BOREL-SMITH FUNCTIONS AND THE DADE GROUP

SERGE BOUC AND ERGÜN YALÇIN

Abstract. We show that there is an exact sequence of biset functors over p-groups

0 → C_{b} ⟶^{j} B^{∗} ⟶^{Ψ} D^{Ω} → 0

where C_{b} is the biset functor for the group of Borel-Smith functions, B^{∗} is the dual of the Burnside ring functor, D^{Ω} is the functor for the subgroup of the Dade group generated by relative syzygies, and the natural transformation Ψ is the transformation recently introduced by the first author in [5]. We also show that the kernel of mod 2 reduction of Ψ is naturally equivalent to the functor B^{×} of units of the Burnside ring and obtain exact sequences involving the torsion part of D^{Ω}, mod 2 reduction of C_{b}, and B^{×}.

1. Introduction and Statement of Results

Let C denote the biset category for finite groups : it is defined as the category whose objects are finite groups, and where the morphism set Map_{C}(G, H) from the group G to the group H is the Grothendieck group Γ(H, G) of finite (H, G)-bisets, i.e. the free abelian group on the set of isomorphism classes of finite (H, G)-bisets, quotiented by the subgroup generated by elements of the form [U ⊔ V] − [U] − [V], where U ⊔ V is the disjoint union of the (H, G)-bisets U and V, and [U] denotes the isomorphism class of U. The composition of two morphisms is given by the bilinear map Γ(K, H) × Γ(H, G) → Γ(K, G) defined as the linear extension of the assignment (V, U) ↦ V ×_{H} U. A biset functor is an additive functor from C to the category Ab of abelian groups.

If p is a prime number, then the biset category over p-groups is the full subcategory of C whose objects are finite p-groups. It is denoted by C_{p}. An additive functor F : C_{p} → Ab is called a biset functor over p-groups or briefly a p-biset functor. More details about biset functors can be found in [2] (see also [5], [6], [7], or [9]).

An important example of a biset functor is the Burnside group functor which sends each finite group G to its Burnside group B(G), where the Burnside group B(G) is defined as the Grothendieck group of isomorphism classes of finite left G-sets. For each finite (H, G)- biset U, the group homomorphism B(U) : B(G) → B(H) is defined as the linear map sending the isomorphism class of the left G-set X to the isomorphism class of the left Hset U ×_{G} X. The other well-known examples are the representation ring functors R_{k} over a field k of characteristic 0 (equal to ℚ or ℝ in this paper). The biset functor structure for R_{k} is defined in a similar way to the Burnside ring using tensor product instead of cartesian product. We usually use the same notation for biset functors and for their associated p-biset functors obtained by restriction to the subcategory of p-groups.

Another interesting biset functor is the functor B^{×} of units of the Burnside ring. This is a functor which assigns to each group G, the unit group B(G)^{×} of the Burnside ring B(G). The biset functor structure for B^{×} is rather complicated, involving multiplicative induction instead of the usual induction. The details of this structure can be found in Section 5 of [8] and Section 2 of [14].

Date: December 21, 2006.

1991 Mathematics Subject Classification. Primary: 19A22; Secondary: 20C20, 57S17.

Key words and phrases. Burnside ring, Borel-Smith functions, Dade group, representation rings.

The second author is partially supported by TÜBİTAK through BDP program and by TÜBA through Young Scientist Award Program (TÜBA-GEBİP/2005-16).

In this paper, we are particularly interested in two other p-biset functors. One is the dual of the Burnside ring functor B^{∗} which assigns to each p-group P the dual group B^{∗}(P) = Hom(B(P), ℤ) and to each (Q, P)-biset U, the transpose of the linear map B(U^{op}) : B(Q) → B(P). Here U^{op} denotes the (P, Q)-biset which is isomorphic to U as a set and whose (P, Q)-action is given by g · u · h = h^{−1}ug^{−1}. The second functor is the functor D^{Ω} which sends each p-group P to the subgroup D^{Ω}(P) of the Dade group D_{k}(P) generated by relative syzygies. Here k can be taken as any field of characteristic p, and the choice does not affect the structure of the group D^{Ω}(P). The biset functor structure of D^{Ω} is described in [3]. The definition of tensor induction and verification of composition rule is particularly difficult. The following theorem is proved in [5].

Theorem 1.1 (Bouc [5]). There is a unique natural transformation Ψ : B^{∗} → D^{Ω} of p-biset functors with the property that

Ψ_{P} (ω_{X}) = Ω_{X}

for any finite p-group P and any finite P -set X.

The element ω_{X} ∈ B^{∗}(P) denotes the homomorphism B(P) → ℤ such that ω_{X}(P/H) is equal to 1 if H fixes a point on X and equal to 0 if it does not. The element Ω_{X} ∈ D^{Ω}(P) is the equivalence class of the endo-permutation module Δ_{k}(X), where Δ_{k}(X) is the kernel of the augmentation map ε : kX → k (in the cases where Δ_{k}(X) is not a capped endopermutation module, we take Ω_{X} = 0). The fact that Ψ_{P} is a well-defined homomorphism is a nontrivial fact requiring verification that both ω_{X} and Ω_{X} are subject to the same set of relations. We refer the reader to [3] and [5] for details.

Note that the dual of the Burnside ring B^{∗}(P) can be naturally identified with the group of super class functions C(P) where a super class function is a function from the set of subgroups of P to the integers which is constant on the conjugacy classes. The identification comes from the duality pairing C(P) × B(P) → ℤ defined by (f, [G/H]) = f(H). Under the identification B^{∗} ≅ C, the kernel of Ψ_{P} can be described as a subset of super class functions formed by super class functions satisfying a certain set of conditions. We observe that these conditions are exactly the same as the conditions known as the Borel-Smith conditions (see Definition 3.1). We obtain the following:

Theorem 1.2. The kernel of Ψ : B^{∗} → D^{Ω} is naturally equivalent to the biset functor C_{b} of Borel-Smith functions under the identification of B^{∗} with the functor C of super class functions. Hence, there is an exact sequence of p-biset functors of the form

0 → C_{b} ⟶^{j} B^{∗} ⟶^{Ψ} D^{Ω} → 0.

The Borel-Smith conditions are the conditions which the dimension function of a homotopy representation satisfies. This suggests that the exact sequence given in Theorem 1.2 has some geometric meaning. One probably needs to extend the concept of homotopy representation in a suitable way so that it includes G-CW-complexes which are homotopy equivalent to a wedge of spheres. If this can be done, then it may lead to a more natural description of the transformation Ψ. At this point we do not know how to do this and we leave it as an open problem.

In [5], the first author considers another subfunctor of B^{∗}, namely the dual of rational representations functor R_{ℚ}^{∗}. Note that by the Ritter-Segal theorem, the linearization map Lin_{ℚ} : B(P) → R_{ℚ}(P) is surjective for every p-group P, so the dual of the natural transformation Lin_{ℚ} gives an injective natural transformation i : R_{ℚ}^{∗} → B^{∗}. In [5], it is shown that the image of the natural transformation Ψ ∘ i is equal to the torsion part D_{tors}^{Ω} of D^{Ω}. This gives an exact sequence of the form

0 → R_{ℚ}^{∗} ⟶^{i} B^{∗} ⟶^{\overline{Ψ}} D^{Ω}/D_{tors}^{Ω} → 0

where \overline{Ψ} is the composition of Ψ with the quotient transformation D^{Ω} → D^{Ω}/D_{tors}^{Ω}.

The first step in our proof of Theorem 1.2 will be to show that j(C_{b}) is a subfunctor of i(R_{ℚ}^{∗}), so that after some identifications Theorem 1.2 is equivalent to the following :

Theorem 1.3. There is an exact sequence of p-biset functors of the following form

0 → C_{b} ⟶^{j} R_{ℚ}^{∗} \underrightarrow{Ψ̃} D_{tors}^{Ω} → 0

where Ψ̃ is the composition Ψ ∘ i.

