Gluing torsion endo-permutation modules

Serge Bouc and Jacques Thévenaz

1. Introduction

Endo-permutation modules for finite p-groups play an important role in the representation theory of finite groups. Recently, the combined efforts of several authors lead to the complete classification of all endo-permutation modules (see in particular [Al], [CaTh2], [CaTh3], [BoMa], [Bo3]). Now new problems appear and the purpose of the present paper is to consider one of them, namely the question of gluing compatible families of endo-permutation modules.

More precisely, given a finite p-group P and a field k of characteristic p, let M_{Q} be an endo-permutation k[N_{P} (Q)/Q]-module, for every subgroup Q of P belonging to a suitable class X of subgroups of P, with some straightforward compatibility conditions between the M_{Q}’s. We wish to find an endo-permutation kP -module M which ’glues’ those data, in the sense that

Defres_{N_{P} (Q)/Q}^{P}([M]) = [M_{Q}] ,

where [M] denotes the class of M in the Dade group D(P) of P, and Defres is the deflation–restriction map. We note that we are dealing in fact always with classes in the Dade group rather than endo-permutation modules.

This problem is important in block theory, particularly when X is either the class of centric subgroups of P or the class of all non-trivial subgroups of P. When P is abelian, the problem was completely solved by Puig [Pu2] and he used the result to construct suitable stable equivalences between blocks.

Here we address this question for an arbitrary p-group P with p odd and for the class X of all non-trivial subgroups of P (the class of centric subgroups is briefly treated in Section 6). Also, we consider only the subgroup D_{t}(P) consisting of torsion elements of the Dade group, because on the one hand D_{t}(P) requires special treatment, and on the other hand all known endo-permutation modules arising from block theory are either known or expected to be torsion elements of the Dade group. The problem of gluing non-torsion endo-permutation modules will be treated in another paper [Bo4].

Our main result asserts that, if p is odd, it is almost always possible to glue torsion endo-permutation modules. The only exception occurs for groups having maximal elementary abelian subgroups of rank 2 and in this case we control entirely the obstruction, which lies in the zero-th cohomology group of the poset A_{≥2}(P) of elementary abelian subgroups of P of rank at least 2. More precisely, our main result takes the following form:

1.1. Theorem. If p is an odd prime number and P is a non cyclic p-group, then there is an exact sequence

0 ⟶ D_{t}(P) ⟶^{r_{P}} lim ; ⟵_{1<Q≤P} D_{t}(N_{P} (Q)/Q) \underrightarrow{d̃_{P}} H̃^{0}(A_{≥2}(P), 𝔽_{2})^{P} ⟶ 0 .

In order to explain the connection with the problem described above, note that a compatible family is an element of the middle term of the sequence and that it can be ’glued’ precisely when it is in the image of the map r_{P}.

The problem of gluing arbitrary endo-permutation modules (not necessarily torsion) is solved in [Bo4] and it is interesting to notice that it is the first cohomology group of the same poset A_{≥2}(P) which plays a role. This shows the relevance of this poset, which will be studied in [BoTh2].

The most important ingredients for the proof of our main result are the following. First we need the fact that the torsion subgroup of the group of endo-trivial modules is trivial when p is odd and P is not cyclic, a result proved in [CaTh2]. We also use a consequence of this result wich asserts that the torsion subgroup D_{t}(P) is annihilated by 2 (i.e. a direct sum of copies of ℤ/2ℤ). Finally, another ingredient is concerned with the detection map to all elementary abelian sections

Defres : D(P) ⟶ ∏_{S \underline{◁} T ≤P T/S elementary abelian} D(T/S) ,

which is known to be injective when p is odd [CaTh2]. By adapting a method used in [Bo2], we prove in Section 4 that, given any compatible family on the right hand side, it is possible to construct an element of D(P) whose image is equal to |P | times the family we started with. For the torsion subgroup D_{t}(P) when p is odd, multiplication by |P | is the identity and we thus obtain an isomorphism between D_{t}(P) and the image of the above detection map restricted to the torsion subgroup.

We are grateful to Nadia Mazza for helpful comments on a preliminary version of this paper.

2. Limits of Dade groups

We first recall some basic facts concerning the Dade group and we fix notation (see [BoTh1] for details). Throughout this paper, we fix a prime number p, a finite pgroup P, and a field k of characteristic p. We denote by D(P) the Dade group of P, i.e. the group of equivalence classes of endo-permutation kP -modules, and by T (P) the subgroup of D(P) consisting of equivalence classes of endo-trivial modules. We denote by D_{t}(P) the torsion subgroup of D(P) and by T_{t}(P) the torsion subgroup of T (P).

Recall that if Q is a subgroup of P, we have a restriction map Res_{Q}^{P} : D(P) → D(Q) and a tensor induction map Ten_{Q}^{P} : D(Q) → D(P). If Q is a normal subgroup of P, then there is an inflation map Inf_{P/Q}^{P} : D(P/Q) → D(P) and a deflation map Def_{P/Q}^{P} : D(P) → D(P/Q). If R \underline{◁} Q ≤ P, we write Defres_{Q/R}^{P} = Def_{Q/R}^{Q} Res_{Q}^{P} and similarly

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

A compatible family of elements of D(N_{P} (Q)/Q), where Q runs over the non-trivial subgroups of P, is actually an element of a limit as follows. We denote by

lim ; ⟵_{1<Q≤P} D_{t}(N_{P} (Q)/Q)

the set of sequences (u_{Q})_{1<Q≤P} indexed by non-trivial subgroups Q of P, where u_{Q} ∈ D_{t}(N_{P} (Q)/Q), fulfilling the following two conditions :

1. if Q \underline{◁} 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}) ,

where N_{P} (Q, R) = N_{P} (Q) ∩ N_{P} (R),

2. if Q ≤ P and x ∈ P, then ^{x}u_{Q} = u_{^{x}Q} .

We denote by r_{P} the natural map induced by deflation–restriction

r_{P} : D_{t}(P) ⟶ lim ; ⟵_{1<Q≤P} D_{t}(N_{P} (Q)/Q)

mapping v ∈ D_{t}(P) to the sequence (v_{Q}) defined by v_{Q} = Defres_{N_{P} (Q)/Q}^{P}(v). Our gluing problem can be restated as the question of the surjectivity of the map r_{P}. However, we first settle the question of the injectivity of r_{P}.

