Realising fusion systems

Ian J. Leary^{∗}  Radu Stancu

February 1, 2007

Abstract

We show that every fusion system on a p-group S is equal to the fusion system associated to a discrete group G with the property that every p-subgroup of G is conjugate to a subgroup of S.

1 Introduction

Let p be a prime number. By a p-group we shall mean a finite group whose order is a power of p. A fusion system on a p-group S is a category F whose objects are the subgroups of S, and whose morphisms are injective group homomorphisms, subject to certain axioms. The notion of a fusion system is intended to axiomatize the p-local structure of a discrete group G ≥ S in which every p-subgroup is conjugate to a subgroup of S. Every such G gives rise to a fusion system F_{S}(G) on S, and we say that G realises F if F_{S}(G) = F.

The notion of a saturated fusion system is intended to axiomatize the p-local structure of a finite group in which S is a Sylow p-subgroup. It is known that there are saturated fusion systems F which are not realised by any finite group G, although showing that this is the case is very delicate. In the case when p = 2, the only known examples are certain systems discovered by Ron Solomon [4, 10, 15].

In contrast, we show that every fusion system on any p-group S is realised by some discrete group G ≥ S in which every maximal p-subgroup is conjugate to S. The groups G that are used in our proofs are constructed as graphs of finite groups. In particular each of our groups G contains a free subgroup of finite index. In an appendix we give a brief account of those parts of the theory of graphs of groups that we use.

∗Partially supported by NSF grant DMS-0505471

While preparing this paper, we learned that Geoff Robinson has proved a similar, but not identical result [13]. Since [13] was already submitted when we started to write this paper, we have taken it upon ourselves to compare and contrast the two results. Robinson’s construction realises a large class of fusion systems, including all saturated fusion systems, but does not realise all fusion systems. The groups that Robinson constructs are iterated free products with amalgamation, whereas the groups that we construct are iterated HNN extensions. In both cases the groups may be viewed as graphs of finite groups.

We state and outline the proof of a version of Robinson’s theorem, along the lines of the proof of our main result. We also give examples of fusion systems that cannot be realised by Robinson’s method, we give examples of non-saturated fusion systems that are realised by Robinson’s method, and we prove an analogue of Cayley’s theorem for fusion systems.

The work in this paper grew from the authors’ participation in the Banff conference ‘Homotopy theory and group actions’ and from a VIGRE reading seminar at Ohio State which studied the Aschbacher-Chermak approach to the Solomon fusion systems [3]. The authors thank Andy Chermak and Geoff Robinson for showing them early versions of [3] and [13].

2 Definitions and results

Let p be a prime, and let G be a discrete group. The p-Frobenius category Φ_{p}(G) of the group G is a category whose objects are the p-subgroups of G. If P and Q are p-subgroups of G, or equivalently objects of Φ_{p}(G), the morphisms from P to Q are the group homomorphisms f : P → Q that are equal to conjugation by some element of G. Thus f : P → Q is in Φ_{p}(G) if and only if there exists g ∈ G with f(u) = g^{−1}ug for all u ∈ P. (Note that the element g is not part of the morphism. If g′ = zg for some element z in the centralizer of P, then g and g′ define the same morphism.)

Now suppose that S is a p-subgroup of G that is maximal, and further suppose that every p-subgroup of G is conjugate to a subgroup of S. In this case, every object of Φ_{p}(G) is isomorphic within the category Φ_{p}(G) to a subgroup of S. It follows that the full subcategory F_{S}(G) with objects the subgroups of S is equivalent to Φ_{p}(G). This example motivates Puig’s definition of a fusion system on S [12]. A fusion system on a p-group S is a category F. The objects of F are the subgroups of S, and the morphisms from P to Q form a subset of the set Inj(P, Q) of injective group homomorphisms from P to Q. These are subject to the following axioms:

1. For any s ∈ S, and any P, Q ≤ S with s^{−1}P s ≤ Q, the morphism ϕ : P → Q defined by ϕ : u ↦ s^{−1}us is in F;

2. If f : P → Q is in F, with R = f(P) ≤ Q, then so are f : P → R and f^{−1} : R → P.

It is easily checked that these axioms are satisfied in the case when F = F_{S}(G) as defined above. Note that the first axiom could be rewritten as the statement F_{S}(S) ⊆ F.

Remark 1 Fusion systems arise in other ways. For example, if H is any group and S is any p-subgroup of H, then the full subcategory of Φ_{p}(H) with objects the subgroups of S is a fusion system on S. Another source of fusion systems on a p-group S is the Brauer category of a p-block b [2, 11]. Here H is a finite group, S is the defect group of the p-block b, and the morphisms in the category are those conjugations by elements of H that preserve some extra structure associated to b. In the case when b is the principal block, S is the Sylow p-subgroup of H and this fusion system is just F_{S}(H). One corollary of our Theorem 2 is that every such fusion system is realised by some group G.

There is a fusion system F_{S}^{max} on S, in which the morphisms from P to Q consists of all injective group homomorphisms from P to Q. Any fusion system on S is a subcategory of F_{S}^{max}, and the intersection of a family of fusion systems on S is itself a fusion system. If Φ = {ϕ_{1}, …, ϕ_{r}} is a collection of morphisms in F_{S}^{max}, where ϕ_{i} : P_{i} → Q_{i}, the fusion system generated by Φ is defined to be the smallest fusion system that contains each ϕ_{i}.

Theorem 2 Suppose that F is the fusion system on S generated by Φ = {ϕ_{1}, …, ϕ_{r}}. Let T be a free group with free generators t_{1}, …, t_{r}, and define G as the quotient of the free product S ∗ T by the relations t_{i}^{−1}ut_{i} = ϕ_{i}(u) for all i and for all u ∈ P_{i}. Then S embeds as a subgroup of G, every p-subgroup of G is conjugate to a subgroup of S, and F_{S}(G) = F. Moreover, every finite subgroup of G is conjugate to a subgroup of S, and G has a free normal subgroup of index dividing |S|!.

