Geometry & Topology

Correction to: Construction of 2-local finite groups of a type studied by Solomon and Benson

Ran Levi Bob Oliver

Department of Mathematical Sciences, University of Aberdeen, Meston Building 339,

Aberdeen AB24 3UE, U.K. and LAGA, Institut Galilée, Av. J-B Clément, 93430 Villetaneuse, France

       Email: ran@maths.abdn.ac.uk and bob@math.univ-paris13.fr

Abstract

A p-local finite group is an algebraic structure with a classifying space which has many of the properties of p-completed classifying spaces of finite groups. In our paper [LO], we constructed a family of 2-local finite groups which are “exotic” in the following sense: they are based on certain fusion systems over the Sylow 2-subgroup of Spin_{7}(q) (q an odd prime power) shown by Solomon not to occur as the 2-fusion in any actual finite group. As predicted by Benson, the classifying spaces of these 2-local finite groups are very closely related to the Dwyer-Wilkerson space BDI(4). An error in our paper [LO] was pointed out to us by Andy Chermak, and we correct that error here.

AMS Classification numbers Primary: 55R35

Secondary: 55R37, 20D06, 20D20

Keywords: Classifying space, p-completion, finite groups, fusion.

A saturated fusion system over a finite p-group S is a category whose objects are the subgroups of S, whose morphisms are all monomorphisms between the subgroups, and which satisfy certain axioms first formulated by Puig, and also described at the start of the first section in [LO]. The main result of [LO] is the construction of saturated fusion systems over certain 2-groups, motivated by a theorem of Solomon [So], which implies that these systems cannot be induced by fusion in any finite group. Recently, Andy Chermak has pointed out to us that the fusion systems actually constructed in [LO] are not saturated (do not satisfy all of Puig’s axioms). In this note, we describe how to modify that construction in a way so as to obtain saturated fusion systems of the desired type, and explain why all of the results in [LO] (aside from [LO, Lemma A.10]) are true under this new construction.

The following is the main theorem in [LO]:

Theorem 1.1 ([LO, Theorem 2.1]) Let q be an odd prime power, and fix S ∈ Syl_{2}(Spin_{7}(q)). Let z ∈ Z(Spin_{7}(q)) be the central element of order 2. Then there is a saturated fusion system F = F_{Sol}(q) which satisfies the following conditions:

(a) C_{F}(z) = F_{S}(Spin_{7}(q)) as fusion systems over S.

(b) All involutions of S are F -conjugate.

Furthermore, there is a unique centric linking system L = L_{Sol}^{c}(q) associated to F.

We have, in fact, found two errors in our proof of this theorem which we correct here. The more serious one is in [LO, Lemma A.10], which is not true as stated: the last sentence in its proof is wrong. This has several implications on the rest of our construction, all of which are systematically treated here. There is also an error in the statement of [LO, Lemma 2.8(b)] which is corrected below (Lemma 1.8).

We first state and prove here a corrected version of [LO, Lemma A.10], and then state a modified version of the main technical proposition, [LO, Proposition 1.2], used to prove saturation. Afterwards, we describe the changes which are needed in [LO, §2] to prove the main theorem. In the following table, we list the correspondence between results and proofs in [LO, §2] and those here. This is intended as a guide to the reader who is not yet familiar with [LO], and who wants to read it simultaneously with this correction.

Reference in [LO]  Reference here Remarks
Proposition 1.2  Proposition 1.3  more general statement
Theorem 2.1  Theorem 1.1  unchanged
Definition 2.2, Lemma 2.3 —  unchanged
Definition 2.4 — omit definition of Γ_{n}
Prp. 2.5, Def. 2.6, Lem. 2.7 —  unchanged
—  Lem. 1.4, Prp. 1.5 added
—  Definition 1.6  new definition of Γ_{n}
—  Lemma 1.7  added
Lemma 2.8  Lemma 1.8  (b) restated,

new proofs of (b), (e)

Proposition 2.9  Proposition 1.9  partly new proof
Lemma 2.10  Lemma 1.10  partly new proof
Proposition 2.11  Proposition 1.11  partly new proof

The only difference between [LO, Lemma A.10] and the corrected version shown here is that in [LO], we claimed that the “correction factor” Z must lie in a certain subgroup of order 2 in SL_{3}(ℤ/2^{k}), which is definitely not the case. Also, for convenience, we state this lemma here for matrices over the 2-adic integers ℤ̂_{2}, instead of for matrices over the finite rings ℤ/2^{k}.

Lemma 1.2 (modified [LO, Lemma A.10]) Let T_{1} and T_{2} be the two maximal parabolic subgroups of GL_{3}(2):

T_{1} = GL_{2}^{1}(ℤ/2) = {(a_{ij}) ∈ GL_{3}(2) | a_{21} = a_{31} = 0}

and

T_{2} = GL_{1}^{2}(ℤ/2) = {(a_{ij}) ∈ GL_{3}(2) | a_{31} = a_{32} = 0}.

Set T_{0} = T_{1} ∩ T_{2}: the group of upper triangular matrices in GL_{3}(2). Assume, for some k ≥ 2, that

μ_{i}: T_{i} ⟶ SL_{3}(ℤ̂_{2})

are lifts of the inclusions T_{i} ⟶ GL_{3}(2) = SL_{3}(2) (for i = 1, 2) such that μ_{1}|_{T_{0}} = μ_{2}|_{T_{0}} . Then there is a homomorphism

μ: GL_{3}(2) ⟶ SL_{3}(ℤ̂_{2}),

and an element Z ∈ C_{SL_{3}(ℤ̂_{2})}(μ_{1}(T_{0})), such that μ|_{T_{1}} = μ_{1}, and μ|_{T_{2}} = c_{Z} ∘ μ_{2}.

Proof By [DW1, Lemma 4.4], there is a lifting μ′: GL_{3}(2) → SL_{3}(ℤ̂_{2}) of the identity on GL_{3}(2); and any two liftings to SL_{3}(ℤ̂_{2}) of the inclusion of T_{1} or of T_{2} into GL_{3}(2) differ by conjugation by an element of SL_{3}(ℤ̂_{2}). In particular, there are elements Z_{1}, Z_{2} ∈ SL_{3}(ℤ̂_{2}) such that μ_{i} = c_{Z_{i}} ∘ μ′|_{T_{i}} (for i = 1, 2). Set μ = c_{Z_{1}} ∘μ′, and Z = Z_{1}Z_{2}^{−1}. Then μ_{1} = μ|_{T_{i}}, and μ|_{T_{2}} = c_{Z} ∘μ_{2}. Since μ_{1}|_{T_{0}} = μ_{2}|_{T_{0}} , conjugation by Z is the identity on μ_{1}(T_{0}) = μ_{2}(T_{0}), and thus Z ∈ C_{SL_{3}(ℤ̂_{2})}(μ_{1}(T_{0})).

Whenever G is a finite group, S ∈ Syl_{p}(G), S_{0} ◁ S, and Γ ≤ Aut(S_{0}), then

⟨F_{S}(G); F_{S_{0}}(Γ)⟩

denotes the smallest fusion system over S which contains all G-fusion, and which also contains all restrictions of automorphisms in Γ. In other words, if F denotes this fusion system, then for each P, Q ≤ S, Hom_{F}(P, Q) is the set of all composites

P = P_{0} ⟶^{φ_{1}} P_{1} ⟶^{φ_{2}} P_{2} ⟶ ⋯ ⟶ P_{k−1} ⟶^{φ_{k}} P_{k} = Q,

where for each i, P_{i} ≤ S, and either φ_{i} ∈ Hom_{G}(P_{i−1}, P_{i}), or (if P_{i−1}, P_{i} ≤ S_{0}) φ_{i} = ψ_{i}|_{P_{i−1}} for some ψ_{i} ∈ Γ such that ψ_{i}(P_{i−1}) = P_{i}.

Whenever G is a finite group and S ∈ Syl_{p}(G), an automorphism φ ∈ Aut(S) is said to preserve G-fusion if for each P, Q ≤ S and each α ∈ Iso(P, Q), α ∈ Iso_{G}(P, Q) if and only if φαφ^{−1} ∈ Iso_{G}(φ(P), φ(Q)). The proof of Theorem 1.1 is based on the following proposition.

Proposition 1.3 (modified [LO, Proposition 1.2]) Fix a finite group G, a prime p dividing |G|, and a Sylow p-subgroup S ∈ Syl_{p}(G). Fix a normal subgroup Z ◁ G of order p, an elementary abelian subgroup U ◁ S of rank two containing Z such that C_{S}(U) ∈ Syl_{p}(C_{G}(U)), and a group Γ ≤ Aut(C_{S}(U)) of automorphisms which preserve all C_{G}(U)-fusion, and such that γ(U) = U for all γ ∈ Γ. Set

S_{0} = C_{S}(U)  and  F = ⟨F_{S}(G); F_{S_{0}}(Γ)⟩,

and assume the following hold.

