S_{3}-FREE 2-FUSION SYSTEMS

Michael Aschbacher

California Institute of Technology

Let G be a finite group. A section of G is a group of the form H/K, where K ⊴ H ≤ G. Given a positive integer n, write S_{n} for the symmetric group on a set of order n. The group G is S_{n}-free if G has no section isomorphic to S_{n}.

One of the striking results in finite group theory from the decade before the classification of the finite simple groups, was the classification of the S_{4}-free finite simple groups. The first result in this direction was Thompson’s proof (cf. [T]) that the Suzuki groups Sz(2^{2m+1}) are the only nonabelian finite simple groups of order prime to 3. Next in [Gl], Glauberman proved a triple factorization theorem for constrained S_{4}-free groups, and used this result to show each S_{4}-free group has a nontrivial strongly closed abelian 2-subgroup. As a corollary to this result and Goldschmidt’s theorem in [Go] classifying finite groups with such a subgroup, Glauberman classified the S_{4}-free and S_{3}-free nonabelian finite simple groups. Later in [St], Stellmacher showed that in a constrained S_{4}-free group G, there is a nontrivial characteristic subgroup of a Sylow 2-subgroup of G normal in G; this theorem can be used to give an alternate treatment of S_{4}-free groups.

In this paper we give yet another treatment of S_{4}-free groups, this time from the point of view of 2-fusion. Actually we consider 2-fusion systems rather than groups, and prove that all saturated S_{3}-free 2-fusion systems are constrained. Then we derive our theorems about groups from this theorem about fusion systems, and Goldschmidt’s theorem. Indeed from our point of view, the main object of the paper is to put in place machinery to study fusion systems of even characteristic, building on earlier results in [A3]. Given this machinery, the proof of the theorem on S_{3}-free fusion systems takes only a few paragraphs, making possible a very elegant treatment of S_{4}-free groups.

Fusion systems were defined and first studied by L. Puig, although Puig calls these objects Frobenius categories rather than fusion systems; see in particular [P1] and [P2]. Our introduction to the subject was from [BLO], and we adopt the notation and terminology found there.

This work was partially supported by NSF-0504852

The reader is directed to [FGT] for notation and terminology involving finite groups, and to Section 1 and Appendix A of [BLO] for notation and terminology involving fusion systems. Section 1 of this paper also contains some background on fusion systems.

Let F be a saturated fusion system on a finite 2-group S. We say that F is S_{3}-free if for each subgroup U of S, the group Aut_{F}(U) is S_{3}-free. Here is our result on S_{3}-free fusion systems:

Theorem 1. Let F be a saturated S_{3}-free fusion system on a finite 2-group S. Then F is constrained.

The result on fusion systems, together with Goldschmidt’s Theorem, leads almost immediately to the following corollaries for groups:

Corollary 2. Assume G is a finite S_{4}-free nonabelian finite simple group. Then G is a Goldschmidt group.

A finite simple group G is a Goldschmidt group if either G is a group of Lie type of Lie rank 1 in characteristic 2, or G has abelian Sylow 2-subgroups. The groups of the first type are the groups L_{2}(2^{n}), Sz(2^{n}), and U_{3}(2^{n}), n > 1. The nonabelian simple groups with abelian Sylow 2-subgroups (other than the groups L_{2}(2^{n})) are the groups L_{2}(q), q ≡ ±3 mod 4, ^{2}G_{2}(3^{2m+1}), m ≥ 1, and J_{1}.

Corollary 3. Assume G is a finite S_{3}-free nonabelian finite simple group. Then G is Sz(2^{2m+1}) or L_{2}(3^{2m+1}), with m ≥ 1.

Corollary 4. Assume G is a finite nonabelian simple group of order prime to 3. Then G ≅ Sz(2^{2m+1}) for some m ≥ 1.

Section 1 contains a brief discussion of fusion systems. In sections 2 through 4 we put in place machinery for studying saturated fusions systems F on finite 2-groups S with F of even characteristic (as defined in section 1). Then in section 5, we use this machinery to prove a version of Stellmacher’s qrc-Lemma for fusion systems, together with a few supporting lemmas which make the qrc-Lemma more effective. Section 6 contains some lemmas on representations of finite groups over fields of even characteristic that are needed to apply the qrc-Lemma. Finally the main theorem and its corollaries are proved in section 7.

Section 1. Fusion systems

In this section p is a prime, S is a finite p-group, and F is a saturated fusion system on S.

The definition of a fusion system appears in Definition 1.1 in [BLO], and the definition of a saturated fusion system appears in Definition 1.2 in [BLO]. Roughly speaking, a fusion system on S is a category whose objects are the subgroups of S, and such that the set hom_{F}(P, Q) of morphisms between subgroups P and Q of S is a set of injective group homomorphisms from P into Q satisfying some weak axioms. If G is a finite group and S ∈ Syl_{p}(G), then F_{S}(G) is the fusion system on F such that hom_{F_{S}(G)}(P, Q) consists of the conjugation maps c_{g} : x ↦ x^{g} for g ∈ G with P^{g} ≤ Q. Again, roughly speaking, F is saturated if it satisfies some axioms that are easily verified for F_{S}(G) using Sylow’s Theorem.

The reader is referred to Section 1 and Appendix A of [BLO] for notation, terminology, and basic results about fusion systems. However we also record some of this notation and terminology in this section, plus introducing some new notation.

Let P ≤ S, and let P^{F} = {P ϕ : ϕ ∈ hom_{F}(P, S)} be the set of F-conjugates of P. Recall P is fully normalized, centric if for all Q ∈ P^{F}, |N_{S}(P)| ≥ |N_{S}(Q)|, C_{S}(Q) ≤ Q, respectively. Write F^{f} for the set of nontrivial fully normalized subgroups of S.

Let S = {F_{i} : i ∈ I) be a set of subcategories F_{i} of F. Define the subsystem of F generated by S to be the smallest fusion system contained in F and containing each member of S. Write ⟨S⟩ for this subsystem. Thus ⟨S⟩ is the intersection of all fusion subsystems of F containing each member of S.

Recall that for P ∈ F^{f}, N_{F}(P), C_{F}(P) is the fusion system T on T = N_{S}(P), C_{S}(P) such that for Q ≤ T, hom_{T} (Q, T) consists of those ϕ ∈ hom_{F}(Q, T) such that ϕ extends to ϕ̂ ∈ hom_{F}(P Q, T) normalizing, centralizing P, respectively. By a result of Puig (appearing as Proposition A.6 in [BLO]), T is a saturated fusion system on T.

We say P is normal in F if F = N_{F}(P). There is a largest subgroup O_{p}(F) of S normal in F.

Further F is constrained if F has a normal centric subgroup. If F is constrained then by Proposition C in [BCGLO], there is a finite group G with S ∈ Syl_{p}(G), F^{∗}(G) = O_{p}(G), and F = F_{S}(G). Moreover G is unique up to an isomorphism extending the identity map on S, and we refer to any such group as a model for F. In particular if U ∈ F^{f} with N_{F}(U) constrained, then there exists a model for N_{F}(U), which we denote by G(U). Write 𝔊(F) for the set of models G(U), as U varies over the members of F^{f} with N_{F}(U) constrained. For G ∈ 𝔊(F), write S(G) for the Sylow group S ∩ G of G.

The system F is of characteristic p-type if for each 1 ≠ U ∈ F^{f}, N_{F}(U) is constrained. When p = 2, F is of even characteristic if N_{F}(U) is constrained for each 1 ≠ U ⊴ S. Finally F is a local CK-system if for each U ∈ F^{f}, Aut_{F}(U) is a K-group; here a finite group H is a K-group if each simple section of H is on the list K of “known” simple groups appearing in the statement of the theorem classifying the finite simple groups.

(1.1) Let F be a saturated constrained fusion system on a finite p-group S, and G a model for F. Assume E is a saturated constrained subsystem of F on S. Then there exists an overgroup H of S in G with E = F_{S}(H).

Proof. Set Q = O_{p}(E). As E is saturated and constrained, C_{S}(Q) = Z(Q) and there exists a model G_{1} for E. Further Q ⊴ E ≤ F, so E ≤ N_{F}(Q). Now N_{G}(Q) is a model for N_{F}(Q), which is a constrained and saturated fusion system, so replacing F by N_{F}(Q), we may assume Q ⊴ F.

Next Aut_{E}(Q) ≤ Aut_{F}(Q) = Aut_{G}(Q) ≅ G/Z(Q); let G_{2} be the preimage in G of Aut_{E}(Q). Observe τ_{i} = (G_{i}, S, Q), i = 1, 2, are triples in the set T of Definition 2.1 in [A1]. Let c_{i} : G_{i} → Aut(Q) be the conjugation map, and ι : S → S the identity map. As Aut_{G_{1}}(Q) = Aut_{E}(Q) = Aut_{G_{2}}(Q),

G_{1}c_{1}ι^{∗} = Aut_{G_{1}}(Q)ι^{∗} = Aut_{G_{1}}(Q) = Aut_{G_{2}}(Q) = G_{2}c_{2},

so ι : τ_{1} → τ_{2} is a morphism, as defined in Definition 2.1 of [A1]. Hence by 2.4.1 in [A1], there exists an isomorphism β : G_{1} → G_{2} extending ι, such that βc_{2} = c_{1}ι^{∗}. In particular F_{S}(G_{2}) = F_{S}(G_{1}) = E, so we can take H = G_{2}.