If f : S′ → S is an injective group homomorphism between p-groups, and F′ is a fusion system on S′, then there is a functor f_{∗} from F′ to F_{S}^{max}, which sends P′ ≤ S′ to f(P′) and ϕ′ : P′ → Q′ to

f ∘ ϕ′ ∘ f^{−1} : f(P′) → f(Q′).

Theorem 3 (Robinson [13]) Suppose that F is the fusion system on S generated by the images (f_{i})_{∗}(F_{S_{i}}(G_{i})) for injective group homomorphisms f_{i} : S′_{i} → S for 1 ≤ i ≤ r, where G_{i} is a finite group with S′_{i} as a Sylow p-subgroup. Define G as the quotient of the free product S ∗ G_{1} ∗ ⋯ ∗ G_{r} by the relations s = f_{i}(s) for all i and for all s ∈ S′_{i}. Then S embeds as a subgroup of G, every p-subgroup of G is conjugate to a subgroup of S, and F_{S}(G) = F. Moreover, every finite subgroup of G is conjugate to a subgroup of one of the G_{i}, or to a subgroup of S, and G has a free normal subgroup of index dividing N!, where N is the least common multiple of |S| and the |G_{i}|.

Remark 4 The above theorem can be obtained from theorem 1 of [13] by induction. The main result of [13] is theorem 2, which is similar to the above statement except that extra conditions are put on the G_{i}. These extra conditions allow Robinson to improve the bound on the index of a free normal subgroup, and to deduce some information about the finite quotient by such a subgroup.’ Another slight difference is that Robinson describes his group as a free product with amalgamation G_{1} ∗ ⋯ ∗ G_{r}, where G_{1} has S as a Sylow p-subgroup. The groups that arise in this way are the same groups as those that arise from our statement, since if S is a subgroup of G_{1}, then S ∗_{S} G_{1} = G_{1}.

Theorem 5 Let Σ denote the group of all permutations of the elements of a p-group S, and identify S with a subgroup of Σ via the Cayley embedding. Every fusion system on S is equal to a subcategory of the Frobenius category Φ_{p}(Σ) of Σ.

3 Saturated fusion systems

In this section we present the definition of a saturated fusion system, due to Puig [12], although we shall describe an equivalent definition due to Broto, Levi and Oliver [6]. There are two additional axioms as well as the axioms for a fusion system. These axioms necessitate some preliminary definitions.

As usual, if G is a group and H is a subgroup of G, we write C_{G}(H) for the centralizer of H in G and N_{G}(H) for the normalizer of H in G.

Suppose that F is a fusion system on S. Say that P ≤ S is fully Fcentralized if

|C_{S}(P)| ≥ |C_{S}(P′)|

for every P′ which is isomorphic to P as an object of F. Suppose that F = F_{S}(G) for some discrete group G in which every p-subgroup is conjugate to a subgroup of S. In this case, if P is fully F-centralized, one sees that C_{S}(P) is a p-subgroup of C_{G}(P) of maximal order.

Similarly, say that P is fully F-normalized if

|N_{S}(P)| ≥ |N_{S}(P′)|

for every P′ which is isomorphic to P as an object of F. If F = F_{S}(G) as above and P is fully F-normalized, one sees that N_{S}(P) is a p-subgroup of N_{G}(P) of maximal order.

Now suppose that F = F_{S}(G) for some finite group G, and that P ≤ S is fully F-normalized. In this case, N_{S}(P) must be a Sylow p-subgroup of the finite group N_{G}(P). Moreover, C_{G}(P) ∩ N_{S}(P) = C_{S}(P) must be a Sylow p-subgroup of C_{G}(P), and Aut_{S}(P) = N_{S}(P)/C_{S}(P) must be a Sylow p-subgroup of Aut_{F}(P) = N_{G}(P)/C_{G}(P). This gives the first of two extra axioms for a saturated fusion system:

3. If P is fully F-normalized, then P is also fully F-centralized, and Aut_{S}(P) is a Sylow p-subgroup of Aut_{F}(P).

Next, suppose that F = F_{S}(G) for some finite group G and that f : P → Q ≤ S is an isomorphism in F such that Q is fully F-centralized. This implies that C_{S}(Q) is a Sylow p-subgroup of C_{G}(Q). Pick an element h ∈ G so that f is equal to conjugation by h, i.e., so that f(u) = c_{h}(u) = h^{−1}uh for all u ∈ P. The image c_{h}(C_{S}(P)) is a p-subgroup of C_{G}(c_{h}(P)) = C_{G}(Q), and so there exists h′ ∈ C_{G}(Q) so that c_{h′} ∘ c_{h}(C_{S}(P)) ≤ C_{S}(Q). Since c_{h′} acts as the identity on Q, if we define k = hh′, we see that c_{k} extends f and c_{k}(C_{S}(P)) ≤ C_{S}(Q).

The map c_{k} clearly extends to a map from N_{f} = N_{S}(P) ∩ c_{k}^{−1}(N_{S}(Q)) to N_{S}(Q). But since C_{S}(P) is a subgroup of c_{k}^{−1}(N_{S}(Q)), we may rewrite this as

N_{f} = {g ∈ N_{S}(P): c_{k} ∘ c_{g} ∘ c_{k}^{−1} ∈ Aut_{S}(Q)}
= {g ∈ N_{S}(P): f ∘ c_{g} ∘ f^{−1} ∈ Aut_{S}(Q)},

which does not depend on choice of k. This leads to the second extra axiom:

4. If f : P → Q is an isomorphism in F and Q is fully F-centralized, then f extends in F to a map from N_{f} to N_{S}(Q), where

N_{f} = {g ∈ N_{S}(P): f ∘ c_{g} ∘ f^{−1} ∈ Aut_{S}(Q)}.

Remark 6 It has been shown [8] that the axioms for a saturated fusion system can be simplified to:

3′. Aut_{S}(S) is a Sylow p-subgroup of Aut_{F}(S).