2.1. Lemma. (a) The kernel of the map r_{P} is equal to T_{t}(P).

(b) If p is odd and P is not cyclic, then r_{P} is injective. Proof. (a) Recall that T (P) is equal to the intersection of all the kernels of the deflation–restriction maps

Defres_{N_{P} (Q)/Q}^{P} : D(P) ⟶ D(N_{P} (Q)/Q)

where Q runs over all non-trivial subgroups of P (see 2.1.2 in [Pu1]). It follows immediately that the kernel of r_{P} is equal to the torsion subgroup of T (P).

(b) If p is odd and P is not cyclic, then T (P) is torsion–free by [CaTh2].

In our next lemma, we consider one case where it is easy to prove that r_{P} is an isomorphism. It is a special case of the result of Puig mentioned in the introduction (see Prop. 3.6 in [Pu2]).

2.2. Lemma. Let E be an elementary abelian p-group of rank at least 2. Then

r_{E} : D_{t}(E) ⟶ lim ; ⟵_{1<F ≤E} D_{t}(E/F)

is an isomorphism. Proof. By the previous lemma, r_{E} is injective. For the surjectivity of r_{E}, we do not need to restrict to the torsion subgroup and we can argue as follows. Given

(u_{F}) ∈ lim ; ⟵_{1<F ≤E} D(E/F), define

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

where μ(1, F) is the Möbius function of the poset of subgroups of F. It is easy to check, using basic properties of this Möbius function, that Def_{E/F}^{E} (v) = u_{F} for any non trivial subgroup F of E, proving the surjectivity of r_{E}.

Another way of proving the surjectivity of r_{E} (for the torsion subgroup) is to notice that, by Dade’s theorem for the abelian case, any torsion element of D(P) is a sum of elements inflated from quotients E/M where M runs over the maximal subgroups of P. Therefore we have

D_{t}(E) = ∏_{M maximal} D_{t}(E/M) = lim ; ⟵_{1<F ≤E} D_{t}(E/F) ,

by direct inspection.

Note that Lemma 2.2 holds if p = 2 but yields an uninteresting result because D_{t}(E) = 0 when E is an elementary abelian 2-group.

If H is any subgroup of P, we denote by

Res_{H}^{P} : lim ; ⟵_{1<Q≤P} D_{t}(N_{P} (Q)/Q) ⟶ lim ; ⟵_{1<Q≤H} D_{t}(N_{H}(Q)/Q)

the map defined by

(Res_{H}^{P}(u))_{Q} = Res_{N_{H}(Q)/Q}^{N_{P} (Q)/Q}(u_{Q}) ,

for any (u_{Q}) ∈ lim ; ⟵_{1<Q≤P} D_{t}(N_{P} (Q)/Q). Similarly, if x ∈ P, there is a conjugation map

u ∈ lim ; ⟵_{1<Q≤H} D_{t}(N_{H}(Q)/Q) ↦ ^{x}u ∈ lim ; ⟵_{1<Q≤^{x}H} D_{t}(N_{^{x}H}(Q)/Q)

defined by (^{x}u)_{Q} = ^{x}u_{Q^{x}}.

These operations of restriction and conjugation have usual properties of transitivity and commutation. For example, if K ≤ H ≤ P, then Res_{K}^{H} ∘ Res_{H}^{P} = Res_{K}^{P}. Moreover :

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

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

Res_{H}^{P} ↓  ↓ Res_{H}^{P}

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

is commutative. Proof. This is straightforward.

3. The poset of elementary abelian subgroups

We denote by A_{≥2}(P) the poset of elementary abelian subgroups of P of rank at least 2, ordered by inclusion of subgroups. The cohomology group H^{0}(A_{≥2}(P), 𝔽_{2}) of A_{≥2}(P), with values in the field 𝔽_{2}, is the set of functions f : A_{≥2}(P) → 𝔽_{2} such that f(F) = f(E) whenever F and E are elements of A_{≥2}(P) such that F ≤ E. The reduced cohomology group H̃^{0}(A_{≥2}(P), 𝔽_{2}) is the quotient of H^{0}(A_{≥2}(P), 𝔽_{2}) by the one-dimensional subspace of all constant functions on A_{≥2}(P).

The group P acts on A_{≥2}(P) by conjugation, hence on H̃^{0}(A_{≥2}(P), 𝔽_{2}), and we denote by H̃^{0}(A_{≥2}(P), 𝔽_{2})^{P} the set of fixed points under this action. It is the set of functions as above which are moreover invariant by P -conjugation, modulo constant functions again.

The dimension of the cohomology group H^{0}(A_{≥2}(P), 𝔽_{2})^{P} is the number of conjugacy classes of connected components of A_{≥2}(P). The case where A_{≥2}(P) is not connected is well understood and we now recall the result. Clearly the isolated vertices of A_{≥2}(P) are precisely the maximal elementary abelian subgroups of rank 2. Let B be the subposet of A_{≥2}(P) consisting of all elementary abelian subgroups of P of rank at least 3 as well as their subgroups of rank 2. The following group-theoretic result is well-known and not very hard to prove (see Lemma 10.21 in [GLS] and Lemma 2.2 in [CaTh3]).

3.1. Lemma. Assume that A_{≥2}(P) is disconnected.

(a) P has a unique central subgroup Z of order p. Moreover Z is contained in every maximal elementary abelian subgroup of P.

(b) If every maximal elementary abelian subgroup has rank 2, then A_{≥2}(P) consists of isolated vertices.

(c) If there exists a maximal elementary abelian subgroup of rank at least 3, then B is a connected component of A_{≥2}(P) and all other connected components are isolated vertices.

(d) If E is an isolated vertex of A_{≥2}(P) and if S is a non-central subgroup of E, then N_{P} (S)/S is cyclic or generalized quaternion.

(e) If E is an isolated vertex of A_{≥2}(P), then all non-central subgroups of E are conjugate in P.

Assume p is odd and P is non trivial, and choose some central subgroup Z of order p in P. In the case where A_{≥2}(P) is disconnected (which will turn out to be the only important case for the following construction), the subgroup Z is unique by the above lemma. In the other cases, we simply choose Z arbitrarily. Recall that D(Z) = D_{t}(Z) ≅ 𝔽_{2} because p is odd. If u = (u_{Q}) ∈ lim ; ⟵_{1<Q≤P} D_{t}(N_{P} (Q)/Q), let

d_{P} (u) : A_{≥2}(P) → 𝔽_{2}

be the function defined by

d_{P} (u)(E) = γ_{Z} (Res_{Z}^{EZ} r_{EZ}^{−1} Res_{EZ}^{P}(u)) ,

where γ_{Z} : D_{t}(Z) → 𝔽_{2} is the canonical isomorphism and where r_{EZ}^{−1} is the inverse of the isomorphism of Lemma 2.2 (note that EZ is elementary abelian). We let d̃_{P} (u) be the class of d_{P} (u) in the quotient modulo the constant functions.

