Dade’s Projective Conjecture for p-Solvable Groups Geoffrey R. Robinson School of Mathematics and Statistics University of Birmingham Edgbaston Birmingham B15 2TT England

INTRODUCTION: At MSRI, in late 1990, E.C. Dade stated his ordinary and projective conjectures, and announced that strengthened versions of these conjectures could be made which were amenable to induction and could be reduced to questions about simple groups. See also [3],[4],[5]. The conjectures concern numbers of complex irreducible characters in certain p-blocks for a prime number p.

At present, it is our understanding that a statement of the “final” form of the “inductive” conjecture which admits the reduction mentioned above has not yet been published ( Dade has informed us that the version which appeared in [5] needs further refinement). It also seems likely that the reductions announced by Dade will not appear for several more years.

In view of this, we feel that it would be a service to certain sections of the representationtheoretic community to present a self-contained proof of the conjecture of the title for p-solvable groups, as there is no published proof of this case currently available ( or even a pre-print version, as far as we are aware).

The reader should take note of the fact that p-solvable groups are more amenable to the Cliffordtheoretic reductions of block-theory than general finite groups ( though they already throw up problems of formidable difficulty- for example, we are fortunate here to be able to assume the results of Külshammer-Puig [10] as one of our starting points), and should not be misled into believing that the necessary Clifford-theoretic reductions in the general case will be as straightforward.

SECTION 1: STATEMENT OF THE CONJECTURE

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 algeba 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}.

Now let Y be a central p-subgroup of X. We denote by N (X, Y) the set of all normal chains of p-subgroups of X of the form

σ = (Q_{0}(= Y) < Q_{1} < … < Q_{n}),

where Q_{i} ⊲ Q_{n} for i = 1, …, n ( we include the singleton chain (Y)). For such a chain σ, we let |σ| = n, the number of p-subgroups appearing in σ which strictly contain Y ( this is consistent with Knörr-Robinson [9] in the case that Y is trivial). 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, Y). We define the sum of blocks b_{σ} of RX_{σ} as in [9] ( as we are dealing with normal chains, this is just the sum of the usual Brauer correspondents of b for X_{σ}). As is now customary, the notation L < M for finite groups L and M indicates that L is a proper subgroup of M.

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).

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).

We suppose from now on that O_{p}(G), the unique largest normal p-subgroup 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.

Dade’s projective conjecture, in the version we prefer to treat, 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: The reader may wonder why we only seem to consider irreducible characters lying over linear characters of central p-subgroups. We are dealing with a “blockwise” conjecture so, 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. The reader may also wonder why we do not use radical chains. As explained in [9], radical chains may be interchanged with chains associated to various p-subgroup complexes ( strictly, we are identifying chains which begin with Z with chains from the p-subgroup complex of G/Z ). It will become apparent that normal chains are more suitable for the proof of our main theorem.

The following observation (noted by Hassan and Horvath in special cases in [8]) is useful. Suppose that the formula

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

( which we henceforth call the “weak” version of Dade’s projective conjecture for Ḡ) 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

∑_{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.

We outline a justification. Fix a choice of such d and λ. By induction, if λ is not faithful, we may suppose that the appropriate alternating sum ( with d′ = d − log_{p}(|kerλ|) in place of d ) is 0 for Ḡ = G/kerλ. Hence we may, and do, suppose that λ is faithful, and that Z is cyclic. Next, considering G/Ω_{1}(Z), we may delete from the alternating sum of the “weak” version (for G) the contribution from characters which lie over non-faithful linear characters of Z and still obtain 0. As usual, Ω_{1}(Z) denotes the subgroup of Z generated by the elements of order p in Z. All faithful linear characters of Z are p-conjugate, so that the number of irreducible characters of fixed defect of any B_{σ} which lie over any one of them is independent of the particular faithful linear character chosen.

SECTION 2: PROOF OF THE THEOREM

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 O_{p′}(G), the unique largest normal p′-subgroup of G, is central in G.

Theorem 1 (or, more precisely, its proof) yields a proof of Dade’s projective conjecture for p-solvable groups. A p-solvable finite group H such that O_{p}(H)O_{p′}(H) ≤ Z(H) is Abelian. Dade’s projective conjecture is vacuously true for Abelian groups, so the proof of Theorem 1 ( with the arguments restricted to p-solvable groups) shows that no p-solvable counterexample to Dade’s projective conjecture exists.

