̂(1) = |N|/ϕ̂(1) ∉ J(R),

we conclude as in [16], page 242, that each coset of N in G contains a coset representative x such that

[H : C_{H}(x)]ϕ̂(x)/ϕ̂(1) ∉ J(R),

where H is the full pre-image in G of C_{L}(x̄). This is the condition that is really required for the subsequent arguments to work in [16].

Now if (P, b_{P}) and (Q, b_{Q}) are B-subpairs whose canonical images are conjugate via an element of L, then P^{g}N = QN for some g ∈ G, and P^{g}, Q are both defect groups for the unique block of RQN covering b, so they are conjugate via an element of N. Hence the bijection of subpairs constructed respects fusion, and extends to a fusion-respecting canonical bijection between chains of B-subpairs and chains of B̄-subpairs. Furthermore, the stabilizer of a given chain of B̄-subpairs is the image in G/N of the stabilizer of the corresponding chain of B-subpairs.

Let us now return to the case that the possibility that Z is non-trivial is allowed, and that L is a central extension of G/N by Z. The construction above has given us a canonical fusionrespecting bijection between chains of B^{∗}-subpairs and chains of B̄-subpairs. Using the earlier general remarks, this yields a canonical bijection between N_{B}(G, Z)/G and NB̃(L, Z)/L.

Now let (σ, b′) be a chain in N_{B}(G, Z) whose maximal B-subpair (V^{σ}, b_{V^{σ}}) is contained in the maximal B-subpair (D, b^{∗}). Consider N_{NG_{σ}}(V^{σ}) = G_{σ}N_{N}(V^{σ}). Now [V^{σ}, N_{N}(V^{σ})] ≤ N ∩ V^{σ} ≤ N ∩ D = Z. Hence N_{N}(V^{σ}) ≤ G_{σ}, so that we have N_{NG_{σ}}(V^{σ}) = G_{σ}. Now b_{V^{σ}} ( viewed as a block of V^{σ}C_{G}(V^{σ})) is the Brauer correspondent of a unique block of NV^{σ}C_{G}(V^{σ}) with defect group containing V^{σ} lying over the block b of N. We denote this block by b^{σ}.

Let I_{σ} denote the inertial subgroup in NG_{σ} of b^{σ} ( this subgroup certainly contains N). Then I_{σ} ∩ N_{G}(V^{σ}) stabilizes the unique Brauer correspondent of b^{σ} for V^{σ}C_{G}(V^{σ}), namely b_{V^{σ}}. Hence N_{N}(V^{σ}) stabilizes b_{V^{σ}}, so we may deduce from above that N_{NG_{(σ,b′)}}(V^{σ}) = G_{(σ,b′)}.

Now there is a unique block, β′_{(σ,b′)} say, of RG_{(σ,b′)} which covers the block b_{V^{σ}}. The Brauer correspondent of β′_{(σ,b′)} for NG_{(σ,b′)} is the unique block of RNG_{(σ,b′)} lying over b^{σ} and having a defect group containing V^{σ}. We denote this block by β_{(σ,b′)}. Now β_{(σ,b′)} covers a nilpotent block, bσ̃ say, of RNV^{σ} ( the unique extension of b to a block of RNV^{σ}), and bσ̃ is stable under RNG_{σ}. Hence we may conclude from the Külshammer-Puig results, as explained in section 3.1, that the blocks β_{(σ,b′)} and β′_{(σ,b′)} are Morita equivalent. The block β_{(σ,b′)} is Morita equivalent to the corresponding block β̃_{(σ̃,b̃′)}of RL_{(σ̃,b̃′)}, where (σ̃, b̃′) is the chain of B̃-subpairs of L corresponding to (σ, b′). This is because L_{(σ̃,b̃′)} can play the role of the extension of NG_{(σ,b′)}/N by Z which would occur if the Külshammer-Puig results were applied with NG_{(σ,b′)} in place of G and β_{(σ,b′)} in place of B.

Hence we obtain for each non-negative integer d,

∑_{(σ,b′)∈N_{B}(G,Z)/G}(−1)^{|σ|}k_{d}(β′_{(σ,b′)}) = ∑_{(σ̃,b̃)∈NB̃(L,Z)/L}(−1)^{|σ̃|}k_{d}(β̃_{(σ̃,b̃′)}).

