Title : Gluing endo-permutation modules

Author : Serge Bouc.

Address : CNRS - UMR 6140 - LAMFA, Université de Picardie - Jules Verne, 33, rue St Leu, F-80039 Amiens Cedex 1, France.

email : serge.bouc@u-picardie.fr

Abstract : In this paper, I show that if p is an odd prime, and if P is a finite p-group, then there exists an exact sequence of abelian groups

0 → T (P) → D(P) → lim ; ⟵_{1<Q≤P} D(N_{P} (Q)/Q) → H^{1}(A_{≥2}(P), ℤ)^{(P)} ,

where D(P) is the Dade group of P and T (P) is the subgroup of endo-trivial modules. Here lim ; ⟵_{1<Q≤P} D(N_{P} (Q)/Q) is the group of sequences of compatible elements in the

Dade groups D(N_{P} (Q)/Q) for non trivial subgroups Q of P. The poset A_{≥2}(P) is the set of elementary abelian subgroups of rank at least 2 of P, ordered by inclusion. The group H^{1}(A_{≥2}(P), ℤ)^{(P)} is the subgroup of H^{1}(A_{≥2}(P), ℤ) consisting of classes of P -invariant 1-cocycles.

A key result to prove that the above sequence is exact is a characterization of elements of 2D(P) by sequences of integers, indexed by sections (T, S) of P such that T/S ≅ (ℤ/pℤ)^{2}, fulfilling certain conditions associated to subquotients of P which are either elementary abelian of rank 3, or extraspecial of order p^{3} and exponent p.

AMS Subject classification : 20C20

Keywords : endo-permutation module, Dade group, gluing.

Gluing endo-permutation modules

Serge Bouc

Abstract : In this paper, I show that if p is an odd prime, and if P is a finite p-group, then there exists an exact sequence of abelian groups

0 → T (P) → D(P) → lim ; ⟵_{1<Q≤P} D(N_{P} (Q)/Q) → H^{1}(A_{≥2}(P), ℤ)^{(P)} ,

where D(P) is the Dade group of P and T (P) is the subgroup of endo-trivial modules.

Here lim ; ⟵_{1<Q≤P} D(N_{P} (Q)/Q) is the group of sequences of compatible elements in the Dade groups

D(N_{P} (Q)/Q) for non trivial subgroups Q of P. The poset A_{≥2}(P) is the set of elementary abelian subgroups of rank at least 2 of P, ordered by inclusion. The group H^{1}(A_{≥2}(P), ℤ)^{(P)} is the subgroup of H^{1}(A_{≥2}(P), ℤ) consisting of classes of P -invariant 1-cocycles.

A key result to prove that the above sequence is exact is a characterization of elements of 2D(P) by sequences of integers, indexed by sections (T, S) of P such that T/S ≅ (ℤ/pℤ)^{2}, fulfilling certain conditions associated to subquotients of P which are either elementary abelian of rank 3, or extraspecial of order p^{3} and exponent p.

AMS Subject classification : 20C20

Keywords : endo-permutation module, Dade group, gluing.

1. Introduction

The classification of all endo-permutation modules for finite p-groups has been completed recently, thanks to the work of several authors (see in particular [1], [11], [12], [5], [3]). This paper addresses the question of gluing arbitrary endo-permutation modules, and it is intended to be a complement to our previous joint work with Jacques Thévenaz ([7]), where the case of torsion endo-permutation modules was handled.

The gluing problem is the following : let p be an odd prime, let P be a finite pgroup, and let k be a field of characteristic p. If v is an element of the Dade group D(P) of endo-permutation kP -modules, and if Q is a non trivial subgroup of P, denote by v_{Q} the image of v by the deflation-restriction map Defres_{N_{P} (Q)/Q}^{P}. Then the v_{Q}’s are subject to some obvious compatibility conditions. Conversely, if Q is a non-trivial subgroup of P, let u_{Q} be an element of the Dade group D_{k}(N_{P} (Q)/Q), and assume that these compatibility conditions between the u_{Q}’s are fulfilled. Is there an element u ∈ D(P) such that for any non trivial subgroup Q of P

Defres_{N_{P} (Q)/Q}^{P}(u) = u_{Q} ?

Such an element u is called a solution to the gluing problem for the gluing data (u_{Q})_{1<Q≤P}.

When P is abelian, the gluing problem was completely solved by Puig [16] (see also Lemma 2.3 below), and he used the result to construct suitable stable equivalences between blocks.

The main result of the present paper is that if p is an odd prime, and if P is a finite p-group, then there exists an exact sequence of abelian groups

0 ⟶ T (P) ⟶ D(P) ⟶ lim ; ⟵_{1<Q≤P} D(N_{P} (Q)/Q) ⟶^{h_{P}} H^{1}(A_{≥2}(P), ℤ)^{(P)} ,

where D(P) is the Dade group of P and T (P) is the subgroup of endo-trivial modules. Here lim ; ⟵_{1<Q≤P} D(N_{P} (Q)/Q) is the group of gluing data for P, i.e. the group of sequences

of compatible elements in the Dade groups D(N_{P} (Q)/Q) for non trivial subgroups Q of P. The poset A_{≥2}(P) is the set of elementary abelian subgroups of rank at least 2 of P, ordered by inclusion. The group H^{1}(A_{≥2}(P), ℤ)^{(P)} is the subgroup of H^{1}(A_{≥2}(P), ℤ) consisting of classes of P -invariant 1-cocycles.

The main consequence of this result is that if H^{1}(A_{≥2}(P), ℤ) = {0}, then the gluing problem always has a solution. Unfortunately, the map h_{P} is not surjective in general, so when H^{1}(A_{≥2}(P), ℤ) ≠ {0}, not much can be said at the time for the gluing problem. In Section 6, the example of the extraspecial group of order p^{5} and exponent p is described in details. In this case, the group H^{1}(A_{≥2}(P), ℤ)^{(P)} is a free group of rank p^{4}, and the image of h_{P} has finite index in this group. In particular it is non zero, and the gluing problem does not always have a solution.

It could be true in general that h_{P} always has finite cokernel, and this would be enough to show that if H^{1}(A_{≥2}(P), ℤ)^{(P)} ≠ {0}, then the image of h_{P} is non zero, hence that the gluing problem does not always have a solution : it is known indeed that the group H^{1}(A_{≥2}(P), ℤ)^{(P)} is a free abelian group, since the poset A_{≥2}(P) has the homotopy type of a wedge of spheres (see [8]).

1.1. Notation. Throughout this paper, the symbol p denotes an odd prime number, and P denotes a finite p-group. Inclusion of subgroups will be denoted by ≤, and strict inclusion by <. Inclusion up to P -conjugation will be denoted by ≤_{P}.

A section (T, S) of P is a pair of subgroups of P with S ⊴ T. The factor group T/S is the corresponding subquotient of P. If (T, S) is a section of P, then N_{P} (T, S) denotes N_{P} (T) ∩ N_{P} (S).

A class Y of p-groups is said to be closed under taking subquotients if for any Y ∈ Y and any section (T, S) of Y, any group isomorphic to T/S belongs to Y. If Y is such a class, and P is a finite p-group, let Y(P) be the set of sections (T, S) of P such that T/S ∈ Y.

The symbol X_{p^{3}} denotes an extraspecial p-group of order p^{3} and exponent p. The symbol X_{3} denotes the class of p-groups which are either elementary abelian of rank at most 3, or isomorphic to X_{p^{3}}. Thus, the symbol X_{3}(P) denotes the set of sections (T, S) of P such that T/S is elementary abelian of rank at most 3, or isomorphic to X_{p^{3}}. Let moreover E_{2}^{♯}(P) denote the set of sections (T, S) of P such that T/S ≅ (ℤ/pℤ)^{2}.

If P is a finite p-group, and k is a field of characteristic p, let D(P) denote the Dade group of endo-permutation kP -modules. The field k does not appear in this notation, because it turns out that D(P) is independent of k, at least when p is odd (see [3] Theorem 9.5 for details).

When (T, S) is a section of P, there is a deflation-restriction map Defres_{T/S}^{P} : D(P) → D(T/S), which is the group homomorphism obtained by composing the restriction map Res_{T}^{P} : D(P) → D(T), followed by the deflation map Def_{T/S}^{T} : D(T) → D(T/S).

Recall that if X is a finite P -set, there is a corresponding element Ω_{X} of the Dade group of P, called the syzygy of the trivial module relative to X (or the X-relative syzygy for short) : it is defined as the class of the kernel of the augmentation map kX → k when this does make sense, and by 0 otherwise (see e.g. [2] for details). When X is the set P itself, on which P acts by multiplication, the corresponding element will be denoted by Ω_{P/1} or Ω_{P}.

1.2. Contents. This paper is organized as follows :

• In Section 2, I state the main theorem (Theorem 2.15), and this requires in particular the definition of some objects and maps between them.

• Section 3 recalls some notation on biset functors, forgetful functors between categories of biset functors, and corresponding adjoint functors.

• Section 4 is devoted to the main tool (Theorem 4.5) used in the proof of Theorem 2.15, namely a characterization by linear equations of the image of the group 2D(P) by the deflation-restriction maps to all subquotients T/S of P which are elementary abelian of rank 2. This characterization may be a result of independent interest.

• Section 5 exposes the proof of Theorem 2.15.

• Finally, Section 6 focuses on the example of the extraspecial p-group of order p^{5} and exponent p : the reason for choosing this particular group is twofold : it is is one of the smallest p-groups P for which H^{1}(A_{≥2}(P), ℤ) ≠ {0}, and moreover the Dade group of this p-group is rather well known, thanks to our joint work with Nadia Mazza ([5]).