(1.2) Let F be a saturated fusion system on a finite p-group S, and set Q = O_{p}(F). Then

(1) If Q ≤ Z(F) then F/Q is saturated and O_{p}(F/Q) = 1.

(2) C_{F}(Q) is a saturated fusion system on C_{S}(Q), and if C_{F}(Q) is constrained then F is constrained.

Proof. Assume Q ≤ Z(F). The fusion system F/Q on S/Q is defined in section 8 of [A1], and by 8.10 in [A1], F/Q is saturated, and writing P for the preimage of O_{p}(F/Q) in S, P ⊴ F. Thus P ≤ Q = O_{p}(F), so O_{p}(F/Q) = P/Q = 1, establishing (1).

Set C = C_{F}(Q). Recall that C is a saturated system on T = C_{S}(Q). By 10.2.1 in [A2], C ⊴ F. Then by 7.4 and 7.10 in [A2], Z = O_{p}(C) ⊴ F, so Z ≤ Q. Therefore Z = Q ∩ T = Z(Q). Assume C is constrained. Then C_{T} (Z) ≤ Q. But C_{T} (Z) = T, so T ≤ Q. Thus C_{S}(Q) ≤ Q, so F is constrained, completing the proof of (2).

Section 2. An equivalence relation and ordering

In this section we assume:

Hypothesis 2.1. p is a prime and F is a saturated fusion system on a finite p-group S. Write U for the set of nontrivial normal subgroups of S, and assume for each U ∈ U that N_{F}(U) is constrained.

Notation 2.2. Let ι be the identity map on S. Write H for the set of pairs (H, U) such that U ∈ U and H ∈ O_{G(U)}(S). Denote by �_{S}(H) the set of nontrivial H-invariant subgroups of S. Write ℭ for the set of constrained, saturated fusion subsystems of F on S.

(2.3) For i = 1, 2, let U_{i} ∈ U and set G_{i} = G(U_{i}). Then (1) ι extends to an isomorphism ι_{1,2} : N_{G_{1}}(U_{2}) → N_{G_{2}}(U_{1}).

(2) ι_{1,2} is determined up to conjugacy by an element of Z(S), so for each H ∈ O_{N_{G_{1}} (U_{2})}(S), the image Hι_{1,2} of H in G_{2} is independent of the choice of ι_{1,2}. Proof. Observe that for i = 1, 2, U_{3−i} ≤ S = N_{S}(U_{i}), so (1) follows from 2.2.4 in [A3]. Further by 2.5.3 in [A1], if μ and ν are two isomorphisms extending ι, then ν = c_{z}μ for some z ∈ Z(S). Thus Hν = Hc_{z}μ = Hμ as z ∈ S ≤ H, so H = Hc_{z}. Thus (2) holds.

(2.4) Let I = {1, 2, 3} and for i ∈ I set G_{i} = G(U_{i}). For σ ∈ Sym(I), set G_{1σ,2σ,3σ} = N_{G_{1σ}}(U_{2σ}) ∩ N_{G_{1σ}}(U_{3σ}). Then

(1) G_{1,2,3}ι_{1,2} = G_{2,1,3}.

(2) G_{1,2,3}ι_{1,2}ι_{2,3} = G_{3,1,2}.

(3) (ι_{1,2}ι_{2,3})_{|G_{1,2,3}} = (c_{z}ι_{1,3})_{|G_{1,2,3}} for some z ∈ Z(S).

(4) For each H ∈ O_{G_{1,2,3}}(S), Hι_{1,2}ι_{2,3} = Hι_{1,3}. Proof. Set X = G_{1,2,3}. As ι_{1,2} extends ι, Xι_{1,2} ≤ G_{2,1,3}. By symmetry, G_{2,1,3}ι_{2,1} ≤ X, so (1) follows. Then by two applications of (1), Xι_{1,2}ι_{2,3} = G_{2,1,3}ι_{2,3} = G_{3,1,2}, establishing (2).

Set ξ = ι_{1,2}ι_{2,3}. By (1) and (2), ξι_{1,3}^{−1} is an automorphism of G_{1,2,3} extending ι, so (3) follows from 2.5 in [A1]. Then (3) implies (4), as in the proof of 2.3.2.

Definition 2.5. Define a relation ≡ on H by (H_{1}, U_{1}) ≡ (H_{2}, U_{2}) if U_{2} ∈ �_{S}(H_{1}) and H_{1}ι_{1,2} = H_{2}. By 2.3.2, this definition is independent of the choice of ι_{1,2}. We find in the next lemma that ≡ is an equivalence relation on H. Write ℌ for the set of equivalence classes of ≡, and write [(H, U)] for the equivalence class of (H, U) ∈ H. Sometimes we write H for [(H, U)], since the group H is determined up to an isomorphism extending ι.

(2.6) (1) The relation ≡ is an equivalence relation.

(2) If (H_{1}, U_{1}) ≡ (H_{2}, U_{2}) then �_{S}(H_{1}) = �_{S}(H_{2}).

(3) (H_{1}, U_{1}) ≡ (H_{2}, U_{2}) iff F_{S}(H_{1}) = F_{S}(H_{2}).

(4) The map H ↦ F_{S}(H) is a bijection between ℌ and ℭ.

Proof. Suppose (H_{1}, U_{1}) ≡ (H_{2}, U_{2}). Then ι_{1,2}H_{1} = H_{2}, so as ι_{1,2} extends ι, (2) follows and F_{S}(H_{1}) = F_{S}(H_{2}).

Assume next that F_{S}(H_{1}) = F_{S}(H_{2}) = E, and set U_{3} = O_{p}(E). Then U_{3} ∈ �_{S}(H_{i}), so (H_{i}, U_{i}) ≡ (H_{i}ι_{i,3}, U_{3}), and hence replacing H_{i} by H_{i}ι_{i,3}, we may assume U_{1} = U_{2} = U_{3}. Next Aut_{H_{1}}(U_{3}) = Aut_{E}(U_{3}) = Aut_{H_{2}}(U_{3}), so H_{1} and H_{2} are each the preimages in G(U_{3}) of Aut_{E}(U_{3}). This completes the proof of (3).

Finally (3) implies (1), and says that φ : H ↦ F_{S}(H) is a well defined injection of ℌ into ℭ. Suppose D ∈ ℭ. Then U = O_{p}(D) ∈ U, and D ≤ N_{F}(U), so D has a model H ≤ G(U) by 1.1. Then D = F_{S}(H) = Hφ, so φ : ℌ → ℭ is a surjection, completing the proof of (4) and the lemma.

Definition 2.7. Define a relation ≤ on ℌ by [(H_{1}, U_{1})] ≤ [(H_{2}, U_{2})] if U_{2} ∈ �_{S}(H_{1}) and H_{1}ι_{1,2} ≤ H_{2}. By 2.3.2, this definition is independent of the choice of ι_{1,2}. Further the proof of part (1) of the next lemma shows that U_{2} ∈ �_{S}(H_{1}) and H_{1}ι_{1,2} ≤ H_{2} iff F_{S}(H_{1}) ≤ F_{S}(H_{2}); hence by 2.6.3, the definition of the relation ≤ is independent of the choice of representatives for the equivalence classes.

(2.8) (1) For H_{i} ∈ ℌ, i = 1, 2, H_{1} ≤ H_{2} iff F_{S}(H_{1}) ≤ F_{S}(H_{2}).

(2) The relation ≤ is a partial ordering of ℌ.

(3) The map H ↦ F_{S}(H) is an isomorphism of the poset ℌ with ℭ partially ordered by inclusion.

Proof. Assume [(H_{1}, U_{1})] ≤ [(H_{2}, U_{2})]. Then U_{2} ∈ �_{S}(H_{1}) and H_{1}ι_{1,2} ≤ H_{2}. Hence by 2.6.3, F_{S}(H_{1}) = F_{S}(H_{1}ι_{1,2}) ≤ F_{S}(H_{2}).

Next suppose instead that F_{S}(H_{1}) ≤ F_{S}(H_{2}). As U_{2} ⊴ H_{2}, also U_{2} ⊴ F_{S}(H_{1}), so U_{2} ∈ �_{S}(H_{1}). By 2.6.3, F_{S}(H_{1}ι_{1,2}) = F_{S}(H_{1}) ≤ F_{S}(H_{2}). Hence

Aut_{H_{1}ι_{1,2}}(U_{2}) = Aut_{F_{S}(H_{1})}(U_{2}) ≤ Aut_{F_{S}(H_{2})}(U_{2}) = Aut_{H_{2}}(U_{2}),

so the preimage H_{1}ι_{1,2} in G(U_{2}) of Aut_{H_{1}ι_{1,2}}(U_{2}) is contained in the preimage H_{2} of Aut_{H_{2}}(U_{2}). This establishes (1).

