ON THE GLAUBERMAN AND WATANABE CORRESPONDENCES

FOR BLOCKS OF FINITE p−SOLVABLE GROUPS

M. E. Harris, M. Linckelmann

Abstract. If G is a finite p-solvable group for some prime p, A a solvable subgroup of the automorphism group of G of order prime to |G| such that A stabilises a p-block b of G and acts trivially on a defect group P of b, then there is a Morita equivalence between the block b and its Watanabe correspondent w(b) of C_{G}(A), given by a bimodule M with vertex ΔP and an endo-permutation module as source, which on the character level induces the Glauberman correspondence (and which is an isotypy by Watanabe’s results in [24]).

1 Introduction

The Glauberman correspondence [8] for a finite group G and a solvable group A of automorphisms of G of order prime to |G| is a bijection between the set of A−stable ordinary irreducible characters of G and the set of ordinary irreducible characters of the centraliser C = C_{G}(A) of A in G.

Watanabe showed in [24], that under suitable hypotheses, the Glauberman correspondence gives rise to isotypys - a concept due to Broué [2] - between A−stable blocks of G and C in characteristic prime to the order of A. What we want to show here is, that if G is p−solvable, the perfect isometries arising in this way are induced by Morita equivalences. In order to explain this more precisely, we introduce the following notation that we maintain throughout the paper.

We denote by O a complete discrete valuation ring having an algebraically closed residue field k of prime characteristic p and a quotient field K of characteristic zero which will always assumed to be large enough for any of the finite groups that we consider in the sequel.

If p does not divide the order of A and b is an A−stable block of OG having a defect group P contained in C, then by [24, Prop. 1], every ordinary irreducible character of G belonging to the block b is A−stable, and the main result of [24] shows then that the Glauberman correspondence maps the set Irr_{K}(G, b) onto the set Irr_{K}(C, w(b)) for a uniquely determined block w(b) of OC, which is then shown still to have P as defect group and a p−local structure equivalent to that of b. Together with the signs occurring in the Glauberman correspondence in [24], this yields an isotypy between the blocks b and w(b).

Typeset by AMS-TEX

2000 Mathematics Subject Classification 20C20

In the light of Broué’s theorem [2, 3.1] asserting that any derived equivalence between two block algebras induces a perfect isometry, and more precisely, that every splendid derived equivalence induces an isotypy, it is natural to ask whether there is a splendid derived equivalence between the blocks b and w(b) behind this isotypy. What we show in this paper is that if G is p−solvable, this isotypy is actually induced by a Morita equivalence between the block algebras of b and w(b) with an endopermutation source. Following [10], [20], such a Morita equivalence induces a splendid derived equivalence whenever the occurring endo-permutation source has a fusionstable p−permutation resolution. Our main result is a consequence of Puig’s structure theorem [19] on blocks of p−solvable groups, and has independently been observed by Puig himself. In the case where the defect groups are abelian, the existence of a Morita equivalence between the block algebras of b and w(b) over the residue field k has also been shown by Koshitani and Michler [13].

Theorem 1.1. Let G be a finite p−solvable group, A a solvable group of automorphisms of G of order prime to |G|p. Set C = C_{G}(A). Let b be an A−stable block of OG with a defect group P contained in C. Denote by w(b) the block of OC corresponding to b through the Glauberman correspondence.

There is a Morita equivalence between the block algebras OGb and OCw(b) given by an indecomposable OGb − OCw(b)−bimodule M with the following properties:

(i) When viewed as an O(G × C)−module, M has ΔP = {(u, u)}_{u∈P} as vertex and a fusion-stable endo-permutation OΔP −module W as source.

(ii) The bijection between the sets of ordinary irreducible characters Irr_{K}(G, b) and Irr_{K}(C, w(b)) induced by the Morita equivalence given by M is precisely the Glauberman correspondence.

(iii) If b is the principal block of G, then w(b) is the principal block of C and OGb = OCw(b).

“Fusion-stable” means that for a suitable choice of a maximal b−Brauer pair (P, e_{P}) we have Res_{R}^{P}(W) ≅ Res_{φ}(W) for any subgroup R of P and any group homomorphism φ : R → P for which there is x ∈ G satisfying φ(r) = xrx^{−1} for all r ∈ R and ^{x}(R, e_{R}) ⊂ (P, e_{P}), where e_{R} is the unique block of kC_{G}(R) such that (R, e_{R}) ⊂ (P, e_{P}). We freely use some standard results and concepts such as the Brauer correspondence for blocks of finite groups, relative traces (introduced by J. A. Green), the Brauer homomorphism, induction of interior algbras (due to L. Puig) and the inclusion of Brauer pairs (due to Alperin and Broué) which can, for instance, be found in J. Thévenaz’ book [23]. Recall that an interior P −algebra, where P is a finite group, is an O−algebra A endowed with a group homomorphism σ : P ⟶ A^{×}. The opposite algebra A^{0} of A is then considered as interior P −algebra via the group homomorphism sending u ∈ P to σ(u^{−1}).

Remark 1.2. It will be apparent from the proof of Theorem 1.1 that it suffices to require k to be perfect and large enough.

GLAUBERMAN CORRESPONDENCE FOR BLOCKS OF p−SOLVABLE GROUPS 3

2 Quoted results on character correspondences

We recall in this section the definition and main properties of the Glauberman correspondence, as well as Watanabe’s results in [24].

Theorem 2.1. (Glauberman [8], [11]) For every pair (G, A) consisting of a finite group G and a finite solvable group A acting on G such that G and A have coprime orders, there is a bijection π(G, A) : Irr_{K}(G)^{A} ⟶ Irr_{K}(C_{G}(A)) satisfying, for any such pair (G, A), the following properties:

(i) For any normal subgroup B of A, the bijection π(G, B) maps Irr_{K}(G)^{A} to Irr_{K}(C_{G}(B))^{A}, and we have π(G, A) = π(C_{G}(B), A/B) ∘ π(G, B) on Irr_{K}(G)^{A}.

(ii) If A is a q-group for some prime q, for any χ ∈ Irr_{K}(G)^{A} the corresponding irreducible character π(G, A)(χ) of C_{G}(A) is the unique irreducible constituent of Res_{C_{G}(A)}^{G}(χ) occurring with a multiplicity prime to q.

Since A is assumed to be solvable, the composition factors of A are cyclic of prime order. The strategy of the proof of Theorem 2 is to first show (ii) and then proceed inductively along a composition series of A (cf. [11,13.3]). For any χ ∈ Irr_{K}(G)^{A}, there is a unique character χ̂ to G ⋊ A extending χ such that A ⊂ ker(det(χ̂)), called the canonical extension of χ. If A is cyclic, its value χ̂(a) at any generator a of A does not depend on the choice of a, and this value is plus or minus the degree of the Glauberman correspondent of χ. In fact, more precisely, we have χ̂(ac) = ϵ_{χ}π(G, A)(χ)(c) for any generator a of A, any c ∈ C_{G}(A) and a sign ϵ_{χ} depending only on χ (cf. [11, 13.6]).

If in the situation of Theorem 2.1, C_{G}(A) contains a defect group of an A−stable block b of OG, Watanabe showed in [24], that the Glauberman correspondence induces an isotypy between b and some block w(b) of OC_{G}(A):

Theorem 2.2. (Watanabe [24, Theorem 1, Proposition 5]) Let G be a finite group, A a solvable subgroup of Aut(G) of order prime to |G| and set C = C_{G}(A). Let b be an A−stable block of OG having a defect group P contained in C.

(i) The group A acts trivially on Irr_{K}(G, b), and there is a unique block w(b) of OC such that the Glauberman correspondence π(G, A) maps Irr_{K}(G, b) onto Irr_{K}(C, w(b)).

(ii) The block w(b) has again P as defect group, and the Brauer categories of b, w(b) are equivalent.

(iii) There is an isotypy ℤIrr_{K}(G, b) ≅ ℤIrr_{K}(C, w(b)) mapping χ ∈ Irr_{K}(G, b) to ϵ_{χ}π(G, A)(χ).

Our approach will yield an alternative proof of Watanabe’s theorem in the case where G is p−solvable. We require the following induction and restriction theorem for the Glauberman correspondence, due to Isaacs and Navarro [12]. Since the results in [12] are in fact more general than what we need here, we include proofs for the convenience of the reader.

