THE DADE GROUP OF A FUSION SYSTEM

MARKUS LINCKELMANN AND NADIA MAZZA

Abstract. We define a notion of a Dade group of a fusion system and show that some of the gluing and detection results for Dade groups of finite p-groups due to Bouc and Thévenaz in [8],

[9] extend to Dade groups of fusion systems.

1. Introduction

The Dade group D(P, F) of a fusion system F on a finite p-group P, for p a prime, is defined as the subgroup of F-stable elements in the Dade group D(P) of P. The motivation for considering this notion comes from block theory: any choice of a maximal b-Brauer pair (P, e) of a block b of a finite group G over an algebraically closed field k of characteristic p determines a fusion system F on P - see e.g. [16] for details. Moreover, for any subgroup Q of P there is by [1] a unique block e_{Q} of kC_{G}(Q) such that (Q, e_{Q}) ⊆ (P, e). Then, for any pair (R, Q) of subgroups of Q, R of P such that Q is normal in R and such that the block e_{Q} is a nilpotent block in the sense of [3], the unique (up to isomorphism) simple kC_{G}(Q)e_{Q}-module V_{Q} is invariant under R by the uniqueness of inclusion of Brauer pairs, and hence becomes an endo-permutation kR/Q-module. In this way, any block gives rise to a fusion compatible family of endo-permutation modules for all sections R/Q of P for which the block e_{Q} is nilpotent. One of the questions, first raised by Puig in [21], is whether this family can be “glued together” to an endo-permutation kP -module V such that Defres_{R/Q}^{P}(V) is equal to V_{Q} in the Dade group D(R/Q) for any section R/Q of P as above. Following [21], this gluing problem has an affirmative answer if P is abelian, which in turn is used to show the existence of a stable equivalence of Morita type between the block and its Brauer correspondent under the assumption that the inertial quotient acts freely on the non trivial elements of P. In [8], Bouc and Thévenaz gave a more general criterion for when the gluing problem has a solution provided that the involved endo-permutation modules are torsion in their respective Dade groups and p is an odd prime. The purpose of the present paper is to investigate compatibility issues with respect to an arbitrary fusion system F on a finite p-group P. After a brief review on fusion systems in §2, we define in §3 the Dade group D(P, F) of a fusion system F on a finite p-group P and relate this to the definition of endo-p-permutation modules, due to Urfer [24]. In §4 and §5, we extend some results of Bouc and Thévenaz [8], [9] to this context, and in §6, we describe some examples of Dade groups of fusion systems.

2. Generalities on fusion systems

We review the definition of a fusion system, a concept due to Puig [19]. Our terminology follows [2] or [17].

Date: January 23, 2008.

Research of the second author was supported by Swiss National Research Fellowship PA002-113164/1.

Definition 2.1. Let P be a finite p-group. A fusion system on P is a category F having as objects the subgroups of P; for any two subgroups Q, R of P the morphism set Hom_{F}(Q, R) is a set of injective group homomorphisms with the following properties:

(1) composition of morphisms in F is the usual composition of group homomorphisms;

(2) if φ : Q → R is a morphism in F then so is the induced isomorphism Q ≅ φ(Q) as well as its inverse;

(3) Hom_{F}(Q, R) contains the set Hom_{P} (Q, R) of group homomorphisms φ : Q → R for which there exists an element y ∈ P satisfying φ(u) = yuy^{−1} for all u ∈ Q;

(4) (I-S) if |N_{P} (Q)| ≥ |N_{P} (φ(Q))| for any φ ∈ Hom_{F}(Q, P) then Aut_{P} (Q) is a Sylow psubgroup of Aut_{F}(Q);

(5) (II-S) if φ : Q → P is a morphism in F such that |N_{P} (φ(Q))| ≥ |N_{P} (τ(Q))| for any τ ∈ Hom_{F}(Q, P) then φ extends to a morphism ψ : N_{φ} → P in F where N_{φ} is the subgroup of N_{P} (Q) consisting of all y ∈ N_{P} (Q) for which there exists z ∈ N_{P} (φ(Q)) with the property φ(yuy^{−1}) = zφ(u)z^{−1} for all u ∈ Q.

Given a fusion system F on a finite p-group P, a subgroup Q of P is called fully F-normalised if |N_{P} (Q)| ≥ |N_{P} (φ(Q))| for any φ ∈ Hom_{F}(Q, P); similarly, Q is called fully F-centralised if |C_{P} (Q)| ≥ |C_{P} (φ(Q))| for any φ ∈ Hom_{F}(Q, P). A subgroup Q of P is called F-centric if C_{P} (φ(Q)) = Z(φ(Q)) for any φ ∈ Hom_{F}(Q, P), and Q is called F-radical if Aut_{Q}(Q) = O_{p}(Aut_{F}(Q)), the largest normal p-subgroup of Aut_{F}(Q). A subgroup Q of P is called F-essential if Q is F-centric and if the partially ordered set of non-trivial p-subgroups of Aut_{F}(Q)/ Aut_{Q}(Q) is not connected. If Q is F-essential then Q is F-centric radical. Alperin’s fusion theorem implies that F is generated by the automorphism groups Aut_{F}(Q) of F-essential subgroups Q of P. Following Stancu [23], the axioms (I-S) and (II-S) imply the a priori stronger axioms used in work of Broto-Levi-Oliver [2], where fusion systems are called saturated fusion systems.

Remark 2.2. Let G be a finite group. For any two subgroups Q, R of G denote by Hom_{G}(Q, R) the set of injective group homomorphisms φ : Q → R for which there exists an element x ∈ G satisfying φ(u) = xux^{−1} for all u ∈ Q. Let p be a prime and let P be a Sylow p-subgroup of G. The fusion system of G on P is the category denoted by F_{P} (G) having the subgroups of P as objects and the sets Hom_{G}(Q, R) as morphism sets, for any two subgroups Q, R of G. The category F_{P} (G) is a fusion system on P in the sense of the definition 2.1. Axiom (3) of that definition implies that any fusion system F on P contains the “trivial” fusion system F_{P} (P) of P on itself.

Proposition 2.3. [2, Proposition 2.5] Let F be a fusion system on a finite p-group P. There is a finite P -P -biset X with the following properties:

(1) Every transitive P -P -subbiset of X is of the form P_{Q}×_{φ}P for some subgroup Q of P and some φ ∈ Hom_{F}(Q, P).

(2) \frac{|X|}{|P |} is congruent to 1 modulo p.

(3) For any subgroup Q of P and any φ : Q → P we have an isomorphism of Q-P -bisets φX ≅ _{Q}X and an isomorphism of P -Q-bisets X_{φ} ≅ X_{Q}.

Here P_{Q}×_{φ}P is the P -P -biset of equivalence classes in P × P with respect to the equivalence relation (yu, z) ∼ (y, φ(u)z), where y, z ∈ P and u ∈ Q, and _{φ}X is the Q-P -biset equal to X as right P -set with u ∈ Q acting on x ∈ X, as φ(u)x; similarly for X_{φ}. Note that P acts freely on the left and on the right of a biset X as in Proposition 2.3.

3. The Dade group of a fusion system

Let P be a finite p-group and let k be a field of characteristic p. Following Dade [13], a kP - module M is called an endo-permutation kP -module if End_{k}(M) ≅ M ⊗_{k} M^{∗} is a permutation kP -module. Here M^{∗} = Hom_{k}(M, k) denotes the k-dual of M. Furthermore, M is called an endotrivial kP -module if End_{k}(M) ≅ k ⊕ Y for some projective kP -module Y.

