THE BURNSIDE RING OF FUSION SYSTEMS

ANTONIO DÍAZ AND ASSAF LIBMAN

Abstract. Given a saturated fusion system F on a finite p-group S we show that after closing the “centric orbit category” O(F^{c}) of F to coproducts, the resulting category is also closed to products and pullbacks. There results a ring A(F) modeled on the Burnside ring A(G) of finite groups. We show that these rings have several properties in common. When F is the fusion system of G we describe the relationship between these rings. Our results also set up a suitable framework to define and study Mackey functors for F.

1. Introduction

Let G be a finite group. The category of finite G-sets is closed under formation of disjoint unions X ⊔ Y and products X × Y. The set of isomorphism classes of finite G-sets therefore forms a commutative monoid under the operation ⊔. Its Grothendieck group completion is denoted A(G). Disjoint unions (coproducts) of G-sets distribute over products whence products of G-sets render A(G) a commutative ring. This is the Burnside ring of G, and it is one of the fundamental representation rings of G (see [1] for a survey on the subject).

The category of G-sets contains the full subcategory O_{G} of the transitive G-sets called the “orbit category of G”. Every object of O_{G} is isomorphic to the G-set G/H of the left cosets of H for some H ≤ G. Also, G/H is isomorphic to G/K if and only if H and K are conjugate in G. It is a fact that every finite G-set is a coproduct of finitely many orbits. This gives the following description of A(G). Its underlying group is a free abelian group whose basis is the set of isomorphism classes of the transitive G-sets. This set can be identified with the set of the conjugacy classes [H] of the subgroups H ≤ G. The product of basis elements [H] and [K] is given by the double coset formula:

(1.1)  [H] · [K] = ∑_{g∈K\G/H}[K^{g} ∩ H]

where K\G/H denotes the set of double-cosets KgH and K^{g} = g^{−1}Kg.

One may wish to reverse this procedure in the following way. Given a small category C, its additive extension is a category C_{⊔} which is obtained from C by introducing formal finite coproducts. See e.g. [11]. The reader should have in mind the inclusion of O_{G} in the category of finite G-sets above. An explicit construction of C_{⊔} appears in Section 4 below. One would then define A(C) as the Grothendieck group completion of the commutative monoid of the isomorphism classes of the objects of C_{⊔}. It is a free abelian group whose basis is the set of the isomorphism classes of the objects of C. In general, however, C_{⊔} need not have products and therefore there is no natural way in which A(C) can be given a ring structure as it is the case when C = O_{G}.

Date: November 26, 2007.

A finite group G and a choice of a Sylow p-subgroup S in G give rise to a fusion system F_{S}(G) over S. This is a small category whose objects are the subgroups of S and it contains all the p-local information of G. The more general concept of a “saturated fusion system F on a finite p-group S” was introduced by Lluis Puig in unpublished notes and will be recalled in Section 2. A saturated fusion system F over S is associated with an orbit category O(F), see e.g. [3] and Definition 2.4. Even when F is the fusion system of a finite group G, O(F) is very different from O_{G} or any of its subcategories (to some extent, the choice of name for O(F) is a little unfortunate.) We shall prove the following result which is contained in the more elaborate statements of Corollary 4.8 and Theorem 4.12.

1.2. Theorem. Let F be a saturated fusion system over a finite p-group S. Let O(F^{c}) be the full subcategory of the orbit category O(F) whose objects are the Fcentric subgroups of S. Then O(F^{c})_{⊔} has finite products and pullbacks.

As a consequence we can define the Burnside ring A(F) of any saturated fusion system as the Grothendieck group completion of the monoid of the isomorphism classes of the objects of O(F^{c})_{⊔}. We shall describe A(F) in Proposition 4.10 and prove that A(F) has similar properties to the Burnside ring of a finite group (so the choice of name for O(F) is not so inappropriate after all).

1.3. Theorem. Let F be a saturated fusion system over a finite p-group S. Then ℚ ⊗ A(F) is a commutative semisimple ℚ-algebra with one primitive idempotent for every isomorphism class of objects of O(F^{c}).

In fact, we will prove this result and describe the primitive idempotents of ℚ ⊗ A(F) in Theorem 5.3.

Now let G be a finite group. It is easy to see that the subgroup of A(G) generated by the G-sets all of whose isotropy groups are finite p-groups forms an ideal A(G; p) in A(G). We will show this in Section 5. The identity in A(G) corresponds to the G-set with one element. This G-set is not present in A(G; p) and in general this subring of A(G) does not have a unit. Given a saturated fusion system let A(F)_{(p)} denote ℤ_{(p)} ⊗ A(F). We will prove in Corollary 5.5

1.4. Theorem. For any saturated fusion system F the ring A(F)_{(p)} has a unit.

The ring A(G; p) is not local in general as it exhibits 𝔽_{q} as a quotient ring for several primes q (see [7]). However, we will give a description of the prime spectrum of A(F)_{(p)} in Corollary 5.10 which in particular includes the following statement.

1.5. Theorem. For any saturated fusion system F the ring A(F)_{(p)} is a local ring.

When F is the fusion system of a finite group G we expect to find a relationship between the Burnside rings A(G) and A(F). To do this we now define a certain section of the ring A(G).

Let A(G; ¬p - cent) denote the subgroup of A(G) generated by the G-sets all of whose isotropy groups are p-subgroups which are not p-centric subgroups of G (see Section 2 below). We will see that this is an ideal of A(G) which is contained in A(G; p). The quotient ring A(G; p)/A(G, ¬p - cent) is denoted A^{p - cent}(G). We shall write A^{p - cent}(G)_{(p)} for ℤ_{(p)} ⊗ A^{p - cent}(G).

1.6. Theorem. Let S be a Sylow p-subgroup of a finite group G and let F denote the associated fusion system. Then the rings A(F)_{(p)} and A^{p - cent}(G)_{(p)} are isomorphic.

2. Saturated fusion systems

A fusion system F on a finite p-group S is a category whose objects are the subgroups of S, and whose morphism sets Hom_{F}(P, Q), where P, Q ≤ S, consist of group monomorphisms which satisfy the following two conditions:

(a) The set Hom_{S}(P, Q) of all the homomorphisms P → Q which are induced by conjugation by elements of S is contained in Hom_{F}(P, Q). In particular all the inclusions P ≤ Q are morphisms in F.

(b) Every morphism in F factors as an isomorphism in F followed by an inclusion. It easily follows that φ: P → Q in F is an isomorphism in F if and only if it is an

isomorphism of groups. In this case we say that P and Q are F-conjugate. Another consequence is that the endomorphisms of every object of F are automorphisms and we write Aut_{F}(P) for Hom_{F}(P, P).

Fusion systems form a convenient framework to study the p-local structure of finite groups. Let G be a finite group. For subgroup H, K ≤ G denote

N_{G}(H, K) = {g ∈ G : gHg^{−1} ≤ K}.

Every element g ∈ N_{G}(H, K) gives rise to a group monomorphism c_{g}: H → K where c_{g}(h) = ghg^{−1}. That is, c_{g} is a restriction of the inner automorphism c_{g} of G to H and K.

A Sylow p-subgroup S of G gives rise to a fusion system F_{S}(G) over S. Its objects are the subgroups of S. The morphisms P → Q in F_{S}(G) for P, Q ≤ S are the group monomorphisms c_{g}: P → Q for all g ∈ N_{G}(P, Q). That is, the set of morphisms P → Q in F_{S}(G) is N_{G}(P, Q)/C_{G}(P). The fusion system F_{S}(G) satisfies several crucial axioms which lead L. Puig to consider the class of saturated fusion systems.

2.1. Definition. Let F be a fusion system over a p-group S. A subgroup P ≤ S is called fully centralized in F if |C_{S}(P)| ≥ |C_{S}(P′)| for all P′ which is F-conjugate to P. It is called fully normalized in F if |N_{S}(P)| ≥ |N_{S}(P′)| for all P′ which is F-conjugate to P.

The fusion system F is called saturated if the following two conditions hold: (I) If P ≤ S is fully normalized then it is fully centralized and Aut_{S}(P) is a

Sylow p-subgroup of Aut_{F}(P). (II) For every P ≤ S and φ ∈ Hom_{F}(P, S) set

N_{φ} = {g ∈ N_{S}(P) | φ ∘ c_{g} ∘ φ^{−1} ∈ Aut_{S}(φP)},

If φ(P) is fully centralized then there is \overline{φ} ∈ Hom_{F}(N_{φ}, S) such that \overline{φ}|_{P} = φ.

Using the inclusion Q ≤ S, every morphism φ: P → Q in F can be thought of as a morphism P → S. We shall write N_{φ} for N_{incl_{Q}^{S}∘φ}. It is also clear that N_{φ} contains P and C_{S}(P) and that N_{φ} is the largest subgroup R of N_{S}(P) such that φ extends to a morphism Q → S in F. The fusion system F = F_{S}(G) associated to a finite group G is saturated by [3, Proposition 1.3].

2.2. Definition. Let F be a fusion system over a p-group S. A subgroup P ≤ S is F-centric if P and all its F-conjugates contain their S-centralizers.

Note that P is F-centric if and only if C_{S}(P′) = Z(P′) for any P′ ≤ S which is F-conjugate to P. In particular if P is F-centric then all its F-conjugates are fully centralized in F. In addition any subgroup of S which contains P must also be F-centric.

The collection of the F-centric subgroups has an alternative group theoretic description.

2.3. Definition. A p-subgroup P ≤ G is p-centric if Z(P) is a Sylow p-subgroup of C_{G}(P).

Equivalently, P is p-centric in G if C_{G}(P) = Z(P) × C′_{G}(P) where C′_{G}(P) is a subgroup of C_{G}(P) of order prime to p which is generated by the elements of C_{G}(P) of order prime to p. In particular C′_{G}(P) is characteristic in C_{G}(P). It is easy to see that P ≤ S is F_{S}(G)-centric if and only if it is p-centric in G.

2.4. Definition. The orbit category O(F) of a fusion system F over S has the same object set as F and the set of morphisms P → Q is the set Hom_{F}(P, Q) modulus the action of Inn(Q) by postcomposition. The category F^{c} is the full subcategory of F on the set of the F-centric subgroups. The category O(F^{c}) is the full subcategory of O(F) on the object set of F^{c}.

The morphisms P → Q in O(F) will be denoted by [φ] for some φ: P → Q in F. Thus, [φ] = [ψ] in O(F) if and only if there exists some x ∈ Q such that ψ = c_{x} ∘ φ as morphisms in F. It is easy to see that every endomorphism of P in O(F) is an isomorphism and we write Out_{F}(P) for the automorphism group of P in O(F). The following Lemma will be used extensively throughout this paper

2.5. Lemma ([3, Lemma A.8]). Let F be a saturated fusion system over S. Consider an F-centric subgroup P and two morphisms φ, ψ: Q → S in F such that P ≤ Q and φ|_{P} = ψ|_{P}. Then there exists z ∈ C_{S}(P) such that φ = ψ ∘ c_{z}.

2.6. Corollary. Let F be a saturated fusion over S. Then every morphism in O(F^{c}) is an epimorphism (in the category-theory sense.)

Proof. Let [α]: P → Q and [φ], [ψ]: Q → R be morphisms in O(F^{c}) such that [φ] ∘ [α] = [ψ] ∘ [α]. We have to show that [φ] = [ψ]. By definition, there exists some x ∈ R such that φ ∘ α = c_{x} ∘ ψ ∘ α in F. Since α is a group monomorphism this is equivalent to

φ|_{α(P)} = c_{x} ∘ ψ|_{α(P)} as homomorphisms α(P) → S.

Now, α(P) is F-centric because P is F-centric and Lemma 2.5 implies that there exists z ∈ C_{S}(α(P)) = Z(α(P)) = α(Z(P)) ≤ Q such that

φ = c_{x} ∘ ψ ∘ c_{z} = c_{x} ∘ c_{ψ(z)} ∘ ψ.

Hence [φ] = [ψ] because ψ(z), x ∈ Q. □

Another consequence of Lemma 2.5 is the following useful result

2.7. Proposition. Let F be a saturated fusion system over S and let P, Q ≤ S be F-centric subgroups. Consider a morphism φ: Q → P in F and an element s ∈ N_{S}(Q) such that c_{x} ∘ φ = φ ∘ c_{s} for some x ∈ P. Then there exists a subgroup Q′ ≤ S and a morphism φ′: Q′ → P in F such that Q ≤ Q′ and s ∈ Q′ and φ′|_{Q} = φ.

Proof. By axiom (II) of saturation there exists a morphism φ̃ ∈ F(N_{φ}, S) which extends incl_{P}^{S} ∘ φ. Our hypothesis on φ and s implies that s ∈ N_{φ}. Let Q′ be the subgroup of N_{φ} generated by Q and s and set ψ = φ̃|_{Q′}.

Consider now c_{x} ∘ψ and ψ ∘c_{s} in F(Q′, S). By hypothesis, their restrictions to Q are equal and by Lemma 2.5 there exists some z ∈ C_{S}(Q) such that c_{x}∘ψ = ψ∘c_{s}∘c_{z}. Now, C_{S}(Q) = Z(Q) because Q is F-centric and therefore sz ∈ Q′. It follows that ψ ∘ c_{sz} = c_{ψ(sz)} ∘ ψ, hence c_{x} ∘ ψ = c_{ψ(sz)} ∘ ψ. Therefore

x^{−1}ψ(sz) ∈ C_{S}(ψ(Q′)) ≤ C_{S}(ψ(Q)) = C_{S}(φ(Q)) ≤ φ(Q) ≤ P.

