SIMPLICITY OF FUSION SYSTEMS OF FINITE SIMPLE GROUPS

BOB OLIVER AND ALBERT RUIZ

Abstract. We determine for which known finite simple groups G and which primes p the p-fusion system of G is simple. This means first collecting together the results that were already known (and correcting two errors made in an earlier study of this question), and then handling the remaining cases. At the same time, we develop some new tools to use when determining O^{p′}(F) for arbitrary saturated fusion systems.

For a prime p and a finite group G, the fusion system of 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 those homomorphisms between subgroups induced by conjugation by elements of G. The fusion system of G thus encodes its local structure: the conjugacy relations among its p-subgroups and p-elements. Motivated by connections with modular representation theory, Puig in the 1990s defined the concept of abstract fusion systems (published later in [Pg]): an (abstract) fusion system F over a finite p-group S is a category whose objects are the subgroups of S, and whose morphisms are injective homomorphisms satisfying certain conditions motivated by properties of finite groups such as the Sylow theorems. (See Section 1 for more detail.)

Normal fusion subsystems and simple fusion systems are defined by analogy to those of finite groups. Our main result here is to determine for exactly which known finite simple groups G and which primes p the p-fusion system of G is simple. This question was studied and answered in most cases by Aschbacher in Chapter 16 of [A2], but a few cases (all involving groups of Lie type in cross characteristic) were left open, and it is on those that we focus here. We also correct two errors found among Aschbacher’s conclusions (explained in the proof of Proposition 4.1(d)). One problem of particular interest is that of for which G and p, the p-fusion system of G contains a normal subsystem of index prime to p that is exotic, and we are also able to describe this situation quite precisely.

Our detailed group-by-group results are stated in Propositions 4.1 (alternating and sporadic groups and groups of Lie type in characteristic p), 4.2 (groups of Lie type in odd characteristic when p = 2), 4.5 (classical groups in characteristic different from p when p is odd), and 4.6 (exceptional groups of Lie type in characteristic different from p for p odd). They are then summarized by the following theorem, which will be restated with more details as Theorem 4.8.

Theorem A. Fix a prime p, a known finite simple group G, and S ∈ Syl_{p}(G). Set F = F_{S}(G), and assume that S ⋬ F. Then either

(a) 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 exotic; 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) is realized by SL_{3}^{±}(q); or (c) O^{p′}(F) is simple, and it is realized by a known finite simple group G^{∗}.

2000 Mathematics Subject Classification. Primary 20D05. Secondary 20D20, 20D45.

Key words and phrases. finite simple groups, fusion systems, automorphisms.

B. Oliver is partially supported by UMR 7539 of the CNRS.

A. Ruiz is partially supported by MICINN grant MTM2016-80439-P and AGAUR grant 2017-SGR-1725.

Moreover, in case (a), 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 realizable if and only if it contains F_{0}.

The statements in Theorem A that certain fusion systems are exotic are proved using the classification of finite simple groups. It has been formulated so that the other statements are independent of this.

Here, by analogy with the notation used for finite groups, O^{p′}(F) denotes the smallest (saturated) fusion subsystem of index prime to p (see Definition 2.1 and Proposition 2.4(b)). Also, a fusion system is called realizable if it is isomorphic to the fusion system of some finite group, and is called exotic otherwise.

Among the tools developed to determine O^{p′}(F) in these cases, we note in particular the following, which reduce the information needed in two different directions:

• Although O^{p′}(F) is characterized by its restriction to F-centric subgroups, we can, in fact, get the information we need from the smaller family of subgroups that are F-centric and F-radical, as described in Proposition 2.8.

• Although subsystems of index prime to p are determined by their automizers on S, it is sometimes more convenient to deal with the automizer of some subgroup A ⊴ S that is F-centric and weakly closed in F (such as the p-power torsion in a maximal torus in a group of Lie type). Tools for doing this are developed in Section 3.

This paper was originally motivated by our work (still continuing) with Carles Broto and Jesper Møller on tameness of realizable fusion systems (see [AKO, Definition III.6.3]). While trying to determine whether or not all realizable fusion systems are tame, we found that it is first necessary to understand more precisely the normal fusion subsystems of fusion systems of simple groups. Independently of that, some of our results are used in recent work of Radha Kessar, Gunter Malle, and Jason Semeraro to calculate weights (in the context of the Alperin weight conjecture in modular representation theory) attached to exotic fusions arising from homotopy fixed points of p-compact groups.

We begin the paper with a brief introduction to fusion systems in Section 1, and an introduction to (normal) subsystems of index prime to p in Section 2. We then develop tools, in the last part of Section 2 and in Section 3, to help determine O^{p′}(F) for a given fusion system F. Finally, in Section 4, we go through the list of all fusion systems of known finite simple groups at primes dividing their order, starting with a survey of the cases already handled by Aschbacher in [A2, Chapter 16], to determine which of them are simple.

Notation: As usual, for a finite group G, O_{p}(G) and O_{p′}(G) denote the largest normal subgroups of p-power order and of order prime to p, respectively, while O^{p}(G) and O^{p′}(G) are the smallest normal subgroups of p-power index and of index prime to p. We follow the standard conventions by setting (P)SL_{n}^{+}(q) = (P)SL_{n}(q) and (P)SL_{n}^{−}(q) = (P)SU_{n}(q), as well as E_{6}^{+}(q) = E_{6}(q) and E_{6}^{−}(q) = ^{2}E_{6}(q).

1. Definitions and earlier results

We begin by listing some of the basic definitions and properties of fusion systems. We use [AKO, Part I] as our main reference, but most or all of the following definitions are due originally to Puig: first in unpublished notes and then in [Pg].

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) =^{def} {c_{g} = (x ↦ ^{g}x) | g ∈ S, ^{g}P ≤ Q}; and

• for each φ ∈ Hom_{F}(P, Q), φ^{−1} ∈ Hom_{F}(φ(P), P).

Thus Hom_{F}(P, Q) denotes the set of morphisms in F from P to Q. We also write 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).

Since we will need to refer to the Sylow and extension axioms on several occasions, the following version of the definition of a saturated fusion system seems the most convenient one to give 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) The fusion system F is saturated if it satisfies the following two conditions:

(I) (Sylow axiom) For each subgroup P ≤ S fully normalized in F, P is fully centralized and Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)).

(II) (extension axiom) For each isomorphism φ ∈ Iso_{F}(P, Q) in F such that Q is fully centralized in F, φ extends to a morphism φ ∈ Hom_{F}(N_{φ}, S) where

N_{φ} = {g ∈ N_{S}(P) | φc_{g}φ^{−1} ∈ Aut_{S}(Q)}.

If G is a finite group and P, Q ≤ G, then Hom_{G}(P, Q) ⊆ Hom(P, Q) denotes the set of (injective) homomorphisms induced by conjugation in G. If S ∈ Syl_{p}(G), then F_{S}(G) denotes the fusion system over S where Hom_{F_{S}(G)}(P, Q) = Hom_{G}(P, Q) for P, Q ≤ S, and F_{S}(G) is saturated by [AKO, Theorem I.2.3]. A saturated fusion system F over 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.

We next list some of the terminology used to describe certain subgroups in a fusion system.

Definition 1.2. Let F be a fusion system over a finite p-group S, and let P ≤ S be a subgroup.

(a) P is F-centric if C_{S}(Q) ≤ Q for each Q ∈ P^{F}.

(b) P is F-radical if O_{p}(Out_{F}(P)) = 1.

(c) F^{c} ⊇ F^{rc} denote the sets of F-centric, and F-centric F-radical, subgroups of S, and also (depending on context) the full subcategories of F with those objects.

(d) P is weakly closed in F if P^{F} = {P }.

(e) P is strongly closed in F if for each x ∈ P, x^{F} ⊆ P.

(f) P is normal in F (P ⊴ 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) = P.

(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}.

(h) O_{p}(F) ≥ Z(F) denote the (unique) largest normal and central subgroups, respectively, in F.

When X is a set of homomorphisms between subgroups of a given finite p-group S, we let ⟨X⟩ denote the smallest fusion system over S (not necessarily saturated) that contains X among its morphisms. In other words, ⟨X⟩ is the category whose objects are the subgroups of S, and whose morphisms are composites of restrictions of homomorphisms in X ∪ Inn(S). Similarly, when E is a fusion system over S or a subgroup of S, and X is a set of homomorphisms between subgroups of S, we set ⟨E, X⟩ = ⟨Mor(E) ∪ X⟩: the smallest fusion system over S containing E and X.

The following version of Alperin’s fusion theorem for fusion systems will suffice for our purposes here. We refer to Theorem I.3.5 and Proposition I.3.3(a) in [AKO].

Theorem 1.3. For each saturated fusion system F over a finite p-group S,

F = ⟨Aut_{F}(P) | P ∈ F^{rc} and is fully normalized in F⟩.

Equivalently, each morphism in F is a composite of restrictions of F-automorphisms of subgroups that are centric and radical in F.

We finish the section with a much more specialized lemma: one which will be useful in the last section when showing the nonexistence of certain fusion systems.

Lemma 1.4. Let p be an odd prime, and let F be a saturated fusion system over a finite pgroup S. Assume that there is an abelian subgroup A ⊴ S such that |S/A| = p, |[S, A]| ≥ p^{2}, and A ⋬ F. Set G = Aut_{F}(A) and U = Aut_{S}(A) ∈ Syl_{p}(G); then |Aut_{G}(U)| = p − 1.

Proof. Since |[S, A]| > p, A is the unique abelian subgroup of index p in S. By Lemmas 2.2(a,c) and 2.3(a,b) in [COS] and since A ⋬ F, there is a subgroup A ≠ P < S in one of the classes H or B of subgroups defined in [COS, Notation 2.1], maximal among all F-centric F-radical subgroups properly contained in S, and with the properties that |N_{S}(P)/P | = p and O^{p′}(Out_{F}(P)) ≅ SL_{2}(p). Choose α ∈ O^{p′}(Aut_{F}(P)) of order prime to p whose class in O^{p′}(Out_{F}(P)) normalizes Out_{S}(P) ≅ C_{p} and has order p − 1. Then α extends to an element of Aut_{F}(N_{S}(P)) by the extension axiom (Definition 1.1(b)), and hence by the maximality of P and Theorem 1.3 (Alperin’s fusion theorem) to some α ∈ Aut_{F}(S). Then α|_{A} normalizes U, and its class in Aut_{G}(U) has order p − 1. □

2. Subsystems of index prime to p

Subsystems of fusion systems of index prime to p were first studied by Puig [Pg, § 6.5], and later in [5a2, § 3,5]. We begin the section with some basic results, where for simplicity we give references only in [AKO] when possible. After covering the basic properties, we develop some tools which will be used in the next section to determine O^{p′}(F) and its index in F in specific cases.

Note in the following definition that the analogy for a group G is “subgroups of G that contain O^{p′}(G)”, not arbitrary subgroups of index prime to p in G. For example, the fusion system of S is not, in general, of index prime to p in an arbitrary fusion system F over S.

Definition 2.1. Let F be a saturated fusion system over a finite p-group S.

(a) [AKO, Theorem I.7.7.d] Let O_{∗}^{p′}(F) ⊆ F be the fusion subsystem (not necessarily saturated) generated by the groups O^{p′}(Aut_{F}(P)) for all P ≤ S, and set

Aut_{F}^{0}(S) = ⟨α ∈ Aut_{F}(S) | α|_{P} ∈ Hom_{O_{∗}^{p′}(F)}(P, S), some P ∈ F^{c}⟩
Γ_{p′}(F) = Aut_{F}(S)/Aut_{F}^{0}(S).

(b) A fusion subsystem E ≤ F has index prime to p in F if E contains O_{∗}^{p′}(F); equivalently, if E is a fusion system over S and Aut_{E}(P) ≥ O^{p′}(Aut_{F}(P)) for each P ≤ S. For such a subsystem E ≤ F, the index of E in F is defined to be the index of Aut_{E}(S) in Aut_{F}(S).

Note that O_{∗}^{p′}(F) is denoted E_{0} in [AKO, Theorem I.7.7(d)].

Lemma 2.2. Let F be a saturated fusion system over a finite p-group S, and let E ⊆ F be a fusion subsystem (not necessarily saturated) of index prime to p. Then

(a) E^{c} = F^{c} and E^{cr} = F^{cr};

(b) a subgroup of S is fully normalized (fully centralized) in E if and only if it is fully normalized (fully centralized) in F; and

(c) (Frattini condition) for each P, Q ≤ S and φ ∈ Hom_{F}(P, Q), there are

α_{1}, α_{2} ∈ Aut_{F}(S), φ_{1} ∈ Hom_{E}(P, α_{1}^{−1}(Q)), and φ_{2} ∈ Hom_{E}(α_{2}(P), Q)

such that φ = (α_{1}|_{α_{1}^{−1}(Q)}) ∘ φ_{1} = φ_{2} ∘ (α_{2}|_{P}).

Proof. Point (c), and the equality E^{c} = F^{c}, are shown in [AKO, Lemma I.7.6(a)]. For all P ≤ S, we have O^{p′}(Aut_{F}(P)) ≤ Aut_{E}(P) ≤ Aut_{F}(P), and hence

O_{p}(Aut_{F}(P)) ≤ O_{p}(Aut_{E}(P)) ≤ O_{p}(O^{p′}(Aut_{F}(P))) ≤ O_{p}(Aut_{F}(P)),

where the last inclusion holds because O_{p}(G) is characteristic in G for all G. Thus P is F-radical if and only if it is E-radical, and so E^{cr} = F^{cr}, proving (a).

By (c), for each pair P ≤ S and Q ∈ P^{F}, there is Q^{∗} ∈ P^{E} such that Q^{∗} is Aut_{F}(S)- conjugate to Q. Thus N_{S}(Q^{∗}) ≅ N_{S}(Q), so P is fully normalized in F if and only if it is fully normalized in E, and similarly for fully centralized. □

The following terminology will be useful when working with fusion subsystems of index prime to p.

Definition 2.3. Let F be a saturated fusion system over a finite p-group S, and let F_{0} ⊆ F be a full subcategory with S ∈ Ob(F_{0}). A map θ: Mor(F_{0}) ⟶ Γ to a group Γ will be called multiplicative if it sends composites in F_{0} to products in Γ and sends inclusions to the identity. It will be called strongly multiplicative if in addition, θ(O^{p′}(Aut_{F}(P))) = 1 for each P ∈ Ob(F_{0}).

Of course, the condition that θ send composites to products is equivalent to saying that it extends to a functor from F_{0} to B(Γ), where B(Γ) is the category with one object and morphism group Γ. In these terms, the basic properties of the group Γ_{p′}(F) and of saturated fusion subsystems of index prime to p are summarized as follows.

Proposition 2.4 ([AKO, Theorem I.7.7]). The following hold for each saturated fusion system F over a finite p-group S.

(a) There is a unique multiplicative map θ_{F}: Mor(F^{c}) ⟶ Γ_{p′}(F) whose restriction to Aut_{F}(S) is the natural surjection. Also, θ_{F} is strongly multiplicative, and is universal in the following sense: if Γ is a group and θ: Mor(F^{c}) ⟶ Γ is strongly multiplicative, then there is a unique homomorphism f: Γ_{p′}(F) ⟶ Γ such that θ = f ∘ θ_{F}.

(b) There is a bijection

\{subgroups of Γ_{p′}(F)\} ⟶^{≅} \left\{{saturated fusion subsystems \atop E ≤ F of index prime to p}\right\}

that sends H ≤ Γ_{p′}(F) to the fusion subsystem E_{H} = ⟨θ_{F}^{−1}(H)⟩. This subsystem has the properties that (E_{H})^{c} = F^{c} as sets and (E_{H})^{c} = θ_{F}^{−1}(H) as categories. In particular, there is a unique smallest fusion subsystem O^{p′}(F) = ⟨θ_{F}^{−1}(1)⟩ of index prime to p.

Proof. Point (a) follows from [AKO, Theorem I.7.7(d)] (where Γ_{p′}(F) is defined to be the target of the universal strongly multiplicative map on Mor(F^{c})). Point (b) is shown in [AKO, Theorem I.7.7(b,c)]. □

By definition, Γ_{p′}(F) = 1 if Aut_{F}(S) is generated by elements α ∈ Aut_{F}(S) such that α|_{P} ∈ O^{p′}(Aut_{F}(P)) for some P ∈ F^{c}, and in this case, O^{p′}(F) = F by Proposition 2.4. In particular, this always applies if Out_{F}(S) = 1.