Given an endo-permutation kP -module M, its restriction Res_{Q}^{P}(M) to any subgroup Q of P has at most one isomorphism class of indecomposable direct summands with Q as vertex. In particular, M itself has at most one isomorphism class of indecomposable direct summands with P as vertex, and any such summand is then called a cap of M. If M has a cap we say that M is capped. Note that M is capped if and only if End_{k}(M) has a trivial direct summand as kP - module. In particular, any endotrivial kP -module is a capped endo-permutation kP -module. More precisely, an endotrivial kP -module restricts to any subgroup Q of P as the direct sum of one indecomposable endotrivial kQ-module and a projective kQ-module. If M, N are capped endopermutation kP -modules then so is their tensor product M ⊗_{k} N. Two capped endo-permutation kP -modules M, N are called equivalent if their caps are isomorphic. The Dade group of P is the group of equivalence classes of capped endo-permutation kP -modules, with product induced by the tensor product over k. The inverse of the class of a capped endo-permutation kP -module M in this group is the class of its k-dual M^{∗}. The Dade group D(P) is a finitely generated abelian group (cf. [20]). The equivalence classes whose cap is endotrivial form a subgroup of D(P), denoted by T (P). Note that two endotrivial kP -modules are equivalent if and only if they are isomorphic in the stable module category of kP -modules. The Dade group is contravariant functorial in P. If P and Q are two finite p-groups and φ : Q → P is a group homomorphism, we denote for any kP -module M by Res_{φ}(M), or sometimes simply by _{φ}M, the kQ-module which is equal to M as k-vector space, with u ∈ Q acting on m ∈ M by φ(u)m. Restriction along φ defines in this way an exact functor Res_{φ} : mod(kP) → mod(kQ). If M is a capped endo-permutation kP -module then Res_{φ}(M) is a capped endo-permutation kQ-module, and hence the functor Res_{φ} induces a group homomorphism between the corresponding Dade groups, still denoted by Res_{φ} : D(P) → D(Q). This map sends T (P) to T (Q) and the torsion subgroup D^{t}(P) of D(P) to the torsion subgroup D^{t}(Q) of D(Q). If φ is the inclusion morphism of a subgroup Q of P we write Res_{Q}^{P} instead of Res_{φ}. We refer the reader to [7] for more background material and other usual notation concerning the Dade group. In particular, a section of a finite p-group P is a pair (T, S) of subgroups of P such that S ⊴ T ≤ P. If R ≤ S ⊴ T ≤ P, with R ⊴ P, we write

Defres_{T/S}^{P/R} = Def_{T/S}^{T/R} ∘ Res_{T/R}^{P/R} and Teninf_{T/S}^{P/R} = Ten_{T/R}^{P/R} ∘ Inf_{T/S}^{T/R} .

As before, we use this notation for elements of the Dade group as well as for modules.

Definition 3.1. Let P be a finite p-group and let F be a fusion system on P. The Dade group D(P, F) of (P, F) is the abelian group

D(P, F) = \underleftarrow{lim}_{F}D ,

where D denotes the contravariant functor from F to the category of abelian groups sending a subgroup Q of P to the Dade group D(Q) and a morphism φ : Q → R in F to the group homomorphism Res_{φ} : D(R) → D(Q). Similarly, we define

T (P, F) = \underleftarrow{lim}_{F}T ,

D^{t}(P, F) = \underleftarrow{lim}_{F}D^{t} ,

where T and D^{t} are the subfunctors of D sending a subgroup Q of P to T (Q) and D^{t}(Q), respectively.

Remark 3.2. We usually identify D(P, F) with a subgroup of D(P). With this identification, we have T (P, F) = T (P) ∩ D(P, F) and D^{t}(P, F) = D^{t}(P) ∩ D(P, F) , which is the torsion subgroup of D(P, F). Denoting as in 2.2 by F_{P} (P) the “trivial” fusion system of P on itself we have obvious identifications D(P) = D(P, F_{P} (P)) , T (P) = T (P, F_{P} (P)) , and D^{t}(P) = D^{t}(P, F_{P} (P)) .

In order to describe D(P, F) as subgroup of D(P), we use the following terminology.

Definition 3.3. Let P be a finite p-group and let F be a fusion system on P.

(1) A kP -module M is called F-stable if Res_{φ}(M) ≅ Res_{Q}^{P}(M) as kQ-modules for any subgroup Q of P and any φ ∈ Hom_{F}(Q, P).

(2) An element v ∈ D(P) is called F-stable if Res_{φ}(v) = Res_{Q}^{P}(v) in D(Q), for any subgroup Q of P and any φ ∈ Hom_{F}(Q, P).

In other words, the class v in D(P) of a capped endo-permutation kP -module V is F-stable if and only if the endo-permutation kQ-modules Res_{φ}(V) and Res_{Q}^{P}(V) have isomorphic caps, for any subgroup Q of P and any φ ∈ Hom_{F}(Q, P). Thus the notion of F-stability characterises elements in D(P, F):

Proposition 3.4. Let P be a finite p-group and let v ∈ D(P). Then v ∈ D(P, F) if and only if v is F-stable.

Remark 3.5. Let P be a finite p-group and let F be a fusion system on P. As a consequence of Alperin’s fusion theorem, in order to determine whether a kP -module M is F-stable or not, it suffices to check that Res_{φ}(M) ≅ Res_{Q}^{P}(M) for every fully F-normalised F-essential subgroup Q of P and any φ ∈ Aut_{F}(Q), viewed as morphism from Q to P. Also, by property (4) of Definition 2.1, we can furthermore assume that φ has order prime to p.

For endotrivial modules the notions of stability of a module and its class coincide:

Proposition 3.6. Let P be a finite p-group. The class v ∈ T (P) of an endotrivial kP -module V is F-stable if and only if V is F-stable.

Proof. This follows from the fact that for any subgroup Q of P we have Res_{Q}^{P}(V) ≅ V_{Q} ⊕ Y, for some cap V_{Q} of Res_{Q}^{P}(V) and some projective kQ-module Y. □

For the class v ∈ D(P) of an arbitrary capped endo-permutation kP -module V it is still true that if V is F-stable then so is v. The converse, however, need not be true. However, by Green’s Indecomposability Theorem, a capped endo-permutation kP -module V can be written in the form

V ≅ ⨁_{R}(Ind_{R}^{P}(V_{R}))^{n_{R}} ,

where R runs over a set of representatives of the conjugacy classes of subgroups of P and where V_{R} is a cap of Res_{R}^{P}(V) for any subgroup R of P, with uniquely determined integers n_{R}. The class of V is F-stable if Res_{φ}(V_{φ(R)}) ≅ V_{R} for any morphism φ : R → P in F. For the module V itself to be F-stable one would need further compatibility conditions on the integers n_{R}. What we show next is that any F-stable class v ∈ D(P) has at least some F-stable representative.

Proposition 3.7. Let F be a fusion system on a finite p-group P and let v ∈ D(P, F). Then, there exists V ∈ v such that V is F-stable.

