CONSTRUCTION OF 2-LOCAL FINITE GROUPS OF A TYPE STUDIED BY SOLOMON AND BENSON

RAN LEVI AND BOB OLIVER

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 this paper, we construct 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. Thus, the resulting classifying spaces are not homotopy equivalent to the 2-completed classifying space of any finite group. As predicted by Benson, these classifying spaces are also very closely related to the Dwyer-Wilkerson space BDI(4).

As one step in the classification of finite simple groups, Ron Solomon [So] considered the problem of classifying all finite simple groups whose Sylow 2-subgroups are isomorphic to those of the Conway group Co_{3}. The end result of his paper was that Co_{3} is the only such group. In the process of proving this, he needed to consider groups G in which all involutions are conjugate, and such that the centralizer of each involution contains a normal subgroup isomorphic to Spin_{7}(q) with odd index, where q is an odd prime power. Solomon showed that such a group G does not exist. The proof of this statement was also interesting, in the sense that the 2-local structure of the group in question appeared to be internally consistent, and it was only by analyzing its interaction with the p-local structure (where p is the prime of which q is a power) that he found a contradiction.

In a later paper [Be], Dave Benson, inspired by Solomon’s work, constructed certain spaces which can be thought of as the 2-completed classifying spaces which the groups studied by Solomon would have if they existed. He started with the spaces BDI(4) constructed by Dwyer and Wilkerson having the property that

H^{∗}(BDI(4); 𝔽_{2}) ≅ 𝔽_{2}[x_{1}, x_{2}, x_{3}, x_{4}]^{GL_{4}(2)}

(the rank four Dickson algebra at the prime 2). Benson then considered, for each odd prime power q, the homotopy fixed point set of the ℤ-action on BDI(4) generated by an “Adams operation” ψ^{q} constructed by Dwyer and Wilkerson. This homotopy fixed point set is denoted here BDI_{4}(q).

In this paper, we construct a family of 2-local finite groups, in the sense of [BLO2], which have the 2-local structure considered by Solomon, and whose classifying spaces are homotopy equivalent to Benson’s spaces BDI_{4}(q). The results of [BLO2] combined with those here allow us to make much more precise the statement that these spaces have many of the properties which the 2-completed classifying spaces of the groups studied by Solomon would have had if they existed. To explain what this means, we first recall some definitions.

1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R37, 20D06, 20D20.

Key words and phrases. Classifying space, p-completion, finite groups, fusion.

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

A fusion system over a finite p-group S is a category whose objects are the subgroups of S, and whose morphisms are monomorphisms of groups which include all those induced by conjugation by elements of S. A fusion system is saturated if it satisfies certain axioms formulated by Puig [Pu], and also listed in [BLO2, Definition 1.2] as well as at the beginning of Section 1 in this paper. In particular, for any finite group G and any S ∈ Syl_{p}(G), the category F_{S}(G) whose objects are the subgroups of S and whose morphisms are those monomorphisms between subgroups induced by conjugation in G is a saturated fusion system over S.

If F is a saturated fusion system over S, then a subgroup P ≤ S is called F-centric if C_{S}(P′) = Z(P′) for all P′ isomorphic to P in the category F. A centric linking system associated to F consists of a category L whose objects are the F-centric subgroups of S, together with a functor L ⟶ F which is the inclusion on objects, is surjective on all morphism sets and which satisfies certain additional axioms (see [BLO2, Definition 1.6]). These axioms suffice to ensure that the p-completed nerve |L|_{p}^{∧} has all of the properties needed to regard it as a “classifying space” of the fusion system F. A p-local finite group consists of a triple (S, F, L), where S is a finite p-group, F is a saturated fusion system over S, and L is a linking system associated to F. The classifying space of a p-local finite group (S, F, L) is the p-completed nerve |L|_{p}^{∧} (which is p-complete since |L| is always p-good [BLO2, Proposition 1.11]). For example, if G is a finite group and S ∈ Syl_{p}(G), then there is an explicitly defined centric linking system L_{S}^{c}(G) associated to F_{S}(G), and the classifying space of the triple (S, F_{S}(G), L_{S}^{c}(G)) is the space |L_{S}^{c}(G)|_{p}^{∧} ≃ BG_{p}^{∧}.

Exotic examples of p-local finite groups for odd primes p — i.e., examples which do not represent actual groups — have already been constructed in [BLO2], but using ad hoc methods which seemed to work only at odd primes.

In this paper, we first construct a fusion system F_{Sol}(q) (for any odd prime power q) over a 2-Sylow subgroup S of Spin_{7}(q), with the properties that all elements of order 2 in S are conjugate (i.e., the subgroups they generated are all isomorphic in the category), and the “centralizer fusion system” (see the beginning of Section 1) of each such element is isomorphic to the fusion system of Spin_{7}(q). We then show that F_{Sol}(q) is saturated, and has a unique associated linking system L_{Sol}^{c}(q). We thus obtain a 2local finite group (S, F_{Sol}(q), L_{Sol}^{c}(q)) where by Solomon’s theorem [So] (as explained in more detail in Proposition 3.4), F_{Sol}(q) is not the fusion system of any finite group. Let BSol(q) =^{def} |L_{Sol}^{c}(q)|_{2}^{∧} denote the classifying space of (S, F_{Sol}(q), L_{Sol}^{c}(q)). Thus, BSol(q) does not have the homotopy type of BG_{2}^{∧} for any finite group G, but does have many of the nice properties of the 2-completed classifying space of a finite group (as described in [BLO2]).

Relating BSol(q) to BDI_{4}(q) requires taking the “union” of the categories L_{Sol}^{c}(q^{n}) for all n ≥ 1. This however is complicated by the fact that an inclusion of fields 𝔽_{p^{m}} ⊆ 𝔽_{p^{n}} (i.e., m|n) does not induce an inclusion of cenric linking systems. Hence we have to replace the centric linking systems L_{Sol}^{c}(q^{n}) by the full subcategories L_{Sol}^{cc}(q^{n}) whose objects are those 2-subgroups which are centric in F_{Sol}^{c}(q^{∞}) = ⋃_{n≥1} F_{Sol}^{c}(q^{n}),

and show that the inclusion induces a homotopy equivalence BSol′(q^{n}) =^{def} |L_{Sol}^{cc}(q^{n})|_{2}^{∧} ≃ BSol(q^{n}). Inclusions of fields do induce inclusions of these categories, so we can then define L_{Sol}^{c}(q^{∞}) =^{def} ⋃_{n≥1} L_{Sol}^{cc}(q^{n}), and spaces

BSol(q^{∞}) = |L_{Sol}^{c}(q^{∞})|_{2}^{∧} ≃ (⋃_{n≥1} BSol′(q^{n}))_{2}^{∧}.

The category L_{Sol}^{c}(q^{∞}) has an “Adams map” ψ^{q} induced by the Frobenius automorphism x ↦ x^{q} of 𝔽_{q}. We then show that BSol(q^{∞}) ≃ BDI(4), the space of Dwyer and Wilkerson mentioned above; and also that BSol(q) is equivalent to the homotopy fixed point set of the ℤ-action on BSol(q^{∞}) generated by Bψ^{q}. The space BSol(q) is thus equivalent to Benson’s spaces BDI_{4}(q) for any odd prime power q.

The paper is organized as follows. Two propositions used for constructing saturated fusion systems, one very general and one more specialized, are proven in Section 1. These are then applied in Section 2 to construct the fusion systems F_{Sol}(q), and to prove that they are saturated. In Section 3 we prove the existence and uniqueness of a centric linking systems associated to F_{Sol}(q) and study their automorphisms. Also in Section 3 is the proof that F_{Sol}(q) is not the fusion system of any finite group. The connections with the space BDI(4) of Dwyer and Wilkerson is shown in Section 4. Some background material on the spinor groups Spin(V, 𝔟) over fields of characteristic ≠ 2 is collected in an appendix.

We would like to thank Dave Benson, Ron Solomon, and Carles Broto for their help while working on this paper.

1. Constructing saturated fusion systems

In this section, we first prove a general result which is useful for constructing saturated fusion systems. This is then followed by a more technical result, which is designed to handle the specific construction in Section 2.

We first recall some definitions from [BLO2]. A fusion system over a p-group S is a category F whose objects are the subgroups of F, such that

Hom_{S}(P, Q) ⊆ Mor_{F}(P, Q) ⊆ Inj(P, Q)

for all P, Q ≤ S, and such that each morphism in F factors as the composite of an F-isomorphism followed by an inclusion. We write Hom_{F}(P, Q) = Mor_{F}(P, Q) to emphasize that the morphisms are all homomorphisms of groups. We say that two subgroups P, Q ≤ S are F-conjugate if they are isomorphic in F. A subgroup P ≤ S is fully centralized (fully normalized) in F if |C_{S}(P)| ≥ |C_{S}(P′)| (|N_{S}(P)| ≥ |N_{S}(P′)|) for all P′ ≤ S which is F-conjugate to P. A saturated fusion system is a fusion system F over S which satisfies the following two additional conditions:

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

(II) For each P ≤ S and each φ ∈ Hom_{F}(P, S) such that φ(P) is fully centralized in F, if we set

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

then φ extends to a homomorphism φ ∈ Hom_{F}(N_{φ}, S).

For example, if G is a finite group and S ∈ Syl_{p}(G), then the category F_{S}(G) whose objects are the subgroups of S and whose morphisms are the homomorphisms induced by conjugation in G is a saturated fusion system over S. A subgroup P ≤ S is fully centralized in F_{S}(G) if and only if C_{S}(P) ∈ Syl_{p}(C_{G}(P)), and P is fully normalized in F_{S}(G) if and only if N_{S}(P) ∈ Syl_{p}(N_{G}(P)).

For any fusion system F over a p-group S, and any subgroup P ≤ S, the “centralizer fusion system” C_{F}(P) over C_{S}(P) is defined by setting

Hom_{C_{F} (P)}(Q, Q′) = {(φ|_{Q}) | φ ∈ Hom_{F}(P Q, P Q′), φ(Q) ≤ Q′, φ|_{P} = Id_{P} }

for all Q, Q′ ≤ C_{S}(P) (see [BLO2, Definition A.3] or [Pu] for more detail). We also write C_{F}(g) = C_{F}(⟨g⟩) for g ∈ S. If F is a saturated fusion system and P is fully centralized in F, then C_{F}(P) is saturated by [BLO2, Proposition A.6] (or [Pu]).

Proposition 1.1. Let F be any fusion system over a p-group S. Then F is saturated if and only if there is a set 𝔛 of elements of order p in S such that the following conditions hold:

(a) Each x ∈ S of order p is F-conjugate to some element of 𝔛.

(b) If x and y are F-conjugate and y ∈ 𝔛, then there is some ψ ∈ Hom_{F}(C_{S}(x), C_{S}(y)) such that ψ(x) = y.

(c) For each x ∈ 𝔛, C_{F}(x) is a saturated fusion system over C_{S}(x).

Proof. Throughout the proof, conditions (I) and (II) always refer to the conditions in the definition of a saturated fusion system, as stated above or in [BLO2, Definition 1.2].

Assume first that F is saturated, and let 𝔛 be the set of all x ∈ S of order p such that ⟨x⟩ is fully centralized. Then condition (a) holds by definition, (b) follows from condition (II), and (c) holds by [BLO2, Proposition A.6] or [Pu].

Assume conversely that 𝔛 is chosen such that conditions (a–c) hold for F. Define

U = {(P, x) | P ≤ S, |x| = p, x ∈ Z(P)^{T}, some T ∈ Syl_{p}(Aut_{F}(P)) containing Aut_{S}(P)},

where Z(P)^{T} is the subgroup of elements of Z(P) fixed by the action of T. Let U_{0} ⊆ U be the set of pairs (P, x) such that x ∈ 𝔛. For each 1 ≠ P ≤ S, there is some x such that (P, x) ∈ U (since every action of a p-group on Z(P) has nontrivial fixed set); but x need not be unique.

We first check that

(P, x) ∈ U_{0}, P fully centralized in C_{F}(x) ⟹ P fully centralized in F. (1)

Assume otherwise: that (P, x) ∈ U_{0} and P is fully centralized in C_{F}(x), but P is not fully centralized in F. Let P′ ≤ S and φ ∈ Iso_{F}(P, P′) be such that |C_{S}(P)| < |C_{S}(P′)|. Set x′ = φ(x) ≤ Z(P′). By (b), there exists ψ ∈ Hom_{F}(C_{S}(x′), C_{S}(x)) such that ψ(x′) = x. Set P′′ = ψ(P′). Then ψ ∘ φ ∈ Iso_{C_{F} (x)}(P, P′′), and in particular P′′ is C_{F}(x)-conjugate to P. Also, since C_{S}(P′) ≤ C_{S}(x′), ψ sends C_{S}(P′) injectively into C_{S}(P′′), and |C_{S}(P)| < |C_{S}(P′)| ≤ |C_{S}(P′′)|. Since C_{S}(P) = C_{C_{S}(x)}(P) and C_{S}(P′′) = C_{C_{S}(x)}(P′′), this contradicts the original assumption that P is fully centralized in C_{F}(x).

By definition of U, for each (P, x) ∈ U, N_{S}(P) ≤ C_{S}(x) and hence Aut_{C_{S}(x)}(P) = Aut_{S}(P). By assumption, there is T ∈ Syl_{p}(Aut_{F}(P)) such that τ(x) = x for all τ ∈ T; i.e., such that T ≤ Aut_{C_{F} (x)}(P). In particular, it follows that

∀(P, x) ∈ U : Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)) ⟺ Aut_{C_{S}(x)}(P) ∈ Syl_{p}(Aut_{C_{F} (x)}(P)). (2)

We are now ready to prove condition (I) for F; namely, to show for each P ≤ S fully normalized in F that P is fully centralized and Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)). By definition, |N_{S}(P)| ≥ |N_{S}(P′)| for all P′ F-conjugate to P. Choose x ∈ Z(P) such that (P, x) ∈ U; and let T ∈ Syl_{p}(Aut_{F}(P)) be such that T ≥ Aut_{S}(P) and x ∈ Z(P)^{T}. By (a) and (b), there is an element y ∈ 𝔛 and a homomorphism ψ ∈ Hom_{F}(C_{S}(x), C_{S}(y)) such that ψ(x) = y. Set P′ = ψP, and set T′ = ψT ψ^{−1} ∈ Syl_{p}(Aut_{F}(T′)). Since T ≥ Aut_{S}(P) by definition of U, and ψ(N_{S}(P)) = N_{S}(P′) by the maximality assumption, we see that T′ ≥ Aut_{S}(P′). Also, y ∈ Z(P′)^{T′} (T′y = y since T x = x), and this shows that (P′, y) ∈ U_{0}. The maximality of |N_{S}(P′)| = |N_{C_{S}(y)}(P′)| implies that P′ is fully normalized in C_{F}(y). Hence by condition (I) for the saturated fusion system C_{F}(y), together with (1) and (2), P fully centralized in F and Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)).

It remains to prove condition (II) for F. Fix 1 ≠ P ≤ S and φ ∈ Hom_{F}(P, S) such that P′ =^{def} φP is fully centralized in F, and set

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

We must show that φ extends to some φ ∈ Hom_{F}(N_{φ}, S). Choose some x′ ∈ Z(P′) of order p which is fixed under the action of Aut_{S}(P′), and set x = φ^{−1}(x′) ∈ Z(P). For all g ∈ N_{φ}, φc_{g}φ^{−1} ∈ Aut_{S}(P′) fixes x′, and hence c_{g}(x) = x. Thus

x ∈ Z(N_{φ}) and hence N_{φ} ≤ C_{S}(x);  and N_{S}(P′) ≤ C_{S}(x′).  (3)

Fix y ∈ 𝔛 which is F-conjugate to x and x′, and choose

ψ ∈ Hom_{F}(C_{S}(x), C_{S}(y)) and ψ′ ∈ Hom_{F}(C_{S}(x′), C_{S}(y))

such that ψ(x) = ψ′(x′) = y. Set Q = ψ(P) and Q′ = ψ′(P′). Since P′ is fully centralized in F, ψ′(P′) = Q′, and C_{S}(P′) ≤ C_{S}(x′), we have

ψ′(C_{C_{S}(x′)}(P′)) = ψ′(C_{S}(P′)) = C_{S}(Q′) = C_{C_{S}(y)}(Q′).  (4)

Set τ = ψ′φψ^{−1} ∈ Iso_{F}(Q, Q′). By construction, τ(y) = y, and thus τ ∈ Iso_{C_{F} (y)}(Q, Q′). Since P′ is fully centralized in F, (4) implies that Q′ is fully centralized in C_{F}(y). Hence condition (II), when applied to the saturated fusion system C_{F}(y), shows that τ extends to a homomorphism τ ∈ Hom_{C_{F} (y)}(N_{τ}, C_{S}(y)), where

N_{τ} = {g ∈ N_{C_{S}(y)}(Q) | τc_{g}τ^{−1} ∈ Aut_{C_{S}(y)}(Q′)}.

Also, for all g ∈ N_{φ} ≤ C_{S}(x) (see (3)),

c_{τ(ψ(g))} = τc_{ψ(g)}τ^{−1} = (τψ)c_{g}(τψ)^{−1} = (ψ′φ)c_{g}(ψ′φ)^{−1} = c_{ψ′(h)} ∈ Aut_{C_{S}(y)}(Q′)

for some h ∈ N_{S}(P′) such that φc_{g}φ^{−1} = c_{h}. This shows that ψ(N_{φ}) ≤ N_{τ}; and also (since C_{S}(Q′) = ψ′(C_{S}(P′)) by (4)) that

τ(ψ(N_{φ})) ≤ ψ′(N_{C_{S}(x′)}(P′)).

We can now define

φ =^{def} (ψ′)^{−1} ∘ (τ ∘ ψ)|_{N_{φ}} ∈ Hom_{F}(N_{φ}, S),

and φ|_{P} = φ. □

Proposition 1.1 will also be applied in a separate paper of Carles Broto & Jesper Møller [BM] to give a construction of some “exotic” p-local finite groups at certain odd primes.

Our goal now is to construct certain saturated fusion systems, by starting with the fusion system of Spin_{7}(q) for some odd prime power q, and then adding to that the automorphisms of some subgroup of Spin_{7}(q). This is a special case of the general problem of studying fusion systems generated by fusion subsystems, and then showing that they are saturated. We first fix some notation. If F_{1} and F_{2} are two fusion systems over the same p-group S, then ⟨F_{1}, F_{2}⟩ denotes the fusion system over S generated by F_{1} and F_{2}: the smallest fusion system over S which contains both F_{1} and F_{2}. More generally, if F is a fusion system over S, and F_{0} is a fusion system over a subgroup S_{0} ≤ S, then ⟨F; F_{0}⟩ denotes the fusion system over S generated by the morphisms in F between subgroups of S, together with morphisms in F_{0} between subgroups of S_{0} only. In other words, a morphism in ⟨F; F_{0}⟩ is a composite

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

where for each i, either φ_{i} ∈ Hom_{F}(P_{i−1}, P_{i}), or φ_{i} ∈ Hom_{F_{0}}(P_{i−1}, P_{i}) (and P_{i−1}, P_{i} ≤ S_{0}).

As usual, when G is a finite group and S ∈ Syl_{p}(G), then F_{S}(G) denotes the fusion system of G over S. If Γ ≤ Aut(G) is a group of automorphisms which contains Inn(G), then F_{S}(Γ) will denote the fusion system over S whose morphisms consist of all restrictions of automorphisms in Γ to monomorphisms between subgroups of S.

The next proposition provides some fairly specialized conditions which imply that the fusion system generated by the fusion system of a group G together with certain automorphisms of a subgroup of G is saturated.

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 subgroup Γ ≤ Aut(C_{G}(U)) containing Inn(C_{G}(U)) such that γ(U) = U for all γ ∈ Γ. Set

S_{0} = C_{S}(U)  and  F =^{def} ⟨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_{G}(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)

Proposition 1.2 follows from the following three lemmas. Throughout the proofs of these lemmas, references to points (a–e) mean to those points in the hypotheses of the proposition, unless otherwise stated.

Lemma 1.3. Under the hypotheses of Proposition 1.2, for any P ≤ S and any central subgroup Z′ ≤ Z(P) of order p,

Z ≠ Z′ ≤ U ⟹ ∃ φ ∈ Hom_{Γ}(P, S_{0}) such that φ(Z′) = Z  (1)

and

Z′ ≰ U ⟹ ∃ ψ ∈ Hom_{G}(P, S_{0}) such that ψ(Z′) ≤ U.  (2)

Proof. Note first that Z ≤ Z(S), since it is a normal subgroup of order p in a p-group. Assume Z ≠ Z′ ≤ U. Then U = ZZ′, and

P ≤ C_{S}(Z′) = C_{S}(ZZ′) = C_{S}(U) = S_{0}

since Z′ ≤ Z(P) by assumption. By (b), there is α ∈ Γ such that α(Z′) = Z. Since S_{0} ∈ Syl_{p}(C_{G}(U)), there is h ∈ C_{G}(U) such that h·α(P)·h^{−1} ≤ S_{0}; and since

c_{h} ∈ Aut_{N_{G}(U)}(C_{G}(U)) ≤ Γ

by (c), φ =^{def} c_{h} ∘ α ∈ Hom_{Γ}(P, S_{0}) and sends Z′ to Z.

If Z′ ≰ U, then by (a), there is g ∈ G such that gZ′g^{−1} ≤ U∖Z. Since Z is central in S, gZ′g^{−1} is central in gP g^{−1}, and U is generated by Z and gZ′g^{−1}, it follows that gP g^{−1} ≤ C_{G}(U). Since S_{0} ∈ Syl_{p}(C_{G}(U)), there is h ∈ C_{G}(U) such that h(gP g^{−1})h^{−1} ≤ S_{0}; and we can take ψ = c_{hg} ∈ Hom_{G}(P, S_{0}). □

We are now ready to prove point (1) in Proposition 1.2.

Lemma 1.4. Assume the hypotheses of Proposition 1.2, 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).

For each i, using 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 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 ≰ 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 (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}(−, −), and hence φ_{2} is of type (Γ) by (c). It follows that φ ∈ Iso_{Γ}(P_{0}, P_{3}) sends Z to itself, and is of type (G) by (c) again.

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 (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 with the property that C_{S}(E) ∈ Syl_{p}(C_{H}(E)). Since C_{S}(E) ≤ C_{S}(U) = S_{0} ∈ Syl_{p}(H), the last condition implies that E is fully centralized in the fusion system F_{S_{0}}(H). If E ≤ S is any rank three elementary abelian subgroup which contains U, then there is some a ∈ H such that E′ = aEa^{−1} ∈ E_{3}, since F_{S_{0}}(H) is saturated and U ◁ H. Then c_{a} ∈ Iso_{G}(E, E′) ∩ Iso_{Γ}(E, E′) by (c). So upon composing with such isomorphisms, we can assume that E_{i} ∈ E_{3} for all i, and also that φ_{i}(C_{S}(E_{i−1})) = C_{S}(E_{i}) for each i.

In this way, φ can be assumed to extend to an F-isomorphism φ from C_{S}(E_{0}) to C_{S}(E_{5}) which sends Z to itself. By (e), the rank three subgroups E_{i} are all Gconjugate 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 (d), c_{hg} ∘ φ ∈ Aut_{F}(C_{S}(E_{0})) is of type (G); and thus φ ∈ Iso_{G}(P_{0}, P_{5}). □

To finish the proof of Proposition 1.2, it remains only to show:

Lemma 1.5. Under the hypotheses of Proposition 1.2, the fusion system F generated by F_{S}(G) and F_{S_{0}}(Γ) is saturated.

Proof. We apply Proposition 1.1, by letting 𝔛 be the set of generators of Z. Condition (a) of the proposition (every x ∈ S of order p is F-conjugate to an element of 𝔛) holds by Lemma 1.3. Condition (c) holds since C_{F}(Z) is the fusion system of the group C_{G}(Z) by Lemma 1.4, and hence is saturated by [BLO2, Proposition 1.3].

It remains to prove condition (b) of Proposition 1.1. We must show that if y, z ∈ S are F-conjugate and ⟨z⟩ = Z, then there is ψ ∈ Hom_{F}(C_{S}(y), C_{S}(z)) such that ψ(y) = z. If y ∉ U, then by Lemma 1.3(2), there is φ ∈ Hom_{F}(C_{S}(y), S_{0}) such that φ(y) ∈ U. If y ∈ U∖Z, then by Lemma 1.3(1), there is φ ∈ Hom_{F}(C_{S}(y), S_{0}) such that φ(y) ∈ Z. We are thus reduced to the case where y, z ∈ Z (and are F-conjugate).

In this case, then by Lemma 1.4, there is g ∈ G such that z = gyg^{−1}. Since Z ◁ G, [G:C_{G}(Z)] is prime to p, so S and gSg^{−1} are both Sylow p-subgroups of C_{G}(Z), and hence are C_{G}(Z)-conjugate. We can thus choose g such that z = gyg^{−1} and gSg^{−1} = S. Since C_{S}(y) = C_{S}(z) = S (Z ≤ Z(S) since it is a normal subgroup of order p), this shows that c_{g} ∈ Iso_{G}(C_{S}(y), C_{S}(z)), and finishes the proof of (b) in Proposition 1.1. □

2. A fusion system of a type considered by Solomon

The main result of this section and the next is the following theorem:

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.

Theorem 2.1 will be proven in Propositions 2.11 and 3.3. Later, at the end of Section 3, we explain why Solomon’s theorem [So] implies that these fusion systems are not the fusion systems of any finite groups, and hence that the spaces BSol(q) are not homotopy equivalent to the 2-completed classifying spaces of any finite groups.

Background results needed for computations in Spin(V, 𝔟) have been collected in Appendix A. We focus attention here on SO_{7}(q) and Spin_{7}(q). In fact, since we want to compare the constructions over 𝔽_{q} with those over its field extensions, most of the constructions will first be made in the groups SO_{7}(𝔽_{q}) and Spin_{7}(𝔽_{q}).