Theorem 2.3. ([12, Theorem A(a), Theorem (3.1)] Let G be a finite group, A a solvable subgroup of Aut(G) of order prime to |G|, and let H be an A−stable subgroup of G. Set C = C_{G}(A). Let ψ ∈ Irr_{K}(H)^{A} and χ ∈ Irr_{K}(G)^{A}.

(i) If Res_{H}^{G}(χ) is irreducible, then π(H, A)(Res_{H}^{G}(χ)) = Res_{C∩H}^{C}(π(G, A)(χ)) . (ii) If Ind_{H}^{G}(ψ) is irreducible, then π(G, A)(Ind_{H}^{G}(ψ)) = Ind_{C∩H}^{C}(π(H, A)(ψ)) .

Proof. If ψ is A−stable, so is Ind_{H}^{G}(ψ); similarly, if χ is A−stable, so is Res_{H}^{G}(χ). We proceed for both statements by induction on the order of A.

(i) Assume that A is cyclic of prime order q. Set β = Res_{H}^{G}(χ) and assume that β is irreducible. Denote by χ̂ and β̂ the canonical extensions of χ and β to G⋊A and H⋊A, respectively. Then Res_{H⋊A}^{G⋊A}(χ̂) = β̂. By [11, 13.6] there are signs ϵ, ϵ′ ∈ {+1, −1} such that, for any generator a ∈ A, we have χ̂(ac) = ϵπ(G, A)(χ)(c) for all c ∈ C and β̂(ad) = ϵ′π(H, A)(β)(d) for all d ∈ C ∩ H. For d = c = 1 we get ϵ = ϵ′, which implies (i) in this case. In general, since A is solvable, there is a normal subgroup B of A such that A/B is cyclic of prime order. Set D = C_{G}(B). If χ is A−stable, it is also B−stable. By induction, π(H, B)(β) = Res_{D∩H}^{D}(π(G, B)(χ)). Moreover, π(H, B)(β) and π(G, B)(χ) are A/B−stable. Applying π(D ∩ H, A/B) to the previous equality yields the statement.

(ii)Assume again that A is cyclic of prime order q. Set α = Ind_{H}^{G}(ψ) and assume that α is irreducible. Denote by α̂ and ψ̂ the canonical extensions of α and ψ to G ⋊ A and H ⋊ A, respectively. By [11, 13.6] there are signs δ, ϵ ∈ {+1, −1} such that, for any generator a of A, we have α̂(ca) = δπ(G, A)(α)(c) for all c ∈ C, and ψ̂(da) = ϵπ(H, A)(ψ)(d) for all d ∈ C ∩H. Moreover, since Res_{H}^{H⋊A}(ψ̂) = ψ, applying Mackey’s formula yields

Res_{G}^{G⋊A}(Ind_{H⋊A}^{G⋊A}(ψ̂)) = Ind_{H}^{G}(Res_{H}^{H⋊A}(ψ̂)) = α ,

which shows that Ind_{H⋊A}^{G⋊A}(ψ̂) is also an irreducible character extending α. Whence α̂ = λInd_{H⋊A}^{G⋊A}(ψ̂) for some linear character λ of A inflated to G ⋊ A.

Fix a generator a of A, and let c ∈ C. Let R be a set of representatives of the different cosets y(C ∩ H) of C ∩ H in C for which we have c^{y} ∈ C ∩ H. The formula [22, 3.3, Theorem 12] to compute induced characters yields

Ind_{C∩H}^{C}(π(H, A)(ψ))(c) = ∑_{t∈R}π(H, A)(ψ)(c^{t}) .

The key observation at this point is that R is also a system of representatives of the different cosets y(H ⋊A) of H ⋊A in G⋊A for which we have (ca)^{y} ∈ H ⋊A. Indeed, on one hand, different elements in R are easily seen to represent different cosets of H ⋊ A. On the other hand, if (ca)^{y} ∈ H ⋊ A, then actually (ca)^{y} ∈ Ha, and whence, by Glauberman’s Lemma [11, 13.8], there is h ∈ H such that (ca)^{yh} = da for some d ∈ C ∩ H. As the order q of a is prime to the orders of c and d, we have c^{yh} = d and a^{yh} = a, thus yh ∈ C. This shows that indeed R represents the cosets y(H ⋊ A) such that (ca)^{y} ∈ H ⋊ A. Therefore, again by the formula [22, 3.3, Theorem 12], we have

Ind_{H⋊A}^{G⋊A}(ψ̂)(ca) = ∑_{t∈R}ψ̂(c^{t}a) = ϵInd_{C∩H}^{C}(π(H, A)(ψ))(c) .

Whence π(G, A)(α)(c) = δλ(a)ϵInd_{C∩H}^{C}(π(H, A)(ψ))(c). This holds for any c ∈ C, and thus, by the linear independence of irreducible characters, we have δλ(a)ϵ = 1. This proves the Theorem in the case where A is cyclic of prime order.

In the general case, as A is solvable, there is a normal subgroup B of A of index q for some prime q. Set D = C_{G}(B). Then D ∩ H is A−invariant. By induction, we have π(G, B)(α) = Ind_{D∩H}^{D}(π(H, B)(ψ)). Moreover, π(G, B)(α) and π(H, B)(ψ) are A/B−stable. Applying π(D, A/B) to the previous equality concludes the proof. □

Alperin showed in [1] that the Glauberman correspondence can be interpreted in terms of the Brauer correspondence, if the group A is a q−group for some prime q. We restate this in the following form, adapted to our needs in Section 4 below:

Proposition 2.4. (cf. [1]) Let G be a finite group and A a group of prime order q acting on G. Suppose that q does not divide |G|. Let O′ be a subring of K containing a primitive |G|−th root of unity in which |G| is invertible and let O′ → k′ be a surjective ring homomorphism from O′ onto a field k′ of characteristic q. Let χ be an A−stable irreducible character of G with values in O′. Denote by e and f the images in k′G and k′C_{G}(A) of the central primitive idempotents e(χ) and e(π(G, A)(χ)). We have Br_{A}(e) = f.

Proof. Since |G| is prime to q and χ is absolutely irreducible, the algebra k′Ge is a matrix algebra over k′. The action of A on G induces an action on k′Ge. As k′Ge is a direct summand of k′G and char(k′) = q = |A|, it follows that k′Ge has an A−stable k′−basis. Thus [4] (or [23, (28.7)]) implies that k′C_{G}(A)Br_{A}(e) is a matrix algebra over k′. Thus Br_{A}(e) determines an absolutely irreducible character η of C_{G}(A). The argument in [1] shows that η occurs with multiplicity prime to q in the restriction to C_{G}(A) of χ. Thus η is the Glauberman correspondent of χ by 2.1(ii), which implies the result. □

The next result is known as the Fourier inversion formula:

Proposition 2.5. (cf. [22, 6.2, Prop. 11]) Let G be a finite group. Assume that K contains a primitive |G|-th root of unity. Let χ ∈ Irr_{K}(G) and let e(χ) be its associated central primitive idempotent in KG. For any s ∈ KGe(χ) we have

s = \frac{χ(1)}{|G|} ∑_{x∈G} χ(x^{−1}s)x .

Proof. It suffices to check this for an element s of the form s = ye(χ) for some y ∈ G. Since e(χ) is the central primitive idempotent associated with χ, we have

χ(x^{−1}ye(χ)) = χ(x^{−1}y). Thus ∑_{x∈G}χ(x^{−1}ye(χ))x = ∑_{x∈G}χ(x^{−1}y)xy^{−1}y = \frac{|G|}{χ(1)}e(χ)y,

from which the result follows. □

If H is a subgroup of a finite group G, we say that a function λ : H → O is Gstable, if λ(h) = λ(h′) for any two elements h, h′ in H which are conjugate in G. We conclude this section with the following statement on the extension of stable linear characters.

Proposition 2.6. Let G be a finite group and let P be a Sylow-p-subgroup of G. Let λ : P → O^{×} be a G−stable linear character of P. Then λ extends to a linear character of G.

Proof. By [9, Ch. 7, Theorem 3.4], the intersection P ∩[G, G] of P with the commutator subgroup of G is generated by the set of commutators xux^{−1}u^{−1}, where u ∈ P and x ∈ G such that xux^{−1} ∈ P. Since λ is G−stable, we have P ∩ [G, G] ⊆ ker(λ). Thus λ induces a linear character of the Sylow-p-subgroup P/(P ∩ [G, G]) of the abelian group G/[G, G], which extends therefore to a character of G/[G, G], and its inflation to G in turn extends λ. □

3 Quoted results on O^{×}−groups

As in [10], we follow the terminology of [18, section 5]. An O^{×}−group is a group Ĝ endowed with a group homomorphism O^{×} ⟶ Ĝ whose image lies in Z(Ĝ); we usually denote by λ̂ the image of λ ∈ O^{×} in Ĝ, if no confusion arises, and by Ǧ the opposite O^{×}−group, which as an abstract group is equal to Ĝ, but endowed with the homomorphism O^{×} ⟶ Ǧ sending λ ∈ O^{×} to λ̌ = (λ̂)^{−1}.

