INTRODUCTION TO FUSION SYSTEMS. Markus Linckelmann

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1 INTRODUCTION TO FUSION SYSTEMS Markus Linckelmann Abstract. Fusion systems were introduced by L. Puig in the early 1990 s as a common framework for fusion systems of finite groups or p-blocks of finite groups. Benson [3] suggested that every fusion system should give rise to a p-complete topological space, generalising the notion of a classifying space of a finite group. A criterion for the existence and uniqueness of such spaces was given by Broto, Levi and Oliver, who developed in [5] the homotopy theory of a class of topological spaces, called p-local finite groups, which use fusion systems as their underlying algebraic structure. The present notes, while not touching upon the subject of p-local finite groups directly, are intended to provide a detailed introduction to fusion systems as needed in the structure theory of finite groups, p-blocks and p-local finite groups. As motivation we give a brief review of some aspects, including classical theorems of Burnside and Frobenius, of the interplay between the local and global structure of finite groups in Section 1. We describe the concept of an abstract fusion system in Section 2. Section 3 carries over to fusion systems the notions of normalisers and centralisers of subgroups in a finite group. Using terminology introduced in Section 4, we give in Section 5 a proof of Alperin s fusion theorem for fusion systems. In Section 6 we show that one can take quotients of fusion systems by certain subgroups, and the Sections 7 and 8 develop the analogues for fusion systems of normal subgroups and simple groups. The last Section generalises a control of fusion result from block theory to arbitrary fusion systems. Most of the material in this introduction to fusion system has appeared in print elsewhere; we give references as we go along. Contents 1 Local structure of finite groups 2 2 Fusion systems 4 3 Normalisers and centralisers 9 4 Centric subgroups 16 5 Alperin s fusion theorem 19 6 Quotients of fusion systems 24 7 Normal fusion systems 26 8 Simple fusion systems 28 9 Normal subsystems and control of fusion 31 1 Typeset by AMS-TEX

2 2 MARKUS LINCKELMANN 1 Local structure of finite groups Throughout these notes we denote by p a prime number. The local structure of a finite group G at the prime p can be understood as the structure of a Sylow-p-subgroup P together with some information about the way in which P is embedded into G. The following definition makes this precise, and encodes the relevant information in a category, called fusion system: Definition 1.1. Let G be a finite group and let P be a Sylow-p-subgroup of G. The fusion system of G on P is the category denoted by F P (G), with the set of subgroups of P as objects and group homomorphisms induced by conjugation in G as morphisms; more precisely, for any two subgroups Q, R of P, we set Hom FP (G)(Q, R) = Hom G (Q, R), the set of group homomorphisms ϕ : Q R for which there is an element x G such that ϕ(u) = x u = xux 1 for all u Q. The composition of morphisms in F P (G) is the usual composition of group homomorphisms. With the notation of 1.1, if H is a subgroup of G containing P then P is a Sylow-psubgroup of H and F P (H) F P (G). We will say that H controls G-fusion in P if F P (H) = F P (G). If P is abelian, the following theorem of Burnside states that N G (P) controls G-fusion in P: Theorem 1.2. (Burnside) Let G be a finite group and let P be a Sylow-p-subgroup of G. If P is abelian then F P (G) = F P (N G (P)). Proof. Let Q be a subgroup of P and let ϕ : Q P be a morphism in F P (G). That is, there is an element x G such that x Q P and ϕ(u) = xux 1 for all u Q. Since P is abelian, both P and x P are Sylow-p-subgroups of C G ( x Q). Thus there is c C G ( x Q) such that P = cx P. Then cx N G (P), and we still have cx u = c(xux 1 )c 1 = xux 1 = ϕ(u) for all u Q, because c commutes with all elements of the form xux 1 x Q, where u Q. Thus ϕ is induced by conjugation with an element in N G (P), which proves the Theorem. Example 1.3. Let G = S 4 be the symmetric group on four letters and let P = x, t be the subgroup generated by the cycle x = (1, 2, 3, 4) and the involution t = (1, 2)(3, 4). Then P is a Sylow-2-subgroup of S 4, and clearly P is a dihedral group of order 8. The group P has a Klein four subgroup V = {(1), (1, 2)(3, 4), (1,3)(2,4),(1,4)(2,3)} such that Aut G (V ) is cyclic of order 3, having conjugation by (1, 2, 3) as generator. This automorphism cannot be extended to an automorphism of P because the automorphism group of a dihedral group is easily seen to be a 2-group again. Thus N G (P) does not control fusion in this case.

3 INTRODUCTION TO FUSION SYSTEMS 3 A finite group G is called p-nilpotent if P has a normal complement K; that is, K is a normal subgroup of G such that G = K P. Note that then K = O p (G), the largest normal subgroup of G of order prime to p. The following theorem of Frobenius illustrates that being p-nilpotent is a property which can be read off the p-local structure of G: Theorem 1.4. (Frobenius) Let G be a finite group and let P be a Sylow-p-subgroup of G. The following are equivalent: (i) G is p-nilpotent. (ii) N G (Q) is p-nilpotent for any non-trivial subgroup Q of P. (iii) We have F P (G) = F P (P). (iv) For any Q P the group Aut G (Q) = N G (Q)/C G (Q) is a p-group. Sketch of proof. Suppose that G is p-nilpotent; say G = K P, where K = O p (G) and P is a Sylow-p-subgroup of P. Since P = G/K is a p-group, K is equal to the set of all p -elements in G. Thus, for any subgroup H, the intersection H K is the set of all p -elements in H, and hence H is p-nilpotent with H K = O p (H). In particular, N G (Q) is p-nilpotent for any subgroup Q of P. Thus (i) (ii). Let x G, u P such that x u P. Write x = yz with y K and z P. Then [y, z u] P K = {1}, hence x u = z u, which proves that any morphism in F P (G) is induced by conjugation with an element in P. Thus (i) (iii). The implication (iii) (iv) is trivial. Let Q be a subgroup of P. Since Q and O p (N G (Q)) are normal subgroups of N G (Q) of coprime orders, they centralise each other and hence O p (N G (Q)) = O p (C G (Q)). Thus if N G (Q) is p-nilpotent then N G (Q)/C G (Q) is a p-group. This shows the implication (ii) (iv). In order to show that (iv) implies (i) we proceed by induction over the order of G. The hypothesis (iv) passes down to subgroups of G, and hence we may assume that every proper subgroup of G is p-nilpotent. Let Q be a non-trivial subgroup of P. If N G (Q) G then N G (Q) is p-nilpotent by induction. If G = N G (Q) then G/Q is p-nilpotent by induction. Let L be the subgroup of G containing Q such that L/Q = O p (G/Q). Then Q is a normal Sylow-p-subgroup of L, hence L = Q K for some p -subgroup K of L. The assumption that N G (Q)/C G (Q) is a p-group implies that K centralises Q, hence L = Q K. Then K is a normal p-complement in G = N G (Q). We next show that F P (G) = F P (P). One can do this directly, but this verification is also a trivial consequence of Alperin s fusion theorem 5.2 below. In particular, Z = Z(P) has the property that if x G such that x Z P then x Z = Z. Set N = N G (Z). Since N is p-nilpotent, the quotient N/[N, N] has a non-trivial p-group as quotient. By a theorem of Grün this implies that G/[G, G] has a non-trivial p-group as quotient. Thus the smallest normal subgroup O p (G) of G with G/O p (G) a p-group is a proper subgroup of G, hence p-nilpotent, and a normal p-complement of O p (G) is also one for G itself. This completes the proof of the implication (iv) (i). Glauberman and Thompson sharpened this for odd p as follows. The Thompson