We see that ψ(sz) ∈ P. Also z ∈ Z(Q) ≤ Q so ψ(z) = φ(z) ∈ P. We deduce that ψ(s) ∈ P and therefore ψ(Q′) ≤ P because ψ(Q) = φ(Q) ≤ P. Thus, ψ = incl_{P}^{S} ∘φ′ for some φ′ ∈ F(Q′, P) and it follows that ψ|_{Q} = incl_{P}^{S} ∘ φ, whence φ′|_{Q} = φ. □

Fix a saturated fusion system F over S and let C denote O(F^{c}); See Definition 2.4. Consider F-centric subgroups K, P ≤ S and observe that Out_{F}(K) acts on C(K, P) by precomposition. The fixed point set of [α] ∈ Out_{F}(K) is denoted as usual by C(K, P)^{[α]}. In particular, Out_{S}(K) whose elements are denoted [c_{s}] for s ∈ N_{S}(K), acts on C(K, P) and the set of orbits is denoted C(K, P)/ Out_{S}(K).

2.8. Proposition. Let F be a saturated fusion system over S and let C denote O(F^{c}). Consider F-centric subgroups K, P ≤ S and let H be the subgroup of S which is generated by K and some s ∈ N_{S}(K). Then there is a bijection C(H, P) ≈ C(K, P)^{[c_{s}]} which is induced by the assignment [φ] ↦ [φ|_{K}].

Proof. If [φ] ∈ C(H, P) then

[φ|_{K}] ∘ [c_{s}] = [φ ∘ c_{s}|_{K}] = [c_{φ(s)} ∘ φ|_{K}] = [φ|_{K}]

because s normalises K and φ(s) ∈ P. This shows that restriction [φ] ↦ [φ|_{K}] induces a well defined map r: C(H, P) → C(K, P)^{[c_{s}]}. It is injective because by Corollary 2.6 the inclusion K ≤ H is an epimorphism in C.

If [φ] ∈ C(K, P)^{[c_{s}]} then there exists some x ∈ P such that c_{x} ∘ φ = φ ∘ c_{s}. Proposition 2.7 implies that φ extends to a morphism φ′: H → P and in particular r([φ′]) = [φ]. This shows that r is also surjective. □

As a corollary we obtain:

2.9. Proposition. For any P ≤ P′ in F^{c} we have |C(P, P′)| = | Out_{F}(P′)| mod p. In particular, |C(P, S)| = | Out_{F}(S)| mod p for any P ∈ F^{c}.

Proof. We induct on n = |P′ : P |. When n = 1 the result is trivial. If n > 1 choose some Q ≤ P′ such that P ◁ Q and Q/P = p. Now, Q/P is a cyclic group of order p which acts on C(P, P′) by precomposition. By Proposition 2.8 restriction gives rise to a bijection

C(Q, P′) ≈ C(P, P′)^{Q/P} .

It follows that |C(Q, P′)| = |C(P, P′)| mod p because every orbit of Q/P have either p elements or exactly one element. By induction hypothesis |C(Q, P′)| = | Out_{F}(P′)| mod p and the result follows. □

3. A lemma on the orbit category of a saturated fusion system

In this section we address the question of extending morphisms in the orbit category of a saturated fusion system.

3.1. Definition. Let C be a small category. A system of pseudo-inclusions in C consists of a poset structure (C, ≤) on the object set of C together with a choice of morphisms ι_{A}^{B} ∈ C(A, B) for any pair of objects A ≤ B such that ι_{A}^{A} = id_{A} and ι_{B}^{C} ∘ ι_{A}^{B} = ι_{A}^{C} if A ≤ B ≤ C.

Given A ≤ B and a morphism φ: B → T the restriction of φ to A denoted φ|_{A} is the morphism φ ∘ ι_{A}^{B}. An extension of a morphism α: A → T is a morphism β: B → T such that A ≤ B and β|_{A} = α. In this case we write α ⪯ β.

An extension α ⪯ β is called atomic if whenever γ is a morphism in C such that α ⪯ γ ⪯ β then either γ = α or γ = β.

The set Mor(C) of all the morphisms in C is partially ordered by ⪯ and induces a partial ordering on the set Ext(α) of all the extensions of a given morphism α.

From the definitions it clearly follows that if the domain of a morphism φ is A then φ|_{A} = φ. Also, if C ≤ B ≤ A then φ|_{C} = (φ|_{B})|_{C}. Thus, the relation ⪯ is transitive and reflexive.

3.2. Example. Let F be a fusion system on a finite p-group S. The orbit category O(F) has a natural choice of a sytem of pseudo-inclusions by partially ordering the F-centric subgroups of S by inclusion and by setting ι_{P}^{Q} to be the image in O(F) of the inclusion P ≤ Q.

3.3. Lemma. Let C be a small category with a system of pseudo-inclusions such that the following hold.

(1) There is a “height function” h: obj C → ℕ with the following properties. For any two objects A and B, if there exists a morphism f: A → B then h(A) ≤ h(B). If A ≤ B and h(A) = h(B) then A = B.

(2) Fix a morphism α in C. If β and γ are atomic extensions of α then there

exists a morphism ϵ in C which extends β and γ. Then for any morphism α in C the set Ext(α) has a maximum.

Proof. We begin by showing that any non trivial extension contains a non-trivial atomic one. More precisely Sublemma 1. Consider a non-trivial extension φ ⪵ ψ in Mor(C). Then there exists a morphism φ′ such that φ ⪵ φ′ ⪯ ψ and such that φ ⪵ φ′ is atomic. Proof of Sublemma 1. The hypothesis that ψ: V → T is an extension of φ: U → T means that U ≤ V and φ = ψ|_{U}. Moreover, U ⪇ V because φ ≠ ψ and therefore h(U) < h(V) by hypothesis (1). The set of all the objects X ∈ C such that U ⪇ X ≤ V is not empty since it contains V and we choose in it an element U′ such that h(U′) is minimal. Set φ′ = ψ|_{U′} and note that φ′ ≠ φ because U ≠ U′. It is clear that φ ⪵ φ′ ⪯ ψ. Also φ ⪵ φ′ is atomic by the minimality of h(U′). Q.E.D.

Next, we strengthen the statement in hypothesis (2) by showing that if two elements of Mor(C) have a lower bound then they have an upper bound. More precisely Sublemma 2. For any morphisms α, β, γ in C such that α ⪯ β, γ there exists a morphism ϵ such that β, γ ⪯ ϵ.

Proof of Sublemma 2. For every morphism φ: X → Y in C set |φ| = h(Y) − h(X). By hypothesis (1) we see that |φ| ≥ 0. Also note that if φ ⪯ ψ then |ψ| ≤ |φ| because dom(φ) ≤ dom(ψ) and both φ and ψ have the same codomain. Hypothesis (1) also implies that if φ ⪵ ψ then |ψ| < |φ|.

We shall prove Sublemma 2 by induction on |α|. Assume that the Sublemma holds whenever |α| < d for some integer d ≥ 0. Consider a morphism α such that |α| = d and morphisms β, γ such that α ⪯ β, γ. If α = β then set ϵ = γ and note that β, γ ⪯ ϵ as needed. Similarly, if α = γ we choose ϵ = β.

We therefore assume that α ⪵ β and α ⪵ γ. By Sublemma 1, there are morphisms β′ ⪯ β and γ′ ⪯ γ such that α ⪵ β′ and α ⪵ γ′ are atomic extensions. By hypothesis (2) there exists a morphism ϵ_{1} in C such that

β′, γ′ ⪯ ϵ_{1}.

Now, |β′| < |α| = d because α ⪵ β′. Since β′ ⪯ β and β′ ⪯ ϵ_{1} we may apply the induction hypothesis to find a morphism ϵ_{2} such that

β, ϵ_{1} ⪯ ϵ_{2}.

Since α ⪵ γ′ we deduce that |γ′| < |α| = d. Also, γ′ ⪯ γ and γ′ ⪯ ϵ_{1} ⪯ ϵ_{2}. By induction hypothesis there exists a morphism ϵ in C such that γ, ϵ_{2} ⪯ ϵ. Since β ⪯ ϵ_{2} ⪯ ϵ we deduce that β, γ ⪯ ϵ. This completes the induction step and the prood of the Sublemma. Q.E.D.

We now complete the proof of the Lemma. Choose an element ϵ ∈ Ext(α) such that |ϵ| is smallest possible. This is clearly a maximal element in Ext(α) because if ϵ ⪵ ϵ′ then |ϵ′| < |ϵ|. By Sublemma 2, any two elements in Ext(α) have an upper bound. It follows that ϵ must be the maximum of Ext(α). □

Let F be a saturated fusion system and let O(F) denote its orbit category. There is a projection functor π: F^{c} → O(F^{c}) and the image of a morphism φ in F is denoted [φ]. By Example 3.2, O(F) has a natural system of pseudo-inclusions. There results a notion of restriction of morphisms [φ]: Q → T in O(F) where [φ]|_{P} = [φ] ∘ [incl_{P}^{Q}] = [φ|_{P}]. An extension of a morphism [φ]: P → T in O(F) is a morphism [ψ]: Q → T such that P ≤ Q and [φ] = [ψ|_{P}].

We shall now prove the following result about morphisms in the full subcategory O(F^{c}) of O(F) generated by the F-centric subgroups of S.

3.4. Proposition. Let [φ]: P → T be a morphism in O(F^{c}). Then the set of all the extensions of [φ] has a maximum. That is, there exists a unique morphism [ψ]: Q → T which extends any extension of [φ].

Proof. This is a direct consequence of Lemma 3.3 once we show that its hypotheses hold in O(F^{c}).

(1) Set h : O(F^{c}) → ℕ by h(A) = log_{p}(|A|). The required properties of h are a straightforward consequence of the axioms of F.

(2) Fix a morphisms [α]: A → T. Let [β]: B → T and [γ]: C → T be atomic extensions of [α], that is, A ≤ B, C and [α] = [β|_{A}] = [γ|_{A}]. Since [α] ⪯ [β] is atomic, there can exist no proper subgroup of B which contains A properly unless A = B. Since B is a finite p-group it follows that either A = B or |B : A| = p. Similarly, since [α] ⪯ [γ] is atomic, either A = C or |C : A| = p. It follows from [10, Theorem 4.3.2] that A ⊴ B, C, namely B, C ≤ N_{S}(A).

Since [β|_{A}] = [α] and [γ|_{A}] = [α], there are t_{β}, t_{γ} ∈ T such that α = c_{t_{β}} ∘ (β|_{A}) and α = c_{t_{γ}} ∘ (γ|_{A}). Since [β] = [c_{t_{β}} ∘ β] and [γ] = [c_{t_{γ}} ∘ γ], by possibly replacing β and γ with their composition with c_{t_{β}} and c_{t_{γ}} we may assume that

β|_{A} = α  and  γ|_{A} = α.

Let j : T → S denote the inclusion as a morphism in F^{c}. Since A is F-centric, the image of j ∘ α is fully F-centralized and therefore j ∘ α extends to a morphism φ: N_{j∘α} → S in F. Since B, C ≤ N_{S}(A) and β and γ extend α it follows that B, C ≤ N_{j∘α}.

Let E denote the subgroup of S generated by B and C and set λ = φ|_{E}. Note that E is F-centric because it contains A which is itself F-centric, that is, E is an object in O(F^{c}). By precomposing the following morphisms in F

B ⟶^{incl} E ⟶^{λ} S  and  B ⟶^{β} T ⟶^{j} S

with the inclusion A ≤ B we obtain the homomorphism j ∘ α in F. We deduce from Lemma 2.5 that there exists x ∈ Z(A) ≤ B such that

λ|_{B} = j ∘ β ∘ c_{x} = j ∘ c_{β(x)} ∘ β.

In particular λ(B) ≤ T because β(x) ∈ T. Similarly, since (λ|_{C})|_{A} = j ∘ α = (j ∘ γ)|_{A}, there exists y ∈ Z(A) ≤ C such that

λ|_{C} = j ∘ γ ∘ c_{y} = j ∘ c_{γ(y)} ∘ γ

and therefore λ(C) ≤ T because γ(y) ∈ T. Since E is generated by B and C we deduce that λ(E) ≤ T. By the factorization axiom in F we deduce that λ = j ∘ ϵ for some ϵ: E → T in F. Thus,

j ∘ ϵ|_{B} = λ|_{B} = j ∘ c_{β(x)} ∘ β  and  j ∘ ϵ|_{C} = λ|_{C} = j ∘ c_{γ(y)} ∘ γ.

Since j is a group monomorphism, we conclude that ϵ|_{B} = c_{β(x)} ∘ β and ϵ|_{C} = c_{β(y)}∘γ. Let [ϵ] denote the image of ϵ in O(F^{c}). Then [ϵ|_{B}] = [β] and [ϵ|_{C}] = [γ]. □

3.5. Remark. By [3, Proposition 1.11] any p-local finite group (S, F, L) can be equipped with a choice of morphisms ι_{P}^{Q} in L for any inclusion P ≤ Q of F-centric subgroups. These morphisms are lifts for the inclusions P ≤ Q and equip L with a system of pseudo-inclusions. Thus, we have a notion of extension of morphisms in L. It can be deduced from Proposition 3.4 that the set of all the extensions of a given morphism in L contains a maximum. We omit the proof.

