SIMPLE FUSION SYSTEMS AND

THE SOLOMON 2-LOCAL GROUPS

Markus Linckelmann

Abstract. We introduce a notion of simple fusion systems which imitates the corresponding notion for finite groups and show that the fusion system on the Sylow-2-subgroup of a 7-dimensional spinor group considered by Ron Solomon [15] is simple in this sense.

The bigger picture which motivates the content of the present paper is the intuition, formulated by D. J. Benson in [3], that associated with each fusion system on a finite p-group in the sense of Puig [12] there should be a p-complete topological space which generalises the concept of a classifying space of a finite group. Broto, Levi and Oliver developed in [4] a theory describing how such a space should look, leading to the notion of a p-local finite group, and they give in particular a cohomological criterion for the existence and uniqueness of p-local finite groups. Using this criterion, Levi and Oliver show in [10] that there is up to homotopy equivalence a unique 2-local group associated with Solomon’s fusion system and they show further that this coincides indeed with the space constructed earlier by Benson in [3]. Put in these terms, Solomon’s fusion system provides an example of a simple 2-local finite group which is not the 2-complete classifying space of any finite group. Besides the obvious question - can one classify simple fusion systems? - one might wonder, whether the problem of the existence and uniqueness of a p-local finite group associated with any fusion system can be reduced to simple fusion systems.

Section 1 contains a brief account of Puig’s abstract notion of a fusion system and we recall in Section 2 how fusion systems occur in block theory. The following two sections introduce our notions of normal and simple fusion systems. Sections 5 to 8 are dedicated to developing the material needed in the proof of the simplicity of Solomon’s fusion system in Section 9. Throughout this paper, p denotes a prime.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 20C20.

1 Background material on fusion systems

We recall basic material on Puig’s axiomatisation of the local structure of blocks [12]. If P, Q, R are subgroups of a finite group G, we denote by Hom_{P} (Q, R) the set of group homomorphisms φ : Q → R for which there is y ∈ P satisfying φ(u) = yuy^{−1} for all u ∈ Q; we write Aut_{P} (Q) = Hom_{P} (Q, Q). Thus Aut_{P} (Q) is canonically isomorphic to N_{P} (Q)/C_{P} (Q); in particular Aut_{Q}(Q) ≅ Q/Z(Q) is the group of inner automorphisms of Q.

Definition 1.1. A category on a finite p-group P is a category F whose objects are the subgroups of P and whose morphism sets Hom_{F}(Q, R) consist, for any two subgroups Q, R of P, of injective group homomorphisms with the following properties:

(i) if Q is contained in R then the inclusion Q ⊆ R is a morphism in F; (ii) for any φ ∈ Hom_{F}(Q, R), the induced isomorphism Q ≅ φ(Q) and its inverse

are morphisms in F;

(iii) composition of morphisms in F is the usual composition of group homomorphisms.

Definition 1.2. Let F be a category on a finite p-group P. A subgroup Q of P is called fully F-centralised if |C_{P} (R)| ≤ |C_{P} (Q)| for any subgroup R of P such that R ≅ Q in F, and Q is called fully F-normalised if |N_{P} (R)| ≤ |N_{P} (Q)| for any subgroup R of P such that R ≅ Q in F.

The following definition is due to Broto, Levi and Oliver [4].

Definition 1.3. Let F be a category on a finite p-group P, and let Q be a subgroup of P. For any morphism φ : Q → P in F, we set N_{φ} = {y ∈ N_{P} (Q)| there is z ∈ N_{P} (φ(Q)) such that φ(^{y}u) = ^{z}φ(u)for all u ∈ Q}.

In other words, N_{φ} is the inverse image in N_{P} (Q) of the group Aut_{P} (Q) ∩ (φ^{−1} ∘ Aut_{P} (φ(Q)) ∘ φ). Note that in particular QC_{P} (Q) ⊆ N_{φ} ⊆ N_{P} (Q). Broto, Levi and Oliver use the groups N_{φ} in [4] to give a definition of fusion systems (called saturated fusion systems in [4]) which is equivalent to Puig’s original definition (called full Frobenius systems there), which in turn has been simplified by Stancu [17]; we present here Stancu’s version:

Definition 1.4. A fusion system on a finite p-group P is a category F on P satisfying in addition the following properties.

(i) For any two subgroups Q, R of P we have Hom_{P} (Q, R) ⊆ Hom_{F}(Q, R).

(ii) Aut_{P} (P) is a Sylow-p-subgroup of Aut_{F}(P).

(iii) every morphism φ : Q → P in F such that φ(Q) is fully F-normalised extends to a morphism ψ : N_{φ} → P (that is, ψ|_{Q} = φ).

The extension axiom (iii) relates the role of N_{φ} as object of F to its image N_{φ}/Q in Aut_{F}(Q). We briefly sketch in the following three Propositions that definition 1.4 is equivalent to the definition given in [4]. The first Proposition says that fully Fcentralised implies fully F-normalised.

Proposition 1.5. ([17]) Let F be a fusion system on a finite p-group P and let Q be a subgroup of P. If Q is fully F-normalised then Q is fully F-centralised.

Proof. Let φ : R → Q be an isomorphism in F. Assume that Q is fully F-normalised and that R is fully F-centralised. By 1.4 (iii) there is a morphism ψ : RC_{P} (R) → P in F such that ψ|_{R} = φ. Hence ψ maps C_{P} (R) to C_{P} (Q), which implies that |C_{P} (R)| ≤ |C_{P} (Q)|, hence equality since R is fully F-centralised. Thus Q is fully F-centralised. □

Proposition 1.6. Let F be a fusion system on a finite p-group P and let Q be a subgroup of P. Then Q is fully F-normalised if and only if Q is fully F-centralised and Aut_{P} (Q) is a Sylow-p-subgroup of Aut_{F}(Q).

Proof. Assume that Q is fully F-normalised. Then Q is fully CF -centralised by 1.5. Choose Q to be of maximal order such that Aut_{P} (Q) is not a Sylow-p-subgroup of Aut_{F}(Q). Then Q is a proper subgroup of P by 1.4(ii). Choose a p-subgroup S of Aut_{F}(Q) such that Aut_{P} (Q) is a proper normal subgroup of S. Let φ ∈ S. Since φ normalises Aut_{P} (Q), for every y ∈ N_{P} (Q) there is z ∈ N_{P} (Q) such that φ(^{y}u) = ^{z}φ(u) for all u ∈ Q. In other words, N_{φ} = N_{P} (Q). Since Q is fully F-normalised, it follows from 1.4(iii) that there is an automorphism ψ of N_{P} (Q) in F such that ψ|_{Q} = φ. Since φ has p-power order, by decomposing ψ into its p-part and its p′-part we may in fact assume that ψ has p-power order. Let τ : N_{P} (Q) → P be a morphism in F such that τ(N_{P} (Q)) is fully F-normalised. Now τψτ^{−1} is a p-element in Aut_{F}(τ(N_{P} (Q))), thus conjugate to an element in Aut_{P} (τ(N_{P} (Q))). Therefore we may choose τ in such a way that there is y ∈ N_{P} (τ(N_{P} (Q))) satisfying τψτ^{−1}(v) = ^{y}v for any v ∈ τ(N_{P} (Q)). Since ψ|_{Q} = φ, the automorphism τψτ^{−1} of τ(N_{P} (Q)) stabilises τ(Q). Thus y ∈ N_{P} (τ(Q)). Since Q is fully F-normalised we have N_{P} (τ(Q)) ⊆ τ(N_{P} (Q)), hence ψ(u) = ^{τ^{−1}(y)}u for all u ∈ N_{P} (Q). But then in particular φ ∈ Aut_{P} (Q), contradicting our initial choice of φ. The converse is easy since |N_{P} (Q)| = |Aut_{P} (Q)| · |C_{P} (Q)|. □