4 4 MARKUS LINCKELMANN subgroup J(P) of a finite p-group P is the subgroup of P generated by the set of abelian subgroups of P of maximal order. Theorem 1.5. (Glauberman-Thompson, [10, Ch. 8, Theorem 3.1]) Let p be an odd prime, let G be a finite group and let P be a Sylow-p-subgroup of G. Then G is p- nilpotent if and only if N G (Z(J(P))) is p-nilpotent. Glauberman s ZJ-theorem in [8] is a sufficient criterion for F P (G) to be controlled by the normaliser of the center of the Thompson subgroup J(P). Theorem 1.6. (Glauberman, [8]) Let p be an odd prime. If no subquotient of G is isomorphic to Qd(p) = (C p C p ) SL 2 (p) then F P (G) = F P (N G (Z(J(P)))). The semi-direct product Qd(p) is understood with the natural action of SL 2 (p) on C p C p. One easily checks that the Theorem does not hold for the group G = Qd(p). The relevance of the following theorem, due to R. Solomon, lies in the fact that had there been a finite group G with a 2-local structure as stated below, then G could have been chosen simple, and thus would have given rise to a new finite simple group. Theorem 1.7. (R. Solomon, [24]) There is no finite group G having a Sylow-2-subgroup P of Spin 7 (3) such that F P (Spin 7 (3)) F P (G) and such that all involutions in P are G-conjugate. Alperin and Broué showed in [2] that, extending ideas of Brauer, every p-block of a finite group admits a local structure which can be described in terms of Brauer pairs and which has formal properties very similar to those of the categories F P (G) above; see [12] (this volume) for details on fusion systems of blocks. 2 Fusion systems Puig s abstract notion of fusion systems on p-groups, which we are going to describe in this section, captures the common properties of fusion systems of finite groups and p- blocks. If P, Q, R are subgroups of a finite group G, we denote as before by Hom P (Q, R) the set of group homomorphisms ϕ : Q R for which there is y P satisfying ϕ(u) = yuy 1 for all u Q; we write Aut P (Q) = Hom P (Q, Q). Thus Aut P (Q) is canonically isomorphic to N P (Q)/C P (Q); in particular Aut Q (Q) = Q/Z(Q) is the group of inner automorphisms of Q. Definition 2.1. A category on a finite p-group P is a category F whose objects are the subgroups of P and whose morphism sets Hom F (Q, R) consist, for any two subgroups Q, R of P, of injective group homomorphisms with the following properties: (i) if Q is contained in R then the inclusion Q R is a morphism in F;

5 INTRODUCTION TO FUSION SYSTEMS 5 (ii) for any ϕ Hom F (Q, R), the induced isomorphism Q = ϕ(q) and its inverse are morphisms in F; (iii) composition of morphisms in F is the usual composition of group homomorphisms. Definition 2.2. Let F be a category on a finite p-group P. A subgroup Q of P is called fully F-centralised if C P (R) C P (Q) for any subgroup R of P such that R = Q in F, and Q is called fully F-normalised if N P (R) N P (Q) for any subgroup R of P such that R = Q in F. If G is a finite group with Sylow-p-subgroup P, we will see in 2.9 below that a subgroup Q of P is fully F P (G)-centralised if and only if C P (Q) is a Sylow-p-subgroup of C G (Q); similarly, Q is fully F P (G)-normalised if and only if N P (Q) is a Sylow-p-subgroup of N G (Q). The following definition is due to Broto, Levi and Oliver [5]. Definition 2.3. Let F be a category on a finite p-group P, and let Q be a subgroup of P. For any morphism ϕ : Q P in F, we set N ϕ = {y N P (Q) there is z N P (ϕ(q)) such that ϕ( y u) = z ϕ(u) for all u Q}. In other words, N ϕ is the inverse image in N P (Q) of the group Aut P (Q) (ϕ 1 Aut P (ϕ(q)) ϕ) Note that in particular QC P (Q) N ϕ N P (Q). Definition 2.4. A fusion system on a finite p-group P is a category F on P such that Hom P (Q, R) Hom F (Q, R) for any two subgroups Q, R of P, and such that the following two properties hold: (I-S) Aut P (P) is a Sylow-p-subgroup of Aut F (P). (II-S) every morphism ϕ : Q P in F such that ϕ(q) is fully F-normalised extends to a morphism ψ : N ϕ P (that is, ψ Q = ϕ). This terminology is, of course, coherent with the previous section - we will prove in 2.10 that fusion systems of finite groups as defined in 1 are indeed fusion systems in the sense of 2.4. As mentioned before, it follows from work of Alperin and Broué [2], that every p-block of a finite group gives rise to a fusion system on any of its defect groups; see [12] for details. There are examples, due to Levi and Oliver [17], Ruiz and Viruel [22], of fusion systems which cannot be fusion systems of any finite group, and it has been shown by Kessar [11], Kessar and Stancu [15], that these exotic fusion systems cannot even occur as fusion systems of blocks. It is not known at present whether every fusion system of a block of some finite group occurs as fusion system of some (other) finite group. As far as terminology goes, Broto, Levi and Oliver [5] use the term saturated fusion system for what is called fusion systems in 2.4, and Puig calls fusion systems (full)