3.2. Lemma. The assignement u ↦ d̃_{P} (u) defines a group homomorphism

d̃_{P} : lim ; ⟵_{1<Q≤P} D_{t}(N_{P} (Q)/Q) ⟶ H̃^{0}(A_{≥2}(P), 𝔽_{2})^{P} .

Proof. Let u ∈ lim ; ⟵_{1<Q≤P} D_{t}(N_{P} (Q)/Q). In order to check that

d̃_{P} (u) ∈ H̃^{0}(A_{≥2}(P), 𝔽_{2})^{P}

it suffices to check that d_{P} (u) lies in H^{0}(A_{≥2}(P), 𝔽_{2})^{P}. If F ≤ E are elements of A_{≥2}(P), then

d_{P} (u)(E) = γ_{Z} (Res_{Z}^{EZ} r_{EZ}^{−1} Res_{EZ}^{P}(u)) ,

whereas

d_{P} (u)(F) = γ_{Z} (Res_{Z}^{F Z} r_{F Z}^{−1} Res_{F Z}^{P}(u)) .

By Lemma 2.4, we have

Res_{Z}^{F Z}r_{F Z}^{−1} Res_{F Z}^{P}(u) = Res_{Z}^{F Z}r_{F Z}^{−1} Res_{F Z}^{EZ} Res_{EZ}^{P}(u)
= Res_{Z}^{F Z} Res_{F Z}^{EZ} r_{EZ}^{−1} Res_{EZ}^{P}(u)
= Res_{Z}^{EZ} r_{EZ}^{−1} Res_{EZ}^{P}(u) ,

so d_{P} (u)(F) = d_{P} (u)(E). Now if x ∈ P, we have

Res_{Z}^{^{x}EZ} r_{^{x}EZ}^{−1} Res_{^{x}EZ}^{P}(u) = ^{x}(Res_{Z}^{EZ} r_{EZ}^{−1} Res_{EZ}^{P}(u)) = Res_{Z}^{EZ} r_{EZ}^{−1} Res_{EZ}^{P}(u) ,

so d_{P} (u)(^{x}E) = d_{P} (u)(E).

Finally d̃_{P} is a group homomorphism because it is constructed by composing group homomorphisms.

4. Inverting the detection map

A section of P is by definition a pair (T, S) of subgroups of P such that S \underline{◁} T. We denote by E(P) the set of sections (T, S) of P such that T/S is elementary abelian. When p is odd, by Theorem 13.1 in [CaTh2], we can detect elements in D(P) by means of the following injective map

Defres : D(P) ⟶ ∏_{(T,S)∈E(P)} D(T/S) ,

which we call the detection map. Clearly the image of the detection map consists of compatible families, so that we actually obtain an injective homomorphism

Defres : D(P) ⟶ lim ; ⟵_{(T,S)∈E(P)} D(T/S) .

The purpose of this section is to show that, on restriction to the torsion subgroup, we obtain an isomorphism. However, we shall prove a much more general result.

We shall need the following lemma, which is best understood by drawing the socalled ’butterfly’ diagram going back to Zassenhaus (see Chap. 4 in [La]).

4.1. Lemma. Let (S, F) and (B, A) be two sections of P. Assume either that p is odd or that S/F is abelian. Then, for any u ∈ D(S/F), we have

Defres_{B/A}^{P} Teninf_{S/F}^{P} (u)

= ∑_{x∈[B\P/S]} Teninf_{(B∩^{x}S)A/(B∩^{x}F)A}^{B/A} Iso Defres_{(B∩^{x}S)^{x}F /(A∩^{x}S)^{x}F}^{^{x}S/^{x}F} (^{x}u)

where [B\P/S] denotes a set of representatives of the double cosets B\P/S and where Iso : D((B ∩ ^{x}S)^{x}F /(A ∩ ^{x}S)^{x}F) → D((B ∩ ^{x}S)A/(B ∩ ^{x}F)A) denotes the isomorphism induced by the group isomorphism of the Zassenhaus lemma. Proof. The result could be proved by first applying the Mackey formula (in order to exchange tensor induction and restriction), then exchanging restriction and inflation (in a suitable fashion) and finally applying Corollary 3.12 of [BoTh1] in order to exchange tensor induction and deflation. However, in order to make this argument complete, one has in fact to go back to the bisets which induce the various homomorphisms involved in the statement, as explained in [BoTh1]. Therefore, we give here the complete proof using bisets.

Recall that, if Q and R are p-groups, then any (Q, R)-biset X induces a group homomorphism D(X) : D(R) → D(Q) (see Corollary 2.13 of [BoTh1]). Moreover, the composition of such homomorphisms corresponds here to the usual product of bisets. In general the composition involves a Galois action (see Proposition 3.10 of [BoTh1]), but this is the identity whenever we start from an abelian group. When p is odd, since D(P) is detected on abelian sections of P, the Galois action is also the identity. Thus our assumption guarantees that no Galois action is involved.

Since Inf_{S/F}^{S} is induced by the (S, S/F)-biset S/F and Ten_{S}^{P} by the (P, S)-biset P, the composite Teninf_{S/F}^{P} is induced by the (P, S/F)-biset P ×_{S} (S/F) ≅ P/F. Similarly, Defres_{B/A}^{P} is induced by the (B/A, P)-biset A\B ×_{B} P ≅ A\P. Therefore Defres_{B/A}^{P} Teninf_{S/F}^{P} is induced by the (B/A, S/F)-biset

A\P ×_{P} P/F ≅ A\P/F .

This decomposes into orbits as follows

A\P/F = ⨆_{x∈[B\P/S]} A\BxS/F ≅ ⨆_{x∈[B\P/S]} A\B·^{x}S/^{x}F ×_{^{x}S/^{x}F} Conj_{x} ,

where Conj_{x} denotes the (^{x}S/^{x}F , S/F)-biset S/F (which induces of course conjugation by x).

Taking x = 1 for simplicity, we analyze further the term A\BS/F, which is a transitive (B/A, S/F)-biset. The stabilizer Σ of its element AF is the set of pairs (bA, sF) in (B/A) × (S/F) such that bs^{−1} ∈ AF and

A\BS/F ≅ ((B/A) × (S/F))/Σ .