There is also a geometric interpretation of universal multiplicative maps. For each fusion system F over a p-group S, and each full subcategory F_{0} ⊆ F containing S, let θ: Mor(F_{0}) ⟶ π_{1}(|F_{0}|, ∗) be the map that sends φ ∈ Hom_{F}(P, Q) to the loop (based at ∗ = [S]) formed by the edges corresponding to morphisms incl_{P}^{S}, φ, and incl_{Q}^{S}. Then θ is the universal multiplicative map on Mor(F_{0}) (see, e.g., [AKO, Proposition III.2.8]). By [AKO, Theorem III.4.19], θ_{F} is universal among all multiplicative maps defined on Mor(F^{c}) (not only those that are strongly multiplicative), and hence Γ_{p′}(F) ≅ π_{1}(|F^{c}|).

We next recall some standard notation for isomorphisms and automorphisms of fusion systems. If β: S ⟶^{≅} T is an isomorphism between finite p-groups, and F is a fusion system over S, then we set ^{β}F = βFβ^{−1}: the fusion system over T for which Hom_{^{β}F}(β(P), β(Q)) (for P, Q ≤ S) is the set of all composites (β|_{Q}) ∘ φ ∘ (β|_{P})^{−1} for φ ∈ Hom_{F}(P, Q). When E and F are fusion systems over finite p-groups T and S, respectively, E ≅ F means that E = ^{β}F for some β ∈ Iso(S, T). Using this notation, we define, for a fusion system F over S,

Aut(F) = {β ∈ Aut(S) |^{β}F = F} and Out(F) = Aut(F)/Aut_{F}(S).

This notation is useful when classifying extensions of index prime to p.

Rather than define normal fusion subsystems here, we note for the purpose of the following lemma that if E ≤ F has index prime to p, where both are saturated fusion systems over S, then E ⊴ F if and only if ^{α}E = E for each α ∈ Aut_{F}(S). This follows immediately from [AKO, Definition I.6.1] and Lemma 2.2(c) (the Frattini condition).

Lemma 2.5. Let F_{0} be a saturated fusion system over a finite p-group S. Then the map

Ψ: \{saturated fusion systems F over S with F_{0} ⊴ F normal of index prime to p\} ⟶^{≅} \left\{{subgroups of Out(F_{0}) \atop of order prime to p}\right\}

which sends F to Aut_{F}(S)/Aut_{F_{0}}(S) ≤ Out(F_{0}) is a bijection.

Proof. If F_{1} and F_{2} are two such extensions of F_{0}, and Aut_{F_{1}}(S) = Aut_{F_{2}}(S), then F_{1} = F_{2} since F_{i} = ⟨F_{0}, Aut_{F_{i}}(S)⟩ (the Frattini condition holds for F_{0} ≤ F_{i} by Lemma 2.2(c)). Thus Ψ is injective.

If π = π̃/Aut_{F_{0}}(S) ≤ Out(F_{0}) has order prime to p, then by [5a2, Theorem 5.7(a)], the fusion system F =^{def} ⟨F_{0}, π̃⟩ is saturated and contains F_{0} as a normal subsystem, and Ψ(F) = Γ/Aut_{F_{0}}(S). So Ψ is onto. □

The injectivity of Ψ in Lemma 2.5 implies that if F_{1} and F_{2} are two saturated fusion systems over the same p-group S, then F_{1} = F_{2} if O^{p′}(F_{1}) = O^{p′}(F_{2}) and Aut_{F_{1}}(S) = Aut_{F_{2}}(S). Later, in Lemma 3.4, we will replace this by a more general hypothesis.

If F_{1} and F_{2} are fusion systems over finite p-groups S_{1} and S_{2}, respectively, then

F_{1} × F_{2} = ⟨(α_{1}, α_{2}) ∈ Hom(P_{1} × P_{2}, Q_{1} × Q_{2}) | α_{i} ∈ Hom_{F_{i}}(P_{i}, Q_{i}) for i = 1, 2⟩.

By [AKO, Theorem I.6.6], F_{1} × F_{2} is saturated if F_{1} and F_{2} are both saturated.

Lemma 2.6. (a) If G_{1} and G_{2} are finite groups with S_{i} ∈ Syl_{p}(G_{i}) for i = 1, 2, then F_{S_{1}×S_{2}}(G_{1} × G_{2}) = F_{S_{1}}(G_{1}) × F_{S_{2}}(G_{2}).

(b) For each pair of fusion systems F_{1} and F_{2} over S_{1} and S_{2},

O^{p′}(F_{1} × F_{2}) = O^{p′}(F_{1}) × O^{p′}(F_{2}) and hence Γ_{p′}(F_{1} × F_{2}) = Γ_{p′}(F_{1}) × Γ_{p′}(F_{2}).

Proof. We leave the proof of (a) as an exercise.

To prove (b), let ρ_{1} : Mor(F_{1}^{c}) ⟶ Mor((F_{1} × F_{2})^{c}) be the map that sends a morphism φ to the product φ × Id_{S_{2}}, and similarly for ρ_{2}. Then ρ_{1} and ρ_{2} send composites to composites and inclusions to inclusions, and hence by Proposition 2.4(a) induce a commutative diagram of groups and homomorphisms

\xymatrix{Aut_{F_{1}}(S_{1}) \ar[r]^{ρ_{1}^{0}} \ar@{->>}[d]_{θ_{F_{1}}^{0}} & Aut_{F_{1}×F_{2}}(S_{1} × S_{2}) \ar@{->>}[d]_{θ_{F_{1}×F_{2}}^{0}} & Aut_{F_{2}}(S_{2}) \ar[l]_{ρ_{2}^{0}} \ar@{->>}[d]_{θ_{F_{2}}^{0}} \\ Γ_{p′}(F_{1}) \ar[r]^{ρ̂_{1}} & Γ_{p′}(F_{1} × F_{2}) & Γ_{p′}(F_{2}) , \ar[l]_{ρ̂_{2}}}

where (−)^{0} means restriction to the given automorphism group. Thus

Aut_{O^{p′}(F_{1})}(S_{1}) × Aut_{O^{p′}(F_{2})}(S_{2}) = Ker(θ_{F_{1}}^{0} × θ_{F_{2}}^{0}) ≤ Ker(θ_{F_{1}×F_{2}}^{0}) = Aut_{O^{p′}(F_{1}×F_{2})}(S_{1} × S_{2})

and hence O^{p′}(F_{1}) × O^{p′}(F_{2}) ≤ O^{p′}(F_{1} × F_{2}) by Proposition 2.4(b). The opposite inclusion is clear. □

The next lemma says that when checking whether a fusion subsystem of F has index prime to p, it suffices to look at F-centric F-radical subgroups.

Lemma 2.7. Let F be a saturated fusion system over a finite p-group S, and assume F_{0} ⊆ F is a fusion subsystem over S (not necessarily saturated) such that F ⊆ ⟨F_{0}, Aut(F_{0})⟩. Then F_{0} ⊇ O_{∗}^{p′}(F) and thus has index prime to p in F. Thus

O_{∗}^{p′}(F) = ⟨O^{p′}(Aut_{F}(R)) | R ∈ F^{cr}⟩.

Proof. Assume F_{0} ⊆ F is such that F ⊆ ⟨F_{0}, Aut(F_{0})⟩. We claim that for each P, Q ≤ S and each φ ∈ Hom_{F}(P, Q), conjugation by φ sends Aut_{F_{0}}(P) onto Aut_{F_{0}}(φ(P)). This is clear when φ ∈ Hom_{F_{0}}(P, Q), and holds by definition when φ is the restriction of an element of Aut(F_{0}) ≤ Aut(S). So it holds for all φ ∈ Hom_{F}(P, Q), since φ is a composite of such morphisms.

In particular, Aut_{F_{0}}(P) ⊴ Aut_{F}(P). If P is fully normalized in F, then Aut_{F_{0}}(P) ≥ Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)) since Aut_{F_{0}}(S) ≥ Inn(S), and so Aut_{F_{0}}(P) contains the normal closure O^{p′}(Aut_{F}(P)) of Aut_{S}(P) in Aut_{F}(P). We just saw that Aut_{F_{0}}(Q) is isomorphic to Aut_{F_{0}}(P) whenever Q ∈ P^{F}, and hence Aut_{F_{0}}(P) ≥ O^{p′}(Aut_{F}(P)) for all P ≤ S. Thus F_{0} ⊇ O_{∗}^{p′}(F), and has index prime to p in F.

We now apply this to F_{0} =^{def} ⟨O^{p′}(Aut_{F}(R)) | R ∈ F^{cr}⟩. In this case,

F = ⟨Aut_{F}(P) | P ∈ F^{cr} fully normalized in F⟩

= ⟨O^{p′}(Aut_{F}(P)) · N_{Aut_{F} (P)}(Aut_{S}(P)) | P ∈ F^{cr} fully normalized in F⟩

= ⟨O^{p′}(Aut_{F}(P)), Aut_{F}(N_{S}(P)) | P ∈ F^{cr} fully normalized in F⟩ = ⟨F_{0}, Aut_{F}(S)⟩ :

the first equality by Theorem 1.3 (Alperin’s fusion theorem), the second by the Frattini argument and the Sylow axiom (Definition 1.1), the third by the extension axiom, and the fourth by a downward induction on |P | (and since N_{S}(P) > P for P < S). Also, Aut_{F}(S) ≤ Aut(F_{0}) (i.e., Aut_{F}(S) normalizes F_{0}) since each α ∈ Aut_{F}(S) permutes the members of F^{cr}. So F_{0} ⊇ O_{∗}^{p′}(F), and the opposite inclusion is clear. □

Note that by using the stronger form of Alperin’s fusion theorem stated in [AKO, Theorem I.3.5], we could replace F^{cr} by the set consisting of essential subgroups and S.

The following proposition says that restrictions of θ_{F} to certain full subcategories are also universal among strongly multiplicative maps.

Proposition 2.8. Let F be a saturated fusion system over a finite p-group S, let 𝒫 ⊆ F^{c} be a family of subgroups closed under F-conjugacy and containing F^{cr}, and let F^{𝒫} ⊆ F^{c} be the full subcategory with object set 𝒫. Then

θ_{F}^{𝒫} = θ_{F}|_{Mor(F^{𝒫} )} : Mor(F^{𝒫} ) ⟶ Γ_{p′}(F)

is universal among all strongly multiplicative maps Mor(F^{𝒫} ) ⟶ Γ.

Proof. Let θ: Mor(F^{𝒫} ) ⟶ Γ be the universal strongly multiplicative map, and let f: Γ ⟶ Γ_{p′}(F) be such that f ∘ θ = θ_{F}^{𝒫} . In particular, θ(O^{p′}(Aut_{F}(P))) = 1 for all P ∈ 𝒫 and Γ = ⟨θ(Mor(F^{𝒫} ))⟩. Also, f is onto since θ_{F}^{𝒫} is onto.

Set E = ⟨θ^{−1}(1)⟩: the smallest fusion system over S containing all morphisms in θ^{−1}(1). Thus E ⊇ O_{∗}^{p′}(F) by Lemma 2.7 and since 𝒫 ⊇ F^{cr}, and so E has index prime to p in F. By Lemma 2.2(b), E and F have the same fully normalized and fully centralized subgroups. By Lemma 2.2(c), the Frattini condition holds: for each P ≤ S and each φ ∈ Hom_{F}(P, S), there are α ∈ Aut_{F}(S) and φ_{0} ∈ Hom_{E}(P, S) such that φ = α ∘ φ_{0}.

By the Frattini condition, each φ ∈ Mor(F^{𝒫} ) is a composite of restrictions of morphisms in θ^{−1}(1) and elements of Aut_{F}(S). Hence θ(Aut_{F}(S)) = Γ. Since f is onto, to show that it is an isomorphism and hence that θ_{F}^{𝒫} is universal, it remains to show that Ker(θ|_{Aut_{F} (S)}) ≥ Aut_{F}^{0}(S). We do this by first proving that E is saturated.

By construction, E is 𝒫-generated in the sense of [5a1, Definition 2.1]. We claim that E is also 𝒫-saturated: that the Sylow and extension axioms hold for members of 𝒫. If P ∈ 𝒫 is fully normalized in E, hence in F, then Aut_{S}(P) ∈ Syl_{p}(Aut_{E}(P)) since Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)). Thus the Sylow axiom holds in E for members of 𝒫.

Fix φ ∈ Hom_{E}(P, Q), where P, Q ∈ 𝒫 and φ(P) is fully centralized in E (hence in F), and let N_{φ} ≤ N_{S}(P) be as in Definition 1.1(b). (Note that N_{φ} is the same, whether we are working in E or in F.) By the extension axiom for F, φ extends to some φ ∈ Hom_{F}(N_{φ}, S). By the Frattini condition, φ = α ∘ φ_{0}, where α ∈ Aut_{F}(S) and φ_{0} ∈ Hom_{E}(P, S). Thus

1 = θ(φ) = θ(α ∘ φ_{0}|_{P}) = θ(α) · θ(φ_{0}|_{P}) = θ(α)

where θ(φ) = φ(φ_{0}|_{P}) = 1 since φ and φ_{0}|_{P} are both in E = ⟨θ^{−1}(1)⟩. Hence α ∈ Aut_{E}(S), and so φ ∈ Hom_{E}(N_{φ}, S). The extension axiom for E thus holds for members of 𝒫, and so E is 𝒫-saturated.

If P ∈ F^{c} ∖ 𝒫 is fully normalized in F, then

Aut_{S}(P) ∩ O_{p}(Aut_{E}(P)) = Aut_{S}(P) ∩ O_{p}(Aut_{F}(P)) = O_{p}(Aut_{F}(P)) > Inn(P)

since P is not F-radical. Since each E-conjugacy class in F^{c} ∖ 𝒫 contains such subgroups, E satisfies condition (∗) in [5a1, Theorem 2.2]. Since E is 𝒫-generated and 𝒫-saturated, E is saturated by that theorem. So E ≥ O^{p′}(F), and hence

Ker(θ|_{Aut_{F} (S)}) = Aut_{E}(S) ≥ Aut_{O^{p′}(F)}(S) = Aut_{F}^{0}(S).  □

Proposition 2.8 is similar to a theorem of Grodal [Gr, Theorem 7.5.1], though that theorem is stated only when F is the fusion system of a group (and for a slightly different definition of radical subgroups).

The following very elementary family of examples shows that θ_{F}|_{F^{cr}} need not, in general, be universal among all multiplicative maps on Mor(F^{cr}): only among those that are strongly multiplicative. Recall [AKO, Definition I.4.8] that a saturated fusion system F over a finite p-group S is constrained if O_{p}(F) is F-centric.

Example 2.9. Let F be a constrained saturated fusion system over S, and set Q = O_{p}(F). Then Q ∈ F^{cr} and is contained in all other members of F^{cr}. The universal group for multiplicative maps defined on Mor(F^{cr}) is isomorphic to Aut_{F}(Q), while that for strongly multiplicative maps is isomorphic to Γ_{p′}(F) ≅ Aut_{F}(Q)/O^{p′}(Aut_{F}(Q)).

In fact, the universal group for multiplicative maps on Mor(F^{cr}) need not even be finite.

Example 2.10. Set G = SL_{2}(9) and p = 2, and choose S ∈ Syl_{2}(G). Thus S ≅ Q_{16} and contains two subgroups Q_{1}, Q_{2} ≤ S that are quaternion of order 8. Set F = F_{S}(G); then F^{cr} = {S, Q_{1}, Q_{2}}, and hence the category F^{cr} is the union of the two categories N_{F}(Q_{1})^{cr} and N_{F}(Q_{2})^{cr}, with intersection N_{F}(S)^{cr}. Since these three normalizers are all constrained, we see, using Example 2.9, that the universal group for multiplicative maps on Mor(F^{cr}) is isomorphic to

Aut_{F}(Q_{1})*_{Aut_{F} (S)}Aut_{F}(Q_{2}) ≅ Σ_{4} *_{D_{8}}Σ_{4}.

We now list some more tools that will be useful when working with fusion subsystems of index prime to p.

When E is a saturated fusion subsystem of F, O^{p′}(E) is not, in general, a fusion subsystem of O^{p′}(F). The next two lemmas describe some cases when this is, in fact, true.

Lemma 2.11. Let F be a saturated fusion system over a finite p-group S, and let E ≤ F be a saturated fusion subsystem over T ≤ S.

(a) If E^{cr} ⊆ F^{c} as sets, then O^{p′}(E) ≤ O^{p′}(F).

