The Dade group of a p-group

Serge Bouc

CNRS UMR 7586. Institut de Mathématiques de Jussieu. 2 Place Jussieu, F 75251 - Paris Cedex 05 - France -

                      email : bouc@math.jussieu.fr
                web : http://www.math.jussieu.fr/∼bouc

Abstract

Let p be a prime number. This paper solves the question of the structure of the group D(P) of endo-permutation modules over an arbitrary finite p-group P, that was open after Dade’s original papers in 1978 ([19], [20]), and it gives a proof of the conjectures proposed in [4] and [10]. This leads to a presentation of D(P) by explicit generators and relations, generalizing the presentation obtained by Dade when P is abelian.

A key result of independent interest is the explicit description of the kernel of the natural map from the Burnside group to the group of rational characters, in terms of the extraspecial group of order p^{3} and exponent p if p ≠ 2, or of all dihedral groups of order at least 8 if p = 2.

Keywords : Endo-permutation, Dade group, Burnside biset functor rational representation, p-group

Mathematics Subject Classification : 20C20, 20D15, 18B99

1. Introduction

This paper describes the structure of the Dade group D(P) of a finite pgroup P, that was defined by E. C. Dade in 1978 ([19], [20]), in order to classify endo-permutation kP -modules, where k is a field of characteristic p > 0, or more generally, a “p-local” ring.

Endo-permutation modules for p-groups appear as a crucial tool in many aspects of the p-modular representation theory of finite groups, e.g. as source modules of simple modules, or in the description of source algebras of blocks (L. Puig [22]), or the description of derived or stable equivalences between blocks (L. Puig [24]). They are a generalization of the notion of endo-trivial module or invertible module, studied in particular by J. L. Alperin ([1], [3]) and J. F. Carlson ([15], [16]).

In his original papers, Dade determined the structure of D(P) when P is abelian. Then Puig proved ([23]) that for an arbitrary finite p-group P, the group D(P) is a finitely generated (abelian) group. Next J. Thévenaz and I showed ([12]) that the free rank of D(P) is equal to the number of conjugacy classes of non cyclic subgroups of P.

Recently, in a series of remarkable papers ([13], [14], [17]), J. F. Carlson and J. Thévenaz completed the classification of endo-trivial modules, and proved detection theorems for the Dade group. Their results are the first essential ingredient for the present work. Their third paper ([17]) also gave methods to start an induction procedure, by which N. Mazza and I determined ([11]) the structure of D(P), when P is any (almost) extraspecial p-group.

The second ingredient of the present paper is the notion of biset functor for finite groups, introduced in [5], and specialized to p-groups in [12]. Biset functors over p-groups seem specially well suited to study the subgroup D^{Ω}(P) of D(P) generated by the relative syzygies Ω_{X} associated to (non empty) finite P -sets X, where the module Ω_{X} is defined as the kernel of the augmentation map kX → k (see [2] or [6] for details).

In [6], a formula for tensor induction of relative syzygies in the Dade group was stated, showing that the correspondence sending a p-group P to D^{Ω}(P) is a biset functor. In [9], it was shown that this functor D^{Ω} is a quotient of the ℤ-dual B^{∗} of the Burnside biset functor B, and that there is an exact sequence of biset functors over p-groups

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

where R_{ℚ}^{∗} is the ℤ-dual of the functor of rational representations, and D_{tors}^{Ω} is the torsion subfunctor of D^{Ω}.

In this sequence, the embedding R_{ℚ}^{∗} → B^{∗} is the transpose of the natural transformation χ : B → R_{ℚ}, whose evaluation at P maps the P -set X to the corresponding ℚP -module ℚX. A key result in this paper is Theorem 6.12, which may be of independent interest : it shows that the functor K = Ker χ is generated by its values at the extraspecial group of order p^{3} and exponent p for p ≠ 2, or at the dihedral groups of order at least 8 for p = 2. In other words, it gives an explicit way to build all the “virtual P -sets with zero character” from specific ones for these extraspecial or dihedral p-groups.

On the other hand, the surjectivity of χ is known since 1972 by the Ritter-Segal theorem, which was stated in a more explicit form in [7]. This was the starting point of the study of the functor R_{ℚ} of rational representations of p-groups ([8]), which is the third ingredient of this paper. In particular, the notion of genetic section of a p-group was defined there, and was developed in [10], where the notion of rational biset functor was also introduced.

This led to precise conjectures on the torsion part of the Dade group (Conjectures 6.2 and 6.3 of [10]). In my talk at Oberwolfach in March 2003 ([4]), I formulated another conjecture on the Dade group, saying that D = D^{Ω} if p ≠ 2, and that D/D^{Ω} is isomorphic to a specific subfunctor of 𝔽_{2}R_{ℚ} if p = 2, associated to quaternion 2-groups. In the present paper, I will give a proof of all these conjectures, in Theorem 7.7, Theorem 8.2, and Theorem 10.2.

The main consequence will be presentation of D(P) by explicit generators and relations (Theorem 9.5), generalizing the presentation obtained by Dade when P is abelian.

The paper is organized as follows : Section 2 recalls some definitions and results from rational representation theory of p-groups. Section 3 is devoted to definitions and results on biset functors. Section 4 is a brief presentation of the Dade group, its functorial properties, and theorems of Dade and Carlson-Thévenaz. In Section 5, I will state some results on genetic bases, in particular Lemma 5.2, that is the key to both Theorem 6.12 and Theorem 8.2. The key result on the functor K is stated in Section 6. Section 7 is an application to the Dade group, showing that D = D^{Ω} + D_{tors}, and that D = D^{Ω} if p ≠ 2. Section 8 is devoted to the structure of the torsion part of the Dade group. Section 9 gives a presentation of D(P) by generators and relations. Finally Section 10 describes the functorial structure of D/D^{Ω} for p = 2.

2. Rational representations

Throughout this paper, the symbol p will denote a prime number. All p-groups will be finite ones. If P is a p-group, then Φ(P) denotes its Frattini subgroup, and Z(P) its center. The largest elementary abelian subgroup of Z(P) is denoted by Ω_{1}Z(P).

If n is a positive integer, the symbol C_{n} will denote a cyclic group of order n. If n ≥ 2, then D_{2^{n}} denotes a dihedral group of order 2^{n}, with the convention that D_{4} is the Klein four group. If n ≥ 3, then Q_{2^{n}} denotes a generalized quaternion group of order 2^{n}. If n ≥ 4, then SD_{2^{n}} denotes a semi dihedral group of order 2^{n}.

2.1. Basic subgroups and associated simple modules : Recall some notation and definitions from [8] :

2.2. Notation : if P is a group, denote by R_{ℚ}(P) the Grothendieck group of finitely generated ℚP -modules, and by Irr_{ℚ}(P) a set of representatives of isomorphism classes of irreducible ℚP -modules. There is a natural bilinear form on R_{ℚ}(P), with values in ℤ, defined by

⟨V, W ⟩_{P} = dim_{ℚ} Hom_{ℚP}(V, W)

for ℚP -modules V and W. If V is a simple ℚP -module, there is a unique linear form m(V, −) on R_{ℚ}(P), with values in ℤ, sending the ℚP -module W to the multiplicity m(V, W) of V as a summand of W.

2.3. Definition : ([8] 2.3 ) Let P be a finite p-group. A subgroup Q of P is called \underline{basic} if the following two conditions hold :

1. The quotient N_{P} (Q)/Q is cyclic or generalized quaternion.

2. If R is any subgroup of P and if R ∩ N_{P} (Q) ⊆ Q, then |R| ≤ |Q|.

If Q is a proper basic subgroup of P, then there is a unique subgroup Q̃ ⊃ Q of P with |Q̃ : Q| = p, and the kernel of the projection map

ℚP/Q → ℚP/Q̃

is an irreducible ℚP -module, denoted by V_{Q}.

The group P itself is a basic subgroup of P, and by convention V_{P} is the trivial ℚP -module ℚ. With this notation, any irreducible ℚP -module is isomorphic to V_{Q}, for some basic subgroup Q of P.

If Q and Q′ are basic subgroups of P, then the ℚP -modules V_{Q} and V_{Q′} are isomorphic if and only if Q ≑_{P} Q′, where ≑_{P} is the relation defined in [8] 2.7 by

Q ≑_{P} Q′ ⇔ \{ \begin{matrix}|Q| = |Q′| \\ ∃x ∈ P, Q^{x} ∩ N_{P} (Q′) ⊆ Q′ \end{matrix} .

2.4. Remark : In particular, a normal subgroup N of P is basic if and only if P/N is cyclic or generalized quaternion. Moreover, if Q is a basic subgroup such that Q ≑_{P} N, then Q = N.

2.5. Definition and Notation : ( [8] Proposition 3.7) A p-group P has \underline{normal p-rank 1} if it does not have any normal subgroup isomorphic to (C_{p})^{2}. Up to isomorphism, such a group has a unique rational faithful irreducible representation, denoted by Φ_{P}.

Recall (Theorem 5.4.10 of [21]) that if P is a p-group of normal p-rank 1 and order p^{n}, then P ≅ C_{p^{n}} if p ≠ 2, and if p = 2, then P ≅ C_{2^{n}}, or P ≅ Q_{2^{n}} (n ≥ 3), or P ≅ D_{2^{n}} (n ≥ 4), or P ≅ SD_{2^{n}} (n ≥ 4).

2.6. Definition : ([8] Definition 3.5) A \underline{section} (T, S) of the group P is a pair of subgroups of P such that S \underline{◁} T ⊆ P. A section (T, S) of P is \underline{proper} if (T, S) ≠ (P, 1). If R is a group, the section (T, S) will be said of \underline{type} R if the factor group T/S is isomorphic to R.

The section (T, S) of the p-group P will be called \underline{genetic} if the following three conditions hold :

1. The group T/S has normal p-rank 1.

2. The ℚP -module V (T, S) = Ind_{T}^{P} Inf_{T/S}^{T}Φ_{T/S} is simple.

3. ⟨V (T, S), V (T, S)⟩_{P} = ⟨Φ_{T/S}, Φ_{T/S}⟩_{T/S}.

The simple ℚP -module V (T, S) will be called the simple module associated to (T, S). It was shown in Theorem 3.4 of [8] that for any simple ℚP - module V, there exists a genetic section (T, S) of P such that V ≅ V (T, S).

Moreover ([8] Theorem 7.11 and Proposition 7.1), if (T, S) and (T′, S′) are genetic sections of P, the corresponding modules V (T, S) and V (T′, S′) are isomorphic if and only if (T, S) _{P} (T′, S′), i.e. if there exists x ∈ P such that

T^{x}·S′ = S^{x}·T′  and  T^{x} ∩ S′ = S^{x} ∩ T′ .

In particular, the relation _{P} (“linked modulo P ”) is an equivalence relation on the set of genetic sections of P. Observe that if (T, S) _{P} (T′, S′), then the groups T/S and T′/S′ are isomorphic. In other words (T, S) and (T′, S′) have the same type.

2.7. Definition : If V is a simple ℚP -module, and (T, S) is a genetic section of P such that V ≅ V (T, S), then T/S is called the \underline{type} of V.

The genetic sections of P can be characterized as follows :

2.8. Proposition : ([10] Proposition 4.4) Let P be a finite p-group, and let (T, S) be a section of P. Let Z_{P} (S) denote the subgroup of P defined by

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

Then the following conditions are equivalent :

1. The section (T, S) is an genetic section of P.

2. The group N_{P} (S)/S has normal p-rank 1, the group T is equal to N_{P} (S), and if x ∈ P is such that S^{x} ∩ Z_{P} (S) ⊆ S, then x ∈ N_{P} (S).

In particular, if (T, S) is a genetic section of P, then T = N_{P} (S). Hence (T, S) is actually determined by S. This leads to the following :

2.9. Definition and Notation : A subgroup S of the p-group P will be called \underline{genetic} if the section (N_{P} (S), S) is a genetic section of P. The simple module V (N_{P} (S), S) will be denoted by V (S).

Similarly, if S and S′ are genetic subgroups of P, I will say that S _{P} S′ if (N_{P} (S), S) _{P} (N_{P} (S′), S′).

The relation _{P} is an equivalence relation on the set of genetic subgroups of P. By Theorem 3.11 of [8], the correspondence S ↦ V (S) is a one to one correspondence between the set of equivalence classes of genetic subgroups modulo _{P} , and the set of isomorphism classes of rational irreducible representations of P. Moreover by Theorem 7.11 of [8], if S and S′ are genetic subgroups of P such that S _{P} S′, the set of elements x ∈ P such that S ^{x}S′ is a single double coset N_{P} (S)yN_{P} (S′) in P. In particular if S ^{x}S, then x ∈ N_{P} (S).

2.10. Definition : A subset S of the set of genetic subgroups of P will be called a \underline{genetic basis} of P if it is a set of representatives of the set of equivalence classes of genetic subgroups of P for the relation _{P} .

2.11. Remark : A normal subgroup N of P is genetic if and only if P/N has normal p-rank 1. In that case, if S is any genetic subgroup of P such that S _{P} N, then S = N. In particular, the group N belongs to every genetic basis of P.

2.12. Remark : (see Proposition 7.4 in [8]) If S is a genetic subgroup of P, and if R_{S}/S is a basic subgroup of N_{P} (S)/S, intersecting trivially the center of N_{P} (S)/S, then R_{S} is a basic subgroup of P, corresponding to the same simple QP -module (i.e. with the above notation V_{R_{S}} ≅ V (S)).

If N_{P} (S)/S is cyclic or generalized quaternion, then R_{S} = S. If N_{P} (S))/S is dihedral or semi dihedral, then R_{S}/S is any non central subgroup of order 2 of N_{P} (S)/S.

Basic subgroups obtained from genetic ones by this operation were called origins in [8]. In general, not all basic subgroups are origins, but there is at least an origin in each equivalence class of basic subgroups for the relation ≑_{P} (see Corollary 7.5 in [8] for details).

3. Biset functors

3.1. Notation and Definition : Denote by C_{p} the following category :

• The objects of C_{p} are the finite p-groups.

• If P and Q are finite p-groups, then Hom_{C_{p}}(P, Q) = B(Q × P^{op}) is the Burnside group of finite (Q, P)-bisets. An element of this group is called a \underline{virtual} (Q, P)-biset.

• The composition of morphisms is ℤ-bilinear, and if P, Q, R are finite p-groups, if U is a finite (Q, P)-biset, and V is a finite (R, Q)-biset, then the composition of (the isomorphism classes of) V and U is the (isomorphism class) of V ×_{Q} U. The identity morphism Id_{P} of the p-group P is the class of the set P, with left and right action by multiplication.

Let F_{p} denote the category of additive functors from C_{p} to the category ℤ-Mod of abelian groups. An object of F_{p} is called a \underline{biset functor} (defined over pgroups, with values in ℤ-Mod).

If F is an object of F_{p}, if P and Q are finite p-groups, and if φ ∈ Hom_{C_{p}}(P, Q), then the image of w ∈ F (P) by the map F (φ) will generally be denoted by φ(w). The composition ψ ∘ φ of morphisms φ ∈ Hom_{C_{p}}(P, Q) and ψ ∈ Hom_{C_{p}}(Q, R) will also be denoted by ψ ×_{Q} φ.

3.2. Notation : The \underline{Burnside} biset functor (defined e.g. as the Yoneda functor Hom_{C_{p}}(1, −)), will be denoted by B. The functor of rational representations (see Section 1 of [8]) will be denoted by R_{ℚ}.

3.3. Examples : Recall that this 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 Q is a subgroup of P, then let Ind_{Q}^{P} ∈ Hom_{C_{p}}(Q, P) denote the set P, with left action of P and right action of Q by multiplication.