Observe part (1) and 2.6.3 implies part (2), while (1), (2), and 2.6.4 imply (3).

Definition 2.9. For [(H_{i}, U_{i})] ∈ ℌ, i = 1, 2, define [(H_{1}, U_{1})]∩[(H_{2}, U_{2})] (or just H_{1}∩H_{2}) to be [(N_{H_{1}}(U_{2})ι_{1,2} ∩ H_{2}, U_{2})].

Define F_{S}(H_{1}) ∧ F_{S}(H_{2}) to be the subfusion system of F on S generated by the maps in Aut_{N_{H_{1}} (U_{2})}(P)∩Aut_{N_{H_{2}} (U_{1})}(P), as P varies over the subgroups of S containing C_{S}(P). By the next lemma and 2.6.3, the definition of [(H_{1}, U_{1})] ∩ [(H_{2}, U_{2})] is independent of the choice of representatives of the classes.

(2.10) For [(H_{i}, U_{i})] ∈ ℌ, i = 1, 2, F_{S}(N_{H_{1}}(U_{2})ι_{1,2} ∩ H_{2}) = F_{S}(H_{1}) ∧ F_{S}(H_{2}).

Proof. Set F_{i} = F_{S}(H_{i}), E = F_{1} ∧ F_{2}, and D = F_{S}(N_{H_{1}}(U_{2})ι_{1,2} ∩ H_{2}). Let P ≤ S with C_{S}(P) ≤ P and α ∈ Aut_{D}(P). Then α = c_{h|P} for some h ∈ N_{H_{1}}(U_{2})ι_{1,2} ∩ N_{H_{2}}(P). By 2.6.2, h acts on U_{1} and U_{2}, so α ∈ Aut_{N_{H_{2}} (U_{1})}(P) and

α ∈ Aut_{N_{H_{1}ι_{1,2}} (U_{2})}(P) = Aut_{F_{S}(N_{H_{1}} (U_{2})ι_{1,2})}(P) = Aut_{F_{S}(N_{H_{1}} (U_{2}))}(P) = Aut_{N_{H_{1}} (U_{2})}(P),

by 2.6.3. Therefore Aut_{D}(P) ≤ Aut_{N_{H_{1}} (U_{2})}(P) ∩ Aut_{N_{H_{2}} (U_{1})}(P). Further as D is saturated, Alperin’s Fusion Theorem (cf. A.10 in [BLO]) says that D is generated by the subcategories Aut_{D}(P), as P varies over the subgroups of S containing C_{S}(P). Therefore D ≤ E.

Conversely assume β ∈ Aut_{N_{H_{1}} (U_{2})}(P) ∩ Aut_{N_{H_{2}} (U_{1})}(P). Then, as above, β = c_{h_{i}|P} for some h_{1} ∈ N_{H_{1}}(U_{2})ι_{1,2} and h_{2} ∈ N_{H_{2}}(U_{1}), so h_{1}h_{2}^{−1} ∈ C_{G(U_{2})}(P) = Z(P), as F^{∗}(N_{G(U_{2})}(P)) = O_{p}(N_{G(U_{2})}(P)) and C_{S}(P) ≤ P. Thus h_{1} = c_{z}h_{2} for some z ∈ Z(P), so h_{1} ∈ N_{H_{1}}(U_{2})ι_{1,2} ∩ N_{H_{2}}(U_{1}). Therefore β ∈ Aut_{D}(P), so E ≤ D.

(2.11) Let H_{i} ∈ ℌ, 1 ≤ i ≤ n, for some n ≥ 2. Then (1) H_{1} ∩ H_{2} = H_{2} ∩ H_{1} is the greatest lower bound for H_{1} and H_{2} in the poset ℌ.

(2) There exists a greatest lower bound H_{1} ∩ ⋯ ∩ H_{n} for {H_{1}, …, H_{n}} in ℌ.

(3) (H_{1} ∩ H_{2}) ∩ H_{3} = H_{1} ∩ H_{2} ∩ H_{3} = H_{1} ∩ (H_{2} ∩ H_{3}).

(4) For E, D ∈ ℭ, E ∧ D is the greatest lower bound for E and D in ℭ.

Proof. Set F_{i} = F_{S}(H_{i}). By definition, F_{1} ∧ F_{2} ≤ F_{i} for i = 1, 2, so by 2.8.1 and 2.10, H_{1} ∩H_{2} ≤ H_{i}. Suppose H_{3} ≤ H_{i} for i = 1, 2. Then by 2.8.1, F_{S}(H_{3}) ≤ F_{S}(H_{i}), so using Alperin’s Fusion Theorem (cf. A.10 in [BLO]), F_{S}(H_{3}) ≤ F_{1} ∧ F_{S}(H_{2}). Then by 2.8.1 and 2.10, H_{3} ≤ H_{1} ∩ H_{2}. Therefore H_{1} ∩ H_{2} is the greatest lower bound for H_{1} and H_{2} in ℌ. By symmetry, H_{2} ∩ H_{1} is also the greatest lower bound, so H_{1} ∩ H_{2} = H_{2} ∩ H_{1}, establishing (1). Then (1) and elementary lattice theory imply (2) and (3). Finally 2.8.3 and (1) imply (4).

(2.12) Let H_{i} = [(H_{i}, U_{i})] ∈ ℌ, for i = 1, 2, and assume U_{3} ∈ �_{S}(H_{1})∩�_{S}(H_{2}). Set G_{3} = G(U_{3}). Then

(1) H_{i} ≤ [(G_{3}, U_{3})] for i = 1, 2.

(2) Set H_{3} = ⟨H_{1}ι_{1,3}, H_{2}ι_{2,3}⟩. Then F_{S}(H_{3}) is the smallest member F_{S}(H_{1})∨F_{S}(H_{2}) of ℭ containing F_{S}(H_{i}) for i = 1, 2.

(3) H_{1} ∨ H_{2} = [(H_{3}, U_{3})] is the least upper bound for H_{1} and H_{2} in ℌ.

(4) If E, D ∈ ℭ have a common nontrivial normal subgroup, then E ∨ D is the least upper bound for E and D in ℭ.

(5) If [(K, U)] ∈ ℌ and K_{i} ∈ O_{K}(S) for 1 ≤ i ≤ n with K = ⟨K_{1}, …, K_{n}⟩, then [(K, U)] is the least upper bound K_{1} ∨ ⋯ ∨ K_{n} for [(K_{1}, U)], …, [(K_{n}, U)] in the poset ℌ. In particular if H ∈ ℌ with [(K_{i}, U)] ≤ H for each 1 ≤ i ≤ n, then [(K, U)] ≤ H.

Proof. By hypothesis, U_{3} ∈ �_{S}(H_{i}) for i = 1, 2, so (1) follows. By construction, H_{i} ≤ H_{3} for i = 1, 2, so F_{i} = F_{S}(H_{i}) ≤ F_{3} = F_{S}(H_{3}) by 2.8.1. Thus F_{3} is an upper bound for F_{1} and F_{2} in ℭ.

Suppose F_{4} is an upper bound for F_{1} and F_{2} in ℭ. By 2.8.3, F_{4} = F_{S}(H_{4}) for some H_{4} ∈ ℌ, and H_{i}ι_{i,4} ≤ H_{4}. Thus H_{5} = ⟨H_{1}ι_{1,4}, H_{2}ι_{2,4}⟩ ≤ H_{4} and by 2.8.3, F_{i} ≤ F_{S}(H_{5}) = F_{5} ≤ F_{4}. As U_{3} ∈ �_{S}(H_{i}), also U_{3} ∈ �_{S}(H_{5}) by 2.6.2. By 2.4.4, H_{i}ι_{i,4}ι_{4,3} = H_{i}ι_{i,3}, so H_{5}ι_{4,3} = H_{3}, and hence F_{4} = F_{5} by 2.6.3. Thus F_{3} = F_{5} ≤ F_{4}, completing the proof of (2). Then applying the isomorphism of 2.8.3, (2) implies (3) and (4).

Assume the hypothesis of (5). Proceeding by induction on n, we may take n = 2. Then by (3), K = K_{1} ∨ K_{2}. Hence if K_{i} ≤ H for i = 1, 2, then K ≤ H, completing the proof of (5).

Section 3. More orderings of ℌ

In this section we continue to assume Hypothesis 2.1, and adopt the notation established in section 2, such as Notation 2.2, and Definitions 2.5, 2.7, and 2.9.

Definition 3.1. Let Z = Ω_{1}(Z(S)) and pick 1 ≠ E ≤ Z. Given H_{1} = [(H_{1}, U_{1})] ∈ ℌ, define

V (H_{1}) = V_{E}(H_{1}) = ⟨E^{H_{1}}⟩ ≤ S.

Observe that if (H_{1}, U_{1}) ≡ (H_{2}, U_{2}) then

V (H_{1}) = V (H_{1})ι_{1,2} = ⟨E^{H_{1}}⟩ι_{1,2} = ⟨Eι_{1,2}^{H_{1}ι_{1,2}}⟩ = ⟨E^{H_{2}}⟩ = V (H_{2}),

so the definition of V (H_{1}) is independent of the choice of representative for the equivalence class.