(b) If E = C_{F}(U) for some abelian subgroup 1 ≠ U ≤ S fully centralized in F, then E^{c} ⊆ F^{c} as sets, and hence O^{p′}(E) ≤ O^{p′}(F).

Proof. (a) Since E^{cr} ⊆ F^{c} as categories, θ_{F} restricts to a strongly multiplicative map Mor(E^{cr}) ⟶ Γ_{p′}(F). By the universal property of θ_{E}|_{Mor(E^{cr})} (Proposition 2.8), there is a (unique) homomorphism ω: Γ_{p′}(E) ⟶ Γ_{p′}(F) such that ω ∘ θ_{E}|_{Mor(E^{cr})} = θ_{F}|_{Mor(E^{cr})}. Hence

Mor(O^{p′}(E)^{cr}) = (θ_{E}|_{Mor(E^{cr})})^{−1}(1) ≤ θ_{F}^{−1}(1) = Mor(O^{p′}(F)^{c})

where the first equality holds by Lemma 2.2(a) (O^{p′}(E)^{cr} and E^{cr} have the same objects). Since each morphism in O^{p′}(E) is a composite of restrictions of morphisms in O^{p′}(E)^{cr}, we conclude that O^{p′}(E) ≤ O^{p′}(F).

(b) Assume E = C_{F}(U) and hence T = C_{S}(U), where U ≠ 1 is abelian and fully centralized in F. Fix P ∈ E^{c}; we must show that P ∈ F^{c}. Note that P ≥ C_{T} (P) ≥ U.

Choose P^{∗} ∈ P^{F} fully centralized in F and φ ∈ Iso_{F}(P, P^{∗}), and set U^{∗} = φ(U). Since U is fully centralized in F, there is ψ ∈ Hom_{F}(C_{S}(U^{∗}), T) (recall T = C_{S}(U)) such that ψ|_{U^{∗}} = φ^{−1}|_{U^{∗}} ∈ Iso_{F}(U^{∗}, U). Also, P^{∗} ≤ C_{S}(U^{∗}) since P ≤ C_{S}(U). Set R = ψ(P^{∗}); then ψφ ∈ Iso_{E}(P, R) since its restriction to U is the identity. So R ∈ P^{E} ⊆ E^{c}, hence C_{S}(R) ≤ C_{S}(U) = T, and thus C_{S}(R) = C_{T} (R) ≤ R. But then C_{S}(P^{∗}) ≤ P^{∗} since ψ(C_{S}(P^{∗})) ≤ C_{S}(R), and P, P^{∗} ∈ F^{c} since P^{∗} ∈ P^{F} is fully centralized in F. □

We will also need to deal with “subsystems of p-power index” in a few cases.

Definition 2.12 ([5a2, Definitions 2.1 & 3.1]). Let F be a saturated fusion system over a finite p-group S.

(a) The hyperfocal subgroup of F is defined as follows:

𝔥𝔶𝔭(F) = ⟨g^{−1}α(g) | g ∈ P ≤ S, α ∈ O^{p}(Aut_{F}(P))⟩.

(b) A fusion subsystem E ≤ F over T ≤ S has p-power index in F if T ≥ 𝔥𝔶𝔭(F), and for all P ≤ T, Aut_{E}(P) ≥ O^{p}(Aut_{F}(P)).

By [5a2, Theorem 4.3] or [AKO, Theorem I.7.4], for each saturated fusion system F over S and each T ≤ S containing 𝔥𝔶𝔭(F), there is a unique saturated fusion subsystem F_{T} over T of p-power index in F, and F_{T} = ⟨Inn(S), O^{p}(Aut_{F}(P)) | P ≤ T ⟩ by the second reference.

Lemma 2.13. Let E ≤ F be saturated fusion systems over finite p-groups T ≤ S, where E has p-power index in F and T is normal in S. Then O^{p′}(E) ≤ O^{p′}(F), and Γ_{p′}(F) is isomorphic to a subquotient of Γ_{p′}(E).

Proof. Set F_{0} = O^{p′}(F). Then T ≥ 𝔥𝔶𝔭(F) ≥ 𝔥𝔶𝔭(F_{0}), so by [AKO, Theorem I.7.4], there is a unique saturated fusion subsystem

E_{0} = ⟨Inn(T), O^{p}(Aut_{F_{0}}(P)) | P ≤ T ⟩ ≤ F_{0}

over T of p-power index in F_{0}. Also, E_{0} ≤ E since E = ⟨Inn(T), O^{p}(Aut_{F}(P)) | P ≤ T ⟩. We first prove that O^{p′}(E) ≤ O^{p′}(F). By [5a2, Lemma 3.4(b)],

F = ⟨O_{∗}^{p}(F), Aut_{F}(S)⟩ = ⟨O_{∗}^{p}(F), Inn(S)⟩ where O_{∗}^{p}(F) = ⟨O^{p}(Aut_{F}(P)) | P ≤ S⟩

(as fusion systems over S), where the second equality holds since Inn(S) ∈ Syl_{p}(Aut_{F}(S)). Also, T is strongly closed in O_{∗}^{p}(F) by definition of 𝔥𝔶𝔭(F) and since T ≥ 𝔥𝔶𝔭(F). So T is also strongly closed in F = ⟨O_{∗}^{p}(F), Inn(S)⟩ (recall T ⊴ S).

Let F_{0}|_{T} ⊆ F_{0} and F|_{T} ⊆ F be the full subcategories with objects the subgroups of T. Also, F = ⟨F_{0}, Aut_{F}(S)⟩ by Lemma 2.2(c) and F_{0}|_{T} = ⟨E_{0}, Aut_{F_{0}}(T)⟩ by [5a2, Lemma 3.4(b)] again, so

F|_{T} = ⟨F_{0}|_{T} , α|_{T} | α ∈ Aut_{F}(S)⟩ = ⟨E_{0}, Aut_{F}(T)⟩

since T is strongly closed. Also, Aut_{F}(T) normalizes F_{0}|_{T} since Aut_{F_{0}}(T) and Aut_{F}(S) both normalize it, and hence Aut_{F}(T) also normalizes E_{0} by definition of E_{0}. Thus Aut_{F}(T) ≤ Aut(E_{0}), so E ≤ F|_{T} ≤ ⟨E_{0}, Aut(E_{0})⟩, and E_{0} has index prime to p in E by Lemma 2.7. Since E_{0} is saturated, O^{p′}(E) ≤ E_{0} ≤ F_{0} = O^{p′}(F).

It remains to show that Γ_{p′}(F) is isomorphic to a subquotient of Γ_{p′}(E). Consider the homomorphisms

Γ_{p′}(F) = Aut_{F}(S)/Aut_{F_{0}}(S) ⟶^{f_{1}} Aut_{F}(T)/Aut_{F_{0}}(T) ≅ Out_{F}(T)/Out_{F_{0}}(T)
Γ_{p′}(E) = Out_{E}(T)/Out_{O^{p′}(E)}(T) −↠^{f_{2}} Out_{E}(T)/Out_{E_{0}}(T) ⟶_{≅}^{f_{3}} Out_{F}(T)/Out_{F_{0}}(T)

where f_{1} is induced by restriction to T (recall that T is strongly closed in F), f_{2} is the natural surjection, and f_{3} is induced by the inclusion Out_{E}(T) ≤ Out_{F}(T). Here, f_{3} is an isomorphism since Out_{E}(T) = O^{p}(Out_{F}(T)) and Out_{E_{0}}(T) = O^{p}(Out_{F_{0}}(T)) by the above descriptions of E and E_{0}, both have order prime to p, and Out_{F_{0}}(T) ≥ O^{p′}(Out_{F}(T)) since F_{0} = O^{p′}(F) ⊇ O_{∗}^{p′}(F).

If α ∈ Aut_{F}(S) is such that α|_{T} ∈ Aut_{F_{0}}(T), then by the extension axiom (Definition 1.1(b)), there is β ∈ Aut_{F_{0}}(S) such that β|_{T} = α|_{T}, so β^{−1}α is the identity on T, induces an automorphism of p-power order on S/T, and hence has p-power order and lies in Inn(S). So f_{1} is injective, and Γ_{p′}(F) is isomorphic to a subquotient of Γ_{p′}(E). □

The next lemma is useful when comparing subsystems of index prime to p in a simple group with those in its quasisimple coverings.

Lemma 2.14. Let F be a saturated fusion system over a finite p-group S. Then for each central subgroup Z ≤ Z(F), we have O^{p′}(F/Z) = O^{p′}(F)/Z and Γ_{p′}(F/Z) ≅ Γ_{p′}(F). Thus if F = F_{S}(G) for a finite group G, then for Z ≤ Z(G), we have Γ_{p′}(F_{SZ/Z}(G/Z)) ≅ Γ_{p′}(F).

Proof. Assume P ∈ F^{cr}, and consider the subgroup Δ = {α ∈ Aut_{F}(P) | [α, P] ≤ Z}. All elements of Aut_{F}(P) restrict to the identity on Z, so Δ ⊴ Aut_{F}(P), and each α ∈ Δ induces the identity on Z and on S/Z and hence has p-power order [G, Corollary 5.3.3]. Thus Δ ≤ O_{p}(Aut_{F}(P)) = Inn(P) since P is F-radical. So if gZ ∈ C_{S/Z}(P/Z), then c_{g} ∈ Inn(P), and g ∈ C_{S}(P) ≤ P. Since this applies to all members of P^{F}, we see that P/Z ∈ (F/Z)^{c}.

Consider the following diagram:

\xymatrix{Mor(F^{cr}) \ar@{->>}[r]^{θ_{F}^{cr}} \ar[d]_{P ↦P/Z} & Γ_{p′}(F) \ar@{->>}[d]^{f} \\ Mor((F/Z)^{c}) \ar@{->>}[r]^{θ_{F/Z}} & Γ_{p′}(F/Z)}

where θ_{F}^{cr} is the restriction of θ_{F}, and where f exists (making the square commute) by the universality of θ_{F}^{cr} (Proposition 2.8). This commutativity implies that (O^{p′}(F))^{cr}/Z ≤ O^{p′}(F/Z), and hence that O^{p′}(F)/Z ≤ O^{p′}(F/Z) by Alperin’s fusion theorem (Theorem 1.3). The opposite inclusion holds since O^{p′}(F)/Z is normal of index prime to p in F/Z.

We next show that f is an isomorphism. Let α ∈ Aut_{F}(S) be such that θ_{F}(α) ∈ Ker(f), and let α′ ∈ Aut_{F/Z}(S/Z) be the induced automorphism of S/Z. Thus

α′ ∈ θ_{F/Z}^{−1}(1) = Aut_{O^{p′}(F/Z)}(S/Z) = Aut_{O^{p′}(F)/Z}(S/Z),

so there is β ∈ Aut_{O^{p′}(F)}(S) such that β^{−1}α induces the identity on S/Z. But β^{−1}α also induces the identity on Z ≤ Z(F), so it has p-power order, and θ_{F}(α) = θ_{F}(β) = 1 since Γ_{p′}(F) has order prime to p. This proves that Ker(f) = 1, and hence that f is an isomorphism.

The last statement is now immediate. □

The next lemma deals with normal subgroups of subsystems of index prime to p.

Lemma 2.15. Assume E ≤ F are saturated fusion systems over S, where E has index prime to p in F. Then for Q ⊴ S, Q ⊴ F if and only if Q ⊴ E and Q is weakly closed in F.

Proof. Assume Q ⊴ E and Q is weakly closed in F. By the Frattini condition (Lemma 2.2(c)), for each φ ∈ Hom_{F}(P, R), there are α ∈ Aut_{F}(S) and φ_{0} ∈ Hom_{E}(P, α^{−1}(R)) such that φ = (α|_{α^{−1}(R)}) ∘ φ_{0}. Then α(Q) = Q since Q is weakly closed, and φ_{0} extends to a morphism φ_{0} ∈ Hom_{E}(P Q, α^{−1}(RQ)) such that φ_{0}(Q) = Q since Q ⊴ E. So (α|_{α^{−1}(RQ)})∘φ_{0} ∈ Hom_{F}(P Q, RQ) extends φ. This proves that Q ⊴ F, and the converse is clear. □

The next lemma, which gives one criterion for proving that O^{p′}(F) is simple, will be applied in Section 4.

Lemma 2.16. Fix a prime p and a finite p-group S, let F be a saturated fusion system over S, and assume that no nontrivial proper subgroup of S is strongly closed in F. Then either O^{p′}(F) is simple; or there are subgroups T_{1}, …, T_{k} < S (for 2 ≤ k ≤ |Γ_{p′}(F)|) that are strongly closed in O^{p′}(F) and Aut_{F}(S)-conjugate to each other, and such that S = T_{1} × ⋯ × T_{k}. In particular, F is simple if F = O^{p′}(F).

Proof. Assume O^{p′}(F) is not simple, and let 1 ≠ E ⊴ O^{p′}(F) be a nontrivial proper normal subsystem over 1 ≠ T ⊴ S. In particular, T is strongly closed in O^{p′}(F) (one of the conditions for E to be normal). If T = S, then E has index prime to p in O^{p′}(F) by [AOV, Lemma 1.26], which is impossible since E < O^{p′}(F) and O^{p′}(O^{p′}(F)) = O^{p′}(F). So T < S.

Thus there are proper nontrivial subgroups of S strongly closed in O^{p′}(F), and so O^{p′}(F) < F by assumption. Let 1 ≠ U < S be minimal among all such subgroups. Set m = |Γ_{p′}(F)| = |Aut_{F}(S)/Aut_{O^{p′}(F)}(S)| (Proposition 2.4), choose coset representatives α_{1}, …, α_{m} ∈ Aut_{F}(S) for Aut_{O^{p′}(F)}(S), and set U_{i} = α_{i}(U). Then each U_{i} is a minimal strongly closed subgroup in O^{p′}(F), and Û = U_{1} ⋯ U_{m} is strongly closed in O^{p′}(F) by [A2, Theorem 2] or [Cr, Theorem 5.22]. Also, Û is normalized by Aut_{F}(S) by construction. Since each morphism in F is the composite of a morphism in O^{p′}(F) followed by the restriction of some element of Aut_{F}(S), Û is strongly closed in F. So Û = S by assumption.

For each 1 ≤ i ≤ m − 1, U_{1} ⋯ U_{i} is strongly closed in O^{p′}(F) by [A2, Theorem 2] again, so (U_{1} ⋯ U_{i}) ∩ U_{i+1} is also strongly closed. So by the minimality of U among strongly closed subgroups, either (U_{1} ⋯ U_{i}) ∩ U_{i+1} = 1 or U_{i+1} ≤ U_{1} ⋯ U_{i}. Hence there is a subset I ⊆ {1, …, m} such that S is the direct product of the U_{i} for i ∈ I, and |I| ≥ 2 since U < S. □

3. Fusion systems with centric and weakly closed subgroups

We saw in the last section that when F is a saturated fusion system over a finite p-group S, its subsystems and extensions of index prime to p are determined by their automizers on S. In this section, we show that in fact, it suffices to consider the automizers of some subgroup A ⊴ S that is F-centric and weakly closed in F. In particular, this works in many cases when A is the p-power torsion in the maximal torus of a finite group of Lie type in defining characteristic different from p, and allows us to work with Aut_{F}(A) rather than Aut_{F}(S).

We begin with three lemmas that describe the relationship between Aut_{F}(A) and Γ_{p′}(F) for such a subgroup A ⊴ S, and show how this can be used to get upper and lower bounds for |Γ_{p′}(F)|.

Lemma 3.1. Let F be a saturated fusion system over a finite p-group S, and assume A ⊴ S is a subgroup that is F-centric and weakly closed in F. Let

θ_{F}^{(A)} : Aut_{F}(A) ⟶ Γ_{p′}(F)

be the restriction of θ_{F}: Mor(F^{c}) ⟶ Γ_{p′}(F) to Aut_{F}(A). Then

(a) θ_{F}^{(A)} is surjective and Ker(θ_{F}^{(A)}) = Aut_{O^{p′}(F)}(A);

(b) Γ_{p′}(F) = 1 if O^{p′}(Aut_{F}(A)) = Aut_{F}(A); and

(c) for E ≤ F of index prime to p, E = F if and only if Aut_{E}(A) = Aut_{F}(A).

Proof. Since A is weakly closed, each α ∈ Aut_{F}(S) restricts to an element of Aut_{F}(A), and so the surjectivity of θ_{F}^{(A)} follows from that of θ_{F}|_{Aut_{F} (S)} (Proposition 2.4(a)). Point (b) is then immediate.