4′. If f : P → Q is an isomorphism in F and Q is fully F-normalized, then f extends in F to a map from N_{f} to N_{S}(Q), where N_{f} is as defined in axiom 4.

Remark 7 In the case when S is abelian, axioms 3 and 4 simplify. In this case, every subgroup of S is fully F-centralized and fully F-normalized for any fusion system F, and for any f ∈ F, N_{f} = S. Hence a fusion system F on an abelian p-subgroup S is saturated if and only if Aut_{F}(S) is a p′-group and every morphism f : P → S in F extends to an automorphism of S.

Remark 8 As mentioned in the introduction, there are saturated fusion systems which are not realised by any finite group. One source of saturated fusion systems is the fusion systems associated to p-blocks of finite groups [2, 11]. The question of whether every such fusion system can be realised by a finite group is a long-standing open problem.

4 Examples

Let E be an elementary abelian p-group of rank at least three, i.e., a direct product of at least three copies of the cyclic group of order p. Let A = Aut(E) be the full group of automorphisms of E, which is of course isomorphic to a general linear group over the field of p elements. Let B be a subgroup of A of order a power of p, and let C be a non-trivial subgroup of A of order coprime to p. Note that A is generated by its subgroups of order coprime to p.

Each of A, B and C may be viewed as a collection of morphisms in the fusion system F_{E}^{max}. For X = A, B or C, let F_{E}(X) denote the fusion system generated by all the morphisms in X.

Example 9 The fusion system F_{E}(C) is saturated, and is equal to the fusion system F_{E}(G), where G is the semi-direct product G = E⋊C.

Example 10 The fusion system F_{E}(A) is not saturated, since in F_{E}(A) the automorphism group of the object E does not have E/Z(E) as a Sylow psubgroup. However, F_{E}(A) can be realised by the procedure of Theorem 3. Let C_{1}, …, C_{r} be p′-subgroups of A that together generate A. If we put G_{i} = E⋊C_{i} with f_{i} the identity map of E, then the fusion system generated by all of the (f_{i})_{∗}(F_{E}(G_{i})) is equal to F_{E}(A).

Example 11 The fusion system F_{E}(B) cannot be realised by the procedure used in Theorem 3. For suppose that G_{1}, …, G_{r} are finite groups with Sylow p-subgroups E_{1}, …, E_{r}, each of which is isomorphic to a subgroup of E, and suppose that F_{E}(B) is generated by the fusion systems (f_{i})_{∗}F_{E_{i}}(G_{i}). Those G_{i} for which f_{i} : E_{i} → E is not an isomorphism do not contribute any morphisms to Aut_{F}(E). If f_{i} : E_{i} → E is an isomorphism, then either Aut_{G_{i}}(E_{i}) contains non-identity elements of p′ order, implying that F ≠ F_{E}(B), or E_{i} is central in G_{i} and G_{i} does not contribute any morphisms to Aut_{F}(E).

Next we consider some examples of fusion systems F on an abelian pgroup E in which Aut_{F}(E) is a p′-group, but for which some isomorphisms between proper subgroups of E do not extend to elements of Aut_{F}(E).

Example 12 Let F and F′ be distinct order p subgroups of E, and let ϕ : F → F′ be an isomorphism. Let F_{E}(ϕ) be the fusion system generated by ϕ. Every morphism in F_{E}(ϕ) is equal to either an inclusion map or the composite of either ϕ or ϕ^{−1} with an inclusion map. In particular, in F_{E}(ϕ), the automorphism group of each object E′ ≤ E is trivial. The fusion system F_{E}(ϕ) cannot be realised by the procedure of Theorem 3, as will be explained below.

In view of Remark 7, F_{E}(ϕ) is not a saturated fusion system, since the morphism ϕ : F → F′ does not extend to an automorphism in F_{E}(ϕ) of the group E.

Now suppose that F is a fusion system on E generated by the images (f_{i})_{∗}F_{E_{i}}(G_{i}) of some fusion systems for finite groups. If ϕ : F → F′ is a morphism in F, then there exists i so that F, F′ ≤ f_{i}(E_{i}) and ϕ ∈ (f_{i})_{∗}F_{E_{i}}(G_{i}). But then (by the same argument as used above) there is a morphism ϕ̃ : f_{i}(E_{i}) → f_{i}(E_{i}) extending ϕ : F → F′. Thus F cannot be equal to the fusion system F_{E}(ϕ), since this fusion system contains no such ϕ̃.

Example 13 Let F be a proper subgroup of E, and suppose that D is a non-trivial p′-group of automorphisms of F. Let F ⋊D denote the semidirect product of F and D, let G be the free product with amalgamation G = E∗_{F} (F ⋊D), and let F be the fusion system F_{E}(G). From this definition one sees that F can be obtained by the procedure of Theorem 3. On the other hand, since Aut_{F}(E) is trivial, one sees that the non-trivial automorphisms of F do not extend to automorphisms of E, and hence F is not saturated.

As remarked earlier, Robinson does not consider all fusion systems that can be built by the procedure of Theorem 3, but only those fusion systems that he calls Alperin fusion systems [13]. With the notation of Theorem 3 (and bearing in mind Remark 4), a fusion system is Alperin if the following conditions hold:

1. Inside each G_{i} there is a subgroup E_{i} which is the largest normal psubgroup of G_{i}, and the centralizer of this subgroup is as small as possible, in the sense that C_{G_{i}}(E_{i}) = Z(E_{i});

2. The quotient G_{i}/E_{i} is isomorphic to Out_{F}(E_{i}) := Aut_{F}(E_{i})/Aut_{E_{i}}(E_{i});

3. Inside S, the image of the subgroup S′_{i} (the Sylow p-subgroup of G_{i} which is to be identified with a subgroup of S) is equal to the normalizer of the image of E_{i}, i.e., f_{i}(S′_{i}) = N_{S}(f_{i}(E_{i})).

