Some Refinements of Dade’s Projective Conjecture

Adam Glesser

Mathematical Sciences King’s College University of Aberdeen Aberdeen AB24 3UE, UK

December 2006

Abstract

In this paper a further refinement of Dade’s projective conjecture, due to Boltje, is presented. This new statement includes ideas first published by Isaacs and Navarro as well as the recent contractibility version of Alperin’s conjecture introduced by Boltje. Leaning heavily on the work of Robinson, weaker forms of the conjecture are proved in the case of p-solvable groups.

1 Introduction

Let G be a finite group, p a prime, and (K, R, F) a p-modular system with ℚ_{p}(ν) ⊆ K ⊆ \overline{ℚ_{p}}, where \overline{ℚ_{p}} is an algebraic closure of the p-adic numbers and ℚ_{p}(ν) is the field extension of ℚ_{p} containing all p′-roots of unity. This implies that F is an algebraically closed field with characteristic p. We also assume that K contains a primitive |G|-th root of unity, i.e., that it is large enough for G.

For a central subgroup Z of G, a linear character λ of Z, a block B of G such that the defect group of B properly contains Z_{p} (the subgroup of all p-elements of Z) and a non-negative integer d, Dade conjectured that

∑_{C∈𝒩(G,Z_{p})/G} (−1)^{|C|}k(N_{G}(C), B, λ, d) = 0  (1)

where 𝒩(G, Z_{p})/G denotes the set of G-conjugacy classes of chains of nontrivial p-subgroups Q_{0} < Q_{1} < ⋯ < Q_{n} such that Q_{i} ⊴ Q_{n} for all i = 1, …, n and Z_{p} < Q_{0} (see [5]). Here k(N_{G}(C), B, d, λ) is the number of irreducible Kcharacters of N_{G}(C) lying in blocks that induce to B, have p-defect d and whose restriction to Z contains λ as an irreducible constituent.

Inspired by the work of Isaacs and Navarro (see [10] and [16]), Uno suggested two refinements of Dade’s projective conjecture (see [20] and [21]), one that considers what we call the p-residue of the character, and another that only counts characters that are stabilized (under the typical Galois conjugation action) by a fixed subgroup of 𝒢 := Gal(\overline{ℚ_{p}}/ℚ_{p}). Note that we choose a slightly different Galois group than Navarro in [16] so that we do not have to change the Galois group if a larger group suddenly comes into consideration. While both of these refinements strengthen Dade’s original conjecture by adding parameters to the above ”k-function”, Boltje recently offered (see [1]) a refinement of a different sort, namely, instead of putting extra conditions on the characters, he only considers chains whose length is greater than a fixed integer. The interesting result, which is backed by experimental evidence, is that while the alternating sums may cease to be 0, they will always be non-negative. Putting these refinements together, we get the conjecture within whose framework we will work. To the notation above, we add an integer n, an integer r ∈ {1, …, p−1} and a subgroup ℐ of 𝒢. Then Boltje conjectures that:

∑_{C∈𝒩(G,Z_{p})_{≥n}/G} (−1)^{|C|−n}k(N_{G}(C), B, λ, d, r, ℐ) ≥ 0  (2)

where 𝒩(G, Z_{p})_{≥n} denotes the subset of 𝒩(G, Z_{p}) whose chains have length greater than or equal to n and k(N_{G}(C), B, λ, d, r, ℐ) is the number of irreducible K-characters of N_{G}(C) lying in blocks that induce to B, have p-defect d, presidue congruent to ±r modulo p, whose restriction to Z contains λ as an irreducible constituent, and whose 𝒢-stabilizer equals ℐ.

In [18], Robinson proved Dade’s projective conjecture (1) in the case of p-solvable groups. Following these ideas, we will prove some weaker versions of (2) for p-solvable groups. In fact, we obtain two results that strengthen Robinson’s work. In both, we add the p-residue of the character to the list of parameters. However, we then make a trade-off between the linear character λ of Dade’s projective conjecture and Boltje’s ’contractibility’ refinement. We use an argument of Robinson, based on a result of Hassan and Horváth (see Theorem 3.7 in [7]), to ’suppress’ the linear character. Unfortunately, this argument does not work in Boltje’s generality.

Theorem 1.1 Let G be a p-solvable group, Z a central subgroup of G and λ a linear character of Z. Let B be a p-block of G such that Z_{p} is not a defect group of B, d a non-negative integer, n an integer, and r ∈ {1, …, p − 1}. Then

∑_{C∈𝒩(G,Z_{p})_{≥n}/G} (−1)^{|C|−n}k(N_{G}(C), B, d, r) ≥ 0  (3)

and

∑_{C∈𝒩(G,Z_{p})/G} (−1)^{|C|}k(N_{G}(C), B, λ, d, r) = 0  (4)

2 Reductions

The aim of this section is to show that (2) will follow from considering only the cases where Z = O_{p}(G) (the largest normal p-subgroup of G) and λ is faithful (hence Z is cyclic). These reductions are already well known in the case of Dade’s projective conjecture, but proofs are scattered about the literature and the reductions are mostly taken for granted at this point. We offer them here both as a courtesy to the reader and because Boltje’s refinement requires some extra care.

For any group H and any irreducible character χ of H, the integers d(χ) and r(χ) are defined to be the unique integers such that

\frac{|H|}{χ(1)} = p^{d(χ)}r(χ)

and p ∤ r(χ). The integers d(χ) and r(χ) are called the p-defect and p-residue of χ, respectively.

As is standard, we define 𝒩(G) to be the set of normal chains of p-subgroups of G (including the empty chain), i.e., chains of the form

C = (Q_{0} < Q_{1} < ⋯ < Q_{n})

where Q_{i} is a non-trivial p-subgroup of G and Q_{i} ⊴ Q_{n} for all i. The length of the chain C is defined to be |C| = n (with the length of the empty chain ∅ equal to -1) and we denote the subset of chains with length n by 𝒩(G)_{n}. For a central p-subgroup Z, a block B of G and an integer n, we define:

𝒩(G, Z) := {C = (Q_{0} < ⋯ < Q_{m}) ∈ 𝒩(G) | Z < Q_{0}}
𝒩(G, Z)_{≥n} := {C ∈ 𝒩(G, Z) | |C| ≥ n}.

We should explain some of the subtleties of the above definitions. First, the reader will note that we do not anchor our chains, but merely assume that their initial term lies above some central p-subgroup. Second, the set 𝒩(G) is a G-set under conjugation (in the natural way), as are all of the subsets given above. Hence, it makes sense to talk about the stabilizer, N_{G}(C), for a chain C in any of the above sets. Third, block induction is defined for blocks of N_{G}(C) by Lemma 3.2 of [11].