4. The Burnside ring of saturated fusion systems

Let C be a small category. Assume that C has coproducts ∐ and products × and that coproducts are distributive over the product. The Burnside ring of C is the ring A(C) whose underlying group is the Grothendieck group completion of the commutative monoid formed by the set of isomorphism classes of the objects of C under the operation ∐. Thus, as an abelian group A(C) is the free abelian group whose basis is the equivalence classes [A] of the objects of C modulus the relation [A ∐ B] = [A] + [B]. The ring structure of A(C) is induced by the product in C. See e.g. [11].

To get round the problem that not every category has coproducts we consider the following construction. First, observe that if D is a category with coproducts and C is a full subcategory whose objects are closed under coproducts (resp. products) in D then C has coproducts (resp. products). If the coproduct distributes over the product in D then this is the case also in C.

Let C be a small category and let Ĉ denote the category of the contravariant functors C → Sets. Recall that morphisms in Ĉ are the natural transformations of functors. Furthermore, Yoneda constructed an embedding of C as a full subcategory of Ĉ via the functor 𝔶: C → Ĉ where

𝔶: C ↦ C(−, C).

Yoneda’s Lemma asserts that the functor 𝔶(C) := C(−, C) has the property that for any other functor F: C^{op} → Sets there is a natural bijection Nat(𝔶(C), F) ≈ F (C) and this shows that 𝔶 is a full embedding of C in Ĉ.

The category Ĉ has coproducts and products induced from these constructions in the category of sets. In particular the coproduct in Ĉ is distributive over the product. The image of 𝔶 consists of the functors {C(−, C)}_{C∈C} and we let C_{⊔} denote the full subcategory of Ĉ whose objects are the finite coproducts of the image of 𝔶. By construction C_{⊔} has finite coproducts and we call it the additive extension of C. We remark that C is a full subcategory of C_{⊔} and every object of C_{⊔} is a coproduct of finitely many objects in C. We shall therefore write ⊔_{i}C_{i}, where C_{i} are finitely many objects in C, for a typical object of C_{⊔}. Also Yoneda’s Lemma implies that for an object A ∈ C and an object ⊔_{i}B_{i} ∈ C_{⊔}

(4.1)  C_{⊔}(A, ⊔_{i}B_{i}) = ∐_{i} C(A, B_{i}).

Clearly C_{⊔} is also closed to products in Ĉ if and only if for any objects A, B ∈ C there are objects C_{1}, …, C_{n} ∈ C and an isomorphism of functors

(4.2)  C(−, A) × C(−, B) ≅ ∐_{i=1}^{n} C(−, C_{i}).

For example let C denote the orbit category O_{G} of a finite group G. We have seen in the introduction that C is equivalent to a small category and that C_{⊔} is equivalent to the category of finite G-sets. Thus, postpriori, we see that C_{⊔} has products. But in general, for an arbitrary small category C, there is no reason for this to happen.

For the remainder of this section we fix a saturated fusion system F over S. The goal is to prove that the additive extension of O(F^{c}) has products. Throughout C denotes O(F^{c}).

The image in O(F) of a morphism α in F is denoted [α]. Recall from Example 3.2 that C = O(F^{c}) has a system of pseudo-inclusions where the object set is partially ordered by inclusions of groups and restriction of a morphism [α]: A → Q in C to B ≤ A is [α]|_{B} = [α|_{B}]. We say that [γ] : C → Q extends [α] if A ≤ C and [α] = [γ|_{A}]. We write [α] ⪯ [γ].

Our goal now is to show that C_{⊔} where C = O(F^{c}) has finite products. By the earlier discussion, given A, B ∈ C, we have to find objects C_{1}, …, C_{n} ∈ C which fit into Equation (4.2). This is the content of Theorem 4.6 below. The next definition is motivated by the fact that in Equation (1.1) the subgroup K^{g} ∩ H is precisely the maximal subgroup of H such that conjugation by g^{−1} lands in K.

4.3. Definition. Let P, Q ≤ S be F-centric subgroups of S. Denote the set of all the morphisms [α]: A → Q in O(F^{c}) where A ≤ P is F-centric by K_{P,Q}.

The set K_{P,Q} is partially ordered by the relation ⪯ of extension. The set of the maximal elements of K_{P,Q} under this relation is denoted K_{P,Q}^{max}.

Fix [α]: A → Q in K_{P,Q} and an element x ∈ P. Clearly A^{x} = x^{−1}Ax is an F-centric subgroup of P and we define an element [α] · x in K_{P,Q} by

[α] · x = [α ∘ c_{x}], where c_{x}: A^{x} → A is conjugation.

There results an action of P on K_{P,Q} which is easily seen to be order preserving. In particular P acts on K_{P,Q}^{max} and we define

4.4. Definition. Consider F-centric subgroups P, Q of S. Let Q\F/P denote the set of orbits K_{P,Q}^{max}/P. The orbit of [α] ∈ K_{P,Q} is denoted {[α]}.

Since the category F is finite, it is clear that the sets P \F/Q are finite.

4.5. Lemma. For every [α] ∈ K_{P,Q} there exists a unique maximal element [β] in K_{P,Q}^{max} such that [α] ⪯ [β].

Proof. Fix [α]: A → Q in K_{P,Q}. In particular A ≤ P is F-centric. By Lemma 3.4 the set of all possible extensions of [α] has a maximum [ϵ]: E → Q. Set B = E ∩ P and [β] = [ϵ|_{B}]. Clearly, [β] ∈ K_{P,Q} and α ⪯ β. Assume that [α] ⪯ [γ] for some [γ] ∈ K_{P,Q}. The maximality of [ϵ] implies that [γ] = [ϵ|_{C}] where C ≤ P is the domain of γ. Therefore C ≤ E ∩ P = B, whence [γ] = [ϵ|_{C}] ⪯ [ϵ|_{B}] = [β]. □

4.6. Theorem. Let F be a saturated fusion system over S and let C denote O(F^{c}). Fix F-centric subgroups P, Q ≤ S and fix representatives [α_{i}]: A_{i} → Q for the elements {[α]} in P \F/Q. Then there exists an isomorphism of functors

λ: ∐_{\{[α_{i}]\}∈Q\F/P} C(−, A_{i}) \underrightarrow{([incl_{A_{i}}^{P}]_{∗} , [α_{i}]_{∗})} C(−, P) × C(−, Q).

That is, λ is induced by postcomposition of morphisms [θ]: H → A_{i} with [incl_{A_{i}}^{P}] and [α_{i}].

Proof. For every i, the morphisms [incl_{A_{i}}^{P}] and [α_{i}] induce natural transformations

C(−, A_{i}) ⟶^{[incl_{A_{i}}^{P}]_{∗}} C(−, P)  and  C(−, A_{i}) ⟶^{[α_{i}]_{∗}} C(−, Q)

via postcomposition. In turn, they give rise to natural transformations

λ^{{[α_{i}]}}: C(−, A_{i}) → C(−, P) × C(−, Q), [θ] ↦ ([incl_{A_{i}}^{P}] ∘ [θ] , [α_{i}] ∘ [θ]).

Their sum gives rise to the natural transformation λ.

In order to show that λ is a natural isomorphism of functors we need to show that for every H ∈ C the function

(1)  λ_{H}: ∐_{{[α_{i}]}∈Q\F/P} C(H, A_{i}) → C(H, P) × C(H, Q)

is a bijection. Claim. λ_{H} is surjective.

Fix [φ] ∈ C(H, P) and [ψ] ∈ C(H, Q). Set B = φ(H) and let φ̄: H → B denote the restriction of φ to an isomorphism onto its image. By the axioms of F we know that φ̄ ∈ F(H, B) and since H is F-centric, then B ≤ P is also F-centric.

Now, [ψ ∘φ̄^{−1}] ∈ K_{P,Q} and since the [α_{i}]’s form a complete set of representatives for K_{P,Q}^{max}/P,

(2)  [ψ ∘ φ̄^{−1}] · x ⪯ [α_{i}]  for some x ∈ P and some i.

In particular B^{x} ≤ A_{i} where B^{x} = x^{−1}Bx, and by definition

(3)  [ψ ∘ φ̄^{−1} ∘ c_{x}] = [α_{i}|_{B^{x}}].

Consider the morphism θ ∈ F(H, A_{i}) defined by

θ = incl_{B^{x}}^{A_{i}} ∘ c_{x^{−1}} ∘ φ̄.

Since x ∈ P we see that

[incl_{A_{i}}^{P}] ∘ [θ] = [incl_{A_{i}}^{P} ∘ incl_{B^{x}}^{A_{i}} ∘ c_{x^{−1}} ∘ φ̄] = [incl_{B^{x}}^{P} ∘ c_{x^{−1}} ∘ φ̄]

= [c_{x^{−1}} ∘ incl_{B}^{P} ∘ φ̄] = [c_{x^{−1}} ∘ φ] = [φ].

Taking (3) into account we also see that

[α_{i}] ∘ [θ] = [α_{i} ∘ incl_{B^{x}}^{A_{i}} ∘ c_{x^{−1}} ∘ φ̄] = [α_{i}|_{B^{x}}] ∘ [c_{x^{−1}} ∘ φ]

= [ψ ∘ φ̄^{−1} ∘ c_{x}] ∘ [c_{x^{−1}} ∘ φ̄] = [ψ].

This finally shows that λ_{H}([θ]) = ([φ], [ψ]) and that λ_{H} is surjective. Claim. λ_{H} is injective.

Consider morphisms θ_{1} : H → A_{i_{1}} and θ_{2} : H → A_{i_{2}} in F where H ≤ S is F-centric, such that λ_{H}([θ_{1}]) = λ_{H}([θ_{2}]). That is,

(4)  [incl_{A_{i_{1}}}^{P}] ∘ [θ_{1}] = [incl_{A_{i_{2}}}^{P}] ∘ [θ_{2}]  and

[α_{i_{1}}] ∘ [θ_{1}] = [α_{i_{2}}] ∘ [θ_{2}]. (5)

We have to show that [θ_{1}] = [θ_{2}].

From (4) it follows that there exists some x ∈ P such that

incl_{A_{i_{1}}}^{P} ∘ θ_{1} = c_{x} ∘ incl_{A_{i_{2}}}^{P} ∘ θ_{2}.

Let θ̄_{1} and θ̄_{2} denote the restriction of θ_{1} and θ_{2} to isomorphisms onto their images B_{1} ≤ A_{i_{1}} and B_{2} ≤ A_{i_{2}}, respectively. It follows that

(6)  θ̄_{1} = c_{x} ∘ θ̄_{2}  (x ∈ P).

In particular B_{2} = B_{1}^{x}. Now (5) and (6) imply

[α_{i_{1}}|_{B_{1}}] · x = [α_{i_{1}}|_{B_{1}} ∘ c_{x}] = [α_{i_{1}}|_{B_{1}} ∘ θ̄_{1} ∘ θ̄_{2}^{−1}] =

[α_{i_{1}}|_{B_{1}} ∘ θ̄_{1}] ∘ [θ̄_{2}^{−1}] = [α_{i_{1}} ∘ θ_{1}] ∘ [θ̄_{2}^{−1}] =

[α_{i_{2}} ∘ θ_{2}] ∘ [θ̄_{2}^{−1}] = [α_{i_{2}}|_{B_{2}} ∘ θ̄_{2}] ∘ [θ̄_{2}^{−1}] = [α_{i_{2}}|_{B_{2}}].

Since the action of P on K_{P,Q} is order preserving it now follows that [α_{i_{2}}|_{B_{2}}] = [α_{i_{1}}|_{B_{1}}] · x ⪯ [α_{i_{1}}] · x. By definition [α_{i_{2}}|_{B_{2}}] ⪯ [α_{i_{2}}] so Lemma 4.5 implies that [α_{i_{2}}] = [α_{i_{1}}] · x. Since the [α_{i}]’s are representatives for the P -orbits of K_{P,Q}^{max} we deduce that i_{1} = i_{2} which we now simply denote by i. Thus, we see that [α_{i}] = [α_{i}]·x which in particular implies that

x ∈ N_{P} (A_{i}).

We examine further [θ_{1}], [θ_{2}] ∈ C(H, A_{i}). Equation (5) implies that there exists some y ∈ Q such that α_{i} ∘ θ_{1} = c_{y} ∘ α_{i} ∘ θ_{2}. Equivalently,

(α_{i}|_{B_{1}}) ∘ θ̄_{1} = c_{y} ∘ (α_{i}|_{B_{2}}) ∘ θ̄_{2}, (y ∈ Q).

Applying (6) we deduce

(7)  (α_{i}|_{B_{1}}) ∘ c_{x} = c_{y} ∘ α_{i}|_{B_{2}}  as morphisms in F(B_{2}, Q).

Since x ∈ N_{P} (A_{i}), we may also consider the morphisms

α_{i} ∘ c_{x}  and  c_{y} ∘ α_{i}  in F(A_{i}, Q).