The next Proposition shows that the “extension axiom” 1.4(iii) holds more generally whenever φ(Q) is fully F-centralised:

Proposition 1.7. ([17]) Let F be a fusion system on a finite p-group P, let Q be a subgroup of P and let φ : Q → P be a morphism in F such that φ(Q) is fully F-centralised. Then there is a morphism ψ : N_{φ} → P in F such that ψ|_{Q} = φ.

Proof. Let ρ : φ(Q) → P be a morphism in F such that R = ρ(φ(Q)) is fully Fnormalised. Then ρ ∘ Aut_{P} (φ(Q)) ∘ ρ^{−1} is a p-subgroup of Aut_{F}(R). Moreover, by 1.6, the group Aut_{P} (R) is a Sylow-p-subgroup of Aut_{F}(R). Thus there is α ∈ Aut_{F}(R) such that α ∘ ρ ∘ Aut_{P} (φ(Q)) ∘ ρ^{−1} ∘ α^{−1} ⊆ Aut_{P} (R). This means that after replacing ρ by α ∘ ρ, we may assume that N_{ρ} = N_{P} (φ(Q)). In particular, ρ extends to a morphism σ : N_{P} (φ(Q)) → P. But then N_{φ} ⊆ N_{ρ∘φ}, hence ρ ∘ φ extends to a morphism τ : N_{φ} → P. Then τ(N_{φ}) ⊆ σ(N_{P} (φ(Q)), and hence we get a morphism σ^{−1}|_{τ(N_{φ})} ∘ τ : N_{φ} → P which extends φ as required. □

Definition 1.8. Let F be a fusion system on a finite p-group P and let Q be a subgroup of P.

(i) Q is F-centric if C_{P} (R) = Z(R) for any subgroup R of P such that R ≅ Q in F. (ii) Q is F-radical if O_{p}(Aut_{F}(Q)/Aut_{Q}(Q)) = 1. (iii) Q is F-essential if Q is F-centric, Q ≠ P, and Aut_{F}(Q)/Aut_{Q}(Q) has a

strongly p-embedded proper subgroup M (that is, M contains a Sylow-p-subgroup S of Aut_{F}(Q)/Aut_{Q}(Q) such that ^{φ}S ∩ S = {1} for every φ ∈ Aut_{F}(Q)/Aut_{Q}(Q) − M).

(iv) Q is weakly F-closed if for every morphism φ : Q → P in F we have φ(Q) = Q.

(v) Q is strongly F-closed, if for any subgroup R of P and any morphism φ : R → P in F we have φ(R ∩ Q) ⊆ Q.

If Q is F-centric, then Q is fully F-centralised, and if Q is F-essential, then Q is F-radical. If Q is strongly F-closed then Q is weakly F-closed. One easily checks that if Q is strongly F-closed then for any subgroup R of P and any morphism φ : R → P in F we have in fact φ(R ∩ Q) = φ(R) ∩ Q. Indeed, the left side is contained in the right side by the above definition, and the other inclusion is obtained by applying this inclusion to φ(R) and the morphism φ^{−1} viewed as morphism from φ(R) to P.

Definition 1.9 Let F be a category on a finite p-group P, and let Q be a subgroup of P. We define the category N_{F}(Q) on N_{P} (Q) by Hom_{N_{F} (Q)}(R, R′) = {φ : R → R′| φ extends to a morphism ψ : QR → QR′ in F such that ψ(Q) = Q}, for any two subgroups R, R′ of N_{P} (Q). Similarly, we define the category C_{F}(Q) on C_{P} (Q) by Hom_{C_{F} (Q)}(R, R′) = {φ : R → R′| φ extends to a morphism ψ : QR → QR′ in F such that ψ|_{Q} = Id_{Q}}.

We have clearly inclusions of categories C_{F}(Q) ⊆ N_{F}(Q) ⊆ F. If F = N_{F}(Q) for some subgroup Q of P, then clearly Q is strongly F-closed. The converse of this statement is not true, in general. If F = C_{F}(Z) for some subgroup Z of P then the category on P/Z induced by F is a fusion system on P/Z, denoted by F/Z. In that case, if F′ is a fusion system on P contained in F we have F′ = F if and only if F′/Z = F/Z; this follows from Alperin’s fusion theorem 1.11 below together with the fact that if Q is a subgroup of P then the canonical map Aut_{F}(Q) → Aut_{F/Z}(Q/Z) has a p-group as kernel because any p′-automorphism of Q/Z lifts to a p′-automorphism of Q.

Proposition 1.10. ([12]) Let F be a fusion system on a finite p-group P, and let Q be a subgroup of P. If Q is fully F-centralised, then C_{F}(Q) is a fusion system on C_{P} (Q); if Q is fully normalised, then N_{F}(Q) is a fusion system on N_{P} (Q).

A proof of this Proposition can be found in [4, A6] (applied to the cases where the group K occurring in the statement of [4, A6] is either trivial or equal to Aut(Q)). By the previous remarks, Proposition 1.10 implies that if Q is fully F-centralised then C_{F}(Q)/Z(Q) is a fusion system on C_{P} (Q)/Z(Q). The following result is Alperin’s fusion theorem [1], refined by Goldschmidt [8], and extended to arbitrary fusion systems by Puig [12].

Theorem 1.11. Let F be a fusion system on a finite p-group P. Every isomorphism in F can be written as a composition of finitely many isomorphisms φ : Q ≅ R in F such that either φ = α|_{Q} for some α ∈ Aut_{F}(P) or there is an F-essential subgroup E of P containing both Q, R, and an automorphism β ∈ Aut_{F}(E) such that φ = β|_{Q}.

Lemma 1.12. ([12]) Let F be a fusion system on a finite p-group P. Let Q, R be F-centric subgroups of P such that Q ⊆ R, and let φ ∈ Aut_{F}(R). We have φ|_{Q} = Id_{Q} if and only if φ ∈ Aut_{Z(Q)}(R).