The p-blocks of F G will be denoted by Bl(G) and for B ∈ Bl(G), the identity of B will be denoted by e_{B}. If H is a subgroup of G and χ ∈ Irr(H) := Irr_{K}(H) (the set of irreducible K-characters of H), then the block of H containing χ will be denoted by b(χ). At several points, we will also need the Brauer homomorphism. For a p-subgroup Q of G and a subgroup H of G such that C_{G}(Q) ≤ H ≤ N_{G}(Q), we define the map on F G that projects onto F C_{G}(Q). If we restrict the domain of this map to Z(F G), then we get an F -algebra homomorphism

Br_{Q} : Z(F G) ⟶ Z(F H), L^{+} ↦ ∑_{x∈L∩C_{G}(Q)} x

where L^{+} denotes the class sum for a G-conjugacy class L. Recall that if QC_{G}(Q) ≤ H ≤ N_{G}(Q) and b ∈ Bl(H), then the induced block b^{G} is defined and if B is a block of G that has a defect group containing Q, then Br_{Q}(e_{B}) = ∑_{b^{G}=B} e_{b}

(see Theorem 4.14 in [15]).

Lemma 2.1 Let H be a normal p′-subgroup of G with a G-stable irreducible character ξ, C ∈ 𝒩(G) and b ∈ Bl(N_{G}(C)). Then b covers ξ if and only if b^{G} covers ξ.

Proof: Note first that since H is a normal p′-subgroup of G, it makes sense to refer to ξ as a block covered by b^{G} or b. Write C := (Q_{0} < ⋯ < Q_{n}). Assume that b^{G} covers ξ. Since ξ is G-stable, we have e_{b^{G}} · e_{ξ} = e_{b^{G}}. Also, as Q_{n}C_{G}(Q_{n}) ≤ N_{G}(C) ≤ N_{G}(Q_{n}), we have e_{b} · Br_{Q_{n}}(e_{b^{G}}) = e_{b}. Combining these two equalities along with the fact that Br_{Q_{n}}(e_{ξ}) = e_{ξ} we get:

e_{b} · e_{ξ} = e_{b} · Br_{Q_{n}}(e_{b^{G}}) · e_{ξ} = e_{b} · Br_{Q_{n}}(e_{b^{G}} · e_{ξ}) = e_{b} · Br_{Q_{n}}(e_{b^{G}}) = e_{b} ≠ 0.

Therefore, b covers ξ. Conversely, if b covers ξ and e_{b^{G}} · e_{ξ} = 0, then e_{b} · e_{ξ} = e_{b} · Br_{Q_{n}}(e_{b^{G}} · e_{ξ}) = 0, a contradiction. Hence, b^{G} covers ξ. ■

For a central subgroup Z of G, we may write Z = Z_{p} × Z_{p′}. Therefore, for any irreducible character λ of Z, there exist unique irreducible characters λ_{p} ∈ Irr(Z_{p}) and λ_{p′} ∈ Irr(Z_{p′}) such that λ = λ_{p} × λ_{p′}.

Lemma 2.2 Let Z ≤ Z(G), λ ∈ Hom(Z, K^{×}), C ∈ 𝒩(G) and let χ ∈ Irr(N_{G}(C)). The following statements are equivalent:

(a) χ lies over λ,

(b) χ lies over λ_{p} and λ_{p′}. Moreover, either of the above implies the following equivalent statements:

(c) b(χ) covers λ_{p′},

(d) b(χ)^{G} covers λ_{p′}.

Proof: That (a) is equivalent to (b) follows from the usual Clifford Theory for characters (see Theorem 6.2 in [9]). If χ lies over λ_{p′}, then b(χ) covers λ_{p′} and, by the previous lemma, (c) is equivalent to (d). ■2

The above lemma implies that if χ ∈ Irr(N_{G}(C)) such that b(χ)^{G} does not cover λ_{p′}, then χ does not lie over λ_{p′}. We summarize this in the following:

Proposition 2.3 Let Z ≤ Z(G), λ ∈ Hom(Z, K^{×}), B ∈ Bl(G) such that B does not lie over λ_{p′}. Then for any C ∈ 𝒩(G) we have k(N_{G}(C), B, λ) = 0.

The previous proposition implies that, for the purposes of (2), we may assume B covers λ_{p′}. With this assumption, Lemma 2.2 implies that χ lies over λ if and only if χ lies over λ_{p}. Therefore, without loss of generality, we may assume Z = Z_{p}, i.e., that Z is a p-group. We will assume this for the rest of the paper. Next, we will show that λ may be assumed to be faithful. In this case, we have an injective homomorphism of Z into K^{×} and hence Z is cyclic.

Let N ⊴ G and set \overline{G} = G/N. Recall that a block \overline{B} ∈ Bl(\overline{G}) is dominated by the block B ∈ Bl(G) if Irr(\overline{B}) ⊆ Irr(B) (where we identify Irr(\overline{G}) ⊆ Irr(G)). For the properties of block domination, we refer the reader to §5.8 in [14].

The following proposition is a slightly more general statement than Proposition 3.4 of [7]. We offer a proof for the convenience of the reader.

Proposition 2.4 Let N ⊴ G such that domination gives a bijection between Bl(G/N) and Bl(G). For a subgroup N ≤ H ≤ G let \overline{b} ∈ Bl(H/N), b ∈ Bl(H) such that b dominates \overline{b} and such that b^{G} and \overline{b}^{G/N} are defined. Then b^{G} dominates \overline{b}^{G/N}.

Proof: Let s_{H} : Z(F G) ⟶ Z(F H) denote the F -homomorphism given by C^{+} ↦ (C ∩ H)^{+} where C denotes a conjugacy class of G and C^{+} the corresponding class sum. Also, for any block A, let ω_{A} denote the central character corresponding to A and for any subgroup L of G containing N, let π_{N}^{L} : F L ⟶ F [L/N], x ↦ xN denote the domination map. By the definition of block induction, ω_{b^{G}} = ω_{b} ∘ s_{H} and ω\overline{b}_{G/N} = ω\overline{b} ∘ s_{H/N}. By assumption, ω_{b} = ω\overline{b} ∘ π_{N}^{H}. Putting this together, along with the easy identity π_{N}^{H} ∘ s_{H} = s_{H/N} ∘ π_{N}^{G} gives:

ω_{b^{G}} = ω_{b} ∘ s_{H} = ω\overline{b} ∘ π_{N}^{H} ∘ s_{H} = ω\overline{b} ∘ s_{H/N} ∘ π_{N}^{G} = ω\overline{b}_{G/N} ∘ π_{N}^{G},

and hence, b^{G} dominates \overline{b}^{G/N}. ■

The following useful lemma is elementary and its proof is left as an exercise.