6 6 MARKUS LINCKELMANN Frobenius systems. The extension axiom (II-S) relates the role of N ϕ as object of F to its image N ϕ /Q in Aut F (Q). Definition 2.4 is equivalent to the definition given in Broto, Levi, Oliver [5] which uses a priori stronger axioms; the following two Propositions, due to Stancu [25], show that these two versions are equivalent. Proposition 2.5. ([25]) Let F be a fusion system on a finite p-group P. A subgroup Q of P is fully F-normalised if and only if Q is fully F-centralised and Aut P (Q) is a Sylow-p-subgroup of Aut F (Q). Proof. Assume that Q is fully F-normalised. We first show that then Q is also fully F-centralised. Let ϕ : R Q be an isomorphism in F such that R is fully F-centralised. By the extension axiom (II-S) in 2.4 there is a morphism ψ : RC P (R) P in F such that ψ R = ϕ. Hence ψ maps C P (R) to C P (Q), which implies that C P (R) C P (Q), hence equality since R is fully F-centralised. Thus Q is fully F-centralised. Choose Q to be of maximal order such that Q is fully F-normalised but Aut P (Q) is not a Sylow-psubgroup of Aut F (Q). Then Q is a proper subgroup of P by the Sylow axiom 2.4.(I-S). Choose a p-subgroup S of Aut F (Q) such that Aut P (Q) is a proper normal subgroup of S. Let ϕ S Aut P (Q). Since ϕ normalises Aut P (Q), for every y N P (Q) there is z N P (Q) such that ϕ( y u) = z ϕ(u) for all u Q. In other words, N ϕ = N P (Q). Since Q is fully F-normalised, it follows from 2.4.(II-S) that there is an automorphism ψ of N P (Q) in F such that ψ Q = ϕ. Since ϕ has p-power order, by decomposing ψ into its p-part and its p -part we may in fact assume that ψ has p-power order. Let τ : N P (Q) P be a morphism in F such that τ(n P (Q)) is fully F-normalised. Now τψτ 1 is a p-element in Aut F (τ(n P (Q))), thus conjugate to an element in Aut P (τ(n P (Q))). Therefore we may choose τ in such a way that there is y N P (τ(n P (Q))) satisfying τψτ 1 (v) = y v for any v τ(n P (Q)). Since ψ Q = ϕ, the automorphism τψτ 1 of τ(n P (Q)) stabilises τ(q). Thus y N P (τ(q)). Since Q is fully F-normalised we have N P (τ(q)) τ(n P (Q)), hence ψ(u) = τ 1 (y) u for all u N P (Q). But then in particular ϕ Aut P (Q), contradicting our initial choice of ϕ. Thus Aut P (Q) is a Sylow-p-subgroup in Aut F (Q). The converse is easy since N P (Q) = Aut P (Q) C P (Q). This proves the result. Lemma 2.6. Let F be a fusion system on a finite p-group P let Q, R be subgroups of P and let ϕ : Q R be an isomorphism in F such that R is fully F-normalised. Then there is an isomorphism ψ : Q R in F such that N ψ = N P (Q), or equivalently, such that ψ can be extended to a morphism from N P (Q) to P in F. Proof. The group ϕ Aut P (Q) ϕ 1 is a p-subgroup of Aut F (R). Since R is fully F-normalised, Aut P (R) is a Sylow-p-subgroup of Aut F (R) by 2.5. Thus there is β Aut F (R) such that β ϕ Aut P (Q) ϕ 1 β 1 Aut P (R). Set ψ = β ϕ. The above inclusion means precisely that for any y N P (Q) there is z N P (R) such that ψ c y ψ 1 = c z, where c y and c z are the automorphisms of Q and R induced by conjugation with y and z, respectively. Equivalently, for any y N P (Q) there is z N P (R) such that ψ c y = c z ψ, which in turn means that N ψ = N P (Q). The

7 INTRODUCTION TO FUSION SYSTEMS 7 extension axiom (II-S) implies that ψ can be extended to a morphism from N P (Q) to P in F. Proposition 2.7. ([25]) Let F be a fusion system on a finite p-group P. Let Q be a subgroup of P. Every morphism ϕ : Q P such that ϕ(q) is fully F-centralised extends to a morphism ψ : N ϕ P in F (that is, ψ Q = ϕ). Proof. Let ϕ : Q P be a morphism in F such that ϕ(q) is fully F-centralised. Let ρ : ϕ(q) P be a morphism in F such that R = ρ(ϕ(q)) is fully F-normalised. By 2.6 we may choose ρ in such a way that that N ρ = N P (ϕ(q)). In particular, ρ extends to a morphism σ : N P (ϕ(q)) P. But then N ϕ N ρ ϕ, hence ρ ϕ extends to a morphism τ : N ϕ P. Thus τ(n ϕ ) σ(n P (ϕ(q))), and hence we get a morphism σ 1 τ(nϕ ) τ : N ϕ P which extends ϕ as required. The result follows. Definition 2.8. Let F, F be a fusion systems on finite p-groups P, P, respectively. A morphism of fusion systems from F to F is a pair (α, Φ) consisting of a group homomorphism α : P P and a covariant functor Φ : F F with the following properties: (i) for any subgroup Q of P we have α(q) = Φ(Q); (ii) for any morphism ϕ : Q R in F we have Φ(ϕ) α = α ϕ. Note that Φ, if it exists, is determined by α. Thus the set of morphisms from F to F can be viewed as a subset of the set of group homomorphisms from P to P. In particular, the fusion systems F, F are isomorphic if there is a group isomorphism α : P = P such that Hom F (α(q), α(r)) = α Hom F (Q, R) α 1 α(q) for all subgroups Q, R of P. If one takes the view that understanding a fusion system F on a given finite p-group P amounts to understanding what morphisms beyond those induced by P itself are morphisms in F, one ends up looking at a category obtained from dividing out by inner automorphisms : Definition 2.9. Let F be a fusion system on a finite p-group P. The orbit category of F is the category F having the subgroups of P as objects and, for any two subgroups Q, R of P we have Hom F(Q, R) = Aut R (R)\Hom F (Q, R), with composition of morphisms induced by that in F. This makes sense: with the notation above, the group Aut R (R) acts on Hom F (Q, R) by composition of group homomorphisms, and if ϕ, ϕ Hom F (Q, R) are in the same Aut R (R)-orbit and ψ, ψ Hom F (R, S) in the same Aut S (S)-orbit, a trivial verification shows that ψ ϕ, ψ ϕ Hom F (Q, S) are in the same Aut S (S)-orbit as well, for any subgroups Q, R, S of P. In order to show that fusion systems of finite groups are indeed fusion systems in the sense of 2.4 we collect a few technicalities in the following Lemma.

8 8 MARKUS LINCKELMANN Lemma Let G be a finite group, let P be a Sylow-p-subgroup of G and set F = F P (G). Then F is a category on P, and for any subgroup Q of P the following hold: (i) Q is fully F-centralised if and only if C P (Q) is a Sylow-p-subgroup of C G (Q). (ii) Q is fully F-normalised if and only if N P (Q) is a Sylow-p-subgroup of N G (Q). Proof. Any inclusion morphism is in F because it can be viewed as conjugation by the unit element. If ϕ : Q R is a morphism in F then there is x G such that ϕ(u) = x u for all u Q. Then the induced isomorphism Q = x Q is still given by conjugation with x, hence also a morphism in F. The composition of morphisms in F is the usual composition of group homomorphisms by definition 1.1. Thus F is a category on P in the sense of 2.1. Let Q be a subgroup of P. Let S be a Sylow-psubgroup of C G (Q) containing C P (Q). Then there is x in G such that x (QS) P. Then conjugation by x is an isomorphism ϕ : Q = x Q in F, and x S C P ( x Q). It follows that C P (Q) S C P ( x Q). Thus Q is fully F-centralised if and only if C P (Q) = S, or if and only if C P (Q) = S. This proves (i). The same argument with normalisers instead of centralisers proves (ii). Theorem Let G be a finite group and let P be a Sylow-p-subgroup of G. The category F P (G) is a fusion system. Proof. Set F = F P (G). Clearly F is a category on P, and we have F P (P) F. Thus we only have to check the two axioms (I-S) and (II-S) from 2.4. We have Aut F (P) = N G (P)/C G (P), and the image of P in this quotient group is a Sylow-p-subgroup, which implies that Aut P (P) = P/Z(P) is a Sylow-p-subgroup of Aut F (P). This proves (I-S). Let now Q be a subgroup of P and let ϕ : Q P be a morphism in F. Set R = ϕ(q) and suppose that R is fully F-normalised. Let x G such that ϕ(u) = x u for all u Q. We have N ϕ = {y N P (Q) z N P (R) : ϕ( y u) = z ϕ(u) ( u Q)}. Since ϕ is given by conjugation with x, this translates to N ϕ = {y N P (Q) z N P (R) : xy u = zx u ( u Q)}. The equation xy u = zx u for all u Q means that x 1 z 1 xy centralises Q, hence z 1 xyx 1 centralises x Q = R, and so we can write xyx 1 = zc for some c C G (R). This shows that we have x N ϕ N P (R)C G (R). Since R is fully F-normalised, by 2.10 the group N P (R) is a Sylow-p-subgroup of N G (R), hence of N P (R)C G (R). Thus there is an element d C G (R) such that dx N ϕ N P (R). Define ψ : N ϕ P by ψ(y) = dx y for all y N ϕ. We claim that ψ extends ϕ. Indeed, if u Q then ψ(u) = dx u = d ϕ(u) = ϕ(u) because d centralises R = ϕ(q). This proves (II-S).

