The Dade group of (almost) extraspecial p-groups

Serge Bouc and Nadia Mazza

Abstract: In this paper, we determine a presentation by explicit generators and relations for the Dade group of all (almost) extraspecial p-groups. The proof of the main result uses the cohomological properties of the Tits building corresponding to the natural geometric structure of the lattice of subgroups of such p-groups.

AMS Subject Classification : 20C20, 20D15

1. Introduction

Let p be a prime number, and k be a field of characteristic p. If P is a finite p-group, a finitely generated kP -module M is called an endo-permutation module if End_{k}(M) is a permutation module, i.e. admits a P -invariant k-basis. This notion was introduced at the end of the ’70s, as a generalization of the notion of endo-trivial, or invertible module, and it is now considered as a crucial tool in many aspects of the modular representation theory of finite groups.

In [8], E.C. Dade considered an equivalence relation on the set of all endo-permutation modules whose endomorphism algebra is a permutation kP -module having a trivial direct summand. This allowed him to define a group structure on the set of these equivalence classes, giving rise to the group D_{k}(P) that we now call the Dade group of P. Hence, the classification of endo-permutation kP -modules comes down to the computation of D_{k}(P). Moreover, in these papers, Dade determined the structure of D_{k}(P) when P is abelian.

Shortly afterwards, L. Puig ([14]) proved that for an arbitrary finite p-group P, the group D_{k}(P) is a finitely generated (abelian) group. This paper only appeared in 1990, and there was no major progress on the structure of the Dade group until J. L. Alperin ([1]), using relative syzygies of the trivial module, determined the torsion-free rank of the subgroup T_{k}(P) of D_{k}(P) consisting of the classes of endo-trivial modules. Next (but this was published before Alperin’s result), J. Thévenaz and the first author ([4]) determined the torsion free rank of D_{k}(P), using tensor induction to build endopermutation modules.

Since that time, J. Carlson and J. Thévenaz ([5], [6], [7]) completed the hard task of finding the structure of T_{k}(P), using deep cohomological methods. They also determined the structure of D_{k}(P) when P is a dihedral, semi-dihedral, or generalized quaternion 2-group.

Following that, in her thesis ([13]), the second author determined the structure of D_{k}(P), when P is a metacyclic p-group for an odd prime p ([12]), or when P is an extraspecial 2-group of a particular type.

In this paper, we will extend this latter result to the case of all extraspecial pgroups and almost extraspecial p-groups, by giving a presentation of the Dade group of these groups by explicit generators and relations. The reason for considering those particular groups is twofold : firstly, our method makes an essential use of the geometric structure associated to (almost) extraspecial p-groups, more precisely the cohomological properties of the corresponding Tits building. We proceed by induction, referring to [7] for some of the initial small cases. Secondly, (almost) extraspecial groups appear as a cornerstone in many problems related to p-groups, and our hope is that our results can be used to solve the general problem of determining D_{k}(P) for an arbitrary p-group, although we have no clear idea at the moment of how this could be achieved.

Date : January 29, 2004

The paper is organized as follows : first we recall the definition and basic properties of (almost) extraspecial p-groups. In Section 3, we recall some notation and results on the Dade group. In Section 4, we state some combinatorial results on the geometry of (almost) extraspecial p-groups. Section 5 is devoted to the proof of a particular relation between relative syzygies of the trivial module that holds for this class of groups. In section 6, we state a cohomological property of a particular subposet E of the poset of subgroups of P (and that is where the Tits building mentioned in the abstract appears). In section 7 we describe three coefficient systems related to the poset E, and in the next section, we let them fit into an exact sequence. Sections 9 and 10 give the inductive step of our proof, whereas Section 11 handles the initial steps of induction.

2. (Almost) extraspecial p-groups

In this section, we are going to state some useful facts concerning the structure of (almost) extraspecial p-groups. Let us start by fixing the notation we consider throughout. Let G be a finite group. Denote by Z(G) the center of G, by G′ the commutator subgroup, i.e. the subgroup of G generated by the commutators [x, y] for x, y ∈ G, where [x, y] = xyx^{−1}y^{−1}, ∀ x, y ∈ G, and write Φ(G) for the Frattini subgroup of G. If H, K and L are subgroups of G, we write H ≤ K, respectively H ≤_{L} K, if H is contained in K, respectively if H is contained in a L-conjugate of K. If x, y ∈ G, we set ^{x}y = xyx^{−1} and y^{x} = x^{−1}yx. For our purpose, it is also convenient to recall the definition of the central product of groups (see 2.5.3 [10]).

Definition 2.1 Let H, K and M be groups such that M ≤ Z(H) and such that there exists an injective map θ : M → Z(K). If we identify M with θ(M), then there exists a group G such that G = HK and such that H centralizes K. Such a group is called the central product of H and K and we write G = H ∗ K. If H is isomorphic to K and M = Z(H), we simply write G = H^{∗2}. More generally, for all integer r ≥ 0, we write G = H^{∗r} for the central product of r copies of the group H, where M = Z(H), and with the convention that H^{∗0} = Z(H), and that H^{∗1} = H.

We can now state the main definition of this section.

Definition 2.2 Let p be a prime number and P be a finite p-group.

• We say that P is extraspecial if Z(P) = P′ = Φ(P) has order p.

• We say that P is almost extraspecial if P′ = Φ(P) has order p, and if Z(P) is cyclic of order p^{2}.

Hence, if P is an almost extraspecial p-group, then there exists a subgroup Q of index p of P which does not contain the center of P. Then, we can easily verify that Q is extraspecial, and that P is the central product of Q and Z(P). Conversely, if P is an extraspecial p-group, then P ∗ C_{p^{2}} is almost extraspecial. In particular, if p = 2, then the dihedral group D_{8} of order 8 and the quaternion group Q_{8} are extraspecial 2-groups. While the central product D_{8} ∗ C_{4}, which is isomorphic to the group Q_{8} ∗ C_{4}, is almost extraspecial of order 16 (where C_{n} denotes the cyclic group of order n, for all integer n). In order to state the classification theorem of (almost) extraspecial p-groups, it will be convenient to introduce the following notation. For p ≠ 2, we set

M = <x, y | x^{p} = y^{p} = 1, ^{y}x = x[y, x], ^{y}[y, x] = [y, x]> and
N = <x, y | x^{p^{2}} = y^{p} = 1, ^{y}x = x^{1+p}>.

Theorem 2.3 Let p be a prime number and P be an (almost) extraspecial p-group. Then, the following holds:

• If P is extraspecial, then there exists an integer r ≥ 1 such that P has order p^{2r+1} and P is isomorphic to one of the following non-isomorphic groups:

\begin{matrix}− D_{8}^{∗r} \\ − Q_{8} ∗ D_{8}^{∗(r−1)} \end{matrix} \} if p = 2;
\begin{matrix}− M^{∗r} \\ − N ∗ M^{∗(r−1)} \end{matrix} \} if p ≠ 2.

• If P is almost extraspecial, then there exists an integer r ≥ 1 such that P has order p^{2r+2}, and P is unique, up to isomorphism. More precisely, P is isomorphic to

– D_{8}^{∗r} ∗ C_{4}, if p = 2;

– M^{∗r} ∗ C_{p^{2}}, if p ≠ 2.

The part concerning extraspecial p-groups is a classical result and a proof of this can be found in section 5.5 of [10], or also in section III.13 of [11]. For almost extraspecial pgroups, one can refer to sections 2 and 4 of [5]. In particular, for each prime number p, we have exactly 3 isomorphism types of (almost) extraspecial p-groups. First of all, notice that we distinguish between almost extraspecial and extraspecial p-groups by looking at the parity of the exponent of p in |P |. Then, if p is odd, the exponent of the group characterizes the isomorphism type of extraspecial p-groups. These remarks lead us to introduce the following notation:

Definition 2.4 Let p be a prime number, P be an (almost) extraspecial p-group and r be the integer defined as in the above theorem. We say that P has type:

• T 1 if P ≅ D_{8}^{∗r}, or if P ≅ M^{∗r};

• T 2 if P ≅ Q_{8} ∗ D_{8}^{∗(r−1)}, or if P ≅ N ∗ M^{∗(r−1)};

• T 3 if P is almost extraspecial.

One of the most remarkable properties of (almost) extraspecial p-groups lies in the geometrical structure that these groups carry. Indeed, let P be an (almost) extraspecial p-group, let Φ be its Frattini subgroup, generated by an element z of order p, and set E for the quotient P/Φ, and π for the canonical surjection π : P → E. Then, since Φ = Φ(P), the finite p-group E is elementary abelian, and thus, it is also a finite dimensional 𝔽_{p}-vector space. Moreover, this vector space is endowed with the following two forms.

•

b : E × E ⟶ 𝔽_{p} , (u, v) ⟼ b(u, v) ,

where b(u, v) is the element of 𝔽_{p} satisfying [ũ, ṽ] = z^{b(u,v)}, ∀ ũ ∈ u, ∀ ṽ ∈ v, and ∀ u, v ∈ E.

•

q : E ⟶ 𝔽_{p} , u ⟼ q(u) ,

where q(u) is the element of 𝔽_{p} satisfying ũ^{p} = z^{q(u)}, ∀ ũ ∈ u, and ∀ u ∈ E.

Section 20 in [9] is concerned with extraspecial p-groups. However, their results generalize easily to (almost) extraspecial, and we get the following statements (expressed in the terminology used in chapter 7 in [15]).

• If p = 2, then q is a quadratic form, and b is the corresponding symmetric bilinear form, i.e. we have q(u + v) = q(u) + q(v) + b(u, v), ∀ u, v ∈ E. Moreover, the quadratic form q is non-degenerate, since E^{⊥} ∩ q^{−1}(0) = {0}.

• If p is odd, then b is a symplectic form on E, and q is a linear form on E. Moreover, b is non-degenerate if and only if P is extraspecial. If P is almost extraspecial, then the radical E^{⊥} = π(Z(P)) is a line in E. Here again we have E^{⊥} ∩ q^{−1}(0) = {0}.

Recall that, if p = 2, a subspace V of E is said totally singular if q(V) = {0}, and, if p is odd, we say that a subspace V of E is totally isotropic if V ≤ V^{⊥}, i.e. if b(u, v) = 0, ∀ u, v ∈ V.

Notation 2.5 We set

E = \{V ≤ E | \{0\} ≠ V ≤ V^{⊥} , q(V) = \{0\}\} , and \underline{E} = E ∪ \{0\} .

We denote by Q be the set of non-trivial subgroups Q of P such that Q ∩ Φ = 1, and \underline{Q} = Q ∪ \{1\}, where 1 denotes the trivial group. Set also Q_{P} and \underline{Q}_{P} for the respective sets of conjugacy classes of subgroups in P.

If V ∈ \underline{E}, we set

N(V) = π^{−1}(V^{⊥})  and  m(V) = \{Q ∈ \underline{Q} | π(Q) = V \} .

Note that \underline{E} is a poset for the inclusion of subspaces, and that \underline{Q} is preordered by the relation ≤_{P} of inclusion up to conjugacy in P. The associated poset is the set \underline{Q}_{P}, for the order given by C ≤ D for C, D ∈ \underline{Q}_{P} if there exist subgroups Q ∈ C and R ∈ D, such that Q ≤ R. The link between the posets \underline{E} and \underline{Q}_{P} is a direct consequence of the following lemma.

Lemma 2.6 Let P be an (almost) extraspecial p-group, and let Q be a non-trivial subgroup of P. Then,

1. Q ◁ P ⟺ Φ ≤ Q.

2. Q ◁̸ P ⟹ N_{P} (Q) = C_{P} (Q). In particular, it follows that in this case Q is elementary abelian of rank at most r, for the integer r defined as in Theorem 2.3, and we have |Q|·|C_{P} (Q)| = |P |. Moreover, C_{P} (Q) = Q × U, where U is (almost) extraspecial of order \frac{|P |}{|Q|^{2}} , or U = Z(P).