Lemma 2.5 Let A and B be groups and let α : A ⟶ B be a surjective group homomorphism. Let X be an A-set, Y a B-set and f : X ⟶ Y an injective α-map, i.e., f(a·x) = α(a)·f(x) for all a ∈ A and x ∈ X. Then for any x ∈ X, stab_{B}(f(x)) = α(stab_{A}(x)).

Lemma 2.6 Let Z be a central p-subgroup of G and Z_{0} ≤ Z. For any subgroup H of G, set \overline{H} := HZ_{0}/Z_{0}. For a chain C = (Q_{0} < ⋯ < Q_{n}) ∈ 𝒩(G, Z), let \overline{C} denote the quotient chain (\overline{Q_{0}} < ⋯ < \overline{Q_{n}}) ∈ 𝒩(\overline{G}, \overline{Z}). Then the map C ⟶ \overline{C} induces a length preserving bijection

𝒩(G, Z)/G ⟷ 𝒩(\overline{G}, \overline{Z})/\overline{G}.  (5)

Moreover, if C ∈ 𝒩(G, Z), then N\overline{G}(\overline{C}) = \overline{N_{G}(C)} and block domination gives bijections:

Bl(G) ⟷ Bl(\overline{G})
Bl(N_{G}(C)) ⟷ Bl(\overline{N_{G}(C)}) = Bl(N\overline{G}(\overline{C}))

Proof: The correspondence theorem, along with the fact that Z_{0} ≤ Z ≤ Q_{0} gives the bijection in (5). The second statement follows immediately from Lemma 2.5. As Z_{0} is a central p-subgroup of G and N_{G}(C), block domination gives the desired bijections (see Theorem 8.11 of Chapter 5 in [14]). ■

Proposition 2.7 Let Z be a central p-subgroup of G, Z_{0} ≤ Z, and for any subgroup H of G, set \overline{H} := HZ_{0}/Z_{0}. Choose C ∈ 𝒩(G, Z) and let \overline{C} ∈ 𝒩(\overline{G}, \overline{Z}) correspond to C under the bijection in (5). Then

k(N_{G}(C), B, λ, d, r, ℐ) = k(N\overline{G}(\overline{C}), \overline{B}, \overline{λ}, d − log_{p}(|Z_{0}|), r, ℐ)

where \overline{B} is the unique block of \overline{G} dominated by B and \overline{λ} is the character of \overline{Z} whose inflation is λ.

Proof: Let χ ∈ Irr(N_{G}(C) | λ) (the set of irreducible characters of N_{G}(C) lying over λ). Then ker(λ) ≤ ker(χ) and so χ is the inflation of some character of N\overline{G}(\overline{C}). Thus, inflation gives a bijection

Irr(N_{G}(C) | λ) ⟷ Irr(N\overline{G}(\overline{C}) | \overline{λ}).

Moreover, this bijection respects block domination. That is, inflation gives a bijection

Irr(b | λ) ⟷ Irr(\overline{b} | \overline{λ})

where b ∈ Bl(N_{G}(C)) and \overline{b} ∈ Bl(N\overline{G}(\overline{C})) such that b dominates \overline{b}.

By Proposition 2.4, b(χ)^{G} = B if and only if b(\overline{χ})\overline{G} = \overline{B}. Also, the equalities

\frac{|N\overline{G}(\overline{C})|}{\overline{χ}(1)} = \frac{|\overline{N_{G}(C)}|}{\overline{χ}(1)} = \frac{|N_{G}(C)|}{|Z_{0}|χ(1)} = \frac{1}{|Z_{0}|} · \frac{|N_{G}(C)|}{χ(1)}

along with the fact that Z_{0} is a p-subgroup imply that d(\overline{χ}) = d(χ) − log_{p}(|Z_{0}|) and r(\overline{χ}) = r(χ). Finally, since χ(g) = \overline{χ}(Z_{0}g) for all g ∈ N_{G}(C), if σ ∈ 𝒢, then σχ = χ if and only if ^{σ}\overline{χ} = \overline{χ} and hence stab_{𝒢} (χ) = stab_{𝒢} (\overline{χ}). ■

Remark 2.8 Combining (5) and Proposition 2.7, we obtain

∑_{C∈𝒩(G,Z)_{≥n}/G} (−1)^{|C|−n}k(N_{G}(C), B, λ, d, r, ℐ)

= ∑_{\overline{C}∈𝒩(\overline{G},\overline{Z})_{≥n}/\overline{G}} (−1)^{|\overline{C}|−n}k(N\overline{G}(\overline{C}), \overline{B}, \overline{λ}, d − log_{p}(|Z_{0}|_{p}), r, ℐ)

In particular, we may always assume that λ is faithful and that Z is cyclic. In the next proposition, we will show that the value of the above sum is independent of the chosen faithful linear character of Z.

Proposition 2.9 Let Z ≤ Z(G) be a cyclic p-subgroup, ϑ_{1}, ϑ_{2} ∈ Hom(Z, K^{×}) faithful, B ∈ Bl(G), d ∈ ℕ_{0}, r ∈ {1, …, p − 1}, ℐ ≤ 𝒢. Then, for any C ∈ 𝒩(G, Z), there is a bijection, induced by a Galois automorphism, between

Irr(N_{G}(C), B, ϑ_{1}, d, r, ℐ) ⟷ Irr(N_{G}(C), B, ϑ_{2}, d, r, ℐ).

Proof: For any natural number n, let ω_{n} denote a primitive n-th root of unity. It is well-known that Gal(ℚ(ω_{|G|_{p}})/ℚ) ≅ Gal(ℚ(ω_{|G|})/ℚ(ω_{|G|_{p′}})) ≅ Gal(ℚ_{p}(ω_{|G|})/ℚ_{p}(ω_{|G|_{p′}})) acts transitively on the set of faithful irreducible characters of Z. Via elementary Galois theory, we may then view this as a transitive action of 𝒢 on the faithful irreducible characters of Z. In particular, there is a τ ∈ 𝒢 such that ^{τ} ϑ_{1} = ϑ_{2}. As the p′-roots of unity are fixed by τ, so are the blocks of G. If χ ∈ N_{G}(C), then the degree of χ is preserved by τ and since on the character values τ commutes with the elements of 𝒢, the 𝒢-stabilizer of χ is also preserved. Thus, conjugation by τ gives the desired bijection. ■

So far we have reduced the situation to the one in which Z is a central cyclic p-group. Next we will show that, in this case, conjecture (2) holds for all Z satisfying Z < O_{p}(G) and hence we may assume that G is a group where O_{p}(G) is central, cyclic and Z = O_{p}(G). To this end, we first recall some of the ideas developed by Boltje in [1].