In terms of this definition, the content of Alperin’s fusion theorem with some later embellishments [1, 7] is that the fusion system for any finite group is Alperin. Robinson remarks [13] that work of Broto, Castellana, Grodal, Levi and Oliver implies that every saturated fusion system is Alperin [5]. It is easy to see that a fusion system on an abelian p-group is Alperin if and only if it is saturated. We finish this section by giving an example of a fusion system that is Alperin but not saturated.

Example 14 Let p be an odd prime, let A = (C_{p})^{3}, and let B be a subgroup of Aut(A) of order p such that A is indecomposable as a B-module. (Equivalently, the action of a generator for B on A should be a single Jordan block.) Let S be the semi-direct product S = A⋊B. The centre Z of S has order p. Let E = Z × B ≤ S, a subgroup isomorphic to C_{p} × C_{p}. It is readily seen that C_{S}(E) = E and that P = N_{S}(E) is isomorphic to a semi-direct product (C_{p})^{2}⋊C_{p}, the unique non-abelian group of order p^{3} and exponent p. Let G_{1} be the semi-direct product G_{1} = E⋊Aut(E). Since the Sylow p-subgroups of Aut(E) are cyclic of order p, there is an isomorphism between P and a Sylow p-subgroup of G_{1} that extends the inclusion of E.

By construction, the fusion system F for the free product with amalgamation S ∗_{P} G_{1} is Alperin in the sense of Robinson [13], but this fusion system is not saturated. For example, there are non-identity self-maps of Z inside F, and if F were saturated, any self-map of Z inside F would extend to a self-map of S. But in F, S has only inner automorphisms, and these restrict to Z as the identity.

5 Proofs

Proof. (of Theorem 5.) As in the statement, let Σ be the group of all permutations of S, and identify S with a subgroup of Σ. Let P and Q be subgroups of S ≤ Σ, and let ϕ : P → Q be any injective group homomorphism. It suffices to show that there is some σ ∈ Σ such that for all u ∈ P, σ^{−1}uσ = ϕ(u). Let Ω denote the group S viewed as a set with a left S-action. There are two ways to view Ω as a set with a left P -action, via P ≤ S and via ϕ : P → Q ≤ S. Denote these two P -sets by Ω and ^{ϕ}Ω respectively. Each of Ω and ^{ϕ}Ω is isomorphic as a P -set to the disjoint union of |S : P | copies of P. In particular, there is an isomorphism of P -sets σ : ^{ϕ}Ω → Ω. Viewing σ as an element of Σ, one has that σϕ(u)ω = uσω for all u ∈ P and ω ∈ Ω. Hence σ^{−1}uσ = ϕ(u) for all u as required. □

Remark 15 A version of Theorem 5 appeared in [9], although fusion systems were not mentioned there.

Before proving Theorem 2 we give a result concerning extending group homomorphisms, and two corollaries, one of which will be used in the proof.

Lemma 16 Let S and G be as in the statement of Theorem 2, let j : S → G be the natural map from S to G, let H be a group and let f : S → H be a group homomorphism. There is a group homomorphism f̃: G → H with f = f̃ ∘ j if and only if for each i, the homomorphisms f : P_{i} → H and f ∘ ϕ_{i} : P_{i} → H differ by an inner automorphism of H.

Proof. Given a homomorphism f̃ as in the statement, one has that for each i and for each u ∈ P_{i}, fϕ_{i}(u) = h_{i}^{−1}f(u)h_{i}, where h_{i} = f̃(t_{i}). For the converse, suppose that there exists, for each i, an element h_{i} satisfying the equation fϕ_{i}(u) = h_{i}^{−1}f(u)h_{i} for all u ∈ P_{i}. In this case one may define f̃ on the generators of G by f̃(s) = f(s) for all s ∈ S and f̃(t_{i}) = h_{i}. □

Corollary 17 With notation as in the statement of Theorem 2, there is a homomorphism from G to Σ, the group of all permutations of the set S, extending the Cayley representation of S.

Proof. The argument used in the proof of Theorem 5 shows that the conditions of Lemma 16 hold. □

Remark 18 Corollary 17 gives an alternative way to prove Corollary 24, at least in the special case of a rose-shaped graph.

Corollary 19 With notation as in the statement of Theorem 2, a complex representation of S with character χ extends to a complex representation of G if and only if for each i and for each u ∈ P_{i}, χ(u) = χ(ϕ_{i}(u)).

Remark 20 Of course, a representation of S will extend to G in many different ways if it extends at all.

Proof. (of Theorem 2.) As in Appendix 6.2, one sees that the group G presented in the statement is the fundamental group of a graph of groups with one vertex group, S, and one edge group P_{i} for each ϕ_{i}, 1 ≤ i ≤ r. From Corollary 24 it follows that S is a subgroup of G. From Corollary 28, it follows that any finite subgroup of G, and in particular any p-subgroup of G, is conjugate to a subgroup of S. By Theorem 26, there is a cellular action of G on a tree T, with one orbit of vertices and r orbits of edges. By suitable choice of orbit representatives, we may choose a vertex v whose stabilizer is S, and edges e_{1}, …, e_{r} so that the stabilizer of e_{i} is P_{i}, and so that the initial vertex of e_{i} is v while the final vertex is t_{i}·v.

Since every p-subgroup of G is conjugate to a subgroup of S, there is a fusion system F_{S}(G) associated to G. By construction F_{S}(G) contains each ϕ_{i}, which corresponds to conjugation by t_{i}.

Conversely, suppose that g ∈ G has the property that g^{−1}P g ≤ Q for some subgroups P, Q of S. It suffices to show that conjugation by g, as a map from P to Q, is equal to a composite of (restrictions of) the maps ϕ_{j} and their inverses with conjugation maps by elements of S.

Consider the action of P on the tree T. By hypothesis, the action of P fixes both the vertex v and the vertex g·v. Since T is a tree, P must fix all the vertices and edges on the unique shortest path from v to g·v. Let this path have length n. Define g_{0} = 1_{G}, g_{n} = g, and for 1 ≤ i ≤ n − 1, choose g_{i} ∈ G so that g_{0}·v, g_{1}·v, …, g_{n}·v is the shortest path in T from v to g·v. For each i, P is contained in the stabilizer of the vertex g_{i}·v, and so P ≤ g_{i}Sg_{i}^{−1}, or equivalently g_{i}^{−1}P g_{i} ≤ S.