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

• If N \underline{◁} P, and Q = P/N, then let Inf_{Q}^{P} ∈ Hom_{C_{p}}(Q, P) denote the set Q, with left action of P by projection and multiplication, and right action of Q by multiplication.

• If N \underline{◁} P, and Q = P/N, then let Def_{Q}^{P} ∈ Hom_{C_{p}}(P, Q) denote the set Q, with left action of Q by multiplication, and right action of P by projection and multiplication.

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

3.4. Remark : If P and Q are p-groups, then any element φ ∈ Hom_{C_{p}}(P, Q) is a ℤ-linear combination of transitive (Q, P)-bisets, and by Lemma 7.4 of [12], every transitive (Q, P)-biset can be factored in the category C_{p} as a composition

Ind_{V}^{Q} ∘ Inf_{V/U}^{V} ∘ Iso_{T/S}^{V/U}(φ) ∘ Def_{T/S}^{T} ∘ Res_{T}^{P}

where (T, S) is a section of P, and (V, U) is a section of Q, such that there exists a group isomorphism φ : T/S → V/U.

3.5. Notation : If (T, S) is a section of P, set

Indinf_{T/S}^{P} = Ind_{T}^{P} Inf_{T/S}^{T}  and  Defres_{T/S}^{P} = Def_{T/S}^{T}Res_{T}^{P} .

Then Indinf_{T/S}^{P} ≅ P/S as (P, T/S)-biset, and Defres_{T/S}^{P} ≅ S\P as (T/S, P)- biset.

3.6. Opposite bisets : If P and Q are finite p-groups, and if U is a finite (Q, P)-biset, then let U^{op} denote the opposite biset : as a set, it is equal to U, and it is a (P, Q)-biset for the following action

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

This definition can be extended by linearity, to give an isomorphism

φ ↦ φ^{op} : Hom_{C_{p}}(P, Q) → Hom_{C_{p}}(Q, P) .

It is easy to check that (φ ∘ ψ)^{op} = ψ^{op} ∘ φ^{op}, with obvious notation, and the functor

\{ \begin{matrix}P ↦ P \\ φ ↦ φ^{op} \end{matrix}

is an equivalence of categories from C_{p} to the dual category.

3.7. Example : if P is a p-group, and (T, S) is a section of P, then

(Indinf_{T/S}^{P})^{op} ≅ Defres_{T/S}^{P}

as (T/S, P)-bisets.

3.8. Definition and Notation : If F is a biset functor, the \underline{dual} biset functor F^{∗} is defined by

F^{∗}(P) = Hom_{ℤ}(F (P), ℤ) ,

for a p-group P, and by

F^{∗}(φ)(α) = α ∘ F (φ^{op}) ,

for any α ∈ F^{∗}(P), any p-group Q, and any φ ∈ Hom_{C_{p}}(P, Q).

3.9. Some idempotents in End_{C_{p}}(P) : Let P be a finite p-group, and let N \underline{◁} P. Then it is clear from the definitions that

Def_{P/N}^{P} ∘ Inf_{P/N}^{P} = (P/N) ×_{P} (P/N) = Id_{P/N} .

It follows that the composition e_{N}^{P} = Inf_{P/N}^{P} ∘ Def_{P/N}^{P} is an idempotent in End_{C_{p}}(P). Moreover e_{1}^{P} = Id_{P}, and if M and N are normal subgroups of P, then e_{N}^{P} ∘ e_{M}^{P} = e_{NM}^{P}.

3.10. Lemma : ([10] Lemma 2.5) If N \underline{◁} P, define f_{N}^{P} ∈ End_{C_{p}}(P) by

f_{N}^{P} = ∑_{{M \underline{◁} P \atop N⊆M}} μ_{\underline{◁} P} (N, M)e_{M}^{P} ,

where μ_{\underline{◁} P} denotes the Möbius function of the poset of normal subgroups of P. Then the elements f_{N}^{P}, for N \underline{◁} P, are orthogonal idempotents of End_{C_{p}}(P), and their sum is equal to Id_{P}.

Moreover, it is easy to check from the definition that for N \underline{◁} P,

(3.11)  f_{N}^{P} = Inf_{P/N}^{P} ∘ f_{1}^{P/N} ∘ Def_{P/N}^{P} ,

and

e_{N}^{P} = Inf_{P/N}^{P} ∘ Def_{P/N}^{P} = ∑_{{M \underline{◁} P \atop M⊇N}} f_{M}^{P} .

3.12. Lemma : If N is a non trivial normal subgroup of P, then

f_{1}^{P} ∘ Inf_{P/N}^{P} = 0 and Def_{P/N}^{P} ∘ f_{1}^{P} = 0 .

Proof: Indeed by 3.11

f_{1}^{P} ∘ Inf_{P/N}^{P} = f_{1}^{P} ∘ Inf_{P/N}^{P} ∘ Def_{P/N}^{P} ∘ Inf_{P/N}^{P}
= ∑_{{M \underline{◁} N \atop M⊇N}} f_{1}^{P} f_{M}^{P}Inf_{P/N}^{P} = 0 ,

since M ≠ 1 when M ⊇ N. The other equality of the lemma follows by taking opposite bisets.

3.13. Remark : It was also shown in Section 2.7 of [10] that

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

where μ is the Möbius function of the poset of subgroups of N.

3.14. Notation and Definition : If F is a a biset functor, and if P is a p-group, then the idempotent f_{1}^{P} of End_{C_{p}}(P) acts on F (P). Its image

∂F (P) = f_{1}^{P} F (P)

is a direct summand of F (P) as ℤ-module : it will be called the set of \underline{faithful} elements of F (P).

The reason for this name is that any element u ∈ F (P) which is inflated from a proper quotient of P is such that F (f_{1}^{P})u = 0. From Lemma 3.12, it is also clear that

∂F (P) = ⋂_{1≠N \underline{◁} P} Ker Def_{P/N}^{P} .

3.15. Notation : If F is a biset functor, and P is a p-group, set

\overline{F} (P) = F (P)/ ∑_{{(T,S) \atop |T/S|<|P |}} Indinf_{T/S}^{P}F (T/S) ,

and dually

\underline{F} (P) = ⋂_{{(T,S) \atop |T/S|<|P |}} Ker Defres_{T/S}^{P} .

where in both expressions (T, S) runs through proper sections of P.

4. The Dade group

In this section, I will briefly recall some basic definitions and constructions on the Dade group of endo-permutation modules. Most of them go back to Dade’s paper ([19]), and are also exposed in Sections 28-29 of Thévenaz’s book ([25]). The “functorial approach” to the Dade group was introduced in Sections 2 and 3 of [12].

4.1. Let k be a field of characteristic p. If P is a p-group, the Dade group D(P) = D_{k}(P) can be defined as the group of equivalence classes of capped endo-permutation modules, or as the group of equivalence classes of Dade P -algebras over k, for a suitable equivalence relation in each case. It is an abelian group, and Puig ([23]) has shown that it is finitely generated.

The (equivalence classes of ) endo-trivial modules form a subgroup T (P) of D(P), called the group of endo-trivial modules. Another crucial subgroup of D(P) is the subgroup D^{Ω}(P) generated by relative syzygies (see [2] or [6] for details).

4.2. Functorial properties : The Dade group has some important functorial properties : if Q is a subgroup of P, and if N is a normal subgroup of P, or if φ : P → P′ is a group isomorphism, then there are maps of restriction, induction, deflation, inflation, and transport by isomorphism

(4.3)  \{ \begin{matrix}D(P) \underrightarrow{Res_{Q}^{P}} D(Q) \underrightarrow{Ten_{Q}^{P}} D(P) \\ D(P) \underrightarrow{Def_{P/N}^{P}} D(P/N) \underrightarrow{Inf_{P/N}^{P}} D(P) \\ D(P) \underrightarrow{Iso_{P}^{P′}} D(P′) \end{matrix}

coming respectively from the restriction, tensor induction, Brauer quotient, inflation, and transport by group isomorphism of Dade algebras (note the change of notation for induction).

These five operations can be unified in a single formalism using bisets : if P and Q are p-groups, if U is a finite (Q, P)-biset, then in Corollary 2.3 of [12], we introduced a map

D(U) : D(P) → D(Q) ,

such that the maps in 4.3 are equal to D(U), where U is the corresponding biset Res_{Q}^{P}, Ind_{Q}^{P}, Def_{P/N}^{P}, Inf_{P/N}^{P}, and Iso_{Q}^{P} defined in Section 3.3. The definition of this map D(U) associated to an arbitrary finite (Q, P) biset U followed from the existence of a corresponding functor T_{U} : \underline{P erm}_{k}(P) → \underline{P erm}_{k}(Q), which is a sort of generalized tensor induction, where \underline{P erm}_{k}(P) is a category equivalent to the category of finitely generated permutation kP -modules (see Section 2 of [12], in particular Lemma (2.2) and Corollary (2.13), for details).

Moreover the correspondence U ↦ D(U) is additive (by Proposition 2.10 of [12]), so it can be extended to a correspondence

φ ∈ Hom_{C_{p}}(P, Q) ↦ D(φ) ∈ Hom_{ℤ}(D(P), D(Q)) .

However, these constructions do not define a biset functor structure : the reason is that in general, with obvious notation

D(ψ) ∘ D(φ) ≠ D(ψ ∘ φ) ,

as can be read from Proposition 3.5 and Proposition 3.10 of [12] : in general, the right hand side is equal to a linear combination ∑_{i} r_{i}D(U_{i}), where the U_{i}’s are transitive bisets, and the r_{i}’s are integers, whereas the left hand side is equal to ∑_{i} r_{i}γ_{a_{i}}D(U_{i}), for the same U_{i}’ and r_{i}’s, but with a new kind of operation inserted, namely the γ_{a_{i}}’s. These operations are called for short “Galois torsions”, and they are precisely defined in Section 3 of [12] : if a is any endomorphism of the ground field k, then for any p-group P, there is a linear map γ_{a} = γ_{a,P} : D(P) → D(P). They only appear when one composes a tensor induction followed by a deflation (see Proposition 3.10 of [12] for details).

This remark and Lemma 3.10 show in particular that the maps D(f_{N}^{P}), for N \underline{◁} P, which only involve deflation and inflation maps, are orthogonal idempotents of End_{ℤ}(D(P)), and their sum is the identity.

4.4. Notation : If P and Q are p-groups, and if φ ∈ Hom_{C_{p}}(P, Q), the map D(φ) : D(P) → D(Q) will simply be denoted by φ, and the image of u ∈ D(P) by this map will be denoted by φ(u) or φu. The faithful part f_{1}^{P} D(P) of the Dade group will be denoted by ∂D(P).

It is easy to see that the maps γ_{a} restrict to the identity on D^{Ω}(P) (see Section 1.6 of [9]). Moreover, if φ ∈ Hom_{C_{p}}(P, Q), then D(φ)(D^{Ω}(P)) ⊆ D^{Ω}(Q), by Section 4 and 5 of [6]. The above discussion now shows that the correspondence P ↦ D^{Ω}(P) is a biset functor.

4.5. Some known Dade groups : In his original papers, Dade determined D(P), when P is abelian :

4.6. Theorem : (Dade [19], [20]) Let P be an abelian p-group. Then D(P) is generated by the elements Ω_{P/Q}, for Q ⊆ P, subject to the relations τ_{Q}Ω_{P/Q} = 0, if P/Q is cyclic, where τ_{Q} = 1 if |P/Q| ≤ 2, and τ_{Q} = 2 otherwise.

In particular D(P) = D^{Ω}(P) if P is abelian.

The structure of D(P) is also known for any 2-group P of normal 2-rank 1 : when P is generalized quaternion, the result is due to Dade, and the other cases have been solved by Carlson and Thévenaz :

4.7. Theorem : (Dade [18], Carlson-Thévenaz [13] Theorem 10.3)

1. D(D_{2^{n}}) ≅ ℤ^{2n−3}.

2. D(SD_{2^{n}}) ≅ ℤ^{2n−4} ⊕ ℤ/2ℤ.

3. D(Q_{2^{n}}) ≅ ℤ^{2n−5} ⊕ ℤ/4ℤ ⊕ ℤ/2ℤ, for n ≥ 4.

4. D(Q_{8}) ≅ ℤ ⊕ ℤ/4ℤ ⊕ ℤ/2ℤ, if the ground field contains all cubic roots of unity, and D(Q_{8}) ≅ ℤ ⊕ ℤ/4ℤ otherwise.

This result is actually more precise : Lemma 2.10 of [13] shows that if P is one of these 2-groups, then ∂D(P) = T (P) = Ker Def_{P/Z}^{P}, and D(P) = T (P) + Ind_{P/Z}^{P}D(P/Z), where Z is the center of P. This allows induction, since P/Z is dihedral. Then Theorem 5.4 and Theorem 7.1 of [13] show that D(P) = D^{Ω}(P) when P is dihedral, or semi dihedral.

If P is generalized quaternion of order 2^{n}, then Theorem 6.3 and Theorem 6.5 of [13] show that T (P) is equal to the torsion part D_{tors}(P) of D(P), and that T (P) ≅ ℤ/4ℤ ⊕ Z/2ℤ if n ≥ 4, or if n ≥ 3 and the ground field has all cubic roots of unity. If n = 3 and the ground field has no non trivial cubic roots of unity, then T (P) = D_{tors}(P) = D_{tors}^{Ω}(P) ≅ ℤ/4ℤ.

4.8. Lemma : Let Q be a generalized quaternion 2-group of order 2^{n}. Suppose that n ≥ 4, or that n = 3, and the ground field has all cubic roots of unity. Then

1. There are exactly two elements η_{Q} and η′_{Q} of order 2 in the set D(Q) −

D^{Ω}(Q), and

η_{Q} + η′_{Q} = 2Ω_{Q/1} .

2. If R is a proper subgroup of Q, then Res_{R}^{Q}η_{Q} ∈ D^{Ω}(R).

3. If R is a generalized quaternion group containing Q, then

Ten_{Q}^{R}η_{Q} − η_{R} ∈ D^{Ω}(R) .

4. If a is any endomorphism of the ground field, then γ_{a}(η_{Q}) = η_{Q} if n ≥ 4.

Proof: Assertion 1 follows from the fact that Ω_{Q/1} generates a cyclic summand of order 4 in D_{tors}(P) ≅ ℤ/4ℤ⊕Z/2ℤ. And in the group ℤ/4ℤ⊕Z/2ℤ, the elements of order 2 are (0, 0), (2, 0), (0, 1), and (2, 1).

Assertion 2 is trivial if n = 3, because by construction in that case all proper restrictions of η_{Q} are 0. If n ≥ 4 and R is cyclic, then Res_{R}^{Q}η_{Q} = 2Ω_{R/1} = 0 by construction (see Lemma 6.4 and Theorem 6.5 of [13]). Suppose now that R is quaternion, and |Q : R| = 2. There are two conjugacy classes of quaternion subgroups of order 8 in Q, and one of them is contained in R. Let H and H′ denote representatives for these classes, and suppose H ⊆ R. Then Res_{H}^{Q}η_{Q} is equal to 0 or 2Ω_{H}, by the proof of Theorem 6.5 of [13], and then Res_{H}^{Q}η′_{Q} is respectively equal to 2Ω_{H} or 0. Replacing η_{Q} by η′_{Q} if necessary, I can assume Res_{H}^{Q}η_{Q} = 0. Now the element u = Res_{R}^{Q}η_{Q} is such that Res_{L}^{R}u = 0, for any quaternion subgroups L of order 8 of R. Then u = 0 by Theorem 2.8 of [13] (see also the proof of Theorem 6.5 there). This shows Assertion 2, since η_{Q} + η′_{Q} = 2Ω_{Q/1}.