Set E_{H} = ⟨θ_{F}^{−1}(H)⟩ ≤ F for each subgroup H ≤ Γ_{p′}(F). Then Aut_{E_{H}}(A) = (θ_{F}^{(A)})^{−1}(H), since each α ∈ Aut_{E_{H}}(A) is a composite of restrictions of morphisms in θ_{F}^{−1}(H) (and inclusions are sent to the identity). In particular, Aut_{O^{p′}(F)}(A) = Ker(θ_{F}^{(A)}) (the case H = 1), finishing the proof of (a). For each E ≤ F of index prime to p, E = E_{H} for some H ≤ Γ_{p′}(F) by Proposition 2.4(b), and from the surjectivity of θ_{F}^{(A)}, we see that Aut_{E}(A) = Aut_{F}(A) implies H = Γ_{p′}(F), and hence E = F. □

One immediate consequence of Lemma 3.1 is the following upper bound for |Γ_{p′}(F)|.

Lemma 3.2. Let F be a saturated fusion system over a finite p-group S, and let E ≤ F be a saturated fusion subsystem over T ≤ S. Assume that O^{p′}(E) ≤ O^{p′}(F), that A ⊴ S is F-centric and weakly closed in F, and also that A ≤ T. Then Aut_{O^{p′}(E)}(A) is contained in the kernel of the homomorphism θ_{F}^{(A)} from Aut_{F}(A) onto Γ_{p′}(F), and hence

|Γ_{p′}(F)| ≤ |Aut_{F}(A) : O^{p′}(Aut_{F}(A)) · Aut_{O^{p′}(E)}(A)|.

In particular, O^{p′}(F) = F if Aut_{F}(A) = O^{p′}(Aut_{F}(A)) · Aut_{O^{p′}(E)}(A).

Proof. Since O^{p′}(E) ≤ O^{p′}(F), we have Aut_{O^{p′}(E)}(A) ≤ Aut_{O^{p′}(F)}(A) = Ker(θ_{F}^{(A)}) by Lemma 3.1(a). The other claims now follow from the surjectivity of θ_{F}^{(A)} and since p ∤ |Γ_{p′}(F)|. □

If A is an abelian subgroup that is F-centric and weakly closed, then we can say more. The next lemma will be our main tool for getting lower bounds on |Γ_{p′}(F)| when it is nontrivial.

Lemma 3.3. Let F be a saturated fusion system over the p-group S, and let A ⊴ S be an abelian subgroup which is F-centric and weakly closed in F. Set

X = {t ∈ A | t^{F} ⊆ A},

and assume X ∩ Z(S) ≠ {1}. Choose 1 ≠ Z ≤ ⟨X ∩ Z(S)⟩, set E = C_{F}(Z) and E_{0} = O^{p′}(E), and let N_{0} ≤ N ⊴ Aut_{F}(A) be the normal closures of Aut_{E_{0}}(A) and Aut_{E}(A) in Aut_{F}(A). Then the following hold.

(a) There is a surjective homomorphism f: Γ_{p′}(F) ⟶ Aut_{F}(A)/N with the following property: for each morphism φ ∈ Hom_{F}(P, Q) in F^{c}, there is α ∈ Aut_{F}(A) such that α|_{Z} = φ|_{Z}, and for each such α, f(θ_{F}(φ)) = αN.

(b) We have O^{p′}(Aut_{F}(A))N_{0} ≤ Ker(θ_{F}^{(A)}) ≤ N.

Proof. Since E_{0} ≤ O^{p′}(F) by Lemma 2.11(b), Aut_{E_{0}}(A) ≤ Aut_{O^{p′}(F)}(A) = Ker(θ_{F}^{(A)}), and hence N_{0} ≤ Ker(θ_{F}^{(A)}). Since Im(θ_{F}^{(A)}) has order prime to p, this proves the first inclusion in (b).

Before proving the other claims, we check that

for each P ≤ A and φ ∈ Hom_{F}(P, A), φ extends to some φ ∈ Aut_{F}(A).  (3.1)

To see this, set Q = φ(P) ≤ A, choose R ∈ P^{F} = Q^{F} that is fully centralized in F, and fix χ ∈ Iso_{F}(Q, R). By the extension axiom, χ extends to χ ∈ Hom_{F}(C_{S}(Q), S) and χφ extends to φ ∈ Hom_{F}(C_{S}(P), S), and χ(A) = A = φ(A) since A is weakly closed. Then χ^{−1}φ|_{A} ∈ Aut_{F}(A), and (χ^{−1}φ)|_{P} = χ^{−1}(χφ) = φ.

Let θ_{0} : Aut_{F}(A) ⟶ Aut_{F}(A)/N be the natural map. We first extend θ_{0} to a strongly multiplicative map defined on Mor(F^{c}) (see Definition 2.3).

Fix φ ∈ Mor_{F}(P, Q), where P, Q ∈ F^{c}. In particular, P, Q ≥ Z(S) ≥ Z, and φ(Z) ⊆ φ(X ∩ Z(S)) ⊆ A by definition of X. By (3.1), φ|_{Z} is the restriction of some automorphism α ∈ Aut_{F}(A). If α′ ∈ Aut_{F}(A) is another automorphism for which α′|_{Z} = φ|_{Z}, then α^{−1}α′ ∈ C_{Aut_{F} (A)}(Z) ≤ N, and so we can define θ(φ) = θ_{0}(α) = θ_{0}(α′). We thus have a well defined map of sets θ: Mor(F^{c}) ⟶ Aut_{F}(A)/N, and θ|_{Aut_{F} (A)} = θ_{0}. Clearly, θ sends all inclusions to the identity, and it is surjective since θ_{0} is surjective. Also, Aut_{F}(A)/N has order prime to p since N ≥ Aut_{S}(A) ∈ Syl_{p}(Aut_{F}(A)), and hence θ(O^{p′}(Aut_{F}(P))) = 1 for each P ∈ F^{c}.

It remains to check that θ sends composites in Mor(F^{c}) to products in Aut_{F}(A)/N. Assume φ ∈ Hom_{F}(P, Q) and ψ ∈ Hom_{F}(Q, R), where P, Q, R ∈ F^{c}, and set P^{∗} = ⟨P ∩ X⟩ and Q^{∗} = ⟨Q ∩ X⟩. Then φ(P ∩ X) ⊆ Q ∩ X, so φ(P^{∗}) ≤ Q^{∗}. By (3.1), there are α, β ∈ Aut_{F}(A) which are extensions of φ|_{P^{∗}} and ψ|_{Q^{∗}}, respectively. Then β ∘ α is an extension of (ψ ∘ φ)|_{P^{∗}}, hence of (ψ ∘ φ)|_{Z}, and thus θ(ψ ∘ φ) = β ∘ α = θ(ψ) ∘ θ(φ).

Since θ_{F} is universal among strongly multiplicative maps (Proposition 2.4(a)), there is a unique homomorphism f: Γ_{p′}(F) ⟶ Aut_{F}(A)/N such that θ = f ∘ θ_{F}, and f is surjective since θ is surjective. This proves (a). Also, N = Ker(θ_{0}) ≥ Ker(θ_{F}^{(A)}), which proves the second inclusion in (b). □

We next note that two extensions of index prime to p of the same fusion system over S are equal, or at least isomorphic, if they have the same automizer on some centric and weakly closed subgroup A ⊴ S.

Lemma 3.4. Let F_{1} and F_{2} be saturated fusion systems over the same finite p-group S. Assume that A ⊴ S is F_{i}-centric and weakly closed in F_{i} for i = 1, 2, and also that

O^{p′}(F_{1}) = O^{p′}(F_{2})  and  Aut_{F_{1}}(A) = Aut_{F_{2}}(A).

Then there is β ∈ Aut(O^{p′}(F_{2})) such that β|_{A} = Id_{A} and F_{1} = ^{β}F_{2}. If in addition, H^{1}(Out_{O^{p′}(F_{1})}(A); Z(A)) = 0, then F_{1} = F_{2}.

Proof. Set F_{0} = O^{p′}(F_{1}) = O^{p′}(F_{2}), and consider the subgroup

H = {β ∈ Aut(S) | β|_{A} = Id}.

Each β ∈ H induces the identity on S/A since C_{S}(A) ≤ A, so H is a p-group by, e.g., [G, Corollary 5.3.3]. Each α ∈ Aut_{F_{i}}(S) (for i = 1, 2) normalizes A since A is weakly closed, and hence also normalizes H. For each α ∈ Aut_{F_{1}}(S), α|_{A} ∈ Aut_{F_{1}}(A) = Aut_{F_{2}}(A) and normalizes Aut_{S}(A), and hence extends to some α′ ∈ Aut_{F_{2}}(S) by the extension axiom (Definition 1.1(b)). Then α′ ∈ αH, and this together with a similar argument in the opposite direction shows that Aut_{F_{1}}(S)H = Aut_{F_{2}}(S)H. Set G = Aut_{F_{1}}(S)H = Aut_{F_{2}}(S)H.

For i = 1, 2, Inn(S)H ⊴ Aut_{F_{i}}(S)H since Inn(S) ⊴ Aut_{F_{i}}(S) and both normalize H, and Aut_{F_{i}}(S)H/Inn(S)H ≅ Aut_{F_{i}}(S)/Inn(S) has order prime to p by the Sylow axiom (Definition 1.1(b) again). So by the Schur-Zassenhaus theorem (see [G, Theorem 6.2.1(i)]), all splittings of the extension

1 ⟶ Inn(S)H/Inn(S) ⟶ G/Inn(S) ⟶ G/Inn(S)H ⟶ 1

are conjugate. Since Aut_{F_{1}}(S)/Inn(S) and Aut_{F_{2}}(S)/Inn(S) are two such splittings (since Aut_{F_{i}}(S) ∩ H ≤ Inn(S) by the Sylow axiom), there is β ∈ H such that ^{β}(Aut_{F_{2}}(S)) = Aut_{F_{1}}(S). Thus β|_{A} = Id_{A} and

Aut_{F_{1}}(S) = ^{β}(Aut_{F_{2}}(S)) = Aut_{^{β}F_{2}}(S),

and so F_{1} = ^{β}F_{2} by Lemma 2.5 (and since O^{p′}(F_{1}) = F_{0} = O^{p′}(F_{2})).

Now assume that H^{1}(Out_{F_{0}}(A); Z(A)) = 0, and let G be a model for N_{F_{0}}(A) (see [AKO, Theorem I.4.9]). Thus A ⊴ G, S ∈ Syl_{p}(G), O_{p′}(G) = 1, and F_{S}(G) = N_{F_{0}}(A). Then C_{G}(A) ≤ A, so G/A ≅ Out_{F_{0}}(A). Since β ∈ Aut(F_{0}) ≤ Aut(S) and β|_{A} = Id_{A}, β is also an automorphism of N_{F_{0}}(A) and extends to an automorphism β̂ ∈ Aut(G) by the uniqueness condition in the model theorem [AKO, Lemma II.4.3(a)]. Also, β̂ induces the identity on G/A since β̂|_{A} = Id and C_{G}(A) ≤ A, and β̂ is conjugation by some a ∈ Z(A) since H^{1}(Out_{F_{0}}(A); Z(A)) = 0 (see, e.g., [OV, Lemma 1.2]). But then β ∈ Inn(S) ≤ Aut(F_{2}), and so F_{1} = ^{β}F_{2} = F_{2}. □

The last lemma implies as a special case that when G is a finite group of Lie type in characteristic different from p, adding diagonal automorphisms of order prime to p doesn’t change its p-fusion system.

Lemma 3.5. Let p be a prime, and let G ⊴ Ĝ be finite groups such that p ∤ |O^{p}(Ĝ/G)|. Choose Ŝ ∈ Syl_{p}(Ĝ), set S = Ŝ∩G ∈ Syl_{p}(G), and set F = F_{S}(G) and F̂ = FŜ(Ĝ). Assume A ⊴ S is F-centric and weakly closed in F, and is such that Ĝ = GCĜ(A). Then F is normal of p-power index in F̂, and F = F̂ if p ∤ |Ĝ/G|.

Proof. Set G = G·O^{p}(Ĝ). Then S ∈ Syl_{p}(G) since p ∤ |G/G|, and F_{S}(G) is normal of p-power index in F̂. It remains to show that F = F_{S}(G). Since F is normal of index prime to p in F_{S}(G), it suffices by Lemma 3.1(c) to show that Aut_{F}(A) = Aut_{G}(A), and this holds since N_{G}(A) = N_{G}(A)C_{G}(A). □

4. Simplicity of fusion systems of known simple groups

We are now ready to explicitly describe O^{p′}(F) and Γ_{p′}(F), when F is the fusion system of a known simple group and S ⋬ F.

When r is a prime, let 𝔏𝔦𝔢(r) denote the class of finite groups of Lie type in defining characteristic r. Since this will be used only when restricting attention to simple groups, we needn’t go into details as to exactly which groups are included in this class.

In most cases, the simplicity of F was determined by Aschbacher in [A2, Chapter 16]. His conclusions are summarized in the following proposition.

Proposition 4.1. Fix a prime p, a known simple group G, and S ∈ Syl_{p}(G). Set F = F_{S}(G), and assume that S ⋬ F.

(a) If G ≅ A_{n}, then O^{p′}(F) = F if n ≡ 0, 1 (mod p), and |Γ_{p′}(F)| = 2 otherwise. In all cases, O^{p′}(F) is simple, and is realized by A_{n′}, where n′ = p · [\frac{n}{p}].

(b) If G ∈ 𝔏𝔦𝔢(p), or if p = 2 and G ≅ ^{2}F_{4}(2)′, then G has Lie rank at least two, O^{p′}(F) = F, and F is simple.

(c) If G is a sporadic simple group, then either F is simple, or O^{p′}(F) has index 2 in F and is simple, and is realized by a known simple group H. See Table 4.1 for details in the individual cases.

G  p = 2  p = 3  p = 5  p = 7  p = 11  p = 13
M_{11} simple  ab  ab  ab
M_{12} simple  simple  ab  ab
M_{22} simple  ab  ab  ab  ab
M_{23} simple  ab  ab  ab  ab
M_{24} simple  M_{12} : 2  ab  ab  ab
J_{1}  ab  ab  ab  ab  ab
J_{2}  simple  S ⊴ F  ab  ab
J_{3}  simple  S ⊴ F  ab
J_{4}  simple ^{2}F_{4}(2)′ : 2  ab  ab  S ⊴ F
Co_{3} simple  simple  S ⊴ F  ab  ab
Co_{2} simple  simple  S ⊴ F  ab  ab
Co_{1} simple  simple  simple  ab  ab
HS simple  ab  S ⊴ F  ab  ab
McL simple  simple  S ⊴ F  ab  ab
Suz simple  simple  ab  ab  ab
He simple  M_{12} : 2  ab  simple
Ly simple  simple  simple  ab  ab
Ru simple ^{2}F_{4}(2)′ : 2 SL_{3}(5) : 2  ab  ab
O’N simple  ab  ab  simple  ab
Fi_{22} simple  simple  ab  ab  ab  ab
Fi_{23} simple  simple  ab  ab  ab  ab
Fi′_{24} simple  simple  ab  simple  ab  ab
F_{5}  simple  simple  simple  ab  ab
F_{3}  simple  simple  simple  ab  ab
F_{2}  simple  simple  simple  ab  ab  ab
F_{1}  simple  simple  simple  simple  ab  simple

Table 4.1. In all cases, G is a sporadic simple group, S ∈ Syl_{p}(G), and F = F_{S}(G). An entry “H : 2” means that O^{p′}(F) is simple, has index 2 in F, and is realized by the simple group H. A blank entry means that p ∤ |G|, and “ab” means that S is abelian. A box indicates that S is extraspecial of order p^{3} and exponent p.

Proof. (a) See [A2, 16.5] or [AOV, Proposition 4.9].

(b) Assume G ∈ 𝔏𝔦𝔢(p), or p = 2 and G ≅ ^{2}F_{4}(2)′. Then G has Lie rank at least 2 by [A2, Theorem 15.6.b], and so F is simple by [A2, 16.3].

