7 Realising fusion systems 0 0 2 n Ian J. Leary∗ Radu Stancu a J February 2, 2008 0 3 ] R G Abstract . h We show that every fusion system on a p-group S is equal to the t a fusion system associated to a discrete group G with the property that m every p-subgroup of G is conjugate to a subgroup of S. [ 1 v 1 Introduction 2 9 8 Let p be a prime number. By a p-group we shall mean a finite group whose 1 order is a power of p. A fusion system on a p-group S is a category F whose 0 7 objects are the subgroups of S, and whose morphisms are injective group 0 homomorphisms, subject to certain axioms. The notion of a fusion system / h is intended to axiomatize the p-local structure of a discrete group G ≥ S t a in which every p-subgroup is conjugate to a subgroup of S. Every such G m gives rise to a fusion system F (G) on S, and we say that G realises F if S : v F (G) = F. S i X The notion of a saturated fusion system is intended to axiomatize the r p-local structure of a finite group in which S is a Sylow p-subgroup. It is a 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 arecertain systems discovered by Ron Solomon [4, 10, 15]. In contrast, we show that every fusion system on any p-group S is re- alised 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 ∗Partially supported by NSF grant DMS-0505471 1 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. 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 compareandcontrastthetworesults. Robinson’sconstructionrealisesalarge 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 theSolomonfusionsystems [3]. Theauthors thankAndy Chermak andGeoff 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 (G) of the group G is a category whose objects are the p-subgroups of p G. If P and Q are p-subgroups of G, or equivalently objects of Φ (G), the p 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 (G) if p and only if there exists g ∈ G with f(u) = g−1ug 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 (G) is isomorphic within the category Φ (G) to a p p subgroup of S. It follows that the full subcategory F (G) with objects the S 2 subgroups of S is equivalent to Φ (G). This example motivates Puig’s defini- p tionofafusion system on S [12]. Afusionsystemonap-groupS isacategory 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−1Ps ≤ Q, the morphism φ : P → Q defined by φ : u 7→ s−1us 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 (G) as defined above. Note that the first axiom could be rewritten as the S statement F (S) ⊆ F. S 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 (H) p 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 (H). One corollary of our Theorem 2 is that every such fusion system S is realised by some group G. There is a fusion system Fmax on S, in which the morphisms from P to S Q consists of all injective group homomorphisms from P to Q. Any fusion system on S is a subcategory of Fmax, and the intersection of a family of S fusionsystemsonS isitselfafusionsystem. IfΦ = {φ ,...,φ }isacollection 1 r of morphisms in Fmax, where φ : P → Q , the fusion system generated by S i i i 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 ,...,φ }. Let T be a free group with free generators t ,...,t , and define 1 r 1 r G as the quotient of the free product S∗T by the relations t−1ut = φ (u) for i i i all i and for all u ∈ P . Then S embeds as a subgroup of G, every p-subgroup i of G is conjugate to a subgroup of S, and F (G) = F. Moreover, every S 3 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 Fmax, which ∗ S 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 ) (F (G )) for injective group homomorphisms i ∗ Si i f : S′ → S for 1 ≤ i ≤ r, where G is a finite group with S′ as a Sylow i i i i p-subgroup. Define G as the quotient of the free product S ∗ G ∗ ··· ∗ G 1 r by the relations s = f (s) for all i and for all s ∈ S′. Then S embeds as a i i subgroup of G, every p-subgroup of G is conjugate to a subgroup of S, and F (G) = F. Moreover, every finite subgroup of G is conjugate to a subgroup S of one of the G , or to a subgroup of S, and G has a free normal subgroup of i 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 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 (Σ) of Σ. p 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 (H) for G the centralizer of H in G and N (H) for the normalizer of H in G. G Suppose that