Definition 2.10 Let A be an abelian group, A_{≥0} ⊆ A a submonoid. We call A_{≥0} a positivity area of A if A_{≥0} ∩ −A_{≥0} = {0}. In this case, if (a_{n})_{n∈ℤ} is a sequence in A such that a_{n} = 0 for all but finitely many n ∈ ℤ we call (a_{n})_{n∈ℤ} A_{≥0}-contractible if for all n ∈ ℤ, there exists x_{n} ∈ A_{≥0} such that a_{n} = x_{n}+x_{n−1}.

Remark 2.11 We now recall some properties of contractible sequences.

(a) Let A and B be abelian groups with positivity areas A_{≥0} and B_{≥0}, respectively. Also, let f ∈ Hom(A, B) such that f(A_{≥0}) ⊆ B_{≥0} and assume that (a_{n})_{n∈ℤ} is an A_{≥0}-contractible sequence in A. Then (f(a_{n}))_{n∈ℤ} is a B_{≥0}- contractible sequence in B.

(b) Let A be an abelian group, A_{≥0} ⊆ A a positivity area and (a_{n})_{n∈ℤ} a finite sequence in A_{≥0}. Then (a_{n})_{n∈ℤ} is A_{≥0}-contractible if and only if for each n ∈ ℤ one has a_{n} − a_{n+1} + a_{n+2} − + … ∈ A_{≥0}. □

The two examples with which we will most often deal are the integers with positivity area ℕ_{0} and the Burnside ring, Ω(X), of a finite group X with positivity area the classes of X-sets, which we will denote by Ω(X)_{≥0}. For an X-set S, [S] will denote the (X-set) isomorphism class of S.

Proposition 2.12 Let X be a finite group with O_{p}(X) > 1. Then ([𝒩(X)_{n}])_{n∈ℤ} is Ω(X)_{≥0}-contractible.

Proof: See Proposition 4.3 from [1]. ■

Corollary 2.13 Let Z be a central p-subgroup of G. If Z < O_{p}(G), then (a) ([𝒩(G/Z)_{n}])_{n∈ℤ} is Ω(G/Z)_{≥0}-contractible.

(b) ([𝒩(G, Z)_{n}])_{n∈ℤ} is Ω(G)_{≥0}-contractible.

Proof: (a) Since O_{p}(G/Z) = O_{p}(G)/Z > 1, the result follows immediately from Proposition 2.12.

(b) The inflation map inf : Ω(G/Z) ⟶ Ω(G) is a group homomorphism and clearly inf(Ω(G/Z)_{≥0}) ⊆ Ω(G)_{≥0}. Therefore, by part (a) and Remark 2.11(a), ([𝒩(G, Z)_{n}])_{n∈ℤ} = ([inf(𝒩(G/Z)_{n})])_{n∈ℤ} is Ω(G)_{≥0}-contractible. ■

Proposition 2.14 Let G be a finite group, p a prime, Z ≤ Z(G) a cyclic psubgroup such that Z < O_{p}(G), λ ∈ Hom(Z, K^{×}) faithful, B ∈ Bl(G), d ∈ ℕ_{0}, r ∈ {1, …, p − 1}, n ∈ ℤ, ℐ ≤ 𝒢. Then

∑_{C∈𝒩(G,Z)_{≥n}/G} (−1)^{|C|−n}k(N_{G}(C), B, λ, d, r, ℐ) ≥ 0.

Proof: Define a group homomorphism

κ : Ω(G) ⟶ ℤ, [G/H] ↦ \begin{cases}k(H, B, λ, d, r, ℐ), & if H = N_{G}(C) for some C ∈ 𝒩(G, Z) \\ 0, & otherwise. \end{cases}

Then κ(Ω(G)_{≥0}) ⊆ ℕ_{0} and so Remark 2.11 and Corollary 2.13(b) imply that (κ([𝒩(G, Z)_{m}]))_{m∈ℤ} is ℕ_{0}-contractible. Hence,

0 ≤ ∑_{m≥n}(−1)^{m−n}κ([𝒩(G, Z)_{m}])
= ∑_{m≥n}(−1)^{m−n}κ(  ∑_{C∈𝒩(G,Z)_{m}/G} [G/N_{G}(C)])
= ∑_{m≥n}(−1)^{m−n} ∑_{C∈𝒩(G,Z)_{m}/G} κ([G/N_{G}(C)])
= ∑_{C∈𝒩(G,Z)_{≥n}/G} (−1)^{|C|−n}k(N_{G}(C), B, λ, d, r, ℐ)

= ∑_{C∈𝒩(G,Z)_{≥n}/G} (−1)^{|C|−n}k(N_{G}(C), B, λ, d, r, ℐ),

and this completes the proof. ■

Putting together Propositions 2.3, 2.7 and 2.14, we see that to prove (2), it is enough to show that

∑_{C∈𝒩(G,Z)_{≥n}/G} (−1)^{|C|−n}k(N_{G}(C), B, λ, d, r, ℐ) ≥ 0  (6)

whenever G is a finite group, p a prime such that Z = O_{p}(G) ≤ Z(G) is cyclic, λ is a faithful linear character of Z, B is a block of G whose defect groups properly contain Z, d, n ∈ ℤ, r ∈ {1, …, p − 1} and ℐ ≤ 𝒢.

Before proving Theorem 1 in [18], Robinson introduces a weak version of Dade’s projective conjecture that does not involve the linear character λ. We will now proceed in this same direction, generalizing Robinson’s result to suit our needs.

Proposition 2.15 Let G be a finite group, p a prime, Z ≤ Z(G) a cyclic psubgroup such that Z = O_{p}(G), λ ∈ Hom(Z, K^{×}) faithful, B ∈ Bl(G), d ∈ ℕ_{0}, r ∈ {1, …, p − 1}, ℐ ≤ 𝒢. Moreover, assume that

∑_{\overline{C}∈𝒩(\overline{G},\overline{Z})/\overline{G}} (−1)^{|\overline{C}|}k(N\overline{G}(\overline{C}), \overline{B}, d′, r′, ℐ′) = 0

whenever d′ ∈ ℕ_{0}, r′ ∈ \{1, …, p − 1\}, ℐ′ ≤ 𝒢, \overline{G} = G/Z_{0} for some subgroup Z_{0} of Z and \overline{B} corresponds to B via the bijection in Lemma 2.6. Then

∑_{C∈𝒩(G,Z)/G} (−1)^{|C|}k(N_{G}(C), B, λ, d, r, ℐ) = 0.