Assertion 3 is equivalent to Ten_{Q}^{R}η_{Q} ∉ D^{Ω}(R). Proceed by induction on |R : Q|, and suppose that |R : Q| = 2. If u = Ten_{Q}^{R}η_{Q} ∈ D^{Ω}(R), then u is equal to 0 or 2Ω_{R/1}. By Mackey formula, the restriction of u to the generalized quaternion subgroup Q′ ≠ Q of R is equal to Ten_{Q∩Q′}^{Q′}Res_{Q∩Q′}^{Q}η_{Q}. This is equal to 0 by the above argument, since Q ∩ Q′ is cyclic. It follows that u = 0, since Res_{Q′}^{R}2Ω_{R/1} = 2Ω_{Q′/1} ≠ 0. Now Res_{Q}^{R}u = η_{Q} +^{r}η_{Q}, where r ∈ R − Q. Moreover ^{r}η_{Q} ≠ η_{Q} : if n ≥ 4, this is because r exchanges the conjugacy classes of quaternion subgroups of order 8 of Q. And if n = 3, it can be checked directly from the construction : with the notation of Theorem 6.3 of [13], the action of the automorphism of Q_{8} which exchanges the generators x and y is equivalent to replacing the cubic root ω by its conjugate ω^{2}. So the elements of the outer automorphism group Out(Q_{8}) ≅ S_{3} with odd signature exchange η_{Q} and η′_{Q}. In both cases Res_{Q}^{R}u ≠ 0, which is a contradiction.

Finally Assertion 4 is clear from the construction Theorem 6.5 of [13].

5. Genetic bases

5.1. Lemma : Let P be a p-group, and Z be a central subgroup of order p in P. If S is a genetic subgroup of P, then

Def_{P/Z}^{P}V (S) = {0} ⇔ S ⊉ Z .

Proof: If S = P, the result is trivial. Suppose that S ≠ P, and let R/S be a basic subgroup of N_{P} (S)/S, intersecting trivially the center of N_{P} (S)/S. Then R is a basic subgroup of P for V (S), and V (S) is isomorphic to the kernel of the projection map ℚP/R → ℚP/R̃, where R̃ is the unique subgroup of P such that |R̃ : R| = p. Hence V (S) = ℚP/R − ℚP/R̃ in R_{ℚ}(P).

Then either S ⊇ Z, and then Def_{P/Z}^{P}V (S) = ℚP/R − ℚP/R̃ ≠ 0 in R_{ℚ}(P/Z), or S ⊉ Z, and then R ⊉ Z because ZS/S = Ω_{1}Z(N_{P} (S)/S), thus R̃ = RZ and Def_{P/Z}^{P}V (S) = ℚP/RZ − ℚP/RZ = 0 in R_{ℚ}(P/Z).

5.2. Lemma : Let P be a p-group, and let E be a non-central normal elementary abelian subgroup of order p^{2} of P. Set Z = E ∩Z(P), where Z(P) is the center of P.

1. If R is a genetic subgroup of C_{P} (E) such that Z ⊈ R, then R is a genetic subgroup of P, and N_{P} (R) ⊆ C_{P} (E). Moreover if V is the simple ℚP - module associated to R, and W is the simple ℚC_{P} (E)-module associated to R, then V ≅ Ind_{C_{P} (E)}^{P}W, and ⟨V, V ⟩_{P} = ⟨W, W ⟩_{C_{P} (E)}.

2. If Q is a genetic subgroup of P such that Z ⊈ Q, then there exists a genetic subgroup R of C_{P} (E) such that Z ⊈ R and R _{P} Q.

Proof: Set H = C_{P} (E). Then |P : H| = p, and |Z| = p, since E is not central in P, and since the p-Sylow subgroups of Aut(E) have order p.

Let R be a genetic subgroup of H such that Z ⊈ R, and let W be the corresponding rational irreducible representation of H. Then Def_{H/Z}^{H}W = {0} by Lemma 5.1.

Now set V = Ind_{H}^{P}W. Then since Z ⊆ H

Def_{P/Z}^{P}V = Ind_{H/Z}^{P/Z}Def_{H/Z}^{H}W = {0} .

Restriction to H gives

Res_{H}^{P}V ≅ ⊕_{x∈P/H} ^{x}W .

Let I denote the stabilizer of W in P, i.e. the group of elements x ∈ P such that the representations W and ^{x}W of H are isomorphic. Then I ⊇ H, so I = H or I = P.

If I = H, then the representations ^{x}W, for x ∈ P/H, are non isomorphic to each other. In this case V is an irreducible representation of P : indeed if V_{1} is any simple summand of V, then some conjugate of W is a direct summand of W_{1} = Res_{H}^{P}V_{1}. Since W_{1} ≅ ^{x}W_{1} for any x ∈ P, is follows that all the conjugates of W in P are direct summands of W_{1}. Thus W_{1} = Res_{H}^{P}V, and V_{1} = V. Moreover

⟨V, V ⟩_{P} = ⟨W, Res_{H}^{P}V ⟩_{H} = ∑_{x∈P/H}⟨W,^{x}W ⟩_{H} = ⟨W, W ⟩_{H} .

This means in particular that the section (N_{H}(R), R) is a genetic section of P for the irreducible representation V. Thus N_{H}(R) = N_{P} (R) by Proposition 2.8, and R is a genetic subgroup of P.

If I = P, then ^{x}W ≅ W for any x ∈ P, and Res_{H}^{P}V ≅ pW. Let

V = V_{1} ⊕ … ⊕ V_{n}

be a decomposition of V as a direct sum of irreducible rational representations of P. Then Def_{P/Z}^{P}V_{i} = {0}, for i = 1, … n. Fix i in {1, …, n}, and let S_{i} denote any simple summand of Res_{E}^{P}V_{i}. Then either S_{i} ≅ ℚ, or there exists a subgroup F_{i} of index p in E, such that S_{i} is isomorphic to the kernel of the projection map ℚE/F_{i} → ℚ. Now Def_{E/Z}^{E}S_{i} = {0}, thus S_{i} ≅ ℚ̸, and the subgroup F_{i} is not equal to Z. It follows that N_{P} (F_{i}) = C_{P} (F_{i}) = C_{P} (E), since F_{i} has order p and E = F_{i}Z. Hence the stabilizer of S_{i} in P is equal to H, and by Clifford theory

V_{i} ≅ Ind_{H}^{P} S̃_{i} ,

where S̃_{i} is the S_{i}-isotypic component of Res_{E}^{P}V_{i}. By restriction to H, this gives

Res_{H}^{P}V_{i} ≅ ⊕_{x∈P/H} ^{x}S̃_{i} ,

and the representations ^{x}S̃_{i} appearing in the right hand side are mutually non isomorphic irreducible representations of H, since the stabilizer of S̃_{i} in P is equal to the stabilizer of S_{i}, i.e. the normalizer of F_{i} in P. It follows that

Res_{H}^{P}V ≅ pW ≅ ⊕_{i=1}^{n} ⊕_{x∈P/H} ^{x}S̃_{i} .

This gives a contradiction, since W has to be isomorphic to some conjugate of some S̃_{i}, but W is invariant by P, whereas the stabilizer of S̃_{i} in P is equal to H. This proves Assertion 1 of the lemma.

For Assertion 2, consider a genetic subgroup Q of P, such that Z ⊈ Q, and let V = V (S) denote the corresponding rational irreducible representation of P. Then Def_{P/Z}^{P}V = {0} by Lemma 5.1.

Let S be a simple summand of Res_{E}^{P}V. Then Def_{E/Z}^{E}S = {0}, thus as above S is isomorphic to the kernel of the projection map ℚE/F → ℚ, for some subgroup F of index p in E, and F ≠ Z. Hence the stabilizer of S in P is equal to N_{P} (F) = C_{P} (F) = C_{P} (E) = H, and

V ≅ Ind_{H}^{P}W ,

where W is equal to the S-isotypic component of Res_{E}^{P}V.

Moreover ⟨V, V ⟩_{P} = ⟨W, W ⟩_{H}, since the stabilizer of W in P is equal to H, and Def_{H/Z}^{H}W = {0}. Let R be any genetic subgroup of H associated to the representation W. Then R ⊉ Z, and (N_{H}(R), R) is a genetic section of P for V. Thus N_{H}(R) = N_{P} (R), and (N_{P} (R), R) _{P} (N_{P} (Q), Q), as was to be shown.

5.3. Corollary : In the situation of Lemma 5.2, there exists a genetic basis S of P and a decomposition

S = S_{1} ⊔ S_{2}

in disjoint union such that 1. if Q ∈ S_{2}, then Q ⊇ Z, and

2. the set {^{x}Q | Q ∈ S_{1}, x ∈ [P/C_{P} (E)]} is a set of representatives of genetic subgroups R of C_{P} (E) such that R ⊉ Z, for the relation _{C_{P} (E)} , where [P/C_{P} (E)] is any chosen set of representatives of C_{P} (E)-cosets in P.

Proof: Let S be a genetic basis of P. Set S_{2} = {Q ∈ S | Q ⊇ Z}, and S_{1} = S − S_{2}. By Lemma 5.2, I can assume that if Q ∈ S_{1}, then N_{P} (Q) ⊆ C_{P} (E), and Q is a genetic subgroup of C_{P} (E). Then the set

Γ = {^{x}Q | Q ∈ S_{1}, x ∈ [P/C_{P} (E)]}

is a set of genetic subgroups of C_{P} (E), not containing Z, and they are not equivalent to each other for the relation _{C_{P} (E)} : indeed, if there exist Q, Q′ ∈ S_{1} and elements x, x′ ∈ [P/C_{P} (E)] such that ^{x′}Q′ ^{hx}Q for some h ∈ C_{P} (E), then in particular Q′ ^{x^{′−1}hx}Q, thus Q′ = Q since S is a genetic basis of P. Moreover Q′ = Q = ^{x^{′−1}hx}Q since Q is genetic in P. Thus x^{′−1}hx ∈ N_{P} (Q) ⊆ C_{P} (E), thus x ∈ h^{−1}x′C_{P} (E) ⊆ C_{P} (E)x′C_{P} (E) = x′C_{P} (E), since C_{P} (E) \underline{◁} P, and x = x′.

Conversely, if R is a genetic subgroup of C_{P} (E) such that R ⊉ Z, then R is a genetic subgroup of P, and there exists an element y ∈ P and an element Q of S such that R ^{y}Q. This implies in particular that

Q ∩ Z = Q ∩ Z ∩ N_{P} (R^{y}) ⊆ Z ∩ R^{y} = (Z ∩ R)^{y} = 1 ,

hence Q ∈ S_{1}. Now y is equal to ux, for some unique element x in [P/C_{P} (E)], and some u ∈ C_{P} (E), because C_{P} (E) \underline{◁} P. Thus R _{C_{P} (E)} ^{x}Q, and ^{x}Q is an element of Γ.

6. The kernel of B → R_{ℚ}

6.1. Notation : Let B denote the Burnside biset functor (over ℤ), and R_{ℚ} denote the functor of rational representations. Let K denote the kernel of the natural morphism from B to R_{Q}.

By the Ritter-Segal Theorem, there is an exact sequence of biset functors

0 → K → B → R_{ℚ} → 0 .

6.2. Remark : For any p-group P, the corresponding sequence

0 → K(P) → B(P) → R_{ℚ}(P) → 0 .

is a split exact sequence, since the group R_{ℚ}(P) is a free group with basis Irr_{ℚ}(P) and rank equal to the number of conjugacy classes of cyclic subgroups of P. It follows that K(P) is a free group, of rank equal to the number of conjugacy classes of non cyclic subgroups of P. In particular K(P) = {0} if P is cyclic.

6.3. Notation : If P is a p-group, and B is a set of representatives of proper basic subgroups of P, for the relation ≑_{P}, set \overline{B} = B ⊔ \{P \}.

Denote by [s_{P}] a set of representatives of conjugacy classes of subgroups of P. The elements P/Q, for Q ∈ [s_{P}], form a ℤ-basis of B(P). If Q is a subgroup of P, set

S_{Q}^{B} = P/Q − P/P − ∑_{R∈B} m(V_{R}, ℚP/Q)(P/R − P/R̃) .

6.4. Lemma : Let κ_{B} the endomorphism of B(P) defined by

κ_{B}(P/Q) = S_{Q}^{B} ,

for Q ∈ [s_{P}]. Then κ_{B} is an idempotent endomorphism of B(P), whose image is equal to K(P).

In other words, the elements S_{Q}^{B}, for Q ∈ [s_{P}], generate K(P). Moreover the element u = ∑_{Q∈[s_{P}]} u_{Q}P/Q of B(P), where u_{Q} ∈ ℤ, is in K(P) if and only if u = ∑_{Q∈[s_{P}]} u_{Q}S_{Q}^{B}. Proof: Let u = ∑_{Q∈[s_{P}]} u_{Q}P/Q be an element of B(P), where u_{Q} ∈ ℤ. Its image in R_{ℚ}(P) is equal to

v = ∑_{Q∈[s_{P}]} u_{Q}ℚP/Q .

Now m(ℚ, ℚP/Q) = 1 for any subgroup Q of P, thus m(ℚ, v) = ∑_{Q∈[s_{P}]} u_{Q}. Moreover m(V_{R}, v) = ∑_{Q∈[s_{P}]} u_{Q}m(V_{R}, ℚP/Q), for R ∈ B. Since V_{R} is isomorphic to the kernel of the projection map ℚP/R → ℚP/R̃, it follows that V_{R} is equal to the image in R_{ℚ}(P) of the element P/R − P/R̃ of B(P). Since ℚ is the image of P/P in R_{ℚ}(P), it follows that v is also equal to the image in R_{ℚ}(P) of the element

u′ = ( ∑_{Q∈[s_{P}]} u_{Q})P/P + ∑_{R∈B} ∑_{Q∈[s_{P}]} u_{Q}m(V_{R}, ℚP/Q)(P/R − P/R̃) .

Thus u − u′ = ∑_{Q∈[s_{P}]} u_{Q}S_{Q}^{B} = κ_{B}(u) is in K(P). Moreover u is in K(P) if and only if u′ = 0, i.e. if u = κ_{B}(u). The lemma follows.

6.5. Corollary : Let E be an elementary abelian p-group of order p^{2}. Then K(E) = ∂K(E) is free of rank one, generated by the element

(6.6)  ε_{E} = E/1 − ∑_{{F ⊆E \atop |F |=p}} E/F + pE/E

of B(E). Proof: Indeed K(E) is free of rank 1 by Remark 6.2. In this case, by Remark 2.4, there is a unique set B of representatives of proper basic subgroups of E, consisting of all subgroups of E of order p, and its easy to check that κ_{B}(E/Q) = 0 if Q ≠ 1, and κ_{B}(E/1) = ε_{E}.

6.7. Notation : Fix an elementary abelian p-group E_{p^{2}} of order p^{2},and set ε = ε_{E_{p^{2}}} , defined in 6.6. Let K_{ε} denote the subfunctor of K generated by ε, i.e. the intersection of all subfunctors L of K such that L(E_{p^{2}}) ∋ ε.

Thus for any p-group P

K_{ε}(P) = Hom_{C_{p}}(E, P) ×_{E} ε .

6.8. Lemma : Let P be a p-group.

1. If Q is a subgroup of P such that Q ∩ Z(P) ≠ 1, then f_{1}^{P} P/Q = 0.

2. If the center of P is not cyclic, then ∂B(P) ⊆ K_{ε}(P).

Proof: If Q is a subgroup of P and Z = Q ∩ Z(P) ≠ 1, then P/Q = Inf_{P/Z}^{P}(P/Z)/(Q/Z), thus f_{1}^{P} P/Q = 0 by Lemma 3.12. This proves Assertion 1.