To prove Theorem 1.3, we first observe that all the p-biset functors involved in the sequence are rational in the sense of Section 7 of [6]. Thus to show the exactness of this sequence, it is enough to check the exactness only at p-groups of normal p-rank one. Then, the proof follows from an inspection of this sequence in the case of these groups.

In the rest of the paper, we consider the mod 2 reduction of the exact sequence given in Theorem 1.2. Let us denote the mod 2 reduction 𝔽_{2} ⊗_{ℤ}F of a biset functor F by 𝔽_{2}F and the torsion group Tor_{1}^{ℤ}(𝔽_{2}, F) by Tor_{𝔽_{2}}F. Applying the mod 2 reduction to the sequence given in Theorem 1.2, we obtain a long exact sequence of the form

0 → Tor_{𝔽_{2}}D^{Ω} ⟶ 𝔽_{2}C_{b} ⟶ 𝔽_{2}B^{∗} ⟶^{Ψ_{2}} 𝔽_{2}D^{Ω} → 0

which gives us two short exact sequences involving the kernel of Ψ_{2}.

Theorem 1.4. The kernel of Ψ_{2} : 𝔽_{2}B^{∗} → 𝔽_{2}D^{Ω} is naturally equivalent to the functor B^{×} of units of the Burnside ring.

As an immediate consequence, we obtain

Corollary 1.5. The following sequences of p-biset functors are exact :

0 → B^{×} ⟶ 𝔽_{2}B^{∗} ⟶^{Ψ_{2}} 𝔽_{2}D^{Ω} → 0

0 → Tor_{𝔽_{2}}D^{Ω} → 𝔽_{2}C_{b} → B^{×} → 0.

The second sequence is closely related to the following exact sequence of p-biset functors recently given by the first author in [8] :

0 → B^{×} → 𝔽_{2}R_{ℚ}^{∗} → 𝔽_{2}D_{tors}^{Ω} → 0.

By taking Yoneda splice, we can view these sequences as parts of a long exact sequence of the form

0 → Tor_{𝔽_{2}}D^{Ω} → 𝔽_{2}C_{b} → 𝔽_{2}R_{ℚ}^{∗} \underrightarrow{Ψ̃_{2}} 𝔽_{2}D_{tors}^{Ω} → 0

where the kernel of the last natural transformation is equal to B^{×}. We prove in Section 5 that this long exact sequence is nothing but the mod 2 reduction of the exact sequence in Theorem 1.3 (see Proposition 5.3).

In [9], Bouc and Thévenaz gave an exact sequence of p-biset functors

0 → \overline{D}_{tors} → 𝔽_{2}R_{ℚ} → Γ_{𝔽_{2}} → 0

for odd primes, where Γ_{𝔽_{2}} is the constant functor with values 𝔽_{2} and \overline{D}_{tors} is a quotient of the torsion part of the Dade group. Later Carlson and Thévenaz ([11] Theorem 13.3) proved that actually \overline{D}_{tors} = D_{tors}. These two results together with Theorem 6.2 of [9] show that D_{tors} = D_{tors}^{Ω} when p is an odd prime. So Tor_{𝔽_{2}}D^{Ω} can be identified with the functor D_{tors}. In this case also B^{×} is naturally equivalent to the constant functor Γ_{𝔽_{2}}, and it is interesting to ask whether 𝔽_{2}C_{b} can be identified with 𝔽_{2}R_{ℚ} when p is an odd prime. We show this is true if and only if p is congruent to 3 modulo 4 (Theorem 5.4 and Remark 5.5). Thus, in this case, the second exact sequence given in Corollary 1.5 is the same as the exact sequence given by Bouc-Thévenaz in [9].

The paper is organized as follows : Section 2 is a quick exposition of the notion of rational p-biset functor. In Section 3, we give the definition of Borel-Smith functions and explain their basic properties. Section 4 is devoted to the proof of Theorem 1.2 and Theorem 1.3, and in Section 5, we prove Theorem 1.4 and discuss its consequences.

2. Rational p-biset functors

Since the notion of rational biset functor is an essential tool in the present paper, we will quickly recall the basic definitions and properties related to this particular class of p-biset functors.

2.1. Some particular bisets. Recall that the formalism of bisets allows for a unified description of the operations of induction, restriction, inflation, deflation, and transport by isomorphism :

• If H is a subgroup of the finite group G, the induction biset Ind_{H}^{G} is the (G, H)-biset equal to G as a set, with biset structure given by left multiplication by elements of G and right multiplication by elements of H. The restriction biset Res_{H}^{G} is the (H, G)-biset G, with biset structure given by left multiplication by elements of H and right multiplication by elements of G.

• If N is a normal subgroup of G, then the inflation biset Inf_{G/N}^{G} is the (G, G/N)-biset equal to G/N as a set, with right G/N-action by multiplication, and left G-action by projection onto G/N, and next multiplication in G/N. The deflation biset Def_{G/N}^{G} is the (G/N, G)-biset equal to G/N as a set, with left G/N-action by multiplication, and right G-action by projection and multiplication.

• If φ : G → G′ is a group isomorphism, then the transport by isomorphism biset Iso(φ) or Iso_{G}^{G′} is the (G′, G)-biset equal to G′ as a set, with left action of G′ by multiplication, and right action of G by first taking image by φ, and then multiplying in G′.

• If (T, S) is a section of G, i.e. if S \underline{◁} T ≤ G, we denote by Indinf_{T/S}^{G} the composition Ind_{T}^{G} ×_{T} Inf_{T/S}^{T}. As a (G, T/S) biset, it is isomorphic to the set G/S, with the obvious biset structure. Similarly, we denote by Defres_{T/S}^{G} the composition Def_{T/S}^{T} ×_{T} Res_{T}^{G}. As a (T/S, G)-biset, it is isomorphic to S\G, with the obvious biset structure.

• When F is a biset functor, and U is one of the above (H, G)-bisets, we will also denote by U the map F (U) : F (G) → F (H) : e.g., when (T, S) is a section of G, we will write Indinf_{T/S}^{G} for the map F (T/S) → F (G) obtained by composition of the maps Inf_{T/S}^{T} : F (T/S) → F (T) and Ind_{T}^{G} : F (T) → F (G).

• If G and H are finite groups, and U is an (H, G)-biset, then U^{op} denotes the (G, H)-biset equal to U as a set, with biset structure defined by

∀(g, u, h) ∈ G × U × H, g.u.h (in U^{op}) = h^{−1}ug^{−1} (in U) .

For example, one checks easily that if (T, S) is a section of the finite group G, then the (G, T/S)-bisets (Defres_{T/S}^{G})^{op} and Indinf_{T/S}^{G} are isomorphic.

• If F is a biset functor, the dual biset functor F^{∗} is the biset functor defined by F^{∗}(G) = Hom_{ℤ}(F (G), ℤ) for any finite group G, and by F^{∗}(U) = ^{t}F (U^{op}), for any finite (H, G)- biset U, where ^{t}F (U^{op}) denotes the transposed map of F (U^{op}).

In particular, if (T, S) is a section of the finite group G, if f ∈ F^{∗}(G), then Defres_{T/S}^{G}f is the element of F^{∗}(T/S) defined by

∀u ∈ F (T/S), (Defres_{T/S}^{G}f)(u) = f(Indinf_{T/S}^{G}u) .

