June 1998

The group of endo-permutation modules

Serge Bouc^{1}, Jacques Thévenaz^{2}

1 UFR de Mathématiques, Université de Paris VII, 2 place Jussieu, F–75251 Paris, France (e-mail: bouc@math.jussieu.fr) 2 Institut de Mathématiques, Université de Lausanne, CH–1015 Lausanne, Switzerland (e-mail: Jacques.Thevenaz@ima.unil.ch)

Abstract. The group D(P) of all endo-permutation modules for a finite p-group P is a finitely generated abelian group. We prove that its torsion-free rank is equal to the number of conjugacy classes of non-cyclic subgroups of P . We also obtain partial results on its torsion subgroup. We determine next the structure of ℚ⊗D(−) viewed as a functor, which turns out to be a simple functor S_{E,ℚ} , indexed by the elementary group E of order p^{2} and the trivial Out(E)-module ℚ . Finally we describe a rather strange exact sequence relating ℚ⊗D(P) , ℚ⊗B(P) , and ℚ⊗R(P) , where B(P) is the Burnside ring and R(P) is the Grothendieck ring of ℚP -modules.

Contents

1. The Dade group

2. Tensor induction and some functorial constructions

3. Composition

4. The torsion-free rank of the Dade group

5. The torsion-free rank of the endo-trivial group

6. Partial results on the torsion subgroup

7. Functorial approach

8. The Burnside functor

9. A characterization of some simple functors

10. The torsion-free part of the Dade functor

11. Partial results on the torsion functor

Introduction

Endo-permutation modules for p-groups are ubiquitous in the p-modular representation theory of finite groups. They appear as sources of simple modules for p-soluble groups (see Section 30 in [Th1]), in Puig’s description of the source algebra of a nilpotent block (see [Pu1] or Section 50 in [Th1]), and also in the local analysis of Morita or derived equivalences between blocks [Pu3]. So they are fundamental objects which need to be fully understood, the final goal being the complete classification of endo-permutation modules. This paper is a contribution to this classification program.

Recall that if P is a p-group and if k is a field of characteristic p, then a kP -module M is an endo-permutation module if End_{k}(M) is a permutation kP -module. One is particularly interested in indecomposable endo-permutation modules with vertex P . Twenty years ago, in his seminal papers on endo-permutation modules [Da2, Da3], E.C. Dade introduced most of the basic techniques and defined the group D(P) = D_{k}(P) of endo-permutation kP -modules, which we call the Dade group of P . This can be viewed either as a group of equivalence classes of suitable endo-permutation modules (with multiplication induced by tensor product) or as the group of indecomposable endo-permutation kP -modules with vertex P (because there is one such indecomposable module in each equivalence class, up to isomorphism).

Dade classified all endo-permutation modules when P is abelian and described explicitly the structure of D(P) in that case. Shortly afterwards, L. Puig proved that D(P) is always finitely generated (but his paper [Pu2] only appeared in 1990). So we have

D(P) ≅ ℤ^{r} ⊕ D^{t}(P) , with D^{t}(P) finite .

The question of finding the torsion-free rank r and describing the torsion subgroup D^{t}(P) remained completely open in general. Our first result gives an answer to the first question.

Theorem A. The torsion-free rank of the Dade group D(P) is equal to the number of conjugacy classes of non-cyclic subgroups of P .

In fact we define a natural map ϕ : ℚ ⊗_{ℤ} D(P) → ℚ^{r} which we prove to be an isomorphism, and by means of the inverse of ϕ we produce an explicit basis of ℚ ⊗_{ℤ} D(P) . One of the main ingredients (which follows from deep results of Carlson, Dade, and Puig) is the injectivity of the restriction–deflation map

ℚ ⊗_{ℤ} D(P) ⟶ ∏_{E} ℚ ⊗_{ℤ} D(E) ,

where E runs over all elementary abelian sections of P of rank 2, together with the fact that ℚ⊗_{ℤ}D(E) ≅ ℚ by Dade’s classification in the abelian case. Another main ingredient is tensor induction, a tool which is well adapted because endo-permutation modules behave nicely with respect to tensor products. This tool was first considered by Puig [Pu4] who used it to show that sources of simple modules for p-soluble groups always lie in the torsion subgroup of the Dade group.

One is also particularly interested in the torsion subgroup D^{t}(P) . Indeed the naturally occuring endo-permutation modules mentioned in the first paragraph define interesting elements of D(P) which are actually expected to lie in D^{t}(P) . The cyclic group C_{p} of order p plays a crucial role here (at least for p odd), because D(C_{p}) ≅ ℤ/2ℤ. Unfortunately, one does not know if the restriction–deflation map

ψ_{P} : D^{t}(P) ⟶ ∏_{C} D(C)

is injective, where C runs over all cyclic sections of P of order p. If it were so, we could determine D^{t}(P) by methods analogous to those used for Theorem A. Thus we simply bypass the difficulty by taking the quotient \overline{D^{t}}(P) = D^{t}(P)/Ker(ψ_{P}) . This quotient can be explicitly described:

Theorem B. If p is odd, the group \overline{D^{t}}(P) is isomorphic to the direct sum of s copies of ℤ/2ℤ, where s is the number of conjugacy classes of non-trivial cyclic subgroups of P .

We also obtain explicit generators for \overline{D^{t}}(P) . It is our hope that ψ_{P} is actually injective when p is odd. This question was mentioned to us several years ago by J.F. Carlson and J.L. Alperin. If the answer was positive, Theorem B would completely describe D^{t}(P) and would also imply that every torsion element of D(P) has order 2 (i.e. is self-dual). This is a non-trivial fact which would be very interesting to obtain.

It is known that the situation is more complicated when p = 2 . First of all D(C_{2}) is trivial (so one expects instead that D(C_{4}) ≅ ℤ/2ℤ should play a role), but there are also unexpected endo-permutation modules for the quaternion groups (see [Da1]). For instance D(Q_{8}) ≅ ℤ/2ℤ ⊕ ℤ/4ℤ ⊕ ℤ. Thus the assumption in Theorem B cannot be removed.

Our methods can also be used to analyse the subgroup T (P) consisting of the endo-trivial modules, that is, indecomposable kP -modules M such that End_{k}(M) ≅ k ⊕ (free) . Those modules also play an important role in modular representation theory, for instance in self-equivalences of stable module categories (see [Li] and [CR]). We prove a theorem analogous to Theorem A, which gives the torsion-free rank of T (P) ; this is a recent unpublished theorem of J.L. Alperin but our proof is different (we use tensor induction instead of relative syzygies). In contrast, in analogy with Theorem B, one only has a nearly trivial statement concerning the torsion subgroup of T (P) .

Our analysis of D(P) takes place in a functorial framework. We view D(−) as a functor defined on the category of all finite p-groups, the morphisms being compositions of five types of basic morphisms, namely restriction to a subgroup, tensor induction to an overgroup, inflation from a quotient group, deflation to a quotient group (this is Dade’s construction of slash modules), and isomorphisms. The necessary machinery for dealing with such functors was introduced in [Bo1]. Our original proofs of Theorems A and B used heavily this functorial theory (and in particular some non-trivial results of [Bo1]), but a more elementary approach emerged later from the functorial point of view. For the convenience of the reader, we first give here these non-functorial proofs (Sections 4 and 6). The only result from [Bo1] which we need is a technical lemma on the Möbius function.

However, the functorial approach yields more and we use it in the second part of the paper. The basic functorial constructions are already introduced in Sections 2 and 3, but the more difficult functorial results only appear in the last five sections. For technical reasons, the Dade group D(P) does not in general give rise to a functor in the sense of [Bo1], but ℚ ⊗_{ℤ} D(P) and \overline{D^{t}}(P) do. Recall that simple functors S_{H,V} over a field K are parametrized by pairs (H, V) where H is a finite group (a p-group in our situation) and V is a simple K[Out(H)]-module.

Theorem C. The functor ℚ ⊗_{ℤ} D is simple. More precisely, it is isomorphic to S_{E,ℚ} , where E is elementary abelian of rank 2 and ℚ is the trivial module.

Our fourth main result asserts that there is an exact sequence relating the Dade group D(P) , the Burnside ring B(P) , and the Grothendieck ring R_{ℚ}(P) of ℚP -modules. The composition factors of the Burnside functor ℚ ⊗_{ℤ} B(−) were analysed in [Bo1] (for the category of all finite groups). For functors defined only on p-groups, we prove a similar result for K ⊗_{ℤ} B , where K is a field of characteristic different from p (see Section 8). When K = ℚ, there are just two composition factors S_{1,ℚ} and S_{E,ℚ} , fitting in a non-split exact sequence of functors

0 ⟶ S_{E,ℚ} ⟶ ℚ ⊗_{ℤ} B ⟶ S_{1,ℚ} ⟶ 0 .

But in fact S_{1,ℚ} ≅ ℚ ⊗_{ℤ} R_{ℚ} and S_{E,ℚ} ≅ ℚ ⊗_{ℤ} D by Theorem C. Thus we obtain the following result.

Theorem D. There is an exact sequence of functors

0 ⟶ ℚ ⊗_{ℤ} D ⟶^{α} ℚ ⊗_{ℤ} B ⟶^{ε} ℚ ⊗_{ℤ} R_{ℚ} ⟶ 0 ,

where ε is the natural homomorphism mapping a P -set to the corresponding permutation ℚP -module.

We also give an explicit description of the map α on a basis of ℚD(P) , for every p-group P .

When p is odd, there are analogous results for the functor \overline{D^{t}}(−) appearing in Theorem B. We prove that it is a simple functor over the field 𝔽_{2} , isomorphic to S_{C_{p},𝔽_{2}} . Moreover, there is an exact sequence of functors

0 ⟶ \overline{D^{t}} ⟶ 𝔽_{2} ⊗_{ℤ} R_{ℚ} ⟶^{dim} Γ_{𝔽_{2}} ⟶ 0 ,

where Γ_{𝔽_{2}} denotes the constant functor with values 𝔽_{2} .

1. The Dade group

We first recall some standard facts concerning endo-permutation modules and the Dade group. Details can be found in Dade’s original paper [Da2] or in Sections 28–29 of [Th]. Let k be a field of characteristic p and let P be a finite p-group. An endo-permutation kP -module M is said to be capped if M has at least one indecomposable direct summand with vertex P . By assumption, the P -algebra A = End_{k}(M) has a P -invariant basis X and the condition that M be capped is equivalent to the requirement that the Brauer quotient A[P] = A^{P} / ∑_{Q<P} A_{Q}^{P} is non-zero, or equivalently that X has at least one P -fixed point (because A[P] has basis X^{P} ). The endo-permutation modules we shall consider will always be capped and we shall often omit to mention it. The main examples of endo-permutation modules are the kP -modules Ω_{P/Q}^{n}(k) where Q is any normal subgroup of P and Ω_{P/Q} denotes the Heller operator for the quotient group P/Q.

A k-algebra A is called a Dade P -algebra if it is a P -algebra (i.e. P acts on A by algebra automorphisms) which is central simple and split (i.e. isomorphic to a matrix algebra over k) and if A has a P -invariant basis with at least one P -fixed point. Thus if M is a capped endo-permutation kP -module, then A = End_{k}(M) is a Dade P -algebra. Conversely, if A is a Dade P -algebra, then A ≅ End_{k}(M) for some k-vector space M and the P -action on A lifts uniquely to a kP -module structure on M (thanks to the fact that P is a p-group), so that M becomes a capped endo-permutation module. Thus the concepts of capped endo-permutation module and Dade P -algebra are equivalent. By the very definition of an endopermutation module, the emphasis is put on the corresponding Dade P -algebra, so we shall often use the algebra point of view.

The following result is basic for understanding the Dade group.

(1.1) Lemma. If M is a capped endo-permutation kP -module, then any two indecomposable direct summands of M with vertex P are isomorphic.

Such an indecomposable direct summand is called a cap of M and is uniquely determined up to isomorphism. It is again a capped endo-permutation kP -module. Two capped endo-permutation modules M and N are said to be equivalent if they have isomorphic caps, in which case we write M ∼ N . The set of equivalence classes of capped endo-permutation kP -modules is written D(P) , or D_{k}(P) when we need to emphasize the base field k. The tensor product of kP -modules induces a group structure on D(P) . The unit element is the class of the trivial module and turns out to consist of all permutation modules (which are capped, i.e. have a summand isomorphic to k). The inverse of the class of M is the class of the dual module M^{∗} . The group D(P) is abelian and is called the Dade group of P .

In any equivalence class, there is a unique indecomposable module up to isomorphism (namely the cap of any module in the class), so it is often convenient to view D(P) as the set of isomorphism classes of indecomposable endo-permutation kP -modules with vertex P . However, the tensor product of two such indecomposable modules need not be indecomposable, so one needs to take its cap to recover the group structure in terms of indecomposable modules.

One can also view the equivalence relation in terms of Dade P -algebras as follows. Two Dade P -algebras A and B are equivalent (written also A ∼ B ) if and only if iAi ≅ jBj , where i is a primitive idempotent of A^{P} with br_{P} (i) ≠ 0 and j is a primitive idempotent of B^{P} with br_{P} (j) ≠ 0 (both are unique up to conjugation by Lemma 1.1). Here br_{P} : A^{P} → A[P] is the quotient map, called the Brauer homomorphism. Thus D(P) is also the group of equivalence classes of Dade P -algebras and we shall freely use either point of view. The group structure is induced by the tensor product of algebras.

There is yet another way of viewing the equivalence relation: M ∼ N if and only if there exist modules V and W in the trivial class (i.e. permutation modules with a trivial summand) such that M ⊗_{k} V ≅ N ⊗_{k} W . In terms of Dade P -algebras, A ∼ B if and only A ⊗_{k} End_{k}(V) ≅ B ⊗_{k} End_{k}(W) for some V and W as above.

Despite the fact that the group structure on D(P) is induced by the tensor product, we shall use an additive notation, which is much more convenient in the formulas and in many computations. Thus, for any endo-permutation modules M and N , we define

[M] + [N] = [M ⊗_{k} N] ,

and similarly in terms of Dade P -algebras. For simplicity, we shall write Ω_{P/Q} for the class of Ω_{P/Q}^{1}(k) . For example, the class of Ω_{P}^{n} (k) is equal to nΩ_{P} , since we have Ω_{P}^{r} (k) ⊗ Ω_{P}^{s} (k) ≅ Ω_{P}^{r+s}(k) ⊕ (free) .

There is an obvious restriction map

Res_{Q}^{P} : D(P) ⟶ D(Q) ,

where Q is a subgroup of P . If Q is a normal subgroup, there is also an obvious inflation map

Inf_{P/Q}^{P} : D(P/Q) ⟶ D(P) .

For instance Ω_{P/Q} is the image under Inf_{P/Q}^{P} of the class of the Heller translate of the trivial module for the quotient group P/Q. Finally we define the deflation map

Def_{P/Q}^{P} : D(P) ⟶ D(P/Q)

as follows. If A is a Dade P -algebra, then the Brauer quotient A[Q] = A^{Q}/ ∑_{R<Q} A_{R}^{Q} turns out to be a Dade (P/Q)-algebra and we set Def_{P/Q}^{P}([A]) = [A[Q]] where [A] denotes the class of A. This is the slash construction in Dade’s terminology. If X is a P -invariant basis of A, then the Q-fixed points X^{Q} form a (P/Q)-invariant basis of A[Q] . The three above maps are group homomorphisms and are part of the functorial framework which will be introduced in Section 2 and 3 (and fully used in Sections 7–11). The fourth important map is tensor induction and will be introduced in Section 2.

If Q is a subgroup of P and R is a normal subgroup of Q, then Q/R is a section of P and we define

Defres_{Q/R}^{P} = Def_{Q/R}^{Q} Res_{Q}^{P} : D(P) ⟶ D(Q/R) .

An important subgroup of D(P) is the kernel of all these maps. We consider

∏_{Q≠1} Defres_{N_{P} (Q)/Q}^{P} : D(P) ⟶ ∏_{Q≠1} D(N_{P} (Q)/Q) ,

where Q runs over all non-trivial subgroups of P (up to conjugation). We define

T (P) = Ker( ∏_{Q≠1} Defres_{N_{P} (Q)/Q}^{P}) = ⋂_{Q≠1} Ker(Defres_{N_{P} (Q)/Q}^{P}) .

Recall that a kP -module M is called endo-trivial if End_{k}(M) ≅ k ⊕ (free) as a kP -module. In that case M is a capped endo-permutation module. The connection with T (P) is provided by the following result.

(1.2) Lemma. (Puig [Pu2, 2.1.2]). Let M be an indecomposable endo-permutation kP -module with vertex P . Then [M] ∈ T (P) if and only if M is endo-trivial.

Thus the subgroup T (P) coincides with the group of endo-trivial modules. Another characterization of endo-trivial modules has recently been obtained by J. Carlson [Ca2].

In view of its essential role in the sequel, we recall the following result about Ω_{P} .

(1.3) Lemma. Let Q be a subgroup of P .

(a) Res_{Q}^{P}(Ω_{P}) = Ω_{Q} .

(b) If Q is a non-trivial normal subgroup of P , then Def_{P/Q}^{P}(Ω_{P}) = 0 . In other words, Ω_{P} is endo-trivial.

Proof. (a) We have Res_{Q}^{P}(Ω_{P}^{1} (k)) = Ω_{Q}^{1}(k) ⊕ (free) .

(b) We have End_{k}(Ω_{P}^{1} (k)) ≅ Ω_{P}^{1} (k) ⊗_{k} Ω_{P}^{1} (k)^{∗} ≅ Ω_{P}^{1} (k) ⊗_{k} Ω_{P}^{−1}(k) ≅ k ⊕ (free) as kP -modules.

We can now recall the two main theorems about the Dade group.

(1.4) Theorem. (Dade [Da3, 10.1 and 12.5]). Assume that P is abelian.

(a) T (P) is generated by Ω_{P} . Thus T (P) is trivial if |P | = 2 , cyclic of order 2 if P is cyclic of order ≥ 3 , and infinite cyclic otherwise.

(b) D(P) is isomorphic to the direct sum of the groups T (P/Q) , where Q runs over the subgroups of P . Thus D(P) ≅ ℤ^{r} ⊕ (ℤ/2ℤ)^{s} , where r is the number of non-cyclic quotients of P and s is the number of cyclic quotients of P of order ≥ 3 .

Of course T (P/Q) is identified with a subgroup of D(P) via inflation. Note that (a) is hard to prove, but that (b) follows rather easily from (a) by induction. Another proof of the crucial step for (a) (namely when P has order p^{2} ) is due to Carlson [Ca1].

(1.5) Theorem. (Puig [Pu2, 2.2 and 2.4]).

(a) The kernel of the restriction map T (P) ⟶ ∏_{E} T (E) , where E runs over the elementary abelian subgroups of P , is finite.

(b) The group D(P) is finitely generated.

The proof of (a) uses a non-trivial cohomological result of Carlson and some algebraic geometry (though Puig’s version is explained in terms of commutative algebra). By induction, (b) follows easily from (a) and Dade’s theorem above.

The previous results imply the following injectivity theorem, which is one of our main tools for finding the torsion-free ranks of D(P) and T (P) . For simplicity, we write ℚD(P) = ℚ ⊗_{ℤ} D(P) and ℚT (P) = ℚ ⊗_{ℤ} T (P) .

(1.6) Theorem. The following maps are injective:

(a) Res : ℚT (P) ⟶ ∏_{E} ℚT (E) , where E runs over the elementary abelian subgroups of P of rank 2.

(b) Defres : ℚD(P) ⟶ ∏_{Q/R} ℚD(Q/R) , where Q/R runs over the elementary abelian sections of P of rank 2.

Proof. (a) By Theorem 1.5 (a), the restriction map to all elementary abelian subgroups is injective after tensoring with ℚ. But if E_{k} denotes the elementary abelian group of rank k, we have ℚT (E_{1}) = 0 , while the restriction map Res_{E_{2}}^{E_{k}} : T (E_{k}) → T (E_{2}) is an isomorphism if k ≥ 2 , by Theorem 1.4 (a) and the fact that Res_{E_{2}}^{E_{k}}(Ω_{E_{k}}) = Ω_{E_{2}} . The result follows.

(b) We prove the injectivity of the map by induction on |P |, starting with the trivial case of a cyclic group. If now P is not cyclic, let a ∈ ℚD(P) lie in the kernel of all maps Defres_{Q/R}^{P} for sections Q/R which are elementary abelian of rank 2. If H is a non-trivial subgroup of P , then a_{H} = Defres_{N_{P} (H)/H}^{P}(a) also lies in the kernel of all maps Defres_{Q/R}^{N_{P} (H)/H} for sections Q/R of rank 2 in N_{P} (H)/H , by transitivity of these maps (which easily follows from the definitions). By induction, we have a_{H} = 0 . Since this holds for every non-trivial subgroup H , it follows that a ∈ ℚT (P) , by definition of T (P) . Now for every elementary abelian subgroup E of rank 2, Res_{E}^{P}(a) = 0 by assumption. Therefore a = 0 by (a).

2. Tensor induction and some functorial constructions

The purpose of this section is to introduce a construction which contains tensor induction as a special case. In fact, it turns out that the five natural operations on Dade P -algebras, namely restriction, inflation, deflation, tensor induction, and isomorphisms, can be recovered by means of a single mechanism. We define this construction here, not only because of its simplicity and its use later in Section 7, but also because the proofs of the results which we need about tensor induction of Dade P -algebras are actually simpler if we adopt this approach. Those results will be proved in the next section.

Throughout this section and the next, it is much more convenient to work with Dade P -algebras rather than endo-permutation modules. However, we first recall the notion of tensor induction of modules, and then of algebras. Let H be a subgroup of a finite group G and let G/H denote the set of right cosets gH where g ∈ G (i.e. right orbits under the action of H ). If M is a kH-module, recall that the tensor induction of M is the kG-module Ten_{H}^{G}(M) defined as follows. As a k-vector space, we have

Ten_{H}^{G}(M) = ⨂_{C∈G/H}(kC ⊗_{kH} M) ,

and the kG-module structure is given by

g · [ ⨂_{C∈G/H}(s_{C} ⊗ m_{C})] = ⨂_{C∈G/H}(gs_{g^{−1}C} ⊗ m_{g^{−1}C})

where g ∈ G and, for each C ∈ G/H , s_{C} ∈ C and m_{C} ∈ M . If [G/H] denotes a set of coset representatives, then we have Ten_{H}^{G}(M) = ⨂_{s∈[G/H]}(s ⊗ M) and the action of g is given by

g · [ ⨂_{s∈[G/H]}(s ⊗ m_{s})] = ⨂_{s∈[G/H]}(τ(s) ⊗ h_{s}m_{s}) = ⨂_{s∈[G/H]}(s ⊗ h_{τ^{−1}(s)}m_{τ^{−1}(s)}) ,

where gs = τ(s)h_{s} , h_{s} ∈ H and τ is a permutation of [G/H] . The basic properties of tensor induction (e.g. transitivity, multiplicativity, Mackey formula) can be found in [Be, 3.15], but we shall actually recover them by means of our functorial formalism (at least for permutation modules).