(c) Assume G is sporadic. If p = 2, then G ≅ J̸_{1} since S ⋬ F, and hence F is simple in all cases by [A2, 16.8]. If p is odd, then since S ⋬ F, G is not “p-Goldschmidt” in the terminology of [A2], and hence is not one of the groups listed in [A2, Theorem 15.6]. By [A2, 16.10], either F is simple, or (G, p) is one of the pairs listed there (and marked as such in Table 4.1), and F is realized by the almost simple group given in the same reference. In particular, when F is not simple, O^{p′}(F) is realized by one of the simple groups M_{12} or 2F_{4}(2)′ (when p = 3) or SL_{3}(5) (when p = 5), and hence is simple by [A2, 16.10] again or by (b), respectively.

Note, however, the following two errors in the statement of [A2, Theorem 15.6]. When p = 3 and G ≅ He, F is isomorphic to the fusion system of M_{24} and hence that of Aut(M_{12}): this follows from the tables on pp. 47–48 of [RV]. The error in the proof in [A2] came from mis-identifying D = N_{G}(S) (see line 3 on p. 102 in [A2]).

When p = 5 and G ≅ Co_{1}, F is simple by Theorem 2.8 and Table 2.2 in [OR]. In this case, the error in the proof of [A2, 16.9.5] occurred because the F-radical subgroups of order 5^{2} were overlooked.

In addition, when p = 3 and G ≅ Fi′_{24}, argument (ii) on p. 101 in [A2] cannot be used to prove that O^{3′}(F) = F, since there is no maximal subgroup H ≤ G such that O_{3}(H) = F^{∗}(H), H = O^{3′}(H), and H ≥ N_{G}(S). Instead, argument (iii) (on the same page) can be applied. □

It remains to consider the cases where G ∈ 𝔏𝔦𝔢(q) for q ≠ p, using Lemmas 2.11, and 3.2, and 3.3. We first look at the case p = 2.

Proposition 4.2. Let G be a finite simple group of Lie type in odd characteristic, fix S ∈ Syl_{2}(G), and set F = F_{S}(G). Assume S ⋬ F. Then O^{2′}(F) = F, and F is simple.

Proof. We will prove that O^{p′}(F) = F. Once we know this, then since no proper nontrivial subgroup of S is strongly closed with respect to G by [FF, Theorem 1.1], it follows by Lemma 2.16 that F is simple.

By [BMO2, Proposition 6.2], we can assume that G satisfies Case (III.1) of [BMO2, Hypotheses 5.1]. Hence G ≅ 𝔾(q) for q ≡ 1 (mod 4) (i.e., a Chevalley group), or G ≅ ^{2}𝔾(q) for 𝔾 = A_{n}, D_{n}, or E_{6} and q ≡ 1 (mod 4). So by [BMO2, Lemma 5.3], there is a centric abelian subgroup A ⊴ S such that Aut_{G}(A) is generated by reflections; in particular, such that O^{2′}(Aut_{G}(A)) = Aut_{G}(A). (These results in [BMO2] are stated for the groups of universal type, but they carry over easily to the adjoint case.) By [BMO2, Proposition 5.13], in all cases except when G ≅ Sp_{2n}(q) for some n ≥ 2 and q ≡ 5 (mod 8), A is weakly closed in F, and hence Γ_{p′}(F) = 1 and O^{2′}(F) = F by Lemma 3.1(b).

Now assume that G ≅ PSp_{2n}(q), where n ≥ 2 and q ≡ 5 (mod 8), and set G̃ = Sp_{2n}(q). More generally, we use tildes for subgroups and elements of G̃, and the same symbol without tilde for their images in G. Thus S̃ ∈ Syl_{2}(G̃) and also F̃ = FS̃(G̃). Let (V, 𝔟) be the symplectic space on which G̃ acts as the group of isometries, choose an orthogonal decomposition V = V_{1} ⊥ ⋯ ⊥ V_{n} where dim_{𝔽_{q}}(V_{i}) = 2 for each i, and let z̃_{i} ∈ G̃ act via −Id on V_{i} and the identity on the other summands. Set Z̃ = ⟨z̃_{1}, …, z̃_{n}⟩: then CG̃(Z̃) is a product of n copies of Sp_{2}(q) ≅ SL_{2}(q) and NG̃(Z̃) ≅ Sp_{2}(q) ≀ Σ_{n}. We can assume that S ∈ Syl_{2}(G) was chosen so that S̃ ≤ NG̃(Z̃) (hence so that Z̃ ≤ S̃).

Set B̃ = CS̃(Z̃) ≅ (Q_{8})^{n}. Each g ∈ z̃F̃ ; 1 acts on V with 2-dimensional (−1)-eigenspace, hence centralizes the z_{i} or permutes them via a 2-cycle, and no element that permutes them via a 2-cycle can be a square in S̃. So each φ ∈ HomF̃(B̃, S̃) sends Z̃ to itself, and thus sends B̃ = CS̃(Z̃) to itself. This proves that B̃ is weakly closed in F̃.

Set W̃ = ⟨z̃_{1}z̃_{2}⟩. We can assume S̃ (or the ordering of the z̃_{i}) was chosen so that W̃ is fully centralized in F̃. Set Ẽ = CF̃(W̃): the fusion system of CG̃(W̃) ≅ Sp_{4}(q) × Sp_{2n−4}(q). By Lemma 2.6, Ẽ = Ẽ_{1} × Ẽ_{2} and O^{2′}(Ẽ) = O^{2′}(Ẽ_{1}) × O^{2′}(Ẽ_{2}), where Ẽ_{1} and Ẽ_{2} are the fusion systems of Sp_{4}(q) and Sp_{2n−4}(q), respectively. The fusion system of PSp_{4}(q) is simple by [O2, Proposition 5.1], so O^{2′}(Ẽ_{1}) = Ẽ_{1} by Lemma 2.14, and hence Aut_{O^{2′}(Ẽ)}(B̃) contains (C_{3} ≀ Σ_{2}) × 1 as a subgroup of AutF̃(B̃) ≅ C_{3} ≀ Σ_{n}. Thus Aut_{O^{2′}(Ẽ)}(B̃) and O^{2′}(AutF̃(B̃)) generate AutF̃(B̃), so Γ_{p′}(F̃) = 1 by Lemma 3.2, and hence Γ_{p′}(F) = 1 by Lemma 2.14 again. □

When handling the cases where p is odd, the following list of isomorphisms between fusion systems of groups of Lie type will be helpful. It is convenient to write G ∼_{p} H to mean that G and H are finite groups whose p-fusion systems are isomorphic.

For each prime p and each q prime to p, ord_{p}(q) denotes the multiplicative order of q modulo p.

Lemma 4.3. Fix an odd prime p and a prime power q such that p ∤ q. Then each of the following inclusions of finite groups induces an isomorphism of p-fusion systems:

(a) GL_{n}(q) ≤ GL_{n+1}(q) when n ≥ 2 and ord_{p}(q) ∤ (n + 1);

(b) Sp_{2n}(q) ≤ GL_{2n}(q) (all n ≥ 1) when ord_{p}(q) is even;

(c) SO_{2n+1}(q) ≤ GL_{2n+1}(q) (all n ≥ 1) when ord_{p}(q) is even;

(d) GO_{2n}^{ε}(q) ≤ GL_{2n}(q) when ε = ±1, n ≥ 1, ord_{p}(q) is even, and q^{n} ≢ −ε (mod p); and

(e) SO_{2n−1}(q) ≤ SO_{2n}^{ε}(q) when ε = ±1, n ≥ 2, and q^{n} ≢ ε (mod p).

Proof. By [BMO1, Lemma A.2], for each of the inclusions (a,b,c,d,e), the p-fusion system of the subgroup is a full subcategory of the p-fusion system of the larger group. So it remains only to check that each subgroup has index prime to p in the larger group, and this holds since by the well known formulas for the orders of these groups,

v_{p}(|GL_{n+1}(q) : GL_{n}(q)|) = v_{p}(q^{n+1} − 1)
v_{p}(|GL_{2n}(q) : Sp_{2n}(q)|) = ∑_{i=1}^{n} v_{p}(q^{2i−1} − 1)
v_{p}(|GL_{2n+1}(q) : SO_{2n+1}(q)|) = ∑_{i=1}^{n+1} v_{p}(q^{2i−1} − 1)
v_{p}(|GL_{2n}(q) : GO_{2n}^{ε}(q)|) = v_{p}(q^{n} + ε) · ∑_{i=1}^{n} v_{p}(q^{2i−1} − 1)
v_{p}(|SO_{2n}^{ε}(q) : SO_{2n−1}(q)|) = v_{p}(q^{n} − ε)

(see, e.g., [Ta, pp. 19, 70, 141]). □

Lemma 4.4. Fix an odd prime p and a prime power q such that p ∤ q. Let q^{∨} be a prime power such that q^{∨} ≡ −q (mod p) and v_{p}((q^{∨})^{p−1} − 1) = v_{p}(q^{p−1} − 1). Then

(a) PSU_{n}(q) ∼_{p} PSL_{n}(q^{∨}) for all n;

(b) Ω_{2n}^{ε}(q) ∼_{p} Ω_{2n}^{(−1)^{n}ε}(q^{∨}) for all ε ∈ {±1} and all n;

(c) E_{6}(q) ∼_{p} ^{2}E_{6}(q^{∨}) ∼_{p} F_{4}(q^{∨}); and

(d) 𝔾(q) ∼_{p} 𝔾(q^{∨}) when 𝔾 = Ω_{2n+1} or Sp_{2n} for n ≥ 2, or 𝔾 = G_{2}, F_{4}, E_{7}, or E_{8}.

Proof. These are all special cases of [BMO1, Theorem A(c,d)], together with [BMO1, Example 4.4(b)] (for the second equivalence in (c)). □

We now look at the classical groups when p is odd. We use the following, standard notation to describe certain automizers. For a given prime p, k | m | (p − 1) and n ≥ 1, and a commutative ring R whose group of m-th roots of unity is cyclic of order m, define G(m, k, n) ≤ GL_{n}(R) as follows:

G(m, k, n) = {diag(u_{1}, …, u_{n}) ∈ GL_{n}(R) | u_{i}^{m} = 1 ∀ i, (u_{1} ⋯ u_{n})^{m/k} = 1} · 𝔓𝔢𝔯𝔪(n)

where 𝔓𝔢𝔯𝔪(n) ≅ Σ_{n} is the group of permutation matrices: the matrices that permute the canonical basis. Thus, for example, G(m, 1, n) ≅ C_{m} ≀ Σ_{n}, and G(m, m, n) ≅ (C_{m})^{n−1} ⋊ Σ_{n}. This notation will be used in particular when R = ℤ/p^{ℓ} (some ℓ ≥ 1) and m | (p − 1).

Proposition 4.5. Fix an odd prime p, and let G be a finite simple linear, unitary, symplectic, or orthogonal group in defining characteristic different from p. Fix S ∈ Syl_{p}(G), assume S is nonabelian, and set F = F_{S}(G). Then O^{p′}(F) is simple, and Γ_{p′}(F) ≅ Aut_{F}(S)/Aut_{O^{p′}(F)}(S) is as follows:

(a) If G ≅ PSL_{n}^{ε}(q) for some ε ∈ {±1} and n ≥ 2, then Γ_{p′}(F) is cyclic of order ord_{p}(εq).

(b) If G ≅ PSp_{2n}(q) or Ω_{2n+1}(q), then Γ_{p′}(F) is cyclic of order lcm(2, ord_{p}(q)).

(c) If G ≅ PΩ_{2n}^{ε}(q) where ε = ±1 and q^{n} ≢ ε (mod p), then Γ_{p′}(F) is cyclic of order lcm(2, ord_{p}(q)).

(d) If G ≅ PΩ_{2n}^{ε}(q) where ε = ±1 and q^{n} ≡ ε (mod p), then Γ_{p′}(F) is cyclic of order \frac{1}{2} · lcm(2, ord_{p}(q)). If, in addition, ord_{p}(q) is even, then F is isomorphic to a normal subsystem of index 2 in the p-fusion system of SL_{n}(q).

In each of the above cases, there is an abelian subgroup A ⊴ S that is F-centric and weakly closed in F. More precisely, if we define m, μ, θ, and κ as in Table 4.2 and set ℓ = v_{p}(q^{μ}−θ), then A has exponent p^{ℓ}, and A, Aut_{F}(A), and Aut_{O^{p′}(F)}(A) are as described in Table 4.2.

case  m  μ  θ  κ  Aut_{F}(A) Aut_{O^{p′}(F)}(A)
(a) ord_{p}(εq)  m  1  [n/μ]  C_{μ} ≀ Σ_{κ}  G(μ, μ, κ)
(b) ord_{p}(q) lcm(2, m) (−1)^{m+1}  [2n/μ]  C_{μ} ≀ Σ_{κ}  G(μ, μ, κ)
(c) ord_{p}(q) lcm(2, m) (−1)^{m+1} [2(n − 1)/μ] C_{μ} ≀ Σ_{κ}  G(μ, μ, κ)
(d) ord_{p}(q) lcm(2, m) (−1)^{m+1}  [2n/μ]  G(μ, 2, κ)  G(μ, μ, κ)
Table 4.2. In all cases except (a) when p | (q − ε), A ≅ (C_{p^{ℓ}})^{κ}.

Proof. We first show that O^{p′}(F) is simple, assuming the other claims hold. If G = PSL_{n}^{ε}(q) where ε = ±1 and p | (q − ε), then no proper nontrivial subgroup of S is strongly closed in F by [FF, Theorem 1.2] and since S is nonabelian. So once we know that O^{p′}(F) = F, then Lemma 2.16 implies that F is simple.

In all other cases, by Table 4.2, Ω_{1}(A) is a simple 𝔽_{p}Aut_{O^{p′}(F)}(A)-module, and |A|^{2} > |S| since |A| > |S/A|. If S = T_{1} × ⋯ × T_{k}, where k ≥ 2, T_{i} ≅ T_{j} for each i, j, and each T_{i} is strongly closed in Aut_{O^{p′}(F)}(A), then either A is contained in one of the factors, or it is sent by the projection injectively into each factor. But this is impossible since |A|^{2} > |S|. So by Lemma 2.16 and since no proper nontrivial subgroup of S is strongly closed in F by [FF, Theorem 1.2] again, O^{p′}(F) is simple.

(a) By Lemma 4.4(a), it suffices to prove this when G ≅ PSL_{n}(q): the case handled by Ruiz in [Ru]. We sketch a slightly different argument, based on Lemmas 3.2 and 3.3. Recall that m = μ = ord_{p}(q).

Case a.1: Assume m = 1; i.e., p | (q − 1). Set G = SL_{n}(q), and let A ≤ G be the subgroup of diagonal matrices of p-power order and determinant 1. Thus A ≅ (C_{p^{ℓ}})^{n−1} (recall that ℓ = v_{p}(q − 1)). Choose S ∈ Syl_{p}(G) such that A ≤ S ≤ N_{G}(A), and set F = F_{S}(G). By Lemma 2.14, Γ_{p′}(F) ≅ Γ_{p′}(F), and hence it suffices to show that Γ_{p′}(F) = 1. Set

Γ = Aut_{F}(A) ≅ Σ_{n} and Γ_{0} = Aut_{O^{p′}(F)}(A).

If Γ_{p′}(F) ≠ 1, then by Lemma 3.1(b), Γ_{0} < Γ and hence Γ_{0} ≅ A_{n} (the alternating group).

If p ≤ n ≤ 2p − 1, then |S/A| = p, and |[A, S]| ≥ p^{2}. (Note that if p = n = 3, then ℓ > 1 since S ∈ Syl_{p}(PSL_{n}(q)) is assumed to be nonabelian.) If Γ_{p′}(F) ≠ 1, then Γ_{0} ≅ A_{p}, so that |Aut_{Γ_{0}}(U)| = (p−1)/2 for U ∈ Syl_{p}(Γ_{0}). So A ⊴ O^{p′}(F) by Lemma 1.4, hence A ⊴ F, which is impossible since each element of G of p-power order is diagonalizable.

Now assume n ≥ 2p. Let A_{0} ≤ A be the subgroup of all diagonal matrices diag(a_{1}, …, a_{n}) of p-power order such that a_{1} = a_{2} = ⋯ = a_{p+1}. Thus C_{G}(A_{0}) ≅ G∩(GL_{p+1}(q)×(𝔽_{q}^{×})^{n−p−1}). Set G_{0} = O^{p′}(O^{p}(C_{G}(A_{0}))) ≅ SL_{p+1}(q), so that G_{0} ⊴ C_{G}(A_{0}) and C_{G}(A_{0})/G_{0} ≅ (𝔽_{q}^{×})^{n−p−1}.