Their restriction to B_{2} are equal by (7), so it follows from Lemma 2.5 that there exists some z ∈ Z(B_{2}) such that α_{i} ∘ c_{x} ∘ c_{z} = c_{y} ∘ α_{i}. Set w = xz and note that w ∈ N_{P} (A_{i}) because z ∈ B_{2} ≤ A_{i}. Then

(8)  α_{i} ∘ c_{w} = c_{y} ∘ α_{i}  in F(A_{i}, Q).

Proposition 2.7 implies that there exists some γ: C → Q in F where C = ⟨A_{i}, w⟩ which extends α_{i}. In particular C is contained in P and it is F-centric because it contains A_{i}. Therefore [γ] ∈ K_{P,Q}. Also, by construction [α_{i}] ⪯ [γ]. Maximality of [α_{i}] in K_{P,Q} implies that [α_{i}] = [γ] and as a consequence A_{i} = C, whence w ∈ A_{i}. Since w = xz and z ∈ B_{2} ≤ A_{i} it also follows that x ∈ A_{i} and now (6) implies that [θ_{1}] = [θ_{2}]. □

4.7. Remark. Clearly, in the category Ĉ there is an isomorphism

C(−, P) × C(−, Q) ≅ C(−, Q) × C(−, P).

In terms of the isomorphism provided in Theorem 4.6, this isomorphism is obtained from a bijection K_{P,Q}^{max} /P ≈ K_{Q,P} /Q where {[α_{i}]: A → Q} in K_{P,Q}^{max} /P corresponds to {[α_{i}^{−1}]: α_{i}(A_{i}) → P } in K_{Q,P}^{max} /Q.

Recall that O(F^{c}) is a full subcategory of O(F^{c})_{⊔} and that it is natural to write the objects of the latter in the form ⊔_{i}P_{i} for finitely many objects P_{i} ∈ O(F^{c}).

4.8. Corollary. Let F be a saturated fusion system over S and let C denote the centric orbit category O(F^{c}). Then C_{⊔} has finite products. In fact, the product of P, Q ∈ C in C_{⊔} is isomorphic to

⨆_{{[α]}∈Q\F/P}[dom(α)]

and the natural maps to P and Q are induced by the inclusion [incl_{dom(α)}^{P}] and the morphism [α].

Proof. The product of the images of two objects P, Q ∈ C in Ĉ = Funct(C^{op}, Sets) is an object of C_{⊔} by Theorem 4.6. The description of the product of P and Q also follows from this theorem by using Yoneda’s Lemma. □

The following definition now makes sense

4.9. Definition. The Burnside ring A(F) of a saturated fusion system F over S is the Burnside ring of the category O(F^{c})_{⊔}.

From Corollary 4.8 we immediately obtain the following description of A(F).

4.10. Proposition. Let F be a saturated fusion system over S. Then the additive group underlying A(F) is a free abelian group whose basis is the set of the Fconjugacy classes [P] of the F-centric subgroup P ≤ S. The product of two basis elements is given by the formula [P] · [Q] = ∑_{{[α]}∈Q\F/P} [dom(α)].

4.11. Definition. Let P, Q, R be F-centric subgroups of F such that P, Q ≤ R. Define (Q\R/P)_{F} as the set of double cosets QxP where x ∈ R, such that Q^{x} ∩ P is F-centric. This is well defined because for any y ∈ Q and u ∈ P we have Q^{yxu} ∩ P = (Q^{x} ∩ P)^{u}. Hence Q^{x} ∩ P is F-centric if and only if Q^{yxu} ∩ P is F-centric.

Now consider c_{x}: Q^{x} ∩ P → P. Clearly [c_{x}] ∈ K_{P,Q}. There results a function (Q\R/P)_{F} → K_{P,Q}/P where QxP ↦ {[c_{x}]} which is well defined because for any y ∈ Q and u ∈ P

{[c_{yxu}]} = {[c_{y} ∘ c_{x}] ∘ [c_{u}]} = {[c_{x}] · u} = {[c_{x}]} ∈ [Q\F/P]_{R}.

4.12. Theorem. Let F be a saturated fusion system over S. Then all pullbacks in

O(F^{c})_{⊔} exist. The pullback of P ⟶^{[incl_{P}^{R}]} R ⟵^{[incl_{Q}^{R}]} Q in O(F^{c})_{⊔}, where P, Q ≤ R are F-centric subgroups of S, is isomorphic to

∐_{x∈(Q\R/P)_{F}} Q^{x} ∩ P

Proof. Let C denote O(F^{c}) and recall from the remark after Definition 2.4 that morphisms in C are denoted by [φ] where φ is a morphism in F^{c}. Since C_{⊔} has products by Corollary 4.8, it is enough to prove that C_{⊔} has equalizers [12, V.2,Theorem 2].

By construction C_{⊔} is a full subcategory of the category Ĉ of all the contravariant functors C → Sets. This category has equalizers which are taken at every object of C. Thus, C_{⊔} has equalizers if given a diagram

(1)  \xymatrix{X \ar[r]^{φ} \ar[r]_{ψ} & Y}

in C_{⊔}, its equalizer in Ĉ belongs to C_{⊔}.

By construction X ≅ ∐_{i} C(−, P_{i}) and Y ≅ ∐_{j} C(−, Q_{j}) where P_{i} and Q_{j} are finitely many objects of C. Then φ = ∑_{i} φ_{i} and ψ = ∑_{i} ψ_{i} where φ_{i} and ψ_{i} are natural transformations C(−, P_{i}) → Y. By inspection of equalizers in the category of sets, it is clear that

equalizer [\xymatrix{X \ar[r]^{φ} \ar[r]_{ψ} & Y}] ≅ ∐_{i} equalizer [\xymatrix{C(−, P_{i}) \ar[r]^{φ_{i}} \ar[r]_{ψ_{i}} & Y}].

We may therefore assume that X = C(−, P) in the equalizer diagram (1).

By Yoneda’s Lemma the natural transformation φ: C(−, P) → Y factors through C(−, Q_{j_{1}}) ⊆ Y and it is induced by a morphism [φ]: P → Q_{j_{1}} in C. Similarly ψ: C(−, P) → Y factors through C(−, Q_{j_{2}}) ⊆ Y and is induced by a morphism [ψ]: P → Q_{j_{2}} in C. Clearly, if j_{1} ≠ j_{2} then the equalizer of (1) is empty. It is therefore left to prove that the equalizer of

(2)  \xymatrix{C(−, P) \ar[r]^{[φ]_{∗}} \ar[r]_{[ψ]_{∗}} & C(−, Q)}

where φ, ψ ∈ C(P, Q), is an element of C_{⊔}. Claim 1. The equalizer of (2) is equal to C(−, P) if [φ] = [ψ] and it is empty if [φ] ≠ [ψ]. Proof. If the equalizer of (2) is not empty then there exists some H ∈ C and some [θ] ∈ C(H, P) such that [φ]_{∗}([θ]) = ψ_{∗}([θ]). That is, [φ] ∘ [θ] = [ψ] ∘ [θ]. It follows from Corollary 2.6 that [φ] = [ψ], hence the equalizer of (2) is C(−, P). Q.E.D.

We have thus completed the proof that pullbacks in C_{⊔} exist. The pullback of

P ⟶^{[incl_{P}^{R}]} R ⟵^{[incl_{Q}^{R}]} Q in C_{⊔} is the equalizer of

\xymatrix{C(−, P) × C(−, Q) \ar[r]^{[incl_{P}^{R}]_{∗}∘(proj_{1})} \ar[r]_{[incl_{Q}^{R}]_{∗}∘(proj_{2})} & C(−, R).}

We now fix representatives [α]: A → Q for the elements {[α]} of Q\F/P. Thus, in the remainder of the proof the symbol [α] refers to one of these finitely many representatives. Strictly speaking we should have indexed this set, but doing so would introduce an unnecessary clutter. By Theorem 4.6 we can replace the equalizer diagram above with

(3)  [figure: ∐_{{[α]}∈Q\F/P} C(−, A); [incl_{A}^{R}]_{∗}; [incl_{Q}^{R}∘α]_{∗}; C(−, R).]

Let [Q\F/P]_{R} denote the subset of Q\F/P consisting of all {[α]} such that [α] = [c_{r}] for some r ∈ R. Claim 2. Let [α]: A → Q be a representative for some {[α]} ∈ [Q\F/P]_{R}. Then A = Q^{x} ∩ P and α = c_{x}: A → Q for some x ∈ R. Proof. By definition [α] = [c_{r}] for some r ∈ R and since Q ≤ R it follows that α = c_{x}: A → Q for some x ∈ R. In particular, c_{x}(A) ≤ Q whence A ≤ Q^{x} ∩ P. It then follows that c_{x}: Q^{x} ∩ P → Q is an extension of α and since [α] is maximal in K_{P,Q} we deduce that A = Q^{x} ∩ P. Q.E.D Claim 3. The equalizer of (3) is equal to

∐_{{[α]}∈[Q\F/P]_{R}} C(−, A)

as a subfunctor of the left-hand-side of (3). Proof. From Claim 1 we see that the equalizer of (3) is equal to the coproduct of the C(−, A)’s indexed by {[α]} such that [incl_{Q}^{R} ∘ α] = [incl_{A}^{R}]. This is equivalent to [α] = [c_{x}] for some x ∈ R. Q.E.D. Claim 4. Given x ∈ R such that Q^{x} ∩ P is F-centric, then [c_{x}]: Q^{x} ∩ P → Q is a maximal element of K_{P,Q}. Proof. Let [ϵ]: E → Q be a maximal element of K_{P,Q} which extends [c_{x}], that is, Q^{x} ∩ P ≤ E and [ϵ|_{Q^{x}∩P}] = [c_{x}]. By definition there exists y ∈ Q such that ϵ|_{Q^{x}∩P} = c_{y} ∘ c_{x} = c_{yx}. Consider the following morphisms in F

c_{yx}: E → S  and  incl_{Q}^{S} ∘ ϵ: E → S.

Their restrictions to Q^{x} ∩ P, which is F-centric, are equal to c_{yx} and Lemma 2.5 implies that there exists some z ∈ Z(Q^{x} ∩ P) such that incl_{Q}^{S} ∘ ϵ = c_{yx} ∘ c_{z} = c_{yxz}.

In particular, c_{yxz}(E) ≤ Q which implies that E ≤ Q^{yxz} ∩P = (Q^{x} ∩P)^{z} = Q^{x} ∩P. Thus, E = Q^{x} ∩ P and [c_{x}] = [ϵ], namely [c_{x}] ∈ K_{P,Q}^{max} . Q.E.D.

Claim 4 and the construction in 4.11 now imply the existence of a well defined function

Φ: [Q\R/P]_{F} → [Q\F/P]_{R}, QxP ↦ {[c_{x}]: Q^{x} ∩ P → Q}.

Claim 2 shows that Φ is, in fact, surjective. Given {[α]: A → Q} in [Q\F/P]_{R}, suppose that {[α]} = {[c_{x}]} = {[c_{x′}]} for some x, x′ ∈ R. Then by definition c_{x′} = c_{y} ∘ c_{x} ∘ c_{u} for some u ∈ P and y ∈ Q, that is, c_{x′} = c_{yxu}. As a consequence x′ = yxu · z for some z ∈ C_{S}(A) = Z(A) because A is F-centric. We see that QxP = Qx′P because A ≤ P and this shows that Φ is also injective.

Combining the bijection Φ with Claim 3 we deduce that the equalizer of (3) is

∐_{x∈[Q\R/P]_{F}} C(−, Q^{x} ∩ P)

with the maps to C(−, P) × C(−, Q) induced by the inclusion Q^{x} ∩ P ≤ P and the homomorphism c_{x}: Q^{x} ∩ P → Q. This completes the description of the pullback of

P ⟶^{[incl_{P}^{R}]} R ⟵^{[incl_{Q}^{R}]} Q. □

4.13. Remark. The pullback of the G-sets G/P → G/R ← G/Q induced by the inclusions P, Q ≤ R is isomorphic to the ∐_{g∈Q\R/P} G/Q^{g} ∩ P. The formula for the corresponding pullback in O(F^{c})_{⊔} is strikingly similar. The difference is that the coproduct is only taken over the terms that leave F-centric subgroup.

4.14. Mackey functors. Given Corollary 4.8 and Theorem 4.12 we are now able to apply the machinery in [11]. Let C denote O(F^{c}). By construction C satisfies the condition (PB ×∐) of [11, p. 123] and therefore by [11, Lemma 5.12] C_{⊔} satisfies the conditions (× ∐) and (PB) of [11, p. 121]. A Mackey functor on C_{⊔} consists of a functor M: C_{⊔}^{op} → ℤ_{(p)} -mod and a functor M_{∗}: C_{⊔} → ℤ_{(p)} -mod such that the following hold. First, M(C) = M_{∗}(C) for any object of C_{⊔}. Second, every pullback diagram in C_{⊔} displayed below on the left gives rise to the commutative square on the right:

\xymatrix{X \ar[r]^{ϕ̄} \ar[d]_{ψ̄} & C_{1} \ar[d]^{ψ} & M(X) \ar[r]^{M_{∗}(ϕ̄)} & M(C_{1}) \\ C_{2} \ar[r]_{ϕ} & C_{0} & M(C_{2}) \ar[u]^{M(ψ̄)} \ar[r]_{M_{∗}(ϕ)} & M(C_{0}) \ar[u]_{M(ψ)}}

Third, the natural map M(X ⊔ Y) → M(X) ⊕ M(Y) is an isomorphism.