If A is an H-algebra over k, then kC⊗_{kH}A is an algebra (isomorphic to A), via (s⊗a)(s⊗a′) = s⊗aa′ where s ∈ C is fixed (and this is independent of the choice of s). Therefore the tensor product of algebras ⨂_{C∈G/H}(kC ⊗_{kH} A) has an algebra structure. It follows that the kG-module Ten_{H}^{G}(A) has a G-algebra structure, since it is easy to check that G acts on Ten_{H}^{G}(A) by algebra automorphisms. Moreover tensor induction of modules and algebras are related in the following way.

(2.1) Lemma. Let M be a kH-module and consider the corresponding H-algebra End_{k}(M) . Then there is an isomorphism of G-algebras Ten_{H}^{G}(End_{k}(M)) ≅ End_{k}(Ten_{H}^{G}(M)).

Proof. Let [G/H] be a set of coset representatives. Then there is a canonical isomorphism of k-algebras

ϕ : ⨂_{s∈[G/H]} (s ⊗ End_{k}(M)) ⟶^{∼} End_{k}( ⨂_{s∈[G/H]}(s ⊗ M))

and therefore Ten_{H}^{G}(End_{k}(M)) ≅ End_{k}(Ten_{H}^{G}(M)) as k-algebras. In order to check that ϕ commutes with the action of G, we choose g ∈ G and write gs = τ(s)h_{s} as before. We also have g^{−1}s = τ^{−1}(s)h_{τ^{−1}(s)}^{−1} . Since H acts on End_{k}(M) by conjugation, the action of g on Ten_{H}^{G}(End_{k}(M)) is given by

g · [ ⨂_{s∈[G/H]}(s ⊗ a_{s})] = ⨂_{s∈[G/H]} (τ(s) ⊗ h_{s}a_{s}h_{s}^{−1}) ,

where a_{s} ∈ End_{k}(M) for all s. We have to prove that this is the same action as the conjugation action of g on End_{k}(Ten_{H}^{G}(M)). Given ⨂_{s∈[G/H]}(s ⊗ m_{s}) ∈ Ten_{H}^{G}(M) , we have

g [ ⨂_{s∈[G/H]}(s ⊗ a_{s})] g^{−1}[ ⨂_{s∈[G/H]}(s ⊗ m_{s})] = g [ ⨂_{s∈[G/H]}(s ⊗ a_{s})] [ ⨂_{s∈[G/H]}(s ⊗ h_{s}^{−1}m_{τ(s)})]

= g [ ⨂_{s∈[G/H]}(s ⊗ a_{s}h_{s}^{−1}m_{τ(s)})] = ⨂_{s∈[G/H]} (τ(s) ⊗ h_{s}a_{s}h_{s}^{−1}m_{τ(s)})

= [ ⨂_{s∈[G/H]} (τ(s) ⊗ h_{s}a_{s}h_{s}^{−1})] [ ⨂_{s∈[G/H]} (τ(s) ⊗ m_{τ(s)})]

= [ ⨂_{s∈[G/H]} (τ(s) ⊗ h_{s}a_{s}h_{s}^{−1})] [ ⨂_{s∈[G/H]}(s ⊗ m_{s})] .

The result follows.

Now we introduce our functorial formalism. The main idea is to change the point of view about Dade P -algebras. Instead of viewing them as P -algebras in which there exists a P -invariant basis, we consider a permutation kP -module kX with a fixed basis X and we then look at all possible algebra structures on kX . For Dade P -algebras, we actually only need to consider central simple split algebra structures.

We first have to introduce our construction for permutation modules. Let p be a prime number and let k be a field of characteristic p. If P is a finite p-group, we denote by P erm_{k}(P) the category of finitely generated permutation kP -modules. The objects of P erm_{k}(P) are the kP -modules M which admit a P -invariant k-basis and the morphisms in P erm_{k}(P) are the morphisms of kP -modules. Let \underline{P erm}_{k}(P) denote the following category: the objects of \underline{P erm}_{k}(P) are the finite P -sets, and a morphism in \underline{P erm}_{k}(P) from Y to X is a matrix m(x, y) , indexed by X × Y , with coefficients in k, which is P -invariant, i.e. such that m(gx, gy) = m(x, y) for all (x, y) ∈ X × Y and all g ∈ P . The composition of morphisms is given by usual matrix multiplication.

There is an obvious functor e_{P} : \underline{P erm}_{k}(P) → P erm_{k}(P) , mapping the P -set X to the module kX , and the matrix m indexed by X × Y to the morphism from kY to kX defined on the k-basis Y of kY by

y ∈ Y ↦ ∑_{x∈X} m(x, y)x ∈ kX .

The functor e_{P} is fully faithful by construction, and it is essentially surjective since any permutation module admits a P -invariant k-basis X , hence is isomorphic to kX . Thus e_{P} is an equivalence of categories, and this allows us to define our constructions on permutation P -algebras only in the category \underline{P erm}_{k}(P) .

It is clear for instance that the direct sum functor in P erm_{k}(P) , mapping a pair of modules M and M′ to their direct sum M ⊕ M′ , corresponds to the disjoint union functor, mapping a pair of P -sets (X, X′) to their disjoint union X ⊔ X′ , and a pair of morphisms (m, m′) with m : Y → X and m′ : Y′ → X′ to the morphism m ⊔ m′ : Y ⊔ Y′ → X ⊔ X′ defined by

(m ⊔ m′)(a, b) = \{ \begin{matrix}m(a, b) & if a ∈ X, b ∈ Y , \\ m′(a, b) & if a ∈ X′, b ∈ Y′ , \\ 0 & if a ∈ X, b ∈ Y′ or a ∈ X′, b ∈ Y . \end{matrix}

In other words, we have an isomorphism

e_{P} (X ⊔ X′) ≅ e_{P} (X) ⊕ e_{P} (X′)

which is natural in X and X′ . Similarly, the tensor product of modules corresponds to the cartesian product: define the cartesian product of m : Y → X and m′ : Y′ → X′ to be the morphism

m × m′ : Y × Y′ → X × X′ , (m × m′)(x, x′, y, y′) = m(x, y)m′(x′, y′) .

Then clearly e_{P} (X × X′) ≅ e_{P} (X) ⊗_{k} e_{P} (X′) , and this is functorial in X and X′ .

Let P and Q be finite p-groups, and let U be a finite biset, which is a left Q-set and a right P -set, such that the actions of P and Q commute. Such a biset will be called a Q-set-P for short. The opposite biset U^{op} is the P -set-Q obtained as usual from the underlying set U by reversing the actions

g·u·h ( in U^{op} ) = h^{−1}·u·g^{−1} ( in U ) , ∀g ∈ P, ∀h ∈ Q, ∀u ∈ U .

If X is a finite P -set, then the set Hom_{P} (U^{op}, X) of all P -equivariant maps U^{op} → X is a finite Q-set. We define a functor T_{U} from \underline{P erm}_{k}(P) to \underline{P erm}_{k}(Q) on objects by

T_{U}(X) = Hom_{P} (U^{op}, X) .

If m is a P -invariant matrix indexed by X × Y , let T_{U}(m) be the matrix indexed by T_{U}(X) × T_{U}(Y) , defined by

T_{U}(m)(ϕ, ψ) = ∏_{u∈[U/P]} m(ϕ(u), ψ(u)) .

It is clear that this does not depend on the choice of the set of orbit representatives of U/P , since m is P -invariant.

(2.2) Lemma. The correspondence T_{U} is a functor from \underline{P erm}_{k}(P) to \underline{P erm}_{k}(Q) .

Proof. First it is clear that T_{U}(m) is Q-invariant, since for every h ∈ Q we have

T_{U}(m)(hφ, hψ) = ∏_{u∈[U/P]} m(φ(h^{−1}u), ψ(h^{−1}u)) = T_{U}(m)(φ, ψ) ,

because the image by h^{−1} of a set of orbit representatives of U/P is another set of representatives.

Now of course, if m represents the identity morphism, then T_{U}(m)(φ, ψ) is non-zero if and only if φ(u) = ψ(u) for all u ∈ U , or equivalently if φ = ψ. So T_{U}(m) is the identity morphism.

Finally, let Z be another P -set and let n be a matrix representing a morphism in \underline{P erm}_{k}(P) from Z to Y . The product matrix m · n is defined by

(m · n)(x, z) = ∑_{y∈Y} m(x, y)n(y, z) .

Let θ ∈ Hom_{P} (U^{op}, Z) and φ ∈ Hom_{P} (U^{op}, X) . Then

(2.3)  T_{U}(m · n)(φ, θ) = ∏_{u∈[U/P]} ( ∑_{y∈Y} m(φ(u), y)n(y, θ(u))) .

Now for a given u ∈ [U/P] , we have

∑_{y∈Y} m(φ(u), y)n(y, θ(u)) = ∑_{{y∈[P_{u}\Y] \atop g∈[P_{u}/P_{u,y}]}} m(φ(u), gy)n(gy, θ(u))

= ∑_{{y∈[P_{u}\Y] \atop g∈[P_{u}/P_{u,y}]}} m(φ(ug), y)n(y, θ(ug))

= ∑_{y∈[P_{u}\Y]} |P_{u} : P_{u,y}| m(φ(u), y)n(y, θ(u)) .

As P_{u} is a p-group and k is of characteristic p, the coefficient |P_{u} : P_{u,y}| is zero unless P_{u,y} = P_{u} , or equivalently y ∈ Y^{P_{u}} .

Now expanding the product in Equation (2.3) is equivalent to choosing for each u ∈ [U/P] an element y_{u} ∈ Y^{P_{u}} . This in turn is equivalent to defining a P -morphism ψ from U to Y (by ψ(u′) = gy_{u} if u ∈ [U/P] , g ∈ P , and u′ = ug). This gives finally

\begin{aligned}T_{U}(m · n)(φ, θ) & = ∑_{ψ∈Hom_{P} (U^{op},Y)} ∏_{u∈[U/P]} m(φ(u), ψ(u))n(ψ(u), θ(u)) \\ & = ∑_{ψ∈Hom_{P} (U^{op},Y)} T_{U}(m)(φ, ψ) T_{U}(n)(ψ, θ) . \end{aligned}

This proves that m ↦ T_{U}(m) is multiplicative, and the lemma follows.

(2.4) Remark. The previous lemma still holds in a more general framework: we only need to suppose that k is a commutative ring of characteristic p and that P is a finite group such that, for any u ∈ U , the stabilizer P_{u} of u in P is a p-group. The construction of the functor T_{U} is taken from [Bo2], where it is studied in a much more general framework. In fact, it is possible to define T_{U}(M) for any kP -module M (and also for Mackey functors) but the definition is more complicated. Our presentation here is easier thanks to our use of invariant bases.

(2.5) Example: Restriction. Let Q be a subgroup of P , and set U = P , viewed as a Q-set-P by left and right multiplication. Then if X is a P -set, it is clear that

T_{U}(X) = Hom_{P} (U^{op}, X) = Hom_{P} (P, X) ≅ Res_{Q}^{P}X .

Similarly, if m is a morphism in \underline{P erm}_{k}(P) from X to Y , then since U/P = \{1\} we have

T_{U}(m)(φ, ψ) = m(φ(1), ψ(1)) .

In other words, the functor T_{U} is just the restriction functor in this case.

(2.6) Example: Tensor induction. In the same situation, set V = P , viewed as a P -set-Q by left and right multiplication. Then the functor T_{V} is a functor from \underline{P erm}_{k}(Q) to \underline{P erm}_{k}(P) . Let S be a set of coset representatives of P/Q. If X is any Q-set, there is a bijection

T_{V} (X) = Hom_{Q}(V^{op}, X) = Hom_{Q}(P, X) ≅ ∏_{s∈S} X ,

mapping φ ∈ Hom_{Q}(P, X) to the sequence (φ(s))_{s∈S} . Conversely, if (x_{s})_{s∈S} is a sequence in ∏_{s∈S} X , then we can define a morphism of Q-sets φ from V to X by

φ(sh) = h^{−1}x_{s} , ∀s ∈ S, ∀h ∈ Q .

With this identification, the action of g ∈ P on ∏_{s∈S} X is given as follows. There is a permutation τ of S and h_{s} ∈ Q such that gs = τ(s)h_{s} for all s ∈ S . Then we have g^{−1}s = τ^{−1}(s)h_{τ^{−1}(s)}^{−1} and it follows that the action of g on the sequence (x_{s})_{s∈S} produces the sequence (h_{τ^{−1}(s)}x_{τ^{−1}(s)})_{s∈S} .

Now the module k ( ∏_{s∈S} X) is isomorphic to the n-th tensor power of kX , where n = |S|, and the previous description of the action of P on Ten_{Q}^{P}(kX) shows that

k T_{V} (X) ≅ k ( ∏_{s∈S} X) ≅ Ten_{Q}^{P}(kX)

as kP -modules. Moreover, if m is a morphism in \underline{P erm}_{k}(P) from X to Y , then for φ ∈ Hom_{Q}(V^{op}, X) and ψ ∈ Hom_{Q}(V^{op}, Y) we have

T_{V} (m)(ψ, φ) = ∏_{s∈S} m(ψ(s), φ(s)) .

Identifying k T_{V} (X) with Ten_{Q}^{P}(kX) , this shows that the morphism T_{V} (m) is given by

T_{V} (m)(x_{s_{1}} ⊗…⊗x_{s_{n}}) = ∑_{t_{1},...,t_{n}∈S}( ∏_{i=1}^{n} m(y_{t_{i}}, x_{s_{i}})) t_{1}⊗…⊗t_{n} = ( ∑_{t_{1}∈S} m(t_{1}, s_{1})t_{1})⊗…⊗( ∑_{t_{n}∈S} m(t_{n}, s_{n})t_{n}) .

In other words T_{V} (m) is the |S|-th tensor power of m. So the functor T_{V} gives rise to the tensor induction functor (up to some obvious composition with the equivalences e_{P} : \underline{P erm}_{k}(P) → P erm_{k}(P) and e_{Q} : \underline{P erm}_{k}(Q) → P erm_{k}(Q) ).

(2.7) Example: Inflation. Let R be a normal subgroup of P and let Q = P/R. Let U = Q, viewed as a P -set-Q by projection and multiplication on the left, and by multiplication on the right. If X is any Q-set, then T_{U}(X) = Hom_{Q}(U^{op}, X) is just the set X with its structure of P -set obtained by inflation from Q to P . If m is a morphism of Q-sets from X to Y , as U/Q = {1}, we have

T_{U}(m)(φ, ψ) = m(φ(1), ψ(1)) ,

as in Example 2.5 above. So T_{U} is just the inflation functor in this case.

(2.8) Example: Deflation. In the same situation as in Example 2.7, set V = Q, viewed as a Q-set-P . Then if X is a P -set and if X^{R} denotes the set of R-fixed points, there is an isomorphism of Q-sets

T_{V} (X) = Hom_{P} (V^{op}, X) ≅ X^{R} ,

mapping φ ∈ Hom_{P} (V^{op}, X) to φ(1) ∈ X^{R} . The inverse isomorphism maps x ∈ X^{R} to the morphism φ ∈ Hom_{P} (V^{op}, X) defined by φ(\overline{g}) = gx where \overline{g} ∈ Q is the image of g ∈ P . It follows that k T_{V} (X) is isomorphic as kQ-module to the Brauer quotient (kX)[R] . Here again, as V/P is a single point, it is clear that this isomorphism is functorial in X .

(2.9) Example: Isomorphisms. Let θ : P → Q be a group isomorphism. Then we can view U = Q as a Q-set-P , by left multiplication, and right multiplication twisted by θ. It should be clear in that case that the functor T_{U} is just the functor “change of group via θ ”. We shall denote it by Iso_{P}^{Q} , for it will always be clear in the context which isomorphism is used.

Before introducing algebra structures on our permutation modules, we need to know the basic properties of the functors T_{U} .

(2.10) Proposition. Let P and Q be finite p-groups and let U be a finite Q-set-P .

(a) If X is the trivial P -set, then T_{U}(X) is the trivial Q-set.

(b) If X and Y are P -sets, then there are isomorphisms

T_{U}(X × Y) ≅ T_{U}(X) × T_{U}(Y)

which are natural in X and Y in the category \underline{P erm}_{k}(P) . Moreover if Y = X and σ is the switch endomorphism of X ×X (defined by σ(x_{1}, x_{2}, x′_{1}, x′_{2}) = δ_{x_{1},x′_{2}}δ_{x_{2},x′_{1}} , where δ is the Kronecker symbol), then T_{U}(σ) is the switch endomorphism of T_{U}(X) × T_{U}(X) .

(c) If U′ is another finite Q-set-P , then there is an isomorphism of functors T_{U⊔U′} ≅ T_{U} × T_{U′} .

Proof. (a) If X is the trivial P -set, then T_{U}(X) is the trivial Q-set, since there is a unique morphism of P -sets from any P -set to X .

(b) There is a canonical isomorphism of Q-sets

T_{U}(X × Y) = Hom_{P} (U^{op}, X × Y) ≅ Hom_{P} (U^{op}, X) × Hom_{P} (U^{op}, Y) = T_{U}(X) × T_{U}(Y) .

To see that it is natural in X and Y in the category \underline{P erm}_{k}(P) , suppose m : X → X′ (resp. n : Y → Y′ ) are morphisms in \underline{P erm}_{k}(P) . Then m is a P -invariant matrix indexed by X′ ×X , and n is a P -invariant matrix indexed by Y′ × Y . The associated morphism from X × Y to X′ × Y′ is the matrix m × n indexed by (X′ × Y′) × (X × Y) , given by

(m × n)(x′, y′, x, y) = m(x′, x) n(y′, y) .

Now let φ : U^{op} → (X × Y) and φ′ : U^{op} → (X′ × Y′) be morphisms of P -sets. Denote by φ_{X} (respectively φ_{Y} ) the composition of φ with the projection X × Y → X (respectively with the projection X × Y → Y ), and define similarly φ′_{X′} and φ′_{Y′} . The morphism T_{U}(m × n) is then defined by

T_{U}(m × n)(φ′, φ) = ∏_{u∈[U/P]} m(φ′_{X′}(u), φ_{X}(u)) n(φ′_{Y′}(u), φ_{Y} (u)) .

This can also be written as

T_{U}(m × n)(φ′, φ) = ( ∏_{u∈[U/P]} m(φ′_{X′}(u), φ_{X}(u)))( ∏_{u∈[U/P]} n(φ′_{Y′}(u), φ_{Y} (u))) ,

so that

T_{U}(m × n)(φ′, φ) = T_{U}(m)(φ′_{X′}, φ_{X}) T_{U}(n)(φ′_{Y′}, φ_{Y}) ,

which is equivalent to T_{U}(m × n) = T_{U}(m) × T_{U}(n) .

Now when Y = X , the switch morphism σ is defined by

σ(x_{1}, x_{2}, x′_{1}, x′_{2}) = δ_{x_{1},x′_{2}}δ_{x′_{1},x_{2}}

where δ is the Kronecker symbol. So if φ = (φ_{1}, φ_{2}) and φ′ = (φ′_{1}, φ′_{2}) are morphisms from U^{op} to X×X , then

T_{U}(σ)(φ, φ′) = ∏_{u∈[U/P]} δ_{φ_{1}(u),φ′_{2}(u)}δ_{φ′_{1}(u),φ_{2}(u)} .

This is non-zero if and only if φ_{1} = φ′_{2} and φ′_{1} = φ_{2} . Therefore T_{U}(σ) is the switch endomorphism of T_{U}(X) × T_{U}(X) .

(c) If X is a P -set, there is a canonical isomorphism of Q-sets

T_{U⊔U′}(X) = Hom_{P} ((U ⊔ U′)^{op}, X) ≅ Hom_{P} (U^{op}, X) × Hom_{P} (U′^{op}, X) = T_{U}(X) × T_{U′}(X) ,

mapping a morphism φ : (U ⊔ U′)^{op} → X to the pair (φ_{|U}, φ_{|U′}) . Moreover, if m is a morphism from Y to X , then T_{U⊔U′}(m) is defined by

T_{U⊔U′}(m)(φ, ψ) = ∏ ; u∈[(U⊔U′)/P] m(φ(u), ψ(u))

where φ ∈ Hom_{P} ((U ⊔ U′)^{op}, X) and ψ ∈ Hom_{P} ((U ⊔ U′)^{op}, Y) . This product can also be written as

\begin{aligned}T_{U⊔U′}(m)(φ, ψ) & = ( ∏_{u∈[U/P]} m(φ_{|U}(u), ψ_{|U}(u))) ( ∏_{u′∈[U′/P]} m(φ_{|U′}(u′), ψ_{|U′}(u′))) \\ & = T_{U}(m)(φ_{|U}, ψ_{|U}) T_{U′}(m)(φ_{|U′}, ψ_{|U′}) . \end{aligned}

This proves that T_{U⊔U′}(m) = T_{U}(m) × T_{U′}(m) .

Now we introduce algebra structures on our permutation modules. Suppose we are given a permutation P -algebra A over k, that is, a P -algebra with a P -invariant k-basis X . Equivalently, A is a permutation kP -module together with a multiplication map and a unit map

μ : A ⊗_{k} A → A  and  ε : k → A ,

satisfying conditions of associativity and unitarity, which can be expressed in terms of commutative diagrams in the category P erm_{k}(P) as follows:

[figure: A ⊗_{k} A ⊗_{k} A; id⊗μ; μ⊗id; A ⊗_{k} A; μ; A ⊗_{k} A; μ; A; k ⊗_{k} A; ε⊗id; ε; A ⊗_{k} A; μ; A ⊗_{k} k; id⊗ε; ε; A]

Using the basis X , we have morphisms \underline{μ} : X × X → X and \underline{ε} : • → X in \underline{P erm}_{k}(P) and the following diagrams are commutative:

[figure: X × X × X; id×\underline{μ}; \underline{μ}×id; X × X; \underline{μ}; X × X; \underline{μ}; X; • × X; \underline{ε}×id; \underline{ε}; X × X; \underline{μ}; X × •; id×\underline{ε}; \underline{ε}; X]

We can take the images of \underline{μ} and \underline{ε} by the functor T_{U} . Since T_{U} maps the trivial set to the trivial set and since it commutes with direct products, we obtain maps

T_{U}(\underline{μ}) : T_{U}(X) × T_{U}(X) → T_{U}(X)  and  T_{U}(\underline{ε}) : • → T_{U}(X) ,

which define a multiplication map and a unit map on k T_{U}(X) . Taking the images of the previous diagrams by the functor T_{U} , we see that k T_{U}(X) is a k-algebra. Moreover, since k T_{U}(X) is a permutation kQ-module, it is a permutation Q-algebra, which we denote by T_{U}(A) .