2.2. Idempotents. Let G be a finite group, and N be a normal subgroup of G. It is easy to check that Def_{G/N}^{G} ×_{G} Inf_{G/N}^{G} is isomorphic to the identity (G/N, G/N)-biset G/N. It follows that the composition

j_{N}^{G} = Inf_{G/N}^{G} ×_{G/N} Def_{G/N}^{G}

is an idempotent endomorphism of G in the category C. Moreover, j_{1}^{G} is the identity of End_{C}(G), and if N and M are normal subgroups of G, then one checks easily that j_{M}^{G} ∘ j_{N}^{G} = j_{MN}^{G} in End_{C}(G). A classical inversion procedure now shows that if for N \underline{◁} G, we set

f_{N}^{G} = ∑_{{M \underline{◁} G \atop M⊇N}} μ_{\underline{◁} G}(N, M)j_{M}^{G} ,

we get a complete set of orthogonal (non primitive in general) idempotents in End_{C}(G), as N runs through the set of normal subgroups of G. Here μ_{\underline{◁} G} denotes the Möbius function of the poset of normal subgroups of G.

It follows in particular that for any biset functor F and any finite group G, the subgroup

∂F (G) = F (f_{1}^{G})(F (G))

is a direct summand of F (G). It is called the set of faithful elements of F (G). It is also the set of elements of F (G) mapping to zero by any proper deflation.

2.3. Genetic subgroups. Let p be a prime, and P be a finite p-group. A subgroup Q of P is called genetic if the following two conditions hold :

• The group N_{P} (Q)/Q has normal p-rank 1, i.e. all its abelian normal subgroups are cyclic.

• Let Z_{P} (Q) be the subgroup of N_{P} (Q) defined by

Z_{P} (Q)/Q = Z(N_{P} (Q)/Q) .

Then for any x ∈ P, the intersection Q^{x} ∩ Z_{P} (Q) is contained in Q if and only if Q^{x} = Q.

Two genetic subgroups Q and R are said to be linked modulo P (notation Q _{P} R) if there exist elements x and y in P such that Q^{x} ∩ Z_{P} (R) ⊆ R and R^{y} ∩ Z_{P} (Q) ⊆ Q. It was shown in [4] that this is an equivalence relation on the set of genetic subgroups of P, and that the equivalence classes are in one to one correspondence with the isomorphism classes of rational irreducible representations of P.

A genetic basis is a set of representatives of genetic subgroups of P for the relation _{P} .

2.4. Rational p-biset functors. Let p be a prime, and let F be a p-biset functor. If P is a finite p-group, and G is a genetic basis of P, one can show ([6] Theorem 3.2, see also Remark 4.6 of [8]) 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.

The p-biset functor F is called rational if for any p-group P, there exists a genetic basis G of P such that the map I_{G} is an isomorphism. Equivalently, for any genetic basis I_{G} of P, the map I_{G} is an isomorphism.

The use of the word rational here comes from the fact that R_{ℚ} is a rational p-biset functor (Example 7.2 of [6]). Rational p-biset functors have two important properties : the first one is that by definition, proving a result involving only rational p-biset functors and morphisms between them, generally amounts to checking that the desired result holds when evaluated at p-groups of normal p-rank 1. The second one is that the full subcategory of the abelian category of p-biset functors, whose objects are rational p-biset functors, is a Serre subcategory, i.e. if F′ ⊆ F are p-biset functors, then F is rational if and only if F′ and F/F′ are. Moreover, any dual functor of a rational p-biset functor is rational (see Proposition 7.4 of [6] for details).

3. Borel-Smith functions

Let G be a finite group, and let C(G) denote the set of super class functions. Recall that a super class function is a function from the set of subgroups of G to integers which is constant on conjugacy classes. Borel-Smith functions are defined as follows:

Definition 3.1. A function f ∈ C(G) is called a Borel-Smith function if it satisfies the following conditions:

(i) If H ⊴ L ≤ G, L/H ≅ ℤ/pℤ, and p is odd, then f(H) − f(L) is even.

(ii) If H ⊴ L ≤ G, L/H ≅ ℤ/pℤ × ℤ/pℤ, H_{i}/H the subgroups of order p in L/H, then

f(H) − f(L) = ∑_{i=0}^{p}(f(H_{i}) − f(L)).

(iii) If H⊴L⊴N ≤ N_{G}(H) and L/H ≅ ℤ/2ℤ, then f(H)−f(L) is even if N/H ≅ ℤ/4ℤ, and f(H) − f(L) is divisible by 4 if N/H is the quaternion group of order 8.

These conditions are usually referred as Borel-Smith conditions. They were first discovered as the conditions satisfied by the dimension function of a homology mod p sphere with a G-action. The set of Borel-Smith functions is an additive subgroup of C(G) which we denote by C_{b}(G) (see page 210 in [12] for more details).

Remark 3.2. Condition (iii) is usually stated in stronger terms, since one requires that f(H) − f(L) should be divisible by 4 for any H ⊴ L ⊴ N ≤ N_{G}(H) such that N/H is a generalized quaternion group of order 2^{k}, for k ≥ 3. But clearly, it is enough to consider the quaternion group of order 8, since any larger quaternion group contains such a group of order 8, which contains its unique subgroup of order 2.

Given a real representation V, we define the super class function DimV as the function with values DimV (H) = dim_{ℝ} V^{H} for all H ≤ G. It is an easy exercise to show that the dimension function of a real representation satisfies the Borel-Smith conditions, and hence DimV is a Borel-Smith function. The key result on Borel-Smith functions is that when G is a nilpotent group, for every f ∈ C_{b}(G), there exist real representations V and W such that f = DimV − DimW. In particular, we have the following:

Theorem 3.3. (Theorem 5.4 on page 211 of [12]) Let G be a nilpotent group, and let R_{ℝ}(G) denote the real representation ring of G. Consider the group homomorphism Dim : R_{ℝ}(G) → C(G) defined as the linear extension of the assignment V ↦ DimV. Then, the image of Dim is exactly equal to the group of Borel-Smith functions C_{b}(G).

The assignment G ↦ C_{b}(G) together with appropriate action of bisets is a biset functor, and the assignment V ↦ DimV is a morphism of biset functors. To show this, we first need to describe the biset functor structure of the group of super class functions, and for this we will identify C(G) with B^{∗}(G).

Recall that the biset functor B^{∗}, the dual of the Burnside group functor, is defined as the functor which sends every finite group G to B^{∗}(G) = Hom(B(G), ℤ), and every morphism U ∈ Γ(H, G), to the homomorphism B^{∗}(U) : B^{∗}(G) → B^{∗}(H) where B^{∗}(U) is the transpose of the linear map B(U^{op}) : B(H) → B(G). Now the group C(G) of super class functions can be identified with B^{∗}(G) via the duality pairing C(G) × B(G) → ℤ defined by (f, [G/L]) = f(L), and we will freely use this identification throughout the paper. In particular, we will use it for considering the assignment G ↦ C(G) as a biset functor. It is easy to check that if G and H are finite groups, if U is a finite (H, G)-biset, and if f ∈ C(G), then the value of the superclass function C(U)(f) at the subgroup K of H is equal to

(3.4)  C(U)(f)(K) = ∑_{u∈[K\U/G]} f(K^{u}) ,

where [K\U/G] is a set of representatives of (K, G)-orbits on U, and K^{u} is the subgroup of G defined by

K^{u} = {g ∈ G | ∃k ∈ K, ku = ug} .

Note that for a real representation V of G, the value of the element DimV of B^{∗}(G) on X ∈ B(G) is equal to dim_{ℚ} Hom_{ℚG}(ℚX, V). It follows easily that the assignment V ↦ DimV is a morphism of biset functors from R_{ℝ} to B^{∗}.