Proof. Assume that φ|_{Q} = Id_{Q}. We proceed by induction over [R : Q]. Consider first the case where Q is normal in R. Let u ∈ Q and v ∈ R. Then ^{v}u ∈ Q, hence vu = φ(^{v}u) = ^{φ(v)}u, and thus v^{−1}φ(v) ∈ C_{R}(Q) = Z(Q), or equivalently, φ(v) = vz for some z ∈ Z(Q). If φ has order prime to p in Aut(R) this forces φ = Id_{R}. Therefore we may assume that the order of φ is a power of p. Upon replacing R by a fully F-normalised F-conjugate we may assume that φ ∈ Aut_{P} (R). Since φ restricts to Id_{Q} and since Q is F-centric this implies that φ ∈ Aut_{Z(Q)}(R). This proves 1.12 if Q is normal in R. In general, if φ|_{Q} = Id_{Q} then φ(N_{R}(Q)) = N_{R}(Q). Thus φ|_{N_{R}(Q)} ∈ Aut_{Z(Q)}(N_{R}(Q)) by the previous paragraph. Hence there is z ∈ Z(Q) such that c_{z} ∘ φ|_{N_{R}(Q)} = Id_{N_{R}(Q)}, where c_{z} is the automorphism of R given by conjugation with z. By induction we get c_{z} ∘ φ ∈ Aut_{Z(N_{R}(Q))}(Q). As all involved groups are F-centric we have Z(N_{R}(Q)) ⊆ Z(Q), and thus φ ∈ Aut_{Z(Q)}(R) as claimed. The converse is trivial. □

Lemma 1.13. Let F be a fusion system on a finite p-group P, let Q, R be F-centric subgroups of P such that Q ⊆ R, and let φ, φ′ ∈ Hom_{F}(R, P) such that φ|_{Q} = φ′|_{Q}. Then φ(R) = φ′(R).

Proof. Let v ∈ N_{R}(Q). For every u ∈ Q we have φ(^{v}u) = φ′(^{v}u), hence φ(v)^{−1}φ′(v) ∈ C_{P} (φ(Q)) = Z(φ(Q)). It follows that φ(N_{R}(Q)) = φ′(N_{R}(Q)), hence φ(R) = φ′(R) by induction. □

Given a fusion system F on a finite p-group P, we denote by F^{c} the full subcategory of F-centric subgroups of P; we denote by F̄ the orbit category of F, which has the same objects as F but whose sets of morphisms are the quotient sets HomF̄(Q, R) = Aut_{R}(R)\Hom_{F}(Q, R) of morphisms in F modulo inner automorphisms of the corresponding subgroups of P. We denote by F̄^{c} the image in F̄ of F^{c}. The category F has the property that every morphism is a monomorphism, and every endomorphism is an automorphism. The orbit category F̄ has still the property that every endomorphism is an automorphism, but not every morphism is a monomorphism, in general. As observed in [11] in the context of fusion systems of finite groups, the straightforward consequence of 1.12 is that in the opposite category (F̄^{c})^{0} every morphism is a monomorphism, or equivalently:

Proposition 1.14. Let F be a fusion system on a finite p-group P. Every morphism in the category F̄^{c} is an epimorphism.

Proof. Let Q, R, S be F-centric subgroups of P, let φ ∈ Hom_{F}(Q, R) and let ψ, ψ′ ∈ Hom_{F}(R, S). Assume that the images of ψ ∘ φ and ψ′ ∘ φ in HomF̄^{c}(Q, S) coincide. Up to replacing ψ′ by some S-conjugate, we may assume that ψ ∘ φ = ψ′ ∘ φ. Thus the restrictions to φ(Q) of ψ, ψ′ coincide. It follows from 1.13 that ψ(R) = ψ′(R). Thus ψ^{−1} ∘ ψ′ is an automorphism of R which restricts to the identity on φ(Q), hence ψ^{−1} ∘ ψ′ ∈ Aut_{Z(φ(Q))}(R) by 1.12. Thus the images of ψ, ψ′ in the orbit category are equal. □

2 Fusion systems of finite groups and p-blocks

For expository purpose, we describe in this section briefly the well-known examples which motivate Puig’s definition of a fusion system.

Definition 2.1 Let G be a finite group, and let P be a Sylow-p-subgroup of G. We denote by F_{P} (G) the category on P whose morphisms are the group homomorphisms φ : Q → R for which there is an element x ∈ G such that φ(u) = xux^{−1} for all u ∈ Q.

Equivalently, Hom_{F_{P} (G)}(Q, R) = Hom_{G}(Q, R); in particular, Aut_{F_{P} (G)}(Q) = Aut_{G}(Q) ≅ N_{G}(Q)/C_{G}(Q). We leave the elementary proof of the following well-known statement to the reader.

Theorem 2.2. Let G be a finite group, and let P be a Sylow-p-subgroup of G. (i) The category F_{P} (G) is a fusion system on P. (ii) A subgroup Q of P is fully F_{P} (G)-centralised if and only if C_{P} (Q) is a Sylow-

p-subgroup of C_{G}(Q).

(iii) A subgroup Q of P is fully F_{P} (G)-normalised if and only if N_{P} (Q) is a Sylowp-subgroup of N_{G}(Q).

Following Alperin-Broué [2], there is a fusion system on a defect group of a p-block of a finite group which generalises the definition of F_{P} (G) above in the sense, that it coincides with F_{P} (G) if the considered block is the principal p-block of G. In order to describe this briefly, let k be a field of characteristic p, let G be a finite group, and let b be a block of kG; that is, b is a primitive idempotent in Z(kG). A b-Brauer pair is a pair (Q, f) consisting of a p-subgroup Q of G and a block f of kC_{G}(Q) such that Br_{Q}(b)f = f. Here Br_{Q} : (kG)^{Q} → kC_{G}(Q) is the Brauer homomorphism mapping any element of C_{G}(Q) to itself and any non trivial Q-conjugacy class sum of elements in G to zero. By [2], the set of b-Brauer pairs admits a partial order “ ⊆ ” which is compatible with the action of G by conjugation on this set, such that the maximal b-Brauer pairs form a single G-conjugacy class. Given a maximal b-Brauer pair (P, e), for every subgroup Q of P there is a unique block e_{Q} of kC_{G}(Q) such that (Q, e_{Q}) ⊆ (P, e), and the group P is called a defect group of the block b. The choice of a maximal b-Brauer pair gives rise to a category on P (we follow the notation of [9]):

Definition 2.3. Let G be a finite group, let b be a block of kG, and let (P, e) be a maximal b-Brauer pair. For any subgroup Q of P, denote by e_{Q} the unique block of kC_{G}(Q) such that (Q, e_{Q}) ⊆ (P, e). We denote by F_{(P,e)}(G, b) the category on P whose morphisms are the group homomorphisms φ : Q → R for which there is an element x ∈ G such that φ(u) = xux^{−1} for all u ∈ Q such that xe_{Q}x^{−1} = e_{xQx^{−1}}, or equivalently, sucht that ^{x}(Q, e_{Q}) ⊆ (R, e_{R}), where Q, R are subgroups of P.