(a) All subgroups of order p in S different from Z are G-conjugate.

(b) Γ permutes transitively the subgroups of order p in U.

(c) {φ ∈ Γ | φ(Z) = Z} = Aut_{N_{G}(U)}(C_{S}(U)).

(d) For each E ≤ S which is elementary abelian of rank three, contains U, and is fully centralized in F_{S}(G),

{α ∈ Aut_{F}(C_{S}(E)) | α(Z) = Z} = Aut_{G}(C_{S}(E)).

(e) For all E, E′ ≤ S which are elementary abelian of rank three and contain U, if E and E′ are Γ-conjugate, then they are G-conjugate.

Then F is a saturated fusion system over S. Also, for any P ≤ S such that Z ≤ P,

{φ ∈ Hom_{F}(P, S) | φ(Z) = Z} = Hom_{G}(P, S).  (1)

This proposition is slightly more general than [LO, Proposition 1.2], in that Γ is assumed only to be a group of automorphisms of C_{S}(U) which preserves C_{G}(U)-fusion, and not a group of automorphisms of C_{G}(U) itself. This extra generality is necessary when proving that the fusion systems F_{Sol}(q), under our modified construction, are saturated. The changes needed to prove this more general version of [LO, Proposition 1.2] are described in Section 2.

Proposition 1.3 is applied with G = Spin_{7}(q), Z = Z(G), and S ∈ Syl_{2}(G), U ≤ S, and Γ ≤ Aut(C_{G}(U)) to be chosen shortly. The error in [LO] arose in the choice of Γ, as will be explained in detail below.

We first recall some of the definitions and notation used in [LO]. Throughout, we fix an odd prime power q, let 𝔽_{q} be a field with q elements, and let 𝔽_{q} be its algebraic closure. We write SL_{2}(q^{∞}) = SL_{2}(𝔽_{q}), Spin_{7}(q^{∞}) = Spin_{7}(𝔽_{q}), etc., for short. For each n, ψ^{q^{n}} denotes the automorphism of Spin_{7}(q^{∞}) or of SL_{2}(q^{∞}) induced by the field isomorphism (x ↦ x^{q^{n}}). We then fix elements z, z_{1} ∈ Spin_{7}(q) of order 2, where ⟨z⟩ = Z(Spin_{7}(q)), set U = ⟨z, z_{1}⟩, and construct an explicit homomorphism

ω: SL_{2}(q^{∞})^{3} ⟶ Spin_{7}(q^{∞})

such that

Im(ω) = C_{Spin_{7}(q^{∞})}(U)  and  Ker(ω) = ⟨(−I, −I, −I)⟩.

Write

H(q^{∞}) =^{def} ω(SL_{2}(q^{∞})^{3}) = C_{Spin_{7}(q^{∞})}(U) and [[X_{1}, X_{2}, X_{3}]] = ω(X_{1}, X_{2}, X_{3})

for short. In particular,

z = [[I, I, −I]]  and  z_{1} = [[−I, I, I]],

and thus

U = {[[±I, ±I, ±I]]}

(with all combinations of signs). By [LO, Lemma 2.3 & Proposition 2.5], there is an element τ ∈ N_{Spin_{7}(q)}(U) of order 2 such that

τ·[[X_{1}, X_{2}, X_{3}]]·τ^{−1} = [[X_{2}, X_{1}, X_{3}]]

for all X_{1}, X_{2}, X_{3} ∈ SL_{2}(q^{∞}), and such that

N_{Spin_{7}(q^{∞})}(U) = H(q^{∞})·⟨τ⟩.

We next fix elements A, B ∈ SL_{2}(q) of order 4, such that ⟨A, B⟩ ≅ Q_{8} (a quaternion group of order 8). Most of the following notation is taken from [LO, Definition 2.6]. We set

Â = [[A, A, A]]  and  B̂ = [[B, B, B]];
C(q^{∞}) = {X ∈ C_{SL_{2}(q^{∞})}(A) | X^{2^{k}} = I, some k} ≅ ℤ/2^{∞};

and

Q(q^{∞}) = ⟨C(q^{∞}), B⟩.

Here, ℤ/2^{∞} means a union of cyclic 2-groups ℤ/2^{n} for all n; equivalently, the group ℤ[\frac{1}{2}]/ℤ. We then define

A(q^{∞}) = ω(C(q^{∞})^{3}) ≅ (ℤ/2^{∞})^{3},
S_{0}(q^{∞}) = ω(Q(q^{∞})^{3}) ≤ H(q^{∞})
S(q^{∞}) = S_{0}(q^{∞})·⟨τ⟩ ≤ H(q^{∞})·⟨τ⟩ ≤ Spin_{7}(q^{∞}).

In all cases, whenever a subgroup Θ(q^{∞}) ≤ Spin_{7}(q^{∞}) has been defined, we set

Θ(q^{n}) = Θ(q^{∞}) ∩ Spin_{7}(q^{n}).

Since Spin_{7}(q^{n}) is the fixed subgroup of ψ^{q^{n}} acting on Spin_{7}(q^{∞}) (cf. [LO, Lemma A.3]), H(q^{n}) ≤ H(q^{∞}) is the subgroup of all elements of the form [[X_{1}, X_{2}, X_{3}]], where either X_{i} ∈ SL_{2}(q^{n}) for each i, or ψ^{q^{n}}(X_{i}) = −X_{i} for each i. By [LO, Lemma 2.7], for all n,

S(q^{n}) ∈ Syl_{2}(Spin_{7}(q^{n})).

The following lemma is what is needed to tell us how to choose a subgroup Γ_{n} ≤ Aut(S_{0}(q^{n})) so that the fusion system ⟨F_{S(q^{n})}(Spin_{7}(q^{n})); F_{S_{0}(q^{n})}(Γ_{n})⟩ is saturated. Note that since each element of C(q^{∞}) has 2-power order, it makes sense to write X^{u} ∈ C(q^{∞}) for X ∈ C(q^{∞}) and u ∈ ℤ̂_{2}.

Lemma 1.4 Assume α ∈ Aut(A(q^{∞})) centralizes Aut_{S(q^{∞})}(A(q^{∞})). Then α has the form

α([[X_{1}, X_{2}, X_{3}]]) = [[X_{1}^{v}, X_{2}^{v}, X_{3}^{u}]]

for some u, v ∈ (ℤ̂_{2})^{∗}.

Proof Set

Δ_{0} = Aut_{S(q^{∞})}(A(q^{∞})) = ⟨c_{[[B,I,I]]}, c_{[[I,B,I]]}, c_{[[I,I,B]]}, c_{τ} ⟩

for short. The second equality follows since S(q^{∞}) is by definition generated by A(q^{∞}) and the four elements listed. Set

A_{1} =^{def} ⟨z, z_{1}, Â⟩ ≅ C_{2}^{3},

the 2-torsion subgroup in A(q^{∞}). The image of Δ_{0} ≤ Aut(A(q^{∞})) in the group Aut(A_{1}) ≅ GL_{3}(2) (the image under restriction) is the group of all automorphisms which leave ⟨z⟩ and U = ⟨z, z_{1}⟩ invariant (i.e., the group of upper triangular matrices with respect to the ordered basis {z, z_{1}, Â}).

By assumption, [α, Δ_{0}] = 1, and in particular, αΔ_{0}α^{−1} = Δ_{0}. Since each element of Δ_{0} sends U to itself, this means that each element of Δ_{0} also sends α(U) to itself. Also, U is the only subgroup of rank 2 left invariant by all elements of Δ_{0} (since Δ_{0} contains all automorphisms which leave ⟨z⟩ and U invariant), and hence α(U) = U.

It follows that α induces an automorphism α′ of A(q^{∞})/U = (C(q^{∞})/⟨ − I⟩)^{3}. Also, α′ commutes with the following automorphisms of (C(q^{∞})/⟨ − I⟩)^{3}:

(Y_{1}, Y_{2}, Y_{3}) ↦ (Y_{1}^{±1}, Y_{2}^{±1}, Y_{3}^{±1})  and  (Y_{1}, Y_{2}, Y_{3}) ↦ (Y_{2}, Y_{1}, Y_{3})

since these are induced by automorphisms in Δ_{0}. Hence α′ has the form α′(Y_{1}, Y_{2}, Y_{3}) = (Y_{1}^{v}, Y_{2}^{v}, Y_{3}^{u}) for some u, v ∈ (ℤ̂_{2})^{∗}. Thus for all X_{1}, X_{2}, X_{3} ∈ C(q^{∞}),

α([[X_{1}, X_{2}, X_{3}]]) = [[±X_{1}^{v}, ±X_{2}^{v}, ±X_{3}^{u}]].