Notation 3.5. If H is a finite p-group, define an element ε_{H} of B(H) by

ε_{H} = ∑_{E≤Ω_{1}Z(H)} μ(1, E) [H/E] ,

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

It follows easily from this definition that Def_{H/N}^{H}ε_{H} = 0 for any nontrivial normal subgroup N of H (for details, note that with the notation of Lemma 3.12 and Remark 3.13 of [7], one has that ε_{H} = f_{1}^{H}[H/1] in B(H)).

Notation 3.6. Let Ξ denote the class of p-groups which are cyclic of order p with p > 2, cyclic of order 4, quaternion of order 8, or elementary abelian of rank 2. If H ∈ Ξ, define the integer m_{H} by m_{H} = 2 if H is cyclic, by m_{H} = 4 if H is quaternion, and by m_{H} = 0 if H is elementary abelian of rank 2.

With this notation, we can rephrase the Borel-Smith conditions in the following way : observe first that ε_{H} = H/1 − H/Z, if H is a nontrivial cyclic p-group or a generalized quaternion 2-group, where Z is the unique central subgroup of order of p, and that

ε_{H} = H/1 − ∑_{|H:K|=p} H/K + pH/H = (H/1 − H/H) − ∑_{|H:K|=p}(H/K − H/H) ,

if H ≅ ℤ/pℤ × ℤ/pℤ. Now the Borel-Smith conditions can be expressed by saying that an element f ∈ B^{∗}(G) is in C_{b}(G) if and only if (Defres_{T/S}^{G}f)(ε_{T/S}) ∈ m_{T/S}ℤ, whenever (T, S) is a section of G such that T/S ∈ Ξ. This leads to the following :

Proposition 3.7. For a finite group G, let C_{b}(G) denote the group of Borel-Smith functions. Set

β(G) = {f ∈ B^{∗}(G) | ∀H ∈ Ξ, ∀ψ ∈ Map_{C}(G, H), B^{∗}(ψ)(f)(ε_{H}) ∈ m_{H}ℤ} .

Then β(G) = C_{b}(G), up to the identification B^{∗}(G) = C(G). In particular, the assignment G ↦ C_{b}(G) ⊆ C(G) defines a subfunctor of C.

Proof. First β(G) ⊆ C_{b}(G), because if (T, S) is a section of G with T/S ∈ Ξ, then the set S\G is a (T/S, G)-biset, i.e. an element of Map_{C}(G, T/S), whose action is precisely Defres_{T/S}^{G}.

Conversely, let G be any finite group, and let f ∈ C_{b}(G). We prove that for any H ∈ Ξ and any ψ ∈ Map_{C}(G, H), the value B^{∗}(ψ)(f)(ε_{H}) is a multiple of m_{H}.

We can assume that ψ is some transitive (H, G)-biset, so that, by [2] Lemme 3, there exists a section (Y, X) of H and a section (T, S) of G, and a group isomorphism θ : T/S → Y/X with

B^{∗}(ψ)(f) = Indinf_{Y/X}^{H}Iso(θ)Defres_{T/S}^{G}f ,

thus

B^{∗}(ψ)(f)(ε_{H}) = (Iso(θ)Defres_{T/S}^{G}f)(Defres_{Y/X}^{H}ε_{H}) .

If the section (Y, X) is the section (H, 1), then

B^{∗}(ψ)(f)(ε_{H}) = (Iso(θ)Defres_{T/S}^{G}f)(Defres_{Y/X}^{H}ε_{H}) = (Defres_{T/S}^{G}f)(Iso(θ^{−1})ε_{H})
= (Defres_{T/S}^{G}f)(ε_{T/S}) ∈ m_{H}ℤ ,

since f ∈ C_{b}(G) and T/S ≅ H. So we can assume that (Y, X) is a proper section of H.

Suppose first that H ≅ ℤ/pℤ × ℤ/pℤ. Then it is easily checked that any proper restriction and any proper deflation of ε_{H} is equal to 0. So B^{∗}(ψ)(f)(ε_{H}) = 0 as was to be shown in this case.

Now, if H is cyclic, then m_{H} = 2. If H has odd prime order p, then the proper deflation Def_{H/H}^{H}ε_{H} is zero, and the proper restriction Res_{1}^{H}ε_{H} is (p − 1)1/1, which is a multiple of 2. So B^{∗}(ψ)(f)(ε_{H}) is even. If H has order 4, then any proper deflation of ε_{H} is zero, and the restriction of ε_{H} to its subgroup K of order 2 is 2(K/1 − K/K), again a multiple of m_{H}. It follows that the restriction of ε_{H} to the trivial group is also a multiple of m_{H}. Hence B^{∗}(ψ)(f)(ε_{H}) is a multiple of m_{H} in this case.

Finally, suppose H is the quaternion group of order 8. Since any proper deflation of ε_{H} is zero and every subgroup of H is normal,

Defres_{Y/X}^{H}ε_{H} = Res_{Y/X}^{H/X}Def_{H/X}^{H}ε_{H} = 0

for every section (Y, X) with X ≠ 1. So, we can assume X = 1. Note that for every subgroup Y of H, we have Res_{Y}^{H}ε_{H} = 2^{|H:Y |}ε_{Y} which is obviously a multiple of m_{H} = 4 if |H : Y | ≥ 4. In the case |H : Y | = 2, we have Y ≅ ℤ/4ℤ, and hence

B^{∗}(ψ)(f)(ε_{H}) = 2(Defres_{T/S}^{G}f)(ε_{T/S})

is multiple of 4 since (Defres_{T/S}^{G}f)(ε_{T/S}) is even, for f ∈ C_{b}(G) and T/S ≅ ℤ/4ℤ. Hence β(G) = C_{b}(G).

Now if G and G′ are two finite groups, and if φ ∈ Map_{C}(G, G′), and if f ∈ β(G), then for any H ∈ Ξ and any ψ ∈ Map_{C}(G′, H), one has that

B^{∗}(ψ)B^{∗}(φ)(f)(ε_{H}) = B^{∗}(ψφ)(f)(ε_{H}) ∈ m_{H}ℤ ,

since ψφ ∈ Map_{C}(G, H) and f ∈ β(G). Thus B^{∗}(φ)(f) ∈ β(G′), and β is a biset subfunctor of B^{∗}. This completes the proof. □

4. Proof of Theorem 1.2 and Theorem 1.3

Let P be a p-group, and k be a field of characteristic p. Given a non empty finite P -set X, consider the kG-module Δ_{k}(X) = ker{ε : kX → k} where ε is the k-linear homomorphism which takes every element x ∈ X to 1 ∈ k. It has been shown by Alperin [1] that Δ_{k}(X) is an endo-permutation module which is capped in most of the cases (when P does not have a single fixed point on X). Recall that a kP -module M is called an endopermutation module if End_{k}(M) is a permutation module, and it is called capped if it has an indecomposable summand with vertex P. When Δ_{k}(X) is a capped module, then we define the relative syzygy Ω_{X} as the equivalence class of the endo-permutation module Δ_{k}(X) in the Dade group D_{k}(P) (see [7] for the definition of the Dade group). When Δ_{k}(X) is not capped, or when X = ∅, we take Ω_{X} = 0. The group D^{Ω}(P) is defined to be the subgroup of the Dade group D_{k}(P) generated by relative syzygies.

By Theorem 1.1, there is a surjective natural transformation Ψ : B^{∗} → D^{Ω} of bisets functors which is defined by

Ψ_{P} (ω_{X}) = Ω_{X}

for any finite p-group P and any finite P -set X. There is also an injective natural transformation j : C_{b} → B^{∗} defined as the composition of the inclusion of the subfunctor C_{b} into C with the identification C ≅ B^{∗}.

Recall that the transpose of the linearization map B → R_{ℚ} is an injection i : R_{ℚ}^{∗} → B^{∗}. It was shown in Theorem 1.8 of [5] that the image of R_{ℚ}^{∗}(P) by the map Ψ_{P} is precisely the torsion part D_{tors}^{Ω}(P) of D^{Ω}(P).