By Lemma 7.4 of [BoTh1] (or Lemma 3 in [Bo1]), this biset can be decomposed according to the projections and intersections of Σ. The projection of Σ on the group S/F is equal to (B ∩ S)F/F and the intersection of Σ with 1 × (S/F) is 1 × ((A ∩ S)F/F). Similarly, the projection of Σ on B/A is equal to (B ∩ S)A/A and the intersection of Σ with (B/A) × 1 is equal to ((B ∩ F)A/A) × 1. It follows that (B/A × S/F)/Σ is the product of the following sequence of bisets:

• the (B/A, (B ∩ S)A/A)-biset (B ∩ S)A/A, which induces Ten_{(B∩S)A/A}^{B/A} .

• the ((B ∩ S)A/A, (B ∩ S)A/(B ∩ F)A)-biset (B ∩ S)A/(B ∩ F)A, which induces Inf_{(B∩S)A/(B∩F)A}^{(B∩S)A/A} ,

• the ((B ∩S)A/(B ∩F)A, (B ∩S)F/(A∩S)F )-biset (B ∩S)F/(A∩S)F, which induces the Zassenhaus isomorphism Iso_{(B∩S)F/(A∩S)F}^{(B∩S)A/(B∩F)A} ,

• the ((B ∩ S)F/(A ∩ S)F, (B ∩ S)F/F )-biset (B ∩ S)F/(A ∩ S)F, which induces Def_{(B∩S)F/(A∩S)F}^{(B∩S)F/F} ,

• and the ((B ∩ S)F/F, S/F )-biset (B ∩ S)F/F, which induces Res_{(B∩S)F/F}^{S/F} , The other terms of the orbit decomposition are treated similarly, replacing S by ^{x}S

and F by ^{x}F. Thus each such term induces the following sequence of homomorphisms between Dade groups:

Teninf_{(B∩^{x}S)A/(B∩^{x}F)A}^{B/A} Iso_{(B∩^{x}S)^{x}F /(A∩^{x}S)^{x}F}^{(B∩^{x}S)A/(B∩^{x}F)A} Defres_{(B∩^{x}S)^{x}F /(A∩^{x}S)^{x}F}^{^{x}S/^{x}F} Conj_{x} .

The result follows.

Recall that our purpose is to show that the detection map is an isomorphism on restriction to the torsion subgroup when p is odd. However, the result is much more general and works as well when p = 2. It is inspired by Section 7.4 of [Bo2], in particular by formula 7.4.7.

4.2. Theorem. Let P be a p-group. Let u = (u_{T,S})_{(T,S)∈E(P)} ∈ lim ; ⟵_{(T,S)∈E(P)} D(T/S). In

other words, for each (T, S) ∈ E(P), the elements u_{T,S} ∈ D(T/S) satisfy the following two conditions :

1. if (T, S), (T′, S′) ∈ E(P) with S ≤ S′ ≤ T′ ≤ T, then Defres_{T′/S′}^{T/S}(u_{T,S}) = u_{T′,S′} ,

2. if x ∈ P and (T, S) ∈ E(P), then ^{x}u_{T,S} = u_{^{x}T ,^{x}S} .

Then there exists an element v ∈ D(P) such that Defres_{T/S}^{P}(v) = |P | · u_{T,S} for any (T, S) ∈ E(P).

Moreover, if all the elements u_{T,S} are torsion elements, then v can be chosen to be a torsion element. Proof. Set

v = ∑_{(T,S)∈E(P)} |S| μ(S, T) Teninf_{S/Φ(T)}^{P}u_{S,Φ(T)} ,

where Φ(T) denotes the Frattini subgroup of T and where μ denotes the Möbius function of the poset of subgroups of P. Note that T/S is elementary abelian if and only if Φ(T) ≤ S ≤ T.

We have to compute Defres_{B/A}^{P}(v) for every (B, A) ∈ E(P) and so we first apply Lemma 4.1. By our two assumptions, the conjugation and deflation–restriction involved in the formula can be replaced by the following:

Defres_{(B∩^{x}S)^{x}Φ(T)/(A∩^{x}S)^{x}Φ(T)}^{^{x}S/^{x}Φ(T)}(^{x}u_{S,Φ(T)}) = u_{(B∩^{x}S)^{x}Φ(T),(A∩^{x}S)^{x}Φ(T)}
= u_{(B∩^{x}S)Φ(^{x}T),(A∩^{x}S)Φ(^{x}T)} = u_{x} ,

where we define u_{x} in this way for simplicity of notation.

Thus, by Lemma 4.1 we obtain that Defres_{B/A}^{P}(v) is equal to

∑_{(T,S)∈E(P)} |S| μ(S, T) ∑_{x∈[B\P/S]} Teninf_{(B∩^{x}S)A/(B∩^{x}Φ(T))A}^{B/A} Iso (u_{x})

= ∑_{(T,S)∈E(P)} μ(S, T) ∑_{x∈[B\P/S]} |S| Teninf_{(B∩^{x}S)A/(B∩^{x}Φ(T))A}^{B/A} Iso (u_{x})
= ∑_{(T,S)∈E(P)} μ(S, T) ∑_{x∈[B\P]} |B ∩ ^{x}S| Teninf_{(B∩^{x}S)A/(B∩^{x}Φ(T))A}^{B/A} Iso (u_{x})
= ∑_{x∈[B\P]} ∑_{(T,S)∈E(P)} μ(^{x}S, ^{x}T)|B ∩ ^{x}S| Teninf_{(B∩^{x}S)A/(B∩^{x}Φ(T))A}^{B/A} Iso (u_{x})
= |P : B| ∑_{(T,S)∈E(P)} μ(S, T)|B ∩ S| Teninf_{(B∩S)A/(B∩Φ(T))A}^{B/A} Iso (u_{1})

using the invariance of the Möbius function by conjugation and the fact that (^{x}T , ^{x}S) runs over E(P) whenever (T, S) does. Here u_{1} = u_{(B∩S)Φ(T),(A∩S)Φ(T)} .

Grouping together the terms for which T is given and B ∩ S is a given subgroup R of B ∩ T, we get that Defres_{B/A}^{P}(v) is equal to

|P : B| ∑_{{T ≤P \atop R≤B∩T}} s_{R,T} |R| Teninf_{RA/(R∩Φ(T))A}^{B/A}Iso_{RΦ(T)/(A∩R)Φ(T)}^{RA/(R∩Φ(T))A} (u_{RΦ(T),(A∩R)Φ(T)}) ,

where s_{R,T} = ∑_{{Φ(T)≤S≤T \atop S∩B=R}} μ(S, T). Since μ(S, T) = 0 if S does not contain Φ(T) (see

Lemma 4.6 in [BoTh1]), we can write

s_{R,T} = ∑_{{S≤T \atop S∩(B∩T)=R}} μ(S, T) .

