SATURATED FUSION SYSTEMS OVER 2-GROUPS

BOB OLIVER AND JOANA VENTURA

Abstract. We develop methods for listing, for a given 2-group S, all nonconstrained centerfree saturated fusion systems over S. These are the saturated fusion systems which could, potentially, include minimal examples of exotic fusion systems: fusion systems not arising from any finite group. To test our methods, we carry out this program over four concrete examples: two of order 2^{7} and two of order 2^{10}. Our long term goal is to make a wider, more systematic search for exotic fusion systems over 2-groups of small order.

For any prime p and any finite p-group S, a saturated fusion system over S is a category F whose objects are the subgroups of S, whose morphisms are injective group homomorphisms between the objects, and which satisfy certain axioms due to Puig and described here in Section 2. Among the motivating examples are the categories F = F_{S}(G) where G is a finite group with Sylow p-subgroup S: the morphisms in F_{S}(G) are the group homomorphisms between subgroups of S which are induced by conjugation by elements of G. A saturated fusion system F which does not arise in this fashion from a group is called “exotic”.

When p is odd, it seems to be fairly easy to construct exotic fusion systems over p-groups (see, e.g., [BLO2, §9], [RV], and [Rz]), although we are still very far from having any systematic understanding of how they arise. But when p = 2, the only examples we know are those constructed by Levi and Oliver [LO], based on earlier work by Solomon [Sol] and Benson [Bs]. The smallest such example known is over a group of order 2^{10}, and it is possible that there are no exotic examples over smaller groups. Our goal in this paper is to take a first step towards developing techniques for systematically searching for exotic fusion systems, a search which eventually can be carried out in part using a computer.

A fusion system F is constrained (Definition 2.3) if it contains a normal p-subgroup which contains its centralizer. Any constrained fusion system is the fusion system of a unique finite group with analogous properties [BCGLO1, Proposition C]. A fusion system F over S is centerfree (Definition 2.3) if there is no element 1 ≠ z ∈ Z(S) such that each morphism in F extends to a morphism between subgroups containing z which sends z to itself. By [BCGLO2, Corollary 6.14], if there is such a z, and if F is exotic, then there is a smaller exotic fusion system F/⟨z⟩ over S/⟨z⟩. Thus all minimal exotic fusion systems must be nonconstrained and centerfree, and these conditions provide a convenient class of fusion systems to search for and list.

If F is a saturated fusion system over any p-group S, then the F-essential subgroups of S are the proper subgroups P ≨ S which “contribute new morphisms” to the category F: it is the smallest set of objects such that each morphism in S is a composite of restrictions of automorphisms of essential subgroups and of S itself. We refer to Definition 2.3, Proposition 2.5, and Corollary 2.6 for more details. We define a critical subgroup of S to be one which could, potentially, be essential in some fusion system over F. The precise definition (Definition 3.1) is somewhat complicated (and stated without reference to fusion systems), and involves the existence of subgroups of Out(P) which contain strongly embedded subgroups. Thus Bender’s classification of groups with strongly embedded subgroups (at the prime 2) plays a central role in our work. In addition, one important thing about critical subgroups is that the 2-groups we have studied contain very few of them (even those 2-groups which support many “interesting” saturated fusion systems), and we have developed some fairly efficient techniques for listing them.

2000 Mathematics Subject Classification. Primary 20D20. Secondary 20D45, 20D08.

Key words and phrases. finite groups, 2-groups, fusion, simple groups.

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

J. Ventura is partially supported by FCT/POCTI/FEDER and grant PDCT/MAT/58497/2004.

Thus, the first step when trying to find all saturated fusion systems over a 2-group S is to list its critical subgroups. Afterwards, for each critical P (and for P = S), one computes Out(P), and determines which subgroups of Out(P) can occur as Aut_{F}(P) if P is F-essential. The last step is then to put this all together: to see which combinations of essential subgroups and their automorphism groups can generate a nonconstrained centerfree saturated fusion system F.