Our proof of Theorem 1.2 is as follows : we first show that C_{b} is a subfunctor of the image of R_{ℚ}^{∗} in B^{∗}. In other words, the injection j : C_{b} → B^{∗} factors through an injection we also denote by j : C_{b} → R_{ℚ}^{∗}, so Theorem 1.2 is equivalent to Theorem 1.3, i.e. to the exactness of the following sequence of biset functors

(4.1)  0 → C_{b} ⟶^{j} R_{ℚ}^{∗} \underrightarrow{Ψ̃} D_{tors}^{Ω} → 0

where Ψ̃ = Ψ ∘ i.

Now all functors in this sequence are rational p-biset functors in the sense of Section 7 of [6]. In particular, the evaluation of this sequence at some p-group P is determined in a precise way by its evaluations at p-groups of normal p-rank one, and the proof of its exactness comes down to an inspection of this sequence in the case of these groups.

Lemma 4.2. Let P be a p-group, and f ∈ B^{∗}(P). Then f ∈ R_{ℚ}^{∗}(P) if and only if for any section (T, S) of P with T/S ≅ ℤ/pℤ × ℤ/pℤ, one has that (Defres_{T/S}^{P}f)(ε_{T/S}) = 0, i.e.

f(P/S) − f(P/T) = ∑_{S<X<T} (f(P/X) − f(P/T)) .

Proof. The proof is similar to the proof of Lemma 3.2 of [5] (which gives another more complicated criterion for f to belong to R_{ℚ}^{∗}(P)), and we refer to this lemma for details. Since R_{ℚ}(P) and B(P) are free abelian groups, the commutative diagram

R_{ℚ}^{∗}(P) ⟶ B^{∗}(P)
↓  ↓
ℚ ⊗_{ℤ} R_{ℚ}^{∗}(P) ⟶ ℚ ⊗_{ℤ} B^{∗}(P)

is a pullback diagram, where all the maps are injective. Moreover ℚ ⊗_{ℤ}R_{ℚ}^{∗}(P) = ℚR_{ℚ}^{∗}(P) identifies with Hom_{ℚ}(ℚR_{ℚ}(P), ℚ), and ℚB^{∗}(P) identifies with Hom_{ℚ}(ℚB(P), ℚ). So the assertion of the lemma is equivalent to the same assertion with R_{ℚ}^{∗}(P) replaced by ℚR_{ℚ}^{∗}(P) and B^{∗}(P) replaced by ℚB^{∗}(P).

Now ℚB(P) has a basis over ℚ consisting of its primitive idempotents e_{Q}^{P}, which are indexed by subgroups Q of P up to P -conjugation. The kernel of the linearization map ℚB(P) → ℚR_{ℚ}(P) consists of the linear combinations of idempotents e_{Q}^{P} indexed by noncyclic subgroups Q of P. Thus an element f of ℚB^{∗}(P) lies in ℚR_{ℚ}^{∗}(P) if and only if f(e_{Q}^{P}) = 0 for any noncyclic subgroup Q of P.

Suppose first that f ∈ ℚR_{ℚ}^{∗}(P). Then since ℚR_{ℚ}^{∗} is a biset subfunctor of ℚB^{∗}, it follows that Defres_{T/S}^{P}f ∈ ℚR_{ℚ}^{∗}(T/S) for any section (T, S) of P. In particular if T/S

is elementary abelian of rank 2, this amounts to saying that (Defres_{T/S}^{P}f)(e_{T/S}^{T/S}) = 0, since the only noncyclic subgroup of T/S is T/S itself. Moreover one checks easily that ε_{T/S} = pe_{T/S}^{T/S}. Hence (Defres_{T/S}^{P}f)(ε_{T/S}) = 0.

Conversely, suppose that this condition holds for any section (T, S) of P such that T/S is elementary abelian of rank 2. Then (Defres_{T/S}^{P}f)(e_{T/S}^{T/S}) = 0. We prove that f(e_{Q}^{P}) = 0 for any noncyclic subgroup Q of P by induction on |Q|. If |Q| = p^{2}, then Q is elementary abelian of rank 2, so (Res_{Q}^{P}f)(e_{Q}^{Q}) = 0, using the hypothesis for the section (Q, 1) of P. But (Res_{Q}^{P}f)(e_{Q}^{Q}) = f(Ind_{Q}^{P}e_{Q}^{Q}), and Ind_{Q}^{P}e_{Q}^{Q} is a nonzero multiple of e_{Q}^{P}. Thus f(e_{Q}^{P}) = 0 in this case, and this starts induction.

Suppose that Q is a noncyclic subgroup of P, such that f(e_{X}^{P}) = 0 for any noncyclic subgroup X of P with |X| < |Q|. Choose a normal subgroup S of Q such that Q/S is elementary abelian of rank 2. Such a subgroup exists since Q is noncyclic. Then

0 = (Defres_{Q/S}^{P}f)(e_{Q/S}^{Q/S}) = f(Ind_{Q}^{P}Inf_{Q/S}^{Q}e_{Q/S}^{Q/S}) .

Moreover Inf_{Q/S}^{Q}e_{Q/S}^{Q/S} = ∑_{X} e_{X}^{Q}, where the summation is over all subgroups X of Q for which

XS = Q, up to Q-conjugation. Such subgroups are noncyclic, since X/X ∩ S ≅ Q/S is noncyclic, and all of them except Q itself have order less than |Q|. Now Ind_{Q}^{P}e_{X}^{Q} is a nonzero multiple of e_{X}^{P}, thus f(Ind_{Q}^{P}e_{X}^{Q}) = 0 for X ≠ Q. It follows that f(Ind_{Q}^{P}e_{Q}^{Q}) = 0, hence f(e_{Q}^{P}) = 0, completing the inductive step of the proof. □

Corollary 4.3. Let P be a p-group. Then C_{b}(P) ⊆ R_{ℚ}^{∗}(P). So C_{b} is a p-biset subfunctor of R_{ℚ}^{∗}. In particular, the functor C_{b} is a rational p-biset functor.

Proof. The inclusion C_{b}(P) ⊆ R_{ℚ}^{∗}(P) follows from the lemma, and from the second Borel-Smith condition. So C_{b} is a p-biset subfunctor of R_{ℚ}^{∗}, hence it is rational, since R_{ℚ} is rational, and since the dual as well as any subfunctor of a rational p-biset functor are rational. □

Proof of Theorem 1.3. Proving that the sequence (4.1) is exact amounts to showing that for each p-group P, the sequence

0 → C_{b}(P) ⟶^{j_{P}} R_{ℚ}^{∗}(P) \underrightarrow{Ψ̃_{P}} D_{tors}^{Ω}(P) → 0

is an exact sequence of abelian groups. Choose a genetic basis G of P. We have a diagram

0 →  C_{b}(P)  ⟶^{j_{P}}  R_{ℚ}^{∗}(P)  \underrightarrow{Ψ̃_{P}}  D_{tors}^{Ω}(P)  → 0
↑  ↑  ↑
0 → ⊕_{Q∈G} ∂C_{b}(N_{P} (Q)/Q) → ⊕_{Q∈G} ∂R_{ℚ}^{∗}(N_{P} (Q)/Q) → ⊕_{Q∈G} ∂D_{tors}^{Ω}(N_{P} (Q)/Q) → 0

where the vertical arrows are the maps ⊕_{Q∈G} Indinf_{N_{P} (Q)/Q}^{P}, and where for any p-biset

functor F and any p-group P, the group ∂F (R) is the subgroup of faithful elements of F (R), i.e. the image of the map F (f_{1}^{R}) associated to the idempotent f_{1}^{R} ∈ End_{C}(R).