Also U_{3} = V (H_{1}) ∈ �_{S}(H_{1}) and (H_{1}, U_{1}) ≡ (H_{1}ι_{1,3}, U_{3}). Therefore our convention in this section will be that our canonical representative for 𝔥 ∈ ℌ will be (H, U), where U = V (𝔥) and H is the unique subgroup of G(U) such that (H, U) ∈ 𝔥.

Write ℌ_{E} for the set of H ∈ ℌ such that H = G(V (H)).

Define a relation ≲ = ≲_{E} on ℌ by H_{1} ≲ H_{2} if H_{1} = (N_{H_{2}}(U_{1})ι_{2,1} ∩ H_{1})C_{H_{1}}(U_{1}). For H ∈ ℌ, let O(H) = {K ∈ ℌ : H ≤ K} and denote by M(H) the set of maximal

members of O(H). Notice M(H) ⊆ ℌ_{E}.

(3.2) Let H_{i} ∈ ℌ, for i = 1, 2. Then H_{1} ≲ H_{2} iff Aut_{F_{S}(H_{1})∧F_{S}(H_{2})}(V (H_{1})) = Aut_{F_{S}(H_{1})}(V (H_{1})).

Proof. Let V = V (H_{1}), and recall from 3.1 that, by convention, V = U_{1}. Set H = N_{H_{2}}(U_{1})ι_{2,1} ∩ H_{1} and for i = 1, 2, set F_{i} = F_{S}(H_{i}). By 2.10, F_{1} ∧ F_{2} = F_{S}(H), so Aut_{H}(V) = Aut_{F_{1}∧F_{2}}(V). Therefore as Aut_{H_{1}}(V) = Aut_{F_{1}}(V), it follows that Aut_{F_{1}∧F_{2}}(V) = Aut_{F_{1}}(V) iff Aut_{H}(V) = Aut_{H_{1}}(V) iff H_{1} = HC_{H_{1}}(V) iff H_{1} ≲ H_{2}.

(3.3) Suppose H_{i} ∈ ℌ, for i = 1, 2, with H_{2} ∈ ℌ_{E}. Then (1) N_{H_{2}}(U_{1})ι_{2,1} ∩ H_{1} = N_{H_{1}}(U_{2}).

(2) H_{1} ≲ H_{2} iff H_{1} = N_{H_{1}}(U_{2})C_{H_{1}}(U_{1}).

Proof. As H_{2} ∈ ℌ_{E}, H_{2} = G(U_{2}), so N_{H_{2}}(U_{1})ι_{2,1} = N_{G(U_{1})}(U_{2}) by 2.3.1. Thus (1) follows, and (1) implies (2).

(3.4) If H_{i} ∈ ℌ, for i = 1, 2, with H_{1} ≲ H_{2} then:

(1) U_{1} = V (H_{1}) ≤ V (H_{2}) = U_{2}.

(2) C_{H_{2}}(U_{2}) ≤ C_{H_{2}}(U_{1}).

(3) If H_{2} ∈ ℌ_{E} then C_{H_{2}}(U_{1})ι_{2,1} ∩ H_{1} = N_{H_{1}}(U_{2}) ∩ C_{H_{1}}(U_{1}), so C_{H_{2}}(U_{2})ι_{2,1} ∩ H_{1} ≤ C_{H_{1}}(U_{1}).

Proof. As H_{1} ≲ H_{2}, H_{1} = (N_{H_{2}}(U_{1})ι_{2,1} ∩ H_{1})C_{H_{1}}(U_{1}). Thus

U_{1} = ⟨E^{H_{1}}⟩ = ⟨E^{N_{H_{2}} (U_{1})ι_{2,1}∩H_{1}}⟩ ≤ ⟨E^{N_{H_{2}} (U_{1})ι_{2,1}}⟩ = ⟨E^{N_{H_{2}} (U_{1})}⟩ι_{2,1} = ⟨E^{N_{H_{2}} (U_{1})}⟩ ≤ ⟨E^{H_{2}}⟩ = U_{2},

establishing (1). Then (1) implies (2).

Assume H_{2} ∈ ℌ_{E}. By 3.3.1, N_{H_{2}}(U_{1})ι_{2,1} ∩ H_{1} = N_{H_{1}}(U_{2}), so as ι_{2,1} extends ι, the first statement in (3) follows. Then the second statement of (3) follows from the first statement and (2).

(3.5) Assume H_{i} ∈ ℌ_{E} for i = 1, 2. Then the following are equivalent:

(1) H_{1} = H_{2}.

(2) V (H_{1}) = V (H_{2}).

(3) H_{1} ≲ H_{2} ≲ H_{1}.

Proof. Trivially, (1) implies (3). As H_{i} ∈ ℌ_{E}, H_{i} = G(U_{i}), so (2) implies (1). Finally by 3.4.1, (3) implies (2).

(3.6) ≲ is a partial ordering of ℌ_{E}.

Proof. Trivially, ≲ is reflexive. By 3.5, ≲ is antisymmetric. Assume H_{1} ≲ H_{2} ≲ H_{3}. By 3.3.2, H_{2} = N_{H_{2}}(U_{3})C_{H_{2}}(U_{2}), and by 3.4.3 and as H_{1} = G(U_{1}), C_{H_{2}}(U_{2})ι_{2,1} ≤ C_{H_{1}}(U_{1}). Therefore C_{H_{2}}(U_{2}) ≤ N_{H_{2}}(U_{1}). Then

N_{H_{2}}(U_{1}) = N_{H_{2}}(U_{1}) ∩ N_{H_{2}}(U_{3})C_{H_{2}}(U_{2}) = (N_{H_{2}}(U_{1}) ∩ N_{H_{2}}(U_{3}))C_{H_{2}}(U_{2}),

so by 2.3.1,

N_{H_{1}}(U_{2}) = N_{H_{2}}(U_{1})ι_{2,1} = (N_{H_{2}}(U_{1}) ∩ N_{H_{2}}(U_{3}))ι_{2,1}C_{H_{2}}(U_{2})ι_{2,1} ≤ N_{H_{1}}(U_{3})C_{H_{1}}(U_{1}),

and hence H_{1} ≲ H_{3}, so that ≲ is transitive.

(3.7) Let M be maximal in M(S) with respect to ≲. Then (1) {M} = M(X) for each X ∈ ℌ with X ≤ M and M = XC_{M}(V (M)).

(2) Let R = C_{S}(V (M)). Then R = O_{2}(N_{M}(R)), Aut_{M}(V (M)) = Aut_{N_{M} (R)}(V (M)), and M(N_{M}(R)) = {M}.

(3) Let H ∈ ℌ_{E}. Then H is maximal in ℌ_{E} with respect to ≲ iff H ∈ M(S) and H is maximal in M(S) with respect to ≲.

Proof. Assume H is maximal in either ℌ_{E} or M(S) with respect to ≲, and set U = V (H). Suppose X ∈ ℌ with X ≤ H and H = XC_{M}(U). If M_{2} ∈ M(X) then as H = XC_{H}(U), then also H = N_{H}(V (M_{2}))C_{H}(U), so H ≲ M_{2} by 3.3.2. Therefore H = M_{2} by maximality of H and 3.6. This proves (1), and shows that if H is maximal in ℌ_{E} with respect to ≲, then for M_{3} ∈ M(H), H = M_{3}. Thus one implication of (3) is also established.

Let V = V (M). By a Frattini argument, M = N_{M}(R)C_{M}(V), so Aut_{M}(V) = Aut_{N_{M} (R)}(V). Now V = ⟨E^{M}⟩, so as M = N_{M}(R)C_{M}(V), V = ⟨E^{N_{M} (R)}⟩. Hence it follows from B.2.14 in [ASm] that O_{2}(N_{M}(R)) ≤ C_{S}(V) = R, so R = O_{2}(N_{M}(R)). Then by (1) {M} = M(N_{M}(R)) by (1), completing the proof of (2).

Let M ≲ K with K maximal in ℌ_{E} with respect to ≲. By paragraph one of the proof, K ∈ M(S), so M = K as M is maximal in M(S) with respect to ≲. This completes the proof of (3).

(3.8) Let H_{i} ∈ ℌ for i = 1, 2. Then (1) If H_{1} ≤ H_{2} then H_{1} ≲ H_{2}.

(2) If H_{i} ≲ H_{3−i} for i = 1 and 2, then V (H_{1}) = V (H_{2}).

Proof. Part (1) is immediate from the definitions, while part (2) follows from 3.4.1.

Section 4. Further results on ℌ

In this section we assume the following hypothesis:

Hypothesis 4.1. Assume Hypothesis 2.1, and in addition assume F = ⟨N_{F}(U) : U ∈ U⟩ and F is not constrained. For H ∈ ℌ, let X (H) be the set of minimal parabolics of H over S. We say that H is a uniqueness group if |M(H)| = 1.

If G is a finite group with Sylow 2-subgroup T, a minimal parabolic of G over T is an overgroup H of T in G such that T is not normal in H, and T is contained in a unique maximal subgroup of H.