2. Statement of the main theorem

2.1. Notation. If P is a finite p-group, then A_{≥2}(P) denotes the poset of elementary abelian subgroups of P of rank at least 2. Let A_{=2}(P) denote the set of elementary abelian subgroups of rank 2 of P.

Recall that if the p-rank of P is at least equal to 3, then all the elementary abelian subgroup of P of rank at least 3 are in the same connected component of A_{≥2}(P). This component is called the big component. It is obviously invariant under P -conjugation. Each of the other connected components, if there are any, consists of a single maximal elementary abelian subgroup of rank 2.

2.2. Notation. Denote by lim ; ⟵_{1<Q≤P} D(N_{P} (Q)/Q) the set of sequences (u_{Q})_{1<Q≤P},

indexed by non trivial subgroups of P, where u_{Q} ∈ D(N_{P} (Q)/Q), such that :

• If x ∈ P, then ^{x}u_{Q} = u_{^{x}Q}.

• If Q ⊴ R, then Defres_{N_{P} (Q,R)/R}^{N_{P} (Q)/Q}u_{Q} = Res_{N_{P} (Q,R)/R}^{N_{P} (R)/R}u_{R}.

Denote by r_{P} : D(P) → lim ; ⟵_{1<Q≤P} D(N_{P} (Q)/Q) the map sending v ∈ D(P) to the

sequence (Defres_{N_{P} (Q)/Q}^{P}v)_{1<Q≤P}.

If E is an abelian p-group, denote by σ_{E} the map lim ; ⟵_{1<F ≤E} D(E/F) → D(E) defined

by

σ_{E}(u) = − ∑_{1<F ≤E} μ(1, F)Inf_{E/F}^{E} u_{F} ,

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

It has been shown by Puig ([15] 2.1.2) that the kernel of r_{P} is equal to the group T (P) of endo-trival modules. Moreover, when E is an abelian group, the map r_{E} is surjective ([16] Proposition 3.6). More precisely :

2.3. Lemma. Let E be an abelian p-group. Then σ_{E} is a section of r_{E}, i.e. r_{E}σ_{E} is equal to the identity map of lim ; ⟵_{1<F ≤E} D(E/F).

Proof. Let 1 < G ≤ E. Then

Def_{E/G}^{E}σ_{E}(u) = − ∑_{1<F ≤E} μ(1, F)Def_{E/G}^{E}Inf_{E/F}^{G} u_{F}
= − ∑_{1<F ≤E} μ(1, F)Inf_{E/F G}^{E/G}Def_{E/F G}^{E/F}u_{F}
= − ∑_{1<F ≤E} μ(1, F)Inf_{E/F G}^{E/G}u_{F G}
= − ∑_{G≤R≤E} ( ∑_{{1<F ≤R \atop F G=R}} μ(1, F))Inf_{E/R}^{E/G}u_{R} .

Now if G < R

∑_{{1<F ≤R \atop F G=R}} μ(1, F) = ∑_{{1≤F ≤R \atop F G=R}} μ(1, F) ,

and this is equal to zero, by a classical combinatorial lemma, since G ≠ 1. And if G = R

∑_{{1<F ≤R \atop F G=R}} μ(1, F) = −μ(1, 1) + ∑_{1≤F ≤R} μ(1, F) = −1 .

Thus Def_{E/G}^{E}σ_{E}(u) = Inf_{E/G}^{E/G}u_{G} = u_{G}, as was to be shown.

2.4. Lemma. Let E be an elementary abelian group of rank at least 2. Then the map r_{E} is surjective, and its kernel is the free abelian group of rank one generated by Ω_{E/1}. Proof. The kernel of r_{E} is the group T (E) of endo-trivial modules. Since E is elementary abelian, this group is free of rank one, generated by Ω_{E/1}, by Dade’s Theorem ([13] [14]). The surjectivity of r_{E} follows from Lemma 2.3.

2.5. Restriction and conjugation. The following construction has been introduced in [7], for the torsion subgroup of the Dade group, but it works as well for the whole Dade group : let P be a finite p-group, and H be a subgroup of P. If u ∈ lim ; ⟵_{1<Q≤P} D(N_{P} (Q)/Q),

then the sequence (v_{Q})_{1<Q≤H} defined by

v_{Q} = Res_{N_{H}(Q)/Q}^{N_{G}(Q)/Q}u_{Q}

is an element of lim ; ⟵_{1<Q≤H} D(N_{H}(Q)/Q), denoted by Res_{H}^{P}u. The map u ↦ Res_{H}^{P}u

is a linear map lim ; ⟵_{1<Q≤P} D(N_{P} (Q)/Q) to lim ; ⟵_{1<Q≤Q} D(N_{Q}(Q)/Q). The following is the

analogue of Lemma 2.4 of [7] :

2.6. Lemma. Let H be a subgroup of P. The diagram

D(P) \underrightarrow{r_{P}} lim ; ⟵_{1<Q≤P} D(N_{P} (Q)/Q)

Res_{H}^{P} ↓  ↓ Res_{H}^{P}

D(H) \overrightarrow{r_{H}} lim ; ⟵_{1<Q≤H} D(N_{H}(Q)/Q)

is commutative. Proof. This is straightforward.

Similarly, if x ∈ P, denote by c_{x,H} : D(H) → D(^{x}H) the conjugation by x, sending v to ^{x}v. If u ∈ lim ; ⟵_{1<Q≤H} D(N_{H}(Q)/Q), then the sequence (v_{R})_{1<R≤^{x}H} de-

fined by v_{R} = c_{x,R^{x}}(u_{R^{x}}) is an element of lim ; ⟵_{1<R≤^{x}H} D(N_{^{x}H}(R)/R), that will be de-

noted by ^{x}u. The assignment u ↦ ^{x}u is a linear map from lim ; ⟵_{1<Q≤H} D(N_{H}(Q)/Q) to

lim ; ⟵_{1<R≤^{x}H} D(N_{^{x}H}(R)/R), also denoted by c_{x,H}. Then :

2.7. Lemma. Let H be a subgroup of P, and let E be an abelian subgroup of P. The following diagrams are commutative :

\xymatrix{D(H) \ar[r]^{r_{H}} \ar[d]_{c_{x,H}} & lim ; ⟵_{1<Q≤H} D(N_{H}(Q)/Q) \ar[d]^{c_{x,H}} & D(E) \ar[d]_{c_{x,E}} & lim ; ⟵_{1<R≤E} D(E/R) \ar[d]^{c_{x,E}} \ar[l]_{σ_{E}} \\ D(^{x}H) \ar[r]^{r_{^{x}H}} & lim ; ⟵_{1<R≤^{x}H} D(N_{^{x}H}(R)/R) & D(^{x}E) & lim ; ⟵_{1<R≤^{x}E} D(E/R) \ar[l]_{σ_{^{x}E}}}

Proof. This is also straightforward.

2.8. Construction of a map. Let E and F be elements of A_{≥2}(P) such that E < F. If v ∈ lim ; ⟵_{1<Q≤P} D(N_{P} (Q)/Q), consider the element

d_{E,F} = Res_{E}^{F}σ_{F} Res_{F}^{P} v − σ_{E}Res_{E}^{P}v

of D(E). Then by Lemma 2.4 and Lemma 2.3

r_{E}(d_{E,F}) = r_{E}Res_{E}^{F}σ_{F} Res_{F}^{P} v − r_{E}σ_{E}Res_{E}^{P}v
= Res_{E}^{F}r_{F} σ_{F} Res_{F}^{P} v − r_{E}σ_{E}Res_{E}^{P}v
= Res_{E}^{F}Res_{F}^{P} v − Res_{E}^{P}v = 0 .

By Lemma 2.4, there exists a unique integer w_{E,F} such that

d_{E,F} = w_{E,F} · Ω_{E/1} .

If x ∈ P, then it is clear from Lemma 2.7 that ^{x}d_{E,F} = d_{^{x}E,^{x}F}, and it follows that w_{^{x}E,^{x}F} = w_{E,F}. Moreover, if E, F, G ∈ A_{≥2}(P) with E < F < G, then

d_{E,F} + Res_{E}^{F}d_{F,G} = d_{E,G} ,

hence w_{E,F} + w_{F,G} = w_{E,G}. In other words the function sending the pair (E, F) of elements of A_{≥2}(P), with E < F, to w_{E,F}, is a P -invariant 1-cocycle on A_{≥2}(P), with values in ℤ :

2.9. Notation. Let P be a finite p-group. A P -invariant 1-cocycle on A_{≥2}(P), with values in ℤ, is a function sending a pair (E, F) of elements of A_{≥2}(P), with E < F, to an integer w_{E,F}, with the following two properties :

1. If x ∈ P and E < F in A_{≥2}(P), then w_{^{x}E,^{x}F} = w_{E,F}.

2. If E < F < G in A_{≥2}(P), then w_{E,F} + w_{F,G} = w_{E,G}.

The set (Z^{1}(A_{≥2}(P)))^{P} of P -invariants 1-cocycles is a group for addition of functions.

Denote by (B^{1}(A_{≥2}(P)))^{P} the subgroup of (Z^{1}(A_{≥2}(P)))^{P} consisting of cocycles w for which there exists a P -invariant function E ↦ m_{E} from A_{≥2}(P) to ℤ such that

∀E < F ∈ A_{≥2}(P), w_{E,F} = m_{F} − m_{E} .