The algebra T_{U}(A) only depends on A up to isomorphism. Indeed, if A′ is a permutation P -algebra isomorphic to A, with multiplication μ′ and unit ε′ , then A′ admits a P -invariant k-basis X′ , and there exists an isomorphism \underline{m} : X → X′ in \underline{P erm}_{k}(P) such that the diagrams

\begin{matrix}X × X & ⟶^{\underline{μ}} & X \\ \underline{m}×\underline{m} ↓ & & ↓ \underline{m} \\ X′ × X′ & ⟶_{\underline{μ′}} & X′ \end{matrix}  \begin{matrix}• & ⟶^{\underline{ε}} & X \\ 1 ↓ & & ↓ \underline{m} \\ • & ⟶_{\underline{ε′}} & X′ \end{matrix}

are commutative. Taking the images of those diagrams under T_{U} , we see that T_{U}(\underline{m}) induces an algebra isomorphism T_{U}(A) ≅ T_{U}(A′) .

The endomorphism algebra End_{k}(kX) of a permutation kP -module kX is a special case of a permutation P -algebra. In terms of the basis X , End_{k}(kX) has basis X × X where (x, y) is the endomorphism of kX mapping z ∈ X to δ_{y,z} · x. This is the standard basis of a matrix algebra. Moreover the algebra structure of End_{k}(kX) is given by (x, y)(z, t) = δ_{y,z} · (x, t) .

(2.11) Lemma. Let X be a P -set and let U be a finite Q-set-P . Then the Q-algebra T_{U}(End_{k}(kX)) is isomorphic to End_{k}(k T_{U}(X)).

Proof. In view of the above observations, T_{U}(End_{k}(kX)) has basis T_{U}(X × X) while End_{k}(k T_{U}(X)) has basis T_{U}(X) × T_{U}(X) . So we have to prove that the natural isomorphism T_{U}(X × X) ≅ T_{U}(X) × T_{U}(X) gives rise to an algebra isomorphism of the corresponding permutation P -algebras. The multiplication in k(X × X) corresponds to the matrix m indexed by X^{4} × X^{2} given by

m(x_{1}, x_{2}, x_{3}, x_{4}, y_{1}, y_{2}) = \{ \begin{matrix}1 & if x_{2}=x_{3}, x_{1}=y_{1}, x_{4}=y_{2} , \\ 0 & otherwise. \end{matrix}

Therefore the multiplication in T_{U}(X × X) is given by

m(ϕ_{1}, ϕ_{2}, ϕ_{3}, ϕ_{4}, ψ_{1}, ψ_{2}) = ∏_{u∈[U/P]} m(ϕ_{1}(u), ϕ_{2}(u), ϕ_{3}(u), ϕ_{4}(u), ψ_{1}(u), ψ_{2}(u))

= \{ \begin{matrix}1 & if ϕ_{2}(u)=ϕ_{3}(u), ϕ_{1}(u)=ψ_{1}(u), ϕ_{4}(u)=ψ_{2}(u) , ∀u ∈ [U/P] , \\ 0 & otherwise, \end{matrix}

= \{ \begin{matrix}1 & if ϕ_{2}=ϕ_{3}, ϕ_{1}=ψ_{1}, ϕ_{4}=ψ_{2} , \\ 0 & otherwise. \end{matrix}

This corresponds precisely to the multiplication in k(T_{U}(X) × T_{U}(X)).

Note that, in the special case of tensor induction (Example 2.6), Lemma 2.11 gives a version of Lemma 2.1 (only for permutation modules).

(2.12) Proposition. Let A and B be two Dade P -algebras over k and let U be a finite Q-set-P .

(a) T_{U}(A) is a Dade Q-algebra.

(b) If A and B are equivalent, then T_{U}(A) and T_{U}(B) are equivalent.

Proof. (a) We first observe that if we apply T_{U} to End_{k}(kX) where kX is a permutation kP -module, we obtain a Q-algebra T_{U}(End_{k}(kX)) which is again isomorphic to the endomorphism algebra of a permutation module, by Lemma 2.11.

Let X be a P -invariant basis of A. By construction, T_{U}(A) is a permutation Q-algebra, with Q-invariant basis T_{U}(X) . Since X has a P -fixed point x by definition, T_{U}(X) = Hom_{P} (U^{op}, X) has a Q-fixed point, namely the constant map on x.

Now A is isomorphic to a matrix algebra, so that we have A ⊗_{k} A^{op} ≅ End_{k}(A) . As the product for the opposite algebra A^{op} is obtained by composing the multiplication μ with the switch endomorphism of A ⊗_{k} A, it follows from Proposition 2.10 that T_{U}(A^{op}) ≅ T_{U}(A)^{op} as kQ-algebras. Therefore, by Proposition 2.10 again,

T_{U}(A) ⊗_{k} T_{U}(A)^{op} ≅ T_{U}(A ⊗_{k} A^{op}) ≅ T_{U}(End_{k}(A)) ,

and by the observation above, T_{U}(End_{k}(A)) is again isomorphic to the endomorphism algebra of a (permutation) module, hence to a matrix algebra. It follows that T_{U}(A) is a central simple algebra over k.

Finally we have to see that T_{U}(A) is split. This is a technical point which we shall prove in full generality in the next section (see Lemma 3.12), but we mention here that it obviously holds if k is either algebraically closed or a finite field, because the Brauer group of such a field is trivial. So T_{U}(A) is a Dade Q-algebra.

(b) If A and B are equivalent, then

A ⊗_{k} End_{k}(kX) ≅ B ⊗_{k} End_{k}(kY)

for some permutation modules kX and kY . Therefore, by Proposition 2.10 and Lemma 2.11, we obtain

T_{U}(A) ⊗_{k} End_{k}(k T_{U}(X)) ≅ T_{U}(B) ⊗_{k} End_{k}(k T_{U}(Y)) ,

and this shows that T_{U}(A) and T_{U}(B) are equivalent.

(2.13) Corollary. If U is a finite Q-set-P, the functor T_{U} induces a group homomorphism D(P) → D(Q), mapping the class of a Dade P -algebra A to the class of the Dade Q-algebra T_{U}(A) .

This map will be denoted by D(U) : D(P) → D(Q) . In the five examples mentioned before (Examples 2.5–2.9), we obtain the fundamental operations of restriction, tensor induction, inflation, deflation, and isomorphisms.

3. Composition

We need to know how the natural operations introduced in the previous section compose with each other. A particularly important result for the sequel describes the composition of tensor induction and deflation. But our uniform formalism allows us to prove a single result for the composition of the functors T_{U} , which can be specialized to various situations.

It turns out that the composition of two functors T_{U} cannot be expressed only in terms of other such functors. There is still another natural operation on Dade algebras and endo-permutation modules which appears, that relies on the action of field endomorphisms (Galois actions for perfect fields).

If a is an endomorphism of k, we denote by

γ_{a} : \underline{P erm}_{k}(P) ⟶ \underline{P erm}_{k}(P)

the functor which is the identity on objects and which maps the morphism m : Y → X to the morphism γ_{a}(m) : Y → X defined by

γ_{a}(m)(x, y) = a(m(x, y)) , ∀x ∈ X, ∀y ∈ Y .

We shall be interested in the special case where a(λ) = λ^{p^{n}} for all λ ∈ k (where n ≥ 0 is an integer), in which case we simply write γ_{p^{n}} for this functor. Note that this endomorphism is a (Galois) automorphism if and only if k is perfect. So in the general case we have to consider endomorphisms.

Let A be a permutation P -algebra, with multiplication μ and unit ε. Choosing a P -invariant k-basis X of A, we have morphisms \underline{μ} : X × X → X and \underline{ε} : • → X in \underline{P erm}_{k}(P) and commutative diagrams:

[figure: X × X × X; id×\underline{μ}; \underline{μ}×id; X × X; \underline{μ}; X × X; \underline{μ}; X; • × X; \underline{ε}×id; \underline{ε}; X × X; \underline{μ}; X × •; id×\underline{ε}; \underline{ε}; X]

We can take the image of those diagrams by our functor γ_{a} . It is clear that if m and n are morphisms in \underline{P erm}_{k}(P) , then γ_{a}(m × n) = γ_{a}(m) × γ_{a}(n) , so the previous diagrams become

[figure: X × X × X; id×γ_{a}(\underline{μ}); γ_{a}(\underline{μ})×id; X × X; γ_{a}(\underline{μ}); X × X; γ_{a}(\underline{μ}); X; • × X; γ_{a}(\underline{ε})×id; γ_{a}(\underline{ε}); X × X; \underline{μ}; X × •; id×γ_{a}(\underline{ε}); γ_{a}(\underline{ε}); X]

and this shows that we have defined another algebra structure on kX , which we will denote by γ_{a}(A) . This algebra structure is obtained by twisting by the endomorphism a the multiplication constants of A with respect to the fixed P -invariant k-basis X .

The algebra γ_{a}(A) only depends on A up to isomorphism. Indeed, if A′ is a permutation P -algebra isomorphic to A, with multiplication μ′ and unit ε′ , then A′ admits a P -invariant k-basis X′ , and there exists an isomorphism \underline{m} : X → X′ in \underline{P erm}_{k}(P) such that the diagrams

\begin{matrix}X × X & ⟶^{\underline{μ}} & X \\ \underline{m}×\underline{m} ↓ & & ↓ \underline{m} \\ X′ × X′ & ⟶_{\underline{μ′}} & X′ \end{matrix}  \begin{matrix}• & ⟶^{\underline{ε}} & X \\ 1 ↓ & & ↓ \underline{m} \\ • & ⟶_{\underline{ε′}} & X′ \end{matrix}

are commutative. The images of those diagrams under γ_{a} show that γ_{a}(\underline{m}) is an isomorphism of P -algebras from γ_{a}(A) to γ_{a}(A′) . It is also clear that if A and B are permutation P -algebras, then

γ_{a}(A ⊗_{k} B) ≅ γ_{a}(A) ⊗ γ_{a}(B)  and  γ_{a}(A^{op}) ≅ γ_{a}(A)^{op} .

(3.1) Lemma. Let a be an endomorphism of the field k and let A and B be two Dade P -algebras over k.

(a) γ_{a}(A) is a Dade P -algebra.

(b) If A and B are equivalent, then γ_{a}(A) and γ_{a}(B) are equivalent.

Proof. (a) As γ_{a}(A) has the same P -invariant basis as A, there is a fixed point in this basis. In order to show that γ_{a}(A) is a matrix algebra, it suffices to restrict the group action to the trivial group (in which case we have many more bases available). Now A is isomorphic to M_{n}(k) for some n and so γ_{a}(A) ≅ γ_{a}(M_{n}(k)) by the observations above. Choosing the canonical basis of the matrix algebra M_{n}(k) , the multiplication constants have values 0 or 1 and so are fixed under a. It follows that γ_{a}(M_{n}(k)) = M_{n}(k) .

(b) If A and B are equivalent, then A⊗_{k} End_{k}(kX) ≅ B ⊗_{k} End_{k}(kY) for some permutation modules kX and kY . Since γ_{a} preserves tensor products, it suffices to show that γ_{a}(End_{k}(kX)) is again the endomorphism algebra of a permutation module. But since End_{k}(kX) has P -invariant basis X × X with multiplication constants having values 0 or 1, it is fixed under γ_{a} .

It follows that γ_{a} induces a well defined map γ_{a} : D(P) ⟶ D(P) , mapping the class of a Dade P -algebra A to the class of γ_{a}(A) .

(3.2) Lemma. Let P and Q be p-groups, let U be a Q-set-P , and let a be an endomorphism of k.

(a) There is an isomorphism of functors θ : T_{U} ∘ γ_{a} → γ_{a} ∘ T_{U} from \underline{P erm}_{k}(P) to \underline{P erm}_{k}(Q) .

(b) We have D(U) ∘ γ_{a} = γ_{a} ∘ D(U) as maps D(P) → D(Q) .

Proof. Note that we use the same notation γ_{a} : \underline{P erm}_{k}(P) → \underline{P erm}_{k}(P) for every p-group P , without mentioning P . It is clear that T_{U} ∘ γ_{a} and γ_{a} ∘ T_{U} coincide on objects since γ_{a} is the identity on objects. Thus the map θ_{X} is equal to the identity for any P -set X . Now if m : Y → X is a morphism in \underline{P erm}_{k}(P) , we have

\begin{aligned}(T_{U} ∘ γ_{a})(m)(φ, ψ) & = ∏_{u∈[U/P]} γ_{a}(m)(φ(u), ψ(u)) = ∏_{u∈[U/P]} a(m(φ(u), ψ(u))) \\ & = a ( ∏_{u∈[U/P]} m(φ(u), ψ(u)) = (γ_{a} ∘ T_{U})(m)(φ, ψ) , \end{aligned}

and (a) follows. Now if X and Y are P -sets, the diagram

(T_{U} ∘ γ_{a})(X × Y) ⟶^{≃} (T_{U} ∘ γ_{a})(X) × (T_{U} ∘ γ_{a})(Y)
θ_{X×Y} ↓  ↓ θ_{X}×θ_{Y}
(γ_{a} ∘ T_{U})(X × Y) ⟶_{≃} (γ_{a} ∘ T_{U})(X) × (γ_{a} ∘ T_{U})(Y)

is commutative, since the vertical arrows are both the identity maps, and the horizontal ones are the same map. Applying this to the multiplication map for a permutation P -algebra A, it follows that the Q-algebras (T_{U} ∘ γ_{a})(A) and (γ_{a} ∘ T_{U})(A) are isomorphic, proving (b).

(3.3) Example. For every endomorphism a of k, we have γ_{a}(Ω_{P}) = Ω_{P} . Indeed Ω_{P} is defined over the prime field 𝔽_{p} and a is the identity on 𝔽_{p} . It follows that if P is abelian, then γ_{a} : D(P) → D(P) is the identity because, by Dade’s Theorem 1.4, D(P) is generated by elements of the form Ω_{P/Q} . By the injectivity of the map in Theorem 1.6, this implies that γ_{a} is always the identity on ℚD(P) , for every p-group P , because γ_{a} commutes with restriction and deflation by Lemma 3.2.

(3.4) Example. The only example we know of an endo-permutation module which is not defined over the prime field 𝔽_{p} occurs for p = 2 and the quaternion group Q_{8} . By a construction of Dade [Da1], there is a 3-dimensional endo-trivial 𝔽_{4}Q_{8}-module M , which has order 4 in the Dade group. If a denotes the non-trivial Galois automorphism of 𝔽_{4} , it turns out that γ_{a}(M) ≅ M^{∗} and this shows that γ_{a} is not the identity of the Dade group.

Despite the fact that γ_{a} is often the identity, we are forced to introduce it in general in order to describe the composition of the functors T_{U} . Suppose that P , Q, and R are p-groups. If U is a finite Q-set-P and V is a finite R-set-Q, we denote by V ×_{Q} U the quotient of V × U by the right Q-action defined by

(v, u)·h = (vh, h^{−1}u) , ∀v ∈ V, ∀u ∈ U, ∀h ∈ Q .

In other words V ×_{Q} U = (V × U^{op})/Q. It is an R-set-P for the following double action:

l·(v, u)·g = (lv, ug) , ∀v ∈ V, ∀u ∈ U, ∀l ∈ R, ∀g ∈ P .

If (u, v) ∈ V ×_{Q}U , let Q_{v} denote the right stabilizer of v in Q and Q_{v,uP} the stabilizer of the pair (v, uP) , that is,

Q_{v,uP} = {h ∈ Q | vh = v, ∃g ∈ P such that hu = ug} .

Now we can describe the composition of the functors T_{V} and T_{U} .

(3.5) Proposition. Let U be a finite Q-set-P and let V be a finite R-set-Q, where P , Q, and R are finite p-groups.

(a) If X is a finite P -set, there is a canonical isomorphism of R-sets

(T_{V} ∘ T_{U})(X) = Hom_{Q}(V^{op}, Hom_{P} (U^{op}, X)) ≅ Hom_{P} (U^{op} ×_{Q} V^{op}, X) = T_{V ×_{Q}U}(X) .

In other words the functors T_{V} ∘ T_{U} and T_{V ×_{Q}U} are isomorphic on objects of \underline{P erm}_{k}(P) .

(b) If m is a morphism in \underline{P erm}_{k}(P) from X to Y , then

\begin{aligned}T_{V ×_{Q}U}(m)(ψ, φ) & = ∏_{(v,u)∈[(V ×_{Q}U)/P]} m(ψ(v, u), φ(v, u)) , \\ (T_{V} ∘ T_{U})(m)(ψ, φ) & = ∏_{(v,u)∈[(V ×_{Q}U)/P]} (m(ψ(v, u), φ(v, u)))^{|Q_{v}:Q_{v,uP} |} , \end{aligned}

for all ψ ∈ Hom_{P} ((V ×_{Q} U)^{op}, Y ) and φ ∈ Hom_{P} ((V ×_{Q} U)^{op}, X).

Proof. Statement (a) is straightforward to show, so we prove (b). The expression for T_{V ×_{Q}U}(m)(ψ, φ) is just the definition. On the other hand, using the isomorphism of part (a), the matrix (T_{V} ∘ T_{U})(m) can be expressed as

(3.6)  (T_{V} ∘ T_{U})(m)(ψ, φ) = ∏_{v∈[V/Q]} ∏_{u∈[U/P]} m(ψ(v, u), φ(v, u)) ,

where we have fixed sets of representatives [U/P] and [V/Q] of U/P and V/Q respectively. In general, the sets (V ×_{Q} U)/P and V/Q × U/P are not isomorphic. So the functors T_{V ×_{Q}U} and T_{V} ∘ T_{U} are not isomorphic on morphisms. To see how much they differ, consider the map

θ : [V/Q] × [U/P] → (V ×_{Q} U)/P , θ(v_{0}, u_{0}) = (v_{0}, u_{0})P .

This map is surjective, since if (v, u) ∈ V ×_{Q} U , there exist v_{0} ∈ [V/Q] (unique) and h ∈ Q such that v = v_{0}h. Now in V ×_{Q} U , we have (v, u) = (v_{0}, hu) , and there exist u_{0} ∈ [U/P] (unique) and g ∈ P such that hu = u_{0}g. Then we clearly have

θ(v_{0}, u_{0}) = (v_{0}, u_{0})P = (v_{0}, u_{0}g)P = (v_{0}, hu)P = (v_{0}h, u)P = (v, u)P .

Now two pairs (v_{0}, u_{0}) and (v_{1}, u_{1}) have the same image under θ if and only if there exist h ∈ Q and g ∈ P such that

v_{1} = v_{0}h and h^{−1}u_{1} = u_{0}g .

The first equality gives v_{1} = v_{0} = v_{0}h, since v_{1} and v_{0} are in the set of representatives [V/Q] . The second one gives u_{1} = hu_{0}g. In other words u_{1}P is in the left orbit of u_{0}P under the right stabilizer Q_{v_{0}} of v_{0} . So the inverse image θ^{−1}((v, u)P ) has cardinality |Q_{v} : Q_{v,uP} | and we can rewrite Equation (3.6) as

(T_{V} ∘ T_{U})(m)(ψ, φ) = ∏_{(v,u)∈[(V ×_{Q}U)/P]} (m(ψ(v, u), φ(v, u)))^{|Q_{v}:Q_{v,uP} |} ,

as was to be shown.

(3.7) Corollary. Let U be a finite Q-set-P and let V be a finite R-set-Q, where P , Q, and R are finite p-groups.

(a) If the right action of Q on V is free, then the functors T_{V} ∘ T_{U} and T_{V ×_{H}U} are isomorphic.

(b) If the right action of P on U is transitive, then the functors T_{V} ∘ T_{U} and T_{V ×_{H}U} are isomorphic.

Proof. For (a), we have Q_{v} = 1 for every v ∈ V , while for (b), U/P is a singleton, so that Q_{v} = Q_{v,uP} . So in both cases [Q_{v} : Q_{v,uP}] = 1 and Proposition 3.5 shows that (T_{V} ∘ T_{U})(m) = T_{V ×_{Q}U}(m) for every morphism m.

The assumption in (a) is satisfied if T_{V} is the restriction Res_{R}^{Q} where R ≤ Q (because V = Q as an R-set-Q), if T_{V} is the tensor induction Ten_{Q}^{R} where Q ≤ R (because V = R as an R-set-Q), if T_{V} is the inflation Inf_{Q}^{R} where Q = R/S (because V = Q as an R-set-Q), and also if T_{V} is an isomorphism Iso_{Q}^{R} . Similarly, the assumption in (b) is satisfied if T_{U} is a restriction, an inflation, a deflation, or an isomorphism.

Many standard results follow from this, including transitivity properties and the Mackey formula. We illustrate the mechanism with the latter formula. Let R and S be subgroups of Q. Let _{R}Q_{Q} = Q viewed as an R-set-Q (restriction) and similarly _{Q}Q_{S} = Q viewed as a Q-set-S (tensor induction). Making the product over Q, we obtain the R-set-S

RQ_{Q} ×_{Q} _{Q}Q_{S} = _{R}Q_{S} = ∐_{u∈[R\Q/S]} RuS ≅ ∐_{u∈[R\Q/S]} R ×_{R∩^{u}S} ^{u}S ×_{(^{u}S)} S ,

from which it follows that

Res_{R}^{Q} ∘ Ten_{S}^{Q} = T_{_{R}Q_{Q}} ∘ T_{_{Q}Q_{S}} = T_{_{R}Q_{Q}×_{Q}_{Q}Q_{S}} = ∏_{u∈[R\Q/S]} Ten_{R∩^{u}S}^{R} ∘ Res_{R∩^{u}S}^{^{u}S} ∘ Iso_{S}^{^{u}S} ,

which is the Mackey formula. The product here is the product in \underline{P erm}_{k}(R) , which corresponds to the tensor product of permutation modules.

All the results above have been expressed in terms of the functors T_{V} and T_{U} . When these functors are applied to a permutation P -algebra A, we obtain corresponding isomorphisms of R-algebras, as follows.

(3.8) Corollary. Let U be a finite Q-set-P , let V be a finite R-set-Q, where P , Q, and R are finite p-groups, and let A be a permutation P -algebra. If either the right action of Q on V is free, or the right action of P on U is transitive, then the R-algebras (T_{V} ∘ T_{U})(A) and T_{V ×_{H}U}(A) are isomorphic. In particular, D(V) ∘ D(U) = D(V ×_{H} U) as maps D(P) → D(R) .

As explained above, this applies whenever D(V) is either a restriction, a tensor induction, an inflation, or an isomorphism, or when D(U) is a restriction, an inflation, a deflation, or an isomorphism. The only remaining case is when D(V) is a deflation and D(U) is a tensor induction. Indeed, if U is an arbitrary transitive Q-set-P , then U decomposes as a product of those five special types of Q-sets-P (see Lemma 7.4).