(4.2) O_{p}(F) = 1.

Proof. If U = O_{p}(F) ≠ 1 then U ∈ U, so by Hypothesis 2.1, F = N_{F}(U) is constrained, contrary to Hypothesis 4.1.

(4.3) Let M ∈ M(S) and H ∈ ℌ. Then (1) There exists K ∈ ℌ with K ≰ M.

(2) Either N_{K}(S) ≰ M or there exists X ∈ X (K) with X ≰ M.

(3) H ≤ M iff �_{S}(M)∩�_{S}(H) ≠ ∅.

Proof. Assume (1) fails. Then by 2.8.3, for each U ∈ U, N_{F}(U) ≤ F_{S}(M). But then by 4.1, F = ⟨N_{F}(U) : U ∈ U⟩ ≤ F_{S}(M), so 1 ≠ O_{p}(M) ⊴ F, contrary to 4.2. This establishes (1).

By McBride’s Theorem (cf. B.6.5 in [ASm]), K = ⟨N_{K}(S), X (K)⟩, so (2) follows from 2.12.5.

If H ≤ M then O_{p}(M) ∈ �_{S}(M)∩�_{S}(H). Conversely if �_{S}(M)∩�_{S}(H) ≠ ∅ then by 2.12.1 there exists H ∈ ℌ with M, K ≤ H. But then K ≤ H = M by maximality of M.

(4.4) Let M ∈ M(S),and let H ∈ ℌ. Then (1) Assume Y ≤ M is a uniqueness group. Then H ≤ M iff �_{S}(Y)∩�_{S}(H) ≠ ∅.

(2) Set R = C_{S}(V (M)). Then R = O_{2}(N_{M}(R)).

Proof. Assume the hypothesis of (1). One implication in (1) follows from 4.3.3. Suppose �_{S}(Y)∩�_{S}(H) ≠ ∅. Then by 2.12.1, there exists M′ ∈ M(S) with Y, H ≤ M′. Then M′ = M as Y is a uniqueness group, completing the proof of (1).

By B.2.14 in [ASm], O_{2}(N_{M}(R)) ≤ C_{S}(V) = R, so (2) holds.

(4.5) Let M be maximal in M(S) with respect to ≲, and let H ∈ ℌ. Set R = C_{S}(V (M)) and Y = N_{M}(R). Then

(1) M(Y) = {M} and R = O_{2}(Y).

(2) H ≤ M iff �_{S}(Y)∩�_{S}(H) ≠ ∅.

Proof. Part (1) follows from 3.7.2. Then (1) and 4.4.1 imply (2).

(4.6) Let H ∈ ℌ. Then the following are equivalent:

(1) J(S) centralizes V (H).

(2) Baum(S) centralizes V (H).

(3) H = N_{H}(Baum(S))C_{H}(V (H)).

(4) H ≲ G(Baum(S)).

Proof. Set V = V (H), D = Ω_{1}(Z(J(S))), B = Baum(S), and K = G(B). If J(S) centralizes V then V ≤ D, so B = C_{S}(D) ≤ C_{S}(V). Thus (1) implies (2).

Next as B is weakly closed in S, (2) implies (3) by a Frattini argument.

Assume (3). By 2.3.1, there exists an isomorphism ξ : N_{G(V (K)}(V) → N_{G(V)}(V (K)) extending ι. We may take K = N_{G(V (K))}(B), so N_{K}(V) = N_{G(V (K))}(V) ∩ N_{G(V (K))}(B) and hence N_{K}(V)ξ = N_{G(V)}(V (K)) ∩ N_{G(V)}(B). Then N_{H}(B) ≤ N_{K}(V)ξ ∩ H, so H ≲ K; that is (3) implies (4).

Finally assume (4) and set H^{∗} = H/C_{H}(V). Then H = (N_{K}(V)ξ ∩ H)C_{H}(V) and B ⊴ N_{K}(V)ξ ∩ H, so B^{∗} ≤ O_{p}(H^{∗}) = 1 as E ≤ Z. Therefore J(S) ≤ B ≤ C_{H}(V), so (4) implies (1).

(4.7) Assume J(S) centralizes each member of {V (H) : H ∈ ℌ}. Then (1) M(G(Baum(S))) = {M} is of order 1, where M = G(V (G(Baum(S)))).

(2) M is the unique maximal member of M(S) under ≲.

Proof. Set B = Baum(S) and K = G(B). By 4.6, for each H ∈ ℌ, H ≲ K. On the other hand for M′ ∈ M(K), 3.8 says K ≲ M′ and V (K) = V (M′). Thus (1) follows from 3.5. Further for M′′ ∈ M(S), M′′ ≲ K ≲ M, so if M′′ ≠ M then M′′ is not maximal in M(S) with respect to ≲ by 3.6, establishing (2).

Section 5. Systems of even characteristic

In this section we assume one of the following two hypotheses:

Hypothesis 5.1. Assume Hypothesis 4.1 with p = 2.

Hypothesis 5.2. Assume Hypothesis 2.1 with p = 2. Assume H_{i} ∈ ℌ for i = 1, 2 and set R_{i} = O_{2}(H_{i}) and V_{i} = V (H_{i}). Assume in addition that:

(1) R_{1} ∈ Syl_{2}(C_{H_{1}}(V_{1})).

(2) H_{2} is a minimal parabolic over S.

(3) �_{S}(H_{1})∩�_{S}(H_{2}) = ∅.

(5.3) Assume Hypothesis 5.1. Assume M ∈ M(S) and G(S) ≤ M. Let ℌ_{M} = {H ∈ ℌ : H ≰ M}, and for H ∈ ℌ_{M} let X_{M}(H) = {X ∈ X (H) : X ≰ M} and X_{M}^{∗}(H) the minimal members of X_{M}(H) under inclusion. Then (1) ℌ_{M} ≠ ∅ and for each H ∈ ℌ_{M}, X_{M}(H) ≠ ∅.

(2) Let H ∈ ℌ_{M} and X ∈ X_{M}(H). Set R = C_{S}(V (M)) and Y = N_{M}(R). If Y is a uniqueness group then the pair Y, X satisfies Hypothesis 5.2.

(3) Let H ∈ ℌ_{M} and X ∈ X_{M}^{∗}(H). Then N_{X}(V (M)) is the unique maximal overgroup of S in X.

Proof. Part (1) is a consequence of parts (1) and (2) of 4.3.

Assume the hypothesis of (2) with Y a uniqueness group. Then the pair Y, X satisfies condition (1) and (3) of Hypothesis 5.2 by 4.4, and as X is a minimal parabolic, the pair satisfies condition (2) of Hypothesis 5.2. Thus (2) holds.

Assume that X ∈ X_{M}^{∗}(H) and set V = V (M). Then S ≤ N_{X}(V) < X, so N_{X}(V) ≤ Y_{X} the unique maximal overgroup of S in X. Suppose N_{X}(V) ≠ Y_{X}. Then as M = G(V), Y_{X} ∈ ℌ_{M}, so by (1) there is X′ ∈ X_{M}(Y_{X}). Then X′ ≤ Y_{X} < X, contrary to the minimality of X. This establishes (3).

If G is a finite group and V is a faithful F_{2}G-module then the parameters q(G, V) and q̂(G, V) are defined in Definitions B.1.1 and B.4.1 in [ASm].

Theorem 5.4. (Stellmacher qrc-Lemma for fusion systems) Assume Hypothesis 5.2, and set q = q(H_{1}/C_{H_{1}}(V_{1}), V_{1}) and U = ⟨V_{1}^{H_{2}}⟩. Then one of the following holds:

(1) V_{1} ≰ R_{2}.

(2) q ≤ 1.

(3) The dual of V_{1} is an FF-module for H_{1}/C_{H_{1}}(V_{1}).

(4) q ≤ 2, U is abelian, and H_{2} has c ≥ 2 noncentral chief factors on U. If q = 2 then c = 2 and for each noncentral H_{2}-chief factor W on U, q(H_{2}/C_{H_{2}}(W), W) = 1.

(5) R_{1}∩R_{2} ⊴ H_{2}, U is abelian, H_{2} has one noncentral chief factor W, q(H_{2}/C_{H_{2}}(W), W) = 1, [U, O^{2}(H_{2})] ≤ Z(R_{2}), O^{2}(H_{2}) = [O^{2}(H_{2}), J(R_{1})], and J(R_{1}) = J(S).

Proof. This follows from the qrc-Lemma for groups, which appears in D.1.5 in [ASm]. For example we can choose representatives (H_{i}, V_{i}) for the equivalence classes H_{i}, i = 1, 2, set H_{1,2} = N_{H_{1}}(V_{2}) ∩ N_{H_{2}}(V_{1})ι_{2,1}, and form the amalgam α = (α_{i} : H_{1,2} → H_{i} : i = 1, 2), where α_{1} is the inclusion map, and α_{2} = ι_{1,2} : H_{1,2} → H_{2}. Then we can form the free amalgamated product G of α and identify H_{J}, J ⊆ {1, 2} with the corresponding subgroups of G, to obtain Hypothesis D.1.1 of [ASm], the hypothesis of D.1.5 in [ASm].

