Rational p-biset functors

Serge Bouc

Abstract : In this paper, I give several characterizations of rational biset functors over p-groups, which are independent of the knowledge of genetic bases for p-groups. I also introduce a construction of new biset functors from known ones, which is similar to the Yoneda construction for representable functors, and to the Dress construction for Mackey functors, and I show that this construction preserves the class of rational p-biset functors.

This leads to a characterization of rational p-biset functors as additive functors from a specific quotient category of the biset category to abelian groups. Finally, I give a description of the largest rational quotient of the Burnside p-biset functor : when p is odd, this is simply the functor R_{ℚ} of rational representations, but when p = 2, it is a non split extension of R_{ℚ} by a specific uniserial functor, which happens to be closely related to the functor of units of the Burnside ring.

AMS Subject Classification (2000): 20J15, 19A22. Keywords : biset functor, rational, Burnside, p-group.

1. Introduction

The formalism of biset functors for finite groups yields a unified framework for operations of induction, restriction, inflation, deflation and transport by group isomorphism, which appear in various areas of group representation theory. It was used recently as an essential tool to the solution of some open problems, such as the classification of endo-permutation modules for an arbitrary finite p-group ([7]), or the structure of the group of units of the Burnside ring of such a p-group ([6]).

This formalism seems indeed particularly well suited for studying p-groups (where p is a fixed prime number), the main reason for this being that pgroups have many normal subgroups, so for these groups, the operations of inflation and deflation play a prominent rôle. Also, some important biset functors are only defined on p-groups, and cannot be seen in a natural way as the restriction to p-groups of a functor defined on arbitrary finite groups : an example of this situation is the Dade functor for p-groups. Such biset functors are called p-biset functors.

Among p-biset functors, the subclass of rational functors was introduced in [5] (see 2.10), the name rational being chosen because these functors behave like the functor of rational representations, in regard to the decomposition of their evaluations at a given p-group with respect to faithful elements. These rational biset functors are generally easy to describe, since they are determined by their knowledge at p-groups of normal p-rank 1 : this was in particular the key argument to the structure of the group of units of Burnside rings of p-groups. Another important property of rational p-biset functors is that they form a Serre subclass of all p-biset functors.

This paper originates in stimulating and fruitful conversations I had with Laurence Barker and Ergün Yalçın, during my visit at Bilkent University in April 2006. The original question arose when L. Barker and I understood that we had two a priori different definitions of what a rational p-biset functor should be : L. Barker ([1], see also 5.6) used a definition akin to the notion of functor on the category RG-Morita introduced by I. Hambleton, L. R. Taylor, and E. B. Williams in [10]. This definition is much more natural that mine, since in particular it applies to arbitrary finite groups, and not only to pgroups.

So the question was to know whether the two notions could be equivalent. My very first feeling was that the answer should be negative. Also, it seemed clear to me that rational functors in the sense of L. Barker should be rational in mine, but the converse was not obvious. Shortly afterwards, independently of each other, and using different methods, E. Yalçın, L. Barker and I gave a proof of this fact.

This paper gives a complete answer to the question of the equivalence of these definitions (Theorem 5.3), showing that my first intuition was almost always wrong : the two definitions of rational p-biset functors are equivalent if p is an odd prime number. But for p = 2, the two definitions are not equivalent, the typical example being the largest rational quotient (in my sense) of the Burnside functor, which is not rational for the second definition. In particular, this definition does not lead to a Serre subclass for p = 2.

This paper is organized as follows : in Section 2, I recall the basic definitions and results on biset functors. In Section 3, I state a necessary and sufficient condition (Theorem 3.1) for a biset functor to be rational (in my sense), along the lines of Proposition 8.1 of [7]. In Section 4, I introduce a construction for biset functors (4.3), which is similar both to the Dress construction for Mackey functors, and to Yoneda construction for arbitrary functors. This leads, in Section 5, to a characterization (Theorem 5.3) of the category of rational biset functors as a subcategory of all biset functors over p-groups. In particular, Corollary 5.8 is a kind of geometric characterization of rational p-biset functors. In Section 6, I describe the largest rational quotient of the Burnside functor : if p is odd, this functor is equal to R_{ℚ}, but if p = 2, this functor is a non-split extension of R_{ℚ} by a 2-torsion functor which happens to be isomorphic to a quotient of the functor of units of the Burnside ring, more precisely the cokernel of the exponential map. At the time, this is just a coincidence, and I could find no natural a priori isomorphism between those two objects.

2. Biset functors

2.1. Definition : The biset category C on finite groups is defined as follows :

• The objects of C are finite groups.

• If G and H are finite groups, then

Hom_{C}(G, H) = B(H × G^{op})

is the Burnside group of (G, H)-bisets.

• If G, H, and K are finite groups, then the composition

Hom_{C}(G, H) × Hom_{C}(H, K) → Hom_{C}(G, K)

is the bilinear extension of the product (V, U) ↦ V ×_{H} U, where V is a (K, H)-biset, and U is an (H, G)-biset. Moreover, the identity morphism of the group G in C is the class of the (G, G)-biset G, for left and right multiplication.

A biset functor is an additive functor from C to the category Ab of abelian groups. A morphism of biset functors is a natural transformation of functors. The category of biset functors is denoted by F.

When p is a prime number, the p-biset category C_{p} is the full subcategory whose objects are finite p-groups. A p-biset functor is an additive functor from C_{p} to Ab. The category of p-biset functors is denoted by F_{p}.

2.2. Elementary morphisms. The formalism of bisets gives a single framework for the usual operations of induction, restriction, inflation, deflation, and transport by isomorphism via the following correspondences :

• If H is a subgroup of G, then let Ind_{H}^{G} ∈ Hom_{C}(H, G) denote the set G, with left action of G and right action of H by multiplication.

• If H is a subgroup of G, then let Res_{H}^{G} ∈ Hom_{C}(G, H) denote the set G, with left action of H and right action of G by multiplication.

• If N ⊴ G, and H = G/N, then let Inf_{H}^{G} ∈ Hom_{C}(H, G) denote the set H, with left action of G by projection and multiplication, and right action of H by multiplication.

• If N ⊴ G, and H = G/N, then let Def_{H}^{G} ∈ Hom_{C}(G, H) denote the set H, with left action of H by multiplication, and right action of G by projection and multiplication.

• If φ : G → H is a group isomorphism, then let Iso_{G}^{H} = Iso_{G}^{H}(φ) ∈ Hom_{C}(G, H) denote the set H, with left action of H by multiplication, and right action of G by taking image by φ, and then multiplying in H.

2.3. Definition and Notation : A section (T, S) of the group G is a pair of subgroups of G such that S ⊴ T. The corresponding subquotient of G is the group T/S. If (T, S) is a section of G, set

Δ_{T,S} = {(x, y) ∈ G × G | x ∈ T, xy^{−1} ∈ S} .

It is a subgroup of G × G. If (T, S) is a section of the group G, set

Indinf_{T/S}^{G} = Ind_{T}^{G} ∘ Inf_{T/S}^{T} Defres_{T/S}^{G} = Def_{T/S}^{T} ∘ Res_{T}^{G} .

Then Indinf_{T/S}^{G} is isomorphic to the (G, T/S)-biset G/S, and Defres_{T/S}^{G} is isomorphic to the (T/S, G)-biset S\G.

2.4. Notation : When F is a biset functor, and φ ∈ Hom_{C}(G, H), the image of the element u of F (G) by the map F (φ) will be also be denoted by φ(u) or even φu.

Similarly, the maps F (Ind_{H}^{G}), F (Res_{H}^{G}),…, will be denoted by Ind_{H}^{G}, Res_{H}^{G},…, and they will be called induction, restriction,…

When G and H are finite groups, and U is a finite (H, G)-biset, then U^{op} denotes the set U, endowed with the (G, H)-biset structure given by

g · u · h (in U^{op}) = h^{−1}ug^{−1} (in U) ,

for g ∈ G, u ∈ U, and h ∈ H. This construction extends linearly to a map φ ↦ φ^{op} from Hom_{C}(G, H) to Hom_{C}(H, G).

2.5. Factorization of morphisms. Any morphism form G to H in C is a linear combination with integer coefficients of classes of transitive (H, G)- bisets, and isomorphism classes of transitive (H, G)-bisets are in one to one correspondence with conjugacy classes of subgroups of the direct product H × G : the (H, G)-biset corresponding to the subgroup L of H × G is the set of cosets (H × G)/L, for biset action defined by

∀(g, x) ∈ G^{2}, ∀(h, y) ∈ H^{2}, g.(x, y)L.h = (gx, h^{−1}y)L .

Denote by p_{1}(L) the projection of L on H, and by p_{2}(L) its projection on G. Set moreover

k_{1}(L) = {h ∈ H | (h, 1) ∈ L} k_{2}(L) = {g ∈ G | (1, g) ∈ L} .

Then k_{i}(L) ⊴ p_{i}(L), for i = 1, 2, the group k_{1}(L)×k_{2}(L) is a normal subgroup of L, and there are canonical group isomorphisms

p_{1}(L)/k_{1}(L) ≅ L/(k_{1}(L) × k_{2}(L)) ≅ p_{2}(L)/k_{2}(L) .