3. Let Q, R ◁̸ P. Then,

Q <_{P} R ⟺ Q <_{P} R < N_{P} (R) < N_{P} (Q) , and Q =_{P} R ⟺ QΦ = RΦ .

Moreover, if Q ≤_{P} R, then there exists a unique u ∈ [P/N_{P} (Q)] such that^{u}Q ≤ R.

Proof: Note that ∀ x, y ∈ P, we have ^{y}x = x, or Φ = <[x, y]>.

1. If 1 < Q ≤ Z(P), then the assertion holds. Assume 1 < Q ≰ Z(P) and Q ◁ P. Then, there exist x ∈ Q and y ∈ P such that Φ = <[x, y]>. Since ^{y}x ∈ Q, we have [x, y] ∈ Q and hence Φ ≤ Q. The converse is obvious, since the quotient P/Φ is abelian.

2. If Q ◁̸ P, then Q is elementary abelian, since Φ(Q) ≤ Q ∩ Φ = 1, by the previous assertion. Moreover, the inclusion [Q, N_{P} (Q)] ≤ Q ∩ Φ = 1 implies the equality N_{P} (Q) = C_{P} (Q). Set I = π(Q). Then C_{P} (Q) = π^{−1}(I^{⊥}), and I ∩ E^{⊥} = {0}. Hence we have |E| = |I|·|I^{⊥}|, and it follows that |P | = |Q|·|C_{P} (Q)|. The subspace π(Q) of E is totally isotropic if p is odd, respectively totally singular if p = 2. Thus, its dimension is at most equal to the Witt index r of the form b, since dim(E) = 2r and dim(E^{⊥}) ≤ 1 (see section 7 in [15]). Set G = C_{P} (Q)/Q. Then Φ(G) ≤ QΦ/Q has order 1 or p. In the first case, C_{P} (Q) is abelian, since [G, G] = 1 implies [C_{P} (Q), C_{P} (Q)] ≤ Q ∩ Φ = 1. Hence, C_{P} (Q) ≤ C_{P} C_{P} (Q) = Q·Z(P), and so C_{P} (Q) = Q·Z(P). In the second case, Φ(G) has order p. Let H be the subgroup of P containing Q and such that H/Q = Z(G). Since [H/Q, G] = 1, we have [H, C_{P} (Q)] ≤ Q ∩ Φ = 1, and hence H ≤ C_{P} C_{P} (Q) = Q·Z(P). Thus H = Q·Z(P), and it follows that H/Q is isomorphic to Z(P). This shows that the p-group G is (almost) extraspecial. Notice now that the p-group C_{P} (Q)/Φ is elementary abelian, and contains Q·Φ/Φ. Therefore, there exists U ≤ P, such that π(C_{P} (Q) = π(Q) ⊕ π(U). It follows that

C_{P} (Q) = Q·Φ·U = Q·U , and that Q ∩ U ≤ Q ∩ Φ = 1 ,

since Q·Φ ∩ U = Φ. In other words, C_{P} (Q) is the semi-direct product of U by Q, and in fact this product is direct, since Q is a central subgroup of C_{P} (Q).

3. The previous assertion implies N_{P} (Q) = C_{P} (Q), ∀ Q ◁̸ P. Hence, if Q, R ◁̸ P, then the assertion Q <_{P} R ⟺ Q <_{P} R < N_{P} (R) < N_{P} (Q) is immediate. Set V = π(Q). Then Q ∈ m(V), and m(V) is the set of complements of Φ in the elementary abelian group Q·Φ. Moreover, the group P acts transitively (by conjugation) on m(V). Indeed, since Q∩Φ = 1, we have that |P : N_{P} (Q)| = |Q| is equal to the number of complements of Φ in Q·Φ. In other words, two subgroups Q and R of P are conjugate if, and only if Q·Φ = R·Φ, as was to be shown. Finally, let Q ≤_{P} R, where Q, R ◁̸ P. Then [P/N_{P} (Q)] acts transitively on the set of all conjugates of Q, and so there exists x ∈ [P/N_{P} (Q)] such that Q^{x} ≤ R. Let y ∈ [P/N_{P} (Q)] such that Q^{y} ≤ R. Then, Q^{x}·Q^{y} ≤ R, since R is abelian. Now, if Q^{x} ≠ Q^{y}, then Φ ≤ Q^{x}·Q^{y}. Hence Q^{x} = Q^{y}, since Φ ≰ R.

Corollary 2.7 The canonical surjection π : P → E induces an isomorphism of posets π : \underline{Q}_{P} → \underline{E}, which restricts to an isomorphism of posets π : Q_{P} → E. That is, we have ∀ C, D ∈ \underline{Q}_{P}

C ≤ D in \underline{Q}_{P} ⟺ QΦ ≤ RΦ , ∀ Q ∈ C , ∀ R ∈ D.

Moreover, we have π(N_{P} (Q)) = (π(Q))^{⊥}, ∀ Q ∈ C, and ∀ C ∈ \underline{Q}_{P}.

This fact has the nice property that it will allow us later to translate the traditional operations of restriction-deflation on the Dade groups of (almost) extraspecial p-groups into a geometrical language.

3. The Dade group

In this section, we consider all notations of the previous section, and we want to recall some essential facts concerning the Dade group of a finite p-group. Then, we will apply them to the particular case of an (almost) extraspecial p-group, in order to get a first reduction of the question we are concerned with in this paper, that is, the classification of capped endo-permutation modules (up to equivalence) in the case of an (almost) extraspecial p-group.

Fix a prime number p and a field k of characteristic p. The Dade group D(P) = D_{k}(P) of a finite p-group P is the finitely generated abelian group formed by the equivalence classes of capped endo-permutation kP -modules, with composition law induced by tensor product (over k) of the modules. As usual, we let T (P) be the subgroup of D(P) generated by the classes of endo-trivial modules, and we write D^{t}(P), resp. T^{t}(P), for the torsion subgroup of D(P), resp. T (P). We will also denote by D^{Ω}(P), resp. T^{Ω}(P), the subgroups of D(P), resp. T (P), generated by the classes of the relative syzygies of the trivial module k. Recall that if X is a non empty finite P -set, then the relative syzygy Ω_{X}^{1}(k) of the trivial module k, relatively to X, is the kernel of the k-linear map kX ⟶ k, sending each x ∈ X to 1. We will then just write Ω_{X} for the class of Ω_{X}^{1}(k) in D(P). We assume that the reader is familiar with the usual operations of restriction, tensor induction, inflation, and deflation acting on the Dade group.

Notation 3.1 Let P be a finite p-group. We denote by:

• nc(P) the number of conjugacy classes of non-cyclic subgroups of P, and

• c′(P) the number of conjugacy classes of cyclic subgroups of order at least 3 of P.

Next lemma collects the known results that we will need, and we let the reader refer to [1], [3], [4], [5], [6], [8] and [14] for the proofs.

Lemma 3.2 1. ([8]) If P is abelian, then

T (P) ≅ \{ \begin{matrix}1 & if |P | ≤ 2 \\ ℤ/2ℤ & if P is cyclic and |P | ≥ 3 \\ ℤ & otherwise . \end{matrix}

Moreover, in that case, we have

D(P) = D^{Ω}(P) ≅ ⨁_{1≤Q<P} T (P/Q) ≅ ℤ^{nc(P)} ⊕ (ℤ/2ℤ)^{c′(P)}.

2. ([14]) The subgroups T (P) and ⋂_{1<Q≤P} Ker(Defres_{N_{P} (Q)/Q}^{P}) of D(P) are equal.

3. ([6]) If P is not cyclic, quaternion or semi-dihedral, then T (P) is detected on restriction to all elementary abelian subgroups of rank 2. In particular, in that case, the torsion subgroup T^{t}(P) is trivial.

4. ([1],[4]) If n is the number of conjugacy classes of maximal elementary abelian subgroups of rank 2, then the dimension of the ℚ-vector space ℚ ⊗_{ℤ} T (P) is :

(a) 0 if P is cyclic,

(b) n if P is not cyclic, and if P has no elementary abelian subgroup of rank 3, (c) n + 1 otherwise.

Moreover, if any maximal elementary abelian subgroup of P has rank at least 3, then T (P) =< Ω_{P} >≅ ℤ.

5. ([4]) Let S be the set of non-cyclic subgroups of P, and for every Q ∈ S choose a normal subgroup Q_{0} ◁ Q such that Q/Q_{0} is elementary abelian of rank 2. Then, the ℚ-linear map

∏_{Q∈[S/P]} Id ⊗ Defres_{Q/Q_{0}}^{P} : ℚ ⊗_{ℤ} D(P) ⟶ ∏_{Q∈[S/P]} ℚ ⊗_{ℤ} D(Q/Q_{0})

is an isomorphism.

6. ([3]) Let A denote the set of all elementary abelian sections of P, and set

K(P) = ⋂_{A∈A} Ker(Defres_{A}^{P}).

Then K(P) is a subgroup of D^{t}(P), and the quotient

D(P)/(D^{Ω}(P) + K(P)) is a finite group with exponent dividing |P |.

7. ([5], [6]) If p is odd, let X denote the set of all subgroups Q of P such that the quotient N_{P} (Q)/Q is cyclic. If p = 2, let X denote the set of all subgroups Q of P such that the quotient N_{P} (Q)/Q is cyclic of order at least 4, or generalized quaternion, or semi-dihedral. Then the map

∏_{Q∈X} (ρ_{N_{P} (Q)/Q} ∘ Defres_{N_{P} (Q)/Q}^{P}) : D^{t}(P) ⟶ ∏_{Q∈X} T^{t}(N_{P} (Q)/Q)

is injective, where ρ_{H} : D^{t}(H) ⟶ T^{t}(H) is the canonical surjection (that is well defined for all subgroups H that we consider here). More precisely, if p is odd, and [C/P] is a set of representatives of conjugacy classes of all cyclic non-trivial subgroups of P. Then, the map

∏_{C∈[C/P]} Defres_{C/Φ(C)}^{P} : D^{t}(P) ⟶ ∏_{C∈[C/P]} D(C/Φ(C)) ,

is an isomorphism, and the set {Teninf_{C/Φ(C)}^{P}Ω_{C/Φ(C)} | C ∈ [C/P]} is an 𝔽_{2}-basis of D^{t}(P). In particular, the torsion subgroup D^{t}(P) is contained in D^{Ω}(P).

Consider now an (almost) extraspecial p-group P, and let π : P → E be the canonical surjection, where E = P/Φ(P), as defined in the previous section.

Notation 3.3 We set:

1. ∂D(P) = Ker(Def_{E}^{P}).

2. ∂D^{Ω}(P) = ∂D(P) ∩ D^{Ω}(P).

Proposition 3.4 Let P be an (almost) extraspecial p-group. Then,

1. T (P) ≤ ∂D(P).

2. D(P) = Inf_{E}^{P}(D(E)) ⊕ ∂D(P), and D(P) = D^{Ω}(P) + ∂D(P).

3. The groups D(P)/D^{Ω}(P) and ∂D(P)/∂D^{Ω}(P) are isomorphic.

Proof:

1. We have T (P) = ⋂_{1<Q≤P} Ker(Defres_{N_{P} (Q)/Q}^{P}) =

= Ker(Def_{E}^{P}) ∩ ( ⋂_{Ṽ ∈m(V)^{V ∈E}} Ker(Defres_{N(V)/Ṽ}^{P} )) ≤ ∂D(P).

2. The inflation Inf_{E}^{P} is a section of the surjective map Def_{E}^{P} : D(P) ⟶ D(E), whose kernel is the subgroup ∂D(P), by definition. Thus D(P) = Inf_{E}^{P}(D(E))⊕∂D(P). Since E is abelian, we have D(E) = D^{Ω}(E), and thus it can be identified to a subgroup of D^{Ω}(P), by inflation.