If S is a Sylow-p-subgroup of G containing the defect group P of b, then clearly F_{(P,e)}(G, b) is a subcategory of F_{S}(G), but it is not in general a full subcategory, because the elements x in G used to define the morphisms in F_{(P,e)}(G, b) have to fulfill the additional compatibility property ^{x}(Q, e_{Q}) ⊆ (R, e_{R}). If b is the principal block of kG (that is, b is the unique block of kG not contained in the augmentation ideal of kG), then P is a Sylow-p-subgroup of G and e_{Q} is the principal block of kC_{G}(Q) for any subgroup Q of P, and hence F_{(P,e)}(G, b) = F_{P} (G) in this case. The following statement, which generalises 2.2, is essentially a reformulation of results in [2]; we sketch a proof for the convenience of the reader:

Theorem 2.4. Let G be a finite group, let b be a block of kG, and let (P, e) be a maximal b-Brauer pair. For every subgroup Q of P, denote by e_{Q} the unique block of kC_{G}(Q) such that (Q, e_{Q}) ⊆ (P, e).

(i) The category F_{(P,e)}(G, b) is a fusion system on P.

(ii) A subgroup Q of P is fully F_{(P,e)}(G, b)-centralised if and only if C_{P} (Q) is a defect group of kC_{G}(Q)e_{Q}.

(iii) A subgroup Q of P is fully F_{(P,e)}(G, b)-normalised if and only if N_{P} (Q) is a defect group of kN_{G}(Q, e_{Q})e_{Q}.

Note that the last statement makes sense, as e_{Q} remains a block for the group N_{G}(Q, e_{Q}) by [2]. The automorphism group in F_{(P,e)}(G, b) of a subgroup Q of P is isomorphic to N_{G}(Q, e_{Q})/C_{G}(Q). Thanks to the preceding Theorem, we can apply Alperin’s fusion thereom to the fusion system F_{(P,e)}(G, b), which implies in particular, that F_{(P,e)}(G, b) is completely determined by the automorphism groups N_{G}(Q, e_{Q})/C_{G}(Q) for the F_{(P,e)}(G, b)-essential subgroups Q of P. Specialising Theorem 2.4 to the case where b is the principal block of kG yields Theorem 2.3.

Proof of 2.4. By [2, 4.5], for every subgroup Q of P the group C_{P} (Q) is contained in a defect group of e_{Q} as block of kC_{G}(Q), and there is x ∈ G such that ^{x}(Q, e_{Q}) ⊆ (P, e) and such that C_{P} (^{x}Q) is a defect group of ^{x}e_{Q} as block of kC_{G}(^{x}Q). From this follows (ii). By [2, (2.9)], e_{Q} remains a block of kN_{G}(Q, e_{Q}). As before, N_{P} (Q) is contained in a defect group of e_{Q} as block of kN_{G}(Q, e_{Q}), and there is x ∈ G such that ^{x}(Q, e_{Q}) ⊆ (P, e) and such that N_{P} (^{x}Q) is a defect group of ^{x}e_{Q} as block of kN_{G}(^{x}(Q, e_{Q})). Moreover, if D is a defect group of e_{Q} as block of kN_{G}(Q, e_{Q}) then the image of D in N_{G}(Q, e_{Q})/C_{G}(Q) is a Sylow-p-subgroup of this quotient. This implies (iii). In order to see (i), observe first that F_{(P,e)}(G, b) is clearly a category on P in the sense of 1.1. By the previous remarks, F_{(P,e)}(G, b) has the properties 1.4.(i),(ii),(iii). It remains to verify that F_{(P,e)}(G, b) has also the property 1.4.(iv). Let Q, R be subgroups of P such that C_{P} (R) is a defect group of e_{R} as block of kC_{G}(R), and let x ∈ G such that ^{x}(Q, e_{Q}) = (R, e_{R}). Denote by φ : Q → P the morphism in F_{(P,e)}(G, b) defined by φ(u) = ^{x}u for all u ∈ Q. Then N_{φ} = {y ∈ N_{P} (Q) | there is z ∈ N_{P} (R) such that^{xy}u = ^{zx}u for all u ∈ Q}. Thus^{x}N_{φ} ⊆ N_{P} (R)C_{G}(R). Since R is fully F_{(P,e)}(G, b)-centralised, (N_{P} (R), e_{N_{P} (R)}) is a maximal (N_{P} (R)C_{G}(R), e_{R})-Brauer pair. Thus there is c ∈ C_{G}(R) such that ^{cx}(N_{φ}, e_{N_{φ}}) ⊆ (N_{P} (R), e_{N_{P} (R)}). Hence ψ : N_{φ} → P defined by ψ(n) = ^{cx}n for all n ∈ N_{φ} is a morphism in F_{(P,e)}(G, b) which extends φ. □

3 Normal fusion systems

Definition 3.1 Let F be a category on a finite p-group P, and let F′ be a category on a subgroup P′ of P. We say that F normalises F′ if P′ is strongly F-closed and if for every isomorphism φ : Q → Q′ in F and any two subgroups R, R′ of Q ∩ P′ we have

φ ∘ Hom_{F′}(R, R′) ∘ φ^{−1}|_{φ(R)} ⊆ Hom_{F′}(φ(R), φ(R′)) .

We say that F′ is normal in F and write F′ ⊴ F if F′ is contained in F and F normalises F′.

In other words, F normalises F′ if for any isomorphism φ : Q → Q′ in F and any morphism ψ : R → R′ in F′ such that < R, R′ >⊆ Q, we have < φ(R), φ(R′) >⊆ P′ and the induced morphism φ ∘ ψ ∘ φ^{−1} : φ(R) → φ(R′) is a morphism in F′. Note that this implies that we have in fact an equality

φ ∘ Hom_{F′}(R, R′) ∘ φ^{−1}|_{φ(R)} = Hom_{F′}(φ(R), φ(R′)) .

Indeed, the left side is contained in the right side by the definition, and the other inclusion follows from applying this inclusion to φ^{−1}, φ(R), φ(R′) instead of φ, R, R′, respectively. Applied to R = R′ and S = φ(R) and making use of Alperin’s fusion theorem this implies in particular that if R, S are subgroups of P′ which are isomorphic in F then Aut_{F′}(R) ≅ Aut_{F′}(S).

The unique category on the trivial subgroup {1} of P is a fusion system which is normal in any fusion system F on P. The obvious motivating example for the definition of normal fusion systems is this:

Proposition 3.2. Let G be a finite group, let P be a Sylow-p-subgroup of G, and let N be a normal subgroup of G. We have F_{P ∩N}(N) ⊴ F_{P} (G).

Proof. Trivial. □

Proposition 3.3. Let F be a fusion system on a finite p-group P. Then F_{P} (P) is normal in F if and only if F = N_{F}(P).

Proof. Suppose that F_{P} (P) ⊴ F. Then in particular for any morphism φ : R → P in F and any u ∈ N_{P} (R) there is v ∈ N_{P} (φ(R)) such that φ(^{u}r) = ^{v}φ(r) for all r ∈ R. Whenever φ(R) is fully F-centralised, φ extends to a morphism ψ : N_{P} (R) → P in F. In particular, this holds if R, and hence φ(R), are F-centric. But then also N_{P} (R) and ψ(N_{P} (R)) are F-centric. Inductively, it follows that φ can be extended to an automorphism of P belonging to F. Thus, by Alperin’s fusion theorem, we get F = N_{F}(P). The converse is easy. □

Proposition 3.4. Let F be a fusion system on a finite p-group P. If Q is a strongly F-closed abelian subgroup of P then F_{Q}(Q) is normal in F.