Now (see Lemma 7.4 of [8]) the biset (H × G)/L can be factored as

Ind_{p_{1}(L)}^{H} ∘ Inf_{p_{1}(L)/k_{1}(L)}^{p_{1}(L)} ∘ Iso(φ) ∘ Def_{p_{2}(L)/k_{2}(L)}^{p_{2}(L)} ∘ Res_{p_{2}(L)}^{G} ,

where φ : p_{2}(L)/k_{2}(L) → p_{1}(L)/k_{1}(L) is the above isomorphism, sending gk_{2}(L), for g ∈ G, to hk_{1}(L), where h ∈ H is chosen such that (h, g) ∈ L. Note that if G and H are p-groups, then this factorization holds within the category C_{p}, since the groups p_{i}(L) and p_{i}(L)/k_{i}(L) are p-groups in this case, for i = 1, 2.

2.6. Faithful elements. (see Lemma 2.7 of [5]) Let N be a normal subgroup of G. Set

f_{N}^{G} = ∑_{{M ⊴ G \atop N⊆M}} μ_{⊴ G}(N, M) Inf_{G/M}^{G} ∘ Def_{G/M}^{G} ,

where μ_{⊴ G} is the Möbius function of the poset of normal subgroups of G. This is an element of End_{C}(G). It is an idempotent, and the elements f_{N}^{G}, for N ⊴ G, are mutually orthogonal idempotents, whose sum is equal to the identity of End_{C}(G).

If F is a biset functor, set

∂F (G) = F (f_{1}^{G})F (G) .

It is a direct summand of F (G), called the set of faithful elements of F (G). It is equal to the intersection of the kernel of all deflation maps Def_{G/M}^{G}, where M is a non trivial normal subgroup of G.

2.7. Remark : If P is a p-group, then

f_{1}^{P} = ∑_{Z⊆Ω_{1}Z(P)} μ(1, Z) Inf_{P/Z}^{P} ∘ Def_{P/Z}^{P} ,

where Ω_{1}Z(P) is the largest elementary abelian subgroup of the center of P, and μ is the Möbius function of the poset of its subgroups.

2.8. Simple functors : Recall some notation and basic results on simple p-biset functors : the category of p-biset functors is an abelian category. Its simple objects are parametrized by pairs (H, V), where H is a finite p-group, and V is a simple ℤOut(H)-module, where Out(H) is the group of outer automorphisms of H (see e.g. [2], Proposition 2, page 678, or [8] Proposition 7.10 for details). All the simple functors appearing in the present paper (in Section 6) correspond to the case where V is the quotient ℤ/2ℤ (with obviously trivial Out(H)-action). The corresponding simple functor will be denoted by S_{H,𝔽_{2}}.

2.9. Genetic subgroups of p-groups : ([7] Section 2.9, and [5] Proposition 4.4 and Lemma 4.5) Let P be a p-group. A subgroup Q of P is called genetic if the two following conditions are fulfilled :

• The group N_{P} (Q)/Q has normal p-rank 1, i.e. it is cyclic, generalized quaternion, or dihedral or semi-dihedral of order at least 16.

• If x ∈ P, then Q^{x} ∩ Z_{P} (Q) ⊆ Q if and only if Q^{x} = Q, where Z_{P} (Q) is the subgroup of N_{P} (Q) defined by Z_{P} (Q)/Q = Z(N_{P} (Q)/Q).

Two genetic subgroups Q and R of P are said to be linked modulo P if there exist x and y in P such that

Q^{x} ∩ Z_{P} (R) ⊆ R  and  R^{y} ∩ Z_{P} (Q) ⊆ Q .

This is an equivalence relation on the set of genetic subgroups of P. A genetic basis of P is a set of representatives of equivalence classes for this relation. A genetic basis is in natural one to one correspondence with the set of isomorphism classes of rational irreducible representations of P. If Q and R are genetic subgroups of P, and if Q and R are linked modulo P, then the groups N_{P} (Q)/Q and N_{P} (R)/R are isomorphic. This group is called the type of the equivalence class of genetic subgroups, or of the corresponding rational irreducible representation.

2.10. Rational p-biset functors. (see Section 7 of [5]) Let P be a p group, and G be a genetic basis of P. It was shown in Theorem 3.2. of [5] that the map

I_{G} : ⊕_{Q∈G} Indinf_{N_{P} (Q)/Q}^{P} : ⊕_{Q∈G} ∂F (N_{P} (Q)/Q) → F (P)

is split injective. A left inverse to it is the map

D_{G} : ⊕_{Q∈G} f_{1}^{N_{P} (Q)/Q}Defres_{N_{P} (Q)/Q}^{P} : F (P) → ⊕_{Q∈G} ∂F (N_{P} (Q)/Q) .

In particular, setting

γ_{Q} = Indinf_{N_{P} (Q)/Q}^{P}f_{1}^{N_{P} (Q)/Q}Defres_{N_{P} (Q)/Q}^{P} ,

for Q ∈ G, then the elements γ_{Q} are orthogonal idempotents of End_{C_{p}}(P).

The functor F is called rational if for any p-group P and any genetic basis G of P, the map I_{G} is an isomorphism. Equivalently, for any p-group P, there exists a genetic basis with this property. Equivalently again, the image by F of ∑_{Q∈G} γ_{Q} is the identity map of F (P).

3. A characterization of rational p-biset functors

3.1. Theorem : Let F be a p-biset functor. Then F is rational if and only if the following two conditions are satisfied :

(i) If P is a p-group with non cyclic center, then ∂F (P) = {0}.

(ii) If P is any p-group, if E ⊴ P is a normal elementary abelian subgroup of rank 2, if Z is a central subgroup of order p of P contained in E, then the map

Res_{C_{P} (E)}^{P} ⊕ Def_{P/Z}^{P} : F (P) ⟶ F (C_{P} (E)) ⊕ F (P/Z)

is injective.

Proof: Suppose first that F is rational. Let P be a p-group, and G be a genetic basis of P. Since F is rational, the map

D_{G} : F (P) ⟶^{⊕ b_{Q}^{P}} ⊕_{Q∈G} ∂F (N_{P} (Q)/Q)

is injective. Recall that the component b_{Q}^{P} of the map D_{G} indexed by the genetic subgroup Q ∈ G − {P } is equal to Defres_{N_{P} (Q)/Q}^{P} − Defres_{N_{P} (Q)/Q̂}^{P}, where Q̂ is the subgroup defined by Q̂/G = Ω_{1}Z(N_{P} (Q)/Q). If Q = P, then the corresponding component b_{P}^{P} is equal to Def_{P/P}^{P}.

Suppose that the center Z(P) of P is not cyclic. Then for any Q ∈ G, the group Z = Q ∩ Z(P) is non trivial : indeed, the group Z(P)/Z is isomorphic to a subgroup of the center of N_{P} (Q)/Q, which is cyclic since N_{P} (Q)/Q has normal p-rank 1. Now for Q ≠ P, the maps Defres_{N_{P} (Q)/Q}^{P} and Defres_{N_{P} (Q)/Q̂}^{P} factor as

Defres_{N_{P} (Q)/Q}^{P} = Defres\overline{P} ; N_{\overline{P}} (\overline{Q})/\overline{Q}Def_{P/Z}^{P}
Defres_{N_{P} (Q)/Q̂}^{P} = Defres\overline{P} ; N_{\overline{P}} (\overline{Q})/\overline{Q̂}Def_{P/Z}^{P}

where \overline{P} = P/Z, where \overline{Q} = Q/Z, and \overline{Q̂} = Q̂/Z. Moreover if Q = P, then Def_{P/P}^{P} = Def\overline{P} ; \overline{P} /\overline{P} Def_{P/Z}^{P}. It follows that if u ∈ ∂F (P), then D_{G}(u) = 0, since u is mapped to 0 by the proper deflation Def_{P/Z}^{P}. Thus u = 0, and ∂F (P) = {0} when the center of P is non cyclic. So if F is rational, then it satisfies condition (i) of the theorem.

For condition (ii), there is nothing to check if E is central in P, since in that case the map Res_{C_{P} (E)}^{P} is the identity map, hence it is injective. And if E is not central in P, then Z = E ∩ Z(P). In this situation, by Corollary 5.3 of [7], there exists a genetic basis G which admits a decomposition G = G_{1}⊔G_{2} as a disjoint union, where

(3.2)

• G_{2} = {Q ∈ G | Q ⊇ Z}, and

• the set H_{1} = {^{x}Q | Q ∈ G_{1}, x ∈ [P/C_{P} (E)]} is the subset of a genetic basis of C_{P} (E) consisting of subgroups not containing Z.

Let u ∈ F (P) be an element such that Res_{C_{P} (E)}^{P}u = 0 and Def_{P/Z}^{P}u = 0. If Q ∈ G_{2}, then b_{Q}^{P} = b\overline{P} ; \overline{Q} ∘ Def_{P/Z}^{P}, where \overline{P} = P/Z and \overline{Q} = Q/Z. It follows that b_{Q}^{P}(u) = 0. Similarly if Q ∈ G_{1}, then N_{P} (Q) ⊆ C_{P} (E), and b_{Q}^{P} = b_{Q}^{C_{P} (E)} ∘ Res_{C_{P} (E)}^{P}, so b_{Q}^{P}(u) = 0 again. Hence D_{G}(u) = 0, and u = 0 since D_{G} is injective when F is rational. Thus F satisfies (ii).

Conversely, suppose that F is a p-biset functor satisfying (i) and (ii). Showing that F is rational amounts to showing that the map D_{G} is injective, for any p-group P and any genetic basis G of P. Actually, by Lemma 7.3 of [5], it is enough to show that D_{G} is injective, for any p-group P and for one particular genetic basis G of P. I will prove this by induction on the order of P.