Set S_{0} = G_{0} ∩ C_{S}(A_{0}) ∈ Syl_{p}(G_{0}) and F_{0} = F_{S_{0}}(G_{0}). By Lemma 3.5, F_{0} is normal of p-power index in C_{F}(A_{0}). So by Lemmas 2.13 and 2.11(b),

O^{p′}(F_{0}) ≤ O^{p′}(C_{F}(A_{0})) ≤ O^{p′}(F).

We already saw that Aut_{O^{p′}(F_{0})}(A) ≅ Σ_{p+1}, so Aut_{F}(A) = O^{p′}(Aut_{F}(A))·Aut_{O^{p′}(F_{0})}(A), and hence Γ_{p′}(F) = 1 by Lemma 3.2. Case a.2: Now assume m > 1. Fix a vector space V ≅ (𝔽_{q})^{n}, and set Γ = Aut_{𝔽_{q}}(V) ≅ GL_{n}(q) and Γ_{0} = O^{p′}(Γ) ≅ SL_{n}(q). Note that F = F_{S}(G) is isomorphic to those of Γ_{0} and of Γ (Lemmas 2.14 and 3.5).

Fix an irreducible polynomial f | (X^{p} − 1) of degree m in 𝔽_{q}[X], and identify 𝔽_{q^{m}} ≅ 𝔽_{q}[X]/(f). Choose a decomposition V = V_{1}⊕⋯⊕V_{κ}⊕W, where dim(V_{i}) = m for each i and dim(W) = n−mκ < m (recall κ = [n/m]). Thus p ∤ |Aut_{𝔽_{q}}(W)|. Set V̂ = V_{1}⊕⋯⊕V_{κ}. Choose z_{i} ∈ Aut_{𝔽_{q}}(V_{i}) with characteristic polynomial f, and set z = z_{1} ⊕ ⋯ ⊕ z_{κ} ⊕ Id_{W} ∈ C_{Γ}(W). Thus z determines an 𝔽_{q^{m}}-vector space structure on V̂, and we set

H = C_{Γ}(z) = Aut_{𝔽_{q^{m}}}(V̂) × Aut_{𝔽_{q}}(W) ≅ GL_{κ}(q^{m}) × GL_{n−mκ}(q)
H_{0} = O^{p′}(H) ≅ SL_{κ}(q^{m}) ⋊ C_{p^{ℓ}}.

Now set

A = O_{p}(Aut_{𝔽_{q^{m}}}(V_{1}) × ⋯ × Aut_{𝔽_{q^{m}}}(V_{κ})) ≅ O_{p}((𝔽_{q^{m}}^{×})^{κ}) ≅ (C_{p^{ℓ}})^{κ}.

Thus Ω_{1}(A) = ⟨z_{1}, …, z_{κ}⟩. Each element of N_{Γ}(A) permutes the subgroups ⟨z_{i}⟩, and so

N_{Γ}(A) ≅ (𝔽_{q^{m}}^{×} ⋊ C_{m}) ≀ Σ_{κ}, Aut_{Γ}(A) = Aut_{Γ_{0}}(A) ≅ C_{m} ≀ Σ_{κ}, and Aut_{H}(A) ≅ Σ_{κ}.

Choose S_{0} ∈ Syl_{p}(Σ_{κ}), and set S = AS_{0}. Then S ∈ Syl_{p}(N_{Γ}(A)), and from the formula for |Γ| = |GL_{n}(q)|, we see that

v_{p}(|Γ|) = ∑_{i=1}^{n} v_{p}(q^{i} − 1) = ∑_{i=1}^{κ} v_{p}(q^{mi} − 1) = ∑_{i=1}^{κ}(v_{p}(q^{m} − 1) + v_{p}(i)) = κℓ + v_{p}(κ!) = v_{p}(|S|)

and hence that S ∈ Syl_{p}(Γ) = Syl_{p}(Γ_{0}). We can thus identify F = F_{S}(Γ_{0}).

Set X = {t ∈ A | t^{F} ⊆ A}. Each element in S ∖ A acts on the set {V_{i}} via a permutation of order p^{i} for some i ≥ 1, and hence acts on V with at least p distinct eigenvalues. In particular, each element of A with fewer than p distinct eigenvalues lies in X. Since Ω_{1}(A) is generated by elements z_{1}^{q^{i_{1}}} ⊕ ⋯ ⊕ z_{κ}^{q^{iκ}} for i_{1}, …, i_{κ} ∈ ℤ, each of which has eigenvalues the m roots of f in 𝔽_{q^{m}}, and m ≤ (p − 1), this shows that Ω_{1}(A) = ⟨X⟩. So Ω_{1}(A), and A = C_{S}(Ω_{1}(A)) are weakly closed in F. In particular, z ∈ X ∩ Z(S) since it has exactly m distinct eigenvalues.

Consider the surjective homomorphism

θ_{F}^{(A)} : Aut_{F}(A) = Aut_{Γ}(A) −↠ Γ_{p′}(F)

of Lemma 3.1. Set E = C_{F}(z) = F_{S}(H): the p-fusion system of C_{Γ}(z) and hence of GL_{κ}(q^{m}). We showed in Case a.1 that O^{p′}(E) contains the fusion system of SL_{κ}(q^{m}), and hence Aut_{O^{p′}(E)}(A) = Aut_{E}(A) = Aut_{H}(A) ≅ Σ_{κ}. So Ker(θ_{F}^{(A)}) = O^{p′}(Aut_{F}(A))Aut_{H}(A) ≅ G(m, m, κ) by Lemma 3.3(b) applied with ⟨z⟩ in the role of Z, and thus Γ_{p′}(F) ≅ Aut_{F}(A)/Ker(θ_{F}^{(A)}) ≅ C_{m}.

(b,c,d) If m = ord_{p}(q) is odd, then by Lemma 4.4(b,d) (and Dirichlet’s theorem), there is a prime power q^{∨} ≡ −q (mod p) such that for each n, Sp_{2n}(q) ∼_{p} Sp_{2n}(q^{∨}), Ω_{2n+1}(q) ∼_{p} Ω_{2n+1}(q^{∨}), and Ω_{2n}^{ε}(q) ∼_{p} Ω_{2n}^{(−1)^{n}ε}(q^{∨}) (for ε = ±1). Also, ord_{p}(q^{∨}) = 2m is even, and q^{n} ≡ ε (mod p) if and only if (q^{∨})^{n} ≡ (−1)^{n}ε (mod p). Thus the claims in (b), (c), and (d) and in Table 4.2 all hold when ord_{p}(q) is odd if they hold when ord_{p}(q) is even.

(b) Assume ord_{p}(q) is even. If G ≅ PSp_{2n}(q), then G ∼_{p} Sp_{2n}(q) ∼_{p} GL_{2n}(q) ∼_{p} SL_{2n}(q) by Lemmas 4.3(b) and 3.5, and hence Γ_{p′}(F) is cyclic of order ord_{p}(q) by (a). If G ≅ Ω_{2n+1}(q), then G ∼_{p} SO_{2n+1}(q) ∼_{p} GL_{2n+1}(q) ∼_{p} SL_{2n+1}(q) by Lemmas 4.3(c) and 3.5, and hence Γ_{p′}(F) is cyclic of order ord_{p}(q). In both cases, the claims in Table 4.2 about A, Aut_{F}(A), and Aut_{O^{p′}(F)}(A) all follow from (a).

(c) Assume G ≅ PΩ_{2n}^{ε}(q), where ε = ±1 and q^{n} ≢ ε (mod p). Then G ∼_{p} SO_{2n}^{ε}(q) ∼_{p} SO_{2n−1}(q) by Lemmas 3.5 and 4.3(e). So by (b), Γ_{p′}(F) is cyclic of order lcm(2, ord_{p}(q)), and the claims in Table 4.2 about A, Aut_{F}(A), and Aut_{O^{p′}(F)}(A) all hold.

(d) Assume G ≅ PΩ_{2n}^{ε}(q), where ε = ±1 and q^{n} ≡ ε (mod p), and where m = ord_{p}(q) is even. Set G = GO_{2n}^{ε}(q) and let F be the fusion system of G; thus F contains F as a normal subsystem of index 1 or 2. Then G ∼_{p} GL_{2n}(q) ∼_{p} SL_{2n}(q) by Lemmas 4.3(d) and 3.5, so Γ_{p′}(F) is cyclic of order m, and Γ_{p′}(F) is cyclic of order m/2 or m.

Recall that μ = lcm(2, m) = m and κ = [2n/μ]. Set λ = m/2 = μ/2, so that κ = [n/λ]. Since q^{n} ≡ ε = ±1 by assumption, λ | n, and so n = κλ. Let (x ↦ x) be the field automorphism of order 2 in 𝔽_{q^{m}}; thus x = x^{q^{λ}}. Set V = (𝔽_{q^{m}})^{κ} as an 𝔽_{q}-vector space, and define 𝔮_{0} : 𝔽_{q^{m}} ⟶ 𝔽_{q} and 𝔮: V ⟶ 𝔽_{q} by

𝔮_{0}(x) = Tr(xx) and 𝔮(x_{1}, …, x_{κ}) = ∑_{i=1}^{κ} 𝔮_{0}(x_{i}) = Tr(x_{1}x_{1} + ⋯ + x_{κ}x_{κ}),

where Tr: 𝔽_{q^{λ}} ⟶ 𝔽_{q} is the trace for the field extension. Then |𝔮_{0}^{−1}(0)| = (q^{λ}+1)(q^{λ−1}−1)+1, and by comparison with [Ta, Theorem 11.5], we see that 𝔮_{0} has minus type as a quadratic form over 𝔽_{q}. So 𝔮 has (−1)^{κ} type, where (−1)^{κ} ≡ (q^{λ})^{κ} ≡ ε (mod p), and GO(V, 𝔮) ≅ GO_{2λ}^{ε}(q) = G.

We claim, for u_{1}, …, u_{κ} ∈ 𝔽_{q^{m}}^{×}, that

diag(u_{1}, …, u_{κ}) is orthogonal ⟺ u_{i}u_{i} = 1 for each i.  (4.1)

This means showing, for each u ∈ 𝔽_{q^{m}}^{×}, that 𝔮_{0}(ux) = 𝔮_{0}(x) for all x ∈ 𝔽_{q^{m}}, or equivalently Tr(uuxx) = Tr(xx), if and only if uu = 1. Since u ↦ uu = u^{1+q^{λ}} sends 𝔽_{q^{m}} = 𝔽_{q^{2λ}} surjectively onto 𝔽_{q^{λ}}, we must prove, for all v ∈ 𝔽_{p^{λ}}^{×}, that Tr((v − 1)𝔽_{q^{λ}}) = 0 only if v = 1, and this is clear since v ≠ 1 implies (v − 1)𝔽_{q^{λ}} = 𝔽_{q^{λ}}.

Thus the subgroup

A = {diag(u_{1}, …, u_{κ}) | u_{i}^{p^{ℓ}} = 1 for all i} ≅ (C_{p^{ℓ}})^{κ}

is contained in GO(V, 𝔮) (recall ℓ = v_{p}(q^{m} − 1) = v_{p}(q^{λ} + 1)). We saw in the proof of (a) that N_{GL(V)}(A) ≅ (𝔽_{q^{m}}^{×} ⋊ C_{m}) ≀ Σ_{κ}. A diagonal matrix is orthogonal only if it has order dividing q^{λ} + 1, and the field automorphisms and permutations are clearly orthogonal. So

N_{G}(A) = N_{GO(V,𝔮)}(A) ≅ (C_{q^{λ}+1} ⋊ C_{m}) ≀ Σ_{κ}

and hence Aut_{G}(A) ≅ C_{m} ≀ Σ_{κ} as expected (recall G ∼_{p} GL_{2n}(q) ∼_{p} SL_{2n}(q)).

If q is odd, then the diagonal matrices of order dividing q^{λ} + 1 have determinant 1 (over 𝔽_{q}), and the permutations of coordinates have determinant 1 since m is even. By the normal basis theorem (see [Ja, § 4.14]), the field automorphism (x ↦ x^{q}) acting on just one factor 𝔽_{q^{m}} permutes cyclically an 𝔽_{q}-basis for 𝔽_{q^{m}}, and hence has determinant −1. It follows that Aut_{G}(A) ≅ G(μ, 2, κ) has index 2 in Aut_{G}(A), and hence that Γ_{p′}(F) is cyclic of order λ = μ/2.

If q is a power of 2, then a similar argument applies upon replacing the determinant by the Dickson invariant D(α) = dim([V, α]) (mod 2) for α ∈ GO(V, 𝔮) (see, e.g., [Ta, Theorem 11.43]). The diagonal matrices and the permutations all lie in Ker(D) = Ω(V, 𝔮) since dim([V, α]) ∈ mℤ ≤ 2ℤ for all such α, while if α is the field automorphism (x ↦ x^{q}) acting on one coordinate, then dim([V, α]) = m − 1 is odd. □

It remains to consider the exceptional groups of Lie type.

Proposition 4.6. Fix an odd prime p, and let G be a finite simple group of exceptional Lie type in defining characteristic different from p. Fix S ∈ Syl_{p}(G), assume S ⋬ F, and set F = F_{S}(G). Then either

(a) p = 3 and G ≅ ^{3}D_{4}(q) where 3 ∤ q, ^{2}F_{4}(q) where q = 2^{2k+1} for k ≥ 1, or ^{2}F_{4}(2)′; or

(b) G ≅ F_{4}(q), E_{6}(q), ^{2}E_{6}(q), E_{7}(q), or E_{8}(q), where q ≡ ±1 (mod p) and (G, p) is one of the pairs listed in Table 4.3; or

(c) p = 5 and G ≅ E_{8}(q), where q ≡ ±2 (mod 5); or

(d) p = 3, G ≅ G_{2}(q) where q ≡ ±1 (mod 9), |O_{3}(F)| = 3, and O^{3′}(F) has index 2 in F and is realized by SL_{3}^{±}(q).

In all cases except when p = 3 and G ≅ ^{2}F_{4}(2)′ (and S ≅ 3_{+}^{1+2}), there is a unique abelian subgroup A ⊴ S of maximal order, and A and Aut_{G}(A) are as described in Table 4.3. In all of the cases (a), (b), and (c), F is simple, and in particular, O^{p′}(F) = F.

Proof. Assume p is an odd prime and G is a finite simple group of exceptional Lie type in defining characteristic different from p. Choose S ∈ Syl_{p}(G), and set F = F_{S}(G).

case  p  G  q  A  Aut_{F}(A)
(a)  3  ^{2}F_{4}(q)  q = 2^{2n+1} ≥ 8  C_{3^{ℓ}} × C_{3^{ℓ}}  GL_{2}(3)
(a)  3  ^{3}D_{4}(q)  3 ∤ q  C_{3^{ℓ+1}} × C_{3^{ℓ}}  Σ_{3} × C_{2}
(b)  3  F_{4}(q)  q ≡ ±1 (mod p)  (C_{p^{ℓ}})^{4}  W (F_{4})
(b)  3  E_{6}^{ε}(q)  q ≡ ε (mod p) (C_{3^{ℓ}})^{5} × C_{3^{ℓ−1}}  W (E_{6})
(b)  3  E_{6}^{ε}(q)  q ≡ −ε (mod p)  (C_{p^{ℓ}})^{4}  W (F_{4})
(b)  5  E_{6}^{ε}(q)  q ≡ ε (mod p)  (C_{5^{ℓ}})^{6}  W (E_{6})
(b) 3, 5, 7 E_{n}(q) (n = 7, 8) q ≡ ±1 (mod p)  (C_{p^{ℓ}})^{n}  W (E_{n})
(c)  5  E_{8}(q)  q ≡ ±2 (mod p)  (C_{p^{ℓ}})^{4}  (C_{4} ∘ 2^{1+4}).Σ_{6}
(d)  3  G_{2}(q)  q ≡ ±1 (mod 9)  (C_{3^{ℓ}})^{2}  Σ_{3} × C_{2}

Table 4.3. Here, ℓ = v_{p}(q^{2} − 1), except in case (c) where ℓ = v_{5}(q^{2} + 1), and ε = ±1.

• If G is a Suzuki or Ree group or G ≅ ^{3}D_{4}(q), then either S is abelian and hence S ⊴ F, or (G, p) is one of the pairs appearing in (a).