In our context, O^{×}−groups arise from the action of a finite group G on a matrix algebra S over O by algebra automorphisms (we write ^{x}s for the action of x ∈ G on s ∈ S): by the Skolem-Noether Theorem, any automorphism of S is inner, and hence, for any x ∈ G there is an s ∈ S such that ^{x}t = sts^{−1} for all t ∈ S. The group Ĝ of all such pairs (x, s) ∈ G × S^{×} satisfying ^{x}t = sts^{−1} for all t ∈ S becomes an O^{×}−group with the group homomorphism sending λ ∈ O^{×} to λ̂ = (1_{G}, λ1_{S}). We call Ĝ the O^{×}−group defined by the action of G on S; clearly Ĝ is a central O^{×}−extension of G, since there is an obvious short exact sequence of groups

1 ⟶ O^{×} ⟶ Ĝ ⟶ G ⟶ 1 .

The twisted group algebra O_{∗}Ĝ associated with the O^{×}−group Ĝ is the quotient of the group algebra OĜ by the ideal generated by the set of elements λ.g − 1_{O}.(λ̂g), where λ runs over O^{×} and g runs over Ĝ (cf. [18, 5.12] or [23, 10.4]). If for any x ∈ G we choose some inverse image x̂ of x in Ĝ, the image of the set {x̂}_{x∈G} in O_{∗}Ĝ is an O−basis; in particular, the algebra O_{∗}Ĝ is O−free of rank |G|. Moreover, for any two x, y ∈ G, there is a unique λ_{x,y} ∈ O^{×} such that x̂ŷ = λ̂_{x,y}x̂y, because both x̂ŷ and x̂y are inverse images of xy in Ĝ. The map λ : G × G ⟶ O^{×} sending (x, y) to λ_{x,y} is a 2−cocycle which determines Ĝ as an O^{×}−group. While λ itself is not uniquely determined by Ĝ - it obviously depends on the choice of the x̂ - its class \overline{λ} ∈ H^{2}(G, O^{×}) is uniquely determined by Ĝ (see e.g. [25, 6.6] for standard properties relating central group extensions of a finite group to its second cohomology groups). We use the analogous terminology for k^{×} instead of O^{×}. Since k is perfect, by [21, Ch. II, Section 4, Prop. 8],

3.1. there is a canonical group isomorphism O^{×} ≅ (1 + J(O)) × k^{×} compatible with the inclusion 1 + J(O) ⊂ O^{×} and the canonical surjection O^{×} → k^{×}.

Suppose now that S = M_{n}(O) for some positive integer n which is prime to p, let G be a finite group acting on S and let Ĝ be the O^{×}−group defined by the action of G on S. Then the following hold:

3.2. For any x ∈ G there is s_{x} ∈ S^{×} such that det(s_{x}) = 1 and x̂ = (x, s_{x}) ∈ Ĝ. The order of s_{x} in S^{×} is then finite and divides mn, where m is the order of x. Moreover, if there is any other s′_{x} ∈ S^{×} such that det(s′_{x}) = 1 and (x, s′_{x}) ∈ Ĝ, there is a unique n^{th} root of unity ζ such that s′_{x} = ζs_{x}.

Proof. If s is some element in S^{×} such that (x, s) ∈ Ĝ, then by [21, Ch. II, Section 4, Prop. 7] there is τ ∈ O^{×} such that τ^{n} = det(s). It suffices to take s_{x} = τ^{−1}s. As (s_{x})^{m} acts trivially on S, we have (s_{x})^{m} = μ1_{S} for some μ ∈ O^{×}. Then 1 = det((s_{x})^{m}) = μ^{n} implies (s_{x})^{mn} = 1_{S}. Similarly, s_{x}^{−1}s′_{x} = ζ1_{S} for a unique ζ ∈ O^{×}, and whence 1 = det(s_{x}^{−1}s′_{x}) = ζ^{n}. □

Consider now the 2−cocycle λ defined by x̂ŷ = λ̂_{x,y}x̂y for this particular choice of inverse images x̂ of x ∈ G in Ĝ. Denote again by \overline{λ} its image in H^{2}(G, O^{×}).

3.3. For any two x, y ∈ G, we have (λ_{x,y})^{n} = 1.

In other words, the 2−cocycle λ has values in the group μ_{n} of n^{th} roots of unity in O^{×}. This is because if we look at what the equality x̂ŷ = λ̂_{x,y}x̂y means in the second component, we get s_{x}s_{y} = λ_{x,y}s_{xy}. As both s_{x}s_{y} and s_{xy} have determinant 1, this forces (λ_{x,y})^{n} = 1. In terms of cohomology groups, statement 3.3 is equivalent to:

3.4. The class \overline{λ} ∈ H^{2}(G, O^{×}) is in the image of the map H^{2}(G, μ_{n}) ⟶ H^{2}(G, O^{×}) induced by the inclusion of the group μ_{n} of n^{th} roots of unity in O^{×}.

The map H^{2}(G, μ_{n}) → H^{2}(G, O^{×}) need not be injective; its kernel is the image of the connecting homomorphism H^{1}(G, O^{×}) → H^{2}(G, μ_{n}) of the long exact cohomology sequence associated with the short exact “Kummer” sequence

1 ⟶ μ_{n} ⟶ O^{×} ⟶^{(−)^{n}} O^{×} ⟶ 1 ,

where (−)^{n} is the group homomorphisms taking an element of O^{×} to its n^{th} power; this is a surjective map as k is algebraically closed and n is prime to p. Moreover, the subgroup μ_{n} of O^{×} is contained in the canonical image of k^{×} in O^{×}. This implies that the image \overline{λ} of the 2−cocycle λ in H^{2}(G, O^{×}) is trivial if and only if its image in H^{2}(G, k^{×}) is trivial. In other words, the O^{×}−group Ĝ is already determined by the k^{×}−group G̃ obtained from the action of G on the matrix algebra k ⊗_{O} S. Precisely:

3.5. We have an isomorphism of O^{×}−groups Ĝ ≅ (1 + J(O)) × G̃ , where the O^{×}−group structure on the right side is given via the canonical isomorphism 3.1.

Any central extension of a finite p−group P by the p′−group μ_{n} splits canonically. Thus 3.2, 3.3 imply the following statement (cf. [23, (21.5)]):

3.6. If P is a finite p−group acting on S, there is a unique group homomorphism σ : P → S^{×} lifting the action of P on S with Im(σ) ⊂ ker(det : S^{×} → O^{×}); in particular, the O^{×}−group P̂ defined by the action of P on S is canonically isomorphic to the trivial extension O^{×} × P.

The canonical isomorphism of O^{×}−groups O^{×} × P ≅ P̂ in 3.6 maps (λ, u) ∈ O^{×} × P to (u, λσ(u)). If we choose in the situation of 3.6 a free O−module V of rank n and an isomorphism S ≅ End_{O}(V), then V becomes an OP −module through σ. Recall from [4] that V is called an endo-permutation OP −module, if S is a permutation OP −module; that is, if S has a P −stable O−basis (with respect to the action ^{u}s = σ(u)sσ(u^{−1}) for any u ∈ P, s ∈ S). In that case, the Brauer quotient S(P) = S^{P} /( ∑_{Q<P}S_{Q}^{P} + J(O)S^{P}) is again a matrix algebra over k. Whence if G acts

on both S and P in such a way that σ becomes a G−map, then the action of G on S induces an action of G on S(P) which in turn defines a central k^{×}−extension of G. Dade’s theorem on the splitting of fusion in endo-permutation modules, in its version given by Puig [17], implies that:

3.7. If S is endowed with a group homomorphism σ : P → S^{×} for some finite p−group P on which G acts such that σ(^{x}u) = ^{x}σ(u) for all x ∈ G and u ∈ P and such that k ⊗_{O} S has a P −stable k−basis, then the k^{×}−groups defined by the actions of G on

k ⊗_{O} S and S(P) are isomorphic.