We now fix, for the rest of the section, an odd prime power q. It will be convenient to write Spin_{7}(q^{∞}) =^{def} Spin_{7}(𝔽_{q}), etc. In order to make certain computations more explicit, we set

V_{∞} = M_{2}(𝔽_{q}) ⊕ M_{2}^{0}(𝔽_{q}) ≅ (𝔽_{q})^{7} and 𝔟(A, B) = det(A) + det(B)

(where M_{2}^{0}(−) is the group of (2 × 2) matrices of trace zero), and for each n ≥ 1 set V_{n} = M_{2}(𝔽_{q^{n}}) ⊕ M_{2}^{0}(𝔽_{q^{n}}) ⊆ V_{∞}. Then 𝔟 is a nonsingular quadratic form on V_{∞} and on V_{n}. Identify SO_{7}(q^{∞}) = SO(V_{∞}, 𝔟) and SO_{7}(q^{n}) = SO(V_{n}, 𝔟), and similarly for Spin_{7}(q^{n}) ≤ Spin_{7}(q^{∞}). For all α ∈ Spin(M_{2}(𝔽_{q}), det) and β ∈ Spin(M_{2}^{0}(𝔽_{q}), det), we write α ⊕ β for their image in Spin_{7}(q^{∞}) under the natural homomorphism

ι_{4,3} : Spin_{4}(q^{∞}) × Spin_{3}(q^{∞}) ⟶ Spin_{7}(q^{∞}).

There are isomorphisms

ρ̃_{4} : SL_{2}(q^{∞}) × SL_{2}(q^{∞}) ⟶^{≅} Spin_{4}(q^{∞}) and ρ̃_{3} : SL_{2}(q^{∞}) ⟶^{≅} Spin_{3}(q^{∞})

which are defined explicitly in Proposition A.5, and which restrict to isomorphisms

SL_{2}(q^{n}) × SL_{2}(q^{n}) ≅ Spin_{4}(q^{n}) and SL_{2}(q^{n}) ≅ Spin_{3}(q^{n})

for each n. Let

z = ρ̃_{4}(−I, −I) ⊕ 1 = 1 ⊕ ρ̃_{3}(−I) ∈ Z(Spin_{7}(q))

denote the central element of order two, and set

z_{1} = ρ̃_{4}(−I, I) ⊕ 1 ∈ Spin_{7}(q).

Here, 1 ∈ Spin_{k}(q) (k = 3, 4) denotes the identity element. Define U = ⟨z, z_{1}⟩.

Definition 2.2. Define

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

by setting

ω(A_{1}, A_{2}, A_{3}) = ρ̃_{4}(A_{1}, A_{2}) ⊕ ρ̃_{3}(A_{3})

for A_{1}, A_{2}, A_{3} ∈ SL_{2}(q^{∞}). Set

H(q^{∞}) = ω(SL_{2}(q^{∞})^{3})  and  [[A_{1}, A_{2}, A_{3}]] = ω(A_{1}, A_{2}, A_{3}) .

Since ρ̃_{3} and ρ̃_{4} are isomorphisms, Ker(ω) = Ker(ι_{4,3}), and thus

Ker(ω) = ⟨(−I, −I, −I)⟩.

In particular, H(q^{∞}) ≅ (SL_{2}(q^{∞})^{3})/{±(I, I, I)}. Also,

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

and thus

U = {[[±I, ±I, ±I]]}

(with all combinations of signs).

For each 1 ≤ n < ∞, the natural homomorphism

Spin_{7}(q^{n}) ⟶ SO_{7}(q^{n})

has kernel and cokernel both of order 2. The image of this homomorphism is the commutator subgroup Ω_{7}(q^{n}) ◁ SO_{7}(q^{n}), which is partly described by Lemma A.4(a). In contrast, since all elements of 𝔽_{q} are squares, the natural homomorphism from Spin_{7}(q^{∞}) to SO_{7}(q^{∞}) is surjective.

Lemma 2.3. There is an element τ ∈ N_{Spin_{7}(q)}(U) of order 2 such that

τ·[[A_{1}, A_{2}, A_{3}]]·τ^{−1} = [[A_{2}, A_{1}, A_{3}]]  (1)

for all A_{1}, A_{2}, A_{3} ∈ SL_{2}(q^{∞}).

Proof. Let τ ∈ SO_{7}(q) be the involution defined by setting

τ(X, Y) = (−θ(X), −Y)

for (X, Y) ∈ V_{∞} = M_{2}(𝔽_{q}) ⊕ M_{2}^{0}(𝔽_{q}), where

θ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ( \begin{matrix}d & −b \\ −c & a \end{matrix} ).

Let τ ∈ Spin_{7}(q^{∞}) be a lifting of τ. The (−1)-eigenspace of τ on V_{∞} has orthogonal basis

\{(I, 0) , (0, ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} )) , (0, ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )) , (0, ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ))\},

and in particular has discriminant 1 with respect to this basis. Hence by Lemma A.4(a), τ ∈ Ω_{7}(q), and so τ ∈ Spin_{7}(q). Since in addition, the (−1)-eigenspace of τ is 4-dimensional, Lemma A.4(b) applies to show that τ^{2} = 1.

By definition of the isomorphisms ρ̃_{3} and ρ̃_{4}, for all A_{i} ∈ SL_{2}(q^{∞}) (i = 1, 2, 3) and all (X, Y) ∈ V_{∞},

[[A_{1}, A_{2}, A_{3}]](X, Y) = (A_{1}XA_{2}^{−1}, A_{3}Y A_{3}^{−1}).

Here, Spin_{7}(q^{∞}) acts on V_{∞} via its projection to SO_{7}(q^{∞}). Also, for all X, Y ∈ M_{2}(𝔽_{q}),

θ(X) = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )·X^{t}·( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )^{−1} and in particular θ(XY) = θ(Y)·θ(X);

and θ(X) = X^{−1} if det(X) = 1. Hence for all A_{1}, A_{2}, A_{3} ∈ SL_{2}(q^{∞}) and all (X, Y) ∈ V_{∞},

(τ·[[A_{1}, A_{2}, A_{3}]]·τ^{−1})(X, Y) = τ(−A_{1}·θ(X)·A_{2}^{−1}, −A_{3}Y A_{3}^{−1})
= (A_{2}XA_{1}^{−1}, A_{3}Y A_{3}^{−1}) = [[A_{2}, A_{1}, A_{3}]](X, Y).

This shows that (1) holds modulo ⟨z⟩ = Z(Spin_{7}(q^{∞})). We thus have two automorphisms of H(q^{∞}) ≅ (SL_{2}(q^{∞})^{3})/{±(I, I, I)} — conjugation by τ and the permutation automorphism — which are liftings of the same automorphism of H(q^{∞})/⟨z⟩. Since H(q^{∞}) is perfect, each automorphism of H(q^{∞})/⟨z⟩ has at most one lifting to an automorphism of H(q^{∞}), and thus (1) holds. Also, since U is the subgroup of all elements [[±I, ±I, ±I]] with all combinations of signs, formula (1) shows that τ ∈ N_{Spin_{7}(q)}(U). □

Definition 2.4. For each n ≥ 1, set

H(q^{n}) = H(q^{∞}) ∩ Spin_{7}(q^{n})  and  H_{0}(q^{n}) = ω(SL_{2}(q^{n})^{3}) ≤ H(q^{n}).

Define

Γ_{n} = Inn(H(q^{n})) ⋊ Σ̂_{3} ≤ Aut(H(q^{n})),

where Σ̂_{3} denotes the group of permutation automorphisms

Σ̂_{3} = {[[A_{1}, A_{2}, A_{3}]] ↦ [[A_{σ1}, A_{σ2}, A_{σ3}]] | σ ∈ Σ_{3}} ≤ Aut(H(q^{n})) .

For each n, let ψ^{q^{n}} be the automorphism of Spin_{7}(q^{∞}) induced by the field isomorphism (q ↦ q^{p^{n}}). By Lemma A.3, Spin_{7}(q^{n}) is the fixed subgroup of ψ^{q^{n}}. Hence each element of H(q^{n}) is of the form [[A_{1}, A_{2}, A_{3}]], where either A_{i} ∈ SL_{2}(q^{n}) for each i (and the element lies in H_{0}(q^{n})), or ψ^{q^{n}}(A_{i}) = −A_{i} for each i. This shows that H_{0}(q^{n}) has index 2 in H(q^{n}).

The goal is now to choose “compatible” Sylow subgroups S(q^{n}) ∈ Syl_{2}(Spin_{7}(q^{n})) (all n ≥ 1) contained in N(H(q^{n})), and let F_{Sol}(q^{n}) be the fusion system over S(q^{n}) generated by conjugation in Spin_{7}(q^{n}) and by restrictions of Γ_{n}.

Proposition 2.5. The following hold for each n ≥ 1.

(a) H(q^{n}) = C_{Spin_{7}(q^{n})}(U).

(b) N_{Spin_{7}(q^{n})}(U) = N_{Spin_{7}(q^{n})}(H(q^{n})) = H(q^{n})·⟨τ⟩, and contains a Sylow 2-subgroup of Spin_{7}(q^{n}).

Proof. Let z_{1} ∈ SO_{7}(q) be the image of z_{1} ∈ Spin_{7}(q). Set V_{−} = M_{2}(𝔽_{q}) and V_{+} = M_{2}^{0}(𝔽_{q}): the eigenspaces of z_{1} acting on V. By Lemma A.4(c),

C_{Spin_{7}(q^{∞})}(U) = C_{Spin_{7}(q^{∞})}(z_{1})

is the group of all elements α ∈ Spin_{7}(q^{∞}) whose image α ∈ SO_{7}(q^{∞}) has the form

α = α_{−} ⊕ α_{+} where α_{±} ∈ SO(V_{±}).

In other words,

C_{Spin_{7}(q^{∞})}(U) = ι_{4,3}(Spin_{4}(q^{∞}) × Spin_{3}(q^{∞})) = ω(SL_{2}(q^{∞})^{3}) = H(q^{∞}).

Furthermore, since

τz_{1}τ^{−1} = τ[[−I, I, I]]τ^{−1} = [[I, −I, I]] = zz_{1}

by Lemma 2.3, and since any element of N_{Spin_{7}(q^{∞})}(U) centralizes z, conjugation by τ generates Out_{Spin_{7}(q^{∞})}(U). Hence

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

Point (a), and the first part of point (b), now follow upon taking intersections with Spin_{7}(q^{n}).

If N_{Spin_{7}(q^{n})}(U) did not contain a Sylow 2-subgroup of Spin_{7}(q^{n}), then since every noncentral involution of Spin_{7}(q^{n}) is conjugate to z_{1} (Proposition A.8), the Sylow 2subgroups of Spin_{7}(q) would have no normal subgroup isomorphic to C_{2}^{2}. By a theorem of Hall (cf. [Go, Theorem 5.4.10]), this would imply that they are cyclic, dihedral, quaternion, or semidihedral. This is clearly not the case, so N_{Spin_{7}(q^{n})}(U) must contain a Sylow 2-subgroup of Spin_{7}(q), and this finishes the proof of point (b).

Alternatively, point (b) follows from the standard formulas for the orders of these groups (cf. [Ta, pp.19,140]), which show that

\frac{|Spin_{7}(q^{n})|}{|H(q^{n})·⟨τ⟩|} = \frac{q^{9n}(q^{6n} − 1)(q^{4n} − 1)(q^{2n} − 1)}{2·[q^{n}(q^{2n} − 1)]^{3}} = q^{6n}(q^{4n} + q^{2n} + 1)(\frac{q^{2n} + 1}{2})

is odd. □

We next fix, for each n, a Sylow 2-subgroup of Spin_{7}(q^{n}) which is contained in H(q^{n})·⟨τ⟩ = N_{Spin_{7}(q^{n})}(U).

Definition 2.6. Fix elements A, B ∈ SL_{2}(q) such that ⟨A, B⟩ ≅ Q_{8} (a quaternion group of order 8), and set Â = [[A, A, A]] and B̂ = [[B, B, B]]. Let C(q^{∞}) ≤ C_{SL_{2}(q^{∞})}(A) be the subgroup of elements of 2-power order in the centralizer (which is abelian), and set Q(q^{∞}) = ⟨C(q^{∞}), B⟩. Define

S_{0}(q^{∞}) = ω(Q(q^{∞})^{3}) ≤ H_{0}(q^{∞}) and S(q^{∞}) = S_{0}(q^{∞})·⟨τ⟩ ≤ H(q^{∞}) ≤ Spin_{7}(q^{∞}).

Here, τ ∈ Spin_{7}(q) is the element of Lemma 2.3. Finally, for each n ≥ 1, define

C(q^{n}) = C(q^{∞}) ∩ SL_{2}(q^{n}), Q(q^{n}) = Q(q^{∞}) ∩ SL_{2}(q^{n}),

S_{0}(q^{n}) = S_{0}(q^{∞}) ∩ Spin_{7}(q^{n}),  and  S(q^{n}) = S(q^{∞}) ∩ Spin_{7}(q^{n}).

Since the two eigenvalues of A are distinct, its centralizer in SL_{2}(q^{∞}) is conjugate to the subgroup of diagonal matrices, which is abelian. Thus C(q^{∞}) is conjugate to the subgroup of diagonal matrices of 2-power order. This shows that each finite subgroup of C(q^{∞}) is cyclic, and that each finite subgroup of Q(q^{∞}) is cyclic or quaternion.

Lemma 2.7. For all n, S(q^{n}) ∈ Syl_{2}(Spin_{7}(q^{n})).

Proof. By [Sz, 6.23], A is contained in a cyclic subgroup of order q^{n} − 1 or q^{n} + 1 (depending on which of them is divisible by 4). Also, the normalizer of this cyclic subgroup is a quaternion group of order 2(q^{n}±1), and the formula |SL_{2}(q^{n})| = q^{n}(q^{2n}− 1) shows that this quaternion group has odd index. Thus by construction, Q(q^{n}) is a Sylow 2-subgroup of SL_{2}(q^{n}). Hence ω(Q(q^{n})^{3}) is a Sylow 2-subgroup of H_{0}(q^{n}), so ω(Q(q^{∞})^{3})∩Spin_{7}(q^{n}) is a Sylow 2-subgroup of H(q^{n}). It follows that S(q^{n}) is a Sylow 2-subgroup of H(q^{n})·⟨τ⟩, and hence also of Spin_{7}(q^{n}) by Proposition 2.5(b). □

Following the notation of Definition A.7, we say that an elementary abelian 2subgroup E ≤ Spin_{7}(q^{n}) has type I if its eigenspaces all have square discriminant, and has type II otherwise. Let E_{r} be the set of elementary abelian subgroups of rank r in Spin_{7}(q^{n}) which contain z, and let E_{r}^{I} and E_{r}^{II} be the sets of those of type I or II, respectively. In Proposition A.8, we show that there are two conjugacy classes of subgroups in E_{4}^{I} and one conjugacy class of subgroups in E_{4}^{II}. In Proposition A.9, an invariant x_{C}(E) ∈ E is defined, for all E ∈ E_{4} (and where C is one of the conjugacy classes in E_{4}^{I}) as a tool for determining the conjugacy class of a subgroup. More precisely, E has type I if and only if x_{C}(E) ∈ ⟨z⟩, and E ∈ C if and only if x_{C}(E) = 1. The next lemma provides some more detailed information about the rank four subgroups and these invariants.

Recall that we define Â = [[A, A, A]] and B̂ = [[B, B, B]].

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) E_{4}^{U} = {E_{ijk}, E′_{ijk} | i, j, k ∈ ℤ} (a finite set), where

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

(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(H(q^{n})) (see Definition 2.4), if E′, E′′ ∈ E_{4}^{U} are such that φ(E′) = E′′, then φ(x_{C}(E′)) = x_{C}(E′′).

Proof. (a) The set

{(I, 0) , (A, 0) , (B, 0) , (AB, 0) , (0, A) , (0, B) , (0, AB)}

is a basis of eigenvectors for the action of E_{∗} on V_{n} = M_{2}(𝔽_{q^{n}}) ⊕ M_{2}^{0}(𝔽_{q^{n}}). (Since the matrices A, B, and AB all have order 4 and determinant one, each has as eigenvalues the two distinct fourth roots of unity, and hence they all have trace zero.) Since all of these have determinant one, E_{∗} has type I by definition.

(b) Consider the subgroups

R_{0} = ω(C(q^{∞})^{3}) ∩ S(q^{n}) = {[[X^{i}, X^{j}, X^{k}]], [[X^{i}Y, X^{j}Y, X^{k}Y]] | i, j, k ∈ ℤ}

and

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

Clearly, each subgroup E ∈ E_{4}^{U} is contained in

C_{S(q^{n})}(U) = S_{0}(q^{n}) = R_{0}·⟨[[B^{i}, B^{j}, B^{k}]]⟩.

All involutions in this subgroup are contained in R_{1} = R_{0}·⟨[[B, B, B]]⟩, and thus E ≤ R_{1}. Hence E ∩ R_{0} has rank 3, which implies that E ≥ ⟨z, z_{1}, Â⟩ (the 2-torsion in R_{0}). Since all elements of order two in the coset R_{0}·B̂ have the form

[[X^{i}B, X^{j}B, X^{k}B]] or [[X^{i}Y B, X^{j}Y B, X^{k}Y B]]

for some i, j, k, this shows that E must be one of the groups E_{ijk} or E′_{ijk}. (Note in particular that E_{∗} = E_{000}.)

(c) By Proposition A.9(a), the element x_{C}(E) ∈ E is characterized uniquely by the property that x_{C}(E) = g^{−1}ψ^{q^{n}}(g) for some g ∈ Spin_{7}(q^{∞}) such that gEg^{−1} ∈ C. We now apply this explicitly to the subgroups E_{ijk} and E′_{ijk}.

For each i, Y^{−i}(X^{i}B)Y^{i} = Y^{−2i}X^{i}B = B. Hence for each i, j, k,

[[Y^{i}, Y^{j}, Y^{k}]]^{−1}·E_{ijk}·[[Y^{i}, Y^{j}, Y^{k}]] = E_{∗}

and

ψ^{q^{n}}([[Y^{i}, Y^{j}, Y^{k}]]) = [[Y^{i}, Y^{j}, Y^{k}]]·[[(−I)^{i}, (−I)^{j}, (−I)^{k}]].

Hence

x_{C}(E_{ijk}) = [[(−I)^{i}, (−I)^{j}, (−I)^{k}]].

Similarly, if we choose Z ∈ C_{SL_{2}(q^{∞})}(A) such that Z^{2} = Y, then for each i,

(Y^{i}Z)^{−1}(X^{i}Y B)(Y^{i}Z) = B.

Hence for each i, j, k,

[[Y^{i}Z, Y^{j}Z, Y^{k}Z]]^{−1}·E′_{ijk}·[[Y^{i}Z, Y^{j}Z, Y^{k}Z]] = E_{∗}.

Since ψ^{q^{n}}(Z) = ±ZA,

ψ^{q^{n}}([[Y^{i}Z, Y^{j}Z, Y^{k}Z]]) = [[Y^{i}Z, Y^{j}Z, Y^{k}Z]]·[[(−I)^{i}A, (−I)^{j}A, (−I)^{k}A]],

and hence

x_{C}(E′_{ijk}) = [[(−I)^{i}A, (−I)^{j}A, (−I)^{k}A]].

(d) This now follows immediately from point (c) and Proposition A.9(b,c).

(e) By Definition 2.4, Γ_{n} is generated by Inn(H(q^{n})) and the permutations of the three factors in H(q^{∞}) ≅ (SL_{2}(q^{∞})^{3})/{±(I, I, I)}. If φ ∈ Γ_{n} is a permutation automorphism, then it permutes the elements of E_{4}^{U}, and preserves the elements x_{C}(−) by the formulas in (c). If φ ∈ Inn(H(q^{n})) and φ(E′) = E′′ for E′, E′′ ∈ E_{4}^{U}, then φ(x_{C}(E′)) = x_{C}(E′′) by definition of x_{C}(−); and so the same property holds for all elements of Γ_{n}. □

Following the notation introduced in Section 1, for P, Q ≤ S(q^{n}), Hom_{Spin_{7}(q^{n})}(P, Q) denotes the set of homomorphisms from P to Q induced by conjugation by some element of Spin_{7}(q^{n}). Also, if P, Q ≤ S(q^{n}) ∩ H(q^{n}), Hom_{Γ_{n}}(P, Q) denotes the set of homomorphisms induced by restriction of an element of Γ_{n}. Let F_{n} = F_{Sol}(q^{n}) be the fusion system over S(q^{n}) generated by Spin_{7}(q^{n}) and Γ_{n}. In other words, for each P, Q ≤ S(q^{n}), Hom_{F_{n}}(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 P_{i} ≤ S(q^{n}) for all i, and each φ_{i} lies in Hom_{Spin_{7}(q^{n})}(P_{i−1}, P_{i}) or (if P_{i−1}, P_{i} ≤ H(q^{n})) Hom_{Γ_{n}}(P_{i−1}, P_{i}). This clearly defines a fusion system over S(q^{n}).

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. Consider the subgroups

R_{0} = R_{0}(q^{n}) =^{def} ω(C(q^{∞})^{3}) ∩ S and R_{1} = R_{1}(q^{n}) =^{def} C_{S}(⟨U, Â⟩) = ⟨R_{0}, B̂⟩.

Here, R_{0} is generated by elements of the form [[X_{1}, X_{2}, X_{3}]], where either X_{i} ∈ C(q^{n}), or X_{1} = X_{2} = X_{3} = X ∈ C(q^{2n}) and ψ^{q^{n}}(X) = −X. Also, C(q^{n}) ∈ Syl_{2}(C_{SL_{2}(q^{n})}(A)) is cyclic of order 2^{k} ≥ 4, where 2^{k} is the largest power which divides q^{n} ± 1; and C(q^{2n}) is cyclic of order 2^{k+1}. So

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

where B̂ = [[B, B, B]] has order 2 and acts on R_{0} via (g ↦ g^{−1}). Note that

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

is the 2-torsion subgroup of R_{0}.

We claim that

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

To see this, let R′ ≤ S be any subgroup isomorphic to (C_{2^{k}})^{3}, and let E′ ≅ C_{2}^{3} be its 2-torsion subgroup. Recall that for any 2-group P, the Frattini subgroup Fr(P) is the subgroup generated by commutators and squares in P. Thus

E′ ≤ Fr(R′) ≤ Fr(S) ≤ ⟨R_{0}, [[B, B, I]]⟩

(note that [[B, B, I]] = (τ·[[B, I, I]])^{2}). Any elementary abelian subgroup of rank 4 in Fr(S) would have to contain ⟨U, Â⟩ (the 2-torsion in R_{0} ≅ C_{2^{k}}^{3}), and this is impossible since no element of the coset R_{0}·[[B, B, I]] commutes with Â. Thus, rk(Fr(S)) = 3. Hence U ≤ E′, since otherwise ⟨U, E′⟩ would be an elementary abelian subgroup of Fr(S) of rank ≥ 4. This in turn implies that R′ ≤ C_{S}(U), and hence that E′ ≤ Fr(C_{S}(U)) ≤ R_{0}. Thus E′ = ⟨U, Â⟩ (the 2-torsion in R_{0} again). Hence R′ ≤ C_{S}(⟨U, Â⟩) = ⟨R_{0}, B̂⟩, and it follows that R′ = R_{0}. This finishes the proof of (2).

Choose generators x_{1}, x_{2}, x_{3} ∈ R_{0} as follows. Fix X ∈ C_{SL_{2}(q^{∞})}(A) of order 2^{k}, and Y ∈ C_{SL_{2}(q^{2n})}(A) of order 2^{k+1} such that Y^{2} = X. Set x_{1} = [[I, I, X]], x_{2} = [[X, I, I]], and x_{3} = [[Y, Y, Y]]. Thus, x_{1}^{2^{k−1}} = z, x_{2}^{2^{k−1}} = z_{1}, and (x_{3})^{2^{k−1}} = Â.

Now 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}(E)). In particular, E ≤ R_{1} = C_{S(q^{n})}(U). 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})⟩ for i = 0, 1.  (4)

By Proposition A.8, Aut_{Spin}(E) is the group of all automorphisms of E which send z to itself. In particular, since H(q^{n}) = C_{Spin}(U), Aut_{H(q^{n})}(E) is the group of all automorphisms of E which are the identity on U. Also, Γ = Inn(H(q^{n}))·Σ̂_{3}, where Σ̂_{3} sends Â = [[A, A, A]] to itself and permutes the nontrivial elements of U = {[[±I, ±I, ±I]]}. Hence Aut_{Γ}(E) is the group of all automorphisms which send U to itself. So if we identify Aut(E) ≅ GL_{3}(ℤ/2) via the basis {z, z_{1}, Â}, then

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

and

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

By (2) (and since E is the 2-torsion in R_{0}),

N_{Spin}(E) = N_{Spin}(R_{0}) and {γ ∈ Γ | γ(E) = E} = {γ ∈ Γ | γ(R_{0}) = R_{0}}.

Since C_{Spin}(E) = C_{Spin}(R_{0})·⟨B̂⟩, the only nonidentity element of Aut_{Spin}(R_{0}) or of Aut_{Γ}(R_{0}) which is the identity on E is conjugation by B̂, which is −I. Hence restriction from R_{0} to E induces isomorphisms

Aut_{Spin}(R_{0})/{±I} ≅ Aut_{Spin}(E) and Aut_{Γ}(R_{0})/{±I} ≅ Aut_{Γ}(E).

Upon identifying Aut(R_{0}) ≅ GL_{3}(ℤ/2^{k}) via the basis {x_{1}, x_{2}, x_{3}}, these can be regarded as sections

μ_{i}: T_{i} ⟶ GL_{3}(ℤ/2^{k})/{±I} = SL_{3}(ℤ/2^{k}) × {λI | λ ∈ (ℤ/2^{k})^{∗}}/{±I}

of the natural projection from GL_{3}(ℤ/2^{k})/{±I} to GL_{3}(ℤ/2), which agree on the group T_{0} = T_{1} ∩ T_{2} of upper triangular matrices.