Since all elements of C(q^{∞}) are squares, these signs must all be positive, and α has the form α([[X_{1}, X_{2}, X_{3}]] = [[X_{1}^{v}, X_{2}^{v}, X_{3}^{u}]].

For each u ∈ (ℤ̂_{2})^{∗}, let δ_{u} ∈ Aut(A(q^{∞})) be the automorphism

δ_{u}([[X_{1}, X_{2}, X_{3}]]) = [X_{1}, X_{2}, X_{3}^{u}].

Define γ, γ_{u} ∈ Aut(A(q^{∞})) by setting

γ([[X_{1}, X_{2}, X_{3}]]) = [[X_{3}, X_{1}, X_{2}]]  and  γ_{u} = δ_{u}γδ_{u}^{−1}.

Proposition 1.5 There is an element u ∈ ℤ̂_{2} such that u ≡ 1 (mod 4), and such that the subgroup Ω_{u} ≤ Aut(A(q^{∞})) given by

Ω_{u} =^{def} ⟨ Aut_{Spin_{7}(q^{∞})}(A(q^{∞})), γ_{u}⟩

is isomorphic to C_{2} × GL_{3}(2). Furthermore, the following hold:

(a) The subgroup of elements of Ω_{u} which act via the identity on all 2-torsion in A(q^{∞}) has order 2, and contains only the identity and the automorphism (g ↦ g^{−1}).

(b) For each n ≥ 1,

⟨ Aut_{Spin_{7}(q^{n})}(A(q^{n})), γ_{u}⟩ ≅ C_{2} × GL_{3}(2).

Proof For each k ≥ 1, let A_{k} ≤ A(q^{∞}) denote the 2^{k} -torsion subgroup. In particular,

A_{1} = ⟨z, z_{1}, Â⟩ ≅ C_{2}^{3},  where  Â = [[A, A, A]].

Let R_{1}, R_{2}, ⋯ ∈ C(q^{∞}) ≅ ℤ/2^{∞} be elements such that R_{1} = −I, R_{2} = A, and (R_{i})^{2} = R_{i−1} for all k ≥ 2. Thus, |R_{i}| = 2^{i} for all i. For each k ≥ 1, let {r_{1}^{(k)}, r_{2}^{(k)}, r_{3}^{(k)}} be the basis of A_{k} defined by

r_{1}^{(k)} = [[I, I, R_{k}]],  r_{2}^{(k)} = [[R_{k}, I, I]],  and  r_{3}^{(k)} = [[R_{k+1}, R_{k+1}, R_{k+1}]].

In particular, r_{1}^{(1)} = z, r_{2}^{(1)} = z_{1}, and r_{3}^{(1)} = Â. Using these bases, we identify Aut(A_{k}) = GL_{3}(ℤ/2^{k}) and Aut(A(q^{∞})) = GL_{3}(ℤ̂_{2}).

Set

Δ_{0} = Aut_{S(q^{∞})}(A(q^{∞})), Δ_{1} = Aut_{Spin_{7}(q^{∞})}(A(q^{∞})), and Δ_{2} = ⟨Δ_{0}, γ⟩.

In particular, Δ_{2} is the group of all signed permutations

[[X_{1}, X_{2}, X_{3}]] ↦ [[X_{σ(1)}^{±1}, X_{σ(2)}^{±1}, X_{σ(3)}^{±1}]]

for σ ∈ Σ_{3}.

For each i = 0, 1, 2 and each k ≥ 1, let Δ_{i}^{(k)} ≤ Aut(A_{k}) be the image of Δ_{i} under restriction. By [LO, Proposition A.8], Δ_{1}^{(1)} = Aut_{Spin}(A_{1}) is the group of all elements of Aut(A_{1}) ≅ GL_{3}(ℤ/2) which send z to itself. Also, Δ_{0}^{(1)} was seen in the proof of Lemma 1.4 to be the group of all automorphisms of A_{1} which leaves both z and U = ⟨z, z_{1}⟩ invariant; and a similar argument shows that Δ_{2}^{(1)} is the group of all automorphisms of A_{1} which leaves U invariant. Hence, with respect to the ordered basis {z, z_{1}, Â} of A_{1}, each group Δ_{i}^{(1)} ≤ Aut(A_{1}) (i = 0, 1, 2) can be identified with the subgroup T_{i} ≤ GL_{3}(ℤ/2) of Lemma 1.2.

By [LO, Proposition 2.5],

C_{Spin_{7}(q^{∞})}(U) = H(q^{∞}) ≅ (SL_{2}(q^{∞})^{3})/⟨(−I, −I, −I)⟩.

An element [[X_{1}, X_{2}, X_{3}]] ∈ H(q^{∞}) (X_{i} ∈ SL_{2}(q^{∞})) centralizes Â =^{def} [[A, A, A]] if and only if [X_{i}, A] = 1 for each i, or X_{i}AX_{i}^{−1} = −A = A^{−1} for each i.

Set C = C_{SL_{2}(q^{∞})}(A). This is an abelian group (the union of the finite cyclic groups C_{SL_{2}(q^{n})}(A)), and N_{SL_{2}(q^{∞})}(A) = C·⟨B⟩. Hence, since A_{1} = ⟨U, Â⟩ and B̂ = [[B, B, B]], we have

C_{Spin_{7}(q^{∞})}(A_{1}) = C_{H(q^{∞})}(Â) = ω(C^{3})·⟨B̂⟩.  (1)

Since ω(C^{3}) is abelian (thus centralizes A(q^{∞})), this shows that the kernel of each of the projection maps Δ_{i} −↠ Δ_{i}^{(1)} is generated by conjugation by B̂; i.e., by the automorphism (g ↦ g^{−1}).

Since each Δ_{i} is finite, their elements all have determinant of finite order in (ℤ̂_{2})^{∗}, hence are ±1 in all cases. Also, for each i, Δ_{i} surjects onto Δ_{i}^{(1)} = T_{i} with kernel generated by the automorphism (g ↦ g^{−1}) of determinant (−1). Hence the elements of determinant one in Δ_{i} are sent isomorphically to T_{i}, and define a lift

μ_{i}: T_{i} ⟶ SL_{3}(ℤ̂_{2})

with respect to the given bases. In particular, μ_{1}|_{T_{0}} = μ_{2}|_{T_{0}} = μ_{0}.

For all i, j, k ∈ (ℤ̂_{2})^{∗}, we define ψ_{i,j,k} ∈ Aut(A(q^{∞})) by setting

ψ_{i,j,k}([[X_{1}, X_{2}, X_{3}]]) = [[X_{1}^{i}, X_{2}^{j}, X_{3}^{k}]]

for all [[X_{1}, X_{2}, X_{3}]] ∈ A(q^{∞}). Recall that c_{τ} ([[X_{1}, X_{2}, X_{3}]]) = [[X_{2}, X_{1}, X_{3}]] (by choice of τ). Since Δ_{0} = Aut_{S(q^{∞})}(A(q^{∞})) is generated by c_{[[B,I,I]]}, c_{[[I,B,I]]}, c_{[[I,I,B]]}, and c_{τ} (corresponding to generators of S(q^{∞})/A(q^{∞})), μ_{0} has image

Im(μ_{0}) = μ_{1}(T_{0}) = ⟨ψ_{−1,−1,1}, ψ_{1,−1,−1}, c_{τ} ⟩,

where ψ_{−1,−1,1} = c_{[[B,B,I]]} and ψ_{1,−1,−1} = c_{[[I,B,B]]}.

By Lemma 1.2, there is a homomorphism

μ: GL_{3}(2) ⟶ SL_{3}(ℤ̂_{2}),

and an element Z ∈ Aut(A(q^{∞})) ≅ GL_{3}(ℤ̂_{2}) which commutes with all elements of μ_{1}(T_{0}), such that μ|_{T_{1}} = μ_{1} and μ|_{T_{2}} = c_{Z} ∘ μ_{2}. By Lemma 1.4, Z = ψ_{v,v,u} (using the above notation) for some u, v ∈ (ℤ̂_{2})^{∗}. Since ψ_{v,v,v} lies in the center of Aut(A(q^{∞})) (it sends every element to its v-th power), we can assume that v = 1 (without changing c_{Z}), and thus that Z = ψ_{1,1,u} = δ_{u}. Finally, since δ_{−1} = ψ_{1,1,−1} ∈ Δ_{0}, we can replace δ_{u} by δ_{−u} if necessary, and assume that u ≡ 1 (mod 4).