F is a fusion system on S. Say that P ≤ S is fully F- centralized if ′ |C (P)| ≥ |C (P )| S S 4 for every P′ which is isomorphic to P as an object of F. Suppose that F = F (G) forsome discrete groupG inwhich every p-subgroupisconjugate S to a subgroup of S. In this case, if P is fully F-centralized, one sees that C (P) is a p-subgroup of C (P) of maximal order. S G Similarly, say that P is fully F-normalized if ′ |N (P)| ≥ |N (P )| S S for every P′ which is isomorphic to P as an object of F. If F = F (G) as S above and P is fully F-normalized, one sees that N (P) is a p-subgroup of S N (P) of maximal order. G Now suppose that F = F (G) for some finite group G, and that P ≤ S S is fully F-normalized. In this case, N (P) must be a Sylow p-subgroup of S the finite group N (P). Moreover, C (P) ∩ N (P) = C (P) must be a G G S S Sylow p-subgroup of C (P), and Aut (P) = N (P)/C (P) must be a Sylow G S S S p-subgroup of Aut (P) = N (P)/C (P). This gives the first of two extra F G G axioms for a saturated fusion system: 3. If P is fully F-normalized, then P is also fully F-centralized, and Aut (P) is a Sylow p-subgroup of Aut (P). S F Next, suppose that F = F (G) for some finite group G and that f : S P → Q ≤ S is an isomorphism in F such that Q is fully F-centralized. This implies that C (Q) is a Sylow p-subgroup of C (Q). Pick an element h ∈ G S G so that f is equal to conjugation by h, i.e., so that f(u) = c (u) = h−1uh h for all u ∈ P. The image c (C (P)) is a p-subgroup of C (c (P)) = C (Q), h S G h G and so there exists h′ ∈ CG(Q) so that ch′ ◦ ch(CS(P)) ≤ CS(Q). Since ch′ acts as the identity on Q, if we define k = hh′, we see that c extends f and k c (C (P)) ≤ C (Q). k S S The map c clearly extends to a map from N = N (P)∩c−1(N (Q)) to k f S k S N (Q). But since C (P) is a subgroup of c−1(N (Q)), we may rewrite this S S k S as N = {g ∈ N (P): c ◦c ◦c−1 ∈ Aut (Q)} f S k g k S = {g ∈ N (P): f ◦c ◦f−1 ∈ Aut (Q)}, S g S 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 to N (Q), where f S N = {g ∈ N (P): f ◦c ◦f−1 ∈ Aut (Q)}. f S g S 5 Remark 6 It has been shown [8] that the axioms for a saturated fusion system can be simplified to: 3′. Aut (S) is a Sylow p-subgroup of Aut (S). S F 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 to N (Q), where N is as defined in f S f 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 = S. Hence a fusion system F f on an abelian p-subgroup S is saturated if and only if Aut (S) is a p′-group F 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 productofatleastthreecopiesofthecyclicgroupoforderp. LetA = 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 fusionsystem Fmax. ForX = A, B orC, let F (X) denotethefusion system E E generated by all the morphisms in X. Example 9 Thefusionsystem F (C)issaturated, andisequaltothefusion E system F (G), where G is the semi-direct product G = E⋊C. E 6 Example 10 The fusion system F (A) is not saturated, since in F (A) the E E automorphism group of the object E does not have E/Z(E) as a Sylow p- subgroup. However, F (A) can be realised by the procedure of Theorem 3. E Let C ,...,C be p′-subgroups of A that together generate A. If we put 1 r G = E⋊C with f the identity map of E, then the fusion system generated i i i by all of the (f ) (F (G )) is equal to F (A). i ∗ E i E Example 11 The fusion system F (B) cannot be realised by the procedure E used in Theorem 3. For suppose that G ,...,G are finite groups with Sylow 1 r p-subgroups E ,...,E , each of which is isomorphic to a subgroup of E, and 1 r suppose that F (B) is generated by the fusion systems (f ) F (G ). Those E i ∗ Ei i G for which f : E → E is not an isomorphism do not contribute any i i i morphisms to Aut (E). If f : E → E is an isomorphism, then either F i i Aut (E ) contains non-identity elements of p′ order, implying that F =6 Gi i F (B), or E is central in G and G does not contribute any morphisms to E i i i Aut (E). F Next we consider some examples of fusion systems F on an abelian p- group E in which Aut (E) is a p′-group, but for which some isomorphisms F between proper subgroups of E do not extend to elements of Aut (E). F Example 12 Let F and F′ be distinct order p subgroups of E, and let φ : F → F′ be an isomorphism. Let F (φ) be the fusion system generated E by φ. Every morphism in F (φ) is equal to either an inclusion map or the E composite of either φ or φ−1 with