Denote by H^{1}(A_{≥2}(P), ℤ)^{(P)} the factor group (Z^{1}(A_{≥2}(P)))^{P} /(B^{1}(A_{≥2}(P)))^{P}

2.10. Remark: One can show that the group (B^{1}(A_{≥2}(P)))^{P} is also equal to the set

of elements w of (Z^{1}(A_{≥2}(P)))^{P} for which there exists a (not necessarily P -invariant) function E ↦ m_{E} such that w_{E,F} = m_{F} − m_{E} for any E < F in A_{≥2}(P). This is because if E < F in A_{≥2}(P), then E and F are the “big component”, which is P -invariant. Since w is P -invariant, it follows that the function

x ∈ P ↦ m_{^{x}E} − m_{E}

does not depend on the choice of E, and that it is a group homomorphism from P to ℤ (i.e. an element of H^{1}(P, ℤ)). There are no non zero such homomorphisms, so m is actually P -invariant.

2.11. Remark: On the other hand, one can consider the ordinary first cohomology group H^{1}(A_{≥2}(P), ℤ) of A_{≥2}(P) over ℤ, which is defined similarly to H^{1}(A_{≥2}(P), ℤ)^{(P)}, but forgetting all conditions of P -invariance. Then the group P acts on H^{1}(A_{≥2}(P), ℤ),

and it follows from Remark 2.10 that H^{1}(A_{≥2}(P), ℤ)^{(P)} is a subgroup of the group H^{1}(A_{≥2}(P), ℤ)^{P} of P -invariant elements in H^{1}(A_{≥2}(P), ℤ). It might happen however that this inclusion is proper : an argument similar to the one used in Remark 2.10 yields an element in H^{2}(P, ℤ), and this group need not be zero.

2.12. Notation. Let P be a finite p-group. Denote by

h_{P} : lim ; ⟵_{1<Q≤P} D(N_{P} (Q)/Q) → H^{1}(A_{≥2}(P), ℤ)^{(P)}

the map sending v ∈ lim ; ⟵_{1<Q≤P} D(N_{P} (Q)/Q) to the class of the 1-cocycle w defined by the

following equality, for E < F in A_{≥2}(P) :

(2.13)  w_{E,F} · Ω_{E/1} = Res_{E}^{F}σ_{F} Res_{F}^{P} v − σ_{E}Res_{E}^{P}v .

2.14. Proposition. Let P be a finite p-group. Then h_{P} is a group homomorphism, and the composition

D(P) ⟶^{r_{P}} lim ; ⟵_{1<Q≤P} D(N_{P} (Q)/Q) ⟶^{h_{P}} H^{1}(A_{≥2}(P), ℤ)^{(P)}

is equal to 0. Proof. Clearly, the definition of h_{P} implies that it is a group homomorphism. Observe next that for any E ∈ A_{≥2}(P), since r_{E}σ_{E} is the identity map, the image of the map σ_{E}r_{E} − Id_{D(E)} is contained in the kernel of r_{E}. By Lemma 2.4, it follows that there is a unique linear form s_{E} on D(E), with values in ℤ, such that

σ_{E}r_{E}(u) = u + s_{E}(u) · Ω_{E/1} ,

for any u ∈ D(E). By Lemma 2.7, this definition clearly implies that if x ∈ P, then s_{^{x}E}(^{x}u) = s_{E}(u), for any u ∈ D(E).

Now if E < F in A_{≥2}(P), and if v = r_{P} (t), for t ∈ D(P), Equation 2.13 becomes

w_{E,F} · Ω_{E/1} = Res_{E}^{F}σ_{F} Res_{F}^{P} r_{P} (t) − σ_{E}Res_{E}^{P}r_{P} (t)
= Res_{E}^{F}σ_{F} r_{F} Res_{F}^{P} t − σ_{E}r_{E}Res_{E}^{P}t (by Lemma 2.6)
= Res_{E}^{F}(Res_{F}^{P} t + s_{F} (Res_{F}^{P} t) · Ω_{F/1}) − (Res_{E}^{P}t + s_{E}(Res_{E}^{P}t) · Ω_{E/1})
= (s_{F} (Res_{F}^{P} t) − s_{E}(Res_{E}^{P}t)) · Ω_{E/1} .

Setting m_{E} = s_{E}(Res_{E}^{P}t), for E ∈ A_{≥2}(P), yields

w_{E,F} = m_{F} − m_{E} ,

hence w ∈ (B^{1}(A_{≥2}(P)))^{P} (the P -invariance of m follows easily from the above remark, or from Remark 2.10). Thus h_{P} r_{P} (u) = 0, as was to be shown.

The main theorem of this paper is the following :

2.15. Theorem. Let P be a finite p-group. Then the sequence of abelian groups

0 ⟶ T (P) ⟶ D(P) ⟶^{r_{P}} lim ; ⟵_{1<Q≤P} D(N_{P} (Q)/Q) ⟶^{h_{P}} H^{1}(A_{≥2}(P), ℤ)^{(P)}

is exact.

The key point in this theorem is to show that the kernel of h_{P} is equal to the image of r_{P}. This will be done in two steps : first take an element u ∈ Ker h_{P}, and show that 2u ∈ r_{P} (2D(P)). This amounts to replacing D by 2D, which is easier to handle, since it is torsion free. Next, write 2u = r_{P} (2v), for some v ∈ D(P). Then u − r_{P} (v) is an element in lim ; ⟵_{1<Q≤P} D_{t}(N_{P} (Q)/Q), and it has been shown in [7] that such a sequence

of compatible torsion elements can always be glued (i.e. it always lies in r_{P} (D(P)), though possibly not in r_{P} (D_{t}(P))).

3. Biset functors

The main ingredient in the proof of Theorem 2.15 is the formalism of biset functors. A short exposition of the notation and main results on this subject can be found in Section 2 of [9], Section 3 of [4], or Section 3 of [3].

Recall in particular that if (T, S) is a section of the group P, and if M is a biset functor, then the set P/S is a (P, T/S)-biset, and the corresponding induction-inflation morphism

M(P/S) : M(T/S) → M(P)

is denoted by Indinf_{T/S}^{P}. Similarly, the set S\P is a (T/S, P)-biset, and the corresponding deflation-restriction map

M(S\P) : M(P) → M(T/S)

is denoted by Defres_{T/S}^{P}.

This notation was already used for the Dade group, and it is coherent : it was shown more generally in [6] that, if P and Q are finite p-groups, and U is a finite (Q, P)-biset, one can define a natural group homomorphism D(U) : D(P) → D(Q). This construction yields a structure of biset functor on the correspondence sending a p-group P to the subgroup D^{Ω}(P) of D(P) generated by all the relative syzygies Ω_{X} obtained for various P -sets X. In the case p ≠ 2, it was shown in [3] that D = D^{Ω}, so D is a biset functor in this case. For p = 2, the biset functor structure on D^{Ω} cannot in general be extended to the whole of D, because of Frobenius twists (also called Galois twists) (see [6] or [3] for details).

The following additional notation was introduced in [9] : a class Y of p-groups is said to be closed under taking subquotients if for any Y ∈ Y and any section (T, S) of Y, the corresponding subquotient T/S belongs to Y. If Y is such a class, and P is a finite p-group, denote by Y(P) the set of sections (T, S) of P such that T/S ∈ Y. One can consider biset functors defined only on Y, with values in abelian groups. Let F_{Y} be the category of all such functors, and let F denote the category of functors defined on all finite p-groups. These categories are abelian categories.

The obvious forgetful functor

O_{Y} : F → F_{Y}

admits left and right adjoints

L_{Y} : F_{Y} → F and R_{Y} : F_{Y} → F ,

whose evaluations can be computed as follows (cf. [9] Theorem 1.2 for details, in particular on the direct and inverse limits appearing in this statement) : 3.1. Theorem. With the notation above, for any functor M ∈ F_{Y}, we have :

L_{Y}M(P) ≅ lim ; ⟶_{(T,S)∈Y(P)} M(T/S)  and  R_{Y}M(P) ≅ lim ; ⟵_{(T,S)∈Y(P)} M(T/S) .

Moreover ([9] Corollary 6.17), for any biset functor M and any p-group P, the unit map

η_{M,P} : M(P) → R_{Y}O_{Y}(P) = lim ; ⟵_{(T,S)∈Y(P)} M(T/S)

is given by

η_{M,P} (u)_{T,S} = Defres_{T/S}^{P}u ,

for any u ∈ M(P) and any (T, S) ∈ Y(P).

4. Image in subquotients of rank two

4.1. Lemma. Let X be a non-empty class of finite p-groups, closed under taking subquotients. Let A be any abelian group, and denote by B̂ the functor Hom_{ℤ}(B(−), A), where B is the Burnside functor. Then the unit map

β : B̂ → R_{X} O_{X} B̂

is an isomorphism. Proof. Indeed if P is a p-group, if φ ∈ Hom_{ℤ}(B(P), A), if (T, S) is a section of P and X is a subgroup such that S ≤ X ≤ T, then

(Defres_{T/S}^{P}φ)((T/S)/(X/S)) = φ(Indinf_{T/S}^{P}(T/S)/(X/S)) = φ(P/X) .

Suppose that φ ∈ Ker β. Since the class X is non empty and closed under taking subquotients, then it contains the trivial group, and for any subgroup X of P, the section (X, X) is in X (P). It follows in particular that

0 = (Defres_{X/X}^{P}φ)((X/X)/(X/X)) = φ(P/X) .

Thus φ = 0, and β is injective.