We claim that μ_{1} and μ_{2} both map trivially to the second factor. Since this factor is abelian, it suffices to show that T_{0} is generated by [T_{1}, T_{1}] ∩ T_{0} and [T_{2}, T_{2}] ∩ T_{0}, and that each T_{i} is generated by [T_{i}, T_{i}] and T_{0} — and this is easily checked. (Note that T_{1} ≅ T_{2} ≅ Σ_{4}.)

By carrying out the above procedure over the field 𝔽_{q^{2n}}, we see that both of these sections μ_{i} can be lifted further to SL_{3}(ℤ/2^{k+1}) (still agreeing on T_{0}). So by Lemma A.10, there is a section

μ: GL_{3}(ℤ/2) ⟶ SL_{3}(ℤ/2^{k})

which extends both μ_{1} and μ_{2}. By (4), Aut_{F}(R_{0}) = Im(μ)·⟨ − I⟩.

We next identify Aut_{F}(R_{1}). By Lemma 2.8(a), E_{∗} =^{def} ⟨z, z_{1}, Â, B̂⟩ ≤ Spin_{7}(q^{n}) is a subgroup of rank 4 and type I. So by Proposition A.8, Aut_{Spin}(E_{∗}) contains all automorphisms of E_{∗} ≅ C_{2}^{4} which send z ∈ Z(Spin) to itself. Hence for any x ∈ N_{Spin}(R_{1}), since c_{x}(z) = z, there is x_{1} ∈ N_{Spin}(E_{∗}) such that c_{x_{1}}|_{E} = c_{x}|_{E} (i.e., xx_{1}^{−1} ∈ C_{Spin}(E)) and c_{x_{1}}(B̂) = B̂ (i.e., [x_{1}, B̂] = 1). Set x_{2} = xx_{1}^{−1}.

Since C_{Spin}(U) = H(q^{n}) ≤ Im(ω), we see that C_{Spin}(E) = K_{0}·⟨B̂⟩, where

K_{0} = ω(C_{SL_{2}(q^{∞})}(A)^{3}) ∩ Spin

is abelian, R_{0} ∈ Syl_{2}(K_{0}), and B̂ acts on K_{0} by inversion. Upon replacing x_{1} by B̂x_{1} and x_{2} by x_{2}B̂^{−1} if necessary, we can assume that x_{2} ∈ K_{0}. Then

[x_{2}, B̂] = x_{2}·(B̂x_{2}B̂^{−1})^{−1} = x_{2}^{2},

while by the original choice of x, x_{1} we have

[x_{2}, B̂] = [xx_{1}^{−1}, B̂] = [x, B̂] ∈ R_{0}.

Thus x_{2}^{2} ∈ R_{0} ∈ Syl_{2}(K_{0}), and hence x_{2} ∈ R_{0} ≤ R_{1}. Since x = x_{2}x_{1} was an arbitrary element of N_{Spin}(R_{1}), this shows that N_{Spin}(R_{1}) ≤ R_{1}·C_{Spin}(B̂), and hence that

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

Since Aut_{Γ}(R_{1}) is generated by its intersection with Aut_{Spin}(R_{1}) and the group Σ̂_{3} which permutes the three factors in H(q^{∞}) (and since the elements of Σ̂_{3} all fix B̂), we also have

Aut_{Γ}(R_{1}) = Inn(R_{1})·{φ ∈ Aut_{Γ}(R_{1}) | φ(B̂) = B̂}.

Together with (4) and (5), this shows that Aut_{F}(R_{1}) is generated by Inn(R_{1}) together with certain automorphisms of R_{1} = R_{0}·⟨B̂⟩ which send B̂ to itself. In other words,

Aut_{F}(R_{1}) = Inn(R_{1})·{φ ∈ Aut(R_{1}) | φ(B̂) = B̂, φ|_{R_{0}} ∈ Aut_{F}(R_{0})}
= Inn(R_{1})·{φ ∈ Aut(R_{1}) | φ(B̂) = B̂, φ|_{R_{0}} ∈ μ(GL_{3}(ℤ/2))}.

Thus

{φ ∈ Aut_{F}(R_{1}) | φ(z) = z}

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

the last equality by (5); and (1) now follows.

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). Since C_{S}(E) is not isomorphic to R_{1} = C_{S}(⟨z, z_{1}, Â⟩) (by (2)), this shows that E is not Spin-conjugate to ⟨z, z_{1}, Â⟩. By Proposition A.8, Spin contains exactly two conjugacy classes of rank 3 subgroups containing z, and thus E must have type II. Hence by 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 2.8(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 2.8(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}(−) (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 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. □

One more technical result is needed.

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. By [Sz, 3.6.3(ii)], −I is the only element of order 2 in SL_{2}(q^{∞}). 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}}). All elements in J_{1} are SL_{2}(q)-conjugate (this follows, for example, from [Sz, 3.6.23]), and we claim the same is true for elements of J_{2}.

Let SL_{2}^{∗}(q^{n}) be the group of all elements X ∈ SL_{2}(q^{2n}) such that ψ^{q^{n}}(X) = ±X. This is a group which contains SL_{2}(q^{n}) with index 2. Let k be such that the Sylow 2-subgroups of SL_{2}(q^{n}) have order 2^{k}; then k ≥ 3 since |SL_{2}(q^{n})| = q^{n}(q^{2n} − 1). Any S ∈ Syl_{2}(SL_{2}^{∗}(q^{n})) is quaternion of order 2^{k+1} ≥ 16 (see [Go, Theorem 2.8.3]) and its intersection with SL_{2}(q^{n}) is quaternion of order 2^{k}, so all elements in S ∩ J_{2} are S-conjugate. It follows that all elements of J_{2} are SL_{2}^{∗}(q^{n})-conjugate. If X, X′ ∈ J_{2} and X′ = gXg^{−1} for g ∈ SL_{2}^{∗}(q^{n}), then either g ∈ SL_{2}(q^{n}) or gX ∈ SL_{2}(q^{n}), and in either case X and X′ are conjugate by an element of SL_{2}(q^{n}).

By Proposition 2.5(a),

E, E′ ≤ 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, since E and E′ are Γ_{n}-conjugate (and each element of Γ_{n} leaves U = ⟨z, z_{1}⟩ invariant), the X_{i} and X′_{i} must all be in the same set J_{1} or J_{2}. Hence they are all SL_{2}(q^{n})-conjugate, and so E and E′ are Spin_{7}(q^{n})-conjugate. □

We are now ready to show that the fusion systems F_{n} are saturated, and satisfy the conditions listed in Theorem 2.1.

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 1.2, where p = 2, G = Spin_{7}(q^{n}), S = S(q^{n}), Z = ⟨z⟩ = Z(G); and U and C_{G}(U) = H(q^{n}) are as defined above. Also, Γ = Γ_{n} ≤ Aut(H(q^{n})). Condition (a) in Proposition 1.2 (all noncentral involutions in G are conjugate) holds since all subgroups in E_{2} are conjugate (Proposition A.8), and condition (b) holds by definition of Γ. Condition (c) holds since

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

by definition, since H(q^{n}) = C_{G}(U), and by Proposition 2.5(b). Condition (d) was shown in Proposition 2.9, and condition (e) in Lemma 2.10. So by Proposition 1.2, F_{n} is a saturated fusion system, and C_{F_{n}}(Z) = F_{S(q^{n})}(Spin_{7}(q^{n})).

The last statement is clear. □

3. Linking systems and their automorphisms

We next show the existence and uniqueness of centric linking systems associated to the F_{Sol}(q), and also construct certain automorphisms of these categories analogous to the automorphisms ψ^{q} of the group Spin_{7}(q^{n}). One more technical lemma about elementary abelian subgroups, this time about their F-conjugacy classes, is first needed.

Lemma 3.1. Set F = F_{Sol}(q). For each r ≤ 3, there is a unique F-conjugacy class of elementary abelian subgroups E ≤ S(q) of rank r. There are two F-conjugacy classes of rank four elementary abelian subgroups E ≤ S(q): one is the set C of subgroups Spin_{7}(q)-conjugate to E_{∗} = ⟨z, z_{1}, Â, B̂⟩, while the other contains the other conjugacy class of type I subgroups as well as all type II subgroups. Furthermore, Aut_{F}(E) = Aut(E) for all elementary abelian subgroups E ≤ S(q) except when E has rank four and is not F-conjugate to E_{∗}, in which case

Aut_{F}(E) = {α ∈ Aut(E) | α(x_{C}(E)) = x_{C}(E)}.

Proof. By Lemma 2.8(d), the three subgroups

E_{∗} = ⟨z, z_{1}, Â, [[B, B, B]]⟩, E_{001} = ⟨z, z_{1}, Â, [[B, B, XB]]⟩, E_{100} = ⟨z, z_{1}, Â, [[XB, B, B]]⟩

(where X is a generator of C(q)) represent the three Spin_{7}(q)-conjugacy classes of rank four subgroups. Clearly, E_{100} and E_{001} are Γ_{1}-conjugate, hence F-conjugate; and by Lemma 2.8(e), neither is Γ_{1}-conjugate to E_{∗}. This proves that there are exactly two F-conjugacy classes of such subgroups.

Since E_{∗} and E_{001} both are of type I in Spin_{7}(q), their Spin_{7}(q)-automorphism groups contain all automorphisms which fix z (see Proposition A.8). By Lemma 2.8(e), z is fixed by all Γ-automorphisms of E_{001}, and so Aut_{F}(E_{001}) is the group of all automorphisms of E_{001} which send z = x_{C}(E_{001}) to itself. On the other hand, E_{∗} contains automorphisms (induced by permuting the three coordinates of H) which permute the three elements z, z_{1}, zz_{1}; and these together with Aut_{Spin}(E_{∗}) generate Aut(E_{∗}).

It remains to deal with the subgroups of smaller rank. By Proposition A.8 again, there is just one Spin_{7}(q)-conjugacy class of elementary abelian subgroups of rank one or two. There are two conjugacy classes of rank three subgroups, those of type I and those of type II. Since E_{100} is of type II and E_{001} of type I, all rank three subgroups of E_{001} have type I, while some of the rank three subgroups of E_{100} have type II. Since E_{001} is F-conjugate to E_{100}, this shows that some subgroup of rank three and type II is F-conjugate to a subgroup of type I, and hence all rank three subgroups are conjugate to each other. Finally, Aut_{F}(E) = Aut(E) whenver rk(E) ≤ 3 since any such group is F-conjugate to a subgroup of E_{∗} (and we have just seen that Aut_{F}(E_{∗}) = Aut(E_{∗})). □

To simplify the notation, we now define

F_{Spin}(q^{n}) =^{def} F_{S(q^{n})}(Spin_{7}(q^{n}))

for all 1 ≤ n ≤ ∞: the fusion system of the group Spin_{7}(q^{n}) at the Sylow subgroup S(q^{n}). By construction, this is a subcategory of F_{Sol}(q^{n}). We write

O_{Sol}(q^{n}) = O(F_{Sol}(q^{n})) and O_{Spin}(q^{n}) = O(F_{Spin}(q^{n}))

for the corresponding orbit categories: both of these have as objects the subgroups of S(q^{n}), and have as morphism sets

Mor_{O_{Sol}(q^{n})}(P, Q) = Hom_{F_{Sol}(q^{n})}(P, Q)/ Inn(Q) ⊆ Rep(P, Q)

and

Mor_{O_{Spin}(q^{n})}(P, Q) = Hom_{F_{Spin}(q^{n})}(P, Q)/ Inn(Q) .

Let O_{Sol}^{c}(q^{n}) ⊆ O_{Sol}(q^{n}) and O_{Spin}^{c}(q^{n}) ⊆ O_{Spin}(q^{n}) be the centric orbit categories; i.e., the full subcategories whose objects are the F_{Sol}(q^{n})- or F_{Spin}(q^{n})-centric subgroups of S(q^{n}). (We will see shortly that these in fact have the same objects.)

The obstructions to the existence and uniqueness of linking systems associated to the fusion systems F_{Sol}(q^{n}), and to the existence and uniqueness of certain automorphisms of those linking systems, lie in certain groups which were identified in [BLO2] and [BLO1]. It is these groups which are shown to vanish in the next lemma.

Lemma 3.2. Fix a prime power q, and let

Z_{Sol}(q): O_{Sol}^{c}(q) ⟶ Ab  and  Z_{Spin}(q): O_{Spin}^{c}(q) ⟶ Ab

be the functors Z(P) = Z(P). Then for all i ≥ 0,

\underleftarrow{lim}^{i}_{O_{Sol}^{c}(q)}(Z_{Sol}(q)) = 0 = \underleftarrow{lim}^{i}_{O_{Spin}^{c}(q)}(Z_{Spin}(q)).

Proof. Set F = F_{Sol}(q) for short. Let P_{1}, …, P_{k} be F-conjugacy class representatives for all F-centric subgroups P_{i} ≤ S(q), arranged such that |P_{i}| ≤ |P_{j}| for i ≤ j. For each i, let Z_{i} ⊆ Z_{Sol}(q) be the subfunctor defined by setting Z_{i}(P) = Z_{Sol}(q)(P) if P is conjugate to P_{j} for some j ≤ i and Z_{i}(P) = 0 otherwise. We thus have a filtration

0 = Z_{0} ⊆ Z_{1} ⊆ ⋯ ⊆ Z_{k} = Z_{Sol}(q)

of Z_{Sol}(q) by subfunctors, with the property that for each i, the quotient functor Z_{i}/Z_{i−1} vanishes except on the conjugacy class of P_{i} (and such that (Z_{i}/Z_{i−1})(P_{i}) = Z_{Sol}(q)(P_{i})). By [BLO2, Proposition 3.2],

\underleftarrow{lim}^{∗}(Z_{i}/Z_{i−1}) ≅ Λ^{∗}(Out_{F}(P_{i}); Z(P_{i}))

for each i. Here, Λ^{∗}(Γ; M) are certain graded groups, define in [JMO, §5] for all finite groups Γ and all finite ℤ_{(p)}[Γ]-modules M. We claim that Λ^{∗}(Out_{F}(P_{i}); Z(P_{i})) = 0 except when P_{i} = S(q) or S_{0}(q) (see Definition 2.6).

Fix an F-centric subgroup P ≤ S(q). For each j ≥ 1, let Ω_{j}(Z(P)) = {g ∈ Z(P) | g^{2^{j}} = 1}, and set E = Ω_{1}(Z(P)) — the 2-torsion in the center of P. For each j ≥ 1, let Ω_{j}(Z(P)) = {g ∈ Z(P) | g^{2^{j}} = 1}, and set E = Ω_{1}(Z(P)) — the 2-torsion in the center of P. We can assume E is fully centralized in F (otherwise replace P and E by appropriate subgroups in the same F-conjugacy classes).

Assume first that Q =^{def} C_{S(q)}(E) ≩ P, and hence that N_{Q}(P) ≩ P. Then any x ∈ N_{Q}(P)∖P centralizes E = Ω_{1}(Z(P)). Hence for each j, x acts trivially on Ω_{j}(Z(P))/Ω_{j−1}(Z(P)), since multiplication by p^{j−1} sends this group N_{Q}(P)/P -linearly and monomorphically to E. Since c_{x} is a nontrivial element of Out_{F}(P) of p-power order,

Λ^{∗}(Out_{F}(P); Ω_{j}(Z(P))/Ω_{j−1}(Z(P))) = 0

for all j ≥ 1 by [JMO, Proposition 5.5], and thus Λ^{∗}(Out_{F}(P); Z(P)) = 0.

Now assume that P = C_{S(q)}(E) = P, the centralizer in S(q) of a fully F-centralized elementary abelian subgroup. Since there is a unique conjugacy class of elementary abelian subgroup of any rank ≤ 3, C_{S(q)}(E) always contains a subgroup C_{2}^{4}, and hence P contains a subgroup C_{2}^{4} which is self centralizing by Proposition A.8(a). This shows that Z(P) is elementary abelian, and hence that Z(P) = E.

We can assume P is fully normalized in F, so

Aut_{S(q)}(P) ∈ Syl_{2}(Aut_{F}(P))

by condition (I) in the definition of a saturated fusion system. Since P = C_{S(q)}(E) (and E = Z(P)), this shows that

Ker[Out_{F}(P) ⟶ Aut_{F}(E)]

has odd order. Also, since E is fully centralized, any F-automorphism of E extends to an F-automorphism of P = C_{S(q)}(E), and thus this restriction map between automorphism groups is onto. By [JMO, Proposition 6.1(i,iii)], it now follows that

Λ^{i}(Out_{F}(P); Z(P)) ≅ Λ^{i}(Aut_{F}(E); E).  (1)

By Lemma 3.1, Aut_{F}(E) = Aut(E), except when E lies in one certain F-conjugacy class of subgroups E ≅ C_{2}^{4}; and in this case P = E and Aut_{F}(E) is the group of automorphisms fixing the element x_{C}(E). In this last (exceptional) case, O_{2}(Aut_{F}(E)) ≠ 1 (the subgroup of elements which are the identity on E/⟨x_{C}(E)⟩), so

Λ^{∗}(Out_{F}(P); Z(P)) = Λ^{∗}(Aut_{F}(E); E) = 0  (2)

by [JMO, Proposition 6.1(ii)]. Otherwise, when Aut_{F}(E) = Aut(E), by [JMO, Proposition 6.3] we have

Λ^{i}(Aut_{F}(E); E) ≅ \begin{cases}ℤ/2 & if rk(E) = 2, i = 1 \\ ℤ/2 & if rk(E) = 1, i = 0 \\ 0 & otherwise. \end{cases}  (3)

By points (1), (2), and (3), the groups Λ^{∗}(Out_{F}(P); Z(P)) vanish except in the two cases E = ⟨z⟩ or E = U, and these correspond to P = S(q) or P = N_{S(q)}(U) = S_{0}(q).

We can assume that P_{k} = S(q) and P_{k−1} = S_{0}(q). We have now shown that \underleftarrow{lim}^{∗}(Z_{k−2}) = 0, and thus that Z_{Sol}(q) has the same higher limits as Z_{k}/Z_{k−2}. Hence \underleftarrow{lim}^{j}(Z_{Sol}(q)) = 0 for all j ≥ 2, and there is an exact sequence

0 ⟶ \underleftarrow{lim}^{0}(Z_{Sol}(q)) ⟶ \underleftarrow{lim}^{0}(Z_{k}/Z_{k−1})_{≅ℤ/2} ⟶ \underleftarrow{lim}^{1}(Z_{k−1}/Z_{k−2})_{≅ℤ/2}

⟶ \underleftarrow{lim}^{1}(Z_{Sol}(q)) ⟶ 0.

One easily calculates that \underleftarrow{lim}^{0}(Z_{Sol}(q)) = 0, and hence we also get \underleftarrow{lim}^{1}(Z_{Sol}(q)) = 0.

The proof that \underleftarrow{lim}^{i}(Z_{Spin}(q)) = 0 for all i ≥ 1 is similar, but simpler. If F = F_{Spin}(q), then for any F-centric subgroup P ≨ S(q), there is an element x ∈ N_{S}(P)∖P such that [x, P] = ⟨z⟩, and c_{x} is a nontrivial element of O_{2}(Out_{F}(P)). Thus

Λ^{∗}(Out_{F}(P); Z(P)) = 0

for all such P by [JMO, Proposition 6.1(ii)] again. □

We are now ready to construct classifying spaces BSol(q) for these fusion systems F_{Sol}(q). The following proposition finishes the proof of Theorem 2.1, and also contains additional information about the spaces BSol(q).

To simplify notation, we write L_{Spin}^{c}(q^{n}) = L_{S(q^{n})}^{c}(Spin_{7}(q^{n})) (n ≥ 1) to denote the centric linking system for the group Spin_{7}(q^{n}). The field automorphism (x ↦ x^{q}) induces an automorphism of Spin_{7}(q^{n}) which sends S(q^{n}) to itself; and this in turn induces automorphisms ψ_{F}^{q} = ψ_{F}^{q}(Sol), ψ_{F}^{q}(Spin), and ψ_{L}^{q}(Spin) of the fusion systems F_{Sol}(q^{n}) ⊇ F_{Spin}(q^{n}) and of the linking system L_{Spin}^{c}(q^{n}).

Proposition 3.3. Fix an odd prime q, and n ≥ 1. Let S = S(q^{n}) ∈ Syl_{2}(Spin_{7}(q^{n})) be as defined above. Let z ∈ Z(Spin_{7}(q^{n})) be the central element of order 2. Then there is a centric linking system

L = L_{Sol}^{c}(q^{n}) ⟶^{π} F_{Sol}(q^{n})

associated to the saturated fusion system F =^{def} F_{Sol}(q^{n}) over S, which has the following additional properties.

(a) A subgroup P ≤ S is F-centric if and only if it is F_{Spin}(q^{n})-centric.

(b) L_{Sol}^{c}(q^{n}) contains L_{Spin}^{c}(q^{n}) as a subcategory, in such a way that π|_{L_{Spin}^{c}(q^{n})} is the usual projection to F_{Spin}^{c}(q^{n}), and that the distinguished monomorphisms

P ⟶^{δ_{P}} Aut_{L}(P)

for L = L_{Sol}^{c}(q^{n}) are the same as those for L_{Spin}^{c}(q^{n}).

(c) Each automorphism of L_{Spin}^{c}(q^{n}) which covers the identity on F_{Spin}^{c}(q^{n}) extends to an automorphism of L_{Sol}^{c}(q^{n}) which covers the identity on F_{Sol}^{c}(q^{n}). Furthermore, such an extension is unique up to composition with the functor

C_{z}: L_{Sol}^{c}(q^{n}) ⟶ L_{Sol}^{c}(q^{n})

which is the identity on objects and sends α ∈ Mor_{L_{Sol}^{c}(q^{n})}(P, Q) to ẑ ∘ α ∘ ẑ^{−1} (“conjugation by z”).

(d) There is a unique automorphism ψ_{L}^{q} ∈ Aut(L_{Sol}^{c}(q^{n})) which covers the automorphism of F_{Sol}(q^{n}) induced by the field automorphism (x ↦ x^{q}), which extends the automorphism of L_{Spin}^{c}(q^{n}) induced by the field automorphism, and which is the identity on π^{−1}(F_{Sol}(q)).

Proof. By Proposition 2.11, F = F_{Sol}(q^{n}) is a saturated fusion system over S = S(q^{n}) ∈ Syl_{2}(Spin_{7}(q^{n})), with the property that C_{F}(z) = F_{Spin}(q^{n}). Point (a) follows as a special case of [BLO2, Proposition 2.5(a)].

Since \underleftarrow{lim}^{i}_{O_{Sol}^{c}(q^{n})}(Z_{Sol}(q^{n})) = 0 for i = 2, 3 by Lemma 3.2, there is by [BLO2, Propo-

sition 3.1] a centric linking system L = L_{Sol}^{c}(q^{n}) associated to F, which is unique up to isomorphism (an isomorphism which commutes with the projection to F_{Sol}(q^{n}) and with the distinguished monomorphisms). Furthermore, π^{−1}(F_{Spin}(q^{n})) is a linking system associated to F_{Spin}(q^{n}), such a linking system is unique up to isomorphism since \underleftarrow{lim}^{2}(Z_{Spin}(q^{n})) = 0 (Lemma 3.2 again), and this proves (b).

(c) By [BLO1, Theorem 6.2] (more precisely, by the same proof as that used in [BLO1]), the vanishing of \underleftarrow{lim}^{i}(Z_{Sol}(q^{n})) for i = 1, 2 (Lemma 3.2) shows that each automorphism of F = F_{Sol}(q^{n}) lifts to an automorphism of L, which is unique up to a natural isomorphism of functors; and any such natural isomorphism sends each object P ≤ S to a isomorphism ĝ for some g ∈ Z(P). Similarly, the vanishing of \underleftarrow{lim}^{i}(Z_{Spin}(q^{n})) for i = 1, 2 shows that each automorphism of F_{Spin}(q^{n}) lifts to an automorphism of L_{Spin}^{c}(q^{n}), also unique up to a natural isomorphism of functors. Since L_{Sol}^{c}(q^{n}) and L_{Spin}^{c}(q^{n}) have the same objects by (a), this shows that each automorphism of L_{Spin}^{c}(q^{n}) which covers the identity on F_{Spin}^{c}(q^{n}) extends to a unique automorphism of L_{Sol}^{c}(q^{n}) which covers the identity on F_{Sol}(q^{n}).

It remains to show, for any Φ ∈ Aut(L_{Sol}^{c}(q^{n})) which covers the identity on F_{Sol}^{c}(q^{n}) and such that Φ|_{L_{Spin}^{c}(q^{n})} = Id, that Φ is the identity or conjugation by z. We have already noted that Φ must be naturally isomorphic to the identity; i.e., that there are elements γ(P) ∈ Z(P), for all P in L_{Sol}^{c}(q^{n}), such that

Φ(α) = γ(Q) ∘ α ∘ γ(P)^{−1} for all α ∈ Mor_{L_{Sol}^{c}(q^{n})}(P, Q), all P, Q.

Since Φ is the identity on L_{Spin}^{c}(q^{n}), the only possibilities are γ(P) = 1 for all P (hence Φ = Id), or γ(P) = z for all P (hence Φ is conjugation by z).

(d) Now consider the automorphism ψ_{F}^{q} ∈ Aut(F_{Sol}(q^{n})) induced by the field automorphism (x ↦ x^{q}) of 𝔽_{q^{n}}. We have just seen that this lifts to an automorphism ψ_{L}^{q} of L_{Sol}^{c}(q^{n}), which is unique up to natural isomorphism of functors. The restriction of ψ_{L}^{q} to L_{Spin}^{c}(q^{n}), and the automorphism ψ_{L}^{q}(Spin) of L_{Spin}^{c}(q^{n}) induced directly by the field automorphism, are two liftings of ψ_{F}^{q}|_{F_{Spin}(q^{n})}, and hence differ by a natural isomorphism of functors which extends to a natural isomorphism of functors on L_{Sol}^{c}(q^{n}). Upon composing with this natural isomorphism, we can thus assume that ψ_{L}^{q} does restrict to the automorphism of L_{Spin}^{c}(q^{n}) induced by the field automorphism.