Under the identification GL_{3}(ℤ̂_{2}) = Aut(A(q^{∞})), we now have

Aut_{Spin_{7}(q^{∞})}(A(q^{∞})) = (g ↦ g^{−1}) × μ(T_{1})

and

γ_{u} = δ_{u}γδ_{u}^{−1} = c_{Z}(γ) = μ (( \begin{matrix}0 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )) ,

where the matrix is that of γ|_{A_{1}} with respect to the basis {r_{1}^{(1)}, r_{2}^{(1)}, r_{3}^{(1)}}. Also, T_{1} is a maximal subgroup of GL_{3}(2) — the subgroup of invertible matrices which send r_{1}^{(1)} to itself — and so T_{1} together with this matrix generate GL_{3}(2). Thus

Ω_{u} = ⟨ Aut_{Spin_{7}(q^{∞})}(A(q^{∞})), γ_{u}⟩ = ⟨g ↦ g^{−1}⟩×μ(GL_{3}(2)) ≅ C_{2} ×GL_{3}(2). (2)

This proves the first claim in the proposition. Point (a) follows by construction, since each nonidentity element of μ(GL_{3}(2)) acts nontrivially on A_{1}. Point (b) follows from (2), once we know that each element of Spin_{7}(q^{n}) which normalizes A_{1} (hence which normalizes A(q^{n})) also normalizes A(q^{∞}) — and this follows from (1).

We are now in a position to define the fusion systems we want. Roughly, they are generated by the fusion systems of Spin_{7}(q^{n}) together with one extra automorphism: the cyclic permutation [[X_{1}, X_{2}, X_{3}]] ↦ [[X_{3}, X_{1}, X_{2}]] “twisted” by the automorphism δ_{u} of the last proposition. By comparison, the construction in [LO] was similar but without the twisting (i.e., done with u = 1), and the resulting fusion system is, in fact, not saturated.

We regard Q(q^{∞}) = C(q^{∞})⋊⟨B⟩ as an infinite quaternion group: BA′B^{−1} = A′^{−1} for each A′ ∈ C(q^{∞}), and each element of the coset C(q^{∞})·B has order 4. Hence any automorphism of C(q^{∞}) extends to a unique automorphism of Q(q^{∞}) which sends B to itself.

Definition 1.6 Let u ∈ (ℤ̂_{2})^{∗} be as in Proposition 1.5. Let

γ̂, δ̂_{u} ∈ Aut(S_{0}(q^{∞}))

be the automorphisms

γ̂([[X_{1}, X_{2}, X_{3}]]) = [[X_{3}, X_{1}, X_{2}]]

and

δ̂_{u}([[X_{1}, X_{2}, A′B^{j}]]) = [[X_{1}, X_{2}, (A′)^{u}B^{j}]]

for all X_{i} ∈ Q(q^{∞}), A′ ∈ C(q^{∞}), and i, j ∈ ℤ; and set γ̂_{u} = δ̂_{u}γ̂δ̂_{u}^{−1} ∈ Aut(S_{0}(q^{∞})). For each n ≥ 1, set

Γ_{n} = ⟨ Inn(S_{0}(q^{n})), c_{τ} , γ̂_{u}⟩ ≤ Aut(S_{0}(q^{n}));

and set

F_{n} = F_{Sol}(q^{n}) = ⟨F_{S(q^{n})}(Spin_{7}(q^{n})), F_{S_{0}(q^{n})}(Γ_{n})⟩.

In order to be able to apply Proposition 1.3, it is important to know that Γ_{n} is fusion preserving. This follows immediately from the following lemma.

Lemma 1.7 For each n ≥ 1, the automorphisms c_{τ}, γ̂, δ̂_{u}, and γ̂_{u} all preserve H(q^{n})-fusion in S_{0}(q^{n}).

Proof The fusion in SL_{2}(q^{n}) is generated by inner automorphisms of its Sylow subgroup Q(q^{n}), together with the groups Aut(P) for subgroups P ≤ Q(q^{n}) isomorphic to Q_{8}. This follows from Alperin’s fusion theorem, since these are the only subgroups whose automorphism group is not a 2-group. Thus any automorphism of Q(q^{n}) — in particular, the automorphism φ_{u} defined by φ_{u}(A′) = A′^{u}, φ_{u}(B) = B for A′ ∈ C(q^{n}) — preserves fusion.

Set H_{0}(q^{n}) = {[[X_{1}, X_{2}, X_{3}]] | X_{i} ∈ SL_{2}(q^{n}), ∀i}. Fix a generator Y of C(q^{2n}); then C(q^{n}) = ⟨Y^{2}⟩, and ψ^{q^{n}}(Y) = −Y . For each g = [[X_{1}, X_{2}, X_{3}]] ∈ H(q^{n}), [[ψ^{q^{n}}(X_{1}), ψ^{q^{n}}(X_{2}), ψ^{q^{n}}(X_{3})]] = [[X_{1}, X_{2}, X_{3}]]; and hence there is some fixed ϵ = ±1 for which ψ^{q^{n}}(X_{i}) = ϵ·X_{i} for i = 1, 2, 3. When ϵ = 1, this means that X_{i} ∈ SL_{2}(q^{n}) for each i; while if ϵ = −1 it means that X_{i} ∈ SL_{2}(q^{n})·Y for each i. So every element of H(q^{n}) either lies in H_{0}(q^{n}), or has the form g·[[Y, Y, Y]] for some g ∈ H_{0}(q^{n}).

Assume g ∈ H(q^{n}) and P, Q ≤ S_{0}(q^{n}) are such that gP g^{−1} = Q. Clearly, P ≤ H_{0}(q^{n}) if and only if Q ≤ H_{0}(q^{n}) (H_{0}(q^{n}) is normal in H(q^{n})). We claim that there is h ∈ H(q^{n}) such that

hδ̂_{u}(P)h^{−1} = δ̂_{u}(Q)  and  c_{h} ∘ δ̂_{u}|_{P} = δ̂_{u} ∘ c_{g}|_{P} .  (1)

Let P_{i}, Q_{i} ≤ SL_{2}(q^{2n}) be the projections of P and Q to the i-th factor (i = 1, 2, 3), and write g = [[X_{1}, X_{2}, X_{3}]] (thus X_{i}P_{i}X_{i}^{−1} = Q_{i}). Consider the following cases.

(a) Assume g ∈ H_{0}(q^{n}) and P, Q ≤ H_{0}(q^{n}). Since φ_{u} preserves fusion in SL_{2}(q^{n}), there is Y_{3} ∈ SL_{2}(q^{n}) such that Y_{3}φ_{u}(P_{3})Y_{3}^{−1} = φ_{u}(Q_{3}) and

c_{Y_{3}} ∘ φ_{u}|_{P_{3}} = φ_{u} ∘ c_{X_{3}}|_{P_{3}} . Then h =^{def} [[X_{1}, X_{2}, Y_{3}]] satisfies (1).

(b) Assume g ∉ H_{0}(q^{n}) and P, Q ≤ H_{0}(q^{n}). Write g = g′·[[Y, Y, Y]], where g′ ∈ H_{0}(q^{n}). Choose h′ as in (a), so that (1) is satisfied with g, h replaced by g′, h′. Then the element h = h′·[[Y, Y, Y^{u}]] satisfies (1).

(c) Finally, assume that P, Q ≰ H_{0}(q^{n}). Then none of the subgroups P_{i}, Q_{i} ≤ SL_{2}(q^{2n}) is contained in SL_{2}(q^{n}). By the same procedure as was used in (a), we can find h ∈ H_{0}(q^{2n}) which satisfies (1); the problem is to do this so that h ∈ H(q^{n}).

As noted above, fusion in SL_{2}(q^{2n}) is generated by inner automorphisms of Q(q^{2n}) and automorphisms of subgroups isomorphic to Q_{8}. Hence if P_{i} is not isomorphic to Q_{8} or one of its subgroups, then there is X′_{i} ∈ Q(q^{2n}) such that c_{X_{i}}|_{P_{i}} = c_{X′_{i}}|_{P_{i}}. If, on the other hand, P_{i} (and hence Q_{i}) is isomorphic to Q_{8} or C_{4}, then P_{i} ≤ ⟨A, Y^{r}B⟩ and Q_{i} ≤ ⟨A, Y^{s}B⟩ for some odd r, s ∈ ℤ, and we can choose k ∈ ℤ such that X′_{i} =^{def} Y^{k} ∈ Q(q^{2n}) has the same conjugation action as X_{i}. Thus in all cases, we can write X_{i} = X′_{i}X′′_{i} for some X′_{i} ∈ Q(q^{2n}) and some X′′_{i} ∈ C_{SL_{2}(q^{2n})·⟨Y ⟩}(P_{i}).