Conversely, let ψ = (ψ_{T,S})_{(T,S)∈X (P)} be an element of lim ; ⟵_{X (P)} Hom_{ℤ}(B(T/S), A).

Equivalently, for each section (T, S) ∈ X (P) and each subgroup X with S ≤ X ≤ T, we have an element ψ_{T,S}((T/S)/(X/S)) of A, fulfilling the following two conditions :

i) If x ∈ P, then ψ_{^{x}T ,^{x}S}((^{x}T /^{x}S)/(^{x}X/^{x}S)) = ψ_{T,S}((T/S)/(X/S)).

ii) If (T, S) ∈ X (P) and (T′, S′) ∈ X (P), and if X is a subgroup of P such that S ≤ S′ ≤ X ≤ T′ ≤ T, then ψ_{T,S}((T/S)/(X/S)) = ψ_{T′,S′}((T′/S′)/(X/S′)).

Condition i) implies in particular that the element ψ_{X,X}((X/X)/(X/X)) is constant on the conjugacy class of X in P. Hence we can define an element φ ∈ Hom_{ℤ}(B(P), A) by setting

φ(P/X) = ψ_{X,X}((X/X)/(X/X)) ,

for any subgroup X of P. Now if (T, S) ∈ X (P) and if S ≤ X ≤ T

(Defres_{T/S}^{P}φ)((T/S)/(X/S)) = φ(P/X) = ψ_{X,X}((X/X)/(X/X))
= (Defres_{X/X}^{T/S}ψ_{T,S})((X/X)/(X/X))
= ψ_{T,S}(Indinf_{(T/S)/(X/X)}^{T/S}(X/X)/(X/X))
= ψ_{T,S}((T/S)/(X/S)) .

Thus Defres_{T/S}^{P}φ = ψ_{T,S} for any (T, S) ∈ X (P). Equivalently β(φ) = ψ, so β is surjective, hence it is an isomorphism.

4.2. Lemma. When p is odd, the map

ε = ∏_{(T,S)∈X_{3}(P)} Defres_{T/S}^{P} : 2D(P) ⟶ lim ; ⟵_{(T,S)∈X_{3}(P)} 2D(T/S)

is an isomorphism.

Proof. Consider the short exact sequence of p-biset functors

0 → 2D → D → 𝔽_{2}D → 0 .

Applying the functor lim ; ⟵_{X_{3}(P)} yields the bottom line of the following commutative diagram

with exact lines

(4.3)  \xymatrix{0 \ar[r] & 2D(P) \ar[r] \ar[d]^{ε} & D(P) \ar[r] \ar[d]^{δ} & 𝔽_{2}D(P) \ar[d]^{γ} \ar[r] & 0 \\ 0 \ar[r] & lim ; ⟵_{X_{3}(P)} 2D \ar[r] & lim ; ⟵_{X_{3}(P)} D \ar[r] & lim ; ⟵_{X_{3}(P)} 𝔽_{2}D}

where the map δ is an isomorphism, by Theorem 1.1 of [9]. Moreover, by Corollary 1.5 of [10], there is an exact sequence of p-biset functors

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

where B^{×} is the functor of units of the Burnside ring, which is isomorphic to the constant functor Γ_{𝔽_{2}} for p odd. Moreover D^{Ω} = D in this case. Applying the functor lim ; ⟵_{X_{3}(P)} to this sequence yields the bottom line of the following commutative diagram

with exact lines

(4.4)  \xymatrix{0 \ar[r] & 𝔽_{2} \ar[r] \ar[d]^{α} & 𝔽_{2}B^{∗}(P) \ar[r] \ar[d]^{β} & 𝔽_{2}D(P) \ar[d]^{γ} \ar[r] & 0 \\ 0 \ar[r] & lim ; ⟵_{X_{3}(P)} Γ_{𝔽_{2}} \ar[r] & lim ; ⟵_{X_{3}(P)} 𝔽_{2}B^{∗} \ar[r] & lim ; ⟵_{X_{3}(P)} 𝔽_{2}D}

Since 𝔽_{2}B^{∗} is naturally isomorphic to Hom_{ℤ}(B(−), 𝔽_{2}), the map β is an isomorphism, by Lemma 4.1. Now the group lim ; ⟵_{X_{3}(P)} Γ_{𝔽_{2}} is the set of sequences (u_{T,S})_{(T,S)∈X_{3}(P)}

fulfilling the two following conditions :

i) If (T, S) ∈ X_{3}(P) and x ∈ P, then u_{^{x}T ,^{x}S} = u_{T,S}.

ii) If (T, S) ∈ X_{3}(P) and (T′, S′) ∈ X_{3}(P) are such that S ≤ S′ ≤ T′ ≤ T, then u_{T,S} = u_{T′,S′}.

Applying this for the case T′ = S′ = S, and next for the case T = T′ = S′, it follows that u_{T,S} = u_{S,S} = u_{T,T} for any (T, S) ∈ X_{3}(P). Thus u_{T,T} = u_{S,S} if T/S ∈ X_{3}. Since X_{3} contains the cyclic group of order p, and since P is a p-group, it follows that u_{T,T} = u_{1,1} for any subgroup T of P. Hence u_{T,S} is constant, and the map α is an isomorphism.

Now the Snake’s Lemma, applied to Diagram 4.4, shows that the map γ is injective. And another application of this Lemma to Diagram 4.3 shows that the map ε is an isomorphism.

Recall that if E is an elementary abelian group of rank 2, then 2D(E) is free of rank one, generated by 2Ω_{E/1}. Thus if u ∈ 2D(P), and if (T, S) ∈ E_{2}^{♯}(P), then Defres_{T/S}^{P}u is a multiple of 2Ω_{T/S}. The following theorem characterizes the sequences of integers (v_{T,S})_{(T,S)∈E_{2}^{♯}(P)} which can be obtained that way from an element of 2D(P) :

4.5. Theorem. Let P be a p-group (with p > 2). The map

D_{P} : 2D(P) → ∏_{(T,S)∈E_{2}^{♯}(P)} ℤ

sending an element u ∈ 2D(P) to the sequence D_{P} (u)_{T,S} of integers defined by

Defres_{T/S}^{P}(u) = D_{P} (u)_{T,S} · 2Ω_{T/S} ,

is injective, and its image is equal to the set of sequences (v_{T,S})_{(T,S)∈E_{2}^{♯}(P)} fulfilling the following conditions :

1. If (T, S) ∈ E_{2}^{♯}(P) and x ∈ P, then v_{T,S} = v_{^{x}T ,^{x}S}.

2. If (T, S) and (T′, S) are in E_{2}^{♯}(P), if T′ ≤ N_{P} (T), and T T′/S ≅ (ℤ/pℤ)^{3}, then

v_{T,S} + ∑_{S<X<T} v_{T T′,X} = v_{T′,S} + ∑_{S<X<T′} v_{T T′,X} .

3. If (T, S) and (T′, S) are in E_{2}^{♯}(P), if T′ ≤ N_{P} (T), and T T′/S ≅ X_{p^{3}}, then

v_{T,S} ≡ v_{T′,S} (mod p) .

Proof. Let u ∈ Ker D_{P}. Then Defres_{T/S}^{P}u = 0 for any (T, S) ∈ E_{2}^{♯}(P). Moreover Defres_{T/S}^{P}u ∈ 2D(ℤ/pℤ) = {0} when (T, S) is a section of P with T/S ≅ ℤ/pℤ. The detection theorem of Carlson and Thévenaz ([11] Theorem 13.1) shows that u = 0. Hence D_{P} is injective.

Now Condition 1 of Theorem 4.5 holds obviously for the elements of the image of D_{P}. To check that Conditions 2 and 3 also hold, suppose that (T, S), (T′, S) ∈ E_{2}^{♯}(P), with T′ ≤ N_{P} (T) and |T T′ : P | = p^{3}, and observe that setting R = T T′, the diagram

[figure: D(P); D_{P}; _{Defres{P \atop R/S}}; ∏_{(T′′,S′′)∈E_{2}^{♯}(P)} ℤ; π_{R,S}; D(R/S); D_{P}; ∏_{(T′′/S,S′′/S)∈E_{2}^{♯}(R/S)} ℤ]

is commutative, where π_{R,S} is the projection map obtained by identifying sections (T′′, S′′) of P such that S ≤ S′′ ≤ T′′ ≤ R with sections (T′′/S, S′′/S) of R/S. This shows that it is enough to suppose that the group P is either elementary abelian of rank 3, or isomorphic to X_{p^{3}}. These special cases are detailed below.

To prove that conversely, Conditions 1, 2, and 3 characterize the image of D_{P}, observe first that by Theorem 1.1 of [9], the map

∏_{(T,S)} Defres_{T/S}^{P} : D(P) ⟶ lim ; ⟵_{(T,S)} D(T/S)

is an isomorphism, where T/S runs through all sections of P which are either elementary abelian p-groups of rank ≤ 3 or extraspecial groups of order p^{3} and exponent p.

Suppose that Theorem 4.5 is true when P is elementary abelian of rank at most 3, or extraspecial of exponent p. Let P be an arbitrary p-group, and consider a sequence v = (v_{T,S})_{(T,S)∈E_{2}^{♯}(P)} fulfilling the conditions of Theorem 4.5.