Now consider the action of ψ_{L}^{q} on Aut_{L}(S_{0}(q)), which by assumption is the identity on Aut_{L_{Spin}^{c}(q)}(S_{0}(q)), and in particular on δ(S_{0}(q)) itself. Thus, with respect to the extension

1 ⟶ S_{0}(q) ⟶ Aut_{L}(S_{0}(q)) ⟶ Σ_{3} ⟶ 1,

ψ_{L}^{q} is the identity on the kernel and on the quotient, and hence is described by a cocycle

η ∈ Z^{1}(Σ_{3}; Z(S_{0}(q))) ≅ Z^{1}(Σ_{3}; (ℤ/2)^{2}).

Since H^{1}(Σ_{3}; (ℤ/2)^{2}) = 0, η must be a coboundary, and thus the action of ψ_{L}^{q} on Aut_{L}(S_{0}(q)) is conjugation by an element of Z(S_{0}(q)). Since it is the identity on Aut_{L_{Spin}^{c}(q)}(S_{0}(q)), it must be conjugation by 1 or z. If it is conjugation by z, then we can replace ψ_{L}^{q} (on the whole category L) by its composite with z; i.e., by its composite with the functor which is the identity on objects and sends α ∈ Mor_{L}(P, Q) to ẑ∘ α ∘ ẑ.

In this way, we can assume that ψ_{L}^{q} is the identity on Aut_{L}(S_{0}(q)). By construction, every morphism in F_{Sol}(q) is a composite of morphisms in F_{Spin}(q) and restrictions of automorphisms in F_{Sol}(q) of S_{0}(q). Since ψ_{L}^{q} is the identity on π^{−1}(F_{Spin}(q)), this shows that it is the identity on π^{−1}(F_{Sol}(q)).

It remains to check the uniqueness of ψ_{L}^{q}. If ψ′ is another functor with the same properties, then by (e), (ψ′)^{−1} ∘ ψ_{L}^{q} is either the identity or conjugation by z; and the latter is not possible since conjugation by z is not the identity on π^{−1}(F_{Sol}(q)). □

This finishes the construction of the classifying spaces BSol(q) = |L_{Sol}(q)|_{2}^{∧} for the fusion systems constructed in Section 2. We end the section with an explanation of why these are not the fusion systems of finite groups.

Proposition 3.4. For any odd prime power q, there is no finite group G whose fusion system is isomorphic to that of F_{Sol}(q).

Proof. Let G be a finite group, fix S ∈ Syl_{2}(G), and assume that S ≅ S(q) ∈ Syl_{2}(Spin_{7}(q)), and that the fusion system F_{S}(G) satisfies conditions (a) and (b) in Theorem 2.1. In particular, all involutions in G are conjugate, and the centralizer of any involution z ∈ G has the fusion system of Spin_{7}(q). When q ≡ ±3 (mod 8), Solomon showed [So, Theorem 3.2] that there is no finite group whose fusion system has these properties. When q ≡ ±1 (mod 8), he showed (in the same theorem) that there is no such G such that Ĥ =^{def} C_{G}(z)/O_{2′}(C_{G}(z)) is isomorphic to a subgroup of Aut(Spin_{7}(q)) which contains Spin_{7}(q) with odd index. (Here, O_{2′}(−) means largest odd order normal subgroup.)

Let G be a finite group whose fusion system is isomorphic to F_{Sol}(q), and again set Ĥ =^{def} C_{G}(z)/O_{2′}(C_{G}(z)) for some involution z ∈ G. Set H = O^{2′}(Ĥ/⟨z⟩): the smallest normal subgroup of Ĥ/⟨z⟩ of odd index. Then H has the fusion system of Ω_{7}(q) ≅ Spin_{7}(q)/Z(Spin_{7}(q)). We will show that H ≅ Ω_{7}(q′) for some odd prime power q′. It then follows that O^{2′}(Ĥ) ≅ Spin_{7}(q′), thus contradicting Solomon’s theorem and proving our claim.

The following “classification free” argument for proving that H ≅ Ω_{7}(q′) for some q′ was explained to us by Solomon. We refer to the appendix for general results about the groups Spin_{n}^{±}(q) and Ω_{n}^{±}(q). Fix S ∈ Syl_{2}(H). Thus S is isomorphic to a Sylow 2-subgroup of Ω_{7}(q), and has the same fusion.

We first claim that H must be simple. By definition (H = O^{2′}(Ĥ/⟨z⟩)), H has no proper normal subgroup of odd index, and H has no proper normal subgroup of odd order since any such subgroup would lift to an odd order normal subgroup of Ĥ = C_{G}(z)/O_{2′}(C_{G}(z)). Hence for any proper normal subgroup N ◁ H, Q =^{def} N ∩ S is a proper normal subgroup of S, which is strongly closed in S with respect to H in the sense that no element of Q can be H-conjugate to an element of S∖Q. Using Lemma A.4(a), one checks that the group Ω_{7}(q) contains three conjugacy classes of involutions, classified by the dimension of their (−1)-eigenspace. It is not hard to see (by taking products) that any subgroup of S which contains all involutions in one of these conjugacy classes contains all involutions in the other two classes as well. Furthermore, S is generated by the set of all of its involutions, and this shows that there are no proper subgroups which are strongly closed in S with respect to H. Since we have already seen that the intersection with S of any proper normal subgroup of H would have to be such a subgroup, this shows that H is simple.

Fix an isomorphism

S ⟶_{≅}^{φ} S′ ∈ Syl_{2}(Ω_{7}(q))

which preserves fusion. Choose x′ ∈ S′ whose (−1)-eigenspace is 4-dimensional, and such that ⟨x′⟩ is fully centralized in F_{S′}(Ω_{7}(q)). Then

C_{O_{7}(q)}(x′) ≅ O_{4}^{+}(q) × O_{3}(q)

by Lemma A.4(c). Since Ω_{4}^{+}(q) ≤ O_{4}^{+}(q) and Ω_{3}(q) ≤ O_{3}(q) both have index 4, C_{Ω_{7}(q)}(x′) is isomorphic to a subgroup of O_{4}^{+}(q) × O_{3}(q) of index 4, and contains a normal subgroup K′ ≅ Ω_{4}^{+}(q) × Ω_{3}(q) of index 4. Since ⟨x′⟩ is fully centralized, C_{S′}(x′) is a Sylow 2-subgroup of C_{Ω_{7}(q)}(x′), and hence S′_{0} =^{def} S′ ∩ K′ is a Sylow 2-subgroup of K′.

Set x = φ^{−1}(x′) ∈ S. Since S ≅ S′ have the same fusion in H and Ω_{7}(q), C_{S}(x) ≅ C_{S′}(x′) have the same fusion in C_{H}(x) and C_{Ω_{7}(q)}(x′). Hence H_{1}(C_{H}(x); ℤ_{(2)}) ≅ H_{1}(C_{Ω_{7}(q)}(x′); ℤ_{(2)}) (homology is determined by fusion), both have order 4, and thus C_{H}(x) also has a unique normal subgroup K ◁ H of index 4. Set S_{0} = K ∩ S. Thus φ(S_{0}) = S′_{0}, and using Alperin’s fusion theorem one can show that this isomorphism is fusion preserving with respect to the inclusions of Sylow subgroups S_{0} ≤ K and S′_{0} ≤ K′.

Using the isomorphisms of Proposition A.5:

Ω_{4}^{+}(q) ≅ SL_{2}(q) ×_{⟨x⟩} SL_{2}(q) and Ω_{3}(q) ≅ P SL_{2}(q),

we can write K′ = K′_{1} ×_{⟨x′⟩} K′_{2}, where K′_{1} ≅ SL_{2}(q) and K′_{2} ≅ SL_{2}(q) × P SL_{2}(q). Set S′_{i} = S′∩K′_{i} ∈ Syl_{2}(K′_{i}); thus S′_{0} = S′_{1}×_{⟨x′⟩}S′_{2}. Set S_{i} = φ^{−1}(S′_{i}), so that S_{0} = S_{1}×_{⟨x⟩}S_{2} is normal of index 4 in C_{S}(x). The fusion system of K thus splits as a central product of fusion systems, one of which is isomorphic to the fusion system of SL_{2}(q).

We now apply a theorem of Goldschmidt, which says very roughly that under these conditions, the group K also splits as a central product. To make this more precise, let K_{i} be the normal closure of S_{i} in K ◁ C_{H}(x). By [Gd, Corollary A2], since S_{1} and S_{2} are strongly closed in S_{0} with respect to K,

[K_{1}, K_{2}] ≤ ⟨x⟩·O_{2′}(K).

Using this, it is not hard to check that S_{i} ∈ Syl_{2}(K_{i}). Thus K_{1} has same fusion as SL_{2}(q) and is subnormal in C_{H}(x) (K_{1} ◁ K ◁ C_{H}(x)), and an argument similar to that used above to prove the simplicity of H shows that K_{1}/(⟨x⟩·O_{2′}(K_{1})) is simple. Hence K_{1} is a 2-component of C_{H}(x) in the sense described by Aschbacher in [As1]. By [As1, Corollary III], this implies that H must be isomorphic to a Chevalley group of odd characteristic, or to M_{11}. It is now straightforward to check that among these groups, the only possibility is that H ≅ Ω_{7}(q′) for some odd prime power q′. □

4. Relation with the Dwyer-Wilkerson space

We now want to examine the relation between the spaces BSol(q) which we have just constructed, and the space BDI(4) constructed by Dwyer and Wilkerson in [DW1]. Recall that this is a 2-complete space characterized by the property that its cohomology is the Dickson algebra in four variables over 𝔽_{2}; i.e., the ring of invariants 𝔽_{2}[x_{1}, x_{2}, x_{3}, x_{4}]^{GL_{4}(2)}. We show, for any odd prime power q, that BDI(4) is homotopy equivalent to the 2-completion of the union of the spaces BSol(q^{n}), and that BSol(q) is homotopy equivalent to the homotopy fixed point set of an Adams map from BDI(4) to itself.

We would like to define an infinite “linking system” L_{Sol}^{c}(q^{∞}) as the union of the finite categories L_{Sol}^{c}(q^{n}), and then set BSol(q^{∞}) = |L_{Sol}^{c}(q^{∞})|_{2}^{∧}. The difficulty with this approach is that a subgroup which is centric in the fusion system F_{Sol}(q^{m}) need not be centric in a larger fusion system F_{Sol}(q^{n}) (for m|n). To get around this problem, we define L_{Sol}^{cc}(q^{n}) ⊆ L_{Sol}^{c}(q^{n}) to be the full subcategory whose objects are those subgroups of S(q^{n}) which are F_{Sol}(q^{∞})-centric; or equivalently F_{Sol}(q^{k})-centric for all k ∈ nℤ. Similarly, we define L_{Spin}^{cc}(q^{n}) to be the full subcategory of L_{Spin}^{c}(q^{n}) whose objects are those subgroups of S(q^{n}) which are F_{Spin}(q^{∞})-centric. We can then define L_{Sol}^{c}(q^{∞}) and L_{Spin}^{c}(q^{∞}) to be the unions of these categories.

For these definitions to be useful, we must first show that |L_{Sol}^{cc}(q^{n})|_{2}^{∧} has the same homotopy type as |L_{Sol}^{c}(q^{n})|_{2}^{∧}. This is done in the following lemma.

Lemma 4.1. For any odd prime power q and any n ≥ 1, the inclusions

|L_{Sol}^{cc}(q^{n})|_{2}^{∧} ⊆ |L_{Sol}^{c}(q^{n})|_{2}^{∧}  and  |L_{Spin}^{cc}(q^{n})|_{2}^{∧} ⊆ |L_{Spin}^{c}(q^{n})|_{2}^{∧}

are homotopy equivalences.

Proof. It clearly suffices to show this when n = 1.

Recall, for a fusion system F over a p-group S, that a subgroup P ≤ S is F-radical if Out_{F}(P) is p-reduced; i.e., if O_{p}(Out_{F}(P)) = 1. We will show that

all F_{Sol}(q)-centric F_{Sol}(q)-radical subgroups of S(q) are also F_{Sol}(q^{∞})-centric (1)

and similarly

all F_{Spin}(q)-centric F_{Spin}(q)-radical subgroups of S(q) are also F_{Spin}(q^{∞})-centric. (2)

In other words, (1) says that for each P ≤ S(q) which is an object of L_{Sol}^{c}(q) but not of L_{Sol}^{cc}(q), O_{2}(Out_{F_{Sol}(q)}(P)) ≠ 1. By [JMO, Proposition 6.1(ii)], this implies that

Λ^{∗}(Out_{F_{Sol}(q)}(P); H^{∗}(BP; 𝔽_{2})) = 0.

Hence by [BLO2, Propositions 3.2 & 2.2] (and the spectral sequence for a homotopy colimit), the inclusion L_{Sol}^{cc}(q) ⊆ L_{Sol}^{c}(q) induces an isomorphism

H^{∗}(|L_{Sol}^{c}(q)|; 𝔽_{2}) ⟶^{≅} H^{∗}(|L_{Sol}^{cc}(q)|; 𝔽_{2}),

and thus |L_{Sol}^{cc}(q)|_{2}^{∧} ≃ |L_{Sol}^{c}(q)|_{2}^{∧}. The proof that |L_{Spin}^{cc}(q)|_{2}^{∧} ≃ |L_{Spin}^{c}(q)|_{2}^{∧} is similar, using (2).

Point (2) is shown in Proposition A.12, so it remains only to prove (1). Set F = F_{Sol}(q), and set F_{k} = F_{Sol}(q^{k}) for all 1 ≤ k ≤ ∞. Let E ≤ Z(P) be the 2-torsion in the center of P, so that P ≤ C_{S(q)}(E). Set

E′ = \begin{cases}⟨z⟩ & if rk(E) = 1 \\ ⟨z, z_{1}⟩ & if rk(E) = 2 \\ ⟨z, z_{1}, Â⟩ & if rk(E) = 3 \\ E & if rk(E) = 4 \end{cases}

in the notation of Definition 2.6. In all cases, E is F-conjugate to E′ by Lemma 3.1. We claim that E′ is fully centralized in F_{k} for all k < ∞. This is clear when rk(E′) = 1 (E′ = Z(S(q^{k}))), follows from Proposition 2.5(a) when rk(E′) = 2, and from Proposition A.8(a) (all rank 4 subgroups are self centralizing) when rk(E′) = 4. If rk(E′) = 3, then by Proposition A.8(d), the centralizer in Spin_{7}(q^{k}) (hence in S(q^{k})) of any rank 3 subgroup has an abelian subgroup of index 2; and using this (together with the construction of S(q^{k}) in Definition 2.6), one sees that E′ is fully centralized in F_{k}.

If E′ ≠ E, choose φ ∈ Hom_{F}(E, S(q)) such that φ(E) = E′; then φ extends to φ ∈ Hom_{F}(C_{S(q)}(E), S(q)) by condition (II) in the definition of a saturated fusion system, and we can replace P by φ(P) and E by φ(E). We can thus assume that E is fully centralized in F_{k} for each k < ∞. So by [BLO2, Proposition 2.5(a)], P is F_{k}-centric if and only if it is C_{F_{k}}(E)-centric; and this also holds when k = ∞. Furthermore, since Out_{C_{F} (E)}(P) ◁ Out_{F}(P), O_{2}(Out_{C_{F} (E)}(P)) is a normal 2-subgroup of Out_{F}(P), and thus

O_{2}(Out_{C_{F} (E)}(P)) ≤ O_{2}(Out_{F}(P)).

Hence P is C_{F}(E)-radical if it is F-radical. So it remains to show that

all C_{F}(E)-centric C_{F}(E)-radical subgroups of S(q) are also C_{F_{∞}}(E)-centric. (3)

If rk(E) = 1, then C_{F}(E) = F_{Spin}(q) and C_{F_{∞}}(E) = F_{Spin}(q^{∞}), and (3) follows from (2). If rk(E) = 4, then P = E = C_{S(q^{∞})}(E) by Proposition A.8(a), so P is F_{∞}-centric, and the result is clear.

If rk(E) = 3, then by Proposition A.8(d), C_{F}(E) ⊆ C_{F_{∞}}(E) are the fusion systems of a pair of semidirect products A⋊C_{2} ≤ A_{∞}⋊C_{2}, where A ≤ A_{∞} are abelian and C_{2} acts on A_{∞} by inversion. Also, E is the full 2-torsion subgroup of A_{∞}, since otherwise rk(A_{∞}) > 3 would imply A_{∞}⋊C_{2} ≤ Spin_{7}(q^{∞}) contains a subgroup C_{2}^{5} (contradicting Proposition A.8). If P ≤ A, then either Out_{C_{F} (E)}(P) has order 2, which contradicts the assumption that P is radical; or P is elementary abelian and Out_{C_{F} (E)}(P) = 1, in which case P ≤ Z(A⋊C_{2}) is not centric. Thus P ≰ A; P ∩ A ≥ E contains all 2-torsion in A_{∞}, and hence P is centric in A_{∞}⋊C_{2}.

If rk(E) = 2, then by Proposition 2.5(a), C_{F_{∞}}(E) and C_{F}(E) are the fusion systems of the groups

H(q^{∞}) ≅ SL_{2}(q^{∞})^{3}/{±(I, I, I)}  (4)

and

H(q) = H(q^{∞}) ∩ Spin_{7}(q) ≥ H_{0}(q) =^{def} SL_{2}(q)^{3}/{±(I, I, I)}.

If P ≤ S(q) is centric and radical in the fusion system of H(q), then by Lemma A.11(c), its intersection with H_{0}(q) ≅ SL_{2}(q)^{3}/{±(I, I, I)} is centric and radical in the fusion system of that group. So by Lemma A.11(a,f),

P ∩ H_{0}(q) ≅ (P_{1} × P_{2} × P_{3})/{±(I, I, I)}  (5)

for some P_{i} which are centric and radical in the fusion system of SL_{2}(q). Since the Sylow 2-subgroups of SL_{2}(q) are quaternion [Go, Theorem 2.8.3], the P_{i} must be nonabelian and quaternion, so each P_{i}/{±I} is centric in P SL_{2}(q^{∞}). Hence P ∩ H_{0}(q) is centric in SL_{2}(q)^{3}/{±(I, I, I)} by (5), and so P is centric in H(q^{∞}) by (4). □

We would like to be able to regard BSpin_{7}(q) as a subcomplex of BSol(q), but there is no simple natural way to do so. Instead, we set

BSpin′_{7}(q) = |L_{Spin}^{cc}(q)|_{2}^{∧} ⊆ |L_{Sol}^{cc}(q)|_{2}^{∧} ⊆ BSol(q);

then BSpin′_{7}(q) ≃ BSpin_{7}(q)_{2}^{∧} by [BLO1, Proposition 1.1] and Lemma 4.1. Also, we write

BSol′(q) = |L_{Sol}^{cc}(q)|_{2}^{∧} ⊆ BSol(q) =^{def} |L_{Sol}^{c}(q)|_{2}^{∧}

to denote the subcomplex shown in Lemma 4.1 to be equivalent to BSol(q); and set BSpin′_{7}(q^{∞}) = |L_{Spin}^{c}(q^{∞})|_{2}^{∧}. From now on, when we talk about the inclusion of BSpin_{7}(q) into BSol(q), as long as it need only be well defined up to homotopy, we mean the composite

BSpin_{7}(q) ≃ BSpin′_{7}(q) ⊆ BSol′(q)

(for some choice of homotopy equivalence). Similarly, if we talk about the inclusion of BSol(q^{m}) into BSol(q^{n}) (for m|n) where it need only be defined up to homotopy, we mean these spaces identified with their equivalent subcomplexes BSol′(q^{m}) ⊆ BSol′(q^{n}).

Lemma 4.2. Let q be any odd prime. Then for all n ≥ 1,

\begin{aligned}H^{∗}(BSol(q^{n}); 𝔽_{2}) & \overline{→} & H^{∗}(BH(q^{n}); 𝔽_{2})^{C_{3}} \\ H^{∗}(BSpin_{7}(q^{n}); 𝔽_{2}) & ↓ & \overline{→} & H^{∗}(BH(q^{n}); 𝔽_{2}) & ↓ \end{aligned}  (1)

(with all maps induced by inclusions of groups or spaces) is a pullback square.

Proof. By [BLO2, Theorem B], H^{∗}(BSol(q^{n}); 𝔽_{2}) is the ring of elements in the cohomology of S(q^{n}) which are stable relative to the fusion. By the construction in Section 2, the fusion in Sol(q^{n}) is generated by that in Spin_{7}(q^{n}), together with the permutation action of C_{3} on the subgroup H(q^{n}) ≤ Spin_{7}(q^{n}), and hence (1) is a pullback square. □

Proposition 4.3. For each odd prime q, there is a category L_{Sol}^{c}(q^{∞}), together with a functor

π: L_{Sol}^{c}(q^{∞}) ⟶ F_{Sol}(q^{∞}),

such that the following hold:

(a) For each n ≥ 1, π^{−1}(F_{Sol}(q^{n})) ≅ L_{Sol}^{cc}(q^{n}).

(b) There is a homotopy equivalence

BSol(q^{∞}) =^{def} |L_{Sol}^{c}(q^{∞})|_{2}^{∧} ⟶_{≃}^{η} BDI(4)

such that the following square commutes up to homotopy

BSpin′_{7}(q^{∞})_{2}^{∧} \overline{^{δ(q^{∞})}→} BSol(q^{∞})

BSpin(7)_{2}^{∧} ↓^{η_{0} ≃}  \overline{δ̂ →} BDI(4) . ↓^{η ≃}

(1)

Here, η_{0} is the homotopy equivalence of [FM], induced by some fixed choice of embedding of the Witt vectors for 𝔽_{q} into ℂ, while δ(q^{∞}) is the union of the inclusions |L_{Spin}^{cc}(q^{n})|_{2}^{∧} ⊆ |L_{Sol}^{cc}(q^{n})|_{2}^{∧}, and δ̂ is the inclusion arising from the construction of BDI(4) in [DW1].

Furthermore, there is an automorphism ψ_{L}^{q} ∈ Aut(L_{Sol}^{c}(q^{∞})) of categories which satisfies the conditions:

(c) the restriction of ψ_{L}^{q} to each subcategory L_{Sol}^{cc}(q^{n}) is equal to the restriction of ψ_{L}^{q} ∈ Aut(L_{Sol}^{c}(q^{n})) as defined in Proposition 3.3(d);

(d) ψ_{L}^{q} covers the automorphism ψ_{F}^{q} of F_{Sol}(q^{∞}) induced by the field automorphism (x ↦ x^{q}); and

(e) for each n, (ψ_{L}^{q})^{n} fixes L_{Sol}^{cc}(q^{n}).

Proof. By Proposition 2.11, the inclusions Spin_{7}(q^{m}) ≤ Spin_{7}(q^{n}) for all m|n induce inclusions of fusion systems F_{Sol}(q^{m}) ⊆ F_{Sol}(q^{n}). Since the restriction of a linking system over F_{Sol}^{cc}(q^{n}) is a linking system over F_{Sol}^{cc}(q^{m}), the uniqueness of linking systems (Proposition 3.3) implies that we get inclusions L_{Sol}^{cc}(q^{m}) ⊆ L_{Sol}^{cc}(q^{n}). We define L_{Sol}^{c}(q^{∞}) to be the union of the finite categories L_{Sol}^{cc}(q^{n}). (More precisely, fix a sequence of positive integers n_{1}|n_{2}|n_{3}| ⋯ such that every positive integer divides some n_{i}, and set

L_{Sol}^{c}(q^{∞}) = ⋃_{i=1}^{∞} L_{Sol}^{cc}(q^{n_{i}}).

Then by uniqueness again, we can identify L_{Sol}^{cc}(q^{n}) for each n with the appropriate subcategory.)

Let π: L_{Sol}^{c}(q^{∞}) ⟶ F_{Sol}(q^{∞}) be the union of the projections from L_{Sol}^{cc}(q^{n_{i}}) to F_{Sol}(q^{n_{i}}) ⊆ F_{Sol}(q^{∞}). Condition (a) is clearly satisfied. Also, using Proposition 3.3(d) and Lemma 4.1, we see that there is an automorphism ψ_{L}^{q} of L_{Sol}^{c}(q^{∞}) which satisfies conditions (c,d,e) above. (Note that by the fusion theorem as shown in [BLO2, Theorem A.10], morphisms in L_{Sol}^{c}(q^{n}) are generated by those between radical subgroups, and hence by those in L_{Sol}^{cc}(q^{n}).)

It remains only to show that |L_{Sol}^{c}(q^{∞})|_{2}^{∧} ≃ BDI(4), and to show that square (1) commutes. The space BDI(4) is 2-complete by its construction in [DW1]. By Lemma 4.1,

H^{∗}(BSol(q^{∞}); 𝔽_{2}) ≅ \underleftarrow{lim}_{n} H^{∗}(|L_{Sol}^{c}(q^{n})|; 𝔽_{2}) = \underleftarrow{lim}_{n} H^{∗}(BSol(q^{n}); 𝔽_{2}).

Hence by Lemma 4.2 (and since the inclusions BSpin_{7}(q^{n}) ⟶ BSol(q^{n}) commute with the maps induced by inclusions of fields 𝔽_{q^{m}} ⊆ 𝔽_{q^{n}}), there is a pullback square

\begin{aligned}H^{∗}(BSol(q^{∞}); 𝔽_{2}) & \overline{→} & H^{∗}(BH(q^{∞}); 𝔽_{2})^{C_{3}} \\ H^{∗}(BSpin_{7}(q^{∞}); 𝔽_{2}) & ↓ & \overline{→} & H^{∗}(BH(q^{∞}); 𝔽_{2}) . & ↓ \end{aligned}  (2)

Also, by [FM, Theorem 1.4], there are maps

BSpin_{7}(q^{∞}) ⟶ BSpin(7) and BH(q^{∞}) ⟶ B(SU(2)^{3}/{±(I, I, I)})

which induce isomorphisms of 𝔽_{2}-cohomology, and hence homotopy equivalences after 2-completion. So by Propositions 4.7 and 4.9 (or more directly by the computations in [DW1, §3]), the pullback of the above square is the ring of Dickson invariants in the polynomial algebra H^{∗}(BC_{2}^{4}; 𝔽_{2}), and thus isomorphic to H^{∗}(BDI(4); 𝔽_{2}).