Proof: We consider the case where Z_{0} = 1. By assumption,

0 = ∑_{C∈𝒩(G,Z)/G} (−1)^{|C|}k(N_{G}(C), B, d, r, ℐ)

= ∑_{ϑ∈Irr(Z)} ∑_{C∈𝒩(G,Z)/G} (−1)^{|C|}k(N_{G}(C), B, ϑ, d, r, ℐ)

= \underbrace{∑_{ϑ,non-faithful} ∑_{C∈𝒩(G,Z)/G} (−1)^{|C|}k(N_{G}(C), B, ϑ, d, r, ℐ)}_{(∗)}

+ \underbrace{∑_{ϑ,faithful} ∑_{C∈𝒩(G,Z)/G} (−1)^{|C|}k(N_{G}(C), B, ϑ, d, r, ℐ)}_{(∗∗)}

If ϑ ∈ Irr(Z) is non-faithful, then there exists an element of order p in ker(ϑ). In particular, since Z is cyclic, Ω_{1}(Z), the subgroup of Z generated by the elements of Z of order p, is non-trivial and contained in ker(ϑ). Setting \overline{G} := G/Ω_{1}(Z), if follows by induction on |G| and Remark 2.8 that

0 = ∑_{\overline{C}∈𝒩(\overline{G},\overline{Z})/\overline{G}} (−1)^{|\overline{C}|}k(N\overline{G}(\overline{C}), \overline{B}, \overline{ϑ}, d − log_{p}(|Ω_{1}(Z)|), r, ℐ)

= ∑_{C∈𝒩(G,Z)/G} (−1)^{|C|}k(N_{G}(C), B, ϑ, d, r, ℐ).

Summing over all non-faithful irreducible characters of Z, we conclude that (∗) = 0 and consequently that (∗∗) = 0. The result now follows from Proposition 2.9. ■

3 The Stable and Unstable Reductions

Throughout this section, G is a finite p-solvable group, Z := O_{p}(G) ≤ Z(G), λ ∈ Hom(Z, K^{×}) is faithful, B ∈ Bl(G) such that Z is not a defect group of B, n, d ∈ ℤ, r ∈ {1, …, p − 1} and ℐ ≤ 𝒢. Finally, set N := O_{p′}(G) (the largest normal p′-subgroup of G).

We will follow, quite closely, the proof of Robinson in [18]. One immediate difference is that Robinson takes N to be any normal p′-subgroup of G properly containing O_{p′}(Z(G)), whereas we take N = O_{p′}(G). This is done so that we may employ a theorem of Külshammer and avoid checking whether Brauer correspondence commutes with Fong correspondence, a crucial step in Robinson’s paper.

As (6) holds vacuously for abelian groups, we have the base case for our proof, which will be by induction on the index of Z(G) in G. Moreover, if N ≤ Z(G), then G is abelian (see Lemma 30.1 in [19]), so we assume that N ≰ Z(G).

Choose μ ∈ Irr(N) covered by B. We first consider the case where μ is G-stable (under conjugation). Külshammer proved (see Proposition J in [12]) that e_{μ} is a block idempotent for G and so that, in particular, B covers μ if and only if e_{B} = e_{μ}. If e_{B} = e_{μ}, then, of course, μ is the unique G-stable irreducible character of N covered by B. Furthermore, Külshammer showed that, in this case, every Sylow p-subgroup of G is a defect group of B. An argument of Alperin (see Theorem 5.12.1 in [14]) shows that Brauer’s first main theorem induces a bijection between the Q-stable irreducible characters of N (for any psubgroup Q) and the irreducible characters of C_{N}(Q). The irreducible character of C_{N}(Q) corresponding to μ will be called the Glauberman correspondent of μ with respect to Q.

Proposition 3.1 Let C = (Q_{0} < ⋯ < Q_{n}) ∈ 𝒩(G), and let θ ∈ Irr(C_{N}(Q_{n})) denote the Glauberman correspondent of μ with respect to Q_{n} (i.e., e_{θ} = Br_{Q_{n}}(e_{μ})). Then:

(a) O_{p′}(N_{G}(C)) = C_{N}(Q_{n}).

(b) O_{p′}(N · N_{G}(C)) = N.

(c) There exists a unique block b of N_{G}(C) covering θ. Moreover, b^{G} = B.

(d) N_{N·N_{G}(C)}(Q_{n}) = N_{G}(C).

(e) There exists a unique block of N · N_{G}(C) covering μ, namely b^{N·N_{G}(C)}.

(f) b and b^{N·N_{G}(C)} have the same defect groups.

Proof: (a) Clearly C_{N}(Q_{n}) ⊴ N_{G}(C) and so C_{N}(Q_{n}) ≤ O_{p′}(N_{G}(C)). To show the reverse inclusion, note first that as O_{p′}(N_{G}(C)), Q_{n} ⊴ N_{G}(C) have coprime order, O_{p′}(N_{G}(C)) centralizes Q_{n}. Thus, it remains to show that O_{p′}(N_{G}(C)) ≤ N. We prove this by induction on n, the length of the chain C. If n = 0 then we may apply Proposition X.1.6 in [8]. Now, by induction O_{p′}(N_{G}(C)) = O_{p′}(N_{N_{G}(Q_{0}<···<Q_{n−1})}(Q_{n})) ≤ O_{p′}(N_{G}(Q_{0} < ⋯ < Q_{n−1})) ≤ N.

(b) Since N is a p′-group, 1 = O_{p′}(N_{G}(C))/C_{N}(Q_{n}) = O_{p′}(N_{G}(C)/C_{N}(Q_{n})) = O_{p′}(N_{G}(C)/N ∩ N_{G}(C)) ≅ O_{p′}(N · N_{G}(C)/N) = O_{p′}(N · N_{G}(C))/N and so O_{p′}(N · N_{G}(C)) = N.

(c) Since μ is G-stable, it follows that θ is N_{G}(C)-stable. By (a) and the above result of Külshammer, there exists a unique block b of N_{G}(C) covering θ and satisfying e_{b} = e_{θ}. Moreover, as Q_{n}C_{G}(Q_{n}) ≤ N_{G}(C) ≤ N_{G}(Q_{n}), it follows that b^{G} is defined and b^{G} = B if and only if Br_{Q_{n}}(e_{B})e_{b} ≠ 0. But Br_{Q_{n}}(e_{B})e_{b} = Br_{Q_{n}}(e_{μ})e_{θ} = e_{θ}e_{θ} = e_{θ} ≠ 0.