The edge joining g_{i}·v and g_{i+1}·v is an edge of the form g_{i}·e_{j} or g_{i+1}·e_{j} for some j depending on i. Consider the two cases separately, first supposing that the edge is of the form g_{i}·e_{j}. In this case it follows that P ≤ g_{i}P_{i}g_{i}^{−1}, since P stabilizes the edge g_{i}·e_{j}. Also one sees that g_{i+1}·v = g_{i}t_{j}·v, and hence g_{i+1}^{−1}g_{i}t_{j} ∈ S. Hence conjugation by g_{i}^{−1}g_{i+1}, viewed as a map from g_{i}^{−1}P g_{i} to g_{i+1}^{−1}P g_{i+1} is equal to the composite of the map ϕ_{j} (restricted to g_{i}^{−1}P g_{i} ≤ P_{i}) followed by conjugation by an element of S.

The other case is similar. Here it follows that P ≤ g_{i+1}P_{i+1}g_{i+1}^{−1}, and one has that g_{i}·v = g_{i+1}t_{j}·v, from which g_{i}^{−1}g_{i+1}t_{j} = s ∈ S. In this case conjugation by g_{i}^{−1}g_{i+1}, as a map from g_{i}^{−1}P g_{i} to g_{i+1}^{−1}P g_{i+1}, is equal to the composite map given by conjugation by s followed by the map ϕ_{j}^{−1} (restricted to s^{−1}g_{i}^{−1}P g_{i}s ≤ ϕ_{j}(P_{i+1})).

Thus conjugation by g = g_{n} as a map from P to Q can be expressed as a composite of maps inside the fusion system generated by the ϕ_{i}, and so F_{S}(G) is equal to this fusion system.

It remains to show that the group G contains a free normal subgroup of index at most |S|!. Let Σ denote the symmetric group on the set S. By Corollary 17, there is a homomorphism G → Σ which extends the natural injection S → Σ. By Corollary 29, the kernel of this homomorphism is a free normal subgroup of G, and its index is a factor of |Σ| = |S|!. □

Proof. (of Theorem 3—sketch.) In this case, the group G is the fundamental group of a star-shaped graph of groups, with one central vertex labelled S and r outer vertices labelled G_{1}, …, G_{r}. The edge from G_{i} to S is labelled by the group S′_{i}. By Theorem 26, there is a cellular action of G on a tree T, with r + 1 orbit of vertices and r orbits of edges. We may choose orbit representatives v_{0}, v_{1}, …, v_{r} of vertices and e_{1}, …, e_{r} of edges so that the stabilizer of v_{0} is S, and for 1 ≤ i ≤ r, the stabilizer of v_{i} is G_{i} (resp. of e_{i} is S′_{i}). Moreover, we may assume that e_{i} has initial vertex v_{i} and terminal vertex v_{0}.

In this case, one sees that any finite subgroup of G is conjugate to either a subgroup of S or to a subgroup of G_{i} for some i. Since S′_{i} is a Sylow psubgroup of G_{i}, it follows that any p-subgroup of G is conjugate to a subgroup of S as required.

As in the previous proof, it is clear that the fusion system F_{S}(G) contains the image of each F_{S′_{i}}(G_{i}), but an argument is needed to show that these images generate F_{S}(G). Given g ∈ G and P, Q ≤ S so that g^{−1}P g ≤ Q, one argues that the action of P fixes the vertices v_{0} and g·v_{0} in the tree T, and hence fixes the shortest path (necessarily of even length, say 2n) that joins these vertices.

Let g_{0} = 1_{G}, g_{2n} = g, and pick group elements so that the vertices on the shortest path from v_{0} to g·v_{0} are:

g_{0}·v_{0} , g_{1}·v_{j(1)} , g_{2}·v_{0} , g_{3}·v_{j(2)} , …, g_{2n−1}·v_{j(n)} , g_{2n}·v_{0}

for some function j : {1, …, n} → {1, …, r}. If i is even, then g_{i}^{−1}P g_{i} ≤ S, and if i is odd then g_{i}^{−1}P g_{i} ≤ G_{j((i+1)/2)} . Since P stabilizes each edge, one sees that P ≤ g_{i}^{−1}S_{k}g_{i}, where S_{k} denotes the image of S′_{k} inside S, and k = j((i + 1)/2) if i is odd and k = j(i/2) if i is even. In particular, each g_{i}^{−1}P g_{i} is a subgroup of S.

One may show that in the case when i is odd, g_{i}^{−1}g_{i+1} ∈ G_{j((i+1)/2)} and that in the case when i is even, g_{i}^{−1}g_{i+1} ∈ S. Thus the map from g_{i}^{−1}P g_{i} to g_{i+1}^{−1}P g_{i+1} given by conjugation by g_{i}^{−1}g_{i+1} is a map inside the fusion system generated by the images of the F_{S′_{i}}(G_{i}), and conjugation by g = g_{2n} as a map from P to Q ≤ S is expressed as a composite of maps of the required form.

Finally, if Ω is a finite set so that |Ω| is divisible by |S| and by each |G_{i}|, one may define free actions of S and each G_{i} on Ω which give rise to the same (free) action of S_{i} = f_{i}(S′_{i}). This gives rise to a group homomorphism from G to Σ, the symmetric group on Ω, whose kernel is free by Corollary 29.

□

6 Appendix: graphs of groups

In this section we give proofs of those results about graphs of groups that we use. Our treatment of graphs of groups follows that of Scott and Wall [14].

For the purposes of this paper, a graph Γ consists of two sets, the vertices V and the directed edges E, together with two functions ι, τ : E → V. For e ∈ E, ι(e) is called the initial vertex of e and τ(e) is the terminal vertex of e. Multiple edges and loops are allowed in this definition. Γ is connected if the only equivalence relation on V that contains all pairs (ι(e), τ(e)) is the relation with just one class.