an inclusion map. In particular, in F (φ), E the automorphism group of each object E′ ≤ E is trivial. The fusion system F (φ) cannot be realised by theprocedure of Theorem 3, aswill be explained E below. In view of Remark 7, F (φ) is not a saturated fusion system, since the E morphism φ : F → F′ does not extend to an automorphism in F (φ) of the E group E. Now suppose that F is a fusion system on E generated by the images (f ) F (G ) of some fusion systems for finite groups. If φ : F → F′ is a mor- i ∗ Ei i phism in F, then there exists i so that F,F′ ≤ f (E ) and φ ∈ (f ) F (G ). i i i ∗ Ei i ˜ But then (by the same argument as used above) there is a morphism φ : f (E ) → f (E ) extending φ : F → F′. Thus F cannot be equal to the i i i i ˜ fusion system F (φ), since this fusion system contains no such φ. E 7 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 semi- direct product of F and D, let G be the free product with amalgamation G = E∗ (F⋊D),andletF bethefusionsystemF (G). Fromthisdefinition F E oneseesthatF canbeobtainedbytheprocedureofTheorem3. Ontheother hand, since Aut (E) is trivial, one sees that the non-trivial automorphisms F 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, a fusion system is Alperin if the following conditions hold: 1. Inside each G there is a subgroup E which is the largest normal p- i i subgroup of G , and the centralizer of this subgroup is as small as i possible, in the sense that C (E ) = Z(E ); Gi i i 2. ThequotientG /E isisomorphictoOut (E ) := Aut (E )/Aut (E ); i i F i F i Ei i 3. Inside S, the image of the subgroup S′ (the Sylow p-subgroup of G i i whichistobeidentified withasubgroupofS)isequaltothenormalizer of the image of E , i.e., f (S′) = N (f (E )). i i i S i 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 )3, and let B be a subgroup p of Aut(A) of order p such that A is indecomposable as a B-module. (Equiva- lently, 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 ×C . It is readily seen p p that C (E) = E and that P = N (E) is isomorphic to a semi-direct product S S (C )2⋊C , the unique non-abelian group of order p3 and exponent p. Let G p p 1 be the semi-direct product G = E⋊Aut(E). Since the Sylow p-subgroups 1 8 of Aut(E) are cyclic of order p, there is an isomorphism between P and a Sylow p-subgroup of G that extends the inclusion of E. 1 By construction, the fusion system F for the free product with amal- gamation S ∗ G is Alperin in the sense of Robinson [13], but this fusion P 1 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 per- mutations 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 homomor- phism. It suffices to show that there is some σ ∈ Σ such that for all u ∈ P, σ−1uσ = 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 φΩ respectively. Each of Ω and φΩ 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 : φΩ → Ω. Viewing σ as an element of Σ, one has that σφ(u)ω = uσω for all u ∈ P and ω ∈ Ω. Hence σ−1uσ = φ(u) for all u as required. (cid:3) Remark 15 A version of Theorem 5 appeared in [9], although fusion sys- tems 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 f : G → H with ˜ f = f ◦ j if and only if for each i, the homomorphisms f : P → H and i f ◦φ : P → H differ by an inner automorphism of H. i i ˜ Proof. Given a homomorphism f as in the statement, one has that for each i and for each u ∈ P , fφ (u) = h−1f(u)h , where h = f˜(t ). For the converse, i i i i i i suppose that there exists, for each i, an element h satisfying the equation i 9 fφ (u) = h−1f(u)h for all u ∈ P . In this case one may define f˜ on the i i i i generators of G by f˜(s) = f(s) for all s ∈ S and f˜(t ) = h . (cid:3) i 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 condi- (cid:3) tions 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 , χ(u) = χ(φ (u)). i i 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 for each φ , 1 ≤ i ≤ r. i i 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 ,...,e so that the stabilizer of e is P , and so that the initial 1 r i i vertex of e is v while the final vertex is t ·v. i i Since every p-subgroup of G is conjugate to a subgroup of S, there is a fusion system F (G) associated to G. By construction F (G) contains each S S φ , which corresponds to conjugation by t . i i Conversely, suppose that g ∈ G has the property that g−1Pg ≤ 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 10