3. From 2., we have D(P)/D^{Ω}(P) ≅ ∂D(P)/(∂D(P) ∩ D^{Ω}(P)).

4. Some combinatorial properties

Recall that P is an (almost) extraspecial p-group with Frattini subgroup Φ, and that \underline{Q} is the set of subgroups Q of P such that Q ∩ Φ = 1. The elements of \underline{Q} are elementary abelian.

Lemma 4.1 1. Let Q and R be subgroups of P. Then

C_{P} (Q ∩ C_{P} (R)) = C_{P} (Q)·R .

2. If moreover R is abelian, then

C_{P} (R·(Q ∩ C_{P} (R))) = R·(C_{P} (Q) ∩ C_{P} (R)) .

3. Let Q and R be subgroups of P such that Q ∩ C_{P} (R) ≤ R ≤ C_{P} (R). Then

C_{P} (R)C_{P} (Q) = C_{P} (Q ∩ C_{P} (R)) = C_{P} (Q ∩ R) ,

and

C_{P} (R) = (C_{P} (R) ∩ C_{P} (Q))·R .

4. If Q and R are elements of \underline{Q} such that Q∩C_{P} (R) ≤ R, then Q·(C_{P} (Q)∩R) ∈ \underline{Q},

and

|Q·(C_{P} (Q) ∩ R)| = |R| .

Proof: For any subgroup R of P, one has that C_{P} C_{P} (R) = R.Z, where Z = Z(P). Moreover C_{P} (R) ≥ Z, hence C_{P} (R) \underline{◁} P. Now

C_{P} C_{P} (Q ∩ C_{P} (R)) = (Q ∩ C_{P} (R)·Z = Q·Z ∩ C_{P} (R)
= C_{P} C_{P} (Q) ∩ C_{P} (R)
= C_{P} (C_{P} (Q)·R)

Taking centralizers once more gives

C_{P} C_{P} C_{P} (Q ∩ C_{P} (R)) = C_{P} C_{P} (C_{P} (Q)·R)
C_{P} (Q ∩ C_{P} (R))·Z = (C_{P} (Q)·R)·Z

and Assertion 1 follows, since Z ≤ C_{P} (Q ∩ C_{P} (R)) and Z ≤ C_{P} (Q)·R. Now

C_{P} (R·(Q ∩ C_{P} (R))) = C_{P} (R) ∩ C_{P} (Q ∩ C_{P} (R))
= C_{P} (R) ∩ C_{P} (Q)·R
= (C_{P} (R) ∩ C_{P} (Q))·R ,

since R ≤ C_{P} (R), proving Assertion 2.

For Assertion 3, the inclusion C_{P} (Q)·C_{P} (R) ≤ C_{P} (Q∩R) is obvious. Conversely the assumption implies Q ∩ C_{P} (R) ≤ Q ∩ R ≤ Q ∩ C_{P} (R), hence Q ∩ R = Q ∩ C_{P} (R). Thus C_{P} (Q ∩ R) = C_{P} (Q)·R ≤ C_{P} (Q)·C_{P} (R) by Assertion 1. In particular C_{P} (Q)·C_{P} (R) = C_{P} (Q)·R, thus R ≤ C_{P} (R) ≤ C_{P} (Q)·R, hence C_{P} (R) = (C_{P} (R) ∩ C_{P} (Q))·R.

For Assertion 4, let z ∈ Q·(C_{P} (Q) ∩ R) ∩ Φ. Then z = qr, for elements q ∈ Q and r ∈ C_{P} (Q) ∩ R. Thus q = zr^{−1} ∈ Q ∩ C_{P} (R), because R ≤ C_{P} (R), and q ∈ Q ∩ R. It follows that z = qr ∈ Φ ∩ R = 1, showing that Q·(C_{P} (Q) ∩ R) ∩ Φ = 1. Now

|Q·(C_{P} (Q) ∩ R)| = \frac{|Q||C_{P} (Q) ∩ R|}{|Q ∩ R|}
= \frac{|Q|}{|Q ∩ R|} \frac{|P |}{|C_{P} (C_{P} (Q) ∩ R)|} ,

since C_{P} (Q) ∩ R ∈ \underline{Q}, and since |S||C_{P} (S)| = |P | for any S ∈ \underline{Q}. Hence

|Q·(C_{P} (Q) ∩ R)| = \frac{|Q|}{|Q ∩ R|} \frac{|P |}{|Q·C_{P} (R)|}
= \frac{|Q|}{|Q ∩ R|} \frac{|P ||Q ∩ C_{P} (R)|}{|Q||C_{P} (R)|}
= \frac{|P ||Q ∩ C_{P} (R)|}{|Q ∩ R||C_{P} (R)|}
= \frac{|R||Q ∩ C_{P} (R)|}{|Q ∩ R|} = |R| ,

since Q ∩ C_{P} (R) = Q ∩ R.

Corollary 4.2 Let Q ∈ \underline{Q}, and let Q_{0} be any maximal element of \underline{Q}. Then :

1. There exists a unique double coset C_{P} (Q)xC_{P} (Q_{0}) of elements y ∈ P such that Q^{y} ∩ C_{P} (Q_{0}) ≤ Q_{0}.

2. The cardinality of the set Q\C_{P} (Q)xC_{P} (Q_{0})/C_{P} (Q_{0}) is equal to |Q_{0}|/|Q|.

3. The group Q^{x}·(C_{P} (Q^{x}) ∩ Q_{0}) is a maximal element of \underline{Q}.

Proof: Since Q_{0} is a maximal element of \underline{Q}, it is contained in a unique maximal elementary abelian subgroup Q_{0}·Φ of P, and it follows that (Q ∩ C_{P} (Q_{0}))·Q_{0} ≤ Q_{0}·Φ. Set R = Q ∩ C_{P} (Q_{0}). There exists a maximal element Q_{1} of \underline{Q} containing R and contained in Q_{0}·Φ. Since Q_{1}·Φ = Q_{0}·Φ, it follows Q_{1} = ^{x}Q_{0} for some x ∈ P. Hence R^{x} = Q^{x}∩C_{P} (Q_{0}) ≤ Q_{0}. Clearly if y ∈ C_{P} (Q)xC_{P} (Q_{0}), then R^{y} = Q^{y}∩C_{P} (Q_{0}) ≤ Q_{0}.

Conversely, if y is any element of P such that R^{y} ≤ Q_{0}, then R is contained in xQ_{0} and ^{y}Q_{0}. Hence there exists z ∈ C_{P} (R) such that ^{y}Q_{0} = ^{zx}Q_{0}. Thus x^{−1}z^{−1}y ∈ C_{P} (Q_{0}), and y ∈ C_{P} (R)xC_{P} (Q_{0}). But

C_{P} (R)x = xC_{P} (R^{x}) = xC_{P} (Q^{x})C_{P} (Q_{0}) ,

since Q^{x}∩C_{P} (Q_{0}) ≤ Q_{0}. Hence C_{P} (R)xC_{P} (Q_{0}) = C_{P} (Q)xC_{P} (Q_{0}), as was to be shown for Assertion 1.

There is a surjective map

\begin{matrix}φ : C_{P} (Q^{x}) & → & Q\C_{P} (Q)xC_{P} (Q_{0})/C_{P} (Q_{0}) \\ z & ↦ & QxzC_{P} (Q_{0}). \end{matrix}

Now for z, t ∈ C_{P} (Q^{x}), one has that φ(z) = φ(t) if and only if

t ∈ (Q^{x}zC_{P} (Q_{0})) ∩ C_{P} (Q^{x}) = zQ^{x}·(C_{P} (Q_{0}) ∩ C_{P} (Q^{x})) .

Hence all the fibers of φ have the same cardinality, and

|Q\C_{P} (Q)xC_{P} (Q_{0})/C_{P} (Q_{0})| = |C_{P} (Q^{x})|/|Q^{x}·(C_{P} (Q_{0}) ∩ C_{P} (Q^{x}))| .

Now R^{x} = Q^{x} ∩ C_{P} (Q_{0}), and so C_{P} (R^{x}) = C_{P} (Q^{x})C_{P} (Q_{0}), by Lemma 4.1. Thus

|C_{P} (R^{x})| = \frac{|C_{P} (Q^{x})||C_{P} (Q_{0})|}{|C_{P} (Q^{x}) ∩ C_{P} (Q_{0})|} .

It follows that

|Q\C_{P} (Q)xC_{P} (Q_{0})/C_{P} (Q_{0})| = \frac{|C_{P} (Q^{x})||Q^{x} ∩ C_{P} (Q_{0})|}{|Q^{x}||C_{P} (Q^{x}) ∩ C_{P} (Q_{0})|}
= \frac{|C_{P} (Q^{x})||R^{x}||C_{P} (R^{x})|}{|Q^{x}||C_{P} (Q^{x})||C_{P} (Q_{0})|}
= \frac{|C_{P} (Q^{x})|}{|C_{P} (Q_{0})|}
= \frac{|Q_{0}|}{|Q^{x}|}

since |S||C_{P} (S)| = |P | for any S ∈ \underline{Q}. Assertion 2 follows, since |Q^{x}| = |Q|.

Finally the group Q^{x}·(C_{P} (Q^{x}) ∩ Q_{0}) is an element of \underline{Q} of order equal to |Q_{0}|, by Lemma 4.1. Hence it is maximal in \underline{Q}, and this completes the proof of Corollary 4.2.

5. Some relations

Theorem 5.1.2 of [3] states a formula for the effect of tensor induction on relative syzygies in the Dade group : if Q is a subgroup of a finite p-group P, if X is a finite Q-set, then

Ten_{Q}^{P}Ω_{X} = ∑_{{R,S∈[s_{P}] \atop R≤_{P} S}} μ_{P} (R, S) |\{a ∈ S\P/Q | X^{S^{a}∩Q} ≠ ∅\}| Ω_{P/R} ,

where [s_{P}] is a set of representatives of the set s_{P} of conjugacy classes of subgroups of P, and μ_{P} is the Möbius function of s_{P}, partially ordered by the relation ≤_{P}. By convention Ω_{∅} = 0.

We apply this formula for the (almost) extraspecial group P : choose any maximal element Q_{0} of \underline{Q}, and consider the subgroup Q = C_{P} (Q_{0}) and the finite Q-set X = Q/Q_{0}. The previous formula gives

Ten_{Q}^{P}Ω_{X} = ∑_{{R,S∈[s_{P}] \atop R≤_{P} S}} μ_{P} (R, S) |\{a ∈ S\P/Q | S^{a} ∩ Q ≤ Q_{0}\}| Ω_{P/R} ,

since X^{S^{a}∩Q} ≠ ∅ if and only if S^{a} ∩ Q ≤ Q_{0}. This implies in particular that S^{a} ∩ Φ ≤ Q_{0} ∩ Φ = 1, thus S ∈ \underline{Q}. Conversely, if S ∈ \underline{Q}, by Corollary 4.2

|{a ∈ S\P/Q | S^{a} ∩ Q ≤ Q_{0}}| = |Q_{0}|/|S| .

Moreover if R ≤ S, and if x is an element of P such that R^{x} ≤ S, then R = R^{x}. It follows in particular that μ_{P} (R, S) is equal to the ordinary Möbius function μ(R, S) of the poset of subgroups of P for the inclusion relation (recall that μ(R, S) = (−1)^{k}p^{\binom{k}{2}} if |S : R| = p^{k}).

This gives finally

Ten_{Q}^{P}Ω_{X} = ∑_{{R,S∈[s_{P}]∩\underline{Q} \atop R≤_{P} S}} μ(R, S) \frac{|Q_{0}|}{|S|} Ω_{P/R}

Notation 5.1 Let τ denote the order of Ω_{C_{P} (Q_{0})/Q_{0}} in the Dade group D(C_{P} (Q_{0})/Q_{0}).

Then τ is equal to 1, 2, or 4 according to C(Q_{0})/Q_{0} being cyclic of order 2, cyclic of order at least 3, or quaternion of order 8. Obviously, the order of Ten_{Q}^{P}Ω_{X} in D(P) divides τ.