The proof of 3.7 in [17] is far from being elementary; it goes by induction on the order of P, using Dade’s classification [5] of endo-permutation modules over finite abelian p−groups. It is therefore worthwhile to point out, that the proof is easy in the particular case where G acts trivially on P. In that case, if we consider again for any x ∈ G an element x̂ = (x, s_{x}) belonging to the O^{×}−group Ĝ defined by the action of G on S, we have s_{x} ∈ S^{P}. Denote by \overline{s}_{x} the image of s_{x} in S(P) under the Brauer homomorphism Br_{P} : S^{P} → S(P). Since Br_{P} is an algebra homomorphism, the equality s_{x}s_{y} = λ_{x,y}s_{x,y} implies the equality \overline{s}_{x}\overline{s}_{y} = \overline{λ}_{x,y}\overline{s}_{x,y}, for any x, y ∈ G, where \overline{λ}_{x,y} is the canonical image of λ_{x,y} in k. It is only this particular case of 3.7 which is used implicitly in the proof of 4.5. In view of uniqueness issues, we recall the following obvious statement:

3.8. For any O^{×}−automorphism α of Ĝ inducing the identity on G there is a unique linear character λ : G → O^{×} of G such that α(x̂) = λ(x)x̂ for all x ∈ G and any inverse image x̂ of x in Ĝ.

If P is a p-subgroup of G, then by 3.6, the inverse image P̂ of P in Ĝ is canonically isomorphic to P × O^{×}, and through this isomorphism, we view O_{∗}Ĝ as OP -OP - bimodule.

3.9. As OP -OP -bimodules, O_{∗}Ĝ and OG are isomorphic.

Proof. Let y ∈ G and denote by ŷ some inverse image of y in Ĝ. If u ∈ P is such that v = y^{−1}uy ∈ P, then ŷ^{−1}(u, σ(u))ŷ = (v, σ(v)) by the uniqueness of σ. Thus O[P ŷP] ≅ O[P yP] as OP -OP -bimodules. By taking the direct sum of these isomorphisms with y running over a set of representatives of the P -P -double cosets in L one gets 3.9. □

4 The reduced case

The proof of Theorem 1.1 goes essentially in two steps, following Puig’s algebra theoretic version [16] of Fong’s reduction in [7]: first, we consider the “reduced” case, where the A−stable block b of the p−solvable group G is actually a block of the maximal normal subgroup O_{p′}(G) of order prime to p. Second, we show in section 5, that the general case can be obtained from the “reduced” case by an induction argument.

Let G be a finite p−solvable group and b a (G ⋊ A)−stable block of O_{p′}(G). By [10, 3.3], b is a block of OG having the Sylow-p−subgroups of G as defect groups. Therefore, if C contains some Sylow-p−subgroup P of G, Watanabe’s theorem 2.2 applies and determines a block w(b) of OC by the condition π(G, A)(Irr_{K}(G, b)) = Irr_{K}(C, w(b)). But the Glauberman correspondence yields another way to produce a block of OC: since b is a block of the p′−group O_{p′}(G), it corresponds to a unique (G⋊A)−stable irreducible character β of O_{p′}(G). Thus its Glauberman correspondent γ = π(O_{p′}(G), A)(β) is an irreducible character of O_{p′}(G) ∩ C. The latter group is shown to be equal to O_{p′}(C), and hence γ determines a C−stable block e of O_{p′}(C). Again by [10, 3.3], e is then also a block of OC having P as defect group. The next result shows that e = w(b), providing an alternative proof of the statements (i), (ii) of Watanabe’s theorem in this particular case.

Theorem 4.1. Let G be a finite p−solvable group, A a solvable subgroup of Aut(G) of order prime to |G|. Set C = C_{G}(A) and suppose that C contains a Sylow-p-subgroup P of G. Let b be a G ⋊ A−stable block of O_{p′}(G). Denote by β the unique irreducible character of O_{p′}(G) associated with b, set γ = π(O_{p′}(G), A)(β), and denote by e the block of O_{p′}(G) ∩ C = O_{p′}(C) determined by γ.

(i) b is a block of OG having P as defect group, and b is of principal type (i.e., Br_{Q}(b) is a block of kC_{G}(Q) for any subgroup Q of P).

(ii) e is a block of OC having P as defect group, and e is of principal type.

(iii) The Brauer categories of b, e, viewed as blocks of OG, OC, respectively, are equivalent.

(iv) We have π(G, A)(Irr_{K}(G, b)) = Irr_{K}(C, e), or equivalently, e = w(b).

(v) If b is the principal block of OG then e is the principal block of OC, and the algebras OGb and OCe are equal.

Before we prove 4.1, we recall some well-known results on p−solvable groups.

Lemma 4.2. ([9]) Let G be a finite p−solvable group.

(i) For any Sylow-p−subgroup Q of O_{p′,p}(G) we have C_{G}(Q) ⊂ O_{p′,p}(G).

(ii) For any p−subgroup R of G we have O_{p′}(N_{G}(R)) = O_{p′}(C_{G}(R)) = O_{p′}(G) ∩ C_{G}(R).

Proof. (i) is well-known (see [9, Ch. 6, Theorem 3.3]). (ii) The image of N_{G}(R) in G/O_{p′}(G) is precisely the normaliser of the image of R in G/O_{p′}(G), because the orders of R and O_{p′}(G) are coprime. Thus we may assume that O_{p′}(G) = {1}. Note that then C_{G}(Q) = Z(Q) by (i). Setting X = O_{p′}(N_{G}(R)), we have to show that X = {1}. The group XR normalises Q, and C_{Q}(R) is normal in N_{G}(R). Thus X and C_{Q}(R) normalise each other, and hence commute elementwise as their orders are coprime. By Thompson’s A × B−Lemma, X and Q commute elementwise. But C_{G}(Q) = Z(Q), and so X = {1}, which proves (ii). □

A finite group satisfying 4.2(i) is called p−constrained. It is easy to check, that the proof of the following Proposition works under the weaker hypothesis that G and thus G ⋊ A are p−constrained. Thompson’s “A × B−Lemma” and the Schur-Zassenhaus Theorem, that are used in the proof of this Proposition, can be found in [9].

Proposition 4.3. Let G be a finite p−solvable group and let A be a subgroup of Aut(G) of order prime to |G|. Set C = C_{G}(A). Denote by Q a Sylow-p−subgroup of O_{p′,p}(G). Suppose that Q ⊂ C. Then the following hold.

(i) For any p−subgroup R of C we have N_{G}(R) = O_{p′}(C_{G}(R))N_{C}(R).

(ii) G = O_{p′}(G)N_{C}(Q).

(iii) O_{p′}(C) = O_{p′}(G) ∩ C.

(iv) The inclusions of the subgroups N_{C}(Q), N_{G}(Q) and C in G induce isomorphisms N_{C}(Q)/O_{p′}(N_{C}(Q)) ≅ C/O_{p′}(C) ≅ G/O_{p′}(G) ≅ N_{G}(Q)/O_{p′}(N_{G}(Q)) .

Proof. Observe that all statements in the Proposition hold trivially if p does not divide the order of G. Thus we may assume that p divides the order of G, and hence p does not divide the order of A. We show now, that then O_{p′}(G ⋊ A) = O_{p′}(G)A. Since A acts trivially on Q and G ⋊ A is again p−solvable, by 4.2(i) we have A ⊂ C_{G⋊A}(Q) ⊂ O_{p′,p}(G ⋊ A), thus A ⊂ O_{p′}(G ⋊ A). Therefore O_{p′}(G ⋊ A) = (O_{p′}(G ⋊ A) ∩ G)A = O_{p′}(G)A.

(i) By 4.2(ii), applied to G⋊A, we have O_{p′}(N_{G⋊A}(R)) = O_{p′}(G⋊A)∩C_{G⋊A}(R) = (O_{p′}(G)A)∩(C_{G}(R)A) = (O_{p′}(G)∩C_{G}(R))A = O_{p′}(C_{G}(R))A, where the last equality comes again from 4.2(ii). Both R, A are characteristic in RA since their orders are coprime. Therefore N_{G⋊A}(RA) = N_{G⋊A}(R)∩N_{G⋊A}(A) = N_{G⋊A}(R)∩(CA) = N_{C}(R)A.

By the Schur-Zassenhaus Theorem applied to RA in O_{p′}(C_{G}(R))(RA), together with the Frattini argument, we get N_{G⋊A}(R) = O_{p′}(C_{G}(R))(N_{C}(R)A). Intersecting this equality with G yields (i).

