key: cord-0214260-z6kwgtr1
authors: Broto, Carles; Moller, Jesper; Oliver, Bob; Ruiz, Albert
title: Realizability and tameness of fusion systems
date: 2021-02-16
journal: nan
DOI: nan
sha: a4bfe56b18178573ed6caafaa38850724004724d
doc_id: 214260
cord_uid: z6kwgtr1

A saturated fusion system over a finite $p$-group $S$ is a category whose objects are the subgroups of $S$ and whose morphisms are injective homomorphisms between the subgroups satisfying certain axioms. A fusion system over $S$ is realized by a finite group $G$ if $S$ is a Sylow $p$-subgroup of $G$ and morphisms in the category are those induced by conjugation in $G$. One recurrent question in this subject is to find criteria as to whether a given saturated fusion system is realizable or not. One main result in this paper is that a saturated fusion system is realizable if all of its components (in the sense of Aschbacher) are realizable. Another result is that all realizable fusion systems are tame: a finer condition on realizable fusion systems that involves describing automorphisms of a fusion system in terms of those of some group that realizes it. Stated in this way, these results depend on the classification of finite simple groups, but we also give more precise formulations whose proof is independent of the classification.

Let p be a prime. The fusion system of a finite group G over a Sylow p-subgroup S of G is the category F S (G) whose objects are the subgroups of S and whose morphisms are the homomorphisms between subgroups induced by conjugation in G, thus encoding G-conjugacy relations among subgroups and elements of S. With this as starting point and also motivated by questions in representation theory, Puig defined the concept of abstract fusion systems (see [Pg] and Definition 1.1) and showed that they behave in many ways like finite groups.

By analogy with finite groups, a component C of a fusion system F is a subnormal fusion subsystem that is quasisimple (i.e., O p (C) = C and C/Z(C) is simple). The basic properties of components were shown by Aschbacher [A2, Theorem 6 ] (see also Lemma 4.1 below).

A fusion system F over a finite p-group S is realized by a finite group G if S ∈ Syl p (G) and F ∼ = F S (G) , and is realizable if it is realized by some finite group. One of our main theorems is the following:

Theorem A. Let p be a prime, let F be a saturated fusion system over a finite p-group, and let E F be a normal fusion subsystem that contains all components of F . If E is realizable, then F is also realizable.

Corollary B. Let p be a prime, and let F be a saturated fusion system over a finite p-group. If all components of F are realizable, then F is realizable.

Corollary B is just the special case of Theorem A where E is the generalized Fitting subsystem of F : the central product of the components of F and O p (F ). Note, however, that a fusion system can be realizable even when some of its components are not.

For each component C of F , C/Z(C) is simple, and is a composition factor of F (see [AKO, § II.10] ). Hence one consequence of Corollary B is that F is realizable if all of its composition factors are realizable. However, the converse of this is not true either: F can be realizable without all of its composition factors being realizable.

In order to prove Theorem A, we need to work with linking systems and tameness. The concept of linking systems associated to fusion systems was first proposed by Benson in [Be3] and in unpublished notes, and was developed in detail by Broto, Levi, and Oliver [BLO2] . See Definition 1.6 for precise definitions. This was originally motivated by questions involving classifying spaces of fusion systems and of the finite groups that they realize, but also turns out to be important when studying many of the purely algebraic properties of fusion systems.

A fusion system F is tamely realized by G if it is realized by G, and in addition, the natural homomorphism from Out (G) to Out(L c S (G)) is split surjective (Definitions 2.7 and 2.8). Here, L c S (G) is the linking system associated to G and to F . We say that F is tame if it is tamely realized by some finite group.

Tameness was originally defined in [AOV, §2] , motivated by questions of realizability and extensions of fusion systems, and that is how it is used here in the proof of Theorem A. In this way, it also plays a role in Aschbacher's program for classifying simple fusion systems over 2-groups and reproving certain parts of the classification of finite simple groups. See [AO, §2.4 ] for more detail.

Tameness can also be interpreted topologically. For a finite group G, let BG ∧ p be the classifying space of G completed at p in the sense of Bousfield and Kan, and let Out(BG ∧ p ) be the set of homotopy classes of self homotopy equivalences of BG ∧ p . Then for S ∈ Syl p (G), the fusion system F S (G) is tamely realized by G if and only if the natural map from Out (G) to Out(BG ∧ p ) is split surjective. We refer to [BLO1, Theorem B] , [BLO2, Lemma 8.2] , and [AOV, Lemma 1.14] for the proof that Out(L c S (G)) ∼ = Out(BG ∧ p ). We can now state our second main theorem.

Theorem C. For each prime p, every realizable fusion system over a finite p-group is tame.

One of the original motivations for defining tameness in [AOV] was the hope that it might provide a new way to construct exotic fusion systems; i.e., fusion systems not realized by any finite group. By [AOV, Theorem B] , if F is a reduced fusion system that is not tame, then there is an extension of F whose reduction is isomorphic to F and is exotic. However, Theorem C tells us that this procedure does not give us any new exotic examples, since if F is not tame, then it is itself exotic. A saturated fusion system F is reduced if O p (F ) = 1 and O p (F ) = F = O p ′ (F ) (see Definitions 1.3 and 3.6). The reduction red(F ) of an arbitrary saturated fusion system F is the fusion system obtained by taking C F (O p (F ))/Z(O p (F )), and then alternately taking O p (−) or O p ′ (−) until the sequence becomes constant. By [AOV, Theorem A] , F is tame if red(F ) is tame. So one immediate consequence of Theorem C is:

Corollary D. If F is a saturated fusion system over a finite p-group S, and red(F ) is realizable, then F is also realizable.

The proofs of Theorems A and C as formulated above, as well as those of Corollaries B and D, require the classification of finite simple groups. But they will be reformulated in Section 5 in a a way so as to be independent of the classification. Our main theorem there, Theorem 5.4, is independent of the classification and includes Theorems A and C as special cases (the latter is reformulated as Theorem 5.6).

The first two sections of the paper contain mostly background material: some basic definitions and properties of fusion and linking systems are in Section 1, and those of automorphism groups and tameness in Section 2. We then deal with products in Section 3 and components of fusion systems in Section 4. Theorems A and C, as well as some other applications, are shown in Section 5, as Theorems 5.4 and 5.6.

The authors would like to thank the Universitat Autònoma de Barcelona and the University of Copenhagen for their hospitality while the four authors met during early stages of this work; and also the French CNRS and BigBlueButton for helping us to meet virtually at frequent intervals to discuss this work during the covid-19 pandemic.

Notation: The notation used in this paper is mostly standard, with a few exceptions. Composition of functions and functors is always from right to left. Also, C n denotes a (multiplicative) cyclic group of order n. When G is a (multiplicative) group, 1 ∈ G always denotes its identity element.

When f : C −→ D is a functor, then for objects c, c ′ in C, we let f c,c ′ be the induced map from Mor C (c, c ′ ) to Mor D (f (c), f (c ′ )), and also set f c = f c,c for short.

When G is a group, we indicate conjugation by setting g x = c g (x) = gxg −1 and g H = c g (H) = gHg −1 for g, x ∈ G and H ≤ G. Also, Hom G (P, Q) (for P, Q ≤ G) is the set of (injective) homomorphisms from P to Q induced by conjugation in G.

Throughout the paper, p will always be a fixed prime.

This is a background section intended to provide the reader with the necessary basic definitions and properties of fusion and linking systems that will be used throughout the paper. Fusion systems and saturation were originally introduced by Puig, first in unpublished notes, and then in [Pg] . Abstract linking systems were defined in [BLO2] . As general references for the subject we refer to [AKO] and [Cr] .

1.1. Fusion systems. For a prime p, a fusion system over a finite p-group S is a category whose objects are the subgroups of S, and whose morphisms are injective homomorphisms between subgroups such that for each P, Q ≤ S:

• Hom F (P, Q) ⊇ Hom S (P, Q); and

• for each ϕ ∈ Hom F (P, Q), ϕ −1 ∈ Hom F (ϕ(P ), P ).

Here, Hom F (P, Q) denotes the set of morphisms in F from P to Q. We also write Aut S (P ) for Hom S (P, P ), Iso F (P, Q) for the set of isomorphisms, Aut F (P ) = Iso F (P, P ), and Out F (P ) = Aut F (P )/Inn(P ). For P ≤ S and g ∈ S, we set P F = {ϕ(P ) | ϕ ∈ Hom F (P, S)} and g F = {ϕ(g) | ϕ ∈ Hom F ( g , S)} (the sets of subgroups and elements F -conjugate to P and to g).

The following version of the definition of a saturated fusion system is the most convenient one to use here. (See Definitions I.2.2 and I.2.4 and Proposition I.2.5 in [AKO] .) Definition 1.1. Let F be a fusion system over a finite p-group S.

(a) A subgroup P ≤ S is fully normalized (fully centralized) in F if |N S (P )| ≥ |N S (Q)| (|C S (P )| ≥ |C S (Q)|) for each Q ∈ P F .

(b) A subgroup P ≤ S is fully automized in F if Aut S (P ) ∈ Syl p (Aut F (P )).

(c) A subgroup P ≤ S is receptive in F if each isomorphism ϕ ∈ Iso F (Q, P ) in F extends to a morphism ϕ ∈ Hom F (N ϕ , S), where N ϕ = {g ∈ N S (Q) | ϕc g ϕ −1 ∈ Aut S (P )}.

(d) The fusion system F is saturated if it satisfies the following two conditions: (I) (Sylow axiom) each subgroup P ≤ S fully normalized in F is also fully automized and fully centralized; and

(II) (extension axiom) each subgroup P ≤ S fully centralized in F is also receptive.

The above definition is motivated by fusion systems of finite groups. When G is a finite group and S ∈ Syl p (G), the p-fusion system of G is the category F S (G) whose objects are the subgroups of S, and where Mor F S (G) (P, Q) = Hom G (P, Q) for each P, Q ≤ S. For a proof that F S (G) is saturated, see, e.g., [AKO, Lemma I.1.2] . In general, a saturated fusion system F over a finite p-group S will be called realizable if F = F S (G) for some finite group G with S ∈ Syl p (G), and will be called exotic otherwise.

The following lemma lists relations between some of these conditions that hold for all fusion systems, not just those that are saturated.

Lemma 1.2 ( [AKO, Lemma I.2.6] ). If F is a fusion system over a finite p-group S, then each receptive subgroup of S is fully centralized, and each subgroup that is fully automized and receptive is fully normalized.

We next list some of the terminology used to describe certain subgroups in a fusion system. Definition 1.3. Let F be a fusion system over a finite p-group S. For a subgroup P ≤ S,

(c) P is F -quasicentric if for each Q ∈ P F which is fully centralized in F , the centralizer fusion system C F (Q) (see Definition 1.5(b)) is the fusion system of the group C S (Q);

extends to a morphism ϕ ∈ Hom F (P Q, P R) such that ϕ(P ) = P ; and (g) P is central in F if each ϕ ∈ Hom F (Q, R) (for Q, R ≤ S) extends to a morphism ϕ ∈ Hom F (P Q, P R) such that ϕ| P = Id P .

Let F cr ⊆ F c ⊆ F q denote the sets of F -centric F -radical, F -centric, and F -quasicentric subgroups of S, respectively, or (depending on the context) the full subcategories of F with those objects. Let O p (F ) ≥ Z(F ) denote the (unique) largest normal and central subgroups, respectively, in F .

The following result is one of the versions of Alperin's fusion theorem for fusion systems.

Theorem 1.4. Let F be a saturated fusion system over a finite p-group S. Then each morphism in F is a composite of restrictions of automorphisms of subgroups that are Fcentric, F -radical, and fully normalized in F .

Proof. This follows from [AKO, Theorem I.3 .6] (the same statement but for F -essential subgroups), together with [AKO, Proposition I.3.3(a) ] (all F -essential subgroups are Fcentric and F -radical). Alternatively, the result as stated here is shown directly (without mention of essential subgroups) in [BLO2, Theorem A.10 ].

Definition 1.5. Let F be a saturated fusion system over a finite p-group S, and let Q ≤ S be a subgroup.

) is a saturated fusion system (see [BLO2, Proposition A.6] or [AKO, Theorem I.5 .5]). There is a similar condition (see [AKO, Theorem I.5.5] 

1.2. Linking systems. Before defining linking systems, we need to introduce more notation. If P, Q ≤ G are subgroups of a finite group G, the transporter set T G (P, Q) is defined by setting T G (P, Q) = {g ∈ G | g P ≤ Q}. The transporter category of G is the category T (G) whose objects are the subgroups of G, and whose morphisms sets are the transporter sets:

Composition in T (G) is given by multiplication in G. If H is a set of subgroups of G, then T H (G) ⊆ T (G) denotes the full subcategory with object set H.

The following definition of linking system taken from [AKO, Definition III.4 .1].