Now proving Assertion 2 amounts to showing that f_{1}^{P} P/Q ∈ K_{ε}(P), for any subgroup Q of P. Consider first the case Q = 1. Since the center Z(P) of P is not cyclic, there exists a subgroup E ⊆ Z(P) which is elementary abelian of order p^{2}. In particular the element

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

of B(E) is the image of ε under any isomorphism E_{p^{2}} → E. Thus ε_{E} ∈ K_{ε}(E). Inducing up to P gives the element

e = P/1 − ∑_{{F ∈E \atop |F |=p}} P/F + pP/E

of K_{ε}(P). But f_{1}^{P} e = f_{1}^{P} P/1 by Assertion 1, since E ⊆ Z(P). Hence f_{1}^{P} P/1 ∈ K_{ε}(P).

I will now show that f_{1}^{P} P/Q ∈ K_{ε}(P) for any subgroup Q of P by induction on the index |P : Q|. If Q = P, then f_{1}^{P} P/P = 0 ∈ K_{ε}(P), since the center of P is non trivial. Now let Q be any subgroup of P, and suppose that f_{1}^{P} P/R ∈ K_{ε}(P) for any subgroup R of P with |R| > |Q|. If Q ∩ Z(P) ≠ 1, then f_{1}^{P} P/Q = 0 ∈ K_{ε}(P). If Q ∩ Z(P) = 1, then Z(P) embeds in the center of \overline{N} = N_{P} (Q)/Q, so this center is not cyclic. The special case above shows that the element

f\overline{N} ; 1 \overline{N}/1 = ∑_{\overline{Z}⊆Ω_{1}Z(\overline{N})} μ(1, \overline{Z})\overline{N}/\overline{Z}

belongs to K_{ε}(\overline{N}). Taking inflation from \overline{N} to N, and then induction from N to P gives the element

w = Indinf_{N_{P} (Q)/Q}^{P}f\overline{N} ; 1 \overline{N}/1 = ∑_{Q′/Q⊆Ω_{1}Z(\overline{N})} μ(1, Q′/Q)P/Q′

of K_{ε}(P). It follows that

f_{1}^{P} w = f_{1}^{P} P/Q + ∑_{{Q′/Q⊆Ω_{1}Z(\overline{N}) \atop Q′/Q≠1}} μ(1, Q′/Q)f_{1}^{P} P/Q′ ∈ K_{ε}(P) .

By induction hypothesis, all terms in the summation are in K_{ε}(P), and by difference f_{1}^{P} P/Q ∈ K_{ε}(P), as was to be shown.

6.9. Notation :

1. If p ≠ 2, denote by X_{p^{3}} an extraspecial group of order p^{3} and exponent p, and by Z its center. Choose two non conjugate non central subgroups I and J of order p in X_{p^{3}}. Let δ be the element of B(X_{p^{3}}) defined by

δ = (X_{p^{3}}/I − X_{p^{3}}/IZ) − (X_{p^{3}}/J − X_{p^{3}}/JZ) .

2. If p = 2, and if n ≥ 3 is an integer, denote by D_{2^{n}} a dihedral group of order 2^{n}, and by Z its center. Choose two non conjugate non central subgroups I_{n} and J_{n} of order 2 in D_{2^{n}}. Let δ_{n} be the element of B(D_{2^{n}}) defined by

δ_{n} = (D_{2^{n}}/I_{n} − D_{2^{n}}/I_{n}Z) − (D_{2^{n}}/J_{n} − D_{2^{n}}/J_{n}Z) .

6.10. Remark : Let P be one of the groups X_{p^{3}} or D_{2^{n}}, for n ≥ 3. Then the center Z of P is cyclic of prime order, and P has a unique faithful rational representation : if P is dihedral of order at least 16, this follows from Proposition 3.7 in [8]. If P = D_{8}, this follows from the remark preceding Notation 3.8, in the same paper. And if P = X_{p^{3}}, this follows from the fact that P has p + 3 conjugacy classes of cyclic subgroups, and that P/Z ≅ (C_{p})^{2} has p + 2 such subgroups.

Now in each case, it is easy to see that any non central subgroup Q of prime order of P is basic, and the corresponding simple module is isomorphic to the kernel of the projection map ℚP/Q → ℚP/QZ. This shows that the elements (P/Q − P/QZ) − (P/R − P/RZ), for any non-central subgroups Q and R of prime order of P, are in K(P). In particular δ ∈ K(X_{p^{3}}), and δ_{n} ∈ K(D_{2^{n}}), for n ≥ 3.

6.11. Lemma :

1. If p ≠ 2, then JZ ≅ E_{p^{2}}, and Res_{JZ}^{X_{p^{3}}} δ = ε_{JZ}.

2. If p = 2, then J_{3}Z ≅ E_{p^{2}}, and Res_{J_{3}Z}^{D_{8}}δ_{3} = ε_{J_{3}Z}

Proof: This is a straightforward consequence of the Mackey formula.

6.12. Theorem :

1. If p ≠ 2, then the functor K is generated by δ : in other words for any

p-group P

K(P) = Hom_{C_{p}}(X_{p^{3}}, P) ×_{X_{p^{3}}} δ .

In particular \overline{K}(P) = \{0\} except if P is isomorphic to X_{p^{3}}, or if P is elementary abelian of order p^{2}.

2. If p = 2, then the functor K is generated by the elements δ_{n}, for n ≥ 3 : in other words for any 2-group P

K(P) = ∑_{n≥3} Hom_{C_{2}}(D_{2^{n}}, P) ×_{D_{2^{n}}} δ_{n} .

In particular \overline{K}(P) = \{0\}, except if P is dihedral of order at least 4.

Proof: If p ≠ 2, denote by L the subfunctor of K generated by δ, and if p = 2, denote by L the subfunctor of K generated by the elements δ_{n}, for n ≥ 3. I will show that L(P) = K(P) for any p-group P by induction on the order of |P |.

Step 1 : The induction starts with the case where P is cyclic : since K(P) = {0} in this case, the result is trivial.

Suppose that P is a p-group such that K(P′) = L(P′) for any p-group with |P′| < |P |. By Lemma 3.10 there is a decomposition

K(P) ≅ ⊕_{N \underline{◁} P} f_{N}^{P} K(P) ,

and f_{N}^{P} = Inf_{P/N}^{P} ∘ f_{1}^{P/N} ∘ Def_{P/N}^{P}. Now by induction hypothesis, it follows that f_{N}^{P} K(P) ⊆ L(P) if N ≠ 1, since K(P/N) = L(P/N). Hence in order to show that K(P) = L(P), it suffices to show that f_{1}^{P} K(P) = ∂K(P) ⊆ L(P).

Now if the center of P is not cyclic, then ∂K(P) ⊆ K_{ε}(P) by Lemma 6.8, and K_{ε}(P) ⊆ L(P) by Lemma 6.11. Thus ∂K(P) ⊆ L(P), and I can suppose that the center of P is cyclic.

Step 2 : Suppose that P admits a normal elementary abelian subgroup E of rank 2. Set H = C_{P} (E) and Z = E ∩ Z(P). Then |P : H| = |Z| = p, since E is not central in P. According to Corollary 5.3, there is a genetic basis S of P and a decomposition S = S_{1} ⊔ S_{2} with the following properties :

1. if S ∈ S_{2}, then S ⊇ Z, and

2. the set {^{x}S | S ∈ S_{1}, x ∈ [P/H]} is a set of representatives of genetic subgroups R of H such that R ⊉ Z, for the relation _{H} , where [P/H] is any chosen set of representatives of H-cosets in P.

For each S ∈ S, choose a basic subgroup R_{S}/S of N_{P} (S)/S, intersecting trivially the center of N_{P} (S)/S, and denote by \overline{B} the set of subgroups R_{S}, for S ∈ S. Then \overline{B} is a set of representatives of basic subgroups of P for the relation ≑_{P}, and in particular it contains P. Set moreover B = \overline{B} − \{P \}.

Observe now that R_{S} ⊇ Z if S ⊇ Z. Conversely, if S ⊉ Z, then ZS/S is the only subgroup of the center of N_{P} (S)/S of order p, thus R_{S} ∩ ZS = S, and R_{S} ⊉ Z. This gives a decomposition of \overline{B} as

\overline{B} = \overline{B}_{1} ⊔ \overline{B}_{2}

where

\overline{B}_{1} = \{R ∈ \overline{B} | R ⊉ Z\} = \{R_{S} | S ∈ S_{1}\} .
\overline{B}_{2} = \{R ∈ \overline{B} | R ⊇ Z\} = \{R_{S} | S ∈ S_{2}\}

By Lemma 6.4, the group K(P) is generated by the elements

S_{Q}^{B} = P/Q − P/P − ∑_{R∈B} m(V_{R}, ℚP/Q)(P/R − P/R̃) ,

for Q ∈ [s_{P}]. Hence in order to show that f_{1}^{P} K(P) = ∂K(P) ⊆ L(P), it is enough to show that f_{1}^{P} S_{Q}^{B} ∈ L(P) for any subgroup Q of P. Moreover f_{1}^{P} P/Q = P/Q − P/QZ for any Q ⊆ P, since Z = Ω_{1}Z(P) is the only central subgroup of order p of P. Thus if R ∈ B and R ⊇ Z, then R̃ ⊇ Z, and f_{1}^{P} P/R = f_{1}^{P} P/R̃ = 0. And if R ⊉ Z, then R̃ = RZ, and f_{1}^{P} (P/R − P/R̃) = P/R − P/RZ in this case. Since f_{1}^{P} P/P = 0, one has that

f_{1}^{P} S_{Q}^{B} = P/Q − P/QZ − ∑_{R∈\overline{B}_{1}} m(V_{R}, ℚP/Q)(P/R − P/RZ) .

6.13. Remark : This shows in particular f_{1}^{P} S_{Q}^{B} = 0 if Q ⊇ Z. Indeed in that case if R ∈ B is such that m(V_{R}, ℚP/Q) ≠ 0, then there exists x ∈ P such that Q^{x} ∩ N_{P} (R) ⊆ R (see Proposition 2.5 in [8]). Thus

Z = Q^{x} ∩ Z ⊆ Q^{x} ∩ N_{P} (R) ⊆ R .

Step 3 : Suppose first that Q ⊆ H. Since \overline{B}_{1} = \{R_{S} | S ∈ S_{1}\}, if R = R_{S} ∈ \overline{B}_{1}, then S ⊆ H by choice of S, and then by Lemma 5.2, the corresponding simple module V_{R} = V (S) is isomorphic to Ind_{H}^{P}W_{R}, where W_{R} = W (S) is the simple ℚH-module corresponding to the genetic subgroup S of H. Moreover in that case ⟨V_{R}, V_{R}⟩_{P} = ⟨W_{R}, W_{R}⟩_{H}.

Let V be any ℚP -module. Then

⟨V_{R}, V ⟩_{P} = m(V_{R}, V)⟨V_{R}, V_{R}⟩_{P}
= ⟨Ind_{H}^{P}W_{R}, V ⟩_{P} = ⟨W_{R}, Res_{H}^{P}V ⟩_{H}
= m(W_{R}, Res_{H}^{P}V)⟨W_{R}, W_{R}⟩_{H} ,

thus m(V_{R}, V) = m(W_{R}, Res_{H}^{P}V). This gives

f_{1}^{P} S_{Q}^{B} = P/Q − P/QZ − ∑_{R∈\overline{B}_{1}} m(W_{R}, Res_{H}^{P}ℚP/Q)(P/R − P/RZ) ,

and since Res_{H}^{P}ℚP/Q ≅ ⊕_{x∈[P/H]}ℚH/Q^{x}, where [P/H] is some set of representatives of H-cosets in P, this leads to

(6.14) f_{1}^{P} S_{Q}^{B} = P/Q − P/QZ − ∑_{{R∈\overline{B}_{1} \atop x∈[P/H]}} m(W_{R}, ℚH/Q^{x})(P/R − P/RZ)

By Corollary 5.3 again, it is possible to choose a genetic basis T of H such that

{^{x}S | S ∈ S_{1}, x ∈ [P/H]} = {S ∈ T | S ⊉ Z} .

This also means that the set

D_{1} = \{^{x}R_{S} | S ∈ S_{1}, x ∈ [P/H]\} = \{^{x}R | R ∈ \overline{B}_{1}, x ∈ [P/H]\}

can be completed to a set \overline{D} of representatives of basic subgroups of H for the relation ≑_{H}. Set D = \overline{D} − \{H\}, and for R ∈ \overline{D}, denote by W_{R} the corresponding simple ℚH-module (so in particular, this notation is consistent for R ∈ \overline{B}_{1}). Then the elements

S_{T}^{D} = H/T − H/H − ∑_{R∈D} m(W_{R}, ℚH/T)(H/R − H/R̃) ,

for T ⊆ H, generate K(H), hence by induction hypothesis, they belong to L(H). This shows that the element

S_{Q}^{D} − S_{QZ}^{D} = H/Q − H/QZ − ∑_{R∈D} m(W_{R}, ℚH/Q − ℚH/QZ)(H/R − H/R̃)

belongs to L(H). Now if R ∈ D and R ⊇ Z, then W_{R} = Inf_{H/Z}^{H}W′_{R}, where W′_{R} = Def_{H/Z}^{H}W_{R}. Moreover ⟨W_{R}, W_{R}⟩_{H} = ⟨W′_{R}, W′_{R}⟩_{H/Z}, thus

m(W_{R}, ℚH/Q) = m(W′_{R}, ℚH/QZ) = m(W_{R}, ℚH/QZ) .

This shows that m(W_{R}, ℚH/Q − ℚH/QZ) = 0 if R ⊇ Z, i.e. if R ∉ D_{1}. And if R ∈ D_{1}, then R̃ = RZ, and

⟨W_{R}, ℚH/QZ⟩_{H} = ⟨Def_{H/Z}^{H}W_{R}, ℚ(H/Z)/(QZ/Z)⟩_{H/Z} = 0 ,

since Def_{H/Z}^{H}W_{R} = {0}. Hence m(W_{R}, ℚH/QZ) = 0, and this gives finally

S_{Q}^{D} − S_{QZ}^{D} = H/Q − H/QZ − ∑_{R∈D_{1}} m(W_{R}, ℚH/Q)(H/R − H/RZ) .

Comparing with expression 6.14, since obviously

m(W_{R}, ℚH/Q^{x}) = m(W_{^{x}R}, ℚH/Q) ,

it follows that

f_{1}^{P} S_{Q}^{B} = Ind_{H}^{P}(S_{Q}^{D} − S_{QZ}^{D}) .

showing that f_{1}^{P} S_{Q}^{B} ∈ L(P) if Q ⊆ H.

Step 4 : Suppose now that Q ⊈ H. This case will be handled by the following lemma :

6.15. Lemma : Let P be a p-group, and suppose that E is a normal subgroup of P, which is elementary abelian of rank 2, and not contained in the center of P. Set Z = E ∩ Z(P). Let S be a subgroup of P, such that S ⊉ Z, and S ⊈ C_{P} (E).

1. The group C_{S}(E) = S ∩ C_{P} (E) is a normal subgroup of SE, and the quotient SE/C_{S}(E) is isomorphic to X_{p^{3}} if p ≠ 2, and to D_{8} if p = 2.

2. There exist a subgroup T of C_{P} (E) and a morphism φ in C_{p} from X_{p^{3}} to P if p ≠ 2 (resp. from D_{8} to P if p = 2), such that the element

(P/S − P/SZ) − (P/T − P/T Z)

is the image by φ of the element δ of K(X_{p^{3}}) (resp. of the element δ_{3} of K(D_{8})).

