ON p-SOLUBLE GROUPS AND THE NUMBER OF SIMPLE MODULES ASSOCIATED WITH A GIVEN BRAUER PAIR

By LAURENCE BARKER

1: Statement and discussion of the main result

Throughout, we fix an algebraically closed field k of prime characteristic p, and a finite group G. All the results stated in this section will be proved in Section 5.

Theorem 1.1: Suppose that G is p-soluble, let P be a p-subgroup of G, and let b be a block of kN_{G}(P). Then the number of isomorphism classes of simple kN_{G}(P)b-modules with vertex P equals the number of isomorphism classes of simple kG-modules X with vertex P such that b fixes the Green correspondent of X.

Theorem 1.1 further refines Okuyama’s refinement, Corollary 1.3 below, of the p-soluble case of the block form of Alperin’s (weight) conjecture. More to the point, Theorem 1.2 below, a reformulation of Theorem 1.1, tells us that when G is p-soluble, the number of isomorphism classes of simple kG-modules associated with a given self-centralising Brauer pair is a local invariant.

Recall that Sibley [28] assigned, to any indecomposable kG-module X, a Brauer pair, well-defined up to G-conjugation, called a vertex pair of X. The vertex pairs of any simple kG-module are self-centralizing.

Theorem 1.2: Suppose that G is p-soluble, and let (P, w) be a self-centralizing Brauer pair on kG. Then the number of isomorphism classes of simple kGmodules with vertex pair (P, w) equals the number of isomorphism classes of simple kN_{G}(P)-modules with vertex pair (P, w).

The conclusions of Theorems 1.1 and 1.2 are both false for arbitrary G. If, however, instead of assuming G to be p-soluble, we assume G to have abelian Sylow p-subgroups, it is easy to show, using Knörr [11, 3.7], that the conclusions of Theorems 1.1 and 1.2 are both equivalent to the assertion that the block form of Alperin’s conjecture [1, page 371] holds for G. Theorem 1.2 may indicate some virtue in seeking generalisations of Alperin’s conjecture in terms of self-centralizing Brauer pairs. In Puig’s letter [25], he suggests such a generalisation, and shows that it is susceptible to the “inversion” techniques in Knörr–Robinson [12] (suitably adapted for the G-poset of Brauer pairs). These remarks, together with the Külshammer– Puig generalisation [13, 1.20.3] of Dade’s theorem [6, 10] perhaps suggest the possibility of extending the machinery in this paper to reduce, to the almost simple cases, local assertions relating to Alperin’s conjecture.

Corollary 1.3: (Okuyama) Suppose that G is p-soluble.

(a) Let P be a p-subgroup of G, and let a be a block of kG. Then the number of isomorphism classes of simple kGa-modules with vertex P equals the number of isomorphism classes of simple kN_{G}(P)-modules Y with vertex P such that a fixes the Green correspondent of Y.

(b) In particular, the block form of Alperin’s conjecture holds for G.

Manz and Wolf’s remark [16, page 284] “... it has been widely rumored for many years that Okuyama has verified the weight conjecture for p-solvable groups...” only hints at the extent to which the result — its proof and the history of its proof — has been shrouded in mystery. Alperin [1, page 371] states Corollary 1.3(b), citing Okuyama [19], but [19, 4.1] is actually the special case of Corollary 1.3(a) where P is a defect group of a. Corollary 1.3(a) is Okuyama [18, Theorem 5] (reference specifications indicate this undated typescript to have been written in 1980 or 1981). The last line of Okuyama’s argument elides over the need to check that Dade’s equivalence [6, 10] can be chosen compatibly with the Brauer correspondence. However, Puig’s improvement [23, (e)] of Dade [6, 10] establishes this compatibility property; see also Theorem 4.1. Corollary 1.3(b) has a sketch proof in Robinson– Staszewski[27, page 241], and appears as Gres′ [9, 2.17]. In both works, Okuyama’s omission is repeated, and in [9], an error is introduced. The rider to Corollary 1.4 below, which was communicated to me by Isaacs, shows that the first part of [9, 2.1(ii)] is false; the non-trivial 2-weight of S_{4} provides an obvious counter-example to the second part of [9, 2.1(ii)].

The argument in Okuyama [18] can also be interpreted (without invoking Puig [23]) as a proof that if G is p-soluble with p-subgroup P, then the number of isomorphism classes of simple kG-modules with vertex P equals the number of isomorphism classes of simple kN_{G}(P)-modules with vertex P. Isaacs–Navarro [10, Theorem A] generalises this result. Dade [8] announces that a generalisation of Alperin’s conjecture can be reduced to the case of an almost simple group, and he informs me that his reduction subsumes a proof of Corollary 1.3(b). Puig’s manuscript [26], considerably more complicated than this paper but, equally, more profound, obtains a bijection χ_{ω,P,G} from the isomorphism classes of simple kG-modules with vertex P to the isomorphism classes of simple kN_{G}(P)-modules with vertex P. Here, the parameter ω (independent of P and G) is a construction based on the Dade groups of the finite p-groups. Puig informs me of the following: for fixed ω, and varying P, G, the bijections χ_{ω,P,G} are natural (in the sense that they commute with group isomorphisms); ω may be chosen canonically when |G| is odd or p = 2; for each ω, P, G, the bijection χ_{ω,P,G} is compatible with blocks in such a way that Corollary 1.3(a) is a consequence, and χ_{ω,P,G} is compatible with defect pairs (defined below), thus giving another proof of Theorems 1.1 and 1.2.

The rider of the following result was communicated to me by Isaacs, and shows the sense in which Theorem 1.1 is strictly stronger than Corollary 1.3.

Corollary 1.4: Suppose that G is p-soluble, and let P be a p-subgroup of G. Then the following two conditions are equivalent:

(1) There exist two simple kN_{G}(P)-modules with vertex P in distinct blocks of kN_{G}(P) but with Green correspondents in the same block of kG.

(2) There exist two simple kG-modules with vertex P in the same block of kG but with Green correspondents in distinct blocks of kN_{G}(P).

Moreover, when p = 2, there exists a finite soluble group G with a 2subgroup P satisfying these conditions.

If G is p-soluble, and X is a simple kG-module with vertex P and source kP -module S, then the isomorphism class of S is N_{G}(P)-stable, and the twisted group algebra associated with a defect multiplicity module of X splits. That is to say, in the notation of Section 2, there is an equivalence η̃ : k_{#}N_{G}(P, S)˜→kN_{G}(P) of twisted group algebras over N_{G}(P). This discovery, due to Puig, is an immediate consequence of Külshammer–Puig [13, 2.12.4] and Puig’s result in Thévenaz [29, 30.5]. The only use we make, in this paper, of the existence of η̃ is the following illustration of a problem which we shall have to face in Section 4. The defect multiplicity k_{#}N_{G}(P, S)-module V_{G}(P, S) of X, being the inflation of a simple projective k_{#}\overline{N}_{G}(P, S)-module (see Section 2), may be regarded, via η̃, as a simple kN_{G}(P)-module with vertex P. Unfortunately, it is unclear whether or not it is possible thus or otherwise to obtain, for all p-soluble G, a canonical bijection from the isomorphism classes of simple kG-modules with vertex P to the isomorphism classes of simple kN_{G}(P)-modules with vertex P. No canonical choice of η̃ is known. Navarro [17, Theorem B] exhibits such a canonical bijection when |G| is odd. Puig informs me that η̃ may be chosen canonically when |G| is odd or p = 2.

Another problem (more precisely, another manifestation of the same problem) arises in connection with Dade’s theorem [6, 10] (proved in the unpublished typescript Dade [7, 9.1], and also in Puig [23, (e)]), which asserts the existence of an equivalence of twisted group algebras

ν̃ : k_{#}N_{G}(P)/C_{R}(P) ≅ k_{#}N_{G}(P)R/R

via the canonical group isomorphism N_{G}(P)/C_{R}(P) ≅ N_{G}(P)R/R. Here, G is arbitrary with p-subgroup P and normal p′-subgroup R, and the two twisted group algebras involved are defined in terms of a G-stable block e of kR such that kGe has an indecomposable module with vertex P; see Theorem 4.1 for details. Dade mistakenly stated in [6] that ν̃ is canonical, but he corrected this in [7], giving an example to show that “there can be several [equivalences]... each just as natural as the others, with no obvious way to choose one among them in a suitable functorial fashion.” (Note that Dade’s correction underlines the need, in Okuyama’s proof of Corollary 1.3(a), to check that ν̃ may be chosen compatibly with the Brauer correspondence.)

Both of these problems (or rather, this single essential problem) may be circumvented by restricting to C_{G}(P). More precisely, it is well-known (see Section 2) that for G arbitrary, and for any indecomposable kG-moule X with vertex P and source kP -module S, the preimage of C_{G}(P) in k_{#}N_{G}(P, S) splits canonically. The defect multiplicity module V_{G}(X, S) thus restricts to a (well-defined) kC_{G}(P)-module. Also, by Puig [23, (e)] (see Theorem 4.1), ν̃ may be chosen so as to restrict to a canonical equivalence

ν̃_{C} : k_{#}C_{G}(P)/C_{R}(P) ≅ k_{#}C_{G}(P)R/R

and moreover, ν̃_{C} is compatible with the Brauer correspondence.

The preparatory material in Section 2, and the reviews, at the beginnings of Sections 3 and 4, of Fong’s correspondences, may strike some as pedantic. However, we have good reason for paying careful attention to detail. Initial work on refining the p-soluble case of Alperin’s conjecture, begun with Puig in 1992 (see ACKNOWLEDGEMENTS), was beset by a difficulty in keeping track of equivalences of twisted group algebras. The different approaches of Puig and Thévenaz to the parametrisation of primitive interior G-algebras — see the references cited in Thévenaz [29, Section 26] — reveal this difficulty to run deep in G-algebra theory. It may be illuminating to note that the result [3, 8.3], although intended specifically for strengthening Corollary 1.3(a), transpired to be of no help towards this objective because, at the time [3] was written, I had not understood the need to establish more than the mere existence of the implicit equivalence of twisted group algebras. Rather than entangling ourselves in the intricacies of this equivalence itself, which is probably non-canonical anyway, our policy of restricting to the centralizer C_{G}(P) of the p-subgroup P concerned “cuts the Gordian knot” by replacing the original equivalence with a canonical and more tractable one which is eventually realised, in the proof of Proposition 4.8, after making suitable identifications, as simply the identity equivalence kC_{G}(P) ≅ kC_{G}(P). Nevertheless, to achieve this realisation, relationships between other twisted group algebras of C_{G}(P) will need to be examined, and the two above-mentioned problems (the single essential problem) will be present in the background. It seems worth making an effort to avoid eliding over these subtleties which, perhaps, I am not alone in having been confused by.

Any primitive interior G-algebra A (over k or over a suitable discrete valuation ring) may be assigned a pair (P, W), well-defined up to G-conjugacy, called a defect pair of A, where P is a p-subgroup of G, and W is a simple kC_{G}(P)-module. Defect pairs will be more thoroughly investigated in [4]. In Section 2, we shall discuss defect pairs only in the context of kGmodules. Generally, the notion of a defect pair is a refinement of that of a vertex pair. However, of particular significance for our purposes here, given a simple kG-module X with vertex P and source kP -module S, the defect pairs and vertex pairs of X (that is, of End_{k}(X) ) are interchangable, and are determined by the restriction to kC_{G}(P) of the defect multiplicity module V_{G}(X, S). Shifting attention from vertex pairs to defect pairs will not increase the information at our disposal, but it will allow us to make use of module-theoretic operations such as inflation, induction, and Morita equivalence, which would be awkward to express in terms of vertex pairs.