Definition 1.6. Let F be a fusion system over a finite p-group S. A linking system associated to F is a triple (L, δ, π) where L is a finite category, and δ and π are a pair of functors

satisfy the following conditions: (A1) Ob(L) is a set of subgroups of S closed under F -conjugacy and overgroups, and contains F cr . Each object in L is isomorphic (in L) to one which is fully centralized in F .

(A2) δ is the identity on objects, and π is the inclusion on objects. For each P, Q ∈ Ob(L) such that P is fully centralized in F , C S (P ) acts freely on Mor L (P, Q) via δ P and right composition, and π P,Q : Mor L (P, Q) → Hom F (P, Q) is the orbit map for this action.

(B) For each P, Q ∈ Ob(L) and each g ∈ T S (P, Q), π P,Q sends δ P,Q (g) ∈ Mor L (P, Q) to c g ∈ Hom F (P, Q).

(C) For all ψ ∈ Mor L (P, Q) and all g ∈ P , the diagram

When the functors δ and π are understood, we refer directly to the category L as a linking system. A centric linking system associated to F is a linking system L associated to F such that Ob(L) = F c .

Linking systems associated to a fusion system were originally motivated by centric linking systems of finite groups. For a finite group G, a p-subgroup P ≤ G is p-centric in G if Z(P ) ∈ Syl p (C G (P )); equivalently, if C G (P ) = Z(P ) × O p (C G (P )) and O p (C G (P )) has order prime to p. For S ∈ Syl p (G), the centric linking system of G over S is consists of the category L c S (G) whose objects are the subgroups of S which are p-centric in G and whose morphism sets are given by

When G is a finite group and S ∈ Syl p (G), then P ≤ S is F -centric (see Definition 1.3) if and only if P is p-centric in G (see [BLO1, Lemma A.5] ). Moreover, F S (G) is always saturated (see [AKO, Theorem I.2 .3]), and (L c S (G), δ, π) is a centric linking system associated to F S (G) .

Some of the basic properties of linking systems are listed in the next proposition.

Proposition 1.7. Let (L, δ, π) be a linking system associated to a saturated fusion system F over a finite p-group S. Then (a) δ is injective on all morphism sets; and

(b) all morphisms in L are monomorphisms and epimorphisms in the categorical sense.

Conditions for the existence of restrictions and extensions of morphisms are as follows:

(c) For every morphism ψ ∈ Mor L (P, Q), and every P 0 , Q 0 ∈ Ob(L) such that P 0 ≤ P ,

(d) Let P, Q, P , Q ∈ Ob(L) and ψ ∈ Mor L (P, Q) be such that P P , Q ≤ Q, and for each g ∈ P there is h ∈ Q such that ψ • δ P (g) = δ Q,Q * (h) • ψ (Q * = hQh −1 ). Then there is a unique morphism ψ ∈ Mor L (P , Q) such that ψ| P,Q = ψ.

Proof. See points (c), (f), (b), and (e), respectively, in [O2, Proposition 4] .

Lemma 1.8. Let G be a finite group with S ∈ Syl p (G), and set F = F S (G).

(a) For each Q ≤ S, Q is fully normalized (fully centralized) if and only if

Proof. Point (a) is shown in [AKO, Proposition I.5.4 ]. In particular, when

We note here the existence and uniqueness of linking systems shown by Chermak, Oliver, and Glauberman-Lynd. Two linking systems (L 1 , δ 1 , π 1 ) and (L 2 , δ 2 , π 2 ) associated to the same fusion system F are isomorphic if there is an isomorphism of categories ρ : L 1 ∼ = − − → L 2 such that ρ • δ 1 = δ 2 and π 2 • ρ = π 1 . Theorem 1.9 ( [Ch, O3, GLn] ). Let F be a saturated fusion system over a finite p-group S, and let H be a set of subgroups of S such that F cr ⊆ H ⊆ F q , and such that H is closed under F -conjugacy and overgroups. Then up to isomorphism, there is a unique linking system L H associated to F with object set H.

Proof. The existence and uniqueness of a centric linking system associated to F was shown by Chermak. See [Ch, Main theorem] and [O3, Theorem A] for two versions of his original proof, and [GLn, Theorem 1.2] for the changes to the proof in [O3] needed to make it independent of the classification of finite simple groups.

More generally, if F cr ⊆ H ⊆ F c , the uniqueness of an H-linking system follows by the same obstruction theory (shown to vanish in [O3, Theorem 3.4] and [GLn, Theorem 1.1] ) as that used in the centric case (by the same argument as in the proof of [BLO2, Proposition 3.1]). For arbitrary H ⊆ F q containing F cr , the existence and uniqueness now follows from [AKO, Proposition III.4.8] , applied with H ⊇ H ∩ F c in the role of H ⊇ H.

1.3. Normal fusion and linking subsystems. Let F be a fusion system over a finite p-group S. A fusion subsystem of F is a subcategory E ⊆ F which is itself a fusion system over a subgroup T ≤ S (in particular, Ob(E) is the set of subgroups of T ). We write E ≤ F when E is a fusion subsystem, and also sometimes say that E ≤ F is a pair of fusion systems over T ≤ S. Definition 1.10. Let F be a fusion system over a finite p-group S.

(a) Let R be another finite p-group and let α : S −→ R be an isomorphism. We denote by α F the fusion system over R with morphism sets

for each pair of subgroups P, Q ≤ R.

(b) Let E be another fusion system over a finite p-group T . We say that E and F are isomorphic fusion systems if there is an isomorphism α : S −→ T such that E = α F .

A more general concept of morphism between fusion systems is given in [AKO, Definition II.2.2] .

Consider now the following definition from [AKO, Definition I.6 .1].

Definition 1.11. Fix a saturated fusion system F over a finite p-group S.

(a) A fusion subsystem E ≤ F over T S is weakly normal if T is strongly closed in F and the following conditions hold: • (invariance condition) α E = E for each α ∈ Aut F (T ), and

(c) A saturated fusion system F over a finite p-group S is simple if it contains no proper nontrivial normal fusion subsystem.

It will be convenient to say that "E F is a normal pair of fusion systems over T S" to mean that F is a fusion system over S and E F is a normal subsystem over T .

Note that if F is a saturated fusion system over a finite p-subgroup S, and P ≤ S, then P F if and only if F P (P ) F [A2, (7.9) ].

Note also that what are called "normal fusion subsystems" in [AOV, Definition 1.18 ] are what we are calling "weakly normal" subsystems here.

When (L, δ, π) is a linking system associated to the fusion system F over S, and F 0 ≤ F is a fusion subsystem over S 0 ≤ S, then a linking subsystem associated to F 0 is a linking system (L 0 , δ 0 , π 0 ) associated to F 0 , where L 0 is a subcategory of L and

are the restrictions of δ and π. In this situation, we write L 0 ≤ L, and sometimes say that L 0 ≤ L is a pair of linking systems. Note in particular the special case where S 0 = S and F 0 = F but Ob(L 0 ) ⊆ Ob(L): a pair of linking systems with different objects associated to the same fusion system. Notice that not every normal pair of fusion systems has an associated normal pair of linking systems. Definition 1.12 differs from Definition 1.27 in [AOV] in that there is no "Frattini condition" in the definition we give here. We have omitted it since it follows from the Frattini condition for normal fusion subsystems, as shown in the next lemma.

Lemma 1.13. If M L is a normal pair of linking systems associated to fusion systems E F over finite p-groups T S, then for all P, Q ∈ Ob(M) and all ψ ∈ Mor L (P, Q), there are morphisms γ ∈ Aut L (T ) and ψ 0 ∈ Mor M (γ(P ), Q) such that ψ = ψ 0 • γ| P,γ(P ) .

Proof. Let ψ ∈ Mor L (P, Q) be as above, and assume first that P is fully centralized in F . Then π(ψ) ∈ Hom F (P, Q), and by the Frattini condition on E F , there are α ∈ Aut F (T ) and ϕ ∈ Hom E (α(P ), Q) such that π(ψ) = ϕ • α| P,α(P ) . Choose lifts ϕ ∈ Mor M (α(P ), Q) of ϕ and α ∈ Aut L (T ) of α. By axiom (A2) for the linking system L, there is z ∈ C S (P ) such that ϕ • α| P,α(P ) = ψ • δ P (z)| P,P . Set γ = α • δ S (z) −1 ∈ Aut L (T ), and then ψ = ϕ • γ| P,γ(P ) .

If P is not fully centralized in F , then choose P * ∈ P F that is fully centralized, and fix ω ∈ Iso L (P * , P ). Then P * ≤ T since T is strongly closed in F , and so P * ∈ Ob(M).

We just showed that there are morphisms γ 1 , γ 2 ∈ Aut L (T ), ψ 1 ∈ Mor M (γ 1 (P * ), Q), and ψ 2 ∈ Mor M (γ 2 (P * ), P ) such that

and ω = ψ 2 • γ 2 | P * ,γ 2 (P * ) .

Also, ψ 2 is an isomorphism since ω is an isomorphism. Set γ = γ 1 γ −1 2 ∈ Aut L (T ). Then

where γψ 2 γ −1 ∈ Iso M (γ 1 (P * ), γ(P )) by Definition 1.12(b).

1.4. Quotient fusion systems. We begin with the basic definition and properties.

Definition 1.14. Let F be a fusion system over a finite p-group S, and assume Q S is strongly closed in F . Then F /Q is defined to be the fusion system over S/Q where

for P, R ≤ S containing Q. Here, ϕ/Q ∈ Hom(P/Q, R/Q) sends gQ to ϕ(g)Q.

Note that by definition, [Cr, Theorem 5 .20]). Lemma 1.15. Let F be a saturated fusion system over a finite p-group S, and assume Q S is strongly closed in F . Then F /Q is saturated. If E F is a normal fusion subsystem over T S such that T ≥ Q, then E/Q F /Q.

Proof. For a proof that F /Q is saturated, see [Cr, Proposition 5.11] or [AKO, Lemma II.5.4 ]. If E F over T ≥ Q, then T /Q is strongly closed in F /Q, and the three conditions for normality of E/Q ≤ F /Q follow immediately from those for E F . We refer to [AKO, § II.5] and [Cr, § 5.2] for some of the other properties of these quotient systems.

The next lemma involves normal fusion subsystems of index prime to p (see Definition 3.6).

Lemma 1.16. Let F be a saturated fusion system over a finite p-group S, and let Z ≤ Z(F ) be a central subgroup.

For each P ∈ F c H that is fully normalized in F , P/Z is fully normalized in F /Z, and hence C S/Z (P/Z) P/Z. Choose x ∈ S P such that xZ ∈ C S/Z (P/Z), and consider the automorphism c x ∈ Aut S (P ). Then c x / ∈ Inn(P ) since P ∈ F c , and c x induces the identity on Z and on P/Z. Since {α ∈ Aut F (P ) | [α, P ] ≤ Z} is a normal p-subgroup of Aut F (P ) (see [G, Corollary 5.3.3] ), this proves that c x ∈ O p (Aut F (P )), and hence that for each P ∈ F c H there is P * ∈ P F with Out S (P * ) ∩ O p (Out F (P * )) = 1.

(1.1)

By [AKO, Theorem I.7 .7], there is a finite group Γ of order prime to p, and a map θ : Mor((F /Z) c ) −→ Γ that sends composites to products and inclusions to the identity, and is such that θ(Aut F /Z (S/Z)) = Γ and O p ′ (F /Z) = θ −1 (1) . Let F H ⊆ F be the full subcategory with object set H, and let Φ be the natural map from Mor(F H ) to Mor((F /Z) c ). Set F 0 = (θΦ) −1 (1) : a fusion subsystem of F over S. By [O4, Lemma 1.6] and (1.1),

Conversely, O p ′ (F )/Z has index prime to p in F /Z since for each P/Z ≤ S/Z, we have

The following construction is needed when we want to look at the image of E F in F /Q but E does not contain Q.

Definition 1.17. Let E ≤ F be a pair of saturated fusion systems over T ≤ S, and let Z ≤ Z(F ) be a central subgroup. Define ZE ≤ F to be the fusion subsystem over ZT where for each P, Q ≤ ZT ,

If E F , then the above definition is a special case of a construction of Aschbacher [A2, § 8] . But the definition and arguments in this very restricted case are much more elementary.

Lemma 1.18. Let E ≤ F be a pair of saturated fusion systems over finite p-groups T ≤ S. Let Z ≤ Z(F ) be a central subgroup. Then ZE is saturated, and ZE F if E F . Proof. A subgroup P ≤ ZT is fully normalized or fully centralized in ZE if and only if P ∩ T is fully normalized or fully centralized in E. The saturation axioms for ZE follow easily from those for E: note, for example, that Aut ZE (P ) ∼ = Aut E (P ∩ T ) for P ≤ ZT . So ZE is saturated.