Moreover, if Y is any finite P -set, and if H is any subgroup of P, one has that Defres_{N_{P} (H)/H}^{P}Ω_{Y} = Ω_{Y^{H}}, where the set Y^{H} on the right hand side of the equality is considered as a N_{P} (H)/H-set. It follows in particular that

Defres_{Q/Q_{0}}^{P}Ten_{Q}^{P}Ω_{X} = Ω_{(P/Q_{0})^{Q_{0}}} = Ω_{X} ,

since for R ≤_{P} S ∈ \underline{Q}, one has that (P/R)^{Q_{0}} ≠ ∅ if and only if R =_{P} S =_{P} Q_{0}, and since moreover (P/Q_{0})^{Q_{0}} = Q/Q_{0} = X. Hence the order of Ten_{Q}^{P}Ω_{X} in D(P) is equal to τ. In particular

(5.2)  τ· ( ∑_{{R,S∈[s_{P}]∩\underline{Q} \atop R≤_{P} S}} μ(R, S)\frac{|Q_{0}|}{|S|}Ω_{P/R}) = 0

in D(P).

Notation 5.3 If V ∈ \underline{E}, let Ṽ be an element of m(V), and set

r_{V} = τ· ( ∑_{{S∈[s_{P}]∩\underline{Q} \atop S≥_{P} Ṽ}} μ(Ṽ , S)\frac{|Q_{0}|}{|S|})

5.4. Remark: There is a slight abuse in the notation here : the expression μ(Ṽ , S) actually means μ(Ṽ , S^{x}), where x is some element of P such that S^{x} ≥ Ṽ. Since the elements of m(V) are conjugate in P, the integer r_{V}, i.e. the coefficient of Ω_{P/Ṽ} in 5.2, does not depend on the choice of Ṽ in m(V). Now relation 5.2 can be written as follows :

(5.5)  ∑_{V ∈\underline{E}} r_{V} Ω_{P/Ṽ} = 0

6. The cohomology of E

If p ≠ 2, recall that E is the set of non zero subspaces V of E which are contained in H = Ker(q), and such that V ≤ V^{⊥}, where V^{⊥} is the orthogonal of V for the bilinear symplectic form b. In other words E is the set of subspaces V of H such that c(V, V) = 0, where c is the restriction of b to H. Since H ∩ E^{⊥} = {0}, it follows that the form c is non-degenerate.

If p = 2, the set E is the set of non zero subspaces V of W such that q(V) = {0}, where q is the quadratic form, with associated bilinear form b. In this case, the form q is non-degenerate.

The set E is partially ordered by the inclusion relation.

Proposition 6.1 Let H be a finite dimensional vector space over a field k of characteristic p, and let c be a non-degenerate bilinear form on H if p ≠ 2, or a non-degenerate quadratic form on H if p = 2, with associated bilinear form c. Let E be the set of non zero totally isotropic subspaces of H for c if p ≠ 2, or the set of non zero totally singular subspaces of H for q if p = 2, ordered by inclusion. Then the cohomology groups H^{n}(E, ℤ) are torsion-free abelian groups, for any integer n ∈ ℕ.

Proof: This follows from the Solomon-Tits theorem, since the cohomology of E is the same as the cohomology of the associated Tits building, which has the homotopy type of a bouquet of spheres of a fixed dimension. Since we are only interested in cohomology groups, and since the full strength of the Solomon-Tits theorem is not required here, we give a more elementary proof of this proposition.

Lemma 6.2 Let (X, ≤) be a poset. Assume that there exists an acyclic subset Y of X such that X − Y consists of minimal elements of X. Then for any k ∈ ℤ, there is an isomorphism

H̃^{k}(X) ≅ ∏_{x∈X−Y} H̃^{k−1}(]x, .[_{X}) ,

where ]x, .[_{X}= {z ∈ X | x < z}, and H̃^{k}(X) is the k^{th} integral reduced cohomology group of X.

Proof: Let C̃_{∗}(X) denote the reduced chain complex of X over ℤ. Then the reduced integral cohomology of X is the cohomology of the reduced cochain complex

C̃^{∗}(X) = Hom_{ℤ}(C̃_{∗}(X), ℤ) .

Let i ∈ ℤ. Then C̃_{i}(X) is equal to {0} if i < −1, to ℤ if i = −1, and if i > 0, it is the free abelian group with basis the set of strictly increasing sequences x_{0} < x_{1} < ⋯ < x_{i} of elements of X. Such a sequence has at most one element outside Y, namely x_{0}, since X − Y consists of minimal elements of X. It follows that

C̃_{i}(X) = C̃_{i}(Y) ⊕ ( ⊕_{x∈X−Y} C̃_{i−1}(]x, .[_{X})) ,

and it is easy to check that this decomposition leads to an exact sequence of complexes

0 → C̃_{∗}(Y) → C̃_{∗}(X) → ⊕_{x∈X−Y} C̃_{∗−1}(]x, .[_{X}) → 0 .

Since these are complexes of free ℤ-modules, taking ℤ-duals gives an exact sequence of cochain complexes

(6.3)  0 → ∏_{x∈X−Y} C̃^{∗−1}(]x, .[_{X}) → C̃^{∗}(X) → C̃^{∗}(Y) → 0 .

But the complex C̃^{∗}(Y) is acyclic if Y is, and the long exact sequence of cohomology groups associated to the exact sequence 6.3 gives the required isomorphisms.

The following lemma (see for instance [2] Corollary 4.1.3) is a weak form of a theorem of Quillen. Recall that two maps of posets h, h′ : X → Y are said to be comparable if h(x) ≤ h′(x) for any x ∈ X, or if h(x) ≥ h′(x) for any x ∈ X.

Lemma 6.4 1. Let X and Y be posets, and let f : X → Y and g : Y → X be maps of posets. If g ∘ f is comparable to Id_{X}, and if f ∘ g is comparable to Id_{Y}, then the maps of complexes C̃_{∗}(f) and C̃_{∗}(g) are mutual inverse homotopy equivalences between the complexes C̃_{∗}(X) and C̃_{∗}(Y). In particular f and g induce inverse group isomorphisms H̃^{k}(Y) ≅ H̃^{k}(X), for any k ∈ ℤ.

2. If the poset X has a largest element, or a smallest element, then the chain complex C̃_{∗}(X) is contractible. In particular H̃^{k}(X) = {0} for any k ∈ ℤ.

Corollary 6.5 Let f : X → Y be a map of posets. Suppose that for any y ∈ Y, the set f^{y} = {x ∈ X | f(x) ≤ y} has a largest element, or that for any y ∈ Y, the set f_{y} = {x ∈ X | f(x) ≥ y} has a smallest element. Then f induces an isomorphism H̃^{k}(Y) ≅ H̃^{k}(X), for any k ∈ ℤ.

Proof: If the first assertion holds, let y ∈ Y, and call g(y) the largest element of f^{y}. Then g : Y → X is a map of posets. Moreover f ∘ g ≤ Id_{Y}, and g ∘ f ≥ Id_{X}. Similarly, if the second assertion holds, call g(y) the smallest element of f_{y}. Then g : Y → X is a map of posets. Moreover g ∘ f ≤ Id_{X} and f ∘ g ≥ Id_{Y}.

Proof of Proposition 6.1: By induction on the dimension of H. If H = 0, there is nothing to prove : in that case indeed E = ∅, and the reduced cohomology group of ∅ are all zero except H̃^{−1}(∅) ≅ ℤ. Suppose that the result holds for spaces of dimension lower than dim_{k} H. If E = ∅, again there is nothing to prove. And if E ≠ ∅, let d be a minimal element of E. Then d is a line. Consider the following subposets of E :

[d, .[_{E} = {V ∈ E | V ≥ d}
]., d^{⊥}] ∩ E = {V ∈ E | V ≤ d^{⊥}}
E_{d} = {V ∈ E | V ∩ d^{⊥} ≠ {0}} ,

where d^{⊥} is the orthogonal of d for the bilinear form c. There are obvious inclusion maps i : [d, .[_{E}↪]., d^{⊥}] ∩ E and j :]., d^{⊥}] ∩ E ↪ E_{d}. Let V ∈]., d^{⊥}] ∩ E. Then the set

i_{V} = {W ∈ [d, .[_{E}| W ≥ V }

has a smallest element d + V : indeed d + V is a non-zero totally isotropic/singular subspace of H if V is a totally isotropic/singular subspace of H contained in d^{⊥}. By Corollary 6.5, the map i induces an isomorphism of reduced cohomology groups H̃^{k}([d, .[_{E}) ≅ H̃^{k}(]., d^{⊥}]∩E), for any k ∈ ℤ. But [d, .[_{E} is an acyclic poset by Lemma 6.4, since it has a smallest element. Thus ]., d^{⊥}] ∩ E is also acyclic.

Now let V ∈ E_{d}. The set

j^{V} = {W ∈]., d^{⊥}] ∩ E | W ≤ V }

has a largest element V ∩ d^{⊥}. Again by Corollary 6.5 and Lemma 6.4, this shows that E_{d} is an acyclic poset. The set E − E_{d} is the set of elements V of E such that V ∩ d^{⊥} = {0}. Since d^{⊥} is an hyperplane of H (because d ∩ H^{⊥} = {0} if d ∈ E), it follows that E − E_{d} consists of lines, i.e. of minimal elements of E. By Lemma 6.2, there are group isomorphisms

H̃^{k}(E) ≅ ∏_{{V ∈E \atop V ∩d^{⊥}=\{0\}}} H̃^{k−1}(]V, .[_{E}) ,

for any k ∈ ℤ. But for V ∈ E −E_{d}, the poset ]V, .[_{E} is isomorphic to the poset of non zero totally isotropic/singular subspaces for the non-degenerate bilinear (and quadratic) form induced by c (and q) on the space V^{⊥}/V. Since dim_{k}(V^{⊥}/V) < dim H, by induction hypothesis, the cohomology groups H̃^{k−1}(]V, .[_{E}) are torsion-free. Hence the groups H̃^{k}(E) are torsion-free, as was to be shown.

6.6. Remark: The previous proof gives actually a more precise result : the cohomology groups H̃^{k}(E) are products of copies of ℤ.

7. The coefficient system

Recall from Notation 2.5 that if π : P → P/Φ denotes the canonical projection, and if V ∈ \underline{E}, we set N(V) = π^{−1}(V^{⊥}) and m(V) = \{Q ∈ \underline{Q} | π(Q) = V \}. Recall that if Q ∈ m(V), then N_{P} (Q) = C_{P} (Q) = N(V), by Lemma 2.6, and that the group P acts transitively on m(V) by conjugation.

It follows that P acts on the group Σ_{V} = ⊕_{Q∈m(V)} ∂D(N(V)/Q), and we denote by

F (V) = ( ⊕_{Q∈m(V)} ∂D(N(V)/Q))_{P} ,

the co-invariants of P on this group, i.e. the largest quotient of Σ_{V} on which P acts trivially. Since the action of P on m(V) is transitive, and since the stabilizer N_{P} (Q) = N(V) of Q ∈ m(V) acts trivially on ∂D(N(V)/Q), the group F (V) is isomorphic to

∂D(N(V)/Q), for any Q ∈ m(V). The consideration of F (V) avoids the choice of a particular element in m(V). Also note that F ({0}) = ∂D(P).

Now if V ≤ W in \underline{E}, then W^{⊥} ≤ V^{⊥}, thus N(W) ≤ N(V). Moreover π^{−1}(V) ≤ π^{−1}(W). If Q ∈ m(V), then there is an R ∈ m(W) such that R ≥ Q, and any two R’s are conjugate by N_{P} (Q). In this case Q ≤ R ≤ N(W) ≤ N(V), and the map

Defres_{N(W)/R}^{N(V)/Q} : ∂D(N(V)/Q) → ∂D(N(W)/R)

gives a map

∂D(N(V)/Q) → F (W)

which does not depend on the choice of R ≥ Q. This leads to a well defined map

F_{V}^{W} : F (V) → F (W)

whose restriction to ∂D(N(V)/Q) is the previous map.