G-Algebra theory, while extending the local theory of Brauer and Green, also encapsulates much of Clifford theory, and is conducive to expressing, in a manner easily susceptible to local reduction, the Morita equivalences inherent in Fong reduction. The use of G-algebra theory to elucidate Fong reduction, well-known to some but not already explicit in the literature, has its origins in Cliff [5] and Puig [21]. Fong reduction, recall, is applicable in the presence of a normal p′-subgroup R of G, and a block e of kR. We shall decompose Fong reduction into two parts: Fong’s first correspondence, reducing to the case where e is G-stable, and Fong’s second correspondence, replacing R with a central cyclic p′-subgroup. In Section 3, we investigate the behaviours, under Fong’s first correspondence, of defect multiplicity modules and defect pairs of indecomposable modules. Section 4 is partially analogous, first reinterpreting the construction in Puig [23] of Dade’s equivalence (denoted ν̃ above), then showing how Fong’s second correspondence respects the defect multiplicity modules and defect pairs of a fairly broad class of indecomposable modules, namely, the simply defective indecomposable modules (which, defined in Section 2, include the simple modules). Our main concern, in this work, is with simple modules, but the wider scope incurs little extra hardship, and allows Sections 2 and 3 to attempt a general exposition of a fundamental connection between G-algebra theory and some traditional Clifford theory.

The general theoretical material having been developed, we shall then be ready, in Section 5, to demonstrate all the results stated in this section. The main effort will be to prove Theorem 1.2, reformulated in terms of defect pairs. It is interesting to note that if, in the inductive hypothesis, the conclusion of Corollary 1.3(a) is substituted for the conclusion of Theorem 1.2, then the argument falters at the application of Fong’s first correspondence (Step 5.2). The reason for this hitch is explianed in the second paragraph of Section 3. Okuyama [18] circumvents this hitch by using different induction parameters, and some group-theoretic reasoning. We might speculate, however, that an attempt to adapt results in this section to situations where fewer group-theoretic constraints are available could perhaps benefit from the more ring-theoretic approach effected by the notion of a defect pair.

2: Multiplicity modules and defect pairs

Recall that J.A.Green defined a G-algebra over k to be an algebra over k upon which G acts as automorphisms. To comprehend the technical manoeuvres in subsequent sections, a little familiarity with Puig’s theory of G-algebras, as established in, for instance, Puig [22], [24] will be advantageous. Thévenaz [29, Chapters 2,3,6] gives a more leisurely exposition of all the relevant G-algebra theory. In this section, we give an account of some very well-known theory of twisted group algebras, and then some fundamental G-algebra theory in the easy special case which concerns us, that of kG-modules and their endomorphism algebras. The presentation below of the constructions we require is more-or-less self-contained except in that we shall refer to other sources for some proofs. Our justification for devoting space to a simplification of material already in the literature is not only to set notation in a way which will turn out to be convenient later, but more importantly, to introduce some terminology enabling us, in subsequent sections, to discuss equivalences of twisted group algebras precisely. Indeed, the concept of an R-normal twisted group algebra of G (see below), and the use of this concept when dealing with defect multiplicity modules or with Fong’s second correspondence are crucial to the way we examine, in Section 4, the effect of Fong’s second correspondence on defect multiplicity modules. The only novelty in this section is the notion of a defect pair, but we shall demonstrate only a few properties of defect pairs here, postponing more thorough discussion to [4].

Let us begin with some generalities on twisted group algebras. Given an algebra A over k (all algebras deemed finite-dimensional), we write A^{∗} for the group of units of A. We take the view that a twisted group algebra of G (over k) is an algebra k_{#}G over k equipped with a k-linear decomposition k_{#}G = ⨁_{g∈G}(k_{#}G)_{g} such that each vector space (k_{#}G)_{g} is 1-dimensional, and (k_{#}G)_{g}(k_{#}G)_{h} = (k_{#}G)_{gh} for all g, h ∈ G. Let us fix twisted group algebras k_{#}G and k_{#}G′ of G. We say that k_{#}G and k_{#}G′ are equivalent provided there exists an equivalence k_{#}G ≅ k_{#}G′, that is, an algebra isomorphism k_{#}G ≅ k_{#}G′ which restricts, for each g ∈ G, to a k-linear isomorphism (k_{#}G)_{g} ≅ (k_{#}G′)_{g}. (We avoid the usual term “isomorphism” in place of “equivalence” because any non-trivial automorphism of G induces an isomorphism kG ≅ kG in the category of twisted group algebras of finite groups, but this isomorphism is not an equivalence.) We define a proper basis for kG to be a basis of the form G̃ = {g̃ : g ∈ G} such that each g̃ ∈ (k_{#}G)_{g}. Given such a proper basis G̃, writing α for the factor set G × G → k^{∗} given by g̃h̃ = α(g, h)g̃h, and writing [k_{#}G] for the image of α in the second-degree cohomology group H^{2}(G, k^{∗}) (where G acts trivially on k^{∗}), it is well-known that [k_{#}G] is independent of the choice of proper basis G̃, and that [k_{#}G] uniquely determines k_{#}G up to equivalence. The subgroup Δ(k_{#}G) of (k_{#}G)^{∗} generated by k^{∗} and G̃ is also independent of the choice of G̃, and k_{#}G is canonically isomorphic to k ⊗_{k(k^{∗})} kΔ(k_{#}G).

We define a k_{#}G-algebra to be an algebra A over k equipped with a structural map k_{#}G → A. The interior G-algebras are precisely the kGalgebras. Any k_{#}G-algebra is a G-algebra whereby each g ∈ G acts as conjugation by g̃. All the twisted group algebras used to prove the results in Section 1 will arise in the following way. Given a simple G-algebra S over k, then by the Noether–Skolem theorem and the fact that k is algebraically closed, the G-algebra structure of S lifts to a k_{#}G^{S}-algebra structure for some twisted group algebra k_{#}G^{S} of G. (We mean that g acts on S as conjugation by any non-zero element of (k_{#}G)_{g}.) Moreover, S determines k_{#}G^{S} up to a unique equivalence; if the G-algebra structure of S lifts to a k_{#}G^{′S}-algebra structure for another twisted group algebra k_{#}G^{′S} of G, then there exists a unique equivalence k_{#}G^{′S} ≅ k_{#}G^{S} commuting with the structural maps k_{#}G^{′S} → S and k_{#}G^{S} → S. Indeed, this follows from the uniqueness property of pull-backs because the short exact sequence 1 → k^{∗} → Δ(k_{#}G^{S}) → G → 1 is determined up to an isomorphism condition by pull-back from the short exact sequence 1 → k^{∗} → Aut(S) → S^{∗} → 1.

Continuing with the twisted group algebras k_{#}G, k_{#}G′ of G, the twisted group algebra k_{#}G∗k_{#}G′ of G is defined to be the subalgebra ⨁_{g∈G}(k_{#}G)_{g} ⊗ (k_{#}G′)_{g} of k_{#}G ⊗ k_{#}G′ with (k_{#}G ∗ k_{#}G′)_{g} = (k_{#}G)_{g} ⊗ (k_{#}G′)_{g}. (All our tensor products, unless otherwise indicated, are over k.) It is easy to check that [k_{#}G ∗ k_{#}G′] = [k_{#}G] + [k_{#}G′]. Given modules V, V′ of k_{#}G, k_{#}G′, respectively, then V ⊗V′ is evidently a k_{#}G∗k_{#}G′-module. Likewise, given a k_{#}G-algebra A, and a k_{#}G′-algebra A′, then A ⊗ A′ is a k_{#}G ∗ k_{#}G′-algebra.

We define the opposite twisted group algebra of k_{#}G to be the twisted group algebra k_{#}G^{∘} of G such that k_{#}G and k_{#}G^{∘} have the same underlying rings, (k_{#}G)_{g} = (k_{#}G^{∘})_{g} as sets for each g ∈ G, and the vector space structures of k_{#}G and k_{#}G^{∘} are related by the condition that the action of any element κ ∈ k^{∗} on k_{#}G coincides with the action of κ^{−1} on k_{#}G^{∘}. Given a proper basis G̃ = {g̃ : g ∈ G} of k_{#}G, we define the opposite basis of G̃ to be the proper basis G̃^{∘} = {g̃^{∘} : g ∈ G} such that each g̃^{∘} coincides with g̃ as an element of the underlying ring. The two factor sets G × G → k^{∗} associated with G̃ and G̃^{∘} commute with inversion in k^{∗}. For any function λ : G → k^{∗}, the proper basis {λ(G)g̃ : g ∈ G} has opposite basis {λ(G)^{−1}g̃^{∘} : g ∈ G}. From this, it is easy to see that there is a canonical equivalence kG ≅ k_{#}G ∗ k_{#}G^{∘} whereby each g ∈ G, regarded as an element of kG, corresponds to g̃ ⊗ g̃^{∘}. If there exists an equivalence μ̃ : k_{#}G˜→k_{#}G′, then we define the opposite equivalence of μ̃ to be the equivalence μ̃^{∘} : k_{#}G^{∘}˜→k_{#}G^{′∘} such that μ̃ and μ̃^{∘} coincide as ring isomorphisms. The following lemma is a special case of Puig’s result in [3, 8.2]. We restate it to be clear about the equivalences involved, but the proof remains unchanged.

Lemma 2.1 (Puig): Let A be an interior G-algebra, let S be a G-stable simple subalgebra of A such that 1_{S} = 1_{A}, let C be the centralizer of S in A, and let k_{#}G be a twisted group algebra of G such that the G-algebra structure of S lifts to a k_{#}G-algebra structure. Then the G-algebra structure of C lifts to a k_{#}G^{∘}-algebra structure in a unique way such that, via the canonical equivalence kG ≅ k_{#}G ∗ k_{#}G^{∘}, we have an interior G-algebra isomorphism A ≅ S ⊗ C given by sc ↔ s ⊗ c for s ∈ S and c ∈ C. ⊓ ⊔

Given a subgroup H ≤ G, the subalgebra k_{#}H := ⨁_{f∈H}(k_{#}G)_{f} of k_{#}G is evidently a twisted group algebra of H. We call k_{#}H the preimage of H in k_{#}G. For a k_{#}G-module X, and a k_{#}H-module Y, we write Res_{H}^{G}(X) for the restriction of X to k_{#}H, and write Ind_{H}^{G}(Y) for the induced k_{#}G-module k_{#}G ⊗_{k_{#}H} Y.