If (V, U) ∈ X_{3}(P), then the correspondence (T, S) ↦ (T/U, S/U) is a one to one correspondence between the set of elements (T, S) of E_{2}^{♯}(P) such that U ≤ S ≤ T ≤ V, and E_{2}^{♯}(V/U). Through this bijection, the sequence of integers v_{T,S}, for U ≤ S ≤ T ≤ V, yields a sequence of integers fulfilling the conditions of Theorem 4.5 for the group V/U, hence an element in the image of the map D_{V/U}. In other words, there is a unique element w_{V,U} ∈ 2D(V/U) such that

Defres_{T/S}^{V/U}w_{V,U} = v_{T,S} · 2Ω_{T/S} ,

for all (T, S) ∈ E_{2}^{♯}(P) with U ≤ S ≤ T ≤ V.

Now the uniqueness of w_{V,U} shows that w_{^{x}V ,^{x}U} = ^{x}w_{V,U} for any x ∈ P, and that Defres_{V′/U′}^{V/U}w_{V,U} = w_{V′,U′} whenever (V, U) and (V′, U′) are in X_{3}(P) and U ≤ U′ ≤ V′ ≤ V. In other words, the sequence (w_{V,U})_{(V,U)∈X_{3}(P)} is an element of lim ; ⟵_{X_{3}(P)} 2D. By

Lemma 4.2, there exists an element t ∈ 2D(P) such that

w_{V,U} = Defres_{V/U}^{P}t ,

for any (V, U) ∈ X_{3}(P). Then obviously D_{P} (t) = v, and v lies in the image of D_{P}, as was to be shown.

So the only thing left to check is that Theorem 4.5 holds when P is elementary abelian of rank at most 3, or isomorphic to X_{p^{3}}. This is a case by case verification, using the following lemma :

4.6. Lemma. Let P be a finite p-group, and X be a finite P -set.

1. If T/S is a section of P, then

Defres_{T/S}^{P}Ω_{X} = Ω_{X^{S}} ,

where X^{S} denotes the set of fixed points of S on X, viewed as a T/S-set.

2. If moreover (T, S) ∈ E_{2}^{♯}(P), then

Defres_{T/S}^{P}(2Ω_{X}) = ( ∑_{{S≤V ≤T \atop X^{V} ≠∅}} μ(S, V))2Ω_{T/S} .

In other words D_{P} (2Ω_{X})_{T,S} = ∑_{{S≤V ≤T \atop X^{V} ≠∅}} μ(S, V).

Proof. Assertion 1 follows from Section 4 of [2]. For Assertion 2, note that by Assertion 1 and Lemma 5.2.3 of [2], since T/S is abelian,

Defres_{T/S}^{P}2Ω_{X} = ∑_{{S≤U≤V ≤T \atop X^{V} ≠∅}} μ(U, V) · 2Ω_{T/U} ,

and that 2Ω_{T/U} = 0 in D(T/S) unless U = S.

Now there are four cases :

• If |P | ≤ p, there is nothing to do, since the map D_{P} is an isomorphism {0} → {0}.

• If P ≅ (ℤ/pℤ)^{2}, then 2D(P) ≅ ℤ, and E_{2}^{♯}(P) = {(P, 1)}. In this case, there is no condition on the image of D_{P}, and D_{P} is an isomorphism ℤ → ℤ. So Theorem 4.5 holds in this case.

• If P ≅ (ℤ/pℤ)^{3}, then E_{2}^{♯}(P) consists of p^{2} + p + 1 sections (P, R), for |R| = p, and p^{2} + p + 1 sections (Q, 1), for |Q| = p^{2}. The group 2D(P) is a free abelian group, with basis

{2Ω_{P/1}} ⊔ {2Ω_{P/R} | |R| = p} .

The following arrays gives the values of the sequence v = D_{P} (u) for the element u in its first column on the left :

v_{P,R}  v_{Q,1}
2Ω_{P/1}  0  1
2Ω_{P/R′}  \{ \begin{matrix}0 & if R′ ≠ R \\ 1 & if R′ = R \end{matrix} \{ \begin{matrix}0 & if R′ < Q \\ 1 & if R′ ≮ Q \end{matrix}

The image of the element u = m_{1} · 2Ω_{P/1} + ∑_{|R′|=p} m_{R′} · 2Ω_{P/R′} by the map D_{P} is equal

to the sequence v = (v_{T,S}), where

(4.7)  v_{P,R} = m_{R}  v_{Q,1} = m_{1} + ∑_{R≮Q} m_{R}

If Q ≠ Q′ are subgroups of order p^{2} of P, then QQ′ = P, and

v_{Q,1} + ∑_{1<X<Q} v_{P,X} = m_{1} + ∑_{|X|=p} m_{X} = v_{Q′,1} + ∑_{1<X<Q′} v_{P,X} ,

so Condition 2 of 4.5 holds for the sections (Q, 1) and (Q′, 1) of P. Since P is abelian, Conditions 1 and 3 of 4.5 are obviously satisfied.

Conversely, suppose that Condition 2 hold for a sequence v = (v_{T,S})_{(T,S)∈E_{2}^{♯}(P)}. This sequence is in the image of D_{P} if and only if there exist integers m_{1}, m_{R′}, for |R′| = p, such that 4.7 hold.

The first equation gives m_{R} = v_{P,R}, and then the second one gives

m_{1} = v_{Q,1} − ∑_{R≮Q} v_{P,R} .

This is consistent if the right hand side does not depend on Q, i.e. if for any subgroups Q ≠ Q′ of order p^{2} of P

v_{Q,1} − ∑_{R≮Q} v_{P,R} = v_{Q′,1} − ∑_{R≮Q′} v_{P,R} ,

or equivalently

v_{Q,1} + ∑_{R<Q} v_{P,R} = v_{Q′,1} + ∑_{R<Q′} v_{P,R} .

This is precisely Condition 2 of 4.5 for the section (Q, 1) and (Q′, 1), since QQ′ = P in this case. Thus Theorem 4.5 holds for P ≅ (ℤ/pℤ)^{3}.

• If P ≅ X_{p^{3}}, then E_{2}^{♯}(P) consists of the section (P, Z), where Z is the centre of P, and of p + 1 sections (Q, 1), where Q is a subgroup of index p in P. The group D(P) is equal to D^{Ω}(P), since p ≠ 2, so it is generated by the elements Ω_{P/1}, Ω_{P/X}, for |X| = p, and Ω_{P/Q}, for |Q| = p^{2}, which have order 2 in D(P). Thus 2D(P) is generated by the elements 2Ω_{P/1} and 2Ω_{P/X}, for |X| = p. The following array gives the values of the sequence v = D_{P} (u) for the element u in its first column on the left, where R denotes a non central subgroup of order p of P :

v_{P,Z}  v_{Q,1}
2Ω_{P/1}  0  1
2Ω_{P/Z}  1  0
2Ω_{P/R} 0 \{ \begin{matrix}1 & if R ≮ Q \\ 1 − p & if R < Q \end{matrix}

The values in this table can be computed using Lemma 4.6 : for example

Res_{Q}^{P}2Ω_{P/R} = ( ∑_{{1≤V ≤Q \atop V ≤_{P} R}} μ(1, V))2Ω_{Q/1} .

If R ≰ Q, then there is only one term in the summation, for V = 1, and μ(1, V) = 1 in this case. And if R ≤ Q, then there are p additional terms, obtained for the p distinct conjugates V of R in P, and μ(1, V) = −1 for each of them. This gives the value 1 − p in this case.

Now if u = m_{1} · 2Ω_{P/1} + m_{Z} · 2Ω_{P/Z} + ∑_{[R]} m_{R} · 2Ω_{P/R} (where the brackets around R

mean that R runs through a set of representatives of conjugacy classes of non central subgroups of order p of P), then the sequence v = D_{P} (u) is given by :

(4.8)  v_{P,Z} = m_{Z}  v_{Q,1} = m_{1} + ∑_{[R]≮Q} m_{R} + (1 − p) ∑_{[R]<Q} m_{R} ,

The second equation is equivalent to

(4.9)  v_{Q,1} = m_{1} + ∑_{[R]} m_{R} − p ∑_{[R]<Q} m_{R} .

It follows that v_{Q,1} ≡ v_{Q′,1} (mod p), for any subgroups Q and Q′ of order p^{2} in P. This shows that Condition 3 of 4.5 holds for the sections (Q, 1) and (Q′, 1) of P. Condition 2 is obviously satisfied in this case, since P has no subquotient isomorphic to (ℤ/pℤ)^{3}.

Suppose now conversely that a sequence v = (v_{T,S})_{(T,S)∈E_{2}^{♯}(P)} is given, and that Conditions 1 and 3 of 4.5 hold. Then v lies in the image of D_{P} if and only if there exist integers m_{1}, m_{Z}, m_{R} (invariant by P -conjugation), such that Equations 4.8 hold.

The first equation in 4.8 gives m_{Z} = v_{P,Z}, and the second one gives

m_{1} − v_{Q,1} + ∑_{[R]} m_{R} = p ∑_{[R]<Q} m_{R} .

All subgroups of order p of Q different from Z are conjugate in P. Denoting by R_{Q} one of them, this equation becomes

(4.10)  m_{1} − v_{Q,1} + ∑_{[R]} m_{R} = pm_{R_{Q}} .

Summing this relation over Q yields

(p + 1)m_{1} − ∑_{Q} v_{Q,1} + (p + 1) ∑_{[R]} m_{R} = p ∑_{[R]} m_{R} ,

thus

∑_{[R]} m_{R} = ∑_{Q} v_{Q,1} − (p + 1)m_{1} .

Now 4.10 yields

pm_{R_{Q}} = ∑_{Q′≠Q} v_{Q′,1} − pm_{1} .