8. A sequence of coefficient systems

We will consider the constant coefficient system Z on \underline{E}, defined by Z(V) = ℤ for any V ∈ \underline{E}, and Z_{V}^{W} = Id_{ℤ} for any V ≤ W ∈ \underline{E}.

If V ∈ \underline{E}, set

\underline{E}(V) = \{V′ ∈ \underline{E} | V ≤ V′\} .

Define another coefficient system Z\underline{E} on \underline{E} by

Z\underline{E}(V) = ℤ^{\underline{E}(V)} ,

for V ∈ \underline{E}. It is the free abelian group on the set \underline{E}(V) : an element of Z\underline{E}(V) is a linear combination with integer coefficients of elements of \underline{E}(V). If V ≤ W ∈ \underline{E}, the transition map Z\underline{E}_{V}^{W} : Z\underline{E}(V) → Z\underline{E}(W) is the canonical projection map corresponding to the inclusion \underline{E}(W) ⊆ \underline{E}(V).

Fix V ∈ \underline{E}. Let i_{V} : ℤ → ℤ^{\underline{E}(V)} be the linear map sending 1 ∈ ℤ to the element ∑_{U∈\underline{E}(V)} r_{U}U. Let s_{V} : ℤ^{\underline{E}(V)} → F (V) be the map sending U ∈ \underline{E}(V) to the image in

F (V) of the element Ω_{(N(V)/Ṽ)/(Ũ/Ṽ)}, where Ũ is any element of m(U) containing Ṽ.

Lemma 8.1 The maps i_{V} and s_{V}, for V ∈ \underline{E}, define maps of coefficient systems

i : Z → Z\underline{E}  and  s : Z\underline{E} → F ,

such that s ∘ i = 0.

Proof: Saying that i is a map of coefficient systems is equivalent to saying that for any V ≤ W in \underline{E}, the square

Z(V) \underrightarrow{i_{V}} Z\underline{E}(V)
Z_{V}^{W} ↓  ↓ Z\underline{E}_{V}^{W}
Z(W) \overrightarrow{i_{W}} Z\underline{E}(W)

is commutative. Equivalently, this amounts to checking that

Z\underline{E}_{V}^{W} ( ∑_{U∈\underline{E}(V)} r_{U}U) = ∑_{U∈\underline{E}(W)} r_{U}U

which is obvious. Thus i is a map of coefficient systems.

Similarly, for the map s, we must check that the square

Z\underline{E}(V) \underrightarrow{s_{V}} F (V)
Z\underline{E}_{V}^{W} ↓  ↓ F_{V}^{W}
Z\underline{E}(W) \overrightarrow{s_{W}} F (W)

is commutative. Equivalently, we must check that

(8.2)  F_{V}^{W} ∘ s_{V} (U) = s_{W} ∘ Z\underline{E}_{V}^{W} (U) ,

for any U ∈ \underline{E}(V). There are two cases : first if U ≱ W, then Z\underline{E}_{V}^{W} (U) = 0. But s_{V} (U) = Ω_{(N(V)/Ṽ)/(Ũ/Ṽ)}, where Ũ is any element of m(U) containing Ṽ. Thus F_{V}^{W} ∘

s_{V} (U) is equal to the image in F (W) of

Defres_{N(W)/W̃}^{N(V)/Ṽ} Ω_{(N(V)/Ṽ)/(Ũ/Ṽ)} ,

where W̃ is an element of m(W) containing Ṽ. This is equal to Ω_{Y}, where Y is the set of fixed points of the subgroup W̃ /Ṽ of N(V)/Ṽ on the set (N(V)/Ṽ )/(Ũ/Ṽ). This is zero unless W̃ /Ṽ is contained in Ũ/Ṽ up to conjugation by N(V)/Ṽ. In this case W̃ ≤ Ũ^{n}, for some n ∈ N(V), hence W = π(W̃) ≤ π(Ũ^{n}) = U. Hence both sides of equation 8.2 are zero if W ≰ U.

Suppose now that W ≤ U. Choose W̃ ∈ m(W) such that W̃ ≥ Ṽ, and Ũ ∈ m(U) such that Ũ ≥ W̃. Then Z\underline{E}_{V}^{W} (U) = U, and s_{W} ∘ Z\underline{E}_{V}^{W} (U) is equal to the image of Ω_{(N(W)/W̃)/(Ũ/W̃)} in F (W). On the other hand F_{V}^{W} ∘ s_{V} (U) is equal to the image in

F (W) of Ω_{Y}, where Y is the set of fixed points of the subgroup W̃ /Ṽ of N(V)/Ṽ on the set (N(V)/Ṽ )/(Ũ/Ṽ). But if W̃^{n} ≤ Ũ for some n ∈ N(V), then W̃^{n} = W̃. Hence Y = (N(W)/W̃ )/(Ũ/Ṽ), and equation 8.2 also holds in this case.

It remains to check that s ∘ i = 0, or equivalently that

∑_{U∈\underline{E}(V)} r_{U}Ω_{(N(V)/Ṽ)/(Ũ/Ṽ)} = 0 .

But this is relation 5.5 for the group N(V)/Ṽ.

In the proof of our main theorem, we will have to consider inverse limits of the above coefficient systems on the poset E, and we summarize the results we need in the following lemma :

Lemma 8.3 1. The group lim ; ⟵_{V ∈E} Z(V) is isomorphic to ℤ^{Γ}, where Γ is the set of

connected components of the poset E.

2. The group lim ; ⟵_{V ∈E} Z\underline{E}(V) is isomorphic to ℤ^{E}.

Proof: Assertion 1 is well known (and straightforward). Now for assertion 2 : an element of lim ; ⟵_{V ∈E} Z\underline{E}(V) is a sequence of elements l_{V} ∈ Z\underline{E}(V), for V ∈ E, such that

Z\underline{E}_{V}^{W} (l_{V}) = l_{W} whenever V ≤ W in E. If l_{V} = ∑_{U∈\underline{E}(V)} s_{V,U}U, for coefficients s_{V,U} ∈ ℤ, this gives

s_{V,U} = s_{W,U}

whenever V ≤ W ≤ U in E. Set t_{V} = s_{V,V}, for V ∈ E. Then l_{V} = ∑_{U∈\underline{E}(V)} t_{U}U, and the sequence (l_{V})_{V ∈E} is determined by the element l = ∑_{U∈E} t_{U}U. Conversely, if such an element l = ∑_{U∈E} t_{U}U is given, then setting l_{V} = ∑_{U∈\underline{E}(V)} t_{U}U for any V ∈ E gives an element of the inverse limit.

Notation 8.4 Let a : F ({0}) → lim ; ⟵_{V ∈E} F (V) be the map sending u ∈ F ({0}) to the

sequence (F_{\{0\}}^{V}(u))_{V ∈E}. Let b : ℤ\underline{E} = Z\underline{E}(\{0\}) → lim ; ⟵_{V ∈E} Z\underline{E}(V) ≅ ℤ^{E} denote the corre-

sponding map for the coefficient system Z\underline{E}, and let c : ℤ = Z(\{0\}) → lim ; ⟵_{V ∈E} Z(V) ≅ ℤ^{Γ}

the corresponding map for the constant coefficient system Z. Let t = lim ; ⟵_{V ∈E} s_{V} : ℤ\underline{E} →

lim ; ⟵_{V ∈E} F (V), and let j = lim ; ⟵_{V ∈E} i_{V} : ℤ^{Γ} ≅ lim ; ⟵_{V ∈E} Z → lim ; ⟵_{V ∈E} Z\underline{E}(V) ≅ ℤ^{E}.

It is easy to see that b : ℤ\underline{E} → ℤ^{E} is the canonical projection map corresponding to the inclusion E ⊆ \underline{E}, that c : ℤ → ℤ^{Γ} is the “diagonal” inclusion, and that j : ℤ^{Γ} → ℤ^{E} is the linear map sending the basis vector γ ∈ Γ to the sum ∑_{U∈γ} r_{U}U.

Proposition 8.5 The diagram

0 ⟶ ℤ ⟶^{i} ℤ\underline{E} ⟶^{s} F (\{0\})
c ↓  b ↓  a ↓
0 ⟶ ℤ^{Γ} ⟶^{j} ℤ^{E} ⟶^{t} lim ; ⟵_{V ∈E} F (V)

is commutative. Moreover s ∘ i = 0 and t ∘ j = 0.

Proof: This is a straightforward consequence of Lemma 8.1, since taking inverse limit is functorial with respect to coefficient systems.

9. The main theorem in the general case

Theorem 9.1 Let P be an (almost) extraspecial p-group. If p = 2 and if the ground field contains cubic roots of unity, assume moreover that P is not isomorphic to D_{8}^{∗n}∗Q_{8}. Then :

1. The sequence

0 → ℤ →^{i} ℤ\underline{E} →^{s} ∂D(P) → 0

is exact.

In other words the group ∂D(P) is generated by the elements Ω_{P/Ṽ}, for V ∈ \underline{E}, subject to the single relation

(9.2)  ∑_{V ∈\underline{E}} r_{V} Ω_{P/Ṽ} = 0 .

In particular, the group ∂D(P) is isomorphic to ℤ^{|E|} ⊕ ℤ/τℤ.

2. The group D(P) admits the following presentation : it is generated by the elements Ω_{P/Q}, where Q ⊇ Φ with |P : Q| ≥ 3, and the elements Ω_{P/Ṽ}, for V ∈ \underline{E}, subject to the above relation 9.2, together with the relations 2Ω_{P/Q} = 0 if |P : Q| = p.

Proof: Since D(P) = ∂D(P) ⊕ D(E), the second assertion follows from the first one, and from Dade’s Theorem giving generators and relations for the Dade group of the elementary abelian group E.

To prove Assertion 1, we proceed by induction on the order of P. The initial case of induction will be checked in section 11. Now let P be an (almost) extraspecial group, and assume that the theorem holds for all (almost) extraspecial groups of order strictly smaller than |P |. Assume further that E = \underline{E} − \{\{0\}\} is a connected poset.

With these assumptions, the restriction to E of the coefficient systems Z, Z\underline{E}, and F fit together in an exact sequence of coefficient systems

0 → Z →^{i} Z\underline{E} →^{s} F → 0 .

Since taking inverse limits of coefficients systems on E is a covariant left exact functor, we get a long exact sequence starting with

0 → lim ; ⟵_{V ∈E} Z(V) → lim ; ⟵_{V ∈E} Z\underline{E}(V) → lim ; ⟵_{V ∈E} F (V) →^{δ} H^{1}(E, Z) → …,

where δ is the connecting homomorphism of the long exact sequence of cohomology groups. Now lim ; ⟵_{V ∈E} Z(V) ≅ ℤ since E is a connected poset. Moreover lim ; ⟵_{V ∈E} Z\underline{E}(V) ≅ ℤ^{E}

by Lemma 8.3. By Proposition 8.5, the above exact sequence fits into the commutative diagram

0 ⟶ ℤ ⟶^{i} ℤ\underline{E} ⟶^{s} F (\{0\})
Id ↓  b ↓  a ↓
0 ⟶ ℤ ⟶^{j} ℤ^{E} ⟶^{t} lim ; ⟵_{V ∈E} F (V) ⟶^{δ} H^{1}(E, Z)

In particular the images of a ∘ s and t ∘ b are equal. Now the image of s is clearly equal to ∂D^{Ω}(P), and since b is surjective, it follows that

a(∂D^{Ω}(P)) = Im(t) = Ker(δ) .