A functor M: C^{op} → ℤ_{(p)} -mod is called a proto-Mackey functor if its obvious extension to C_{⊔}^{op} (indeed, this is the right Kan extension along C^{op} ⊆ C_{⊔}^{op}) admits a suitable M_{∗} rendering it a Mackey functor.

Let S ∈ C denote the Sylow of F. From the axioms for a saturated fusion system in Definition 2.1, Out_{F}(S) is a p′-group. Then Proposition 2.9 shows that |C(P, S)| ≠ 0 mod p for any P ∈ C. Finally, [11, Corollary 5.16] shows that every proto-Mackey functor M: C^{op} → ℤ_{(p)} -mod is acyclic in the sense that \underleftarrow{lim}^{i} M = 0 for all i > 0.

For example, let A: O(F^{c})^{op} → ℤ_{(p)} -mod denote the constant functor with value 𝔽_{p}. It is a proto-Mackey functor by defining the covariant part (the “transfer maps”) to be the identity on isomorphisms in C and the zero maps for non-isomorphisms in C. It follows that A has vanishing higher derived functors. This result can be easily obtained using the “Λ-functor technology” set up in [3, Section 3].

Here is another example. Let S = p_{+}^{1+2} be the extraspecial group of order p^{3} and exponent p, where p is odd, and let F be a saturated fusion system over S (see e.g. [14]). Inspection of the pullback formula in Theorem 4.12 and the doublecoset formula for group cohomology easily shows that the functor P ↦ H^{i}(P; 𝔽_{p}) is a proto-Mackey functor on O(F^{c}). To see this one only needs to observe that if Q < P is an inclusion of subgroups of S of order p and p^{2} respectively, then P is F-centric, is elementary abelian and by [11, Proof of Lemma 7.5] the transfer map H^{i}(Q; 𝔽_{p}) → H^{i}(P; 𝔽_{p}) is the zero map. We leave the details to the reader. It follows that \underleftarrow{lim}_{O(F^{c})^{op}}^{j}(P ↦ H^{i}(P; 𝔽_{p})) = 0 for all j > 0. This proves that the collection of F-centric subgroups is “subgroup sharp” in the sense of [8] and[3, Proposition 2.2] for all the p-local finite groups defined on S = p_{+}^{1+2}. In particular H^{∗}(F; 𝔽_{p}) = \underleftarrow{lim}_{P ∈O(F^{c})} H^{∗}(P; 𝔽_{p}) for these p-local finite groups.

5. Properties of the Burnside ring

We shall now fix a saturated fusions system F over S and let C denote O(F^{c}); See Definition 2.4. We shall write [C] for the set of the isomorphism classes of the objects of C, that is, [C] is the set of the F-conjugacy classes of the F-centric subgroups of S. The elements of [C] are denoted [P] for an F-centric P ≤ S. Obviously, [C] is a finite set.

We now consider the ring ∏_{[C]} ℤ. As an abelian group it is free with the set [C] as a natural choice of a basis. Thus, every element in ∏_{[C]} ℤ has the form ∑_{[Q]∈[C]} n_{Q}·[Q] and we shall sometimes abbreviate by writing (n_{Q}) for this element. The product in this ring is defined coordinatewise, namely (n_{Q}) · (m_{Q}) = (n_{Q}m_{Q}). It is clearly a commutative ring with unit and every basis element [Q] is a primitive idempotent.

We have seen in Proposition 4.10 that as an abelian group A(F) = ⊕_{[C]}ℤ and we let [C] be a basis for this free abelian group. By determining its values on basis elements, we obtain a homomorphism of groups

(5.1)  Φ: A(F) → ∏_{[Q]∈[C]} ℤ,  [P] ↦ ∑_{[Q]∈[C]} |C(Q, P)| · [Q].

In fact, this is a ring homomorphism. To see this, consider basis elements [P], [Q] of A(F), then by Theorem 4.6 and Proposition 4.10, their product [P] · [Q] is equal to ∑_{i}[A_{i}] where ⨆_{i} A_{i} is the categorical product of P and Q in C_{⊔}. As a consequence, since C is a full subcategory of C_{⊔} and taking (4.1) into account, for any F-centric H ≤ S we have a bijection

∐_{i} C(H, A_{i}) = C_{⊔}(H, ⊔_{i}A_{i}) ≈ C_{⊔}(H, P) × C_{⊔}(H, Q) = C(H, P) × C(H, Q).

It now follows that Φ([P] · [Q]) = Φ(∑_{i}[A_{i}]) = ∑_{i} Φ[A_{i}] = Φ([P]) · Φ([Q]).

Using the basis [C] for the free abelian groups underlying A(F) and ∏_{[C]} ℤ, the homomorphism Φ is represented by a matrix m whose entries are

m([Q], [P]) = |C(Q, P)|, [Q], [P] ∈ [C].

In the language of tom-Dieck, this is the analogue of the “table of marks” for the Burnside ring of a finite group.

It is clearly possible to totally order the set [C] in such a way that [H] ⪯ [K] implies |H| ≤ |K|. Using this total order the matrix m becomes upper triangular and its diagonal entries are | Out_{F}(Q)| for [Q] ∈ [C].

The rational Burnside ring. We shall use the symbol Q ≃_{F} P for the statement that Q and P are F-conjugate subgroups in a fusion system F.

5.2. Proposition. Let F be a saturated fusion system over S and let C denote O(F^{c}). Then, for every F-centric subgroups Q, P ≤ S

|C(Q, P)| = \frac{|Z(Q)| · |Aut_{F}(Q)|}{|P |} · |\{T ≤ P : T ≃_{F} Q\}|

Proof. By definition C(Q, P) is the set of orbits of the action of P on Hom_{F}(Q, P) by conjugation. By the “orbit-stabilizer property”

|C(Q, P)| = \frac{1}{|P |} · ∑_{φ∈Hom_{F} (Q,P)} |P_{φ}|

where P_{φ} is the stabilizer (or the isotropy subgroup) of φ under the action of P. Since φ is a group monomorphism, it follows that P_{φ} = C_{P} (φ(Q)). Since Q is Fcentric, |P_{φ}| = Z(φ(Q))| = |Z(Q)|. Plugging this result into the previous formula we conclude that

(1)  |C(Q, P)| = \frac{|Z(Q)|}{|P |} · |Hom_{F}(Q, P)|.

The assignment φ ↦ φ(Q) defines a surjective function

Hom_{F}(Q, P) → {T ≤ P : T ≃_{F} Q}.

The fibre of this function over an element T of its codomain is clearly Iso_{F}(Q, T) which is, in turn, equipotent to Aut_{F}(Q). Therefore

(2)  |Hom_{F}(Q, P) = |Aut_{F}(Q)| · |{T ≤ P : T ≃_{F} Q}|.

Combining (1) and (2) yields the result. □

Let P be a finite poset. The Möbius function of P is a function μ: P × P → ℤ which is defined recursively by the requirement that μ(a, b) = 0 unless a ≤ b and the following equivalent equalities hold for any a ≤ b,

∑_{a≤c≤b} μ(a, c) = δ_{a,b},  and  ∑_{a≤c≤b} μ(c, b) = δ_{a,b}

where δ is the Kronecker delta function (See e.g. Solomon [15] or Rota [13].)

The set of F-centric subgroups of S forms a poset and we let μ_{F} denote its Möbius function. Using the homomorphism (5.1) we are now ready to describe the the rational Burnside ring of F, cf. e.g. Solomon [15], Gluck [9] or [17].

5.3. Theorem. Let F be a saturated fusion system over S and let [C] denote the set of the F-conjugacy classes of the F-centric subgroups of S. Then ℚ ⊗ Φ is an isomorphism of rings ℚ ⊗ A(F) ≈ ∏_{[C]} ℚ. In particular ℚ ⊗ A(F) is a semisimple algebra with one primitive idempotent e_{P} for every element [P] of [C]. In fact, using [C] as a basis for ℚ ⊗ A(F),

e_{P} = \frac{1}{|P | · | Out_{F}(P)|} · ∑_{Q≤P, Q∈F^{c}} (|Q| · μ_{F}(Q, P)) · [Q].

The summation is over Q ≤ P such that Q ∈ F^{c}.

Proof. Consider the ring homomorphism Φ defined in (5.1). We have already remarked that by appropriately ordering the elements of [C], the matrix m which represents Φ becomes an upper triangular with non-zero values on the diagonal. Therefore, ℚ ⊗ Φ is an isomorphism. In particular ℚ ⊗ A(F) is semisimple with primitive idempotents e_{P} for every basis element [P] of ∏_{[C]} ℚ.

To avoid clutter we shall write Φ for ℚ ⊗ Φ and for every [Q] ∈ [C] we write Φ_{Q} for the projection of Φ onto the factor of [Q] in ∏_{[C]} ℚ. It remains to show that Φ(e_{P}) = [P] for all [P]. For every [Q] ∈ [C] use Proposition 5.2 and the definition of Φ to deduce that

Φ_{Q}( ∑_{H≤P,H∈F^{c}} |H| · μ_{F}(H, P) · [H]) = ∑_{H≤P,H∈F^{c}} |H| · μ_{F}(H, P) · |C(Q, H)| = (1)

∑_{H≤P, H∈F^{c}} (|H| · μ_{F}(H, P)) · \frac{|Z(Q)| · |Aut_{F}(Q)|}{|H|} · |\{T ≤ H : T ≃_{F} Q\}| =

|Z(Q)| · |Aut_{F}(Q)| · ∑_{H≤P, H∈F^{c}} ∑_{T ≤H, T ≃_{F} Q} μ_{F}(H, P) =

|Z(Q)| · |Aut_{F}(Q)| · ∑_{T ≤P, T ≃_{F} Q} ∑_{H≤P, T ≤H} μ_{F}(H, P).

Here we used the fact that if T ≃_{F} Q then T is F-centric, hence so is every subgroup H ≤ S containing T. By the recursive relation of μ_{F} and Proposition 5.2 we see that (1) is equal to

|Z(Q)| · |Aut_{F}(Q)| · ∑_{T ≤P,T ≃_{F} Q} δ_{T,P} = \{ \begin{matrix}0 & if Q ≄_{F}P \\ |P | · | Out_{F}(P)| & if Q ≃_{F} P \end{matrix}

Therefore, Φ_{Q}(e_{P}) = δ_{[Q],[P]} i.e. Φ(e_{P}) = [P]. □

The p-local Burnside ring. We shall now study A(F)_{(p)} = ℤ_{(p)} ⊗ A(F) and prove that it has a unit.

We denote O(F^{c}) by C and let [C] denote the set of the isomorphism classes of its objects. Consider Φ from (5.1) and denote Φ_{(p)} = ℤ_{(p)} ⊗Φ. By Proposition 4.10 its domain A(F)_{(p)} and codomain ∏_{[C]} ℤ_{(p)} are free ℤ_{(p)}-modules with basis [C].

5.4. Theorem. Let F be a saturated fusion system over S. For F-centric subgroups H, K ≤ S such that K is fully normalized in F set

n(K, H) = |{[c_{s}] ∈ Out_{S}(K) : ⟨s, K⟩ ≃_{F} H}|.

Then an element (y_{H}) ∈ ∏_{[H]∈[C]} ℤ_{(p)} is in the image of Φ_{(p)} if and only if the following congruences hold for any fully F-normalized F-centric subgroup K ≤ S

(1)  ∑_{[H]∈[C]} n(K, H) · y_{H} ≡ 0 mod (| Out_{S}(K)|)  in ℤ_{(p)}.

Proof. First, note that n(K, H) is well defined because if s, s′ ∈ N_{S}(K) define the same element in Out_{S}(K) then s^{−1}s′ ∈ K because K is F-centric so C_{S}(K) ≤ K. It follows that ⟨s, K⟩ = ⟨s′, K⟩. It is also clear that n(K, H) = n(K, H′) if H ≃_{F} H′.

We shall now fix once and for all representatives H for the elements [H] ∈ [C] which are fully normalized in F. We also totally order [C] in such a way that [H] ⪯ [K] implies that |H| ≤ |K|.

With respect to the ordered basis [C] of A(F)_{(p)} and ∏_{[C]} ℤ_{(p)}, the homomorphism Φ is is represented by an upper triangular matrix m whose entries are

m([H], [P]) = |C(H, P)|.

Using the choice of representatives H for the elements of [C] we now define matrices n and t with the same dimensions as m and with the set [C] as basis whose entries are

n([K], [H]) = n(K, H) ,  [K], [H] ∈ [C] and
t([K], [P]) = |C(K, P)/ Out_{S}(K)| ,  [K], [P] ∈ [C].

In addition let d be the diagonal matrix whose diagonal entries are

d([K], [K]) = | Out_{S}(K)|, [K] ∈ [C].

We now note that if K and K′ are F-conjugate subgroups of S which are fully normalized in F, then axiom (I) of saturation (Definition 2.1) and Sylow’s theorems imply that there is an isomorphism ψ: K → K′ such that ψ Out_{S}(K)ψ^{−1} = Out_{S}(K′). Therefore N_{ψ} = N_{S}(K) and ψ extends to ψ̃: N_{S}(K) → N_{S}(K′). This shows that the definition of n, t and d is independent of our choice of the fully F-normalized representatives K for the elements [K] of [C].