(d) We have N_{N·N_{G}(C)}(Q_{n}) = N_{G}(Q_{n}) ∩ N · N_{G}(C) and since N_{G}(C) ≤ N_{G}(Q_{n}), Dedekind’s modular law implies that N_{G}(Q_{n})∩N ·N_{G}(C) = (N_{G}(Q_{n})∩ N)N_{G}(C) = N_{N}(Q_{N}) · N_{G}(C). As N is a normal p′-subgroup, any element of N that normalizes Q_{n} necessarily centralizes Q_{n}. Hence, N_{N}(Q_{n}) · N_{G}(C) = C_{N}(Q_{N}) · N_{G}(C) = N_{G}(C) and the proposition is proved.

(e) By part (b) and Külshammer’s result, there exists a unique block of N · N_{G}(C) covering μ (with block idempotent e_{μ}). Since Q_{n}C_{N·N_{G}(C)}(Q_{n}) ≤ N_{G}(C) = N_{N·N_{G}(C)}(Q_{n}), we may apply the same argument that was used in (c) to see that the block obtained is b^{N·N_{G}(C)}.

(f) This follows immediately from the Külshammer result since the Sylow p-subgroups of N_{G}(C) and N · N_{G}(C) are the same. ■

We summarize the above proposition in the following diagram:

[figure: b^{G} = B, e_{μ}; G; b^{N·N_{G}(C)}, e_{μ}; N · N_{G}(C); N_{G}(C) = N_{N·N_{G}(C)}(Q_{n}); b, e_{θ}; e_{μ}; N · Q_{n}; Q_{n} · C_{N}(Q_{n}); e_{θ}; μ; N; C_{N}(Q_{n}); θ; induction; BFMT; Glauberman]

Next, we need a theorem of Dade to make the transition from counting characters of N_{G}(C) to counting characters of N · N_{G}(C). Reference to this statement first appeared in the introduction to [3]. The first formal statement occurred in [4] where an outline of a proof is given. The key step in the proof can be found in [17] as well as in [2]. A full proof can be found in [13]. Before stating the theorem, we set up a little notation. Let H be a finite group and M a normal p′-subgroup of H. For any Φ ∈ Irr(M) that is H-stable and any p-subgroup Q of H, Q is a defect group of the unique block of MQ covering Φ. Denote by ϕ the Glauberman correspondent of Φ in C_{M}(Q).

Theorem 3.2 (Dade’s Theorem) Assume MQ ⊴ H. Then

Mat_{ϕ(1)}(e_{Φ}RH) ≅ Mat_{Φ(1)}(e_{ϕ}RN_{H}(Q)) (as R-algebras).

In particular, e_{Φ}RH is Morita equivalent to e_{ϕ}RN_{H}(Q) and so induces a bijection

Irr(H | Φ) ˜ ⟷ Irr(N_{H}(Q) | ϕ).

Lemma 3.3 Φ(1) ≡ ±ϕ(1) (mod p).

Proof: By Glauberman’s theorem, ϕ is the unique irreducible character of C_{M}(Q) such that p ∤ (res_{C_{M} (Q)}^{M}(Φ), ϕ) and (res_{C_{M} (Q)}^{M}(Φ), ϕ) ≡ ±1 (mod p). Therefore, if Irr(C_{M}(Q)) = {ϕ, ϕ_{1}, …, ϕ_{t}) and res_{C_{M} (Q)}^{M}(Φ) = (res_{C_{M} (Q)}^{M}(Φ), ϕ)·

ϕ + ∑_{i=1}^{t} a_{i} · ϕ_{i}, then Φ(1) = (res_{C_{M} (Q)}^{M}(Φ), ϕ) · ϕ(1) + ∑_{i=1}^{t} a_{i} · ϕ_{i}(1) ≡ ±ϕ(1) (mod

p). ■

Corollary 3.4 If χ ∈ Irr(H | Φ) and θ ∈ Irr(N_{H}(Q) | ϕ) correspond via the bijection in Dade’s theorem, then d(χ) = d(θ) and r(χ) ≡ ±r(θ) (mod p).

Proof: The isomorphism of algebras in Dade’s theorem implies that θ(1)· \frac{Φ(1)}{ϕ(1)} = χ(1). The result now follows from Lemma 3.3. ■

Proposition 3.5 Let G be a finite p-solvable group for some prime p, d ∈ ℤ, r ∈ {1, … p − 1} and C ∈ 𝒩(G). If μ is a G-stable irreducible character of N := O_{p′}(G) covered by B ∈ Bl(G), then there is a bijection:

Irr(N_{G}(C), B, d, r) ˜ ⟷ Irr(N · N_{G}(C), B, d, r).

Proof: This follows immediately from Proposition 3.1(c,d,e), Dade’s theorem and Corollary 3.4. ■

From here, we make use of the Fong correspondence for stable blocks, first studied by Fong in [6]. Though the results are well known, we give a brief summary both for the convenience of the reader and since we need to work in a slightly different situation as that which is normally found in the literature.

Definition 3.6 (a) A Fong-tuple is a triple (G, N, μ) where G is a finite group, N a normal p′-subgroup of G and μ a G-stable irreducible character of N.

(b) Let (G, N, μ) be a Fong-tuple. A Fong correspondent of (G, N, μ) is a 5-tuple (G̃, Ñ, μ̃, α, β) where

• G̃ is a central extension of G/N by a cyclic p′-group Ñ

• μ̃ is a G̃-stable irreducible character of Ñ

• α : G/N ⟶ G̃/Ñ is a group isomorphism

• β : RGe_{μ} ⟶ b ⊗_{R} RG̃eμ̃ is an α-graded R-algebra isomorphism.

Let bars denote the images of elements or subsets of G under the canonical epimorphism ν : G ↠ G/N. If H ≤ G, then the correspondence theorem gives a subgroup of G̃ containing Ñ corresponding to α(\overline{H}). We denote this subgroup by H̃.

Remark 3.7 (a) RGe_{μ} is a G/N-crossed product and RG̃eμ̃ is a G̃/Ñ-crossed

product. Note that RNe_{μ} ⊗_{R} RG̃eμ̃ = ⨁_{g̃Ñ∈G̃/Ñ} RNe_{μ} ⊗_{R} R[g̃Ñ]eμ̃ and so

RNe_{μ} ⊗_{R} RG̃eμ̃ is a G̃/Ñ-crossed product. Typically, the map β is only assumed to be an R-algebra isomorphism. However, if we take, for instance, the map constructed in Chapter 5.7 of [14], we see that β is even α-graded.

(b) It is important to note that there may be more than one Fong correspondent for a Fong-tuple. However, Fong proved that for any Fong-tuple there exists a Fong correspondent.