Point (b), including the commutativity of (1), now follows from the following lemma. □

Lemma 4.4. Let X be a 2-complete space such that H^{∗}(X; 𝔽_{2}) is the Dickson algebra in 4 variables. Assume further that there is a map BSpin(7) ⟶^{f} X such that H^{∗}(f|_{BC_{2}^{4}}; 𝔽_{2}) is the inclusion of the Dickson invariants in the polynomial algebra H^{∗}(BC_{2}^{4}; 𝔽_{2}). Then X ≃ BDI(4). More precisely, there is a homotopy equivalence between these spaces such that the composite

BSpin(7) ⟶^{f} X ≃ BDI(4)

is the inclusion arising from the construction in [DW1].

Proof. In fact, Notbohm [Nb, Theorem 1.2] has proven that the lemma holds even without the assumption about BSpin(7) (but with the more precise assumption that H^{∗}(X; 𝔽_{2}) is isomorphic as an algebra over the Steenrod algebra to the Dickson algebra). The result as stated above is much more elementary (and also implicit in [DW1]), so we sketch the proof here.

Since H^{∗}(X; 𝔽_{2}) is a polynomial algebra, H^{∗}(ΩX; 𝔽_{2}) is isomorphic as a graded vector space to an exterior algebra on the same number of variables, and in particular is finite. Hence X is a 2-compact group. By [DW3, Theorem 8.1] (the centralizer decomposition for a p-compact group), there is an 𝔽_{2}-homology equivalence

\underrightarrow{hocolim}_{A}(α) ⟶^{≃} X.

Here, A is the category of pairs (V, φ), where V is a nontrivial elementary abelian 2-group, and φ : BV ⟶ X makes H^{∗}(BV; 𝔽_{2}) into a finitely generated module over H^{∗}(X; 𝔽_{2}) (see [DW2, Proposition 9.11]). Morphisms in A are defined by letting Mor_{A}((V, φ), (V′, φ′)) be the set of monomorphisms V ⟶ V′ of groups which make the obvious triangle commute up to homotopy. Also,

α: A^{op} ⟶ Top is the functor α(V, φ) = Map(BV, X)_{φ}.

By [DW1, Lemma 1.6(1)] and [La, Théorème 0.4], A is equivalent to the category of elementary abelian 2-groups E with 1 ≤ rk(E) ≤ 4, whose morphisms consist of all group monomorphisms. Also, if BC_{2} ⟶^{φ} X is the restriction of f to any subgroup C_{2} ≤ Spin(7), then in the notation of Lannes,

T_{C_{2}}(H^{∗}(X; 𝔽_{2}); φ^{∗}) ≅ H^{∗}(BSpin(7); 𝔽_{2})

by [DW1, Lemmas 16.(3), 3.10 & 3.11], and hence

H^{∗}(Map(BC_{2}, X)_{φ}; 𝔽_{2}) ≅ H^{∗}(BSpin(7); 𝔽_{2})

by Lannes [La, Théorème 3.2.1]. This shows that

(Map(BC_{2}, X)_{φ})_{2}^{∧} ≃ BSpin(7)_{2}^{∧},

and thus that the centralizers of other elementary abelian 2-groups are the same as their centralizers in BSpin(7)_{2}^{∧}. In other words, α is equivalent in the homotopy category to the diagram used in [DW1] to define BDI(4). By [DW1, Proposition 7.7] (and the remarks in its proof), this homotopy functor has a unique homotopy lifting to spaces. So by definition of BDI(4),

X ≃ (\underrightarrow{hocolim}_{A}(α))_{2}^{∧} ≃ BDI(4).  □

Set Bψ^{q} =^{def} |ψ_{L}^{q}|, a self homotopy equivalence of BSol(q^{∞}) ≃ BDI(4). By construction, the restriction of Bψ^{q} to the maximal torus of BSol(q^{∞}) is the map induced by x ↦ x^{q}, and hence this is an “Adams map” as defined by Notbohm [Nb]. In fact, by [Nb, Theorem 3.5], there is an Adams map from BDI(4) to itself, unique up to homotopy, of degree any 2-adic unit.

Following Benson [Be], we define BDI_{4}(q) for any odd prime power q to be the homotopy fixed point set of the ℤ-action on BSol(q^{∞}) ≃ BDI(4) induced by the Adams map Bψ^{q}. By “homotopy fixed point set” in this situation, we mean that the following square is a homotopy pullback:

BDI_{4}(q)  \overline{→} BSol(q^{∞})

BSol(q^{∞}) ↓  \overline{→^{(Id,Bψ^{q})}} BSol(q^{∞}) × BSol(q^{∞}). ↓^{Δ}

The actual pullback of this square is the subspace BSol(q) of elements fixed by Bψ^{q}, and we thus have a natural map BSol(q) ⟶^{δ_{0}} BDI_{4}(q).

Theorem 4.5. For any odd prime power q, the natural map

BSol(q) ⟶_{≃}^{δ_{0}} BDI_{4}(q)

is a homotopy equivalence.

Proof. Since BDI(4) is simply connected, the square used to define BDI_{4}(q) remains a homotopy pullback square after 2-completion by [BK, II.5.3]. Thus BDI_{4}(q) is 2complete. Also, BSol(q) =^{def} |L_{Sol}^{c}(q)|_{2}^{∧} is 2-complete since |L_{Sol}^{c}(q)| is 2-good [BLO2, Proposition 1.11], and hence it suffices to prove that the map between these spaces is an 𝔽_{2}-cohomology equivalence. By Lemma 4.2, this means showing that the following commutative square is a pullback square:

\begin{aligned}H^{∗}(BDI_{4}(q); 𝔽_{2}) & \overline{→} & H^{∗}(BH(q); 𝔽_{2})^{C_{3}} \\ H^{∗}(BSpin_{7}(q); 𝔽_{2}) & ↓ & \overline{→} & H^{∗}(BH(q); 𝔽_{2}) . & ↓ \end{aligned}  (1)

Here, the maps are induced by the composite

BSpin_{7}(q) ≃ BSpin′_{7}(q)_{2}^{∧} ⊆ BSol(q) ⟶ BDI_{4}(q)

and its restriction to BH(q). Also, by Proposition 4.3(b), the following diagram commutes up to homotopy:

BSpin_{7}(q) \overline{→^{incl}} BSpin_{7}(q^{∞}) \overline{^{η_{0}}→} BSpin(7)

BSol(q) ↓^{δ(q)}  \overline{^{incl}→} BSol(q^{∞}) ↓^{δ(q^{∞})}  \overline{^{η} →} BDI(4) δ̂ ; ↓

(2)

By [Fr, Theorem 12.2], together with [FM, §1], for any connected reductive Lie group 𝔾 and any algebraic epimorphism ψ on 𝔾(𝔽_{q}) with finite fixed subgroup, there is a homotopy pullback square

B(𝔾(𝔽_{q})^{ψ})_{2}^{∧}  \overline{^{incl} →} B𝔾(𝔽_{q})_{2}^{∧}

B𝔾(𝔽_{q})_{2}^{∧} ↓^{incl}  \overline{→^{(Id,Bψ)}} B𝔾(𝔽_{q})_{2}^{∧} × B𝔾(𝔽_{q})_{2}^{∧}. ↓^{Δ}

(3)

We need to apply this when 𝔾 = Spin_{7} or 𝔾 = H = (SL_{2})^{3}/{±(I, I, I)}. In particular, if ψ = ψ^{q} is the automorphism induced by the field automorphism (x ↦ x^{q}), then Spin_{7}(𝔽_{q})^{ψ} = Spin_{7}(q) by Lemma A.3, and H(𝔽_{q})^{ψ} = H(q) =^{def} H(𝔽_{q}) ∩ Spin_{7}(q). We thus get a description of BSpin_{7}(q) and BH(q) as homotopy pullbacks.

By [FM, Theorem 1.4], B𝔾(𝔽_{q})_{2}^{∧} ≃ B𝔾(ℂ)_{2}^{∧}. Also, we can replace the complex Lie groups Spin_{7}(ℂ) and H(ℂ) by maximal compact subgroups Spin(7) and H =^{def} SU(2)^{3}/{±(I, I, I)}, since these have the same homotopy type.

If we set ℜ = H^{∗}(B𝔾(𝔽_{q}); 𝔽_{2}) ≅ H^{∗}(B𝔾(ℂ); 𝔽_{2}), then there are Eilenberg-Moore spectral sequences

E_{2} = Tor_{ℜ⊗ℜ^{op}}^{∗}(ℜ, ℜ) ⟹ H^{∗}(B(𝔾(𝔽_{q})^{ψ}); 𝔽_{2});

where the (ℜ⊗ℜ^{op})-module structure on ℜ is defined by setting (a⊗b)·x = a·x·Bψ(b). When 𝔾 = Spin_{7} or H, then ℜ is a polynomial algebra by Proposition 4.7 and the above remarks, and Bψ acts on ℜ via the identity. The above spectral sequence thus satisfies the hypotheses of [Sm, Theorem II.3.1], and hence collapses. (Alternatively, note that in this case, E_{2} is generated multiplicatively by E_{2}^{0,∗} and E_{2}^{−1,∗} by (5) below.) Similarly, when ℜ = H^{∗}(BDI(4); 𝔽_{2}), there is an analogous spectral sequence which converges to H^{∗}(BDI_{4}(q); 𝔽_{2}), and which collapses for the same reason. By the above remarks, these spectral sequences are natural with respect to the inclusions BH(−) ⊆ BSpin_{7}(−), and (using the naturality of ψ^{q} shown in Proposition 3.3(d)) of BSpin_{7}(−) into BSol(−) or BDI(4).

To simplify the notation, we now write

𝔄 =^{def} H^{∗}(BDI(4); 𝔽_{2}), 𝔅 =^{def} H^{∗}(BSpin(7); 𝔽_{2}), and ℭ =^{def} H^{∗}(H; 𝔽_{2})

to denote these cohomology rings. The Frobenius automorphism ψ^{q} acts via the identity on each of them. We claim that the square

Tor_{𝔄⊗𝔄^{op}}^{∗}(𝔄, 𝔄) \overline{→} Tor_{ℭ⊗ℭ^{op}}^{∗}(ℭ, ℭ)^{C_{3}}

Tor_{𝔅⊗𝔅^{op}}^{∗}(𝔅, 𝔅) ↓  \overline{→} Tor_{ℭ⊗ℭ^{op}}^{∗}(ℭ, ℭ) ↓

(4) is a pullback square. Once this has been shown, it then follows that in each degree, square (1) has a finite filtration under which each quotient is a pullback square. Hence (1) itself is a pullback.

For any commutative 𝔽_{2}-algebra ℜ, let Ω_{ℜ/𝔽_{2}} denote the ℜ-module generated by elements dr for r ∈ ℜ with the relations dr = 0 if r ∈ 𝔽_{2},

d(r + s) = dr + ds and d(rs) = r·ds + s·dr.

Let Ω_{ℜ/𝔽_{2}}^{∗} denote the ring of Kähler differentials: the exterior algebra (over ℜ) of Ω_{ℜ/𝔽_{2}} = Ω_{ℜ/𝔽_{2}}^{1}. When ℜ is a polynomial algebra, there are natural identifications

Tor_{ℜ⊗ℜ^{op}}^{∗}(ℜ, ℜ) ≅ HH_{∗}(ℜ; ℜ) ≅ Ω_{ℜ/𝔽_{2}}^{∗}.  (5)

The first isomorphism holds for arbitrary algebras, and is shown, e.g., in [Wb, Lemma 9.1.3]. The second holds for smooth algebras over a field [Wb, Theorem 9.4.7] (and polynomial algebras are smooth as shown in [Wb, §9.3.1]). In particular, the isomorphisms (5) hold for ℜ = 𝔄, 𝔅, ℭ, which are shown to be polynomial algebras in Proposition 4.7 below. Thus, square (4) is isomorphic to the square

\begin{aligned}Ω_{𝔄/𝔽_{2}}^{∗} & → & (Ω_{ℭ/𝔽_{2}}^{∗})^{C_{3}} \\ Ω_{𝔅/𝔽_{2}}^{∗} & ↓ & \overline{→} & Ω_{ℭ/𝔽_{2}}^{∗} , & ↓ \end{aligned}  (6)

which is shown to be a pullback square in Propositions 4.7 and 4.9 below. □

It remains to prove that square (6) in the above proof is a pullback square. In what follows, we let D_{i}(x_{1}, …, x_{n}) denote the i-th Dickson invariant in variables x_{1}, …, x_{n}. This is the (2^{n}−2^{n−i})-th symmetric polynomial in the elements (equivalently in the nonzero elements) of the 𝔽_{2}-vector space ⟨x_{1}, …, x_{n}⟩_{𝔽_{2}}. We refer to [Wi] for more detail. Note that what he denotes c_{n,i} is what we call D_{n−i}(x_{1}, …, x_{n}).

Lemma 4.6. For any n,

D_{1}(x_{1}, …, x_{n+1}) = ∏_{x∈⟨x_{1},...,x_{n}⟩_{𝔽_{2}}}(x_{n+1} + x) + D_{1}(x_{1}, …, x_{n})^{2}

= x_{n+1}^{2^{n}} + ∑_{i=1}^{n} x_{n+1}^{2^{n−i}}D_{i}(x_{1}, …, x_{n}) + D_{1}(x_{1}, …, x_{n})^{2}.

Proof. The first equality is shown in [Wi, Proposition 1.3(b)]; here we prove them both simultaneously. Set V_{n} = ⟨x_{1}, …, x_{n}⟩_{𝔽_{2}}. Since σ_{i}(V_{n}) = 0 whenever 2^{n} − i is not a power of 2 (cf. [Wi, Proposition 1.1]),

D_{1}(x_{1}, …, x_{n+1}) = ∑_{i=0}^{2^{n}} σ_{i}(V_{n})·σ_{2^{n}−i}(x_{n+1} + V_{n})

= ∏_{x∈V_{n}}(x_{n+1} + x) + ∑_{i=1}^{n} D_{i}(x_{1}, …, x_{n})·σ_{2^{n−i}}(x_{n+1} + V_{n}).

Also, since σ_{i}(V_{n}) = 0 for 0 < i < 2^{n−1} as noted above,

σ_{k}(x_{n+1} + V_{n}) = ∑_{i=0}^{k} x_{n+1}^{k−i}·( \begin{matrix}2^{n}−i \\ k−i \end{matrix} )σ_{i}(V_{n}) = \begin{cases}0 & if 0 < k < 2^{n−1} \\ D_{1}(x_{1}, …, x_{n}) & if k = 2^{n−1}. \end{cases}

This proves the first equality, and the second follows since

∏_{x∈V_{n}}(x_{n+1} + x) = x_{n+1}^{2^{n}} + ∑_{i=1}^{2^{n}} x_{n+1}^{2^{n}−i}σ_{i}(V_{n}) = x_{n+1}^{2^{n}} + ∑_{i=1}^{n} x_{n+1}^{2^{n−i}}D_{i}(x_{1}, …, x_{n}). □

In the following proposition (and throughout the rest of the section), we work with the polynomial ring 𝔽_{2}[x, y, z, w], with the natural action of GL_{4}(𝔽_{2}). Let

GL_{2}^{2}(𝔽_{2}), GL_{1}^{3}(𝔽_{2}) ≤ GL_{4}(𝔽_{2})

be the subgroups of automorphisms of V =^{def} ⟨x, y, z, w⟩_{𝔽_{2}} which leave invariant the subspaces ⟨x, y⟩ and ⟨x, y, z⟩, respectively. Also, let GL_{2′}^{2}(𝔽_{2}) ≤ GL_{2}^{2}(𝔽_{2}) be the subgroup of automorphisms which are the identity modulo ⟨x, y⟩. Thus, when described in terms of block matrices (with respect to the given basis {x, y, z, w}),

GL_{1}^{3}(𝔽_{2}) = \{( \begin{matrix}A & X \\ 0 & 1 \end{matrix} )\} ,  GL_{2}^{2}(𝔽_{2}) = \{( \begin{matrix}B & Y \\ 0 & C \end{matrix} )\} ,  and GL_{2′}^{2}(𝔽_{2}) = \{( \begin{matrix}B & Y \\ 0 & I \end{matrix} )\} ,

for A ∈ GL_{3}(𝔽_{2}), X a column vector, B, C ∈ GL_{2}(𝔽_{2}), and Y ∈ M_{2}(𝔽_{2}).

We need to make more precise the relation between V (or the polynomial ring 𝔽_{2}[x, y, z, w]) and the cohomology of Spin(7). To do this, let W ≤ Spin(7) be the inverse image of the elementary abelian subgroup

⟨ diag(−1, −1, −1, −1, 1, 1, 1), diag(−1, −1, 1, 1, −1, −1, 1), diag(−1, 1, −1, 1, −1, 1, −1)⟩ ≤ SO(7).

Thus, W ≅ C_{2}^{4}. Fix a basis {η, η′, ξ, ζ} for W, where ζ ∈ Z(Spin(7)) is the nontrivial element. Identify V = W^{∗} in such a way that {x, y, z, w} ⊆ V is the dual basis to {η, η′, ξ, ζ}. This gives an identification

H^{∗}(BW; 𝔽_{2}) = 𝔽_{2}[x, y, z, w],

arranged such that the action of N_{Spin(7)}(W)/W on V = ⟨x, y, z, w⟩ consists of all automorphisms which leave ⟨x, y, z⟩ invariant, and thus can be identified with the action of GL_{1}^{3}(𝔽_{2}). Finally, set

H = C_{Spin(7)}(ξ) ≅ Spin(4) ×_{C_{2}} Spin(3) ≅ SU(2)^{3}/{±(I, I, I)}

(the central product). Then in the same way, the action of N_{H}(W)/W on H^{∗}(BW; 𝔽_{2}) can be identified with that of GL_{2′}^{2}(𝔽_{2}).

Proposition 4.7. The inclusions

BW ⟶ BH ⟶ BSpin(7) ⟶ BDI(4)

as defined above, together with the identification H^{∗}(BW; 𝔽_{2}) = 𝔽_{2}[x, y, z, w], induce isomorphisms

𝔄 =^{def} H^{∗}(BDI(4); 𝔽_{2}) = 𝔽_{2}[x, y, z, w]^{GL_{4}(𝔽_{2})} = 𝔽_{2}[a_{8}, a_{12}, a_{14}, a_{15}]
𝔅 =^{def} H^{∗}(BSpin(7); 𝔽_{2}) = 𝔽_{2}[x, y, z, w]^{GL_{1}^{3}(𝔽_{2})} = 𝔽_{2}[b_{4}, b_{6}, b_{7}, b_{8}]
ℭ =^{def} H^{∗}(BH; 𝔽_{2}) = 𝔽_{2}[x, y, z, w]^{GL_{2′}^{2}(𝔽_{2})} = 𝔽_{2}[c_{2}, c_{3}, c′_{4}, c′′_{4}] ;

(∗)

where

a_{8} = D_{1}(x, y, z, w), a_{12} = D_{2}(x, y, z, w), a_{14} = D_{3}(x, y, z, w), a_{15} = D_{4}(x, y, z, w);

b_{4} = D_{1}(x, y, z), b_{6} = D_{2}(x, y, z), b_{7} = D_{3}(x, y, z), b_{8} = ∏_{α∈⟨x,y,z⟩}(w + α);

and

c_{2} = D_{1}(x, y), c_{3} = D_{2}(x, y), c′_{4} = ∏_{α∈⟨x,y⟩}(z + α), c′′_{4} = ∏_{α∈⟨x,y⟩}(w + α).

Furthermore,

(a) the natural action of Σ_{3} on H ≅ SU(2)^{3}/{±(I, I, I)} induces the action on ℭ which fixes c_{2}, c_{3} and permutes {c′_{4}, c′′_{4}, c′_{4} + c′′_{4}}; and

(b) the above variables satisfy the relations

a_{8} = b_{8} + b_{4}^{2}  a_{12} = b_{8}b_{4} + b_{6}^{2}  a_{14} = b_{8}b_{6} + b_{7}^{2}  a_{15} = b_{8}b_{7}
b_{4} = c′_{4} + c_{2}^{2}  b_{6} = c_{2}c′_{4} + c_{3}^{2}  b_{7} = c_{3}c′_{4}  b_{8} = c′′_{4}(c′_{4} + c′′_{4}) .

Proof. The formulas for 𝔄 = H^{∗}(BDI(4); 𝔽_{2}) are shown in [DW1]. From [DW1, Lemmas 3.10 & 3.11], we see there are (some) identifications

H^{∗}(BSpin(7); 𝔽_{2}) ≅ 𝔽_{2}[x, y, z, w]^{GL_{1}^{3}(𝔽_{2})} and H^{∗}(BH; 𝔽_{2}) ≅ 𝔽_{2}[x, y, z, w]^{GL_{2′}^{2}(𝔽_{2})}.

From the explicit choices of subgroups W ≤ H ≤ Spin(7) as described above (and by the descriptions in Proposition A.8 of the automorphism groups), the images of H^{∗}(BSpin(7); 𝔽_{2}) and H^{∗}(BH; 𝔽_{2}) in 𝔽_{2}[x, y, z, w] are seen to be contained in the rings of invariants, and hence these isomorphisms actually are equalities as claimed.

We next prove the equalities in (∗) between the given rings of invariants and polynomial algebras. The following argument was shown to us by Larry Smith. If k is a field and V is an n-dimensional vector space over k, then a system of parameters in the polynomial algebra k[V] is a set of n homogeneous elements f_{1}, …, f_{n} such that k[V]/(f_{1}, …, f_{n}) is finite dimensional over k. By [Sm2, Proposition 5.5.5], if V is an n-dimensional k[G]-representation, and f_{1}, …, f_{n} ∈ k[V]^{G} is a system of parameters the product of whose degrees is equal to |G|, then k[V]^{G} is a polynomial algebra with f_{1}, …, f_{n} as generators. By [Sm2, Proposition 8.1.7], 𝔽_{2}[x, y, z, w] is a free finitely generated module over the ring generated by its Dickson invariants (this holds for polynomial algebras over any 𝔽_{p}), and thus 𝔽_{2}[x, y, z, w]/(a_{8}, a_{12}, a_{14}, a_{15}) is finite. (This can also be shown directly using the relation in Lemma 4.6.) So assuming the relations in point (b), the quotients 𝔽_{2}[x, y, z, w]/(b_{4}, b_{6}, b_{7}, b_{8}) and 𝔽_{2}[x, y, z, w]/(c_{2}, c_{3}, c′_{4}, c′′_{4}) are also finite. In each case, the product of the degrees of the generators is clearly equal to the order of the group in question, and this finishes the proof of the last equality in the second and third lines of (∗).

It remains to prove points (a) and (b). Using Lemma 4.6, the c_{i} are expressed as polynomials in x, y, z, w as follows:

\begin{aligned}c_{2} & = D_{1}(x, y) = x^{2} + xy + y^{2} \\ c_{3} & = D_{2}(x, y) = xy(x + y) \\ c′_{4} & = D_{1}(x, y, z) + D_{1}(x, y)^{2} = z^{4} + z^{2}D_{1}(x, y) + zD_{2}(x, y) = z^{4} + z^{2}c_{2} + zc_{3} \\ c′′_{4} & = D_{1}(x, y, w) + D_{1}(x, y)^{2} = w^{4} + w^{2}D_{1}(x, y) + wD_{2}(x, y) = w^{4} + w^{2}c_{2} + wc_{3} . \end{aligned}(1)

In particular,

c′_{4} + c′′_{4} = (z + w)^{4} + (z + w)^{2}D_{1}(x, y) + (z + w)D_{2}(x, y) = ∏_{α∈⟨x,y⟩}(z + w + α). (2)

Furthermore, by (1), we get

\begin{aligned}Sq^{1}(c_{2}) & = c_{3} \\ Sq^{1}(c_{3}) & = Sq^{1}(c′_{4}) = Sq^{1}(c′′_{4}) = 0 \\ Sq^{2}(c_{3}) & = x^{2}y^{2}(x + y) + xy(x + y)^{3} = c_{2}c_{3} \\ Sq^{2}(c′_{4}) & = z^{4}c_{2} + z^{2}c_{2}^{2} + zc_{2}c_{3} = c_{2}c′_{4} \\ Sq^{3}(c′_{4}) & = Sq^{1}(c_{2}c′_{4}) = c_{3}c′_{4} \\ Sq^{2}(c′′_{4}) & = c_{2}c′′_{4} \\ Sq^{3}(c′′_{4}) & = c_{3}c′′_{4}. \end{aligned}  (3)

The permutation action of Σ_{3} on H ≅ SU(2)^{3}/{±(I, I, I)} permutes the three elements ζ, ξ, ζ + ξ of Z(H) ⊆ W, and thus (via the identification V = W^{∗} described above) induces the identity on x, y ∈ V and permutes the elements {z, w, z + w} modulo ⟨x, y⟩. Hence the induced action of Σ_{3} on ℭ = 𝔽_{2}[V]^{GL_{2′}^{2}(𝔽_{2})} is the restriction of the action on 𝔽_{2}[V] = 𝔽_{2}[x, y, z, w] which fixes x, y and permutes {z, w, z + w}. So by (1) and (2), we see that this action fixes c_{2}, c_{3} and permutes the set {c′_{4}, c′′_{4}, c′_{4} + c′′_{4}}. This proves (a).