9 INTRODUCTION TO FUSION SYSTEMS 9 Theorem Let G be a finite group, let P be a Sylow-p-subgroup of G and let K be a normal p -subgroup of G. Set Ḡ = G/K and denote by P the image of P in Ḡ. The canonical group homomorphism α : G Ḡ induces an isomorphism of fusion systems F P (G) = F P(Ḡ). Proof. Since K is a p -group the map α induces an isomorphism P = P. Thus, for any two subgroups Q, R of P the map Hom F (Q, R) Hom F( Q, R) induced by α is injective, where Q, R are the canonical images of Q, R in P. In order to show that it is surjective, let ψ : Q R be a morphism in F. Let S be the inverse image in P of ψ( Q). Then there is x G such that conjugation by x induces ψ; in other words, x Q KS and hence both x Q and S are Sylow-p-subgroups of KS. Thus there is y K such that yx Q = S. Then ϕ : Q R defined by ϕ(u) = yx u for u Q induces the morphism ψ as required. We note an elementary group theoretic consequence for future reference: Corollary Let G be a finite group, let K be a normal p -subgroup of G and let α : G G/K be the canonical surjection. Let T be a Sylow-p-subgroup of G/K and let P be a Sylow-p-subgroup of G such that α(p) = T. Let y G/K. Then there is x G such that α(x) = y and such that α(p x P) = T y T. Proof. Set R = T y T. Then y 1 R T. Thus conjugation by y 1 is a morphism R y 1 R in F T (G/K). By 2.12 this lifts to a morphism Q x 1 Q for some x G satisfying α(x) = y, where Q is the inverse image of R in P. Then Q P x P, hence α(q) α(p x P) α(p) α( x P) = T y T = α(q) and hence equality. The following theorem by Levi and Oliver, using Solomon s work mentioned in 1.7 above, implies that there are exotic fusion systems which are not of the form F P (G) for any finite group G with Sylow-p-subgroup P: Theorem (Levi, Oliver [17]) Let P be a Sylow-2-subgroup of Spin 7 (3). There is a fusion system F on P such that F P (Spin 7 (3)) F and such that all involutions in P are F-conjugate. Ruiz and Viruel classified in [22] all possible fusion systems on extraspecial p-groups of order p 3 and, using the classification of finite simple groups, showed that some of them are exotic. 3 Normalisers and centralisers The concept of K-normalisers in fusion systems as well as the main result of this section, Theorem 3.6, are due to Puig [21]. Our presentation follows partly [5, Appendix].

10 10 MARKUS LINCKELMANN We start with some particular cases of K-normalisers. As in group theory, one can define normalisers and centralisers in fusion systems: Definition 3.1. Let F be a fusion system on a finite p-group P and let Q be a subgroup of P. (i) The normaliser of Q in F is the category N F (Q) on N P (Q) having as morphisms all group homomorphisms ϕ : R S, for R, S subgroups of N P (Q), for which there exists a morphism ψ : QR QS in F satisfying ψ(q) = Q and ψ R = ϕ. (ii) The centraliser of Q in F is the category C F (Q) on C P (Q) having as morphisms all group homomorphisms ϕ : R S, for R, S subgroups of C P (Q), for which there exists a morphism ψ : QR QS in F satisfying ψ Q = Id Q and ψ R = ϕ. (iii) We denote by QC F (Q) the subcategory of N F (Q) on QC P (Q) having as morphisms all group homomorphisms ϕ : R S, for R, S subgroups of QC P (Q), for which there exists a morphism ψ : QR QS and v Q such that ψ Q = c v and ψ R = ϕ, where c v is the automorphism of Q given by conjugation with v. (iv) We denote by N P (Q)C F (Q) the subcategory of N F (Q) on N P (Q) having as morphisms all group homomorphisms ϕ : R S, for R, S subgroups of N P (Q), for which there exists a morphism ψ : QR QS and v N P (Q) such that ψ Q = c v and ψ R = ϕ. Theorem 3.2. Let F be a fusion system on a finite p-group P and let Q be a subgroup of P. If Q is fully F-normalised then N F (Q) is a fusion system on N P (Q), and if Q is fully F-centralised then C F (Q) is a fusion system on C P (Q), QC F (Q) is a fusion system on QC P (Q) and N P (Q)C F (Q) is a fusion system on N P (Q). Note that we have inclusions of categories C F (Q) QC F (Q) N P (Q)C F (Q) N F (Q). Definition 3.1 and Theorem 3.2 are part of the more general concept: Definition 3.3. ([21]) Let P be a finite p-group, let Q be a subgroup of P and let K be a subgroup of Aut(Q). The K-normaliser of Q in P is the subgroup N K P (Q) = {y N P(Q) α K : α(u) = yuy 1 u Q}. We set Aut K P (Q) = K Aut P(Q) and Aut K F (Q) = K Aut F(Q). In other words, N K P (Q) consists of all elements in N P(Q) which induce, by conjugation, an automorphism of Q belonging to the automorphism subgroup K. Note that C P (Q) N K P (Q), and if K contains all inner automorphisms of Q then also Q NK P (Q). We have Aut K P (Q) = N K P (Q)C P(Q)/C P (Q). There are various special cases of this construction we will encounter most frequently: if K = {Id Q } then N K P (Q) = C P(Q) and Aut K P (Q) = AutK F (Q) = {Id Q}; if K = Aut Q (Q) then N K P (Q) = QC P(Q) and Aut K P (Q) = AutK F (Q) = Aut Q(Q); finally, if K = Aut(Q) then N K P (Q) = N P(Q), Aut K P (Q) = Aut P (Q) and Aut K F (Q) = Aut F (Q). For ϕ : Q P any injective group