If P is the trivial group, then the map D_{G} is the identity map, so there is nothing to prove in this case, and this starts induction. Suppose that P is a p-group, such that the map D_{H} is injective for any p-group P′ with |P′| < |P | and any genetic basis H of P′. Let G be a genetic basis of P, and u ∈ Ker D_{G}. Let N be a non-trivial normal subgroup of P. Then the set H = {Q/N | Q ∈ G, ⊇ N} is a genetic basis of P/N. Moreover one checks easily that b\overline{P} ; \overline{Q} ∘ Def_{P/N}^{P} = b_{Q}^{P} for Q ⊇ N, where overlines denote quotients by N, as above. Hence D_{H}(Def_{P/N}^{P}u) = 0, and by induction hypothesis, it follows that Def_{P/N}^{P}u = 0, and u ∈ ∂F (P). In other words Ker D_{G} ⊆ ∂F (P).

Hence if the center of P is not cyclic, then condition (i) implies that Ker D_{G} = {0}, and I can suppose that the center of P is cyclic.

If P has normal p-rank 1, then the trivial subgroup is a genetic normal subgroup of P, hence is belongs to every genetic basis of P. The corresponding map b_{1}^{P} : F (P) → ∂F (P) is equal to F (f_{1}^{P}). Hence it is equal to the identity map on ∂F (P). Thus u = F (f_{1}^{P})(u) = b_{1}^{P} (u) = 0, and the map D_{G} is injective in this case.

Finally I can suppose that P admits a normal and non central subgroup E which is elementary abelian of rank 2. In this case the group Z = E ∩ Z(P) has order p, and by Corollary 5.3 of [7], there exists a genetic basis G of P with the two properties (3.2) above. With the same notation, let H be a genetic basis of C_{P} (E) containing H_{1}. Set v = Res_{C_{P} (E)}^{P}u. Then for any Q ∈ G_{1} and any x ∈ [P/C_{P} (E)], one has N_{P} (Q) ⊆ C_{P} (E) by Lemma 5.2 of [7], and it follows that

b_{^{x}Q}^{C_{P} (E)}(v) = b_{^{x}Q}^{C_{P} (E)}Res_{C_{P} (E)}^{P}u
= Defres_{N_{P} (^{x}Q)/^{x}Q}^{P}u − Defres_{N_{P} (^{x}Q)/^{x}Q̂}^{P}u
= ^{x}(Defres_{N_{P} (Q)/Q}^{P}u − Defres_{N_{P} (Q)/Q̂}^{P}u)
= ^{x}b_{Q}^{P}u = 0 ,

since ^{x}u = u for x ∈ P and u ∈ F (P). Thus b_{R}^{C_{P} (E)}(v) = 0 for any R ∈ H_{1}.

And if R ∈ H − H_{1}, then R ⊇ Z, and

b_{R}^{C_{P} (E)}(v) = b_{R/Z}^{C_{P} (E)/Z}Res_{C_{P} (E)/Z}^{P/Z}Def_{P/Z}^{P}u = 0 .

It follows that D_{H}(v) = 0, hence v = 0 by induction hypothesis. Now Res_{C_{P} (E)}^{P}u = 0 and Def_{P/Z}^{P}u = 0. Condition (ii) implies that u = 0, so the map D_{G} is injective.

3.3. Remark : The argument in the last part of the proof is essentially the same as the one used in the proof of Theorem 8.2 of [7].

Recall some notation from [7] :

3.4. Notation :

• Let E be an elementary abelian p-group of rank 2. Define an element ε_{E}

of B(E) by

ε = ε_{E} = E/1 − ∑_{{F ⊆E \atop |F |=p}} E/F + pE/E .

Denote by B_{ε} the subfunctor of B generated by ε, i.e. the smallest subfunctor F of B such that F (E) ∋ ε. Equivalently, for any p-group P

B_{ε}(P) = Hom_{C_{p}}(E, P)(ε) .

• Let X be either an extraspecial group of order p^{3} and exponent p if p is odd, or a dihedral group of order 8 if p = 2. Let Z denote the center of X. Let I and J be two subgroups of order p in X, different from Z, and not conjugate in X. Define an element δ of B(X) by

δ = δ_{X,I,J} = (X/I − X/IZ) − (X/J − X/JZ) .

Denote by B_{δ} the subfunctor of B generated by δ. Thus, for any pgroup P

B_{δ}(P) = Hom_{C_{p}}(X, P)(δ) .

• Let K denote the kernel of the natural transformation B → R_{ℚ}. Thus K is a biset subfunctor of B.

3.5. Remark : These elements ε and δ have been introduced in Corollary 6.5 and Notation 6.9 of [7]. In the case p = 2, the element δ was denoted by δ_{3} there. It is easy to check that ε_{E} ∈ K(E) and that δ ∈ K(X). So B_{ε} and B_{δ} are also the subfunctors of K generated by ε and δ respectively, that is B_{ε} = K_{ε} and B_{δ} = K_{δ}. Moreover Lemma 6.11 of [7] shows that Res_{JZ}^{X}δ = ε_{JZ}, so in particular B_{ε} ⊆ B_{δ}.

3.6. Remark : If E and E′ are elementary abelian p-groups of rank 2, and if φ is any group isomorphism from E to E′, then φ(ε_{E}) = ε_{E′}. This shows that the subfunctor B_{ε} does not depend on the choice of a particular group E. Similarly, if X′ is another extraspecial group of order p^{3} and exponent p, when p is odd, or another dihedral group of order 8, when p = 2, and if I′ and J′ are two non conjugate non central subgroups of order p of X′, then there exists a group isomorphism φ : X → X′ such that φ(I) = I′ and φ(J) = J′. Thus φ(δ_{X,I,J}) = δ_{X′,I′,J′}, and it follows that B_{δ} does not depend on the choice of the groups X, I and J.

It was shown in Theorem 6.12 of [7] that if p is odd, then K = B_{δ}. The following provides an alternative proof for this theorem :

3.7. Theorem : The functor B/B_{δ} is rational. Proof: Let A be any injective cogenerator for the category of ℤ-modules (e.g. A = ℚ/ℤ). Recall that the A-dual F^{0} of a biset functor F is defined by

F^{0}(P) = Hom_{ℤ}(F (P), A) F^{0}(φ) = ^{t}F (φ^{op})

for any p-group P and any morphism φ in the category C_{p}. Since A is an injective cogenerator, the canonical map F → (F^{0})^{0} is injective. Taking F = B/B_{δ}, I will show that F^{0} is rational. By Proposition 7.4 of [5], it will follow that (F^{0})^{0} and any subfunctor of it are rational, so F will be rational.

Suppose first that P is a p-group with non cyclic center. Then Lemma 6.8 of [7] shows that ∂B(P) ⊆ B_{ε}(P) ⊆ B_{δ}(P). Hence ∂B/B_{δ}(P) = {0}. Since ∂F^{0}(P) ≅ Hom_{ℤ}(∂F (P), A), it follows that ∂F^{0}(P) = {0}, so condition (i) of Theorem 3.1 is satisfied by F^{0}.

Now let P be any p-group, and let E be a normal subgroup of P, which is elementary abelian of rank 2. Let moreover Z be a central subgroup of order p of P, contained in E. Let f ∈ F^{0}(P), such that Res_{C_{P} (E)}^{P}(f) = 0 and Def_{P/Z}^{P}f = 0. Checking condition (ii) of Theorem 3.1 amounts to showing that f = 0.

If E is central in P, there is nothing to do, since C_{P} (E) = P and Res_{C_{P} (E)}^{P}(f) = f. So I can assume that C_{P} (E) has index p in P. Let f̃ : B(P) → I be the homomorphism obtained by composing f with the projection B(P) → B/B_{δ}(P). The hypothesis on f means that f̃(P/Q) = 0 if Q ⊆ C_{P} (E) or if Q ⊇ Z. Let Q be any subgroup of P with Q ⊈ C_{P} (E) and Q ⊉ Z. Then Q ⊉ E. Moreover QC_{P} (E) = P. If the intersection Q ∩ E has order p, it is normalized, hence centralized by Q, and also by C_{P} (E), hence Q ∩ E ⊆ Z(P). Since Q ∩ E ≠ Z, it follows that E = Z·(Q ∩ E) is central in P, a contradiction. Hence Q ∩ E = 1.

Now the group C_{Q}(E) = Q ∩ C_{P} (E) is a normal subgroup of QE. The factor group R = QE/C_{Q}(E) has order p^{3}, and is the semidirect product of the group \overline{Q} = Q/C_{Q}(E), of order p, by the normal elementary abelian subgroup group \overline{E} = EC_{Q}(E)/C_{Q}(E) ≅ E. Moreover R is not abelian, since otherwise [Q, E] ⊆ C_{Q}(E) ∩ E = 1, so Q ⊆ C_{P} (E). It follows that R is extraspecial of order p^{3} and exponent p if p is odd, or dihedral of order 8 if p = 2.

The center of R is the group \overline{Z} = ZC_{Q}(E)/C_{Q}(E). The group \overline{Q} is a non central subgroup of order p of R. If F is any subgroup of order p of E, different from Z, then \overline{F} = F C_{Q}(E)/C_{Q}(E) is another non central subgroup of order p of R, not conjugate to \overline{Q} in R. So in the group R, the element

d = (R/\overline{Q} − R/\overline{Q}\overline{Z}) − (R/\overline{F} − R/\overline{E})

is in B_{δ}(R), since it is the image of δ by a suitable group isomorphism. Taking induction-inflation from the section (QE, C_{Q}(E)) of P gives the element

d̃= Indinf_{QE/C_{Q}(E)}^{P}d = (P/Q − P/QZ) − (P/F C_{Q}(E) − P/EC_{Q}(E)) ,

of B_{δ}(P). Now f̃ vanishes on B_{δ}(P), thus f̃(d̃) = 0, and

f̃(P/Q) = f̃(P/QZ) + f̃(P/F C_{Q}(E)) − f̃(P/EC_{Q}(E)) = 0 ,

since QZ ⊇ Z and since F C_{Q}(E) ⊆ EC_{Q}(E) ⊆ C_{P} (E).