Let R be a normal subgroup of G, let \underline{G} := G/R, and for each g ∈ G, let \underline{g} denote the image of g in \underline{G}. Writing k_{#}R for the preimage of R in k_{#}G, let us suppose that the twisted group algebra k_{#}G is equipped with an equivalence ν̃ : kR˜→k_{#}R such that ν̃(R) is normal in Δ(k_{#}G). Then we call k_{#}G an R-normal twisted group algebra of G. We identify kR with k_{#}R via ν̃, and in particular, write ν̃(x) = x for x ∈ R. We define an R-normal basis for k_{#}G to be a proper basis {g̃ : G ∈ G} such that x = x̃ and xg̃ = x̃g for all x ∈ R and g ∈ G. Then the identification of kR with k_{#}R allows us to regard R as a normal subgroup of Δ(k_{#}G), whereupon the conjugation actions of g and of g̃ on R coincide, and g̃x = g̃xg̃^{−1}g̃ = gxg^{−1}g̃ = g̃x. Suppose that k_{#}G′ is another R-normal twisted group algebra of G. We define an R-normal equivalence k_{#}G˜→k_{#}G′ to be an equivalence k_{#}G˜→k_{#}G′ which restricts to the identity automorphism of kR. Clearly, given an R-normal equivalence μ̃ : k_{#}G˜→k_{#}G′, then μ̃^{∘} : k_{#}G^{∘}˜→k_{#}G^{′∘} is also R-normal.

The following cautionary example should make it clear that the notion of an R-normal twisted group algebra of G is not just a reinvention of a “graded Clifford system” or a “crossed product”. Suppose that p is odd, let G := D_{8}/Z(D_{8}), and let R be the subgroup of G generated by the image of a reflection. Let k_{#}G := k ⊗_{kZ(D_{8})} kD_{8}, where the central involution in D_{8} acts on k as negation. Despite k_{#}R being equivalent to kR, it is impossible to give k_{#}G an R-normal structure.

To indicate the use we shall make of R-normal twisted group algebras of G, suppose that the simple G-algebra S, as above, is also an interior Ralgebra in such a way that the action of R by conjugation coincides with the restriction to R of the action of G. By the pull-back construction of Δ(k_{#}G^{S}), there exists a unique equivalence ν̃ : kR˜→k_{#}R^{S} such that θ_{R} = θ_{G} ν̃, where k_{#}R^{S} is the preimage of R in k_{#}G^{S}, and θ_{R} : kR → S and θ_{G} : k_{#}G^{S} → S are the structural maps. Given a non-zero element g̃ ∈ (k_{#}G^{S})_{g}, and x ∈ R, then g̃ν̃(x)g̃^{−1} is clearly a k^{∗}-multiple of ν̃(^{g}x). But θ_{G}(g̃ν̃(x)g̃^{−1}) = ^{g}θ_{R}(x) = θ_{G}(ν̃(^{g}x)), so g̃ν̃(x)g̃^{−1} = ν̃(^{g}x). Therefore ν̃(R) is normal in Δ(k_{#}G^{S}). We have shown that k_{#}G^{S}, equipped with ν̃, is R-normal.

Still supposing k_{#}G to be an R-normal twisted group algebra of G, let k_{#}\underline{G} be a twisted group algebra of \underline{G}, and let θ be an algebra epimorphism k_{#}G → k_{#}\underline{G} such that θ((k_{#}G)_{g}) = (k_{#}\underline{G})_{g} and θ(x) = 1 for all g ∈ G and x ∈ R. Then we call k_{#}G an inflation of k_{#}\underline{G} to an R-normal twisted group algebra of G. We regard k_{#}\underline{G} as a k_{#}G-algebra via θ. The following facts are easy to check. The structural map θ provides a bijective correspondence between the R-normal bases G̃ = {g̃ : g ∈ G} of k_{#}G, and the proper bases ˜ \underline{G} = \{˜\underline{g} : \underline{g} ∈ \underline{G}\} of k_{#}\underline{G} whereby G̃ ↔ ˜ \underline{G} provided each θ(g̃) = ˜ \underline{g}. When G̃ and ˜ \underline{G} thus correspond, we call G̃ the lift of ˜ \underline{G} to k_{#}G. Then the factor sets α : G × G → k^{∗} and \underline{α} : \underline{G} × \underline{G} → k^{∗} associated with G̃ and ˜ \underline{G} determine each other by the condition that α(g, h) = \underline{α}(\underline{g}, \underline{h}) for all g, h ∈ G. Any twisted group algebra of \underline{G} inflates to an R-normal twisted group algebra of G, and any R-normal twisted group algebra of G is an inflation of a twisted group algebra of \underline{G}. Furthermore, given another twisted group algebra k_{#}\underline{G}′ of \underline{G}, supposing k_{#}G′ to be an inflation of k_{#}\underline{G}′ to an R-normal twisted group algebra of G, and letting θ′ be the structural map k_{#}G′ → k_{#}\underline{G}′, then there is a bijective correspondence between the R-normal equivalences μ̃ : k_{#}G˜→k_{#}G′ and the equivalences ˜ \underline{μ} : k_{#}\underline{G}˜→k_{#}\underline{G}′ whereby μ̃ ↔ ˜ \underline{μ} provided θ′μ̃ = ˜ \underline{μ}θ. Thus the equivalence class of a twisted group algebra of \underline{G} uniquely determines the R-normal equivalence class of its inflations.

Let us turn now to some G-algebra theory, particularly representation theory of kG. Given a subgroup H ≤ G, we define \overline{N}_{G}(H) := N_{G}(H)/H. Given a G-algebra A over k, we write Tr_{H}^{G} for the relative trace map A^{H} → A^{G}, and define A_{H}^{G} := Tr_{H}^{G}(A^{H}). For a p-subgroup P ≤ G, we write Br_{P}^{G} for the Brauer map (kG)^{P} → kC_{G}(P). The following version of Nagao’s theorem is immediate from Landrock [15, III.1.12, III.5.1].

Theorem 2.2 (Nagao): Let e be a central idempotent of kG, let P be a psubgroup of G, and let c := Br_{P}^{G}(e). Then c ≠ 0 if and only if there exists an indecomposable kGe-module with vertex P. When these equivalent conditions hold, c is a central idempotent of kN_{G}(P), and the Green correspondence restricts to a bijective correspondence between the isomorphism classes of indecomposable kGe-modules with vertex P, and the isomorphism classes of indecomposable kN_{G}(P)c-modules with vertex P. ⊓ ⊔

As in Puig [24] and Thévenaz [29, Chapter 2], a multiplicity module can be constructed for any pointed group on any G-algebra. We review this construction in the case of the endomorphism algebra of a kG-module, customizing notation for our application, and dispensing with some abstractions. Let H ≤ G, let U be an indecomposable kH-module, and let X be any kG-module such that U is a direct factor of the restriction Res_{H}^{G}(X). The linear endomorphism algebra End_{k}(X) is an interior G-algebra whose structural map is the representation of X. (It is via this observation that our constructions here are specialisations of those in [24] and [29].) The stabilizer N_{G}(H, U) of U in N_{G}(H) contains HC_{G}(H). The idempotents i of End_{kH}(X) such that U ≅ i Res_{H}^{G}(X) comprise a conjugacy class β of primitive idempotents of End_{kH}(X). Let M_{β} be the maximal ideal of End_{kH}(X) not containing β, let Π_{G}(X, U) be the quotient algebra End_{kH}(X)/M_{β}, and let s_{X,U} be the canonical projection End_{kH}(X) → Π_{G}(X, U). Since N_{G}(H, U) stabilizes M_{β}, we may regard Π_{G}(X, U) as an N_{G}(H, U)-algebra by insisting that s_{X,U} is an N_{G}(H, U)-algebra epimorphism. Now Π_{G}(X, U) is a simple algebra, so there exists a twisted group algebra k_{#}N_{G}(H, U) of N_{G}(H, U) such that the N_{G}(H, U)-algebra structure of Π_{G}(X, U) lifts to a k_{#}N_{G}(H, U)-algebra structure. Let θ_{X,U} denote the stuctural map k_{#}N_{G}(H, U) → Π_{G}(X, U). We define the multiplicity module V_{G}(X, U) of U in X to be the k_{#}N_{G}(H, U)- module obtained by restricting a simple Π_{G}(X, U)-module via θ_{X,U}. The action of Π_{G}(X, U) on V_{G}(X, U) engenders an identification of Π_{G}(X, U) with End_{k}(V_{G}(X, U)), and then θ_{X,U} is the representation of V_{G}(X, U).

The multiplicity module V_{G}(X, U) is well-defined up to isomorphism because k_{#}N_{G}(H, U) is determined up to a unique equivalence. Since an equivalence k_{#}N_{G}(H, U) ≅ k_{#}N_{G}(H, U) may induce a non-trivial permutation of the isomorphism classes of k_{#}N_{G}(H, U)-modules, we emphasise Puig’s injunction [24, 5.4] against replacing k_{#}N_{G}(H, U) with its equivalence class.

Given another kG-module X′ such that U is a direct factor of Res_{H}^{G}(X′), we can regard End_{k}(X) ⊕ End_{k}(X′) as a subalgebra of End_{k}(X ⊕ X′) with End_{k}(X) annihilating X′ and with End_{k}(X′) annihilating X. From this, we see that k_{#}N_{G}(H, U) can be chosen independently of X, and such that V_{G}(X ⊕ X′, U) ≅ V_{G}(X, U) ⊕ V_{G}(X′, U). For any kG-module Y, we can still define V_{G}(Y, U) by taking V_{G}(Y, U) to be the zero k_{#}N_{G}(H, U)-module when U is not a direct factor of Res_{H}^{G}(Y). Given another kG-module Y′, we again have V_{G}(Y ⊕ Y′, U) ≅ V_{G}(Y, U) ⊕ V_{G}(Y′, U). (In fact, Puig has observed that V_{G}(−, U) is an additive functor Mod(kG) → Mod(k_{#}N_{G}(H, U)) .)

Now End_{kH}(X) inherits an interior C_{G}(H)-algebra structure from End_{k}(X), and Π_{G}(X, U) inherits an interior C_{G}(H)-algebra structure from End_{kH}(X) via s_{X,U}. Then k_{#}N_{G}(H, U) becomes a C_{G}(H)-normal twisted group algebra of N_{G}(H, U) whereby each x ∈ C_{G}(H) corresponds to (and will be identified with) the element of (k_{#}N_{G}(H, U))_{x} which has the same image as x in Π_{G}(X, U). Thus V_{G}(X, U) restricts to a kC_{G}(H)-module.