Now by a classical combinatorial lemma (see Cor. 3.9.3 in [St]), this is zero unless T = B ∩ T, i.e. T ≤ B. In this case s_{R,T} = μ(R, T) and this is zero unless R ≥ Φ(T). Moreover Φ(T) ≤ Φ(B) since B is a p-group, and Φ(B) ≤ A since B/A is elementary abelian. Thus Φ(T) ≤ A, and this gives

Defres_{B/A}^{P}(v) = |P : B| ∑_{R≤T ≤B} μ(R, T) |R| Teninf_{RA/A}^{B/A} Iso_{R/A∩R}^{RA/A} (u_{R,A∩R}) .

Now for a given R, the sum ∑_{R≤T ≤B} μ(R, T) is zero unless R = B. Therefore

Defres_{B/A}^{P}(v) = |P : B| |B| Teninf_{B/A}^{B/A} Iso_{B/A}^{B/A} (u_{B,A}) = |P | · u_{B,A} ,

as was to be shown. If every u_{T,S} is a torsion element, then v is obviously also a torsion element.

4.3. Corollary. If p is odd, then the detection map

Defres : D_{t}(P) ⟶ lim ; ⟵_{(T,S)∈E(P)} D_{t}(T/S) .

is an isomorphism. Proof. When p is odd, any element u_{T,S} ∈ D_{t}(T/S) has order 2 (see Theorem 1.5 in [CaTh2]) and hence multiplication by |P | is the identity. Therefore, with the same notation as in Theorem 4.2, there exists an element v ∈ D_{t}(P) such that

Defres_{T/S}^{P}(v) = |P | · u_{T,S} = u_{T,S}

for any (T, S) ∈ E(P). This shows that the detection map is surjective.

5. The main theorem

Using the map r_{P} of Section 2 and the map d̃_{P} of Section 3, we can now state the main result.

5.1. Theorem. Let p be an odd prime number and let P be a non cyclic p-group. Then the sequence

0 ⟶ D_{t}(P) ⟶^{r_{P}} lim ; ⟵_{1<Q≤P} D_{t}(N_{P} (Q)/Q) \underrightarrow{d̃_{P}} H̃^{0}(A_{≥2}(P), 𝔽_{2})^{P} ⟶ 0

is exact.

In particular r_{P} is an isomorphism if every maximal elementary abelian subgroup of P has rank ≥ 3. Proof. Recall that r_{P} is injective by Lemma 2.1. Recall also that A_{≥2}(P) is connected if every maximal elementary abelian subgroup of P has rank ≥ 3 (by Lemma 3.1), in which case the right hand side vanishes. Thus the additional statement follows immediately from the exactness of the sequence.

We first check that d̃_{P} ∘ r_{P} = 0. Let v ∈ D_{t}(P). Then for Q ≤ P, we have

r_{P} (v)_{Q} = Defres_{N_{P} (Q)/Q}^{P}(v) .

Thus if E ∈ A_{≥2}(P), by Lemma 2.4

d_{P} r_{P} (v)(E) = γ_{Z} (Res_{Z}^{EZ} r_{EZ}^{−1} Res_{EZ}^{P} r_{P} (v))
= γ_{Z} (Res_{Z}^{EZ} r_{EZ}^{−1}r_{EZ} Res_{EZ}^{P}(v))
= γ_{Z}(Res_{Z}^{EZ} Res_{EZ}^{P}(v))
= γ_{Z}(Res_{Z}^{P}v) ,

and this is independent of E. Therefore d_{P} r_{P} (v) is a constant function and it follows that d̃_{P} ∘ r_{P} = 0.

We next prove that d̃_{P} is surjective. There is nothing to do if A_{≥2}(P) is connected, for the right hand side is zero in that case. If A_{≥2}(P) is not connected, then by Lemma 3.1, the group H̃^{0}(A_{≥2}(P), 𝔽_{2})^{P} is generated by the classes of the characteristic functions χ_{A} where A runs over the set of isolated points up to conjugacy. Explicitly χ_{A} denotes the function having value 1 on A and its conjugates and zero elsewhere. We now fix such an isolated point A and we want to prove that χ_{A} is in the image of d_{P}. By Lemma 3.1 again, P has a unique central subgroup Z of order p, we can write A = SZ where S is a non-central subgroup of order p, and the group N_{P} (S)/S is cyclic. If Q is any non trivial subgroup of P, we define

a_{Q} = \{ \begin{matrix}Ω_{N_{P} (Q)/Q} & if Q is conjugate to S , \\ 0 & otherwise , \end{matrix}

where for any p-group X, we write Ω_{X} for the class of the Heller translate Ω_{X}1(k) in the Dade group D(X). Recall that Ω_{X} is endo-trivial, hence satisfies Defres_{U/V}^{X} (Ω_{X}) = 0 whenever V \underline{◁} U ≤ X and V ≠ 1. Our next lemma shows that the map d̃_{P} is surjective.

5.2. Lemma. Let A = SZ be an isolated point of A_{≥2}(P) and let a = (a_{Q})_{1<Q≤P} be the sequence defined above.

(a) The sequence a = (a_{Q})_{1<Q≤P} is an element of lim ; ⟵_{1<Q≤P} D_{t}(N_{P} (Q)/Q).

(b) d̃_{P} (a) = χ_{A}. Proof. (a) By construction, it is clear that ^{x}a_{Q} = a_{^{x}Q} for any Q ≤ P and any x ∈ P. Now if 1 < Q \underline{◁} R, we have to check that

Defres_{N_{P} (Q,R)/R}^{N_{P} (Q)/Q}(a_{Q}) = Res_{N_{P} (Q,R)/R}^{N_{P} (R)/R}(a_{R}) .

If Q = R, there is nothing to check. If Q < R, the only case where the above equality is not identically 0 is when Q is conjugate to S. In that case, the right hand side is 0 and the left hand side is

Defres_{N_{P} (Q,R)/R}^{N_{P} (Q)/Q}(a_{Q}) = Defres_{N_{P} (Q,R)/R}^{N_{P} (Q)/Q}(Ω_{N_{P} (Q)/Q}) = 0

by the above remark.

(b) We now compute d_{P} (a), and check that if E ∈ A_{≥2}(P), then

d_{P} (a)(E) = \{ \begin{matrix}1 & if E is conjugate to A , \\ 0 & otherwise . \end{matrix}

By definition

d_{P} (a)(E) = γ_{Z} (Res_{Z}^{EZ} r_{EZ}^{−1} Res_{EZ}^{P}(a)) .