The subgroups of Q(q^{2n}) which are centralized by elements in the coset SL_{2}(q^{n})·Y are precisely the cyclic subgroups. (The quaternion subgroups of order ≥ 8 are all centric in SL_{2}(q^{2n}).) Hence we can choose elements Y′′_{i} as follows: Y′′_{i} = 1 if X′′_{i} ∈ SL_{2}(q^{n}), and Y′′_{i} ∈ SL_{2}(q^{n})·Y and centralizes φ_{u}(P_{i}) if X′′_{i} ∈ SL_{2}(q^{n})·Y . We now define

h = [[X′_{1}, X′_{2}, φ_{u}(X′_{3})]]·[[Y′′_{1}, Y′′_{2}, Y′′_{3}]];

then h ∈ H(q^{n}) and satisfies (1).

This shows that δ̂_{u} preserves H(q^{n})-fusion as an automorphism of S_{0}(q^{n}). Also, γ̂ and c_{τ} preserve fusion, since both extend to automorphisms of H(q^{n}); and hence γ̂_{u} = δ̂_{u}γ̂δ̂_{u}^{−1} also preserves fusion.

Let ψ = ψ^{q^{n}} ∈ Aut(Spin_{7}(q^{∞})) be induced by the field automorphism x ↦ x^{q^{n}}. By [LO, Proposition A.9(a)], if E ≤ Spin_{7}(q^{n}) is an arbitrary elementary abelian 2-subgroup of rank 4, then there is an element a ∈ Spin_{7}(q^{∞}) such that aEa^{−1} = E_{∗}, and we define

x_{C}(E) = a^{−1}ψ(a).

Then x_{C}(E) ∈ E, and is independent of the choice of a.

In the following lemma, we correct the statement and proof of points (b) and (e). The proof of (e) is affected by both the changes in the statement of (b) and those in the definition of Γ_{n}.

Lemma 1.8 ([LO, Lemma 2.8]) Fix n ≥ 1, set E_{∗} = ⟨z, z_{1}, Â, B̂⟩ ≤ S(q^{n}), and let C be the Spin_{7}(q^{n})-conjugacy class of E_{∗}. Let E_{4}^{U} be the set of all elementary abelian subgroups E ≤ S(q^{n}) of rank 4 which contain U = ⟨z, z_{1}⟩. Fix a generator X ∈ C(q^{n}) (the 2-power torsion in C_{SL_{2}(q^{n})}(A)), and choose Y ∈ C(q^{2n}) such that Y^{2} = X. Then the following hold.

(a) E_{∗} has type I.

(b) Each subgroup in E_{4}^{U} which contains Â is of the form

E_{ijk} = ⟨z, z_{1}, Â, [[X^{i}B, X^{j}B, X^{k}B]]⟩ or

E′_{ijk} = ⟨z, z_{1}, Â, [[X^{i}Y B, X^{j}Y B, X^{k}Y B]]⟩.

Each subgroup in E_{4}^{U} is H(q^{n})-conjugate to one of these subgroups E_{ijk} or E′_{ijk} for some i, j, k ∈ ℤ.

(c) x_{C}(E_{ijk}) = [[(−I)^{i}, (−I)^{j}, (−I)^{k}]] and x_{C}(E′_{ijk}) = [[(−I)^{i}, (−I)^{j}, (−I)^{k}]]·Â.

(d) All of the subgroups E′_{ijk} have type II. The subgroup E_{ijk} has type I if and only if i ≡ j (mod 2), and lies in C (is conjugate to E_{∗}) if and only if i ≡ j ≡ k (mod 2). The subgroups E_{000}, E_{001}, and E_{100} thus represent the three conjugacy classes of rank four elementary abelian subgroups of Spin_{7}(q^{n}) (and E_{∗} = E_{000}).

(e) For any φ ∈ Γ_{n} ≤ Aut(S_{0}(q^{n})) and any E ∈ E_{4}^{U}, φ(x_{C}(E)) = x_{C}(φ(E)).

Proof We prove only points (b) and (e) here, and refer to [LO] for the proofs of the other points.

(b) Assume first that Â ∈ E; i.e., that E ≥ A_{1} = ⟨z, z_{1}, Â⟩. By definition, S(q^{n}) is generated by A(q^{n}), whose elements clearly centralize A_{1}; and elements [[B^{i}, B^{j}, B^{k}]] for i, j, k ∈ {0, 1}. Since an element of this form centralizes Â only if i = j = k, this shows that

E ≤ C_{S(q^{n})}(⟨z, z_{1}, Â⟩) = A(q^{n})·⟨B̂⟩.

Since A(q^{n}) is a finite abelian 2-group of rank 3, we have E = ⟨z, z_{1}, Â, gB̂⟩ for some g ∈ A(q^{n}). Also,

A(q^{n}) = ω(C(q^{∞})^{3}) ∩ Spin_{7}(q^{n}) = {[[X^{i}, X^{j}, X^{k}]], [[X^{i}Y, X^{j}Y, X^{k}Y]] | i, j, k ∈ ℤ},

and hence E = E_{ijk} or E′_{ijk} for some i, j, k ∈ ℤ.

Now let E ∈ E_{4}^{U} be arbitrary. Each element of E has the form [[X_{1}, X_{2}, X_{3}]], where either X_{i} ∈ SL_{2}(q^{n}) for all i, or X_{i} ∈ SL_{2}(q^{n})·Y for all i — and the elements of the first type (X_{i} ∈ SL_{2}(q^{n})) form a subgroup of index at most 2. Since U has index 4 in E, this means that there is some g = [[X_{1}, X_{2}, X_{3}]] ∈ E∖U for which X_{i} ∈ SL_{2}(q^{n}) for all i. Also, |X_{i}| = 4 for all i, since g ∉ U = {[[±I, ±I, ±I]]}. Since all elements of order 4 in SL_{2}(q^{n}) are conjugate (cf. [Sz, 3.6.23]), this implies that g is H(q^{n})-conjugate to Â, and hence that E is H(q^{n})-conjugate to one of the above subgroups E_{ijk} or E′_{ijk}.

(e) By construction, x_{C}(−) is preserved under conjugation by elements of the group Spin_{7}(q^{n}). Since Γ_{n} is generated by γ̂_{u} and conjugation by elements of Spin_{7}(q^{n}), it suffices to prove the result when φ = γ̂_{u}. Since u ≡ 1 (mod 4) by Proposition 1.5, γ̂_{u}(Â) = Â, γ̂_{u}(E_{ijk}) = E_{k′,i′,j′}, and γ̂_{u}(E′_{ijk}) = E′_{k′′,i′′,j′′} for some i′ ≡ i′′ ≡ i (mod 2), and similarly for the other indices. Hence by (c), γ̂_{u}(x_{C}(E)) = x_{C}(γ̂_{u}(E)) whenever E = E_{ijk} or E = E′_{ijk} for some i, j, k ∈ ℤ.

Now assume E ∈ E_{4}^{U} is not one of the subgroups E_{ijk} or E′_{ijk}. By (b), there

is g ∈ H(q^{n}) such that E′ =^{def} gEg^{−1} is of this form. Since γ̂_{u} preserves H(q^{n})-fusion by Lemma 1.7, there is h ∈ H(q^{n}) such that the following square commutes:

\begin{matrix}E & \overline{\underrightarrow{γ̂_{u}}} & γ̂_{u}(E) \\ E′ & ↓^{c_{g}} & \overline{\underrightarrow{γ̂_{u}}} & γ̂_{u}(E′) . & ↓^{c_{h}} \end{matrix}

We have seen that γ̂_{u}(x_{C}(E′)) = x_{C}(γ̂_{u}(E′)); and also that c_{g} and c_{h} preserve x_{C}(−). Hence γ̂_{u}(x_{C}(E)) = x_{C}(γ̂_{u}(E)) by the commutativity of the square.

The following is the crucial result needed to apply Proposition 1.3. The statement is exactly the same as that in [LO], but the proof has to be modified slightly due to the changed definition of Γ_{n} (hence of F_{n}).

Proposition 1.9 ([LO, Proposition 2.9]) Fix n ≥ 1. Let E ≤ S(q^{n}) be an elementary abelian subgroup of rank 3 which contains U, and such that C_{S(q^{n})}(E) ∈ Syl_{2}(C_{Spin_{7}(q^{n})}(E)). Then

{φ ∈ Aut_{F_{n}}(C_{S(q^{n})}(E)) | φ(z) = z} = Aut_{Spin_{7}(q^{n})}(C_{S(q^{n})}(E)).  (1)

Proof Set

Spin = Spin_{7}(q^{n}),  S = S(q^{n}),  Γ = Γ_{n},  and  F = F_{n}

for short, and consider the subgroups

R_{0} = R_{0}(q^{n}) =^{def} A(q^{n})  and  R_{1} = R_{1}(q^{n}) =^{def} C_{S}(⟨U, Â⟩) = ⟨R_{0}, B̂⟩.