For a p-subgroup P of G, the inclusion \overline{C}_{G}(P) ↪ P C_{G}(P) induces a group isomorphism C_{G}(P)/Z(P) ≅ P C_{G}(P)/P, via which we shall identify C_{G}(P)/Z(P) with P C_{G}(P)/P, denoting both of these quotient groups by \overline{C}_{G}(P). It is well-known (see Landrock [15, III.3.9], for instance) that the canonical epimorphism kC_{G}(P) → k\overline{C}_{G}(P) induces a bijective correspondence between the blocks of kC_{G}(P) and the blocks of k\overline{C}_{G}(P), and also induces a bijective correspondence between the isomorphim classes of simple kC_{G}(P)-modules and the isomorphism classes of simple k\overline{C}_{G}(P)-modules. If D is a defect group of a given block of kC_{G}(P), then D ≥ Z(P), and D/Z(P) is a defect group of the corresponding block of k\overline{C}_{G}(P). Likewise, if Q is a vertex of a given simple kC_{G}(P)-module W, then Q/Z(P) is a vertex of the simple k\overline{C}_{G}(P)-module inflating to W. Recall that, for a psubgroup P of G, and a block w of kC_{G}(P), we call (P, w) a Brauer pair on kG. We say that (P, w) is self-centralizing provided w has defect group Z(P). (Self-centralizing Brauer pairs were the pairs originally introduced by R.Brauer.) Given a simple kC_{G}(P)-module W, we call (P, W) a local pair on kG, and we say that (P, W) is self-centralizing provided W has vertex Z(P). Two local pairs (P, W) and (P′, W′) are considered to be the same provided P = P′ and W ≅ W′. We allow G to permute the Brauer pairs on kG and the local pairs on kG by ^{g}(P, w) = (^{g}P, ^{g}w) and ^{g}(P, W) = (^{g}P, ^{g}W) for g ∈ G. By comments above, given a Brauer pair (P, w) and a local pair (P, W) such that w fixes W, then (P, w) is self-centralizing if and only if (P, W) is self-centralizing, and when these conditions hold, W is the isomorphically unique simple kC_{G}(P)w-module. So there is an evident G-invariant bijective correspondence between the self-centralizing Brauer pairs on kG and the self-centralizing local pairs on kG. (To further motivate the nomenculture, we note that, using a comment in Puig [22, page 267], it is not hard to construct a canonical G-invariant bijective correspondence between the local pairs on kG and the local pointed groups on kG. In case this is unclear, details will be given in [4].)

Let us now suppose that X is an indecomposable kG-module with vertex P and source kP -module S. Let \overline{N}_{G}(P, S) := N_{G}(P, S)/P. Given any z ∈ Z(P), then θ_{X,S}(z) ∈ Z(Π_{G}(X, S)). But z is a p-element, and Π_{G}(X, S) is a simple algebra, so θ_{X,S}(z) = 1. Therefore, we may regard k_{#}N_{G}(P, S) as a P C_{G}(P)-normal twisted group algebra of N_{G}(P, S) in such a way that θ_{X,S}(x) = 1 for all x ∈ P. Hence k_{#}N_{G}(P, S) is P -normal, is an inflation of a \overline{C}_{G}(P)-normal twisted group algebra k_{#}\overline{N}_{G}(P, S) of \overline{N}_{G}(P, S), and Π_{G}(X, S) becomes a k_{#}\overline{N}_{G}(P, S)-algebra with structural map \overline{θ}_{X,S} such that θ_{X,S} and \overline{θ}_{X,S} commute with the structural map k_{#}N_{G}(P, S) → k_{#}\overline{N}_{G}(P, S). Note that the k_{#}\overline{N}_{G}(P, S)-algebra structure of Π_{G}(X, S) induces the \overline{N}_{G}(P, S)- algebra structure inherited from End_{kP} (X) via s_{X,S}. Let \overline{V}_{G}(X, S) denote the k_{#}\overline{N}_{G}(P, S)-module obtained by restricting a simple Π_{G}(X, S)-module via \overline{θ}_{X,S}. Then V_{G}(X, S) is isomorphic to the inflation of \overline{V}_{G}(X, S) via the structural map of k_{#}\overline{N}_{G}(P, S). By Puig [24, 6.4], or Thévenaz [29, 19.3], \overline{V}_{G}(X, S) is indecomposable and projective. In particular, the head hd(V_{G}(X, S)) of V_{G}(X, S) is simple. By Clifford’s theorem (suitably adapted in a well-known way to twisted group algebras),

W_{G}(X, S) := Res_{C_{G}(P)}^{N_{G}(P,S)}(hd(V_{G}(X, S)))

is a direct sum of mutually N_{G}(P, S)-conjugate simple kC_{G}(P)-modules. Given a simple kC_{G}(P)-module W, we call (P, W) a defect pair of X provided W occurs in the semisimple kC_{G}(P)-module W_{G}(X, S′) for some source kP -module S′ of X.

The multiplicity k_{#}N_{G}(P, S)-module V_{G}(X, S) is called a defect multiplicity module of X. Let P′ be another vertex of X, and S′ a source kP′- module of X. As is well-known, there exists an element g ∈ G such that P′ = gP and S′ ≅ ^{g}S. The conjugation actions of g on G, on kG, and on End_{k}(X) provide, respectively, a group isomorphism μ_{g} : N_{G}(P, S)˜→N_{G}(P′, S′), an algebra isomorphism σ_{g} : kC_{G}(P)˜→kC_{G}(P′), and an algebra isomorphism ρ_{g} : Π_{G}(X, S)˜→Π_{G}(X, S′). Regarding Π_{G}(X, S′) as an N_{G}(P, S)-algebra via μ_{g}, and as an interior C_{G}(P)-algebra via σ_{g}, then ρ_{g} is both an N_{G}(P, S)- algebra isomorphism and an interior C_{G}(P)-algebra isomorphism. By the uniqueness of pull-backs, there exists an isomorphism of twisted group algebras μ̃_{g} : k_{#}N_{G}(P, S)˜→k_{#}N_{G}(P′, S′) inducing μ_{g}, restricting to σ_{g}, and such that, regarding Π_{G}(X, S′) as a k_{#}N_{G}(P, S)-algebra via μ̃_{g}, then ρ_{g} is a k_{#}N_{G}(P, S)-algebra isomorphism. In particular, V_{G}(P, S) and V_{G}(P′, S′) are isomorphic via μ̃_{g}. This shows a sense in which defect multiplicity modules are unique up to conjugation. It also shows that W_{G}(X, S) is isomorphic to W_{G}(X, S′) via σ_{g}. So the defect pairs of X comprise a G-orbit of local pairs on kG.

Let Y be the indecomposable kN_{G}(P)-module with vertex P in Green correspondence with X. Of course, Y has source kP -module S, and N_{G}(P, S) = N_{N_{G}(P)}(P, S). Thévenaz [29, 20.1(c)(ii)] describes an isomorphism Π_{G}(X, S) ≅ Π_{N_{G}(P)}(Y, S) which is both an N_{G}(P, S)-algebra isomorphism and an interior C_{G}(P)-algebra isomorphism. By the uniqueness of pull-backs, we may identify k_{#}N_{N_{G}(P)}(P, S) with k_{#}N_{G}(P, S) as C_{G}(P)-normal twisted group algebras of N_{G}(P, S) in such a way that V_{N_{G}(P)}(Y, S) ≅ V_{G}(X, S). Thus, any defect multiplicity module of Y may be regarded as a defect multiplicity module of X. It follows that any defect pair of Y is a defect pair of X.

After Sibley[28, 2.6], for any block w of kC_{G}(P) such that θ_{X,S}(w) ≠ 0, we call (P, w) a vertex pair of X. It is well-known (see, for instance, Landrock [15, III.3.2]) that the blocks of kN_{G}(P) coincide with the primitive idempotents of (kC_{G}(P))^{N_{G}(P)}. Furthermore, there is a bijective correspondence between the blocks b of kN_{G}(P) and the N_{G}(P)-orbits of blocks w of kC_{G}(P) whereby b corresponds to the orbit of w provided b is the sum of the N_{G}(P)-conjugates of w. Now let b be the block of kN_{G}(P) fixing Y. Then θ_{Y,S}(b) = 1. In view of the way the Green correspondence preserves defect multiplicity modules, θ_{X,S}(b) = 1. So, given a block w of kC_{G}(P), the condition that (P, w) is a vertex pair of Y, and the condition that (P, w) is a vertex pair of X are both equivalent to the condition that b is the sum of the N_{G}(P)-conjugates of w. Since G acts transitively on the vertices of X, we have recovered Sibley’s observation [28, 2.5] that the vertex pairs of X comprise a G-orbit of Brauer pairs on kG.

Let (P, W) be a defect pair of X. Choosing the source kP -module S such that W occurs in W_{G}(X, S), then b fixes W because θ_{X,S}(b) = 1. Letting w be the block of kC_{G}(P) fixing W, then bw = w, and (P, w) must be a vertex pair of X. Comments above and Theorem 2.2 give:

Lemma 2.3: Given an indecomposable kG-module X with vertex pair (P, w), defect pair (P, W), and source kP -module S, then:

(a) The block w of kC_{G}(P) fixes some N_{G}(P)-conjugate of the simple kC_{G}(P)-module W.

(b) Let Y be the indecomposable kN_{G}(P)-module with vertex P in Green correspondence with X. Then Y has vertex pair (P, w), defect pair (P, W), and source kP -module S.

(c) Let b be the block of kN_{G}(P) fixing Y. Then b is the sum of the N_{G}(P)-conjugates of w. Also, b fixes both V_{G}(X, S) and W.

(d) Let a be the block of kG fixing X. Then Br_{P}^{G}(a)w = w, and Br_{P}^{G}(a) fixes both V_{G}(X, S) and W. ⊓ ⊔

We say that X is simply defective provided V_{G}(X, S) is simple (and then all the defect multiplicity modules of X are simple). Sufficient criteria for X to be simply defective are, by Picaronny–Puig [20, Proposition 1], that X is simple, or by [2, 1.2], that the vertices of X coincide with the defect groups of the block of kG fixing X. If X is simply defective, and W is a simple kC_{G}(P)-module, then (P, W) is a defect pair of X provided some N_{G}(P)-conjugate of W occurs in the semisimple kC_{G}(P)-module

W_{G}(X, S) = Res_{C_{G}(P)}^{N_{G}(P,S)}(V_{G}(X, S)).

Lemma 2.4 (Picaronny–Puig) Let X be a simply defective indecomposable kG-module.

(a) Each defect pair of X, and each vertex pair of X, is self-centralizing.

(b) Let P be a vertex of X. There is a bijective correspondence between the defect pairs of X taking the form (P, W), and the vertex pairs of X taking the form (P, w), whereby (P, W) ↔ (P, w) provided W is the isomorphically unique simple kC_{G}(P)-module.

PROOF: Part (a) is shown in Picaronny–Puig[20, page 71]; see also Puig’s clarification presented in the proof of [2, 1.1]. Part (b) follows. ⊓ ⊔

Given a p-group Q, we define a local kQ-module to be an indecomposable kQ-module with vertex Q. Given a local pair (Q, W) on kG, and a local kQmodule T, then by Puig [24, 2.10.3, 9.12], there exists an indecomposable kG-module with defect pair (Q, W) and source kQ-module T.

3: Multiplicity modules and Fong’s first correspondence

Fixing a normal p′-subgroup R of G, and a block e of kR with stabilizer T in G, then e may be regarded as a central idempotent of kT, and the primitive idempotent f := Tr_{T}^{G}(e) of (kR)^{G} may be regarded as a central idempotent of kG.