Note the second statement in (4) appears D.1.3 of [ASm]; more precisely the fact that q(H_{2}/C_{H_{2}}(W), W) = 1 appears in the last paragraph of the proof of that lemma. Similarly the corresponding statement in (5) appears in D.1.4 of [ASm].

(5.5) Assume Hypothesis 5.2, with V_{1} ≰ R_{2}. Assume in addition that N_{H_{2}}(V_{1}) is the unique maximal overgroup of S in H_{2} and q(H_{1}/C_{H_{1}}(V_{1}), V_{1}) > 1. Let Γ = Γ(H_{2}, V_{1}) be the set of subgroups ⟨V_{1}, V_{1}^{g}⟩ of H_{2} which are not 2-groups, and let Γ_{∗} = Γ_{∗}(H_{2}, V) be the set of minimal members of Γ.

(1) Γ ≠ ∅. Pick X ∈ Γ_{∗} and set U_{1} = V_{1} ∩ R_{2}.

(2) N_{X}(V_{1}) is the unique maximal overgroup M of V_{1} in X, and V_{1} is weakly closed in M with respect to X. Pick g ∈ X − M and set U = U_{1}U_{1}^{g}, and I = U_{1} ∩ U_{1}^{g}.

(3) X = ⟨V_{1}, V_{1}^{g}⟩, I ≤ Z(X), and U ⊴ X. Set X̄ = X/I and m = m(Ū_{1}).

(4) Ū = Ū_{1} ⊕ Ū_{1}^{g} is elementary abelian of rank 2m, for some positive integer m.

(5) Let W_{1} = V_{1} ∩ O_{2}(X), W = W_{1}W_{1}^{g}, and J = V_{1} ∩ V_{1}^{g}. Then J ≤ Z(X) and W ⊴ X. Set X̃ = X/J. Then W̃ = W̃_{1} ⊕ W̃_{1}^{g} is elementary abelian of rank 2k, where k = m(W̃_{1}) ≥ m, and W̃_{1} = [W̃ , V_{1}] = CW̃ (v) for each v ∈ V_{1} − W_{1}.

(6) Set e = m_{2}(V_{1}/W_{1}). Then either e = 1 and X/W ≅ D_{2n} for some odd integer n, or e > 1 and X/W ≅ L_{2}(2^{e}) or Sz(2^{e}).

(7) If e ≥ k then e = k, q̂(H_{1}/C_{H_{1}}(V_{1}), V_{1}) ≤ 2, X/W ≅ L_{2}(2^{e}), and q(X/C_{X}(W̃), W̃) ≥ 1.

(8) If e < k then q̂(H_{1}/C_{H_{1}}(V_{1}), V_{1}) < 2.

Proof. As V_{1} ≰ R_{2}, (1) follows from the Baer-Suzuki Theorem (cf. 39.6 in [FGT]). Pick X ∈ Γ_{∗}. Applying the Baer-Suzuki Theorem again, there is g ∈ X with ⟨V_{1}, V_{1}^{g}⟩ not a 2-group, and by minimality of X, X = ⟨V_{1}, V_{1}^{g}⟩.

As V_{1} is abelian, I ≤ Z(X). As R_{2} and V_{1} are normal in S, [V_{1}, R_{2}] ≤ U_{1}. Then as R_{2} ⊴ H_{2}, also [V_{1}^{g}, R_{2}] ≤ U_{1}^{g}. Hence U ⊴ X, completing the proof of (3), modulo (2), which shows that for each g ∈ X − M, ⟨V_{1}, V_{1}^{g}⟩ = X.

Next as U_{1} ⊴ S and, by hypothesis, N_{H_{2}}(V_{1}) is the unique maximal overgroup of S in H_{2}, it follows that N_{H_{2}}(U_{1}) ≤ N_{H_{2}}(V_{1}). Therefore U_{1} ≠ I, so m = m(Ū_{1}) is positive. Now (4) follows.

By minimality of X:

(a) For each x ∈ X, either X = ⟨V_{1}, V_{1}^{x}⟩, or ⟨V_{1}, V_{1}^{x}⟩ is a 2-group. Then it follows from (a) and Baer-Suzuki that:

(b) If V_{1} ≤ Y < X then V_{1} ≤ O_{2}(Y). For V_{1} ≤ Y < X, set P (Y) = ⟨V_{1}^{X} ∩ Y ⟩. It follows from (b) that:

(c) P (Y) is a normal 2-subgroup of Y. Let V_{1} ≤ T ∈ Syl_{2}(X) and M = N_{X}(P (T)). We claim:

(d) M is the unique maximal overgroup of V_{1} in X.

If not, choose V_{1} ≤ Y < X with Y ≰ M and P = P (Y ∩ M) maximal subject to this constraint. If P (T) = P then also P = P (Y), so Y ≤ N_{X}(P (Y)) ≤ M, a contradiction. Thus P < P (N_{P (T)}(P)) and as V_{1} ≤ P but V_{1} ≰ O_{2}(X), N_{X}(P) < X, so by maximality of P, N_{X}(P) ≤ M. Then N_{P (Y)}(P) ≤ M, so N_{P (Y)}(P) = P, and hence P (Y) = P. But then Y ≤ N_{X}(P) ≤ M, for our final contradiction establishing (d).

Next:

(e) For each involution t ∈ X − M, CŪ_{1}(t) = 1, so m_{2}(CŪ(t)) = m.

Namely by (d), X = ⟨V_{1}, t⟩, so X centralizes CŪ_{1}(t), and hence its preimage U_{2} in X is normal in X. Thus U_{2} ≤ U_{1} ∩ U_{1}^{g} = I, so U_{2} = I. Then as m_{2}(Ū) = 2m and m_{2}(CŪ(t)) ≥ m_{2}(Ū)/2, (e) follows.

(f) For each v ∈ V_{1} − O_{2}(X), CŪ(v) = Ū_{1}.

Again by Baer-Suzuki there is x ∈ X with ⟨v, v^{x}⟩ not a 2-group, so as V_{1} ≤ P (T) ≤ O_{2}(M), v^{x} ∉ M. Hence by (e), m_{2}(CŪ(v)) = m_{2}(CŪ(v^{x})) = m = m_{2}(Ū_{1}), so as v centralizes U_{1}, (f) follows.

(g) M = N_{X}(V_{1}) and V_{1} is weakly closed in M with respect to X.

If V_{1} is weakly closed in M then P (T) = V_{1}, so M = N_{X}(V_{1}). Thus we may assume V_{1} is not weakly closed in M, so as ⟨V_{1}^{X} ∩ M⟩ = P (T), V_{1} is not weakly closed in P (T). Then by Alperin’s Fusion Theorem (cf. A.10 in [BLO]), there is an overgroup Q of V_{1} in P (T) such that V_{1} ⊴ Q, and h ∈ N_{X}(Q) such that V_{1} ≠ V_{1}^{h}. Suppose 1 ≠ [V_{1}, V_{1}^{h}]. Then replacing h by h^{−1} if necessary, we may assume m(V^{h}/C_{V^{h}}(V)) ≥ m(V/C_{V} (V^{h})), contradicting q(H_{1}/C_{H_{1}}(V_{1}), V_{1}) > 1.

Therefore [V_{1}, V_{1}^{h}] = 1. But by (f), CŪ(V_{1}^{h}) = Ū_{1}^{h}, so U_{1} = U_{1}^{h}. However we saw during the proof of (4) that N_{H_{2}}(U_{1}) ≤ N_{H_{2}}(V_{1}). Thus V_{1} = V_{1}^{h}, contrary to the choice of h. This completes the proof of (g).

Notice (d) and (g) prove (2), and hence complete the proof of (3).

As M = N_{X}(V_{1}), W_{1} = V_{1} ∩ O_{2}(X) ⊴ M, so M = N_{X}(W_{1}) or X by (d). However if W_{1} ⊴ X then as U_{1} ≤ W_{1}, also U ≤ W_{1} ≤ V_{1}, so U ≤ V_{1} ∩ R_{2} = U_{1}, contrary to (4). Thus M = N_{X}(W_{1}). Then we can apply various arguments above to W_{1} in place of U_{1}, to establish (5).

Set X^{∗} = X/W. If W_{1} < U_{2} ≤ V_{1} and x ∈ X with U_{2}^{x} ≤ M, then as CW̃ (U_{2}) = W̃_{1} by (5), and as CW̃_{1}(U_{2}^{x}) ≠ 1, it follows that x ∈ M. Therefore V_{1}^{∗} is a strongly closed elementary abelian TI-subgroup of M^{∗}. Then as X^{∗} = ⟨V_{1}^{∗}, V_{1}^{∗g}⟩, (6) follows (cf. I.8.3 in [ASm]).