This shows that f̃ = 0, hence f = 0, and the functor F^{0} satisfies both conditions of Theorem 3.1. Hence F^{0} is rational, and F is rational as well, since it is a subfunctor of the A-dual of F^{0}.

3.8. Corollary : [[7], Theorem 6.12-1] If p is odd, then K = B_{δ}. Proof: Indeed, since B/B_{δ} is rational, its subfunctor K/B_{δ} is also rational. Then for any p-group P, and for any genetic basis G of P, the map

⊕_{Q∈G} Indinf_{N_{P} (Q)/Q}^{P} : ⊕_{Q∈G} ∂(K/B_{δ})(N_{P} (Q)/Q) → (K/B_{δ})(P)

is an isomorphism. But the groups N_{P} (Q)/Q have normal p-rank 1, hence they are cyclic since p is odd. Now if R is cyclic, then K(R) = {0}. It follows that K/B_{δ}(P) = {0}, hence K = B_{δ}.

3.9. Remark : The structure of the functor K/B_{δ} in the case p = 2 will be described in Section 6.

3.10. Remark : The key point in the proof of Theorem 3.7 is the construction of a section (T, S) of P such that T/S is extraspecial of order p^{3} and exponent p, or dihedral of order 8. The same idea was used in Lemma 6.15 of [7].

4. Yoneda-Dress constructions and rational functors

In this section I will introduce a construction which is connected both to the Yoneda functors in the case of the biset category C_{p}, and to the Dress construction for Mackey functors (see [9], or [3] 1.2).

4.1. Notation : Let H be a finite p-group. Let π_{H} : C_{p} → C_{p} denote the correspondence sending a p-group P to the direct product P × H. If P and Q are p-groups, and if U is a (Q, P)-biset, denote by U ×H the (Q×H, P ×H)- biset equal to the cartesian product of U and H as a set, with double action defined by

∀y ∈ Q, ∀x ∈ P, ∀(h, k, l) ∈ H^{3}, (y, l)(u, h)(x, k) = (yux, lhk) .

This correspondence extends linearly to an additive map

B(Q × P^{op}) → B((Q × H) × (P × H)^{op})

also denoted by f ↦ π_{H}(f).

4.2. Lemma : The correspondence π_{H} is an endofunctor of C_{p}, which is additive on morphisms. Proof: This is straightforward.

4.3. Notation : If H is a p-group, and F is a p-biset functor, I denote by F_{H} the p-biset functor obtained by precomposition with π_{H}.

In other words, for any p-group P, one has F_{H}(P) = F (P × H). If U is a (Q, P)-biset, then π_{H}(U) = F (U × H).

4.4. Lemma : Let P, Q and H be p-groups. If U is a (P, P)-biset and V is a (P × H, Q) biset, then there is an isomorphism of (P × H, Q)-bisets

(U × H) ×_{(P ×H)} V ≅ U ×_{P} V ,

where the double action on U ×_{P} V is induced by

(x, h)(u, v)q = (xu, hvq) ,

for x ∈ P, h ∈ H, y ∈ Q, u ∈ U, and v ∈ V.

In particular, if for any v ∈ V, the intersection of the stabilizer of v in P × H with Z(P) × {1} is non trivial, then

(f_{1}^{P} × H) ×_{(P ×H)} V = 0 .

Proof: The first assertion is straightforward. The second one follows from the fact that V is a union of P -sets which are inflated from proper quotients of P.

4.5. Proposition : Let H be a p-group, and let F be a rational p-biset functor. Then F_{H} is rational. Proof: To prove the proposition, I will check that F_{H} fulfills the conditions (i) and (ii) of Theorem 3.1.

Let P be a p-group with non cyclic center. Let Q be a genetic subgroup of P × H. Since Z(P) × {1} ⊆ Z(P × H), the group Z(P) × {1} is contained in the center of N_{P ×H}(Q), hence it maps to the center of N_{P ×H}(Q)/Q, which is cyclic. Thus if the center of P is not cyclic, then Z(P) × {1} intersects Q non trivially. In other words Z(P) ∩ k_{1}(Q) ≠ 1.

Now, setting R = P × H, there are two cases : either Q = R. In this case the idempotent γ_{Q} introduced in 2.10 is just an (R, R) biset of cardinality 1. Thus

(f_{1}^{P} × H) ×_{R} γ_{Q} = f_{1}^{P} ×_{P} (P/P) = 0 ,

since P is non trivial. And if Q ≠ R, then

γ_{Q} = (R × R)/Δ_{N_{R}(Q),Q} − (R × R)/Δ_{N_{R}(Q),Q̂} ,

where Q̂ is defined by the condition that Q̂/Q is the only subgroup of order p in the center of N_{R}(Q)/Q.

The left stabilizer in R of the point Δ_{N_{R}(Q),Q} of (R×R)/Δ_{N_{R}(Q),Q} is equal to Q, which intersects Z(P) × {1} non trivially by the previous remarks.

Similarly, the left stabilizer in R of the point Δ_{N_{R}(Q),Q̂} of (R × R)/Δ_{N_{R}(Q),Q̂} is equal to Q̂, which also intersects Z(P) × {1} non trivially. By Lemma 4.4, it follows that

(f_{1}^{P} × H) ×_{(P ×H)} γ_{Q} = 0 ,

for any genetic subgroup Q of R. But since F is rational, the sum of idempotents γ_{Q}, for Q in a genetic basis of P × H, is the identity map of F (P × H). Thus f_{1}^{P} × H acts by 0 on F (P × H). In other words, the idempotent f_{1}^{P} acts by 0 on F_{H}(P). Hence ∂F_{H}(P) = 0, and F_{H} fulfills condition (i) of Theorem 3.1.

For condition (ii), suppose that E is a normal subgroup of P, and that E is elementary abelian of rank 2. Let Z be a central subgroup of order p in P, contained in E. It is easy to see that the restriction map from F_{H}(P) to F_{H}(C_{P} (E)) is equal to the restriction map Res_{C_{P} (E)×H}^{P ×H}, after the identifications F_{H}(P) = F (P × H) and F_{H}(C_{P} (E)) = F (C_{P} (E) × H). Similarly, the deflation map from F_{H}(P) to F_{H}(P/Z) is equal to the map Def_{(P ×H)/(Z×{1})}^{P ×H} from F (P × H) to F ((P/Z) × H), after the identification (P × H)/(Z × {1}) = (P/Z) × H. Now E′ = E × {1} is a normal subgroup of P × H, which is elementary abelian of rank 2. Its centralizer in P × H is C_{P} (E) × H. Moreover E′ contains the subgroup Z′ = Z × {1}, which is central of order p in P × H. Condition (ii) for the functor F implies that

Ker Res_{C_{P} (E)×H}^{P ×H} ∩ Ker Def_{(P/Z)×H}^{P ×H} = {0} .

Thus F_{H} fulfills condition (ii) of Theorem 3.1, and F_{H} is rational.

5. The category of rational p-biset functors

In this section, I will show that the rational p-biset functors are exactly the additive functors C_{p} → ℤ-Mod which factor through the quotient category C_{p}/δ obtained by factoring out every morphism f : P → Q in C_{p} which lies in B_{δ}(Q × P^{op}). The first thing to check for doing this is that this class of morphisms is a two sided ideal in C_{p} :

5.1. Lemma : Let P, Q, R, S be p-groups. Then

B(S × Q^{op}) ×_{Q} B_{δ}(Q × P^{op}) ×_{P} B(P × R^{op}) ⊆ B_{δ}(S × R^{op}) .

Proof: Since B_{δ}(Q × P^{op}) = Hom_{C_{p}}(P, Q)(δ), an element in B_{δ}(Q × P^{op}) is linear combination of isomorphism classes of (Q, P)-bisets of the form U ×_{X} δ, where U is some (Q × P^{op}, X)-biset, i.e. a (Q, P × X)-biset. Now if V is an (S, Q)-biset, and W is a (P, R) biset, then

V ×_{Q} (U ×_{X} δ) ×_{P} W ≅ (V ×_{Q} U ×_{P} W) ×_{X} δ

in B(S × R^{op}), where the (S, R × X)-biset structure on V ×_{Q} U ×_{P} W is induced by

s(v, u, w)(r, x) = (sv, u(1, x), wr) ,

for s ∈ S, r ∈ R, v ∈ V, u ∈ U, w ∈ W, and x ∈ X. The lemma follows.

5.2. Definition : Let C_{p}/δ denote the following category :

• The objects are finite p-groups.

• If P and Q are finite p-groups, then

Hom_{C_{p}/δ}(P, Q) = B(Q × P^{op})/B_{δ}(Q × P^{op}) .

• Composition in C_{p}/δ is induced by the product

− ×_{Q} − : B(R × Q^{op}) × B(Q × P^{op}) → B(R × P^{op})

defining the composition in C_{p}.

Let ρ_{δ} denote the natural functor from C_{p} to C_{p}/δ : so ρ_{δ} is equal to the identity on objects, and to the natural projection on morphisms.

5.3. Theorem : Let F be a p-biset functor. The following conditions are equivalent :

(i) The functor F is rational.

(ii) The functor F factors through ρ_{δ}. In other words, if f ∈ Hom_{C_{p}}(P, Q) lies in B_{δ}(Q × P^{op}), then F (f) = 0.