So finally we come to the formula which describes the last case and which will play a crucial role in the next sections. The result holds when Def_{P/R}^{P} ∘ Ten_{Q}^{P} is applied to any permutation Q-algebra, but for simplicity we only state the result for the corresponding maps between the Dade groups.

(3.9) Proposition. Let P be a p-group, let Q be a subgroup of P , and let R be a normal subgroup of P . Then

Def_{P/R}^{P} ∘ Ten_{Q}^{P} = γ_{|R:Q∩R|} ∘ Ten_{QR/R}^{P/R} ∘ Iso_{Q/Q∩R}^{QR/R} ∘ Def_{Q/Q∩R}^{Q} ,

as maps from D(Q) to D(P/R) .

Proof. Let V = P/R, viewed as a (P/R)-set-P , let U = P , viewed as a P -set-Q, and let A be a Dade Q-algebra with a P -invariant basis X . Then by Examples 2.6 and 2.8, we have

Def_{P/R}^{P} Ten_{Q}^{P}(A) = (T_{V} ∘ T_{U})(A) .

Now V ×_{P} U is the set P/R, viewed as a (P/R)-set-Q. The stabilizer in P of every v ∈ V is P_{v} = R (because R is a normal subgroup), and if u ∈ U , then

P_{v,uQ} = {g ∈ P_{v} | ∃h ∈ Q, gu = uh} = ^{u}Q ∩ R .

Since R is normal in P , we have |R : ^{u}Q ∩ R| = |R : Q ∩ R| and this shows that the index appearing as an exponent in Proposition 3.5 is constant. Therefore, if we apply Proposition 3.5 to any morphism m : Z → Y in \underline{P erm}_{k}(Q) , we obtain

T_{V ×_{P} U}(m)(ψ, φ) = ∏_{(v,u)∈[(V ×_{P} U)/Q]} m(ψ(v, u), φ(v, u)) ,
and  (T_{V} ∘ T_{U})(m)(ψ, φ) = ∏_{(v,u)∈[(V ×_{P} U)/Q]} (m(ψ(v, u), φ(v, u)))^{|R:^{u}Q∩R|}

= γ_{|R:Q∩R|} ( ∏_{(v,u)∈[(V ×_{P} U)/Q]} m(ψ(v, u), φ(v, u)) ) ,

for all ψ ∈ Hom_{P} ((V ×_{Q} U)^{op}, Y ) and φ ∈ Hom_{P} ((V ×_{Q} U)^{op}, Z). Applying this to the morphism m : X × X → X which represents the multiplication in A, we see that the multiplication in (T_{V} ∘ T_{U})(A) is obtained by applying γ_{|R:Q∩R|} to the multiplication constants in T_{V ×_{P} U}(A) . In other words, this shows that

Def_{P/R}^{P} Ten_{Q}^{P}(A) ≅ (T_{V} ∘ T_{U})(A) ≅ γ_{|R:Q∩R|}(T_{V ×_{P} U}(A)) .

Now let

Z = P/R ,  viewed as a (P/R)-set-(QR/R) (tensor induction),
I = QR/R ,  viewed as a (QR/R)-set-(Q/Q ∩ R) (isomorphism),
W = Q/Q ∩ R ,  viewed as a (Q/Q ∩ R)-set-Q (deflation).

Then we have

V ×_{P} U = Z ×_{QR/R} I ×_{Q/Q∩R} W .

As the right action of Q/Q ∩ R on I and the right action of QR/R on Z are free, Corollary 3.8 implies that this product can be replaced by a composition. So we obtain

T_{V ×_{P} U}(A) ≅ (T_{Z} ∘ T_{I} ∘ T_{W})(A) = Ten_{QR/R}^{P/R} Iso_{Q/Q∩R}^{QR/R} Def_{Q/Q∩R}^{Q}(A) ,

and the proposition follows.

(3.10) Remark. When applied to a permutation P -algebra A, it is possible to express the general result of Proposition 3.5 using the twists γ_{p^{n}} , thus obtaining a single result including Corollary 3.8 and Proposition 3.9 as special cases. It suffices to notice that the index |Q_{v} : Q_{v,uP} | only depends on the orbit of (v, u) ∈ V ×_{Q} U under the action of R × P^{op} . Therefore (T_{V} ∘ T_{U})(A) decomposes as

(T_{V} ∘ T_{U})(A) = ⨂_{(v,u)∈[R\(V ×_{Q}U)/P]} γ_{|Q_{v}:Q_{v,uP} |}(A_{v,u}) ,

for suitable R-algebras A_{v,u} , while

(T_{V ×_{Q}U})(A) = ⨂_{(v,u)∈[R\(V ×_{Q}U)/P]} A_{v,u} ,

without any twists.

As mentioned in Example 3.3, the twist γ_{|R:Q∩R|} is the identity if we apply it to the Dade group of an abelian group. Therefore, Proposition 3.9 can be simplified in the following way.

(3.11) Corollary. Let P be a p-group, let Q be a subgroup of P , and let R be a normal subgroup of P .

(a) If Q_{0} is a normal subgroup of Q such that Q/Q_{0} is abelian, then

Def_{P/R}^{P} ∘ Ten_{Q}^{P} ∘ Inf_{Q/Q_{0}}^{Q} = Ten_{QR/R}^{P/R} ∘ Iso_{Q/Q∩R}^{QR/R} ∘ Def_{Q/Q∩R}^{Q} ∘ Inf_{Q/Q_{0}}^{Q} ,

as maps from D(Q/Q_{0}) to D(P/R) .

(b) If Q is abelian, then

Def_{P/R}^{P} ∘ Ten_{Q}^{P} = Ten_{QR/R}^{P/R} ∘ Iso_{Q/Q∩R}^{QR/R} ∘ Def_{Q/Q∩R}^{Q} ,

as maps from D(Q) to D(P/R) .

Proof. Clearly (b) follows from (a) by taking Q_{0} = 1 . For (a) we apply Proposition 3.9 and compose everything with Inf_{Q/Q_{0}}^{Q} . Since γ_{|R:Q∩R|} commutes with tensor induction, isomorphisms, deflation, and inflation (Lemma 3.2) and since it is the identity of D(Q/Q_{0}) because Q/Q_{0} is abelian, the result follows.

We close this section with the result which was needed in the proof of Proposition 2.12.

(3.12) Lemma. Let U be a finite Q-set-P , where P and Q are finite p-groups, and let A be a Dade P -algebra. Then the algebra T_{U}(A) is split (hence is a Dade Q-algebra).

Proof. It suffices to restrict to the trivial group, so we compose T_{U} with Res_{1}^{Q} = T_{V} where V = Q, viewed as a 1-set-Q. Since (T_{V} ∘ T_{U})(A) ≅ T_{V ×_{Q}U}(A) by Corollary 3.8, we have reduced to the case of a 1-set-P . We can also assume that this set is transitive, since a disjoint union corresponds to a tensor product and a tensor product of split algebras is split. So U has the form R\P for some subgroup R of P and therefore T_{U} corresponds to the deflation Def_{P/R}^{P} (followed by Res_{1}^{P/R} since we only consider the 1-action on the left).

So we have to prove that the Brauer quotient Def_{P/R}^{P}(A) = A[R] is split. Recall from the proof of Proposition 2.12 that we already know that A[R] is a central simple k-algebra. Since we only need Res_{R}^{P}(A) in the definition of A[R] (and since the restriction of a split algebra is split!), we can assume that R = P . We proceed by induction on |P |.

If P is cyclic of order p, then up to equivalence of Dade P -algebras, we have either A ≅ k or A ≅ End_{k}(Ω_{P} (k)), in which case the result is easy. Moreover the equivalence relation preserves the splitting property. Alternatively, every module is defined over 𝔽_{p} and indecomposable modules are absolutely indecomposable, so we can work over an algebraically closed field where all central simple algebras are split.

If now |P | > p, we choose a normal subgroup S of index p and we recall that A[P] ≅ A[S][P/S] . By induction, we know that A[S] is split, hence is a Dade (P/S)-algebra. By the cyclic case, we conclude that A[S][P/S] is split.

4. The torsion-free rank of the Dade group

The purpose of this section is to prove Theorem A of the introduction. More precisely, we shall prove the following result.

(4.1) Theorem. Let P be a finite p-group. Let S be the set of non-cyclic subgroups of P and for every Q ∈ S , choose a normal subgroup Q_{0} ◁ Q such that Q/Q_{0} is elementary abelian of rank 2. Then the ℚ-linear map

ψ_{P} = ∏_{Q∈[S/P]} Defres_{Q/Q_{0}}^{P} : ℚD(P) ⟶ ∏_{Q∈[S/P]} ℚD(Q/Q_{0}) ≅ ∏_{Q∈[S/P]} ℚ

is an isomorphism, where [S/P] denotes a set of representatives of the conjugacy classes of subgroups in S . In particular, the torsion-free rank of D(P) is equal to the number of conjugacy classes of non-cyclic subgroups of P .

If E is elementary abelian of rank 2, then by Theorem 1.4 there is an isomorphism ℚD(E) ⟶^{∼} ℚ mapping Ω_{E} to 1 (and this explains the right hand side isomorphism in the statement). This isomorphism is canonical since Ω_{E} is invariant under the group of automorphisms of E . Throughout this section, we shall always identify ℚD(E) with ℚ via this isomorphism. This identification is particularly useful for the statement of the following crucial lemma.

(4.2) Lemma. Let P be an elementary abelian group of rank 3 and let L be a subgroup of P of order p. Then the map

χ = p Def_{P/L}^{P} − ∑_{{E≯L \atop rk(E)=2}}Res_{E}^{P} ,

viewed as a homomorphism ℚD(P) → ℚ, is independent of L.

Proof. By Theorem 1.4, {Ω_{P/C} | |C| = p} ∪ {Ω_{P} } is a basis of ℚD(P) . We compute χ on each basis element. By Lemma 1.3, we have

χ(Ω_{P}) = − ∑_{{E≯L \atop rk(E)=2}}1 = −p^{2} ,

and this is independent of L. If C is a subgroup of order p, we have

Def_{P/L}^{P}(Ω_{P/C}) = \{ \begin{matrix}1 if C = L , \\ 0 if C ≠ L , \end{matrix}

because in the second case, the Brauer quotient corresponding to L is equal to the Brauer quotient corresponding to CL since C acts trivially, and Def_{P/CL}^{P/C}(Ω_{P/C}) = 0 by Lemma 1.3. On the other hand, if E is a subgroup of rank 2,

Res_{E}^{P}(Ω_{P/C}) = \{ \begin{matrix}1 if E ≯ C , \\ 0 if E > C , \end{matrix}

because in the first case, the action of E coincides with the action of EC/C = P/C , while in the second we have Res_{E/C}^{P/C}(Ω_{P/C}) = Ω_{E/C} which is zero since E/C has order p and ℚD(E/C) = 0 (in fact Ω_{E/C} is a torsion element in D(E/C) ). It follows that

if C = L , then χ(Ω_{P/L}) = p · 1 − ∑_{{E≯L \atop rk(E)=2}}1 = p − p^{2} ,
if C ≠ L , then χ(Ω_{P/C}) = 0 − ∑_{{E≯L, E≯C \atop rk(E)=2}}1 = −(p^{2} − p) .

So χ(Ω_{P/C}) = p − p^{2} is independent of L.

(4.3) Remark. Lemma 4.2 is in fact a special case of a result which holds for an arbitrary p-group P : for every normal subgroup N such that P/N is elementary abelian of rank 2, the map

∑_{{X≤P \atop XN=P}} |X| μ(X, P) Defres_{X/(X∩N)}^{P} : ℚD(P) ⟶ ℚ

is independent of N . Here μ(X, P) denotes the Möbius function of the poset of subgroups of P . This fact will be proved in Remark 10.2. Even for an elementary abelian group of rank 4, it is a rather tedious exercise to give a direct proof using the same method as above.

(4.4) Proposition. The homomorphism ψ_{P} is injective.

Proof. By Theorem 1.6, we know that

(4.5)  Defres : ℚD(P) ⟶ ∏_{Q/R} ℚD(Q/R) ≅ ∏_{Q/R} ℚ

is injective, where Q/R runs over all elementary abelian sections of P of rank 2. But the various maps Defres_{Q/R}^{P} are not linearly independent (as Lemma 4.2 shows). We prove the injectivity of the map ψ_{P} of the statement by induction on |P |. The result is clear if |P | = p because ℚD(P) = 0 .

Assume |P | ≥ p^{2} and let a ∈ Ker(ψ_{P}) . By the injectivity of the map (4.5), it suffices to show that Defres_{Q/R}^{P}(a) = 0 for every elementary abelian section Q/R of rank 2. For every proper subgroup T of P , we have Res_{T}^{P} (a) ∈ Ker(ψ_{T}) , provided we make a consistent choice ot the sections Q/Q_{0} for the subgroup T , that is, we keep the same choice of Q_{0} for every Q ≤ T . Note also that the choice of representatives of conjugacy classes does not play any role here since Defres_{^{g}Q/^{g}R}^{P} = Conj_{g}Defres_{Q/R}^{P} , where Conj_{g} is the conjugation map, so that Ker(Defres_{^{g}Q/^{g}R}^{P}) = Ker(Defres_{Q/R}^{P}) .

Since ψ_{T} is injective by induction, we obtain Res_{T}^{P} (a) = 0 for every proper subgroup T and consequently Defres_{Q/R}^{P}(a) = 0 whenever Q is a proper subgroup of P . So we are left with the proof that Def_{P/R}^{P}(a) = 0 for every elementary abelian quotient P/R of rank 2. We know by assumption that Def_{P/P_{0}}^{P}(a) = 0 . The group P/(P_{0} ∩R) is elementary abelian (of rank at most 4) and so it is clearly possible to find a sequence of subgroups S_{i} ( 1 ≤ i ≤ m) such that

S_{1} = P_{0} , S_{m} = R , S_{i} ≥ P_{0} ∩ R , |P : S_{i}| = p^{2} , |P : (S_{i} ∩ S_{i+1})| = p^{3} (1 ≤ i ≤ m − 1) .

So it suffices to prove that if Def_{P/S_{i}}^{P}(a) = 0 , then Def_{P/S_{i+1}}^{P}(a) = 0 . Let N = S_{i} ∩ S_{i+1} so that P/N has rank 3. Applying Def_{P/N}^{P} and using Lemma 4.2 for the group P/N , we obtain

p Def_{P/S_{i+1}}^{P} − ∑_{{E≯S_{i+1} \atop rk(E/N)=2}}Defres_{E/N}^{P} = p Def_{P/S_{i}}^{P} − ∑_{{E≯S_{i} \atop rk(E/N)=2}}Defres_{E/N}^{P} .

Since we already know that Defres_{E/N}^{P}(a) = 0 for every such subgroup E (because E is a proper subgroup of P ), we are left with

p Def_{P/S_{i+1}}^{P}(a) = p Def_{P/S_{i}}^{P}(a) = 0 ,

so that Def_{P/S_{i+1}}^{P}(a) = 0 as required.

We shall not prove directly the surjectivity of ψ_{P} . We shall construct a map α_{P} in the reverse direction and then show that ψ_{P} α_{P} is an isomorphism. If Q/R is a section of P , in analogy with the notation Defres_{Q/R}^{P} , we define

Teninf_{Q/R}^{P} = Ten_{Q}^{P}Inf_{Q/R}^{Q} .

Now for every elementary abelian section Q/N of P of rank 2, we define

β_{Q,N}^{P} = \frac{1}{|Q|} ∑_{{X≤Q \atop XN=Q}} |X| μ(X, Q) Teninf_{X/(X∩N)}^{P} : ℚ ⟶ ℚD(P) .

This homomorphism is dual to the one mentioned in Remark 4.3. Note that the conditions on X imply that X/(X ∩ N) ≅ Q/N is elementary abelian of rank 2, so that ℚD(X/(X ∩ N)) ≅ ℚ and the domain of the map is indeed ℚ. Note also that if we needed an integral linear combination, it would suffice to multiply β_{Q,N}^{P} by p, because p |X| μ(X, Q) is always a multiple of |Q|. There is a close connection between the idempotents of the Burnside ring and the formula for β_{Q,N}^{P} (as well as the formula in Remark 4.3 above). This will appear in Sections 10 and gives a reason why such linear combinations play a crucial role.

Now we define

α_{P} = ∑_{Q∈[S/P]} β_{Q,Q_{0}}^{P} : ∏_{Q∈[S/P]} ℚ ⟶ ℚD(P) ,

where S and Q_{0} are defined as in the statement of Theorem 4.1. In order to be able to compute ψ_{P} α_{P} , we shall collect properties of the maps β_{Q,N}^{P} , but we first need a combinatorial lemma.

We recall that, for p-groups, the Möbius function μ(X, P) vanishes unless X is a normal subgroup with P/X elementary abelian, and if P/X is elementary abelian of rank r, then μ(X, P) = (−1)^{r}p^{\binom{r}{2}} (see [KT, 2.4]). As in [Bo1], whenever N is a normal subgroup of Q, we define

(4.6)  m_{Q,N} = \frac{1}{|Q|} ∑_{{X≤Q \atop XN=Q}} |X| μ(X, Q) .

We need the following technical property about this number.

(4.7) Lemma. Let N be a normal subgroup of Q with Q/N elementary abelian of rank 2.

(a) m_{Q,N} is independent of N .

(b) If M is another normal subgroup of Q with Q/M elementary abelian of rank 2, then

m_{Q,M} = m_{Q,N} = \frac{1}{|Q|} ∑_{{XM=XN=Q \atop X∩M=X∩N}} |X| μ(X, Q) = (1 − p)(1 − p^{2}) … (1 − p^{n−2}) ,

where n is the rank of Q/Φ(Q) (the largest elementary abelian quotient of Q).

Proof. (a) follows from (b). The expression of m_{Q,N} as a sum is a special case of the formula appearing at the bottom of page 703 of [Bo1]. This uses the fact that Q/M is a b-group, which means that for every non-trivial normal subgroup U of Q/M , we have m_{Q/M,U} = 0 (as it is easy to check, or see example 4, page 709 of [Bo1]). It is clear from the formula that m_{Q,M} = m_{Q,N} .

For the expression of m_{Q,N} as a product, one notices first that in (4.6) every subgroup X contains the Frattini subgroup Φ(Q) (otherwise μ(X, Q) = 0 ), so that one can assume that Q is elementary abelian. Writing E_{n} for the elementary abelian group of rank n, we prove by induction on k that

(4.8)  m_{E_{n},E_{k}} = ∏_{i=1}^{k}(1 − p^{n−1−i}) ,

for 1 ≤ k ≤ n−2 . If k = 1 , we have m_{E_{n},E_{1}} = (1−p^{n−2}) by a direct computation (in (4.6), either we have X = E_{n} and μ(E_{n}, E_{n}) = 1 or X has index p and does not contain E_{1} in which case μ(X, E_{n}) = −1 and there are p^{n−1} such subgroups). The induction step is obtained by using the transitivity formula m_{G,N} = m_{G,M} m_{G/M,N/M} where M ≤ N ≤ G. This formula appears on page 703 of [Bo1]. Thus we have

m_{E_{n},E_{k}} = m_{E_{n},E_{k−1}} m_{E_{n−k+1},E_{1}} = [ ∏_{i=1}^{k−1}(1 − p^{n−1−i}) ](1 − p^{n−k+1−2}) = ∏_{i=1}^{k}(1 − p^{n−1−i}) ,

as required. Now the result follows by taking k = n − 2 .

We recall that the notation R ≥_{P} Q means that R contains a P -conjugate of R. We write T (Q, R) for the set of all g ∈ P such that ^{g}Q ≤ R.

(4.9) Proposition. Let Q be a subgroup of a p-group P and let N be a normal subgroup of Q with Q/N elementary abelian of rank 2.

(a) If R is a subgroup of P , then

Res_{R}^{P} β_{Q,N}^{P} = \{ \begin{matrix}∑_{g∈[R\T (Q,R)/Q]} β_{^{g}Q,^{g}N}^{R} if R ≥_{P} Q , \\ 0 if R ≱_{P} Q . \end{matrix}

(b) If M is a normal subgroup of Q with Q/M elementary abelian of rank 2, then Defres_{Q/M}^{P} β_{Q,N}^{P} is the multiplication by m_{Q,M}|N_{P} (Q)/Q|, where m_{Q,M} is given by (4.6).

(c) β_{Q,N}^{P} is independent of N , that is, β_{Q,M}^{P} = β_{Q,N}^{P} , where M is as in (b).

Proof. (a) We first compute Res_{R}^{Q} β_{Q,N}^{Q} . We apply the Mackey formula, taking into account the fact that each subgroup X is a normal subgroup of Q (since otherwise μ(X, Q) = 0 ), so that R\Q/X = RX\Q.

Since X/(X ∩ N) = E has rank 2, ℚD(E) ≅ ℚ is generated by Ω_{E} which is invariant under the action of Aut(E) , so that the map Conj_{g} : ℚD(E) → ℚD(E) is the identity. Thus we obtain

Res_{R}^{Q} β_{Q,N}^{Q} = \frac{1}{|Q|} ∑_{{X≤Q \atop XN=Q}} |X| μ(X, Q) Res_{R}^{Q} Ten_{X}^{Q} Inf_{X/(X∩N)}^{X}

= \frac{1}{|Q|} ∑_{{X≤Q \atop XN=Q}} |X| μ(X, Q) ∑_{g∈[RX\Q]} Ten_{R∩X}^{R} Res_{R∩X}^{X} Inf_{X/(X∩N)}^{X} Conj_{g}

= \frac{1}{|Q|} ∑_{S≤R} Ten_{S}^{R} ∑_{{XN=Q \atop R∩X=S}} |X| μ(X, Q) |Q : RX| Res_{S}^{X} Inf_{X/(X∩N)}^{X} .

We claim that Res_{S}^{X} Inf_{X/(X∩N)}^{X} = 0 if S does not cover the quotient Q/N (that is, SN ≠ Q). Indeed S does not cover either the quotient X/(X ∩ N) , because XN = Q. Therefore Res_{S}^{X} Inf_{X/(X∩N)}^{X} factorizes through the proper subgroup S(X ∩ N)/(X ∩ N) of X/(X ∩ N) . But X/(X ∩ N) = E has rank 2 and ℚD(Y) = 0 for every proper subgroup Y of E , proving the claim.

Now if SN = Q and S ≤ X ≤ Q, then XN = Q and the inclusion S → X induces an isomorphism S/(S ∩ N) ≅ X/(X ∩ N) . Thus we have Res_{S}^{X} Inf_{X/(X∩N)}^{X} = Inf_{S/(S∩N)}^{S} , using our usual identification ℚD(X/(X ∩ N)) ≅ ℚD(S/(S ∩ N)) ≅ ℚ. It follows that

Res_{R}^{Q} β_{Q,N}^{Q} = \frac{1}{|Q|} ∑_{{S≤R \atop SN=Q}} Ten_{S}^{R} [ ∑_{{S≤X≤Q \atop R∩X=S}} |X| μ(X, Q) |Q : RX| ] Inf_{S/(S∩N)}^{S} .