We will have occasion to deal with Morita equivalences between p-blocks of various finite groups. These Morita equivalences (which are for blocks defined over R) induce defect-preserving bijections between the sets of irreducible characters in the equivalent blocks ( one explanation of this is that such Morita equivalent blocks have the same decomposition matrix D ( under a suitable labelling), so they have the same Cartan matrix C. The matrix DC^{−1}D^{T} ( which is a scalar multiple of the famous Brauer-Feit matrix ) determines the number of irreducible characters of each defect in the equivalent blocks, since its rows may be used to index the irreducible characters in either block, and the defect of the irreducible character (for either block) corresponding to its i-th row is d_{i}, the smallest non-negative integer such that p^{d_{i}} times the i-th row has integer entries). In the case of Morita equivalences obtained via Fong correspondence, this defect-preserving bijection can also be seen directly from the usual character-theoretic constructions.

Before we proceed to the proof of Theorem 1, we mention some consequences. C.Eaton, in [6], has recently proved that Dade’s projective conjecture implies conjecture 4.1 of [12] (the reverse implication being immediate). Hence conjecture 4.1 of [12] is valid for p-solvable groups, so that the consequences of that conjecture derived in [12] are valid for such groups. Perhaps the most interesting of these is : THEOREM 2: Let H be a finite p-solvable group, b be a p-block of H with defect group D. Suppose that b contains an irreducible character of defect d′, lying over the linear character γ of the central p-subgroup, W, of H. Then there is a subgroup, U, of D with U = O_{p}(N_{H}(U)) and C_{D}(U) = Z(U), and there is an irreducible character of U, lying over γ, which has defect d′. REMARK: L.Barker, in [1], proves a result implying most of Theorem 2 ( except that it is not clear that the p-subgroup he produces is radical, so that his result does not seem to give the full strength of the p-solvable case of the consequence of conjecture 4.1 of [12] stated in Theorem 5.1 of [12]).

The proof of the p-solvable case of Dade’s projective conjecture yields another proof of the psolvable case of Alperin’s weight conjecture ( consider the case that Z is trivial, sum over all defects d, and use the formulation of Alperin’s weight conjecture given in [9]). We take the opportunity to point out that the “sketch” proof of Alperin’s weight conjecture for p-solvable groups given in Robinson-Staszewski [11] is not invalid (as L. Barker, in [2], asserts that it is) since it invoked the results of Külshammer-Puig [10] ( then in preprint form). PROOF OF THEOREM 1: Let G, B, Z, λ and d be as in the statement of the Theorem. Suppose further that O_{p′}(Z(G)) < N where N ⊲ G and N has order prime to p.

By the minimality of G and the discussion at the end of Section 1, the formula of the “weak” form of the conjecture fails for B with our chosen defect d. Let μ be an irreducible character of N which lies under B. Whenever Q is a p-subgroup of the inertial subgroup, I_{G}(μ), of μ in G, we let f_{Q}(μ) denote the Glauberman correspondent of μ for C_{N}(Q).

Suppose first that μ is G-stable. Then, as usual, we may construct a finite p′-central extension, say Ĝ, of G, using the projective representation of G associated to μ. Then Ĝ may be assumed to have a finite cyclic central p′-subgroup Ŵ such that Ĝ/Ŵ ≅ G. Also, Ĝ may be assumed to have a normal subgroup, N̂, naturally isomorphic to N, such that N̂ ∩ Ŵ is trivial. Identifying μ with an irreducible character of N̂, we may extend μ to an irreducible character, μ̂ say, of Ĝ.

We set Ĝ/N̂ = G̃ and, more generally, we denote the image in G̃ of a subgroup, Ŷ , of Ĝ by Ỹ . Then G̃/W̃ ≅ G/N, and we note that [G̃ : Z(G̃)] ≤ [G : ZN] < [G : Z(G)].

Fong correspondence now gives a unique block, B̃, of RG̃ corresponding to, and Morita equivalent to, B. More precisely, let γ̂ be the unique linear character of Ŵ covered by μ̂ and identify this naturally with a linear character, γ̃, of W̃ . Then there is a bijection between the set of blocks of RG̃ covering the complex conjugate of γ̃ and the set of blocks of RG covering μ. Furthermore, given one of the former blocks, inflating its irreducible characters to characters of Ĝ, then multiplying them by μ̂, yields ( the inflations to Ĝ of) the irreducible characters in the corresponding block of RG.