• If G ≅ F_{4}(q), E_{n}(q), or^{2}E_{6}(q), where p ∤ q, then by [GLS3, Theorem 4.10.2(a,b,c)] or [GL, (10-1(3))], either S is abelian, or for some m such that p · ord_{p}(q) | m, the cyclotomic polynomial Φ_{m}(q) appears in the formula for |G|. This occurs only when (G, p) is one of the pairs listed in Table 4.3, case (b) or (c).

• If G ≅ G_{2}(q) where p ∤ q, then either p ≥ 5 and S is abelian; or p = 3, v_{3}(q^{2} − 1) = 1, and S ⊴ F by [A2, 16.11.4]; or q ≡ ±1 (mod 9) and (G, p) is as in (d).

Thus whenever S ⋬ F, we are in the situation of one of the cases (a)–(d), and in fact, of one of the cases listed in Table 4.3.

In cases (a), (b), and (c), we will prove below that O^{p′}(F) = F. Once this has been shown, then F is simple by Lemma 2.16, together with [FF, Theorem 1.2] which says that no proper nontrivial subgroup of S is strongly closed in F.

(a) Assume p = 3. If G = ^{2}F_{4}(2)′ and S ∈ Syl_{3}(G), then by [RV, Lemma 4.13], S is extraspecial of order 3^{3} and exponent 3, Out_{G}(S) ≅ D_{8}, and all subgroups of order 9 in S are F-radical. A normal subsystem of index 2 would have the same radical subgroups, and so there is no such subsystem by the same lemma.

If G = ^{2}F_{4}(q) where q = 2^{2k+1} for k ≥ 1, then by the main theorem in [Ma], G contains a maximal torus H ≅ C_{q+1} × C_{q+1} such that Aut_{G}(H) ≅ GL_{2}(3) and N_{G}(H) ≥ S. Set A = O_{p}(H) ≅ C_{3^{ℓ}} ×C_{3^{ℓ}}, where ℓ = v_{3}(q +1) = v_{3}(q^{2} −1): then |S/A| = 3 and Aut_{G}(A) ≅ GL_{2}(3). By Lemma 3.1(a,b), if Γ_{p′}(F) ≠ 1, then it has order 2 and Aut_{O^{3′}(F)}(A) = O^{3′}(Aut_{F}(A)) ≅ SL_{2}(3) ≅ 2A_{4}. But this contradicts Lemma 1.4.

If G = ^{3}D_{4}(q) where 3 ∤ q, then set ℓ = v_{3}(q^{2} − 1), so that v_{3}(|G|) = 2(ℓ + 1). By the main theorem in [Kl], S contains a subgroup A ≅ C_{3^{ℓ+1}} × C_{3^{ℓ}} of index 3, and Aut_{G}(A) ≅ Σ_{3} × C_{2} ≅ D_{12}. (If q ≡ ε (mod 3) for ε ∈ {±1}, then N_{G}(A) is contained inside the maximal subgroup (C_{q^{2}+εq+1} ×_{C_{3}} SL_{3}^{ε}(q))⋊Σ_{3} listed in [Kl].) By the conditions listed in [O1, Theorem 2.8(a)], there can be no normal subsystem of index 2 in F = F_{S}(G).

Alternatively, the result for G = ^{2}F_{4}(q) or ^{3}D_{4}(q) follows from Tables 2 and 4 in [DRV]: when m = v_{3}(|G|), then S ≅ B(3, m; 0, 0, 0) in the notation of [DRV], and those two tables list all saturated fusion systems over these groups.

(b) If q ≡ ε (mod p) where ε = ±1, and q^{∨} ≡ −ε (mod p) is another prime power such that v_{p}(q − ε) = v_{p}(q^{∨} + ε), then E_{6}(q) ∼_{p} ^{2}E_{6}(q^{∨}) ∼_{p} F_{4}(q^{∨}) by Lemma 4.4(c). This, together with similar applications of Lemma 4.4(d), shows that it suffices to prove (b) when G = 𝔾(q) for q ≡ 1 (mod p) and 𝔾 = F_{4} or E_{n}. In these cases, A is homocyclic of rank rk(𝔾) and exponent p^{ℓ} (where ℓ = v_{p}(q − 1)), except when 𝔾 = E_{6} and p = 3, and Aut_{F}(A) ≅ W (𝔾). (The description of A can be found in [GL, 10-1(2)] or [GLS3, Theorem 4.10.2(c)], and more details on Aut_{G}(A) are given in [BMO2, Lemmas 6.1 & 5.3].) Recall that we are working with the simple groups, hence the adjoint forms, which is why the description of A is slightly different when (G, p) = (E_{6}(q), 3).

It remains to show that O^{p′}(F) = F, still assuming G = 𝔾(q) for 𝔾 = F_{4} or E_{n} and q ≡ 1 (mod p) (and S is nonabelian). Since Aut_{F}(A) ≅ W (𝔾) is the usual Weyl group for G, we have

Aut_{F}(A)/O^{p′}(Aut_{F}(A)) ≅ \begin{cases}C_{2} × C_{2} & if 𝔾 = F_{4} and p = 3 \\ C_{2} & if 𝔾 = E_{6}, E_{7}, E_{8} and 3 ≤ p | |W (𝔾)|. \end{cases}  (4.2)

When p = 3, these follow from the “standard presentation” of W (𝔾) as a group of reflections acting on a vector space, with a fundamental system of reflections as generating set [Ca, Theorem 2.4.1]. When p = 5, 7, the result follows from the fact that W (E_{n}) contains a quasisimple subgroup of index 2. In all cases, we set ℓ = v_{p}(q − 1) ≥ 1.

We consider two different cases. Case (b.1) Assume |S/A| = p; equivalently, v_{p}(|W (𝔾)|) = 1. Then either p = 5 and 𝔾 = E_{6} or E_{7}, or p = 7 and 𝔾 = E_{7} or E_{8}, and we are in the situation of [COS, Section 2]. We refer to Notation 2.4 in that paper. By [COS, Lemma 2.6(a)] and since A ⋬ F, Aut_{O^{p′}(F)}^{∨}(A) must contain one of the subgroups Δ_{0} or Δ_{−1}, where for t ∈ ℤ,

Δ_{t} = {(r, r^{t}) | r ∈ (ℤ/p)^{×}} ≤ Δ =^{def} (ℤ/p)^{×} × (ℤ/p)^{×}.

By Table 4.1 in [COS], this would not be the case if Aut_{O^{p′}(F)}(A) were strictly contained in Aut_{F}(A) ≅ W (𝔾). Thus Aut_{O^{p′}(F)}(A) = Aut_{F}(A), and so O^{p′}(F) = F by Lemma 3.1(c).

Case (b.2) Assume |S/A| ≥ p^{2}; equivalently, p^{2} | |W (𝔾)|. Then either p = 3 and 𝔾 = F_{4} or E_{n}, or p = 5 and 𝔾 = E_{8}. In each case, there is an element z ∈ Z(S) whose centralizer H = C_{G}(z) is as described in the Table 4.4. The normalizer N_{G}(z) is described in [LSS, Table 5.1], and the descriptions of O^{p}(H)/⟨z⟩ and H/O^{p}(H) are based on that. A comparison of orders shows that p ∤ [G : H] in each case, and hence that we can choose z ∈ Z(S).

𝔾 p  O^{p}(H)/⟨z⟩ = O^{p}(H/⟨z⟩)  H/O^{p}(H)  H
F_{4} 3  PSL_{3}(q) × PSL_{3}(q)  C_{3}  SL_{3}(𝔽_{q}) ∘ SL_{3}(𝔽_{q})
E_{6} 3 PSL_{3}(q) × PSL_{3}(q) × PSL_{3}(q) C_{3} × C_{3} SL_{3}(𝔽_{q}) ∘ SL_{3}(𝔽_{q}) ∘ SL_{3}(𝔽_{q})
E_{7} 3  E_{6}(q) × C_{(q−1)/3^{ℓ}}  C_{3^{ℓ}}  3E_{6}(𝔽_{q}) ∘ 𝔽_{q}^{×}
E_{8} 3  E_{6}(q) × PSL_{3}(q)  C_{3}  3E_{6}(𝔽_{q}) ∘ SL_{3}(𝔽_{q})
E_{8} 5  PSL_{5}(q) × PSL_{5}(q)  C_{5}  SL_{5}(𝔽_{q}) ∘ SL_{5}(𝔽_{q})

Table 4.4. In all cases, G = 𝔾(q) where ℓ = v_{p}(q − 1) ≥ 1, and H = C_{𝔾(𝔽_{q})}(z) for a certain z ∈ Z(S) of order p. Also, E_{6}(−) means the simple group and 3E_{6}(−) its universal form, and “X ∘ Y ” denotes a central product of X and Y over C_{p}.

The description of the centralizer of z in the algebraic group 𝔾(𝔽_{q}) in the last column of the table (taken from [Gr, Table VI]), is included to better explain the structure of H. For example, when p = 3 and G = F_{4}(q), H is an extension of SL_{3}(q) ∘ SL_{3}(q) by C_{3} acting via diagonal automorphisms of order 3 on each factor.

Set E = C_{F}(z) = F_{S}(H). If ℓ = v_{p}(q−1) ≥ 2 or p = 5, then the p-fusion system of PSL_{p}(q) is simple by Proposition 4.5(a), and O^{p′}(O^{p}(E)/⟨z⟩) is the fusion system of O^{p′}(O^{p}(H)/⟨z⟩) by the table. By Lemmas 2.13 and 2.14, O^{p′}(E) = E, except when (𝔾, p) = (E_{7}, 3), in which case O^{p′}(E) = F_{S}(O^{p′}(H)). So by (4.2), Aut_{F}(A) is generated by O^{p′}(Aut_{F}(A)) and Aut_{O^{p′}(E)}(A), and O^{p′}(F) = F by Lemma 3.2.

Now assume ℓ = 1 and p = 3. Set K = PSL_{3}(q) and fix U ∈ Syl_{3}(K). Then U ≅ C_{3} × C_{3}, so F_{U}(K) = F_{U}(N_{K}(U)) ≅ F_{U}(U ⋊ Q_{8}). If 𝔾 = F_{4}, then E/⟨z⟩ is the fusion system of an extension of (C_{3})^{4} ⋊ (Q_{8} × Q_{8}) by C_{3} acting faithfully on each factor Q_{8}, so O^{3′}(E/⟨z⟩) = E/⟨z⟩, and O^{3′}(E) = E by Lemma 2.14. A similar argument shows that O^{3′}(E) = E when 𝔾 = E_{6}. This, together with the descriptions in Table 4.4, show that Aut_{F}(A) is generated by O^{p′}(Aut_{F}(A)) and Aut_{O^{p′}(E)}(A) in all four cases (also when 𝔾 = E_{7} or E_{8}) and hence that O^{p′}(F) = F by Lemma 3.2 again.

(c) Now assume p = 5, and G = E_{8}(q) where q ≡ ±2 (mod 5). Set ℓ = v_{5}(q^{4}−1) = v_{5}(q^{2}+1). By [BMO2, Lemma 6.7], there is a unique A = (C_{5^{ℓ}})^{4} of index 5 in S, and Aut_{F}(A) ≅ (C_{4} ∘ 2^{1+4}).Σ_{6} (where “∘” denotes the central product over C_{2}). We are thus in the situation of [COS, Section 2]. Since rk(A) ≤ p, we have |Z(S)| = p and hence Aut_{F}^{∨}(S) = Aut_{F}(S) as defined in [COS, Notation 2.4]. So O^{p′}(F) = F by Table 2.2 and Lemma 2.7(c.i) in [COS].

(d) This case, where G = G_{2}(q) and p = 3, was handled by Aschbacher [A2, 16.11]. In particular, he showed there that S ⊴ F if q ≡ ±4, ±5 (mod 9), while O_{3}(F) = Z(S) and O^{3′}(F) = C_{F}(Z(S)) has index 2 in F and is realized by SL_{3}^{±}(q) if q ≡ ±1 (mod 9). □

With the help of Propositions 4.5 and 4.6 and Tables 4.2 and 4.3, we can now check that certain fusion systems are exotic.

Lemma 4.7. Assume the classification of finite simple groups. 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 Fcentric. 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

Aut_{F}(A) ∩ G(m, 1, κ) = G(m, r, κ) ≤ GL_{κ}(ℤ/p^{ℓ}) for some 2 < r | m.

Then either A ⊴ F or F is exotic.

Proof. Assume otherwise: assume A ⋬ F and F is realized by the finite group G. Assume also that |G| is minimal among all such counterexamples (A, F, G) to the lemma. In particular, |G| is minimal among all orders of finite groups realizing F, and so O_{p′}(G) = 1.

If ℓ > 1 and Ω_{1}(A) ⊴ F, then Ω_{1}(A) ⊴ G by the minimality assumption (F is realized by N_{G}(Ω_{1}(A))). So in this case, A/Ω_{1}(A) ⋬ F/Ω_{1}(A) where F/Ω_{1}(A) is realized by G/Ω_{1}(A), which again contradicts our minimality assumption on |G|. Thus Ω_{1}(A) ⋬ F, and hence Ω_{k}(A) ⋬ F for each 1 ≤ k ≤ ℓ.

Since Aut_{F}(A) contains G(m, r, 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, r, 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_{F}(A),

Aut_{F}(A) acts faithfully on Ω_{1}(A), Aut_{S}(A) permutes a basis of Ω_{1}(A), and C_{S}(Ω_{1}(A)) = A.

(4.3)

We claim that

Ω_{1}(A) is the only elementary abelian subgroup of S of rank κ.  (4.4)

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 r = rk(W/(V ∩ W)) since C_{W} (V) = W ∩ A = W ∩ V by (4.3). 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^{m_{s}}. Then m_{1} + ⋯ + m_{s} ≥ r = rk(W), and hence

rk(W) = rk(W) + rk(W ∩ V) ≤ r + rk(C_{V} (W)) = r + s ≤ ∑_{i=1}^{s}(1 + m_{i}) < ∑_{i=1}^{s} p^{m_{i}} = rk(V),

proving (4.4). In particular, Ω_{1}(A) and A = C_{S}(Ω_{1}(A)) are weakly closed in F.

Set G_{0} = O^{p′}(G) and F_{0} = F_{S}(G_{0}) ⊴ F, and let Γ = Aut_{F}(A) and Γ_{0} = Aut_{F_{0}}(A) for short. By Lemma 2.15 and since Ω_{1}(A) is weakly closed in F, we have Ω_{1}(A) ⋬ F_{0}.

Step 1: We first claim that G_{0} is simple. Assume otherwise: let H ⊴ G_{0} be a proper nontrivial normal subgroup, and set T = H∩S ∈ Syl_{p}(H). Thus 1 ≠ T < S since O_{p′}(G_{0}) = 1 and O^{p′}(G_{0}) = G_{0}, and T is strongly closed in F_{0}.

Now, Aut_{H}(A) ⊴ Γ_{0} since H ⊴ G_{0}, and by assumption, Γ_{0} contains G(m, m, κ) with index prime to p. So Aut_{H}(A) ∩ G(m, m, κ) is normal in G(m, m, κ) ≅ (C_{m})^{κ−1} ⋊ Σ_{κ}, where κ ≥ p ≥ 5. Consequently, either Aut_{H}(A) ∩ G(m, m, κ) and hence Aut_{H}(A) have order prime to p, or Aut_{H}(A) contains O^{p′}(G(m, m, κ)). In the latter case, H ≥ [N_{H}(A), A] ≥ [O^{p′}(G(m, m, κ)), A] = A, and hence N_{H}(A) has index prime to p in N_{G}(A). But then T = S, a contradiction.

Thus p ∤ |Aut_{H}(A)|, and so T = O^{p′}(N_{H}(A)) ≤ A. Also, T is normalized by Γ_{0} ≥ O^{p′}(G(m, m, κ)), and so T = Ω_{k}(A) for some 1 ≤ k ≤ ℓ. But then T is abelian and strongly closed in F, so T ⊴ F_{0} by [AKO, Corollary I.4.7(a)], hence Ω_{1}(A) = Ω_{1}(T) ⊴ F_{0}, which we already showed is impossible. So there is no such T, and G_{0} must be simple.

Step 2: It remains to show that G_{0} cannot be any known simple group. 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 O^{p′}(G(m, 1, 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_{Σ_{κ}}(B) is a product of wreath

products of the form GL_{c}(p) ≀ Σ_{κ} for c ≥ 1 and κ ≥ 1. Thus Aut_{A_{κ}}(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)], O^{p′}(N_{G_{0}}(A)/A) ≅ O^{p′}(L_{J}) is a central product of groups in 𝔏𝔦𝔢(p), contradicting the assumption that O^{p′}(Aut_{G}(A)) ≅ G(m, r, κ).