Claim 1. The matrices n and t are invertible over ℤ_{(p)}.

Proof. The choice of the total order of [C] implies n(K, H) = 0 if [H] ⪵ [K] because in this case either |H| < |K| or H and K are not F-conjugate. Therefore n is an upper triangular matrix. Also n(K, K) = 1 because ⟨K, s⟩ = K if and only if s ∈ K. Hence the diagonal entries of n are equal to 1 and therefore n is invertible.

Similarly, t([K], [P]) = 0 if [P] < [K] so t is upper triangular. Its diagonal entries are equal to

| Out_{F}(K) : Out_{S}(K)| ≠ 0 mod p

because the representative K of [K] ∈ [C] is fully normalized in F. They are therefore units in ℤ_{(p)}, hence t is invertible. Q.E.D.

Claim. 2 n · m = d · t.

Proof. Fix some [K], [P] in [C] and recall that Out_{S}(K) acts on C(K, P). Since every subgroup of S which contains K is F-centric, the ([K], [P])-entry of n · m is

∑_{[H]∈[C]}  n([K], [H]) · m([H], [P]) =

∑_{[H]∈[C]} |{[c_{s}] ∈ Out_{S}(K) : ⟨K, s⟩ ≃_{F} H}| · |C(H, P)| =

∑_{[H]∈[C]} ( ∑_{[c_{s}]∈Out_{S}(K), ⟨s,K⟩≃_{F} H} |C(H, P)|) =

∑_{[c_{s}]∈Out_{S}(K)} |C(⟨s, K⟩, P)| = by Proposition 2.8

∑_{[c_{s}]∈Out_{S}(K)} |C(K, P)^{[c_{s}]}| = by Frobenius’s Lemma

| Out_{S}(K)| · |C(K, P)/ Out_{S}(K)| = | Out_{S}(K)| · t([K], [P])

which is the ([K], [P])-entry of d · t. Q.E.D. We now prove that every element in the image of Φ_{(p)} satisfies the congruences (1). By linearity it suffices to prove this for elements of the form

Φ([P]) = ∑_{[H]∈[C]} |C(H, P)| · [H] = ∑_{[H]∈[C]} m([H], [P]) · [H]

which we now denote by (y_{H}) ∈ ∏_{[C]} ℤ_{(p)}, that is y_{H} = |C(H, P)|.

For every K ≤ S which is fully normalized in F we have seen that it may be assumed to be the representative of [K] in the definitions of m, n, t and d and therefore Claim 2 implies

∑_{[H]∈[C]} n(K, H) · y_{H} = ∑_{[H]∈[C]} n([K], [H]) · m([H], [P]) =

| Out_{S}(K)| · t([K], [P]) = 0 mod (| Out_{S}(K)|).

That is, (y_{H}) = Φ_{(p)}([P]) satisfies the congruence (1).

Conversely, assume that (y_{H}) ∈ ∏_{[C]} ℤ_{(p)} satisfies all the congruences (1). We view (y_{H}) as a column vector and note that satisfying these congruences is equivalent to the existence of a column vector (z_{H}) ∈ ∏_{[C]} ℤ_{(p)} such that

n · (y_{H}) = d · (z_{H}).

Since n and t are invertible by Claim 1, we deduce from Claim 2 that

(y_{H}) = n^{−1} · d · (z_{H}) = m · t^{−1} · (z_{H}) ∈ Im m = Im Φ_{(p)}

because the matrix m represents Φ_{(p)}. This completes the proof. □

5.5. Corollary. Let F be a saturated fusion system over S. Then A(F)_{(p)} is a commutative ring with a unit. More precisely, let C denote O(F^{c}) and [C] the set of isomorphism classes of its objects. Then Φ_{(p)} embeds A(F)_{(p)} as a subring of ∏_{[C]} ℤ_{(p)} and moreover, the cokernel of Φ_{(p)} is a finite abelian p-group. Furthermore the unit of ∏_{[C]} ℤ_{(p)} is contained in A(F)_{(p)}.

Proof. First, ker Φ_{(p)} is a free ℤ_{(p)}-module because it is a submodule of the free ℤ_{(p)}- module A(F)_{(p)} (note that ℤ_{(p)} is a principal ideal domain.) Similarly coker Φ_{(p)} is a finitely generated ℤ_{(p)}-module. Since ℚ ⊗ − is an exact functor, Theorem 5.3 implies that ker Φ_{(p)} = 0 and that coker Φ_{(p)} must be a finite abelian p-group. In particular A(F)_{(p)} is a commutative ring because ∏_{[C]} ℤ_{(p)} is commutative.

Now we apply Theorem 5.4 to show that the unit (1)_{H} of ∏_{[C]} ℤ_{(p)} is in the image of Φ_{(p)}. For every Q ∈ F^{c} which is fully normalized we have

∑_{P ∈[C]} n(Q, P) · 1 = ∑_{P ∈[C]} ( ∑_{[c_{s}]∈N_{S}(Q)/Q, ⟨s,Q⟩≃_{F} P} 1) = ∑_{[c_{s}]∈N_{S}(Q)/Q} 1

= |N_{S}(Q)/Q| ≡ 0 mod (| Out_{S}(Q)|),

because every subgroup of S which contains Q must be F-centric. □

5.6. Remark. Recall that for a finite group G the Burnside ring A(G) has as unit the unique G-set of cardinal 1. However, A(F) has no unit in general. To see this notice that the “table of marks” m defining the monomorphism (5.1) is an upper triangular matrix and that m([S], [S]) = | Out_{F}(S)| is the only non-zero entry in its row. Thus if Out_{F}(S) ≠ 1 there are no (integral) idempotents in A(F).

The prime spectrum. We shall now study the set of the prime ideals of A(F)_{(p)}. Here it is subsumed in the definition of prime ideal that a prime ideal is strictly included in the ring. The prime spectrum of the Burnside ring of a finite group was determined by Dress [7]. Here we describe the prime spectrum of the p-localized Burnside ring A(F)_{(p)} of a saturated fusion system F.

Throughout we shall fix a saturated fusion system F over S and let C denote O(F^{c}). The F-conjugacy class of an object H ∈ C is denoted [H] and we let [C] denote the collection of these classes. Clearly C is poset under inclusion of groups and this renders [C] a poset as well.

5.7. Definition. Let [H] be an F-conjugacy class of some H ∈ C and let q denote either the integer p or 0. Define 𝔭_{[H],q} as the kernel of the ring homomorphism

\xymatrix{A(F)_{(p)} \ar[r]^{Φ_{(p)}} & ∏_{[C]} ℤ_{(p)} \ar[r]^{^{proj^{[H]}}} & ℤ_{(p)} \ar@{->>}[r] & ℤ_{(p)}/(q).}

which we denote by π_{[H],q}.

We observe that the the homomorphism in 5.7 must be surjective because A(F)_{(p)} is a unital ring by Corollary 5.5. Its image is therefore either ℤ_{(p)} or 𝔽_{p}, whence 𝔭_{[H],q} are prime ideals of A(F)_{(p)}.

Our next result is that these are the only prime ideals of A(F)_{(p)}. Recall from Proposition 4.10 that an additive basis for A(F)_{(p)} is the set [C].

5.8. Proposition. Let 𝔭 be a prime ideal in A = A(F)_{(p)} and let q be the characteristic of A/𝔭. Then

(a) Among all the classes [K] ∈ [C] whose image under the projection A → A/𝔭 is non-zero, there exists a unique minimal class [H].

(b) Either q = 0 or q = p and moreover 𝔭 = 𝔭_{[H],q}.

Proof. (a) Let π: A → A/𝔭 denote the projection. Since A is generated by the classes [K] ∈ [C] and A/𝔭 ≠ 0 there must exist some [K] such that π([K]) ≠ 0. Choose some [H] which is minimal in the poset [C] with this property. Given an arbitrary [Q] ∈ [C], the product formula in Proposition 4.10 and the minimality of [H] with respect to the property π([H]) ≠ 0 imply

π([H]) · π([Q]) = ∑_{{[α]}∈Q\F/H; dom(α)=H} π([H]).

By Definitions 4.3 and 4.4

{{[α]} ∈ Q\F/H : dom(α) = H} = {[α] ∈ K_{H,Q}^{max} : dom(α) = H}/H = C(H, Q),

where the last equality holds because the action of H on C(H, Q) is trivial. Thus, π([H]) · π([Q]) = |C(H, Q)|π([H]) and, as A/𝔭 is an integral domain with a unit,

(1)  π([Q]) = |C(H, Q)| · 1_{A/𝔭}.

If [Q] ∈ [C] satisfies π([Q]) ≠ 0 then C(H, Q) ≠ ∅, namely [H] ⪯ [Q] in [C]. If, in addition, [Q] is also minimal with respect to this property, then [H] = [Q].

(b) Clearly A/𝔭 is a ℤ_{(p)}-module and since A is generated by the classes [Q], equation (1) implies that A/𝔭 = ℤ_{(p)}/(q). Since A/𝔭 ≠ 0 then either q = p or q = 0. It now follows by inspection of Φ_{(p)} (see (5.1)) that the homomorphism in Definition 5.7 coincides with π: A → A/𝔭 ≅ ℤ_{(p)}/(q). In particular it follows that 𝔭 = ker(π) = 𝔭_{[H],q}. □

It remains to understand the relationship between the ideals 𝔭_{[H],q}.

5.9. Proposition. The following holds in A(F)_{(p)}.

(a) 𝔭_{[H],0} = 𝔭_{[K],0} if and only if [H] = [K].

(b) 𝔭_{[H],0} ⪇ 𝔭_{[H],p} for any [H] ∈ [C].

(c) 𝔭_{[H],p} = 𝔭_{[S],p} for all [H] ∈ [C] where S is the Sylow of F.

Proof. (a) One implication is trivial. Assume that 𝔭_{[h],0} = 𝔭_{[K],0}. Observe that [H] ∉ 𝔭_{[H],0} because π_{[H],0}([H]) = |C(H, H)| ≠ 0 (see Definition 5.7.) Therefore [H] ∉ 𝔭_{[K],0}, namely |C(K, H)| = π_{[K],0}([H]) ≠ 0. Similarly |C(H, K)| ≠ 0 which implies that H and K are F-conjugate.

(b) Clearly 𝔭_{[H],0} ⊆ 𝔭_{[H],p}. Now, π_{[H],q} are surjective so A(F)_{(p)}/𝔭_{[H],0} ≅ ℤ_{(p)} while A(F)_{(p)}/𝔭_{[H],p} ≅ 𝔽_{p}.

(c) Consider some [Q] ∉ 𝔭_{[H],p}, that is |C(H, Q)| = π_{[H],p}([Q]) ≠ 0 mod p. In particular H is F-conjugate to a subgroup of Q. By Proposition 2.9, |C(H, Q)| = | Out_{F}(Q)| mod (p) whence | Out_{F}(Q)| ≠ 0 mod (p). The axioms for saturated fusion system (Definition 2.1) imply that Q = S. That is, the only class [Q] which projects non-trivially in A(F)_{(p)}/𝔭_{[H],p} is [S]. Proposition 5.8 now implies that 𝔭_{[H],p} = 𝔭_{[S],p}. □

5.10. Corollary. Let F be a saturated fusion system over S an let A(F)_{(p)} be its p-localized Burnside ring. Then A(F)_{(p)} is a local ring with a maximal ideal 𝔪 = 𝔭_{[S],p}. The remaining prime ideals in A(F)_{(p)} have the form 𝔭_{[H],0}; They are all distinct and none of them is contained in the other.

Proof. Every prime ideal 𝔭 in A = A(F)_{(p)} has the form 𝔭_{[H],q} by Proposition 5.8. Parts (b) and (c) of Proposition 5.9 show that 𝔭_{[S],p} is a unique maximal ideal in A. The remaining prime ideals have the form 𝔭_{[H],0} and they are all distinct by part (a). Suppose that 𝔭_{[H],0} ⊆ 𝔭_{[K],0}. There results a surjective ring homomorphism

ℤ_{(p)} ≅ A/𝔭_{[H],0} → A/𝔭_{[K],0} ≅ ℤ_{(p)}

which must be an isomorphism, whence 𝔭_{[H],0} = 𝔭_{[K],0}. □

5.1. Relationship with the classical Burnside ring. Let G be a finite group. The set of all its subgroups is denoted S(G). The conjugacy class of H ≤ G is denoted [H]. The Burnside ring A(G) of G is isomorphic to the free abelian group ⊕_{[S(G)]}ℤ where product of basis elements [H] and [K] is given by the double coset formula [H] · [K] = ∑_{g∈K\G/H}[K^{g} ∩ H].

A collection in G is a subset H of S(G) which is closed to conjugation in G. The set of the conjugacy classes of the elements of H is denoted [H]. Let A(G; H) be the subgroup of A(G) generated by the basis elements [H] ∈ [H]. Thus,

A(G; H) = ⨁_{[H]∈[H]} ℤ ≤ A(G).

The double coset formula implies that A(G; H) is an ideal in A(G) if H is closed to formation of subgroups. For example, the collection S_{p}(G) of all the p-subgroups of G has this property and it defines an ideal

A(G; p) = A(G; S_{p}(G)) ◁ A(G).