Since N is a p′-group, there is a natural bijection between N (G, Z)/G and N (G̃, Z̃)/G̃. Let bars denote images in G/N. Firstly, as W̃ is a central p′-subgroup of G̃ with G̃/W̃ ≅ Ḡ, there is an obvious conjugacy-preserving bijection between N (Ḡ, Z̄) and N (G̃, Z̃). We now describe in more detail the bijection between N (G, Z)/G and N (Ḡ, Z̄)/Ḡ.

Clearly, G-conjugate chains in N (G, Z) map to Ḡ-conjugate chains in N (Ḡ, Z̄). On the other hand, let

σ̄ = (Q̄_{0}(= Z̄) < Q̄_{1} < … < Q̄_{n})

be a chain in N (Ḡ, Z̄). For each i, let P_{i} be the full pre-image of Q̄_{i} in G. Let Q_{n} be a Sylow psubgroup of P_{n}. For i < n, set Q_{i} = P_{i}∩Q_{n}. Then Q_{i} ∈ Syl_{p}(P_{i}), as P_{i}⊲P_{n} for each i ( in particular, Q_{0} = Z). Also, Q_{i} ⊲ Q_{n} for each i, so that we have produced a chain (Q_{0} < Q_{1} < … < Q_{n}) in N (G, Z). We denote this chain by σ. It is not unique, but is uniquely determined by σ̄ and the choice of Sylow p-subgroup of P_{n}, so it is determined uniquely up to N-conjugacy by σ̄.

Any chain τ ∈ N (G, Z) which maps onto σ̄ under the natural homomorphism from G to Ḡ has the form τ = (Z < Q_{1}^{x_{1}} < Q_{2}^{x_{2}} … < Q_{n}^{x_{n}}) with each x_{i} ∈ N. We are really only concerned with the N-conjugacy class of τ, so we may assume that x_{n} = 1_{G}, and we do so. Then, for each i, we have [Q_{i}, x_{i}] ≤ Q_{n} ∩ N = 1, so that τ = σ. Hence σ̄ has a unique N-conjugacy class of pre-images in N (G, Z). In particular, Ḡ-conjugate chains in N (Ḡ, Z̄) “lift” to G-conjugate chains in N (G, Z). The argument above also shows that Ḡσ̄ = NG_{σ}/N.

As a matter of interest, we remark that chains which are radical in N (G, Z) need not have radical images in N (Ḡ, Z̄) in general, as can be seen already in p-nilpotent examples.

A key observation for the proof of the Theorem is that G_{σ} = N_{NG_{σ}}(V^{σ}) for all σ ∈ N (G, Z). For ( using Dedekind’s modular law and the fact that N is a normal p′-subgroup of G) we have N_{NG_{σ}}(V^{σ}) = G_{σ}N_{N}(V^{σ}) = G_{σ}C_{N}(V^{σ}) = G_{σ}.

This observation allows us to employ results from Harris-Knörr [7] and Külshammer-Puig [10]. These results both concern p-blocks of a group X which cover an X-stable p-block, b say, of a normal subgroup, T, of X. For our purposes, X may be taken to be a finite p′-central extension of a section of G. The main result of [7] tells us that, for Q a defect group of b, Brauer correspondence gives a bijection between blocks of RX which lie over b and blocks of RN_{X}(Q) which lie over the unique Brauer correspondent of b for RN_{T} (Q). The main result of [10] has the consequence that if the block b is also nilpotent, then there is a Morita equivalence between the corresponding blocks of RX and RN_{X}(Q).

Here (and later in the paper) we make use of the compatibility (first made explicit by Alperin) between Glauberman correspondence and Brauer corrrespondence in the case that the coprime group of operators is a p-group.

Whenever σ is a chain in N (G, Z), each block in B_{σ} has defect groups containing V^{σ} and lies over the Glauberman correspondent f_{V^{σ}}(μ). By [7], ( with NG_{σ} in the role of X and NV^{σ} in the role of T), each block of RNG_{σ} which has a defect group containing V^{σ}, and which lies over the unique block of RNV^{σ} covering μ, has a single Brauer correspondent for G_{σ}. Furthermore, each block of RG_{σ} which lies over f_{V^{σ}}(μ) occurs as such a Brauer correspondent. Moreover, as all p-blocks of p-nilpotent groups are nilpotent, two such Brauer correspondent blocks are Morita equivalent, by [10].