Now we have |X| · |Q : RX| = \frac{|Q|}{|RX : X|} = \frac{|Q|}{|R : S|} , and therefore the inner sum is

∑_{{S≤X≤Q \atop R∩X=S}} |X| μ(X, Q) |Q : RX| = \frac{|Q|}{|R : S|} ∑_{{S≤X≤Q \atop R∩X=S}} μ(X, Q) .

But this is zero by a standard combinatorial lemma, provided R ≠ Q (see [St, 3.9.3]). It follows that Res_{R}^{Q} β_{Q,N}^{Q} = 0 whenever R is a proper subgroup of Q.

Now we compute Res_{R}^{P} β_{Q,N}^{P} . From the definition, we have β_{Q,N}^{P} = Ten_{Q}^{P} β_{Q,N}^{Q} . Therefore, applying again the Mackey formula, we obtain

Res_{R}^{P} β_{Q,N}^{P} = Res_{R}^{P} Ten_{Q}^{P} β_{Q,N}^{Q} = ∑_{g∈[R\P/Q]} Ten_{R∩^{g}Q}^{R} Res_{R∩^{g}Q}^{^{g}Q} Conj_{g} β_{Q,N}^{Q}

= ∑_{g∈[R\P/Q]} Ten_{R∩^{g}Q}^{R} Res_{R∩^{g}Q}^{^{g}Q} β _{^{g}Q,^{g}N}^{^{g}Q} = ∑_{g∈[R\T (Q,R)/Q]} Ten_{^{g}Q}^{R} β _{^{g}Q,^{g}N}^{^{g}Q}

= ∑_{g∈[R\T (Q,R)/Q]} β_{^{g}Q,^{g}N}^{R} ,

using the fact that, by the first part of the proof, Res_{R∩^{g}Q}^{^{g}Q} β _{^{g}Q,^{g}N}^{^{g}Q} = 0 if R ∩ ^{g}Q < ^{g}Q, that is, if g ∉ T (Q, R) . In particular, we obtain Res_{R}^{P} β_{Q,N}^{P} = 0 if T (Q, R) is empty, or in other words if R ≱_{P} Q.

(b) Taking R = Q in (a), we have T (Q, R) = N_{P} (Q) and therefore

Defres_{Q/M}^{P} β_{Q,N}^{P} = Def_{Q/M}^{Q} Res_{Q}^{P} β_{Q,N}^{P} = ∑_{g∈[N_{P} (Q)/Q]} Def_{Q/M}^{Q} β_{Q,^{g}N}^{Q} .

Thus it suffices to show that each Def_{Q/M}^{Q} β_{Q,^{g}N}^{Q} is the multiplication by m_{Q,M} and without loss of generality we can assume that ^{g}N = N . We have

Def_{Q/M}^{Q} β_{Q,N}^{Q} = \frac{1}{|Q|} ∑_{{X≤Q \atop XN=Q}} |X| μ(X, Q) Def_{Q/M}^{Q} Ten_{X}^{Q} Inf_{X/(X∩N)}^{X}

= \frac{1}{|Q|} ∑_{{X≤Q \atop XN=Q}} |X| μ(X, Q) Ten_{XM/M}^{Q/M}Iso_{X/(X∩M)}^{XM/M}Def_{X/(X∩M)}^{X} Inf_{X/(X∩N)}^{X} ,

using Corollary 3.11. Now if X ∩ M ≠ X ∩ N , then Def_{X/(X∩M)}^{X} Inf_{X/(X∩N)}^{X} factorizes through the proper subgroup X/(X ∩ M)(X ∩ N) of X/(X ∩ N) . But X/(X ∩ N) = E has rank 2 and ℚD(Y) = 0 for every proper subgroup Y of E . Therefore the sum only runs over subgroups X satisfying X ∩ M = X ∩ N and we have Def_{X/(X∩M)}^{X} Inf_{X/(X∩N)}^{X} = id (by definition of deflation in terms of Brauer quotients). Moreover

X/(X ∩ N) = X/(X ∩ M) ≅ XM/M ≤ Q/M ,

and since X/(X ∩ N) and Q/M have order p^{2} , we also have XM = Q, hence Ten_{XM/M}^{Q/M} = id. Finally Iso_{X/(X∩M)}^{Q/M} = id by our usual identification ℚD(X/(X ∩ M)) ≅ ℚD(Q/M) ≅ ℚ. It follows that

Def_{Q/M}^{Q} β_{Q,N}^{Q} = \frac{1}{|Q|} ∑_{{XM=XN=Q \atop X∩M=X∩N}} |X| μ(X, Q) · id = m_{Q,M} · id ,

by Lemma 4.7.

(c) Since β_{Q,N}^{P} = Ten_{Q}^{P} β_{Q,N}^{Q} , we only have to prove that β_{Q,N}^{Q} is independent of N . By the injectivity of the restriction-deflation map to elementary abelian sections of rank 2 (Theorem 1.6), it suffices to show that Defres_{R/S}^{Q} β_{Q,N}^{Q} is independent of N for every such section R/S . This is obvious if R < Q because Res_{R}^{Q} β_{Q,N}^{Q} = 0 by (a). When R = Q, we have Defres_{Q/S}^{Q} β_{Q,N}^{Q} = m_{Q,S} · id by (b) and this is independent of N .

(4.10) Corollary. Let A = (a_{Q,R}) be the matrix of the linear map ψ_{P} α_{P} with respect to the canonical basis of ∏_{Q∈[S/P]} ℚ, ordered in a way compatible with the relation ≥_{P} .

(a) A is triangular.

(b) The diagonal entries are a_{Q,Q} = m_{Q,Q_{0}} |N_{P} (Q)/Q|.

(c) ψ_{P} is surjective.

Proof. (a) If R ≱_{P} Q, then Res_{R}^{P} β_{Q,Q_{0}}^{P} = 0 by Proposition 4.9 and therefore Defres_{R/R_{0}}^{P} β_{Q,Q_{0}}^{P} = 0 . This implies that A is triangular.

(b) By Proposition 4.9, Defres_{Q/Q_{0}}^{P} β_{Q,Q_{0}}^{P} = m_{Q,Q_{0}} |N_{P} (Q)/Q| · id.

(c) By Lemma 4.7, we know that the explicit value of m_{Q,Q_{0}} is non-zero. Therefore the matrix A is invertible, so that ψ_{P} α_{P} is an isomorphism. In particular ψ_{P} is surjective.

Together with Proposition 4.4, part (c) of Corollary 4.10 completes the proof of Theorem 4.1. Moreover, we also deduce that the map α_{P} is an isomorphism and we now use this for describing two bases of ℚD(P) .

(4.11) Proposition. Let E denote an elementary abelian group of rank 2 and for each section X/Y isomorphic to E , identify ℚD(X/Y) ≅ ℚ with ℚD(E) .

(a) The set { β_{Q,Q_{0}}^{P}(Ω_{E}) | Q ∈ [S/P] } is a basis of ℚD(P) .

(b) The set { Teninf_{Q/Q_{0}}^{P}(Ω_{Q/Q_{0}}) | Q ∈ [S/P] } is a basis of ℚD(P) .

Proof. (a) This set is the image under α_{P} of the canonical basis of ∏_{Q∈[S/P]} ℚ and α_{P} is an isomorphism.

(b) For every Q ≤ P , let J_{Q} be the subspace of ℚD(P) generated by all elements of the form Teninf_{R/N}^{P}(Ω_{R/N}) where R < Q and R/N ≅ E . We have

β_{Q,Q_{0}}^{P}(Ω_{E}) = \frac{1}{|Q|} ∑_{{X≤Q \atop XQ_{0}=Q}} |X| μ(X, Q) Teninf_{X/(X∩Q_{0})}^{P}(Ω_{X/(X∩Q_{0})}) ≡ Teninf_{Q/Q_{0}}^{P}(Ω_{Q/Q_{0}}) (mod J_{Q}) .

Since β_{Q,N}^{P} = β_{Q,Q_{0}}^{P} whenever Q/N ≅ E (Proposition 4.8), we deduce that

Teninf_{Q/N}^{P}(Ω_{Q/N}) ≡ Teninf_{Q/Q_{0}}^{P}(Ω_{Q/Q_{0}}) (mod J_{Q}) .

Therefore, in the expression for β_{Q,Q_{0}}^{P}(Ω_{E}) , we can replace each Teninf_{X/(X∩Q_{0})}^{P}(Ω_{X/(X∩Q_{0})}) by an element of the form Teninf_{X/X_{0}}^{P}(Ω_{X/X_{0}}) + y where y ∈ J_{X} . It follows by induction that there exist rational numbers λ_{R} such that

β_{Q,Q_{0}}^{P}(Ω_{E}) = Teninf_{Q/Q_{0}}^{P}(Ω_{Q/Q_{0}}) + ∑_{{R∈[S/P] \atop R<Q}} λ_{R} Teninf_{R/R_{0}}^{P}(Ω_{R/R_{0}}) .

Thus we obtain a matrix which is triangular with ones along the diagonal and the result follows from (a).

Since Ω_{E} is defined over the prime field 𝔽_{p} , the results of this section are independent of the base field k and this gives our next result.

(4.12) Corollary. For every field k of characteristic p, there is an isomorphism ℚD_{𝔽_{p}}(P) ≅ ℚD_{k}(P) induced by scalar extension from 𝔽_{p} to k.

(4.13) Remark. Let O be a complete discrete valuation ring of characteristic zero with residue field k. One can define the Dade group D_{O}(P) in a similar fashion using Dade P -algebras over O. The residue map D_{O}(P) → D_{k}(P) is injective (see [Th], Section 29) and it is conjectured that it is an isomorphism. Since Ω_{E} obviously lifts to O, the results of this section also hold over O and it follows that the residue map ℚD_{O}(P) → ℚD_{k}(P) is an isomorphism.

5. The torsion-free rank of the endo-trivial group

The torsion-free rank of the endo-trivial group T (P) has been determined recently by J.L. Alperin, using relative Heller translates of endo-permutation modules (which turn out to be again endo-permutation modules). In this section, we prove Alperin’s theorem by another method, using tensor induction.

Let A be the poset of all elementary abelian subgroups of P of rank ≥ 2 . The torsion-free rank of T (P) will turn out to be the number of conjugacy classes of connected components of A. Therefore we first need to understand better those components. Clearly the isolated vertices of A are precisely the maximal elementary abelian subgroups of rank 2. Let B be the subposet of A consisting of all elementary abelian subgroups of P of rank ≥ 3 as well as their subgroups of rank 2. The following group-theoretic result is well-known and not very hard to prove (see Lemma 10.21 in [GLS]).

(5.1) Lemma. Let m(P) be the maximal rank of an elementary abelian subgroup of P .

(a) If m(P) = 2 , then A consists of isolated vertices.

(b) If m(P) ≥ 3 , then B is a connected component of A and all other connected components (if any) are isolated vertices.

We shall call B the big connected component of A. Let X be the set of connected components of A and let [X /P] be a set of representatives of conjugacy classes in X . Since B is clearly invariant by conjugation, it must belong to [X /P] (whenever m(P) ≥ 3 ).

(5.2) Theorem. For every connected component C ∈ [X /P] , choose E_{C} ∈ C .

(a) The restriction map

ρ = ∏_{C∈[X /P]} Res_{E_{C}}^{P} : ℚT (P) ⟶ ∏_{C∈[X /P]} ℚT (E_{C}) ≅ ∏_{C∈[X /P]} ℚ

is an isomorphism.

(b) If C is a singleton (so that E_{C} is an isolated vertex), let a_{C} = Ten_{E_{C}}^{P}(Ω_{E_{C}}) . For the big connected component B, let a_{B} = Ω_{P} . Then the set { a_{C} | C ∈ [X /P] } is a ℚ-basis of ℚT (P) .

Proof. By Theorem 1.5, we know that

Res : ℚT (P) ⟶ ∏_{E} ℚT (E)

is injective, where E runs over all elementary abelian subgroups of P . Clearly we only need to choose one subgroup in each conjugacy class. Now if E_{k} denotes the elementary abelian group of rank k, we have ℚT (E_{1}) = 0 , while the restriction map Res_{E_{2}}^{E_{k}} : T (E_{k}) → T (E_{2}) is an isomorphism if k ≥ 2 , by Theorem 1.4 (a) and the fact that Res_{E_{2}}^{E_{k}}(Ω_{E_{k}}) = Ω_{E_{2}} . Therefore all restriction maps within the big component B are isomorphisms and so it suffices to consider the chosen subgroup E_{B} . It follows from this that we do not loose injectivity if we only restrict to the subgroups E_{C} for C ∈ [X /P] . In other words ρ is injective.

We shall show that the set { a_{C} | C ∈ [X /P] } belongs to ℚT (P) and maps via ρ to a basis of ∏_{C∈[X /P]} ℚ. This will prove the surjectivity of ρ as well as statement (b).

First we need to check that a_{C} ∈ ℚT (P) . This is clear for the big component since Ω_{P} ∈ T (P) (Lemma 1.3). Now if E = E_{C} is an isolated vertex of A, we have a_{C} = Ten_{E}^{P}(Ω_{E}) ∈ ℚD(P) and we check that it belongs to ℚT (P) by computing all deflations Defres_{N/R}^{P} where R ≠ 1 and N = N_{P} (R) . By the Mackey formula, we have

Def_{N/R}^{N} Res_{N}^{P} Ten_{E}^{P}(Ω_{E}) = Def_{N/R}^{N} ( ∑_{g∈[N\P/E]} Ten_{^{g}E∩N}^{N} Res _{^{g}E∩N}^{^{g}E}(^{g}Ω_{E})) .

But ^{g}Ω_{E} = Ω_{^{g}E} and its restriction to any proper subgroup C is zero since E has rank 2 and ℚD(C) = 0 if |C| = p. So g only runs over the set T (E, N) of all g ∈ P such that ^{g}E ≤ N and we obtain

Defres_{N/R}^{P} Ten_{E}^{P}(Ω_{E}) = ∑_{g∈[N\T (E,N)/E]} Def_{N/R}^{N} Ten_{^{g}E}^{N}(Ω_{^{g}E}) .

We show that each term of the sum is zero. By relabelling ^{g}E , we can assume that g = 1 . By Corollary 3.11, we have

Def_{N/R}^{N} Ten_{E}^{N}(Ω_{E}) = Ten_{ER/R}^{N/R} Iso_{E/E∩R}^{ER/R} Def_{E/E∩R}^{E}(Ω_{E}) .

But Def_{E/E∩R}^{E}(Ω_{E}) = 0 whenever E ∩ R ≠ 1 (Lemma 1.3), so we only need to show that E ∩ R ≠ 1 .

Since E ≤ N , E normalizes R, hence centralizes some non-trivial subgroup Q of R (because E and R are p-groups) and we can assume that Q is elementary abelian. Then EQ is elementary abelian and so EQ = E by maximality of E (since E is an isolated vertex, it is a maximal elementary abelian subgroup). Therefore Q ≤ E and E ∩ R ≥ Q ≠ 1 .

Now we want to prove that { ρ(a_{C}) | C ∈ [X /P] } is a ℚ-basis of ∏_{C∈[X /P]} ℚ. We need to compute the restrictions of a_{C} = Ten_{E}^{P}(Ω_{E}) , where E = E_{C} is an isolated vertex as before. If F is an elementary abelian subgroup, we have

Res_{F}^{P} Ten_{E}^{P}(Ω_{E}) = ∑_{g∈[F \P/E]} Ten_{^{g}E∩F}^{F} Res _{^{g}E∩F}^{^{g}E} (Ω_{^{g}E}) .

Since the restriction of Ω_{^{g}E} to any proper subgroup is zero (because E has rank 2), g only runs over the set T (E, F) of all g ∈ P such that ^{g}E ≤ F . But since E is isolated, T (E, F) is non empty only if F is conjugate to E . Therefore Res_{E_{D}}^{P} Ten_{E}^{P}(Ω_{E}) = 0 for every component D not conjugate to C = {E}. Moreover, taking now F = E , we have T (E, F) = N_{P} (E) and we obtain

Res_{E}^{P} Ten_{E}^{P}(Ω_{E}) = ∑_{g∈[N_{P} (E)/E]} Ω_{^{g}E} = |N_{P} (E)/E| · Ω_{E} .

This shows that ρ(a_{C}) is a multiple of the basis element indexed by C in the canonical basis of ∏_{C∈[X /P]} ℚ, whenever C consists of an isolated vertex. We are left with the big component B (but only if m(P) ≥ 3 ). We have defined a_{B} = Ω_{P} and we have Res_{F} (Ω_{P}) = Ω_{F} for every F . Therefore ρ(a_{B}) is the sum of all basis elements in the canonical basis of ∏_{C∈[X /P]} ℚ. Altogether, it is now clear that the set { ρ(a_{C}) | C ∈ [X /P] } is a ℚ-basis of ∏_{C∈[X /P]} ℚ.

As in the case of the whole Dade group we deduce the following result.

(5.3) Corollary. For every field k of characteristic p, there is an isomorphism ℚT_{𝔽_{p}}(P) ≅ ℚT_{k}(P) induced by scalar extension from 𝔽_{p} to k.

As in Remark 4.13, we also deduce that, if O is a complete discrete valuation ring of characteristic zero with residue field k, the residue map ℚT_{O}(P) → ℚT_{k}(P) is an isomorphism. However, much more is known, since it has been proved recently by J.L. Alperin that the residue map T_{O}(P) → T_{k}(P) is always an isomorphism.

(5.4) Remark. If E is an isolated vertex of A, we have used the fact that Res_{C}(Ω_{E}) = 0 if C < E , because C is cyclic and so ℚD(C) = 0 . If we want to work over ℤ, then D(C) = ℤ/2ℤ if |C| = p and p is odd, so we need to consider 2Ω_{E} instead to obtain Res_{C}(2Ω_{E}) = 0 . It follows that the elements 2a_{C} = Ten_{E_{C}}^{P}(2Ω_{E_{C}}) (where C runs over the isolated components up to conjugacy), together with a_{B} = Ω_{P} for the big component (if m(P) ≥ 3 ), belong to T (P) , are linearly independent over ℤ, and span a full lattice in T (P) .

We also observe that 2a_{C} is the class of the endo-permutation module M = Ten_{E}^{P}(Ω_{E}^{2}(k)), where E = E_{C} . This class belongs to T (P) , but the module M itself may not be endo-trivial (it is only equivalent to an endo-trivial module). The reason is that the endomorphism algebra of Ω_{E}^{2}(k) has the form k ⊕ (free) and when we consider its tensor induction, we obtain k ⊕ X where X is a permutation module which may not be free. For instance the tensor induction of a free module may not be free.

6. Partial results on the torsion subgroup

The purpose of this section is to prove Theorem B of the introduction. Let p be an odd prime and let P be a finite p-group. Recall that we have defined \overline{D^{t}}(P) = D^{t}(P)/Ker(ψ_{P}) , where D^{t}(P) is the torsion subgroup of D(P) and where

ψ_{P} : D^{t}(P) ⟶ ∏_{A} D(A)

is the product of the restriction-deflation maps to all cyclic sections A of order p. Since the subgroup Ker(ψ_{P}) is quite difficult to handle (and might be trivial), we only deal with the quotient \overline{D^{t}}(P) . Note that if A is cyclic of order p, we have D(A) = D^{t}(A) = \overline{D^{t}}(A) ≅ ℤ/2ℤ, generated by Ω_{A} (by direct inspection or by Theorem 1.4). We first need a lemma.

(6.1) Lemma. Let p be an odd prime and let E be an elementary abelian p-group of rank 2. Then the restriction

Res : D^{t}(E) ⟶ ∏_{R<E} D^{t}(R)

to all proper subgroups is injective.

Proof. Let a ∈ D^{t}(E) . By Dade’s Theorem 1.4, we have

a = ∑_{Q∈C} n_{Q} Ω_{E/Q} ,

where C is the set of all non-trivial cyclic subgroups of E and where n_{Q} ∈ ℤ/2ℤ for every Q. If a ∈ Ker(Res) , we have, for every R ∈ C ,

0 = Res_{R}^{E}(a) = ∑_{{Q∈C \atop Q≠R}} n_{Q} Ω_{R} ,

because the action of R on Ω_{E/Q}^{1}(k) is obtained using the isomorphism R ≅ E/Q whenever R ≠ Q (that is, RQ = E ). It follows that

∑_{{Q∈C \atop Q≠R}} n_{Q} = 0 ∈ ℤ/2ℤ ,

for every R ∈ C . Considering R, R′ ∈ C with R ≠ R′ , we obtain

n_{R} + ∑_{{Q∈C \atop Q≠R, Q≠R′}} n_{Q} = 0 = n_{R′} + ∑_{{Q∈C \atop Q≠R, Q≠R′}} n_{Q} ,

and therefore n_{R} = n_{R′} . Thus all the numbers n_{Q} are equal and the sum of p of them is zero in ℤ/2ℤ (because |C| = p + 1 ). Since p is odd, n_{Q} = 0 for every Q.

Theorem B of the introduction is contained in the following more precise result.

(6.2) Theorem. Let P be a finite p-group, where p is an odd prime. Let C be the set of non-trivial cyclic subgroups of P and for every C ∈ C , let Φ(C) be the unique maximal subgroup of C .

(a) The group homomorphism

ϕ_{P} = ∏_{C∈[C/P]} Defres_{C/Φ(C)}^{P} : \overline{D^{t}}(P) ⟶ ∏_{C∈[C/P]} D(C/Φ(C)) ≅ ∏_{C∈[C/P]} ℤ/2ℤ

is an isomorphism, where [C/P] denotes a set of representatives of the conjugacy classes of subgroups in C .

(b) The set \{ Ten_{C}^{P}(Ω_{C/Φ(C)}) | C ∈ [C/P] \} is an 𝔽_{2}-basis of \overline{D^{t}}(P) , mapping via ϕ_{P} to the canonical basis of ∏_{C∈[C/P]} ℤ/2ℤ.

Proof. The injectivity of ϕ_{P} will follow from the injectivity of

ψ_{P} : \overline{D^{t}}(P) ⟶ ∏_{A} D(A) ,

where A runs over all sections of order p. Let a ∈ Ker(ϕ_{P}) . We prove that a = 0 by induction on |P |. For every proper subgroup Q, we clearly have Res_{Q}^{P}(a) ∈ Ker(ϕ_{Q}) and therefore Res_{Q}^{P}(a) = 0 by induction. By the injectivity of the map ψ_{P} above, it suffices to prove that Def_{P/S}^{P}(a) = 0 for every quotient P/S of order p.