If 1 < Q ≤ EZ, then

Res_{EZ}^{P}(a)_{Q} = Res_{EZ/Q}^{N_{P} (Q)/Q}(a_{Q}) .

This is zero unless Q = ^{x}S, for some x ∈ P. In this case EZ is an elementary abelian subgroup containing QZ = ^{x}(SZ) = ^{x}A, and therefore EZ = ^{x}A by maximality of A (since A is an isolated point of A_{≥2}(P)). Since E has rank ≥ 2, this forces the equality E = ^{x}A. Thus d_{P} (a)(E) = 0 unless E is conjugate to A.

Now we can assume E = A and in this case AZ = A. By Equation 2.3

r_{A}^{−1} Res_{A}^{P}(a) = − ∑_{1<F ≤A} μ(1, F) Inf_{A/F}^{A} Res_{A/F}^{N_{P} (F)/F} (a_{F}) .

The non-zero terms in the sum are obtained for the subgroups F of A which are conjugate to S. These are all the subgroups of order p of A, different from Z (by Lemma 3.1). For such a subgroup F, we have a_{F} = Ω_{N_{P} (F)/F}, so Res_{A/F}^{N_{P} (F)/F} (a_{F}) = Ω_{A/F}. Moreover μ(1, F) = −1 and so

r_{A}^{−1} Res_{A}^{P}(a) = ∑_{{|F |=p \atop Z≠F <A}} Inf_{A/F}^{A} (Ω_{A/F}) = ∑_{{|F |=p \atop Z≠F <A}} Ω_{A/F} .

Now Res_{Z}^{A}(Ω_{A/F}) = Ω_{Z/1} for all subgroups F of order p of A, different from Z, and since there are p such subgroups, it follows that

d_{P} (a)(A) = γ_{Z}(p · Ω_{Z/1}) = p · 1 = 1

because p is odd. Hence d̃_{P} (a) is the characteristic function χ_{A}. This proves the lemma.

The only thing left to prove is that the kernel of d̃_{P} is equal to the image of r_{P}. Let w = (w_{Q})_{1<Q≤P} ∈ Ker(d̃_{P}). Our aim is to construct an element u ∈ lim ; ⟵_{(T,S)∈E(P)} D_{t}(T/S)

and use the isomorphism of Corollary 4.3 to produce an element v ∈ D_{t}(P) such that r_{P} (v) = w.

Let (T, S) ∈ E(P). If S ≠ 1, set u_{T,S} = Res_{T/S}^{N_{P} (S)/S}(w_{S}). If T ∈ A_{≥2}(P), set u_{T,1} = r_{T}^{−1} Res_{T}^{P} (w). If T is a subgroup of order p different from Z, set u_{T,1} = Res_{T}^{T Z}(u_{T Z,1}), which makes sense since u_{T Z,1} is already defined for T Z ∈ A_{≥2}(P). Finally, if T = Z, choose any element E ∈ A_{≥2}(P) and set u_{Z,1} = Res_{Z}^{EZ}(u_{EZ,1}). Since w ∈ Ker(d̃_{P}), this does not depend on the choice of E. Indeed the function d_{P} (w) is constant, and by definition

γ_{Z}(Res_{Z}^{EZ}(u_{EZ,1})) = γ_{Z} (Res_{Z}^{EZ} r_{EZ}^{−1} Res_{EZ}^{P}(w)) = d_{P} (w)(E) = constant .

To be complete, we set u_{1,1} = 0, so that the element u_{T,S} is now defined for every (T, S) ∈ E(P).

We want to prove that the elements u_{T,S} fulfill the conditions of Theorem 4.2. The second condition on conjugation is obviously satisfied. For the first one, suppose that (T, S) and (T′, S′) are elements of E(P) such that S ≤ S′ ≤ T′ ≤ T. We have to check that

Defres_{T′/S′}^{T/S}(u_{T,S}) = u_{T′,S′} .

This is trivially true if T′ = S′, or if (T, S) = (T′, S′). So we can assume T′ ≠ S′, and S ≠ S′ or T ≠ T′.

If S ≠ 1, then

Defres_{T′/S′}^{T/S}(u_{T,S}) = Defres_{T′/S′}^{T/S} Res_{T/S}^{N_{P} (S)/S}(w_{S})
= Defres_{T′/S′}^{N_{P} (S)/S}(w_{S})
= Res_{T′/S′}^{N_{P} (S,S′)/S′} Defres_{N_{P} (S,S′)/S′}^{N_{P} (S)/S}(w_{S})
= Res_{T′/S′}^{N_{P} (S,S′)/S′} Res_{N_{P} (S,S′)/S′}^{N_{P} (S′)/S′}(w_{S′})

because w ∈ lim ; ⟵_{1<Q≤P} D_{t}(N_{P} (Q)/Q). Thus we are left with

Defres_{T′/S′}^{T/S}(u_{T,S}) = Res_{T′/S′}^{N_{P} (S′)/S′}(w_{S′}) = u_{T′,S′} .

Now if S = 1 ≠ S′, then T is elementary abelian, and since S′ < T′ ≤ T, the group T has order at least p^{2}, so T ∈ A_{≥2}(P). Hence

Defres_{T′/S′}^{T}(u_{T,1}) = Defres_{T′/S′}^{T} r_{T}^{−1} Res_{T}^{P} (w)
= Def_{T′/S′}^{T′} Res_{T′}^{T} r_{T}^{−1} Res_{T}^{P} (w)
= Def_{T′/S′}^{T′} r_{T′}^{−1}(Res_{T′}^{P}(w))  (by Lemma 2.4)
= (Res_{T′}^{P}(w))_{S′}  (by definition of r_{T′})
= Res_{T′/S′}^{N_{P} (S′)/S′}(w_{S′}) = u_{T′,S′} .

Finally, if S = S′ = 1, then 1 < T′ < T, and both are elementary abelian. Moreover T ∈ A_{≥2}(P). If T′ ∈ A_{≥2}(P), then

Res_{T′}^{T}(u_{T,1}) = Res_{T′}^{T} r_{T}^{−1} Res_{T}^{P} (w) = r_{T′}^{−1} Res_{T′}^{P}(w) = u_{T′,1} ,

by Lemma 2.4. If T′ = Z, then we are done since by definition u_{Z,1} = Res_{Z}^{T}(u_{T,1}). Now if T′ has order p and T′ ≠ Z, then, using Lemma 2.4 again, we obtain

Res_{T′}^{T}(u_{T,1}) = Res_{T′}^{T} r_{T}^{−1} Res_{T}^{P} (w)
= Res_{T′}^{T} r_{T}^{−1} Res_{T}^{T Z} Res_{T Z}^{P}(w)
= Res_{T′}^{T} Res_{T}^{T Z} r_{T Z}^{−1} Res_{T Z}^{P}(w)
= Res_{T′}^{T′Z} Res_{T′Z}^{T Z} r_{T Z}^{−1} Res_{T Z}^{P}(w)
= Res_{T′}^{T′Z} r_{T′Z}^{−1} Res_{T′Z}^{P}(w)
= Res_{T′}^{T′Z}(u_{T′Z,1}) = u_{T′,1} .