In particular, we note that, as G is a counterexample to DPC with [G : Z(G)] minimal, then N is central in G, as otherwise, the subpair version of the formula of WDPC (and hence of DPC) would hold for B̃ and L.

Let us turn now to the proof proper of Theorem 1. We have a finite group G such that DPC holds in all finite central extensions of all proper sections of G/Z(G). We suppose further that Z = O_{p}(G) is central in G, and that the p-block B of G covers a G-stable p-block b of a non-central normal subgroup, N, of G.

Let λ be a linear character of Z, and let d be a positive integer. To prove Theorem 1, we need to prove that chains from N (G, Z)\N (N, Z) make no overall contribution to

∑_{σ∈N (G,Z)/G}(−1)^{|σ|}k_{d}(B_{σ}, λ).

Just as in the formulation of WDPC, it suffices to show that they make no contribution to

∑_{σ∈N (G,Z)/G}(−1)^{|σ|}k_{d}(B_{σ}),

so we can omit λ, and do so. The argument at the beginning of this section allows us to conclude that the defect group of b strictly contains Z.

Taking due account of sign, we can identify chains in N (G, Z) other than (Z) with chains in N^{#}(G, Z) by deleting their first subgroup. This respects the action of G and, in particular, preserves the chain stabilizer As in [14], we would like to use the poset map X → X ∩ N to cancel the contributions from the chains mentioned above. In the elementary language of Knörr-Robinson [11], this means that, given a chain σ = (Q_{1} < Q_{2} < … < Q_{n}) ∈ N^{#}(G, Z) with Q_{n} ≰ N, we define a chain σ′ as follows: choose j minimal such that Q_{j} ≰ N. If Q_{j−1} ≠ Q_{j} ∩ N, then set

σ′ = (Q_{1} < Q_{2} < … < Q_{j−1} < Q_{j} ∩ N < Q_{j} < … < Q_{n}).

If Q_{j−1} = Q_{j} ∩ N, then set

σ′ = (Q_{1} < Q_{2} < … < Q_{j−2} < Q_{j} < … < Q_{n}).

As in [11], this procedure cancels the contribution from these chains as desired, except that, as well as the chains remaining from N^{#}(G, Z), the procedure above does not disturb chains above for which the minimal value j = 1 and Q_{1} ∩ N = Z. In other words, we need to concern ourselves with the contribution from chains σ whose initial subgroup V_{σ} satisfies V_{σ} ∩ N = Z.

Let U > Z be a p-subgroup of G such that U ∩ N = Z. We wish to calculate the contribution to the alternating sum from chains beginning with ( G-conjugates of ) U. This contribution is

∑_{σ∈N (G,U)/N_{G}(U)}(−1)^{|σ|}k_{d}(B_{σ}).

The alternating sum in question is 0 unless U is radical ( strictly, it is necessary to replace N (N_{G}(U), U) by P(N_{G}(U), U) and then use the Quillen contraction T → T O_{p}(N_{G}(U)) → O_{p}(N_{G}(U) in the case that U is not radical. All this is permissible using Proposition 3.3 of [11], identifying P(N_{G}(U), U) with P(N_{G}(U)/U, 1)).

Furthermore, we claim that it follows by the minimality of G that for each non-negative integer d, we have

∑_{σ∈N (G,U)/N_{G}(U)}(−1)^{|σ|}k_{d}(B_{σ}) = ∑_{σ∈N (G,U)/N_{G}(U)} ∑_{μ∈Irr_{d}(U)/G_{σ}}(−1)^{|σ|}f_{0}^{(B)}(I_{G_{σ}}(μ)/U).

In particular, for a given σ as above, we only need to count characters in B_{σ} which are afforded by U-projective irreducible characters of G_{σ}. Properly interpreted, this follows from the equivalence of DPC with Dade’s weight conjecture ( 17.10 of [4]), once it is realized ( as explained in [15], which in turn makes use of Proposition 3.2 of Külshammer-Robinson [13] and the preparatory results before it) that the number of ( Dade’s) weights lying over an irreducible character μ of defect d of V_{σ} in Brauer correspondents of B for G_{σ} is equal to f_{0}^{(B)}(I_{G_{σ}}(μ)/V_{σ}). This is also described in more detail in Eaton [7].