To illustrate how this procedure works in practice, we finish by listing all nonconstrained centerfree saturated fusion systems over two groups of order 2^{7} and two groups of order 2^{10}. We chose them because each is the Sylow subgroup of several “interesting” simple or almost simple groups; in fact, each is the Sylow 2-subgroup of at least one sporadic simple group. The groups we chose are the Sylow 2-subgroups of M_{22}, M_{23}, and McL; J_{2} and J_{3}; He, M_{24}, and GL_{5}(2); and Co_{3}. The last case is particularly interesting because it is also the Sylow subgroup of the only known exotic fusion system over a 2-group of order ≤ 2^{10}.

Not surprisingly, we found no new exotic fusion systems over any of these four groups, and a much wider and more systematic search will be needed to have much hope of finding new exotic examples. For example, over the group S = UT_{5}(2) of upper triangular 5×5 matrices over 𝔽_{2}, we show (Theorem 6.10) that the only nonconstrained centerfree saturated fusion systems are those of the simple groups He, M_{24}, and GL_{5}(2). Likewise, over the Sylow 2-subgroup of Co_{3}, we show (Theorem 7.8) that each such fusion system is either the fusion system of Co_{3}, or that of the almost simple group Aut(P Sp_{6}(3)), or the exotic fusion system Sol(3) constructed in [LO]. Thus in these cases, we repeat in part the well known results of Held [He] and Solomon [Sol], except that we classify fusion systems over these 2-groups, and do not try to list all groups which realize a given fusion system. But the techniques we use are somewhat different, and we hope that they can eventually make possible a more systematic search for exotic fusion systems.

This approach also makes it easy to determine all automorphisms of the fusion systems we classify. We don’t state it here explicitly, but using the information given about the fusion systems over the four 2-groups we study, one can easily determine their automorphisms, and check they all extend to automorphisms of the associated groups.

The paper is organized as follows. The first section contains background results on finite groups, their automorphism groups, and strongly embedded subgroups, while Section 2 contains background results on fusion systems. Then, in Section 3, critical subgroups are defined, and techniques developed for determining the critical subgroups of a given 2-group. Afterwards, in Sections 4–7, we present our examples, describing the nonconstrained centerfree saturated fusion systems over four different 2-groups.

We would like in particular to thank Kasper Andersen, who helped revive our interest in this program by doing a computer search for some critical subgroups; and Andy Chermak, for (among other things) suggesting we look at the Sylow subgroup of the Janko groups J_{2} and J_{3}. We also give special thanks to the referee, whose many detailed suggestions helped us to make considerable improvements in the paper.

Notation : Most of the time, when α ∈ Aut(P) for some group P, we write [α] for the class of α in Out(P). The one exception to this is the case of automorphisms defined via conjugation: c_{g} denotes conjugation by g, as an element of a group Aut(P) or Out(P) (for some P normalized by g) which will be specified each time. Occasionally, the same letter will be used to describe a subgroup of Aut(P) and its image in Out(P), but that will be stated explicitly in each case.

Since the two authors are topologists, some of our notation clashes with that usually used by group theorists. Commutators are defined [g, h] = ghg^{−1}h^{−1}, and c_{g} denotes conjugation by g in the sense c_{g}(x) = gxg^{−1}. Homomorphisms are written on the left and composed from right to left. When a matrix is used to describe a linear map between vector spaces with respect to given bases, each column contains the coordinates of the image of one basis element. Finally, in what is standard notation, we write Z_{n}(P) for the n-th term in the upper central series of a p-group P; thus Z_{1}(P) = Z(P) and Z_{n}(P)/Z_{n−1}(P) = Z(P/Z_{n−1}(P)).

1. Background results

We collect here some results about groups and their automorphisms which will be needed later. Almost all of them are either well known, or follow from well known constructions.

We first recall some standard notation. For any group G and any prime p, O_{p}(G) denotes the largest normal p-subgroup (the intersection of the Sylow p-subgroups of G), and O^{p}(G) denotes the smallest normal subgroup of p-power index. Also, O_{p′}(G) denotes the largest normal subgroup of order prime to p, and O^{p′}(G) denotes the smallest normal subgroup of index prime to p.