By Condition 3, the sum ∑_{Q′≠Q} v_{Q′,1} is congruent to pv_{Q,1} modulo p, i.e. to 0. Since

Q = R_{Q}Z, this gives finally

m_{R} = \frac{1}{p}( ∑_{Q′≠RZ} v_{Q′,1}) − m_{1} .

Conversely, if this holds for any R, then equation 4.9 holds : indeed, in this case

∑_{[R]} m_{R} = ∑_{Q} v_{Q,1} − (p + 1)m_{1} ,

thus

m_{1} + ∑_{[R]} m_{R} − p ∑_{[R]<Q} m_{R} = m_{1} + ∑_{Q} v_{Q,1} − (p + 1)m_{1} − pm_{R_{Q}}
= ∑_{Q} v_{Q,1} − pm_{1} − ( ∑_{Q′≠Q} v_{Q′,1}) + pm_{1}
= v_{Q,1} .

Thus 4.9 holds, and Theorem 4.5 also, when P = X_{p^{3}}.

5. Proof of Theorem 2.15

Let P be a finite p-group. Clearly T (P) is the kernel of r_{P}, and Im r_{P} ≤ Ker h_{P}, by Proposition 2.14. So the only thing to show is that this inclusion is an equality.

Let u ∈ Ker h_{P}. It means that there exists a P -invariant function E ↦ m_{E} from A_{≥2}(P) to ℤ such that for any E < F in A_{≥2}(P)

w_{E,F} = m_{E} − m_{F} ,

where the integer w_{E,F} is defined by the equality

w_{E,F} · Ω_{E/1} = Res_{E}^{F}σ_{F} Res_{F}^{P} u − σ_{E}Res_{E}^{P}u .

In other words

Res_{E}^{F}(m_{F} · Ω_{F/1} + σ_{F} Res_{F}^{P} u) = m_{E} · Ω_{E/1} + σ_{E}Res_{E}^{P}u .

Set w_{E} = m_{E} · Ω_{E/1} + σ_{E}Res_{E}^{P}u, for E ∈ A_{≥2}(P). Then Res_{E}^{F}w_{F} = w_{E}, for any E < F in A_{≥2}(P), and ^{x}(w_{E}) = w_{^{x}E} for any x ∈ P and E ∈ A_{≥2}(P). Moreover , for any E ∈ A_{≥2}(P)

r_{E}(w_{E}) = r_{E}σ_{E}Res_{E}^{P}u = Res_{E}^{P}u ,

since r_{E}(Ω_{E/1}) = 0, and since σ_{E} is a section of r_{E}. It means that for any subgroup Y ≠ 1 of E

Def_{E/Y}^{E} w_{E} = Res_{E/Y}^{N_{P} (Y)/Y} u_{Y} .

If (T, S) ∈ E_{2}^{♯}(P), define an integer v_{T,S} by

(5.1)  \{ \begin{matrix}Res_{T/S}^{N_{P} (S)/S}(2u_{S}) & = & v_{T,S} · 2Ω_{T/S} & if S ≠ 1 \\ 2w_{T} & = & v_{T,1} · 2Ω_{T/1} & if S = 1 \end{matrix}

This sequence of integers (v_{T,S})_{(T,S)∈E_{2}^{♯}(P)} satisfies some of the conditions of Theorem 4.5. Indeed :

• If x ∈ P and (T, S) ∈ E_{2}^{♯}(P), then v_{^{x}T ,^{x}S} = v_{T,S} : this is because ^{x}(w_{E}) = w_{^{x}E} for any E ∈ A_{≥2}(P), and because ^{x}(u_{Q}) = u_{^{x}Q} for any subgroup Q ≠ 1 of P. Thus Condition 1 of Theorem 4.5 holds.

• Suppose that (T, S) and (T′, S) are elements of E_{2}^{♯}(P) such that T ≤ N_{P} (T′). There are two cases to consider :

(a) If S ≠ 1, then for any section (V, U) ∈ E_{2}^{♯}(N_{P} (S)/S)

v_{V,U} · Ω_{V/U} = Res_{V/U}^{N_{P} (U)/U}(2u_{U})
= Res_{V/U}^{N_{P} (S,U)/U}Res_{N_{P} (S,U)/U}^{N_{P} (U)/U}(2u_{U})
= Res_{V/U}^{N_{P} (S,U)/U}Defres_{N_{P} (S,U)/U}^{N_{P} (S)/S}(2u_{S})
= Defres_{V/U}^{N_{P} (S)/S}(2u_{S}) .

It follows that the sequence (v_{V,U})_{(V,U)∈E_{2}^{♯}(N_{P} (S)/S)} is equal to D_{N_{P} (S)/S}(2u_{S}),

hence it is in the image of the map D_{N_{P} (S)/S}. Thus if T T′/S ≅ (ℤ/pℤ)^{3}, then Condition 2 of Theorem 4.5 holds for the sections (T, S) and (T′, S) of N_{P} (S)/S. And if T T′/S ≅ X_{p^{3}}, then Condition 3 of Theorem 4.5 holds, for a similar reason.

(b) If S = 1, then set F = T T′. If F ≅ (ℤ/pℤ)^{3}, then consider a section (V, U) ∈ E_{2}^{♯}(F). If U = 1, then

Defres_{V/U}^{F}2w_{F} = Res_{V}^{F} 2w_{F} = 2w_{V} = v_{V,1} · 2Ω_{V/1} .

And if U ≠ 1, then

Defres_{V/U}^{F}2w_{F} = Res_{V/U}^{F/U}Def_{F/U}^{F}2w_{F}
= Res_{V/U}^{F/U}Res_{F/U}^{N_{P} (U)/U}2u_{U}
= Res_{V/U}^{N_{P} (U)/U}2u_{U}
= v_{V,U} · 2Ω_{V/U} .

It follows that the sequence (v_{V,U})_{(V,U)∈E_{2}^{♯}(F)} is equal to D_{F} (2w_{F}). In particular, Condition 2 of Theorem 4.5 is fulfilled for the sections (T, 1) and (T′, 1) of F.

Hence the sequence (v_{T,S})_{(T,S)∈E_{2}^{♯}(P)} fulfills all the conditions of Theorem 4.5, except possibly Condition 3 for sections (T, 1) and (T′, 1) such that T ≤ N_{P} (T′) and T T′ ≅ X_{p^{3}}. This situation is handled by the following lemma :

5.2. Lemma. Let P be a finite p-group, and (v_{T,S})_{(T,S)∈E_{2}^{♯}(P)} be a sequence of integers such that :

1. If x ∈ P and (T, S) ∈ E_{2}^{♯}(P), then v_{^{x}T ,^{x}S} = v_{T,S}.

2. If (T, S) and (T′, S) are in E_{2}^{♯}(P), if T ≤ N_{P} (T′) and if T T′/S ≅ (ℤ/pℤ)^{3}, then

v_{T,S} + ∑_{S<X<T} v_{T T′,X} = v_{T′,S} + ∑_{S<X<T′} v_{T T′,X} .

3. If S ≠ 1, if (T, S) and (T′, S) are in E_{2}^{♯}(P), if T ≤ N_{P} (T′) and if T T′/S ≅ X_{p^{3}},

then

v_{T,S} ≡ v_{T′,S} (mod p) .

Then :

(i) If T, T′ ∈ A_{=2}(P), if T and T′ are in the same connected component of A_{≥2}(P), if T ≤ N_{P} (T′) and T T′ ≅ X_{p^{3}}, then v_{T,1} ≡ v_{T′,1} (mod p).

(ii) There exists a sequence of integers (y_{T})_{T ∈A_{=2}(P)} such that

(a) If x ∈ P and T ∈ A_{=2}(P), then y_{^{x}T} = y_{T}.

(b) If T, T′ ∈ A_{=2}(P), if T ≤ N_{P} (T′) and T T′ ≅ (ℤ/pℤ)^{3}, then y_{T} = y_{T′}.

(c) If T, T′ ∈ A_{=2}(P), if T ≤ N_{P} (T′) and T T′ ≅ X_{p^{3}}, then

y_{T} + v_{T,1} ≡ y_{T′} + v_{T′,1} (mod p) .

Proof. The proof of Assertion (i) goes by induction on |P |, starting with the case where P is cyclic, where there is nothing to prove. Assume then that Hypotheses 1), 2), and 3) imply Assertion 1, for any p-group of order strictly smaller than |P |. Let T and T′ be elementary abelian subgroups of rank 2 of P, such that T ≤ N_{P} (T′) and T T′ ≅ X_{p^{3}}. Set X = T T′, and denote by Z the centre of X.

If there is a proper subgroup Q of P containing X, and such that T and T′ are in the same connected component of A_{≥2}(Q), then v_{T,1} ≡ v_{T′,1} (mod p), by induction, since Hypotheses 1), 2), and 3) obviously hold for Q is they hold for P. It is the case in particular if A_{≥2}(Q) is connected.

Suppose that there exists a subgroup C of order p in C_{P} (X), not contained in X (i.e. different from Z). Then the center T′′ = C × Z of the subgroup Q = C × X of P is not cyclic. Hence A_{≥2}(Q) is connected, and Q contains T and T′. Thus I can suppose that Q = P, and then T′′ is equal to the centre of P. It is elementary abelian of rank 2. Moreover T T′′ ≅ (ℤ/pℤ)^{3}, since T and T′′ are elementary abelian of rank 2 and centralize each other, and since T ∩ T′′ = T ∩ X ∩ T′′ = Z. Hypothesis 2, applied to the sections (T, 1) and (T′, 1) of P yields

(5.3)  v_{T,1} − v_{T′′,1} = ∑_{1<F <T′′} v_{T T′′,F} − ∑_{1<F <T} v_{T T′′,F} .