Proof. Let X be a P -P -biset with the properties as in Proposition 2.3. Consider the permutation kP -kP -bimodule kX with k-basis X. Let v ∈ D(P, F) and let W ∈ v. We will show that the kP -module V = kX ⊗_{kP} W is a capped endo-permutation kP -module belonging to v and that V is F-stable. By property 2.3 (1) we have

kX ≅ ⨁_{(Q,φ)} k[P_{Q}×_{φ}P] ,

with (Q, φ) running over a suitable family of pairs consisting of a subgroup Q of P and a morphism φ ∈ Hom_{F}(Q, P). By property 2.3 (2) the bimodule kX has a direct summand of the form _{φ}kP for some φ ∈ Aut_{F}(P). Thus, by property 2.3 (3) the bimodule kX has in fact a direct summand isomorphic to kP as kP -kP -bimodule. Thus W is isomorphic to a direct summand of V. The stability of V follows from the stability property 2.3 (3) of the biset X. It remains to prove that V is indeed an endo-permutation module. We have

V ≅ ( ⨁_{(Q,φ)} k[P_{Q}×_{φ}P]) ⊗_{kP} W ≅ ⨁_{(Q,φ)} kP ⊗_{kQ} _{φ}kP ⊗_{kP} W ≅ ⨁_{(Q,φ)} Ind_{Q}^{P}(_{φ}W)

with (Q, φ) as before. Since v is F-stable, each indecomposable summand of V is isomorphic to Ind_{R}^{P}(W_{R}) for some subgroup R of P, where W_{R} is a cap of Res_{R}^{P}(W). Hence, to verify that V is endo-permutation, it suffices to check that Ind_{R}^{P}(W_{R}) ⊗_{k} Ind_{S}^{P} (W_{S}^{∗}) is a permutation kP -module, for all subgroups R, S of P. Now,

Ind_{R}^{P}(W_{R}) ⊗_{k} Ind_{S}^{P} (W_{S}^{∗}) ≅ Ind_{R}^{P} (W_{R} ⊗ Res_{R}^{P}(Ind_{S}^{P} (W_{S}^{∗}))) ≅

≅ Ind_{R}^{P} ( ⨁_{x∈[R\P/S]} W_{R} ⊗ Ind_{^{x}S∩R}^{R} ( Res_{^{x}S∩R}^{^{x}S}(W_{^{x}S}^{∗}))) .

Again by F-stability of W, each indecomposable summand of Ind_{^{x}S∩R}^{R}(Res_{^{x}S∩R}^{^{x}S}(W_{^{x}S}^{∗})) is isomorphic to Ind_{T}^{R}(W_{T}^{∗}) for some subgroup T ≤ R. Thus, the above is a direct sum of modules isomorphic to

Ind_{T}^{P} ( Res_{T}^{R}(W_{R}) ⊗_{k} W_{T}^{∗}) , for some subgroup T ≤ R .

Now, each Res_{T}^{R}(W_{R})⊗_{k} W_{T}^{∗} above is a direct summand of Res_{T}^{P} (W ⊗_{k} W^{∗}), which is a permutation kT -module. Hence, so is Res_{T}^{R}(W_{R}) ⊗ W_{T}^{∗}. Since a direct sum of permutation modules is still a permutation module, we conclude that Ind_{R}^{P}(W_{R}) ⊗_{k} Ind_{S}^{P} (W_{S}^{∗}) is a permutation kP -module, for all subgroups R, S ∈ F . This shows that V is an endo-permutation kP -module with the desired properties. □

We end this section with a comparison between D(P, F) and the group D_{P} (G), defined by Urfer in [24], when P is a Sylow p-subgroup of G and F = F_{P} (G) is the fusion system defined by the action of G on P by conjugation. For sake of completeness, let us recall Urfer’s definition.

Definition 3.8 (Définition 2.15, [24]). Let G be a finite group and let P be a Sylow p-subgroup of G.

(1) A kG-module M is endo-p-permutation if Res_{P}^{G}(M) is endo-permutation. If so, then M is capped if Res_{P}^{G}(M) is capped, i.e. has an indecomposable direct summand with vertex P.

(2) The set D_{P} (G) of equivalence classes of capped endo-p-permutation kG-modules for the relation M ∼ N if [Res_{P}^{G}(M)] = [Res_{P}^{G}(N)] in D(P) is an abelian group. In particular, D_{P} (P) = D(P).

Note that the equivalence relation identifies the endo-p-permutation modules whose caps have isomorphic sources. In Proposition 2.10 of [24], Urfer shows that an indecomposable endo-permutation kP -module V has the property that Ind_{P}^{G}(V) is an endo-p-permutation kG-module if and only if V is G-stable (that is, Res_{^{x}P ∩P}^{^{x}P} (^{x}V) ≅ Res_{^{x}P ∩P}^{P} (V), for all x ∈ G). From this, the author deduces an isomorphism between the subgroup of G-stable elements in D(P) and the group D_{P} (G). Translating the G-stability in terms of F-stability, with F = F_{P} (G), his result becomes:

Proposition 3.9 (Propositions 2.10 and 2.19, [24]). Let G be a finite group and let P be a Sylow p-subgroup of G. The restriction functor Res_{P}^{G} : mod(kG) → mod(kP) induces an isomorphism of abelian groups Res_{P}^{G} : D_{P} (G) → D(P, F), with inverse induced by the induction functor Ind_{P}^{G} applied to F-stable modules.

Note that we reach the same conclusion by applying Proposition 3.7.

Remark 3.10. Let G be a finite group and P a Sylow-p-subgroup of G.

(1) Since two capped endo-p-permutation kG-modules are equivalent if and only if their restrictions to P have isomorphic caps, we have that 0 ∈ D_{P} (G) is the set of all p-permutation modules having an indecomposable direct summand with vertex P.

(2) The definition of the group T (G) of endotrivial kG-modules, as given in [10], does not identify T (G) to a subgroup of D_{P} (G) unless P = G. Indeed, in T (G), the elements are isomorphism classes of modules in the stable module category. Hence, two indecomposable endotrivial modules having isomorphic sources are generally not equivalent. Consequently, the kernel of the restriction map Res_{P}^{G} : T (G) → T (P, F) is a finite group, isomorphic to the group generated by the isomorphism classes of the Scott modules that are endotrivial (i.e. such that Res_{P}^{G}(M) ≅ k ⊕ L for some projective kP -module L, cf. [10, § 2]). Besides this fact, let us point out that it is yet still an open problem to determine whether the map Res_{P}^{G} above is surjective or not. The issue is that for an indecomposable endo-p-permutation kG-module M the property [Res_{P}^{G}(M)] ∈ T (P) implies that M has an endotrivial source and that the Green correspondent of M is an endotrivial kN_{G}(P)-module, but M need not, a priori, be an endotrivial module itself.

4. Detection

“Detection” type results describe families C_{Y} of sections (T, S) of a finite p-group P for which the product of the maps Defres_{T/S}^{P} : D(P) → D(T/S) yields an injective map

Defres_{C_{Y}} : D(P) ⟶ ∏_{(T,S)∈C_{Y}} D(T/S) .

Definition 4.1. Let Y be a class of finite p-groups closed under taking sections and isomorphisms. Let P be a finite p-group, let F be a fusion system on P and let C be a subcategory of F closed under isomorphisms. We denote by C_{Y} the category defined as follows:

• the objects of C_{Y} are the sections (T, S) of P belonging to the subcategory C of F such that the quotient group T/S belongs to Y.

• the morphism set Hom_{C_{Y}} ((T, S), (T′, S′)) is the set of morphisms φ : T → T′ in C, such that S′ ≤ φ(S), for any two objects (T, S), (T′, S′) in C_{Y}, with composition induced by that in F. If C = F_{P} (P) we write P_{Y} instead of F_{P} (P)_{Y}.

With the notation above, given a morphism φ : (T, S) → (T′, S′) in C_{Y}, the groups φ(T), φ(S) belong to C because C is assumed to be closed under isomorphisms, and the section (φ(T), φ(S)) of P can be identified with the section (φ(T)/S′, φ(S)/S′) of T′/S′. Hence the map Defres_{φ(T)/φ(S)}^{T′/S′} : D(T′/S′) → D(φ(T)/φ(S)) is well-defined. Restriction along the isomorphism T/S ≅ φ(T)/φ(S) induces an isomorphism D(φ(T)/φ(S)) ≅ D(T/S) which we will denote abusively by Res_{φ} again. In this way taking Dade groups of the quotients T/S yields a contravariant functor on C_{Y}.