This shows that the group L = a(∂D(P))/a(∂D^{Ω}(P)) maps injectively via δ into H^{1}(E, Z). But this group is a torsion-free abelian group by Proposition 6.1. On the other hand L is a factor group of ∂D(P)/∂D^{Ω}(P), which is a torsion group by Lemma 3.2. Hence L = {0}, and a(∂D(P)) = a(∂D^{Ω}(P)). Note that Ker(a) = T (P) ≤ ∂D^{Ω}(P). Indeed, we have x ∈ Ker(a) if and only if x ∈ ∂D(P) and 0 = a(x) = (F_{{0}}^{V}(x))_{V ∈E}. But, for any Ṽ ∈ m(V), the restriction of F_{{0}}^{V}(x) to the component ∂D(N(V)/Ṽ) is the element Defres_{N(V)/Ṽ}^{P} (x). Hence

x ∈ Ker(a) if and only if x ∈ ∂D(P) ∩ ⋂_{V ∈E} Ker(Defres_{N(V)/Ṽ}^{P} ),

for any choice of subgroups Ṽ ∈ m(V). That is

Ker(a) = ⋂_{1<Q≤P} Ker(Defres_{N_{P} (Q)/Q}^{P}) = T (P),

by Lemma 3.2. Moreover, our assumption on the set E implies that any maximal elementary abelian subgroup of P has rank at least 3, and so T (P) = <Ω_{P} >, by Lemma 3.2. Hence Ker(a) = T (P) ≤ ∂D^{Ω}(P), as asserted.

It follows that ∂D(P) = ∂D^{Ω}(P), showing that the map s is surjective.

Now let u = ∑_{U∈\underline{E}} n_{U}U ∈ Ker(s), for coefficients n_{U} ∈ ℤ. Then a∘s(u) = t∘b(u) = 0, thus b(u) ∈ Ker(t) = Im(j). It follows that there is an integer m such that n_{U} = mr_{U}, for any U ≠ {0}. Then

s(u) = n_{{0}}Ω_{P/1} + m ∑_{U∈E} r_{U}Ω_{P/Ũ}
= n_{{0}}Ω_{P/1} − mr_{{0}}Ω_{P/1}
= (n_{{0}} − mr_{{0}})Ω_{P/1} = 0

It follows that n_{{0}} = mr_{{0}}, since Ω_{P/1} has infinite order in D(P). Hence u = i(m), showing that Ker(s) = Im(i), and completing the proof of the first assertion of the theorem.

The second assertion is just a reformulation of the first one. The last assertion follows from the fact that the g.c.d. of the coefficients r_{V}, for V ∈ \underline{E}, is equal to τ, since actually r_{V_{0}} = τ, for any maximal element V_{0} of \underline{E}.

10. The main theorem in the exceptional case

In this section we suppose that p = 2 and that P is isomorphic to D_{8}^{n} ∗ Q_{8}, for n ≥ 2, and that the ground field contains cubic roots of unity. The remaining ”small” cases n ≤ 1 will be handled in section 11. The case where the ground field does not contain cubic roots of unity is part of Theorem 9.1.

In this situation, let Q_{0} be a maximal element of Q. Then C_{P} (Q_{0})/Q_{0} is isomorphic to the quaternion group Q_{8}. Now by Theorem 6.3 of [5], the group D(Q_{8}) is isomorphic to ℤ/4ℤ ⊕ ℤ/2ℤ. There are exactly three elements of order 2 in D(Q_{8}), one of which is equal to 2Ω_{Q_{8}/1}. The two others are equal to M and M′ = M + 2Ω_{Q_{8}/1}, where M and M′ denote the images in D(Q_{8}) of exceptional endo-trivial modules of dimension 5, with the same names. Note that M and M′ depend on the choice of cubic roots in the field of coefficients : they are permuted by any endomorphism of the ground field exchanging these two non trivial roots, so there is no canonical way to specify which of them should be called M and which should be called M′.

Notation 10.1 Choose any maximal element Q_{0} in Q, and choose one element M of order 2 in D(C_{P} (Q_{0})/Q_{0}) − D^{Ω}(C_{P} (Q_{0})/Q_{0}). Set

u_{0} = Teninf_{C_{P} (Q_{0})/Q_{0}}^{P}M .

This is an element of ∂D(P) = F ({0}). An element of ∂D(P) which can be obtained by this construction will be called an exotic element of ∂D(P).

If V ∈ \underline{E}, set

u_{V} = F_{{0}}^{V}(u_{0}) ,

so that with this notation u_{{0}} = u_{0}.

Lemma 10.2 Let Q be any element of Q. Then Defres_{C_{P} (Q)/Q}^{P}u_{0} is an exotic element in D(C_{P} (Q)/Q).

Proof: Set R = C_{P} (Q_{0})/Q_{0} ≅ Q_{8}. Recall that M ∈ D(R) is the class of a critical endo-trivial module in the sense of [6] Definition 5.1 : this means that Res_{H}^{R}M = 0 for all proper subgroups H of R, and that Def_{R/N}^{R}M = 0 for any non-trivial normal subgroup N of R.

Now setting C = C_{P} (Q) and C_{0} = C_{P} (Q_{0}), we have that

Res_{C_{P} (Q)}^{P}u_{0} = Res_{C}^{P}Ten_{C_{0}}^{P}Inf_{C_{0}/Q_{0}}^{C_{0}}M
= ∑_{x∈C\P/C_{0}}Ten_{C∩^{x}C_{0}}^{C}^{x} (Res_{C^{x}∩C_{0}}^{C_{0}}Inf_{C_{0}/Q_{0}}^{C_{0}}M) ,

by the Mackey formula. Now

Res_{C^{x}∩C_{0}}^{C_{0}}Inf_{C_{0}/Q_{0}}^{C_{0}}M = Inf_{(C^{x}∩C_{0})/(C^{x}∩Q_{0})}^{C^{x}∩C_{0}}Iso_{(C^{x}∩C_{0})·Q_{0}/Q_{0}}^{(C^{x}∩C_{0})/(C^{x}∩Q_{0})}Res_{(C^{x}∩C_{0})·Q_{0}/Q_{0}}^{C_{0}/Q_{0}}M ,

where Iso_{(C^{x}∩C_{0})·Q_{0}/Q_{0}}^{(C^{x}∩C_{0})/(C^{x}∩Q_{0})} denotes the transfer by the canonical isomorphism

(C^{x} ∩ C_{0})·Q_{0}/Q_{0} → (C^{x} ∩ C_{0})/(C^{x} ∩ Q_{0}) .

But Res_{(C^{x}∩C_{0})·Q_{0}/Q_{0}}^{C_{0}/Q_{0}}M = 0, unless (C^{x} ∩ C_{0})·Q_{0} = C_{0}. Hence

Res_{C_{P} (Q)}^{P}u_{0} = ∑_{{x∈C\P/C_{0} \atop (C^{x}∩C_{0})·Q_{0}=C_{0}}} Ten_{C∩^{x}C_{0}}^{C}^{x} (Inf_{(C^{x}∩C_{0})/(C^{x}∩Q_{0})}^{C^{x}∩C_{0}}Iso_{C_{0}/Q_{0}}^{(C^{x}∩C_{0})/(C^{x}∩Q_{0})}M) .

Set M_{x} = Iso_{C_{0}/Q_{0}}^{(C^{x}∩C_{0})/(C^{x}∩Q_{0})}M. This is a critical element in D((C^{x} ∩ C_{0})/(C^{x} ∩ Q_{0})).

Set moreover N_{x} = ^{x} (Inf_{(C^{x}∩C_{0})/(C^{x}∩Q_{0})}^{C^{x}∩C_{0}}M_{x}). We get

Res_{C_{P} (Q)}^{P}u_{0} = ∑_{{x∈C\P/C_{0} \atop (C^{x}∩C_{0})·Q_{0}=C_{0}}} Ten_{C∩^{x}C_{0}}^{C}^{x}N_{x} .

Now by Proposition 3.10 of [4], we have that

Def_{C/Q}^{C}Res_{C}^{P}u_{0} = ∑_{{x∈C\P/C_{0} \atop (C^{x}∩C_{0})·Q_{0}=C_{0}}} L_{x} ,

where

L_{x} = γ_{|Q:Q∩^{x}C_{0}|}Ten_{(C∩^{x}C_{0})·Q/Q}^{C/Q}Iso_{(C∩^{x}C_{0})/(Q∩^{x}C_{0})}^{(C∩^{x}C_{0})·Q/Q}Def_{(C∩^{x}C_{0})/(Q∩^{x}C_{0})}^{C∩^{x}C_{0}}N_{x} ,

and γ_{|Q:Q∩^{x}C_{0}|} denotes the effect on the Dade group of raising the scalars of the ground field to the power |Q : Q ∩^{x}C_{0}|. Now

Def_{(C∩^{x}C_{0})/(Q∩^{x}C_{0})}^{C∩^{x}C_{0}}N_{x} = ^{x} (Def_{(C^{x}∩C_{0})/(Q^{x}∩C_{0})}^{C^{x}∩C_{0}}Inf_{(C^{x}∩C_{0})/(C^{x}∩Q_{0})}^{C^{x}∩C_{0}}M_{x})
= ^{x} (Inf_{S_{x}}^{(C^{x}∩C_{0})/(Q^{x}∩C_{0})}Def_{S_{x}}^{(C^{x}∩C_{0})/(C^{x}∩Q_{0})}M_{x}) ,

where S_{x} = (C^{x} ∩ C_{0})/(Q^{x} ∩ C_{0})(C^{x} ∩ Q_{0}). Since M_{x} is critical, this is zero unless Q^{x}∩C_{0} ≤ C^{x}∩Q_{0}. By Corollary 4.2, there is a unique double coset CxC_{0} in P such that Q^{x} ∩ C_{0} ≤ Q_{0}, and for such a double coset, by Lemma 4.1, we have (C^{x} ∩ C_{0})·Q_{0} = C_{0}. For such an element x, we have that

L_{x} = γ_{|Q:Q∩^{x}C_{0}|}Ten_{(C∩^{x}C_{0})·Q/Q}^{C/Q}Iso_{(C∩^{x}C_{0})/(Q∩^{x}C_{0})}^{(C∩^{x}C_{0})·Q/Q}Inf_{(C^{x}∩C_{0})/(C^{x}∩Q_{0})}^{(C^{x}∩C_{0})/(Q^{x}∩C_{0})} ^{x}M_{x}
= γ_{|Q:Q∩^{x}C_{0}|}Ten_{(C∩^{x}C_{0})·Q/Q}^{C/Q}Inf_{(C∩^{x}C_{0})·Q/(C∩^{x}Q_{0})·Q}^{(C∩^{x}C_{0})·Q/Q}Iso_{(C^{x}∩C_{0})/(C^{x}∩Q_{0})}^{(C∩^{x}C_{0})·Q/(C∩^{x}Q_{0})·Q} ^{x}M_{x}

Finally, we get

Def_{C/Q}^{C}Res_{C}^{P}u_{0} = γ_{|Q:Q∩^{x}C_{0}|}Ten_{(C∩^{x}C_{0})·Q/Q}^{C/Q}Iso_{(C∩^{x}C_{0})/(Q∩^{x}C_{0})}^{(C∩^{x}C_{0})·Q/Q}^{x}M_{x}
= γ_{|Q:Q∩^{x}C_{0}|}Ten_{(C∩^{x}C_{0})·Q/Q}^{C/Q}Iso_{(C∩^{x}C_{0})/(Q∩^{x}C_{0})}^{(C∩^{x}C_{0})·Q/Q}Iso_{^{x}C_{0}/^{x}Q_{0}}^{(C∩^{x}C_{0})/(C∩^{x}Q_{0})} ^{x}M

Now the group Q_{1} = (C ∩ ^{x}Q_{0})·Q is a maximal element of \underline{Q} by Corollary 4.2, thus \overline{Q}_{1} = Q_{1}/Q is a maximal element of \underline{Q}(\overline{C}), where \underline{Q}(\overline{C}) is the set of subgroups of \overline{C} = C_{P} (Q)/Q which intersect trivially the Frattini subgroup of this (almost) extraspecial group. Moreover

C_{\overline{C}}(\overline{Q}_{1}) = N_{\overline{C}}(\overline{Q}_{1}) = N_{C_{P} (Q)/Q}(Q_{1}/Q) = N_{C_{P} (Q)}(Q_{1})/Q = C_{P} (Q_{1})/Q .