Then

R_{0} ≅ (C_{2^{k}})^{3}  and  R_{1} = R_{0} ⋊ ⟨B̂⟩,

where 2^{k} is the largest power which divides q^{n} ± 1, and where B̂ = [[B, B, B]] has order 2 and acts on R_{0} via (g ↦ g^{−1}). Also,

⟨U, Â⟩ = ⟨[[±I, ±I, ±I]], [[A, A, A]]⟩ ≅ C_{2}^{3}

is the 2-torsion subgroup of R_{0}. It was shown in the proof of [LO, Proposition 2.9] that

R_{0} is the only subgroup of S isomorphic to (C_{2^{k}})^{3}.  (2)

Let E ≤ S be an elementary abelian subgroup of rank 3 which contains U, and such that C_{S}(E) ∈ Syl_{2}(C_{Spin}(E)). There are two cases to consider: that where E ≤ R_{0} and that where E ≰ R_{0}.

Case 1: Assume E ≤ R_{0}. Since R_{0} is abelian of rank 3, we must have E = ⟨U, Â⟩, the 2-torsion subgroup of R_{0}, and C_{S}(E) = R_{1}. Also, by (2), neither R_{0} nor R_{1} is isomorphic to any other subgroup of S; and hence

Aut_{F}(R_{i}) = ⟨Aut_{Spin}(R_{i}), Aut_{Γ}(R_{i})⟩ = ⟨Aut_{Spin}(R_{i}), γ_{u}|_{R_{i}}⟩ for i = 0, 1.

(3)

(Recall that Aut_{Γ}(R_{i}) is generated by Aut_{S_{0}(q^{n})}(R_{i}) and the restrictions of c_{τ} and γ_{u}.) Hence by Proposition 1.5(b),

Aut_{F}(R_{0}) = ⟨Aut_{Spin}(R_{0}), γ_{u}|_{R_{0}}⟩ ≅ C_{2} × GL_{3}(2).

In other words, if we let A_{1} = ⟨z, z_{1}, Â⟩ denote the 2-torsion subgroup of R_{0}, then restriction to A_{1} sends Aut_{F}(R_{0}) onto Aut(A_{1}) with kernel ⟨cB̂⟩ of order 2. Since Aut_{Spin}(A_{1}) is the group of all automorphisms of A_{1} which send z to itself [LO, Proposition A.8], this shows that

Aut_{Spin}(R_{0}) = {φ ∈ Aut_{F}(R_{0}) | φ(z) = z}.  (4)

In the proof of [LO, Proposition 2.9] (see formula (5) in that proof), we show the first of the following two equalities:

Aut_{Spin}(R_{1}) = Inn(R_{1})·{φ ∈ Aut_{Spin}(R_{1}) | φ(B̂) = B̂}
= Inn(R_{1})·{φ ∈ Aut(R_{1}) | φ(B̂) = B̂, φ|_{R_{0}} ∈ Aut_{Spin}(R_{0})}. (5)

The second equality holds since R_{1} = ⟨R_{0}, B̂⟩, and since R_{0} is the unique abelian subgroup of R_{1} of index 2. Since γ̂_{u}(R_{0}) = R_{0} and γ̂_{u}(B̂) = B̂, this, together with (3), shows that

Aut_{F}(R_{1}) = Inn(R_{1})·{φ ∈ Aut(R_{1}) | φ(B̂) = B̂, φ|_{R_{0}} ∈ Aut_{F}(R_{0})},

and hence that

{φ ∈ Aut_{F}(R_{1}) | φ(z) = z}

= Inn(R_{1})·{φ ∈ Aut(R_{1}) | φ(B̂) = B̂, φ(z) = z, φ|_{R_{0}} ∈ Aut_{F}(R_{0})}

= Inn(R_{1})·{φ ∈ Aut(R_{1}) | φ(B̂) = B̂, φ|_{R_{0}} ∈ Aut_{Spin}(R_{0})},  (6)

where the second equality follows from (4). If φ ∈ Aut(R_{1}) is such that φ(B̂) = B̂ and φ|_{R_{0}} = c_{x}|_{R_{0}} for some x ∈ N_{Spin}(R_{0}), then x normalizes R_{1} (the centralizer of the 2-torsion in R_{0}), c_{x}(B̂) = c_{y}(B̂) for some y ∈ R_{1} = R_{0}·⟨B̂⟩ by (5), so we can assume y ∈ R_{0}. Since R_{0} = A(q^{∞}) is abelian (so [y, R_{0}] = 1), this implies c_{y^{−1}x}|_{R_{1}} = φ, and hence φ ∈ Aut_{Spin}(R_{1}). So {φ ∈ Aut_{F}(R_{1}) | φ(z) = z} ≤ Aut_{Spin}(R_{1}) by (6), and the opposite inclusion is clear.

Case 2: Now assume that E ≰ R_{0}. By assumption, U ≤ E (hence E ≤ C_{S}(E) ≤ C_{S}(U)), and C_{S}(E) is a Sylow subgroup of C_{Spin}(E). Also, E contains an element of the form g·[[B^{i}, B^{j}, B^{k}]] for g ∈ R_{0} = A(q^{n}) and some i, j, k not all even, and hence A(q^{n}) ≰ C_{S}(E). Hence by (2), C_{S}(E) is not isomorphic to R_{1} = C_{S}(⟨z, z_{1}, Â⟩), and this shows that E is not Spin-conjugate to ⟨z, z_{1}, Â⟩. By [LO, Proposition A.8], Spin contains exactly two conjugacy classes of rank 3 subgroups containing z, and thus E must have type II. So by [LO, Proposition A.8(d)], C_{S}(E) is elementary abelian of rank 4, and also has type II.

Let C be the Spin_{7}(q^{n})-conjugacy class of the subgroup E_{∗} = ⟨U, Â, B̂⟩ ≅ C_{2}^{4}, which by Lemma 1.8(a) has type I. Let E′ be the set of all subgroups of S which are elementary abelian of rank 4, contain U, and are not in C. By Lemma 1.8(e), for any φ ∈ Iso_{Γ}(E′, E′′) and any E′ ∈ E′, E′′ =^{def} φ(E′) ∈ E′, and φ sends x_{C}(E′) to x_{C}(E′′). The same holds for φ ∈ Iso_{Spin}(E′, E′′) by definition of the elements x_{C}(−) ([LO, Proposition A.9]). Since C_{S}(E) ∈ E′, this shows that all elements of Aut_{F}(C_{S}(E)) send the element x_{C}(C_{S}(E)) to itself. By [LO, Proposition A.9(c)], Aut_{Spin}(C_{S}(E)) is the group of automorphisms which are the identity on the rank two subgroup ⟨x_{C}(C_{S}(E)), z⟩; and (1) now follows.

The proof of the following lemma is essentially unchanged.

Lemma 1.10 ([LO, Lemma 2.10]) Fix n ≥ 1, and let E, E′ ≤ S(q^{n}) be two elementary abelian subgroups of rank three which contain U, and which are Γ_{n}-conjugate. Then E and E′ are Spin_{7}(q^{n})-conjugate.

Proof Consider the sets

J_{1} = {X ∈ SL_{2}(q^{n}) | X^{2} = −I}

and

J_{2} = {X ∈ SL_{2}(q^{2n}) | ψ^{q^{n}}(X) = −X, X^{2} = −I}.

Here, as usual, ψ^{q^{n}} is induced by the field automorphism (x ↦ x^{q^{n}}). It was shown in the proof of [LO, Lemma 2.10] that all elements in each of these sets are SL_{2}(q)-conjugate to each other.

Since E and E contain U, E, E′ ≤ C_{Spin_{7}(q^{n})}(U). By [LO, Proposition 2.5(a)],

C_{Spin_{7}(q^{n})}(U) = H(q^{n}) =^{def} ω(SL_{2}(q^{∞})^{3}) ∩ Spin_{7}(q^{n}).