Now T T′′ ⊴ P since |P : T T′′| = p, and T ⊴ P, since T ⊴ X and C ≤ C_{P} (X). Hence P acts by conjugation on the set of subgroups F such that 1 < F < T, and F = Z is the unique fixed point under this action. Now Hypothesis 1 implies that

∑_{1<F <T} v_{T T′′,F} ≡ v_{T T′′,Z} (mod p) ,

and Equation 5.3 yields

(5.4)  v_{T,1} − v_{T′′,1} ≡ ∑_{{1<F <T′′ \atop F ≠Z}} v_{T T′′,F} (mod p) .

The same argument applies with T′ instead of T, so

(5.5)  v_{T′,1} − v_{T′′,1} ≡ ∑_{{1<F <T′′ \atop F ≠Z}} v_{T′T′′,F} (mod p) .

Now if 1 < F < T′′ and F ≠ Z, the group P/F has order p^{3} and exponent p (since P has exponent p), and it is non abelian (since F ≰ [P, P] = Z). Hence P/F ≅ X_{p^{3}}. Since P = (T T′′)(T′T′′), Hypothesis 3, applied to the sections (T T′′, F) and (T′T′′, F) yields v_{T T′′,F} ≡ v_{T′T′′,F} (mod p). This shows that the right hand sides of 5.4 and 5.5 are congruent modulo p. So are the left hand sides, thus v_{T,1}−v_{T′′,1} ≡ v_{T′,1}−v_{T′′,1} (mod p), and v_{T,1} ≡ v_{T′,1} (mod p).

Hence I can suppose that Z is the only subgroup of order p of C_{P} (X). In particular, the centre of P is cyclic, and Z is the only subgroup of order p in this centre. Moreover, since T ≠ T′ and T, T′ are in the same connected component of A_{≥2}(P), the groups T and T′ are not maximal elementary abelian subgroups, thus P has p-rank at least equal to 3, and T and T′ are in the big component C of A_{≥2}(P).

In this case, there is a normal subgroup T_{0} of P which is elementary abelian of rank 2, and T_{0} ∈ C. Moreover T_{0} > Z.

If T_{0} ≰ X, then T_{0} ∩ X = Z. Then |T_{0}X| = p^{4}, and |T_{0}X : X| = p. Thus T_{0} normalizes X. Moreover, if Y is a subgroup of index p of X, then Y > Z, and |T_{0}Y | = p^{3}, thus |T_{0}Y : Y | = p, and T_{0} normalizes Y. It follows that the image of T_{0} in the group Out(X) of outer automorphisms of X, which is isomorphic to GL_{2}(𝔽_{p}), is a p-subgroup stabilizing every line. So this image is trivial, and T_{0} acts on X by inner automorphisms. Let t ∈ T_{0} − X. Then there exists y ∈ X such that y^{−1}t ∈ C_{P} (X). In particular y^{−1}t centralizes y, so t centralizes y, and then (y^{−1}t)^{p} = (y^{−1})^{p}t^{p} = 1. Moreover y ≠ t, since t ∉ X. Hence y^{−1}t has order p. Since Z is the only subgroup of order p of C_{P} (X), it follows that y^{−1}t ∈ Z, so t ∈ X. This contradiction shows that T_{0} ≤ X.

Since the congruences v_{T,1} ≡ v_{T_{0},1} (mod p) and v_{T′,1} ≡ v_{T_{0},1} (mod p) imply the congruence v_{T,1} ≡ v_{T′,1} (mod p), it is enough to suppose that T_{0} = T, thus T ⊴ P. Let F be an elementary abelian subgroup of rank 3 of P containing T′ : such a subgroup exists, since T′ is not a maximal element of A_{≥2}(P). Set T′′ = C_{F} (T). Then |F : T′′| divide p, since F/T′′ is a p-subgroup of Aut(T) ≅ GL_{2}(𝔽_{p}). Moreover F ≰ C_{P} (T), since F > T′. Thus |F : T′′| = p, and T′′ ≅ (ℤ/pℤ)^{2}. Moreover T′ ≠ T′′, since T′ ≰ C_{P} (T), thus F = T′T′′.

Now F centralizes T′, and normalizes T. Thus F normalizes T T′ = X. Moreover F ∩ X = T′, since T′ ≤ F ∩ X, and since F and X are distinct subgroups of order p^{3} of P, for F is abelian and X is not. Hence |F X : F | = p, so F X normalizes F. Thus X normalizes F, and X also normalizes C_{P} (T) since T ⊴ P. It follows that X normalizes F ∩ C_{P} (T) = C_{F} (T) = T′′. Obviously X also normalizes its subgroup T′.

Hypothesis 2, applied to the sections (T′, 1) and (T′′, 1) of P, yields

(5.6)  v_{T′,1} − v_{T′′,1} = ∑_{1<Y <T′′} v_{T′T′′,Y} − ∑_{1<Y <T′} v_{T′T′′,Y} .

Since X normalizes T′ and T′′, and since any subgroup of order p normalized by X is centralized by X, Hypothesis 1 yields

∑_{1<Y <T′′} v_{T′T′′,Y} ≡ ∑_{{1<Y <T′′ \atop Y ≤C_{P} (X)}} v_{T′T′′,Y} (mod p) .

But T′′ ∩ C_{P} (X) = Z, since Z ≤ T′′ ∩ C_{P} (X), and since Z is the only subgroup of order p of C_{P} (X). Thus

(5.7)  ∑_{1<Y <T′′} v_{T′T′′,Y} ≡ v_{T′T′′,Z} (mod p) .

The same argument, applied with T′ instead of T′′, since T′ ∩ C_{P} (X) = Z, yields

(5.8)  ∑_{1<Y <T′} v_{T′T′′,Y} ≡ v_{T′T′′,Z} (mod p) .

Now it follows from 5.6, 5.7 and 5.8 that v_{T′,1} − v_{T′′,1} ≡ 0 (mod p), i.e.

(5.9)  v_{T′,1} ≡ v_{T′′,1} (mod p) .

Now the group T T′′ is also elementary abelian of rank 3 : indeed, the group T′′ centralizes T, and T′′ ∩ T = Z since T′′ ∩ T ≥ Z and T′′ ≠ T for T ≰ F ≤ C_{P} (T′). Then Hypothesis 2, for the sections (T, 1) and (T′′, 1), yields

(5.10)  v_{T,1} − v_{T′′,1} = ∑_{1<Y <T′′} v_{T T′′,Y} − ∑_{1<Y <T} v_{T T′′,Y} .

The group X normalizes T and T′′, and T ∩ C_{P} (X) = Z = T′′ ∩ C_{P} (X). The same argument as above yields

(5.11)  v_{T,1} ≡ v_{T′′,1} (mod p) .

Thus v_{T,1} ≡ v_{T′,1} (mod p), by 5.9 and 5.11, and this completes the proof of Assertion (i).

For Assertion 2, there is nothing to do if P has no normal subgroup T_{0} ≅ (ℤ/pℤ)^{2}, since then P is cyclic, and A_{≥2}(P) = ∅. If P is not cyclic, fix such a normal subgroup T_{0} of P, and denote by C the connected component of T_{0} in A_{≥2}(P). Thus C is the big component if P has p-rank at least 3, and C = {T_{0}} otherwise. Define the sequence (y_{T})_{T ∈A_{=2}(P)} by

y_{T} = \{ \begin{matrix}0 & if T ∈ C \\ v_{T_{0},1} − v_{T,1} & otherwise \end{matrix}

This sequence obviously fulfills Condition (a) of Lemma 5.2, by Hypothesis 1, and since C is invariant by P -conjugation. Now it T, T′ ∈ A_{=2}(P), if T ≤ N_{P} (T′) and T T′ ≅ (ℤ/pℤ)^{3}, it follows that P has p-rank at least 3, that C is the big component, and that T, T′ ∈ C. Thus y_{T} = y_{T′} = 0, so Condition (b) of Lemma 5.2 holds. Finally, if T, T′ ∈ A_{=2}(P), if T ≤ N_{P} (T′) and T T′ ≅ X_{p^{3}}, then there are three cases :

• if T and T′ are in C, then y_{T} = y_{T′} = 0, thus y_{T} + v_{T,1} = v_{T,1}, and y_{T′} + v_{T′,1} = v_{T′,1}. But v_{T,1} ≡ v_{T′,1} (mod p) in this case, by Assertion 1. Thus Condition (c) holds in this case.

• if T ∈ C and T′ ∉ C, then y_{T} + v_{T,1} = v_{T,1}, and y_{T′} + v_{T′,1} = v_{T_{0},1}. But now T and T_{0} are both in C, so v_{T,1} ≡ v_{T_{0},1} (mod p) in this case, by Assertion 1. Thus Condition (c) holds in this case also. The case T ∉ C and T′ ∈ C is similar.

• if T ∉ C and T′ ∉ C, then y_{T} + v_{T,1} = v_{T_{0},1} = y_{T′} + v_{T′,1}, so Condition (c) holds in this case also.