(ii) By the Frattini argument, we have G = O_{p′}(G)N_{G}(Q), and by (i), we have N_{G}(Q) = O_{p′}(C_{G}(Q))N_{C}(Q). Since C_{G}(Q) ⊂ O_{p′,p}(G), it follows that O_{p′}(C_{G}(Q)) ⊂ O_{p′}(G), which proves (ii).

(iii) Intersecting the equality in (ii) with C yields C = (O_{p′}(G) ∩ C)N_{C}(Q), and dividing the equality in (ii) by O_{p′}(G) yields O_{p′}(N_{C}(Q)) = O_{p′}(G) ∩ N_{C}(Q), from which (iii) follows.

(iv) is a straightforward consequence of the previous statements. □

Proof of 4.1. (i) is a restatement of the (well-known) result [10, 3.3]. Since the argument is short, we give it for the convenience of the reader.

Set N = O_{p′,p}(G). Since any block of N is a linear combination of p′−elements of N it lies in OO_{p′}(G), and therefore b is a block of N having Q as a defect group. If b′ is any block of G such that b′b = b′, then Q is contained in a defect group of b′ (see Dade [3]); in particular, Br_{Q}(b′) ≠ 0. By the same argument, if b − b′ ≠ 0 then Br_{Q}(b − b′) ≠ 0. However, Br_{Q}(b) is an N_{G}(Q)−stable block of kN_{N}(Q), thus still a block of kN_{G}(Q), since any block of kN_{G}(Q) lies in kC_{G}(Q) = kC_{N}(Q). Therefore Br_{Q}(b′) = Br_{Q}(b), which forces b′ = b by the previous argument, and so b is indeed a block of G. Now P acts by conjugation on OO_{p′}(G)b, which is a matrix algebra of rank prime to p having a P −stable O−basis (as it is a direct summand of OO_{p′}(G)) and therefore P stabilises at least one element of any such P −stable O−basis of OO_{p′}(G)b. This means Br_{P} (OO_{p′}(G)b) ≠ 0, and hence Br_{P} (b) ≠ 0. It follows that P is a defect group of b as a block of G.

In order to see that b is of principal type, let R be a subgroup of P. Then Br_{R}(b) is a block of N_{O_{p′} (G)R}(R) = C_{O_{p′} (G)}(R)R = O_{p′}(C_{G}(R))R. Again, as Br_{R}(b) involves only p′−elements, it is in fact a block of O_{p′}(C_{G}(R)) which is C_{G}(R)−stable as b is G−stable. Therefore, by the first statement applied to Br_{R}(b) and C_{G}(R) instead of b and G, respectively, Br_{R}(b) is a block of C_{G}(R).

(ii) We show that γ is C−stable. In view of 2.1(i) we proceed by induction on |A|, reducing the situation to the case where A is cyclic of prime order q. Then γ is C−stable, as it is the unique irreducible constituent of Res_{C∩O_{p′} (G)}^{O_{p′} (G)}(β) occurring with a multiplicity prime to q. Thus the block e of O_{p′}(C) = C ∩ O_{p′}(G) is C−stable, and whence (ii) follows from (i).

(iii) is a direct consequence of Proposition 4.3(i) together with the fact, that both b and e are of principal type.

(iv) Since A acts trivially on P, all irreducible characters of G belonging to b and of C belonging to e are A−stable (cf. [24, Prop. 1]). Let χ ∈ Irr_{K}(G)^{A} and set ψ = π(G, A)(χ). By [11,13.29], we have < Ind_{O_{p′} (C)}^{C}(γ), ψ >≠ 0 if and only if < Ind_{O_{p′} (G)}^{G}(β), χ >≠ 0. Using Frobenius’ reciprocity, this means that < γ, Res_{O_{p′} (C)}^{C}(ψ) >≠ 0 if and only if < β, Res_{O_{p′} (G)}^{G}(χ) >≠ 0. This in turn means that χ belongs to b if and only if ψ belongs to e, since the restriction to O_{p′}(G) of any irreducible character of G belonging to b is a multiple of β, while β occurs in the restriction of no other irreducible character of G; similarly for C and e. This concludes the proof of (iv).

(v) If b is the principal block of OG, then e is the principal block of OC, and thus (v) follows from 4.3.(ii). □

Using Puig’s method in [16, Proposition 2] (which generalises Fong’s reduction in [7]), one gets the structure of the block algebra OGb in 4.1.

Theorem 4.4. Let G be a finite group, let N be a normal p′−subgroup of G, let b be a G−stable block of N and set S = ONb. Denote by Ĝ the O^{×}−group opposite to that obtained from the action of G on S. Set L̂ = Ĝ/N, where we identify the element y of N to its canonical image (y, yb) in Ĝ. There is a unique algebra isomorphism

OGb ≅ S ⊗_{O} O_{∗}L̂

mapping xb to s ⊗ \overline{(x, s)} with inverse mapping t ⊗ \overline{(x, s)} to ts^{−1}x, where x ∈ G, t ∈ S and s ∈ S^{×} such that (x, s) ∈ Ĝ, and where \overline{(x, s)} is the image of (x, s) in L̂.

Proof. Straightforward verification. □

With the notation of the previous Theorem, let P be a Sylow-p-subgroup of G. By 3.6, for any u ∈ P, there is a canonical σ(u) ∈ S^{×} with determinant 1 such that (u, σ(u)) ∈ Ĝ. Thus S becomes an interior P −algebra with structural mapping sending u ∈ P to σ(u), and O_{∗}L̂ becomes an interior P −algebra with structural homomorphism sending u ∈ P to \overline{(u, σ(u))}. In this way, the algebra isomorphism OGb ≅ S ⊗_{O} O_{∗}L̂ in 4.4 becomes an isomorphism of interior P −algebras. If we write

S = End_{O}(U) for some O−module U, then U becomes an OP −module through σ. In fact, U is then an endo-permutation-OP -module because S is a direct summand of ON, hence a permutation OP −module with respect to the action of P by conjugation. Since S is stable under G−conjugation it follows that U is fusion-stable; more precisely, Res_{R}^{P}(U) ≅ Res_{φ}(U) for any subgroup R of P and any group homomorphism φ : R → P for which there is an element x ∈ G satisfying φ(r) = xrx^{−1} for all r ∈ R.

Of course, Theorem 4.4 applies also to the block OCe in 4.1, yielding an isomorphism OCe ≅ T ⊗ O_{∗}L̃, where T = OO_{p′}(C)e and L̃ = C̃/O_{p′}(C) is obtained from the O^{×}−group C̃ opposite to that defined by the action of C on T. What we will show next, is that L̂ ≅ L̃ in such a way, that the canonical images of P in L̂ and L̃ are preserved. The proof consists of an explicit choice of 2−cocycles λ, μ defining the central O^{×}− extension L̂, L̃. The point is that it is not sufficient to show that the classes of these two 2−cocycles coincide in H^{2}(G, O^{×}), because even though this would imply that there is an isomorphism of O^{×}−groups L̂ ≅ L̃, it would not give information about what happens to the elements of P under such an isomorphism.

Theorem 4.5. With the notation of 4.1, let Q be a Sylow-p- subgroup of O_{p′,p}(G) contained in P. Set S = OO_{p′}(G)b and T = OO_{p′}(C)e. Denote by Ĝ and C̃ the O^{×}−groups opposite to that defined by the actions of G on S and of C on T, respectively. Set L̂ = Ĝ/O_{p′}(G) and L̃ = C̃/O_{p′}(C). Set m = |G||O_{p′}(G)| and let μ_{m} be the subgroup of O^{×} of all m − th roots of unity.

For any x ∈ G, choose s_{x} ∈ S^{×} such that (x, s_{x}) ∈ Ĝ and det(s_{x}) = 1; for any y ∈ C, choose t_{y} ∈ T^{×} such that (y, t_{y}) ∈ C̃ and det(t_{y}) = 1. Set

G′ = {(x, λs_{x})|x ∈ G, λ ∈ μ_{m}} and C′ = {(y, λt_{y})|y ∈ C, λ ∈ μ_{m}} .

Denote by σ : P → S^{×} and τ : P → T^{×} the unique group homomorphisms such that (u, σ(u)) ∈ Ĝ, (u, τ(u)) ∈ C̃, and det(σ(u)) = 1 = det(τ(u)) for all u ∈ P. Then the action of A on G and S induces an action of A on Ĝ, and the following hold.