Proof. Since Q is abelian, the only morphisms in F_{Q}(Q) are inclusions R ⊆ R′ of subgroups R, R′ of Q. Since Q is strongly F-closed, the result follows. □

Proposition 3.5. Let F, F′ be fusion systems on a finite p-group P such that F′ is normal in F. Then for every subgroup Q of P the index [Aut_{F}(Q) : Aut_{F′}(Q)] is prime to p.

Proof. Let Q be a subgroup of P, and let φ : Q → R be an isomorphism in F such that the subgroup R of P is fully F-normalised. Then Aut_{P} (R) is a Sylow-p-subgroup of Aut_{F}(R) by 1.5, and Aut_{P} (Q) ⊆ Aut_{F′}(R). Since F′ is normal in F, it follows that the Sylow-p-subgroup φ^{−1} ∘ Aut_{P} (R) ∘ φ of Aut_{F}(Q) is contained in Aut_{F′}(Q). Thus the index of Aut_{F′}(Q) in Aut_{F}(Q) is prime to p. □

4 Simple fusion systems

Definition 4.1 A fusion system F on a non trivial finite p-group P is called simple if F has no proper non trivial normal fusion subsystem.

In view of work of Broto, Levi, Oliver [4] - introducing p-local finite groups as a generalisation of classifying spaces associated with fusion systems - we extend this terminology in the obvious way: a p-local finite group is called simple if its underlying fusion system is simple. In order to avoid confusion we point out that this definition is different from previous similar definitions such as fusion-simple groups (in a group theoretic context) or the notion of simple fusion systems introduced in [12].

Certainly the fusion system F_{P} (G) of a finite simple group G (with Sylow-p-subgroup P) does not have to be simple, but conversely, if a simple fusion system F on a finite p-group P is equal to F_{P} (G) for some finite group G containing P as Sylow-p-subgroup, then G can be chosen to be simple:

Proposition 4.2. Let F be a simple fusion system on some finite p-group P. Suppose that F = F_{P} (G) for some finite group G having P as Sylow-p-subgroup. If O_{p′}(G) = 1 and if F_{P} (H) ≠ F_{P} (G) for any proper subgroup H of G containing P, then G is simple. In particular, if G has minimal order such that P is a Sylow-p-subgroup of G and such that F = F_{P} (G), then G is simple.

Proof. Suppose that O_{p′}(G) = 1 and that F_{P} (H) ≠ F_{P} (G) for any proper subgroup H of G containing P. Let N be a non-trivial normal subgroup of G. Then N ∩ P is a Sylow-p-subgroup of N, and F_{N∩P} (N) is a normal fusion system in F_{P} (G). As O_{p′}(G) = 1, we have N ∩ P ≠ 1. As F_{P} (G) is simple, this forces P ⊆ N and F_{P} (N) = F_{P} (G), hence N = G by the assumptions. Let now G be a finite group of minimal order such that P is a Sylow-p-subgroup of G and such that F = F_{P} (G). Then O_{p′}(G) = 1, because the canonical map G → G/O_{p′}(G) induces an isomorphism of fusion systems. By the minimality of G, we have F_{P} (H) ≠ F_{P} (G) for any proper subgroup H of G containing P. Thus the second statement follows from the first. □

Proposition 4.3. Let P be a finite p-group. Then F_{P} (P) is simple if and only if P is cyclic of order p.

Proof. By 3.4, for every subgroup Z of Z(P) we have F_{Z}(Z) ⊴ F_{P} (P), from which the statement follows. □

Proposition 4.4. Let P be a finite abelian p-group and let F be a fusion system on P. Then F is simple if and only if P has order p and F = F_{P} (P).

Proof. If F is simple, then F = F_{P} (P) by 3.3, and hence |P | = p by 4.3. The converse is clear. □

Proposition 4.5. Let F be a fusion system on a finite p-group P such that Aut_{F}(P) = Aut_{P} (P). Suppose that for any F-essential subgroup Q of P we have O^{p′}(Aut_{F}(Q)) = Aut_{F}(Q). Then for any fusion system F′ on P such that F′ ⊴ F we have F′ = F. In particular, if in addition no proper non trivial subgroup of P is strongly F-closed then F is simple.

Proof. Let F′ be a fusion system on P such that F′ ⊴ F. Let Q be a fully normalised F-essential subgroup of P. Then Aut_{P} (Q) ⊆ Aut_{F′}(Q) ⊴ Aut_{F}(Q), and Aut_{P} (Q) is a Sylow-p-subgroup of Aut_{F}(Q). Thus Aut_{F′}(Q) = Aut_{F}(Q). Since also Aut_{F′}(P) = Aut_{F}(P) = Aut_{P} (P), the equality F′ = F follows from Alperin’s fusion theorem. □

5 Characterisations of fusion systems

The following result shows that a fusion system F on a finite p-group P is determined by its fusion on elements of order p in P and their centralisers in F. If Q is a subgroup of P, we denote by C_{F}(Q)/Z(Q) the category on C_{P} (Q)/Z(Q) whose morphisms are induced by morphisms in C_{F}(Q) via the canonical map C_{P} (Q) → C_{P} (Q)/Z(Q). By the remarks following 1.8, if Q is fully F-centralised, then C_{F}(Q)/Z(Q) is a fusion system on C_{P} (Q)/Z(Q).

Proposition 5.1. Let P be a finite p-group, and let F, F′ be fusion systems on P such that F′ ⊆ F. The following are equivalent.

(i) F = F′.

(ii) For any fully F′-centralised subgroup Z of order p of P we have Hom_{F}(Z, P) = Hom_{F′}(Z, P) and C_{F}(Z) = C_{F′}(Z).

Proof. Suppose that (ii) holds. Let Q be a non trivial subgroup of P and let φ ∈ Aut_{F}(Q). Let Z be a subgroup of order p of Z(Q). Let ψ : Z → P be a morphism in F′ such that ψ(Z) is fully F′-centralised. Since Q ⊆ C_{P} (Z), the morphism ψ extends to a morphism τ : Q → P in F′. In order to show that φ is a morphism in F′, it suffices to show that τ ∘ φ ∘ τ^{−1}|_{τ(Q)} ∈ Aut_{F′}(τ(Q)). Thus, after replacing Q by τ(Q), we may assume that Z is fully F′-centralised. By the assumptions, the morphism φ^{−1}|_{φ(Z)} : φ(Z) → Z belongs to F′, and hence extends to a morphism κ : Q → P in F′ (since Q = φ(Q) ⊆ C_{P} (φ(Z))). Then κ ∘ φ : Q → P restricts to the identity on Z, hence κ ∘ φ is a morphism in C_{F}(Z) = C_{F′}(Z). In particular, κ ∘ φ is a morphism in F′. But then so is φ, because κ is in F′. Alperin’s fusion theorem implies now (i). The converse is trivial. □

Corollary 5.2. Let F, F′ be fusion systems on a finite p-group P such that F′ ⊴ F. If Hom_{F}(Z, P) = Hom_{F′}(Z, P) and C_{F}(Z)/Z is a simple fusion system on C_{P} (Z)/Z for any fully F′-centralised subgroup Z of order p of P, then F′ = F.