A graph Γ may be viewed as a category, with objects the disjoint union of V and E and two non-identity morphisms with domain e for each e ∈ E, one morphism e → ι(e) and one morphism e → τ(e).

A graph of groups is a connected graph Γ together with groups G_{v}, G_{e} for each vertex and edge, and injective group homomorphisms f_{e,ι} : G_{e} → G_{ι(e)} and f_{e,τ} : G_{e} → G_{τ(e)} for each edge e. If Γ is viewed as a category, this is just a functor from Γ to the category of groups and injective group homomorphisms. Without loss of generality, one may assume that each map f_{e,ι} : G_{e} → G_{ι(e)} is the inclusion of a subgroup.

6.1 The fundamental group of a graph of groups

For a topologist, and arguably for anybody, the easiest way to define the fundamental group of a graph of groups is via the notion of a graph of spaces.

A graph of spaces is a connected graph Γ together with topological spaces X_{v}, X_{e} for each vertex and edge, and continuous maps f_{e,ι} : X_{e} → X_{ι(e)} and f_{e,τ} : X_{e} → X_{τ(e)}. A graph of spaces is just a functor from the category Γ to the category of topological spaces and continuous functions. A graph of based spaces is defined similarly: each X_{e} and X_{v} is equipped with a base point, and the maps must preserve base points. Let I denote the closed unit interval [0, 1]. The total space of a graph of spaces is the space X made from the disjoint union

∐_{v∈V} X_{v} ∐ ∐_{e∈E} X_{e} × I

by identifying (x, 0) ∈ X_{e} × I with f_{e,ι}(x) ∈ X_{ι(e)} and identifying (x, 1) ∈ X_{e} × I with f_{e,τ}(x) ∈ X_{τ(e)}. As an example, consider the graph of spaces in which each X_{e} and X_{v} is a single point. For this graph of spaces the total space is the usual topological realization of the graph as a 1-dimensional CWcomplex. The reader who is familiar with the homotopy colimit construction will note that if one views a graph of spaces as a functor X_{(−)} on the category Γ, then the total space X is naturally homeomorphic to the homotopy colimit of the functor X_{(−)}, or in symbols, X = hocolim_{Γ}X_{(−)}.

Given a graph of groups, one may define a graph of connected based spaces by taking classifying spaces as the spaces X_{e} and X_{v}:

X_{e} = BG_{e} = K(G_{e}, 1) X_{v} = BG_{v} = K(G_{v}, 1).

For the continuous map f_{e,ι} : X_{e} → X_{ι(e)} (resp. f_{e,τ} : X_{e} → X_{τ(e)}) one may take any continuous map that induces the given map G_{e} → G_{ι(e)} (resp. G_{e} → G_{τ(e)}) on fundamental groups. Define a total space X as the realization of this graph of spaces.

For discrete groups K and H, the space BK is unique up to based homotopy, and homotopy class of based maps from BK to BH are in bijective correspondence with group homomorphisms from K to H. It follows that the homotopy type of the space X defined above depends only on the graph of groups, rather than on the particular choices of classifying spaces and maps between them. The fundamental group G of the graph of groups can now be defined as the fundamental group of X. This describes the fundamental group of the graph of groups up to isomorphism. The inclusion of each X_{v} in X defines a conjugacy class of homomorphism G_{v} → G (which will be shown to be injective, below). For many purposes one wants a more precise description of G, together with a single choice of homomorphism G_{v} → G. This can be done by choosing a basepoint for the space X, and for each v, a path in X from the basepoint for X to the basepoint for X_{v} ⊆ X.

6.2 Presentations for graphs of groups

We shall only consider presentations for graphs of groups where the underlying graph is either a ‘rose’ or a ‘star’. By a rose we mean a graph with only one vertex, so that every edge has the same initial and terminal vertices. By a star we mean a connected graph with n + 1 vertices and n edges, for some n > 0, with one central vertex, such that all the edges have this vertex as their terminal vertex and so that every other vertex is the initial vertex of exactly one edge.

Suppose one is given a p-group S, subgroups P_{i}, Q_{i} ≤ S, and injective group homomorphisms ϕ_{i} : P_{i} → Q_{i} for 1 ≤ i ≤ r, as in the statement of Theorem 2. Use this data to make a rose-shaped graph of groups with r edges. Let S be the vertex group, let P_{i} be the ith edge group, with the inclusion map P_{i} ≤ S (resp. the composite ϕ_{i} : P_{i} → Q_{i} ≤ S) as the ith initial (resp. terminal) homomorphism. There is a model for BP_{i} having just one 0-cell and one 1-cell for each element of P_{i}. Take a model for BS having just one 0-cell and take this 0-cell as the base point. To make a CW-complex of the homotopy type of the total space of the graph of groups, it suffices to add to BS one 1-cell t_{i} for each i (with both ends at the unique 0-cell), one 2-cell D_{i,u} for 1 ≤ i ≤ r and for each u ∈ P_{i}, and higher dimensional cells (which will not affect the fundamental group). The attaching map for the 2-cell D_{i,u} spells out the word u t_{i} ϕ_{i}(u) t_{i}^{−1}, and so the presentation coming from this CW-structure is the presentation given in the statement of Theorem 2.

Next suppose that one is given a p-group S, groups G_{i} for 1 ≤ i ≤ r with Sylow p-subgroups S_{i}, and injective group homomorphisms f_{i} : S_{i} → S, i.e., the data found in the statement of Theorem 3. In this case, define a star of groups with central vertex group P, other vertex groups G_{1}, …, G_{r}, and edge groups S_{1}, …, S_{r}. The map of each edge group into its initial vertex group is the inclusion S_{i} → G_{i}, and the map of each edge group into its terminal vertex group is f_{i} : S_{i} → S. An argument similar to that given in the previous paragraph shows that the fundamental group of this graph of groups has the presentation given in the statement of Theorem 3. Note that here one can make a space homotopy equivalent to the total space of the graph of spaces by starting from the one-point union of BS and the BG_{i}, without adding any extra 1-cells. This is reflected in the fact that the vertex groups generate the fundamental group of the graph of groups.