Let A = W_{1}^{g}. Then A ≤ M = N_{X}(V_{1}), and by (5), [A, V_{1}] = W_{1} and [A, W_{1}] ≤ J ≤ C_{V_{1}}(A), so A is cubic on V_{1}. Further by (5), m(A/C_{A}(V_{1})) = m(A/J) = k and m(V_{1}/C_{V_{1}}(A) ≤ m(V_{1}/J) = k + e, so if k > e then q̂(H_{1}/C_{H_{1}}(V_{1}), V_{1}) < 2. So (8) holds. On the other hand if k = e then q̂(H_{1}/C_{H_{1}}(V_{1}), V_{1}) ≤ 2.

Suppose e ≥ k. As X^{∗} is faithful on W̃ of rank 2k, it follows from (5) and the representation theory of X^{∗} (cf. B.4.2 in [ASm]) that e = k, X^{∗} ≅ L_{2}(2^{e}), and W̃ is the natural module for X^{∗}. In particular (7) holds.

(5.6) Assume Hypothesis 5.1, and in addition assume:

(i) M ∈ M(S), R = C_{S}(V (M)), and N_{M}(R) is a uniqueness group, and (ii) q(M/C_{M}(V (M)), V (M)) > 1.

Then Baum(S) = Baum(R) and (1) {M} = M(G(Baum(S))).

(2) G(S) ≤ M.

(3) Suppose H ∈ ℌ with H/O_{2}(H) S_{3}-free. Then one of the following holds:

(a) H ≤ M.

(b) q̂(M/C_{M}(V (M))), V (M)) < 2.

(c) V (M) is a dual FF-module for M/C_{M}(V (M)).

Proof. Let V = V (M) and B = Baum(M). By (ii), J(S) centralizes V, so B ≤ R by 4.6. Set Y = N_{M}(R). Then Y ≤ G(B) and M(Y) = {M} by (i), so (1) follows. Then as G(S) ≤ G(B), (1) implies (2).

Assume the hypothesis of (3) and assume H ≰ M. Then by (2) and 5.3.1, there is H′ ∈ X_{M}^{∗}(H), and replacing H by H′, we may assume H ∈ X_{M}^{∗}(H). Hence by (1) and 5.3.2, the pair Y, H satisfies Hypothesis 5.2, and by 5.3.3, N_{H}(V) is the unique maximal overgroup of S in H.

Suppose V ≰ O_{2}(H). Then Y, H satisfies the hypothesis of 5.5, so by 5.5.1 we can pick X ∈ Γ_{∗}(H, V_{1}). As H/O_{2}(H) is S_{3}-free, X/W is not isomorphic to L_{2}(2^{f}) for any f, so by 5.5.7, e < k. Hence (3b) holds in this case by 5.5.8.

Thus we may assume V ≤ O_{2}(H). Hence by the qrc-Lemma 5.4, one of cases (2)-(5) of that result hold. As H/O_{2}(H) is S_{3}-free, case (5) does not hold, and in case (4), q < 2, so as q̂ = q̂(M/C_{M}(V), V) ≤ q, conclusion (3b) of the lemma holds.

Thus we may assume case (2) or (3) of 5.4 holds. But by (ii), q > 1, so case (3) holds. Hence (3c) holds in this case, completing the proof.

Section 6. The parameter q̂

In this section we assume:

Hypothesis 6.1. G is a finite group with O_{2}(G) = 1 and V is a faithful F_{2}G-module.

In addition adopt the notation from Section D.2 of [ASm]. In particular set q̂ = q̂(G, V) and Q̂_{∗} = Q̂_{∗}(G, V). The parameter q̂ is defined in Definition B.4.1. From Definition D.2.1 in [ASm], Q̂_{∗} consists of those nontrivial elementary abelian 2-subgroups A of G such that m(V/C_{V} (A))/m(A) = q̂, A is cubic on V, and A is minimal subject to these constraints.

For X = O^{2}(X) ≤ G and Y ≤ N_{G}(X), see Definition A.1.40 in [ASm] for the definition of Irr_{+}(X, V) and Irr_{+}(X, V, Y).

Definition 6.2. Given a nonabelian finite quasisimple group L and a nontrivial irreducible F_{2}L-module U, define

μ(L, U) = \frac{(dim_{F} (U) − 1) dim_{F_{2}}(U)}{4(m_{2}(Aut(L)) + 1)},

where F = End_{F_{2}L}(U). Define η(L, U) = q̂(N_{GL(U)}(Aut_{L}(U)), U).

Given a nonabelian finite simple group L, define μ(L), η(L) to be the minimum of the μ(L̂, U), η(L̂, U), respectively, as U varies over all nontrivial irreducible F_{2}L̂-modules, where L̂ is the universal covering group of L.

(6.3) Assume q̂ ≤ 2, A ∈ Q̂_{∗}, and L is a component of G with [A, L] ≠ 1. Set H = ⟨L^{A}⟩, Ṽ = V/C_{V} (H), and pick I ∈ Irr_{+}(H, V) with [I, L] ≠ 0. Then:

(1) Replacing I by a suitable I_{1} ∈ Irr_{+}(H, V) with Ĩ ≅ Ĩ_{1} as an F_{2}H-module, we have q̂(Aut_{HA}(Ĩ), Ĩ) ≤ q̂.

(2) Suppose V is an irreducible F_{2}H-module and let U ∈ Irr_{+}(L, V). If μ(L, U) > 1 then H = L.

Proof. Set K = HA, let A ≤ S ∈ Syl_{2}(K), and set I_{S} = ⟨I^{S}⟩. Then S = (S ∩ H)A, so (a) I^{S} = I^{(H∩S)A} = I^{A}. Next by A.1.42.2 in [ASm], replacing I by a suitable member I_{1} of Irr_{+}(H, I_{S}) with

Ĩ_{1} ≅ Ĩ, we may assume I ∈ Irr_{+}(H, I_{S}, S). Then by A.1.42.3 in [ASm]:

(b) Ĩ_{S} is the direct sum of the members of Ĩ^{A}.

Next by D.2.7 in [ASm], Aut_{A}(I_{S}) ∈ Q̂_{r}(Aut_{K}(I_{S}), I_{S}) for some r ≤ q̂. In particular q̂(Aut_{K}(I_{S}), I_{S}) ≤ q̂. Next trivially q̂(Aut_{K}(Ĩ_{S}), Ĩ_{S}) ≤ q̂(Aut_{K}(I_{S}), I_{S}). Let A_{1} ∈ Q̂_{∗}(Aut_{K}(Ĩ_{S}), Ĩ_{S}). Applying D.2.9.1 in [ASm] to the direct sum decomposition of Ĩ_{S} in (b), we conclude that |A_{1} : N_{A_{1}}(Ĩ)| ≤ 2. Set W̃ = ⟨Ĩ^{A_{1}}⟩. By another application of D.2.7 in [ASm], q̂(Aut_{K}(W̃), W̃) ≤ q̂(Aut_{K}(Ĩ_{S}), Ĩ_{S}). Pick A_{2} ∈ Q̂_{∗}(Aut_{K}(W̃), W̃). If W̃ ≠ Ĩ, then applying D.2.9.1 in [ASm] to the direct sum decomposition W̃ = Ĩ ⊕ Ĩ^{t}, t ∈ A_{1} − N_{A_{1}}(Ĩ), we conclude that A_{2} acts on Ĩ. Then by yet another application of D.2.7 in [ASm], q̂(Aut_{K}(Ĩ), Ĩ) ≤ q̂(Aut_{K}(W̃), W̃). Thus in any even we have established (1).

So assume the hypothesis of (2) with r = |L^{A}| > 1. Let L^{A} = {L_{1}, …, L_{r}} with L = L_{1} and set r = 2^{f}. As H is irreducible on V, U is an irreducible F_{2}L-module and V is a homogeneous semisimple F_{2}L-module. Set F = End_{F_{2}}(U) ≅ F_{2^{e}}, write U_{F} for U regarded as an F L-module, and set d = dim_{F} (U_{F}). We argue as in the proof of D.3.7 in [ASm]. In particular by 27.14 in [FGT], we may regard V as an F H-module V_{F}, and V_{F} ≅ U_{1,F} ⊗ ⋯ ⊗ U_{r,F}, where U_{1} = U, for i > 1, U_{i} = U^{a_{i}} for a_{i} ∈ A with L^{a_{i}} = L_{i}, regarded as an F_{2}L_{i}-module, and U_{i,F} = U_{F}^{a_{i}} is an F L_{i}-module.

Let t ∈ A − N_{A}(L) and choose notation so that H = J × J^{t}, where J = L_{1} … L_{s} and s = r/2. Thus V_{F} = W_{F} ⊗ W_{F}^{t} with W_{F} , W_{F}^{t} F -modules for J and J^{t}, respectively. Moreover W_{F} = U_{1,F} ⊗ ⋯ ⊗ U_{s,F} as an F J-module, so dim_{F} (W_{F}) = d^{s}. Now an argument in the proof of D.3.7 in [ASm] shows:

(c) dim_{F} (V_{F} /C_{V_{F}} (t)) = d^{s}(d^{s} − 1)/2, so dim_{F_{2}}(V/C_{V} (t)) = d^{s}(d^{s} − 1)e/2.

Next observe that:

(d) m_{2}(A) ≤ f + k, where k = m_{2}(Aut(L)).