Proof. We have C_{F′}(Z) ⊴ C_{F}(Z) and hence C_{F′}(Z)/Z ⊴ C_{F}(Z)/Z. Thus, if C_{F}(Z)/Z is simple for any fully F′-centralised subgroup Z of order p of P, then C_{F′}(Z)/Z = C_{F}(Z)/Z, hence C_{F′}(Z) = C_{F}(Z) and thus F′ = F by 5.1. □

Lemma 5.3. Let F, F′ be fusion systems on a finite p-group P such that F′ ⊆ F. Let Q ⟶^{φ} R ⟶^{ψ} S be a sequence of two composable morphisms in F such that Q, R, S are F-centric. If any two of the three morphisms φ, ψ, ψ ∘ φ are in F′, so is the third.

Proof. If φ, ψ are in F′, so is ψ∘φ. If ψ, ψ∘φ are in F′, then so is φ = ψ^{−1}|_{Im(ψ∘φ)}∘ψ∘φ. Assume now that φ and ψ ∘ φ are morphisms in F′. Up to replacing Q by φ(Q), we may assume that φ is the inclusion Q ⊆ R. Let v ∈ N_{R}(Q). Then, for any u ∈ Q, we have ψ(^{v}u) = ^{ψ(v)}u. Thus the morphism ψ|_{Q} extends to a morphism τ : N_{R}(Q) → P in F′. By 1.11, we have τ(N_{R}(Q)) = ψ(N_{R}(Q)) and hence ψ^{−1} ∘ τ ∈ Aut_{Z(Q)}(N_{R}(Q)) by 1.10. Thus ψ|_{N_{R}(Q)} is a morphism in F′. It follows inductively, that ψ is a morphism in F′. □

Proposition 5.4. Let F, F′ be fusion systems on a finite p-group P such that F′ ⊆ F. The following are equivalent.

(i) F = F′.

(ii) Hom_{F}(Q, P) = Hom_{F′}(Q, P) for every minimal F-centric subgroup Q of P.

Proof. Assume that (ii) holds. Let R be an F-centric subgroup of P, and let Q be a minimal F-centric subgroup of P contained in R. Let φ ∈ Hom_{F}(R, P). Then φ|_{Q} ∈ Hom_{F}(Q, P) = Hom_{F′}(Q, P). But then φ ∈ Hom_{F′}(R, P) by 5.3. Alperin’s fusion theorem implies (i). The converse is trivial. □

6 Dihedral 2-local groups

In this section we set P =< x > ⋊ < t >, such that x^{2^{n}} = 1 = t^{2} for some integer n ≥ 2 and txt = x^{−1}; that is, P is a dihedral 2-group of order 2^{n+1}.

Then P has three conjugacy classes of involutions, namely the classes of the elements z = x^{2^{n−1}}, t and xt. Besides the trivial fusion system F_{P} = F_{P} (P), there are two other systems, up to isomorphism. We denote by F_{P}^{I} the fusion system on P generated by F_{P} and an automorphism of order 3 of the Klein four group < z > × < t >. Thus z and t are F_{P}^{I} -conjugate, while z and xt are not; hence there are now two F_{P}^{I} -conjugacy classes of involutions in P. We denote by F_{P}^{II} the fusion system on P generated by F_{P} and an automorphism of order 3 on each of the Klein four groups < z > × < t > and < z > × < xt >. Thus all involutions in P are F_{P}^{II}-conjugate. Any fusion system on P is isomorphic to one of F_{P}, F_{P}^{I}, F_{P}^{II} and any of these systems appear as fusion systems F_{P} (G) of some finite group G having P as Sylow-2-subgroup (this follows easily from Erdmann’s list of examples in [7]). Any 2-block of a finite group having P as defect group has 1 or 2 or 3 isomorphism classes of simple modules, and then its fusion system is isomorphic to F_{P} or F_{P}^{I} or F_{P}^{II}, respectively. The fusion systems F_{P}, F_{P}^{I}, F_{P}^{II} correspond to the cases (bb), (ab), (aa), respectively, in [6].

For notational convenience, if Q is a Klein four group, we denote by F_{Q}^{I} and by F_{Q}^{II} the unique fusion system on Q generated by some automorphism of order 3 of Q.

Theorem 6.1. Let F be a fusion system on the dihedral 2-group P. Then F is simple if and only if F = F_{P}^{II}.

For the proof of 6.1 we need the following.

Lemma 6.2. Let Q be the subgroup of index 2 of P generated by x^{2} and t. Then F_{Q}^{II} ⊴ F_{P}^{I}; in particular, Q is strongly F_{P}^{I} -closed and F_{P}^{I} is not simple.

Proof. Observe first that F_{Q}^{II} is contained in F_{P}^{I}, because the three classes of involutions in Q represented by z, t, x^{2}t are all conjugate in F_{P}^{I}. Indeed, this is clear for z and t by the definition of F_{P}^{I}, and moreover, x^{2}t = xtx^{−1}. As F_{Q}^{II} is the unique maximal fusion system on Q, it suffices to show that Q is strongly F_{P}^{I} -closed, which is easy. □

Proof of Theorem 6.1. Let Q be a strongly F_{P}^{II}-closed subgroup of P. Then Q contains all involutions of P, as all involutions of P are F_{P}^{II}-conjugate. Hence Q = P. The fusion system F_{P} cannot be normal in F_{P}^{II} by 3.3. It suffices to show that F_{P}^{I} is not normal in F_{P}^{II}. Let y ∈< x > such that y^{2} = z. Conjugation by y is a morphism < xt >≅< zxt > in F_{P}, thus in F_{P}^{I}. Let α be an automorphism of order 3 of < z > × < xt > in F_{P}^{II}; we may assume that α(z) = xt and α(xt) = zxt. Thus applying α to the morphism < xt >≅< zxt > in F_{P} yields a morphism < zxt >≅< z >; this is however not a morphism in F_{P}^{I}, because zxt is conjugate (via y) to xt, and there is no morphism < xt >≅< z > in F_{P}^{I}. This shows that F_{P}^{II} is simple. Conversely, F_{P} is not simple by 3.3 and F_{P}^{I} is not simple by 6.2. □

Remark 6.3. Let q is an odd prime power. If q ≡ ±1(mod8), then the group P SL(2, q) has a dihedral Sylow-2-subgroup P, and F_{P} (P SL(2, q)) = F_{P}^{II}. In particular, F_{P} (P SL(2, q)) is simple in that case. If q ≡ ±3(mod8) then P SL(2, q) has a Klein four group as Sylow-2-subgroup, and hence F_{P} (P SL(2, q)) cannot be simple.