Moreover C_{P} (Q_{1}) = (C ∩ ^{x}C_{0})·Q, again by Lemma 4.1. Since γ_{|Q:Q∩^{x}C_{0}|} commutes with tensor induction and inflation (by Lemma 3.2 of [4]), it follows that

Def_{C/Q}^{C}Res_{C}^{P}u_{0} = Ten\overline{C} ; C_{\overline{C}}(\overline{Q}_{1})Inf_{C_{\overline{C}}(\overline{Q}_{1})/\overline{Q}_{1}}^{C_{\overline{C}}(\overline{Q}_{1})} \overline{M} ,

where \overline{M} = γ_{|Q:Q∩^{x}C_{0}|}Iso_{C_{0}^{x}/Q_{0}^{x}}^{C_{\overline{C}}(\overline{Q}_{1})/\overline{Q}_{1}}M^{x}.

But \overline{M} is an element of order 2 in D(C_{\overline{C}}(\overline{Q}_{1})/\overline{Q}_{1})−D^{Ω}(C_{\overline{C}}(\overline{Q}_{1})/\overline{Q}_{1}), since it is the image of M by a group isomorphism followed by a Galois twist. Thus Def_{C/Q}^{C}Res_{C}^{P}u_{0} is an exotic element in D(C_{\overline{C}}(\overline{Q}_{1})/\overline{Q}_{1}), as was to be shown.

Let s′_{u_{0}} be the linear map from ℤ to ∂D(P) defined by s′_{u_{0}}(1) = u_{0}, and let i′ : ℤ → ℤ be multiplication by 2. Since 2u_{0} = 0 in D(P), it follows that s′_{u_{0}} ∘ i′ = 0.

Theorem 10.3 Let k be a field of characteristic 2, containing all cubic roots of unity, and let P ≅ D_{8}^{n} ∗ Q_{8}. Let u_{0} be any exotic element in D(P). Then

1. The sequence

0 ⟶ ℤ ⊕ ℤ ⟶^{i⊕i′} ℤ\underline{E} ⊕ ℤ ⟶^{s+s′_{u_{0}}} ∂D(P) ⟶ 0

is exact.

In other words the group ∂D(P) is generated by u_{0} and by the elements Ω_{P/Ṽ}, for V ∈ \underline{E}, subject to the relations

(10.4)  2u_{0} = 0  ∑_{V ∈\underline{E}} r_{V} Ω_{P/Ṽ} = 0 .

In particular the group ∂D(P) is isomorphic to ℤ^{|E|} ⊕ ℤ/4ℤ ⊕ ℤ/2ℤ.

2. The group D(P) admits the following presentation : it is generated by the elements Ω_{P/Q}, where Q ⊇ Φ with |P : Q| ≥ 4, and the elements u_{0} and Ω_{P/Ṽ}, for V ∈ \underline{E}, subject to the above relations 10.4.

Proof: Here again, Assertion 2 follows from Assertion 1 and Dade’s Theorem. We prove Assertion 1 by induction on n ≥ 2. We will check that the result holds for n ≤ 1 in section 11.

By induction hypothesis, and by Lemma 10.2, for any V ∈ E, we have an exact sequence

0 ⟶ ℤ ⊕ ℤ ⟶^{i_{V} ⊕i′_{V}} ℤ^{\underline{E}(V)} ⊕ ℤ ⟶^{s_{V} +s′_{u_{V}}} F (V) ⟶ 0 ,

where i_{V} and s_{V} are defined in section 8, where i′_{V} is multiplication by 2, and s′_{u_{V}} : ℤ → F (V) is obtained from the exotic element u_{V}, after identification F (V) ≅ ∂D(N(V)/Ṽ ).

By construction, these maps actually define an exact sequence of coefficient systems on E

0 ⟶ Z ⊕ Z ⟶^{i⊕i′} Z\underline{E} ⊕ Z ⟶^{s+s′_{u_{0}}} F ⟶ 0 ,

and we can take inverse limits on E. Since n ≥ 2, the poset E is connected, and we get a long exact sequence of cohomology groups starting with

0 ⟶ ℤ ⊕ ℤ ⟶^{j⊕j′} ℤ^{E} ⊕ ℤ ⟶^{t+t′_{u_{0}}} lim ; ⟵_{V ∈E} F (V) ⟶^{δ} H^{1}(E, Z) ,

where δ is the connecting homomorphism. As in the general case, we get a commutative diagram

0 ⟶ ℤ ⊕ ℤ ⟶^{i⊕i′} ℤ\underline{E} ⊕ ℤ ⟶^{s+s′_{u_{0}}} F (\{0\})
Id⊕Id ↓  b⊕Id ↓  a ↓
0 ⟶ ℤ ⊕ ℤ ⟶^{j⊕i′} ℤ^{E} ⊕ ℤ ⟶^{t+t′_{u_{0}}}  lim ; ⟵_{V ∈E} F (V) ⟶^{δ} H^{1}(E, Z) ,

where the top line is a complex of abelian groups, and the bottom line is an exact sequence. Now setting S = s + s′_{u_{0}}, T = t + t′_{u_{0}}, and B = b ⊕ Id, we have a ∘ S = T ∘ B, thus Im(a ∘ S) = Im(T ∘ B). But obviously Im(S) ⊇ ∂D^{Ω}(P), which has finite index in ∂D(P) = F ({0}). Hence the group a(F ({0}))/Im(a ∘ S) is finite. On the other hand Im(a ∘ S) = Im(T ∘ B) = Im(T) = Ker(δ) since B is surjective.

Hence a(F ({0}))/Im(a ∘ S) maps injectively by δ into H^{1}(E, ℤ), which is a torsionfree group by Proposition 6.1. It follows that a(F ({0})) = Im(a ∘ S), hence F ({0}) = Im(S) + Ker(a). But Ker(a) = T (P) ⊆ ∂D^{Ω}(P) ⊆ Im(S). Thus F ({0}) = Im(S), and S is surjective.

Now let (u, m) ∈ Ker(S), for u = ∑_{U∈\underline{E}} n_{U}U ∈ ℤ\underline{E} and m ∈ ℤ. Thus

∑_{U∈\underline{E}} n_{U}Ω_{P/Ũ} + mu_{0} = 0 .

Applying Defres_{C(Q_{0})/Q_{0}}^{P} to both sides gives

n_{U_{0}}Ω_{C(Q_{0})/Q_{0}} + mM = 0 ,

for any maximal element U_{0} of E. It follows that m is even, i.e. m ∈ Im(i′), and that n_{U_{0}} is a multiple of 4. Note that 4 = τ = r_{U_{0}} in this case. Now since 2u_{0} = 0, we have

∑_{V ∈\underline{E}} n_{U}Ω_{P/Ũ} = 0 .

It follows that b(u) ∈ Ker(t) = Im(i), and there exists an integer n such that n_{U} = nr_{U}, for any U ∈ E. Hence

0 = ∑_{V ∈\underline{E}} n_{U}Ω_{P/Ũ} = n_{\{0\}}Ω_{P/1} + ∑_{V ∈E} nr_{U}Ω_{P/Ũ} = (n_{\{0\}} − nr_{\{0\}})Ω_{P/1} ,

hence n_{{0}} = nr_{{0}} since Ω_{P/1} has infinite order in D(P). But now u = i(n), and (u, n) ∈ Im(i ⊕ i′), as was to be shown. This completes the proof of the theorem.

11. The main theorem in the initial cases

The aim of this paper consists in computing the Dade group of all (almost) extraspecial p-groups P for any prime number p. Since the proof of the main result of the previous section proceeds on induction on the order of P, we treat the anchoring points of the induction in a section apart, for the sake of clarity. Let us start by saying which groups constitute the “initial cases”. In the hypothesis of the main theorem, we include the condition that the set E is connected. The reason for this lies in the two following facts. The first one concerns the part of the proof which is based on the structure of T (P). Indeed, for all (almost) extraspecial p-groups, except those of type T 2 when p = 2, the group T (P) turns out to be an infinite cyclic group, generated by Ω_{P}, in case the set E is connected, and hence, it is contained in ∂D^{Ω}(P). Secondly, this assumption on E allows us to apply the cohomological argument that we recalled in section 2, and that we used in order to finish the proof. Recall that the group T (P) has no torsion for an (almost) extraspecial p-group P of order at least 16, by Lemma 3.2. In this case, T (P) is a free abelian group whose rank depends only on the number of maximal elementary abelian subgroups of P of rank 2, that is, on the number of maximal elements of E of dimension 1. It is an easy exercise to calculate this number for each (almost) extraspecial p-group, and so we come to the conclusion that E is not connected if

P ∈ {M, M ∗ C_{p^{2}}, D_{8}, Q_{8}, D_{8} ∗ Q_{8}, D_{8} ∗ C_{4}} ,

in the notation of section 2, that is M is the extraspecial p-group of order p^{3} and exponent p, for an odd prime number p. Indeed, in these cases, the set E is formed respectively by p+1, p+1, 2, 0, 5, and 3 lines exactly. So, for each of these 6 groups we have to check that the theorem of previous section holds, that is, we must prove that the group

∂D(P) = ∂D^{Ω}(P) ≅ ℤ^{|E|} ⊕ ℤ/τℤ

is generated by the set \{Ω_{P/Ṽ} | Ṽ ∈ m(V), V ∈ \underline{E}\}, subject to the single relation

∑_{V ∈\underline{E}} r_{V} Ω_{P/Ṽ} = 0, except maybe in the case p = 2 and the ground field contains a cubic

root of unity, and P has type T 2. Let us also recall that the integer τ is defined as being the order of Ω_{C_{P} (Q)/Q} in the group D(C_{P} (Q)/Q) for a subgroup Q of P such that π(Q) is a maximal element in \underline{E}. In other words, we have

τ = \{ \begin{matrix}1 , & if P = D_{8} \\ 2 , & if P = D_{8} ∗ C_{4}, or if p is odd \\ 4 , & if P = Q_{8}, or if P = D_{8} ∗ Q_{8} . \end{matrix}

Moreover, by definition, we have r_{V} = τ· ( ∑_{{S∈[s_{P}]∩\underline{Q} \atop S≥_{P} Ṽ}} μ(Ṽ , S)\frac{|Q|}{|S|}) . Hence,

r_{{0}} = (p − |E|)·τ and r_{V} = τ , ∀ V ∈ E .

Consider now each “initial case” separately :

• If P has order 8, then the Dade group of P has been computed in [5]. Comparing Proposition 3.4 with Lemma 10.2 in [5], we deduce that the subgroup ∂D(P) coincides with T (P). Then, Theorem 5.4 and Theorem 6.4 in [5] prove the results we want to show, for the dihedral group of order 8, and, respectively, for the quaternion group.

• Let P be one of the groups M, M ∗ C_{p^{2}}, or D_{8} ∗ C_{4}. Then we know by [7] that T (P) = T^{Ω}(P), and hence T (P) ≤ ∂D^{Ω}(P). In this case, the set E is formed by lines, i.e. it is in bijection with the elementary abelian subgroups of P of rank 2, which are then all maximal in P. In other words, the torsion-free rank of ∂D(P) is equal to |E|, and it is also equal to the rank of the free abelian subgroup T (P) of ∂D(P). Hence the quotient ∂D(P)/T (P) is finite. Consider the map

α = ∏_{V ∈E} Defres_{N(V)/Ṽ}^{P} : ∂D(P) ⟶ ∏_{V ∈E} D(N(V)/Ṽ ) .

The quotients N(V)/Ṽ are all cyclic groups of order at least 3, isomorphic to the center Z of P, and thus D(N(V)/Ṽ ) = <Ω_{N(V)/Ṽ} > ≅ ℤ/2ℤ , ∀ V ∈ E. Moreover,

Ker(α) = ∂D(P) ∩ ( ⋂_{V ∈E} Ker(Defres_{N(V)/Ṽ}^{P} )) = T (P) .