If E F , then the subgroup ZT is strongly closed in F since for each x = zt (for z ∈ Z and t ∈ T ), each ϕ ∈ Hom F ( x , S) extends to ϕ ∈ Hom F (Z t , S), and ϕ(x) = zϕ(t) ∈ ZT . The extension condition for ZE follows directly from that for E F , and the invariance and Frattini conditions for ZE follow from those conditions applied to E F and the definition of a central subgroup. Thus ZE F .

The main aim of this section is to introduce the concept of tameness for fusion systems. This was originally defined in [AOV] and it is one of the main subjects of this article.

2.1. Automorphisms of fusion and linking systems. Before defining tameness, we must define automorphism and outer automorphism groups of fusion and linking systems.

Definition 2.1. Fix a saturated fusion system F over a finite p-group S. Then

is the group of outer automorphisms of F ; and (c) for each α ∈ Aut(F ), we let c α : F −→ F denote the functor that sends an object P to α(P ) and a morphism ϕ ∈ Hom F (P, Q) to αϕα −1 ∈ Hom F (α(P ), α(Q)).

Now that we have defined automorphisms, we can define characteristic subsystems:

Definition 2.2. Fix a saturated fusion system F over a finite p-group S. A fusion subsystem

Likewise, a subgroup P of S is characteristic in F if P F and α(P ) = P for all α ∈ Aut(F ); equivalently, if F P (P ) is a characteristic subsystem of F .

For example, when F is a saturated fusion system over a finite p-group S, then the subsystems O p (F ) and O p ′ (F ) (see Definition 3.6(b,c)) and the subgroups O p (F ) and Z(F ) are all characteristic in F .

The following condition for a subnormal fusion system to be normal is due to Aschbacher.

The following definitions of automorphism groups are taken from [AOV, Definition 1.13 & Lemma 1.14].

Definition 2.5. Let F be a fusion system over a finite p-group S and let L be an associated linking system.

(b) Let Aut(L) be the group of automorphisms of the category L that send inclusions to inclusions and distinguished subgroups to distinguished subgroups.

(c) For γ ∈ Aut L (S), let c γ ∈ Aut(L) be the automorphism which sends an object P to c γ (P ) def = π(γ)(P ), and sends ψ ∈ Mor L (P, Q) to c γ (ψ)

The notation in Definitions 2.1 and 2.5 is slightly different from that used in [AOV] and [AKO] , as described in the following table:

By [AOV, Lemma 1.14], the above definition of Out(L) is equivalent to Out typ (L) in [BLO2] , and by [BLO2, Lemma 8.2] , both are equivalent to Out typ (L) in [BLO1] . So by [BLO1, Theorem 4.5(a) ], Out(L c S (G)) ∼ = Out(BG ∧ p ): the group of homotopy classes of self homotopy equivalences of the space BG ∧ p . The next result shows how an automorphism of a linking system automatically preserves the structure functors. For use in the next section, we state this for certain full subcategories of a linking system that need not themselves be linking systems because their objects might not be closed under overgroups. (Compare with Proposition 6 in [O2] .) Proposition 2.6. Let (L, δ, π) be a linking system associated to a fusion system F over a finite p-group S, let L 0 ⊆ L be a full subcategory such that Ob(L 0 ) ⊇ F cr , and let Aut(L 0 ) be the group of automorphisms of the category L 0 that send inclusions to inclusions and distinguished subgroups to distinguished subgroups. Fix α ∈ Aut(L 0 ), and let β ∈ Aut(S) be such that α(δ S (g)) = δ S (β(g)) for all g ∈ S. Then β ∈ Aut(F ), α(P ) = β(P ) for each P ∈ Ob(L 0 ), and the following diagram of functors

Proof. Clearly, α(S) = S, and hence α S sends δ S (S) ∈ Syl p (Aut L (S)) to itself. Thus β is well defined. Since α sends inclusions to inclusions, it commutes with restrictions. So for P, Q ∈ Ob(L 0 ) and g ∈ T S (P, Q) (the transporter set), we have

In particular, the left-hand square in (2.1) commutes.

When Q = P , (2.2) says that δ α(P ) (β(P )) = α P (δ P (P )), and α P (δ P (P )) = δ α(P ) (α(P )) since α sends distinguished subgroups to distinguished subgroups. So α(P ) = β(P ) since δ α(P ) is a monomorphism (Proposition 1.7(a)).

Fix P, Q ∈ Ob(L) and ψ ∈ Mor L (P, Q), and set ϕ = π(ψ) ∈ Hom F (P, Q). For each g ∈ P , consider the following three squares:

The first and third of these squares commute by axiom (C) in Definition 1.6, and the second commutes since it is the image under α of the first. Since morphisms in L are epimorphisms and δ α(Q) is injective (Proposition 1.7(a,b)), this implies β(ϕ(g)) = π(α(ψ))(β(g)). Thus π(α(ψ)) = βϕβ −1 = c β (π(ψ)), proving that the right-hand square in (2.1) commutes.

In particular, since π is surjective on morphism sets (axiom (A2) in Definition 1.6), βϕβ −1 ∈ Hom F (β(P ), β(Q)) for each P, Q ∈ Ob(L 0 ) and each ϕ ∈ Hom F (P, Q). Since Ob(L 0 ) includes all subgroups which are F -centric and F -radical, all morphisms in F are composites of restrictions of morphisms between objects of L 0 by Theorem 1.4. Hence β F ≤ F with equality since F is a finite category, and so β ∈ Aut(F ).

Tameness of fusion systems. We next define a homomorphism κ G that connects the automorphisms of a group to those of its linking system. We refer to [AOV, § 2.2] for more details about κ G and the proof that it is well defined.

Definition 2.7. Let G be a finite group and choose S ∈ Syl p (G). Let

) denote the homomorphism that sends the class of α ∈ Aut(G) such that α(S) = S to the class of the automorphism of L c S (G) induced by α.

In these terms, tameness can be defined as follows.

Definition 2.8. Let F be a saturated fusion system over a finite p-group S. Then (a) F is tamely realized by a finite group G if F ∼ = F S * (G) for some S * ∈ Syl p (G) and the homomorphism κ G : Out(G) −→ Out(L c S * (G)) is split surjective; and (b) F is tame if it is tamely realized by some finite group.

2.3. Centric fusion and linking subsystems. Some of the results in later sections need the hypothesis that a certain fusion or linking subsystem be centric, which we now define.

Definition 2.9. Let E F be a normal pair of saturated fusion systems over finite p-groups T S.

For pairs of linking systems, this is the definition used in [AOV] (Definition 1.27). The term "centric fusion subsystem" was not used in [O4] , but the condition in Definition 2.9(b) appears in Proposition 2.1 and Theorem 2.3 of that paper (and the term is used in [O4c] ).

In the next lemma, we look at the relation between normal centric fusion subsystems and normal centric linking subsystems.

Lemma 2.10. Let E F be a normal pair of saturated fusion systems over finite p-groups T S with associated linking systems M L, and set If, in addition, p is odd, then

Proof. Throughout the proof, "axiom (-)" always refers to one of the axioms in the definition of a linking system (Definition 1.6). We first claim that

The implication "⇐=" is clear. To see the converse, fix x ∈ S such that δ T (x) ∈ Aut M (T ).

Then c x ∈ Aut E (T ), and c x ∈ Inn(T ) ∈ Syl p (Aut E (T )) since it has p-power order. Thus there is t ∈ T such that xt −1 ∈ C S (T ), δ T (xt −1 ) = δ T (x)δ T (t) −1 ∈ Aut M (T ), and so xt −1 ∈ Z(T ) by axiom (A2) applied to M and since δ T is injective (Proposition 1.7(a)). Hence x ∈ T . We next claim that [AKO, Lemma I.4 .2], finishing the proof of (2.4).

(a) Fix α ∈ C Aut L (T ) (M). By axiom (C) for the linking system L, for all g ∈ T , αδ T (g)α −1 = δ T (π(α)(g)), so δ T (g) = δ T (π(α)(g)) since c α = Id M , and g = π(α)(g) since δ T is injective (see Proposition 1.7(a)). So π(α) = Id T , and α = δ T (x) for some x ∈ C S (T ) by axiom (A2) .

This holds for all ψ ∈ Mor(M) whose domain is fully centralized, and hence for all morphisms

Thus Z(E) ≤ C S (M). By a similar argument but working in L and F instead of M and E, we also have Z

, and by Proposition 1.7(d), ψ extends to a unique morphism ψ ∈ Mor L (P , Q). Set ϕ = π(ψ) ∈ Hom F (P , Q). By axiom (C) again (but applied to L) we have ψδ P (x) = δ Q (ϕ(x))ψ, and after restriction to P and Q this gives ψδ P (x) = δ Q (ϕ(x))ψ. Hence δ Q (ϕ(x))ψ = δ Q (x)ψ, so δ Q (ϕ(x)) = δ Q (x) since ψ is an epimorphism in the categorical sense (see Proposition 1.7(b)), and ϕ(x) = x by the injectivity of δ Q . Thus each morphism in E extends to a morphism in F that sends x to itself, and hence C F (x) ≥ E and x ∈ C S (E). 

(2.5)

Thus for each x ∈ Z(F ), δ S (x) acts trivially on L and hence δ T (x) acts trivially on Aut L (T ). So Z(F ) ≤ C C S (M) (Aut L (T )) = C C S (M) (L/M), and it remains to prove the opposite inclusion.

Fix x ∈ C S (M) such that δ T (x) ∈ Z(Aut L (T )). In particular, δ T (x) ∈ Z(δ T (S)), so x ∈ Z(S) by the injectivity of δ T (Proposition 1.7(a)), and c δ T (x) (P ) = x P = P for each P ∈ Ob(L). We must show that δ T (x) acts trivially on L. Fix P, Q ≤ S and ψ ∈ Mor L (P, Q), and set P 0 = P ∩ T , Q 0 = Q ∩ T , and ψ 0 = ψ| P,Q (see Proposition 1.7(c)). By the Frattini condition on a normal linking subsystem (Lemma 1.13), ψ 0 is the composite of the restriction of a morphism γ ∈ Aut L (T ) followed by some χ ∈ Mor(M). Since δ T (x) commutes with γ by assumption and commutes with χ by (a) (Z(F ) ≤ C S (M)), we have

Then ψδ P (x) = δ Q (x)ψ by the uniqueness of extensions in a linking system (Proposition 1.7(d)), and hence c δ T (x) (ψ) = ψ.

Thus c δ S (x) is the identity on all objects and morphisms in L. So x ∈ Z(F ) by (2.5) and the injectivity of δ S (Proposition 1.7(a)).

(d) Assume now that p is odd. Then the natural homomorphism µ M : Out(M) −→ Out(E) induced by µ M (see Proposition 2.6 and [AKO, §III.4.3] or [AOV, §1.3] ) is an isomorphism by [O3, Theorem C] and [GLn, Theorem 1.1] .

Fix

In this section, we first define centric linking systems associated to products of two or more fusion systems (Lemma 3.5). This is followed by a description of the group of automorphisms of such a product linking systems that leave the factors invariant up to permutation, as well as conditions on the fusion systems that guarantee that these are the only automorphisms of the linking system (Proposition 3.8(a,b)). As a consequence, we show that a product of tame fusion systems that satisfy these same conditions is always tame (Proposition 3.8(c)).

Recall that if F 1 and F 2 are fusion systems over finite p-groups S 1 and S 2 , then F 1 × F 2 is the fusion system over S 1 × S 2 generated by all morphisms ϕ 1 × ϕ 2 ∈ Hom(P 1 × P [AKO, Definition I.6 .5] for more details. When F 1 and F 2 are both saturated, then so is F 1 × F 2 [AKO, Theorem I.6.6].

The following notation and hypotheses will be used throughout the section.

Hypotheses 3.1. Let F 1 , . . . , F k be saturated fusion systems over finite p-groups S 1 , . . . , S k (some k ≥ 2), and set S = S 1 × · · · × S k and F = F 1 × · · · × F k . For each i, let pr i : S −→ S i be the projection. For each P ≤ S, we write P i = pr i (P ) (for 1 ≤ i ≤ k) and P = P 1 × · · · × P k ≤ S. Thus P ≥ P for each P .

We first check which subgroups are centric in a product of fusion systems.

Lemma 3.2. Assume Hypotheses 3.1. Then a subgroup P ≤ S is F -centric if and only if P i is F -centric for all i and Z(P 1 ) × · · · × Z(P k ) ≤ P .

Proof. For each P ≤ S, we have C S (P ) = C S 1 (P 1 ) × · · · × C S k (P k ) = C S ( P ). Hence

The result now follows since P is F -centric if and only if P is fully centralized in F and C S (P ) ≤ P .

Note also that under Hypotheses 3.1, if P ≤ S is F -centric and F -radical, then P = P = P 1 × · · · × P k (see, e.g., [AOV, Lemma 3 .1]). But that will not be needed here.