The existence of the bottom horizontal maps in this diagram, and the fact that this diagram is commutative, follow from the fact that the maps j and Ψ̃ are maps of pbiset functors. The vertical maps are, moreover, isomorphisms, because all three p-biset functors are rational : the functor R_{ℚ}^{∗} is dual to a rational p-biset functor, the functor C_{b} is a subfunctor of R_{ℚ}^{∗}, and the functor D_{tors}^{Ω} is a quotient of R_{ℚ}^{∗}, since Ψ̃ : R_{ℚ}^{∗} → D_{tors}^{Ω} is surjective.

In other words, the above diagram is an isomorphism from the bottom line to the top one. Moreover, the bottom line is the direct sum of sequences

(4.4)  0 → ∂C_{b}(S) ⟶^{j_{S}} ∂R_{ℚ}^{∗}(S) \underrightarrow{Ψ̃_{S}} ∂D_{tors}^{Ω}(S) → 0 ,

where S = N_{P} (Q)/Q for Q ∈ G. So all we have to do is to check that this sequence is an exact sequence when S is a group of normal p-rank one, and we do this by a case by case inspection. Let S be a p-group of normal p-rank one. Recall that :

• The group S is cyclic if p ≠ 2, or cyclic, generalized quaternion, dihedral of order at least 16, or semi-dihedral if p = 2.

• (see [14] or [4]) If S is nontrivial, then there is a unique subgroup Z of order p in the center of S. If Q is a subgroup of S not containing Z, then Q = 1 if S is cyclic or generalized quaternion, or |Q| ≤ 2 if S is dihedral or semidihedral. The noncentral subgroups of order 2 form a single conjugacy class of subgroups of S if S is semidihedral, and two conjugacy classes of subgroups if S is dihedral.

• (Proposition 3.7 of [4]) The group S has a unique faithful irreducible rational representation Φ_{S}.

• (Theorem 10.3 of [10]) The group ∂D_{tors}^{Ω}(S) is : – trivial if |S| ≤ 2, or if S is dihedral.

– of order 2 if S is cyclic of order at least 3, generated by Ω_{S/1}.

– cyclic of order 4 if S is generalized quaternion, generated by Ω_{S/1}.

– of order 2 if S is semidihedral, generated by Ω_{S/1} + Ω_{S/I}, where I is a non-

central subgroup of order 2 of S.

Now we observe that for any p-group S, the group ∂R_{ℚ}^{∗}(S) is the subgroup of R_{ℚ}^{∗}(S) with basis the elements V^{∗}, where V is a faithful rational irreducible representation of S, and V^{∗} is the element of B^{∗}(S) defined by setting V^{∗}(S/R) to be equal to the multiplicity m(V, ℚS/R) of V as a summand of ℚS/R, for any subgroup R of S. It follows that if S has normal p-rank one, then ∂R_{ℚ}^{∗}(S) = ℤΦ_{S}^{∗}.

Now for any finite p-group S, the group ∂B^{∗}(S) is the group of linear forms B(S) → ℤ which map to 0 by any proper deflation. But if N \underline{◁} S and f ∈ B^{∗}(S), then for any subgroup X/N of S/N one has that

(Def_{S/N}^{P}f)((S/N)/(X/N)) = f(S/X) .

This means that f ∈ ∂B^{∗}(S) if and only if f(S/X) = 0 whenever X contains a nontrivial normal subgroup of S, or equivalently since S is a p-group, if X intersects the center of S nontrivially. The group ∂C_{b}(S) consists of linear forms which satisfy this condition, together with the Borel-Smith conditions.

Suppose first that S is cyclic of order at most 2. In this case the Borel-Smith conditions are void, so an element f of ∂C_{b}(S) has only one possibly nonzero value, namely f(S/1), and this value is arbitrary. In other words ∂C_{b}(S) = ∂R_{ℚ}^{∗}(S) in this case. But D_{tors}^{Ω}(S) = 0 in this case, so the sequence (4.4) is exact.

Now suppose that S is cyclic of order at least 3 or generalized quaternion. Then any nontrivial subgroup of S intersects the center of S nontrivially. An element f in ∂B^{∗}(S) has only one possibly nonzero value, namely f(S/1). Thus ∂B^{∗}(S) = ℤω_{S/1}. This is also equal to ∂R_{ℚ}^{∗}(S), since the conditions of Lemma 4.2 are trivially true for f = ω_{S/1} in this case.

Now f ∈ ∂C_{b}(S) if and only if the additional Borel-Smith condition coming from a section (T, 1) is fulfilled, where T is cyclic of prime order if S is cyclic of odd order, or cyclic of order 4 if S is a cyclic 2-group, or quaternion of order 8 if S is generalized quaternion, i.e. if f(S/1) is a multiple of 2 if S is cyclic, or 4 if S is generalized quaternion.

This shows that ∂C_{b}(S) is generated by m_{S}ω_{S/1}, where m_{S} = 2 if S is cyclic, or 4 if S is generalized quaternion. But Ψ̃_{S}(ω_{S/1}) = Ω_{S/1}, and the order of Ω_{S/1} is precisely 2 if S is cyclic of order at least 3, or 4 if S is generalized quaternion. Hence the sequence (4.4) is again exact in this case.

Now if S is dihedral of order at least 16, and f ∈ ∂B^{∗}(S), the only possibly nonzero values of f are f(S/1), f(S/I), and f(S/J), where I and J are noncentral subgroups of order 2 of S, not conjugate in S. The Borel-Smith condition coming from the section (E, 1), where E is an elementary abelian subgroup of rank 2 of S containing I, gives f(S/1) = 2f(S/I), because E contains exactly 2 conjugates of I in S. Similarly f(S/1) = 2f(S/J), hence f(S/I) = f(S/J). Thus f ∈ R_{ℚ}^{∗}(S) if and only if f(S/1) = 2f(S/I) = 2f(S/J). The only other nontrivial Borel-Smith condition comes from the section (T, 1), where T is the subgroup of order 4 in S. This condition gives that f(S/1) is even, but this follows from the previous conditions. This shows that ∂C_{b}(S) = ∂R_{ℚ}(S) in this case, generated by the linear form whose nonzero values are 1 at S/I and S/J, and 2 at S/1. But ∂D_{tors}^{Ω}(S) = 0 if S is dihedral, and the sequence (4.4) is exact in this case.

Finally if S is semi-dihedral, and f ∈ B^{∗}(S), then the only possibly nonzero values of f are f(S/1) and f(S/I), where I is a noncentral subgroup of order 2 in S. Now f ∈ ∂R_{ℚ}^{∗}(P) if and only it satisfies the Borel-Smith condition obtained for the section (E, 1), where E is the elementary abelian subgroup of rank 2 containing I. This condition gives f(S/1) = 2f(S/I) as in the dihedral case. Hence the generator Φ_{S}^{∗} of ∂R_{ℚ}^{∗}(S) has value 1 at S/I, 2 at S/1, and zero anywhere else. Now f ∈ ∂C_{b}(S) if and only if the additional Borel-Smith condition coming from the section (T, 1) is fulfilled, where T is a quaternion subgroup of order 8 in S. This condition on f is that f(S/1) is a multiple of 4. So ∂C_{b}(S) is generated by 2Φ_{S}^{∗}. But also Φ_{S}^{∗} = ω_{S/I} + ω_{S/1} in this case, since the multiplicity of Φ_{S} as a summand of ℚP/I and ℚP/1 is equal to 1 and 2, respectively (Lemma 4.1 of [6]). Thus Ψ̃_{S}(Φ_{S}^{∗}) = Ω_{S/I} + Ω_{S/1}, which has precisely order 2 in D^{Ω}(S). Hence the sequence (4.4) is again exact in this case, and this completes the proof of Theorem 1.3, hence also of Theorem 1.2. □

Remark 4.5. It follows from Theorem 1.2 that for any p-group P, the map

∏_{(T,S)∈Ξ(P)} Defres_{T/S}^{P} : D^{Ω}(P) → ∏_{(T,S)∈Ξ(P)} D^{Ω}(T/S)

is injective, where Ξ(P) is the set of sections (T, S) of P such that T/S is in Ξ. This is a weak form of the detection theorem proved by Carlson and Thévenaz for the whole Dade group (Theorem 13.1 of [11]). Conversely, one can give an alternative proof of Theorem 1.2 based on this detection theorem, which comes down to examining the cases of groups in Ξ.