Definition 4.2. Let P be a finite p-group, let F be a fusion system on P, let C be a subcategory of F closed under isomorphisms, and let Y be a class of finite p-groups closed under taking sections and isomorphisms. Denote by A the category of abelian groups. Let D : C_{Y} → A be the contravariant functor sending an object (T, S) in C_{Y} to the group D(T/S) and a morphism φ : (T, S) → (T′, S′) in C_{Y} to the group homomorphism Defres_{φ} = Res_{φ} ∘ Defres_{φ(T)/φ(S)}^{T′/S′} : D(T′/S′) → D(T/S). We set

D_{←}(P, C_{Y}) = \underleftarrow{lim}_{C_{Y}}(D) .

Similarly, we set D_{←}^{t}(P, C_{Y}) = \underleftarrow{lim}_{C_{Y}}(D^{t}) and T_{←}(P, C_{Y}) = \underleftarrow{lim}_{C_{Y}}(T), where D^{t} and T are the subfunctors

of D sending an object (T, S) in C_{Y} to the subgroups D^{t}(T/S) and T (T/S) of D(T/S), respectively. We denote by

Defres_{(C,Y)}^{F} : D(P, F) → D_{←}(P, C_{Y})

the canonical map determined by the family of maps Defres_{T/S}^{P} : D(P, F) → D(T/S), where (T, S) runs over the objects of C_{Y}. If C = F_{P} (P) we write D_{←}(P_{Y}) instead of D_{←}(P, F_{P} (P)_{Y}) and

Defres_{Y} : D(P) → D(P_{Y}) instead of Defres_{(F_{P} (P),Y)}^{F} .

Remark 4.3. We can identify the group D_{←}(P, C_{Y}) with the set of families (v_{(T,S)})_{(T,S)∈C_{Y}} with v_{(T,S)} ∈ D(T/S) such that for any morphism φ : (T, S) → (T′, S′) in C_{Y}, we have that

Defres_{φ}(v_{(T′,S′)}) = v_{(T,S)} .

With this identification, the groups D_{←}^{t}(P, C_{Y}), T_{←}(P, C_{Y}) become subgroups of D_{←}(P, C_{Y}), the group D_{←}(P, F_{Y}) becomes a subgroup of D_{←}(P_{Y}). More generally, if Q is a subgroup of P and G a fusion system on Q such that G ⊆ F, then G_{Y} is a subcategory of F_{Y}, and the inclusion functor G_{Y} ⊆ F_{Y} induces a canonical group homomorphism

D_{←}(P, F_{Y}) → D_{←}(Q, G_{Y}) ,

mapping (v_{(T,S)})_{(T,S)∈F_{Y}} to its canonical projection (v_{(T,S)})_{(T,S)∈G_{Y}}. In particular, we have a canonical map D_{←}(P_{Y}) → D_{←}(Q_{Y}).

Proposition 4.4. Let P be a finite p-group and let Y be a class of finite p-groups closed under taking sections and isomorphisms. Suppose that for any subgroup Q of P the map Defres_{Y} : D(Q) → D(Q_{Y}) is injective. Then, for any fusion system F on P, the diagram

\xymatrix{D(P, F) \ar[r]^{^{Defres{F \atop (F,Y)}}} \ar@{^{(}->}[d] & D_{←}(P, F_{Y}) \ar@{^{(}->}[d] \\ D(P) \ar[r]^{_{Defres_{Y}}} & D_{←}(P_{Y})}

is a pull-back diagram, where the vertical maps are the canonical inclusions.

Proof. The commutativity of the diagram is immediate. By the assumptions, the bottom map Defres_{Y} in the diagram is injective. Since the vertical maps are inclusions, the top horizontal map Defres_{(F,Y)}^{F} is injective. We identify D(P, F) and D_{←}(P, F_{Y}) with their canonical images in D(P) and D_{←}(P_{Y}), respectively. Let v ∈ D(P) such that Defres_{Y}(v) ∈ D_{←}(P, F_{Y}). We need to

show that v ∈ D(P, F). Write Defres_{Y}(v) = (v_{(T,S)})_{(T,S)∈P_{Y}}, with v_{(T,S)} = Defres_{T/S}^{P}(v), for all (T, S) ∈ P_{Y}. Since Defres_{Y}(v) is assumed to be an element of D_{←}(P, F_{Y}), we have v_{(T,S)} = Defres_{φ}(v_{(T′,S′)}) for any morphism φ : (T, S) → (T′, S′) in F_{Y}. Let Q be a subgroup of P and let α ∈ Aut_{F}(Q), viewed as morphism from Q to P. In order to show that v ∈ D(P, F) it suffices to show that Res_{Q}^{P}(v) = Res_{α}(v). The automorphism α of Q induces an automorphism of the category Q_{Y} and hence an automorphism of the group D_{←}(Q_{Y}), mapping an element (w_{(T,S)})_{(T,S)∈Q_{Y}} to (Res_{α}(w_{(α(T),α(S))}))_{(T,S)∈Q_{Y}}. This automorphism fixes (v_{(T,S)})_{(T,S)∈Q_{Y}} by the choice of v. Since the map Defres_{Y} : D(Q) → D_{←}(Q_{Y}) is injective it follows that α fixes Res_{Q}^{P}(v), or equivalently,

Res_{Q}^{P}(v) = Res_{α}(v), whence the result. □

Bouc and Thévenaz proved in [9, Theorem 4.7] that if p is an odd prime and X is the class of all p-groups of order at most p^{3} and exponent p, then for any finite p-group P the group homomorphism Defres_{X} : D(P) → D_{←}(P_{X}) is an isomorphism. Combining this with Proposition 4.4 yields the following consequence.

Corollary 4.5. Let p be an odd prime and let X be the class of all p-groups of order at most p^{3} and exponent at most p. For any finite p-group P and any fusion system F on P the map

Defres_{(F,X)}^{F} : D(P, F) → D_{←}(P, F_{X})

is an isomorphism.

Remark 4.6. Using 4.5 it is easy to check that the group D_{←}(P, P_{X}^{1} ) is isomorphic to the subgroup

of ⨁_{1<Q≤P} D(N_{P} (Q)/Q) defined by Bouc and Thévenaz at the beginning of [8, §2], where P^{1} is the

full subcategory of F_{P} (P) consisting of all non-trivial subgroups of P.

5. Gluing

“Gluing problems” ask which families (v_{(T,S)})_{(T,S)∈C_{Y}} are in the image of the map

Defres_{(C,Y)}^{F} : D(P) ⟶ ∏_{(T,S)∈C_{Y}} D(T/S) .

where the notation is as in the previous section. A subcategory C of a category F is called convex if C is a full subcategory of F with the property that whenever X, Y, Z are three objects in F such that the morphism sets Hom_{F}(X, Y), Hom_{F}(Y, Z) are non-empty and such that X, Z belong to the subcategory C, then also Y belongs to C. A convex subcategory is closed under isomorphisms.

Proposition 5.1. Let p be an odd prime, let P be a finite p-group, let F be a fusion system on P and let C be a convex subcategory of F. Let Y be a class of finite p-groups closed under taking sections and isomorphisms and containing the class X of p-subgroups of order at most p^{3} and exponent at most p. The inclusion C_{X} ⊆ C_{Y} induces an isomorphism

D_{←}(P, C_{Y}) ≅ D_{←}(P, C_{X}) .