Hence, using B_{σ}, we may define the Morita equivalent sum, B′_{σ}, of Harris-Knörr correspondent blocks for RNG_{σ}, with all irreducible characters of B′_{σ} lying over μ. We may then use Fong correspondence to obtain from B′_{σ} a Morita equivalent sum of blocks, B̃_{σ} say, of RG̃σ̃ (note the apparent distinction between this sum of blocks and B̃σ̃).

In fact, B̃_{σ} = B̃σ̃, but we explain this in more detail, as the fact that Brauer correspondence and Fong correspondence commute in this way is not as widely known as it might be ( though the referee informs us that it can be deduced with some effort from Theorem 14.3 of [4]). This part of the argument is quite general, and does not rely on the fact that G is a minimal counterexample.

For the sake of exposition, we consider the case that μ extends to an irreducible character of G (in general, we would need to work with the group Ĝ discussed earlier in the proof). We choose a particular extension of μ, which we denote by μ^{(G)}, to an irreducible character of G. Whenever we are dealing with a subgroup, X, of G which contains N, we set μ^{(X)} = Res_{X}^{G}(μ^{(G)}), an extension of μ to an irreducible character of X.

We let bars denote images in G/N. We choose a defect group, D, for B which contains V^{σ}. We set M = N_{G}(D) and H = NM. Then D̄ is a defect group for the Fong correspondent, B̄, of B. By Brauer’s First Main Theorem, ( as H contains M) there is just one Brauer correspondent, b′ say, of B for RH which has defect group D. Let b be the unique Brauer correspondent of B for M. Then b′ is also the Harris-Knörr correspondent of b for H = NM, so that b′ lies over μ, as b lies over the Glauberman correspondent f_{D}(μ).

We know that Irr(B) = {χμ^{(G)} : χ ∈ Irr(B̄)} and that Irr(b′) = {θμ^{(H)} : θ ∈ Irr(b̄)} for the Fong correspondent block, b̄, of b′ for RH̄ ( identifying characters in Irr(B̄) with their inflations to G, and likewise for Irr(b̄) and H).

We wish to prove that b̄ is the ( unique) Brauer correspondent for RH̄ of B̄ ( note that H̄ = M̄ = NḠ(D̄)). We need to establish the usual congruence for central character values on p-regular class sums of H̄ with defect group D̄.

We claim that for each p-regular x̄ with D̄ ∈ Syl_{p}(CḠ(x̄)), we may find a p-regular pre-image, x, of x̄ in H with D ∈ Syl_{p}(C_{G}(x)) such that μ^{(G)}(x) ∉ J(R).

We choose a ( necessarily p-regular) pre-image, y say, of x̄ in G. Notice ( as N is a p′-group and D̄ is a Sylow p-subgroup of CḠ(x̄)) that ND contains a Sylow p-subgroup of C_{G}(yn) for each n ∈ N, and that any such Sylow p-subgroup is N-conjugate to a subgroup of D. Notice also that D acts by conjugation on the coset yN. Since μ^{(G)} restricts irreducibly to N, we know that ∑_{n∈N} |μ^{(G)}(yn)|^{2} = |N| ∉ J(R). It follows that there is some n′ ∈ N such that [D : C_{D}(yn′)]μ^{(G)}(yn′) ∉ J(R). Hence D must centralize yn′ and μ^{(G)}(yn′) ∉ J(R). Then yn′ is p-regular, D ∈ Syl_{p}(C_{G}(yn′)), and μ^{(G)}(yn′) ∉ J(R), so that we may take x = yn′.

Now we take a p-regular element x̄ in H̄ with D̄ ∈ Syl_{p}(CḠ(x̄)) and we choose a p-regular pre-image, x, of x̄ in G of the type just described.

Let T_{x} denote the full pre-image in G of CḠ(x̄). Then D ∈ Syl_{p}(T_{x}), so that, using Sylow’s theorem, we see easily that [T_{x} : C_{G}(x)] ≡ [T_{x} ∩ H : C_{H}(x)] ( mod p), as M ≤ H.