This completes the proof of Lemma 5.2. End of the proof of Theorem 2.15 : In the beginning of the proof of Theorem 2.15, I started with an element u ∈ lim ; ⟵_{1<Q≤P} D(N_{P} (Q)/Q) such that h_{P} (u) = 0. From this

data, in 5.1, I built a sequence of integers (v_{T,S})_{(T,S)∈E_{2}^{♯}(P)} fulfilling Hypothesis 1, 2 and 3 of Lemma 5.2. Let (y_{T})_{T ∈A_{=2}(P)} denote the sequence of integers provided by this lemma, and define a new sequence of integers (v′_{T,S})_{(T,S)∈E_{2}^{♯}(P)} by

v′_{T,S} = \{ \begin{matrix}v_{T,S} & if S ≠ 1 \\ y_{T} + v_{T,S} & if S = 1 \end{matrix} .

Then this sequence fulfills Conditions 1, 2, and 3 of Theorem 4.5 : indeed, the new sequence is clearly invariant by conjugation, so Condition 1 is fulfilled. Conditions 2 and 3 for sections (T, S) and (T′, S) with S ≠ 1 are obviously fulfilled, since they are for the sequence (v_{T,S}), and since v_{T,S} = v′_{T,S} when S ≠ 1.

Now if T, T′ ∈ A_{=2}(P), if T ≤ N_{P} (T′) and T T′ ≅ (ℤ/pℤ)^{3}, then T and T′ are in the same connected component of A_{≥2}(P), and y_{T} = y_{T′}. Thus

v′_{T,1} + ∑_{1<Y <T} v′_{T T′,Y} = y_{T} + v_{T,1} + ∑_{1<Y <T} v_{T T′,Y}
= y_{T′} + v_{T′,1} + ∑_{1<Y <T′} v_{T T′,Y}
= v′_{T′,1} + ∑_{1<Y <T′} v′_{T T′,Y} ,

so Condition 2 is fulfilled.

Finally if T, T′ ∈ A_{=2}(P), if T ≤ N_{P} (T′) and T T′ ≅ X_{p^{3}}, then

v′_{T,1} = y_{T} + v_{T,1} ≡ y_{T′} + v_{T′,1} (mod p) ,

hence Condition 3 is fulfilled, since y_{T′} + v_{T′,1} = v′_{T′,1}.

By Theorem 4.5, there exists n ∈ D(P) such that D_{P} (2n) = (v′_{T,S}). In other words, for any (T, S) ∈ E_{2}^{♯}(P)

Defres_{T/S}^{P}(2n) = v′_{T,S} · 2Ω_{T/S} .

Thus if S ≠ 1

Defres_{T/S}^{P}(2n) = v_{T,S} · 2Ω_{T/S} = Res_{T/S}^{N_{P} (S)/S}(2u_{S}) .

Set t_{S} = Defres_{N_{P} (S)/S}^{P}(2n) − 2u_{S}. Now for any (V, U) ∈ E_{2}^{♯}(N_{P} (S)/S)

Defres_{V/U}^{N_{P} (S)/S}(t_{S}) = Defres_{V/U}^{P}(2n) − Defres_{V/U}^{N_{P} (S)/S}(2u_{S})
= Defres_{V/U}^{P}(2n) − Res_{V/U}^{N_{P} (S,U)/U}Defres_{N_{P} (S,U)/U}^{N_{P} (S)/S}(2u_{S})
= Defres_{V/U}^{P}(2n) − Res_{V/U}^{N_{P} (S,U)/U}Res_{N_{P} (S,U)/U}^{N_{P} (U)/U}(2u_{U})
= Defres_{V/U}^{P}(2n) − Res_{V/U}^{N_{P} (U)/U}(2u_{U}) = 0 .

It follows that t_{S} is a torsion element of D(N_{P} (S)/S), which is also in 2D(N_{P} (S)/S). Since the latter is torsion free, it follows that t_{S} = 0, i.e. that 2u_{S} = Defres_{N_{P} (S)/S}^{P}(2n), for any S ≠ 1. Equivalently 2u = r_{P} (2n), or 2(u − r_{P} (n)) = 0.

Now u − r_{P} (n) is an element of lim ; ⟵_{1<Q≤P} D_{tors}(N_{P} (Q)/Q). By Proposition 5.5 of [7],

there exists an element m ∈ D(P) such that r_{P} (m) = u − r_{P} (n). It follows that u = r_{P} (m + n), as was to be shown. This completes the proof of Theorem 2.15.

6. Example : the group X_{p^{5}}

Let P be an extraspecial group of order p^{5} and exponent p. The centre Z of P is cyclic of order p, and it is equal to the Frattini subgroup of P. The commutator P × P → Z induces a non degenerate symplectic 𝔽_{p}-valued scalar product on the factor group E = P/Z ≅ (𝔽_{p})^{4}, and the map Q ↦ Q/Z is a poset isomorphism from the poset of elementary abelian subgroups of P strictly containing Z to the poset E of non zero

totally isotropic subspaces of E. There are e = \frac{p^{4} − 1}{p − 1} isotropic lines in E, and the same number of totally isotropic 2-dimensional subspaces. It follows that |E| is equal to 2e.

There is a commutative diagram

[figure: D(P/Z); ≅; d; lim ; ⟵_{Q≥Z} D(P/Q); 0; T (P); D(P); r_{P}; _{Def{P \atop P/Z}}; lim ; ⟵_{1<Q≤P} D(N_{P} (Q)/Q); h_{P}; π; H^{1}(A_{≥2}(P), ℤ)^{(P)}]

In this diagram, the group lim ; ⟵_{Q≥Z} D(P/Q) is the group of sequences (u_{Q})_{Z≤Q≤P}, where

u_{Q} ∈ D(P/Q) (note that Q ⊴ P if Q ≥ Z), such that

∀R ≥ Q ≥ Z, Def_{P/R}^{P/Q}u_{Q} = u_{R} ,

and the map π is the projection map on the components Q ≥ Z. The map d is the product of the deflation maps Def_{P/Q}^{P/Z}, for Q ≥ Z. It is an isomorphism, since the sequence (u_{Q})_{Q≥Z}, where u_{Q} ∈ D(P/Q), is in the group lim ; ⟵_{Q≥Z} D(P/Q) if and only if

u_{Q} = Def_{P/Q}^{P/Z}u_{Z} for any Q ≥ Z.

The kernel of the map Def_{P/Z}^{P} is the set of faithful elements of D(P), and it was denoted by ∂D(P) in [5]. It was shown in that paper (Theorem 9.1) that

∂D(P) ≅ ℤ^{2e} ⊕ ℤ/2ℤ .

The kernel of π consists of the sequences (u_{Q})_{1<Q≤P} in lim ; ⟵_{1<Q≤P} D(N_{P} (Q)/Q) for which

u_{Q} = 0 if Q ≥ Z. It was shown in [5] that this is also the group lim ; ⟵_{{1<Q≤P \atop Q∩Z=1}} ∂D(N_{P} (Q)/Q).

All these facts show that there is an exact sequence

(6.1) 0 ⟶ T (P) ⟶ ∂D(P) ⟶^{r_{P}} lim ; ⟵_{{1<Q≤P \atop Q∩Z=1}} ∂D(N_{P} (Q)/Q) ⟶^{h_{P}} H^{1}(A_{≥2}(P), ℤ)^{(P)} ,

where r_{P} and h_{P} are the restrictions of the previously defined maps with the same names to the corresponding subgroups.

If Q is a subgroup of P such that Q ∩ Z = 1, then Q is elementary abelian of rank at most 2 (see [5] for details). If Q has order p, then N_{P} (Q)/Q ≅ X_{p^{3}}, thus ∂D(N_{P} (Q)/Q) ≅ ℤ^{p+1} ⊕ ℤ/2ℤ ([5] Theorem 9.1 or Section 11). If Q has order p^{2}, then N_{P} (Q)/Q ≅ ℤ/pℤ, thus ∂D(N_{P} (Q)/Q) ≅ ℤ/2ℤ.

It follows easily that the group lim ; ⟵_{{1<Q≤P \atop Q∩Z=1}} ∂D(N_{P} (Q)/Q) has free rank at least equal

to e(p + 1), since it contains the group ⨁_{{|Q|=p \atop Q≠Z}} 2∂D(N_{P} (Q)/Q).

Now the group T (P) is free of rank one, generated by Ω_{P/1}, by Corollary 1.3 of [11],

and the group H^{1}(A_{≥2}(P), ℤ)^{(P)} is isomorphic to H^{1}(E, ℤ). An easy computation, using e.g. Section 6 of [5], shows that this group is free of rank p^{4}.

Now the free rank of the image of h_{P} in the exact sequence 6.1 is at least equal to

1 − 2e + e(p + 1) = 1 + e(p − 1) = p^{4} ,

and since this is equal to the free rank of H^{1}(A_{≥2}(P), ℤ)^{(P)}, it follows that the free rank of the image of h_{P} is actually equal to p^{4}, and that the free rank of lim ; ⟵_{{1<Q≤P \atop Q∩Z=1}} ∂D(N_{P} (Q)/Q)

is equal to e(p + 1).

Moreover the map h_{P} has finite cokernel, and this shows that in this case, the gluing problem does not always have a solution.

6.2. Remark: In this case, a precise description of the map h_{P} shows that its cokernel is a non trivial finite p-group.

6.3. Aknowledgements: I wish to thank Nadia Mazza and Jacques Thévenaz for careful reading of an early version of this paper, and for many suggestions, comments, and stimulating discussions about it.

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Serge Bouc CNRS - UMR 6140 - LAMFA, Université de Picardie - Jules Verne, 33, rue St Leu, F-80039 Amiens Cedex 1, France.

email : serge.bouc@u-picardie.fr
web : http://www.lamfa.u-picardie.fr/bouc/