1.1 Automorphisms of p-groups and group cohomology

We first consider conditions which can be used to show that certain automorphisms of a p-group P lie in O_{p}(Aut(P)). Recall that the Frattini subgroup Fr(P) of a pgroup P is the subgroup generated by commutators and p-th powers; i.e., the smallest normal subgroup whose quotient is elementary abelian. It has the property that if g_{1}, …, g_{k} ∈ P are elements whose classes generate P/Fr(P), then they generate P.

Lemma 1.1. Fix a prime p, a p-group P, and a sequence of subgroups

P_{0} ◁ P_{1} ◁ ⋯ ◁ P_{k} = P

such that P_{0} ≤ Fr(P). Set

A = {α ∈ Aut(P) | x^{−1}α(x) ∈ P_{i−1}, all x ∈ P_{i}, all i = 1, …, k} ≤ Aut(P) :

the group of automorphisms which leave each P_{i} invariant and which induce the identity on each quotient group P_{i}/P_{i−1}. Then A is a p-group. If the P_{i} are all characteristic in P, then A ◁ Aut(P), and hence A ≤ O_{p}(Aut(P)).

Proof. To prove that A is a p-group, it suffices to show that each element α ∈ A has p-power order. This follows, for example, from [G, Theorems 5.1.4 & 5.3.2]. The last statement is then clear. □

We next turn to the problem of determining Out(P) for a p-group P. In the next lemma, for any group G and any normal subgroup H ◁ G, we let Aut(G, H) ≤ Aut(G) denote the group of automorphisms α of G such that α(H) = H, and set Out(G, H) = Aut(G, H)/Inn(G).

Lemma 1.2. Fix a group G and a normal subgroup H ◁ G such that C_{G}(H) ≤ H (i.e., H is centric in G). Then there is an exact sequence

1 ⟶ H^{1}(G/H; Z(H)) ⟶^{η} Out(G, H) ⟶^{R}

N_{Out(H)}(Out_{G}(H))/Out_{G}(H) ⟶^{χ} H^{2}(G/H; Z(H)), (1)

where R is induced by restriction, and where all maps except (possibly) χ are homomorphisms. If, furthermore, H is abelian and the extension of H by G/H is split, then R is onto.

Proof. Throughout the proof, g = gH ∈ G/H denotes the class of g ∈ G. We first prove that there is an exact sequence of the following form:

1 ⟶ Z^{1}(G/H; Z(H)) \underrightarrow{η̃} Aut(G, H) ⟶^{Res}

N_{Aut(H)}(Aut_{G}(H)) \underrightarrow{χ̃} H^{2}(G/H; Z(H)). (2)

Here, Z^{1}(G/H; Z(H)) denotes the group of 1-cocycles for G/H with coefficients in Z(H) (“crossed homomorphisms” in the terminology of [Mc, §IV.4]). Explicitly,

Z^{1}(G/H; Z(H)) = {ω: G/H ⟶ Z(H) | ω(g_{1}g_{2}) = ω(g_{1}) · g_{1}ω(g_{2})g_{1}^{−1} ∀ g_{1}, g_{2} ∈ G}.

For ω ∈ Z^{1}(G/H; Z(H)), η̃(ω) is defined by setting η̃(ω)(g) = ω(g)·g for g ∈ G. For g_{1}, g_{2} ∈ G,

η̃(ω)(g_{1}g_{2}) = ω(g_{1}g_{2}) · g_{1}g_{2} = ω(g_{1}) · g_{1}ω(g_{2})g_{1}^{−1} · g_{1}g_{2} = η̃(ω)(g_{1}) · η̃(ω)(g_{2}),

so η̃(ω) ∈ Aut(G, H). To see that η̃ is a homomorphism, fix ω_{1}, ω_{2} ∈ Z^{1}(G/H; Z(H)); then for g_{1}, g_{2} ∈ G,

(η̃(ω_{1}) ∘ η̃(ω_{2}))(g) = η̃(ω_{1})(ω_{2}(g)g) = ω_{1}(g) · ω_{2}(g)g = (ω_{1}ω_{2})(g) · g = η̃(ω_{1}ω_{2})(g)

(since ω_{2}(g)g = g). Clearly, η̃ is injective, and η̃(ω)|_{H} = Id_{H} for each ω.