The following easy consequence of the Krull-Remak-Schmidt theorem will be needed. Recall that a group is indecomposable if it is not the direct product of two of its proper subgroups.

Proposition 3.3. Assume G 1 , . . . , G k are finite, indecomposable groups, and set G = G 1 × · · · × G k . Then the following hold for each α ∈ Aut (G) .

By the Krull-Remak-Schmidt theorem in the form stated in [Sz1, Theorem 2.4 .8] (and applied with Ω = 1 or Ω = Inn(G)), any two direct product decompositions of G with indecomposable factors have the same number of factors, and there is always a normal automorphism of G that sends one to the other up to a permutation of the factors . Points (a) and (b) follow immediately from this, applied to the decompositions of G as the product of the G i and of the α(G i ).

One immediate consequence of Proposition 3.3 is the following description of Out(G) when G is a product of simple groups.

Proposition 3.4. Assume G = G 1 × · · · × G k , where G 1 , . . . , G k are finite indecomposable groups and |Z(G)|, |G/[G, G]| = 1. Let Γ be the group of all γ ∈ Σ k such that G γ(i) ∼ = G i for each i. Then there is an isomorphism

with the property that Φ G (β 1 , . . . , β k ) = β 1 × · · · × β k for each k-tuple of automorphisms (G) , and hence induces an isomorphism

To define Φ G more precisely, fix isomorphisms λ ij :

ij . Also, set λ ii = Id G i for each i. Then Φ G can be chosen so that for each γ ∈ Γ, Φ G (γ)(g 1 , . . . , g k ) = λ γ −1 (1),1 (g γ −1 (1) ), . . . , λ γ −1 (k),k (g γ −1 (k) ) .

Proof. Without loss of generality, we can assume that G i = G j and λ ij = Id G i for each i, j such that G i ∼ = G j . Thus Φ G (γ)(g 1 , . . . , g k ) = (g γ −1 (1) , . . . , g γ −1 (k) ) for each γ ∈ Γ. Then Φ G is clearly an injective homomorphism, and it factors through a homomorphism Φ G as above since Inn (G) 

Inn (G i ) . Each automorphism of G permutes the factors by Proposition 3.3(b) , and hence Φ G is surjective.

In the next proposition, we describe one way to construct linking systems associated to products of fusion systems.

Proposition 3.5. Assume Hypotheses 3.1. For each 1 ≤ i ≤ k, let L i be a centric linking system associated to F i , with structure functors δ i and π i . Let L be the category whose objects are the F -centric subgroups of S, and where for each P, Q ∈ Ob(L),

all g = (g 1 , . . . , g k ) ∈ T S (P, Q) π P,Q (ϕ) = π 1 (ϕ 1 ), . . . , π k (ϕ k ) all ϕ = (ϕ 1 , . . . , ϕ k ) ∈ Mor L (P, Q).

Then the following hold:

(a) The functors δ and π make L into a centric linking system associated to F .

(b) Define ξ L : L 1 × · · · × L k −→ L by setting ξ L (P 1 , . . . , P k ) = P 1 × · · · × P k and ξ L (ϕ 1 , . . . , ϕ k ) = (ϕ 1 , . . . , ϕ k ).

Then ξ L is an isomorphism of categories from L 1 ×· · ·×L k to the full subcategory L ⊆ L whose objects are those P ∈ F c such that P = P ; equivalently, the products P 1 ×· · ·×P k for P i ∈ F c i . Also, the following square commutes

where η is the natural isomorphism that sends (P 1 , . . . , P k ) to k i=1 P i . (c) Let ρ i : L i −→ L be the functor that sends P i ∈ Ob(L i ) to its product with the S j for j = i, and sends ϕ i ∈ Mor(L i ) to its product with Id S j for j = i. Then ρ i is injective on objects and on morphism sets. If α ∈ Aut(L) is such that α S (δ S (S i )) = δ S (S i ) for each i, then α(ρ i (L i )) = ρ i (L i ) for each i.

Proof. By Lemma 3.2, for each P ∈ Ob(L) = F c , pr i (P ) is F i -centric for each i. So the definitions of Mor L (P, Q) and δ make sense.

(a) Axiom (A1) is clear. Fix P, Q ∈ Ob(L) and set P i = pr i (P ) and Q i = pr i (Q); then C S (P ) = Z(P ) = Z(P 1 ) × · · · × Z(P k ). So by axiom (A2) for the L i , for ϕ, ϕ ′ ∈ Mor L (P, Q), π(ϕ) = π(ϕ ′ ) if and only if ϕ ′ = ϕ • δ P (z) for some z ∈ Z(P ). For each P, Q ∈ Ob(L), each ϕ ∈ Hom F (P, Q) is the restriction of some morphism k i=1 ϕ i ∈ Hom F ( P , Q) (see [AKO, Theorem I.6 .6]), and hence the surjectivity of π on morphism sets follows from that of the π i . The rest of (A2) (the effect of ϕ and π on objects) is clear. Likewise, axioms (B) and (C) for L follow immediately from the corresponding axioms for the L i . Thus L is a centric linking system with structure functors δ and π.

(b) Both statements (ξ L is an isomorphism of categories and the diagram commutes) are immediate from the definitions and since P 1 × · · · × P k is F -centric if P i is F i -centric for each i (Lemma 3.2).

Set β = µ L (α) ∈ Aut(F ), as defined in Proposition 2.6. By that proposition, α(P ) = β(P ) for P ∈ Ob(L), and π • α = c β • π as functors from L to F . By assumption, β(S i ) = S i and

for some ψ ∈ Mor(L i ). So by axiom (A2) in Definition 1.6, α(ρ i (ϕ i )) = ρ i (ψ)δ S (z, z ′ ) for some z ∈ Z(P i ) and z ′ ∈ Z(S i ). Since z ′ has p-power order, this shows that z ′ = 1 and α(ρ i (ϕ i )) ∈ Mor(ρ i (L i )) if ϕ i is an automorphism of order prime to p. Also,

By [AOV, Theorem 1.12] , each morphism in L i is a composite of restrictions of elements of Aut L i (P ) for fully normalized subgroups P ∈ F cr i . Also, when P is fully normalized, δ P (N S (P )) ∈ Syl p (Aut L i (P )) (see [AKO, Proposition III.4 .2(c)]), and so Aut L i (P ) is generated by δ P (N S (P )) and elements of order prime to P . Thus each morphism in L i is a composite of restrictions of automorphisms of order prime to p and elements of δ i (S i ), and so α(ρ i (L i )) = ρ i (L i ).

As one example, if G 1 , . . . , G k are finite groups, S i ∈ Syl p (G i ), and L i = L c S i (G i ), then it is an easy exercise to show that the linking system L defined in Proposition 3.5 is the centric linking system of G 1 × · · · × G k .

We next recall the definition of reduced fusion systems.

Definition 3.6. Let F be a saturated fusion system over a finite p-group S.

(a) Set hyp(F ) = g −1 α(g) g ∈ P ≤ S, α ∈ O p (Aut F (P )) (the hyperfocal subgroup of F ).

For the existence of the minimal subsystems O p (F ) and O p ′ (F ), see, e.g., Theorems I.7.4 and I.7.7 in [AKO] .

Recall also that a saturated fusion system is indecomposable if it is not the direct product of two proper fusion subsystems.

The subcategory L ⊆ L defined in Proposition 3.5(b) is not a linking system, since Ob( L) is not closed under overgroups. However, Ob( L) = P = P 1 × · · · × P k P i ≤ S i , P ∈ F c does include all subgroups of S that are F -centric and F -radical: this is shown in [AOV, Lemma 3 .1] when k = 2 and follows in the general case by iteration. So Proposition 2.6 applies to the automorphism group Aut( L) = α ∈ Aut cat ( L) α(δ P,S (1)) = δ α(P ),S (1), α(δ P (P )) = δ α(P ) (α(P )) ∀ P ∈ Ob( L) .

Lemma 3.7. Assume Hypotheses 3.1. Let L 1 , . . . , L k be centric linking systems associated to F 1 , . . . , F k , respectively, and let L be the centric linking system associated to F defined as in Proposition 3.5. Let L ⊆ L be the full subcategory with objects the subgroups P 1 ×· · ·×P k ≤ S for P i ∈ F c i = Ob(L i ), and let ξ L : (c) if Z(F i ) = 1 and F i is indecomposable for each i, then E L is an isomorphism.

Proof. (a) This formula clearly defines a homomorphism to the group Aut cat ( L) of all automorphisms of L as a category. That ξ L (α 1 × · · · × α k )ξ −1 L (for α i ∈ Aut(L i )) sends inclusions in L to inclusions and sends distinguished subgroups to distinguished subgroups follows from the commutativity of the square in Proposition 3.5(b).

(b) We will show that each α ∈ Aut( L) extends to some α ∈ Aut(L). By the definition in Proposition 3.5, each morphism in L is a restriction of a morphism in L, and hence there is at most one such extension. So upon setting E L ( α) = α, we get a well defined injective homomorphism from Aut( L) to Aut(L).

Fix α ∈ Aut( L), and let β = µ L ( α) ∈ Aut(F ) be the automorphism of Proposition 2.6. Thus π • α = c β • π, and α(P ) = β(P ) for all P ∈ Ob( L). By the definition in Proposition 3.5, for all P, Q ∈ Ob(L), Mor L (P, Q) = ψ ∈ Mor L ( P , Q) π(ψ)(P ) ≤ Q (3.1) (where P and Q are as in Hypotheses 3.1). Also, β( P ) = β(P ) for each P ∈ Ob(L), since P and β(P ) are the unique minimal objects of L containing P and β(P ), and since β permutes the objects of L and of L. (We are not assuming here that β permutes the factors S i .)

For all P, Q ∈ Ob(L) and ψ ∈ Mor L (P, Q) ⊆ Mor L ( P , Q), we have π( α(ψ))(β(P )) = c β (π(ψ))(β(P )) = β(π(ψ)(P )) ≤ β(Q) :

the first equality since π • α = c β • π and the inequality since π(ψ)(P ) ≤ Q by (3.1). So α(ψ) ∈ Mor L (β(P ), β(Q)) by (3.1) again. We can thus define α ∈ Aut(L) extending α by setting α(P ) = β(P ) for all P , and letting α P,Q be the restriction of α P , Q for P, Q ∈ Ob(L).

(c) Now assume that Z(F i ) = 1 and F i is indecomposable for each i. By [O6, Corollary 5.3 ], F has a unique factorization as a product of indecomposable fusion systems.

Fix α ∈ Aut(L), and set β = µ L (α) ∈ Aut(F ) (Proposition 2.6). By the uniqueness of the factorization, there is γ ∈ Σ k such that c β (F i ) = F γ(i) for each 1 ≤ i ≤ k. In particular, β(S i ) = S γ(i) for each i. So for each object P = P 1 × · · · × P k in L, β(P ) = k i=1 β(P i ) is also an object in L. Hence α( L) = L and α = E L (α| L ). Since α ∈ Aut(L) was arbitrary, E L is onto.

We are now ready to prove our main results concerning the automorphism group of a product of linking systems.

Proposition 3.8. Assume Hypotheses 3.1. Let L i be a centric linking system associated to F i for each 1 ≤ i ≤ k, and let L be the centric linking system associated to F defined as in Proposition 3.5. Set

(a) There is an injective homomorphism

with the property that for each (α 1 , . . . ,

(b) If Z(F ) = 1, and F i is indecomposable for each i, then Φ L is an isomorphism, and induces an isomorphism

(c) Assume that Z(F ) = 1, and that F i is indecomposable and tame for each i. Then F is tame. If G 1 , . . . , G k are finite groups such that O p ′ (G i ) = 1 and F i is tamely realized by G i for each i, and such that F i ∼ = F j implies G i ∼ = G j , then F is tamely realized by the product G 1 × · · · × G k .

Proof. Let L ⊆ L be the full subcategory defined in Proposition 3.5(b). Thus Ob( L) is the set of all P ∈ Ob(L) = F c such that P = P .

Without loss of generality, for each pair of indices i, j such that F i ∼ = F j , we can assume that F i = F j and S i = S j . Then L i ∼ = L j by Theorem 1.9, and so we can also assume that L i = L j .

(a) Define Φ L to be the composite

where E L is the homomorphism of Lemma 3.7(b), where the restriction of c ξ to k i=1 Aut(L i ) is the homomorphism c ξ of Lemma 3.7(a), and where c ξ (γ)(ϕ 1 , . . . , ϕ k ) = (ϕ γ −1 (1) , . . . , ϕ γ −1 (k) ) (3.3)

for γ ∈ Γ and ϕ i ∈ Mor(L i ). Then (3.2) holds by the definition of c ξ . One easily checks using (3.2) and (3.3) that Φ L is an injective homomorphism.