Now assume that G_{0} ∈ 𝔏𝔦𝔢(q_{0}) for some prime q_{0} ≠ p. By [GL, 10-2] (and since p ≥ 5), S ∈ Syl_{p}(G_{0}) contains a unique elementary abelian p-subgroup of maximal rank, and by (4.4), 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.

• If G_{0} is a classical group and hence Aut_{F}(A) ≅ G(m̂, r̂, κ) for m̂ = μ or 2μ and r̂ ≤ 2 (see the next-to-last column in Table 4.2, and recall that G(m̂, 1, κ) ≅ Cm̂ ≀ Σ_{κ}), then the identifications Aut_{F}(A) ≅ G(m̂, r̂, 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 | m̂ and r ≤ 2, contradicting our original assumption.

• If G_{0} is an exceptional group, then by 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). □

The results in this section are now summarized in the following theorem, in which the statements that certain fusion systems are exotic depend on the classification of finite simple groups. Recall that a fusion subsystem of F is characteristic in F if it gets sent to itself by each α ∈ Aut(F).

Theorem 4.8. Fix a prime p, a known finite simple group G, and S ∈ Syl_{p}(G). Set F = F_{S}(G), and assume that S ⋬ F. Then Γ_{p′}(F) is cyclic, and either

(a) 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 exotic; or

(b) p = 3 and G ≅ G_{2}(q) for some q ≡ ±1 (mod 9), in which case |O_{3}(F)| = 3, |Γ_{p′}(F)| = 2, and O^{3′}(F) is realized by SL_{3}^{±}(q); or

(c) |Γ_{p′}(F)| ≤ 2, O^{p′}(F) is simple, and it is realized by a known finite simple group G^{∗}.

Moreover, the following hold in case (a).

(a.1) There are a subsystem F_{0} ⊴ F of index at most 2 in F, and a finite group G_{0} realizing F_{0}, with the following properties. For each saturated fusion system E over S such that O^{p′}(E) = O^{p′}(F), either

• E ≥ F_{0}, and E = F_{S}(H) for some group H with G_{0} ⊴ H and p ∤ |H/G_{0}|; or • E ≱ F_{0}, and E is exotic.

(a.2) If E ≥ F is an extension of index prime to p (i.e., E is saturated and F is normal of index prime to p in E), then F is characteristic in E.

Proof. By Propositions 4.1, 4.2, and 4.6, point (b) or (c) holds except possibly when p is odd and G is one of the classical groups PSL_{n}^{±}(q), PSp_{2n}(q), PΩ_{2n+1}(q), or PΩ_{2n}^{±}(q) for p ∤ q. So assume we are in one of those cases. In particular, O^{p′}(F) is simple by Proposition 4.5.

If G ≅ PSU_{n}(q) for some n and q, then by Lemma 4.4(a), G ∼_{p} PSL_{n}(q^{∨}) for some prime power q^{∨} ≡ −q (mod p). So we can assume G is linear, symplectic, or orthogonal. Case 1: Assume q ≡ ±1 (mod p). If G = PSL_{n}(q) and q ≡ 1 (mod p), then O^{p′}(F) = F by Proposition 4.5(a). If G = PSL_{n}(q) and q ≡ −1 (mod p), so that ord_{p}(q) = 2, then set k = [n/2]. We have

G ∼_{p} GL_{n}(q) ∼_{p} GL_{2k}(q) ∼_{p} GO_{2k}^{ε}(q)

for ε = (−1)^{k} by Lemma 4.3(a,d). Also, O^{p′}(F) has index 2 in F by Proposition 4.5(a,d), and it is realized by the simple group PΩ_{2n}^{ε}(q) by Table 4.2 and Lemma 3.1(c).

If G is one of the groups PSp_{2n}(q) or Ω_{2n+1}(q) for q ≡ ±1 (mod p), or PΩ_{2n}^{ε}(q) for q ≡ ±1 and q^{n} ≢ ε (mod p), then a similar argument shows that O^{p′}(F) has index 2 in F and is realizable by a simple orthogonal group. If G = PΩ_{2n}^{ε}(q) where q ≡ ±1 and q^{n} ≡ ε (mod p), then O^{p′}(F) = F by Proposition 4.5(c).

Thus O^{p′}(F) is realizable (and we are in the situation of (c)) whenever G is one of the classical groups defined over 𝔽_{q} and q ≡ ±1 (mod p). Case 2: Assume q ≢ 0, ±1 (mod p), where G is again one of the classical groups PSL_{n}(q), PSp_{2n}(q), PΩ_{2n+1}(q), or PΩ_{2n}^{±}(q). In particular, p ≥ 5. Set m = ord_{p}(q) and ℓ = v_{p}(q^{m} − 1); thus ℓ ≥ 1 and 2 < m | (p − 1). By Table 4.2, there are κ ≥ 1 and A ⊴ S weakly closed in F such that A ≅ (C_{p^{ℓ}})^{κ} and Aut_{O^{p′}(F)}(A) ≅ G(m, m, κ). So O^{p′}(F) is exotic by Lemma 4.7.

Proof of (a.1): Assume E is a saturated fusion system over S such that O^{p′}(E) = O^{p′}(F). Then Aut_{O^{p′}(F)}(A) ⊴ Aut_{E}(A). Let {a_{1}, …, a_{κ}} be a basis for A as a ℤ/p^{ℓ}-module such that with respect to this basis, Aut_{O^{p′}(F)}(A) = G(m, m, κ): the group of monomial matrices generated by the permutation matrices, and the diagonal matrices of order dividing m and of determinant 1.

Now, Aut_{E}(A) ≤ N_{Aut(A)}(Aut_{O^{p′}(F)}(A)) = G(m, 1, κ)⟨(ℤ/p^{ℓ})^{×}·Id_{A}⟩: each α ∈ Aut(A) that normalizes G(m, m, κ) permutes the subgroups ⟨a_{i}⟩ and thus has a monomial matrix whose nonzero entries differ pairwise by m-th roots of unity. Furthermore, since Aut_{O^{p′}(F)}(A) has index prime to p in Aut_{E}(A), we have

Aut_{E}(A) ≤ G(m, 1, κ)⟨ω · Id_{A}⟩,  (4.5)

for ω ∈ ℤ whose class in (ℤ/p^{ℓ})^{×} has order exactly p − 1.

By Dirichlet’s theorem, we can choose a prime q_{0} ≡ ω (mod p^{ℓ}) such that v_{p}(q_{0}^{p−1} −1) = ℓ. Set q^{∗} = q_{0}^{a}, where a is chosen so that p ∤ a and q^{∗} ≡ q (mod p). Upon replacing q by q^{∗} and G = 𝔾^{±}(q) by 𝔾^{±}(q^{∗}), we can arrange that q = q_{0}^{a} without changing the fusion system F (see [BMO1, Theorem A(a)]).

Let G_{0} ⊴ G_{1} be as defined in Table 4.5, and set F_{0} = F_{S}(G_{0}). In the first two cases shown in the table, the isomorphism in the third column follows from Table 4.2, and G_{0} = G and F_{0} = F. In the third case, G ∼_{p} PGO_{2n′}^{ε}(q) by Lemma 4.3 (and Table 4.2), and we identify S as a subgroup of G_{0} = PΩ_{2n′}^{ε}(q) in such a way that F_{0} ≤ F. Then Aut_{F_{0}}(A) = G(m, 2, κ) and hence F_{0} has index 2 in F. In all cases, ψ_{q_{0}} denotes the field automorphism (x ↦ x^{q_{0}}), this acts on A via (a ↦ a^{q_{0}} = a^{ω}), and so Aut_{G_{1}}(A) = G(m, 1, κ)⟨ω · Id⟩.

m  Aut_{F}(A)  G  G_{0}  G_{1}
odd  G(m, 1, κ)  G ≅ PSL_{n}(q)  PSL_{n}(q)  PSL_{n}(q)⟨ψ_{q_{0}}⟩
even G(m, 2, κ)  G ≅ PΩ_{2n′}^{ε}(q)  PΩ_{2n′}^{ε}(q) PGO_{2n′}^{ε}(q)⟨ψ_{q_{0}}⟩
even G(m, 1, κ) G ∼_{p} PGO_{2n′}^{ε}(q) PΩ_{2n′}^{ε}(q) PGO_{2n′}^{ε}(q)⟨ψ_{q_{0}}⟩

Table 4.5. In the last two cases, n′ ≥ p and ε = ±1 are such that q^{n′} ≡ ε (mod p).

Thus in all three cases,

G(m, m, κ) = Aut_{O^{p′}(F)}(A) ≤ Aut_{E}(A) ≤ G(m, 1, κ)⟨ω · Id⟩ = Aut_{G_{1}}(A),  (4.6)

where the second inequality follows from (4.5). Hence

Aut_{E}(A) ∩ G(m, 1, κ) = G(m, r, κ) for some r | m.  (4.7)

If r > 2, then E is exotic by Lemma 4.7, and E ≱ F_{0} since Aut_{F_{0}}(A) = G(m, gcd(m, 2), κ).

Now assume r ≤ 2, and hence that Aut_{E}(A) ≥ Aut_{F_{0}}(A). By (4.6), there is H ≤ G_{1} such that G_{0} ⊴ H and Aut_{H}(A) = Aut_{E}(A). Since O^{p′}(F_{S}(H)) = O^{p′}(F_{0}) = O^{p′}(E), we have E ≅ F_{S}(H) by Lemma 3.4.

We claim that this is, in fact, an equality (and hence that E ≥ F_{0}). Set Γ = Aut_{O^{p′}(F)}(A) = G(m, m, κ) and Γ_{0} = O_{p′}(Γ) ≅ (C_{m})^{κ−1} for short. The five-term exact sequence for the cohomology of Γ as an extension of Γ_{0} by Γ/Γ_{0} (see [Be2, p. 110]) restricts to an exact sequence

0 ⟶ H^{1}(Γ/Γ_{0}; H^{0}(Γ_{0}; A))_{=0} ⟶ H^{1}(Γ; A) ⟶ H^{0}(Γ/Γ_{0}; H^{1}(Γ_{0}; A))_{=0}  (4.8)

in which the first term is zero since H^{0}(Γ_{0}; A) = C_{A}(Γ_{0}) = 1, and the last term is zero since gcd(|Γ_{0}|, |A|) = 1. Thus H^{1}(Γ; A) = 0, and so E = F_{S}(H) by Lemma 3.4 again. In particular, E ≥ F_{0} in this case, and this finishes the proof of (a.1).

Proof of (a.2): Now assume E ≥ F is an extension of index prime to p, and let α be an automorphism of E. Thus α ∈ Aut(S) is such that ^{α}E = E, and in particular, ^{α}(O^{p′}(F)) = O^{p′}(F), α(A) = A, and^{α}(Aut_{E}(A)) = Aut_{E}(A). Since by (4.5), Aut_{F}(A) is generated by Aut_{O^{p′}(F)}(A) together with either the m-torsion in O_{p′}(Aut_{E}(A)) or the unique normal subgroup of index 2 in that subgroup, we also have ^{α}(Aut_{F}(A)) = Aut_{F}(A). Since H^{1}(Aut_{O^{p′}(F)}(A); A) = 0 by (4.8), we have ^{α}F = F by Lemma 3.4. Since α ∈ Aut(E) was arbitary, F is characteristic in E. □

References

[AF] J. Alperin & P. Fong, Weights for symmetric and general linear groups, J. Algebra 131 (1990), 2–22 [AOV] K. Andersen, B. Oliver, & J. Ventura, Reduced, tame, and exotic fusion systems, Proc. London Math.

Soc. 105 (2012), 87–152

[A2] M. Aschbacher, The generalized Fitting subsystem of a fusion system, Memoirs Amer. Math. Soc. 209 (2011), nr. 986

[AKO] M. Aschbacher, R. Kessar, & B. Oliver, Fusion systems in algebra and topology, Cambridge Univ. Press (2011)

[Be2] D. Benson, Representations and cohomology II: cohomology of groups and modules, Cambridge Univ. Press (1991)

[5a1] C. Broto, N. Castellana, J. Grodal, R. Levi, & B. Oliver, Subgroup families controlling p-local finite groups, Proc. London Math. Soc. 91 (2005), 325–354

[5a2] C. Broto, N. Castellana, J. Grodal, R. Levi, & B. Oliver, Extensions of p-local finite groups, Trans. Amer. Math. Soc. 359 (2007), 3791–3858

[BM] C. Broto, & J. Møller, Chevalley p-local finite groups. Algebraic & Geometric Topology 7 (2007), 1809–1919

[BMO1] C. Broto, J. Møller, & B. Oliver, Equivalences between fusion systems of finite groups of Lie type, Jour. Amer. Math. Soc. 25 (2012), 1–20

[BMO2] C. Broto, J. Møller, & B. Oliver, Automorphisms of fusion systems of finite simple groups of Lie type, Memoirs Amer. Math. Soc. 1267 (2020), 1–117

[Ca] R. Carter, Simple groups of Lie type, Wiley (1972)

[Cr] D. Craven, The theory of fusion systems, Cambridge Univ. Press (2011)

[COS] D. Craven, B. Oliver, & J. Semeraro, Reduced fusion systems over p-groups with abelian subgroup of index p: II, Advances in Mathematics 322 (2017), 201–268

[DRV] A. Díaz, A. Ruiz, & A. Viruel, All p-local finite groups of rank two for odd prime p, Trans. Amer. Math. Soc. 359 (2007), 1725–1764

[FF] R. Flores & R. Foote, Strongly closed subgroups of finite groups, Adv. Math. 222 (2009), 453–484 [G] D. Gorenstein, Finite groups, Harper & Row (1968)

[GL] D. Gorenstein & R. Lyons, The local structure of finite groups of charateristic 2 type, Memoirs Amer. Math. Soc. 276 (1983)

[GLS3] D. Gorenstein, R. Lyons, & R. Solomon, The classification of the finite simple groups, nr. 3, Amer. Math. Soc. surveys and monogr. 40 #3 (1997)

[Gr] R. Griess, Elementary abelian p-subgroups of algebraic groups, Geometriae Dedicata 39 (1991), 253– 305

[Gr] J. Grodal, Endotrivial modules for finite groups via homotopy theory, arXiv 1608.00499

[Ja] N. Jacobson, Basic algebra I (2nd ed.), W. H. Freeman and Co. (1985)

[Kl] P. Kleidman, The maximal subgroups of the Steinberg triality groups ^{3}D_{4}(q) and of their automorphism groups, J. Algebra 115 (1988), 182–199

[LSS] M. Liebeck, J. Saxl, & G. Seitz, Subgroups of maximal rank in finite exceptional groups of Lie type, Proc. London Math. Soc. 65 (1992), 297–325

[Ma] G. Malle, The maximal subgroups of ^{2}F_{4}(q^{2}), J. Algebra 139 (1991), 52–69

[O1] B. Oliver, Simple fusion systems over p-groups with abelian subgroup of index p: I, J. Algebra 398 (2014), 527–541

[O2] B. Oliver, Reduced fusion systems over 2-groups of sectional rank at most 4, Memoirs Amer. Math. Soc. 239 (2015), nr. 1131

[O3] B. Oliver, Automorphisms of fusion systems of sporadic simple groups, Memoirs Amer. Math. Soc. 1267 (2020), 119–163

[OR] B. Oliver & A. Ruiz, Reduced fusion systems over p-groups with abelian subgroup of index p: III, Proceedings of the Royal Society of Edinburgh Section A 150 (2020), 1187–1239

[OV] B. Oliver & J. Ventura, Saturated fusion systems over 2-groups, Trans. Amer. Math. Soc. 361 (2009), 6661–6728

[Pg] L. Puig, Frobenius categories, J. Algebra, 303 (2006), 309–357.

[Ru] A. Ruiz, Exotic normal fusion subsystems of general linear groups, J. London Math. Soc. (2) 76 (2007), 181–196

[RV] A. Ruiz & A. Viruel, The classification of p-local finite groups over the extraspecial group of order p^{3} and exponent p, Math. Z. 248 (2004), 45–65

[Ta] D. Taylor, The geometry of the classical groups, Heldermann Verlag (1992)

Université Sorbonne Paris Nord, LAGA, UMR 7539 du CNRS, 99, Av. J.-B. Clément, 93430 Villetaneuse, France.

  Email address: bobol@math.univ-paris13.fr

Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain

  Email address: albert@mat.uab.cat