The collection S_{p}(G) contains the collections S_{p}^{cent}(G) of all the p-centric subgroups and the collection S_{p}^{¬ cent}(G) of all the p-subgroups of G that are not p-centric; See Definition 2.3. The discussion after 2.3 shows that S_{p}^{¬ cent}(G) is closed to formation of subgroups and defines and ideal

A^{¬p - cent}(G) ◁ A(G)

which is clearly contained in A(G; p). There results a quotient ring

A^{p - cent}(G) = A(G; p)/A^{¬ cent}(G)).

As an abelian group it is free with basis [S_{p}^{cent}(G)]. The product of basis elements [P] and [Q] is ∑_{g}[Q^{g} ∩ P] where the sum ranges through the double cosets QgP such that Q^{g} ∩ P is p-centric.

We can tensor the constructions above with ℤ_{(p)}. We denote ℤ_{(p)} ⊗ A(G) by A(G)_{(p)}. Similarly we will consider A(G; p)_{(p)} and A(G; S_{p}^{¬ cent}(G))_{(p)} and A^{p - cent}(G)_{(p)}. We remark that the latter is the free ℤ_{(p)} module with basis [S_{p}^{cent}(G)] with the same formula for the product of basis elements and moreover

A^{p - cent}(G)_{(p)} = A(G; p)_{(p)}/A(G; S_{p}^{¬ cent}(G))_{(p)}.

5.11. Theorem. Let F be the fusion system associated to a finite group G and a Sylow p-subgroup S. Then the rings A(F)_{(p)} and A^{p−cent}(G)_{(p)} are isomorphic.

Proof. Given a G-set X we denote by X^{H} the points of X fixed by H. A subgroup K ≤ G gives rise to a transitive G-set G/K by left translations. The G-sets G/K and G/K′ are isomorphic if and only if K and K′ are conjugate. There is a ring monomorphism, introduced already by Burnside,

χ: A(G)_{(p)} → ∏_{[H]∈[S(G)]} ℤ_{(p)}, [K] ↦ ∑_{[H]∈[S(G)]} |(G/K)^{H}| · [H].

The inclusion [S_{p}^{p - cent}(G)] ⊆ [S_{p}(G)] gives rise to a composite ring homomorphism

Ψ̃: A(G; p)_{(p)} ⟶^{incl} A(G)_{(p)} ⟶^{χ} ∏_{[S_{p}(G)]} ℤ_{(p)} ⟶^{proj} ∏_{[S_{p}^{p - cent}(G)]} ℤ_{(p)}.

Since S_{p}^{¬p - cent}(G) is closed to formation of subgroups, if K ∈ S_{p}^{¬ cent}(G) and H ∈ S_{p}^{p - cent}(G) then (G/K)^{H} is empty. Hence, A^{¬p - cent}(G)_{(p)} is contained in the kernel of Ψ̃ and there results a ring homomorphism (1)

Ψ: A^{p - cent}(G)_{(p)} → ∏_{[Q]∈[S_{p}^{p - cent}(G)]} ℤ_{(p)},  [P] ↦ ∑_{[Q]∈[S_{p}^{p - cent}(G)]} |G/P^{Q}| · [Q].

For subgroups H, K ≤ G consider now

N_{G}(H, K) = {g ∈ G : gHg^{−1} ≤ K}.

Clearly K acts on N_{G}(H, K) by left translations and N_{G}(H) acts on N_{G}(H, K) by right translations. Clearly, the action of H on K\N_{G}(H, K) is trivial.

By construction of F = F_{S}(G) and C = O(F^{c}), see Section 2, we see that C(P, Q) = Q\N_{G}(P, Q)/C_{G}(P) for any P, Q ≤ S which are F-centric. As usual we let [C] denote the set of the isomorphism classes of the objects of C. Claim 1. If P, Q are F-centric subgroups of S then |G/Q^{P} | = |C′_{G}(P)| · |C(P, Q)|. Proof. By inspection

G/Q^{P} = N_{G}(P, Q)^{−1}/Q ≈ Q\N_{G}(P, Q).

Note that P is p-centric in G because it is F-centric. Thus, C_{G}(P) = C′_{G}(P)×Z(P) and since Z(P) acts trivially on Q\N_{G}(P, Q) it follows that

C(P, Q) = Q\N_{G}(P, Q)/C′_{G}(P).

Furthermore, the action of C′_{G}(P) on Q\N_{G}(P, Q) is free because for any x ∈ C′_{G}(P) and any Qg ∈ Q\N_{G}(P, Q), if Qgx = Qg then gxg^{−1} ∈ Q, which implies x = 1 because x has order prime to p. Therefore |C(P, Q)| = |Q\N_{G}(P, Q)| · |C′_{G}(P)| and the result follows. Q.E.D.

By construction P, Q ≤ S are F-conjugate if and only if they are conjugate in G. Also, P ≤ S is F-centric if and only if it is p-centric. It follows that there is a natural one-to-one correspondence between the sets [S_{p}^{cent}(G)] and [C]. There results an isomorphism of free ℤ_{(p)}-modules

λ: A(F)_{(p)} → A^{p - cent}(G)_{(p)}

which is the identity on basis elements under the identification [C] = [S_{p}^{cent}(G)]. Clearly λ is an isomorphism of ℤ_{(p)}-modules (but it is not a ring homomorphism.)

Let η denote the following element in ∏_{[Q]∈[S_{p}^{cent}(G)]} ℤ_{(p)}

(2)  η = ∑_{[Q]∈[S_{p}^{p - cent}(G)]} |C′_{G}(Q)| · [Q].

It follows from the definition of Ψ in (1), from Claim 1 and from the definition of the ring homomorphism Φ in (5.1) that the following square of ℤ_{(p)}-modules is commutative. (Note: this is not a commutative square of rings!)

(3)  \xymatrix{A(F)_{(p)} \ar[r]^{λ}_{≅} \ar[d]_{Φ_{(p)}} & A^{p - cent}(G)_{(p)} \ar[d]^{Ψ} \\ ∏_{[Q]∈[S_{p}^{cent}(G)]} ℤ_{(p)} \ar[r]_{·η}^{≅} & ∏_{[Q]∈[S_{p}^{cent}(G)]} ℤ_{(p)}}

The arrow at the bottom is induced by multiplication by η which is an isomorphism of ℤ_{(p)}-modules because the |C′_{G}(Q)|’s are invertible in ℤ_{(p)}.

Claim 2. η ∈ Im Φ_{(p)}.

Proof. We apply Theorem 5.4, that is, we show that for every K ≤ S which is F-centric (equivalently, K is p-centric) and fully normalized in F, the following congruence hold

∑_{[H]∈[C]} n(K, H) · |C′_{G}(H)| = 0 mod (| Out_{S}(K)|).

In this sum only the terms where, up to F-conjugacy, K ≤ H ≤ N_{S}(K) and H/K is cyclic show up because the numbers n(K, H) vanish for all other H’s. Now fix some s ∈ N_{S}(K). Clearly N_{S}(K) normalizes C_{G}(K) and it therefore normalizes the characteristic subgroup C′_{G}(K). Thus, N_{S}(K)/K acts via conjugation on C′_{G}(K) and for any s ∈ N_{S}(K) the group ⟨s, K⟩ is p-centric because it contains K. Moreover,

C′_{G}(⟨s, K⟩) = C_{G}(⟨s, K⟩) ∩ C′_{G}(K) = C′_{G}(K)^{s},

namely, these are the fixed points of s in its action on C′_{G}(K). Using Frobenius’s formula,

∑_{H∈[C]} n(K, H) · |C′_{G}(H)| = ∑_{H∈[C]} ( ∑_{[s]∈N_{S}(K)/K,⟨s,K⟩≃_{F} H} |C′_{G}(H)|) =

∑_{H∈[C]} ( ∑_{[s]∈N_{S}(K)/K,⟨s,K⟩≃_{F} H} |C′_{G}(⟨s, K⟩)|) =

∑_{[s]∈N_{S}(K)/K} |C′_{G}(⟨s, K⟩)| = ∑_{[s]∈N_{S}(K)/K} |C′_{G}(K)^{[s]}| =

|N_{S}(K)/K| · |{orbits of N_{S}(K)/K on C′_{G}(K)}| ≡ 0 mod | Out_{S}(K)|.

We conclude from Theorem 5.4 that η ∈ Im ϕ. Q.E.D.

Let A denote the image of Φ_{(p)} and B denote the image of Ψ. Both are ℤ_{(p)}- submodules of M = ∏_{[S_{p}^{cent}(G)]} ℤ_{(p)}. Diagram (3) shows that B = η · A. Since A is a subring of M, Claim 2 implies that η · A ⊆ A. Multiplication with η therefore gives rise to a morphism of short exact sequences of ℤ_{(p)}-modules

0 ⟶ A ⟶ M ⟶ M/A ⟶ 0
·η↓  ^{·η}↓≅  ↓^{·η}
0 ⟶ A ⟶ M ⟶ M/A ⟶ 0

It follows that M/A ⟶^{·η} M/A is an epimorphism. By Corollary 5.5 M/A is a finite p-group and Φ_{(p)} is a monomorphism. It follows that M/A ⟶^{·η} M/A is an isomorphism and that Ψ is a ring monomorphism. Application of the five lemma now shows that A ⟶^{·η} A is an isomorphism. In particular η · A = A. Finally, since Φ_{(p)} and Ψ are ring monomorphisms it follows that A(F)_{(p)} ≅ A = η · A = B ≅ A^{cent}(G)_{(p)}. □

6. Examples

The Burnside ring of a finite group is an algebraic invariant which does not characterize the isomorphism type of the group: the first example of two nonisomorphic groups G_{1} ≇ G_{2} with isomorphic Burnside rings A(G_{1}) ≅ A(G_{2}) was given by Thévenaz in [16].

The situation for fusion systems is similar: consider the 2-group S = (ℤ_{2})^{9} and its automorphism group GL_{9}(2). As the symmetric group Σ_{9} acts faithfully on S there are subgroups ℤ_{9} and ℤ_{3} × ℤ_{3} in GL_{9}(2). Now consider the fusion systems F_{1} = F_{S}(S ⋊ ℤ_{9}) and F_{2} = F_{S}(S ⋊ (ℤ_{3} × ℤ_{3})). These two saturated fusion systems are not isomorphic as Out_{F_{1}}(S) = ℤ_{9} ≇ ℤ_{3} × ℤ_{3} = Out_{F_{2}}(S). Moreover, since S is abelian, the only F_{i}-centric group for i = 1, 2 is S itself. Thus the matrix defining the monomorphism (5.1) becomes the scalar | Out_{F_{i}}(S)| for i = 1, 2. As | Out_{F_{1}}(S)| = 9 = | Out_{F_{2}}(S)| we deduce that A(F_{1})_{(2)} ≅ A(F_{2})_{(2)}.

Also notice that the fusion systems of the groups G_{1} = ℤ_{3} ⋊ℤ_{2} and G_{2} = ℤ_{9} ⋊ℤ_{2} at the prime 3 show that we can have isomorphic Burnside rings even with different order Sylow subgroups because A(F_{ℤ/3}(G_{1}))_{(3)} ≅ ℤ_{(3)} ≅ A(F_{ℤ/9}(G_{2}))_{(3)}.

Here are more examples.

6.1. Lemma. Let F be a saturated fusion system over S. Assume that F has exactly n ≥ 1 conjugacy classes of F-centric subgroup P ≤ S all of which have index p or 1 in S. Then A(F)_{(p)} is isomorphic to the subring of ℤ_{(p)}^{n} whose ℤ_{(p)}-basis is p · e_{1}, …, p · e_{n−1} and e_{1} + ⋯ + e_{n} where e_{i} are the standard basis vectors.

Proof. Let P_{1}, …, P_{n−1} be representatives for the F-conjugacy classes of the Fcentric subgroups of S of index p. The matrix representing Φ_{(p)} : A_{(p)}(F) → ℤ_{(p)}^{n} has the form

| Out_{F}(P_{1})|  0  |O(F)(P_{1}, S)|
...  ⋮
| Out_{F}(P_{n−1})| |O(F)(P_{n−1}, S)|
0  | Out_{F}(S)|

Thus A(F)_{(p)} is isomorphic to the image of this matrix, namely the submodule of ℤ_{(p)}^{n} generated by its columns. Since P_{i} is F-centric and |S : P_{i}| = p, it follows that | Out_{F}(P_{i})| = pζ_{i} where ζ_{i} is a unit in ℤ_{(p)}. In particular the submodule U of ℤ_{(p)}^{n} spanned by {p · e_{1}, …, p · e_{n−1}} is contained in the image of Φ_{(p)}. Proposition 2.9 and the fact that | Out_{F}(S)| is a unit in ℤ_{(p)} now imply that the last column of the matrix above is equal modulus U to the column vector e_{1} + ⋯ + e_{n} and therefore the image of Φ_{(p)} is equal to the submodule generated by U and e_{1} + ⋯ + e_{n}. □

6.2. Example. Lemma 6.1 implies that if F is a fusion system over S where |S| = p^{3} then A(F)_{(p)} depends only on the number of conjugacy classes of the F-centric subgroups (because no subgroup of order p can be F-centric.)