It remains to check that Φ L k i=1 Aut(L i ) = Aut 0 (L): the subgroup of those α ∈ Aut(L) such that α S (δ S (S i )) = δ S (S i ) for each i. The inclusion of the first group in Aut 0 (L) is clear. By Proposition 3.5(c), there are embeddings of categories ρ i : L i −→ L sending P i ≤ S i to its product with the S j for all j = i, and α(ρ i (L i )) = ρ i (L i ) for each α ∈ Aut 0 (L). We can thus define Ψ L,i : Aut 0 (L) −→ Aut(L i ) by sending α to ρ −1 i αρ i . Set Ψ 0 L = (Ψ L,1 , . . . , Ψ L,k ); then Φ L • Ψ 0 L is the inclusion of Aut 0 (L) into Aut(L), and so Aut 0 (L) ≤ Im(Φ L ). (b) Fix α ∈ Aut(L), and set β = µ L (α) ∈ Aut(F ) (see Proposition 2.6). By [AOV, Proposition 3 .6] and since Z(F ) = 1 and the F i are indecomposable, c β permutes the factors F i .

Let γ ∈ Σ k be such that c β (F i ) = F γ(i) for each i, and hence also β(S i ) = S γ(i) and α S (δ S (S i )) = δ S (S γ(i) ). In particular, γ ∈ Γ. Then Φ L (γ) −1 • α ∈ Aut 0 (L), and since Aut 0 (L) ≤ Im(Φ L ) by (a), this shows that Aut(L) ≤ Im(Φ L ) and hence that Φ L is onto.

Since Aut L (S) = Aut L 1 (S 1 ) × · · · × Aut L k (S k ), Φ L induces an isomorphism of quotient groups

(c) Assume now that Z(F ) = 1, and that F i is indecomposable and tame for each i. Let G 1 , . . . , G k be such that O p ′ (G i ) = 1 and F i is tamely realized by G i for each i, and such that

Without loss of generality, we can assume that G i = G j whenever G i ∼ = G j and also (by the uniqueness of linking systems again) that

Consider the following diagram

where Φ G is the isomorphism of Proposition 3.4 (as defined when taking λ ij = Id G i for each i < j such that G i = G j ), where s is defined to make the top square commute, and where the commutativity of the bottom square is immediate from the definitions. Thus κ G • s = Id Out(L) , so κ G is split surjective, and F is tamely realized by G.

In this section, we set up some tools that will be used later when proving inductively that all realizable fusion systems are tame. The starting point for the inductive procedure is Theorem 4.5, which summarizes the main results in [OR] . Note that if F is a saturated fusion system such that O p ′ (F ) is simple, then for each finite group G with O p ′ (G) = 1 that realizes F , O p ′ (G) is simple (so G is almost simple), and O p ′ (G) realizes F if F is simple.

Let Comp(G) denote the set of components of a finite group G; i.e., the set of subnormal subgroups of G that are quasisimple. (Recall that a subgroup H 

in the following one, and H is quasisimple if H is perfect and H/Z(H) is simple.) The components of G commute with each other pairwise (see [A1, § 31] or [AKO, Lemma A.12] ). In particular, when O q (G) = 1 for all primes q, they are all simple groups, and the subgroup E(G) def = Comp(G) is their direct product. Similarly, the components of a saturated fusion system F over a finite p-group S are its subnormal fusion subsystems C F that are quasisimple (i.e., O p (C) = C and C/Z(C) is simple). The set of components of F will be denoted Comp(F ), and E(F ) = Comp(F ) .

By analogy with the case for groups, a central product of fusion systems E 1 , . . . , E k is a fusion system E ∼ = (E 1 × · · · × E k )/Z, for some central subgroup Z ≤ k i=1 Z(E i ) that intersects trivially with each factor Z(E i ). More precisely, if F is a fusion system over S and E 1 , . . . , E k ≤ F are fusion subsystems over T 1 , . . . , T k ≤ S, then the subsystems commute in F if the T i commute pairwise, and for each k-tuple of morphisms (ϕ 1 , . . . , ϕ k ), where ϕ i ∈ Hom E i (P i , Q i ), there is a morphism ϕ ∈ Hom F (P 1 · · · P k , Q 1 · · · Q k ) that extends each of the ϕ i . In this case, the (internal) central product of the E i is the fusion subsystem

over T 1 · · · T k ≤ S. See Definition 2.4 and Lemma 2.8 in [O6] for some more details, and see [He1, Proposition 3 .3] for a slightly different approach to defining central products of fusion subsystems.

Lemma 4.1. Let F be a saturated fusion system over a finite p-group S. Then the following hold: Proof. These are all shown in Chapter 9 of [A2] : point (a) in 9.8.1, 9.8.2, and 9.12.3; point (b) in 9.9.1 and 9.9.2; and (c) in 9.13.

In particular, Lemma 4.1(a) implies that when O p (F ) = 1, the subsystem E(F ) is the direct product of the components, all of which are simple. Proof. (a,b) In all cases, each fusion subsystem subnormal in E is subnormal in F , and hence Comp(E) ⊆ Comp(F ). If C ∈ Comp(F ) Comp(E), then by Lemma 4.1(c), F contains a central product of C and E, and in particular, C E. This proves (a), and also shows that E is not centric in F in this case, proving the first part of (b). If E has p-power index in F , then F cannot be a central product of E with a quasisimple system, so Comp(E) = Comp(F ) also in this case. It remains to prove the opposite inclusion. Set E = E(F ): the central product of the C i and a saturated fusion system over U = U 1 · · · U k normal in F (see Lemma 4.1(a)). Set K = {α ∈ Aut F (U) | [α, U] ≤ Z}: a p-group of automorphisms by [G, Corollary 5.3.3] . Each ϕ ∈ Hom C F /Z (ZU/Z) (P/Z, Q/Z) (for Z ≤ P, Q ≤ N K S (ZU)) extends to ϕ ∈ Hom F /Z (P U/Z, QU/Z) such that ϕ| ZU/Z = Id, and this in turn lifts to ψ ∈ Hom F (P U, QU) with ψ| ZU ∈ K. Thus N K F (ZU)/Z = C F /Z (ZU/Z). Recall that K is a p-group. Hence the group Aut C F (ZU ) (P ) is normal of p-power index in Aut N K F (ZU ) (P ) for each P ≤ C S (ZU), and so C F (ZU) has p-power index in N K F (ZU). Also, by [He1, Lemma 4.2(b) , and hence lies in Comp(C F (E)) by Lemma 4.2(a) and since C F (E) is normal.

By Lemma 4.1(a) and since E = E(F ), C F (E) is constrained. Hence C F (E)/Z is also constrained by [He2, Lemma 2.10] . We just saw that Comp(C F (E)/Z) contains all components of F /Z not in Comp 0 (F /Z). Since a constrained fusion system does not have any components, this proves that Comp(F /Z) = Comp 0 (F /Z).

Lemma 4.4. Let F be a saturated fusion system over a finite p-group S, for some prime p ≥ 5, and assume A S is abelian and F -centric. Assume also, for some ℓ ≥ 1, κ ≥ p, and 2 < m | (p − 1), that A is homocyclic of rank κ and exponent p ℓ , and that with respect to some basis {a 1 , . . . , a κ } for A as a Z/p ℓ -module, Aut F (A) contains G(m, m, κ) with index prime to p, and

for some 2 < r | m.

Then either A F , or O p ′ (F ) is simple and F is not realized by any known finite almost simple group.

Proof. Since Aut F (A) contains G(m, m, κ) with index prime to p, some subgroup conjugate to Aut S (A) is contained in G(1, 1, κ) ∼ = Σ κ , and hence Aut S (A) permutes some basis of Ω 1 (A) . Also, G(m, m, κ) acts faithfully on Ω 1 (A), as does each subgroup of Aut F (A) of order prime to p (see [G, Theorem 5.2.4] ). So by the assumptions on Aut This is well known, but the proof is simple enough that we give it here. Set V = Ω 1 (A) for short, let W ≤ S be another elementary abelian subgroup, and set W = Aut W (V ) and r = rk (W ) . Then (4.1) . Let B be a basis for V permuted by W , and assume W acts on B with s orbits (including fixed orbits) of lengths p m 1 , . . . , p ms . Then p r = |W | ≤ p m 1 · · · p ms , and hence m 1 + · · · + m s ≥ r. So

proving (4.2). In particular, Ω 1 (A) and A = C S (Ω 1 (A)) are weakly closed in F . G(m, m, κ) . There are exactly κ 1-dimensional subspaces of Ω 1 (A) invariant under the action of O p ′ (G 0 (m, m, κ)) ∼ = (C m ) κ−1 , and they are permuted transitively by the alternating group A κ . Hence

Step 1: Assume F 0 is not simple, and let E F 0 be a proper nontrivial normal subsystem over 1 = T S. Then T is strongly closed in F 0 , so T ∩ A is normalized by the action of Γ 0 on A, and T ∩ A = Ω k (A) for some 1 ≤ k ≤ ℓ since Ω 1 (A) is simple by (4.3). Also, T /Ω k (A) is normal in S/Ω k (A), so if k < ℓ and T > Ω k (A), then T /Ω k (A) ∩ Z(S/Ω k (A)) = 1. Since Z(S/Ω k (A)) ≤ A/Ω k (A), this implies that T ∩ A > Ω k (A), contradicting the choice of k. Thus either T = Ω k (A) for some 1 ≤ k ≤ ℓ, or T > A.

If T = Ω k (A) for some k ≤ ℓ, then since T is abelian and strongly closed, T = Ω k (A) F by [AKO, Corollary I.4.7(a) ]. Hence for each a ∈ A and each x ∈ a F , there is ϕ ∈ Hom F (Ω k (A) a , S) such that ϕ(a) = x and ϕ(Ω k (A)) = Ω k (A). Then x ∈ C S (Ω 1 (A)) = A by (4.1), so A is strongly closed in this case, and A F by [AKO, Corollary I.4 .7(a)] again.

Thus if F 0 is not simple and A F , then there is a proper normal subsystem E F 0 over T S such that T > A. Set ∆ = Aut E (A) Γ 0 . Then ∆ ≥ Aut T (A) = 1 since T > A, so p | |∆|. Since Γ 0 contains G 0 (m, m, κ) with index prime to p, we have

where κ ≥ p ≥ 5 and p | |∆ ∩ G 0 (m, m, κ)|. Since A κ and hence G 0 (m, m, κ) have no proper normal subgroups of order a multiple of p, it follows that ∆ ≥ G 0 (m, m, κ), and hence that ∆ has index prime to p in Aut F (A). But then Aut T (A) = Aut S (A), so T = S since A is F -centric, and E has index prime to p in F 0 and F by [AOV, Lemma 1.26] . Thus E = F 0 = O p ′ (F ), contradicting our assumption that E is proper.

Step 2: It remains to show, when A F , that F is not realized by any known finite almost simple group. Assume otherwise: assume F = F S (G) where G is almost simple, and

We claim this is impossible. Note that A is a radical p-subgroup of G 0 , since O p (Aut G 0 (A)) = 1 and p ∤ |C G 0 (A)/A| (i.e., A is F 0 -centric). Although we do not know Aut G 0 (A) precisely, we know that it is contained in Aut F (A) and contains G 0 (m, m, κ) ∼ = (C m ) κ−1 ⋊ A κ .

Since p ≥ 5 and rk p (G 0 ) ≥ p, G 0 cannot be a sporadic group by [GLS3, Table 5.6 .1]. By [AF, § 2] , for each abelian radical p-subgroup B ≤ Σ κ , Aut Σκ (B) is a product of wreath products of the form GL c (p) ≀ Σ κ for c ≥ 1 and κ ≥ 1. Thus Aut Aκ (B) can have index 2 in C p−1 ≀ Σ κ for some κ, but not index larger than 2. So G 0 cannot be an alternating group.

If G 0 ∈ Lie(p), then N G 0 (A) is a parabolic subgroup by the Borel-Tits theorem [GLS3, Corollary 3.1.5] and since A is centric and radical. So in the notation of [GLS3, § 2.6], A = U J and N G 0 (A) = P J (up to conjugacy) for some set J of primitive roots for G 0 . Hence by [GLS3, Theorem 2.6.5(f, g) 

is a central product of groups in Lie(p), contradicting the assumption that O p ′ (Aut G (A)) ∼ = G 0 (m, m, κ). Now assume that G 0 ∈ Lie(q 0 ) for some prime q 0 = p. By [GL, (and since p ≥ 5), S ∈ Syl p (G 0 ) contains a unique elementary abelian p-subgroup of maximal rank, and by (4.2), it must be equal to Ω 1 (A). Hence Aut F (A) must be as in one of the entries in Table  4 .2 or 4.3 in [OR] .