Suppose that the alternating sum in question is non-zero. If U were not radical, then the alternating sum would vanish using Quillen’s contraction T → T O_{p}(N_{G}(U)) → O_{p}(N_{G}(U)), so U must be radical. There is a block B′ of N_{G}(U) which is a Brauer correspondent of B such that

∑_{σ∈N (N_{G}(U),U)/N_{G}(U)}(−1)^{|σ|}k_{d}(B′_{σ}) ≠ 0.

Let B′ have defect group D. By the arguments above, there is some σ ∈ N (N_{G}(U), U) such that B′_{σ} contains a U-projective irreducible character. Then, arguing as at the end of section 3.2, we have C_{D}(U) ≤ U. Set T = N_{G}(U). Now U is not a defect group for B, since B covers a block of N with non-central defect group, while U ∩ N = Z. Hence U is not a defect group for B′ either, by Brauer’s first main theorem. More precisely, consider the unique extension of b to a block of RNU. It does not have defect group U by hypothesis, so none of its Brauer correspondents for UN_{N}(U) can have defect group U. Any such Brauer correspondent lies under B′ as b lies under B. Let M = N_{N}(U) = T ∩ N, and let H = O^{p}(M). Now [M, U] ≤ N ∩ U ≤ Z. Hence we see that [M, U, M] = [U, M, M] = 1, so that [U, H] = 1 by standard facts ( eg chapter 5 of Gorenstein [8]) about actions of coprime automorphisms ( we only need to know that pregular elements of M centralize U for our claim). Now B′ covers a block of H with defect group D ∩ H ≤ C_{D}(U) ∩ H = Z(U) ∩ H ≤ Z. Hence B′ covers a block, say b′, of H whose defect group is contained in Z, which is certainly central in T. Now any block of RM covered by B′ must have defect group strictly containing Z, since we have already seen that any block of RUM lying over such a block has defect group strictly containing U. Now Theorem 3.2.2 supplies a contradiction ( with N_{G}(U) in the role of X, UM in the role of Y). PROOF OF THEOREM 2: We proceed by the usual type of inversion argument employed in [11] and [14]. Suppose that DPC ( for the prime p) holds for every finite central extension of every section of the finite group G. Let B be a block of RG, and let N be a normal subgroup of G which contains Z = O_{p}(G) ≤ Z(G), and is such that B covers blocks of RN which have defect group strictly greater than Z. As before, we may reduce to the case that B covers a G-stable block of RN. Theorem 1 then tells us that we have

∑_{σ∈N (N,Z)/G}(−1)^{|σ|}k_{d}(B_{σ}, λ) = 0,

for each positive defect d and each linear character, λ, of Z.

Another interpretation of this equation is that the alternating sum

∑_{σ∈N (N,Z)/G}(−1)^{|σ|+1}k_{d}(B_{σ}, λ)

has value equal to the number of irreducible characters in B which have defect, d, lie over λ, and lie over blocks of N with defect group Z. The advantage of this last interpretation is that it remains valid even when B does cover blocks of N with defect group Z, as B_{σ} = B when σ = Z while B_{σ} = 0 for all other σ ∈ N (N, Z).

Suppose now that we have a normal subgroup M say, of G, which does not contain Z. Set Z_{1} = Z ∩ M, and let μ be a linear character of Z_{1}. Notice that B covers a block of M with defect group Z_{1} if and only if it covers a block of MZ with defect group Z. Now μ has [Z : Z_{1}] distinct extensions to linear characters of Z. There is a natural bijection between N (N, Z) and N (M, Z_{1}) which sends a chain

σ = (Q_{0}(= Z) < Q_{1} < … < Q_{n}) to σ′ = (Q_{0} ∩ M(= Z_{1}) < Q_{1} ∩ M < … < Q_{n} ∩ M), and this respects the conjugation action of G. The type of inductive argument used to replace DPC by WDPC allows us to conclude that for any non-negative integer d,

∑_{σ∈N (M,Z_{1})/G}(−1)^{|σ|+1}k_{d}(B_{σ}, μ)

is the number of irreducible characters in B which have defect, d, lie over μ, and lie over blocks of M with defect group Z_{1}.