The map Res sends α ∈ Aut(G, H) to α|_{H} ∈ Aut(H); this lies in the normalizer of Aut_{G}(H) since for all g ∈ G, (α|_{H})c_{g}(α|_{H})^{−1} = c_{α(g)} ∈ Aut_{G}(H). Assume α ∈ Ker(Res); we want to show that α ∈ Im(η̃). Since α ∈ Aut(G) and α|_{H} = Id_{H}, we have c_{g} = c_{α(g)} ∈ Aut(H) for all g ∈ G. Hence α(g) ≡ g (mod Z(H)), and the element α(g)·g^{−1} depends only on g ∈ G/H. So we can define ω: G/H → Z(H) by setting ω(g) = α(g)·g^{−1} for g ∈ G, and α takes the form α(g) = ω(g)·g. Also, for all g_{1}, g_{2} ∈ G,

ω(g_{1}g_{2}) = α(g_{1}g_{2}) · (g_{1}g_{2})^{−1} = α(g_{1})(α(g_{2})g_{2}^{−1})g_{1}^{−1} = ω(g_{1}) · g_{1}ω(g_{2})g_{1}^{−1},

and hence ω ∈ Z^{1}(G/H; Z(H)). This proves the exactness of (2) at Aut(G, H).

We next define χ̃ and prove that Im(Res) = χ̃^{−1}(0). Fix some φ ∈ N_{Aut(H)}(Aut_{G}(H)), and let ψ ∈ Aut(G/Z(H)) be defined by ψ(gZ(H)) = g′Z(H) if φc_{g}φ^{−1} = c_{g′} in Aut(H). This defines ψ uniquely since C_{G}(H) = Z(H) (H is centric). Intuitively, the obstruction to extending φ and ψ to an automorphism of G is the same as the obstruction to two extensions of H by G/H (with the same outer action of G/H on H) being isomorphic, and thus lies in H^{2}(G/H; Z(H)) (cf. [Mc, Theorem IV.8.8]).

To show this explicitly, we first choose a map of sets φ̂: G ⟶ G such that for all g ∈ G and h ∈ H,

φ(ghg^{−1}) = φ̂(g)φ(h)φ̂(g)^{−1}, φ̂(hg) = φ(h)φ̂(g), and φ̂(gh) = φ̂(g)φ(h) (3)

(any two of these imply the third). To construct φ̂, let X = {g_{1}, …, g_{k}} be a set of representatives for cosets in G/H. For each i, let φ̂(g_{i}) be any element of the coset ψ(g_{i}Z(H)) ∈ G/Z(H). Then φ(g_{i}hg_{i}^{−1}) = φ̂(g_{i})φ(h)φ̂(g_{i})^{−1} for all h ∈ H by definition of ψ. So if we set φ̂(hg_{i}) = φ(h)φ̂(g_{i}) for all h ∈ H, then (3) holds for all g and h.

For all g_{1}, g_{2} ∈ G, φ̂(g_{1}g_{2}) ≡ φ̂(g_{1})φ̂(g_{2}) (mod Z(H)), since the two elements have the same conjugation action on H by (3). So there is a function τ: G/H×G/H ⟶ Z(H) such that

φ̂(g_{1}g_{2}) = φ(τ(g_{1}, g_{2})) · φ̂(g_{1})φ̂(g_{2}).

That τ(g_{1}, g_{2}) depends only on the classes of g_{1} and g_{2} in G/H follows from the last two identities in (3). For each triple of elements g_{1}, g_{2}, g_{3} ∈ G,

φ̂(g_{1}g_{2}g_{3}) = φ(τ(g_{1}g_{2}, g_{3})) · φ̂(g_{1}g_{2})φ̂(g_{3}) = φ(τ(g_{1}g_{2}, g_{3}) · τ(g_{1}, g_{2})) · φ̂(g_{1})φ̂(g_{2})φ̂(g_{3})

= φ(τ(g_{1}, g_{2}g_{3})) · φ̂(g_{1})φ̂(g_{2}g_{3}) = φ(τ(g_{1}, g_{2}g_{3})) · φ̂(g_{1})φ(τ(g_{2}, g_{3}))φ̂(g_{2})φ̂(g_{3})

= φ(τ(g_{1}, g_{2}g_{3}) · g_{1}τ(g_{2}, g_{3})g_{1}^{−1}) · φ̂(g_{1})φ̂(g_{2})φ̂(g_{3});

and hence

τ(g_{1}g_{2}, g_{3}) · τ(g_{1}, g_{2}) = τ(g_{1}, g_{2}g_{3}) · g_{1}τ(g_{2}, g_{3})g_{1}^{−1}.  (4)

This is precisely the relation which implies τ is a 2-cocycle (cf. [Mc, §IV.4, (4.8)]).