Proof: Since E is normal in P, and not central in P, the group C_{P} (E) has index p in P. Now C_{P} (E) \underline{◁} P, hence S normalizes C_{P} (E), hence also S ∩ C_{P} (E). Since E centralizes this latter group, it follows that C_{S}(E) \underline{◁} SE. Moreover E∩C_{S}(E) = E∩S. If this is not trivial, then it has order p, otherwise E ⊆ S, hence Z ⊆ S. Now S normalizes S ∩ E, hence it centralizes it. Thus S centralizes E = (S ∩ E)Z, and this contradicts the hypothesis.

Hence E ∩ C_{S}(E) = 1, thus E embeds as a normal elementary abelian subgroup \overline{E} of order p^{2} in the quotient X = SE/C_{S}(E). Now X is equal to the semi-direct product of \overline{E} by its subgroup \overline{S} = S/C_{S}(E), which has order |S : S ∩ C_{P} (E)| = p. Moreover X is not abelian, since otherwise [S, E] ⊆ E ∩ C_{S}(E) = 1, thus S ⊆ C_{P} (E). Assertion 1 of the lemma follows.

The group X has center of order p, equal to \overline{Z} = ZC_{S}(E)/C_{S}(E). Let θ be any group isomorphism from X_{p^{3}} to X if p ≠ 2 (resp. from D_{8} to X if p = 2) sending the subgroup IZ of X_{p^{3}} (resp. the subgroup I_{3}Z of D_{8}) to \overline{S}\overline{Z}, and the subgroup JZ of X_{p^{3}} (resp. the subgroup J_{3}Z of D_{8}) to \overline{E}. Such an isomorphism exists since the automorphism group of X_{p^{3}} (resp. of D_{8}) acts 2-transitively on its elementary abelian subgroups of order p^{2}. Then θ(I) (resp. θ(I_{3})) is some conjugate of \overline{S} in X, and θ(J) (resp. θ(J_{3})) is some subgroup \overline{T} of \overline{E}. Let w = δ if p ≠ 2 (resp. w = δ_{3} if p = 2). Then

θ(w) = (X/\overline{S} − X/\overline{S}\overline{Z}) − (X/\overline{T} − X/\overline{T} \overline{Z}) .

Taking inflation to SE, and then induction to P gives the element

Indinf_{SE/C_{S}(E)}^{P}θ(w) = (P/S − P/SZ) − (P/T − P/T Z) ,

where T is the preimage in SE of the subgroup \overline{T} of \overline{E} = EC_{S}(E)/C_{S}(E) under the projection SE → SE/C_{S}(E).

In particular T ⊆ E(S ∩C_{P} (E)) = C_{P} (E) (since E ⊆ C_{P} (E) ⊆ SC_{P} (E) = P), proving Assertion 2 of the lemma. Step 5 : It remains to show that f_{1}^{P} S_{Q}^{B} ∈ L(P) if Q is a subgroup of P, such that Q ⊉ Z and Q ⊈ H = C_{P} (E). By Lemma 6.15, there exists a subgroup T of H such that the element

u = (P/Q − P/QZ) − (P/T − P/T Z)

is the image of δ (resp. δ_{3}) by some morphism in the category C_{p}. Since δ ∈ L(X_{p^{3}}) (resp. δ_{3} ∈ L(D_{8})), it follows that u ∈ L(P). Since moreover L(P) ⊆ K(P), the element u is equal to

(S_{Q}^{B} − S_{QZ}^{B}) − (S_{T}^{B} − S_{T Z}^{B})

by Lemma 6.4. It follows that

(f_{1}^{P} S_{Q}^{B} − f_{1}^{P} S_{QZ}^{B}) − (f_{1}^{P} S_{T}^{B} − f_{1}^{P} S_{T Z}^{B}) ∈ L(P) .

Since T ⊆ H, the previous discussion shows that f_{1}^{P} S_{T}^{B} ∈ L(P). Moreover f_{1}^{P} S_{QZ}^{B} = f_{1}^{P} S_{T Z}^{B} = 0 by Remark 6.13. Hence f_{1}^{P} S_{Q}^{B} ∈ L(P), as announced.

Step 6 : To complete the proof of Theorem 6.12, it remains to consider the case where P has no elementary abelian normal subgroup of order p^{2}, i.e. the case where P has normal p-rank 1. If p is odd, there is nothing more to do, since then P is cyclic, and K(P) = {0}. If p = 2, then :

− if P is cyclic, again there is nothing more to do, since K(P) = {0}. − if P is generalized quaternion, then ∂K(P) = {0} : indeed the only

subgroup of P intersecting trivially the center is the trivial group, and the trivial group is a normal basic subgroup of P, hence it belongs to any set B of representatives of basic subgroups of P, by Remark 2.4. Clearly then S_{1}^{B} = 0.

− if P is semi-dihedral, then ∂K(P) is free of rank 1 : up to conjugation, there are only two subgroups of P which intersect trivially the center of P, namely the trivial group and the non central subgroup Q of order 2, which is a basic subgroup of P. Then any set B of representatives of basic subgroups of P modulo the relation ≑_{P} must contain Q (up to conjugation). Then clearly f_{1}^{P} S_{Q}^{B} = 0. On the other hand

f_{1}^{P} S_{1}^{B} = P/1 − P/Z − m(V_{Q}, ℚP/1)(P/Q − P/QZ)
= P/1 − P/Z − 2P/Q + 2P/QZ
= Ind_{QZ}^{P}ε_{QZ}

Thus in this case also ∂K(P) ⊆ K_{ε}(P) ⊆ L(P). − The only remaining case is when P is dihedral, say P ≅ D_{2^{n}}, with

n ≥ 4. In this case up to conjugation, there are 3 subgroups which intersect trivially the center of P, namely the trivial group, and the non central subgroups I_{n} and J_{n} of order 2. Since I_{n} ≑_{P} J_{n}, I can suppose that B contains J_{n} (and not I_{n}). In this case

f_{1}^{P} S_{Q}^{B} = \{ \begin{matrix}P/1 − P/Z − 2(P/J_{n} − P/J_{n}Z) & if Q = 1 \\ (P/I_{n} − P/I_{n}Z) − (P/J_{n} − P/J_{n}Z) & if Q =_{P} I_{n} \\ 0 & otherwise \end{matrix}

Thus f_{1}^{P} S_{1}^{B} = Ind_{J_{n}Z}^{P}ε_{J_{n}Z} is in L(P), by Lemma 6.11, and f_{1}^{P} S_{I_{n}}^{B} = δ_{n} ∈ L(P). Thus ∂K(P) ⊆ L(P) also in this case.

The only thing to prove now is that \overline{K}(P) = \{0\} except if P ≅ X_{p^{3}} or P ≅ E_{p^{2}} if p ≠ 2, or if P is dihedral of order at least 4 if p = 2. But by Remark 3.4, if Q is any p-group, then any transitive (P, Q)-biset factors as a composition of a deflation-restriction to some section of Q, followed by a group isomorphism, followed by an induction-inflation from a section of P.

Since any section of X_{p^{3}} is cyclic, isomorphic to E_{p^{2}} or to X_{p^{3}}, since any section of a dihedral group is cyclic or dihedral of order at least 4 (with the usual convention that a dihedral group of order 4 is elementary abelian), it follows that for any φ ∈ Hom_{C_{p}}(X_{p^{3}}, P) (resp. for any φ ∈ Hom_{C_{2}}(D_{2^{n}}, P)), the element φ(δ) (resp. the element φ(δ_{n})) is a linear combination of elements of the form Indinf_{T/S}^{P}v, for sections T/S of P which are also sections of X_{p^{3}} (resp. of D_{2^{n}}), and elements v ∈ K(T/S). Here of course I can assume that T/S is not cyclic, otherwise K(T/S) = {0}.

Hence \overline{K}(P) = \{0\} if P is not isomorphic to E_{p^{2}} or X_{p^{3}} for p ≠ 2 (resp. if P is not dihedral of order at least 4 if p = 2). This completes the proof of Theorem 6.12.

6.16. Corollary : Let P be a p-group. Then K(P) is equal to the set of linear combinations of elements of the form Indinf_{T/S}^{P}θ(κ), where (T, S) is a section of P, where θ is a group isomorphism from one of the groups X_{p^{3}}, E_{p^{2}}, or D_{2^{n}} (n ≥ 3) to T/S, and κ is respectively equal to δ, ε, δ_{n}.

6.17. Remark : One can check easily that if P is isomorphic to one of the groups E_{p^{2}}, X_{p^{3}}, D_{2^{n}} for n ≥ 3, then \overline{K}(P) is actually non zero, respectively isomorphic to ℤ, ℤ/pℤ, or ℤ/2ℤ, and generated by the image of ε, δ, or δ_{n}.

7. The Dade group modulo torsion

In this section I will show that D(P) = D^{Ω}(P) + D_{tors}(P), for any pgroup P. In the case p ≠ 2, this will be enough to conclude that D = D^{Ω}.

7.1. Theorem : Let n be a positive integer. Then

(nD ∩ D^{Ω}) + D_{tors}^{Ω} = nD^{Ω} + D_{tors}^{Ω} .

Proof: If n = 1, there is nothing to prove, so assume n ≥ 2. By Theorem 1.8 of [9], there is an exact sequence of bisets functors

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

where R_{Q}^{∗} and B^{∗} are the respective ℤ-dual of R_{ℚ} and B, and the inclusion R_{ℚ}^{∗} → B^{∗} is the transpose of the natural morphism B → R_{ℚ}. If P is a p-group, evaluation at P gives the exact sequence of abelian groups

0 → R_{ℚ}^{∗}(P) → B^{∗}(P) → (D^{Ω}/D_{tors}^{Ω})(P) → 0 ,

and this sequence is actually split exact, since (D^{Ω}/D_{tors}^{Ω})(P) is a free abelian group. In particular, this sequence remains exact after tensoring (over ℤ) with Γ_{n} = ℤ/nℤ, and this shows that the sequence

0 → Γ_{n}R_{ℚ}^{∗} → Γ_{n}B^{∗} → Γ_{n}(D^{Ω}/D_{tors}^{Ω}) → 0

is an exact sequence of biset functors, where the ⊗_{ℤ} symbols have been dropped (so e.g. Γ_{n}R_{ℚ} denotes Γ_{n} ⊗_{ℤ} R_{ℚ}). Now there are canonical isomorphisms

Γ_{n}R_{ℚ}^{∗} ≅ Hom_{Γ_{n}}(Γ_{n}R_{ℚ}, Γ_{n})
Γ_{n}B^{∗} ≅ Hom_{Γ_{n}}(Γ_{n}B, Γ_{n}) .

Moreover

Γ_{n}(D^{Ω}/D_{tors}^{Ω}) ≅ (D^{Ω}/D_{tors}^{Ω})/n(D^{Ω}/D_{tors}^{Ω})
≅ (D^{Ω}/D_{tors}^{Ω})/((nD^{Ω} + D_{tors}^{Ω})/D_{tors}^{Ω})
≅ D^{Ω}/(nD^{Ω} + D_{tors}^{Ω}) .

Setting T_{n} = D^{Ω}/(nD^{Ω} + D_{tors}^{Ω}), this gives the exact sequence

(7.2)  0 → Hom_{Γ_{n}}(Γ_{n}R_{ℚ}, Γ_{n}) → Hom_{Γ_{n}}(Γ_{n}B, Γ_{n}) → T_{n} → 0 .

Note that T_{n}(P) is a free Γ_{n}-module, for any p-group P. Now on the other hand the exact sequence of biset functors

0 → K → B → R_{ℚ} → 0

remains exact after tensoring with Γ_{n}, since every evaluation of this sequence is a split exact sequence of abelian groups. Hence there is an exact sequence

0 → Γ_{n}K → Γ_{n}B → Γ_{n}R_{ℚ} → 0 ,

and every evaluation of this sequence is a split exact sequence of Γ_{n}-modules. Now taking Γ_{n}-duals leads to the exact sequence (7.3)

0 → Hom_{Γ_{n}}(Γ_{n}R_{ℚ}, Γ_{n}) → Hom_{Γ_{n}}(Γ_{n}B, Γ_{n}) → Hom_{Γ_{n}}(Γ_{n}K, Γ_{n}) → 0 .

Comparing the sequences 7.2 and 7.3 gives the following natural isomorphism of biset functors

(7.4)  T_{n} ≅ Hom_{Γ_{n}}(Γ_{n}K, Γ_{n}) .

This means that for any p-group P, there is a non-degenerate scalar product

( , )_{P} : T_{n}(P) × Γ_{n}K(P) → Γ_{n}

with the property that for any p-group Q and any φ ∈ Hom_{C_{p}}(Q, P), one has that

(7.5)  ∀u ∈ T_{n}(P), ∀v ∈ Γ_{n}K(Q), (u, φ(v))_{P} = (φ^{op}(u), v)_{Q} .

Now set

F = ((nD ∩ D^{Ω}) + D_{tors}^{Ω})/(nD^{Ω} + D_{tors}^{Ω}) .

Then F is a biset subfunctor of T_{n}. Suppose that F ≠ {0}, and let P be a p-group of minimal order such that F (P) ≠ {0}. Then F (P) is a subset of

\underline{T}_{n}(P) ⊆ \{u ∈ T_{n}(P) | Defres_{T/S}^{P}(u) = 0, ∀S \underline{◁} T ⊆ P, |T/S| < |P |\} .

By the above duality 7.5, it follows that if u ∈ \underline{T_{n}}(P), then for any proper section T/S of P, and any v ∈ Γ_{n}K(P)

(u, Indinf_{T/S}^{P}(v))_{P} = 0 .

Hence (u, w)_{P} = 0 for any w ∈ ∑_{{S \underline{◁} T ⊆P \atop |T/S|<|P |}} Indinf_{T/S}^{P}Γ_{n}K(T/S). By Theo-

rem 6.12, this is the whole of Γ_{n}K(P), unless P is isomorphic to X_{p^{3}} or E_{p^{2}} if p ≠ 2, or if P is dihedral of order at least 4 if p = 2. Thus if P is not isomorphic to one of the groups in this list, then (u, v)_{P} = 0 for any v ∈ Γ_{n}K(P), hence u = 0 since the scalar product is non degenerate. Thus \underline{T}_{n}(P) = \{0\}.

Hence the minimal group P is one of the groups X_{p^{3}}, E_{p^{2}} or D_{2^{n}} (note that E_{2^{2}} ≅ D_{2^{2}}). If F (P) = {0} in each of these cases, this gives a contradiction, proving that F = {0}. But if P is one of the groups X_{p^{3}}, E_{p^{2}} or D_{2^{n}}, then D(P) = D^{Ω}(P) : for P = X_{p^{3}}, this follows from Theorem 10.2 of [11], for P = E_{p^{2}}, this follows from Theorem 4.6, and for P = D_{2^{n}}, this follows from Theorem 4.7. Hence nD(P) ∩ D^{Ω}(P) = nD^{Ω}(P), and F (P) = {0}. This completes the proof of the theorem.

7.6. Remark : Using the fact that Γ_{n} is a self injective ring, one can show that the above isomorphism 7.4 actually restricts to an isomorphism

\underline{T}_{n}(P) ≅ Hom_{Γ_{n}}(\overline{Γ_{n}K}(P), Γ_{n}) ,

for any p-group P. Moreover

\overline{Γ_{n}K}(P) = Γ_{n}K(P)/ ∑_{{S \underline{◁} T ⊆P \atop |T/S|<|P |}} Indinf_{T/S}^{P}Γ_{n}K(T/S)

≅ (K(P)/nK(P))/ ∑_{{S \underline{◁} T ⊆P \atop |T/S|<|P |}} Indinf_{T/S}^{P}(K(T/S)/nK(T/S))

≅ K(P)/( ∑_{{S \underline{◁} T ⊆P \atop |T/S|<|P |}} Indinf_{T/S}^{P}K(T/S) + nK(P))

≅ \overline{K}(P)/n\overline{K}(P) ≅ Γ_{n}\overline{K}(P) ,

so in fact there is always a group isomorphism \underline{T}_{n}(P) ≅ Hom_{Γ_{n}}(Γ_{n}\overline{K}(P), Γ_{n}).