Let χ be an irreducible character in B̄ and let θ be an irreducible character in b̄. Short calculations show that

[G : C_{G}(x)]\frac{χ(x)μ^{(H)}(x)}{χ(1)μ^{(H)}(1)} = [Ḡ : CḠ(x̄)]\frac{χ(x̄)}{χ(1)}[T_{x} : C_{G}(x)]\frac{μ^{(H)}(x)}{μ^{(H)}(1)}

and that

[H : C_{H}(x)]\frac{θ(x)μ^{(H)}(x)}{θ(1)μ^{(H)}(1)} = [H̄ : CH̄(x̄)]\frac{θ(x̄)}{θ(1)}[T_{x} ∩ H : C_{H}(x)]\frac{μ^{(H)}(x)}{μ^{(H)}(1)}.

Also, using Brauer correspondence between B and b′, we have

[G : C_{G}(x)]\frac{χ(x)μ^{(H)}(x)}{χ(1)μ^{(H)}(1)} ≡ [H : C_{H}(x)]\frac{θ(x)μ^{(H)}(x)}{θ(1)μ^{(H)}(1)}(modJ(R)).

Since μ^{(H)}(x)/μ^{(H)}(1) is a unit of R and [T_{x} : C_{G}(x)] ≡ [T_{x} ∩H : C_{H}(x)] ( mod p), we conclude that

[Ḡ : C̄Ḡ(x̄)]\frac{χ(x̄)}{χ(1)} ≡ [H̄ : C̄H̄(x̄)]\frac{θ(x̄)}{θ(1)}(modJ(R)).

Hence b̄ is the ( unique) Brauer correspondent for RH̄ of B̄.

This establishes that Fong correspondence and Brauer correspondence commute in the desired fashion in the case that σ = (Z < D), where D is a defect group for B.

We digress for a moment to discuss the behaviour of Fong correspondence when we have a pair of subgroups T and X of G, with N ≤ T ⊲ X and a block, β, of RX, lying over μ, covering blocks β_{1}, …, β_{n} ( and no others) of RT.

We use bars ( over blocks of RX or RT covering μ ) to denote their Fong correspondents for X̄ or T̄ respectively. Then Irr(β) = {μ^{(X)}ϕ : ϕ ∈ Irr(β̄)}, identifying characters in β̄ with their inflations to characters of X. A similar description of Irr(β_{i}), in terms of μ^{(T)} and Irr(\overline{β_{i}}), may be given for each i. Restricting irreducible characters in β to T, and doing likewise with β̄ and T̄, we deduce that the blocks of T̄ covered by β̄ are precisely the Fong correspondents \overline{β_{1}}, …, \overline{β_{n}} of β_{1}, …, β_{n} respectively.

Returning to the matter at hand, given a B-subpair (Q, b_{0}), we may use b_{0}^{∗} (the unique block of RQC_{G}(Q) lying over b_{0} ) to define the Harris-Knörr correspondent block, b_{0}^{∗′}, of RNQC_{G}(Q) which lies over μ and has a defect group containing Q. Then b_{0}^{∗′} covers a unique block of RNC_{G}(Q), say b′_{0}, which we will (somewhat loosely) refer to as the Harris-Knörr correspondent of b_{0}. Let b̄_{0} denote the Fong correspondent of b′_{0}. By the digression above, b̄_{0} is the unique block of RCḠ(Q̄) covered by the Fong correspondent of b_{0}^{∗′}. This procedure gives us a subpair (Q̄, b̄_{0}).

Let (D, b_{1}) be a maximal B-subpair. We now have a subpair (D̄, b̄_{1}), which we claim is a maximal B̄-subpair. We observe that when we apply the above procedure with (D, b_{1}), the block b̄_{1} has defect group Z(D̄).

We take our previously defined block b for M and its Harris-Knörr correspondent b′ for H. Then b lies over b_{1} and b′ lies over the Harris-Knörr correspondent, b′_{1}, of b_{1}. The Fong correspondent of b′ is b̄, which we already know to be the unique Brauer correspondent of B̄ for NḠ(D̄). ¿From the digression above, we conclude that the blocks of RCḠ(D̄) covered by b̄ are the Fong correspondents of the blocks of RNC_{G}(D) covered by b′. Hence b̄_{1} is one of these blocks and (D̄, b̄_{1}) is indeed a maximal B̄-subpair.