Proof. Identify D_{←}(P, C_{Y}) and D_{←}(P, C_{X}) with the families (v_{(T,S)})_{(T,S)} satisfying

Defres_{φ}(v_{(T′,S′)}) = v_{(T,S)}

for all objects (T, S) and morphisms φ belonging to the categories C_{Y} and C_{X}, respectively. The inclusion C_{X} ⊆ C_{Y} induces a canonical morphism

D_{←}(P, C_{Y}) → D_{←}(P, C_{X})

We use Corollary 4.5 to show that this is an isomorphism. We construct an inverse as follows. Let (v_{(T,S)})_{(T,S)∈C_{X}} be an element in D_{←}(P, C_{X}). Let (R, Q) be an object in C_{Y}. Note that since C is convex, the objects of (R/Q)_{X} belong to the category C_{X}. Indeed, let (T/Q, S/Q) ∈ (R/Q)_{X}. That is, we have a chain Q ⊴ S ⊴ T ≤ R of objects of F, with Q ⊴ R and R/Q ∈ Y. Thus S, T ∈ C since Q, R ∈ C and C is convex. Consequently, (T/Q, S/Q) ≅ (T, S) ∈ C_{X}. Now, define w_{(R,Q)} ∈ D(R/Q) as pre-image of the subfamily (v_{(T,S)})_{(T,S)∈(R/Q)_{X}} under the isomorphism

Defres_{X} : D(R/Q) ⟶ D_{←}((R/Q)_{X})

from [9, Theorem 4.7]. The family (w_{(R,Q)})_{(R,Q)∈C_{Y}} defined in this way belongs to D_{←}(P, C_{Y}). In particular, w_{(T,S)} = v_{(T,S)}, for all (T, S) ∈ C_{X}, whence the composition

D_{←}(P, C_{X}) → D_{←}(P, C_{Y}) → D_{←}(P, C_{X})

is the identity. For the composition of the maps the other way round, let (v_{(R,Q)})_{(R,Q)∈C_{Y}} ∈ D_{←}(P, C_{Y}), and let (w_{(R,Q)})_{(R,Q)∈C_{Y}} be the image of (v_{(R,Q)})_{(R,Q)∈C_{X}} ∈ D_{←}(P, C_{X}) under the map defined above. Identify, via Defres_{X}, any w_{(R,Q)} ∈ D(R/Q) with

(Defres_{T/S}^{R/Q} w_{(R,Q)})_{(T,S)∈(R/Q)_{X}} ∈ D_{←}((R/Q)_{X}) , ∀ (R, Q) ∈ C_{Y} .

By construction of w_{(R,Q)} ∈ D(R/Q), we have, for any (T, S) ∈ C_{X},

Defres_{T/S}^{R/Q} w_{(R,Q)} = w_{(T,S)} = v_{(T,S)} .

Thus, Defres_{X} associates any w_{(R,Q)} ∈ D(R/Q) with (v_{(T,S)})_{(T,S)∈(R/Q)_{X}} ∈ D_{←}((R/Q)_{X}), which is

identified with v_{(R,Q)} under Defres_{X}^{−1}. Hence v_{(R,Q)} = w_{(R,Q)}, for all (R, Q) ∈ C_{Y}. It follows that the composition of maps

D_{←}(P, C_{Y}) → D_{←}(P, C_{X}) → D_{←}(P, C_{Y})

yields the identity, as was left to be shown. □

Definition 5.2. Let p be a prime and let P be a finite p-group. Denote by A_{≥2}(P) the partially ordered set of elementary abelian subgroups of P of rank at least 2. For any fusion system F on P denote by E_{≥2}(F) the full subcategory of F having A_{≥2}(P) as set of objects; if F = F_{P} (P) we write E_{≥2}(P) instead of E_{≥2}(F_{P} (P)). Denote by 𝔼_{F} the quotient of the 𝔽_{2}-vector space of maps E_{≥2}(F) → 𝔽_{2} which are constant on connected components of E_{≥2}(F) by the subspace of constant maps E_{≥2}(F) → 𝔽_{2}; in the case F = F_{P} (P) we write 𝔼_{P} instead of 𝔼_{F_{P} (P)}.

Bouc and Thévenaz proved in [8, Theorem 5.1] that for p an odd prime and P a non cyclic finite p-group there is a short exact sequence of 𝔽_{2}-vector spaces

\xymatrix{0 \ar[r] & D^{t}(P) \ar[r]^{^{Defres{P \atop (P^{1},X)}}} & D_{←}^{t}(P, P_{X}^{1} ) \ar[r]^{Φ_{P}} & 𝔼_{P} \ar[r] & 0} ,

where P^{1} is the full subcategory of F_{P} (P) consisting of all non-trivial subgroups of P (cf. 4.6). In particular, if A_{≥2}(P) is connected we have an isomorphism D^{t}(P) ≅ D_{←}^{t}(P, P_{X}^{1} ). If A_{≥2}(P) is not connected, the map Φ_{P} is constructed explicitly, making use of the very particular structure of P and the connected components of A_{≥2}(P) in that case. This can be generalised to arbitrary fusion systems; the proof is not as straightforward as one might expect:

Theorem 5.3. Let p be an odd prime, P a finite p-group and F a fusion system on P. Denote by F^{1} the full subcategory of F consisting of all non-trivial subgroups of P and let X be the class of finite p-groups of order at most p^{3} and exponent at most p. The map Φ_{P} induces a short exact sequence of 𝔽_{2}-vector spaces

\xymatrix{0 \ar[r] & D^{t}(P, F) \ar[r] & D_{←}^{t}(P, F_{X}^{1} ) \ar[r] & 𝔼_{F} \ar[r] & 0}

As in the proof of [8, Theorem 5.1], the difficult part of the proof of 5.3 arises when A_{≥2}(P) is not connected. By [12, Lemma 2.2], in that case, P has a unique central cyclic subgroup of order p, which we will denote by Z. Furthermore, any connected component of A_{≥2}(P) contains either all elementary abelian subgroups of rank at least 3 or consists of a single elementary abelian subgroup E of rank 2, in which case E is called isolated in A_{≥2}(P). If P has rank at least 3 we denote by B the connected component of A_{≥2}(P) containing all elementary abelian subgroups of rank at least 3 (and their subgroups of rank 2), and in that case, B contains a subgroup E_{0} of rank 2 which is normal in P. If P has rank 2, any connected component of A_{≥2}(P) consists of a single isolated subgroup of rank 2, and there exists an isolated E_{0} which is normal in P; we set B = {E_{0}} in that case. Whenever E is isolated in A_{≥2}(P) then E contains Z, the group N_{P} (E) acts transitively on the set of complements of Z in E, and for any such complement S of Z in E we have N_{P} (S) = S × L for some cyclic subgroup L of P. One of the additional difficulties in the proof of 5.3 is that Z need no longer be strongly F-closed - and hence a connected component of E_{≥2}(F) may contain both isolated and non isolated elementary abelian subgroups of P. The following lemma will be needed to handle this situation.

Lemma 5.4. Let p be an odd prime, P a finite p-group and F a fusion system on P. Let E be an elementary abelian subgroup of rank 2 of P which is isolated in A_{≥2}(P) and let Q be a subgroup of order p of E. Let ψ ∈ Hom_{F}(E, P). If there is an automorphism α ∈ Aut_{F}(E) such that α(Q) ≠ Z then there is an automorphism β ∈ Aut_{F}(ψ(E)) such that β(ψ(Q)) ≠ Z.