It remains to prove the formulas in (b). From (1) and (3) we get

b_{4} = D_{1}(x, y, z) = c′_{4} + c_{2}^{2},
b_{6} = D_{2}(x, y, z) = Sq^{2}(b_{4}) = c_{2}c′_{4} + c_{3}^{2},
b_{7} = D_{3}(x, y, z) = Sq^{1}(b_{6}) = c_{3}c′_{4}.

Also, by (1) and (2),

b_{8} = ∏_{α∈⟨x,y,z⟩}(w + α) = ( ∏_{α∈⟨x,y⟩}(w + α))·( ∏_{α∈⟨x,y⟩}(w + z + α)) = c′′_{4}(c′_{4} + c′′_{4}).

This proves the formulas for the b_{i} in terms of c_{i}. Finally, we have

a_{8} = D_{1}(x, y, z, w) = b_{8} + b_{4}^{2},
a_{12} = D_{2}(x, y, z, w) = Sq^{4}(b_{8} + b_{4}^{2}) = Sq^{4}(c′′_{4}(c′_{4} + c′′_{4}) + (c′_{4} + c_{2}^{2})^{2})
= c′_{4}c′′_{4}(c′_{4} + c′′_{4}) + c_{2}^{2}c′′_{4}(c′_{4} + c′′_{4}) + c_{2}^{2}c′_{4}^{2} + c_{3}^{4} = b_{8}b_{4} + b_{6}^{2}
a_{14} = D_{3}(x, y, z, w) = Sq^{2}(a_{12}) = c_{2}c′_{4}c′′_{4}(c′_{4} + c′′_{4}) + c_{3}^{2}c′′_{4}(c′_{4} + c′′_{4}) + c_{3}^{2}c′_{4}^{2}
= b_{8}b_{6} + b_{7}^{2}
a_{15} = D_{4}(x, y, z, w) = Sq^{1}(a_{14}) = c_{3}c′_{4}c′′_{4}(c′_{4} + c′′_{4}) = b_{8}b_{7} ;

and this finishes the proof of the proposition. □

Lemma 4.8. Let κ ∈ Aut(ℭ) be the algebra involution which exchanges c′_{4} and c′′_{4} and leaves c_{2} and c_{3} fixed. An element of ℭ will be called “κ-invariant” if it is fixed by this involution. Then the following hold:

(a) If β ∈ 𝔅 is κ-invariant, then β ∈ 𝔄.

(b) If β ∈ 𝔅 is such that β·c′_{4}^{i} is κ-invariant, then β = β′·b_{8}^{i} for some β′ ∈ 𝔄.

Proof. Point (a) follows from Proposition 4.7 upon regarding 𝔄, 𝔅, and ℭ as the fixed subrings of the groups GL_{4}(𝔽_{2}), GL_{1}^{3}(𝔽_{2}) and GL_{2′}^{2}(𝔽_{2}) acting on 𝔽_{2}[x, y, z, w], but also follows from the following direct argument. Let m be the degree of β as a polynomial in b_{8}; we argue by induction on m. Write β = β_{0} + b_{8}^{m}·β_{1}, where β_{1} ∈ 𝔽_{2}[b_{4}, b_{6}, b_{7}], and where β_{0} has degree < m (as a polynomial in b_{8}). If m = 0, then β = β_{1} ∈ 𝔽_{2}[b_{4}, b_{6}, b_{7}] ⊆ 𝔽_{2}[c_{2}, c_{3}, c′_{4}], and hence β ∈ 𝔽_{2}[c_{2}, c_{3}] since it is κ-invariant. But from the formulas in Proposition 4.7(b), we see that 𝔽_{2}[b_{4}, b_{6}, b_{7}] ∩ 𝔽_{2}[c_{2}, c_{3}] contains only constant polynomials (hence it is contained in 𝔄).

Now assume m ≥ 1. Then, expressed as a polynomial in c_{2}, c_{3}, c′_{4}, c′′_{4}, the largest power of c′′_{4} which occurs in β is c′′_{4}^{2m}. Since β is κ-invariant, the highest power of c′_{4} which occurs is c′_{4}^{2m}; and hence by Proposition 4.7(b), the total degree of each term in β_{1} (its degree as a polynomial in b_{4}, b_{6}, b_{7}) is at most m. So for each term b_{4}^{r}b_{6}^{s}b_{7}^{t} in β_{1},

b_{4}^{r}b_{6}^{s}b_{7}^{t}b_{8}^{m} − a_{8}^{m−r−s−t}a_{12}^{r}a_{14}^{s}a_{15}^{t}

is a sum of terms which have degree < m in b_{8}, and thus lies in 𝔄 by the induction hypothesis.

To prove (b), note first that since β·c′_{4}^{i} is κ-invariant and divisible by c′_{4}^{i}, it must also be divisible by c′′_{4}^{i}, and hence c′′_{4}^{i}|β. Furthermore, by Proposition 4.7, all elements of 𝔅 as well as c′_{4} are invariant under the involution which fixes c′_{4} and sends c′′_{4} ↦ c′_{4} + c′′_{4}. Thus (c′_{4} + c′′_{4})^{i}|β. Since b_{8} = c′′_{4}(c′_{4} + c′′_{4}), we can now write β = β′·b_{8}^{i} for some β ∈ 𝔅. Finally, since

β·c′_{4}^{i} = β′·c′_{4}^{i}·c′′_{4}^{i}·(c′_{4} + c′′_{4})^{i}

is κ-invariant, β′ is also κ-invariant, and hence β′ ∈ 𝔄 by (a). □

Note that C_{3} ≤ Σ_{3} = GL_{2}(𝔽_{2}) act on ℭ = 𝔽_{2}[x, y, z, w]^{GL_{2′}^{2}(𝔽_{2})}: via the action of GL_{2}^{2}(𝔽_{2})/GL_{2′}^{2}(𝔽_{2}), or equivalently by permuting c′_{4}, c′′_{4}, and c′_{4} + c′′_{4} (and fixing c_{2}, c_{3}). Thus 𝔄 = 𝔅∩ℭ^{C_{3}}, since GL_{4}(𝔽_{2}) is generated by the subgroups GL_{1}^{3}(𝔽_{2}) and GL_{2}^{2}(𝔽_{2}). This is also shown directly in the following lemma.

Proposition 4.9. The following square is a pullback square, where all maps are induced by inclusions between the subrings of 𝔽_{2}[x, y, z, w]:

\begin{aligned}Ω_{𝔄/𝔽_{2}}^{∗} & \overline{→} & (Ω_{ℭ/𝔽_{2}}^{∗})^{C_{3}} \\ Ω_{𝔅/𝔽_{2}}^{∗} & ↓ & \overline{→} & Ω_{ℭ/𝔽_{2}}^{∗} . & ↓ \end{aligned}

Proof. Let κ be the involution of Lemma 4.8: the algebra involution of ℭ which exchanges c′_{4} and c′′_{4} and leaves c_{2} and c_{3} fixed. By construction, all elements in the image of Ω_{𝔅/𝔽_{2}}^{∗} are invariant under the involution which fixes c′_{4} (and c_{2}, c_{3}), and sends c′′_{4} to c′_{4} + c′′_{4}. Hence elements in the image of Ω_{𝔅/𝔽_{2}}^{∗} are fixed by C_{3} if and only if they are fixed by Σ_{3}, if and only if they are κ-invariant. So it will suffice to show that all of the above maps are injective, and that all κ-invariant elements in the image of Ω_{𝔅/𝔽_{2}}^{∗} lie in the image of Ω_{𝔄/𝔽_{2}}^{∗}. The injectivity is clear, and the square is a pullback for Ω_{−/𝔽_{2}}^{0} by Lemma 4.8.

Fix a κ-invariant element

\begin{aligned}ω & = P_{1} db_{4} + P_{2} db_{6} + P_{3} db_{7} + P_{4} db_{8} \\ & = P_{2}c′_{4} dc_{2} + P_{3}c′_{4} dc_{3} + P_{4}c′_{4} dc′′_{4} + (P_{1} + P_{2}c_{2} + P_{3}c_{3} + P_{4}c′′_{4}) dc′_{4} ∈ Ω_{𝔅/𝔽_{2}}^{1}, \end{aligned} (1)

where P_{i} ∈ 𝔅 for each i. By applying κ to (1) and comparing the coefficients of dc_{2} and dc_{3}, we see that P_{2}c′_{4} and P_{3}c′_{4} are κ-invariant. Also, upon comparing the coefficients of dc′_{4}, we get the equation

P_{1} + P_{2}c_{2} + P_{3}c_{3} + P_{4}c′′_{4} = κ(P_{4})c′′_{4}.  (2)

So by Lemma 4.8, P_{2} = P′_{2}b_{8} and P_{3} = P′_{3}b_{8} for some P′_{2}, P′_{3} ∈ 𝔄. Upon subtracting

P′_{2} da_{14} + P′_{3} da_{15} = P_{2} db_{6} + P_{3} db_{7} + (P′_{2}b_{6} + P′_{3}b_{7}) db_{8}

from ω and introducing an appropriate modification to P_{4}, we can now assume that P_{2} = P_{3} = 0. With this assumption and (2), we have

P_{1} + P_{4}c′′_{4} = κ(P_{4}c′_{4}) = κ(P_{4})·c′′_{4},

so that

P_{1}c′_{4} = (P_{4} + κ(P_{4}))c′_{4}c′′_{4}  (3)

is κ-invariant. This now shows that P_{1} = P′_{1}b_{8} for some P′_{1} ∈ 𝔄, and upon subtracting P′_{1} da_{12} from ω we can assume that P_{1} = 0. This leaves ω = P_{4} db_{8} = P_{4} da_{8}. By (3) again, P_{4} is κ-invariant, so P_{4} ∈ 𝔄 by Lemma 4.8 again, and thus ω ∈ Ω_{𝔄/𝔽_{2}}^{1}.

The remaining cases are proved using the same techniques, and so we sketch them more briefly. To prove the result in degree two, fix a κ-invariant element

ω = P_{1} db_{4} db_{6} + P_{2} db_{4} db_{7} + P_{3} db_{4} db_{8} + P_{4} db_{6} db_{7} + P_{5} db_{6} db_{8} + P_{6} db_{7} db_{8}
= P_{4}c′_{4}^{2} dc_{2} dc_{3} + (P_{1}c′_{4} + P_{4}c_{3}c′_{4} + P_{5}c′_{4}c′′_{4}) dc_{2} dc′_{4} + P_{5}c′_{4}^{2} dc_{2} dc′′_{4}
+ (P_{2}c′_{4} + P_{4}c_{2}c′_{4} + P_{6}c′_{4}c′′_{4}) dc_{3} dc′_{4} + P_{6}c′_{4}^{2} dc_{3} dc′′_{4}
+ (P_{3}c′_{4} + P_{5}c_{2}c′_{4} + P_{6}c_{3}c′_{4}) dc′_{4} dc′′_{4} ∈ Ω_{𝔅/𝔽_{2}}^{2}.

Using Lemma 4.8, we see that P_{4} = P′_{4}b_{8}^{2}, and hence can assume that P_{4} = 0. One then eliminates P_{1} and P_{2}, then P_{5} and P_{6}, and finally P_{3}.

If

ω = P_{1} db_{4} db_{6} db_{7} + P_{2} db_{4} db_{6} db_{8} + P_{3} db_{4} db_{7} db_{8} + P_{4} db_{6} db_{7} db_{8}
= (P_{1}c′_{4}^{2} + P_{4}c′_{4}^{2}c′′_{4}) dc_{2} dc_{3} dc′_{4} + (P_{2}c′_{4}^{2} + P_{4}c_{3}c′_{4}^{2}) dc_{2} dc′_{4} dc′′_{4}
+ (P_{3}c′_{4}^{2} + P_{4}c_{2}c′_{4}^{2}) dc_{3} dc′_{4} dc′′_{4} + P_{4}c′_{4}^{3} dc_{2} dc_{3} dc′′_{4} ∈ Ω_{𝔅/𝔽_{2}}^{3}

is κ-invariant, then we eliminate successively P_{1}, then P_{4}, then P_{2} and P_{3}.

Finally, if

ω = P db_{4} db_{6} db_{7} db_{8} = P c′_{4}^{3} dc_{2} dc_{3} dc′_{4} dc′′_{4} ∈ Ω_{𝔅/𝔽_{2}}^{4}

is κ-invariant, then P = P′b_{8}^{3} for some P′ ∈ 𝔄 by Lemma 4.8 again, and so ω = P′ da_{8} da_{12} da_{14} da_{15} ∈ Ω_{𝔄/𝔽_{2}}^{4}. □

Appendix

Appendix A. Spinor groups over finite fields

Let F be any field of characteristic ≠ 2. Let V be a vector space over F, and let 𝔟: V ⟶ F be a nonsingular quadratic form. As usual, O(V, 𝔟) denotes the group of isometries of (V, 𝔟), and SO(V, 𝔟) the subgroup of isometries of determinant 1. We will be particularly interested in elementary abelian 2-subgroups of such orthogonal groups.

Lemma A.1. Fix an elementary abelian 2-subgroup E ≤ O(V, 𝔟). For each irreducible character χ ∈ Hom(E, {±1}), let V_{χ} ⊆ V denote the corresponding eigenspace: the subspace of elements v ∈ V such that g(v) = χ(g)·v for all g ∈ E. Then the restriction of 𝔟 to each subspace V_{χ} is nonsingular, and V is the orthogonal direct sum of the V_{χ}.

Proof. Elementary. □

We give a very brief sketch of the definition of spinor groups via Clifford algebras; for more details we refer to [Di, §II.7] or [As2, §22]. Let T (V) denote the tensor algebra of V, and set

C(V, 𝔟) = T (V)/⟨(v ⊗ v) − 𝔟(v)⟩ :

the Clifford algebra of (V, 𝔟). To simplify the notation, we regard F as a subring of C(V, 𝔟), and V as a subgroup of its additive group; thus the class of v_{1} ⊗ ⋯ ⊗ v_{k} will be written v_{1}⋯v_{k}. Note that vw + wv = 0 if v, w ∈ V and v ⊥ w. Hence if dim_{F} (V) = n, and {v_{1}, …, v_{n}} is an orthogonal basis, then the set of 1 and all v_{i_{1}} ⋯ v_{i_{k}} for i_{1} < ⋯ < i_{k} (1 ≤ k ≤ n) is an F -basis for C(V, 𝔟).

Write C(V, 𝔟) = C_{0} ⊕ C_{1}, where C_{0} and C_{1} consist of classes of elements of even or odd degree, respectively. Let G ≤ C(V, 𝔟)^{∗} denote the group of invertible elements u such that uV u^{−1} = V, and let π: G ⟶ O(V, 𝔟) be the homomorphism

π(u) = \begin{cases}(v ↦ −uvu^{−1}) & if u ∈ C_{1} \\ (v ↦ uvu^{−1}) & if u ∈ C_{0} . \end{cases}

In particular, for any nonisotropic element v ∈ V (i.e., 𝔟(v) ≠ 0), v ∈ G and π(v) is the reflection in the hyperplane v^{⊥}. By [Di, §II.7], π is surjective and Ker(π) = F^{∗}.

Let J be the antiautomorphism of C(V, 𝔟) induced by the antiautomorphism v_{1} ⊗ ⋯ ⊗ v_{k} ↦ v_{k} ⊗ ⋯ ⊗ v_{1} of T (V). Since O(V, 𝔟) is generated by hyperplane reflections, G is generated by F^{∗} and nonisotropic elements v ∈ V. In particular, for any u = λ·v_{1} ⋯ v_{k} ∈ G,

J(u)·u = λ^{2} · v_{k} ⋯ v_{1} · v_{1} ⋯ v_{k} = λ^{2} · 𝔟(v_{1}) ⋯ 𝔟(v_{k}) ∈ F^{∗} = Ker(π);

implying that π(J(u)) = π(u)^{−1} for all u ∈ G. There is thus a homomorphism

θ̃: G ⟶ F^{∗} defined by θ̃(u) = u·J(u).

In particular, θ̃(λ) = λ^{2} for λ ∈ F^{∗} ≤ G, while for any set of nonisotropic elements v_{1}, …, v_{k} of V,

θ̃(v_{1} ⋯ v_{k}) = (v_{1} ⋯ v_{k})(v_{k} ⋯ v_{1}) = 𝔟(v_{1}) ⋯ 𝔟(v_{k}).

Hence θ̃ factors through a homomorphism

θ_{V,𝔟}: O(V, 𝔟) ⟶ F^{∗}/F^{∗2} = F^{∗}/{u^{2} | u ∈ F^{∗}},

called the spinor norm.

Set G^{+} = π^{−1}(SO(V, 𝔟)) = G ∩ C_{0}, and define

Spin(V, 𝔟) = Ker(θ̃|_{G^{+}}) and Ω(V, 𝔟) = Ker(θ_{V,𝔟}|_{SO(V,𝔟)}).

In particular, Ω(V, 𝔟) has index 2 in SO(V, 𝔟) if F is a finite field, and Ω(V, 𝔟) = SO(V, 𝔟) if F is algebraically closed (all units are squares). We thus get a commutative diagram

(A.2) \xymatrix{ & 1 \ar[d] & 1 \ar[d] & 1 \ar[d] \\ 1 \ar[r] & \{±1\} \ar[r] \ar[d] & F^{∗} \ar[r]^{λ↦λ^{2}} \ar[d] & F^{∗2} \ar[d] \ar[r] & 1 \\ 1 \ar[r] & Spin(V, 𝔟) \ar[r] \ar[d] & G^{+} \ar[r]^{θ̃} \ar[d]_{π} & F^{∗} \ar[d] \ar[r] & 1 \\ 1 \ar[r] & Ω(V, 𝔟) \ar[r] \ar[d] & SO(V, 𝔟) \ar[r]^{θ_{V,𝔟}} \ar[d] & F^{∗}/F^{∗2} \ar[r] \ar[d] & 1 \\ & 1 & 1 & 1}

where all rows and columns are short exact, and where all columns are central extensions of groups. If dim(V) ≥ 3 (or if dim(V) = 2 and the form 𝔟 is hyperbolic), then Ω(V, 𝔟) is the commutator subgroup of SO(V, 𝔟) [Di, §II.8].

The following lemma follows immediately from this description of Spin(V, 𝔟), together with the analogous description of the corresponding spinor group over the algebraic closure of F.

Lemma A.3. Let F be the algebraic closure of F, and set V = F ⊗_{F} V and 𝔟 = Id_{F} ⊗𝔟. Then Spin(V, 𝔟) is the subgroup of Spin(V , 𝔟) consisting of those elements fixed by all Galois automorphisms ψ ∈ Gal(F /F). □

For any nonsingular quadratic form 𝔟 on a vector space V, the discriminant of 𝔟 (or of V) is the determinant of the corresponding symmetric bilinear form B, related to 𝔟 by the formulas

𝔟(v) = B(v, v) and B(v, w) = \frac{1}{2}(𝔟(v + w) − 𝔟(v) − 𝔟(w)).

Note that the discriminant is well defined only modulo squares in F^{∗}. When W ⊆ V is a subspace, then we define the discriminant of W to mean the discriminant of 𝔟|_{W}. In what follows, we say that the discriminant of a quadratic form is a square or a nonsquare to mean that it is the identity or not in the quotient group F^{∗}/F^{∗2}.

Lemma A.4. Fix an involution x ∈ SO(V, 𝔟), and let V = V_{+} ⊕ V_{−} be its eigenspace decomposition. Then the following hold.

(a) x ∈ Ω(V, 𝔟) if and only if the discriminant of V_{−} is a square.

(b) If x ∈ Ω(V, 𝔟), then it lifts to an element of order 2 in Spin(V, 𝔟) if and only if dim(V_{−}) ∈ 4ℤ.

(c) If x ∈ Ω(V, 𝔟), and if α ∈ Ω(V, 𝔟) is such that [x, α] = 1, then α = α_{+} ⊕ α_{−}, where α_{±} ∈ O(V_{±}, 𝔟). Also, the liftings of x and α commute in Spin(V, 𝔟) if and only if det(α_{−}) = 1.

Proof. Let {v_{1}, …, v_{k}} be an orthogonal basis for V_{−} (k is even). Then x = π(v_{1}⋯v_{k}) in the above notation, since π(v_{i}) is the reflection in the hyperplane v_{i}^{⊥}. Hence by the commutativity of Diagram (A.2),

θ_{V,𝔟}(x) ≡ 𝔟(v_{1})⋯𝔟(v_{k}) = det(𝔟|_{V_{−}}) (mod F^{∗2}).

Thus x ∈ Ω(V, 𝔟) = Ker(θ_{V,𝔟}) if and only if V_{−} has square discriminant.

In particular, if x ∈ Ω(V, 𝔟), then the product of the 𝔟(v_{i}) is a square, and hence (upon replacing v_{1} by a scalar multiple) we can assume that 𝔟(v_{1})⋯𝔟(v_{k}) = 1. Then x̃ =^{def} v_{1} ⋯ v_{k} ∈ Spin(V, 𝔟) = Ker(θ̃). Since vw = −wv in the Clifford algebra whenever v ⊥ w, and since (v_{i})^{2} = 𝔟(v_{i}) for each i,

x̃^{2} = (−1)^{k(k−1)/2}·(v_{1})^{2}⋯(v_{k})^{2} = (−1)^{k(k−1)/2} = \begin{cases}1 & if k ≡ 0 (mod 4) \\ −1 & if k ≡ 2 (mod 4) . \end{cases}

This proves (b).

It remains to prove (c). The first statement (α = α_{+} ⊕ α_{−}) is clear. Fix liftings α̃_{±} ∈ C(V_{±}, 𝔟)^{∗}. Rather than defining the direct sum of an element of C(V_{+}, 𝔟) with an element of C(V_{−}, 𝔟), we regard the groups C(V_{±}, 𝔟)^{∗} as (commuting) subgroups of C(V, 𝔟)^{∗}, and set

α̃ = α̃_{+} ∘ α̃_{−} = α̃_{−} ∘ α̃_{+} ∈ Spin(V, 𝔟).

Let x̃ = v_{1}⋯v_{k} be as above. Clearly, x̃ commutes with all elements of C(V_{+}, 𝔟). Since

(v_{1}⋯v_{k})·v_{i} = (−1)^{k−1}·v_{i}·(v_{1}⋯v_{k}) = −v_{i}·(v_{1}⋯v_{k})

for i = 1, …, k, we have x̃·β = (−1)^{i}·β·x̃ for all β ∈ C_{i}(V_{−}, 𝔟) (i = 0, 1). In particular, since [α̃_{+}, α̃_{−}] = 1, [x̃, α̃] = [x̃, α̃_{−}] = det(α_{−}), and this finishes the proof. □

We will need explicit isomorphisms which describe Spin_{3}(F) and Spin_{4}(F) in terms of SL_{2}(F). These are constructed in the following proposition, where M_{2}^{0}(F) denotes the vector space of matrices of trace zero. Note that the determinant is a nonsingular quadratic form on M_{2}(F) and on M_{2}^{0}(F), in both cases with square discriminant.

Proposition A.5. Define

ρ_{3} : SL_{2}(F) ⟶ Ω(M_{2}^{0}(F), det)

and

ρ_{4} : SL_{2}(F) × SL_{2}(F) ⟶ Ω(M_{2}(F), det)

by setting

ρ_{3}(A)(X) = AXA^{−1} and ρ_{4}(A, B)(X) = AXB^{−1}.

Then ρ_{3} and ρ_{4} are both epimorphisms, and lift to unique isomorphisms

SL_{2}(F) ρ̃_{3} ; ⟶ ; ≅ Spin(M_{2}^{0}(F), det)

and

SL_{2}(F) × SL_{2}(F) ρ̃_{4} ; ⟶ ; ≅ Spin(M_{2}(F), det).

Proof. See [Ta, pp. 142, 199] for other ways of defining these isomorphisms. By Lemma A.3, it suffices to prove this (except for the uniqueness of the lifting) when F is algebraically closed. In particular, Ω(M_{2}^{0}(F), det) = SO(M_{2}^{0}(F), det) and Ω(M_{2}(F), det) = SO(M_{2}(F), det) in this case.

For general V and 𝔟, the group SO(V, 𝔟) is generated by reflections fixing nonisotropic subspaces (i.e., of nonvanishing discriminant) of codimension 2 (cf. [Di, §II.6(1)]). Hence to see that ρ_{3} and ρ_{4} are surjective, it suffices to show that such elements lie in their images. A codimension 2 reflection in SO(M_{2}^{0}(F), det) is of the form R_{X} (the reflection fixing the line generated by X) for some X ∈ M_{2}^{0}(F) which is nonisotropic (i.e., det(X) ≠ 0). Since F is algebraically closed, we can assume X ∈ SL_{2}(F). Then X^{2} = −I (since Tr(X) = 0 and det(X) = 1), and R_{X} = ρ_{3}(X) since it has order 2 and fixes X. Thus ρ_{3} is onto.

Similarly, any 2-dimensional nonisotropic subspace W ⊆ V has an orthonormal basis {Y, Z}, and ZY^{−1} and Y^{−1}Z have trace zero (since they are orthogonal to the identity matrix) and determinant one. Hence their square is −I, and one repeats the above argument to show that R_{W} = ρ_{4}(ZY^{−1}, Y^{−1}Z). So ρ_{4} is onto.

The liftings ρ̃_{m} exist and are unique since SL_{2}(F) is the universal central extension of P SL_{2}(F) (or universal among central extensions by 2-groups if F = 𝔽_{3}). □

We now restrict to the case F = 𝔽_{q} where q is an odd prime power. We refer to [As2, §21] for a description of quadratic forms in this situation, and the notation for the associated orthogonal groups. If n is odd and 𝔟 is any nonsingular quadratic form on 𝔽_{q}^{n}, then every nonsingular quadratic form is isomorphic to u𝔟 for some u ∈ 𝔽_{q}^{∗}, and hence one can write SO_{n}(q) = SO(𝔽_{q}^{n}, 𝔟) = SO(𝔽_{q}^{n}, u𝔟) without ambiguity. If n is even, then there are exactly two isomorphism classes of quadratic forms on 𝔽_{q}^{n}; and one writes SO_{n}^{+}(q) = SO(𝔽_{q}^{n}, 𝔟) when 𝔟 is the hyperbolic form (equivalently, has discriminant (−1)^{n/2} modulo squares), and SO_{n}^{−}(q) = SO(𝔽_{q}^{n}, 𝔟) when 𝔟 is not hyperbolic (equivalently, has discriminant (−1)^{n/2}·u for u ∈ 𝔽_{q}^{∗} not a square). This notation extends in the obvious way to Ω_{n}^{±}(q) and Spin_{n}^{±}(q).