• If G 0 is a classical group and hence Aut F (A) ∼ = G( m, r, κ) for m = µ or 2µ and r ≤ 2 (see the next-to-last column in [OR, Table 4 .2] and recall that G( m, 1, κ) ∼ = C m ≀ Σ κ ), then the identifications Aut F (A) ∼ = G( m, r, κ) and Aut F (A) ≥ G(m, r, κ) are based on the same decompositions of A as a direct sum of cyclic subgroups, and hence we have m | m and r ≤ 2, contradicting our original assumption.

• If G 0 is an exceptional group, then by [OR, Table 4 .3], either κ = rk(A) < p, or p = 3, or (in case (b)) m κ−1 · κ! does not divide |Aut F (A)| for any m > 2 and hence Aut F (A) cannot contain any such G(m, r, κ).

Recall again that if F is a realizable fusion system such that O p ′ (F ) is simple, then F is realized by a finite almost simple group, and by a simple group if F is simple.

Theorem 4.5. Fix a prime p and a known finite simple group G such that p | |G|. Fix S ∈ Syl p (G), and set F = F S (G) . Then either (a) S F ; or (b) p = 3 and G ∼ = G 2 (q) for some q ≡ ±1 (mod 9), in which case |O 3 (F )| = 3, and O 3 ′ (F ) < F is realized by SL ± 3 (q); or (c) p ≥ 5, G is one of the classical groups PSL ± n (q), PSp 2n (q), Ω 2n+1 (q), or PΩ ± 2n+2 (q) where n ≥ 2 and q ≡ 0, ±1 (mod p), in which case O p ′ (F ) is simple and is not realized by any known finite simple group; or (d) O p ′ (F ) is simple, and it is realized by a known finite simple group G * .

Moreover, in case (c), there is a subsystem F 0 F of index at most 2 in F with the property that for each saturated fusion system E over S such that O p ′ (E) = O p ′ (F ), E is realized by a known finite almost simple group if and only if it contains F 0 .

Proof. This is essentially [OR, Theorem 4.8] , but restated to make its proof independent of the classification of finite simple groups. The only difference between the proof of this version and that of Theorem 4.8 in [OR] is that we replace [OR, Lemma 4 .7] by the above Lemma 4.4.

When we consider the fusion systems of known finite quasisimple groups, Theorem 4.5 takes the following form.

Corollary 4.6. Let L be a known finite quasisimple group with O p ′ (L) = 1. Fix T ∈ Syl p (L), and set E = F T (L). Then either T E and hence E is constrained, or O p ′ (E) is quasisimple.

Proof. Set Z = Z(L). By Theorem 4.5, and since L/Z is a known simple group and E/Z = F T /Z (L/Z), either T /Z E/Z and E/Z is constrained, or O p ′ (E/Z) is simple, or p = 3 and L/Z ∼ = G 2 (q) for some q ≡ ±1 (mod 9). If T /Z E/Z, then T E and E is constrained. If p = 3 and L/Z ∼ = G 2 (q), then Z = 1, and

, and hence is also simple.

Assume O p (E 0 ) < E 0 . Then ZO p (E 0 )/Z has p-power index in the simple fusion system E 0 /Z, so ZO p (E 0 ) = E 0 . Each α ∈ Aut E (T ) normalizes O p (E 0 ) and centralizes Z, so O p (E 0 ), Aut E (T ) is a proper subsystem of p-power index in E, which is impossible since

Proposition 4.7. Let G be a finite group such that O p ′ (G) = 1, fix S ∈ Syl p (G), and set F = F S (G) . Set Comp(G) = {L 1 , . . . , L k }, and assume that each L i is a known quasisimple group. For each 1 ≤ i ≤ k, set

and assume the L i were ordered so that for some

Proof. Set L = L 1 · · · L k and T = T 1 · · · T k . For each 1 ≤ i ≤ k, we have T i ∈ Syl p (L i ) since L i L G, and also

the first normality relation by [AKO, Theorem I.7 .7] and the other two by [AKO, Proposition I.6 .2] and since L i L G. Thus C i F for each i. If i ≤ m, then C i is quasisimple and hence is a component of F . If i ≥ m + 1, then T i E i by Corollary 4.6, and hence

Assume D F is another component. By Lemma 4.1(b), F * (F ) is a central product of O p (F ) and the components of F , including C 1 , . . . , C m and D. Hence D ≤ C F (QT ), where C F (QT ) is the fusion system of C G (QT ) by Lemma 1.8. (Note that QT is fully centralized in F since it is normal in S.)

Conjugation by each element of G permutes the subgroups L i , and since T i = 1 for each i (recall O p (G) = 1), each element of C G (QT ) normalizes each of the L i . So conjugation induces homomorphisms

where Ker(c 1 ) = C G (QL) and Ker(c 2 ) = Aut C L (T ) (L). By [AKO, Theorem A.13(c) ], C G (QL) ≤ QL, and hence C G (QL) = Z(Q): an abelian p-group. Also, C L (T )/Z(T ) has order prime to p since T ∈ Syl p (L), and Out(L i ) is solvable for each i by the Schreier conjecture and since each L i is a known simple group (see [GLS3, Theorem 7 

Thus C F (QT ) = F C S (QT ) (C G (QT )) is solvable in the sense of [AKO, Definition II.12 .1], and its saturated fusion subsystems are all solvable by [AKO, Lemma II.12.8 ]. So D is solvable, and hence is constrained by [AKO, Lemma II.12.5(b) ] (see also [AKO, Definition I.4 .8]), which is impossible since D was assumed to be quasisimple. We conclude that C 1 , . . . , C m are the only components of F .

We need to understand the role played by the components of G and of F S (G) when determining automorphisms of the linking system L c S (G) and tameness. The next proposition is a first step towards doing that. 1, let L 1 , . . . , L k be its components, and set L = L 1 × · · · × L k G. Assume for each i that L i is a known simple group. Fix S ∈ Syl p (G), set F = F S (G), and assume that O p (F ) = 1. For each i, set

, and set T = U 1 × · · · × U k = S ∩ L ∈ Syl p (L) and E = O p ′ (F T (L)). Then (a) there is a unique minimal normal fusion subsystem E * F containing E that is realized by a product of known finite almost simple groups, and E = O p ′ (E * ); 

, and hence permutes the factors C * i in E * . Thus E * F T (L) F and c α (E * ) = E * for each α ∈ Aut F (T ), so E * F by [A2, 7.4] . It remains to show that each normal realizable fusion subsystem of F containing E also contains E * .

Assume that K is a finite group with subgroup R ∈ Syl p (K) such that T ≤ U S and E ≤ F R (K)

F . As usual, we can assume that O p ′ (K) = 1, and O p (K) = 1 by Lemma 1.8(b) and since O p (F R (K)) = 1. Let K 1 , . . . , K m K be the components of K, and set R i = R ∩ K i . The K i are simple since O p (K) = O p ′ (K) = 1, and by Proposition 4.7, the subsystems O p ′ (F R i (K i )) for 1 ≤ i ≤ m are the components of F R (K). Each component of F R (K) is subnormal in F (hence a component) since F R (K) F , and each component of F is contained in and hence a component of F R (K) by assumption. Thus m = k, and we can assume that the indices are chosen so that

by Theorem 4.5 (the last statement) and the minimality assumption. So F R (K) contains E * = F T (H), and thus E * is minimal among normal realizable subsystems.

(c) To see that E * is centric in F , we must show that C S (E * ) ≤ T , where C S (E * ) ≤ C S (T ) is the unique largest subgroup X ≤ C S (T ) such that C F (X) ≥ E * (see Definition 2.9(a,b)). By [A2, Theorem 4 ] (see also [He1, Theorem 2] ), there is a normal fusion subsystem C F (E * ) F over C S (E * ). Also, C S (E * ) ∩ T = Z(E * ) ≤ O p (E * ) = 1, and O p (C F (E * )) ≤ O p (F ) = 1 by Lemma 2.3(b) . So if C S (E * ) = 1, then each minimal subnormal fusion subsystem of C F (E * ) is simple, and hence a component of F not contained in E. Since this is impossible by Proposition 4.7, we conclude that C S (E * ) = 1 and hence that E * is centric in F . By a similar argument, E is also centric in F .

Finally, E is characteristic in F since it is the product of the components of F , and so E * is characteristic in F by the uniqueness property in (a) and since E * F . Proposition 4.9. Let F be a saturated fusion system over a finite group S. Let C 1 , . . . , C k be the components of F , where C i is a fusion system over U i . Assume, for each 1 ≤ i ≤ k, that Z(C i ) = 1 (i.e., that C i is simple). Set

Thus E(F ) is the direct product of the C i , and is a fusion subsystem over U by Lemma 4.1(a). Then the fusion subsystem N is saturated and characteristic in F , E(F ) N , and C i N for each 1 ≤ i ≤ k.

Proof. By Lemma 4.1(a) and since Z(C i ) = 1 for each i, E(F ) is the direct product of the C i and is characteristic in F . In particular, E(F ) F , and U is strongly closed in F .

We first claim that for each P ∈ H and each ϕ ∈ Hom F (P, S), there is δ ∈ ∆ such that

It suffices to prove this when P ≤ U (hence ϕ(P ) ≤ U). Since E(F ) F by Lemma 4.1(a), the Frattini condition implies that ϕ = αϕ ′ for some α ∈ Aut F (U) and some ϕ ′ ∈ Hom E(F ) (P, U). Since α permutes the components of F , there is δ ∈ ∆ such that α(

We next claim that for each δ ∈ ∆, there is α ∈ Aut F (N) such that α(U i ) = U δ(i) for each 1 ≤ i ≤ k. (4.5)

To see this, fix δ ∈ ∆, and choose β ∈ Aut F (U) such that β(U i ) = U δ(i) for each i ∈ I. Since Aut S (U) ∈ Syl p (Aut F (U)) and Aut N (U) Aut F (U), we have that Aut N (U) and β Aut N (U) are both Sylow p-subgroups of Aut N (U). So there is γ ∈ Aut N (U) such that γβ normalizes Aut N (U), and hence by the extension axiom extends to α ∈ Aut F (N). By construction, α(U i ) = U δ(i) for each 1 ≤ i ≤ k.

Step 1 that N is saturated, in Step 2 that N is characteristic, and in Step 3 that E(F ) N and C i N for 1 ≤ i ≤ k.

Step 1:

By definition, N is H-generated. So by [AKO, Theorem I.3 .10], to prove that N is saturated, it suffices to prove that it is H-saturated; i.e., that each P ∈ H is N -conjugate to a subgroup that is fully automized and receptive in N (see [AKO, Definition I.3.9] ).

If P ∈ H is receptive in F and ϕ ∈ Iso N (Q, P ) for some Q ≤ N, then ϕ extends to some ϕ ∈ Hom F (N F ϕ , S), and ϕ(N F ϕ ∩ U i ) ≤ U i for each 1 ≤ i ≤ k by (4.4). Hence ϕ restricts to an element of Hom N (N N ϕ , N). Thus P is receptive in N . Assume P ∈ H is fully automized in F . By (4.4), each β ∈ Aut F (P ) permutes the subgroups P ∩ U i for 1 ≤ i ≤ k, while β ∈ Aut N (P ) if and only if it sends each P ∩ U i to itself. So Aut N (P ) is normal in Aut F (P ). Also, Aut N (P ) = Aut S (P ) ∩ Aut N (P ): if c x ∈ Aut N (P ) for x ∈ N S (P ), then x (P ∩ U i ) ≤ U i for each 1 ≤ i ≤ k and hence x ∈ N. So Aut N (P ) ∈ Syl p (Aut N (P )) since Aut S (P ) ∈ Syl p (Aut F (P )), and we conclude that P is fully automized in N . Now fix P ∈ H, and let χ ∈ Hom F (P, N) be such that χ(P ) is fully normalized in F . Then χ(P ) ∈ H, and we just showed that χ(P ) is fully automized and receptive in N . By (4.4) and (4.5), there is α ∈ Aut F (N) such that αχ ∈ Hom N (P, N). Then αχ(P ) is also fully automized and receptive in N , and is N -conjugate to P . Since P ∈ H was arbitrary, this proves that N is H-saturated, and finishes the proof that it is saturated.

Step 2: We first check that N is strongly closed in F . Set

. Let x, y ∈ S be such that x ∈ N and y ∈ x F ; we claim that y ∈ N. Let P, Q ≤ S and ϕ ∈ Hom F (P, Q) be such that x ∈ P , y ∈ Q, and ϕ(x) = y. Then ϕ(P ∩ U) ≤ Q ∩ U since U is strongly closed in F as noted above, and so ϕ induces a homomorphism ϕ from P U/U ∼ = P/(P ∩ U) to QU/U ∼ = Q/(Q ∩ U). By a theorem of Puig (see [Cr, Theorem 5 .14]), ϕ ∈ Hom F /U (P U/U, QU/U). In other words, there is ψ ∈ Hom F (P U, QU) such that for each g ∈ P , ψ(g) ∈ ϕ(g)U.