(i) G′ is a finite A−stable subgroup of Ĝ; the exponent of G′ divides m; we have Ĝ = O^{×} · G′ and G′ contains {(x, xb)|x ∈ O_{p′}(G)} and {(u, σ(u))|u ∈ P }.

(ii) C_{G′}(A) = {(y, λs_{y})|y ∈ C, λ ∈ μ_{m}} and G′ = O_{p′}(G)C_{G′}(A).

(iii) C′ is a finite subgroup of C̃; the exponent of C′ divides m; we have C̃ = O^{×}·C′ and C′ contains {(y, ye)|y ∈ O_{p′}(C)} and {(u, τ(u))|u ∈ P }.

(iv) There is a group isomorphism Φ : C′ ≅ C_{G′}(A) which makes the following obvious diagram

1 ⟶ μ_{m} ⟶ C′ ⟶ C ⟶ 1
∥  ↓^{Φ}  ∥
1 ⟶ μ_{m} ⟶ C_{G′}(A) ⟶ C ⟶ 1

commutative, such that Φ(y, ye) = (y, yb) for any y ∈ O_{p′}(C) and Φ(u, τ(u)) = (u, σ(u)) for any u ∈ P.

Before we prove this Theorem, let us note a consequence of statement (iv): since the group isomorphism C′ ≅ C_{G′}(A) preserves the canonical images of O_{p′}(C) as well as of P in C′ and G′, we can reformulate (iv) as follows.

Corollary 4.6. With the notation of 4.5, the inclusion C ⊆ G lifts to an O^{×}- isomorphism L̃ ≅ L̂ which preserves the canonical images of P elementwise.

From this and 4.4 follows already that OGb and OCe are Morita equivalent, and that there is a Morita equivalence given by a bimodule with endo-permutation source; this is because the group isomorphism in 4.6 preserves P and S = End_{O}(U), T = End_{O}(V) for some endo-permutation OP −modules U, V (whose module structure is given by σ and τ, respectively). It remains to see, that we can choose this Morita equivalence in such a way, that it induces the Glauberman correspondence on characters.

Proof of Theorem 4.5. If (x, s) ∈ Ĝ and a ∈ A, then ^{a}(x, s) = (^{a}x,^{a}s) ∈ Ĝ, since S is A-stable, and thus A acts on Ĝ.

By 4.3 we have G = CO_{p′}(G); thus Ĝ = ĈO_{p′}(G), where Ĉ is the O^{×}−group opposite to that defined by the action of C on S, and where we identify again O_{p′}(G) to its canonical image {(x, xb)}_{x∈O_{p′} (G)} in Ĝ. We have to show that

4.5.1. there is an isomorphism of O^{×}−groups Ĉ ≅ C̃ which lifts the identity on C and which preserves the canonical images of P and O_{p′}(C).

Proceeding along a composition series of A, we may assume that

4.5.2. A is cyclic of prime order q,

where q is a prime which does not divide the order of G.

For the computations that follow, keep in mind that Ĝ is the O^{×}−group opposite to that defined by the action of G on S; in other words, for any (x, s) ∈ Ĝ and any λ ∈ O^{×} we have λ(x, s) = (x, λ^{−1}s). By 3.3, the associated 2−cocycles have values in groups of units; more precisely,

4.5.3. for any x, y ∈ G there is an |O_{p′}(G)|-th root of unity λ_{x,y} satisfying s_{x}s_{y} = (λ_{x,y})^{−1}s_{xy}.

Also if a ∈ A and x ∈ G, then det(^{a}s_{x}) = 1 since a acts as an inner automorphism on S. Then, from 3.2, we conclude that G′ is a finite A-stable subgroup of Ĝ. The statement on the exponent of G′ follows from 3.2. If x ∈ O_{p′}(G) then xb = λs_{x} for some λ ∈ O^{×}; hence det(xb) = λ^{n} and thus λ^{m} = 1 (cf. 3.2). Thus (i) holds. Similarly,

4.5.4. for any x, y ∈ C there is an |O_{p′}(C)|-th root of unity μ_{x,y} satisfying t_{x}t_{y} = (μ_{x,y})^{−1}t_{xy}.

As above, this implies (iii). To see that (ii) holds, let y ∈ C and a ∈ A. By the Skolem-Noether Theorem, the automorphism of S induced by a is inner, and thus both s_{y} and ^{a}(s_{y}) have determinant one. Since they both act as y on S, they differ by some |O_{p′}(G)|-th root of unity, by 3.2. As a has order dividing q, it follows that s_{y} and ^{a}(s_{y}) have to be equal. Thus (y, s_{y}) ∈ C_{G′}(A), which implies (ii).

Recall that β denotes the unique irreducible character of O_{p′}(G) associated with b and that γ = π(O_{p′}(G), A)(β) is the unique irreducible character of O_{p′}(C) associated with e. The Fourier inversion formula 2.5 applied to O_{p′}(G) and β implies that

4.5.5. for any x ∈ C we have s_{x} = \frac{β(1)}{|O_{p′} (G)|} ∑_{y∈O_{p′} (G)} β(s_{x}y^{−1})y.

We have a similar formula for the elements t_{x}, using the character γ. Given our choice of s_{x}, for any x ∈ C and any y ∈ O_{p′}(G), the element s_{x}y^{−1} has a finite order in S^{×} dividing m (cf. 3.2 applied to the element xy^{−1}), and thus β(s_{x}y^{−1}) ∈ ℤ[ζ], where ζ is a primitive m−th root of unity in K. Let O′ be the local subring of K obtained from localising ℤ[ζ] at a maximal ideal containing q. Clearly O′ has the residue field k′ = 𝔽_{q}[ζ] of characteristic q.

The formula 4.5.5 and its analogue for the elements t_{x} implies that 4.5.6. for any x ∈ C we have s_{x} ∈ (O′O_{p′}(G)b)^{A} and t_{x} ∈ O′O_{p′}(C)e.

It is the statement 4.5.6 which makes it possible to apply the Brauer construction with respect to A. Note that by 2.4 we have Br_{A}(b) = ē, where ē is the image of e in k′O_{p′}(C). We compare for x ∈ C the image Br_{A}(s_{x}) in k′O_{p′}(C)ē to the canonical image of t_{x} in k′O_{p′}(C)ē. Both elements act as x on this matrix algebra, whence differ by a scalar. Since both elements have finite order, they differ actually by a root of unity; that is,

4.5.7. for any x ∈ C there is a unique root of unity ν_{x} of order dividing m such that Br_{A}(s_{x}) = ν_{x}Br_{A}(t_{x})

Any such root lifts uniquely to O′, and we denote this abusively by ν_{x} again. Comparing the equations 4.5.3, 4.5.4 and 4.5.7 we will show that

4.5.8. there is a unique O^{×}−isomorphism C̃ ≅ Ĉ mapping (x, t_{x}) to ν_{x}(x, s_{x}) for any x ∈ C.

To see that this is a group homomorphism, let x, y ∈ C. Write s_{x}s_{y} = (λ_{x,y})^{−1}s_{xy}. Applying Br_{A} to this equation yields, by 4.5.3 and 4.5.7, the equation λ_{x,y}ν_{x}ν_{y}Br_{A}(t_{x}t_{y}) = ν_{xy}Br_{A}(t_{xy}). Using 4.5.4 implies that

4.5.9. for any x, y ∈ C we have ν_{x}ν_{y}λ_{x,y} = ν_{xy}μ_{x,y}.

This shows that the O^{×}−linear map defined in 4.5.8 maps (x, t_{x})(y, t_{y}) = μ_{x,y}(xy, t_{xy}) to ν_{xy}μ_{x,y}(xy, s_{x,y}) = ν_{x}ν_{y}λ_{x,y}(xy, s_{xy}) = ν_{x}ν_{y}(x, s_{x})(y, s_{y}), which is the product of the images of (x, t_{x}), (y, t_{y}). This shows that the map in 4.5.8 is a group homomorphism. We show next, that this group homomorphism preserves the canonical images of O_{p′}(C). Precisely, we are going to show that

4.5.10. the group isomorphism 4.5.8 maps (x, xe) to (x, xb) for any x ∈ O_{p′}(C).