The following lemma does, in fact, hold for for orthogonal representations over arbitrary fields of characteristic ≠ 2. But to simplify the proof (and since we were unable to find a reference), we state it only in the case of finite fields.

Lemma A.6. Assume F = 𝔽_{q}, where q is a power of an odd prime. Let V be an F -vector space, and let 𝔟 be a nonsingular quadratic form on V. Let P ≤ O(V, 𝔟) be a 2-subgroup which is orthogonally irreducible; i.e., such that V has no splitting as an orthogonal direct sum of nonzero P -invariant subspaces. Then dim_{F} (V) is a power of 2; and if dim(V) > 1 then 𝔟 has square discriminant.

Proof. This means showing that each orthogonal group O(𝔽_{q}^{n}, 𝔟), such that either n is not a power of 2, or n = 2^{k} ≥ 2 and the quadratic form 𝔟 has nonsquare discriminant, contains some subgroup O_{m}^{±}(q) × O_{n−m}^{±}(q) (for 0 < m < n) of odd index. We refer to the standard formulas for the orders of these groups (see [Ta, p.165]): if ϵ = ±1 then

|O_{2n}^{ϵ}(q)| = 2q^{n(n−1)}(q^{n} − ϵ) ∏_{i=1}^{n−1}(q^{2i} − 1) and |O_{2n+1}(q)| = 2q^{n^{2}} ∏_{i=1}^{n}(q^{2i} − 1).

We will also use repeatedly the fact that for all 0 < i < 2^{k} (k ≥ 1), the largest powers of 2 dividing (q^{2^{k}+i} − 1) and (q^{i} − 1) are the same. In other words, (q^{2^{k}+i} − 1)/(q^{i} − 1) is invertible in ℤ_{(2)}.

For any n ≥ 1,

\frac{|O_{2n+1}(q)|}{|O_{2n}^{ϵ}(q)|·|O_{1}(q)|} = q^{n}·\frac{q^{n} + ϵ}{2}

is odd for an appropriate choice of ϵ. Thus, there are no irreducibles of odd dimension.

Assume n is not a power of 2, and write n = 2^{k} + m where 0 < m < 2^{k} and k ≥ 1. Then

\frac{|O_{2n}^{ϵ}(q)|}{|O_{2^{k+1}}^{+}(q)|·|O_{2m}^{ϵ}(q)|} = q^{m2^{k+1}}· (∏_{i=1}^{m−1} \frac{q^{2(2^{k}+i)} − 1}{q^{2i} − 1}) · (\frac{q^{2^{k}+m} − ϵ}{q^{m} − ϵ}) · (\frac{q^{2^{k}} + 1}{2}) ,

and each factor is invertible in ℤ_{(2)}. When n = 2m = 2^{k} and k ≥ 1, then O_{2n}^{−}(q) is the orthogonal group for the quadratic form with nonsquare discriminant, and

\frac{|O_{2n}^{−}(q)|}{|O_{2m}^{+}(q)|·|O_{2m}^{−}(q)|} = q^{2m^{2}}· (∏_{i=1}^{m−1} \frac{q^{2(m+i)} − 1}{q^{2i} − 1}) ·\frac{q^{2m} + 1}{2},

and again each factor is invertible in ℤ_{(2)}. Finally,

\frac{|O_{2}^{ϵ}(q)|}{|O_{1}(q)|·|O_{1}(q)|} = \frac{q − ϵ}{2}

is odd whenever q ≡ 1 (mod 4) and ϵ = −1, or q ≡ 3 (mod 4) and ϵ = +1; and these are precisely the cases where the quadratic form on 𝔽_{q}^{2} has nonsquare discriminant. □

We must classify the conjugacy classes of those elementary abelian 2-subgroups of Spin_{7}(q) which contain its center. The following definition will be useful when doing this.

Definition A.7. Fix an odd prime power q, and identify SO_{7}(q) = SO(𝔽_{q}^{7}, 𝔟) and Spin_{7}(q) = Spin(𝔽_{q}^{7}, 𝔟), where 𝔟 is a nonsingular quadratic form with square discriminant. An elementary abelian 2-subgroup of SO_{7}(q) or of Spin_{7}(q) will be called of type I if its eigenspaces all have square discriminant (with respect to 𝔟), and of type II otherwise. Let E_{n} be the set of elementary abelian 2-subgroups in Spin_{7}(q) which contain Z(Spin_{7}(q)) ≅ C_{2} and have rank n. Let E_{n}^{I} and E_{n}^{II} be the subsets of E_{n} consisting of those subgroups of types I and II, respectively.

In the following two propositions, we collect together the information which will be needed about elementary abelian 2-subgroups of Spin_{7}(q). We fix Spin_{7}(q) = Spin(V, 𝔟), where V ≅ 𝔽_{q}^{7}, and 𝔟 is a nonsingular quadratic form with square discriminant. Let z ∈ Z(Spin_{7}(q)) be the generator. For any subgroup H ≤ Spin_{7}(q) or any element g ∈ Spin_{7}(q), let H and g denote their images in Ω_{7}(q) ≤ SO_{7}(q). For each elementary abelian 2-subgroup E ≤ Spin_{7}(q), and each character χ ∈ Hom(E, {±1}), V_{χ} ⊆ V denotes the eigenspace of χ (and V_{1} denotes the eigenspace of the trivial character). Also (when z ∈ E), Aut(E, z) denotes the group of all automorphisms of E which send z to itself.

Proposition A.8. For any odd prime power q, the following table describes the numbers of Spin_{7}(q)-conjugacy classes in each of the sets E_{n}^{I} and E_{n}^{II}, the dimensions and discriminants of the eigenspaces of subgroups in these sets, and indicates in which cases Aut_{Spin_{7}(q)}(E) contains all automorphisms which fix z.

Set of subgroups  E_{2}^{I}  E_{3}^{I}  E_{3}^{II}  E_{4}^{I}  E_{4}^{II}
Nr. conj. classes  1  1  1  2  1
dim(V_{1})  3  1  0
dim(V_{χ}), χ ≠ 1  4  2  1
discr(V_{1}, 𝔟)  square square nonsq. —  —
discr(V_{χ}, 𝔟), χ ≠ 1  square square nonsq. square both
Aut_{Spin_{7}(q)}(E) = Aut(E, z) yes  yes  yes  yes  no

There are no subgroups in E_{2} of type II, and no subgroups of rank ≥ 5. Furthermore, we have:

(a) For all E ∈ E_{4}, C_{Spin_{7}(q)}(E) = E.

(b) If E, E′ ∈ E_{4}^{I}, then E′ = gEg^{−1} for some g ∈ SO_{7}(q), and E and E′ are Spin_{7}(q)- conjugate if and only if g ∈ Ω_{7}(q).

(c) If E ∈ E_{4}^{II}, then there is a unique element 1 ≠ x = x(E) ∈ E with the property that for 1 ≠ χ ∈ Hom(E, {±1}), V_{χ} has square discriminant if χ(x) = 1 and nonsquare discriminant if χ(x) = −1. Also, the image of Aut_{Spin_{7}(q)}(E) in Aut(E) is the group of all automorphisms which send x to itself; and if X ≤ E denotes the inverse image of ⟨x⟩ ≤ E, then Aut_{Spin_{7}(q)}(E) contains all automorphisms of E which are the identity on X and the identity modulo ⟨z⟩.

(d) If E ∈ E_{3}, then C_{Spin_{7}(q)}(E) = A⋊C_{2}, where A is abelian and C_{2} acts on A by inversion. If E ∈ E_{3}^{II}, then the Sylow 2-subgroups of C_{Spin_{7}(q)}(E) are elementary abelian of rank 4 (and type II).

Proof. Write Spin = Spin_{7}(q) for short. Fix an elementary abelian subgroup E ≤ Spin such that z ∈ E.

Step 1: We first show that rk(E) ≤ 4, and that the dimensions of the eigenspaces V_{χ} for χ ∈ Hom(E, {±1}) are as described in the table.

By Lemma A.4, every involution in E has a 4-dimensional (−1)-eigenspace. In particular, if rk(E) = 2, (E ≅ C_{2}), then dim(V_{χ}) = 4 for 1 ≠ χ ∈ Hom(E, {±1}), while dim(V_{1}) = 3.

Now assume rk(E) = n for some n > 2. Assume we have shown, for all E′ ∈ E_{n−1}, that the eigenspace of the trivial character of E′ is r-dimensional. For each 1 ≠ χ ∈ Hom(E, {±1}), let E_{χ} ∈ E_{n−1} be the subgroup such that E_{χ} = Ker(χ); then V_{1} ⊕ V_{χ} is the eigenspace of the trivial character of E_{χ} = Ker(χ), and thus dim(V_{1})+dim(V_{χ}) = r. Hence all nontrivial characters of E have eigenspaces of the same dimension. Since there are 2^{n−1} − 1 nontrivial characters of E, we have dim(V_{1}) + (2^{n−1} − 1) dim(V_{χ}) = 7, and these two equations completely determine dim(V_{1}) and dim(V_{χ}). Using this procedure, the dimensions of the eigenspaces are shown inductively to be equal to those given by the table. Also, this shows that rk(E) ≤ 4, since otherwise rk(E) ≥ 4, so the V_{χ} for χ ≠ 1 must be trivial (they cannot all have dimension ≥ 1), so E acts on V via the identity, which contradicts the assumption that E ≤ Spin_{7}(q).

Step 2: We next show that E_{2}^{II} = ∅, describe the discriminants of the eigenspaces of characters of E for E ∈ E_{n} (for all n), and show that subgroups of rank 4 are self centralizing. In particular, this proves (a) together with the first statement of (c).

If E ∈ E_{2}, then E = ⟨z, g⟩ for some noncentral involution g ∈ Spin_{7}(q), and the eigenspaces of E = ⟨g⟩ have square discriminant by Lemma A.4(a) (and since the ambient space V has square discriminant by assumption). Thus E_{2}^{II} = ∅.

If E ∈ E_{3}, then the sum of any two eigenspaces of E is an eigenspace of g for some g ∈ E∖⟨z⟩. Hence the sum of any two eigenspaces of E has square discriminant, so either all of the eigenspaces have square discriminant (E ∈ E_{3}^{I}), or all of the eigenspaces have nonsquare discriminant (E ∈ E_{3}^{II}).

Assume rk(E) = 4. We have seen that all eigenspaces of E are 1-dimensional. By Lemma A.4(c), for each a ∈ C_{Spin_{7}(q)}(E), a(V_{χ}) = V_{χ} for each χ ≠ 1, and since dim(V_{χ}) = 1 it must act on each V_{χ} via ± Id. Thus a ∈ Ω_{7}(q) has order 2; let V_{±} be its eigenspaces. Then dim(V_{−}) is even since det(a) = 1, and V_{−} has square discriminant by Lemma A.4(a). If dim(V_{−}) = 4, then |a| = 2 (Lemma A.4(b)), and hence a ∈ E since otherwise ⟨E, a⟩ would have rank 5. If dim(V_{−}) = 2, then V_{−} is the sum of the eigenspaces of two distinct characters χ_{1}, χ_{2} of E, there is some g ∈ E such that χ_{1}(g) ≠ χ_{2}(g), hence det(g|_{V_{−}}) = χ_{1}(g)χ_{2}(g) = −1, so [g, a] = z by Lemma A.4(c), and this contradicts the assumption that [a, E] = 1. If dim(V_{−}) = 6, then V_{−} is the sum of the eigenspaces of all but one of the nontrivial characters of E, and this gives a similar contradiction to the assumption [a, E] = 1. Thus, C_{Spin_{7}(q)}(E) = E.

Now assume that E ∈ E_{4}^{II}, and let x ∈ O_{7}(q) be the element which acts via − Id on eigenspaces with nonsquare discriminant, and via the identity on those with square discriminant. Since 𝔟 has square discriminant on V, the number of eigenspaces of E on which the discriminant is nonsquare is even, so x ∈ Ω_{7}(q) by Lemma A.4(a), and lifts to an element x ∈ Spin_{7}(q). Also, for each g ∈ E, the (−1)-eigenspace of g has square discriminant (Lemma A.4(a) again), hence contains an even number of eigenspaces of E of nonsquare discriminant, and by Lemma A.4(c) this shows that [g, x] = 1. Thus x ∈ C_{Spin_{7}(q)}(E) = E, and this proves the first statement in (c).

Step 3: We next check the numbers of conjugacy classes of subgroups in each of the sets E_{n}^{I} and E_{n}^{II}, and describe Aut_{Spin}(E) in each case. This finishes the proof of (b) and (c), and of all points in the above table.

From the above description, we see immediately that if E and E′ have the same rank and type, then any isomorphism α ∈ Iso(E, E′), such that α(x(E)) = x(E′) if E, E′ ∈ E_{4}^{II}, has the property that for all χ ∈ Hom(E′, {±1}), V_{χ} and V_{χ∘α} have the same dimension and the same discriminant (modulo squares). Hence for any such α, there is an element g ∈ O_{7}(q) such that g(V_{χ∘α}) = V_{χ} for all χ; and α = c_{g} ∈ Iso(E, E′) for such g. Upon replacing g by −g if necessary, we can assume that g ∈ SO_{7}(q). This shows that

E, E′ have the same rank and type ⟹ E and E′ are SO_{7}(q)-conjugate (1)

and also that

Aut_{SO_{7}(q)}(E) = \begin{cases}Aut(E) & if E ∉ E_{4}^{II} \\ Aut(E, x(E)) & if E ∈ E_{4}^{II}. \end{cases}  (2)

We next claim that

E ∉ E_{4}^{I} ⟹ ∃γ ∈ SO_{7}(q)∖Ω_{7}(q) such that [γ, E] = 1 .  (3)

To prove this, choose 1-dimensional nonisotropic summands W ⊆ V_{χ} and W′ ⊆ V_{ψ}, where χ, ψ are two distinct characters of E, and where W has square discriminant and W′ has nonsquare discriminant. Let γ ∈ SO_{7}(q) be the involution with (−1)-eigenspace W ⊕ W′. Then [γ, E] = 1, since γ sends each eigenspace of E to itself, and γ ∉ Ω_{7}(q) since its (−1)-eigenspace has nonsquare discriminant (Lemma A.4(a)).

If E has rank 4 and type I, then Aut(E) ≅ GL_{3}(𝔽_{2}) is simple, and in particular has no subgroup of index 2. Hence by (2), each element of Aut(E) is induced by conjugation by an element of Ω_{7}(q). Also, if g ∈ SO_{7}(q) centralizes E, then g(V_{χ}) = V_{χ} for all χ ∈ Hom(E, {±1}), g acts via − Id on an even number of eigenspaces (since it has determinant +1), and hence g ∈ Ω_{7}(q) by Lemma A.4(a). Thus

E ∈ E_{4}^{I} ⟹ N_{SO_{7}(q)}(E) ≤ Ω_{7}(q)  (4)

If E ∉ E_{4}^{I}, then by (3), for any g ∈ SO_{7}(q), there is γ ∈ SO_{7}(q)∖Ω_{7}(q) such that c_{g}|_{E} = c_{gγ}|_{E}, and either g or gγ lies in Ω_{7}(q). Thus Iso_{SO_{7}(q)}(E, E′) = Iso_{Ω_{7}(q)}(E, E′) for any E′. Together with (1), this shows that E is Spin-conjugate to all other subgroups of the same rank and type, and together with (2) it shows that

Im[Aut_{Spin}(E) ⟶ Aut(E)] = \begin{cases}Aut(E) & if E ∉ E_{4}^{II} \\ Aut(E, x(E)) & if E ∈ E_{4}^{II}. \end{cases}  (5)

If E ∈ E_{4}^{I}, then by (4) and (2), Aut_{Ω_{7}(q)}(E) = Aut_{SO_{7}(q)}(E) = Aut(E), and so (5) also holds in this case. Furthermore, for any g ∈ SO_{7}(q)∖Ω_{7}(q), E and gEg^{−1} are representatives for two distinct Ω_{7}(q)-conjugacy classes — since by (4), no element of the coset g·Ω_{7}(q) normalizes E.

We have now determined in all cases the number of conjugacy classes of subgroups of a given rank and type, and the image of Aut_{Spin}(E) in Aut(E). We next claim that if rk(E) < 4 or E ∈ E_{4}^{I}, then

E ∉ E_{4}^{II} ⟹ Aut_{Spin}(E) ≥ {α ∈ Aut(E) | α(z) = z, α ≡ Id (mod ⟨z⟩) } . (6)

Together with (5), this will finish the proof that Aut_{Spin}(E) is the group of all automorphisms of E which send z to itself. We also claim that

E ∈ E_{4}^{II} ⟹ Aut_{Spin}(E) ≥ {α ∈ Aut(E) | α|_{X} = Id_{X}, α ≡ Id (mod ⟨z⟩)} , (7)

where X ≤ E denotes the inverse image of ⟨x(E)⟩ ≤ E, and this will finish the proof of (c).

We prove (6) and (7) together. Fix α ∈ Aut(E) (α ≠ Id) which sends z to itself, induces the identity on E, and such that α|_{X} = Id_{X} if E ∈ E_{4}^{II}. Then there is 1 ≠ χ ∈ Hom(E, {±1}) such that α(g) = g when χ(g) = 1 and α(g) = zg when χ(g) = −1. Choose any character ψ such that V_{ψ} ≠ 0 and V_{ψχ} ≠ 0, and let W ⊆ V_{ψ} and W′ ⊆ V_{ψχ} be 1-dimensional nonisotropic subspaces with the same discriminant (this is possible when E ∈ E_{4}^{II} since x(E) ∈ Ker(χ)). Let g ∈ O_{7}(q) be the involution whose (−1)-eigenspace is W ⊕ W′. Then g ∈ Ω_{7}(q) by Lemma A.4(a), so g lifts to g ∈ Spin_{7}(q), and using Lemma A.4(c) one sees that c_{g} = α.

Step 4: It remains to prove (d). Assume E ∈ E_{3}. Let 1 = χ_{1}, χ_{2}, χ_{3}, χ_{4} be the four characters of E, and set V_{i} = V_{χ_{i}}. Then dim(V_{1}) = 1, dim(V_{i}) = 2 for i = 2, 3, 4, and the V_{i} either all have square discriminant or all have nonsquare discriminant. For each g ∈ C_{Spin}(E), we can write g = ⨁_{i=1}^{4} g_{i}, where g_{i} ∈ O(V_{i}, 𝔟_{i}). For each pair of distinct indices i, j ∈ {2, 3, 4}, there is some g ∈ E whose (−1)-eigenspace is V_{i} ⊕ V_{j}, and hence det(g_{i} ⊕ g_{j}) = 1 by Lemma A.4(c). This shows that the g_{i} all have the same determinant. Let A ≤ C_{Spin}(E) be the subgroup of index 2 consisting of those g such that det(g_{i}) = 1 for all i.

Now, SO_{1}(𝔽_{q}) = 1, while SO_{2}(𝔽_{q}) ≅ 𝔽_{q}^{∗} is the group of diagonal matrices of the form diag(u, u^{−1}) with respect to a hyperbolic basis of 𝔽_{q}^{2}. Thus A is contained in a central extension of C_{2} by (𝔽_{q}^{∗})^{3}, and any such extension is abelian since H_{2}((𝔽_{q}^{∗})^{3}) = 0. Hence A is abelian. The groups O_{2}^{±}(q) are all dihedral (see [Ta, Theorem 11.4]). Hence for any g ∈ C_{Spin}(E)∖A, g has order 2 and (−1)-eigenspace of dimension 4 (its intersection with each V_{i} is 1-dimensional), and hence |g| = 2 by Lemma A.4(b). Thus all elements of C_{Spin}(E)∖A have order 2, so the centralizer must be a semidirect product of A with a group of order 2 which acts on it by inversion.

Now assume that E ∈ E_{3}^{II}; i.e., that the V_{i} all have nonsquare discriminant. Then for i = 2, 3, 4, SO(V_{i}, 𝔟_{i}) has order q ± 1, whichever is not a multiple of 4 (see [Ta, Theorem 11.4] again). Thus if g ∈ A ≤ C_{Spin}(E) has 2-power order, then g_{i} = ±I for each i, the number of i for which g_{i} = Id is even (since the (−1)-eigenspace of g has square discriminant), and hence g ∈ E. In other words, E ∈ Syl_{2}(A). A Sylow 2-subgroup of C_{Spin}(E) is thus generated by E together with an element of order 2 which acts on E by inversion; this is an elementary abelian subgroup of rank 4, and is necessarily of type II. □

We also need some more precise information about the subgroups of Spin_{7}(q) of rank 4 and type II. Let ψ^{q} ∈ Aut(Spin_{7}(𝔽_{q})) denote the automorphism induced by the field automorphism (x ↦ x^{q}). By Lemma A.3, Spin_{7}(q) is precisely the subgroup of elements fixed by ψ^{q}.

Proposition A.9. Fix an odd prime power q, and let z ∈ Z(Spin_{7}(q)) be the central involution. Let C and C′ denote the two conjugacy classes of subgroups E ≤ Spin_{7}(q) of rank 4 and type I. Then the following hold.

(a) For each E ∈ E_{4}, there is an element a ∈ Spin_{7}(𝔽_{q}) such that aEa^{−1} ∈ C. For any

such a, if we set

x_{C}(E) =^{def} a^{−1}ψ^{q}(a),

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

(b) E ∈ C if and only if x_{C}(E) = 1, and E ∈ C′ if and only if x_{C}(E) = z.

(c) Assume E ∈ E_{4}^{II}, and set τ(E) = ⟨z, x_{C}(E)⟩. Then rk(τ(E)) = 2, and

Aut_{Spin_{7}(q)}(E) = {α ∈ Aut(E) | α|_{τ(E)} = Id}.

The four eigenspaces of E contained in the (−1)-eigenspace of x_{C}(E) all have nonsquare discriminant, and the other three eigenspaces all have square discriminant.

Proof. (a) For all E ∈ E_{4}, E has type I as a subgroup of Spin_{7}(q^{2}) since all elements of 𝔽_{q} are squares in 𝔽_{q^{2}}. Hence by Proposition A.8(b), for all E′ ∈ C, there is a ∈ SO_{7}(q^{2}) ≤ Ω_{7}(q^{4}) such that aEa^{−1} = E′. Upon lifting a to a ∈ Spin_{7}(q^{4}), this proves that there is a ∈ Spin_{7}(𝔽_{q}) such that aEa^{−1} ∈ C.

Fix any such a, and set

x = x_{C}(E) = a^{−1}ψ^{q}(a).

For all g ∈ E, ψ^{q}(g) = g and ψ^{q}(aga^{−1}) = aga^{−1} since E, aEa^{−1} ≤ Spin_{7}(q), and hence

aga^{−1} = ψ^{q}(a)·g·ψ^{q}(a^{−1}) = a(xgx^{−1})a^{−1}.

Thus, x ∈ C_{Spin_{7}(𝔽_{q})}(E), and so x ∈ E since it is self centralizing in each Spin_{7}(q^{k}) (Proposition A.8(a)).

We next check that x_{C}(E) is independent of the choice of a. Assume a, b ∈ Spin_{7}(𝔽_{q}) are such that aEa^{−1} ∈ C and bEb^{−1} ∈ C. Then by Proposition A.8(b), there is g ∈ Spin_{7}(q) such that gbE(gb)^{−1} = aEa^{−1}. Set E′ = aEa^{−1} ∈ C, then gba^{−1} ∈ N_{Spin_{7}(𝔽_{q})}(E′). Furthermore, since Aut_{Spin_{7}(q)}(E′) contains all automorphisms which send z to itself, and since E′ is self centralizing in each of the groups Spin_{7}(q^{k}) (both by Proposition A.8 again), we see that N_{Spin_{7}(𝔽_{q})}(E′) is contained in Spin_{7}(q). Thus, ba^{−1} ∈ Spin_{7}(q), so ψ^{q}(ba^{−1}) = ba^{−1}; and this proves that x_{C}(E) = a^{−1}ψ^{q}(a) = b^{−1}ψ^{q}(b) is independent of the choice of a.

(b) If E ∈ C, then we can choose a = 1, and so x_{C}(E) = 1.

If E ∈ C′, then by Proposition A.8(b), there is a ∈ Spin_{7}(q^{2}) such that a ∈ SO_{7}(q)∖Ω_{7}(q) and aEa^{−1} ∈ C. Then ψ^{q}(a) ≠ a since a ∉ Spin_{7}(q) (Proposition A.3), but ψ^{q}(a) = a since a ∈ SO_{7}(q). Thus, x_{C}(E) = a^{−1}ψ^{q}(a) = z in this case.

We have now shown that x_{C}(E) ∈ ⟨z⟩ if E has type I, and it remains to prove the converse. Fix a ∈ Spin_{7}(𝔽_{q}) such that aEa^{−1} ∈ C. If x_{C}(E) ∈ ⟨z⟩, then ψ^{q}(a) ∈ {a, za}, so ψ^{q}(a) = a, and hence a ∈ SO_{7}(q). Conjugation by an element of SO_{7}(q) sends eigenspaces with square discriminant to eigenspaces with square discriminant, so all eigenspaces of E must have square discriminant since all eigenspaces of aEa^{−1} do. Hence E has type I.

(c) Now write Spin = Spin_{7}(q) for short. Assume E ∈ E_{4}^{II}, and set x = x_{C}(E) and τ(E) = ⟨z, x⟩. Then x ∉ ⟨z⟩ by (b), and thus τ(E) has rank 2.