Assume φ̃: G ⟶ G is another map satisfying (3), and let τ̃ be the 2-cocycle defined using φ̃. Then φ̂(g) ≡ φ̃(g) (mod Z(H)) for each g. So using (3) again, we define σ: G/H ⟶ Z(H) so that φ̃(g) = φ(σ(g))φ̂(g) for all g. Then for g_{1}, g_{2} ∈ G,

τ̃(g_{1}, g_{2}) = φ^{−1}(φ̃(g_{1}g_{2})φ̃(g_{2})^{−1}φ̃(g_{1})^{−1})
= φ^{−1}(φ(σ(g_{1}g_{2}))φ̂(g_{1}g_{2})φ̂(g_{2})^{−1}φ(σ(g_{2})^{−1})φ̂(g_{1})^{−1}φ(σ(g_{1})^{−1}))
= σ(g_{1}g_{2}) · τ(g_{1}, g_{2}) · g_{1}σ(g_{2})^{−1}g_{1}^{−1} · σ(g_{1})^{−1}.

In the notation of [Mc, §IV.4], τ̃ = τ·δ(σ)^{−1}, and thus τ̃ and τ represent the same element in H^{2}(G/H; Z(H)). We now define χ̃ by setting χ̃(φ) = [τ] ∈ H^{2}(G/H; Z(H)).

If φ ∈ Im(Res), then φ̂ can be chosen to be an automorphism of G, so τ = 1, and φ ∈ Ker(χ̃). Conversely, if φ ∈ Ker(χ̃), then for any choice of φ̂ (with τ defined as above), there is σ: G/H ⟶ Z(H) such that τ = δ(σ). So if we define φ̃ by setting φ̃(g) = φ(σ(g))φ̂(g) for all g, the above computations show that τ̃ = 1, and thus that φ̃ ∈ Aut(G, H) and φ ∈ Im(Res).

This proves the exactness of (2). The following sequence is clearly exact:

1 ⟶ Aut_{Z(H)}(G) ⟶^{incl} Inn(G) ⟶^{restr.} Aut_{G}(H) ⟶ 1.

So if we replace the first three terms in (2) by their quotients modulo these three subgroups, the resulting sequence

1 ⟶ Z^{1}(G/H; Z(H))/η̃^{−1}(Aut_{Z(H)}(G)) ⟶^{η} Aut(G, H)/Inn(G)

⟶^{R} N_{Aut(H)}(Aut_{G}(H))/Aut_{G}(H) ⟶^{χ} H^{2}(G/H; Z(H)). (5)

is still exact. For z ∈ Z(H), c_{z}(g) = (zgz^{−1}g^{−1})g for g ∈ G, and hence c_{z} = η̃(ω_{z}) where ω_{z}(g) = (zgz^{−1})g^{−1}. The group of all such ω_{z} is precisely the group B^{1}(G/H; Z(H)) of all 1-coboundaries (cf. [Mc, §IV.2]), and hence the first term in (5) is equal to to H^{1}(G/H; Z(H)). This finishes the proof that (1) is defined and exact.

It remains to prove the last statement. Assume H is abelian, and G = HK where H ∩ K = 1. Thus K projects isomorphically onto G/H ≅ Out_{G}(H), and so we can identify these groups. Fix β ∈ Aut(H) = Out(H), and assume β ∈ N_{Aut(H)}(Aut_{G}(H)). Let γ ∈ Aut(K) be the automorphism of K ≅ Aut_{G}(H) induced by conjugation by β. Then there is an automorphism α ∈ Aut(G) such that α|_{H} = β and α|_{K} = γ, and thus R is surjective in this case. □

Lemma 1.2 will frequently be applied in the following special case:

Corollary 1.3. Let G be a finite 2-group with centric characteristic subgroup H ◁ G. Assume Z(H) has exponent two, and has a basis over 𝔽_{2} which is permuted freely under the conjugation action of G/H. Then there is an isomorphism

R: Out(G) ⟶^{≅} N_{Out(H)}(Out_{G}(H))/Out_{G}(H)

which sends the class of α ∈ Aut(G) to the class of α|_{H} ∈ Aut(H).