To see this, observe that if x ∈ O_{p′}(C), then the elements s_{x} and xb differ by a scalar since they both act as x on S; say s_{x} = δ_{x}xb for some δ_{x} ∈ O^{×}. Both s_{x} and xb have finite order dividing m, and therefore δ_{x} is a root of unity belonging to μ_{m} ⊂ O′. Similarly, t_{x} = ϵ_{x}xe for some root of unity ϵ_{x} ∈ μ_{m}. Denote by b̄ and ē the canonical images of b and e in k′O_{p′}(G) and k′O_{p′}(C), respectively. By 2.4, we have Br_{A}(b̄) = Br_{A}(ē), and thus Br_{A}(xb̄) = xē for any x ∈ C. Now 4.5.7 shows that Br_{A}(s_{x}) = ν_{x}Br_{A}(t_{x}) = ν_{x}ϵ_{x}xē. Together we get δ_{x} = ν_{x}ϵ_{x}. Thus the homomorphism in 4.5.8 maps (x, xe) = ϵ_{x}(x, t_{x}) to ϵ_{x}ν_{x}(x, s_{x}) = ϵ_{x}ν_{x}δ_{x}^{−1}(x, xb) = (x, xb), which proves 4.5.10. It remains to investigate the effect of the group homomorphism 4.5.8 on the images of the elements of P in C̃ and Ĉ. For that, we may assume that s_{u} = σ(u) and t_{u} = τ(u) for any u ∈ P. Then λ_{u,v} = 1 = μ_{u,v} for any u, v ∈ P. Hence ν_{u}ν_{v} = ν_{uv} by 4.5.9. In other words, the map ν : P → ℤ[ζ]^{×} sending u ∈ P to ν(u) = ν_{u} is a linear character of P. This character is C−stable: by the uniqueness properties of σ and τ we have ^{y}σ(u) = σ(^{y}u) and ^{y}τ(u) = τ(^{y}u) for any u ∈ P and any y ∈ C such that ^{y}u ∈ P, and so ν(u) = ν(^{y}u) by 4.5.7. Thus ν extends, by 2.6, to a linear character of C, abusively still denoted by ν, such that O_{p′}(C) ⊆ ker(ν). Now the map sending (x, t_{x}) to ν(x)^{−1}ν_{x}(x, s_{x}), where x ∈ C, induces a group isomorphism C̃ ≅ Ĉ which satisfies 4.5.1. □

Proof of 1.1 in the reduced case. We use the notation of 4.1 and 4.5. By 4.5 (iv), we have a commutative diagram of finite groups

1 ⟶ μ_{m} ⟶ C′ ⟶ C ⟶ 1
∥  _{ι′}↓  ↓^{ι}
1 ⟶ μ_{m} ⟶ G′ ⟶ G ⟶ 1

where ι is the inclusion, ι′ is the isomorphism Φ : C′ ≅ C_{G′}(A) from 4.5 (iv) followed by the inclusion C_{G′}(A) ⊆ G′.

By the notation in 4.1, we have γ = π(O_{p′}(G), A)(β) . Consider β as O-linear function on OO_{p′}(G) and set β′(x, s) = β(s) for any (x, s) ∈ G′. Then β′ is an Astable irreducible character of G′ which extends β through the canonical embedding O_{p′}(G) → G′. Similarly, define γ′ ∈ Irr_{K}(C′) by setting γ′(y, t) = γ(t) for any (y, t) ∈ C′; as before, γ′ extends γ through the canonical embedding O_{p′}(C) → C′.

Since β′ extends β, it follows from 2.3 (i) that π(G′, A)(β′) ∘ Φ is a character of C′ which extends γ. Therefore, there is a linear character λ : C′ → O^{×} such that O_{p′}(C) ⊆ ker(λ) and such that

π(G′, A)(β′) ∘ Φ = λγ′ .

We are going to show that μ_{m} ⊆ ker(λ). If ζ ∈ μ_{m}, then β′(1, ζb) = ζ deg(β), and since π(G′, A)(β′) appears in the restriction of β′ to C_{G′}(A), we also have π(G′, A)(β′)(1, ζb) = ζ deg(π(G′, A)(β′)) = ζ deg(γ) = γ′(1, ζe). Since Φ(1, ζe) = (1, ζb), it follows that μ_{m} ⊆ ker(λ). Moreover, since the exponent of C′ divides m, it follows that Im(λ) ⊆ μ_{m}, and thus the map κ = λ^{−1}ι′ : C′ → G′ sending (y, t) ∈ C′ to λ(y, t)^{−1}ι′(y, t) is still an injective group homomorphism with image C_{G′}(A) making the following diagram commutative

\begin{aligned}1 & ⟶ & μ_{m} & ⟶ & C′ & ⟶ & C & ⟶ & 1 \\ & & ∥ & & ^{κ}↓ & & ↓^{ι} \\ 1 & ⟶ & μ_{m} & ⟶ & G′ & ⟶ & G & ⟶ & 1 \end{aligned} .

By the construction of κ, we still have κ(y, ye) = (y, yb) for any y ∈ O_{p′}(C). Note that κ need no longer preserve the images of P, because κ(u, τ(u)) = λ(u, τ(u))^{−1}(u, σ(u)) for any u ∈ P. The reason why we replace ι′ by κ is that we have now

π(G′, A)(β′) ∘ κ = γ′ .

The group homomorhism κ induces still an isomorphism of O^{×}-groups μ : L̃ ≅ L̂, which in turn induces a commutative diagram of O-algebras

OC′ ⟶ O_{∗}L̃
κ↓  ↓^{μ}
OG′ ⟶ O_{∗}L̂

where we denote abusively the induced algebra homomorphisms by κ and μ again, and where the horizontal maps are the canonical surjections.

We have thus a Morita equivalence between OGb and OCe given by composing the Morita equivalence between OGb ≅ S ⊗_{O} O_{∗}L̂ and O_{∗}L̂, the algebra isomorphism

μ : O_{∗}L̃ ≅ O_{∗}L̂, and the Morita equivalence between O_{∗}L̃ and OCe ≅ T ⊗_{O} O_{∗}L̃. We

are going to show that this Morita equivalence satisfies 1.1. The OGb-OCe-bimodule M inducing this equivalence is explicitly given by

M = (U ⊗_{O} (O_{∗}L̂)_{μ}) ⊗_{O_{∗}L̃} (O_{∗}L̃ ⊗_{O} V^{∗}) ≅ U ⊗_{O} (O_{∗}L̂)_{μ} ⊗_{O} V^{∗} ,

where S = End_{O}(U) and T = End_{O}(V), and where the left and right module structure on M is given via the isomorphisms OGb ≅ S ⊗_{O} O_{∗}L̂ and OCe ≅ T ⊗_{O} O_{∗}L̃ from 4.4.

We show that M has ΔP as vertex and a fusion-stable endo-permutation OΔP - source. The O-modules U and V are fusion-stable endo-permutation OP −modules (through the group homomorphisms σ : P → S^{×} and τ : P → T^{×}, respectively, as explained before 4.5). Denote by λ_{P} the rank one OP −module whose character is the restriction of λ to P. All three modules U, V, λ_{P} have ranks prime to p. Thus U ⊗_{O} λ_{P}^{∗} ⊗_{O} V^{∗} has an indecomposable direct summand W with vertex ΔP. Since all

of U, V, λ_{P} are fusion-stable endo-permutation modules, so is W. It suffices to see that W is a source of M. It follows from 3.9 that there is an isomorphism of OP -OP - bimodules O_{∗}L̂ ≅ OL, where L = G/O_{p′}(G). Thus the restriction of M to P × P is isomorphic to the direct sum of the modules U ⊗_{O} O[P yP]_{μ} ⊗_{O} V^{∗}, where we denote

abusively the restriction of μ to P by the same letter, and where y runs over a set of representatives of the P -P -double cosets in L. For any such y, we have O[P yP] ≅ Ind_{R}^{P ×P} (O) as O(P × P)-modules, where R = {(u, y^{−1}uy)|u ∈ P ∩ yP y^{−1}}, and thus

O[P yP]_{μ} ≅ Ind_{R}^{P ×P} (λ_{P}^{∗}). Therefore U ⊗_{O}O[P yP]_{μ} ⊗_{O}V^{∗} ≅ Ind_{R}^{P ×P} (U ⊗_{O}λ_{P}^{∗} ⊗_{O}V^{∗}), and

hence a vertex of M has order at most |P |. For y = 1 we get that Ind_{ΔP}^{P ×P} (U ⊗_{O}λ_{P}^{∗} ⊗_{O}V^{∗})

is isomorphic to a direct summand of M restricted to P × P. Thus ΔP is a vertex and W a source of M.

We show next, that on characters, the equivalence given by M induces the Glauberman correspondence. Arguing by induction, we may assume that A is a q-group for some prime q not dividing the order of G.