This gives a hassle-free calculation of the rings A(F)_{(p)} of the fusion systems on the extraspecial group p_{+}^{1+2} - all of which were classified by Ruiz and Viruel in [14]. For example, the ring A(F)_{(7)} of the exotic examples at the prime 7 listed in rows 8 and 11 of Table 1.2 in [14] are isomorphic to the 7-local Burnside ring of the fusion system of Fi_{24} at the prime 7. The exotic example appearing in the 12th row of this table has a 7-local Burnside ring whose underlying ℤ_{(7)}-module has rank 2, and no other fusion system on 7_{+}^{1+2} has an isomorphic Burnside ring.

Here is a more sophisticated relationship between the p-local Burnside rings of subfusion systems. Recall that if F_{1} is a sub fusion system of F_{2} over the same S then any F_{2}-centric subgroup is also F_{1}-centric.

6.3. Proposition. Let S be a finite p-group and let F_{1} ⊆ F_{2} be saturated fusion systems on S. Assume that the following hold for any P ≤ S which is F_{2}-centric.

(i) The conjugacy class of P in F_{2} is equal to its conjugacy class in F_{1}.

(ii) |Aut_{F_{2}}(P): Aut_{F_{1}}(P)| = 1 mod |R| where R ≤ Out_{F_{1}}(P) is a Sylow psubgroup. Moreover, if Aut_{F_{1}}(P) ≠ Aut_{F_{2}}(P) then P is minimal with respect to the property that it is F_{2}-centric.

Then the ℤ_{(p)}-submodule I of A(F_{1})_{(p)} generated by the elements [P] such that P ≤ S is F_{1}-centric but not F_{2}-centric, is an ideal in A(F_{1})_{(p)} and moreover, A(F_{2})_{(p)} ≅ A(F_{1})_{(p)}/I.

Proof. Let C_{1} and C_{2} denote the sets of the conjugacy classes of the F_{1}-centric and F_{2}-centric subgroups of S. By definition I is the ℤ_{(p)}-submodule of A_{(p)}(F_{1}) generated by the elements [P] ∈ C_{1}\C_{2}. We shall now consider the ring monomorphisms (5.1)

Φ_{1} : A_{(p)}(F_{1}) → ℤ_{(p)}^{C_{1}}, Φ_{2} : A_{(p)}(F_{2}) → ℤ_{(p)}^{C_{2}}.

By hypothesis (i) there is a natural inclusion C_{2} ⊆ C_{1} and there results a ring epimorphism

π: ℤ_{(p)}^{C_{1}} → ℤ_{(p)}^{C_{2}}.

Claim. I = ker(π ∘ Φ_{1}). Proof. Observe that if [P] ∈ C_{1}\C_{2} then O(F_{1})(Q, P) is empty if [Q] ∈ C_{2} (otherwise P must be F_{2}-centric.) It follows immediately that I ⊆ ker(π ∘ Φ_{1}).

Conversely consider an element x = ∑_{[P]∈C_{1}} α_{P} [P] in A(F_{1}) and assume that it is not in I. Then α_{Q} ≠ 0 for some [Q] ∈ C_{2} and we choose Q of maximal order with this property. Recall that Φ_{1}(x) is a function C_{1} → ℤ_{(p)} and the maximality of P implies that

Φ_{1}(x)([Q]) = ∑_{[P]} α_{P} · |O(F_{1})(Q, P)| = α_{Q} · | Out_{F_{1}}(Q)| ≠ 0.

This shows that x ∉ ker(π ∘ Φ_{1}). We deduce that I = ker(π ∘ Φ_{1}). Q.E.D.

From the claim it follows that I ◁ A_{(p)}(F_{1}) and that Im(π ∘ Φ_{1}) ≅ A_{(p)}(F_{1})/I. Clearly Im(π ∘ Φ_{1}) is a subring of ℤ_{(p)}^{C_{2}} and it remains to prove that it is equal to the image of Φ_{2} which is isomorphic to A_{(p)}(F_{2}) by Corollary 5.5 .

Let P_{1}, …, P_{k} be representatives for the F_{2}-conjugacy classes of minimal F_{2}- centric subgroups of S. By hypothesis (ii)

| Out_{F_{2}}(P_{i})| = ζ_{i} · | Out_{F_{1}}(P_{i})|

where ζ_{i} = 1 mod |R_{i}| where R_{i} is a Sylow p-subgroup of Out_{F_{1}}(P_{i}) and also of Out_{F_{2}}(P_{i}). The equality of the F_{1}- and F_{2}-conjugacy classes of P_{i} together with Proposition 5.2 also implies that for any F_{2}-centric Q ≤ S

(1)  |O(F_{2})(P_{i}, Q)| = ζ_{i} · |O(F_{1})(P_{i}, Q)|.

For every P_{i} we consider f_{i} = π ∘ Φ_{1}([P_{i}]) and g_{i} = Φ_{2}([P_{i}]). The minimality of P_{i} implies that f_{i}([Q]) = | Out_{F_{1}}(P_{i})| if [Q] = [P_{i}] and it is zero otherwise. Similarly g_{i}([Q]) = | Out_{F_{2}}(P_{i})| if [Q] = [P_{i}] and it is zero otherwise. Since R_{i} is a Sylow p-subgroup in both Out_{F_{1}}(P_{i}) and Out_{F_{2}}(P_{i}) we see that the ℤ_{(p)}-submodules U of ℤ_{(p)}^{C_{2}} generated by f_{1}, …, f_{k} and g_{1}, …, g_{k} is equal to the submodule generated by |R_{1}| · e_{[P_{1}]}, …, |R_{k}| · e_{[P_{k}]} where e_{[P_{i}]} are the obvious standard-basis elements in ℤ_{(p)}^{C_{2}}.

Now consider any [P] ∈ C_{2} which is not minimal. We now show that f := π ∘ Φ_{1}([P]) and g := Φ_{2}([P]) are equal modulus U. Set f = π ∘ Φ_{1}([P]) and g = Φ_{2}([P]). Given any [Q] ∈ C_{2} observe that if Q is not minimal F_{2}-centric then O(F_{1})(Q, P) = O(F_{2})(Q, P) by hypothesis (ii) and Alperin’s fusion theorem. Thus, by definition f([Q]) − g([Q]) = 0. We deduce that the support of f − g is contained in {[P_{1}], …, [P_{k}]}.

Now fix some [P_{i}]. Then from (1) we deduce that

f([P_{i}]) − g([P_{i}]) = |O(F_{1})(P_{i}, P)| − |O(F_{2})(P_{i}, P)| =

(1 − ζ_{i}) · |O(F_{1})(P_{i}, P)| = 0 mod |R_{i}|.

It follows immediately that f − g ∈ U. This completes the proof that Im(Φ_{2}) = Im(π ∘ Φ_{1}). □

6.4. Example. We recall from [4, Proposition 5.3] a useful method to construct saturated fusion systems. We start with any fusion system F_{S}(G) of a finite group and fix subgroup Q_{1}, …, Q_{m} of S such that Q_{i} is not subconjugate to Q_{j} if i ≠ j. To avoid triviality we assume that Q_{i} ≠ S. We set K_{i} ≤ Out_{G}(Q_{i}) and fix Δ_{i} ≤ Out(Q_{i}) which contain K_{i}. We assume that p |̸|Δ_{i}: K_{i}|. We also assume that Q_{i} is p-centric in G but for any P ⪇ Q_{i} there exists some α ∈ Δ_{i} such that α(P) is not p-centric in G. Furthermore, we assume that for any α ∈ Δ_{i}\K_{i} the order of K_{i} ∩ K_{i}^{α} ≤ Δ_{i} is prime to p. Then the fusion system F generated by F_{S}(G) and F_{Q_{i}}(Δ_{i}) is saturated.

This method of construction was introduced in [2, Proposition 5,1]. It yields many exotic examples, e.g. [3, Example 9.3], [2], [14, Table 1.2] and [6]. We claim that in all these cases Proposition 6.3 applies to the inclusion F_{S}(G) ≤ F.

To see this observe that the conditions under which the construction of F is carried out guarantee that the Q_{i}’s are minimal F-centric subgroups. Thus, the F-centric subgroups are the F_{S}(G)-centric subgroups which are not subconjugate to one of the Q_{i}’s. From the construction it is clear that the F-conjugacy class of any F-centric P ≤ S is equal to its conjugacy class in F_{S}(G). It is also clear that only the automorphism groups of the Q_{i}’s are altered in the passage from F_{S}(G) to F and they are minimal F-centric by construction. Thus it only remains to prove that |Δ_{i} : K_{i}| = 1 mod |R_{i}| where R_{i} is a Sylow p-subgroup of K_{i}.

Fix some α ∈ Δ_{i}\K_{i}. By hypothesis the order of K_{i} ∩ K_{i}^{α} is prime to p and therefore R_{i} ∩ R_{i}^{α} = 1. As a consequence, since Q_{i} ≠ S so R_{i} ≠ 1, we see that N_{Δ_{i}}(R_{i}) = N_{K_{i}}(R_{i}).

We now let R_{i} act by conjugation on Syl_{p}(K_{i}) and on Syl_{p}(Δ_{i}). Clearly Syl_{p}(K_{i}) ⊆ Syl_{p}(Δ_{i}) and both consists of the K_{i}- and Δ_{i}-conjugates of R_{i}. If R_{i}^{α} is not contained in K_{i} then then its stabilizer is

R_{i} ∩ N_{Δ_{i}}(R_{i}^{α}) = R_{i} ∩ N_{Δ_{i}}(R_{i})^{α} = R_{i} ∩ R_{i}^{α} = 1.

Thus, the action of R_{i} on Syl_{p}(Δ_{i})\ Syl_{p}(K_{i}) is free so the number of elements of this set is divisible by |R_{i}|.

Now, | Syl_{p}(K_{i})| = |K_{i}: N_{K_{i}}(R_{i})| and | Syl_{p}(Δ_{i})| = |Δ_{i}: N_{Δ_{i}}(R_{i})| = |Δ_{i}: N_{K_{i}}(R_{i})|. It follows that

|Δ_{i}: K_{i}| = \frac{| Syl_{p}(Δ_{i})|}{| Syl_{p}(K_{i})|} = \frac{| Syl_{p}(K_{i})|}{| Syl_{p}(K_{i})|} mod |R_{i}| = 1 mod |R_{i}|.

This shows that all the conditions of Proposition 6.3 are fulfilled for the inclusion F_{S}(G) ≤ F and therefore A_{(p)}(F) is a quotient ring of A_{(p)}(F_{S}(G)).

For example, all the fusion systems listed in the [3, Example 9.3] have the same p-local Burnside ring as the fusion system of the groups Γ ⋊ A appearing in their construction.

References

[1] S. Bouc, Burnside rings, Handbook of algebra, Vol. 2, 739–804, North-Holland, Amsterdam, 2000.

[2] C. Broto, R. Levi, B. Oliver A geometric construction of saturated fusion systems. An alpine anthology of homotopy theory, 11–40, Contemp. Math., 399, Amer. Math. Soc.

[3] C. Broto, R. Levi, B. Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), 779–856.

[4] C. Broto, R. Levi, R. Oliver, The theory of p-local groups: a survey, Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic K-theory, 51–84, Contemp. Math., 346, Amer. Math. Soc., Providence, RI, 2004.

[5] K. S. Brown, Cohomology of groups, Graduate Texts in Mathematics, 87. Springer-Verlag, New York-Berlin, 1982.

[6] A. Díaz, A. Ruiz, A. Viruel, All p-local finite groups of rank two for odd prime p, Trans. Amer. Math. Soc. 359 (2007), no. 4, 1725–1764.

[7] A. Dress, A characterisation of solvable groups, Math. Z. 110 (1969), 213–217.

[8] Dwyer, W. G. Sharp homology decompositions for classifying spaces of finite groups. Group representations: cohomology, group actions and topology (Seattle, WA, 1996), 197–220, Proc. Sympos. Pure Math., 63, Amer. Math. Soc., Providence, RI, 1998.

[9] Gluck, David Idempotent formula for the Burnside algebra with applications to the psubgroup simplicial complex. Illinois J. Math. 25 (1981), no. 1, 63–67.

[10] M. Hall, Jr., The theory of groups, The Macmillan Co., New York, N.Y. 1959.

[11] S. Jackowski, J. McClure, Homotopy decomposition of classifying spaces via elementary abelian subgroups, Topology 31 (1992), no. 1, 113–132.

[12] S. MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5. Springer-Verlag, New York-Berlin, 1971.

[13] Rota, G. C., On the Foundations of Combinatorial Theory. I, Theory of Möbius Functions, Z. Wahrscheinlichkeitstheorie 2 (1964), 340-368.

[14] A. Ruiz, A. Viruel, The classification of p-local finite groups over the extraspecial group of order p^{3} and exponent p. Math. Z. 248 (2004), no. 1, 45–65.

[15] L. Solomon, The Burnside algebra of a finite group, J. Combinatorial Theory 2, 1967, 603– 615.

[16] J. Thébnvenaz, Isomorphic Burnside rings, Comm. Algebra 16 (1988), no. 9, 1945–1947.

[17] Yoshida, Tomoyuki Idempotents of Burnside rings and Dress induction theorem. J. Algebra 80 (1983), no. 1, 90–105.

(A. Díaz, A. Libman) Department of Mathematical Sciences, King’s College, University of Aberdeen, Aberdeen AB24 3UE , Scotland, U.K.

E-mail address, A. Díaz: a.diaz@maths.abdn.ac.uk

E-mail address, A. Libman: assaf@maths.abdn.ac.uk