11 INTRODUCTION TO FUSION SYSTEMS 11 homomorphism the group ϕ K = ϕ K ϕ 1 is a subgroup of Aut(ϕ(Q)), and it makes thus sense to consider N ϕ K P (ϕ(q)). Definition 3.4. Let F be a fusion system on a finite p-group P, let Q be a subgroup of P and let K be a subgroup of Aut(Q). We say that Q is fully K-normalised in F if N K P (Q) N ϕ K P (ϕ(q)) for any morphism ϕ : Q P in F. Thus Q is fully Aut(Q)-normalised in F if and only if Q is fully normalised in the sense of 2.2, and Q is fully {Id Q }-normalised if and only if Q is fully F-centralised. Note that being fully Aut Q (Q)-normalised is also equivalent to being fully centralised. Definition 3.5 Let F be a fusion system on a finite p-group P, let Q be a subgroup of P and let K be a subgroup of Aut(Q). The K-normaliser of Q in F is the subcategory NF K(Q) of F on NK P (Q) having morphism sets Hom N K F (Q)(R, S) = {ϕ Hom F (R, S) ψ Hom F (QR, QS) : ψ R = ϕ, ψ Q K}. for any two subgroups R, S in N K P (Q). If K = {Id Q } then N K F (Q) = C F(Q) and if K = Aut(Q) then N K F (Q) = N F(Q). Other cases of interest, as mentioned above, include K = Aut Q (Q) which yields N K F (Q) = QC F(Q) and K = Aut P (Q) which yields N K F (Q) = N P(Q)C F (Q). Thus Theorem 3.2 is a consequence of the following more general result. Theorem 3.6. (Puig [21]) Let F be a fusion system on a finite p-group P, let Q be a subgroup of P and let K be a subgroup of Aut(Q). Suppose that Q is fully K-normalised in F. Then N K F (Q) is a fusion system on NK P (Q). The proof requires the following Proposition (part (i) generalises 2.5). Proposition 3.7. Let F be a fusion system on a finite p-group P, let Q be a subgroup of P and let K be a subgroup of Aut(Q). (i) The group Q is fully K-normalised in F if and only if Q is fully F-centralised and Aut K P (Q) is a Sylow-p-subgroup of Aut K F (Q). (ii) Let ϕ : Q R be an isomorphism in F and set L = ϕ K ϕ 1. Suppose that R is fully L-normalised in F. Then there are morphisms τ : Q NP K (Q) P in F and κ K such that τ Q = ϕ κ. (iii) Let ψ : QNP K (Q) P be a morphism in F. If Q is fully K-normalised in F then ψ(q) is fully ψ K ψ 1 -normalised in F. (iv) If Q is fully F-normalised and K Aut P (Q) then Q is fully K-normalised in F. Proof. (i) Suppose that Q is fully K-normalised in F. By 2.6 there is an isomorphism ϕ : Q R in F such that R is fully F-normalised and such that ϕ extends to a morphism

12 12 MARKUS LINCKELMANN ψ : N P (Q) P in F. Set L = ϕ K ϕ 1. Thus ψ maps NP K(Q) to NL P (R). Since Q is fully K-normalised we have NP K(Q) NL P (R) and hence ψ induces in fact an isomorphism NP K(Q) = NP L (R). Any such isomorphism restricts to an isommorphism C P (Q) = C R (Q). Since R is fully F-normalised, R is fully F-centralised by 2.5 and thus Q is fully F-centralised. The group ϕ Aut K P (Q) ϕ 1 is a p-subgroup of the subgroup Aut L F(R) of Aut F (R). By 2.5 the group Aut P (R) is a Sylow-p-subgroup of Aut F (R). Thus some conjugate of Aut P (R) intersected with Aut L F(R) will be a Sylowp-subgroup of Aut L F(R), and this Sylow-p-subgroup can, of course, be chosen to contain the p-subgroup ϕ Aut K F (Q) ϕ 1. Thus there is β Aut F (R) such that ϕ Aut K F (Q) ϕ 1 β Aut P (R) β 1 Aut L F (R) and such that the right side of this inclusion is a Sylow-p-subgroup of Aut L F (R). Then Aut P (R) Aut β 1 L β F (R) = Aut β 1 L β P Since Q is fully K-normalised we have Aut K P (Q) Autβ 1 L β P (R) is a Sylow-p-subgroup of Aut β 1 L β F (R). (R) which implies that Aut K P (Q) is a Sylow-p-subgroup of Aut K F (Q). Suppose conversely that Q is fully F-centralised and that Aut K P (Q) is a Sylowp-subgroup of Aut K F (Q). Let ϕ : Q R be an isomorphism in F and set L = ϕ K ϕ 1. Then C P (R) C P (Q) and since Aut K F (Q) = Aut L F(R) we get that Aut L P(R) Aut K P (Q), which together implies NP L(R) = C P(R) Aut L P(R) C P (Q) Aut K P (Q) = NK P (Q), showing that Q is fully K-normalised in F. (ii) Since R is fully K-normalised in F it is in particular fully F-centralised. Set N = Q NP K(Q). Then ϕ N ϕ 1 is a p-subgroup of Aut L F (R). By (i), AutL P (R) is a Sylow-p-subgroup of Aut L F(R). Thus there is λ Aut L F(R) such that λ ϕ N ϕ 1 λ 1 Aut P (R). This means that N N λ ϕ, and hence λ ϕ extends to a morphism τ : Q NP K(Q) P. Now λ ϕ = ϕ (ϕ 1 λ ϕ) and κ = ϕ 1 λ ϕ Aut K P (Q) is as required. (iii) The map ψ sends NP K (Q) to Nψ K ψ 1 P (ψ(q)). In particular, NP K(Q) N ψ K ψ 1 P (ψ(q)), whence the result. (iv) If K Aut P (Q) then NP K(Q) = C P(Q) K. Thus if Q is fully F-normalised, it is fully F-centralised, whence the statement. Proof of Theorem 3.6. Clearly NF K(Q) is a category on NK P (Q) in the sense of 2.1. For any subgroup R of NP K (Q) and any subgroup I of Aut(R) we set Then N I N K P K I = {α Aut(QR) α Q K, α R I}. (Q)(R) = NK I P (QR) is the subgroup of all y N P (Q) N P (R) such that conjugation by y induces on Q an automorphism in K and on R an automorphism in I. With this notation,