Let χ ∈ Irr_{K}(G, b). Then χ = β ⊗ χL̂ for a unique irreducible character χL̂ of O_{∗}L̂ via the isomorphism OGb ≅ S ⊗_{O} O_{∗}L̂. That is,

χ(x) = β(s)χL̂(\overline{(x, s)}) ,

where x ∈ G and s ∈ S^{×} such that (x, s) ∈ Ĝ, and where \overline{(x, s)} is the canonical image of (x, s) in O_{∗}L̂.

Through the algebra isomorphism μ, the character χL̂ corresponds to an irreducible character χL̃ of O_{∗}L̃. The character η = γ ⊗ χL̃ corresponding to χL̃ via the isomorphism OCe ≅ T ⊗_{O} O_{∗}L̃ is given, as above, by

η(y) = γ(t)χL̃(\overline{(y, t)}) ,

where (y, t) ∈ C̃. Consider now the inflation χ′ of χ to G′ via the canonical surjection G′ → G and the inflation η′ to C′ of η. The preceding equalities read now

χ′(x, s) = β′(x, s)χL̂(\overline{(x, s)})

for any (x, s) ∈ G′, and similarly,

η′(y, t) = γ′(y, t)χL̃(\overline{(y, t)})

for any (y, t) ∈ C′. Here we use now the particular choice of κ: since π(G′, A)(β′)∘κ = γ′, the character γ′ appears with multiplicity prime to q in the restriction β′ ∘ κ of β′ through κ (cf. 2.1(ii)). Since μ is precisely the algebra isomorphism induced by κ, it follows that η′ appears with multiplicity prime to q in the restriction χ′ ∘ κ of χ′ through κ. As χ′, η′ are just the inflations of χ, η, it follows that η appears with multiplicity prime to q in the restriction Res_{C}^{G}(χ). By 2.1(ii), this is equivalent to η = π(G, A)(χ).

The statements (i) and (ii) follow. If b is the principal block of G, then e is the principal block of C, because the Glauberman correspondent of the trivial character of G is the trivial character of C. Moreover, by 4.3 (ii), we have G = CO_{p′}(G), which implies the equality OGb = OCe as O_{p′}(G) is the kernel of the principal block of G. □

Remark 4.7. A Morita equivalence between two blocks having isomorphic defect groups which is given by a bimodule with a fusion-stable endo-permutation source induces always an isotypy. This yields an alternative proof of the fact that the Glauberman correspondence is indeed an isotypy in the situation of Thereom 1.1.

5 The reduction step

The Theorem 5.1 below shows that the “reduction step” as described in [10, 3.1] is both A−equivariant and compatible with the Glauberman correspondence. It shows in particular, that in order to prove Theorem 1.1, we may assume that b is actually a G−invariant block of the maximal normal subgroup O_{p′}(G) of order prime to p in G.

Theorem 5.1. Let G be a finite p−solvable group, A a solvable group of automorphisms of G of order prime to |G|p, and b be an A−stable block of OG having a defect group which is centralised by A.

There is an A−stable subgroup H of G and an H ⋊ A−stable block e of O_{p′}(H) such that the following hold:

(i) We have O_{p′}(G) ⊂ H and OGb ≅ Ind_{H}^{G}(OHe) as interior G−algebras and as A−algebras; in particular, e is an A−stable block of H and we have b = Tr_{H}^{G}(e).

(ii) Any Sylow-p−subgroup of H is a defect group of e as a block of H and of the block b of G; moreover, there is a Sylow-p−subgroup P of H contained in C_{H}(A).

(iii) The Brauer categories of OGb and OHe are equivalent and the block e of H is of principal type; that is, Br_{Q}(e) is a block of kC_{H}(Q) for any subgroup Q of P.

If we set C = C_{G}(A), D = C_{H}(A) = C ∩ H and denote by w(b), w(e) the blocks of OC, OD corresponding to b, e, respectively, through the Glauberman correspondence, then moreover the following hold:

(iv) We have O_{p′}(D) = O_{p′}(H) ∩ C, and w(e) is a D−stable block of OO_{p′}(D).

(v) We have OCw(b) ≅ Ind_{D}^{C}(ODw(e)) as interior C−algebras; in particular, w(b) = Tr_{D}^{C}(w(e)).

(vi) The Brauer categories of OCw(b) and ODw(e) are equivalent; moreover, the block w(e) of OD is of principal type.

(vii) For any ψ ∈ Irr_{K}(H, e) we have π(G, A)(Ind_{H}^{G}(ψ)) = Ind_{D}^{C}(π(H, A)(ψ)) ; that is, the Glauberman correspondence commutes with induction.

(viii) If b is the principal block of OG then w(b) is the principal block of OC, and we have G = H and C = D.

Proof. If d is a block of OO_{p′}(G) such that bd ≠ 0, then b(^{x}d) ≠ 0 for any x ∈ G. Thus, by denoting by H the stabiliser in G of d, the sum Tr_{H}^{G}(d) of all different G−conjugates of d is a G−stable idempotent satisfying bTr_{H}^{G}(d) = b. Thus G acts transitively on the set of blocks d of OO_{p′}(G) satisfying bd ≠ 0. By Glauberman’s lemma [11, 13.8], there is an A−stable block d of OO_{p′}(G) such that bd ≠ 0. Denote still by H the stabiliser of d in G. If G = H then C = D and b = d, and whence all statements of the Theorem follow from 4.1.

We may therefore assume that H is a proper subgroup of G. Then OGTr_{H}^{G}(d) ≅ Ind_{H}^{G}(OHd). Thus the map Tr_{H}^{G} induces an isomorphism of the centers Z(OHd) ≅ Z(OGTr_{H}^{G}(d)), and therefore determines a unique block e of OH such that OGb ≅ Ind_{H}^{G}(OHe). Clearly H and e are A−stable as d is A-stable, and we have O_{p′}(G) ⊂ H. By standard properties of induction of interior algebras (cf. [18]), any defect group of OHe is also a defect group of OGb, and the Brauer categories of OGb, OHe are equivalent. Moreover, H acts transitively on the set of defect groups of OHe, and whence, again by Glauberman’s lemma [11, 13.8], there is an A−stable defect group P of OHe. Since there is some defect group of OGb on which A acts trivially, and since C = C_{G}(A) acts, by [11, 13.9], transitively on the set of A−stable defect groups of OGb, it follows that P ⊆ C_{H}(A).

Therefore the statements (i), (ii), (iii) follow now by induction.

Using 4.3(iii) we also get O_{p′}(D) = O_{p′}(H) ∩ D. By 4.1 applied to e instead of b, the idempotent w(e) is a D−stable block of OO_{p′}(D). This shows (iv). Statement (vii) is clear by the induction theorem 2.3. Since w(b) and w(e) are the sums of the centrally primitive idempotents in KCw(b) and KDw(e), a straightforward computation shows, that this implies w(b) = Tr_{D}^{C}(w(e)), which in turn shows (v). Statement (vi) follws then from (v). By (i), the algebra OGb is isomorphic to the matrix algebra M_{[G:H]}(OHc); in particular, [G : H] divides the degree of any χ ∈ Irr_{K}(G, b). Thus, if b is the principal block, we have G = H. But then we are in the reduced case of 4.1, and so (viii) follows from 4.1 (v). □

Proof of Theorem 1.1 in the general case. The above Theorem 5.1 implies that in order to prove Theorem 1.1, we may assume that we are in the situation of Theorem 4.1. In that case, the proof follows from the developments in section 4, which completes the proof of Theorem 1.1. □

Remark 5.2. By combining 4.4 and 5.1 one gets the main result of Puig [19] on the structure of the source algebras of blocks of p−solvable groups, with the exception of the uniqueness statements in [19]. If G is a finite p−solvable group and b a block of OG, then by 5.1 (with A = 1) there is a subgroup H of G and an H−stable block e of OO_{p′}(H) such that OGb ≅ Ind_{H}^{G}(OHe). By a standard result on the induction of interior algebras due to Puig, the block algebras OHe and OGb have isomorphic source algebras. Thus we may assume that we are in the situation of 4.4 (with O_{p′}(G) instead of N). The structure of the source algebras in that case follows from the structure of the block algebra as described in 4.4.

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Morton E. Harris University of Minnesota School of Mathematics 105 Vincent Hall 206 Church St SE

Minneapolis, MN 55455-0487

and Markus Linckelmann

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

FRANCE