Cliff observed in [5, 1.6(iii)] that Fong’s first correspondence provides an injective function from the isomorphism classes of indecomposable kT emodules with vertex P, and the isomorphism classes of indecomposable kGfmodules with vertex P (see Lemma 3.1). Alas, this function is not surjective when kT e has an indecomposable module with a vertex G-conjugate but not T -conjugate to P. Thus, when studying indecomposable kGf-modules with a given vertex, and seeking a reduction to kT e, there appears to be an irritating consideration of G-fusion in T. (In particular, this issue arises when counting simple kGf-modules with a given vertex.) Fortunately, confining attention to indecomposable kGf-modules with a given defect pair (P, W), a felicitous reduction is provided by a bijective correspondence as follows. Assuming that kGf has an indecomposable module with defect pair (P, W), then by Lemma 2.3(d), Br_{P}^{G}(e) fixes W. Since f is the sum of the G-conjugates of e, and Br_{P}^{G} annihilates (kR)_{Q}^{P} for all Q < P, we have Br_{P}^{G}(f) = ∑_{e′} Br_{P}^{G}(e′), where e′ runs over the P -stable G-conjugates of e. So we can replace e with a suitable G-conjugate so as to ensure that Br_{P}^{G}(e) does not annihilate W. Then (see Proposition 3.3), kT has a defect pair of the form (P, W_{T}) such that Fong’s first correspondence restricts to a bijective correspondence between the isomorphism classes of indecomposable kGf-modules with defect pair (P, W), and the isomorphism classes of indecomposable kT e-modules with defect pair (P, W_{T}). Moreover, W and W_{T} are themselves related by a “local” case of Fong’s first correspondence.

The induced algebra Ind_{T}^{G}(kT e) is, by Puig’s definition [22, 3.3], the interior G-algebra kG ⊗_{kT} kT e ⊗_{kT} kG with structural map given by g ↦ ∑_{yT ⊆G} gy ⊗ 1 ⊗ y^{−1} for g ∈ G, and with multiplication such that, for x_{1}, y_{1}, x_{2}, y_{2} ∈ G and η_{1}, η_{2} ∈ kT e, the product (x_{1} ⊗ η_{1} ⊗ y_{1}^{−1})(x_{2} ⊗ η_{2} ⊗ y_{2}^{−1}) vanishes unless y_{1}T = x_{2}T, in which case it equals x_{1} ⊗ η_{1}y_{1}^{−1}x_{2}η_{2} ⊗ y_{2}^{−1}. Using the fact that ege = 0 for all g ∈ G − T, it is easy to check that there is an interior G-algebra isomorphism

Ind_{T}^{G}(kT e) ≅ kGf

given by x ⊗ η ⊗ y^{−1} ↔ xηy^{−1} for x, y ∈ G and η ∈ kT e. We now see that the induction functor Ind_{T}^{G} on modules restricts to a Morita equivalence

Ind_{T}^{G} : Mod(kT e)˜→Mod(kGf).

This is Fong’s first correspondence; see Cliff[5, Section 1]. The inverse Morita equivalence is evidently given by the condition that, for any kT e-module Y with corresponding kGe-module X = Ind_{T}^{G}(Y), we have Y = e Res_{T}^{G}(X).

We shall also need a twisted group algebra version of Fong’s first correspondence. Let k_{#}G be an R-normal twisted group algebra of G. Then the preimage k_{#}T of T in k_{#}G is also R-normal. We can still regard e, f as central idempotents of k_{#}T, k_{#}G, respectively, and the appropriate induction functor still restricts to a Morita equivalence

Ind_{T}^{G} : Mod(k_{#}T e)˜→Mod(k_{#}Gf)

with inverse Morita equivalence as described before.

We now show that the Glaubermann correspondence (the special case, as in Okuyama [19, page 311], pertaining to our scenario here) can be formulated in terms of the Brauer correspondence. (I have been unable to locate, in the existing literature, a clear discussion of this well-known observation.) Let us fix a p-subgroup P of T such that kT e has an indecomposable module with vertex P, and define c := Br_{P}^{G}(e). Then c = Br_{P}^{T} (e), so by Theorem 2.2, c ≠ 0. Evidently, we may regard c and e as blocks of kP C_{R}(P) and kP R, respectively, in Brauer correspondence with common defect group P. Hence, in fact, c is a block of kC_{R}(P). Furthermore, N_{T} (P) is clearly the stabilizer of c in N_{G}(P). It is easy to check that the isomorphically unique simple kC_{R}(P)c-module is the Glaubermann correspondent of the isomorphically unique simple kRe-module (but we shall not use this fact).

Lemma 3.1: Given an indecomposable kT e-module Y with vertex P and source kP -module S, then the corresponding indecomposable kGf-module X = Ind_{T}^{G}(Y) has vertex P and source kP -module S. Moreover, Y is simply defective if and only if X is simply defective.

PROOF: Of course, X has vertex P and source kP -module S. Evidently, we may identify k_{#}N_{T} (P, S) with the preimage of N_{T} (P, S) in k_{#}N_{G}(P, S) in such a way that V_{T} (Y, S) is a direct factor of Res_{N_{T} (P,S)}^{N_{G}(P,S)}(V_{G}(X, S)). The twisted group algebras k_{#}N_{T} (P, S) and k_{#}N_{G}(P, S) are C_{R}(P)-normal, and N_{T} (P, S) is the stabilizer of c in N_{G}(P, S), so c is a central idempotent of k_{#}N_{T} (P, S), and d_{N} := Tr_{N_{T} (P,S)}^{N_{G}(P,S)}(c) is a central idempotent of k_{#}N_{G}(P, S). We have a case of Fong’s first correspondence

Ind_{N_{T} (P,S)}^{N_{G}(P,S)} : Mod(k_{#}N_{T} (P, S)c)˜→Mod(k_{#}N_{G}(P, S)d_{N}).

By Lemma 2.3(c), c fixes V_{T} (Y, S), so

Ind_{N_{T} (P,S)}^{N_{G}(P,S)}(V_{T} (Y, S)) ≅ V_{G}(X, S). ⊓ ⊔

Now C_{T} (P) is the stabilizer of c in C_{G}(P), and in particular, c is a central idempotent of kC_{T} (P). Let d := Tr_{C_{T} (P)}^{C_{G}(P)}(c), which is a central idempotent of kC_{G}(P). Fong’s first correspondence may be cast as

Ind_{C_{T} (P)}^{C_{G}(P)} : Mod(kC_{T} (P)c)˜→Mod(kC_{G}(P)d).

Lemma 3.2: There is a bijective correspondence between the local pairs on kG of the form (P, W) with W fixed by d, and the local pairs on T of the form (P, W_{T}) with W_{T} fixed by c, whereby (P, W) ↔ (P, W_{T}) provided W ≅ Ind_{C_{T} (P)}^{C_{G}(P)}(W_{T}), in which case, (P, W) is self-centralizing if and only if (P, W_{T}) is self-centralizing.

PROOF: This is clear. ⊓ ⊔

Proposition 3.3: Let Y, X be indecomposable modules of kT e, kGf, respectively, which correspond as in Lemma 3.1. Given local pairs of the form (P, W_{T}), (P, W) on T, G, respectively, which correspond as in Lemma 3.2, then Y has defect pair (P, W_{T}) if and only if X has defect pair (P, W).

PROOF: Using the proof of Lemma 3.1, the uniqueness of defect multiplicity modules up to conjugacy, and a Frattini argument, it is not hard to show that Y has vertex P and source kP -module S if and only if X does. We may assume that these two equivalent conditions hold. Applying the inverse form of Fong’s first correspondence, we get

hd(V_{T} (Y, S)) ≅ c Res_{N_{T} (P,S)}^{N_{G}(P,S)}(hd(V_{G}(X, S))).

Restricting to C_{R}(P), and recognising Fong’s first correspondence again,

Ind_{C_{R}(P)}^{C_{G}(P)}(W_{T} (Y, S)) ≅ W_{G}(X, S). ⊓ ⊔

4: Multiplicity modules and Fong’s second correspondence

We continue to work with a normal p′-subgroup R of G, but now assume that the block e of kR is G-stable. Fong’s second correspondence will be presented as a Morita equivalence of the form

Mod(k ˆ\underline{G}ϵ) ≅ Mod(kGe)

where ˆ \underline{G} is a central extension of \underline{G} := G/R by a finite cyclic p′-group E, and ϵ is a block of kE. Let P be a p-subgroup of G such that kGe has an indecomposable module with vertex P, and let d := Br_{P}^{G}(e). (This is consistent with the notation in Section 3, indeed, in that notation, the Gstability of e forces c = d and e = f.) The discussion, in Section 3, of the Glaubermann correspondence shows that d is an N_{G}(P)-stable block of kC_{R}(P). Fong’s second correspondence, with N_{G}(P), C_{R}(P), d in place of G, R, e, respectively, is a Morita equivalence

Mod(k ˆ\underline{N_{G}(P)}δ) ≅ Mod(kN_{G}(P)d)

where ˆ \underline{N_{G}(P)} is a central extension of \underline{N_{G}(P)} := N_{G}(P)/C_{R}(P) by a finite cyclic p′-subgroup D, and δ is a block of kD. Our first task will be to show how Puig’s improvement of Dade’s theorem, Theorem 4.1, allows us to choose D = E, δ = ϵ, and to choose ˆ \underline{N_{G}(P)} to coincide with the preimage of N_{G}(P)R/R in ˆ \underline{G}. Having set the machinery in this way, we shall find (see the proof of Proposition 4.8) that any simply defective indecomposable kGe-module X with vertex P, and the corresponding indecomposable k ˆ\underline{G}ϵ module ˆ \underline{X}, have defect multiplicity modules which can be related by a formula involving an extension of a simple kC_{R}(P)d-module. Furthermore ˆ \underline{X} is simply defective, has a vertex P̂ canonically isomorphic to P, and the defect pairs of X and of ˆ \underline{X} are related in terms of Fong’s second correspondence cast as a Morita equivalence

Mod(kC_{\underline{G}}(P̂)ϵ) ≅ Mod(kC_{G}(P)d).

For any element g ∈ G, let \underline{g} denote the image of g in \underline{G}. The simple subalgebra kRe of kGe has centralizer (kGe)^{R}, so by Lemma 2.1, there is an algebra isomorphism

ϕ̃_{G,e} : (kGe)^{R} ⊗ kRe˜→kGe

given by μ ⊗ η ↦ μη for μ ∈ (kGe)^{R} and η ∈ kRe. Let us recall how (kGe)^{R} acquires the structure of a twisted group algebra k_{#}\underline{G}^{e} of \underline{G}. Regarding kGe as a k(G × G)-module by left-right translation, the left translation action of kGe on itself induces an algebra isomorphism kGe˜→End_{1×G}(kGe). By restriction, we obtain an algebra isomorphism ρ : (kGe)^{R}˜→End_{R×G}(kGe) given by μ = (ρ(μ))(e) for μ ∈ (kGe)^{R}. Since e is G-stable, the simple k(R × R)-module kRe is G × G-stable. So kGe = ⨁_{gR⊆G} gkRe as a direct sum of mutually isomorphic simple k(R × R)-modules. In particular, each (gkRe)^{R} is 1-dimensional. For each \underline{g} ∈ \underline{G}, we choose a non-zero element ˜\underline{g}^{e} ∈ (gkRe)^{R}. It is easy to check that k_{#}\underline{G}^{e} := (kGe)^{R} is a twisted group algebra of \underline{G} with proper basis ˜ \underline{G}^{e} := \{˜\underline{g}^{e} : \underline{g} ∈ \underline{G}\}.