Proof. Note that E contains Z by the above remarks. If ψ(Q) ≠ Z we take β = Id_{ψ(E)}. Suppose that ψ(Q) = Z. If also Q = Z take β = ψ ∘ α ∘ ψ^{−1}. If Q ≠ Z there is y ∈ N_{P} (E) such that yQ ≠ Q because N_{P} (E) acts transitively on the set of complements of Z in E, again by the above remarks. Thus in this case take β = ψ ∘ c_{y} ∘ ψ^{−1}, where c_{y} is the automorphism of E given by conjugation with y. □

Proof of Theorem 5.3. We identify D^{t}(P, F) and D_{←}^{t}(P, F_{X}^{1} ) with their canonical images in D^{t}(P)

and D_{←}^{t}(P, P_{X}^{1} ), respectively. Clearly the map D^{t}(P) → D_{←}^{t}(P, P_{X}^{1} ) induces an injective map

D^{t}(P, F) → D_{←}^{t}(P, F_{X}^{1} ). We break up the proof in several steps.

We show the exactness at D_{←}^{t}(P, F_{X}^{1} ). Let (v_{(T,S)})_{(T,S)∈F_{X}^{1}} be an element in ker(Φ_{P})∩D_{←}^{t}(P, F_{X}^{1} ).

Explicitly, this means that for any morphism φ : (T, S) → (T′, S′) in F_{X}^{1} we have an equality v_{(T,S)} = Defres_{φ}(v_{(T′,S′)}), where Defres_{φ} is as defined in 4.2. The fact that this element is also in ker(Φ_{P}) implies, by the above exact sequence of Bouc and Thévenaz, that there is an element v ∈ D^{t}(P) such that Defres_{T/S}^{P}(v) = v_{(T,S)} for any object (T, S) in F_{X}^{1} . We need to show that v belongs actually to D^{t}(P, F). By Alperin’s fusion theorem, we need to show that for any Fcentric subgroup Q of P and any φ ∈ Aut_{F}(Q) considered as morphism from Q to P we have Res_{Q}^{P} (v) = Res_{φ}(v) in D^{t}(Q). If Q is cyclic this is trivial. If Q is not cyclic, the exact sequence of Bouc and Thévenaz applies and yields in particular an injective map D^{t}(Q) → D_{←}^{t}(Q, Q_{X}^{1} ).

In other words, Res_{φ}(v) is completely determined by the family ( Defres_{V/U}^{Q} (Res_{φ}(v)))_{(V,U)∈Q_{X}^{1}} .

Now, by transitivity of restriction, Defres_{V/U}^{Q}(Res_{Q}^{P} v) = Defres_{V/U}^{P}(v) = v_{(V,U)} . Moreover, by assumption on v, we have v_{(V,U)} = Res_{φ}(v_{(φ(V),φ(U))}) . Thus, the equality Res_{Q}^{P} (v) = Res_{φ}(v) holds. This shows the exactness at D_{←}^{t}(P, F_{X}^{1} ).

We show next the exactness at 𝔼_{F}. If A_{≥2}(P) is connected then 𝔼_{P} = {0}, hence 𝔼_{F} = {0}, and we get an isomorphism D^{t}(P, F) ≅ D_{←}^{t}(P, F_{X}^{1} ), which proves the theorem in that case. We assume now that A_{≥2}(P) is not connected, and so P has the particular structure described above. The proof of the exactness at 𝔼_{F} breaks up into two parts: we first need to show that Φ_{P} maps D_{←}^{t}(P, F_{X}^{1} ) to 𝔼_{F}, and second that this is onto. Let v = (v_{(T,S)})_{(T,S)∈F_{X}^{1}} be an element in

D_{←}^{t}(P, F_{X}^{1} ). The map Φ_{P} in [8, Theorem 5.1] is constructed as follows: for E an elementary abelian subgroup of P of rank at least 2, the subfamily (v_{(T,S)})_{(T,S)∈(EZ)_{X}^{1}} determines a unique element w_{EZ} in D^{t}(EZ), where we use the isomorphism D^{t}(EZ) ≅ D_{←}^{t}(EZ, (EZ)_{X}^{1} ) from [8, Theorem 5.1] applied to the elementary abelian group EZ. Since Z is cyclic of order p we have D(Z) ≅ 𝔽_{2}. Thus, restriction from EZ to Z sends w_{EZ} to a uniquely determined element ϵ_{v}(E) ∈ 𝔽_{2}. It is shown in [8, Lemma 3.2] that this defines a map ϵ_{v} on A_{≥2}(P) satisfying ϵ_{v}(E) = ϵ_{v}(E′) if E is conjugate to a subgroup of E′, and hence ϵ_{v} is in fact a map E_{≥2}(P) → 𝔽_{2} which is constant on connected components of E_{≥2}(P). That is, ϵ_{v} yields an element ϵ̄_{v} in 𝔼_{P}. We need to show that ϵ̄_{v} belongs to 𝔼_{F}. It suffices to show that the map ϵ_{v} : E_{≥2}(P) → 𝔽_{2} is constant on connected components of E_{≥2}(F). Since ϵ_{v} is invariant with respect to inclusions, it suffices to show that ϵ_{v}(E) = ϵ_{v}(φ(E)), for all φ ∈ Hom_{F}(E, P), with E elementary abelian of rank 2. If E and φ(E) are in the same connected component of E_{≥2}(P) the equality holds by [8, Theorem 5.1]. Thus we may assume that E is isolated. Write E = S × Z, where Z is the unique central subgroup of order p. Then N_{P} (S) = C_{P} (E) = S × L has rank 2 as well. Consider first the case where φ(E) is also isolated for all φ ∈ Hom_{F}(E, P). Choose φ ∈ Hom_{F}(E, P) such that φ(E) is fully F-normalised, so that N_{φ} = N_{P} (E); that is, such that φ extends to a morphism φ̃ : N_{P} (E) → P. Since N_{P} (E) permutes transitively the complements of Z in E, the group φ̃(N_{P} (E)), which is contained in N_{P} (φ(E)), permutes transitively the complements of φ(Z) in φ(E). On the other hand, φ(E) = Z × S′ for some non-central subgroup S′ of P of order p, and N_{P} (φ(E)) permutes transitively the complements of Z in φ(E). This forces φ(Z) = Z. It follows, by definition of Φ_{P}, that w_{E} = Res_{φ}(w_{φ(E)}), and hence ϵ_{v}(E) = ϵ_{v}(φ(E)). Since there is always an isomorphism φ with N_{φ} = N_{P} (E), we conclude that ϵ_{v}(E) = ϵ_{v}(φ(E)), for all φ ∈ Hom_{F}(E, P), in the situation where φ(E) is isolated for all φ ∈ Hom_{F}(E, P). Suppose next that there exists φ ∈ Hom_{F}(E, P) such that φ(E) is contained in an elementary abelian subgroup of rank 3. Denote by ψ : φ(Z) → Z the restriction of the inverse of φ. Clearly, Z is fully F-centralised, and hence ψ extends to a morphism ψ̃ : C_{P} (φ(Z)) → P, such that the composition ψ̃ ∘ φ is the identity on Z. Note that φ(E) ≤ C_{P} (φ(E)) ≤ C_{P} (φ(Z)) and that ψ̃ maps C_{P} (φ(E)) to C_{P} (ψ̃(φ(E))), which has rank at least 3. Thus, ψ̃(φ(E)) belongs to the connected component B of A_{≥2}(P). As before, the equality ψ̃(φ(Z)) = Z implies w_{E} = Res_{φ}(w_{φ(E)}), hence ϵ_{v}(E) = ϵ_{v}(ψ̃(φ(E)). In other words, the value of ϵ_{v} at E coincides with the value of ϵ_{v} on B. This shows that Φ_{P} maps D_{←}^{t}(P, F_{X}^{1} ) to 𝔼_{F}, by Φ_{P} (v) = ϵ_{v}.