7.7. Theorem :

1. D = D^{Ω} + D_{tors}.

2. If p ≠ 2, then D = D^{Ω}.

(Assertion 2 was the first part of Conjecture B in [4]). Proof: Indeed, by Proposition 7.4.9 of [6], for any p-group P, the quotient D(P)/(D^{Ω}(P) + L(P)) is a finite p-group, where L(P) is the intersection of the kernels of the maps Defres_{T/S}^{P}, where T/S is an elementary abelian section of P. Thus if u ∈ D(P), then there is m ∈ ℕ such that p^{m}u ∈ D^{Ω}(P) + L(P).

Now the group L(P) is always a subgroup of D_{tors}(P), by Theorem 1.6 of [12]. Let e denote its exponent. Then ep^{m}u ∈ D^{Ω}(P), hence

ep^{m}u ∈ ep^{m}D(P) ∩ D^{Ω}(P) ⊆ ep^{m}D^{Ω}(P) + D_{tors}^{Ω} ,

by Theorem 8.2. Hence there exists v ∈ D^{Ω}(P) such that the difference ep^{m}u − ep^{m}v = ep^{m}(u − v) is a torsion element of D^{Ω}(P). Hence u − v is a torsion element of D(P), proving Assertion 1.

If p ≠ 2, it has been shown by Carlson and Thévenaz (see Theorem 13.1 of [14]) that L(P) = {0}. Hence with the same notation e = 1, and the difference w = p^{m}(u−v) is in D_{tors}^{Ω}(P). This group is a 2-group of exponent 2 (by Corollary 7.6 of [10], or by Corollary 13.2 of [14]). Thus w = p^{m}w, and p^{m}(u − v − w) = 0. But by Corollary 13.2 of [14], there is no p-torsion in D(P). Thus u = v + w ∈ D^{Ω}(P), and D(P) = D^{Ω}(P) in this case.

8. The torsion part of the Dade group

8.1. Proposition : Let P be a p-group, and suppose that E is a normal subgroup of P, which is elementary abelian of rank 2. Let Z be a subgroup of order p of E ∩ Z(P). Then

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

Proof: by induction on the order of P : suppose that the result holds for all p-groups Q with |Q| < |P |.

Step 1 : If E is central in P, the result is trivial, for in that case C_{P} (E) = P, and Res_{C_{P} (E)}^{P} is the identity map.

If E is not central in P, then |P : C_{P} (E)| = p, and E ∩ Z(P) has order p, hence it is equal to Z. Let u ∈ D_{tors}(P), such that Res_{C_{P} (E)}^{P}u = 0 and Def_{P/Z}^{P}u = 0.

Let H be a proper subgroup of P, containing E, and consider v = Res_{H}^{P}u. Then the hypothesis of the proposition holds for H, its normal subgroup E, and its subgroup Z contained in E ∩ Z(H). Moreover v is in D_{tors}(H), and

Res_{C_{H}(E)}^{H}v = Res_{C_{H}(E)}^{P}u
= Res_{C_{H}(E)}^{C_{P} (E)}Res_{C_{P} (E)}^{P}u = 0 .

Similarly

Def_{H/Z}^{H}v = Def_{H/Z}^{H}Res_{H}^{P}u
= Res_{H/Z}^{P/Z}Def_{P/Z}^{P}u = 0 .

The induction hypothesis now shows that v = 0. Thus Res_{H}^{P}u = 0 for any proper subgroup of P containing E.

Consider now a subgroup Z′ of order p of Z(P), not contained in E (equivalently Z′ ≠ Z, since E is not central in P). The group E embeds in the group \overline{P} = P/Z′, since E ∩ Z′ = 1, and its image \overline{E} = EZ′/Z′ is a normal subgroup of \overline{P}.

If u ∈ P is such that uZ′ ∈ C_{\overline{P}} (\overline{E}), then [u, E] ⊆ Z′, hence [u, E] ⊆ E ∩ Z′ = 1. This shows that C_{\overline{P}} (\overline{E}) = C_{P} (E)/Z′. Moreover \overline{Z} = Z.Z′/Z′ is a central subgroup of order p of \overline{P}, contained in \overline{E}.

Set w = Def_{P/Z′}^{P}u. Then w ∈ D_{tors}(\overline{P}), and

Res\overline{P} ; C_{\overline{P}} (\overline{E})w = Res_{C_{P} (E)/Z′}^{P/Z′}Def_{P/Z′}^{P}u
= Def_{C_{P} (E)/Z′}^{C_{P} (E)}Res_{C_{P} (E)}^{P}u = 0 .

Similarly

Def\overline{P} ; \overline{P} /\overline{Z}w = Def_{P/ZZ′}^{P}u
= Def_{P/ZZ′}^{P/Z}Def_{P/Z}^{P}u = 0 .

Then w = 0 by induction hypothesis. It follows that Def_{P/X}^{P}u = 0 for any central subgroup X of order p of P, and then Def_{P/N}^{P}u = 0 for any non trivial normal subgroup N of P.

On the other hand, Carlson and Thévenaz have shown that the element u of D_{tors}(P) is equal to 0 if and only if Defres_{N_{P} (Q)/Q}^{P}u = 0 for any subgroup Q of P such that N_{P} (Q)/Q is cyclic, if p ≠ 2, or cyclic, generalized quaternion, or semi-dihedral, if p = 2 (see [14] Theorem 13.4). Let Q be such a subgroup. If M = Q ∩ Z(P) is non trivial, then setting \overline{P} = P/M and \overline{Q} = Q/M

Defres_{N_{P} (Q)/Q}^{P}u = Defres\overline{P} ; N_{\overline{P}} (\overline{Q})/\overline{Q}DefP ; \overline{P} u = 0 .

And if M = 1, then Z(P) embeds in the center of the group N_{P} (Q)/Q, which is always cyclic when N_{P} (Q)/Q is cyclic, generalized quaternion, or semi-dihedral. This cannot happen if Z(P) is not cyclic, and u = 0 in this case.

Step 2 : Hence I can assume that Z(P) is cyclic. Then by Assertion (b) of Theorem 13.1 of [14], the element u of D_{tors}(P) is equal to 0 if and only if Defres_{T/S}^{P}u = 0, for any section (T, S) of P such that T/S is cyclic of order p if p ≠ 2, or cyclic of order 4 or quaternion of order 8 if p = 2.

Let (T, S) be such a section. If M = S ∩Z(P) ≠ 1, then setting \overline{P} = P/M, \overline{T} = T/M and \overline{S} = S/M

Defres_{T/S}^{P}u = Defres\overline{P} ; \overline{T} /\overline{S}DefP ; \overline{P} u = 0 .

Similarly, if H = T E ≠ P, then

Defres_{T/S}^{P}u = Defres_{T/S}^{H}Res_{H}^{P}u = 0 ,

since H is a proper subgroup of P containing E.

Step 3 : So I can suppose that T E = P, and S ∩ Z(P) = 1. Set M = S ∩ C_{P} (E). Then M is normalized by T and centralized by E, thus M \underline{◁} P, and M ∩ Z(P) = 1 since M is a subgroup of S. Thus M = 1. Since C_{P} (E) has index p in P, this implies that |S| ≤ p, and there are two cases :

• if |S| = p, then S ⊆ T ⊆ SC_{P} (E) = P, thus T = S(T ∩ C_{P} (E)), and in particular T/S ≅ T ∩ C_{P} (E) is isomorphic to C_{p} for p ≠ 2, or to C_{4} or Q_{8} for p = 2. Moreover S in central in T, thus T = S × (T ∩ C_{P} (E)). The group SE is isomorphic to the (non trivial) semi-direct product of C_{p} by E_{p^{2}}, hence it is isomorphic to X_{p^{3}} if p ≠ 2, and to D_{8} if p = 2. The group E ∩ T has order at most p (otherwise E ⊆ T, thus E ⊆ T ∩ C_{P} (E), and this cannot happen since T ∩ C_{P} (E) is one of the groups C_{p}, C_{4} or Q_{8}). If E ∩ T = 1, then P = SE × (T ∩ C_{P} (E)), since

SE ∩ (T ∩ C_{P} (E)) = S(E ∩ T) ∩ C_{P} (E) = S ∩ C_{P} (E) = 1 .

In that case the center of P cannot be cyclic, since both groups SE and T ∩ C_{P} (E) are non trivial.

It follows that E ∩ T has order p. If E ∩ T ≠ Z, then E = (E ∩ T)Z is centralized by T, hence contained in the center of P. Thus E ∩ T = Z. In that case

SE ∩ (T ∩ C_{P} (E)) = SZ ∩ C_{P} (E) = Z(S ∩ C_{P} (E)) = Z ,

and P is isomorphic to the central product of SE and T ∩ C_{P} (E). Thus if p ≠ 2, then T ∩ C_{P} (E) = T ∩ E = Z, for T ∩ E ⊆ T ∩ C_{P} (E), and both have order p, and P = SE ≅ X_{p^{3}} in this case. And if p = 2, the group P is isomorphic to D_{8} ∗ C_{4} or D_{8} ∗ Q_{8}.

• if S = 1, then T is isomorphic to C_{p} if p ≠ 2, or to C_{4} or Q_{8} if p = 2. Again if E ∩ T = 1 then P = E ⋊ T ≅ E_{p^{2}} ⋊ C_{p} ≅ X_{p^{3}} if p ≠ 2. And if p = 2, then P ≅ E_{4} ⋊ C_{4} or P ≅ E_{4} ⋊ Q_{8}, and in each case Z(P) = Z × Z(T) ≅ C_{2} × C_{2}, which is not cyclic. And if E ∩ T ≠ 1, then E ∩ T = Z as above. If p ≠ 2, then T has order p, hence T ⊆ E, hence P = E, a contradiction. And if p = 2, then P ≅ E ⋊ T/Δ, where Δ is the unique subgroup of order 2 of Z × Z(T) which is neither contained in Z nor in Z(T). If T ≅ C_{4}, then P ≅ D_{8}, and u = 0 since D(D_{8}) is torsion free by Theorem 10.3 of [13]. And if T ≅ Q_{8}, then clearly [P, P] = (Z × [T, T])/Δ, which has order 2, and is also equal to the Frattini subgroup of P, since P/[P, P] is elementary abelian of order 8. In this case also Z(P) is cyclic of order 4, generated by the element et, where e ∈ E − Z, and t is a generator of C_{T} (E). Hence P is almost extraspecial of order 16, isomorphic to P ≅ D_{8} ∗ C_{4} again.

Step 4 : Finally, the only cases left to consider are

• Case 1 : p ≠ 2 and P ≅ X_{p^{3}},

• Case 2 : p = 2 and P ≅ D_{8} ∗ C_{4},

• Case 3 : p = 2 and P ≅ D_{8} ∗ Q_{8}.

By Theorem 9.1 of [11], the group ∂D_{tors}(P) is cyclic of order 2 in Case 1 and Case 2, or cyclic of order 4 in Case 3 and the ground field does not contain primitive cubic roots of unity. In Case 3, when the ground field contains primitive cubic roots of unity, then ∂D_{tors}(P) ≅ ℤ/4ℤ ⊕ ℤ/2ℤ, by Theorem 10.1 of [11].

Note that by Theorem 4.6 and Theorem 4.7, this can be expressed as

∂D_{tors}(P) ≅ \{ \begin{matrix}T_{tors}(C_{p}) & in Case 1 \\ T_{tors}(C_{4}) & in Case 2 \\ T_{tors}(Q_{8}) & in Case 3 \end{matrix} .

Using this observation, these results can be interpreted as follows : the genetic subgroups S of P which do not contain Z = Ω_{1}Z(P) = Φ(P) are precisely the non central subgroups of order p. For such a subgroup set R = N_{P} (S). Then

R/S ≅ \{ \begin{matrix}C_{p} & in Case 1 \\ C_{4} & in Case 2 \\ Q_{8} & in Case 3 \end{matrix} .

By Theorem 6.1 of [10], and with the same notation, the map

Teninf_{R/S}^{P} : T_{tors}(R/S) → D_{tors}(P)

is injective, and its image is contained in ∂D_{tors}(P) = Ker Def_{P/Z}^{P} : indeed

for v ∈ T_{tors}(R/S) = f_{1}^{R/S}D_{tors}(R/S), since Def_{R/SZ}^{R/S}v = 0, it follows that

Teninf_{R/S}^{P}v = a_{R,S}(v) = (P/S − P/SZ)v ,

where P/S − P/SZ is a virtual (P, R/S)-biset. Then clearly

Def_{P/Z}^{P}Teninf_{R/S}^{P}v = (P/SZ − P/SZ)v = 0 ,

since the Galois torsions have no effect here : the group P/Z is elementary abelian, hence D(P/Z) = D^{Ω}(P/Z).

Hence in each case above, the map Teninf_{R/S}^{P} induces an isomorphism from T_{tors}(R/S) to ∂D_{tors}(P). It follows that the (restriction of the) map b_{R,S} = S\P − SZ\P is the inverse isomorphism ∂D_{tors}(P) → T_{tors}(R/S).

Now the elementary abelian subgroups of P of rank 2 are all maximal elementary abelian subgroups of P. If E is one of them, then E = SZ, where S is any non central subgroup of order p of E. In particular C_{P} (E) is equal to R = N_{P} (S).

Now if u ∈ ∂D_{tors}(P) is such that Res_{C_{P} (E)}^{P}u = 0, then

b_{R,S}(u) = (S\P − SZ\P)u
= f_{1}^{R/S}Def_{R/S}^{R}Res_{R}^{P}u = 0 ,

hence u = 0 since b_{R,S} is an isomorphism. This completes the proof of Proposition 8.1.

8.2. Theorem : Let P be a p-group, and let S be a genetic basis of P. Then the map

I_{S} = ⊕_{S∈S} Teninf_{N_{P} (S)/S}^{P} : ⊕_{S∈S} T_{tors}(N_{P} (S)/S) → D_{tors}(P)

is an isomorphism. (This was Conjecture A in [4], or Conjecture 6.2 in [10]). Proof: Let

D_{S} : D_{tors}(P) → ⊕_{S∈S} T_{tors}(N_{P} (S)/S)

be the map in the other direction, defined as in Section 6 of [10] by D_{S} = ⊕_{S∈S} b_{N_{P} (S),S}. Since D_{S}∘I_{S} = Id by Theorem 6.1 of [10], proving Theorem 8.2 is equivalent to proving that the map D_{S} is injective, and I will proceed by induction on the order of P.

Before that, suppose that the map I_{S} is an isomorphism, for a particular genetic basis S of P. It follows that

D_{tors}(P) ≅ ⊕_{S∈S} T_{tors}(N_{P} (S)/S) ,

and up to isomorphism, the right hand side is independent of the choice of S, since the groups N_{P} (S)/S are the types of the irreducible rational representations of P. Now if S′ is another genetic basis of P, then the map I_{S′} is always injective, by Theorem 6.1 of [10]. Hence it is an isomorphism, since it is an injection from a finite group to an isomorphic finite group. In other words, it is enough to prove Theorem 8.2 for any particular genetic basis S of P.

Suppose now that Theorem 8.2 holds for any p-group Q with |Q| < |P |. Let S be a genetic basis of P, and let Z be any central subgroup of order p in P. Set

S′ = {S ∈ S | S ⊇ Z} .