6.3 Properties of graphs of groups

Proposition 21 Let G be the fundamental group of a graph of groups based on a graph Γ. Every subgroup H ≤ G is itself the fundamental group of a graph of groups, indexed by a graph Δ equipped with a map f : Δ → Γ which does not collapse any edges. For each v and e ∈ Δ, the group H_{v} (resp. H_{e}) is a subgroup of G_{f(v)} (resp. G_{f(e)}).

Proof. Use the bijection between connected covering spaces of a connected CW-complex (with a choice of base point) and subgroups of its fundamental group. Let X be the total space of the graph of spaces used in the definition of G, so that there is a covering space of X whose fundamental group is H.

Any connected covering space of X can be expressed as the total space of a graph of spaces indexed by some Δ as in the statement. This gives an expression for the fundamental group of any connected covering space of X as the fundamental group of a graph of groups as claimed. □

Theorem 22 Let X be the total space of the graph of spaces used in the definition of the fundamental group G of a graph of groups. The universal covering space of X is contractible, and hence X is homotopy equivalent to BG.

Proof. We shall build a space Y, in such a way that it is clear that Y is contractible, and that Y is a covering space of X.

For v a vertex, define the subspace X′_{v} of X by

X′_{v} = X_{v} ∪ ⋃_{ι(e)=v} X_{e} × [0, 0.5) ∪ ⋃_{τ(e)=v} X_{e} × (0.5, 1].

Similarly, define for e an edge, X′_{e} = X_{e} × (0, 1). The inclusions X_{v} → X′_{v} and X_{e} ≅ X_{e} × {0.5} → X′_{e} are homotopy equivalences, and it may be useful to think of X′_{v} as a nice open neighbourhood of X_{v} in X. Let Y_{v}, Y′_{v}, Y_{e}, and Y′_{e} be the universal covering spaces of X_{v}, X′_{v}, X_{e} and X′_{e} respectively. Each Y′_{v} (resp. Y′_{e}) is contractible since it is the universal covering space of the classifying space BG_{v} (resp. BG_{e}).

The definition of the space X′_{v} lifts to a description of the space Y′_{v}. The complement Y′_{v} − Y_{v} is identified with a collection of disjoint copies of Y_{e} × (0, 0.5), and Y_{e} × (0.5, 1), for different edges e. There are copies of Y_{e} × (0, 0.5) if and only if ι(e) = v. In this case the copies are in bijective correspondence with the cosets of f_{e,ι}(G_{e}) in G_{v}. Similarly, there are copies of Y_{e} × (0.5, 1) for each e with τ(e) = v, and these copies are indexed by cosets of f_{e,τ}(G_{e}) in G_{v}.

By induction, we shall construct a sequence Y_{0} ⊆ Y_{1} ⊆ Y_{2} ⋯ of spaces so that: each Y_{n} is contractible; there is a map π : Y_{n} → X which is locally a covering map except at some points of X; for any x ∈ X and any n ≥ 0, at least one of π : Y_{n} → X and π : Y_{n+1} → X is locally a covering map at x.

Pick a vertex v of the graph Γ, and define Y_{0} to be the space Y′_{v}. Define a map π : Y′_{v} → X as the composite of the map Y′_{v} → X′_{v} and the inclusion X′_{v} ⊆ X. As remarked earlier, Y′_{v} − Y_{v} consists of lots of subspaces of the form Y_{e} × (0, 0.5) for ι(e) = v and lots of subspaces of the form Y_{e} × (0.5, 1) for τ(e) = v. Define Y_{1} by attaching to each such subspace a copy of Y′_{e}. The map π : Y_{0} → X extends uniquely to π : Y_{1} → X by insisting that on each newly-added Y′_{e} subspace, π is equal to the composite map Y′_{e} → X′_{e} ⊆ X. From the construction of Y_{1}, it is apparent that Y_{1} is contractible.

In constructing Y_{1}, one attached to Y_{0} many spaces of the form Y′_{e}, by identifying one end of Y′_{e} with part of Y_{0}. For each copy of Y′_{e} that was attached via its initial end, take a copy of Y′_{τ(e)}, and attach this at the other end of Y′_{e}. Similarly, for each copy of Y′_{e} that was attached to Y_{0} by its terminal end, take a copy of Y′_{ι(e)} and attach this at the other end of Y′_{e}. This defines a space Y_{2}, which is clearly contractible, and the map π extends uniquely to a map Y_{2} → X which agrees with the covering map Y′_{e} → X′_{e} or Y′_{v} → X′_{v} on each such subspace.

Now suppose that n is even, and that Y_{n} has been constructed from Y_{n−1} by attaching subspaces Y′_{v} in such a way that the intersection of Y_{n−1} and each new Y′_{v} is equal to one of the components of Y′_{v} − Y_{v}. Furthermore, suppose that the map π on each new Y′_{v} is equal to the map Y′_{v} → X′_{v} ⊆ X. Form Y_{n+1} by attaching a copy of Y′_{e} to each other component of Y′_{v} − Y_{v} for each of the copies of Y′_{v}. Extend the map π as before.

In the case when n is odd, suppose that Y_{n} has been constructed from Y_{n−1} by attaching subspaces Y′_{e} in such a way that the intersection of Y_{n−1} and each new Y′_{e} is equal to one of the two components of Y′_{e} − Y_{e} × {0.5}. Suppose also that the map π on each of the new Y′_{e} is equal to the map Y′_{e} → X′_{e} ⊆ X. Form Y_{n+1} by attaching a copy of Y′_{v} to the other component of each Y′_{e} − Y_{e} × {0.5}, where v is either ι(e) or τ(e) depending which component of Y′_{e} − Y_{e} × {0.5} was used. Extend the map π in the same way as before.