Now let U > 1 be a normal p-subgroup of a finite group G, M be a normal subgroup of G with U = O_{p}(M), μ be an irreducible character of U, B be a block of RG, and d be a non-negative integer. Then to calculate

∑_{σ∈N (M,U)/G}(−1)^{|σ|}k_{d}(B_{σ}, μ),

the results of Dade ( [4], section 17), tell us that we may pass to the inertial subgroup I_{G}(μ) and use Brauer correspondents of B. Hence we may (taking due account of the effect on the defect d ) reduce to the case that μ is G-stable. Then we may replace G by a ( p-power) central extension G̃ of G/U and μ by a linear character of a central p-subgroup Z̃. Then Theorem 1 tells us that the remaining alternating sum vanishes unless the block in question lies over a block of M̃ with defect group Z̃. “Undoing” the Clifford theory, and using the results of Külshammer-Robinson [13, especially proposition 3.2], this means that B covers a block of M which contains a U-projective irreducible character lying over μ. If B does cover such a block of M, then by arguments like those used in the proof of Theorem 1, we may replace arbitrary irreducible characters in B lying over μ by irreducible characters lying over U-projective irreducible characters of the respective M_{σ} ( of the same defect) without changing the value of the alternating sum. PROOF OF THEOREM 3: Suppose that DPC holds, and that N is a normal subgroup of G, B is a block of RG with defect group D^{∗} such that D = D^{∗} ∩ N is Abelian. Suppose that B covers the stable block b of N. We use Theorem 2 to calculate k_{d}(B). By that theorem, we only need to count characters ( of the appropriate defect) which lie over V_{σ}-projective irreducible characters of N for chains σ ∈ R(N)/G, and we may assume that V_{σ} ≤ D. But then we may suppose that C_{D}(V_{σ}) ≤ V_{σ} by arguing as we did at the end of section 3.2. Since D is Abelian, we see that ( up to conjugacy) σ is the chain Z ∩ N < D, in which case B_{σ} is the Harris-Knörr correspondent of B for N_{G}(D) and all irreducible characters in B_{σ} lie over D-projective irreducible characters of N_{N}(D).

References

[1] Alperin,J.L., Weights for Finite Groups, Proceedings of Symposia in Pure Mathematics 47, ( AMS, Providence,1987),369-379.

[2] Alperin,J.L.; Broué,M. Local Methods in Block Theory, Annals of Mathematics (2),110(1979), 143-157.

[3] Dade, E.C., Counting Characters in Blocks,I, Invent. Math. 109, 1,(1992), 187-210.

[4] Dade, E.C., Counting Characters in Blocks,II, J. Reine Angew. Math. 448,(1994),97-190.

[5] Dade, E.C., Counting Characters in Blocks,II.9, ( in Representation Theory of Finite Groups ( Columbus,OH,1995)), Ohio State University Math Research Institute Publications,6, de Gruyter, Berlin, (1997),45-59.

[6] Eaton, C.W., On Finite Groups of p-local Rank One and a Conjecture of Robinson, Ph.D. Thesis, University of Leicester, (1999).

[7] Eaton, C.W.,On a Minimal Counterexample to Dade’s Projective Conjecture, submitted.

[8] Gorenstein,D.,Finite Groups,Harper and Row, New York, (1968).

[9] Harris, M;Knörr, R., Brauer correspondence for covering blocks of finite groups, Comm. Alg. 13,(1985),1213-1218.

[10] Knörr, R., On the Vertices of Irreducible Modules, Annals of Mathematics, 110 (1979),487-499.

[11] Knörr, R.; Robinson, G.R., Some remarks on a conjecture of Alperin, J. London Math. Soc., (2),39,(1989), 48-60.

[12] Külshammer, B.; Puig, L., Extensions of Nilpotent Blocks, Invent. Math,102,1,(1990), 17-71.

[13] Külshammer, B.; Robinson, G.R., Characters of Relatively Projective Modules II, J. London Math. Soc.,(2),36,(1987), 59-67.

[14] Robinson, G. R.; Staszewski, R., More on Alperin’s conjecture, Astérisque 181-182 (1990), 237-255.

[15] Robinson, G.R., Local Structure, Vertices and Alperin’s Conjecture, Proc. London Math Soc., 72,(1996),312-330.

[16] Robinson, G.R., Dade’s Projective Conjecture for p-Solvable Groups, J. Algebra, 229,(2000),234-248.