Then the set \overline{S}′ = \{S/Z | S ∈ S′\} is a genetic basis for \overline{P} = P/Z. Let u ∈ Ker D_{S}, and consider v = Def_{P/Z}^{P}u. If \overline{S} = S/Z is an element of \overline{S}′, for S ∈ S, then N_{\overline{P}} (\overline{S}) = \overline{R} = R/S, where R = N_{P} (S) and

b_{\overline{R},\overline{S}}(v) = b_{\overline{R},\overline{S}}Def_{P/Z}^{P}u
= Def_{P/Z}^{P}b_{R,S}(u) = 0 .

Now the induction hypothesis shows that v = 0. Since this holds for any central subgroup Z of order p of P, it follows that u ∈ ∂D_{tors}(P). Thus

Ker D_{S} ⊆ ∂D_{tors}(P) ,

for any genetic basis S of P.

Now let u ∈ ∂D_{tors}(P). Theorem 13.4 of [14] shows that u is equal to 0 if and only if Defres_{N_{P} (Q)/Q}^{P}u = 0, for any subgroup Q of P such that N_{P} (Q)/Q is cyclic, semi dihedral or generalized quaternion. Since Def_{P/Z}^{P}u = 0 for any non-trivial central subgroup of P, it follows that Defres_{N_{P} (Q)/Q}^{P}u = 0 if M = Q ∩ Z(P) ≠ 1, for

Defres_{N_{P} (Q)/Q}^{P}u = Defres\overline{P} ; N_{\overline{P}} (\overline{Q})/\overline{Q}DefP ; \overline{P} u ,

where \overline{P} = P/M and \overline{Q} = Q/M.

And if M = 1, then Z(P) embeds in the center of N_{P} (Q)/Q, which is always cyclic if Q cyclic, semi dihedral or generalized quaternion. This shows that u = 0 if Z(P) is not cyclic, hence ∂D_{tors(P)} = {0} in this case, and Ker D_{S} = {0} also.

Now if Z(P) is cyclic, let Z = Ω_{1}Z(P) be its subgroup of order p. If P has normal p-rank 1, then 1 is a normal genetic subgroup of P, thus 1 ∈ S. Moreover ∂D_{tors}(P) is equal to T_{tors}(P). Thus for u ∈ ∂D_{tors}(P)

b_{P,1}(u) = f_{1}^{P} u = u .

It follows that Ker D_{S} = {0} in this case.

The remaining case is when P admits a normal subgroup E which is elementary abelian of rank 2. Set H = C_{P} (E). Since Z(P) is cyclic, it follows that |P : H| = p, and that Z = E ∩ Z(P). By Corollary 5.3, there exists a genetic basis S of P and a decomposition

S = S_{1} ⊔ S_{2}

in disjoint union such that

1. if S ∈ S_{2}, then S ⊇ Z, and

2. the set Γ = {^{x}S | S ∈ S_{1}, x ∈ [P/H]} is a set of representatives of genetic subgroups R of H such that R ⊉ Z, for the relation _{H} , where [P/H] is any chosen set of representatives of H-cosets in P.

Let u ∈ Ker D_{S}. Then u ∈ Ker Def_{P/Z}^{P} = ∂D_{tors}(P). Set v = Res_{H}^{P}u. Then there is a genetic basis T of H such that Γ = {S ∈ T | S ⊉ Z}. Let S ∈ S_{1}, and x ∈ [P/H]. Then ^{x}S ∈ T , and

N_{H}(^{x}S) = ^{x}N_{H}(S) = ^{x}N_{P} (S) .

Moreover

b_{^{x}R,^{x}S}(v) = (^{x}S\H −^{x}SZ\H)Res_{H}^{P}u
= (^{x}S\P −^{x}SZ\P)u = xb_{R,S}(x^{−1}u) = xb_{R,S}(u) = 0 .

Moreover if S ∈ T −Γ, then S ⊇ Z, so setting R = N_{P} (S) again, and denoting by Ŝ the subgroup of R such that Ŝ/S is central of order p in R/S,

b_{R,S}(v) = (S\H − Ŝ\H)Res_{H}^{P}u
= (\overline{S}\\overline{H} − \overline{Ŝ}\\overline{H})Res\overline{P} ; \overline{H}DefP ; \overline{P} u = 0 ,

where \overline{P} = P/Z, \overline{S} = S/Z, \overline{H} = H/Z, \overline{Ŝ} = Ŝ/Z.

It follows that b_{N_{H}(S),S}(v) = 0 for any S ∈ T , thus v = 0 by induction hypothesis. Now u ∈ Ker Res_{C_{P} (E)}^{P} ∩ Ker Def_{P/Z}^{P}. Hence u = 0 by Proposition 8.1. This completes the proof of Theorem 8.2.

8.3. Corollary : Let P be a p-group. Then

D_{tors}(P) ≅ (ℤ/2ℤ)^{n_{P}} ⊕ (ℤ/4ℤ)^{m_{P}} ,

where m_{P} is equal to the number of isomorphism classes of rational irreducible representations of P of generalized quaternion type, and n_{P} is the number of isomorphism classes of rational irreducible representations of P whose type is

• cyclic of order at least 3, or semi dihedral, or generalized quaternion, if the ground field contains primitive cubic roots of unity.

• cyclic of order at least 3, or semi dihedral, or generalized quaternion of order at least 16 otherwise.

(This was Conjecture 6.3 in [10]).

8.4. Corollary : If P is a 2-group, then D(P) = D^{Ω}(P) +_{2}D(P), where 2D(P) = {u ∈ D(P) | 2u = 0}. Proof: Indeed D(P) = D^{Ω}(P) + D_{tors}(P) by Theorem 7.7. Moreover Corollary 8.3 shows that 2D_{tors}(P) = 2D_{tors}^{Ω}(P) : indeed, it is enough to check this equality when P is cyclic, semi dihedral, or generalized quaternion, since for an arbitrary 2-group P, any element in D_{tors}(P) is a sum of elements obtained by inflation and tensor induction from sections of P of this type. Now by Theorem 4.7, if P is cyclic or semi dihedral, then D_{tors}(P) = D_{tors}^{Ω}(P). And if P is generalized quaternion, then 2D_{tors}(P) is a group of order 2, generated by 2Ω_{P/1}.

Now if P is an arbitrary 2-group, and if u ∈ D(P), there is an element v ∈ D^{Ω}(P) such that w = u − v is a torsion element of D(P). So there exists an element t ∈ D_{tors}^{Ω}(P) such that 2w = 2t, i.e. w − t ∈ _{2}D(P). Now u = v + t + (w − t), and Corollary 8.4 follows.

8.5. Corollary : Let O be a commutative noetherian local ring with maximal ideal 𝔭, complete for the 𝔭-adic topology, such that the residue field k = O/𝔭 has characteristic p. If P is a p-group, then reduction mod 𝔭 induces a group isomorphism D_{O}(P) → D_{k}(P) from the group of Dade P -algebras over O to the corresponding group over k. In other words, every endo-permutation kP -module can be lifted to an endo-permutation OP -lattice. Proof: The assumption is Assumption 2.1 of [25], except that k is not assumed to be algebraically closed. Now with the Notation of Section 29 of [25], reduction mod 𝔭 induces an injection D_{O}(P) → D_{k}(P), by Proposition 29.4 of [25]. This injection obviously restricts to an isomorphism D_{O}^{Ω}(P) → D_{k}^{Ω}(P). Also, this injection is an isomorphism if P has normal p-rank 1, by Dade’s Theorem 4.6, and by Carlson-Thévenaz explicit description of D_{k}(P), when P is a 2-group of normal 2-rank 1.

Now it follows from Theorem 8.2 and Theorem 7.7, that the reduction map D_{O}(P) → D_{k}(P) is always an isomorphism.

9. Generators and relations

9.1. Notation : If P is a p-group, denote by [s_{P}] a set of representatives of conjugacy classes of subgroups of P, ordered by the relation

U ≤_{P} V ⇔ ∃x ∈ P, U ⊆ ^{x}V .

Let μ_{P} denote the Möbius function of the poset ([s_{P}], ≤_{P}). If V ∈ [s_{P}], let Δ_{P/V} denote the element of D^{Ω}(P) defined by

Δ_{P/V} = ∑_{{U∈[s_{P}] \atop U≤_{P} V}} μ_{P} (U, V)Ω_{P/U} .

If Q is any subgroup of P, set Δ_{P/Q} = Δ_{P/V}, if V ∈ [s_{P}] is a conjugate of Q in P.

If Q and S are subgroups of P, recall from Notation 2.4 of [8] that

I_{P} (Q, S) = {x ∈ Q\P/N_{P} (S) | Q^{x} ∩ N_{P} (S) ⊆ S} ,

and set i_{P} (Q, S) = |I_{P} (Q, S)|.

Define similarly

J_{P} (Q, S) = {x ∈ Q\P/N_{P} (S) | |J_{x}(Q, S)| = p, J_{x}(Q, S) ⊈ Z(N_{P} (S)/S)} ,

where J_{x}(Q, S) = (Q^{x} ∩ N_{P} (S))S/S, and set j_{P} (Q, S) = |J_{P} (Q, S)|.

9.2. Remark : If N_{P} (S)/S is cyclic or generalized quaternion, then j_{P} (Q, S) is equal to 0 for any Q.

If Q is a generalized quaternion group, recall (Lemma 4.8) that η_{Q} denotes any element of order 2 in D(Q) − D^{Ω}(Q).

9.3. Notation : If P is a 2-group, and S is a genetic subgroup of P such that N_{P} (S)/S is generalized quaternion, choose such an element η_{N_{P} (S)/S} ∈ D(N_{P} (S)/S), and set

Λ_{S} = Teninf_{N_{P} (S)/S}^{P}η_{N_{P} (S)/S} .

9.4. Remark : By Lemma 4.8, the element Λ_{Q} does not exist if Q ≅ Q_{8} and the ground field does not have primitive roots of unity. In all other cases, there are exactly two elements η_{Q} and η′_{Q} of order 2 in D(Q) − D^{Ω}(Q), and η_{Q} + η′_{Q} = 2Ω_{Q/1} in D(Q).

9.5. Theorem : Let P be a p-group, and let S be a genetic basis of P. Denote by Q the subset of S consisting of elements S such that N_{P} (S)/S is generalized quaternion, if the ground field contains all cubic roots of unity, or generalized quaternion of order at least 16 otherwise.

Then the Dade group D(P) is generated by

• the elements Δ_{P/Q}, for Q ∈ [s_{P}], and

• if p = 2, the elements Λ_{S}, for S ∈ Q.

These generators are subject to the following relations 9.6 and 9.7 :

(9.6)  ∀S ∈ S,  τ_{S} ∑_{Q∈[s_{P}]}(a_{S}i_{P} (Q, S) + j_{P} (Q, S))Δ_{P/Q} = 0 ,

where τ_{S} = \{ \begin{matrix}1 & if |N_{P} (S)/S| ≤ 2 \\ & or if N_{P} (S)/S is dihedral \\ 2 & if N_{P} (S)/S is cyclic of order at least 3 \\ & or semi dihedral \\ 4 & if N_{P} (S)/S is generalized quaternion \end{matrix}

and a_{S} = \{ \begin{matrix}1 & if N_{P} (S)/S is cyclic or generalized quaternion \\ 2 & if N_{P} (S)/S is dihedral or semi dihedral \end{matrix} .

(9.7)  ∀S ∈ Q,  2Λ_{S} = 0 .

Moreover, these generators and relations form a presentation of D(P) as an abelian group : more precisely, the generators Δ_{P/Q}, for Q ∈ [s_{P}], subject to the relations 9.6, form a presentation of D^{Ω}(P) as an abelian group. The elements Λ_{S}, for S ∈ Q generate a subgroup D_{Q}(P), isomorphic to (ℤ/2ℤ)^{|Q|}, and

D(P) = D^{Ω}(P) ⊕ D_{Q}(P) .

Proof: Recall from Theorem 1.7 of [9] that there exists a unique natural transformation of biset functors Θ : B^{∗} → D^{Ω} such that Θ_{P} (ω_{X}) = Ω_{X}, for any p-group P and any finite P -set X, and Θ is surjective. Let L = Ker Θ.

Then L is a biset functor, and since composition of Θ with the projection D^{Ω} → D^{Ω}/D_{tors}^{Ω} leads to the exact sequence

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

it follows that L is a subfunctor of R_{ℚ}^{∗}, hence L is a subfunctor of the dual of a rational biset functor. By Proposition 7.4 of [10], the functor L is rational, and thus for any genetic basis S of P, the map

I_{S} = ⊕_{S∈S} Teninf_{N_{P} (S)/S}^{P} : ⊕_{S∈S} ∂L(N_{P} (S)/S) → L(P)

is an isomorphism.

Now the element Δ_{P/Q} of D^{Ω}(P) is the image by Θ_{P} of the element δ_{P/Q} of the canonical basis of B^{∗}(P) (see Remark 2.3 of [10]). In particular the elements Δ_{P/Q}, for Q ∈ [s_{P}], generate D^{Ω}(P).

Let P be any p-group. Evaluating at P the exact sequence

0 → L → B^{∗} → D^{Ω} → 0 ,

and taking images by the idempotent f_{1}^{P} of End_{C_{p}}(P), leads to the exact sequence of abelian groups

0 → ∂L(P) → ∂B^{∗}(P) → ∂D^{Ω}(P) → 0 .

But R_{ℚ}^{∗} has a ℤ-basis consisting of the elements V^{∗}, where V runs through a set of representatives of isomorphism classes of rational irreducible representation of P, and V^{∗}(W) = m(V, W) for any W ∈ R_{ℚ}(P) (see Lemma 3.2 of [9]). It follows that ∂R_{ℚ}(P) has a basis consisting of the elements V^{∗}, where V runs through a set of representatives of isomorphism classes of faithful rational irreducible representation of P.

Now suppose that P has normal p-rank 1, and set Z = Ω_{1}Z(P). Then there is a unique such representation Φ_{P}, by Proposition 3.7 of [8]. Thus ∂R_{ℚ}^{∗}(P) is free of rank 1, generated by the element Φ_{P}^{∗}. Viewed as an element of B^{∗}(P), it is equal to

Φ_{P}^{∗} = ∑_{Q∈[s_{P}]} m(Φ_{P} , ℚP/Q)δ_{P/Q} .

Then by Lemma 4.1 of [10], if P is cyclic or generalized quaternion, then

m(Φ_{P} , ℚP/Q) = \{ \begin{matrix}0 & if Q ≠ 1 \\ 1 & if Q = 1 \end{matrix} ,

and if P is dihedral or semi dihedral, then

m(Φ_{P} , ℚP/Q) = \{ \begin{matrix}0 & if Q ⊇ Z \\ 1 & if Q ⊉ Z, |Q| = 2 \\ 2 & if Q = 1 \end{matrix} .

If P is cyclic or generalized quaternion, then ∂R_{ℚ}(P) is generated by Φ_{P}^{∗} = δ_{P/1} = ω_{P/1}. The order τ_{P} of Θ_{P} (Φ_{P}^{∗}) = Ω_{P/1} is equal to 1 if |P | ≤ 2, to 2 if P is cyclic of order at least 3, and to 4 if P is generalized quaternion. Hence ∂L(P) is generated by ρ_{P} = τ_{P} ω_{P/1} in this case.

If P is semi dihedral, then ∂R_{ℚ}^{∗}(P) is generated by

Φ_{P}^{∗} = 2δ_{P/1} + δ_{P/R} = ω_{P/1} + ω_{P/R} ,

where R is a non central subgroup of P of order 2. Then Θ_{P} (Φ_{P}^{∗}) is equal to Ω_{P/1} + Ω_{P/R}, which has order 2 in D(P), by Theorem 7.1 of [13]. Hence ∂L(P) is generated by ρ_{P} = 2(ω_{P/1} + ω_{P/R}) in this case.