13 INTRODUCTION TO FUSION SYSTEMS the restriction map AutF K I (QR) AutI NF K (Q)(R) is surjective. Indeed, any β Aut N K F (Q)(R) extends to some α Aut F (QR) with α Q K, and since β = α R, the surjectivity follows. We observe next that for any subgroup R of NP K (Q) and any subgroup I of Aut(R) there is a morphism ϕ : QR QNP K(Q) in F such that ϕ Q K and such that ϕ(qr) = Qϕ(R) is fully ϕ (K I) ϕ 1 -normalised. To see this, let ρ : QR P be a morphism in F such that ρ(qr) is fully ρ (I K) ρ 1 -normalised. Set σ = ρ Q. Since Q is fully K-normalised in F, by 3.7.(ii) applied to σ 1 and σ(q) there is a morphism τ : σ(q)np σ K σ 1 (σ(q)) P and κ σ K σ 1 such that τ σ(q) = σ 1 κ. Set ϕ = τ ρ. Then ϕ Q = τ ρ Q = τ σ = σ 1 κ σ K. Note that ρ(qr)n ρ (K I) ρ 1 P (ρ(qr)) σ(q)np σ K σ 1 (σ(q)) and that ρ(qr) is fully ρ (K I) ρ 1 -normalised in F. It follows from 3.7.(iii) applied to the appropriate restriction of τ and ρ(qr) that RQ is fully K I-normalised in F. This proves We prove now the Sylow axiom I-S from 2.4. Set R = NP K (Q). We have to show that the inner automorphism group I = Aut R (R) is a Sylow-p-subgroup of Aut N K F (Q)(R). Set A = Aut(R). By 3.6.2, the restriction map AutF K A (QR) Aut N K F (Q)(R) is surjective, and it maps AutF K I (QR) onto I = Aut R (R). It suffices to show that AutF K I (QR) contains a Sylow-p-subgroup of AutF K A (QR). Applying with R = NP K (Q) shows that QR is fully ϕ (K A) ϕ 1 -normalised for some ϕ : QR QNP K (Q) in F whose restriction to Q is in K, and hence ϕ (K A) ϕ 1 = K A. Thus the group AutP K A (QR) is a Sylow-p-subgroup of AutF K A (QR) by 3.7.(i). But since R = NP K(Q) we get AutP K A (QR) = AutP K I (QR), which is clearly contained in AutF K I (QR). This proves that the Sylow axiom I-S holds. Before we proceed with the proof of the extension axiom, we show that if a subgroup R of NP K(Q) is fully I-normalised in NK F (Q) for some subgroup I of Aut(R) then QR is fully K I-normalised in F. Indeed, by there exists a morphism ϕ : QR QNP K (Q) in F such that ϕ Q K and such that Qϕ(Q) is fully ϕ (K I) ϕ 1 -normalised. We have NP K I (QR) = N I NP K(Q)(R) by and NI NP K (Q)(R) Nϕ I ϕ 1(ϕ(R)) because R NP K(Q) is fully I-normalised. Now N ϕ I ϕ 1 (ϕ(r)) = N K (ϕ I ϕ 1 ) NP K(Q) P (ϕ(r)) by and since ϕ Q K we have K (ϕ I ϕ 1 ) = ϕ (K I) ϕ 1, from which we get that the last group is equal to N ϕ (K I) ϕ 1 P which proves the statement (ϕ(r)). In particular, N K I P (QR) N ϕ (K I) ϕ 1 P (ϕ(r)), For the proof of the extension axiom II-S, let R be a subgroup of NP K (Q) and let ϕ : R NP K(Q) be a morphism in NK F (Q) such that ϕ(r) is fully NK F (Q)-normalised. We consider the group N ϕ as defined in 2.3 for the morphism ϕ in the category NF K(Q); that is, N ϕ is the subgroup of all y N N K P (Q)(R) for which there exists an element

14 14 MARKUS LINCKELMANN z N N K P (Q)(ϕ(R)) satisfying ϕ(yuy 1 ) = zϕ(u)z 1 for all u R. Set I = Aut Nϕ (R). Note that conjugation by any y N ϕ leaves Q invariant and induces an automorphism of Q belonging to K. Then N ϕ = N I NP K (Q)(R) = NK I P (QR). We have ϕ I ϕ 1 Aut N K P (Q)(ϕ(Q)) Aut N K F (Q)(ϕ(R)), and hence ϕ(r) is fully ϕ I ϕ 1 -normalised in NF K(Q) by 3.7.(iv). Thus Qϕ(R) is fully K (ϕ I ϕ 1 )-normalised by Since ϕ is a morphism in NF K(Q) there exists a morphism ψ : QR P in F such that ψ Q K and ψ R = ϕ. So ψ(qr) = Qϕ(R) is fully K (ϕ I ϕ 1 )-normalised by the above. Now ψ 1 (K (ϕ I ϕ 1 )) ψ = K I because ψ Q K and because ψ 1 R ϕ = Id R. Applying 3.7.(ii) to ψ and QR yields the existence of a morphism τ : QRNP K I (QR) P and κ K I such that τ QR = ψ κ. Since κ R I = Aut R (R) there is y R such that κ R = c y, conjugation by y. Then τ Nϕ c 1 y : N ϕ P is a morphism in NF K(Q), because it extends to the morphism τ QNϕ c y : QN ϕ P whose restriction to Q is τ Q c 1 y Q = ψ Q κ Q c 1 y Q K. Moreover, the morphism τ Nϕ c 1 y restricted to R is equal to ψ R κ R c 1 y R = ψ R = ϕ, and hence this morphism extends ϕ as required in the extension axiom. This completes the proof of 3.6. As in finite group theory, given a fusion system F on a finite p-group P, one of the questions one would like to be able to address is that of control of fusion by normalisers of p-subgroups, a question which we can now rephrase as follows: when do we have an equality F = N F (Q) for some normal subgroup Q of P. The most basic case, Burnside s theorem 1.2, remains true for arbitrary fusion systems: Theorem 3.8. Let F be a fusion system on an abelian finite p-group P. Then F = N F (P). Proof. Let Q be a subgroup of P and let ϕ : Q P be a morphism in F. Since P is abelian, every subgroup of P is fully F-normalised, and hence ϕ extends to a morphism ψ : QC P (Q) P, by 2.4.(II-S). Again, since P is abelian, we have QC P (Q) = P, and thus ψ Aut F (P). We quote further results without proof, referring to the original papers. It has been shown by Stancu that the conclusion of 3.8 remains true for some other classes of finite p-groups: Theorem 3.9. ([26]) Let P be a metacyclic finite p-group for some odd prime p. Then for every fusion system F on P we have F = N F (P). Theorem ([26]) Let P be a finite p-group of the form P = E A, where A is elementary abelian and where E is extra-special of exponent p 2 if p is odd and not isomorphic to D 8 if p = 2. Then for every fusion system F on P we have F = N F (P). The parts of Frobenius s theorem 1.4 related to the fusion system of a finite group remain true as well:

15 INTRODUCTION TO FUSION SYSTEMS 15 Theorem Let F be a fusion system on a finite p-group P. The following are equivalent: (i) For any Q P the group Aut F (Q) is a p-group. (ii) F = F P (P). (iii) For any non-trivial fully F-normalised subgroup Q of P we have N F (Q) = F NP (Q)(N P (Q)). The proof of 3.11 is a trivial adaptation of the ideas of the proof of 1.4. The p- nilpotency criterion 1.5 of Glauberman and Thompson can also be generalised: Theorem ([13, Theorem A]) Let p be an odd prime and let F be a fusion system on a finite p-group P. We have F = F P (P) if and only if N F (Z(J(P))) = F P (P). Gilotti and Serena [7] gave a general criterion when a fusion system of a finite group is of the form N F (Q), and Stancu generalised this to arbitrary fusion systems. Before we state this, we need the following terminology: Definition Let F be a fusion system on a finite p-group P and let Q be a subgroup of P. (iv) Q is weakly F-closed if for every morphism ϕ : Q P in F we have ϕ(q) = Q. (v) Q is strongly F-closed, if for any subgroup R of P and any morphism ϕ : R P in F we have ϕ(r Q) Q. If Q is strongly F-closed then Q is weakly F-closed. One easily checks that if Q is strongly F-closed then for any subgroup R of P and any morphism ϕ : R P in F we have in fact ϕ(r Q) = ϕ(r) Q. Indeed, the left side is contained in the right side by the above definition, and the other inclusion is obtained by applying this inclusion to ϕ(r) and the morphism ϕ 1 viewed as morphism from ϕ(r) to P. If F = N F (Q) for some subgroup Q of P, then clearly Q is strongly F-closed. The converse of this statement is not true, in general. More precisely: Theorem ([26, 6.11]) Let F be a fusion system on a finite p-group P, and let Q be a subgroup of P. The following are equivalent: (i) F = N F (Q). (ii) The subgroup Q is strongly F-closed and there is a central series Q = Q n > Q n 1 > > Q 1 > {1} with Q i weakly F-closed for 1 i n 1. If Q is an abelian and strongly F-closed subgroup of P then Q > {1} is a central series of weakly F-closed subgroups, and hence 3.14 applies:

16 16 MARKUS LINCKELMANN Corollary Let F be a fusion system on a finite p-group P and let Q be an abelian subgroup of P. We have F = N F (Q) if and only if Q is strongly F-closed. We give an alternative proof of 3.15 following 9.2 below. By [13], Glauberman s ZJtheorem 1.6 carries over as well, but it requires some extra effort to define a notion of Qd(p)-free fusion systems, replacing the hypothesis on G having no subquotient isomorphic to Qd(p) in 1.6; see 4.7 and 4.8 below for more details. When specialised to fusion systems of finite groups, normalisers and centralisers in the fusion system correspond to the fusion systems of the relevant normaliser and centralisers: Proposition Let G be a finite group, let P be a Sylow-p-subgroup and set F = F P (G). Let Q be a subgroup of P. (i) If Q is fully F-normalised then N FP (G)(Q) = F NP (Q)(N G (Q)). (ii) If Q is fully F-centralised then C FP (G)(Q) = F CP (Q)(C G (Q)). (iii) If Q is normal in G then F = N F (Q). Proof. The statements make sense: if Q is fully F-normalised then N P (Q) is a Sylowp-subgroup of N G (Q), and if Q is fully F-centralised then C P (Q) is a Sylow-p-subgroup of C G (Q), by 2.8. The rest is a trivial verification. 4 Centric subgroups Definition 4.1. Let F be a fusion system on a finite p-group P. A subgroup Q of P is called F-centric if for any isomorphism ϕ : Q R in F we have C P (R) = Z(R). Note that every F-centric subgroup Q of P is trivially fully F-centralised, because all centralisers C P (R) of subgroups R isomorphic to Q in F have the same order Z(Q). Definition 4.2. Let F be a fusion system on a finite p-group P. We denote by F c the full subcategory of F having as objects all F-centric subgroups of P. We denote by F c the canonical image of F c in the orbit category F of F. Proposition 4.3. Let F be a fusion system on a finite p-group P and let Q be a fully F-centralised subgroup of P. Then QC P (Q) is F-centric. Proof. Let ϕ : QC P (Q) R be an isomorphism in F. Let ψ : ϕ(q) Q be the inverse of ϕ restricted to ϕ(q). Since Q is fully F-centralised, ψ extends, by 2.6, to a morphism τ : ϕ(q)c P (ϕ(q)) P whose image must be contained in QC P (Q). Since ϕ(q) R we have RC P (R) ϕ(q)c P (ϕ(q)), hence RC P (R) QC P (Q) = R, which forces C P (R) R or C P (R) = Z(R). Thus QC P (Q) is F-centric.

17 INTRODUCTION TO FUSION SYSTEMS 17 Proposition 4.4. Let F be a fusion system on a finite p-group P and let Q, R be subgroups of P such that Q R. If Q is F-centric then R is F-centric and we have Z(R) Z(Q). Proof. Let ψ : R P be a morphism in F. If Q is F-centric then C P (ψ(q)) = Z(ψ(Q)), hence C P (ψ(r)) C P (ψ(q)) ψ(r) and hence C P (ψ(r)) = Z(ψ(R)). In particular, Z(R) = C P (R) C P (Q) = Z(Q). Even though elementary, the crucial observation is that thanks to 4.4, taking centers of centric subgroups is a contravariant functor from F c to the category of finitely generated Z (p) -modules, where Z (p) = { a a, b Z, p b}. b Theorem 4.5. Let F be a fusion system on a finite p-group P There is a unique functor Z : F c mod(z (p) ) sending any F-centric subgroup Q of P to Z(Q) and sending the class of any morphism ϕ : Q R between F-centric subgroups Q, R in F to the unique morphism Z(ϕ) : Z(R) Z(Q) which sends z Z(R) to the unique element y Z(Q) satisfying ϕ(y) = z. Proof. If ϕ : Q R is a morphism in F with Q, R F-centric, then the subgroup ϕ(q) of R is F-centric, and hence Z(R) Z(ϕ(Q)) = ϕ(z(q)). Thus ϕ induces a unique map Z(R) Z(Q) as claimed. This map does not depend on the choice of ϕ in its class modulo inner automorphisms of R because any inner automorphism of R is the identity on Z(R). Thus Z is well-defined. The following result appears in work of Külshammer and Puig [16] for fusion systems of p-blocks and in Broto, Castellana, Grodal, Levi, Oliver [4, 4] in general: Theorem 4.6. Let F be a fusion system on a finite p-group P. For any F-centric fully normalised subgroup Q of P there is, up to isomorphism, a unique finite group L = L F Q having N P(Q) as Sylow-p-subgroup such that Q L, C L (Q) = Z(Q) and N F (Q) = F NP (Q)(L). In other words, if F is a fusion system on a finite p-group P such that F = N F (Q) for some F-centric normal subgroup Q of P then F is automatically the fusion system of some finite group. The proof of 4.6 uses some of the cohomological machinery developed in [5], showing that the center functor from 4.5 applied to the fusion system N F (Q) is acyclic. The three properties of the group L = L F Q imply that Aut F (Q) = N L (Q)/C L (Q) = L/Z(Q), and hence L fits into a short exact sequence of finite groups 1 Z(Q) L Aut F (Q) 1