Let k_{#}G^{e} be the inflation of k_{#}\underline{G}^{e} to an R-normal twisted group algebra of G, and let G̃^{e} = \{g̃^{e} : g ∈ G\} be the lift of ˜ \underline{G}^{e} to an R-normal basis of k_{#}G^{e}. This k_{#}\underline{G}^{e} is a k_{#}G^{e}-algebra via the epimorphism k_{#}G^{e} → k_{#}\underline{G}^{e} given by g̃^{e} ↦ ˜ \underline{g}^{e}. We regard kGe as an interior G-algebra via the projection g ↦ ge. By Lemma 2.1, the G-algebra structure of kRe provided by conjugation lifts to a k_{#}G^{e∘}-algebra structure in such a way that

ϕ̃_{G,e} : k_{#}\underline{G}^{e} ⊗ kRe˜→kGe

is an interior G-algebra isomorphism via the canonical equivalence kG ≅ k_{#}G^{e} ∗ k_{#}G^{e∘}. The structural map θ_{G,e} : k_{#}G^{e∘} → kRe is determined by the condition that ge = ˜ \underline{g}^{e}θ_{G,e}(g̃^{e∘}) for g ∈ G. Since k_{#}G^{e} is R-normal, so is k_{#}G^{e∘}. For all y ∈ R, we have ˜ \underline{y}^{e} = 1, hence θ_{G,e}(y) = ye. That is to say, θ_{G,e} restricts to the structural map of the interior R-algebra kRe.

Given a subgroup R ≤ H ≤ G, we define \underline{H} := H/R as a subgroup of \underline{G}. Writing k_{#}\underline{H}^{e}, k_{#}H^{e}, k_{#}H^{e∘} for the preimages of \underline{H}, H, H in k_{#}\underline{G}^{e}, k_{#}G^{e}, k_{#}G^{e∘}, respectively, then k_{#}\underline{H}^{e} = (kHe)^{R} is a k_{#}H^{e}-algebra, kRe is a k_{#}H^{e∘}-algebra, and ϕ̃_{G,e} restricts to the interior H-algebra isomorphism

ϕ̃_{H,e} : k_{#}\underline{H}^{e} ⊗ kRe˜→kHe.

In the parallel “local” situation, k_{#}\underline{N_{G}(P)}^{d} := (kN_{G}(P)d)^{R} is a twisted group algebra of \underline{N_{G}(P)}, we lift k_{#}\underline{N_{G}(P)}^{d} to a C_{R}(P)-normal twisted group algebra k_{#}N_{G}(P)^{d} of N_{G}(P), we realise kC_{R}(P)d as a k_{#}N_{G}(P)^{d∘}-algebra, and we have an interior N_{G}(P)-algebra isomorphism

ϕ̃_{N_{G}(P),d} : k_{#}\underline{N_{G}(P)^{d}} ⊗ kC_{R}(P)d˜→kN_{G}(P)d.

Given a subgroup C_{R}(P) ≤ L ≤ N_{G}(P), we define \underline{L} := L/C_{R}(P) as a subgroup of \underline{N_{G}(P)}. (This is consistent with the notation above because if R ≤ L ≤ N_{G}(P) then C_{R}(P) = R and \underline{L} is defined unambiguously.) As before, ϕ̃_{N_{G}(P),d} restricts to an interior L-algebra isomorphism

ϕ̃_{L,d} : k_{#}\underline{L}^{d} ⊗ kC_{R}(P)d˜→kLd.

We define group epimorphisms π^{d} : N_{G}(P) → \underline{N_{G}(P)} and π^{e} : N_{G}(P) → \underline{N_{G}(P)R}, and define a group isomorphism ν : \underline{N_{G}(P)}˜→\underline{N_{G}(P)R} where π^{d} is the evident canonical epimorphism, ν is induced by the inclusion N_{G}(P) ↪ N_{G}(P)R, and π^{e} := νπ^{d}. Via ν, we regard k_{#}\underline{N_{G}(P)R}^{e} as a twisted group algebra of \underline{N_{G}(P)}, and in particular, regard k_{#}\underline{C_{G}(P)R}^{e} as a twised group algebra of \underline{C_{G}(P)}.

In the following version of Dade’s theorem, the construction of ν̃_{C} is implicit in Külshammer–Robinson [14, page 101], while the extendibility of ν̃_{C} to an equivalence ν̃ as specified is essentially Puig’s improvement [23, (e)] of Dade [6, 10], [7, 9.1].

Theorem 4.1 (Dade, Puig): We have k_{#}\underline{C_{G}(P)R}^{e} = (kC_{G}(P)Re)^{P R}. The Brauer map Br_{P}^{G} restricts to an equivalence ν̃_{C} : k_{#}\underline{C_{G}(P)R}^{e}˜→k_{#}\underline{C_{G}(P)}^{d} of twisted group algebras of \underline{C_{G}(P)}. Moreover, ν̃_{C} extends to an equivalence ν̃ : k_{#}\underline{N_{G}(P)R}^{e}˜→k_{#}\underline{N_{G}(P)}^{d} of twisted group algebras of \underline{N_{G}(P)}.

PROOF: The asserted equality holds because P stabilises, and hence fixes, each of the 1-dimensional summands in

k_{#}\underline{C_{G}(P)R}^{e} = ⨁_{xC_{R}(P)⊆C_{G}(P)}(xkRe)^{R}.

Landrock [15, III.2.1], the identity N_{P R}(P) = P C(R), and the fact thatP is a Sylow p-subgroup of both P R and P C_{R}(P), give Br_{P}^{G}((kC_{G}(P)R)^{P R}) = (kC_{G}(P))^{C_{R}(P)}. So Br_{P}^{G} restricts to an algebra epimorphism ν̃_{C} with domain and codomain as specified. Clearly, ν̃_{C} is an equivalence.

Let k_{#}N_{G}(P)^{e} denote the preimage of N_{G}(P) in k_{#}N_{G}(P)R^{e}. Let N denote the normalizer in (kRe)^{∗} of the image of P in (kRe)^{∗}. Now Br_{P}^{G} restricts to a group epimorphism ((kRe)^{P})^{∗} → (kC_{R}(P)d)^{∗} which, by [23, (e)], extends to a group epimorphism γ : N → (kC_{R}(P)d)^{∗} such that γ(t)Br_{P}^{G}(η)γ(t^{−1}) = Br_{P}^{G}(tηt^{−1}) for all t ∈ N and η ∈ (kRe)^{P}. By the uniqueness property of pull-backs, there exists a unique equivalence

μ̃^{∘} : k_{#}N_{G}(P)^{e∘}˜→k_{#}N_{G}(P)^{d∘}

such that θ_{N_{G}(P),d}(μ̃^{∘}(g̃^{e∘})) = γ(θ_{G,e}(g̃^{e∘})) for all g ∈ N_{G}(P). It is easily checked that μ̃^{∘} is C_{R}(P)-normal. Therefore, we can choose tha proper basis ˜ \underline{N_{G}(P)}^{d} of k_{#}\underline{N_{G}(P)}^{d} in such a way that μ̃^{∘}(g̃^{e∘}) = g̃^{d∘} for all g ∈ N_{G}(P). The opposite equivalence of μ̃^{∘} is the R-normal equivalence μ̃ : k_{#}N_{G}(P)^{e}˜→k_{#}N_{G}(P)^{d} given by μ̃(g̃^{e}) = g̃^{d}. Straightforward manipulations confirm that μ̃ induces, via π^{e} and π^{d}, an equivalence ν̃ with the required properties. ⊓ ⊔

Theorem 1.2 will be proved by an inductive argument making use of the fact that p-soluble groups are closed under subgroups, factor groups, and extension. To prepare for this, the twisted group algebras in Dade’s theorem will be replaced with central extensions by a cyclic p′-group. Let us choose, and fix, a finite cyclic p′-group E of order divisible by |\underline{G}|_{p′}. Let ϵ be any block of kE such that the kEϵ-modules are faithful, so that E acts on any kEϵ-module via a group monomorphism E → k^{∗}. It is well-known that the exponent of the abelian group H^{2}(\underline{G}, k^{∗}) divides |\underline{G}|_{p′}, so we can choose the respective proper bases ˜ \underline{G}^{e}, G̃^{e}, G̃^{e∘} of k_{#}\underline{G}^{e}, k_{#}G^{e}, k_{#}G^{e∘} such that the factor set \underline{α}^{e} : \underline{G} × \underline{G} → k^{∗} associated with ˜ \underline{G}^{e} takes values in the image of E. Then the R-normal bases k_{#}G^{e}, k_{#}G^{e∘} of the R-normal twisted group algebras k_{#}G^{e}, k_{#}G^{e∘} have factor sets α^{e}, α^{e∘}, respectively, satisfying \underline{α}^{e}(\underline{g}, \underline{h}) = α^{e}(g, h) = α^{e∘}(g, h)^{−1} for all g, h ∈ G. Let 1 → E → ˆ \underline{G} → \underline{G} → 1 be a central extension of \underline{G} by E such that \underline{G} has a section \{ˆ\underline{g} : \underline{g} ∈ \underline{G}\} in ˆ \underline{G} with the property that each ˆ \underline{g} ˆ \underline{h} ˆ \underline{gh}^{−1} is the elment of E with image \underline{α}^{e}(\underline{g}, \underline{h}) in k^{∗}. We have an algebra isomorphism ψ_{G,e} : k ˆ\underline{G}ϵ˜→k_{#}\underline{G}^{e} given by ˆ \underline{g}ϵ ↦ ˜ \underline{g}^{e} for all \underline{g} ∈ \underline{G}. We impose a k_{#}G^{e}-algebra structure on k ˆ\underline{G}ϵ by insisting that ψ_{G,e} is a k_{#}G^{e}-algebra isomorphism. The k_{#}G^{e}-algebra structure of k ˆ\underline{G}ϵ, and the interior ˆ \underline{G}-algebra structure of k ˆ\underline{G}ϵ clearly induce the same \underline{G}-algebra structure on k ˆ\underline{G}ϵ. Let ϕ̂_{G,e} := ϕ̃_{G,e}(ψ_{G,e} ⊗ 1), which is an interior G-algebra isomorphism ϕ̂_{G,e} : k ˆ\underline{G}ϵ ⊗ kRe˜→kGe. Since kRe is a simple algebra, ϕ̂_{G,e} provides a Morita equivalence

Φ_{G,e} : Mod(k ˆ\underline{G}ϵ)˜→Mod(kGe).

This is Fong’s second correspondence; see Cliff [5, Section 2].

Given a subgroup R ≤ H ≤ G, we let ˆ \underline{H} denote the preimage of \underline{H} = H/R in ˆ \underline{G}. Note that ϕ̃_{H,e}, ψ_{H,e}, ϕ̂_{H,e} are restrictions of ϕ̃_{G,e}, ψ_{G,e}, ϕ̂_{G,e}, respectively. Given a kˆ\underline{H}ϵ-module ˆ \underline{Y}, we write ˆ \underline{Y}^{e} to denote ˆ \underline{Y} regarded as a k_{#}H^{e}-module via ψ_{H,e} and inflation, that is via the composite map k_{#}H^{e} → k_{#}\underline{H}^{e} → kˆ\underline{H}ϵ. Let S_{R}^{e} be a simple kRe-module, and let S_{H}^{e} denote S_{R}^{e} regarded as a k_{#}H^{e∘}-module via the structural map θ_{H,e}. (This notation makes sense because k_{#}H^{e∘} is R-normal, and θ_{R,e} is the structural map of kRe as an interior R-algebra.) We have Φ_{H,e}( ˆ\underline{Y}) = ˆ \underline{Y}^{e} ⊗ S_{H}^{e}. We can now see that Φ_{H,e}, Φ_{G,e} commute with the restriction functors

Res_{H}^{G} : Mod(kGe) → Mod(kHe) , Res_{\underline{G}} ; _{\underline{H}} : Mod(k ˆ\underline{G}ϵ) → Mod(kˆ\underline{H}ϵ),

and that Φ_{H,e}, Φ_{G,e} commute with the induction functors

Ind_{H}^{G} : Mod(kHe) → Mod(kGe) , Ind_{\underline{G}} ; _{\underline{H}} : Mod(kˆ\underline{H}) → Mod(k ˆ\underline{G}ϵ).