7 Fusion systems of normal subgroups of p-power index

Theorem 7.1. Let G be a finite group such that G = P N for some Sylow-p-subgroup P of G and a normal subgroup N of G. Let F be a fusion system on P such that F_{P ∩N}(N) ⊆ F ⊆ F_{P} (G). Then F = F_{P} (G).

For the proof of 7.1 we need the following Lemma (which is an elementary consequence of standard properties of coprime group actions, but we include a proof for the convenience of the reader).

Lemma 7.2. Let G be a finite group such that G = P N for some Sylow-p-subgroup P of G and a normal subgroup N of G. For any subgroup Q of P, the index [C_{G}(Q ∩ N) ∩ N_{G}(Q) : C_{G}(Q)] is a power of p.

Proof. We argue by induction over [G : N]. Assume that [G : N] = p. Then [G, G] ⊆ N. Let x be a p′-element in C_{G}(Q∩N)∩N_{G}(Q). For every u ∈ Q there is n ∈ Q∩N such that xux^{−1} = nu, thus such that x^{k}ux^{−k} = n^{k}u for any positive integer k. Since the orders of x and n are coprime, this forces n = 1, hence x ∈ C_{G}(Q). Assume now that N ⊴ H ⊴ G for some normal subgroup H of G such that [G : H] = p. Since [G : N] is a power of p, it suffices to show that [C_{N}(Q∩N)∩N_{G}(Q) : C_{N}(Q)] is a power of p. By induction, the two indices [C_{H}(Q∩N)∩N_{G}(Q) : C_{H}(Q)] and [C_{N}(Q∩H)∩N_{H}(Q∩H) : C_{N}(Q ∩ H)] are powers of p. Intersecting the groups occurring in the first of these two indices with N and the those in the second index with N_{G}(Q) shows that the two indices [C_{N}(Q∩H)∩N_{G}(Q) : C_{N}(Q)] and [C_{N}(Q∩N)∩N_{G}(Q) : C_{N}(Q∩H)∩N_{G}(Q)] are powers of p. Their product is the wanted index. □

Lemma 7.3. Let G be a finite group such that G = P N for some Sylow-p-subgroup P of G and a normal subgroup N of G. Then, for any subgroup Q of P, there is an isomorphism φ : Q ≅ R in F_{P} (G) such that φ restricts to an isomorphism φ|_{Q∩N} : Q ∩ N ≅ R ∩ N belonging to F_{P ∩N}(N) and such that both R, R ∩ N are fully F_{P} (G)- normalised.

Proof. We have N_{G}(Q) ⊆ N_{G}(Q ∩ N); thus there is a Sylow-p-subgroup S of N_{G}(Q) and a Sylow-p-subgroup T of N_{G}(Q ∩ N) such that S ⊆ T. Since G = P N there is n ∈ N such that ^{n}S ⊆ ^{n}T ⊆ P. Taking R = ^{n}Q and φ defined by φ(u) = ^{n}u for all u ∈ Q proves the statement. □

Proof of 7.1. By Alperin’s fusion theorem, it suffices to show that Aut_{F}(Q) = Aut_{G}(Q) for any subgroup Q of P such that Q is fully F_{P} (G)-normalised. By 7.3, we may assume that Q ∩ N is fully F_{P} (G)-normalised, too. We show now, that we may assume that Q is normal in G. By 2.2, N_{P} (Q) is a Sylow-p-subgroup of N_{G}(Q) and by 1.8, the category N_{F}(Q) is a fusion system on N_{P} (Q) contained in F_{N_{P} (Q)}(N_{G}(Q)) and containing F_{N_{P} (Q)∩N}(N_{G}(Q) ∩ N). Clearly Aut_{F}(Q) = Aut_{N_{F} (Q)}(Q) and Aut_{G}(Q) = Aut_{N_{G}(Q)}(Q). Moreover, as N contains all p′-elements of G, we have N_{G}(Q) = N_{P} (Q)(N_{G}(Q) ∩ N). Therefore, we indeed may assume that G = N_{G}(Q). Note that the index [Aut_{F_{P} (G)}(Q) : Aut_{F}(Q)] is prime to p by 1.5. Let φ ∈ Aut_{F_{P} (G)}(Q) of order prime to p. Then φ restricts to an automorphism of Q ∩ N belonging to the fusion system F_{P ∩N}(N), because N contains all p′-elements in G. Thus φ|_{Q∩N} belongs to Aut_{F}(Q∩N). Since Q∩N is fully F_{P} (G)-normalised, it is also fully F-normalised. As φ|_{Q∩N} is the restriction of a group automorphism of Q, it follows that Q is contained in the group N_{φ|_{Q∩N}}. Thus φ|_{Q∩N} extends to a morphism τ : Q → P in F. Since Q is normal in G, we have τ(Q) = Q. Then τ^{−1}φ is an automorphism of Q which restricts to the identity on Q ∩ N. By 7.2, the order of τ^{−1}φ is a power of p, and thus τ^{−1}φ belongs to Aut_{F}(Q). But then also φ ∈ Aut_{F}(Q), which implies the result. □

8 The 2-fusion system of Ω(7, q), q ≡ ±3(mod8), is simple

All group theoretic results used in the proof of 8.1 below are due to R. Solomon [13, 14, 15, 16]. See [18] for background material on orthogonal groups.

Theorem 8.1. Let q be an odd prime power such that q ≡ ±3(mod8), and let S be a Sylow-2-subgroup of Ω(7, q). The fusion system F_{S}(Ω(7, q)) is simple.

Proof. The Sylow-2-subgroups of Ω(7, q) are isomorphic to the Sylow-2-subgroups of A_{12}. Thus S = A ⋊ D for some elementary abelian subgroup A of order 2^{6} and some dihedral group D of order 2^{3}. Then A = J(S), the Thompson subgroup of S, and Z(S) is a Klein four group.

Decomposing the 7-dimensional vectorspace on which the orthogonal group O(7, q) acts naturally yields a subgroup of odd index in O(7, q) isomorphic to O^{+}(4, q) × O^{±}(2, q)×O(1, q), where “±” is according to whether q ≡ 1(mod 4) or q ≡ −1(mod 4). Identifying O(1, q) with {±I} yields a subgroup of odd index in SO(7, q) isomorphic to O^{+}(4, q) × O^{±}(2, q). Intersecting this further with Ω(7, q) shows that the Sylow-2-subgroup S of Ω(7, q) can be chosen such that Z(S) is generated by the central involutions z of O^{+}(4, q) and t of O^{±}(2, q). Then z has four eigenvalues −1 and t has two eigenvalues −1; since H = C_{Ω(7,q)}(Z(S)) has to respect the corresponding eigenspaces, H is a subgroup of index 2 in O^{+}(4, q) × O^{±}(2, q). Thus Ω(4, q) × Ω^{±}(2, q) is normal of index 8 in H. Hence X = O_{2′}(Ω^{±}(2, q)) is cyclic of odd order and Ω^{+}(4, q) × X is normal of index 16 in H. Note that Ω^{+}(4, q) ≅ SL(2, q) × SL(2, q)/ < (−I, −I) >

Let F be a fusion system on S such that F ⊴ F_{S}(Ω(7, q)). By [14], the fusion system F_{S}(Ω(7, q)) is generated by the fusion system on S of the group C_{Ω(7,q)}(Z(S)) together with the group of automorphisms N_{Ω(7,q)}(A)/C_{Ω(7,q)}(A) of A, and moreover this group of automorphisms of A is isomorphic to A_{7}. Since A is fully F_{S}(Ω(7, q))- normalised, it follows that Aut_{F}(A) is a normal subgroup of odd index in the simple group Aut_{F_{S}(Ω(7,q))}(A), hence both are equal. Therefore, in order to show the equality F = F_{S}(Ω(7, q)), it suffices to show the equality C_{F}(Z(S)) = F_{S}(H).