By (a) (the uniqueness of x having the given properties), each element of Aut_{Spin}(E) restricts to the identity on τ(E). We have already seen (Proposition A.8(c)) that there is an element x(E) ∈ E such that the image in Aut(E) of Aut_{Spin}(E) is the group of automorphisms which fix x(E), and this shows that x(E) = x: the image in E of x. Since we already showed (Proposition A.8(c) again) that Aut_{Spin}(E) contains all automorphisms which are the identity on τ(E) and the identity modulo ⟨z⟩, this finishes the proof that Aut_{Spin}(E) is the group of all automorphisms which are the identity on τ(E). The last statement (about the discriminants of the eigenspaces) follows directly from the first statement of Proposition A.8(c). □

Throughout the rest of the section, we collect some more technical results which will be needed in Sections 2 and 4.

Lemma A.10. Fix k ≥ 2. Let A = e_{13}(2^{k−1}) ∈ GL_{3}(ℤ/2^{k}) be the elementary matrix which has off diagonal entry 2^{k−1} in position (1, 3). 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 that

μ_{i}: T_{i} ⟶ SL_{3}(ℤ/2^{k})

are lifts of the inclusions (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}(ℤ/2^{k})

such that μ|_{T_{1}} = μ_{1}, and either μ|_{T_{2}} = μ_{2}, or μ|_{T_{2}} = c_{A} ∘ μ_{2}.

Proof. We first claim that any two liftings σ, σ′: T_{2} ⟶ SL_{3}(ℤ/2^{k}) are conjugate by an element of SL_{3}(ℤ/2^{k}). This clearly holds when k = 1, and so we can assume inductively that σ ≡ σ′ (mod 2^{k−1}). Let M_{3}^{0}(𝔽_{2}) be the group of 3×3 matrices of trace zero, and define ρ: T_{2} ⟶ M_{3}^{0}(𝔽_{2}) via the formula

σ′(B) = (I + 2^{k−1}ρ(B))·σ(B)

for B ∈ T_{2}. Then ρ is a 1-cocycle. Also, H^{1}(T_{2}; M_{3}^{0}(𝔽_{2})) = 0 by [DW1, Lemma 4.3] (the module is 𝔽_{2}[T_{2}]-projective), so ρ is the coboundary of some X ∈ M_{3}^{0}(𝔽_{2}), and σ and σ′ differ by conjugation by I + 2^{k−1}X.

By [DW1, Theorem 4.1], there exists a section μ defined on GL_{3}(2) such that μ|_{T_{1}} = μ_{1}. Let B ∈ SL_{3}(ℤ/2^{k}) be such that μ|_{T_{2}} = c_{B} ∘ μ_{2}. Since μ|_{T_{0}} = μ_{2}|_{T_{0}}, B must commute with all elements in μ(T_{0}), and one easily checks that the only such elements are A = e_{13}(2^{k−1}) and the identity. □

Recall that a p-subgroup P of a finite group G is p-radical if N_{G}(P)/P is p-reduced; i.e., if O_{p}(N_{G}(P)/P) = 1. (Here, O_{p}(−) denotes the largest normal p-subgroup.) We say here that P is F_{p}(G)-radical if Out_{G}(P) (= Out_{F_{p}(G)}(P)) is p-reduced. In Section 4, some information will be needed involving the F_{2}(Spin_{7}(q))-radical subgroups of Spin_{7}(q) which are also 2-centric. We first note the following general result.

Lemma A.11. Fix a finite group G and a prime p. Then the following hold for any p-subgroup P ≤ G which is p-centric and F_{p}(G)-radical.

(a) If G = G_{1} × G_{2}, then P = P_{1} × P_{2}, where P_{i} is p-centric in G_{i} and F_{p}(G_{i})-radical.

(b) If P ≤ H ◁ G, then P is p-centric in H and F_{p}(H)-radical.

(c) If H ◁ G has p-power index, then P ∩ H is p-centric in H and F_{p}(H)-radical.

(d) If G ◁ G has p-power index, then P = G ∩ P for some P ≤ G which is p-centric in G and F_{p}(G)-radical.

(e) If Q ◁ G is a central p-subgroup, then Q ≤ P, and P/Q is p-centric in G/Q and F_{p}(G/Q)-radical.

(f) If G̃ −↠^{α} G is an epimorphism such that Ker(α) ≤ Z(G̃), then α^{−1}(P) is p-centric in G̃ and F_{p}(G̃)-radical.

Proof. Point (a) follows from [JMO, Proposition 1.6(ii)]: P = P_{1} ×P_{2} for P_{i} ≤ G_{i} since P is p-radical, and P_{i} must be p-centric in G_{i} and F_{p}(G_{i})-radical since

C_{G}(P) = C_{G_{1}}(P_{1}) × C_{G_{2}}(P_{2}) and Out_{G}(P) ≅ Out_{P_{1}}(G_{1}) × Out_{P_{2}}(G_{2}).

Point (b) holds since C_{H}(P) ≤ C_{G}(P) and O_{p}(Out_{H}(P)) ≤ O_{p}(Out_{G}(P)). It remains to prove the other four points.

(e) Fix a central p-subgroup Q ≤ Z(G). Then P ≥ Q, since otherwise 1 ≠ N_{QP} (P)/P ≤ O_{p}(N_{G}(P)/P). Also, P/Q is p-centric in G/Q, since otherwise there would be x ∈ G∖P of p-power order such that

1 ≠ [c_{x}] ∈ Ker[Out_{G}(P) ⟶ Out_{G/Q}(P/Q) × Out_{G}(Q)] ≤ O_{p}(Out_{G}(P)).

It remains only to prove that P/Q is F_{p}(G/Q)-radical, and to do this it suffices to show that

Out_{G/Q}(P/Q) ≅ Out_{G}(P).

Equivalently, since P/Q and P are p-centric, we must show that

\frac{N_{G/Q}(P/Q)}{C′_{G/Q}(P/Q) × P/Q} ≅ \frac{N_{G}(P)}{C′_{G}(P) × P};

and this is clear once we have shown that

C′_{G/Q}(P/Q) ≅ C′_{G}(P).

Any x ∈ C′_{G/Q}(P/Q) lifts to an element x ∈ G of order prime to p, whose conjugation action on P induces the identity on Q and on P/Q. By [Go, Corollary 5.3.3], all such automorphisms of P have p-power order, and thus x centralizes P. Since Q is a p-group and C′_{G/Q}(P/Q) has order prime to p, this shows that the projection modulo Q sends C′_{G/Q}(P/Q) isomorphically to C′_{G}(P).

(f) Let G̃ −↠^{α} G be an epimorphism whose kernel is central. Clearly, α^{−1}P is pcentric in G̃. It remains only to prove that α^{−1}P is F_{p}(G̃)-radical, and to do this it suffices to show that

OutG̃(α^{−1}P) ≅ Out_{G}(P).

Equivalently, since P and α^{−1}(P) are p-centric, we must show that

\frac{NG̃(α^{−1}P)}{C′ ; G̃(α^{−1}P) × α^{−1}P} ≅ \frac{N_{G}(P)}{C′_{G}(P) × P};

and this is clear once we have shown that

C′ ; G̃(α^{−1}P) ≅ C′_{G}(P).

This follows by exactly the same argument as in the proof of (e).

(c) Set P′ = P ∩ H for short. Let

N_{H}(P′) −↠^{π} Out_{H}(P′) ≅ N_{H}(P′)/(C_{H}(P′)·P′)

be the natural projection, and set

K = π^{−1}(O_{p}(Out_{H}(P′))) ≤ N_{H}(P′).

Then K ≥ O_{p}(N_{H}(P′)) is an extension of C_{H}(P′)·P′ by the p-group O_{p}(Out_{H}(P′)). It suffices to show that p ∤ [K:P′], since this implies that O_{p}(Out_{H}(P′)) = 1 (i.e., P′ is F_{p}(H)-radical), and that any Sylow p-subgroup of C_{H}(P′) is contained in P′ (hence P′ is p-centric in H).

Assume otherwise: that p|[K:P′]. Note first that P′ ◁ N_{G}(P), and that N_{G}(P) ≤ N_{G}(K); i.e., N_{G}(P) normalizes P′ and K. The first statement is obvious, and the second is verified by observing directly that N_{G}(P) normalizes N_{H}(P′) and C_{H}(P′). Thus the action of N_{G}(P) on K induces an action of N_{G}(P), and in particular of P, on K/P′. Let K_{0}/P′ denote the fixed subgroup of this action of P. Since p|[K:P′] by assumption, and since P is a p-group, p||K_{0}/P′|. A straightforward check also shows that K_{0} ◁ N_{G}(P), and therefore that P K_{0} ◁ N_{G}(P). Also, since P′ ≤ K_{0} ≤ H,

P K_{0}/P ≅ K_{0}/(P ∩ K_{0}) = K_{0}/P′

is a normal subgroup of N_{G}(P)/P of order a multiple of p. Since P is p-centric in G by assumption,

Out_{G}(P) = N_{G}(P)/(C_{G}(P)·P) = N_{G}(P)/(C′_{G}(P) × P),

and hence the image of P K_{0}/P in Out_{G}(P) is a normal subgroup which also has order a multiple of p.

By definition of K as an extension of C_{H}(P′)·P′ by a p-group, if x ∈ K has order prime to p, then x ∈ C_{H}(P′). Hence if x ∈ K_{0} has order prime to p, then for every z ∈ P, [x, z] ∈ P′, so x acts trivially on P/P′. Since x also centralizes P′, it follows that x centralizes P. This shows that the image of P K_{0}/P in Out_{G}(P) is a p-group, thus a nontrivial normal p-subgroup of Out_{G}(P), and this contradicts the original assumption that P is F_{p}(G)-radical.

(d) Let G ◁ G be a normal subgroup of p-power index and let P ≤ G be a p-centric and F_{p}(G)-radical subgroup. Let

N_{G}(P) −↠^{π} Out_{G}(P) ≅ N_{G}(P)/(C_{G}(P)·P )

be the natural surjection, and set

K = π^{−1}(O_{p}(Out_{G}(P))) ≤ N_{G}(P).

Then K is an extension of C_{G}(P)·P by O_{p}(Out_{G}(P)). Fix any P ∈ Syl_{p}(K). We will show that P ∩ G = P, and that P is p-centric in G and F_{p}(G)-radical.

For each x ∈ K ∩ G ≤ N_{G}(P),

π(x) ∈ O_{p}(Out_{G}(P)) ∩ Out_{G}(P) ≤ O_{p}(Out_{G}(P)) = 1.

Hence

x ∈ Ker[N_{G}(P) ⟶ Out_{G}(P)] = (C_{G}(P)·P) ∩ G = C_{G}(P) · P ≅ C′_{G}(P) × P,

where C′_{G}(P) ≤ C_{G}(P) is of order prime to p. Since the opposite inclusion is obvious, this shows that K ∩ G = C′_{G}(P) × P, and hence (since P ∈ Syl_{p}(K)) that P ∩ G = P.

Next, note that (K ∩ G) ◁ K and K/(K ∩ G) ≤ G/G, and hence K/C′_{G}(P) has p-power order. Since P ∈ Syl_{p}(K), P is an extension of P by K/(K ∩ G), and N_{K}(P) is an extension of a subgroup of (K ∩ G) = (C′_{G}(P) × P) by K/(K ∩ G). Also, an element x ∈ C′_{G}(P) normalizes P if and only if [x, P] ∈ P ∩ C′_{G}(P) = 1. Hence

N_{K}(P) = C_{K}(P)·P = C′_{G}(P) × P ,  (1)

where C′_{G}(P) = C′_{G}(P) ∩ C_{G}(P) has order prime to p and is normal in N_{K}(P). Since C_{G}(P) ≤ C_{G}(P) ≤ K, (1) shows that C_{G}(P) ≤ C′_{G}(P) × P, and hence that P is p-centric in G.

It remains to show that P is F_{p}(G)-radical. Note first that K ◁ N_{G}(P) by construction, so for any x ∈ N_{G}(P), xP x^{−1} ∈ Syl_{p}(K). Since K is an extension of C′_{G}(P) × P by the p-group K/(K ∩G), and since C′_{G}(P) ◁ K, it follows that K is a split extension of C′_{G}(P) by P. Hence for any x ∈ N_{G}(P), xP x^{−1} = yP y^{−1} for some y ∈ C′_{G}(P). Consequently, the restriction map

N_{G}(P)/C_{G}(P) ≅ Aut_{G}(P) ⟶ Aut_{G}(P) ≅ N_{G}(P)/C_{G}(P)  (2)

is surjective. Also, if x ∈ C_{G}(P) ≤ K normalizes P, then x ∈ N_{K}(P) ≅ P × C′_{G}(P) by (1), and so c_{x} ∈ Inn(P). Thus the kernel of the map in (2) is contained in Inn(P). Consequently,

Out_{G}(P) = Aut_{G}(P)/ Inn(P) ≅ Aut_{G}(P)/ Aut_{P} (P) ≅ Out_{G}(P)/O_{p}(Out_{G}(P)),

and it follows that P is F_{p}(G)-radical. □

This is now applied to show the following:

Proposition A.12. Fix an odd prime power q, and let P ≤ Spin_{7}(q) be any subgroup which is 2-centric and F_{2}(Spin_{7}(q))-radical. Then P is centric in Spin_{7}(𝔽_{q}); i.e., C_{Spin_{7}(𝔽_{q})}(P) = Z(P).

Proof. Let z be the central involution in Spin_{7}(q). By Lemma A.11(e), z ∈ P, and P =^{def} P/⟨z⟩ is 2-centric in Ω_{7}(q) and is F_{2}(Ω_{7}(q))-radical. So by Lemma A.11(d), there is a 2-subgroup P̂ ≤ O_{7}(q) such that P̂ ∩ Ω_{7}(q) = P, and such that P̂ is 2-centric in O_{7}(q) and is F_{2}(O_{7}(q))-radical.

Let V = ⨁_{i=1}^{m} V_{i} be a maximal decomposition of V as an orthogonal direct sum of P̂-representations, and set 𝔟_{i} = 𝔟|_{V_{i}}. We assume these are arranged so that for some k, dim(V_{i}) > 1 when i ≤ k and dim(V_{i}) = 1 when i > k. Let V_{+} be the sum of those 1-dimensional components V_{i} with square discriminant, and let V_{−} be the sum of those 1-dimensional components V_{i} with nonsquare discriminant. We will be referring to the two decompositions

(V, 𝔟) = ⨁_{i=1}^{m}(V_{i}, 𝔟_{i}) = ⨁_{i=1}^{k}(V_{i}, 𝔟_{i}) ⊕ (V_{+}, 𝔟_{+}) ⊕ (V_{−}, 𝔟_{−}),

both of which are orthogonal direct sums. We also write

V^{(∞)} = 𝔽_{q} ⊗_{𝔽_{q}} V and V_{i}^{(∞)} = 𝔽_{q} ⊗_{𝔽_{q}} V_{i},

and let 𝔟^{(∞)} and 𝔟_{i}^{(∞)} be the induced quadratic forms.

Step 1: For each i, set

D_{i} = {± Id_{V_{i}}} ≤ O(V_{i}, 𝔟_{i}),

a subgroup of order 2; and write

D = ∏_{i=1}^{m} D_{i} ≤ O(V, 𝔟), and D_{±} = ∏_{V_{i}⊆V_{±}} D_{i} ≤ O(V ±, 𝔟_{±}).

Thus D and D_{±} are elementary abelian 2-groups of rank m and dim(V_{±}), respectively. We first claim that

P̂ ≥ D,  (1)

and that

P̂ = ∏_{i=1}^{m} P_{i} where ∀ i, P_{i} is 2-centric in O(V_{i}, 𝔟_{i}) and F_{2}(O(V_{i}, 𝔟_{i}))-radical. (2)

Clearly, [D, P̂] = 1 (and D is a 2-group), so D ≤ P̂ since P̂ is 2-centric. This proves (1). The V_{i} are thus distinct (pairwise nonisomorphic) as P̂-representations, since they are pairwise nonisomorphic as D-representations. The decomposition as a sum of V_{i}’s is thus unique (not only up to isomorphism), since HomP̂(V_{i}, V_{j}) = 0 for i ≠ j.

Let Ĉ be the group of elements of O(V, 𝔟) which send each V_{i} to itself, and let N̂ be the group of elements which permute the V_{i}. By the uniqueness of the decomposition of V,

P̂·C_{O(V,𝔟)}(P̂) ≤ Ĉ = ∏_{i=1}^{m} O(V_{i}, 𝔟_{i}) and N_{O(V,𝔟)}(P̂) ≤ N̂.

Since P̂ is 2-centric in O(V, 𝔟) and F_{2}(O(V, 𝔟))-radical, it is also 2-centric in N̂ and F_{2}(N̂)-radical (this holds for any subgroup which contains N_{O(V,𝔟)}(P̂)). So by Lemma A.11(b) (and since Ĉ ◁ N̂), P̂ is 2-centric in Ĉ and F_{2}(Ĉ)-radical. Point (2) now follows from Lemma A.11(a).

Step 2: Whenever dim(V_{i}) > 1 (i.e., 1 ≤ i ≤ k), then by Lemma A.6, dim(V_{i}) is even, and 𝔟_{i} has square discriminant. So by Lemma A.4(a), − Id_{V_{i}} ∈ Ω(V_{i}, 𝔟_{i}) for such i. Together with (1), this shows that

P = P̂ ∩ Ω_{7}(q) ≥ ∏_{i=1}^{k} D_{i} × (Ω(V_{+}, 𝔟_{+}) ∩ D_{+}) × (Ω(V_{−}, 𝔟_{−}) ∩ D_{−}) .  (3)

Also, by Lemma A.4(a) again,

\begin{matrix}(Ω(V_{±}, 𝔟_{±}) ∩ D_{±}) = (SO(V_{±}, & 𝔟_{±}) ∩ D_{±}) \\ = ⟨− Id_{V_{i}⊕V_{j}} | k+1 ≤ i < j ≤ m, V_{i}, V_{j} ⊆ V_{±}⟩. \end{matrix} (4)

Step 3: By (3) and (4), the V_{i} are distinct as P -representations (not only as P̂- representations), except possibly when dim(V_{±}) = 2. We first check that this exceptional case cannot occur. If dim(V_{+}) = 2 and its two irreducible summands are isomorphic as P -representations, then the image of P under projection to O(V_{+}, 𝔟_{+}) is just {± Id_{V_{+}}}. Hence we can write V_{+} = W ⊕ W′, where W ⊥W′ are 1-dimensional, P -invariant, and have nonsquare discriminant. Also, dim(V_{−}) is odd, since V_{+} and the V_{i} for i ≤ k are all even dimensional. So − Id_{V_{−}⊕W} lies in C_{Ω_{7}(q)}(P) but not in P. But this is impossible, since P is 2-centric in Ω_{7}(q). The argument when dim(V_{−}) = 2 is similar.

The V_{i} are thus distinct as P -representations. So for all i ≠ j, Hom_{P} (V_{i}, V_{j}) = 0, and hence

Hom_{𝔽_{q}[P]}(V_{i}^{(∞)}, V_{j}^{(∞)}) ≅ 𝔽_{q} ⊗_{𝔽_{q}} Hom_{𝔽_{q}[P]}(V_{i}, V_{j}) = 0.

Thus any element of O(V^{(∞)}, 𝔟^{(∞)}) which centralizes P sends each V_{i}^{(∞)} to itself. In other words,

C_{Spin_{7}(𝔽_{q})}(P)/⟨z⟩ ≤ C_{Ω_{7}(𝔽_{q})}(P) ≤ ∏_{i=1}^{m} O(V_{i}^{(∞)}, 𝔟_{i}^{(∞)}).

If dim(V_{±}) ≥ 2, then since P contains all involutions in O(V_{±}, 𝔟_{±}) which are P̂- invariant and have even dimensional (−1)-eigenspace (see (3)), Lemma A.4(c) shows that each element of Spin_{7}(𝔽_{q}) which commutes with P must act on V_{±} via ± Id. Also, for 1 ≤ i ≤ k, since − Id_{V_{i}} ∈ P by (3), each element in the centralizer of P acts on V_{i} with determinant 1 (Lemma A.4(c) again). We thus conclude that

C_{Spin_{7}(𝔽_{q})}(P)/⟨z⟩ ≤ ∏_{i=1}^{k} SO(V_{i}^{(∞)}, 𝔟_{i}^{(∞)}) × {± Id_{V_{+}}} × {± Id_{V_{−}}}.  (5)

Step 4: We next show that

C_{Spin_{7}(𝔽_{q})}(P)/⟨z⟩ ≤ ∏_{i=1}^{k}{± Id_{V_{i}}} × {± Id_{V_{+}}} × {± Id_{V_{−}}}.  (6)

Using (5), this means showing, for each 1 ≤ i ≤ k, that

pr_{i}(C_{Spin_{7}(𝔽_{q})}(P)/⟨z⟩) ≤ {± Id_{V_{i}}};  (7)

where pr_{i} denotes the projection of O_{7}(𝔽_{q}) = O(V^{(∞)}, 𝔟^{(∞)}) to O(V_{i}^{(∞)}, 𝔟_{i}^{(∞)}). By Lemma A.6, dim(V_{i}) = 2 or 4. We consider these two cases separately.

Case 4A: If dim(V_{i}) = 4, then by (2) and Lemma A.11(c), P′_{i} =^{def} P_{i} ∩ Ω(V_{i}, 𝔟_{i}) is 2-centric in Ω(V_{i}, 𝔟_{i}) and is F_{2}(Ω(V_{i}, 𝔟_{i}))-radical. Also, by Proposition A.5,

Ω(V_{i}, 𝔟_{i}) ≅ Ω_{4}^{+}(q) ≅ SL_{2}(q) ×_{C_{2}} SL_{2}(q).

By Lemma A.11(a,f), under this identification, we have P′_{i} = Q×_{C_{2}} Q′, where Q and Q′ are 2-centric in SL_{2}(q) and F_{2}(SL_{2}(q))-radical. The Sylow 2-subgroups of SL_{2}(q) are quaternion groups of order ≥ 8, all subgroups of a quaternion 2-group are quaternion or cyclic, and cyclic 2-subgroups of SL_{2}(q) cannot be both 2-centric and F_{2}(SL_{2}(q))- radical. So Q and Q′ must be quaternion of order ≥ 8. By [Sz, 3.6.3], any cyclic 2-subgroup of SL_{2}(𝔽_{q}) of order ≥ 4 is conjugate to a subgroup of diagonal matrices, whose centralizer is the group of all diagonal matrices in SL_{2}(𝔽_{q}). Knowing this, one easily checks that all nonabelian quaternion 2-subgroups of SL_{2}(𝔽_{q}) are centric in SL_{2}(𝔽_{q}). It follows that P′_{i} is centric in

SO(V_{i}^{(∞)}, 𝔟_{i}^{(∞)}) ≅ SL_{2}(𝔽_{q}) ×_{C_{2}} SL_{2}(𝔽_{q}),

and hence that

pr_{i}(C_{Spin_{7}(𝔽_{q})}(P)/⟨z⟩) ≤ C_{SO(V_{i}^{(∞)},𝔟_{i}^{(∞)})}(P′_{i}) = Z(P′_{i}) = {± Id_{V_{i}}}.

Thus (7) holds in this case.

Case 4B: If dim(V_{i}) = 2, then O(V_{i}, 𝔟_{i}) ≅ O_{2}^{±}(q) is a dihedral group of order 2(q ∓ 1) [Ta, Theorem 11.4]. Hence P_{i} ∈ Syl_{2}(O(V_{i}, 𝔟_{i})), since the Sylow subgroups are the only radical 2-subgroups of a dihedral group. Fix V_{j} for any k < j ≤ m, and choose α ∈ O(V_{i}, 𝔟_{i}) of determinant (−1) whose (−1)-eigenspace has the same discriminant as V_{j}. Since P_{i} ∈ Syl_{2}(O(V_{i}, 𝔟_{i})), we can assume (after conjugating if necessary) that α ∈ P_{i}. Then (− Id_{V_{j}}) ⊕ α lies in P = P̂ ∩ Ω_{7}(q). Hence for any g ∈ C_{Spin_{7}(𝔽_{q})}(P)/⟨z⟩,

pr_{i}(g) ∈ O(V_{i}^{(∞)}, 𝔟_{i}^{(∞)}) leaves both eigenspaces of α invariant, and has determinant 1 by (5). Thus pr_{i}(g) = ± Id_{V_{i}}; and so (7) holds in this case.

Step 5: Clearly, − Id_{V_{±}} lies in SO(V_{±}, 𝔟_{±}) if and only if dim(V_{±}) is even (which is the case for exactly one of the two spaces V_{±}), and this holds if and only if − Id_{V_{±}} ∈ Ω(V_{±}, 𝔟_{±}). Also, since each V_{i} for 1 ≤ i ≤ k has square discriminant (Lemma A.6 again), − Id_{V_{i}} ∈ Ω(V_{i}, 𝔟_{i}) for all such i. Thus (6) and (1) imply that

C_{Spin_{7}(𝔽_{q})}(P)/⟨z⟩ ≤ P̂ ∩ Ω_{7}(q) = P ,

and hence that P is centric in Spin_{7}(𝔽_{q}). □

Proposition A.12 does not hold in general if Spin_{7}(−) is replaced by an arbitrary algebraic group. For example, assume q is an odd prime power, and let P ≤ SL_{5}(q) be the group of diagonal matrices of 2-power order. Then P is 2-centric in SL_{5}(q) and F_{2}(SL_{5}(q))-radical, but is definitely not 2-centric in SL_{5}(𝔽_{q}).

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Department of Mathematical Sciences, University of Aberdeen, Meston Building 339, Aberdeen AB24 3UE, U.K.

  E-mail address: ran@maths.abdn.ac.uk

LAGA, Institut Galilée, Av. J-B Clément, 93430 Villetaneuse, France

  E-mail address: bob@math.univ-paris13.fr