Proof: Suppose first that (ii) holds. Let S be a set of representatives of isomorphism classes of finite p-groups. For any P ∈ S, choose a set Γ_{P} of generators of F (P) as an abelian group. By Yoneda lemma, the set of natural transformations from B_{P} to F is in one to one correspondence with F (P), and this correspondence is as follows : if s ∈ F (P), then for any p-group Q, define a map

σ_{P,s,Q} : B_{P} (Q) = B(Q × P^{op}) → F (Q)

by σ_{P,s,Q}(f) = f(s). Then the maps σ_{P,s,Q} define a morphism of functors σ_{P,s} : B_{P} → F. If condition (ii) holds, this morphism of functors factors through the quotient functor (B/B_{δ})_{P}, which is rational by Theorem 3.7 and Proposition 4.5. This gives a morphism

⊕_{P ∈S} ⊕_{s∈Γ_{P}}(B/B_{δ})_{P} → F ,

which is obviously surjective. Now any direct sum of rational p-biset functors is a rational p-biset functor, and any quotient of a rational p-biset functor is rational. So F is rational, and (i) holds.

Conversely, suppose now that F is rational. Let P and Q be p-groups. Any morphism in B_{δ}(Q, P) is a linear combination of morphisms of the form U ×_{X} δ, where U is a (Q × P^{op}, X)-biset, i.e. a (Q, P × X)-biset. So proving that (ii) holds is equivalent to proving that F (U ×_{X} δ) = 0 for any P and Q and any such biset U. This will be a consequence of the next two lemmas, the first of which uses the following notation : let Ũ be the (X × Q, P)-biset equal to U as a set, with double action defined by

\underbrace{(x, h)·u·g}_{in Ũ} = \underbrace{hu(g, x^{−1})}_{in U}

for x ∈ X, h ∈ Q, u ∈ U and g ∈ P. Then : 5.4. Lemma : With this notation, let T ∈ B(X^{op}) ≅ B(1 × X^{op}). So π_{Q}(T) ∈ B(Q × (X × Q)^{op}), and T^{op} ∈ B(X). Then

U ×_{X} T^{op} = π_{Q}(T) ×_{(X×Q)} Ũ

in B(Q × P^{op}). Proof: I can suppose that T is a right X-set. Then it is easily checked that the maps

(u, t) ∈ U ×_{X} T^{op} ↦ ((t, 1), u) ∈ π_{Q}(T) ×_{(X×Q)} Ũ
((t, q), u) ∈ π_{Q}(T) ×_{(X×Q)} Ũ ↦ (qu, t) ∈ U ×_{X} T^{op}

are well defined mutual inverse isomorphisms of (Q, P)-bisets.

5.5. Lemma : If F is a rational p-biset functor, then the map

F (δ^{op}) : F (X) → F (1)

is equal to 0. Proof: Recall that δ = (X/I − X/IZ) − (X/J − X/JZ), i.e.

δ^{op} = (I\X − IZ\X) − (J\X − JZ\X) ,

where Z is the center of X, and where I and J are non central subgroups of order p in X, not conjugate in X.

Let G the set of subgroups of X defined by

G = {Y ⊆ X | |X : Y | ≤ p} ⊔ {I} .

Then G is a genetic basis of X : indeed the set {Y ⊆ X | |X : Y | ≤ p} is obtained by inflation from the unique genetic basis of the elementary abelian group X/Z. This set contains p + 2 subgroups. Moreover, there is a unique faithful irreducible rational representation of X (because there are p + 3 conjugacy classes of cyclic subgroups in X), and it is easy to check that I is a genetic subgroup of X corresponding to this representation : clearly I ∩ Z = 1, and N_{X}(I)/I = IZ/I is cyclic of order p. Moreover Î = IZ, and if x ∈ X is such that I^{x} ∩ IZ ⊆ I, then I^{x} ⊆ I because IZ is a normal subgroup of X. So I^{x} = I, and I is a genetic subgroup of X, and G is a genetic basis of X.

Let Y ∈ G. If Y = P, then the idempotent γ_{Y} (see 2.10) is an (X, X)- biset with one element. So T \X ×_{X} γ_{P} is a set of cardinality 1, for any subgroup T of X. Hence in B(X^{op})

δ^{op} ×_{X} γ_{P} = 0 .

Now if Y has index p in X, then γ_{Y} = X/Y − X/X, thus

δ^{op} ×_{X} γ_{Y} = δ^{op} ×_{X} X/Y .

But T \X ×_{X} X/Y ≅ T Y \X, so T \X ×_{X} X/Y ≅ T Z\X ×_{X} X/Y, for any subgroup T of X, since Y ⊴ X and Y ⊇ Z. Thus again

δ^{op} ×_{X} γ_{Y} = 0

in B(X^{op}), in this case .

Finally if Y = I, then

γ_{Y} = (X/I ×_{IZ} I\X) − (X/IZ ×_{IZ} IZ\X) = (X/I ×_{IZ} I\X) − X/IZ since IZ ⊴ X. By the previous argument, it follows that

δ^{op} ×_{X} γ_{Y} = δ^{op} ×_{X} (X/I ×_{IZ} I\X) .

Now X/I ×_{IZ} I\X ≅ (X × X)/Δ_{IZ,Z}, and by the Mackey formula for the product of bisets (Proposition 1 of [2]), for any subgroup T of X

T \X ×_{X} (X × X)/Δ_{IZ,Z} = ∑_{x∈T \X/IZ} T_{x}\X ,

where

T_{x} = {a ∈ X | ∃t ∈ T, (t^{x}, a) ∈ Δ_{IZ,I}}
= {a ∈ X | ∃t ∈ T, t^{x} ∈ IZ, t^{x}a^{−1} ∈ I}
= I(T^{x} ∩ IZ) .

Moreover T \X/IZ = T IZ\X since IZ ⊴ X. Hence :

I\X ×_{X} (X × X)/Δ_{IZ,Z} = I\X + IZ\X
IZ\X ×_{X} (X × X)/Δ_{IZ,Z} = 2 IZ\X
J\X ×_{X} (X × X)/Δ_{IZ,Z} = I\X
JZ\X ×_{X} (X × X)/Δ_{IZ,Z} = IZ\X

Thus

δ^{op} ×_{X} γ_{Y} = (I\X + IZ\X) − 2 IZ\X − I\X + IZ\X = 0

again, in this case.

But since F is rational, the sum ∑_{Y ∈G} F (γ_{Y}) is equal to the identity map of F (X). Hence

F (δ^{op}) = F (δ^{op}) ∘ ∑_{Y ∈G} F (γ_{Y}) = ∑_{Y ∈G} F (δ^{op} ×_{X} γ_{Y}) = 0

as was to be shown.

End of the proof of Theorem 5.3 : it follows from Lemma 5.4 that

F (U ×_{X} δ) = F (π_{Q}(δ^{op})) ∘ F (Ũ) .

Now F (π_{Q}(δ^{op})) = F_{Q}(δ^{op}), and F_{Q} is rational by Theorem 4.5. Thus F_{Q}(δ^{op}) = 0 by Lemma 5.4, and F (U ×_{X} δ) = 0, completing the proof of Theorem 5.3.

5.6. Remark : Consider the quotient category C_{p}^{lin} of C_{p}, whose objects are finite p-groups, and morphisms are defined by

Hom_{C_{p}^{lin}}(P, Q) = R_{ℚ}(Q × P^{op}) .

Then a rational biset functor for L. Barker’s definition is just an additive functor from C_{p}^{lin} to Ab.

When p is odd, it follows from Corollary 3.8 that the category C_{p}/δ and C_{p}^{lin} coincide. So in this case, this definition of a rational biset functor is equivalent to Definition 2.10. However if p = 2, those two definitions are not equivalent, as will be shown in Section 6.

5.7. Remark : If F is rational, then F_{Q} is also rational for any Q, thus F_{Q}(δ^{op}) = 0. Conversely, if F_{Q}(δ^{op}) = 0 for any Q, then the previous proof shows that F is rational. This leads to the following :

5.8. Corollary : Let S denote a Sylow p-subgroup of P GL(3, 𝔽_{p}). Then S acts on the set ℙ of points and on the set 𝕃 of lines of the projective plane over 𝔽_{p}.

Let F be a p-biset functor. Then F is rational if and only if for any p-group Q,

F (ℙ^{op} × Q) = F (𝕃^{op} × Q)

as maps from F (S×Q) to F (Q), images by F of the (Q, S×Q)-bisets ℙ^{op}×Q and 𝕃^{op} × Q respectively. Proof: Indeed S is isomorphic to the group X, so I can suppose S = X. Now it is easy to check that S has exactly one fixed point on ℙ and on 𝕃, and that for suitable choices of I and J, the permutation representations of S on ℙ and 𝕃 are respectively

ℙ ≅ S/S ⊔ S/IZ ⊔ S/J 𝕃 ≅ S/S ⊔ S/JZ ⊔ S/I ,

so that ℙ − 𝕃 = δ in B(S).

5.9. Remark : Let A be an injective cogenerator for the category of ℤmodules. Recall the duality argument already used in the proof of Theorem 3.7 : if F is a rational p-biset functor, then F^{0} = Hom(F, A) is rational by Proposition 7.4 of [5]. Conversely, if F^{0} is rational, then (F^{0})^{0} is rational, and since the canonical map F → (F^{0})^{0} is injective (because A is an injective cogenerator of ℤ-modules), it follows that F is rational. This duality argument shows that a p-biset functor F is rational if and only if for any p-group Q, one has that

F (ℙ × Q) = F (𝕃 × Q) ,

as maps from F (Q) to F (S × Q), images by F of the (S × Q, Q)-bisets ℙ × Q and 𝕃 × Q respectively. Equivalently F_{Q}(δ) = 0.