Namely let B be a complement to N_{A}(L) in A. Then m_{2}(B) = f, and (module Z(H)) C_{L}(B) is a full diagonal subgroup of H isomorphic to L with N_{A}(L) faithful on C_{L}(B), so m_{2}(A) = f + m_{2}(N_{A}(L)) ≤ f + k.

Now by (c) and (d):

2 ≥ q̂ = \frac{m(V/C_{V} (A)}{m(A)} ≥ \frac{m(V/C_{V} (t))}{f + k} = \frac{d^{s}(d^{s} − 1)e}{2(f + k)},

and hence as m(U) = de we have:

(e) 1 ≥ (d^{s−1}(d^{s} − 1)m(U)/4(f + k).

Further as f ≥ 1, d^{s−1}(d^{s} − 1)/(f + k) ≥ (d − 1)/(k + 1), so it follows from (e) that:

(f) 1 ≥ μ(L, U).

Now (2) follows from (f), completing the proof of (2) and the lemma.

(6.4) Assume q̂ ≤ 2 and L is a component of G which is not centralized by Q̂_{∗}(G, V). Assume further that μ(L/Z(L)) > 1. Then q̂(G, V) ≥ η(L/Z(L)), so in particular η(L/Z(L)) ≤ 2.

Proof. By hypothesis there is A ∈ Q̂_{∗} with [A, L] ≠ 1. Set H = ⟨L^{A}⟩. By 6.3.1, we may assume V is an irreducible F_{2}HA-module. Then as μ(L/Z(L)) > 1, we conclude from 6.3.2 that H = L, so V is an irreducible F_{2}L-module. Hence by definition of η(L/Z(L)), q̂ ≥ η(L/Z(L)), completing the proof of the lemma.

The following result is probably known, but I don’t have a reference, so instead I’ll supply a proof.

Theorem 6.5. Let G be a finite S_{3}-free K-group with O_{2}(G) = 1 and V a faithful F_{2}G-module. Then q̂(G, V) ≥ 2.

Proof. Assume the theorem fails and choose a counter example of minimal order, and, subject to that constraint, with m(V) minimal. Thus q̂ < 2 and we can choose A ∈ Q̂_{∗}. By parts (1) and (2) of D.2.13 in [ASm], A centralizes F (G). Therefore there exists a component L of G with [L, A] ≠ 1. Thus to obtain a counter example and complete the proof, it suffices by 6.4 to show that μ(L/Z(L)) > 1 and η(L/Z(L)) ≥ 2.

As G is an S_{3}-free K-group, it follows that L/Z(L) ≅ Sz(2^{k}) or L_{2}(3^{k}) for some k ≥ 3 odd. In particular m_{2}(Aut(L/Z(L)) = k, 2, in the respective case. Observe from [GLS3] that the Schur multiplier of L/Z(L) is a 2-group, so L is simple. Pick U and F as in 6.2, and set d = dim_{F} (U).

Suppose first that L ≅ Sz(2^{k}). Then by [GM], η(L) = 2. Further d ≥ 4 and dim(U) ≥ 4k, so indeed μ(L) > 1.

Suppose next that L ≅ L_{3}(3^{k}). By [GM], η(L) > 2. Further m(U) ≥ m([B, U]) where B is a Borel subgroup of L, and as B is a Frobenius group which is the split extension of E_{3^{k}} by a cyclic group of order (3^{k} − 1)/2, m([B, U]) ≥ 3^{k} − 1 ≥ 26. Thus

μ(L) ≥ \frac{m(U)}{12} ≥ \frac{26}{12} > 1,

completing the proof.

Section 7. S_{3}-free fusion systems

We begin this section with a proof of Theorem 1. We prove the theorem via a series of reductions. Assume the theorem is false, and choose a minimal counter example F.

(7.1) O_{2}(F) = 1.

Proof. Assume Q = O_{2}(F) ≠ 1. Suppose first that Q ≤ Z(F). Then by 1.2.1, F/Q is a saturated fusion system on S/Q with O_{2}(F/Q) = 1. But by definition of F/Q in section 8 of [A1], for Q ≤ P ≤ S, Aut_{F/Q}(P/Q) = Aut_{F}(P/Q), so F/Q is S_{3}-free. Therefore by minimality of F, F/Q is constrained, so 1 ≠ O_{2}(F/Q), a contradiction.

Therefore Q ≰ Z(F), so C = C_{F}(Q) ≠ F. However by 1.2.2, C is a saturated fusion system on C_{S}(Q). As C ≤ F and F is S_{3}-free, so is C. Hence as C ≠ F, C is constrained by minimality of F. Then F is constrained by 1.2.2, contrary to the choice of F.

(7.2) (1) F is of characteristic 2-type.

(2) F is a local CK-system.

Proof. Let U ∈ F^{f} and N = N_{F}(U). By 7.1, N < F, while as F is S_{3}-free, so is N. Hence by minimality of F, N is constrained, establishing (1). Further, applying Corollary 3 in an inductive context, all composition factors of Aut_{N} (U) are in K, so (2) also holds.

(7.3) F = ⟨N_{F}(U) : U ∈ U⟩.

Proof. By 7.2, F satisfies the hypotheses of Theorem 1 in [A3], so by that theorem, either the lemma holds, or F is an obstruction to pushing up at the prime 2, and we may assume the latter. However by inspection of the list of such obstructions in [A3], none are S_{3}-free.

Remark 7.4. The full strength of Theorem 1 in [A3] is not required in the proof of 7.3; see the discussion in Example 8.6 in [A3], which shows that the weaker Theorem 7.13 in [A3] suffices when F is S_{3}-free. Moreover when F is S_{3}-free, the proof of Theorem 7.13 in [A3] is fairly easy; namely in that case, 7.13 follows from 6.6.3 in [A3].

We are now in a position to complete the proof of Theorem 1. By choice of F, F is a saturated fusion system on a finite 2-group S, which is not constrained. By 7.2.1, F is of characteristic 2-type. Thus Hypothesis 2.1 is satisfied. Then by 7.3, Hypothesis 4.1, and hence also Hypothesis 5.1 are satisfied.

Choose M it be maximal in M(S) with respect to ≲ and set V = V (M). As F is S_{3}-free, so is M/C_{M}(V), and by 7.2.2, M/C_{M}(V) is a K-group. Therefore by Theorem 6.5:

(*)  q̂(M/C_{M}(V), V) ≥ 2.

Thus hypothesis (ii) of 5.6 is satisfied, while hypothesis (i) of that lemma holds by 4.5.1. Therefore G(S) ≤ M by 5.6.2. Next by 5.3.1, there is H ∈ ℌ_{M}. As F is S_{3}-free, so is H/O_{2}(H), so the hypotheses of 5.6.3 are satisfied, and therefore one of conclusions (a)-(c) of that lemma hold. As H ∈ ℌ_{M}, conclusion (a) fails. Further conclusion (b) fails by (*). Finally as M/C_{M}(V) is S_{3}-free, condition (c) fails. This contradiction completes the proof of Theorem 1.

We next prove Corollary 2. Assume G is an S_{4}-free finite simple group, let S ∈ Syl_{2}(G), and F = F_{S}(G). As G is S_{4}-free, F is S_{3}-free. Therefore F is constrained by Theorem 1. Hence there exists a nontrivial abelian subgroup of S, which is strongly closed in S with respect to G. Now G is Goldschmidt group by a theorem of Goldschmidt in [Go], establishing Corollary 2.

Next we observe that Corollary 3 follows from Corollary 2. Namely assume G is a finite nonabelian finite simple group which is S_{3}-free. By Corollary 2, G is a Goldschmidt group. Hence Corollary 3 follows as G is S_{3}-free.

Finally observe that Corollary 4 follows from Corollary 3 and the fact that 3 divides the order of L_{2}(3^{n}).

References

[A1] M. Aschbacher, Normal subsystems of fusion systems, Proc. London Math. Soc. 97 (2008), 239–271.

[A2] M. Aschbacher, The generalized Fitting subsystem of a fusion system, preprint.

[A3] M. Aschbacher, Generation of fusion systems of characteristic 2-type, preprint.

[ASm] M. Aschbacher and S. Smith, The Classification of Quasithin Groups, Mathematical Surveys and Monographs, vol 11, AMS, 2004.

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

[BLO] C. Broto, R. Levi, and Bob Oliver, The homotopy theory of fusion systems, JAMS 16 (2003), 779–856.

[FGT] M. Aschbacher, Finite Group Theory, Cambridge University Press, 1986.

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[GM] R. Guralnick and G. Malle, Classification of 2F-modules, II, Finite Groups 2003: Proceedings of the Gainesville Conference on Finite Groups, Gruyter, 2004, pp. 117–184.

[Go] D. Goldschmidt, 2-Fusion in finite groups, Ann. Math. 99 (1974), 70–117.

[P1] L. Puig, Frobenius categories, J. Alg. 303 (2006), 309–357.

[P2] L. Puig, Frobenius categories versus Brauer blocks (to appear).

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[T] J. Thompson, Simple 3′-groups, Symposia Math. 13 (1974), 517–530.

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