Working with N_{G}(P) in place of G, we choose a finite cyclic p′-group D of order divisible by |\underline{N_{G}(P)}|_{p′}, and choose a block δ of kD such that the kDδ-modules are faithful. As before, we obtain a central extension 1 → D → ˆ \underline{N_{G}(P)} → \underline{N_{G}(P)} → 1 of \underline{N_{G}(P)} = N_{G}(P)/C_{R}(P) by D, a k_{#}N_{G}(P)^{d}-algebra isomorphism ψ_{N_{G}(P),d} : k ˆ\underline{N_{G}(P)}δ˜→k_{#}\underline{N_{G}(P)}^{d}, an interior N_{G}(P)-algebra isomorphism ϕ̂_{N_{G}(P),d} : k ˆ\underline{N_{G}(P)}δ ⊗ kC_{R}(P)d˜→kN_{G}(P)d, and a Morita equivalence Φ_{N_{G}(P),d} : Mod(k ˆ\underline{N_{G}(P)}δ)˜→Mod(kN_{G}(P)d).

We now harness the power of Dade’s equivalence. As in the proof of Theorem 4.1, we may insist that the proper basis ˜ \underline{G}^{d} of k_{#}\underline{G}^{d} satisfies ν(˜\underline{g}^{e}) = π̃^{d}(g)^{d} for all g ∈ N_{G}(P). Then the factor set \underline{α}^{d} : \underline{N_{G}(P)} × \underline{N_{G}(P)} → k^{∗}

associated with ˜ \underline{G}^{d} has the property that \underline{α}^{d}(π^{d}(g), π^{d}(h)) = \underline{α}^{e}(\underline{g}, \underline{h}) for all g, h ∈ N_{G}(P). We are free to choose any finite cyclic P′-group D and block δ of kD satisfying the stipulations above, and |\underline{N_{G}(P)}|_{p′} divides |\underline{G}|_{p′}, so we can insist that D = E, δ = ϵ, ˆ \underline{N_{G}(P)} = ˆ \underline{N_{G}(P)R}, that the epimorphisms ˆ \underline{N_{G}(P)} → \underline{N_{G}(P)} and ˆ \underline{N_{G}(P)R} → \underline{N_{G}(P)R} commute with ν, and that ψ_{N_{G}(P),d} and ψ_{N_{G}(P),e} commute with ν̃.

Given a subgroup C_{R}(P) ≤ L ≤ N_{G}(P), writing ˆ \underline{L} for the preimage of \underline{L} = L/C_{R}(P) in ˆ \underline{N_{G}(P)} = ˆ \underline{N_{G}(P)R}, then ˆ \underline{L} = ˆ \underline{LR}. The k_{#}L^{d}-algebra isomorphism ψ_{L,d} : k ˆ\underline{L}ϵ˜→k_{#}\underline{L}^{d}, commutes with ψ_{LR,e} and ν̃. For any k ˆ\underline{L}ϵmodule ˆ \underline{Y}, we write ˆ \underline{Y}^{d} to denote ˆ \underline{Y} regarded as a k_{#}\underline{L}^{d}-module via ψ_{L,d} and inflation. The Morita equivalence Φ_{L,d} : Mod(k ˆ\underline{L}ϵ)˜→Mod(kLd) is given by Φ_{L,d}( ˆ\underline{Y}) = ˆ \underline{Y}^{d} ⊗ S_{L}^{d}.

We have ˆ \underline{P R} = P̂E, where P̂ := O_{p}( ˆ\underline{P R}). The epimorphisms G → \underline{G} and ˆ \underline{G} → G restrict to isomorphisms P ≅ \underline{P R} ≅ P̂. Let P R ≤ H ≤ G, and let Y, ˆ \underline{Y} be modules of kHe, kˆ\underline{H}ϵ, respectively, which correspond via the Morita equivalence Φ_{H,e}. Since Φ_{P R,e} and Φ_{H,e} commute with induction and restriction, Y is P R-projective if and only if ˆ \underline{Y} is ˆ \underline{P R}-projective. But P, P̂ are Sylow p-subgroups of P R, ˆ \underline{P R}, respectively, so Y is P -projective if and only if ˆ \underline{Y} is P̂-projective. This conclusion still holds when P is replaced with any subgroup of P, so we recover the following version of Cliff [5, 2.5(iii)].

Lemma 4.2 (Cliff): For P R ≤ H ≤ G, the Morita equivalence Φ_{H,e} restricts to a bijective correspondence between the isomorphism classes of indecomposable kHe-modules with vertex P, and the isomorphism classes of indecomposable kˆ\underline{H}ϵ-modules with vertex P̂. ⊓ ⊔

Given a local kP -module S, then since N_{P R}(P) = P C_{R}(P), we can form the indecomposable kN_{P R}(P)d-module S ⊗ S_{C_{R}(P)}^{d}, which has vertex P. We write S_{P R} to denote the indecomposable kP R-module in Green correspondence with S ⊗ S_{C_{R}(P)}^{d}.

Lemma 4.3: There is a bijective correspondence between the isomorphism classes of local kP -modules S, and the isomorphism classes of indecomposable kP Re-modules T with vertex P, whereby S ↔ T provided T ≅ S_{P R}.

PROOF: This is easy to prove by considering the evident algebra isomorphism kN_{P R}(P)d ≅ kP ⊗ kC_{R}(P)d. ⊓ ⊔

For any local kP -module S, we write Ŝ to denote a local kP̂-module such

that Φ_{P R,e}(Ŝ_{\underline{P R}}) ≅ S_{P R}.

Lemma 4.4: There is a bijective correspondence between the isomorphism classes of local kP -modules and the isomorphism classes of local kP̂-modules, whereby each local kP -module S corresponds to Ŝ.

PROOF: This is immediate from Lemmas 4.2 and 4.3. ⊓ ⊔

Lemma 4.5: N_{\underline{G}}(P̂, Ŝ) = ˆ \underline{N_{G}(P, S)} for any local kP -module S.

PROOF: This follows quickly from the fact that Φ_{P R,e} respects the conjugation actions of N\underline{G}(\underline{P R}) on Mod(k ˆ\underline{P R}ϵ) and on Mod(kP Re). ⊓ ⊔

Lemma 4.6: Let S be a local kP -module. The isomorphism classes of indecomposable kGe-modules with vertex P and source kP -module S are, via Φ_{G,e}, in bijective correspondence with the isomorphism classes of indecomposable k ˆ\underline{G}ϵ-modules with vertex P̂ and source kP̂-module Ŝ.

PROOF: Let ˆ \underline{X} be an indecomposable k ˆ\underline{G}ϵ-module, and let X = Φ_{G,e}(ˆ\underline{X}). Lemma 4.2 says that X has vertex P if and only if ˆ \underline{X} has vertex P̂. Suppose now that this condition holds. It is easy to show that X has source kP - module S if and only if S_{P R} is a direct factor of Res_{P R}^{G}(X). Similarly, ˆ \underline{X} has source kP̂-module Ŝ if and only if Ŝ_{\underline{P R}} is a direct factor of Res_{\underline{G}} ; _{\underline{P R}}(ˆ\underline{X}). The assertion follows because Φ_{P R,e} and Φ_{G,e} commute with restriction. ⊓ ⊔

We have C_{\underline{G}}(P̂) = ˆ \underline{C_{G}(P)}, so Φ_{C_{G}(P),d} : Mod(kC_{\underline{G}}(P̂)ϵ)˜→Mod(kC_{G}(P)d) is a Morita equivalence. Let ˆ \underline{W} be a simple kC_{\underline{G}}(P̂)ϵ-module, and let W = Φ_{C_{G}(P),d}(ˆ\underline{W}) be the corresponding simple kC_{G}(P)d-module. Observing that Z(P̂) = O_{p}( ˆ\underline{Z(P)C_{R}(P)}), Lemma 4.2 shows that ˆ \underline{W} has vertex Z(P̂) if and only if W has vertex Z(P). The following result is now clear.

Lemma 4.7: There is a bijective correspondence between the self-centralizing local pairs on kG of the form (P, W) with W fixed by d, and the self-centralizing local pairs on k ˆ\underline{G} of the form (P̂, ˆ \underline{W}) with ˆ \underline{W} fixed by ϵ, whereby (P, W) ↔ (P̂, ˆ \underline{W}) provided the simple modules W, ˆ \underline{W} of kC_{G}(P)d, kC_{\underline{G}}(P̂)ϵ, respectively, correspond via Φ_{C_{G}(P),d}. ⊓ ⊔

Proposition 4.8: Let X be a simply defective indecomposable kGe-module with defect pair (P, W), let ˆ \underline{X} := Φ_{G,e}^{−1}(X), and let (P̂, ˆ \underline{W}) be the selfcentralizing local pair corresponding to (P, W) as in Lemma 4.7. Then ˆ \underline{X} is simply defective with defect pair (P̂, ˆ \underline{W}).

PROOF: We sketch the argument. By Lemma 2.3(d), d fixes W. By Lemma 2.4(a), (P, W) is self-centralizing, so the definition of (P̂, ˆ \underline{W}) makes sense. Let S be a source kP -module of X such that W occurs in the semisimple kC_{G}(P)-module W_{G}(X, S). Lemma 4.6 tells us that ˆ \underline{X} has vertex P̂ and source kP̂-module Ŝ.

Via the equality ˆ \underline{X}^{e} ⊗ S_{G}^{e} = X, and evident isomorphisms End_{k}(ˆ\underline{X}^{e}) ≅ End_{k}(ˆ\underline{X}) of k_{#}G^{e}-algebras, and End_{k}(S_{G}^{e}) ≅ kRe of k_{#}G^{e∘}-algebras, we obtain an interior G-algebra isomorphism τ : End_{k}(ˆ\underline{X}) ⊗ kRe˜→End_{k}(X). Following through the proof of [3, 8.3], and using the simplicity of V_{G}(X, S), we obtain an N_{G}(P, S)-algebra isomorphism

σ : Π_{\underline{G}}(ˆ\underline{X}, Ŝ) ⊗ kC_{R}(P)d˜→Π_{G}(X, S)

given by σ(s_{_{\underline{X}},_{S}}(ζ) ⊗ Br_{P}^{G}(η)) = s_{X,S}(τ(ζ ⊗ η)) for ζ ∈ End_{k_{P}}(ˆ\underline{X}) and η ∈ (kRe)^{P}. The N_{G}(P, S)-algebra structure of the simple algebra Π_{\underline{G}}(ˆ\underline{X}, Ŝ) lifts to a k_{#}N_{G}(P, S)′-algebra structure for some twisted group algebra k_{#}N_{G}(P, S)′ of N_{G}(P, S). The uniqueness property of pull-backs provides an equivalence

σ̃ : k_{#}N_{G}(P, S)′ ∗ k_{#}N_{G}(P, S)^{d∘}˜→k_{#}N_{G}(P, S)

such that σ and σ̃ commute with the two structural maps

k_{#}N_{G}(P, S)′ ∗ k_{#}N_{G}(P, S)^{d∘} → Π_{\underline{G}}(ˆ\underline{X}, Ŝ) ⊗ kC_{R}(P)d,

k_{#}N_{G}(P, S) → Π_{G}(X, S).