If P is cyclic, then necessarily S = Φ(P) and the condition Def_{P/S}^{P}(a) = 0 is part of the assumption that a ∈ Ker(ϕ_{P}) . If P is not cyclic, then there is an elementary abelian quotient P/R of rank 2 with R < S and it suffices to prove that Def_{P/R}^{P}(a) = 0 . But the restriction of Def_{P/R}^{P}(a) to every proper subgroup Q/R is zero since Res_{Q}^{P}(a) = 0 . Therefore, by Lemma 6.1, we have Def_{P/R}^{P}(a) = 0 as required.

We now prove that, for every C ∈ [C/P] , ϕ_{P} (Ten_{C}^{P}(Ω_{C/Φ(C)})) is the basis element indexed by C in the canonical basis of ∏_{C∈[C/P]} ℤ/2ℤ. This will prove the surjectivity of ϕ_{P} as well as statement (b). For every D ∈ C , we have by the Mackey formula

Def_{D/Φ(D)}^{D} Res_{D}^{P} Ten_{C}^{P}(Ω_{C/Φ(C)}) = ∑_{g∈[D\P/C]} Def_{D/Φ(D)}^{D} Ten_{D∩^{g}C}^{D} Res_{D∩^{g}C}^{^{g}C}(Ω_{^{g}C/Φ(^{g}C)}) ,

using the fact that conjugation by g maps Ω_{C/Φ(C)} to Ω_{^{g}C/Φ(^{g}C)} . If D ∩ ^{g}C < ^{g}C , then D ∩ ^{g}C ≤ Φ(^{g}C) and therefore Res_{D∩^{g}C}^{^{g}C}(Ω_{^{g}C/Φ(^{g}C)}) = 0 . So g only runs over the set T (C, D) of all g ∈ P such that gC ≤ D. Thus we obtain

Defres_{D/Φ(D)}^{P} Ten_{C}^{P}(Ω_{C/Φ(C)}) = ∑_{g∈[D\T (C,D)/C]} Def_{D/Φ(D)}^{D} Ten_{^{g}C}^{D}(Ω_{^{g}C/Φ(^{g}C)}) .

By Corollary 3.11, Def_{D/Φ(D)}^{D} Ten_{^{g}C}^{D} factorizes as the composite of Def _{^{g}C/^{g}C∩Φ(D)}^{^{g}C} followed by an isomorphism and a tensor induction. So if ^{g}C < D, that is, ^{g}C ≤ Φ(D) , we have a deflation to the trivial group, hence zero since D(1) = 0 . This proves that Defres_{D/Φ(D)}^{P} Ten_{C}^{P}(Ω_{C/Φ(C)}) = 0 whenever D is not conjugate to C . If now D = C , we are left with the sum over g ∈ [N_{P} (C)/C] and we have Def_{C/Φ(C)}^{C} (Ω_{C/Φ(C)}) = Ω_{C/Φ(C)} . Therefore we obtain

Defres_{C/Φ(C)}^{P} Ten_{C}^{P}(Ω_{C/Φ(C)}) = |N_{P} (C)/C| · Ω_{C/Φ(C)} = Ω_{C/Φ(C)} ,

using the fact that |N_{P} (C)/C| is odd and Ω_{C/Φ(C)} has order 2. This completes the proof that the image ϕ_{P} (Ten_{C}^{P}(Ω_{C/Φ(C)})) is the basis element indexed by C .

It is easy to see by induction (as in the proof of Theorem 1.6) that the injectivity of the map ψ_{P} would follow if the restriction map

Res : T (P) ⟶ ∏_{E} T (E)

was injective, where E runs over elementary abelian subgroups of P . This is one of the important open questions about the Dade group (mentioned to us several years ago by J.F. Carlson and J.L. Alperin). We mention the following consequence.

(6.3) Corollary. Assume that p is odd and that Ker(ψ_{P}) = 0 . Then D^{t}(P) is an 𝔽_{2}-vector space of dimension s, where s is the number of conjugacy classes of non-trivial cyclic subgroups of P . In particular every torsion element of D^{t}(P) is self-dual.

Proof. The first statement follows from Theorem 6.2 and the second from the fact that the self-dual endopermutation modules are precisely the elements of order 2 of the Dade group.

(6.4) Remark. A typical element of order 2 in \overline{D^{t}}(P) is Ω_{P/Q} where Q is a proper normal subgroup with P/Q cyclic. Its expression relative to the 𝔽_{2}-basis of \overline{D^{t}}(P) is

Ω_{P/Q} = ∑_{C} Ten_{C}^{P}(Ω_{C/Φ(C)}) ,

where C runs over the cyclic subgroups (up to conjugation) such that Q has index p in QC . This follows easily from the computation of Defres_{D/Φ(D)}^{P}(Ω_{P/Q}) for every cyclic subgroup D.

(6.5) Remark. For the torsion subgroup T^{t}(P) of the endo-trivial group T (P) , one can define similarly \overline{T^{t}}(P) = T^{t}(P)/Ker(Res) where

Res : T (P) ⟶ ∏_{E} T (E)

is the restriction to all elementary abelian subgroups (since we have Ker(Res) ⊆ T^{t}(P) by Theorem 1.5). However, there is no interesting result here, because \overline{T^{t}}(P) is zero if P is not cyclic. Indeed, if P is not a quaternion group, every maximal elementary subgroup E has rank ≥ 2 and T (E) has no torsion, so that T^{t}(P) is contained in Ker(Res) . If P is a quaternion group, then T (P) = T^{t}(P) = Ker(Res) because the target of the restriction map above is T (C_{2}) = 0 . Note that if P is cyclic of order ≥ 3 , then T (P) = T^{t}(P) = \overline{T^{t}}(P) ≅ ℤ/2ℤ.

7. Functorial approach

Now we start the second part of this paper, which is concerned with the functorial approach to the Dade group. This approach is the original one we used for the proof of Theorems A and B, from which was then extracted the more elementary version of the previous sections. Our purpose in this section is to show how ℚD and \overline{D^{t}} can be viewed as functors (in the sense of [Bo1]). We also recall a number of general facts concerning such functors.

We have seen in Sections 2 and 3 that all the natural operations on the Dade group can be described with a single formalism. If P and Q are p-groups, and if U is a Q-set-P , then there is a functor T_{U} : \underline{P erm}_{k}(P) → \underline{P erm}_{k}(Q) which maps a Dade P -algebra to a Dade Q-algebra. Moreover, this functor induces a group homomorphism

D(U) : D(P) ⟶ D(Q) .

We have seen in Section 3 that when we compose two such functors T_{V} and T_{U} , we don’t get the functor T_{V ×_{Q}U} , but a kind of twisted version of it, using endomorphisms of the field k. It follows that the map D(V) ∘ D(U) is not equal in general to the map D(V ×_{Q} U) . In order to get rid of those twists, we consider a quotient of D(P) defined by

D_{G}(P) = D(P)/G(P) ,

where G(P) is the subgroup consisting of the classes of the algebras γ_{p}(A) ⊗_{k} A^{op} , where A is a Dade P -algebra (see Lemma 3.1). Recall that γ_{p} is the twist induced by the endomorphism of k mapping λ to λ^{p} . Note that G(P) is indeed a subgroup of D(P) since γ_{p} commutes with tensor products and with taking opposite algebras. In D_{G}(P) , we have [γ_{p}(A)] = [A] , and consequently [γ_{p^{n}}(A)] = [A] for all n ∈ ℕ.

(7.1) Proposition. Let P be a p-group.

(a) The subgroup G(P) is finite, that is, G(P) ⊆ D^{t}(P) . In particular ℚD(P) ≅ ℚD_{G}(P) .

(b) More precisely, we have G(P) ⊆ Ker(ψ_{P}) , where ψ_{P} : D^{t}(P) → ∏_{A} D(A) is the product of the restriction-deflation maps to all cyclic sections A of order p.

(c) For every n ∈ ℕ, the functor γ_{p^{n}} induces the identity map on D_{G}(P) and also on both ℚD(P) and \overline{D^{t}}(P) = D^{t}(P)/Ker(ψ_{P}) .

Proof. (a) First recall that, by Example 3.3, if A is an abelian group, then the map γ_{p} : D(A) → D(A) is the identity. By Theorem 1.5, D(P) is a finitely generated abelian group. Therefore, by Theorem 1.6, the restriction-deflation map

Defres : D(P) ⟶ ∏_{E} D(E)

has finite kernel, where E runs over all elementary abelian sections of P of rank 2. Since γ_{p} commutes with Defres (by Lemma 3.2) and is the identity on D(E) because E is abelian, the class of the algebra γ_{p}(A) ⊗_{k} A^{op} lies in Ker(Defres) for every Dade P -algebra A. It follows that G(P) is contained in the finite group Ker(Defres) and so G(P) ⊆ D^{t}(P) .

(b) Exactly the same argument applies to the restriction-deflation map

ψ_{P} : D^{t}(P) → ∏_{A} D(A) ,

where A runs over all cyclic sections of order p. Therefore G(P) ⊆ Ker(ψ_{P}) .

(c) Clearly γ_{p} induces the identity map on D_{G}(P) , by definition of D_{G}(P) , hence also on the subquotient \overline{D^{t}}(P) . Moreover, since G(P) is finite, D(P) → D_{G}(P) induces an isomorphism ℚD(P) ≅ ℚD_{G}(P) and the result follows.

It is precisely part (c) of the proposition which will allow us to put on ℚD and \overline{D^{t}} a structure of functor in the sense of [Bo1]. However, D itself need not be a functor in the sense of [Bo1], because of the presence of the twists γ_{p^{n}} (see Example 3.4).

(7.2) Remark. Though we shall not adopt this point of view in this paper, we would like to mention that it is possible to define a similar formalism which involves the twists by endomorphisms of k and which allows us to view D as a functor, without taking the quotient D_{G} . We define an End(k)-graded Q-set-P to be a pair (U, a) where U is a Q-set-P and a is a function from Q\U/P to End(k) . The disjoint union of (U, a) and (U′, a′) is defined in the obvious way. Now if R is another p-group and if (V, b) is an End(k)-graded R-set-Q, then the product (V, b) ×_{Q} (U, a) is the End(k)-graded R-set-P

(V, b) ×_{Q} (U, a) = (V ×_{Q} U, b ×_{Q} a) ,

where b ×_{Q} a is the function from R\(V ×_{Q} U)/P to End(k) defined by

(b ×_{Q} a)(v, u) = b(v) ∘ a(u) ∘ π_{v,u} ,

where π_{v,u} is raising to the power |Q_{v} : Q_{v,uP} | (see Proposition 3.5). If (U, a) is an End(k)-graded Q-set-P , we define a functor T_{(U,a)} : \underline{P erm}_{k}(P) → \underline{P erm}_{k}(Q) by

T_{(U,a)}(X) = Hom_{P} (U^{op}, X) .

If m is a morphism in \underline{P erm}_{k}(P) from Y to X , we set

T_{(U,a)}(m)(φ, ψ) = ∏_{u∈U/P} a(u)(m(φ(u), ψ(u))) .

Clearly, if we take for a the constant function equal to the identity endomorphism of k, then the functor T_{(U,a)} is just our previous functor T_{U} . With those definitions, is is easy to see that we get a map

D(U, a) : D(P) → D(Q) ,

which is additive in (U, a) . Moreover D(V, b) ∘ D(U, a) = D((V, b) ×_{Q} (U, a)) by Proposition 3.5.

Now we introduce the functorial framework of [Bo1]. If P and Q are p-groups, we denote by Γ(Q, P) the Grothendieck group of finite Q-sets-P , with addition induced by the decomposition into disjoint union. Thus Γ(Q, P) is the free abelian group on transitive bisets up to conjugation. An element of Γ(Q, P) is called a virtual biset. The product of bisets (V, U) ↦ V ×_{Q} P can be extended to a bilinear product

∘ : Γ(R, Q) × Γ(Q, P) → Γ(R, P) .

Let C be the following category:

• The objects of C are the finite p-groups.

• If P and Q are finite p-groups, then Hom_{C}(P, Q) = Γ(P, Q) .

• The composition of morphisms is given by the product ∘ defined above. The category C is a preadditive category (in the sense of [Mc]). Similarly, when K is a commutative ring, we consider the category C_{K} obtained by tensoring everything with K : the objects of C_{K} are the same as the objects of C , but

Hom_{C_{K}} (P, Q) = K ⊗_{ℤ} Γ(P, Q) .

The composition of morphisms is defined by K-linearity.

Let us fix some notation about subgroups of products and bisets. If P and Q are finite groups, then any P -set-Q is a disjoint union of transitive ones, and the isomorphism classes of transitive P -sets-Q are in bijection with the conjugacy classes of subgroups of Q×P : if L is such a subgroup, we denote by (P ×Q)/L the set of right cosets of L in P × Q, viewed as a P -set-Q by

g · (a, b)L · h = (ga, h^{−1}b)L , ∀g, a ∈ P, ∀b, h ∈ Q .

Define the following subgroups of P and Q:

(7.3)  \begin{aligned}_{p}L & = \{g ∈ P | ∃h ∈ Q, (g, h) ∈ L\} ,  _{k}L = \{g ∈ P | (g, 1) ∈ L\} , \\ L_{p} & = \{h ∈ Q | ∃g ∈ P, (g, h) ∈ L\} ,  L_{k} = \{h ∈ Q | (1, h) ∈ L\} , \end{aligned}

where the letter p stands for projection and k for kernel. Then _{k}L \underline{◁}_{p}L, L_{k} \underline{◁} L_{p} , and the quotients _{p}L/_{k}L and L_{p}/L_{k} are naturally isomorphic. Let us recall the following basic decomposition lemma. We use the notation _{L_{p}}Q_{Q} for Q viewed as a L_{p}-set-Q in the obvious way, and similarly in other situations.

(7.4) Lemma. ([Bo1], Lemma 3). With the notation above, the morphism (P ×Q)/L from Q to P is equal to the product of the morphisms _{L_{p}}Q_{Q} (restriction), _{L_{p}/L_{k}}(L_{p}/L_{k})_{L_{p}} (deflation), _{_{p}L/_{k}L}(L_{p}/L_{k})_{L_{p}/L_{k}} (isomorphism), _{_{p}L}(_{p}L/_{k}L)_{_{p}L/_{k}L} (inflation), and _{P} P_{_{p}L} (induction).

If R is another p-group and M is a subgroup of Q × R, let

L ∗ M = { (x, y) ∈ P × R | ∃z ∈ Q, (x, z) ∈ L, (z, y) ∈ M }

This is a subgroup of P × R. With this notation, recall the following “Mackey formula”.

(7.5) Lemma. ([Bo1], Proposition 1). Let P , Q, and R be finite p-groups. If L is a subgroup of P × Q and M is a subgroup of Q × R, then

[(P × Q)/L] ×_{Q} [(Q × R)/M] = ∑_{x∈[L_{p}\Q/_{p}M]}(P × R)/(L ∗^{(x,1)}M) ,

where L_{p} and _{p}M are defined by (7.3).

Now we come to the main fact for our purposes.

(7.6) Proposition. With the notation of the beginning of this section, the following hold.

(a) The correspondence D_{G} which maps the p-group P to D_{G}(P) and the virtual biset U − U′ to the group homomorphism D_{G}(U)−D_{G}(U′) is an additive functor from C to the category of abelian groups.

(b) ℚD is a ℚ-linear functor from C_{ℚ} to the category of ℚ-vector spaces.

(c) \overline{D^{t}} is an 𝔽_{2}-linear functor from C_{𝔽_{2}} to the category of 𝔽_{2}-vector spaces.

Proof. (a) Let U be a Q-set-P and let V be a R-set-Q, where P , Q and R are p-groups. First note that, by Lemma 3.2, D(U) : D(P) → D(Q) induces a group homomorphism

D_{G}(U) : D_{G}(P) → D_{G}(Q) .

Now we have to prove that D(V) ∘ D(U) and D(V ×_{Q} U) induce the same map

D_{G}(V) ∘ D_{G}(U) = D_{G}(V ×_{Q} U) : D_{G}(P) ⟶ D_{G}(R) .

This is a direct consequence of Remark 3.10, since all the twists γ_{p^{n}} act as the identity of D_{G}(R) by Proposition 7.1. Alternatively, one can decompose U and V , reducing first to the case where U and V are transitive bisets (using Proposition 2.10), and then to the case where U and V are among the special types of bisets appearing in Lemma 7.4 above. By Corollary 3.8, we know that D(V) ∘ D(U) = D(V ×_{Q} U) in all cases except one, namely when D(U) corresponds to a tensor induction and D(V) to a deflation. Now in this case, by Proposition 3.9 and its proof, for every Dade P -algebra A, we have

(T_{V} ∘ T_{U})(A) ≅ γ_{p^{n}}(T_{V ×_{Q}U}(A))

for some n ∈ ℕ, so that D_{G}(V) ∘ D_{G}(U) = D_{G}(V ×_{Q} U) since γ_{p^{n}} acts as the identity on D_{G}(R) . (b) This follows from (a) by tensoring with ℚ, since ℚD(P) ≅ ℚD_{G}(P) .

(c) Clearly the torsion subgroup D_{G}^{t}(P) of D_{G}(P) defines a subfunctor of D_{G} . With the notation of Proposition 7.1, we have to prove that Ker(ψ_{P})/G(P) defines a subfunctor of D_{G}^{t}(P) , for the quotient \overline{D^{t}} will then be a functor in the category F_{ℤ} . Note that \overline{D^{t}}(P) is, by construction, an abelian group of exponent 2, hence an 𝔽_{2}-vector space, so \overline{D^{t}} will in fact be an 𝔽_{2}-linear functor from C_{𝔽_{2}} to the category of 𝔽_{2}-vector spaces.

In order to prove that Ker(ψ_{P})/G(P) defines a subfunctor of D_{G}^{t}(P) , it is easy to verify that it is invariant under restriction, tensor induction, inflation, and deflation. For a change, we use a more functorial argument. We prove that Ker(ψ_{P})/G(P) is equal to the kernel of all maps D_{G}^{t}(φ) , for all morphisms φ : P → C in C_{ℤ} . Since this clearly defines a subfunctor of D_{G}^{t}(P) , the result follows.

By linearity, it suffices to consider a morphism φ : P → C in C_{ℤ} defined by a transitive biset, that is, φ = (C × P)/L for some subgroup L of C × P . If there is a subgroup A of C and a subgroup B of P such that L = A × B , then the morphism φ factors through the trivial group (by Lemma 7.4). Therefore

D_{G}^{t}(φ) : D_{G}^{t}(P) ⟶ D_{G}^{t}(C)

is zero since D_{G}^{t}(1) = 0 . Otherwise there is a subgroup B of P and a surjection s : B → C such that L = {(s(b), b) | b ∈ B}. Then the morphism φ is a restriction-deflation map to a section of order p of P , so that D_{G}^{t}(φ) vanishes on Ker(ψ_{P})/G(P) .

We denote by F_{K} the category of K-linear functors from C_{K} to the category of K-modules. Thus the functor K ⊗_{ℤ} D_{G} is an object of F_{K} , but our goal in this paper is to study ℚD as an object of F_{ℚ} and \overline{D^{t}} as an object of F_{𝔽_{2}} . Recall that F_{K} is an abelian category.

We end this section with generalities about the category F_{K} , which are proved in [Bo1]. The emphasis in [Bo1] is put on functors defined on a similar category whose objects are all finite groups. Here instead we consider only finite p-groups, but this does not change anything to the general facts which we now recall. If F is an object of F_{K} , and if P is a p-group, then F (P) is a module for the endomorphism algebra End_{C_{K}} (P) of P in C_{K} . In fact the correspondence

F ↦ F (P)

is a functor from F_{K} to End_{C_{K}} (P)-Mod, called the evaluation functor (at P ). This evaluation functor has a left adjoint, defined as follows: if V is an End_{C_{K}} (P)-module and Q is a p-group, set

(7.7)  L_{P,V} (Q) = Hom_{C_{K}} (P, Q) ⊗_{End_{C_{K}} (P)} V .

This makes sense because Hom_{C_{K}} (P, Q) has a natural structure of right End_{C_{K}} (P)-module, given by right composition of morphisms in C_{K}(P, Q) . If φ is a morphism in C_{K} from Q to Q′ , then composition of morphisms (on the left) defines a map

L_{P,V} (φ) : L_{P,V} (Q) → L_{P,V} (Q′) .

So L_{P,V} is an object of F_{K} . Moreover, the correspondence V ↦ L_{P,V} is clearly a functor from the category of End_{C_{K}} (P)-modules to F_{K} , and this functor is left adjoint to the evaluation functor at P .

If the module V is a simple End_{C_{K}} (P)-module, then L_{P,V} is not simple in general, but it admits a unique maximal (proper) subfunctor J_{P,V} defined by

(7.8)  J_{P,V} (Q) = {∑_{i} ψ_{i} ⊗ v_{i} | ∀ψ ∈ Hom_{C_{K}} (Q, P), _{∑i}(ψψ_{i})v_{i} = 0}

for every p-group Q. The quotient L_{P,V} /J_{P,V} is a simple object of F_{K} and is denoted by S_{P,V} .

Now in the algebra End_{C_{K}} (P) , let I_{P} denote the K-submodule generated by the endomorphisms of P which factor through a group Q of order strictly smaller than the order of P . It is clear that I_{P} is a two-sided ideal of End_{C_{K}} (P) . Moreover, if A is the K-submodule generated by the bisets which are both free and transitive on the right and on the left, then A is a subalgebra isomorphic to the algebra over K of the group Out(P) of outer automorphisms of P . The isomorphism is obtained by mapping the automorphism φ of P to the set P_{φ} , which is the set P , with double action given by

g · x · g′ = g x φ(g′) , ∀x ∈ P, ∀g, g′ ∈ P ,

where the product on the right hand side is the product inside the group P . Finally, there is a decomposition

(7.9)  End_{C_{K}} (P) = A ⊕ I_{P} ,  with A ≅ KOut(P) .

With this decomposition, the quotient algebra End_{C_{K}} (P)/I_{P} is isomorphic to KOut(P) and we can view any KOut(P)-module V as an End_{C_{K}} (P)-module, still denoted by V .

(7.10) Proposition. ([Bo1], Proposition 2). Let S_{P,V} = L_{P,V} /J_{P,V} .

(a) The correspondence (P, V) ↦ S_{P,V} is a bijection between the set of isomorphism classes of pairs (P, V) consisting of a p-group P and a simple KOut(P)-module V , and the set of isomorphism classes of simple objects of F_{K} .

(b) The inverse bijection maps a simple object S to the pair (P, S(P)), where P is a group of minimal order such that S(P) ≠ 0 . This condition forces S(P) to be a KOut(P)-module, which turns out to be simple.

In particular, when K is a field, we denote by S_{P,K} the simple functor associated to the pair (P, K) , where K is the trivial KOut(P)-module.

Recall finally ([Bo1] section 7.1) that there is a morphism X ↦ X̃ of K-algebras (with unity) from the Burnside algebra KB(P) of P over K to the algebra End_{C_{K}} (P) . If X is a P -set, the set

(7.11)  X̃ = X × P

is a P -set-P with double action given by

g_{1} · (x, g) · g_{2} = (g_{1}x, g_{1}gg_{2}) , ∀x ∈ X, ∀g, g_{1}, g_{2} ∈ P .