Proof. The condition on Z(H) implies that H^{i}(G/H; Z(H)) = 0 for all i > 0. Also, Out(G) = Out(G, H) since H is characteristic in G, and so the result follows directly from Lemma 1.2. □

The following slightly related lemma was suggested to us by the referee.

Lemma 1.4. Let G be a finite 2-group, with normal subgroup H ◁ G of index two. Assume there is x ∈ G∖H of order two. Assume also there is a sequence

1 = H_{0} ≤ H_{1} ≤ ⋯ ≤ H_{m} = H

of subgroups all normal in G, such that for each 1 ≤ i ≤ m, V_{i} =^{def} H_{i}/H_{i−1} is elementary abelian and C_{V_{i}}(x) = [x, V_{i}] (equivalently, |V_{i}| = |C_{V_{i}}(x)|^{2}). Then all involutions in xH = G∖H are conjugate by elements of H.

Proof. We show this by induction on m. When m = 1, all involutions in xH are conjugate since H^{1}(⟨x⟩; H) ≅ C_{H}(x)/[x, H] = 1 (cf. [A1, 17.7]).

If m > 1, then by the induction hypothesis applied to G/H_{1}, all involutions in xH are conjugate modulo H_{1}. Thus if y ∈ xH is another involution, then y is H-conjugate to some y′ ∈ xH_{1}, and y′ is H_{1}-conjugate to x by the first part of the proof. □

1.2 Strongly embedded subgroups

Strongly embedded subgroups of a finite group play a central role in this paper. We begin with the definition.

Definition 1.5. Fix a prime p. For any finite group G, a subgroup G_{0} ≨ G is called strongly embedded at p if p||G_{0}|, and for all g ∈ G∖G_{0}, G_{0} ∩ gG_{0}g^{−1} has order prime to p. A subgroup G_{0} ≨ G is strongly embedded if it is strongly embedded at 2.

The classification of all finite groups with strongly embedded subgroups at 2 is due to Bender.

Theorem 1.6 (Bender). Let G be a finite group with strongly embedded subgroup at the prime 2. Fix a Sylow 2-subgroup S ∈ Syl_{2}(G). Then either S is cyclic or quaternion, or O^{2′}(G/O_{2′}(G)) is isomorphic to one of the simple groups P SL_{2}(2^{n}), P SU_{3}(2^{n}), or Sz(2^{n}) (where n ≥ 2, and n is odd in the last case).

Proof. See [Bd]. □

The following lemma about 𝔽_{2}-representations of groups with strongly embedded subgroups (at the prime 2) will be needed in Section 3, and plays a key role in later applications. When α is an automorphism of a vector space V, we write

[α, V] = Im[V ⟶^{α−Id} V].

Lemma 1.7. Let G be a finite group with strongly embedded subgroup at the prime 2, and let V be an 𝔽_{2}-vector space on which G acts linearly and faithfully. Fix some S ∈ Syl_{2}(G), and let 1 ≠ s ∈ S be any nonidentity element. Then the following hold.

(a) If |S| = 2^{k}, then dim_{𝔽_{2}}(V) ≥ 2k.

(b) If Z(S) ≅ C_{2}^{n}, then dim_{𝔽_{2}}([s, V]) ≥ n.

(c) If S is cyclic of order 4, then dim_{𝔽_{2}}([s, V]) ≥ 2. If S is cyclic or quaternion of order 2^{k} ≥ 8, then dim_{𝔽_{2}}(V) ≥ 3·2^{k−2} and dim_{𝔽_{2}}([s, V]) ≥ 2^{k−2}.