All this shows that the family u = (u_{T,S}) belongs to lim ; ⟵_{(T,S)∈E(P)} D_{t}(T/S) and so

we can apply Corollary 4.3. Hence there exists an element v ∈ D_{t}(P) such that Defres_{T/S}^{P}(v) = u_{T,S} for every (T, S) ∈ E(P). We have now to prove that r_{P} (v) = w, that is, we have to show that Defres_{N_{P} (Q)/Q}^{P}(v) = w_{Q} for every non-trivial subgroup Q of P. Since elements of the Dade group of N_{P} (Q)/Q are detected on deflation– restriction to all elementary abelian sections, it suffices to prove that we have equality after applying all the maps Defres_{T/S}^{N_{P} (Q)/Q}, where (T, S) ∈ E(P) such that Q ≤ S ≤ T ≤ N_{P} (Q). This is proved as follows :

Defres_{T/S}^{N_{P} (Q)/Q} Defres_{N_{P} (Q)/Q}^{P}(v) = Defres_{T/S}^{P}(v)
= u_{T,S}
= Res_{T/S}^{N_{P} (S)/S}(w_{S})
= Res_{T/S}^{N_{P} (Q,S)/S} Res_{N_{P} (Q,S)/S}^{N_{P} (S)/S}(w_{S})
= Res_{T/S}^{N_{P} (Q,S)/S} Defres_{N_{P} (Q,S)/S}^{N_{P} (Q)/Q}(w_{Q})
= Defres_{T/S}^{N_{P} (Q)/Q}(w_{Q}) ,

as was to be shown.

5.3. Remark. In view of Lemma 2.1, one can include cyclic groups in the statement of Theorem 5.1 by adding one term in the exact sequence, as follows. Let p be an odd prime number and let P be a p-group. Then the sequence

0 ⟶ T_{t}(P) ⟶ D_{t}(P) ⟶^{r_{P}} lim ; ⟵_{1<Q≤P} D_{t}(N_{P} (Q)/Q) \underrightarrow{d̃_{P}} H̃^{0}(A_{≥2}(P), 𝔽_{2})^{P} ⟶ 0

is exact.

5.4. Remark. Theorem 5.1 does not hold when p = 2 because there are exceptional torsion endo-trivial modules for quaternion groups and semi-dihedral groups. An easy modification would be either to exclude these cases or to try and state a result in the form of the previous remark. However, the method used for the proof when p is odd does not work when p = 2 because of Section 4. For the detection map of Section 4, elementary abelian sections are not sufficient and one must include cyclic groups of order 4 and quaternion groups of order 8. Also, D_{t}(P) is not annihilated by 2, but by 4. But the most important problem is that multiplication by |P | is a power of 2, hence annihilates D_{t}(P), whereas it is the identity when p is odd. Thus the final argument of Section 4 collapses completely.

We can be more explicit about the compatible families which do not glue to an element of D_{t}(P). We now show that they actually glue to a non-torsion element in D(P).

5.5. Proposition. Let a = (a_{Q})_{1<Q≤P} be an element of lim ; ⟵_{1<Q≤P} D_{t}(N_{P} (Q)/Q) which

is not in the image of D_{t}(P) under the map r_{P}. Then there exists a (non-torsion) element b ∈ D(P) such that

Defres_{N_{P} (Q)/Q}^{P}(b) = a_{Q} for every Q ≠ 1 .

Proof. Since a is not in the image of D_{t}(P), it is not in the kernel of d̃_{P}. This forces A_{≥2}(P) to be disconnected and H̃^{0}(A_{≥2}(P), 𝔽_{2})^{P} ≠ 0. By linearity, we can assume that d̃_{P} (a) is a basis element χ_{A} of H̃^{0}(A_{≥2}(P), 𝔽_{2})^{P}, where A is an isolated point of A_{≥2}(P) and χ_{A} is the corresponding characteristic function. We write A = SZ where S is a non-central subgroup of order p and Z is central of order p. By Lemma 5.2 , we can assume that our family a = (a_{Q})_{1<Q≤P} is defined by

a_{Q} = \{ \begin{matrix}Ω_{N_{P} (Q)/Q} & if Q is conjugate to S , \\ 0 & otherwise , \end{matrix}

because this family maps to the basis element χ_{A} under d̃_{P}. Now we show that this family glues to an element of D(P), namely b = Ω_{P/S}. We have to prove that

Defres_{N_{P} (Q)/Q}^{P}(Ω_{P/S}) = a_{Q} for every Q ≠ 1 .

By Section 4 of [Bo2], we have Defres_{N_{P} (Q)/Q}^{P}(Ω_{P/S}) = Ω_{(P/S)^{Q}}, where the set of Qfixed points (P/S)^{Q} is viewed as an N_{P} (Q)/Q-set in the obvious way. Since Q has no fixed point on P/S if Q is not conjugate to S, we obtain Defres_{N_{P} (Q)/Q}^{P}(Ω_{P/S}) = 0 in that case (Lemma 4.2.1 of [Bo2]). If now Q is conjugate to S, then P/S ≅ P/Q as P -sets, (P/Q)^{Q} = N_{P} (Q)/Q and therefore Ω_{(P/Q)^{Q}} = Ω_{N_{P} (Q)/Q}.

5.6. Remark. As mentioned in the introduction, the problem of gluing arbitrary endo-permutation modules (not necessarily torsion) will be solved in [Bo4] for p odd. The result is the existence of an exact sequence

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

5.7. Remark. Theorem 5.1 and the previous remark show the relevance of the poset A_{≥2}(P). It will be proved in [BoTh2] that A_{≥2}(P) has the homotopy type of a bouquet of spheres (of possibly different dimensions). This implies that H^{1}(A_{≥2}(P), ℤ)^{P} is free abelian and this shows again that any torsion compatible family can always be glued to a (possibly non-torsion) element in D(P). Indeed the torsion part of lim ; ⟵_{1<Q≤P} D(N_{P} (Q)/Q) maps to zero under d̃_{P}, hence is in the image of r_{P}.