Therefore ˆ \underline{X} is simply defective.

In the notation of the proof of Theorem 4.1, we regard the domain of σ as an interior C_{G}(P)-algebra by regarding Π_{\underline{G}}(ˆ\underline{X}, Ŝ) as a k_{#}\underline{C_{G}(P)}^{d}-algebra via μ̃. We have

θ_{N_{G}(P),d}(μ̃^{∘}(x̃^{e∘})) = γ(θ_{G,e}(x̃^{e∘})) = Br_{P}^{G}(θ_{G,e}(x̃^{e∘}))

for all x ∈ C_{G}(P). So σ is an interior C_{G}(P)-algebra isomorphism. Identifying Π_{\underline{G}}(ˆ\underline{X}, Ŝ), Π_{G}(X, S), kC_{R}(P)d with End_{k}(W_{\underline{G}}(ˆ\underline{X}, Ŝ)), End_{k}(W_{G}(X, S)), End_{k}(S_{C_{G}(P)}^{d}), respectively, then σ determines a kC_{G}(P)-isomorphism

Φ_{C_{G}(P),d}(W_{\underline{G}}(ˆ\underline{X}, Ŝ)) ≅ W_{G}(X, S).

Therefore ˆ \underline{W} occurs in W_{\underline{G}}(ˆ\underline{X}, Ŝ), and ˆ \underline{X} has defect pair (P̂, ˆ \underline{W}). ⊓ ⊔

I do not know whether the image under Φ_{G,e} of a simply defective indecomposable k ˆ\underline{G}ϵ-module need be simply defective. (Note that the construction of σ in the argument above relies on X being simply defective.)

Corollary 4.9: Let (P, W) be a self-centralizing local pair on kG such that there exists an indecomposable kGe-module with vertex P. Let (P̂, ˆ \underline{W}) be the self-centralizing local pair on k ˆ\underline{G} corresponding to (P, W) as in Lemma 4.7. Then Φ_{G,e} provides a bijective correspondence between the isomorphism classes of simple kG-modules with defect pair (P, W), and the isomorphism classes of simple k ˆ\underline{G}-modules with defect pair (P̂, ˆ \underline{W}).

PROOF: By Lemma 2.3(d), e fixes every simple kG-module with defect pair (P, W). Similarly, ϵ fixes every simple k ˆ\underline{G}ϵ-module with defect pair (P̂, ˆ \underline{W}). The assertion is now an immediate consequence of Proposition 4.8. ⊓ ⊔

5: Proof of the main result

Lemmas 2.3(c) and 2.4(a) tell us that, for any simple kG-module X with vertex P, the vertex pairs of X are self-centralizing, and uniquely determine the block of kN_{G}(P) fixing the Green correspondent of X. So Theorem 1.1 is equivalent to Theorem 1.2. By Theorem 2.2, Theorem 1.1 implies Corollary 1.3(a) which, in turn, clearly implies Corollary 1.3(b). Theorems 1.1 and 2.2 also give the equivalence of conditions (1) and (2) of Corollary 1.4. To complete the proof that Theorem 1.1 implies Corollary 1.4, we put p = 2, and give an example of a finite soluble group G with a 2-subgroup P satisfying condition (1) of Corollary 1.4. This is a slight simplification of an example communicated to me by Isaacs so, while the ideas in the construction are due to him, any mistakes are my responsibility. (Isaacs has also communicated to me another example based on a group of order 2^{7}3^{3}.)

Let A be a copy of A_{4}, let V be the Klein subgroup of A, and let K be the A-module of order 27 obtained by induction from a non-trivial V -module of order 3. Then K is a direct product of three groups U, U′, U′′ of order 3 such that, on each of U, U′, U′′, exactly two of the involutions in V acts nontrivially. Let Q be the group generated by the involution fixing U, and define Q′ similarly for U′, so that V = Q×Q′. Let H be the semidirect product KA. We have N_{H}(Q) = UQ′ × Q and UQ′ ≅ S_{3}. Let H act faithfully on a finite 2-group F. Let A/V act non-trivially on a group E of order 7, and inflate this to an action of H. We construct the semidirect product G := (E ×F)H, and let P := F Q.

Now N_{G}(P) = (E × F)N_{H}(Q) = (E × F)UV, so E is a central Sylow 7-subgroup of N_{G}(P), and N_{G}(P) is the direct product of E and the unique Hall 7′-subgroup L := F N_{H}(Q). We have L/F ≅ N_{H}(Q) ≅ S_{3} × Q, so kL has a unique block, hence kN_{G}(P) has precisely seven blocks, all with Sylow defect group R := F V, and the blocks b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6}, b_{7} of kN_{G}(P) coincide with the blocks of kE. Moreover, kL has a simple module Y with vertex P. For 1 ≤ i ≤ 7, tensoring a simple kEb_{i}-module with Y, we obtain a simple kN_{G}(P)b_{i}-module Y_{i} with vertex P. Using Theorem 2.2, it can be shown that each Y_{i} lies in a block of kG with defect group R. So it suffices to show that kN_{G}(R) has less than seven blocks. Now N_{G}(R) = (E × F)A, and the quotient \overline{N}_{G}(R), being non-abelian of order 21, has precisely four conjugacy classes. Perforce, kN_{G}(R) has at most four blocks.

Having now shown that Theorem 1.2 implies the other three results stated in Section 1, we end with a three-step proof of Theorem 1.2. Suppose that G is p-soluble, and let (P, W) be a self-centralizing local pair on kG. Writing ℓ_{G}(P, W) for the number of isomorphism classes of simple kG-modules with defect pair (P, W), Theorem 1.2 is, by Lemma 2.4(b), the assertion that

ℓ_{G}(P, W) = ℓ_{N_{G}(P)}(P, W).

We argue by a triple induction, firstly on |G : O_{p}(G)|_{p}, secondly on |G|_{p}, thirdly on |G|. If O_{p}(G) ≠ 1, let R := O_{p}(G), otherwise let R := O_{p′}(G). For any R ≤ H ≤ G, let \underline{H} := H/R.

First, let us suppose that 1 < R = O_{p}(G). We may then assume that P ≤ R, becuase otherwise ℓ_{G}(P, W) = 0 = ℓ_{N_{G}(P)}(P, W). Let N_{G}(P) ≤ H ≤ G, and let X be a simple kH-module with vertex P and source kP -module S. We have inflations X = infH ; \underline{H}(\underline{X}) and S = infP ; \underline{P} (\underline{S}) for a simple k\underline{H}-module \underline{X} with vertex \underline{P} and source k\underline{P} -module \underline{S}. Identifying \overline{N}\underline{G}(\underline{P}) with \overline{N}_{G}(P), then \overline{C}_{G}(P) ≤ \overline{C}\underline{G}(\underline{P}). It is not hard to show that

Res_{\overline{C}_{G}(P)}^{\overline{N}_{H}(P,S)}(\overline{V}_{H}(X, S)) = Res_{\overline{C}_{G}(P)}^{\overline{N}\underline{H}(\underline{P} ,\underline{S})}(\overline{V} \underline{H}(\underline{X}, \underline{S})).

This may be rewritten as

W_{H}(X, S) = Res_{C_{G}(P)}^{C_{G}(P)R}(infC_{G}(P)R ; \underline{C_{G}(P)R}(ResC\underline{G}(\underline{P}) ; \underline{C_{G}(P)R}(W\underline{H}(\underline{X}, \underline{S})))).

By Clifford’s theorem, ℓ_{H}(P, W) = ∑\underline{W} ℓ\underline{H}(\underline{P} , \underline{W}), where \underline{W} runs over representatives of the \underline{N_{G}(P)}-orbits of the isomorphism classes of simple kC\underline{G}(\underline{P})- modules with vertex Z(\underline{P}) such that

Res_{C_{G}(P)}^{C_{G}(P)R}(infC_{G}(P)R ; \underline{C_{G}(P)R}(ResC\underline{G}(\underline{P}) ; \underline{C_{G}(P)R}(\underline{W})))

is a sum of N_{G}(P)-conjugates of W. Induction gives ℓ_{G}(P, W) = ℓ_{N_{G}(P)}(P, W) in this case.

We may suppose, henceforth, that 1 = O_{p}(G) < R = O_{p′}(G) < G. Let f be the primitive idempotent of (kR)^{G} such that the central idempotent Br_{P}^{G}(f) of kC_{G}(P) fixes W. Since Br_{P}^{G}(f) = ∑_{e} Br_{P}^{G}(e), where e runs over the P -stable blocks of kR such that fe = e, we can choose a block e of kR with stabilizer T in G such that P ≤ T, f = Tr_{T}^{G}(e), and Br_{P}^{G}(e) does not annihilate W. Let c := Br_{P}^{G}(e) and d := Tr_{C_{T} (P)}^{C_{G}(P)}(c) as in Section 3. Then the block d of kC_{G}(P) fixes W. Let W_{T} := c Res_{C_{T} (P)}^{C_{G}(P)}(W), which is the simple kC_{T} (P)-module such that (P, W_{T}) is the self-centralizing local pair on kT corresponding to (P, W) as in Lemma 3.2. By Lemma 2.3(d), f fixes all the simple kG-modules with defect pair (P, W). Similarly, e fixes all the simple kT -modules with defect pair (P, W_{T}). If T ≠ G, then Proposition 3.3 and induction give

ℓ_{G}(P, W) = ℓ_{T} (P, W_{T}) = ℓ_{N_{T} (P)}(P, W_{T})ℓ_{N_{G}(P)}(P, W) = ℓ_{N_{T} (P)}(P, W_{T}).

On the other hand, if T = G, then by the identity N_{\underline{G}}(P̂) = ˆ \underline{N_{G}(P)}, the inductive hypothesis, and Corollary 4.9,

ℓ_{G}(P, W) = ℓ_{\underline{G}}(P̂, ˆ \underline{W}) = ℓ_{\underline{N_{G}(P)}}(P̂, ˆ \underline{W}) = ℓ_{N_{G}(P)}(P, W). ⊓ ⊔

ACKNOWLEDGEMENTS

The idea of refining the p-soluble case of Alperin’s conjecture by considering multiplicity modules of simple modules originated with Puig. Much of the strategy behind this work arose during long conversations with him in 1992 and 1993. Letters from him and Isaacs in 1995 and 1996 contained several valuable suggestions. I also thank the referee for some corrections and useful comments.

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School of Mathematics, Cardiff University of Wales, Senghennydd Road, PO Box 926, Cardiff CF2 4HY, Britain.

Present address: Department of Mathematics, Bilkent University, 06533 Bilkent, Ankara, Turkey. e-mail: barker@fen.bilkent.edu.tr