If X is the transitive P -set P/Q, then it is easy to check that X̃ is a transitive P -set-P isomorphic to (P × P)/Δ(Q) , where Δ(Q) is the diagonal inclusion of Q in P × P .

We have the following idempotents of ℤ[\frac{1}{p}]B(P) , indexed by subgroups Q of P :

e_{Q}^{P} = \frac{1}{|N_{P} (Q)|} ∑_{R≤Q} |R| μ(R, Q) (P/R) ,

where μ denotes the Möbius function of the poset of subgroups of P . These idempotents are characterized by the fact that if X is a P -set, then

X · e_{Q}^{P} = |X^{Q}|e_{Q}^{P} .

It follows from this characterization that the restriction of e_{Q}^{P} to a subgroup R of P is the sum of the idempotents e_{S}^{R} associated to the subgroups S of R which are conjugate to Q in P , up to conjugation by R. In particular, the restriction of e_{P}^{P} to any proper subgroup of P is zero.

If p is invertible in K , the idempotents e_{Q}^{P} have natural images in KB(P) , still denoted by e_{Q}^{P} . When Q runs over a set of representatives of the conjugacy classes of subgroups of P , the idempotents e_{Q}^{P} are mutually orthogonal, and their sum is the identity of KB(P) . So we obtain the following result.

(7.12) Proposition. If p is invertible in K , the elements ẽ_{Q}^{P} are mutually orthogonal idempotents of sum 1 in End_{C_{K}} (P) .

For any functor F in the category F_{K} , one deduces a decomposition of F (P) as a direct sum

F (P) = ⨁_{Q} F (ẽ_{Q}^{P}) F (P) ,

and this will play a crucial role in Section 9.

8. The Burnside functor

One of the main examples of functors in the category F_{K} is the Burnside functor KB , where KB(P) is the Burnside algebra of the p-group P over the commutative ring K . When K is a field, the simple functors which appear as composition factors of KB are easier to describe than arbitrary simple functors. It turns out that the functors ℚD and \overline{D^{t}} , which we are interested in, are isomorphic to composition factors of ℚB and 𝔽_{2}B respectively. In view of this, we need to understand better the composition factors of KB and this is the purpose of this section.

When K = ℚ, this question was studied in [Bo1] (for all finite groups instead of p-groups). The methods can be modified to find all subfunctors of the functor KB defined on the category C_{K} of p-groups, when K is a field of characteristic q different from p. In particular when K = ℚ, we recover the results of [Bo1], specialized to the category of p-groups. The description of all subfunctors of KB requires some combinatorial computations, involving the constants m_{G,N} defined in (4.6).

It is clear that KB is a functor in the category F_{K} , since if U is a Q-set-P , we have a K-linear homomorphism

KB(U) : KB(P) → KB(Q)

mapping a P -set Y to the Q-set U ×_{P} Y . If such a biset U is transitive, then it decomposes as a product of bisets corresponding to a restriction, a deflation, an isomorphism, an inflation, and an induction (Lemma 7.4).

Since the characteristic q of K is distinct from p, the K-algebra KB(P) is semi-simple for any p-group P and its primitive idempotents are the idempotents e_{Q}^{P} defined in the previous section. It is easy to compute the value of the five natural operations above on the primitive idempotents e_{Q}^{P} , with the single exception of deflation which requires a little work. We first recall this. Details can be found in [Bo1], page 701.

(8.1) Lemma. Let P be a p-group, let Q be a subgroup of P , and let N be a normal subgroup of P . Then

Def_{P/N}^{P}(e_{Q}^{P}) = \frac{|N_{P} (QN)/QN|}{|N_{P} (Q)/Q|} m_{Q,Q∩N} · e_{P N/N}^{P/N} ,

where m_{Q,Q∩N} is the rational number defined in (4.6).

Given a positive integer r, we define the ideal

I_{r}(P) = < e_{Q}^{P} | |Q/Φ(Q)| ≥ p^{r} > ,

where Φ(Q) denotes the Frattini subgroup of Q. The condition |Q/Φ(Q)| ≥ p^{r} means that Q cannot be generated by fewer than r elements, or in other words that the elementary abelian group Q/Φ(Q) has rank ≥ r. For suitable values of r, we want to prove that I_{r} is a subfunctor of KB . These values will be those of the set

I = {0} ∪ {r ∈ ℕ | r > 0, p^{r−2} ≡ 1 (mod q)} ∪ {∞} ,

where I_{∞}(P) denotes the zero ideal of KB(P) . If s denotes the order of p modulo q, then I consists of all positive integers congruent to 2 modulo s (together with 0 and ∞). If q = 0 , then s has to be understood as ∞ and we have I = {0, 2, ∞} in that case. If q divides p − 1 , then s = 1 and I consists of all positive integers (together with 0 and ∞).

For every r ∈ I , r ≠ ∞, we write s(r) for the successor of r for the natural ordering of I . Thus s(0) = 2 (unless q divides p − 1 in which case s(0) = 1 ), and s(r) = r + s if r > 0 . Finally we denote by E_{r} an elementary abelian group of rank r.

(8.2) Theorem. Let K be a field of characteristic q ≠ p.

(a) For every r ∈ I , the family of ideals I_{r}(P) , where P runs over all finite p-groups, forms a subfunctor of KB .

(b) The functors I_{r} (for r ∈ I ) are the only subfunctors of KB . In other words, the functor KB has a unique filtration

KB = I_{0} ⊃ I_{2} ⊃ I_{∞} = {0}  if q = 0,
KB = I_{0} ⊃ I_{1} ⊃ I_{2} ⊃ I_{3} ⊃ …  if q ≠ 0 and q | p − 1 ,
KB = I_{0} ⊃ I_{2} ⊃ I_{2+s} ⊃ I_{2+2s} ⊃ …  if q ≠ 0 and q |̸ p − 1 .

(c) If r ∈ I , r ≠ ∞, the quotient I_{r}/I_{s(r)} is isomorphic to the simple functor S_{E_{r},K} , where K denotes the trivial module.

(d) If P is a p-group, the dimension of S_{E_{r},K}(P) is equal to the number of conjugacy classes of subgroups Q of P such that p^{r} ≤ |Q/Φ(Q)| < p^{s(r)} .

Proof. We first prove (b). If F is a non-zero subfunctor of KB , then F (P) is an ideal of KB(P) for any p-group P . This follows from the isomorphism X̃ ×_{P} Y ≃ X × Y whenever X and Y are P -sets. Here X̃ ∈ End_{C_{K}} (P) is defined by (7.11). It follows that there is a family A(P) such that

F (P) = < e_{Q}^{P} | Q ∈ A(P) > .

If Q is a subgroup of P , we know that Ind_{Q}^{P}(F (Q)) ⊆ F (P) and Res_{Q}^{P}(F (P)) ⊆ F (Q) , since F is a subfunctor of KB . Since e_{Q}^{P} = |N_{P} (Q) : Q|^{−1}Ind_{Q}^{P}(e_{Q}^{Q}) , and since Res_{Q}^{P}(e_{Q}^{P}) = e_{Q}^{Q} , we conclude that Q ∈ A(P) if and only if Q ∈ A(Q) . In other words, there is a family A of p-groups such that, for any p-group P ,

F (P) = < e_{Q}^{P} | Q ∈ A, Q ≤ P > .

If N \underline{◁} P , we must have Inf_{P/N}^{P}(F (P/N)) ⊆ F (P) . Since

Inf_{P/N}^{P}e_{P/N}^{P/N} = ∑_{{XN=P \atop X mod P}} e_{X}^{P} ,

it follows that if P/N ∈ A, then P ∈ A. Finally, let B be the family of elements of A having no proper quotient in A. Then of course A is the family of groups having a quotient in B.

If P ∈ B, we must have m_{P,N} = 0 for every non-trivial normal subgroup N of P . Indeed Def_{P/N}^{P}(e_{P}^{P}) = m_{P,N} e_{P/N}^{P/N} by Lemma 8.1, and if m_{P,N} ≠ 0 , we have P/N ∈ A, which is impossible since P ∈ B. Since m_{P,Φ(P)} is always equal to 1, this proves that P is elementary abelian. If P ≠ {1} is elementary abelian of order p^{r} , and if N is a subgroup of order p of P , we have

m_{P,N} = \frac{1}{|P |} ∑_{XN=P} |X| μ(X, P) = 1 − p^{r−2}

since a proper subgroup X of P such that XN = P is a complement of N , hence is a maximal subgroup of P , so that μ(X, P) = −1 . Moreover, there are p^{r−1} such complements.

In other words, the family B is contained in the family of elementary abelian p-groups of order p^{r} with either r = 0 or p^{r−2} = 1 in K . If B is non-empty, and if r is minimal such that the elementary abelian p-group E_{r} of order p^{r} is in B, it follows that A is the family of p-groups having E_{r} as a quotient, that is, the family of p-groups P such that |P/Φ(P)| ≥ p^{r} . This proves that any non-zero subfunctor of KB is equal to I_{r} with either r = 0 or p^{r−2} ≡ 1 (mod q) .

(a) We have to prove that any such I_{r} is indeed a subfunctor of KB . This is trivial if r = 0 since I_{0} = KB . If now r > 0 , it should be clear from the previous remarks that I_{r} is stable under induction and restriction. It is invariant under inflation, because

Inf_{P/N}^{P}e_{Q/N}^{P/N} = ∑_{{XN=Q \atop X mod N_{P} (U)}} e_{X}^{P} ,

and moreover, if XN = Q and if Q/N has an elementary abelian quotient of rank ≥ r, then so does X since X/X ∩ N ≅ Q/N . In order to check that I_{r} is stable under deflation, let Q be a subgroup of P such that |Q/Φ(Q)| ≥ p^{r} and let N be a normal subgroup of P . By Lemma 8.1, we have

Def_{P/N}^{P}(e_{Q}^{P}) = \frac{|N_{P} (QN)/QN|}{|N_{P} (Q)/Q|} m_{Q,Q∩N} e_{P N/N}^{P/N} ,

and this is non-zero if and only if m_{Q,Q∩N} is non-zero. Finally, all we have to check is that if Q is a p-group such that |Q/Φ(Q)| ≥ p^{r} , and if M is a normal subgroup of Q such that m_{Q,M} ≠ 0 , then setting R = Q/M , we have |R/Φ(R)| ≥ p^{r} . As in the proof of Lemma 4.7, we have

m_{Q,M} = m_{Q,MΦ(Q)} = m_{Q/Φ(Q),MΦ(Q)/Φ(Q)}

so it is enough to suppose that Q is elementary abelian. If |Q| = p^{s} and |M| = p^{t} , then we have s ≥ r by hypothesis, and we can suppose t > 0 . Moreover, by (4.8), we have

m_{Q,M} = (1 − p^{s−2})(1 − p^{s−3}) … (1 − p^{s−t−1}) .

If s − t − 1 ≤ r − 2 , this is zero in K . Hence s − t ≥ r, which means that |R/Φ(R)| ≥ p^{r} .

(c) It follows from (b) that each non-zero subfunctor I_{r} has a unique maximal subfunctor I_{s(r)} and so the quotient I_{r}/I_{s(r)} is a simple functor S_{r} . Clearly E_{r} is a minimal group for S_{r} , and S_{r}(E_{r}) is one-dimensional, generated by e_{E_{r}}^{E_{r}} , which is invariant by any automorphism of E_{r} . So S_{r} is the simple functor S_{E_{r},K} .

(d) If P is a p-group, it is clear that the dimension of I_{r}(P) is equal to the number of conjugacy classes of subgroups Q of P such that p^{r} ≤ |Q/Φ(Q)|, and similarly for I_{s(r)}(P) . Therefore the dimension of

S_{E_{r},K}(P) = I_{r}(P)/I_{s(r)}(P)

is equal to the number of conjugacy classes of subgroups Q of P such that p^{r} ≤ |Q/Φ(Q)| < p^{s(r)} .

(8.3) Corollary. Suppose K is a field of characteristic 0.

(a) There is a non-split exact sequence of K-linear functors

0 ⟶ S_{E_{2},K} ⟶ KB ⟶ S_{1,K} ⟶ 0 .

(b) If P is a p-group, the dimension of S_{1,K}(P) is the number of conjugacy classes of cyclic subgroups of P , and the dimension of S_{E_{2},K}(P) is the number of conjugacy classes of non-cyclic subgroups of P .

(c) The functor S_{1,K} is isomorphic to KR_{ℚ} and the right hand side morphism in (a) can be chosen to be the natural morphism KB → KR_{ℚ} (mapping a P -set to its permutation ℚP -module).

Proof. (a) The set I in Theorem 8.2 is equal to {0, 2, ∞}. This means that the functor KB has a unique proper non-zero subfunctor S_{E_{2},K} , with quotient isomorphic to S_{1,K} .

(b) By Theorem 8.2, the dimension of S_{1,K}(P) is the number of conjugacy classes of subgroups Q such that 1 ≤ |Q/Φ(Q)| < p^{2} , that is, the cyclic subgroups of P . On the other hand the dimension of S_{E_{2},K}(P) is the number of conjugacy classes of subgroups of P having a quotient isomorphic to E_{2} . Those subgroups are precisely the non-cyclic subgroups of P .

(c) There is a natural morphism ε : KB → KR_{ℚ} such that ε(P) : KB(P) → KR_{ℚ}(P) maps a P -set X to the corresponding permutation ℚP -module ℚX . Now it is well-known that the dimension of KR_{ℚ}(P) is equal to the number of conjugacy classes of cyclic subgroups of P , so that

dim_{K} (KR_{ℚ}(P)) = dim_{K} (S_{1,K}(P))

by the theorem. It follows that ε must be surjective (actually a well-known easy fact!) and the functor S_{1,K} must be isomorphic to KR_{ℚ} .

Note that Corollary 8.3 is a special case of the analysis of the composition factors of KB made in [Bo1] for all finite groups (see [Bo1], Propositions 10 and 12, and Example 4 page 709).

(8.4) Corollary. Suppose K is a field of characteristic q dividing p − 1 .

(a) The functor KB has a unique filtration

KB = I_{0} ⊃ I_{1} ⊃ I_{2} ⊃ I_{3} ⊃ …

and for every r ≥ 0 , the simple functor S_{E_{r},K} is a composition factor of KB .

(b) If P is a p-group, the dimension of S_{E_{r},K}(P) is the number of conjugacy classes of subgroups Q of P such that Q/Φ(Q) has order p^{r} . In particular, the dimension of S_{E_{1},K}(P) is the number of conjugacy classes of non-trivial cyclic subgroups of P .

(c) S_{1,K} is isomorphic to the constant functor Γ_{K} with Γ_{K}(P) = K for every P .

(d) KR_{ℚ} is isomorphic to a quotient functor of KB and there is a non-split exact sequence of K-linear

functors

0 ⟶ S_{E_{1},K} ⟶ KR_{ℚ} ⟶ Γ_{K} ⟶ 0 .

Proof. (a) This follows directly from Theorem 8.2 since I = ℕ ∪ {∞} when q divides p − 1 . (b) This follows immediately from Theorem 8.2.

(c) This follows from (b) by taking r = 0 .

(d) There is again a morphism ε : KB → KR_{ℚ} , mapping a P -set to the corresponding permutation ℚP -module. The kernel of this morphism must be one of the subfunctors I_{r} of KB . It cannot be I_{0} = KB since ε is non-zero (the trivial P -set is not mapped to zero). It cannot be I_{1} , since otherwise the image of ε would be the simple functor S_{1,K} = Γ_{K} , which is one dimensional in every evaluation. But for the cyclic group C of order p, the image of ε(C) is KR_{ℚ}(C) and has dimension 2 (since the characters Ind_{1}^{C}1 and 1 are not proportional modulo q). Finally, the kernel of ε cannot be one of the subfunctors I_{r} for r ≥ 3 , for reasons of dimension. Indeed the dimension of KR_{ℚ}(P) is the number of conjugacy classes of cyclic subgroups of P , and this is also the dimension of KB(P)/I_{2}(P) .

It follows that the kernel of ε is I_{2} , and that ε is surjective. Thus KR_{ℚ} is a quotient of KB , having a unique non-zero proper subfunctor, isomorphic to I_{1}/I_{2} ≅ S_{E_{1},K} . The quotient functor is the constant functor Γ_{K} (and the morphism KR_{ℚ} → Γ_{K} is actually the reduction modulo q of the dimension).

We note that the case q | p − 1 is the only one where a constant functor Γ_{K} exists.

9. A characterization of some simple functors

We prove in this section that the simple functors S_{E,K} , where E is an elementary abelian p-group and K is a field of characteristic different from p, can be characterized in some precise way by their values on elementary abelian groups. This is important for our purposes since both ℚD and \overline{D^{t}} turn out to be simple functors of this type.

Recall that, for every p-group A and every KOut(A)-module V , we have

S_{A,V} = L_{A,V} /J_{A,V} ,

where L_{A,V} and J_{A,V} are defined by (7.7) and (7.8). We first need two lemmas.

(9.1) Lemma. Let A be an abelian p-group and let V be a KOut(A)-module. If P is a p-group such that L_{A,V} (P) ≠ 0 , then the group A is isomorphic to a section of P .

Proof. If L_{A,V} (P) ≠ 0 , then there exists elements φ_{i} ∈ Hom_{C_{K}} (A, P) and elements v_{i} ∈ V , for 1 ≤ i ≤ n, such that

∑_{i=1}^{n} φ_{i} ⊗ v_{i} ≠ 0 in L_{A,V} (P) = Hom_{C_{K}} (A, P) ⊗_{End_{C_{K}} (P)} V .

Moreover, by linearity, we can suppose that the elements φ_{i} are transitive P -sets-A. So there exists such a transitive biset φ and an element v ∈ V such that φ ⊗ v is non-zero in L_{A,V} (P) . There is a subgroup L of P × A such that φ = (P × A)/L. Using the notation 7.3, we let

B = L_{p}/L_{k} ≅ _{p}L/_{k}L ,

so that there are surjective morphisms s : _{p}L → B with kernel _{k}L and t : L_{p} → B with kernel L_{k} such that

L = {(g, a) ∈ P × A | g ∈ _{p}L, a ∈ L_{p}, s(g) = t(a)} .

It is easy to see from Lemma 7.5 that

(9.2)  φ = (P × A)/L = (P × B)/M ×_{B} (B × A)/N ,

where

M = {(g, s(g)) | g ∈ _{p}L} ⊆ P × B  and  N = {(t(a), a) | a ∈ L_{p}} ⊆ B × A .

Now the group B = A′/A′′ is a section of the abelian group A, where we have set A′ = L_{p} and A′′ = L_{k} . The groups A, B , A′ , and A′′ are all abelian, hence isomorphic to their dual (or character group). As A′ is a subgroup of A, its dual Â′ ≅ A′ is a quotient of Â ≅ A. Now B is a quotient of A′ , hence also a quotient of A. Thus there exists a surjective morphism u : A → B .

Now set

R = {(a, u(a)) | a ∈ A} ⊆ A × B  and  S = {(u(a), a) | a ∈ A} ⊆ B × A .

It follows easily from Lemma 7.5 that

(B × A)/S ×_{A} (A × B)/R = (B × B)/{(b, b) | b ∈ B} ,

which is the identity endomorphism of B in the category C_{K} (this is inflation followed by deflation). We can insert this relation in (9.2), and this gives

φ = (P × B)/M ×_{B} (B × A)/S ×_{A} (A × B)/R ×_{B} (B × A)/N .

Setting

ψ = (P × B)/M ×_{B} (B × A)/S  and  θ = (A × B)/R ×_{B} (B × A)/N ,

we have φ = ψ ∘ θ. Now if |B| < |A|, the endomorphism θ of A is in the ideal I_{A} defined in (7.9), since it factors through B . Therefore its action on the KOut(A)-module V is zero, and it follows that in L_{A,V} (P) , we have

φ ⊗ v = (ψ ∘ θ) ⊗ v = ψ ⊗ θ · v = 0 .

This shows that |B| = |A|, hence B ≅ A. Now B is also a section of P , and the lemma follows.

(9.3) Lemma. Let K be a field of characteristic different from p, let A be an elementary abelian p-group, and let V be a KOut(A)-module. Then any subfunctor F of L_{A,V} is generated by its values on elementary abelian p-groups, that is,

F (P) = ∑_{E p−elementary abelian φ:E→P} F (φ)(F (E)) ,

for every p-group P .

Proof. We prove the lemma by induction on the order of P , the result being trivial if P itself is elementary abelian. Let P be an arbitrary p-group, and suppose the result holds for all groups of order strictly smaller than |P |. Let f ∈ F (P) . By Proposition 7.12, we can decompose f as

f = ∑_{{Q≤P \atop Q mod P}} F (ẽ_{Q}^{P})(f) .

Recall that

ẽ_{Q}^{P} = \frac{1}{|N_{P} (Q)|} ∑_{R≤Q} |R| μ(R, Q) (P × P)/Δ(R) .

This expression shows that if Q is a proper subgroup of P , then ẽ_{Q}^{P} is a linear combination of elements of type (P × P)/Δ(R) , where R is a subgroup of Q, hence a proper subgroup of P . By Lemma 7.5, the biset (P × P)/Δ(R) factorizes as

(P × P)/Δ(R) = (P × R)/Δ(R) ×_{R} (R × P)/Δ(R) ,

and it follows that

F ((P × P)/Δ(R))(f) ∈ F ((P × R)/Δ(R))(F (R)) .

By induction hypothesis, this gives

F ((P × P)/Δ(R))(f) ∈ F ((P × R)/Δ(R)) ( ∑_{φ:E→R} F (φ)(F (E))) ⊆ ∑_{ψ:E→P} F (ψ)(F (E)) .

It follows that we only have to deal with the term F (ẽ_{P}^{P})(f) , so we can suppose that f = F (ẽ_{P}^{P})(f) . On the other hand, the element f is in F (P) ⊆ L_{A,V} (P) , so it can be written

f = ∑_{i=1}^{n} φ_{i} ⊗ v_{i} ,

for suitable elements v_{i} ∈ V and φ_{i} ∈ Hom_{C_{K}} (A, P) , that we can suppose to be transitive P -sets-A. Now by definition of the functor L_{A,V} , we have

f = F (ẽ_{P}^{P})(∑_{i=1}^{n} φ_{i} ⊗ v_{i}) = ∑_{i=1}^{n}(ẽ_{P}^{P} ×_{P} φ_{i}) ⊗ v_{i} .