Set G = H/Z(S)X and P = S/Z(S). Let N be the image of Ω^{+}(4, q) in G. Then Z(G) has order 2 and G/Z(G)N ≅ O^{+}(4, q)/Ω^{+}(4, q) is a Klein four group. Note that N ≅ P SL(2, q) × P SL(2, q) has index 8 in G. In particular, the Sylow-2-subgroup P ∩ N of N is of the form P ∩ N = V_{1} × V_{2} for some Klein four groups V_{1}, V_{2} and Aut_{N}(P ∩ N) = E_{1} × E_{2} for some automorphism groups E_{i} of order 3 of V_{i}, i = 1, 2.

Set G = C_{F}(Z(S))/Z(S). By standard properties of fusion systems with respect to dividing by central p-subgroups it suffices to show that G = F_{P} (G). By 7.1, it suffices to show that F_{P ∩N}(N) ⊆ G. Since P ∩ N is abelian, N_{N}(P ∩ N) controls N-fusion in P ∩ N, and therefore it suffices to show that Aut_{N}(P ∩ N) ⊆ Aut_{G}(P ∩ N).

Now P ∩ N is fully F_{P} (G)-normalised, and G ⊴ F_{P} (G); thus Aut_{G}(P ∩ N) is a normal subgroup of odd index in Aut_{G}(P ∩ N); in particular, Aut_{G}(P ∩ N) contains the Sylow-2-subgroup Aut_{P} (P ∩ N) ≅ P/C_{P} (P ∩ N) of Aut_{G}(P ∩ N), and by the above remarks, Aut_{P} (P ∩ N) is a Klein four group. Since Aut_{G}(P ∩ N) has to be normalised in particular by the subgroup Aut_{N}(P ∩ N) = E_{1} × E_{2} of Aut_{G}(P ∩ N), the normal subgroup of Aut_{G}(P ∩ N) generated by Aut_{P} (P ∩ N) contains E_{1} × E_{2}. Hence Aut_{N}(P ∩ N) ⊆ Aut_{G}(P ∩ N).

This shows that F = F_{S}(Ω(7, q)). It remains to see that S has no proper non trivial strongly F_{S}(Ω(7, q))-closed subgroup. If Q is a non trivial strongly F_{S}(Ω(7, q))- closed subgroup of S then Q ∩ Z(Q) ≠ 1, hence Q ∩ A ≠ 1. Since Aut_{Ω(7,q)}(A) ≅ A_{7} acts irreducibly on A this implies A ⊆ Q. Since moreover every involution of S is Ω(7, q)-conjugate to an involution in A, this implies Q = S. □

Remark 8.2. As long as q is congruent to either +3 or −3 modulo 8, the isomorphism class of the Sylow-2-subgroup S of Ω(7, q) and the fusion system F_{S}(Ω(7, q)) do not depend on q.

9 The Solomon 2-local finite group Sol(3) is simple

Let q be an odd prime power such that q ≡ ±3(mod8). Let P be a Sylow-2-subgroup of the 7-dimensional spinor group Spin(7, q) over 𝔽_{q}. Then Spin(7, q) has a central involution z such that Spin(7, q)/ < z >≅ Ω(7, q), and hence P/ < z > is isomorphic to a Sylow-2-subgroup S of Ω(7, q). R. Solomon considered in [15] a fusion system F on P which properly contains F_{P} (Spin(7, q)) in which all involutions of P are conjugate and which has the property that C_{F}(z)/ < z >≅ F_{S}(Ω(7, q)). It has been shown by Levi and Oliver [10], that this fusion system is the underlying fusion system of a 2-local finite group; we are going to call this the Solomon 2-local finite group, denoted by Sol(3).

Theorem 9.1. The Solomon 2-local finite group Sol(3) is simple.

Proof. By [15, 2.1, 2.2, 3.1] the underlying fusion system F of Sol(3) fulfills the hypotheses of Theorem 9.2 below. Moreover, C_{F}(z)/ < z >≅ F_{S}(Ω(7, q)) is simple by 8.1, and one easily checks that P is generated by its set of involutions. Thus 9.2(iii) below implies that the underlying fusion system F of Sol(3) is simple. □

Theorem 9.2. Let F be a fusion system on a finite 2-group P. Suppose that there is an involution z ∈ Z(P) and an involution s ∈ P − {z} such that all involutions in P − {z} are C_{F}(z)-conjugate and such that Aut_{F}(< z > × < s >) ≅ S_{3}. Then the following hold.

(i) If F′ is a fusion system on P such that F′ ⊴ F, then all involutions in P are F′-conjugate.

(ii) If F′ is a fusion system on P such that C_{F}(z) ⊆ F′ ⊴ F, then F′ = F.

(iii) If C_{F}(z)/ < z > is simple and if P is generated by its set of involutions, then F is simple.

Proof. (i) Since F′ is normal in F we have Aut_{F′}(< z > × < s >) = Aut_{F}(< z > × < s >) ≅ S_{3} by 3.5 and the hypotheses. Thus < z >≅< s > in F′. If t is any involution in P − {z}, there is an isomorphism < s >≅< t > in C_{F}(z), hence an isomorphism ψ :< z > × < s >≅< z > × < t > in F mapping s to t and z to itself. The isomorphism < z >≅< s > is mapped under ψ to the isomorphism < z >≅< t >, which is still in F′ as F′ is normal in F. Thus s and t are F′-conjugate.

(ii) Since by (i) all involutions in P are F′-conjugate, we have Hom_{F}(< t >, P) = Hom_{F′}(< t >, P) for any involution t in P. In particular, every involution in P is F′-conjugate to z, and so < z > is the only fully F′-centralised subgroup of order 2 of P. Thus F′ = F by 5.1.

(iii) Since C_{F}(z)/ < z > is simple, it follows that every fusion system on P which is normal in F is actually equal to F. Moreover, since P is generated by its set of involutions, P has no proper non trivial strongly F-closed subgroup, since F is transitive on the set of involutions of P. □

Remark 9.3. The existence and uniqueness of a 2-local group is shown in [10] without any hypothesis on q; we expect that the methods of the present paper extend to the general case. This would imply that there is in fact a series of simple 2-local finite groups Sol(q).

Acknowledgements. The author would like to thank Ron Solomon for many helpful comments regarding the group theory needed in the proof of 8.1 and the statement of 9.2.

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