, and K is normal in Aut F (U) since each α ∈ Aut F (U) permutes the U i . So c ψ(x) ∈ K, and hence ψ(x) ∈ N. Hence y ∈ ψ(x)U ⊆ N, finishing the proof that N is strongly closed.

, and so x ∈ N. Thus C S (N) ≤ N, so the extension condition holds for N ≤ F . By (4.4) and (4.5), for each P, Q ≤ N and ϕ ∈ Hom F (P, Q), there is δ ∈ ∆ such that ϕ(P ∩ U i ) ≤ Q ∩ U δ(i) for all 1 ≤ i ≤ k, and α ∈ Aut F (N) such that α(U i ) = U δ(i) for each 1 ≤ i ≤ k. So α −1 ϕ ∈ Hom N (P, α −1 (Q)), and the Frattini condition for normality holds.

Fix α ∈ Aut F (N), and let δ ∈ ∆ be such that α(U i ) = U δ(i) for all i ∈ I. For each P, Q ∈ H and ϕ ∈ Hom N (P, Q), ϕ(P ∩ U i ) ≤ Q ∩ U i for each 1 ≤ i ≤ k, and since δ(J) = J, we have αϕα −1 (α(P ) ∩ U i ) ≤ α(Q) ∩ U i for each 1 ≤ i ≤ k. Thus α ϕ ∈ Mor(N ). So α normalizes the subsystem N , and the invariance condition holds. Thus N F .

For each β ∈ Aut(F ), β permutes the components of F , and hence permutes the subgroups U i and the members of the set H I . So c β (N I ) = N I by the above definition of N I , and N I is characteristic in F .

Step 3: By [A2, 9.8.3] and since N F , we have E(F ) = E(N ) N . For each 1 ≤ i ≤ k, 3, C i E(F ) by [A2, 9.8.2] , and c α (C i ) = C i for each α ∈ Aut N (U) by definition of N . So C i N by Lemma 2.4.

By construction, N is the largest saturated subsystem of F that contains each of the C i for 1 ≤ i ≤ k as a normal subsystem.

We are now ready to show that realizable fusion systems are tame. This has already been shown in earlier papers for fusion systems of known simple groups (see Proposition 5.2). When F is the fusion system of an arbitrary finite group G all of whose components are known, we will show that it is tame via a series of reductions based on an examination of those components.

We first restrict attention to tameness of fusion systems of finite simple groups. This was shown in most cases in earlier papers, and will be summarized below, but there were two cases whose proofs assumed earlier results that were in error:

Lemma 5.1. Let (G, p) be one of the pairs (He, 3) or (Co 1 , 5), choose S ∈ Syl p (G), and set F = F S (G) and L = L c S (G) . Then Out(L) = 1, and so F is tamely realized by G.

Proof. Since p is odd, Out(F ) ∼ = Out(L) in both cases. The simplest proof of this is given in [O1, Theorem C] (and the sporadic groups are handled in Proposition 4.4 of that paper). A more general result is shown in [O3, Theorem C] and [GLn, Theorem 1.1 ].

When G = He and p = 3, the argument in [O5, p. 139 ] claimed (wrongly) that F is simple, but did not actually use this. Since S is extraspecial of order 27 and exponent 3 and Out G (S) ∼ = D 8 , we have

Elements in N Aut(S) (Aut G (S)) Aut G (S) exchange subgroups of S of order 9 with nonisomorphic automizers, and hence do not normalize F . So Aut(F ) = Aut G (S), and Out(F ) = 1.

When G = Co 1 and p = 5, the proof that Out(F ) = 1 in [O5, p. 138] used the incorrect claim that F has a normal subsystem of index 2. So we replace that argument with the following one. By [Cu, Theorem 5 .1] and the correction in [W, p. 145 ], S contains a unique elementary abelian subgroup Q of order 5 3 and index 5, and N G (Q)/Q ∼ = Aut G (Q) ∼ = C 4 ×Σ 5 . Set H = N G (Q). By [O5, Lemma 1.2(b) ] and since C H (Q) = Q, we have |Out(F )| ≤ |Out(H)|. By [OV, Lemma 1.2] , there is an exact sequence

and by [Be2, p. 110] , there is a 5-term exact sequence for the homology of H/Q as an extension of C 4 by Σ 5 that begins with

Since H 0 (C 4 ; Q) = H 1 (C 4 ; Q) = 0 (C 4 acts on Q ∼ = C 5 × C 5 × C 5 via multiplication by scalars), this proves that H 1 (H/Q; Q) = 0. Also,

is trivial since GL 3 (5) ∼ = C 4 × PSL 3 (5) and GO 3 (5) ∼ = Σ 5 is a maximal subgroup of PSL 3 (5) (see, e.g., [GLS3, Theorem 6.5.3] ). So Out(F ) = Out(H) = 1.

We now summarize what we need to know here about tameness of fusion systems of finite simple groups.

Proposition 5.2. Fix a known simple group G, choose S ∈ Syl p (G), and assume that S F S (G). Then F S (G) is tamely realized by some known simple group G * .

Proof. Set F = F S (G) and L = L c S (G) for short. Note that G is nonabelian since S F S (G). Assume first that G ∼ = A n for some n ≥ 5. By [AOV, Proposition 4.8] , if p = 2 and n ≥ 8 or if p is odd and p 2 ≤ n ≡ 0, 1 (mod p), then κ G is an isomorphism. If p is odd and p 2 < n ≡ k (mod p) where 2 ≤ k ≤ p − 1, then F is still tamely realized by A n : Out(L) = 1 since F is isomorphic to the fusion system of Σ n and also that of Σ n−k . If p = 2 and n = 6, 7, then F is tamely realized by A 6 ∼ = PSL 2 (9) (and κ A 6 is an isomorphism). In all other cases, S is abelian and hence S F . If G is of Lie type in defining characteristic p, or if p = 2 and G ∼ = 2 F 4 (2) ′ , then by [BMO, Theorems A and D] , κ G is an isomorphism except when p = 2 and G ∼ = SL 3 (2). In this exceptional case, F is tamely realized by A 6 again.

If G is of Lie type in defining characteristic q 0 for some prime q 0 = p, then by [BMO, Theorem B] , F is tamely realized by some other simple group G * of Lie type. See also Tables 0.1-0.3 in [BMO] for a list of which groups of Lie type do tamely realize their fusion system, and when they do not, which other groups they can be replaced by.

If G is a sporadic simple group (and S F ), then by [O5, Theorem A] and Lemma 5.1, κ G is an isomorphism except when (G, p) is one of the pairs (M 11 , 2) or (He, 3). If (G, p) = (He, 3), then by the same theorem, |Out(G)| = 2 and Out(L) = 1, so F is still tamely realized by G. If (G, p) = (M 11 , 2), then F is the unique simple fusion system over SD 16 , and is tamely realized by G * = PSU 3 (5) (and κ G * is an isomorphism) by [AOV, Proposition 4.4] .

The statements in the next proposition are very similar to results proven in [AOV] , but except for part (a) are not stated there explicitly. Their proof consists mostly of repeating those arguments. Since many of the results referred to in [AOV, §2] require considering linking systems that are not centric, they depend in a crucial way on [AOV, Lemma 1.17] , which states that Out(L 0 ) ∼ = Out(L) whenever L 0 ≤ L are linking systems associated to the same fusion system F and Ob(L 0 ) ⊆ Ob(L) are both Aut(F )-invariant.

Proposition 5.3. Let F be a saturated fusion system over a finite p-group S.

, then F is tamely realized by a finite group G such that G 0 G and C G (S 0 ) ≤ G 0 , and Comp(G) = Comp (G 0 ).

(c) If F 0 F is a characteristic subsystem of index prime to p, and F 0 is tamely realized by a finite group G 0 such that Z(G 0 ) = Z(F 0 ), then F is tamely realized by a finite group G such that G 0 G and Comp(G) = Comp(G 0 ).

(d) If F 0 F is a characteristic subsystem of p-power index, F 0 is tamely realized by a finite group G 0 such that Z(G 0 ) is a p-group, and Z(F ) = 1, then F is tamely realized by a finite group G such that G 0 G and Comp(G) = Comp (G 0 (b) Let H 0 be the set of all P ∈ F c such that P ≤ S 0 , and let H be the set of all P ≤ S such that P ∩ S 0 ∈ H 0 . For each P ∈ H, P ∩ S 0 ∈ F c by assumption, and hence P ∈ F c . Thus H ⊆ F c , and F cr 0 ⊆ H 0 since F cr 0 ⊆ F c . So by [AOV, Lemma 1.30] , there is a normal pair of linking systems L 0 L associated to F 0 F with object sets H 0 and H. Furthermore,

the equality and first inequality by Lemma 2.10(a), the third inequality since S 0 ∈ F cr 0 ⊆ F c , and the other two by definition. So L 0 is centric in L.

Assume F 0 is tamely realized by the finite group G 0 , where S 0 ∈ Syl p (G 0 ) and Z(G 0 ) = Z(F 0 ). Then L 0 ∼ = L H 0 S 0 (G 0 ) (the full subcategory of L c S 0 (G 0 ) with objects the set H 0 ) by the uniqueness of linking systems. By definition, the sets of objects H 0 and H are invariant under the actions of Aut(F 0 ) and Aut(F ), respectively. Also, L 0 is Aut(L)-invariant, since F 0 is characteristic in F , and L 0 = π −1 (F 0 ) by [AOV, Lemma 1.30] (where π : L −→ F is the structure functor for L). All hypotheses in [AOV, Proposition 2.16] are thus satisfied, and so F is tamely realized by some finite group G such that

By the Frattini argument, G = G 0 N G (S 0 ), and hence

Since Aut L (S 0 ) = N G (S 0 )/O p (C G (S 0 )) and similarly for L 0 , this proves that O p (C G (S 0 )) = O p (C G 0 (S 0 )). Also,

the last equality since S 0 ∈ F cr 0 ⊆ F c . So C G (S 0 ) = C G 0 (S 0 ) ≤ G 0 . As usual, we let F * (H) denote the generalized Fitting subgroup of a finite group H: the central product of the subgroups O q (H) (for all primes q) with the components of H. Assume Comp(G) strictly contains Comp(G 0 ), and let K G be the subgroup generated by all components of G not in G 0 . Then [K, F * (G 0 )] = 1 and C G 0 (F * (G 0 )) ≤ F * (G 0 ) (see [AKO, Theorem A.13] ), so the conjugation action of K on G 0 induces a homomorphism

where ρ is induced by restriction (see [OV, Lemma 1.2] ). Since K is perfect, we have χ = 1, and hence K ≤ G 0 C G (G 0 ) ≤ G 0 C G (S 0 ) = G 0 , contradicting the assumption that K is generated by components not in G 0 . Thus Comp(G) = Comp (G 0 ).

(c) By [AKO, Lemma I.7 .6(a)], we have F c 0 = F c . (See also Definition 3.6(c).) Hence (c) is a special case of (b).

(d) Assume F 0 has p-power index in F and Z(F ) = 1. By [BCGLO, Proposition 3.8(b) ], a subgroup P ≤ S 0 is F -quasicentric if and only if it is F 0 -quasicentric. Let H be the set of all P ≤ S such that P ∩ S 0 is F 0 -quasicentric; then H ⊆ F q since overgroups of quasicentric subgroups are quasicentric. By Theorem 1.9, there is a unique linking system L associated to F with Ob(L) = H, and by [BCGLO, Theorem 4.4] , there is a unique linking system L 0 ≤ L associated to F 0 with Ob(L 0 ) = (F 0 ) q . Then L 0 L: the condition on objects (Definition 1.12(a)) holds by construction, and the invariance condition (1.12(b)) holds by the uniqueness of L 0 . By Lemma 2.10(c), there is an action of L/L 0 on C S (L 0 ) such that C C S (F 0 ) (L/L 0 ) = Z(F ), where Z(F ) = 1 by assumption. Also, L/L 0 = Aut L (S 0 )/Aut L 0 (S 0 ) is a p-group since F 0 has p-power index in F , so C S (L 0 ) = 1. Hence C Aut L (S 0 ) (L 0 ) = 1 by Lemma 2.10(a), and so L 0 is centric in L.

If in addition, F 0 is characteristic in F , then each α ∈ Aut(L) induces an element β = µ L (α) ∈ Aut(F ) (Proposition 2.6), and c β (F 0 ) = F 0 by assumption. Hence α(L 0 ) = L 0 by the uniqueness of the linking systems in [BCGLO, Theorems 4.4] . Also, by construction, Ob(L 0 ) and Ob(L) are invariant under the actions of Aut(F 0 ) and Aut(F ).

Set Z 0 = Z(F 0 ) for short. By assumption, 1 = Z(F ) = C Z 0 (Aut F (Z 0 )) where Aut F (Z 0 ) is a p-group since F 0 has p-power index in F . Hence Z 0 = 1.