Thus

E = ⟨z, z_{1}, [[X_{1}, X_{2}, X_{3}]]⟩  and  E′ = ⟨z, z_{1}, [[X′_{1}, X′_{2}, X′_{3}]]⟩,

where the X_{i} are all in J_{1} or all in J_{2}, and similarly for the X′_{i}. Also, E and E′ are Γ_{n}-conjugate, and each element of Γ_{n} leaves U = ⟨z, z_{1}⟩ and ω(SL_{2}(q^{n})^{3}) invariant. Hence either E and E′ are both contained in ω(SL_{2}(q^{n})^{3}), in which case the X_{i} and X′_{i} are all in J_{1}; or neither is contained in ω(SL_{2}(q^{n})^{3}), in which case the X_{i} and X′_{i} are all in J_{2}. This shows that the X_{i} and X′_{i} are all SL_{2}(q^{n})-conjugate, and so E and E′ are Spin_{7}(q^{n})-conjugate.

We are now ready to prove:

Proposition 1.11 ([LO, Proposition 2.11]) For a fixed odd prime power q, let S(q^{n}) ≤ S(q^{∞}) ≤ Spin_{7}(q^{∞}) be as defined above. Let z ∈ Z(Spin_{7}(q^{∞})) be the central element of order 2. Then for each n, F_{n} = F_{Sol}(q^{n}) is saturated as a fusion system over S(q^{n}), and satisfies the following conditions:

(a) For all P, Q ≤ S(q^{n}) which contain z, if α ∈ Hom(P, Q) is such that α(z) = z, then α ∈ Hom_{F_{n}}(P, Q) if and only if α ∈ Hom_{Spin_{7}(q^{n})}(P, Q).

(b) C_{F_{n}}(z) = F_{S(q^{n})}(Spin_{7}(q^{n})) as fusion systems over S(q^{n}).

(c) All involutions of S(q^{n}) are F_{n}-conjugate.

Furthermore, F_{m} ⊆ F_{n} for m|n. The union of the F_{n} is thus a category F_{Sol}(q^{∞}) whose objects are the finite subgroups of S(q^{∞}).

Proof We apply Proposition 1.3, where p = 2, G = Spin_{7}(q^{n}), S = S(q^{n}), Z = ⟨z⟩ = Z(G), U = ⟨z, z_{1}⟩, C_{G}(U) = H(q^{n}), S_{0} = C_{S}(U) = S_{0}(q^{n}), and Γ = Γ_{n} ≤ Aut(S_{0}). By Lemma 1.7, γ̂_{u} preserves H(q^{n})-fusion in S_{0}. Since Γ is generated by γ̂_{u} and certain automorphisms of H(q^{n}), this shows that all automorphisms in Γ preserve H(q^{n})-fusion.

Condition (a) in Proposition 1.3 (all noncentral involutions in G are conjugate) holds since all subgroups in E_{2} are conjugate ([LO, Proposition A.8]), and condition (b) holds by definition of Γ. Condition (c) holds since

{γ ∈ Γ | γ(z) = z} = Inn(S_{0}(q^{n}))·⟨c_{τ} ⟩ = Aut_{N_{G}(U)}(S_{0}(q^{n}))

by [LO, Proposition 2.5(b)]. Condition (d) was shown in Proposition 1.9, and condition (e) in Lemma 1.10. So by Proposition 1.3, F_{n} is a saturated fusion system, and C_{F_{n}}(Z) = F_{S(q^{n})}(Spin_{7}(q^{n})).

The proofs of the other statements remain unchanged.

In Section 3 of [LO], these corrections affect only the proof of Lemma 3.1. In that proof, the groups E_{100} and E_{001} are not Γ_{1}-conjugate under the new definitions; instead E_{100} is Γ_{1}-conjugate to a subgroup which is Spin_{7}(q)- conjugate to E_{001} (and hence the two are F -conjugate). Also, when showing that Aut_{F}(E_{001}) is the group of all automorphisms which fix z = x_{C}(E_{001}), it is important to know that all Γ_{1}-isomorphisms between subgroups in that conjugacy class preserve the elements x_{C}(−) (as shown in Lemma 1.8(e)), and not just that Γ_{1}-automorphisms of E_{001} do so.

The changes do not affect the later sections. Just note that we are able to consider F_{Sol}(q^{n}) as a subcategory of F_{Sol}(q^{kn}) for k > 1, because they were both chosen using the same “correction factor” u ∈ (ℤ̂_{2})^{∗} in the definitions of Γ_{n} and Γ_{kn}.

2 Proof of Proposition 1.3

Proposition 1.3 follows from Lemmas 1.3, 1.4, and 1.5 in [LO], once they are restated to assume the hypotheses of this new proposition, and not those of [LO, Proposition 1.2]. The only one of these lemmas whose proof is affected by the change in hypotheses is Lemma 1.4, and so we restate and reprove it here.

Lemma 2.1 Assume the hypotheses of Proposition 1.3, and let

F = ⟨F_{S}(G); F_{S_{0}}(Γ)⟩

be the fusion system generated by G and Γ. Then for all P, P′ ≤ S which contain Z,

{φ ∈ Hom_{F}(P, P′) | φ(Z) = Z} = Hom_{G}(P, P′).

Proof Upon replacing P′ by φ(P) ≤ P′, we can assume that φ is an isomorphism, and thus that it factors as a composite of isomorphisms

P = P_{0} ⟶_{≅}^{φ_{1}} P_{1} ⟶_{≅}^{φ_{2}} P_{2} ⟶_{≅}^{φ_{3}} ⋯ ⟶_{≅}^{φ_{k−1}} P_{k−1} ⟶_{≅}^{φ_{k}} P_{k} = P′,

where for each i, φ_{i} ∈ Hom_{G}(P_{i−1}, P_{i}) or φ_{i} ∈ Hom_{Γ}(P_{i−1}, P_{i}). Let Z_{i} ≤ Z(P_{i}) be the subgroups of order p such that Z_{0} = Z_{k} = Z and Z_{i} = φ_{i}(Z_{i−1}).

To simplify the discussion, we say that a morphism in F is of type (G) if it is given by conjugation by an element of G, and of type (Γ) if it is the restriction of an automorphism in Γ. More generally, we say that a morphism is of type (G, Γ) if it is the composite of a morphism of type (G) followed by one of type (Γ), etc. We regard Id_{P}, for all P ≤ S, to be of both types, even if P ≰ S_{0}. By definition, if any nonidentity isomorphism is of type (Γ), then its source and image are both contained in S_{0} = C_{S}(U). In particular, if P_{i} ≰ S_{0} for any 0 < i < k, then φ_{i} and φ_{i+1} are both of type (G), and we can remove it and replace the two morphisms by their composite. We can thus assume that [P_{i}, U] = 1 for all 0 < i < k, and hence that Z_{i} ≤ Z(P_{i}U) for all i.

For each i, using [LO, Lemma 1.3], choose some ψ_{i} ∈ Hom_{F}(P_{i}U, S) such that ψ_{i}(Z_{i}) = Z. More precisely, using points (1) and (2) in [LO, Lemma 1.3], we can choose ψ_{i} to be of type (Γ) if Z_{i} ≤ U (the inclusion if Z_{i} = Z), and to be of type (G, Γ) if Z_{i} ≰ U. Set P′_{i} = ψ_{i}(P_{i}). To keep track of the effect of morphisms on the subgroups Z_{i}, we write them as morphisms between pairs, as shown below. Thus, φ factors as a composite of isomorphisms

(P′_{i−1}, Z) ⟶^{ψ_{i−1}^{−1}} (P_{i−1}, Z_{i−1}) ⟶^{φ_{i}} (P_{i}, Z_{i}) ⟶^{ψ_{i}} (P′_{i}, Z).

If φ_{i} is of type (G), then this composite (after replacing adjacent morphisms of the same type by their composite) is of type (Γ, G, Γ). If φ_{i} is of type (Γ), then the composite is again of type (Γ, G, Γ) if either Z_{i−1} ≤ U or Z_{i} ≤ U, and is of type (Γ, G, Γ, G, Γ) if neither Z_{i−1} nor Z_{i} is contained in U. So we are reduced to assuming that φ is of one of these two forms.

Case 1: Assume first that φ is of type (Γ, G, Γ); i.e., a composite of isomorphisms of the form

(P_{0}, Z) ⟶_{(Γ)}^{φ_{1}} (P_{1}, Z_{1}) ⟶_{(G)}^{φ_{2}} (P_{2}, Z_{2}) ⟶_{(Γ)}^{φ_{3}} (P_{3}, Z).

Then Z_{1} = Z if and only if Z_{2} = Z because φ_{2} is of type (G). If Z_{1} = Z_{2} = Z, then φ_{1} and φ_{3} are of type (G) by Proposition 1.3(c), and the result follows.