5. Proof of Theorem 1.4

Let G be a finite group. In our identification C(G) ≅ B^{∗}(G), we have until now only considered the additive group structure on C(G). But C(G) has also a natural ring structure, given by pointwise multiplication of superclass functions. We will now consider the group C(G)^{×} of units of the ring C(G).

Of course, since the units group of the ring ℤ is {±1}, an element f of C(G) is in C(G)^{×} if and only if f takes values in {±1}. It follows that C(G)^{×} is an elementary abelian 2group of rank equal to the number of conjugacy classes of subgroups of G. Hence if we define the exponential map γ : C(G) → C(G)^{×} by

γ(f)(K) = (−1)^{f(K)}

for any f ∈ C(G) and any subgroup K of G, we get a surjective group homomorphism C(G) → C(G)^{×}, which factors as

γ : C(G) → 𝔽_{2}C(G) →^{γ̄} C(G)^{×}

where the left map is the mod 2 reduction C(G) → 𝔽_{2}C(G), and the map γ̄ is a group isomorphism 𝔽_{2}C(G) → C(G)^{×}.

We can now use this isomorphism to endow the assignment G ↦ C(G)^{×} with a biset functor structure, since 𝔽_{2}C ≅ 𝔽_{2}B^{∗} is a biset functor. We denote this biset functor by C^{×} : in other words, for any finite group G, we set C^{×}(G) = C(G)^{×}. If G and H are finite groups, if U is a finite (H, G)-biset, and if f ∈ C^{×}(G), then the value at the subgroup K of H of the superclass function C^{×}(U)(f) is obtained by the following formula, similar to 3.4 :

C^{×}(U)(f)(K) = ∏_{u∈[K\U/G]} f(K^{u}) .

This formula shows in particular that the embedding of B^{×}(G) into C^{×}(G) given by the Mark homomorphism (also called the ghost map) is a morphism of biset functors (see Section 5 of [8] for details).

Now composing γ : C(G) → C^{×}(G) with the dimension function Dim : R_{ℝ}(G) → C(G), we obtain the homomorphism

Θ = γ ∘ Dim : R_{ℝ}(G) → C^{×}(G)

which is known as tom Dieck’s homomorphism. It is shown by tom Dieck that the image of Θ lies in B^{×}(G). Moreover, by a result of Tornehave [13] it is known that tom Dieck’s homomorphism is surjective onto B^{×}(G) when G is a p-group (see also [14] for an alternative proof). One of the consequences of Tornehave’s result is the following :

Proposition 5.1. Let P be a p-group. Then, γ(C_{b}(P)) = B^{×}(P).

Proof. First we show that γ(C_{b}(P)) lies in B^{×}(P). Take u = γ(f) with f ∈ C_{b}(P). Since the image of Dim is equal to C_{b}(P) when P is a p-group, there is a virtual real representation ξ ∈ R_{ℝ}(P) such that Dim(ξ) = f. This gives u = γ(f) = Θ(ξ), so u is in B^{×}(P) as desired. Now, the equality imγ = B^{×}(P) follows from Tornehave’s result that Θ is surjective onto B^{×}(P) for a p-group. □

Now, we are ready to prove Theorem 1.4.

Proof of Theorem 1.4. Let P be a p-group. Consider the following commuting diagram

0 ⟶ C_{b}(P) ⟶^{j_{P}} C(P) ⟶^{Ψ_{P}} D^{Ω}(P) ⟶ 0
↓^{γ|_{C_{b}(P)}}  ↓^{γ}  ↓
0 ⟶ B^{×}(P) ⟶ C^{×}(P) ⟶ Q(P) ⟶ 0

where the first exact sequence is the one given in Theorem 1.2 and Q(P) denotes the quotient group C^{×}(P)/B^{×}(P). Taking the mod 2 reduction of the top sequence, we obtain

0 ⟶ 𝔽_{2}C_{b}(P)/Tor_{𝔽_{2}}D^{Ω}(P) ⟶ 𝔽_{2}C(P) ⟶^{Ψ_{2}} 𝔽_{2}D^{Ω}(P) ⟶ 0
↓^{γ̄|_{C_{b}(P)}}  ↓γ̄  ↓
0 ⟶  B^{×}(P)  ⟶ C^{×}(P) ⟶ Q(P) ⟶ 0 .

Since γ̄|_{C_{b}(P)} is surjective and γ̄ is an isomorphism, all the vertical maps are isomorphisms. Hence,

ker Ψ_{2} ≅ 𝔽_{2}C_{b}(P)/Tor_{𝔽_{2}}D^{Ω}(P) ≅ B^{×}(P).

This completes the proof. □

Remark 5.2. One of the consequences of Theorem 1.4 is that for every p-group P, there is an exact sequence of elementary abelian groups (𝔽_{2}-vector spaces) of the form

0 → B^{×}(P) → C^{×}(P) → 𝔽_{2}D^{Ω}(P) → 0

where the first map is the ghost map. In [14], it has been shown that B^{×}(P) can be characterized as the subspace of C^{×}(P) satisfying certain conditions called Yoshida conditions (see Corollary 2.3 of [14]) and that these conditions can be viewed as a set of 𝔽_{2}-linear forms coming from certain subquotients of P. The exact sequence above gives that 𝔽_{2}D^{Ω}(P) has a presentation by Yoshida conditions as an 𝔽_{2}-vector space.

In the rest of the section, we study the mod 2 reduction of the exact sequence in Theorem 1.3. We prove the following :

Proposition 5.3. Let

0 → Tor_{𝔽_{2}}D^{Ω} → 𝔽_{2}C_{b} → 𝔽_{2}R_{ℚ}^{∗} \underrightarrow{Ψ̃_{2}} 𝔽_{2}D_{tors}^{Ω} → 0

be the mod 2-reduction of the exact sequence given in Theorem 1.3. Then, the kernel of Ψ̃_{2} is naturally equivalent to the functor B^{×} of units of the Burnside ring.

Proof. Consider the commuting diagram of p-biset functors

0 ⟶ ker Ψ̃_{2} ⟶ 𝔽_{2}R_{ℚ}^{∗} \underrightarrow{Ψ̃_{2}} 𝔽_{2}D_{tors}^{Ω} ⟶ 0
↓  ↓^{i}  ↓
0 ⟶ B^{×} ⟶ 𝔽_{2}B^{∗} ⟶^{Ψ_{2}} 𝔽_{2}D^{Ω} ⟶ 0 .

Note that by Theorem 1.8 of [5] there is an exact sequence of the form

0 → R_{ℚ}^{∗} → B^{∗} → D^{Ω}/D_{tors}^{Ω} → 0.