In order to show that Φ_{P} maps D_{←}^{t}(P, F_{X}^{1} ) onto 𝔼_{F}, we define a map σ : 𝔼_{F} → D_{←}^{t}(P, F_{X}^{1} ), such that Φ_{P} ∘ σ is the identity on 𝔼_{F}. Let c be a connected component of E_{≥2}(F). Write ϵ_{c} : E_{≥2}(F) → 𝔽_{2} for the map taking values ϵ_{c}(E) = 1 if E ∈ c and ϵ_{c}(E) = 0 otherwise. The image B̄ = {ϵ̄_{c} | B ⊈ c} under the canonical projection of the set B = {ϵ_{c} | B ⊈ c}, where c runs through the connected components of E_{≥2}(F) not containing B, forms an 𝔽_{2}-basis of 𝔼_{F}. If E_{≥2}(F) is connected, or equivalently, if 𝔼_{F} = {0} there is nothing to prove. Assume that 𝔼_{F} ≠ {0} and let c be a connected component in E_{≥2}(F) not containing B; that is, c consists of subgroups of rank 2 which are isolated in A_{≥2}(P). For any object (R, Q) in F_{X}^{1} set

N (c, R, Q) = {E ∈ c | Q < E ≤ R; ∃φ ∈ Aut_{F}(E) : φ(Q) ≠ Z}

and define an element in D^{t}(R/Q) by setting

σ(ϵ̄_{c})_{(R,Q)} = ∑_{E∈N (c,R,Q)} Ten_{E/Q}^{R/Q}(Ω_{E/Q})

with the convention σ(ϵ̄_{c})_{(R,Q)} = 0 if the set N (c, R, Q) is empty. Note that if N (c, R, Q) is non empty then Q has order p. We will show that the family

σ(ϵ̄_{c}) = (σ(ϵ̄_{c})_{(R,Q)})_{(R,Q)∈F_{X}^{1}}

is an element in the group D_{←}^{t}(P, F_{X}^{1} ); that is, given a morphism ψ : (R, Q) → (R′, Q′) in F_{X}^{1} we need to show that

Defres_{ψ}(σ(ϵ̄_{c})_{(R′,Q′)}) = σ(ϵ̄_{c})_{(R,Q)}

If Q, Q′ have order greater than p both sides are zero, so we may assume that Q′ has order p. If Q has order greater than p then again both sides are zero. Thus we may assume that ψ(Q) = Q′. We consider first the case where ψ(R) = R′; that is, ψ is an isomorphism in F_{X}^{1} . In that case, thanks to 5.4, ψ induces a bijection between the sets N (c, R, Q) and N (c, R′, Q′), and thus the required equality follows. We may therefore assume, after possibly replacing (R, Q) by (ψ(R), ψ(Q)), that Q = Q′ ≤ R < R′ and that ψ is the obvious inlcusion map. We show that this implies Q = Z. Indeed, if Q ≠ Z there is at most one isolated E containing Q, namely E = Q × Z. Thus either N (c, R, Q), N (c, R′, Q) are both empty (in which case we are done) or both equal to {E}, where E = Q×Z. But in that case we have R′ ≤ N_{P} (Q) = Q×L for some cyclic L ≤ P. Since also R′/Q has exponent at most p, this forces E = R = R′, contradicting our current assumption R < R′. Thus Q = Q′Z, and Defres_{ψ} is just restriction Res_{R/Z}^{R′/Z} from R′/Z to R/Z. Thus

Defres_{ψ}(σ(ϵ̄_{c})_{(R′,Z)}) = ∑_{F ∈N (c,R′,Z)} Res_{R/Z}^{R′/Z} Ten_{F/Z}^{R′/Z}(Ω_{F/Z})

Note that if ^{x}F is not contained in R then R/Z ∩^{x}F /Z is trivial because F has order p^{2}. Thus the Mackey formula applied to the summands in the above expression yields

Defres_{ψ}(σ(ϵ̄_{c})_{(R′,Z)}) = ∑_{(F,x)∈A} Ten_{^{x}F /Z}^{R/Z}(Ω_{^{x}F /Z})

where A is the set of all pairs (F, x) in N (c, R′, Z)×[R\R′] satisfying ^{x}F ≤ R. Now ^{x}F ≤ R means precisely that ^{x}F ∈ N (c, R, Z). Thus the summands on the right side of the previous expression are of the form Ten_{E/Z}^{R/Z}(Ω_{E/Z}) for some E ∈ N (c, R, Z). In order to check that this sum is equal to σ(ϵ̄_{c})_{(R,Z)}, all we need to do is to check that the multiplicity of such a summand is odd because they all have order two in the Dade group. Write A as disjoint union of sets A_{E}, where, for E ∈ N (c, R, Q), we set

A_{E} = {(F, x) ∈ A | ^{x}F = E}

and denote by a_{E} the cardinal of A_{E}. Then

Defres_{ψ}(σ(ϵ̄_{c})_{(R′,Z)}) = ∑_{E∈N (c,R,Q)} a_{E} · Ten_{E/Z}^{R/Z}(Ω_{E/Z})

We need to show that the coefficients a_{E} are odd, for E ∈ N (c, R, Z). Every element (F, x) in A_{E} is uniquely determined by its second component x ∈ [R\R′] because ^{x}F = E, hence F = ^{x^{−1}}E. Thus a_{E} = [R′ : R], which is a power of the odd prime p, hence odd. This concludes the proof of the fact that the family σ(ϵ̄_{c}) belongs to the group D_{←}^{t}(P, F_{X}^{1} ).

Finally, we need to show that the map σ : 𝔼_{F} → D_{←}^{t}(P, F_{X}^{1} ) is a section for Φ_{P} restricted to

D_{←}^{t}(P, F_{X}^{1} ). That is, with the notation above, we need to show that Φ_{P} (σ(ϵ̄_{c})) is the image ϵ̄_{c} in 𝔼_{F} of the map ϵ_{c} sending E ∈ c to 1 and any E ∈ E_{≥2}(F) − c to zero. Let E be an elementary abelian subgroup of P of rank at least 2 such that E ∉ c and let (R, Q) ∈ (EZ)_{X}^{1} . Then the set N (c, R, Q) is empty; indeed, any E′ ∈ N (c, R, Q) would be contained in R, hence in EZ, implying that E, E′ belong to the same connnected component in E_{≥2}(F). Thus σ(ϵ̄_{c})_{(R,Q)} = 0 for all (R, Q) ∈ (EZ)_{X}^{1} , and hence this family corresponds to the zero element in D^{t}(EZ) via the isomorphism D^{t}(EZ) ≅ D_{←}^{t}(EZ, (EZ)_{X}^{1} ), hence its image in D^{t}(Z) is zero. By the definition of Φ_{P} this means that Φ_{P} (σ(ϵ̄_{c})) is the image of a map sending E ∈ E_{≥2}(F) − c to zero. Let now E ∈ c. The non trivial objects of E_{X}^{1} are of the form (E, Q), where Q is a subgroup of order p of E, and we have σ(ϵ̄_{c})_{(E,Q)} = Ω_{E/Q}. The image in D^{t}(E) of this family under the isomorphism D^{t}(E) ≅ D_{←}^{t}(E, E_{X}^{1} ) is equal to ∑_{Q;1<Q<E} Ω_{E/Q}. The restriction to Z of this sum is the element

pΩ_{Z} = Ω_{Z} in D^{t}(Z) because E has exactly p subgroups of order p different from Z and because p is odd. This concludes the proof of the fact that Φ_{P} (σ(ϵ̄_{c})) = ϵ̄_{c}, whence the result. □

6. Examples

The Dade group of a arbitrary fusion system on a finite 2-group P whose possible essential subgroups are Klein or quaternion (of order 8) group turns out to be equal to the Dade group D(P) of P itself. This includes the case of all cyclic, generalised quaternion, quasi- and semidihedral 2-groups (cf. [15, Satz I.14.9] for the definitions).