5.10. Proposition : Let F be a p-biset functor.

1. The correspondence sending a p-group P to the image of the map F_{P} (δ^{op}) : F (X × P) → F (P) defines a biset subfunctor of F, and the quotient functor F^{rat} is the largest rational quotient of F. Thus

F^{rat}(P) = F (P)/Im F_{P} (δ^{op}) .

2. The correspondence sending a p-group P to the kernel of the map F_{P} (δ) : F (P) → F (X × P) defines a biset subfunctor F_{rat} of F, which is the largest rational subfunctor of F. Thus

F_{rat}(P) = Ker F_{P} (δ) .

Proof: Let P and Q be finite p-groups, and let U be a finite (Q, P)-biset. It is easy to check that the following diagram

F (X × P) \underrightarrow{F_{P} (δ^{op})} F (P)
F (X × U) ↓  ↓ F (U)
F (X × Q) \overrightarrow{F_{Q}(δ^{op})} F (Q)

is commutative. It shows that the correspondence P ↦ Im F_{P} (δ^{op}) defines a subfunctor of F. Now if N is a subfunctor of F, then F/N is rational if and only if for any p-group P, the map (F/N)_{P} (δ^{op}) is equal to 0. This is equivalent to Im F_{P} (δ^{op}) ⊆ N(P), so F^{rat} is the largest rational quotient of F.

Similarly, for Assertion 2, the diagram

F (P) \underrightarrow{F_{P} (δ)} F (X × P)
F (U) ↓  ↓ F (X × U)
F (Q) \overrightarrow{F_{Q}(δ)} F (X × Q)

is commutative, showing that F_{rat} is a subfunctor of F. And if M is a subfunctor of F, then M is rational if and only if for any p-group P, the map M_{P} (δ) is equal to 0, i.e. if M(P) ⊆ F_{rat}(P).

5.11. Corollary : The functor B/B_{δ} is the largest rational quotient of B. If p is odd, it is equal to R_{ℚ}. Proof: Indeed B/B_{δ} is rational by Theorem 3.7. Conversely, the image of B_{1}(δ) : B(1) → B(X) contains δ, so if B′ is a subfunctor of B such that B/B′ is rational, then B′ ⊇ B_{δ}. Finally B/B_{δ} = R_{ℚ} if p is odd, by Corollary 3.8.

6. The functor K/B_{δ}

Since B/K ≅ R_{ℚ}, there is an exact sequence of p-biset functors

0 → K/B_{δ} → B/B_{δ} → R_{ℚ} → 0 .

Moreover, the functor B/B_{δ} is rational by Theorem 3.7. It follows that its subfunctor K/B_{δ} is also rational, and Corollary 3.8 shows that this functor is equal to zero if p is odd. This section describes the structure of K/B_{δ} for p = 2, but the methods and results exposed here also hold for p odd, and for this reason p still denotes an arbitrary prime number here.

Since K/B_{δ} is rational, then for any p-group P and any genetic basis G of P, the map

⊕_{Q∈G} Indinf_{N_{P} (Q)/Q}^{P} : ⊕_{Q∈G} ∂(K/B_{δ})(N_{P} (Q)/Q) → (K/B_{δ})(P)

is an isomorphism. Hence the evaluation of K/B_{δ} at P is known if the groups ∂(K/B_{δ})(N_{P} (Q)/Q) are known, when Q is a genetic subgroup of P. Since N_{P} (Q)/Q has normal p-rank 1 in this case, this relies on the following lemma :

6.1. Lemma : Let R be a p-group of normal p-rank 1. If R is non trivial, denote by Z the unique subgroup of order p in the center of R. Then :

1. If R is cyclic or generalized quaternion, then ∂K(R) = {0}, hence ∂(K/B_{δ})(R) = {0}.

2. If R is semidihedral, then ∂K(R) is free of rank one, generated by the element

−2(R/W − R/W Z) + (R/1 − R/Z) = Ind_{W Z}^{R}ε_{W Z} ,

where W is a non central subgroup of order 2 in R. In particular ∂(K/B_{δ})(R) = {0}.

3. If R is dihedral, then ∂K(R) is free of rank two, generated by the elements

−2(R/W − R/W Z) + R/1 − R/Z = Ind_{W Z}^{R}ε_{W Z} , and
δ_{R} = (R/W − R/W Z) − (R/W′ − R/W′Z) ,

where W and W′ are non central subgroups of order 2 in R, which are not conjugate in R. In this case ∂(K/B_{δ})(R) ≅ ℤ/2ℤ, generated by the image of δ_{R}.

Proof: If R is trivial, then K(R) = {0}, so the result holds. And if R is non trivial, then the idempotent f_{1}^{R} is equal to R − R/Z, so for any subgroup S of R

f_{1}^{R}R/S = R/S − R/SZ ,

and this is zero if S ⊇ Z.

Now if R is cyclic or generalized quaternion, and if S ⊉ Z, then S = 1. Thus ∂B(R) is free of rank one in this case, generated by R/1 − R/Z. Since |R/1 − R/Z| = |R|/p, and since |Y | = 0 for any Y ∈ K(R), it follows that ∂K(R) = {0} in this case, proving Assertion 1.

If R is semidihedral, and S ⊉ Z, then either S = 1 or S = W, up to conjugation in R. So ∂B(R) is free of rank 2, generated by R/1 − R/Z and R/W − R/W Z. Since |R/1 − R/Z| = |R|/2 and |R/W − R/W Z| = |R|/4, any element in ∂K(R) is a multiple of

u = −2(R/W − R/W Z) + (R/1 − R/Z) ,

which is equal to Ind_{W Z}^{R}ε_{W Z}. Since ε_{W Z} ∈ K(W Z), it follows that ∂K(R) is free of rank one, generated by u. Assertion 2 follows, since moreover u ∈ K_{ε}(R) and K_{ε}(R) ⊆ K_{δ}(R).

Finally if R is dihedral, and S ⊉ Z, then either S = 1, or S has order 2, and S ≠ Z. So ∂B(R) is free of rank 3, generated by the elements R/1−R/Z, R/W −R/W Z, and R/W′ −R/W′Z, where W and W′ are representatives of the two conjugacy classes of non central subgroups of order 2 in R. Looking at cardinalities shows that ∂K(R) is contained in the submodule of ∂B(R) generated by the elements

u = −2(R/W − R/W Z) + (R/1 − R/Z) , and
δ_{R} = (R/W − R/W Z) − (R/W′ − R/W′Z) .

Conversely, since u = Ind_{W Z}^{R}ε_{W Z}, it follows that u ∈ K(R). But δ_{R} ∈ K(R) also, by Remark 6.10 of [7], or by direct computation of the character of δ_{R}. Hence ∂K(R) is equal to the module generated by u and δ_{R}. Since moreover u ∈ K_{ε}(R) ⊆ K_{δ}(R) as before, and since

2δ_{R} = Ind_{W′Z}^{R}ε_{W′Z} − Ind_{W Z}^{R}ε_{W Z} ∈ K_{ε}(R) ⊆ K_{δ}(R) ,

it follows that ∂K/B_{δ}(R) is either trivial or of order 2, generated by the image of δ_{R}. The next lemma will show that δ_{R} ∉ B_{δ}(R), completing the proof of Assertion 3.

6.2. Lemma : Suppose p = 2, and let R be a dihedral group of order at least 16. Then δ_{R} ∉ B_{δ}(R). Proof: Suppose that δ_{R} ∈ B_{δ}(R). Then there exists φ ∈ Hom_{C_{p}}(D_{8}, R) such that δ_{R} = φ(δ). In particular there exists some (R, D_{8})-biset U such that the coefficient of R/W in the expansion of U ×_{D_{8}} δ in the canonical basis of B(R) is odd, and I can assume that U is a transitive biset, of the form U = (R × D_{8})/L, for some subgroup L of R × D_{8}. Such a transitive biset factors as

Indinf_{T/S}^{R}Iso_{B/A}^{T/S}Defres_{B/A}^{D_{8}} ,

where (T, S) is a section of R and (B, A) is a section of D_{8} such that the factor groups T/S and B/A are isomorphic. Now since δ ∈ K(D_{8}), and since K(C) = {0} for a cyclic group C, it follows that Defres_{B/A}^{D_{8}}δ = 0, except if (B, A) = (D_{8}, 1) or if B/A is elementary abelian of rank 2. Moreover any proper deflation of δ is equal to 0, hence in the latter case B cannot be equal to D_{8}. So B is one of the two elementary abelian subgroups of index 2 in D_{8}, and A = 1. In this case Defres_{B/A}^{D_{8}}δ is equal to the restriction of δ to B, i.e. to ±ε_{B}.