Now we consider a more general chain σ, with maximal p-subgroup V^{σ}. Set L_{σ} = NG_{σ}, and recall that \overline{L_{σ}} = Ḡσ̄. We consider a block of RG_{σ}, say b_{σ}, which is a summand of B_{σ}. Let D_{σ} be a defect group for b_{σ} and recall that D_{σ} contains the normal p-subgroup, V^{σ}, of G_{σ}, so that D_{σ}C_{G}(D_{σ}) ≤ G_{σ}. We choose a maximal b_{σ}-subpair, (D_{σ}, b_{2}), which may also be viewed as a B-subpair. We note that Z(D_{σ}) is the unique defect group for b_{2}. We assume ( as we may, after possibly amending our original choice of D) that (D, b_{1}) contains (D_{σ}, b_{2}).

There is a B-subpair (D_{σ}C_{D}(D_{σ}), b_{∞}), contained in (D, b_{1}) and containing (D_{σ}, b_{2}). This too may be regarded as a b_{σ}-subpair. Consequently, we have C_{D}(D_{σ}) ≤ D_{σ} by the maximality of (D_{σ}, b_{2}) as a b_{σ}-subpair.

¿From now on, we regard (D_{σ}, b_{2}) as a B-subpair. As above, we construct from this a subpair (\overline{D_{σ}}, b̄_{2}). Now Ḡσ̄ contains \overline{D_{σ}}CḠ(\overline{D_{σ}}), so that b̄_{2} has a unique Brauer correspondent block for Ḡσ̄, which we denote by b̄σ̄.

Let b′_{σ} be the unique Harris-Knörr correspondent of b_{σ} for RL_{σ}. The earlier argument ( for G, H and B ) may be applied with L_{σ} in the role of G and b′_{σ} in the role of B. We deduce that the unique Brauer correspondent of b̄_{2} for Ḡσ̄ is the Fong correspondent of b′_{σ}. In other words, b̄σ̄ is the Fong correspondent of b′_{σ}. We wish to know that b̄σ̄ is a Brauer correspondent of B̄.

To prove this, it suffices to prove that (\overline{D_{σ}}, b̄_{2}) is a B̄-subpair. Since (D̄, b̄_{1}) is a B̄-subpair, it suffices to prove that (\overline{D_{σ}}, b̄_{2}) is contained in (D̄, b̄_{1}).

If possible, choose a B-subpair (D_{3}, b_{3}), contained in (D, b_{1}) and containing (D_{σ}, b_{2}), which is maximal subject to the subpair (\overline{D_{3}}, b̄_{3}) not being contained in (D̄, b̄_{1}). Notice that D_{3} < D and that, furthermore, C_{D}(D_{3}) = Z(D_{3}), as D_{σ} ≤ D_{3}.

Let E_{3} be a defect group for b_{3}. Then E_{3} ≤ C_{G}(D_{σ}) and certainly Br_{E_{3}}(1_{b_{2}}) ≠ 0. Hence E_{3} is contained in Z(D_{σ}), the unique defect group for b_{2}. Consequently, E_{3} ≤ C_{D}(D_{3}) = Z(D_{3}), so that E_{3} = Z(D_{3}), as Z(D_{3}) is contained in each defect group of every block of RC_{G}(D_{3}).

We set D_{4} = N_{D}(D_{3}) > D_{3} and we work for a while within the group X = NC_{G}(D_{3})D_{4}. Corresponding to the block b_{3} is a block b′′_{3} of RX, lying over μ, having defect group D_{4}, which is the unique block of RX covering the Harris-Knörr correspondent, b′_{3}, of b_{3} for NC_{G}(D_{3}). By the earlier digression, the Fong correspondent of b′′_{3} ( which has defect group \overline{D_{4}}) covers b̄_{3}.

There is a B-subpair (D_{4}, b_{4}) lying between (D_{3}, b_{3}) and (D, b_{1}) and a corresponding subpair (\overline{D_{4}}, b̄_{4}). The block b′′_{3} has defect group D_{4} and (D_{4}, b_{4}) may be viewed as a maximal b′′_{3}-subpair containing the b′′_{3}-subpair (D_{3}, b_{3}). Applying the earlier arguments ( with X in the role of G and b′′_{3} in the role of B) and using the fact that the Fong correspondent of b′′_{3} lies over b̄_{3}, we may conclude that (\overline{D_{4}}, b̄_{4}) contains (\overline{D_{3}}, b̄_{3}).