6. Centric subgroups

A subgroup Q of a p-group P is called centric (or self-centralizing) if C_{P} (Q) ≤ Q. Such subgroups play an important role in block theory (see Chapter 41 in [Th]). In particular, the problem of gluing endo-permutation modules also appears when a compatible family of elements of D(N_{P} (Q)/Q) is given, but Q runs only over the centric subgroups of P.

Let C be the family of centric subgroups of P. Here we address the question of gluing a compatible family of torsion elements, defined for subgroups Q ∈ C only. As before, the question is to know whether or not there is an element of D_{t}(P) which glues the data. The purpose of this section is to show that the phenomenon described in Proposition 5.5 may appear again. We construct an example of a compatible family (a_{Q})_{Q∈C} which is not in the image of D_{t}(P), but is the image of some non-torsion b ∈ D(P), that is, Defres_{N_{P} (Q)/Q}^{P}(b) = a_{Q} for every Q ∈ C.

The group P is defined by a non-split extension

1 ⟶ E ⟶ P ⟶ \overline{A} ⟶ 1 ,

where E = ⟨v, x_{1}, x_{2}, z⟩ is elementary abelian of rank 4 and \overline{A} = ⟨\overline{a}_{1}, \overline{a}_{2}⟩ is elementary abelian of rank 2. We assume that the extension splits on restriction to each of the two cyclic groups ⟨\overline{a}_{i}⟩, that is, we assume that the element \overline{a}_{i} lifts to an element a_{i} ∈ P of order p, for i = 1, 2. The action of \overline{A} on E is described as follows (in multiplicative notation, hence via commutators):

[a_{1}, v] = x_{1} , [a_{1}, x_{1}] = 1 , [a_{1}, x_{2}] = z , [a_{1}, z] = 1 ,
[a_{2}, v] = x_{2} , [a_{2}, x_{2}] = 1 , [a_{2}, x_{1}] = z , [a_{2}, z] = 1 .

In order to describe the extension, we set [a_{1}, a_{2}] = v and it follows that the sequence does not split. Note that Z(P) = ⟨z⟩.

We let F = ⟨x_{1}, x_{2}, z⟩ and we notice that \overline{P} = P/F is extraspecial of order p^{3} and exponent p, because [a_{1}, a_{2}] = v. It is not hard to see that the sequence

1 ⟶ F ⟶ P ⟶ \overline{P} ⟶ 1

does not split. The whole point of the construction is that every subgroup Q containing F properly is a centric subgroup (as one can check easily), but F itself is not centric (because C_{P} (F) = E).

The group \overline{P} = P/F is such that A_{≥2}(\overline{P}) is disconnected and we can apply Proposition 5.5. Consider a family

(a_{Q})_{F <Q≤P} ∈ lim ; ⟵_{F <Q≤P} D_{t}(N_{P} (Q)/Q)

which is not in the image of D_{t}(\overline{P}). By Proposition 5.5, there exists a non-torsion element \overline{b} ∈ D(\overline{P}) such that

Defres_{N_{P} (Q)/Q}^{P}(\overline{b}) = a_{Q} whenever F < Q ≤ P .

We now define b = InfP ; \overline{P} (\overline{b}) and we extend the family to all centric subgroups Q by setting a_{Q} = Defres_{N_{P} (Q)/Q}^{P}(b) for every Q ∈ C. This extends the previously defined family because DefP ; \overline{P} InfP ; \overline{P} = id_{D(\overline{P})}.

6.1. Lemma. For every Q ∈ C, the element a_{Q} belongs to the torsion subgroup D_{t}(N_{P} (Q)/Q).

Assuming this lemma, we obtain a family in lim ; ⟵_{Q∈C} D_{t}(N_{P} (Q)/Q) which can be glued

to an element b ∈ D(P) by construction. However, this family cannot be glued to a torsion element, otherwise the deflation of this element in the group D_{t}(\overline{P}) would glue the family (a_{Q})_{F <Q≤P}, but this is impossible by our choice of (a_{Q})_{F <Q≤P}.

This example shows that the phenomenon of Proposition 5.5 may happen again with the family of centric subgroups. We have a family of torsion elements (defined for centric subgroups) which cannot be glued to a torsion element of D(P), but which can be glued to a non-torsion element of D(P).

Proof of Lemma 6.1. By construction, we know that a_{Q} lies in D_{t}(N_{P} (Q)/Q) whenever F < Q. Let now Q be a centric subgroup not containing F. Then F Q is centric (because every overgroup of a centric subgroup is centric) and therefore F Q contains F properly because F is not centric. By Lemma 4.1 (applied with S = P, B = N_{P} (Q), and A = Q), we have

Defres_{N_{P} (Q)/Q}^{P} InfP ; \overline{P} = Φ Defres_{F N_{P} (Q)/F Q}^{P}

for some homomorphism Φ : D(F N_{P} (F Q)/F Q) → D(N_{P} (Q)/Q) (actually a composition of an isomorphism and an inflation). Moreover, since F N_{P} (Q) ≤ N_{P} (F Q), we clearly have

Defres_{F N_{P} (Q)/F Q}^{P} = Res_{F N_{P} (Q)/F Q}^{N_{P} (F Q)/F Q} Defres_{N_{P} (F Q)/F Q}^{P}

Since b = InfP ; \overline{P} (\overline{b}), it follows that

a_{Q} = Φ Defres_{N_{P} (Q)/Q}^{P} InfP ; \overline{P} (\overline{b})
= Φ Res_{F N_{P} (Q)/F Q}^{N_{P} (F Q)/F Q} Defres_{N_{P} (F Q)/F Q}^{P}(\overline{b})
= Φ Res_{F N_{P} (Q)/F Q}^{N_{P} (F Q)/F Q}(a_{F Q}) .

Since we know that a_{F Q} lies in D_{t}(N_{P} (Q)/Q) because F < F Q, the lemma follows.

Note that, since the group is small enough, it is possible to make a complete list of all centric subgroups, but the proof of the lemma shows that this is not necessary.

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Serge Bouc, Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées (LAMFA), Université de Picardie - Jules Verne, 33, rue St Leu, F-80039 Amiens Cedex 1, France. serge.bouc@u-picardie.fr

Jacques Thévenaz, Institut de Géométrie, Algèbre et Topologie, EPFL, Bâtiment BCH, CH-1015 Lausanne, Switzerland. Jacques.Thevenaz@epfl.ch