For the following theorem, let ℤ_{(p)} denote the localization of the integers at the prime p and for a natural number n and a ring S, let M_{n}(S) denote the complete n × n-matrix ring with coefficients in S.

Theorem 3.8 (Fong Correspondence) Let (G̃, Ñ, μ̃, α, β) be a Fong correspondent for the Fong-tuple (G, N, μ). Then RGe_{μ} is Morita equivalent to RG̃eμ̃. Furthermore, there is a bijection between the blocks of G covering μ and the blocks of G̃ covering μ̃ and corresponding blocks are Morita equivalent.

In particular, if a block summand B of RGe_{μ} corresponds to the block summand B̃ of RG̃eμ̃, then there is a bijection

Irr(G, B, d, r | μ) ˜ ⟷ Irr(G̃, B̃, d, r̃ | μ̃),

where r̃ is the unique element in \{1, …, p − 1\} such that r̃ ≡ μ(1)\frac{|Ñ|}{|N|} · r (mod pℤ_{(p)}).

Proof: By definition, RGe_{μ} ≅ RNe_{μ} ⊗_{R} RG̃eμ̃. As RNe_{μ} ≅ M_{μ(1)}(R), we have RGe_{μ} ≅ M_{μ(1)}(R) ⊗_{R} RG̃eμ̃ ≅ M_{μ(1)}(RG̃eμ̃) which is Morita equivalent to RG̃eμ̃. With this, only the last sentence of the proposition requires any explanation. Let χ ∈ Irr(B) correspond to χ̃ ∈ Irr(B̃) and set μ(1) = m. Then χ(1) = mχ̃(1). We thus have

\frac{|G|}{χ(1)} = \frac{|N| · |G/N|}{mχ̃(1)} = \frac{|N| · |G̃|/Ñ|}{mχ̃(1)} = \frac{|N|}{m|Ñ|} · \frac{|G̃|}{χ̃(1)}.

As p ∤ |N| and p ∤ m · |Ñ|, it follows that d(χ̃) = d(χ) and r(χ̃) = \frac{m|Ñ|}{|N|}r(χ). ■

As we will be interested in a variety of subgroups of our group G, it behooves us to examine how the Fong correspondence behaves with respect to restriction. For any subgroup X of G containing N, let ℱ_{X} denote the category equivalence between _{RXeμ} mod and _{RX̃eμ̃} mod constructed above (as in [14]).

Proposition 3.9 Let (G̃, Ñ, μ̃, α, β) be a Fong correspondent for the Fong-tuple (G, N, μ) and let H be a subgroup of G containing N. Then:

(a) (H, N, μ) is a Fong-tuple.

(b) (H̃, Ñ, μ̃, α|\overline{H}, β|_{RHe_{μ}}) is a Fong correspondent of (H, N, μ).

(c) The diagram

[figure: _{RHeμ} mod; _{RH̃eμ̃} mod; _{RUeμ} mod; _{RŨeμ̃} mod; ℱ_{H}; Res_{U}^{H}; Res_{U}^{H}; ℱ_{U}]

commutes for all N ≤ U ≤ H ≤ G. In particular, ℱ_{N}(μ) = μ̃.

Proof: (a) This statement is trivial.

(b) All of the conditions are easily checked, the only one of much interest being the last one, namely that the restriction of β to RHe_{μ} has as its image RNe_{μ} ⊗_{R} RH̃eμ̃ and this follows immediately from β being α-graded.

(c)The first statement follows easily from the fact that β_{H} (respectively, β_{U}) is α|\overline{H}-graded (respectively, α|\overline{U}-graded). The second by Theorem 3.8. ■

Our next goal is to use the Fong correspondence to pass from N · N_{G}(C) to NG̃(C̃) (here C̃ corresponds to C via the bijection (7) below).

[figure: B̃, G̃; G, B; b̃, NG̃(C̃); N · N_{G}(C), b^{N·N_{G}(C)}; N_{G}(C), b; Fong; induction; Fong; induction; Dade]

Note that (G, N, μ) (as defined in 6.3) is a Fong-tuple. Fix a Fong correspondent (G̃, Ñ, μ̃, α, β) of (G, N, μ).

Robinson proved in [18] that for any finite group G with normal p′-subgroup N and central p-subgroup Z, the map

C = (Q_{0} < ⋯ < Q_{n}) ↦ \overline{C} = (\overline{Q_{0}} < ⋯ < \overline{Q_{n}})

induces a length preserving bijection

𝒩(G, Z)/G ˜ ⟷ 𝒩(\overline{G}, \overline{Z})/\overline{G}.

Moreover, for any C ∈ 𝒩(G, Z), one has N\overline{G}(\overline{C}) = \overline{N · N_{G}(C)}. This bijection, along with the group isomorphism α : G/N ⟶ G̃/Ñ induces a length preserving bijection

𝒩(G, Z)/G ˜ ⟷ 𝒩(G̃, Z̃)/G̃.  (7)

Lemma 3.10 Let C ∈ 𝒩(G, Z). Then { \atop NN_{G}(C)} = NG̃(C̃).

Proof: Let ν̃ : G̃ ↠ G̃/Ñ denote the canonical epimorphism and let bars denote the images of subsets of G in G/N. By the previous remark, we have

{ \atop NN_{G}(C)} = ν^{−1}(α(\overline{NN_{G}(C)})) = ν^{−1}(α(N\overline{G}(\overline{C})))) = NG̃(C̃).■

Recall that μ ∈ Irr(N) and λ ∈ Irr(Z). Thus, μ × λ ∈ Irr(NZ). By Proposition 3.9 and the Fong correspondence, we get a corresponding irreducible character { \atop μ × λ} ∈ Irr(ÑZ). Let Z̃ ∈ Syl_{p}(ÑZ). Then ÑZ = Ñ × Z̃ and since

{ \atop μ × λ} lies over μ̃, there exists a unique irreducible character λ̃ ∈ Irr(Z̃) such that { \atop μ × λ} = μ̃×λ̃. We will say that λ and λ̃ correspond via the Fong correspondence. Finally, inspired by Theorem 3.8, we set r̃ = μ(1)\frac{|Ñ|}{|N|} · r.

We summarize our situation in the following diagram:

B̃, eμ̃ G̃ G/N G B, e_{μ}

b̃, eμ̃ NG̃(C̃) N · N_{G}(C) b^{N·N_{G}(C)}, e_{μ}

{ \atop μ × λ}, ÑZ NZ, μ × λ N_{G}(C) b, e_{μ}

μ̃ Ñ N, μ Z = O_{p}(G), cyclic λ

λ̃ Z̃

˜1

1

[figure]

[figure]

[figure]

p′,central, cyclic

O_{p′} (G)

cyclic, p−group

The following proposition is now an immediate consequence of the Fong correspondence and Proposition 3.9.