If Z_{1} ≠ Z ≠ Z_{2}, then U = ZZ_{1} = ZZ_{2}, and thus φ_{2}(U) = U. Neither φ_{1} nor φ_{3} can be the identity, so P_{i} ≤ S_{0} = C_{S}(U) for all i by definition of Hom_{Γ}(−, −). Let γ_{1}, γ_{3} ∈ Γ be such that φ_{i} = γ_{i}|_{P_{i−1}} , and let g ∈ N_{G}(U) be such that φ_{2} = c_{g}. Then gC_{S}(U)g^{−1} ∈ Syl_{p}(C_{G}(U)), so there is h ∈ C_{G}(U) such that hg ∈ N(C_{S}(U)). Then c_{hg} ∈ Γ by Proposition 1.3(c).

Since γ_{3} preserves C_{G}(U)-fusion among subgroups of S_{0}, there is g′ ∈ C_{G}(U) such that c_{g′} = γ_{3} ∘ c_{h}^{−1} ∘ γ_{3}^{−1}. Thus φ is the composite

P_{0} ⟶_{(Γ)}^{φ_{1}} P_{1} ⟶_{(Γ)}^{c_{hg}} hP_{2}h^{−1} ⟶_{(Γ)}^{γ_{3}} γ_{3}(hP_{2}h^{−1}) ⟶_{(G)}^{c_{g′}} P_{3}.

The composite of the first three isomorphisms is of type (Γ) and sends Z to g′^{−1}Zg′ = Z, hence is of type (G) by Proposition 1.3(c) again, and so φ is also of type (G).

Case 2: Assume now that φ is of type (Γ, G, Γ, G, Γ); more precisely, that it is a composite of the form

(P_{0}, Z) ⟶_{(Γ)}^{φ_{1}} (P_{1}, Z_{1}) ⟶_{(G)}^{φ_{2}} (P_{2}, Z_{2}) ⟶_{(Γ)}^{φ_{3}} (P_{3}, Z_{3})

⟶_{(G)}^{φ_{4}} (P_{4}, Z_{4}) ⟶_{(Γ)}^{φ_{5}} (P_{5}, Z),

where Z_{2}, Z_{3} ≰ U. Then Z_{1}, Z_{4} ≤ U and are distinct from Z, and the groups P_{0}, P_{1}, P_{4}, P_{5} all contain U since φ_{1} and φ_{5} (being of type (Γ)) leave U invariant. In particular, P_{2} and P_{3} contain Z, since P_{1} and P_{4} do and φ_{2}, φ_{4} are of type (G). We can also assume that U ≤ P_{2}, P_{3}, since otherwise P_{2} ∩ U = Z or P_{3} ∩ U = Z, φ_{3}(Z) = Z, and hence φ_{3} is of type (G) by Proposition 1.3(c) again. Finally, we assume that P_{2}, P_{3} ≤ S_{0} = C_{S}(U), since otherwise φ_{3} = Id.

Let E_{i} ≤ P_{i} be the rank three elementary abelian subgroups defined by the requirements that E_{2} = UZ_{2}, E_{3} = UZ_{3}, and φ_{i}(E_{i−1}) = E_{i}. In particular, E_{i} ≤ Z(P_{i}) for i = 2, 3 (since Z_{i} ≤ Z(P_{i}), and U ≤ Z(P_{i}) by the above remarks); and hence E_{i} ≤ Z(P_{i}) for all i. Also, U = ZZ_{4} ≤ φ_{4}(E_{3}) = E_{4} since φ_{4}(Z) = Z, and thus U = φ_{5}(U) ≤ E_{5}. Via similar considerations for E_{0} and E_{1}, we see that U ≤ E_{i} for all i.

Set H = C_{G}(U) for short. Let E_{3} be the set of all elementary abelian subgroups E ≤ S of rank three which contain U, and let E_{3}^{∗} ⊆ E_{3} be the set of all E ∈ E_{3} such that C_{S}(E) ∈ Syl_{p}(C_{G}(E)). Then for all E ∈ E_{3},

C_{S}(E) = C_{S_{0}}(E)  and  C_{G}(E) = C_{H}(E)

since E ≥ U (and S_{0} = C_{S}(U)). Thus E_{3}^{∗} is the set of all subgroups E ∈ E_{3} which are fully centralized in the fusion system F_{S_{0}}(H), and so each subgroup in E_{3} is H -conjugate to a subgroup in E_{3}^{∗}.

Let γ_{1}, γ_{3}, γ_{5} ∈ Γ be such that φ_{i} = γ_{i}|_{P_{i−1}} for odd i. Let g_{2}, g_{4} ∈ G be such that φ_{i} is conjugation by g_{i} for i = 2, 4. We will construct a commutative diagram of the following form

\begin{matrix}(P_{0}, E_{0}) & →^{γ_{1}} & (P_{1}, E_{1}) & →^{c_{g_{2}}} & (P_{2}, E_{2}) & →^{γ_{3}} & (P_{3}, E_{3}) & →^{c_{g_{4}}} & (P_{4}, E_{4}) & →^{γ_{5}} & (P_{5}, E_{5}) \\ (C′_{0}, E′_{0}) & ↓^{c_{a_{0}}} & →^{γ_{1}} & (C′_{1}, E′_{1}) & ↓^{c_{a_{1}}} & →^{c_{h_{2}}} & (C′_{2}, E′_{2}) & ↓^{c_{a_{2}}} & →^{γ_{3}} & (C′_{3}, E′_{3}) & ↓^{c_{a_{3}}} & →^{c_{h_{4}}} & (C′_{4}, E′_{4}) & ↓^{c_{a_{4}}} & →^{γ_{5}} & (C′_{5}, E′_{5}) & ↓^{c_{a_{5}}} \end{matrix}

where E′_{i} ∈ E_{3}^{∗}, C′_{i} = C_{S}(E′_{i}), h_{2}, h_{4} ∈ G, and a_{i} ∈ H = C_{G}(U). To do this, first choose a′_{0}, a′_{2}, a′_{4} ∈ H such that E′_{i} =^{def} a′_{i}E_{i}a′_{i}^{−1} ∈ E_{3}^{∗} for i = 0, 2, 4, and set E′_{i} = γ_{i}(E′_{i−1}) for i = 1, 3, 5. Then C_{S}(E′_{i}) = γ_{i}(C_{S}(E′_{i−1})) for i = 1, 3, 5, so C_{S}(E′_{i}) ∈ Syl_{p}(C_{H}(E′_{i})) and E′_{i} ∈ E_{3}^{∗} for all i. So we can choose x_{0} ∈ C_{H}(E′_{0}) such that x_{0}(a′_{0}P_{0}a′_{0}^{−1})x_{0}^{−1} ≤ C_{S}(E′_{0}), and set a_{0} = x_{0}a′_{0} ∈ H. Since γ_{1} ∈ Aut(S_{0}) preserves H -fusion, there is a_{1} ∈ H which makes the first square in the above diagram commute. Now choose x_{2} ∈ C_{H}(E′_{2}) such that

x_{2}((a′_{2}g_{2}a_{1}^{−1})C_{S}(E′_{1})(a′_{2}g_{2}a_{1}^{−1})^{−1})x_{2}^{−1} ≤ C_{S}(E′_{2}),

and set a_{2} = x_{2}a′_{2} and h_{2} = a_{2}g_{2}a_{1}^{−1}. Upon continuing this procedure we obtain the above diagram.

Let φ̂ ∈ Iso_{F}(C_{S}(E′_{0}), C_{S}(E′_{5})) be the composite of the morphisms in the bottom row of the above diagram. Then φ̂(Z) = Z, since φ and the c_{a_{i}} all send Z to itself. By Proposition 1.3(e), the rank three subgroups E′_{i} are all G-conjugate to each other. Choose g ∈ G such that gE′_{5}g^{−1} = E′_{0}. Then g·C_{S}(E′_{5})·g^{−1} and C_{S}(E′_{0}) are both Sylow p-subgroups of C_{G}(E′_{0}), so there is h ∈ C_{G}(E′_{0}) such that (hg)C_{S}(E′_{5})(hg)^{−1} = C_{S}(E′_{0}). By Proposition 1.3(d), c_{hg} ∘ φ̂ ∈ Aut_{F}(C_{S}(E′_{0})) is of type (G), so φ̂ is of type (G), and hence φ = c_{a_{5}}^{−1} ∘ φ̂ ∘ c_{a_{0}} is also of type (G).

References

[DW1] W. Dwyer & C. Wilkerson, A new finite loop space at the prime two, J. Amer. Math. Soc. 6 (1993), 37–64

[LO] R. Levi & B. Oliver, Construction of 2-local finite groups of a type studied by Solomon and Benson, Geometry & Topology 6 (2002), 917–990

[So] R. Solomon, Finite groups with Sylow 2-subgroups of type .3, J. Algebra 28 (1974), 182–198

[Sz] M. Suzuki, Group theory I, Springer-Verlag (1982)