By the maximal choice of D_{3}, we know that (D̄, b̄_{1}) contains (\overline{D_{4}}, b̄_{4}). This contradicts the fact that (D̄, b̄_{1}) does not contain (\overline{D_{3}}, b̄_{3}). This contradiction shows that (\overline{D_{σ}}, b̄_{2}) is, after all, a B̄-subpair.

The conclusion of this part of the discussion is that the Fong correspondent of b′_{σ} is a Brauer correspondent of B̄. On the other hand, given a block b̄σ̄ which is a summand of B̄σ̄, essentially applying the above arguments “in reverse,” we see that b̄σ̄ occurs as a Fong correspondent of one of the block summands of B′_{σ}.

Returning to our proof of the main theorem, we obtain ( for our fixed “bad” defect d)

∑_{σ∈N (G,Z)/G}(−1)^{|σ|}k_{d}(B_{σ}) = ∑_{σ̃∈N (G̃,Z̃)/G̃}(−1)^{|σ|}k_{d}(B̃σ̃).

If O_{p}(G̃) > Z̃, then the right-hand alternating sum vanishes by the usual elementary argument. Since we know that the left-hand alternating sum does not vanish, we conclude that O_{p}(G̃) = Z̃. But then, by the minimal choice of G, the right-hand alternating sum vanishes, since it is that appearing in the “weak” form of Dade’s projective conjecture for B̃. This contradiction finally shows that μ is not G-stable.

For the remainder of the proof, it is convenient to deal with the conjecture via B-subpairs. 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 the “weak” form of Dade’s projective conjecture 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′.

The inertial subgroup I_{G}(μ) = I satisfies [I : Z(I)] < [G : Z(G)], as I contains Z(G) and μ is not G-stable. Let B′ be the block of RI which lies over μ 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. We choose ( as we may) a maximal B-subpair (D, b_{D}) such that the Glauberman correspondent f_{D}(μ) lies under b_{D} ( implicit here is the assumption that D ≤ I. Then D is a defect group for both B and B′). 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. For each subgroup, Q, of D, the Glauberman correspondent f_{Q}(μ) lies under b_{Q}.

We wish to replace (Q, b_{Q}) by a B′-subpair (Q, b′_{Q}). We obtain b′_{Q} by multiplying the block idempotent of b_{Q} by e_{f_{Q}(μ)}, which is the centrally primitive idempotent of RC_{N}(Q) associated to f_{Q}(μ). This yields a centrally primitive idempotent of RC_{I}(Q) (it is easy to check that C_{I}(Q) is the inertial subgroup in C_{G}(Q) of f_{Q}(μ)) and b′_{Q} is defined as the corresponding block of RC_{I}(Q). To check that this really is a B′-subpair, we need to know that Br_{Q}(1_{B′}) involves Br_{Q}(1_{b′_{Q}}) ( this is just the image of 1_{b′_{Q}} ( mod J(R)G)).

Since 1_{B′} = 1_{B}e_{μ} and Br_{D}(1_{b_{D}}) appears in Br_{D}(1_{B}), we have what we wanted when Q = D ( applying Br_{D} to both sides, then multiplying by Br_{D}(1_{b_{D}})) since we assumed that e_{f_{D}(μ)}.1_{b_{D}} ≠ 0. For Q ⊲ D, we have Br_{D}(e_{f_{D}(μ)}.1_{b_{D}}) = Br_{D}(1_{b_{D}}).Br_{D}(e_{f_{Q}(μ)}.1_{b_{Q}}), so certainly e_{f_{Q}(μ)}.1_{b_{Q}} ≠ 0. Applying Br_{Q} to the equation 1_{B′} = 1_{B}e_{μ}, then multiplying by Br_{Q}(1_{b_{Q}}), we obtain what we wanted when Q ⊲ D. Repeating this argument enough times gives the desired involvement for all subgroups Q of D, as all such subgroups are subnormal in D.

For any subgroup, Q, of D, the argument above also shows that (Q, b′_{Q}) ≤ (D, b′_{D}). Hence we have a canonical bijection between B-subpairs contained in (D, b_{D}) and B′-subpairs contained in (D, b′_{D}), which 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}).