Proposition 3.11 Let C ∈ 𝒩(G, Z), χ ∈ Irr(N · N_{G}(C), B, λ, d, r) and let χ̃ denote the character of NG̃(C̃) corresponding to χ via the Fong correspondence. Then χ̃ ∈ Irr(NG̃(C̃), B̃, λ̃, d, r̃) where B̃, λ̃ and r̃ are given by the Fong correspondence. In particular, there is a bijection

Irr(N · N_{G}(C), B, λ, d, r) ˜ ⟷ Irr(NG̃(C̃), B̃, λ̃, d, r̃).

Corollary 3.12 Let C ∈ 𝒩(G, Z). Then there is a bijection

Irr(N_{G}(C), B, d, r) ˜ ⟷ Irr(NG̃(C̃), B̃, d, r̃)

which only depends on N_{G}(C).

Proof: This is just the composition of the bijections from Proposition 3.5 and Proposition 3.11 where we omit the parameter λ. ■

Combining Corollary 3.12 with the bijection of (7) gives the following theorem.

Theorem 3.13 With the notation as above, one has

∑_{C∈𝒩(G,Z)_{≥n}} (−1)^{|C|−n}k(N_{G}(C), B, d, r)

= ∑_{C̃∈𝒩(G̃,Z̃)_{≥n}} (−1)^{|C̃|−n}k(NG̃(C̃), B̃, d, r̃).

Remark 3.14 The index of the center of G̃ in G̃ is strictly smaller than the index of the center of G in G. In fact, N · Z(G)/N ≤ Z(G/N) implies that { \atop N · Z(G)} ≤ Z(G̃) and so as N ≰ Z(G), we get

[G̃ : Z(G̃)] ≤ [G̃ : { \atop N · Z(G)}] = [G/N : N · Z(G)/N] ≤ [G : N · Z(G)] < [G : Z(G)].

Hence, by induction, we conclude that

∑_{C∈𝒩(G,Z)_{≥n}} (−1)^{|C|−n}k(N_{G}(C), B, d, r) ≥ 0.  (8)

Moreover, if we drop Boltje’s contractibility requirement, we may employ Robinson’s argument to suppress the λ, and we then obtain

∑_{C∈𝒩(G,Z)} (−1)^{|C|}k(N_{G}(C), B, λ, d, r) = 0.  (9)

It would, of course, be nice to write down the bijection in Corollary 3.12 with the λ included, as this would allow one to avoid using Robinson’s argument and, consequently, recover Boltje’s contractibility condition.

Given the above theorem, it remains to consider the situation where no irreducible character of O_{p′}(G) covered by the block B is G-stable. In this case, we may employ the results of Fong and Reynolds. Robinson deftly reduces the situation to one involving chains and normalizers of subpairs and we can hardly improve upon his presentation. The important point of his argument is that all of the reductions involve Morita equivalences that, on the character level, are given by induction. So, if we take Robinson’s arguments as given, then we may work in a more general context and prove the following proposition, which will, essentially, complete the proof of our main theorem. Recall that if X is a finite group, Y a normal subgroup of X with a block b that is not X-stable (under conjugation) and I denotes the subgroup of X that stabilizes b, then for any block β of I covering b the Fong-Reynolds correspondence (see Theorem 5.10 of Chapter 5 in [14]) says that induction gives a bijection

ind_{I}^{X} : Irr(β) ˜ ⟷ Irr(β^{X}).

We now show that our parameters are preserved by induction.

Proposition 3.15 Let X be a finite group, Y ⊴ X and let b ∈ Bl(Y). Set I := I_{G}(b). Let β be a block of I covering b. Fix χ ∈ Irr(β).

(a) d(χ) = d(ind_{I}^{X}(χ)) and r(χ) = r(ind_{I}^{X}(χ)).

(b) Let C ≤ Z(X) and φ ∈ Irr(C). Then χ lies over φ if and only if ind_{I}^{X}(χ) lies over φ.

(c) Let ℋ = stab_{𝒢} (β^{X}) ∩ stab_{𝒢} (β) ≤ 𝒢. Then stab_{ℋ} (χ) = stab_{ℋ} (ind_{I}^{X}(χ)).

Proof: (a) This follows immediately from the equalities:

\frac{|X|}{ind_{I}^{X}(χ)(1)} = \frac{|X|}{[X : I]χ(1)} = \frac{|I|}{χ(1)}.

(b) By the Mackey formula,

(res_{C}^{X}(ind_{I}^{X}(χ)), φ) = ∑_{t∈X/I}(res_{C}^{I^{t}}(χ^{t}), φ).

Since φ is a character for a central subgroup, the above sum is non-zero if and only if (res_{C}^{I}(χ), φ) ≠ 0.

(c) As ℋ acts on Irr(β) and Irr(β^{X}) and commutes with induction, Lemma 2.5 implies the result. ■

Using Proposition 3.15 and Robinson’s arguments we obtain a stronger statement than in the stable case.

Theorem 3.16 Let G be a finite group, p a prime, B ∈ Bl(G) such that Z := O_{p}(G) is a central cyclic subgroup but not a defect group of B, λ a faithful irreducible character of Z, d ∈ ℕ, r ∈ {1, …, p − 1}, n ∈ ℤ, ℐ ≤ 𝒢. Assume μ ∈ Irr(N) is covered by B and is not G-stable. Let I := I_{G}(μ) and let β ∈ Bl(I) be the Fong correspondent of B with respect to μ. Then

∑_{C∈𝒩(G,Z)_{≥n}} (−1)^{|C|−n}k(N_{G}(C), B, λ, d, r, ℐ)

= ∑_{C∈𝒩(I,Z)_{≥n}} (−1)^{|C|−n}k(N_{I}(C), β, d, r, ℐ).

Remark 3.17 As μ is not G-stable, we have Z(G) ≤ Z(I) ≤ I < G and so [I : Z(I)] < [G : Z(G)]. By induction, we therefore conclude that

∑_{C∈𝒩_{B}(G,Z)_{≥n}} (−1)^{|C|−n}k(N_{G}(C), B, λ, d, r, ℐ) ≥ 0.  (10)

With (6) in mind, combining (10) with (8) and (9) gives (3) and (4), respectively, and thus completes the proof of Theorem 1.1.

Acknowledgements

This work is based on my doctoral thesis at the University of California Santa Cruz. I would like to thank Robert Boltje for suggesting the problem and for his dedication in helping me work through it. I would also like to thank Burkhard Külshammer for the help he provided during my several visits to the University of Jena.

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