On the other side, since

Defres_{N(V)/Ṽ}^{P} Ω_{P/Ũ} = δ_{U,V} ·Ω_{N(V)/Ṽ} , ∀ U, V ∈ E ,

we have α(∂D^{Ω}(P)) = ∏_{V ∈E} D(N(V)/Ṽ ). Therefore, we finally get the following com-

mutative diagram with exact rows (where i denotes the inclusion), proving the equality ∂D(P) = ∂D^{Ω}(P), as required.

0 ⟶ T (P) ⟶ ∂D^{Ω}(P) ⟶^{α} ∏_{V ∈E} D(N(V)/Ṽ) ⟶ 0
Id ↓  i ↓  Id ↓
0 ⟶ T (P) ⟶ ∂D(P) ⟶^{α} ∏_{V ∈E} D(N(V)/Ṽ) ⟶ 0

Notice also that the relation 5.2 gives in this case

2·Ω_{P} = 2· ∑_{V ∈E} Ω_{P/Ṽ} ,

but Ω_{P} − ∑_{V ∈E} Ω_{P/Ṽ} ≠ 0, since Ω_{P} − ∑_{V ∈E} Ω_{P/Ṽ} ∉ T (P). Hence, D^{t}(P) is a cyclic group

of order 2, generated by Ω_{P} − ∑_{V ∈E} Ω_{P/Ṽ}. Thus, the statement of the main theorem also holds in these three “initial” cases.

• Finally, consider the case where P = D_{8} ∗ Q_{8}, and the ground field k contains a cubic root of unity. Then E consists in 5 lines, and the group N(V)/Ṽ is a quaternion group of order 8 for each subspace V ∈ E. Hence, we have an exact sequence

0 ⟶ T_{k}(P) ⟶^{i} ∂D_{k}(P) ⟶^{α_{k}} ∏_{V ∈E} T_{k}(N(V)/Ṽ ),

where i denotes the inclusion and where α_{k} is the map

∏_{V ∈E} Defres_{N(V)/Ṽ}^{P} : ∂D_{k}(P) ⟶ ∏_{V ∈E} T_{k}(N(V)/Ṽ ) ,

which is well defined, since we have ∂D_{k}(N(V)/Ṽ ) = T_{k}(N(V)/Ṽ ), ∀ V ∈ E. Since the results depend a priori on the ground field k, we have specified which Dade groups we are dealing with by subscripts.

The group ∂D_{k}(N(V)/Ṽ ) is isomorphic to ℤ/4ℤ⊕ℤ/2ℤ. It is generated by Ω_{N(V)/Ṽ}, of order 4, and by an exotic element Λ_{V}, of order 2 (see sections 6 and 10 in [5]). There are exactly two exotic elements in D_{k}(N(V)/Ṽ ), and the other one is equal to Λ_{V} + 2Ω_{N(V)/Ṽ}.

The group T_{k}(P) = Ker(α_{k}) is a free abelian group of rank 5, and it is generated by Ω_{P} and the five elements 4Ω_{P/Ṽ} , V ∈ E. Indeed 4x ∈ T_{k}(P), ∀ x ∈ ∂D_{k}(P),

since ∏_{V ∈E} T_{k}(N(V)/Ṽ ) has exponent 4. In order to show that these elements generate

the whole group of endo-trivial modules, we need the less obvious results proved in [7]. Consider now V ∈ E and an elementary abelian subgroup A of P of rank 2. That is, there exists U_{A} ∈ E such that A = π^{−1}(U_{A}). Let Ṽ ∈ m(V), and let us determine the element Res_{A}^{P}Ω_{P/Ṽ} of D_{k}(A) = T_{k}(A). We have Res_{A}^{P}Ω_{P/Ṽ} = Ω_{X}, where X is the A-set Res_{A}^{P}(P/Ṽ). Since A is normal in P, the Mackey formula gives us

X = ∐_{g∈[P/Ṽ ·A]} A/A ∩^{g}Ṽ = \{ \begin{matrix}4·A , & if V ≠ U_{A} \\ 2·(A/Ṽ) ∐ 2·(A/^{g}Ṽ) , & if V = U_{A} , \end{matrix}

for any g ∈ P such that ^{g}Ṽ ≠ Ṽ. It follows that

Res_{A}^{P}Ω_{P/Ṽ} = (−1)^{δ_{V,U_{A}}} Ω_{A}.

From this computation, and using some elementary linear algebra, we deduce the relationship between the six elements of T_{k}(P) described above, unique up to multiplication by a scalar. Namely, we get:

−12Ω_{P} + ∑_{V ∈E} 4Ω_{P/Ṽ} = 0 .

Notice that this equality corresponds to the one given in section 5.2. Indeed, in the present case τ = 4 and the maximal subgroups in Q have order 2. Thus, r_{V} = τ = 4 for any V ∈ E and

r_{{0}} = 4(μ(1, 1)· | Q_{0} | + ∑_{V ∈E} μ(1, Ṽ)·1) = 4(2 + 5·(−1)) = −12 .

We now want to show that the image of the map α_{k} is equal to the inverse image Δ_{k} under the canonical surjection

∏_{V ∈E} T_{k}(N(V)/Ṽ ) ⟶ ∏_{V ∈E} T_{k}(N(V)/Ṽ )/T_{k}^{Ω}(N(V)/Ṽ )

of the diagonal subgroup of order 2 of the quotient group (which is isomorphic to (ℤ/2ℤ)^{5}). To prove this, we can make the following assumption on k :

Assumption 11.1 There exists and 𝔽_{2}-endomorphism γ of k exchanging the nontrivial cubic roots of unity.

Indeed there exists a subfield l of k such that the extension l/𝔽_{2} is purely transcendental and the extension k/l is algebraic (such an l can be obtained by considering the subfield of k generated by a transcendence basis of k over 𝔽_{2}). Let L denote an algebraic closure of l containing k, and let f denote the subfield of k generated by l and the cubic roots of unity. Then there is an l-isomorphism γ of f exchanging the cubic roots of unity, and such a γ can be extended to an l-isomorphism of L. Hence the assumption holds for the fields f and L. Now we have a commutative diagram

0 ⟶ T_{L}(P) ⟶ ∂D_{L}(P) ⟶^{α_{L}}  ∏_{V ∈E} T_{L}(N(V)/Ṽ )
↑  ↑  ↑
0 ⟶ T_{k}(P) ⟶ ∂D_{k}(P) ⟶^{α_{k}}  ∏_{V ∈E} T_{k}(N(V)/Ṽ )
↑  ↑  ↑
0 ⟶ T_{f}(P) ⟶ ∂D_{f}(P) ⟶^{α_{f}}  ∏_{V ∈E} T_{f}(N(V)/Ṽ )

where the vertical arrows are given by scalar extension from f to k and from k to L. The left hand side vertical arrows are isomorphisms, since the three groups of endotrivial modules have the same generators by the above computation of T_{k}(P). The right hand side vertical arrows are also isomorphisms, by the above description of the group T_{K}(Q_{8}), for a field K containing cubic roots of unity. If we know that Im(α_{L}) = Δ_{L} and Im(α_{f}) = Δ_{f}, then it follows that Im(α_{k}) = Δ_{k}.

Thus we can suppose that Assumption 11.1 holds for k. Then the endomorphism γ acts on the Dade group of any 2-group P over k. In particular, it maps the element Λ_{V} of D(N(V)/Ṽ ) to the element Λ_{V} + 2Ω_{N(V)/Ṽ}.

Now, the equalities Defres_{N(V)/Ṽ}^{P} (Ω_{P/Ũ}) = δ_{V,U}Ω_{N(V)/Ṽ} , ∀ U, V ∈ E imply that

α(∂D^{Ω}(P)) = ∏_{V ∈E} T^{Ω}(N(V)/Ṽ ) .

Set Λ = Teninf_{N(U)/Ũ}^{P}Λ_{U}, for an arbitrary U ∈ E and an arbitrary exotic element

Λ_{U} in D(N(U)/Ũ). Then, Λ ∈ ∂D(P) and Defres_{N(V)/Ṽ}^{P} Λ is an exotic element of D(N(V)/Ṽ ), ∀ V ∈ E, by Lemma 10.2. It follows that the image of α_{k}(Λ) by the surjection

∏_{V ∈E} T_{k}(N(V)/Ṽ ) ⟶ ∏_{V ∈E} T (N(V)/Ṽ )/T_{k}^{Ω}(N(V)/Ṽ )

generates a diagonal subgroup of order 2 in the quotient group. Hence Im(α_{k}) ⊇ Δ_{k}.

Conversely, suppose that u is an element of ∂D_{k}(P). For each V ∈ E, fix an exotic element λ_{V} of D(N(V)/Ṽ), and set ω_{V} = Ω_{N(V)/Ṽ} for short. Then

α_{k}(u) = (a_{V} ω_{V} + x_{V} λ_{V} )_{V ∈E} ,

for uniquely defined coefficients a_{V} ∈ ℤ/4ℤ and x_{V} ∈ ℤ/2ℤ. Since γ and α_{k} commute, it follows that

α_{k}(γ(u)) = ((a_{V} + 2x_{V})ω_{V} + x_{V} λ_{V} )_{V ∈E} ,

Set v = γ(u) − u − ∑_{V ∈E} 2x_{V} Ω_{P/Ṽ}. Then α_{k}(v) = 0, hence v ∈ T (P), and so there are (non unique) integers m_{V}, for V ∈ E, and a (non unique) integer n such that

(11.2)  v = nΩ_{P/1} + ∑_{V ∈E} 4m_{V} Ω_{P/Ṽ} .

Let U be any element of E. By restriction to A = π^{−1}(U), which is an elementary abelian subgroup of rank 2 of P, equality 11.2 gives

Res_{A}^{P}v = (n − 4m_{U} + 4 ∑_{V ≠U} m_{V})Ω_{A/1} .

On the other hand, since γ commutes with restriction, and since γ acts trivially on D_{k}(A), it follows that Res_{A}^{P}(γ(u) − u) = 0, and the definition of v gives

Res_{A}^{P}v = (2x_{U} − ∑_{V ≠U} 2x_{V})Ω_{A/1} .

Since Ω_{A/1} has infinite order in D_{k}(A), it follows that

2x_{U} − ∑_{V ≠U} 2x_{V} = n − 4m_{U} + 4 ∑_{V ≠U} m_{V} .

Setting x = ∑_{V ∈E} x_{V} and m = ∑_{V ∈E} m_{V}, this can also be written as follows

4x_{U} − 2x = n − 8m_{U} + 4m ,

which gives 4x_{U} + 8m_{U} = n + 2x + 4m, which is independent of U. This shows in particular that for any U and U′ in E, we have 4x_{U} ≡ 4x_{U′} modulo 8. Thus x_{U} ≡ x_{U′} modulo 2, showing that α_{k}(u) ∈ Δ_{k}.

Finally, we conclude that the group ∂D_{k}(P) is isomorphic to ℤ^{5} ⊕ ℤ/4ℤ ⊕ ℤ/2ℤ, and that it is generated by the seven elements Ω_{P} , Ω_{P/Ṽ} , V ∈ E and Λ, submitted to the two following relations

2Λ = 0 and − 12Ω_{P} + ∑_{V ∈E} 4Ω_{P/Ṽ} = 0 .

This ends the proof of the main theorem in the initial cases.

Acknowledgements : We wish to thank Jacques Thévenaz for useful comments on this paper, and for pointing out a gap in our initial proof of the case P = D_{8} ∗ Q_{8}. We also thank Jon Carlson, who solved first the case of extraspecial groups of order p^{3}.

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Serge Bouc, Institut de Mathématiques, Université Paris 7-Denis Diderot, 2 Place Jussieu, 75251 - Paris Cedex 05. FRANCE.

E-mail :  bouc@math.jussieu.fr

Nadia Mazza, Department of Mathematics, University of Georgia, Athens, Georgia 30602, USA.

E-mail :  nmazza@math.uga.edu