Suppose that (Q, b_{Q}) and (Q, b_{Q})^{g} are both contained in (D, b_{D}) for some g ∈ G. We claim that the corresponding B′-subpairs (Q, b′_{Q}) and (Q^{g}, b′_{Q^{g}}) are I-conjugate. For b_{Q^{g}} = (b_{Q})^{g} by the uniqueness of (Q^{g}, b_{Q^{g}}) ≤ (D, b_{D}). The Glauberman correspondent f_{Q^{g}}(μ^{g}) (of the Q^{g}-stable character, μ^{g}, of N) clearly lies under (b_{Q})^{g}. Since all irreducible characters of C_{N}(Q^{g}) which lie under b_{Q^{g}} are conjugate via elements of C_{G}(Q^{g}) to f_{Q^{g}}(μ), we deduce that μ may be conjugated to μ^{g} by an element of C_{G}(Q^{g}). Hence there are elements x ∈ I and c ∈ C_{G}(Q) with g = cx. Now we have b′_{Q^{x}} = e_{f_{Q^{x}} (μ)}(1_{b_{Q}})^{x} = (b′_{Q})^{x}, as μ^{x} = μ. Since Q^{x} = Q^{g}, the claim is established.

On the other hand, if (Q, b′_{Q}) and (Q, b′_{Q})^{x} are B′-subpairs which are both contained in (D, b′_{D}) for some x ∈ I, then (Q, b_{Q}) and (Q, b_{Q})^{x} are B-subpairs which are both contained in (D, b_{D}). First, uniqueness subject to containment in (D, b′_{D}) tells us that b′_{Q^{x}} = (b′_{Q})^{x}. Since

(1_{b′_{Q}})^{x}1_{b′_{Q^{x}}} = (e_{f_{Q}(μ)}1_{b_{Q}})^{x}1_{b_{Q^{x}}} e_{f_{Q^{x}} (μ)} ≠ 0,

we certainly have (1_{b_{Q}})^{x}1_{b_{Q^{x}}} ≠ 0, so that b_{Q^{x}} = (b_{Q})^{x}.

More generally, given g ∈ G and a chain (σ, b_{V^{σ}}) such that (V^{σ}, b_{V^{σ}}) and (V^{σ}, b_{V^{σ}})^{g} are both contained in (D, b_{D}), we may write g = cx for some x ∈ I, and some c ∈ C_{G}(V^{σ}). This implies, in particular, that (σ, b_{V^{σ}})^{g} = (σ, b_{V^{σ}})^{x}, and then that the corresponding chains (σ, b′_{V^{σ}}) and (σ^{g}, b′_{(V^{σ})^{g}}) are I-conjugate ( via x). Conversely, given a chain (σ, b′_{V^{σ}}) and an element x ∈ I,with both (σ, b′_{V^{σ}}) and (σ, b′_{V^{σ}})^{x} contained in (D, b′_{D}), we conclude from the earlier argument that the corresponding chains (σ, b_{V^{σ}}) and (σ, b_{V^{σ}})^{x} are both contained in (D, b_{D}).

We have now demonstrated 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.

Furthermore, for corresponding chains (σ, b) and (σ, b′), we have G_{(σ,b)} ∩ I = I_{(σ,b′)}. There is a defect-preserving bijection between irreducible characters in β_{(σ,b)} and in the analogously defined block β′_{(σ,b′)}, since I ∩ G_{(σ,b)} is the inertial subgroup of f_{V^{σ}}(μ) in G_{(σ,b)}.

We thus obtain ( for our fixed “bad” defect d):

∑_{(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′)}).

If O_{p}(I) > Z, then the usual elementary argument shows that the right-hand alternating sum is zero, a contradiction, as the left-hand alternating sum does not vanish by hypothesis. Hence O_{p}(I) = Z.

By the minimal choice of G, the formula of Dade’s projective conjecture holds for B′, for each linear character of Z and each positive defect. Hence the “weak” form of Dade’s projective conjecture, and its subpair version, both hold for B′, for each positive defect. Thus the right-hand alternating sum above is zero, a contradiction. The proof of Theorem 1 is complete. ACKNOWLEDGEMENTS: I am greatly indebted to Burkhard Külshammer for a number of helpful comments. I also wish to warmly thank the referee for several suggestions which led to improvements in the exposition of the paper.

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