18 18 MARKUS LINCKELMANN whose restriction to the Sylow-p-subgroup N P (Q) of L is of the form 1 Z(Q) N P (Q) Aut P (Q) 1. This characterises the group L up to isomorphism because the restriction map on cohomolgy H 2 (Aut F (Q); Z(Q)) H 2 (Aut P (Q); Z(Q)) is well-known to be injective. Definition 4.7. ([13, 1.1]) A fusion system F on a finite p-group P is called Qd(p)-free if Qd(p) is not involved in any of the groups L F Q, with Q running over the set of F-centric fully F-normalised subgroups of P. Glauberman s ZJ-theorem admits the following version for fusion systems (which for fusion systems of blocks has also been noted by G. R. Robinson): Theorem 4.8. ([13, Theorem B]) Let p be an odd prime and let F be a fusion system on a finite p-group P. If F is Qd(p)-free then F = N F (Z(J(P))). While all morphisms in the fusion system F are monomorphisms, this is no longer true in the orbit category. However, we have the following: Theorem 4.9. Let F be a fusion system on a finite p-group P. Every morphism in the category F c is an epimorphism. The technicalities for the proof of 4.9 are given in the following two well-known lemmas. Lemma Let F be a fusion system on a finite p-group P. Let Q, R be F-centric subgroups of P such that Q R, and let ϕ Aut F (R). We have ϕ Q = Id Q if and only if ϕ Aut Z(Q) (R). Proof. Assume that ϕ Q = Id Q. We proceed by induction over [R : Q]. Consider first the case where Q is normal in R. Let u Q and v R. Then v u Q, hence v u = ϕ( v u) = ϕ(v) u, and thus v 1 ϕ(v) C R (Q) = Z(Q), or equivalently, ϕ(v) = vz for some z Z(Q). If ϕ has order prime to p in Aut(R) this forces ϕ = Id R. Therefore we may assume that the order of ϕ is a power of p. Upon replacing R by a fully F- normalised F-conjugate we may assume that ϕ Aut P (R). Since ϕ restricts to Id Q and since Q is F-centric this implies that ϕ Aut Z(Q) (R). This proves 4.10 if Q is normal in R. In general, if ϕ Q = Id Q then ϕ(n R (Q)) = N R (Q). Thus ϕ NR (Q) Aut Z(Q) (N R (Q)) by the previous paragraph. Hence there is z Z(Q) such that c z ϕ NR (Q) = Id NR (Q), where c z is the automorphism of R given by conjugation with z. By induction we get c z ϕ Aut Z(NR (Q))(Q). As all involved groups are F-centric we have Z(N R (Q)) Z(Q), and thus ϕ Aut Z(Q) (R) as claimed. The converse is trivial.

19 INTRODUCTION TO FUSION SYSTEMS 19 Lemma Let F be a fusion system on a finite p-group P, let Q, R be F-centric subgroups of P such that Q R, and let ϕ, ϕ Hom F (R, P) such that ϕ Q = ϕ Q. Then ϕ(r) = ϕ (R). Proof. Let v N R (Q). For every u Q we have ϕ( v u) = ϕ ( v u), hence ϕ(v) 1 ϕ (v) C P (ϕ(q)) = Z(ϕ(Q)). It follows that ϕ(n R (Q)) = ϕ (N R (Q)). By 4.10, ϕ NR (Q) and ϕ NR (Q) differ by conjugation with an element in Z(Q), and we may therefore assume that their restrictions to N R (Q) actually coincide. The equality ϕ(r) = ϕ (R) follows by induction. Proof of Theorem 4.9. Let Q, R, S be F-centric subgroups of P, let ϕ Hom F (Q, R) and let ψ, ψ Hom F (R, S). Assume that the images of ψ ϕ and ψ ϕ in Hom F c(q, S) coincide. Up to replacing ψ by some S-conjugate, we may assume that ψ ϕ = ψ ϕ. Thus the restrictions to ϕ(q) of ψ, ψ coincide. It follows from 4.11 that ψ(r) = ψ (R). Thus ψ 1 ψ is an automorphism of R which restricts to the identity on ϕ(q), hence ψ 1 ψ Aut Z(ϕ(Q)) (R) by Thus the images of ψ, ψ in the orbit category are equal. We conclude this section with a remark on what the notion of being centric means in the case of fusion systems of finite groups: Proposition Let G be a finite group, let P be a Sylow-p-subgroup of G and set F = F P (G). A subgroup Q of P is F-centric if and only if C G (Q) = Z(Q) O p (C G (Q)). Proof. By 2.8, Q is F-centric if and only if Z(Q) is a Sylow-p-subgroup of C G (Q). Since Z(Q) is a central subgroup of C G (Q) this happens if and only if C G (Q) splits as described, by some standard group theory. Using 4.12 it is easy to directly construct the groups L F Q in the case where F = F P(G): just take L F Q = N G(Q)/O p (C G (Q)). Since C G (Q) = Z(Q) O p (Q) we get a short exact sequence 1 Z(Q) L F Q N G(Q)/C G (Q) = Aut F (Q) 1 as above. 5 Alperin s fusion theorem Alperin s fusion theorem states that a fusion system F on a finite p-group P is completely determined by automorphism groups Aut F (Q) with Q running over a certain subset of subgroups of P. Its original version, due to Alperin [1], is stated for fusion systems of finite groups and the class of so-called radical subgroups of P, refined by Goldschmidt [9] who extrapolates Alperin s methods to show that the potentially smaller class of essential subgroups suffices to determine F. Puig showed in [20] that

20 20 MARKUS LINCKELMANN essential subgroups are essential - that is, no smaller class will determine F - and showed furthermore which subsets of Aut F (Q) are actually necessary to determine F, with Q running over essential subgroups. We need some terminology from finite group theory. A proper subgroup H of a finite group G is called strongly p-embedded if H contains a Sylow-p-subgroup P of G and P 1 but H x P = 1 for any x G H. The existence of a strongly p-embedded subgroup is equivalent to the poset S p (G) of nontrivial p-subgroups of G being disconnected. Indeed, if H is a strongly p-embedded subgroup of G then S p (G) has at least two connected components, namely that of a Sylow-p-subgroup P contained in H and that of x P with x G H. Conversely, if S p (G) has a connected component different from S p (G) then the stabliser H of such a connected component is easily checked to be strongly p-embedded. We denote by O p (G) the unique maximal normal subgroup of G whose order is a power of p, and we denote by O p (G) the unique maximal normal subgroup of G whose order is prime to p. Note that if O p (G) 1 then S p (G) is connected, and hence G has no strongly p-embedded subgroup. If F is a fusion system on a finite p-group P then for any subgroup Q in P we have Aut Q (Q) Aut F (Q), and hence Aut Q (Q) O p (Aut F (Q)). Definition 5.1. Let F be a fusion system on a finite p-group P. (i) A subgroup Q of P is called F-radical if O p (Aut F (Q)) = Aut Q (Q). (ii) A subgroup Q of P is called F-essential if Q is F-centric and if Aut F (Q)/Aut Q (Q) has a strongly p-embedded subgroup. By the remarks preceding this definition, if Q is an F-essential subgroup of P then Q is F-centric radical. The converse need not be true, in general. Note also that an F-essential subgroup has to be a proper subgroup of P because Aut F (P)/Aut P (P) is a p -group. Theorem 5.2. (Alperin s fusion theorem) Let F be a fusion system on a finite p-group P. Every isomorphism in F can be written as a composition of finitely many isomorphisms of the form ϕ : R S in F for which there exists a subgroup Q containing both R, S and an automorphism α Aut F (Q) such that α R = ϕ and either Q = P or Q is fully F-normalised essential. The picture to have in mind is this: given any isomorphism ϕ : R S in F, the above theorem claims that there are finitely many subgroups R 0, R 1,.., R n of P, with isomorphisms ϕ i : R i R i+1 for 0 i n 1, and fully F-normalised essential subgroups Q 1, Q 2,.., Q n such that R i, R i+1 are contained in Q i, with automorphisms α i Aut F (Q i ) satisfying α i Ri = ϕ i for 0 i n 1, and such that R = R 0, R n = S with ϕ = ϕ n 1 ϕ 1 ϕ 0. The crucial ingredient which makes this work is the extension axiom. We separate some of the arguments, and for the sake of providing future reference, allow some redundancy in the following four Lemmas. In many applications it suffices to know that