By construction, each Y_{n} is contractible, and comes equipped with a map π : Y_{n} → X. If n is even, this map is locally a covering except possibly at points of X contained in the union of the images of the X_{v}. If n is odd, this map is a covering except possibly at point of X contained in the union of the images of the X_{e} × {0.5}. Now define Y by Y = ⋃_{n} Y_{n}. This space Y is contractible, and the map π : Y → X is a covering map, since it is locally a covering map at every point of X. It follows that Y is the universal covering space of X. Since the universal covering space of X has been shown to be contractible, it follows that X is a model for BG. □

Remark 23 The above proof relies on the fact that the edge groups map injectively to the vertex groups.

Corollary 24 Each vertex group G_{v} maps injectively into the fundamental group of a graph of groups.

Proof. Given a vertex v, construct the universal covering space as in the proof of Theorem 22, with Y_{0} = Y′_{v}. The group of all deck transformations of Y is naturally isomorphic to G, the fundamental group of X. Under this isomorphism, the subgroup of those deck transformations that preserve Y_{0} is identified with G_{v}. □

Remark 25 There is also an algebraic proof that each G_{v} embeds in G. In the case when the graph is a rose, this argument is given in Corollary 17.

6.4 The action on a tree

Say that an action of a group on a tree is cellular if no element of the group exchanges the ends of any edge.

Theorem 26 Let G be the fundamental group of a graph of groups indexed by the graph Γ. There is a tree T with a cellular G-action and an isomorphism f : T/G ≅ Γ. If x̃ is either a vertex or edge of T, and x = f(x̃) is the image of G·x̃ under f, then the stabilizer of x̃ is conjugate to G_{x}.

Proof. Let X be the total space of the graph of spaces used in defining G. As remarked earlier, the underlying topological space of the graph Γ can be identified with the total space of the constant graph of 1-point spaces indexed by Γ. The unique map from each X_{v} and X_{e} to a point induces a map from X to Γ.

Now let Y be the universal covering space of X, as constructed in the proof of Theorem 22. This Y can be viewed as a graph of spaces over some graph Δ, with vertex spaces copies of the spaces Y_{v} and edges spaces copies of the spaces Y_{e}. The group G acts on Y in such a way that the setwise stabilizer of each copy of Y_{v} is a conjugate of G_{v}, and similarly the setwise stabilizer of each copy of Y_{e} × (0, 1) is a conjugate of G_{e}. Define T to be the total space of the graph of 1-point spaces over the graph Δ. By construction, T is a graph equipped with a G-action, an equivariant map ϕ : Y → T, and an isomorphism f : T/G → Γ. To check that T is a tree, let T_{n} = ϕ(Y_{n}). As in the proof of Theorem 22, one shows inductively that T_{n} is contractible, and T = ⋃_{n} T_{n}. □

Lemma 27 Any cellular action of a finite group H on a tree T fixes a vertex.

Proof. Take any vertex t ∈ T, and define a finite subtree T′ to be the union of all the shortest paths between elements of the orbit H·t. If T′ is not itself fixed by H, remove an H-orbit of ‘leaves’ (i.e., vertices of valency one) from T′, and continue this process until a subtree fixed by H is all that remains.

□

Corollary 28 Every finite subgroup of the fundamental group of a graph of groups is conjugate to a subgroup of a vertex group.

Proof. Let G be the fundamental group of the graph of groups and let T be the corresponding tree. If H is a finite subgroup of G then H fixes some vertex of T. The stabilizer of each vertex of T is a conjugate of one of the vertex groups G_{v}. □

Corollary 29 Let H be a subgroup of a graph of groups whose intersection with each conjugate of each vertex group is trivial. Then H is a free group.

Proof. The hypotheses imply that H acts freely on the tree T, and so the quotient space T/H is a 1-dimensional classifying space for H. □

References

[1] J. L. Alperin, Sylow intersections and fusion, J. Algebra 6 (1967), 222– 241.

[2] J. L. Alperin and M. Broué. Local methods in block theory, Ann. of Math 110 (1979), 143–157.

[3] M. Aschbacher and A. Chermak, A group-theoretic approach to a family of 2-local finite groups constructed by Levi and Oliver, preprint (2005).

[4] D. J. Benson, Conway’s group Co_{3} and the Dickson invariants, Manuscripta Math. 85 (1994), 177–193.

[5] C. Broto, N. Castellana, J. Grodal, R. Levi and R. Oliver, Subgroup families controlling p-local finite groups, Proc. London Math. Soc. 91 (2005), 325–354.

[6] C. Broto, R. Levi and B. Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), no. 4, 779–856.

[7] D. M. Goldschmidt, A conjugation family for finite groups, J. Algebra 16 (1970), 138–142.

[8] R. Kessar and R. Stancu, A reduction theorem for fusion systems of blocks, J. Algebra, to appear.

[9] I. J. Leary, B. Schuster and N. Yagita, On universally stable elements, Quart. J. Math. Oxford 48 (1997) 493–498.

[10] R. Levi and R. Oliver, Construction of 2-local finite groups of a type studied by Solomon and Benson, Geometry and Topology 6 (2002), 917– 990.

[11] M. Linckelmann, Simple fusion systems and the Solomon 2-local groups, J. Algebra, to appear.

[12] Ll. Puig, Full Frobenius systems and their localizing categories, preprint (2001).

[13] G. Robinson, Amalgams, blocks, weights, fusion systems and finite simple groups, preprint (2006).

[14] P. Scott and C. T. C. Wall, Topological methods in group theory. Homological group theory (Proc. Sympos., Durham, 1977), 137–203, London Math. Soc. Lecture Note Ser.,36, Cambridge Univ. Press, Cambridge-New York, 1979.

[15] R. Solomon, Finite groups with Sylow 2-subgroups of type .3, J. Algebra 28 (1974) 182–198.

Authors’ addresses:

Ian J. Leary: leary@math.ohio-state.edu
Radu Stancu: stancu@math.ohio-state.edu

Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210