Thus there is a section (T, S) of R, such that T/S is isomorphic to D_{8} or (C_{2})^{2}, such that U ×_{D_{8}} δ = Indinf_{T/S}^{R}u, where u = δ_{T/S} if T/S ≅ D_{8}, and u = ε_{T/S} if T/S ≅ (C_{2})^{2}. The coefficient of R/W in Indinf_{T/S}^{R}u is equal to 0, except if S ⊆ W^{r} ⊆ T for some r ∈ R. Thus either S = 1 or S is conjugate to W. But N_{R}(W)/W has order 2, so it cannot contain a group isomorphic to T/S. It follows that S = 1. In other words U ×_{D_{8}} δ = Ind_{T}^{R}u, where T is a subgroup of R which is isomorphic to D_{8}, and in this case u = δ_{T}, or isomorphic to (C_{2})^{2}, and u = ε_{T}. In the first case

Ind_{T}^{R}δ = (R/I − R/IZ) − (R/J − R/JZ) ,

where I and J are non central subgroups of order 2 of D_{8}. Since |R| ≥ 16, the groups I and J are conjugate in R, so Ind_{T}^{R}δ = 0. In the second case T ⊇ Z, and

Ind_{T}^{R}ε_{T} = R/1 − (R/A + R/B + R/Z) + 2R/T ,

where A, B and Z are the subgroups of order 2 of T. But A and B are conjugate in R, so in fact

Ind_{T}^{R}ε_{T} = R/1 − (2R/A + R/Z) + 2R/T ,

The coefficient of R/W in this expression is equal to -2 or 0, according to A being conjugate to W in R or not. In any case, it is even, and this completes the proof.

6.3. Corollary : Let P be a p-group, and let G be a genetic basis of P. Then K(P)/B_{δ}(P) is a vector space over 𝔽_{2}, with basis the images of the elements Indinf_{N_{P} (Q)/Q}^{P}δ_{N_{P} (Q)/Q}, where Q runs through the elements of G for which N_{P} (Q)/Q is dihedral.

So K(P)/B_{δ}(P) ≅ (ℤ/2ℤ)^{d}, where d is the number of isomorphism classes of rational irreducible representations of P of dihedral type. This result suggest a little more, in view of the following result :

6.4. Theorem : If P is a p-group, denote by B^{×}(P) the group of units of its Burnside ring B(P).

1. [[6] Theorem 7.2 and Corollary 7.3] The correspondence P ↦ B^{×}(P) is a p-biset functor, which is isomorphic to a subfunctor of 𝔽_{2}R_{ℚ}^{∗}. In particular, it is a rational p-biset functor.

2. [[6] Theorem 8.4] The functor B^{×} is uniserial, and the poset of its proper subfunctors

{0} ⊂ L_{0} ⊂ L_{1} … ⊂ L_{n} ⊂ …

is such that L_{0} is isomorphic to the simple biset functor S_{1,𝔽_{2}}, and L_{i}/L_{i−1}, for i ≥ 1, is isomorphic to the simple functor S_{D_{2^{i+3}},𝔽_{2}}.

3. [[6] Theorem 8.4] There is a map of p-biset functors exp : B → B^{×}, whose image is equal to the socle L_{0} of B^{×}. So for any p-group P, the 𝔽_{2}-dimension of the cokernel of exp_{P} : B(P) → B^{×}(P) is equal to the number of isomorphism classes of rational irreducible representations of P of dihedral type.

This suggest that the biset functors K/B_{δ} and Coker exp = B^{×}/soc(B^{×}) might be isomorphic. And this is indeed the case :

6.5. Theorem : The functor K/B_{δ} is isomorphic to the cokernel of the map exp : B → B^{×}. So K = B_{δ} if p ≠ 2, and if p = 2, the functor K/B_{δ} is uniserial. Its lattice of proper subfunctors is

\{0\} = \overline{K}_{0} ⊂ \overline{K}_{1} ⊂ \overline{K}_{2} ⊂ … ⊂ \overline{K}_{n} ⊂ …

where \overline{K}_{n}, for n ≥ 1, is generated by the image of δ_{D_{2^{n+3}}} ∈ K(D_{2^{n+3}}) in (K/B_{δ})(D_{2^{n+3}}). Moreover \overline{K}_{n}/\overline{K}_{n−1} ≅ S_{D_{2^{n+3}},𝔽_{2}}, for n ≥ 1. Proof: First recall that if R is a p-group of normal p-rank 1, then R has a unique faithful rational irreducible representation Φ_{R}, up to isomorphism, by Proposition 3.7 of [4]. For any p-group P, the group R_{ℚ}(P) is a free abelian group with basis the set Irr_{ℚ}(P) of isomorphism classes of irreducible rational representations of P. So the dual group R_{ℚ}^{∗}(P) has a dual basis (V^{∗})_{V ∈Irr_{ℚ}(P)}, where the value of V^{∗} on some ℚP -module M is equal to the multiplicity of V as a direct summand of M (with the slight abuse of notation identifying an irreducible representation V with its isomorphism class). Denote by \overline{V}^{∗} the image of V^{∗} in 𝔽_{2}R_{ℚ}^{∗}.

Also set K_{♭} = K/B_{δ}, and B_{♭}^{×} = B^{×}/L_{0}. With this notation, and identifying moreover the functor B^{×} with its isomorphic image inside 𝔽_{2}R_{ℚ}^{∗}, the subfunctor L_{n} of B^{×} is generated by the element \overline{Φ}_{D_{2^{n+3}}}^{∗} ∈ 𝔽_{2}R_{ℚ}^{∗}(D_{2^{n+3}}), for n ≥ 1, and the functor L_{0} is generated by the element \overline{Φ}_{1}^{∗} ∈ 𝔽_{2}R_{ℚ}^{∗}(1). So if P is a p-group, and if G is a genetic basis of P, then the group B_{♭}^{×} has a basis over 𝔽_{2} consisting of the elements \overline{V} _{S}^{∗} = Indinf_{N_{P} (S)/S}^{P}\overline{Φ}_{N_{P} (S)/S}^{∗}, where S runs through the set D of elements of G for which N_{P} (S)/S is dihedral.

Moreover, if R is a p-group of normal p-rank 1, then ∂B^{×}(R) is equal to {0}, unless R is dihedral or cyclic of order at most 2, and in each of these cases ∂B^{×}(R) ≅ 𝔽_{2} (see Yalçın [11] Lemma 4.6 and Lemma 5.2, or Corollary 5.9 of [6]). But if |R| ≤ 2, then S_{D_{2^{n+3}},𝔽_{2}}(R) = {0} for n ≥ 1, thus B^{×}(R) = L_{0}(R), and it follows that ∂B_{♭}^{×}(R) = {0}, except if R is dihedral. In this case since ∂S_{R,𝔽_{2}}(R) ≅ 𝔽_{2} ≠ {0}, it follows that ∂B_{♭}^{×}(R) ≅ 𝔽_{2}, generated by the image of the element \overline{Φ}_{R}^{∗}. It will be useful for further computation to notice that \overline{Φ}_{R}^{∗}(ℚR/T) is equal to 2 if T = 1, to 1 if T is a non central subgroup of order 2 of R, and to 0 otherwise.

On the other hand, by Corollary 6.3, the group K_{♭}(P) has an 𝔽_{2}-basis consisting of the images \overline{d}_{S} in K_{♭}(P) of the elements Indinf_{N_{P} (S)/S}^{P}δ_{N_{P} (S)/S}, for S ∈ D. So there is a unique linear map

ψ_{P,G} : K_{♭}(P) → B_{♭}^{×}(P)

sending \overline{d}_{S} to \overline{V} _{S}^{∗}, for S ∈ D, and this map is an isomorphism. In other words, the map ψ_{P,G} is equal to the composition of isomorphisms

(6.6) K_{♭}(P) →^{β} ⊕_{S∈G} ∂K_{♭}(N_{P} (S)/S) →^{γ} ⊕_{S∈G} ∂B_{♭}^{×}(N_{P} (S)/S) →^{α} B_{♭}^{×}(P) ,

where α = ⊕_{S} B_{♭}^{×}(a_{S}) and β = ⊕_{S} K_{♭}(b_{S}), and where γ is the direct sum of the unique isomorphisms γ_{N_{P} (S)/S} : ∂K_{♭}(N_{P} (S)/S) → ∂B_{♭}^{×}(N_{P} (S)/S) : this makes sense because the groups ∂K_{♭}(N_{P} (S)/S) and B_{♭}^{×}(N_{P} (S)/S) are both cyclic of order 2, when S ∈ D, or both trivial if S ∈ G − D.

6.7. Lemma : Let P and Q be p-groups. Let G be a genetic basis of P, and H be a genetic basis of Q. Then if φ ∈ Hom_{C_{p}}(P, Q), the diagram

K_{♭}(P) \underrightarrow{ψ_{P,G}} B_{♭}^{×}(P)
K_{♭}(φ) ↓  ↓ B_{♭}^{×}(φ)
K_{♭}(Q) \overrightarrow{ψ_{Q,H}} B_{♭}^{×}(Q)

is commutative. Proof: Using decomposition 6.6 both for P and Q, this diagram gives

K_{♭}(P) →^{β} ⊕_{S∈G} ∂K_{♭}(N_{P} (S)/S) →^{γ} ⊕_{S∈G} ∂B_{♭}^{×}(N_{P} (S)/S) →^{α} B_{♭}^{×}(P)
K_{♭}(φ) ↓  ↓ B_{♭}^{×}(φ)
K_{♭}(Q) →^{β′} ⊕_{T ∈H} ∂K_{♭}(N_{Q}(T)/T ) →^{γ} ⊕_{T ∈H} ∂B_{♭}^{×}(N_{Q}(T)/T ) →^{α′} B_{♭}^{×}(Q)

where the maps α′ and β′ are the analogues of α and β for the group Q and its genetic basis H. Showing that this diagram is commutative amounts to showing that

B_{♭}^{×}(φ) ∘ α ∘ γ ∘ β = α′ ∘ γ′ ∘ β′ ∘ K_{♭}(φ) ,

or equivalently, that

α^{′−1} ∘ B_{♭}^{×}(φ) ∘ α ∘ γ = γ ∘ β′ ∘ K_{♭}(φ) ∘ β^{−1} .