Assume F 0 is tamely realized by G 0 , where Z(G 0 ) is a p-group. Then Z(G 0 ) ≤ Z(F 0 ) = 1 by Lemma 1.8(b). By Theorem 1.9 (the uniqueness of centric linking systems), L 0 ∼ = L c S 0 (G 0 ). The hypotheses of [AOV, Proposition 2.16 ] thus hold, and so F is tamely realized by some G such that G 0 G and G/G 0 ∼ = L/L 0 . Since L/L 0 = Aut L (S 0 )/Aut L 0 (S 0 ) is a p-group, all components of G are in G 0 , and so Comp(G) = Comp(G 0 ).

We are now ready to prove our main theorem. As explained in the introduction, Theorem 5.4, as well as Theorems 5.5 and 5.6, have been formulated so that their proofs are independent of the classification of finite simple groups.

Theorem 5.4. Let E F be a normal pair of fusion systems over T S such that Comp(E) = Comp(F ). Assume that E is realized by a finite group all of whose components are known quasisimple groups. Then F is tamely realized by a finite group all of whose components are known quasisimple groups.

Note: By Lemma 4.2(b,c), the condition Comp(E) = Comp(F ) in the above statement is satisfied whenever E is centric in F , and these two conditions are in fact equivalent if O p (F ) = 1.

Proof. For convenience, we let K C denote the class of all finite groups whose components are known quasisimple groups. Let S be the set of all triples (F , E, H) such that

F are saturated fusion systems over finite p-groups T S such that Comp(E) = Comp(F );

• H ∈ K C is such that T ∈ Syl p (H) and E = F T (H); and

• F is not tamely realized by any finite group in the class K C .

Assume the theorem does not hold; i.e., that S = ∅. Let (F , E, H) ∈ S be such that (|H|, |Mor(F )|) ∈ N 2 is the smallest possible under the lexicographic ordering. In other words, there are no triples (F * , E * , H * ) in S where |H * | < |H|; and among those where |H * | = |H|, there are none where |Mor(F * )| < |Mor(F )|. In particular, O p ′ (H) = 1.

We show in Step 1 that O p (F ) = 1, and that H is a product of known finite simple groups. We then show in Step 2 that the components of F are all normal in F , and reduce this to a contradiction in Step 3.

Step 1: Let L 1 , . . . , L k be the components of H, and set U i = T ∩ L i ∈ Syl p (L i ). Set C i = O p ′ (F U i (L i )) for each i. Assume that the L i were ordered so that for some m, C i is quasisimple if and only if i ≤ m. Then Comp(F ) = Comp(E) = {C 1 , . . . , C m } by Proposition 4.7 (with U i in the role of T i ).

Set H 0 = E(H) = L 1 · · · L m and E 0 = F T ∩H 0 (H 0 ) ≤ E. Then (F , E 0 , H 0 ) ∈ S , and so E = E 0 and H = H 0 by the minimality of |H|. In particular, m = k, and H is a central product of known finite quasisimple groups.

Set Q = O p (F ), and set S 0 = N Inn(Q) S (Q) = QC S (Q) and F 0 = N Inn(Q) F (Q). Thus F 0 is a fusion subsystem over S 0 . Since Q F = {Q}, the subgroup Q is fully Inn(Q)-normalized in F and hence F 0 is saturated (see Definition I.5.1 and Theorem I.5.5 in [AKO] ). Also, F 0 is weakly normal in F by [AOV, Proposition 1.25(c) ], and is normal since C S (S 0 ) ≤ S 0 (the extension condition holds).

If P ∈ F cr 0 , then P ≥ Q since Q F 0 (see [AKO, Proposition I.4 .5(a⇒b)]). So for each P * ∈ P F , P * ≥ Q, and C S (P * ) ≤ C S (Q) ≤ S 0 . Thus C S (P * ) = C S 0 (P * ) ≤ P * since P * ∈ F c 0 , so P ∈ F c . Thus F cr 0 ⊆ F c . By Proposition 5.3(b) and since F is not tamely realized by any group in K C , the subsystem F 0 is not realized by any group in K C . Thus (F 0 , E, H) ∈ S , and by the minimality assumption, we have F = F 0 . So Aut F (Q) ≤ Inn(Q).

Let Z(Q) = Z 1 (Q) ≤ Z 2 (Q) ≤ · · · be the upper central series of Q. Thus for each i, Z i+1 (Q)/Z i (Q) = Z(Q/Z i (Q)) = Z(F /Z i (Q)). By Proposition 5.3(a), if F /Z i+1 (Q) is tamely realized by a finite group G i+1 such that O p ′ (G i+1 ) = 1 and Z(G i+1 ) = Z(F /Z i+1 (Q)) = Z i+2 (Q)/Z i+1 (Q), then F /Z i (Q) is tamely realized by a finite group G i 

By assumption, F is not tamely realized by any group in K C . So inductively, F /Q ∼ = C F (Q)/Z(Q) is not realized by any group in K C either. Set Z = Z(Q). Then ZE/Z F /Z by Lemma 1.18, where ZE/Z is realized by H/(Z ∩ T ). Also, Comp(ZE/Z) = {ZC 1 /Z, . . . , ZC k /Z} = Comp(C F (Q)/Z) by Lemma 4.3. So (C F (Q)/Z, ZE/Z, H/(Z ∩ T )) ∈ S , and by the minimality assumption on (F , E, H), we have F ∼ = F /Q and thus O p (F ) = Q = 1.

To summarize, we have reduced to the case where (F , E, H) ∈ S satisfies:

O p (F ) = O p (E) = 1, E = F T (H) where H = L 1 × · · · × L k , and each L i is a known finite simple group. Also, Comp(F ) = Comp(E) = {C 1 , . . . , C k } where T = U 1 × · · · × U k , U i ∈ Syl p (L i ), and C i = O p ′ (F U i (L i )).

(5.1)

Step 2: Let N F be the normal subsystem constructed in Proposition 4.9: a subsystem over N = k i=1 N S (U i ) normal in F and containing each component C i as a normal subsystem. In particular, for each α ∈ Aut N (T ), α(U i ) = U i for all 1 ≤ i ≤ k.

For each P ∈ N c and each Q ∈ P F , Q ∈ N c since N F , and so Q ≥ Z(N). For each 1 ≤ i ≤ k, we have U i N by definition of N, and hence Q ∩ U i = 1. Each x ∈ C S (Q) centralizes each Q ∩ U i and hence normalizes each subgroup U i (recall that each element of S permutes the U i ). So C S (Q) = C N (Q) ≤ Q, and P ∈ F c . Thus N c ⊆ F c . By Proposition 5.3(b) and since F is not tame, N is not tame either.

Now, E is weakly normal in N by [Cr, Proposition 8.17 ] and since E F and N ≤ F . By the definition of N in Proposition 4.9, if T ≤ P ≤ N, and α ∈ Aut F (P ) is such that α| T ∈ Aut N (T ), then α ∈ Aut N (P ). So the extension condition for E ≤ N follows from that for E F , and hence E N .

Thus (N , E, H) ∈ S , and F = N by the minimality assumption. In other words, (F , E, H) ∈ S satisfies: (5.1) holds, and C i F for each 1 ≤ i ≤ k.

(5.2)

Step 3: Set C = E(F ) = C 1 × · · · × C k . By Proposition 4.8 (applied with H, E, C, C * in the roles of G, F , E, E * ), there is a unique minimal normal fusion subsystem C * = C * 1 ×· · ·×C * k E containing C that is realized by a product of known finite almost simple groups. Furthermore (by the same proposition), C = O p ′ (C * ), C * is centric and characteristic in E, and for each i,

where H * i is a known finite simple group. Then C * F T (H) = E F , and so C * F by Lemma 2.4 and since C * is characteristic in E. Set H * = H * 1 × · · · × H * k , so that C * = F T (H * ). Thus (F , C * , H * ) ∈ S , and E = C * and H = H * by the minimality assumption.

For each α ∈ Aut(F ), c α (C) = C since C = E(F ) is characteristic in F by Lemma 4.1(a), so c α (C * ) = C * by the uniqueness of the condition on each C * i . Thus C * is characteristic in F .

Let Aut 0 (C * ) ≤ Aut(C * ) be the subgroup of all automorphisms that send each U i to itself. Then Aut F (T ) ≤ Aut 0 (C * ) by (5.2) and since C * F , and so

Out(C * i ).

By assumption, each C * i is realized by a known finite simple group, hence is tamely realized by a known finite simple group by Proposition 5.2. Since Out(K) is solvable for each known finite simple group K (see [GLS3, Theorem 7.1 .1(a) ]), the groups Out(C * i ) are also solvable. So Aut F (T )/Aut * C (T ) is solvable. The hypotheses of [O4c, Theorem 5(b) ] thus hold for the pair C * F . By that theorem, there is a sequence C * = F 0 F 1 · · · F m = F of saturated fusion subsystems, for some m ≥ 0, such that (i) for each 0 ≤ j < m, F j is normal of p-power index or index prime to p in F j+1 and C * F j F ; and (ii) for each 1 ≤ j ≤ m and each α ∈ Aut(F j ) with c α (C * ) = C * , we have c α (F j ′ ) = F j ′ for all 0 ≤ j ′ < j.

Recall that Comp(F ) = {C 1 , . . . , C k }. For each 0 ≤ j ≤ m − 1, Comp(F j ) ⊆ Comp(F ) since F j F , and the opposite inclusion holds since C i C C * F j for each i. Hence C is characteristic in F j . For each α ∈ Aut(F j ), c α (C) = C since C is characteristic, hence c α (C * ) = C * since C * is the unique minimal realizable subsystem containing C, and hence c α (F j ′ ) = F j ′ for all 0 < j ′ < j by condition (ii) above.

In particular, this shows that F j is characteristic in F j+1 for each j. Also, Z(F j ) ≤ O p (F j ) = 1 for each j by Lemma 2.3(b) and since O p (F ) = 1 and F j F . So by Proposition 5.3(c,d), and since F j has index prime to p or p-power index in F j+1 and Z(F j ) = 1, if F j is tamely realized by a finite group G j , then F j+1 is tamely realized by a finite group G j+1 ≥ G j with Comp(G j+1 ) = Comp(G j ).

By assumption, F = F m is not realized by any finite group in K C . Hence the same holds for C * = F 0 , and so (C * , C * , H * ) ∈ C . For each 1 ≤ i ≤ k, C * i is tamely realized by some known finite simple group by Proposition 5.2, and so C * is tamely realized by a product of known finite simple groups by Proposition 3.8(c). This is a contradiction, and we conclude that S = ∅.

Note that Theorem A is just Theorem 5.4 without mentioning tameness. We now list some special cases of Theorem 5.4.

Theorem 5.5. Let F be a saturated fusion system over a finite p-group S. If all components of F are realized by known finite quasisimple groups, then F is realized by a finite group all of whose components are known quasisimple groups.

Proof. This is the special case of Theorem 5.4 where E is the generalized Fitting subsystem of F . Note that E is realizable since it is the central product of its components (which are realizable by Corollary 4.6) and a p-group.

Our third theorem is the special case of Theorem 5.4 where E = F . Theorem 5.6. Let p be a prime, and let F be a fusion system over a finite p-group that is realized by a finite group all of whose components are known quasisimple groups. Then F is tamely realized by a finite group all of whose components are known quasisimple groups.

Weights for symmetric and general linear groups

Reduced, tame, and exotic fusion systems

Finite group theory

The generalized Fitting subsystem of a fusion system

Fusion systems in algebra and topology

Fusion systems

Representations and cohomology II: cohomology of groups and modules

Cohomology of sporadic groups, finite loop spaces, and the Dickson invariants, Geometry and cohomology in group theory

Extensions of p-local finite groups

Homotopy equivalences of p-completed classifying spaces of finite groups

The homotopy theory of fusion systems

Automorphisms of fusion systems of finite simple groups of Lie type

Fusion systems and localities

The theory of fusion systems

On subgroups of ·0 II. Local structure

Control of fixed points and existence and uniqueness of centric linking systems

Finite groups

The local structure of finite groups of charateristic 2 type

The classification of the finite simple groups, nr. 3

Centralizers of normal subsystems revisited

Subcentric linking systems

Equivalences of classifying spaces completed at odd primes

Extensions of linking systems and fusion systems

Existence and uniqueness of linking systems: Chermak's proof via obstruction theory

Reductions to simple fusion systems

Correction to: Reductions to simple fusion systems

Automorphisms of fusion systems of sporadic simple groups

A Krull-Remak-Schmidt theorem for fusion systems

Simplicity of fusion systems of finite simple groups

Saturated fusion systems over 2-groups

Frobenius categories

Group theory I

The maximal subgroups of Conway's group Co 1