Since the cokernel D^{Ω}/D_{tors}^{Ω} is torsion free, this still gives us an exact sequence after tensoring with 𝔽_{2}. Thus the second vertical map in the above diagram is injective with cokernel equal to 𝔽_{2} ⊗_{ℤ} (D^{Ω}/D_{tors}^{Ω}). By a similar argument one can see easily that the third vertical transformation is also injective with the same cokernel. So, the first vertical map is an isomorphism by the Snake lemma. □

Note that as a consequence of Proposition 5.3, we obtain two short exact sequences of p-biset functors :

0 → B^{×} → 𝔽_{2}R_{ℚ}^{∗} → 𝔽_{2}D_{tors}^{Ω} → 0
0 → Tor_{𝔽_{2}}D^{Ω} → 𝔽_{2}C_{b} → B^{×} → 0 .

From the proof of Proposition 5.3, it is easy to see that the second exact sequence is the same as the second exact sequence given in Corollary 1.5. We also observe that the first exact sequence above is the same as the exact sequence given in Proposition 9.11 of [8]. Thus our arguments provide a more natural way to see the exactness of the sequence given there.

Note that when p is an odd prime, the unit group B^{×}(P) is equal to {±1} for every p-group P, so in this case the second exact sequence above reduces to a sequence of the form

0 → Tor_{𝔽_{2}}D^{Ω} → 𝔽_{2}C_{b} → Γ_{𝔽_{2}} → 0

where Γ_{𝔽_{2}} denotes the constant functor with values 𝔽_{2}. This is closely related to an exact sequence given by Bouc and Thévenaz in [9]. The exact sequence given there is of the form

0 → \overline{D}_{tors} → 𝔽_{2}R_{ℚ} → Γ_{𝔽_{2}} → 0

where \overline{D}_{tors} is a quotient of the torsion part of the Dade group. It has been shown by Carlson and Thévenaz ([11] Theorem 13.3) that actually \overline{D}_{tors} = D_{tors}, and by Theorem 6.2 and Theorem 11.2 of [9], it follows that D_{tors} = D_{tors}^{Ω} when p is an odd prime. It is clear from the second sequence above that D_{tors} is an 𝔽_{2}-vector space. So, we can identify Tor_{𝔽_{2}}D^{Ω} with D_{tors}. Thus, it makes sense to ask whether the two above sequences are the same. We have the following :

Theorem 5.4. The p-biset functors 𝔽_{2}C_{b} and 𝔽_{2}R_{ℚ} are naturally equivalent when p is congruent to 3 modulo 4.

Proof. Let P be a p-group with p > 2. Consider the 𝔽_{2}-linear map φ : 𝔽_{2}R_{ℚ}(P) → 𝔽_{2}C_{b}(P) defined as the linear extension of the assignment V ↦ f_{V} where V is a rational representation of P and f_{V} is the Borel-Smith function given by

f_{V} (Q) = \frac{2}{p − 1}(dim_{ℚ} V^{Q} − dim_{ℚ} V^{P} ) + dim_{ℚ} V^{P}

for every subgroup Q ≤ P. To see that f_{V} (Q) is a Borel-Smith function, we first note that when Q ≤ R are two subgroups in P, then p − 1 divides dim_{ℚ} V^{Q} − dim_{ℚ} V^{R}. This can be shown by an easy induction and by noting that it is true when Q has index p in R (for a cyclic group H of order p, it is clear that dim_{ℚ} W − dim_{ℚ} W^{H} is divisible by p − 1 for every ℚH-module W.) This shows in particular that f_{V} (Q) is an integer for all Q ≤ P. It also shows that for every Q ≤ R with |R : Q| = p, we have f_{V} (Q) ≡ f_{V} (R) mod 2. To show that f_{V} also satisfies the Borel-Smith conditions coming from ℤ/pℤ × ℤ/pℤ sections, we just notice that f_{V} is a linear combination (with rational coefficients) of two Borel-Smith functions, namely DimV and a constant function. So, it satisfies these Borel-Smith conditions as well.

Now, we need to verify that φ commutes with biset action. For this, it is enough to show that for any (P, R)-biset U and an ℚR-module V, the equality

C(U)f_{V} (Q) = f_{ℚU⊗_{ℚR}V} (Q)

holds for every Q ≤ P. Since the assignment V → DimV commutes with the biset action, we have

f_{ℚU⊗_{ℚR}V} (Q) − C(U)f_{V} (Q) = \frac{p − 3}{p − 1}(dim_{ℚ}(ℚU ⊗_{ℚR} V)^{P} − ∑_{u∈Q\U/R} dim_{ℚ} V^{R})

= \frac{p − 3}{p − 1}(Dim(ℚU ⊗_{ℚR} V)(P) − |Q\U/R| dim_{ℚ} V^{R}) .

Since p−3 is a multiple of 4, and since 4R_{ℚ}^{∗} ⊆ 2C_{b}, it is enough to show that the difference

D = Dim(ℚU ⊗_{ℚR} V)(P) − |Q\U/R| dim_{ℚ} V^{R}

is divisible by p − 1. But this follows from the fact that the function DimV is constant modulo (p − 1), and that there exists a constant biset functor modulo p − 1. In other words,

Dim(ℚU ⊗_{ℚR} V)(P) = (C(U)DimV )(P) = ∑_{u∈P \U/R} dim_{ℚ} V^{P^{u}}
≡ |P \U/R| dim_{ℚ} V (mod. (p − 1)) .

Since dim_{ℚ} V^{R} is also equal to dim_{ℚ} V modulo p − 1, we have that

D ≡ (|P \U/R| − |Q\U/R|) dim_{ℚ} V

modulo p − 1. But |P \U/R| − |Q\U/R| is divisible by p − 1 : this easily follows from the fact that for a cyclic group H of order p, the difference |W | − |W/H| is divisible by p − 1 for every H-set W. Thus D ≡ 0 modulo p − 1, as was to be shown.

Finally, the fact that φ is an isomorphism follows from the general theory for rational biset functors. It is easy to see that both of these functors are rational and so it is enough to check the isomorphism on genetic sections, i.e. on p-groups of normal p-rank one, which are cyclic, in this case. For cyclic groups it is very easy to verify that φ is an isomorphism by direct calculation. □

Remark 5.5. In the case p ≡ 1 (mod. 4), then it is easy to check that the constant functions form a subfunctor of 𝔽_{2}C_{b}, isomorphic to Γ_{𝔽_{2}}. Since 𝔽_{2}R_{ℚ} has no such subfunctor (Corollary 8.4 of [9]), it follows that the functors 𝔽_{2}R_{ℚ} and 𝔽_{2}C_{b} are not isomorphic in this case.

We conclude the following

Corollary 5.6. If p ≡ 3 (mod. 4), then the following two exact sequences are isomorphic:

0 → Tor_{𝔽_{2}}D^{Ω} → 𝔽_{2}C_{b} → B^{×} → 0,
0 → D_{tors} → 𝔽_{2}R_{ℚ} → Γ_{𝔽_{2}} → 0.

Proof. Since 2 | p − 1, the functor 𝔽_{2}R_{ℚ} has a unique proper non-zero subfunctor, by Corollary 8.4 of [9]. Since each of the functors Tor_{𝔽_{2}}D^{Ω}, B^{×}, D_{tors} and Γ_{𝔽_{2}} is non zero, it follows that the isomorphism 𝔽_{2}R_{ℚ} → 𝔽_{2}C_{b} of Theorem 5.4 maps the subfunctor D_{tors} of 𝔽_{2}R_{ℚ} to the subfunctor Tor_{𝔽_{2}}D^{Ω} of 𝔽_{2}C_{b} , and induces the isomorphism Γ_{𝔽_{2}} → B^{×}. □

Acknowledgements : We wish to thank the referee for her/his detailed report on the first version of this paper, and for making many useful suggestions to improve it.

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

Ergün Yalçin - Department of Mathematics, Bilkent University, Ankara, 06800, Turkey.

E-mail address: serge.bouc@u-picardie.fr, yalcine@fen.bilkent.edu.tr