Now β^{−1} = ⊕_{S∈G} K_{♭}(a_{S}) and α^{′−1} = ⊕_{T ∈H} B_{♭}^{×}(b_{T}). So this equality is equiva-

lent to

(6.8)  B_{♭}^{×}(b_{T} φa_{S}) ∘ γ_{N_{P} (S)/S} = γ_{N_{Q}(T)/T} ∘ K_{♭}(b_{T} φa_{S}) ,

for any S ∈ G and any T ∈ H. Now

b_{T} φa_{S} ∈ f_{1}^{N_{Q}(T)/T} Hom_{C_{p}}(N_{P} (S)/S, N_{Q}(T)/T )f_{1}^{N_{P} (S)/S} ,

so equality (6.8), hence Lemma 6.7, follow from the next lemma.

6.9. Lemma : Let M and N be p-groups of normal p-rank 1, and let f ∈ f_{1}^{N}Hom_{C_{p}}(M, N)f_{1}^{M}, then

B_{♭}^{×}(f) ∘ γ_{M} = γ_{N} ∘ K_{♭}(f) .

Proof: Both sides are maps from ∂K_{♭}(M) to ∂B_{♭}^{×}(N), so if one of the groups M or N is not dihedral, both sides are equal to zero. So I can suppose that M and N are dihedral. I can also suppose that f = f_{1}^{N}Uf_{1}^{P}, for some transitive (N, M)-biset U of the form (N × M)/L, where L is a subgroup of N × M. If f = 0, there is nothing to prove, and otherwise k_{1}(L) ∩ Z(N) = 1 and k_{2}(L) ∩ Z(M) = 1. It follows that k_{1}(L) and k_{2}(L) are trivial or non central of order 2. If one of these groups, say k_{1}(L), is non central of order 2, then its normalizer has order 4, and contains p_{1}(L). Hence the group C = q(L) ≅ p_{1}(L)/k_{1}(L) has order 1 or 2. But f factors through C and both K_{♭}(C) and B_{♭}^{×}(C) are equal to {0}. So again there is nothing to prove in this case.

Hence I can assume that k_{1}(L) = k_{2}(L) = 1. In this case, the morphism (N ×M)/L is the composition of the restriction from M to p_{2}(L), followed by a group isomorphism from p_{2}(L) to p_{1}(L), followed by induction from p_{1}(L) to N. In particular, it factors through H = p_{1}(L). If this group is cyclic, or contained in a dihedral subgroup of order 8 of M, then K_{♭}(H) = B_{♭}^{×}(H) = {0} because H, has no genetic subgroup of dihedral type. So I can suppose that H is dihedral of order at least 16.

Now ∂K_{♭}(M) is generated by the image of the element

δ_{M} = (M/W − M/W Z) − (M/W′ − M/W′Z) ,

where Z is the center of M, and W and W′ are non central subgroups of order 2, non conjugate in M. If M′ is a dihedral group of index 2 in M, containing W, then one checks easily that in B(M′)

Res_{M′}^{M}δ_{M} = δ_{M′} − Ind_{W^{x}Z}^{M′}ε_{W^{x}Z} .

where x ∈ M − M′, so Res_{M′}^{M}δ_{M} = δ_{M′} in K_{♭}(M′) since

Ind_{W^{x}Z}^{M′}ε_{W^{x}Z} ∈ B_{ε}(M′) ⊆ B_{δ}(M′) .

By induction, it follows that Res_{H}^{M}δ_{M} = δ_{H} in K_{♭}(H). This element is sent to δ_{p_{1}(L)} by the isomorphism from H = p_{2}(L) to p_{1}(L), hence finally

K_{♭}(f)(δ_{M}) = f_{1}^{N}Ind_{p_{1}(L)}^{N}δ_{p_{1}(L)}

in K_{♭}(N). Now there are two cases : either p_{1}(L) = N and K_{♭}(f)(δ_{M}) = δ_{N}, or p_{1}(L) is a proper dihedral subgroup of N. In this case, the non central subgroups of order 2 of p_{1}(L) are conjugate in N, so Ind_{p_{1}(L)}^{N}δ_{p_{1}(L)} = 0 in K(p_{1}(L)).

So either p_{1}(L) ≠ N, and γ_{N} ∘ K_{♭}(f)(δ_{M}) = 0, or p_{1}(L) = N, and γ_{N} ∘ K_{♭}(f)(δ_{M}) is equal to the generator \overline{Φ}_{N}^{∗} of ∂B_{♭}^{×}(N).

On the other hand γ_{M}(δ_{M}) is equal to the image of \overline{Φ}_{M}^{∗} in B_{♭}^{×}(M). The restriction of \overline{Φ}_{M}^{∗} to the dihedral subgroup H = p_{2}(L) is easily seen to be equal to \overline{Φ}_{H}^{∗}, hence it is mapped to \overline{Φ}_{p_{1}(L)}^{∗} by the isomorphism p_{2}(L) → p_{1}(L). Finally

B_{♭}^{×} ∘ γ_{M}(δ_{M}) = Ind_{p_{1}(L)}^{N}\overline{Φ}_{p_{1}(L)}^{∗} .

If p_{1}(L) = N, this is equal to \overline{Φ}_{N}^{∗}. Otherwise it is induced from a proper dihedral subgroup. It is easily checked in this case that the values of the linear form Ind_{p_{1}(L)}^{N}Φ_{p_{1}(L)}^{∗} are all even, so Ind_{p_{1}(L)}^{N}\overline{Φ}_{p_{1}(L)}^{∗} = 0 in this case. This completes the proof of Lemma 6.9. End of the proof of Theorem 6.5 : First apply Lemma 6.7 in the case Q = P, and φ = Id_{P} : this shows that if G and H are two genetic bases of P, then ψ_{P,G} = ψ_{P,H}. In other words, the map ψ_{P,G} does not depend on the choice of the genetic basis G of P, and it can be denoted by ψ_{P}.

Now Lemma 6.7 shows that these maps ψ_{P} define a morphism of p-biset functors from K_{♭} to B_{♭}^{×}, and this morphism is an isomorphism. The other assertions in Theorem 6.5 follow from the structure of the functor B^{×} (see Theorem 8.4 of [6]).

6.10. Remark : The proof of Theorem 6.5 is rather complicated. The main reason is that I couldn’t find any obvious natural non zero morphism of biset functors from K_{♭} to B_{♭}^{×}. Once the existence of such a morphism is known, it is rather easy to prove that it has to be an isomorphism. Very recently, L. Barker announced to me that he has made an important step toward such a more conceptual construction, that should allow for substantial simplification of the present proof.

6.11. Remark : It follows from Theorem 6.5 that for p = 2, the class R of additive functors C_{p}^{lin} → Ab defined in Remark 5.6 is not closed under extensions : indeed, in the short exact sequence

0 → K/K_{δ} → B/K_{δ} → R_{Q} → 0 ,

the functors K/K_{δ} and R_{ℚ} are in R, but the functor R_{ℚ} = B/K is clearly the largest quotient of B with this property. So B/K_{δ} is not in the class R.

On the other hand, one can give the following alternative proof of the fact that the class of rational biset functors is closed under extensions : in a short exact sequence

0 → M → L → N → 0 ,

where M and N are rational p-biset functors, i.e. additive functors C_{p}/δ → Ab, a straightforward argument shows that the middle term L is a functor from the quotient category C_{p}/(δ × δ) of C_{p}, to Ab. And it turns out that the categories C_{p}/(δ × δ) and C_{p}/δ coincide : one can indeed find a subgroup Y of X^{3} = X × X^{2} such that

(X^{3}/Y) ×_{X^{2}} (δ × δ) = δ .

A tedious computation shows that using Notation 3.4, one can take

Y = {(xy^{−1}φ(x), x, y) | (x, y) ∈ X^{2}, xy^{−1} ∈ IZ} ,

where φ : X → J is any surjective homomorphism with kernel IZ.

References

[1] L. Barker. Rhetorical biset functors, rational p-biset functors, and their semisimplicity in characteristic zero. Preprint, 2006.

[2] S. Bouc. Foncteurs d’ensembles munis d’une double action. J. of Algebra, 183(0238):664–736, 1996.

[3] S. Bouc. Green-functors and G-sets, volume 1671 of Lecture Notes in Mathematics. Springer, October 1997.

[4] S. Bouc. The functor of rational representations for p-groups. Advances in Mathematics, 186:267–306, 2004.

[5] S. Bouc. Biset functors and genetic sections for p-groups. Journal of Algebra, 284(1):179–202, 2005.

[6] S. Bouc. The functor of units of Burnside rings for p-groups. Preprint, 2005, to appear in Comm. Math. Helv., 82, 2007.

[7] S. Bouc. The Dade group of a p-group. Inv. Math., 164:189–231, 2006.

[8] S. Bouc and J. Thévenaz. The group of endo-permutation modules. Invent. Math., 139:275–349, 2000.

[9] A. Dress. Contributions to the theory of induced representations, volume 342 of Lecture Notes in Mathematics, pages 183–240. Springer-Verlag, 1973.

[10] I. Hambleton, L. R. Taylor, and E. B. Williams. Detection theorems for K-theory and L-theory. J. Pure and Applied Algebra, 63:247–299, 1990.

[11] E. Yalçın. An induction theorem for the unit groups of Burnside rings of 2-groups. J. of Algebra, 289:105–127, 2005.

Serge BOUC CNRS - LAMFA - Université de Picardie-Jules Verne 33 rue St Leu - 80039 - Amiens Cedex 1 - France

email :  serge.bouc@u-picardie.fr