Finally if P is dihedral, then ∂R_{ℚ}(P) is generated by

Φ_{P}^{∗} = 2δ_{P/1} + δ_{P/R} + δ_{P/R′} = ω_{P/R} + ω_{P/R′} ,

where R and R′ are non central subgroups of order 2 of P, not conjugate in P. Then u = Θ_{P} (Φ_{P}^{∗}) = Ω_{P/R} + Ω_{P/R′} is a torsion element of D(P), which is torsion-free by Theorem 4.7. Hence u = 0. So ∂L(P) is generated by ρ_{P} = ω_{P/R} + ω_{P/R′} in this case.

Let S ∈ S. If Q is a subgroup of P, then

Indinf_{N_{P} (S)/S}^{P}Φ_{N_{P} (S)/S}^{∗}(P/Q) = Φ_{N_{P} (S)/S}^{∗}(Defres_{N_{P} (S)/S}^{P}(P/Q))
= m(Φ_{N_{P} (S)/S}, Defres_{N_{P} (S)/S}^{P}(ℚP/Q))
= m(Indinf_{N_{P} (S)/S}^{P}Φ_{N_{P} (S)/S}, ℚP/Q)
= a_{S}i_{P} (Q, S) + j_{P} (Q, S) .

Here the equality in the second line follows from the definition. The equality in the third line follows by Frobenius reciprocity, from the fact that (N_{P} (S), S) is a genetic section of P, and the last equality follows from Lemma 4.1 of [10].

Then

Indinf_{N_{P} (S)/S}^{P}Φ_{N_{P} (S)/S}^{∗} = ∑_{Q∈[s_{P}]}(a_{S}i_{P} (Q, S) + j_{P} (Q, S))δ_{P/Q} ,

and it follows that D^{Ω}(P) has a presentation as an abelian group generated by the element Δ_{P/Q}, for Q ∈ [s_{P}], subject to the relations 9.6.

Now the elements Λ_{S}, for S ∈ Q, are of order 2, since the element η_{N_{P} (S)/S} has order 2, and since the map

Teninf_{N_{P} (S)/S}^{P} : T_{tors}(N_{P} (Q)/Q) → D(P)

is injective, by Theorem 6.1 of [10]. Let D_{Q}(P) denote the subgroup generated by the elements Λ_{S}, for S ∈ Q.

Then D_{tors}(P) = D_{tors}^{Ω}(P) + D_{Q}(P) : to prove this, by Theorem 8.2, it suffices to check that if P is a quaternion group, then D_{tors}(P) is generated by D^{Ω}(P) and one chosen element η_{Q}. But this is obvious, from Lemma 4.8.

Now by Theorem 7.7, it follows that D(P) = D^{Ω}(P) + D_{Q}(P), and

D(P)/D^{Ω}(P) ≅ D_{tors}(P)/D_{tors}^{Ω}(P) ≅ D_{Q}(P)/(D^{Ω}(P) ∩ D_{Q}(P)) .

By Corollary 8.3 and by Corollary 7.6 of [10], the group D_{tors}(P)/D_{tors}^{Ω}(P) is isomorphic to (ℤ/2ℤ)^{q_{P}}, where q_{P} is the number of isomorphism classes of rational irreducible representations of P which are of generalized quaternion type if the ground field contains all cubic roots of unity, and of generalized quaternion type of order at least 16 otherwise. In other words q_{P} = |Q|.

Now D_{Q}(P) is an elementary abelian 2-group generated by |Q| elements, so its rank is at most equal to |Q|. Since its factor group

D_{Q}(P)/(D^{Ω}(P) ∩ D_{Q}(P))

has rank equal to |Q|, it follows that D^{Ω}(P) ∩ D_{Q}(P) = {0}, that the elements Λ_{S}, for S ∈ Q, are linearly independent over 𝔽_{2}, and that

D(P) = D^{Ω}(P) ⊕ D_{Q}(P) .

This completes the proof of the theorem.

9.8. Remark : If the group P is abelian, then by Remark 2.11, there is a unique genetic basis S of P, consisting of subgroups S of P such that P/S is cyclic. If Q is any subgroup of P, and if S ∈ S, then j_{P} (Q, S) = 0 by Remark 9.2, and i_{P} (Q, S) is equal to 1 if Q ⊆ S, and to 0 otherwise. Hence the relations 9.6 become

∀S ∈ S, τ_{S} ∑_{Q⊆S} Δ_{P/Q} = 0 .

Moreover

∑_{Q⊆S} Δ_{P/Q} = ∑_{Q⊆S} ∑_{{U∈[s_{P}] \atop U≤_{P} Q}} μ_{P} (U, Q)Ω_{P/U}

= ∑_{U⊆P} ( ∑_{U⊆Q⊆S} μ(U, Q))Ω_{P/U}
= Ω_{P/S}

by the defining property of the Möbius function, since the poset s_{P} is equal to the poset of all subgroups of P if P is abelian. Thus Theorem 9.5 is another form of Dade’s Theorem (Theorem 4.6) if P is abelian.

10. The functor D/D^{Ω}

In this section, I will show that D/D^{Ω} is a biset functor. Of course, by Theorem 7.7, this is non trivial only if p = 2, which I shall assume throughout this section.

10.1. Notation : (see Notation 4.1 of [8]) If Q is a 2-group of normal 2-rank 1, let H_{Q} denote the subfunctor of 𝔽_{2}R_{ℚ} generated by the image \overline{Φ_{Q}} ∈ 𝔽_{2}R_{ℚ}(Q) of the unique (up to isomorphism) rational irreducible representation Φ_{Q} of Q.

If P is any 2-group, then 𝔽_{2}R_{ℚ}(P) has a canonical basis consisting of the images \overline{V} of the rational irreducible ℚP -modules V, up to isomorphism. If u ∈ 𝔽_{2}R_{ℚ}(P), denote by γ(V, u) the coefficient of \overline{V} in the decomposition of u in this basis.

The next theorem is a precise form of the second part of Conjecture B in [4] :

10.2. Theorem : If P is a 2-group, there is an exact sequence of abelian groups

0 ⟶ D^{Ω}(P) ⟶^{i_{P}} D(P) ⟶^{σ_{P}} H_{Q}(P) ⟶ 0 ,

where i_{P} is the inclusion, and Q ≅ Q_{8} if the ground field contains all cubic root of unity, and Q ≅ Q_{16} otherwise.

Moreover this sequence is functorial in the following sense : if P′ is a 2-group, and if ψ ∈ Hom_{C_{2}}(P, P′), then the diagram

0 ⟶ D^{Ω}(P) ⟶^{i_{P}} D(P) ⟶^{σ_{P}} H_{Q}(P) ⟶ 0
D^{Ω}(ψ) ↓  D(ψ) ↓  H_{Q}(ψ) ↓
0 ⟶ D^{Ω}(P′) ⟶^{i_{P′}} D(P′) ⟶^{σ_{P′}} H_{Q}(P′) ⟶ 0

is commutative.

In other words, there is a natural structure of biset functor on D/D^{Ω}, and D/D^{Ω} ≅ H_{Q}.

In general however, because of “Galois torsion” (see Section 3 in [12]), there is no natural biset functor structure on D, so there is no genuine “exact sequence of biset functors”

0 → D^{Ω} → D → H_{Q} → 0 .

If the ground field does not contain non trivial cubic root of unity, then D is a genuine biset functor, by Lemma 4.8, and this sequence is an exact sequence of biset functors. Proof: If the ground field contains all cubic roots of unity, fix a group Q ≅ Q_{8}, otherwise fix a group Q ≅ Q_{16}. In any case, choose an element η_{Q} of order 2, in D(Q) − D^{Ω}(Q). If P is any 2-group, set

F (P) = Hom_{C_{2}}(Q, P)(η_{Q}) .

Then F (P) is a subgroup of D(P), and 2F (P) = 0. Define a map π_{P} : F (P) → H_{Q}(P) by

(10.3)  π_{P} (φ(η_{Q})) = φ(\overline{Φ_{Q}}) ,

for φ ∈ Hom_{C_{2}}(Q, P).

10.4. Lemma : The map π_{P} is well defined. Proof: This amounts to showing that if φ ∈ Hom_{C_{2}}(Q, P) is such that φ(η_{Q}) = 0, then φ(\overline{Φ_{Q}}) = 0. Now if φ(η_{Q}) = 0, then ψ(φ(η_{Q})) = 0 for any morphism ψ ∈ Hom_{C_{p}}(P, R), where R is a generalized quaternion group, such that |R| ≥ |Q|.

I have to be careful here, since in general, due to “Galois torsion”, it is not true that

ψ(φ(η_{Q})) = (ψφ)(η_{Q}) .

However the quotient D(R)/D^{Ω}(R) is isomorphic to ℤ/2ℤ : indeed, by the choice of Q, the hypothesis |R| > |Q|, and the fact that generalized quaternion groups have a unique rational irreducible representation of quaternion type, up to isomorphism, the set Q of Theorem 9.5 for the group R is of cardinality 1 (more precisely Q = {1}).

Hence there is a linear form

λ_{R} : D(R) → ℤ/2ℤ

defined by λ_{R}(u) = 1 if u ∉ D^{Ω}(R), and λ_{R}(u) = 0 otherwise.

The point now is that ψ(φ(η_{Q})) differs from (ψφ)(η_{Q}) by an element of D^{Ω}(R) (even 2D^{Ω}(R)) : this follows from the case where ψ and φ are bisets, using Proposition 3.10 of [12], and from the fact that for any endomorphism a of the ground field the difference γ_{a}(η_{Q}) − η_{Q} is a multiple of 2Ω_{Q}. In particular, this shows that

λ_{R}(ψ(φ(η_{Q}))) = λ_{R}((ψφ)(η_{Q})) = 0 .

10.5. Lemma : Let Q and R be generalized quaternion groups, and let f ∈ Hom_{C_{2}}(Q, R). Then

λ_{R}(f(η_{Q})) = γ(Φ_{R}, f(\overline{Φ_{Q}})) .

Proof: Since λ_{R}(f(η_{Q})) and γ(Φ_{R}, f(\overline{Φ_{Q}})) are linear in f, it suffices to consider the case where f is a transitive biset, hence (by Remark 3.4) of the form

Indinf_{T/S}^{R}Iso_{Y/X}^{T/S}Defres_{Y/X}^{Q} ,

where T/S is a section of R and Y/X is a section of Q, such that there exists a group isomorphism Y/X → T/S.

If Y is proper subgroup of Q, then Res_{Y}^{Q}η_{Q} is in D^{Ω}(Y), by Lemma 4.8. Thus if Y ≠ Q, then f(η_{Q}) ∈ D^{Ω}(R), hence λ_{R}(f(η_{Q})) = 0. But the restriction of Φ_{Q} to any maximal subgroup H of Q is equal to 2Φ_{H}, by Lemma 3.14 of [8]. Hence f(Φ_{Q}) ∈ 2R_{ℚ}(R), thus γ(Φ_{R}, f(\overline{Φ_{Q}})) = 0 in this case.

Suppose now that Y = Q. If X ≠ 1, then Def_{Q/X}^{Q}η_{Q} = 0 (by construction,

or because if Z is the center of Q, then Def_{Q/Z}^{Q}η_{Q} is in D_{tors}(Q/Z) = 0, since Q/Z is dihedral). Thus if X ≠ 1, then λ_{R}(f(η_{Q})) = 0. On the other hand, any proper deflation of Φ_{Q} is equal to 0, by Lemma 3.12 of [8], hence γ(Φ_{R}, f(\overline{Φ_{Q}})) = 0 in this case also.

Now assume X = 1. Then T/S is isomorphic to Q. But any section of the generalized quaternion group R which is itself a generalized quaternion group is actually a subgroup of R. This shows that S = 1, and I can suppose that Q is a subgroup of R. In this case Ten_{Q}^{R}η_{Q} is equal to η_{R} or η_{R} + 2Ω_{R} , by Lemma 4.8. But on the other hand Ind_{Q}^{R}Φ_{Q} = Φ_{R}, thus γ(Φ_{R}, f(\overline{Φ_{Q}})) = 1, proving the lemma.

10.6. End of proof of Lemma 10.4 : consider the element u = φ(\overline{Φ_{Q}}) of 𝔽_{2}R_{ℚ}(P). The previous discussion and Lemma 10.5 show that γ(Φ_{R}, ψ(u)) is equal to 0 for any ψ ∈ Hom_{C_{2}}(P, R), for any generalized quaternion group R such that |R| ≥ |Q|. Now Corollary 6.7 and Theorem 5.12 of [8] show that H_{Q}(P) has an 𝔽_{2}-basis consisting of the images \overline{V} of the rational irreducible representations V of P, whose type is generalized quaternion of order at least equal to |Q|, up to isomorphism. Moreover, if V is such a representation, and (T, S) is a genetic section of P for V, then γ(V, u) = γ(Φ_{T/S}, Defres_{T/S}^{P}u), by Lemma 4.2 of [8]. It follows that γ(V, u) = 0 for any V ∈ Irr_{ℚ}(P) of quaternion type R with |R| ≥ |Q|. Hence u = 0, as was to be shown for Lemma 10.4.

10.7. End of proof of Theorem 10.2 : since H_{Q}(P) ⊆ Im π_{P} by 10.3, the map π_{P} is surjective. Now F (P) ∩ D^{Ω}(P) ⊆ Ker π_{P} : indeed if φ ∈ Hom_{C_{2}}(Q, P) is such that u = φ(η_{Q}) ∈ D^{Ω}(P), then ψ(φ(η_{Q})) ∈ D^{Ω}(R), for any generalized quaternion group R and any ψ ∈ Hom_{C_{p}}(P, R). The same argument as above, using Lemma 10.5, shows that φ(\overline{Φ_{Q}}) = 0, i.e. π_{P} (u) = 0.

Now F (P)/(F (P) ∩ D^{Ω}(P)) ≅ (F (P) + D^{Ω}(P))/D^{Ω}(P) is isomorphic to a subgroup of D(P)/D^{Ω}(P), which is isomorphic to (ℤ/2ℤ)^{q_{P}}, where q_{P} is equal to the number of isomorphism classes of rational irreducible representations of P whose type is generalized quaternion of order at least equal to |Q|. Thus by the above remarks D(P)/D^{Ω}(P) ≅ H_{Q}(P), and the surjection

F (P)/(F (P) ∩ D^{Ω}(P)) → H_{Q}(P)

induced by π_{P} has to be an isomorphism. Hence F (P) + D^{Ω}(P) = D(P), and F (P) ∩ D^{Ω}(P) = Ker π_{P}. This gives a well defined map

σ_{P} : D(P) → H_{Q}(P)

sending the element u + w of D(P), where u ∈ F (P) and w ∈ D^{Ω}(P), to π_{P} (u).

Suppose that u = φ(η_{Q}), for φ ∈ Hom_{C_{p}}(Q, P). Let P′ be a 2-group, and f ∈ Hom_{C_{2}}(P, P′). Then w′ = f(φ(η_{Q})) − (fφ)(η_{Q}) is in D^{Ω}(P′), so

σ_{P′}D(f)(u + w) = σ_{P′}(f(φ(η_{Q})) + f(w))
= σ_{P′}((fφ)(η_{Q}) + w′ + f(w))
= fφ(\overline{Φ_{Q}}) = H_{Q}(f)π_{P} (u) = H_{Q}(f)σ_{P} (u + w) .

This shows that the square on right hand side in the theorem is commutative. This completes the proof of Theorem 10.2, since the square on the left hand side is also commutative.

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