Theorem 6.1. Let P be a finite 2-group and F a fusion system on P. Assume that all F-essential subgroups of P are Klein four groups or quaternion groups of order 8. Then D(P, F) = D(P).

By Alperin’s fusion theorem (cf. Remark 3.5) it suffices to show that for any F-essential fully F-normalised subgroup Q of P and any automorphism φ ∈ Aut_{F}(Q) we have Res_{φ}(v) = Res_{Q}^{P}(v) in D(Q), for all v ∈ D(P). The proof is based on this in conjunction with the following two observations:

Lemma 6.2. Let P be a finite p-group, where p is any prime, let Q be a characteristic subgroup of P, and let φ ∈ Aut(P). Then, Res_{φ}(Ω_{P/Q}) = Ω_{P/Q}.

Proof. By definition, Res_{φ}(Ω_{P/Q}) = Ω_{φ^{−1}(P)/φ^{−1}(Q)} = Ω_{P/Q}, since Q is characteristic. □

Lemma 6.3. If Q is a Klein four group or a quaternion group of order 8, then for any w ∈ D(Q) and any φ ∈ Aut(Q) of odd order we have Res_{φ}(w) = w.

Proof. By Lemma 6.2, the Lemma holds for w = Ω_{Q}, and w = Ω_{Q/Z} in case Q is quaternion of order 8 and Z is its centre. It remains to consider the ‘exotic’ elements of the Dade group of a quaternion group Q and in case k has cubic roots of unity. By [11, § 6], in that case, the cap of an exotic element v ∈ D(Q) is indecomposable endotrivial of dimension 5, and Q has exactly two isomorphism classes of such modules. Hence an automorphism of odd order must stabilise both classes. □

Proof of Theorem 6.1. By the previous remarks, in order to show the F-stability of all the elements of D(P), it is enough to consider the F-stability with respect to automorphisms φ ∈ Aut_{F}(Q), for Q a fully F-normalised F-essential subgroup of P. Stability with respect to conjugation by elements in P holds trivially, and since Aut_{P} (Q) is a Sylow-2-subgroup of Aut_{F}(Q), we may suppose that φ has odd order. Hence, if all F-essential 2-subgroups of P are Klein groups or quaternion groups of order 8, then the only automorphisms φ of Q of odd order have order 3. By Lemma 6.3, D(Q) is ⟨φ⟩-stable, and hence Res_{Q}^{P}(v) = Res_{φ}(v) for all v ∈ D(P), as was left to be shown. □

Corollary 6.4. If P is a 2-group that is either cyclic, dihedral, semi-dihedral, quasi-dihedral or generalised quaternion, and F is a fusion system on P, then D(P, F) = D(P).

A similar statement for odd p does not hold because Ω_{C_{p}} ≠ 0, whereas Ω_{C_{2}} = 0. More precisely:

Proposition 6.5. Let p be an odd prime and P be an elementary abelian p-group of order p^{2}. Let F be a fusion system on P containing a non trivial automorphism φ ∈ Aut(P) of order dividing p + 1. Then D(P, F) is a proper subgroup of D(P).

Proof. Let C be a subgroup of P of order p such that φ(C) ≠ C. Then, Res_{φ}(Ω_{P/C}) = Ω_{P/φ^{−1}(C)} ≠ Ω_{P/C}. □

Example 6.6. Let p be an odd prime, let G = PSL(3, p), let P be a Sylow-p-subgroup of G and set F = F_{P} (G). Then P is an extraspecial p-group of order p^{3} and exponent p, and its center Z has order p. Set E_{1}, …, E_{p+1} for the p + 1 elementary abelian subgroups of rank 2, and write E_{i} = S_{i} × Z, for some non central subgroup S_{i} of E_{i} of order p, and for all 1 ≤ i ≤ p + 1. By [6],

D(P) = ⟨Ω_{P/Q} | 1 ≤ Q < P ⟩ = ker(Def_{P/Z}^{P}) ⊕ Inf_{P/Z}^{D}(P/Z) ≅ ℤ^{p+2} ⊕ (ℤ/2)^{p+2}.

More precisely, let

e_{0} = Ω_{P} , e_{i} = Ω_{P/S_{i}}, 1 ≤ i ≤ p + 1, e_{p+2} = Ω_{P/Z}, and e_{i} = Ω_{P/E_{i−p−2}}, p + 3 ≤ i ≤ 2p + 3.

Then, ker(Def_{P/Z}^{P}) is generated by e_{i}, 0 ≤ i ≤ p+1, subject to the relation 2e_{0} = ∑_{i=1}^{p+1} 2e_{i}; whereas Inf_{P/Z}^{D}(P/Z) is generated by all the other e_{i}’s, subject to the relations 2e_{i} = 0, for all p + 3 ≤ i ≤ 2p + 3. Moreover, T (P) ≅ ℤ^{p+1} is generated by e_{0} and 2e_{i}, for 1 ≤ i ≤ p + 1. By Alperin’s fusion theorem, the F-stable elements are detected upon F-stability under automorphisms of P, and of the F-essential subgroups. There are two elementary abelian subgroups of order p^{2} that are F-essential. We choose notation such that E_{1} and E_{2} are essential (then the poset of the non trivial p-subgroups of the quotient group N_{G}(E_{i})/E_{i}C_{G}(E_{i}) is not connected, for i = 1, 2). Moreover, E_{3}, …, E_{p+1} form a single F-conjugacy class. The group D(P, F) is generated by the elements

f_{0} = e_{0}, f_{1} = e_{p+4} + e_{1}, f_{2} = e_{p+3} + e_{2}, f_{3} = ∑_{i=3}^{p+1} e_{i} and f_{4} = ∑_{i=p+5}^{2p+3} e_{i},

subject to the relations 2f_{0} − ∑_{i=1}^{3} 2f_{i} = 0 and 2f_{4} = 0.

Example 6.7. Let P be an extraspecial group of order 7^{3} and exponent 7. By [22], there is an exotic fusion system F on P, where the F-centric F-radical subgroups of P are the 8 elementary abelian subgroups of order 49. In the same notation as above, with p = 7, we may assume that E_{1} and E_{2} are F-conjugate, and E_{3}, …, E_{8} form a single F-conjugacy class. Moreover, any morphism in F preserves Z. By [6], D(P) ≅ ℤ^{9} ⊕ (ℤ/2)^{9}, and T (P) ≅ ℤ^{3}. A direct computation yields D(P, F) ≅ ℤ^{3} ⊕(ℤ/2)^{3}, and also T (P, F) = T (P), because Z is normalised by F. Explicitly, ker(Def_{P/Z}^{P}) ∩ D(P, F) ≅ ℤ^{2} is generated by e_{0}, e_{1} + e_{2} and e_{3} + ⋯ + e_{8} subject to the relation 2e_{0} = ∑_{i=1}^{8} 2e_{i}, and Inf_{P/Z}^{P}(P/Z) ∩ D(P, F) ≅ ℤ ⊕ (ℤ/2)^{3} is generated by e_{9}, e_{10} + e_{11} and e_{12} + ⋯ + e_{17} subject to the relations 2e_{10} + 2e_{11} = 0 and ∑_{i=12}^{17} 2e_{i} = 0. Finally, T (P, F) ≅ ℤ^{2} is generated by e_{0}, 2e_{1} + 2e_{2} and 2e_{3} + ⋯ + 2e_{8}.

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THE DADE GROUP OF A FUSION SYSTEM 17

Markus Linckelmann, University of Aberdeen, Aberdeen, UK Nadia Mazza, University of Aberdeen, Aberdeen, UK