There exist subgroups L_{i} of P × A such that φ_{i} = (P × A)/L_{i} . Using the notation 7.3 and setting B_{i} = (L_{i})_{p}/(L_{i})_{k} , there are subgroups M_{i} of P × B_{i} and N_{i} of B_{i} × A such that

φ_{i} = (P × A)/L_{i} = (P × B_{i})/M_{i} ×_{B_{i}} (B_{i} × A)/N_{i} ,

and it follows that

(ẽ_{P}^{P} ×_{P} φ_{i}) ⊗ v_{i} = L_{A,V} (ẽ_{P}^{P} ∘ (P × B_{i})/M_{i})(((B_{i} × A)/N_{i}) ⊗ v_{i}) .

Now the element ((B_{i} ×A)/N_{i})⊗v_{i} is in L_{A,V} (B_{i}) , which is zero if A is not a section of B_{i} by Lemma 9.1. Moreover, if A is a section of B_{i} , as B_{i} is also a section of A, we have A ≅ B_{i} .

In this case there is a subgroup P_{i} of P and a surjective homomorphism s_{i} : P_{i} → A such that

L_{i} = {(g, s_{i}(g)) | g ∈ P_{i}} ⊆ P × A ,

and we have the factorization

φ_{i} = (P × A)/L_{i} = (P × P_{i})/Δ(P_{i}) ×_{P_{i}} (P_{i} × A)/L′_{i}

where L′_{i} is the group L_{i} , viewed as a subgroup of P_{i} × A. Now Proposition 14 of [Bo1] shows that

ẽ_{P}^{P} ×_{P} (P × P_{i})/Δ(P_{i}) = (P × P_{i})/Δ(P_{i}) ×_{P_{i}} R̃es_{P_{i}}^{P}e_{P}^{P} ,

and this is zero if P_{i} ≠ P . So we have P_{i} = P for all i and

φ_{i} = (P × A)/Δ_{s_{i}}(P) , where Δ_{s_{i}}(P) = {(g, s_{i}(g)) | g ∈ P } .

Therefore we obtain

\begin{aligned}ẽ_{P}^{P} ×_{P} φ_{i} & = \frac{1}{|P |} ∑_{R≤P} |R| μ(R, P) (P × P)/Δ(R) ×_{P} (P × A)/Δ_{s_{i}}(P) \\ & = \frac{1}{|P |} ∑_{R≤P} |R| μ(R, P) (P × A)/Δ_{s_{i}}(R) , \end{aligned}

where Δ_{s_{i}}(R) = {(r, s_{i}(r)) | g ∈ R}. The sum only runs over subgroups R such that μ(R, P) ≠ 0 , so that R contains the Frattini subgroup Φ of P . Setting \overline{P} = P/Φ and \overline{R} = R/Φ , the surjection s_{i} : P → A factorizes as the projection π : P → \overline{P} followed by a surjection \overline{s}_{i} : \overline{P} → A, because A is elementary abelian. It is easy to check that

(P × A)/Δ_{s_{i}}(R) = (P × \overline{P})/Δ_{π}(P) ×\overline{P} (\overline{P} × A)/Δ_{\overline{s}_{i}}(\overline{R}) ,

where Δ_{π}(P) = \{(g, π(g)) | g ∈ P \} and Δ_{\overline{s}_{i}}(\overline{R}) = \{(\overline{r}, \overline{s}_{i}(\overline{r})) | \overline{r} ∈ \overline{R}\}. Therefore we obtain

f = ∑_{i=1}^{n}(ẽ_{P}^{P} ×_{P} φ_{i}) ⊗ v_{i}

= ∑_{i=1}^{n} \frac{1}{|P |} ∑_{R≤P} |R| μ(R, P) ((P × A)/Δ_{s_{i}}(R)) ⊗ v_{i}

= ∑_{i=1}^{n} [(P × \overline{P})/Δ_{π}(P) ×\overline{P} (\frac{1}{|P |} ∑_{R≤P} |R| μ(R, P) (\overline{P} × A)/Δ_{\overline{s}_{i}}(\overline{R}))] ⊗ v_{i}

= L_{A,V} ((P × \overline{P})/Δ_{π}(P)) (\overline{f})

= InfP ; \overline{P} (\overline{f}) ,

where \overline{f} ∈ L_{A,V} (\overline{P}) denotes the expression

\overline{f} = \frac{1}{|P |} ∑_{R≤P} |R| μ(R, P) ((\overline{P} × A)/Δ_{\overline{s}_{i}}(\overline{R})) ⊗ v_{i} .

But we have already noticed that inflation followed by deflation is the identity, that is,

(\overline{P} × P)/\{(π(g), g) | g ∈ P \} ×_{P} (P × \overline{P})/\{(g, π(g)) | g ∈ P \} = (\overline{P} × \overline{P})/\{(\overline{g}, \overline{g}) | \overline{g} ∈ \overline{P} \} = id\overline{P} ,

as morphisms in C_{K} . Therefore \overline{f} = DefP ; \overline{P} InfP ; \overline{P} (\overline{f}) = DefP ; \overline{P} (f) , and so \overline{f} ∈ F (\overline{P}) since f ∈ F (P) and F is a subfunctor of L_{A,V} . Hence

f = InfP ; \overline{P} (\overline{f}) ∈ F (φ)(\overline{P}) ,

for some morphism φ : \overline{P} → P , as was to be shown.

Now we can state our characterization of the simple functors S_{E,K} .

(9.4) Theorem. Let K be a field of characteristic different from p, let E be an elementary abelian p-group, and let F be an object of F_{K} .

(a) Suppose F has the following two properties: (1) F (E) ≅ K is the trivial KOut(E)-module.

(2) If P is elementary abelian, dim_{K} (F (P)) = dim_{K} (S_{E,K}(P)). Then the functor F admits a direct summand isomorphic to S_{E,K} .

(b) Suppose moreover that the following condition holds: (3) For any p-group P , the map

∏_{φ} F (φ) : F (P) → ∏_{A p−elementary abelian φ:P →A} F (A)

is injective. Then the functor F is isomorphic to the simple functor S_{E,K} .

Proof. Since S_{E,K} vanishes on proper sections of E , hypothesis (2) implies that F (P) = 0 if P is a proper section of E . Therefore the ideal I_{P} of End_{C_{K}} (P) acts by zero (see (7.9)) and so F (E) is not only isomorphic to K as KOut(E)-module (hypothesis (1)), but also as End_{C_{K}} (P)-module. Therefore, by adjunction, there is a morphism θ : L_{E,K} → F which is the identity of K when evaluated at E . Hence the kernel of θ is contained in the unique maximal proper subfunctor J = J_{E,K} of L_{E,K} . Now the image F′ of θ contains a submodule F′′ = θ(J) such that F′/F′′ ≅ S_{E,K} . Hypothesis (2) implies that for any elementary abelian p-group A, we have F′(A) = F (A) and F′′(A) = 0 .

But for any p-group P , we know from Lemma 9.3 that

J(P) = ∑_{φ:A→P} J(φ)(J(A)) ,

where the sum runs over the elementary abelian groups A, and morphisms φ : A → P in C_{K} . Taking images by θ, we have

F′′(P) = θ_{P} (J(P)) = ∑_{φ:A→P} θ_{P} J(φ)(J(A)) = ∑_{φ:A→P} F′′(φ)θ_{A}(J(A)) = ∑_{φ:A→P} F′′(φ)F′′(A) = 0 .

Hence F′′ = 0 , and F′ is isomorphic to S_{E,K} , so we have an injection i : S_{E,K} → F .

The next observation is that there is a natural notion of duality in F_{K} . If F is any K-linear functor from C_{K} to K-vector spaces, the dual F^{∗} of F is defined on each object P by

F^{∗}(P) = Hom_{K}(F (P), K) .

If Q is another p-group, and φ : P → Q is a Q-set-P , then one defines F^{∗}(φ) = F (φ^{op})^{∗} , where F (φ^{op})^{∗} is the transpose map of F (φ^{op}) . This definition extends by linearity. There is a canonical morphism of functors δ from F to its bidual F^{∗∗} , defined for a p-group P by

δ_{P} (x)(φ) = φ(x) ∀x ∈ F (P), ∀φ ∈ F (P)^{∗} .

The map δ_{P} is an isomorphism if F (P) is finite dimensional over K .

It is clear that the dual of a simple functor is simple, and the classification of simple functors shows that the functor S_{E,K} is self dual. Now hypotheses (1) and (2) hold clearly for F^{∗} if they do for F . Our previous argument shows therefore that S_{E,K} is a subfunctor of F^{∗} . By duality, we obtain a surjective morphism s : F^{∗∗} → S_{E,K} . Now s∘δ∘i is an endomorphism of S_{E,K} , which is non-zero since its evaluation at E is the identity of K . As S_{E,K} is simple, it follows that s ∘ δ ∘ i is an isomorphism, so that S_{E,K} is a direct summand of F , proving (a).

Now we can write F ≅ S_{E,K} ⊕ X . Hypothesis (2) shows that X(P) = 0 if P is elementary abelian. Since hypothesis (3) holds for any subfunctor of F if it does for F , it follows that for any p-group P , the map

∏_{φ} X(φ) : X(P) → ∏_{A p−elementary abelian φ:P →A} X(A)

is injective. As the right hand side is zero, we have X(P) = 0 , hence X = 0 and F is isomorphic to S_{E,K} , proving (b).

10. The torsion-free part of the Dade functor

The purpose of this section is to prove Theorems C and D of the introduction. For convenience, we first restate Theorem C.

(10.1) Theorem. The functor ℚD is simple and is isomorphic to S_{E,ℚ} , where E is an elementary abelian group of order p^{2} .

We shall give two independent proofs of this theorem. The first uses the main theorem of Section 4, which gives the dimension of ℚD(P) . Our second proof is more conceptual and involves more of the functorial formalism. It is based on the characterization of S_{E,ℚ} obtained in Section 9 and it is independent of the results of Section 4. In both cases we need to apply the results of Section 8 giving the dimension of S_{E,ℚ}(P) , though in the second proof this is only used when P is elementary abelian.

First proof of Theorem 10.1. Clearly if we evaluate ℚD at E , we get ℚD(E) ≅ ℚ and this isomorphism is an isomorphism of ℚOut(E)-modules, since ℚD(E) is generated over ℚ by Ω_{E} , which is invariant under the action of any automorphism of E . Moreover, the group E is minimal for ℚD, so in fact the ideal I_{E} of End_{C_{ℚ}}(E) acts by zero and ℚD(E) is isomorphic to ℚ as End_{C_{ℚ}}(E)-module, see (7.9).

By adjunction, we now have a morphism of functors θ : L_{E,ℚ} → ℚD, see (7.7). This morphism is non-zero, so its kernel is contained in the unique maximal proper subfunctor J_{E,ℚ} of L_{E,ℚ} , see (7.8). This shows that the image of θ has a quotient isomorphic to S_{E,ℚ} = L_{E,ℚ}/J_{E,ℚ} .

But we know from Theorem 4.1 that for any p-group P , the dimension of ℚD(P) is equal to the number nc(P) of conjugacy classes of non-cyclic subgroups of P . On the other hand, we have shown in Corollary 8.3 that the dimension of S_{E,ℚ}(P) is also nc(P) . Since the image of θ_{P} has a quotient isomorphic to S_{E,ℚ}(P) , it follows that θ_{P} is surjective, and that ℚD is isomorphic to S_{E,ℚ} .

Second proof of Theorem 10.1. We want to apply Theorem 9.4 for K = ℚ, for the functor F = ℚD and for the elementary abelian p-group E of order p^{2} . So we need to check the three hypotheses of Theorem 9.4. We first note that ℚD(E) ≅ ℚ is the trivial ℚOut(E)-module since it is generated by Ω_{E} , which is invariant under automorphisms of E . Thus hypothesis (1) holds.

If P is elementary abelian, then the dimension of ℚD(P) is given by Dade’s Theorem 1.4, and is equal to the number of non-cyclic quotients of P , which is equal to the number of non-cyclic subgroups of P , by duality. But by Corollary 8.3, the dimension of S_{E,ℚ}(P) is equal to the number of non-cyclic subgroups of P . Hence hypothesis (2) holds for ℚD.

Finally hypothesis (3) also holds for ℚD, by Theorem 1.6. We conclude that ℚD is isomorphic to S_{E,ℚ} , by Theorem 9.4.

It is now straightforward to explain our original proof of Theorem A (giving the dimension of ℚD(P) ), which was based on Theorem 10.1 (and its second proof above). It suffices to notice that the dimension of ℚD(P) is that of S_{E,ℚ}(P) , which is known to be the number of conjugacy classes of non-cyclic subgroups of P , by Corollary 8.3.

(10.2) Remark. In Section 4, we have encountered the homomorphism

β_{Q,N}^{P} = \frac{1}{|Q|} ∑_{{X≤Q \atop XN=Q}} |X| μ(X, Q) Teninf_{X/(X∩N)}^{P} : ℚ → ℚD(P) ,

where P is a p-group, Q ≤ P , and N \underline{◁} Q is such that Q/N is elementary abelian of order p^{2} . Clearly, this is defined by a formula analogous to the expression for the idempotents of the Burnside ring. The explicit connection is made by means of the action of the Burnside algebra KB(Q) on ℚD(Q) (see Proposition 7.12). Indeed we have

β_{Q,N}^{P} = Ten_{Q}^{P} ∘ ẽ_{Q}^{Q} ∘ Inf_{Q/N}^{Q} ,

by a direct computation (using the bisets corresponding to Ten_{Q}^{P} and Inf_{Q/N}^{Q} , together with the fact that ℚD vanishes on proper sections of Q/N ). However, the fact that β_{Q,N}^{P} is independent of N (Proposition 4.9) still requires arguments involving the numbers m_{Q,N} (see the proof of Proposition 4.9 or [Bo1]). On the other hand we also mentioned in Remark 4.3 the homomorphism

δ_{Q,N}^{P} = \frac{1}{|Q|} ∑_{{X≤Q \atop XN=Q}} |X| μ(X, Q) Defres_{X/(X∩N)}^{P} : ℚD(P) → ℚ ,

which is dual to β_{Q,N}^{P} , and we claimed that it is also independent of N . The proof of this is now straightforward. It simply follows from the fact that ℚD is self-dual, because S_{E,ℚ} is self-dual, and the duality transforms δ_{Q,N}^{P} into β_{Q,N}^{P} and conversely.

(10.3) Remark. The self-dual property of ℚD means that there exist non-degenerate bilinear forms

<−, −>_{P} : ℚD(P) × ℚD(P) ⟶ ℚ ,

which satisfy suitable functorial relations (i.e. Ten_{Q}^{P} is adjoint of Res_{Q}^{P} and Def_{P/N}^{P} is adjoint of Inf_{P/N}^{P} ). The forms are symmetric because they are determined by the form <−, −>_{E} (by simplicity of ℚD ≅ S_{E,ℚ} ) and ℚD(E) is one-dimensional. It would be useful to have a natural description of the form <−, −>_{P} on endo-permutation modules, but we can only describe it on the basis of ℚD(P) given in Proposition 4.11.

Now we come to Theorem D of the introduction.

(10.4) Theorem. There is an exact sequence of functors

0 ⟶ ℚD ⟶^{α} ℚB ⟶^{ε} ℚR_{ℚ} ⟶ 0

where for every p-group P , ε(P) : ℚB(P) → ℚR_{ℚ}(P) is the morphism mapping a P -set to the corresponding permutation module over ℚ.

Proof. Corollary 8.3 shows that ℚB has a unique non-zero proper subfunctor I_{2} , isomorphic to S_{E,ℚ} , and that the quotient functor is isomorphic to ℚR_{ℚ} via the morphism ε. The result follows from the isomorphism ℚD ≅ S_{E,ℚ} .

(10.5) Remark. We have no natural description of the map α in terms of arbitrary endo-permutation modules, but we can give an explicit description of the map on a basis of ℚD(P) . We first note that α is unique up to a scalar multiple, because ℚD ≅ S_{E,ℚ} is simple and its endomorphism ring is isomorphic to the endomorphism ring of ℚ as ℚOut(E)-module, by adjunction. In other words, α is completely determined by its evaluation at E

α_{E} : ℚD(E) ≅ ℚ ⟶ ℚB(E) .

Recall that the image of α is the subfunctor I_{2} defined in Section 8 and that I_{2}(E) is one-dimensional, generated by the primitive idempotent e_{E}^{E} of ℚB(E) . Thus we specify α_{E} by mapping Ω_{E} to e_{E}^{E} .

Now for an arbitrary p-group P , we know that ℚD(P) has a basis consisting of the elements Ten_{Q}^{P}Inf_{Q/Q_{0}}^{Q}(Ω_{Q/Q_{0}}) , where Q runs over the non-cyclic subgroups of P up to conjugation, and Q_{0} denotes a fixed normal subgroup of Q such that Q/Q_{0} ≅ E . Since α is functorial, we immediately deduce that

α_{P} (Ten_{Q}^{P}Inf_{Q/Q_{0}}^{Q}(Ω_{Q/Q_{0}})) = Ind_{Q}^{P} Inf_{Q/Q_{0}}^{Q}(e_{Q/Q_{0}}^{Q/Q_{0}}) = Ind_{Q}^{P}( ∑_{{XQ_{0}=Q \atop X mod Q}} e_{X}^{Q})

= Ind_{Q}^{P}( ∑_{{X≤Q \atop XQ_{0}=Q}} \frac{1}{|Q : N_{Q}(X)|} e_{X}^{Q}) = ∑_{{X≤Q \atop XQ_{0}=Q}} \frac{|N_{P} (X)|}{|Q|} e_{X}^{P} .

This can also be written in terms of the standard basis of ℚB(P) by expanding e_{X}^{P} .

11. Partial results on the torsion functor

In this final section, we consider the functor \overline{D^{t}} and prove results analogous to those proved for ℚD in the previous section. We suppose that p is an odd prime.

(11.1) Theorem. If p is odd, the functor \overline{D^{t}} is simple and is isomorphic to S_{C,𝔽_{2}} , where C is a cyclic group of order p.

We shall again give two independent proofs of this theorem. The first uses the main theorem of Section 6, which gives the dimension of \overline{D^{t}}(P) , while the second uses more functorial formalism, since it is based on the characterization of S_{C,𝔽_{2}} obtained in Section 9.

First proof of Theorem 11.1. Clearly if we evaluate \overline{D^{t}} at C , we get \overline{D^{t}}(C) ≅ 𝔽_{2} and this has to be an isomorphism of ℚOut(C)-modules. Moreover, the group C is minimal for \overline{D^{t}} , so in fact the ideal I_{C} of End_{C_{𝔽_{2}}} (C) acts by zero and \overline{D^{t}}(C) is isomorphic to 𝔽_{2} as End_{C_{𝔽_{2}}} (C)-module, see (7.9).

By adjunction, we now have a morphism of functors θ : L_{C,𝔽_{2}} → \overline{D^{t}} , see (7.7). This morphism is non-zero, so its kernel is contained in the unique maximal proper subfunctor J_{C,𝔽_{2}} of L_{C,𝔽_{2}} , see (7.8). This shows that the image of θ has a quotient isomorphic to S_{C,𝔽_{2}} = L_{C,𝔽_{2}}/J_{C,𝔽_{2}} .

But we know from Theorem 6.1 that for any p-group P , the dimension of \overline{D^{t}}(P) is equal to the number c(P) of conjugacy classes of non-trivial cyclic subgroups of P . On the other hand, we have shown in Corollary 8.4 that the dimension of S_{C,𝔽_{2}}(P) is also c(P) (note that 2 divides p − 1 since p is odd!). Since the image of θ_{P} has a quotient isomorphic to S_{C,𝔽_{2}}(P) , it follows that θ_{P} is surjective, and that \overline{D^{t}} is isomorphic to S_{C,𝔽_{2}} .

Second proof of Theorem 11.1. We want to apply Theorem 9.4 for K = 𝔽_{2} , for the functor F = \overline{D^{t}} and for the cyclic group C of order p. So we need to check the three hypotheses of Theorem 9.4. We first note that \overline{D^{t}}(C) ≅ 𝔽_{2} is the trivial ℚOut(C)-module. Thus hypothesis (1) holds.

If P is elementary abelian, then the dimension of \overline{D^{t}}(P) is given by Dade’s Theorem 1.4, and is equal to the number of non-trivial cyclic quotients of P , which is equal to the number of non-trivial cyclic subgroups of P , by duality. But by Corollary 8.4, the dimension of S_{C,𝔽_{2}}(P) is equal to the number of non-trivial cyclic subgroups of P (again note that 2 divides p − 1 since p is odd). Hence hypothesis (2) holds for \overline{D^{t}} .

Finally hypothesis (3) also holds for \overline{D^{t}} , by definition, since \overline{D^{t}}(P) = D^{t}(P)/Ker(ψ_{P}) and the injectivity of the map induced by ψ_{P} guarantees hypothesis (3). We conclude that \overline{D^{t}} is isomorphic to S_{C,𝔽_{2}} , by Theorem 9.4.

There is a short exact sequence for \overline{D^{t}} which is very similar to the one we obtained for ℚD.

(11.2) Theorem. If p is odd, there is an exact sequence of functors

0 ⟶ \overline{D^{t}} ⟶^{β} 𝔽_{2}R_{ℚ} ⟶^{dim} Γ_{𝔽_{2}} ⟶ 0 ,

where Γ_{𝔽_{2}} denotes the constant functor.

Proof. Corollary 8.4 (which can be applied since 2 | p − 1 ) shows that 𝔽_{2}R_{ℚ} has a unique non-zero proper subfunctor, isomorphic to S_{C,𝔽_{2}} , and that the quotient functor is isomorphic to Γ_{𝔽_{2}} via the dimension morphism. The result follows from the isomorphism \overline{D^{t}} ≅ S_{C,𝔽_{2}} .

(11.3) Remark. The situation is the same as for ℚD. We have no natural description of the map β in terms of arbitrary endo-permutation modules of finite order, but we can give an explicit description of the map on a basis of \overline{D^{t}}(P) .

The representation ring 𝔽_{2}R_{ℚ}(C) has dimension 2 with basis {1, Ind_{1}^{C}(1)}, and the kernel of the dimension map modulo 2 is one-dimensional over 𝔽_{2} , generated by Ind_{1}^{C}(1) − 1 . So we necessarily have β_{C}(Ω_{C}) = Ind_{1}^{C}(1) − 1 . Now for an arbitrary p-group P , we know that \overline{D^{t}}(P) has an 𝔽_{2}-basis consisting of the elements Ten_{Z}^{P} Inf_{Z/Φ(Z)}^{Z}(Ω_{Z/Φ(Z)}) , where Z runs over the non-trivial cyclic subgroups of P up to conjugation, and Φ(Z) denotes the unique subgroup of Z of index p. Since β is functorial, we immediately deduce that

β_{P} (Ten_{Z}^{P} Inf_{Z/Φ(Z)}^{Z}(Ω_{Z/Φ(Z)})) = Ind_{Z}^{P} Inf_{Z/Φ(Z)}^{Z}(Ind_{1}^{Z/Φ(Z)}(1) − 1) = Ind_{Φ(Z)}^{P}(1) − Ind_{Z}^{P}(1) .

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