Proof. The result is clear when |S| = 2 (dim(V) ≥ 2· dim([s, V]) ≥ 2), so we assume |S| ≥ 4. If s ∈ S has order ≥ 4, then [s^{2}, V] ⊆ [s, V] ((s^{2} − Id)(v) = (s − Id)(v + s(v))), so it suffices to prove (b) and the statements about [s, V] in (c) when s is an involution. Also, since all involutions in G are conjugate by [Sz2, Lemma 6.4.4], it suffices to prove (b) for just one involution s in S. We handle the case where Z(S) is noncyclic in Case 1, and the case where S is cyclic or quaternion in Case 2.

Case 1: Assume first that O^{2′}(G/O_{2′}(G)) ≅ L where L is simple. There are three subcases to consider. In all cases, we set |S| = 2^{k} and |Z(S)| = 2^{n}.

Case 1A: Assume L ≅ P SL_{2}(q), where q = 2^{k}. Then S ≅ C_{2}^{k}. Also, L contains a dihedral subgroup D of order 2(q + 1). Let h_{1}, h_{2} ∈ L be a pair of involutions which generate D, and let h_{1}, h_{2} ∈ G be a pair of liftings to involutions. Then ⟨h_{1}, h_{2}⟩ is dihedral of order a multiple of 2(q + 1), and in particular, G also has a dihedral subgroup D of order 2(q + 1). Write D = ⟨g, h⟩, where |g| = q + 1 and |h| = 2.

By Zsigmondy’s theorem (see [Z] or [Ar, p.358]), there is a prime p such that p|(2^{2k}−1) and p∤(2^{ℓ}−1) for ℓ < 2k — unless 2k = 6 in which case we take p = 9. Thus p|(q + 1). Since g acts faithfully and p is a prime power, there is at least one eigenvalue λ ∈ 𝔽_{2} of g with order a multiple of p. Since the set of eigenvalues is stable under (λ ↦ λ^{2}), the number of eigenvalues (hence the dimension of V) is at least equal to the order of 2 modulo p; thus dim(V) ≥ 2k. Furthermore, the 2k eigenvalues {λ^{2^{i}}} are permuted in pairs under the action of h, and so dim([h, V]) ≥ k.

Case 1B: Next assume L ≅ Sz(q) for q = 2^{n} ≥ 8, where n is odd. Then |S| = q^{2} = 2^{2n} and Z(S) ≅ C_{2}^{n}. By Zsigmondy’s theorem again, there is a prime p such that 2 has order 4n modulo p. Since

2^{4n} − 1 = q^{4} − 1 = (q + \sqrt{2q} + 1)(q − \sqrt{2q} + 1)(q^{2} − 1),

p divides at least one of the factors m = q ± \sqrt{2q} + 1. By [HB3, Theorem XI.3.10], L contains a dihedral subgroup of order 2m. Hence by the same argument as that used in Case 1A, dim(V) ≥ 4n, and dim([h, V]) ≤ 2n for h ∈ G of order 2.

Case 1C: Now assume L ≅ P SU_{3}(q) for q = 2^{n} with n ≥ 2. Then |S| = q^{3} = 2^{3n} and Z(S) ≅ C_{2}^{n}. Also, L contains a cyclic subgroup of order m = (q^{2} − q + 1) or m = (q^{2} − q + 1)/3, which comes from regarding 𝔽_{q^{6}} as a 3-dimensional 𝔽_{q^{2}}-vector space with hermitian product (x, y) = xy^{q^{3}}.

Using Zsigmondy’s theorem, choose a prime p such that 2 has order 6n modulo p (recall n ≥ 2). Then p|m since q^{6} − 1 = (q^{3} − 1)(q + 1)(q^{2} − q + 1). So by the same arguments as used in Case 1A, dim(V) ≥ 6n.

Let D ≤ L be the dihedral subgroup of order 2(q +1) generated by diagonal matrices diag(u, u^{−1}, 1) for u ∈ 𝔽_{q^{2}} with u^{q+1} = 1, and by the permutation matrix ( \begin{matrix}0 & 1 & 0 \\ −1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ). Since 2 has order 2n modulo q + 1, dim([h, V]) ≥ n by the arguments used earlier.

Case 2: Now assume S is cyclic or quaternion of order 2^{k} with k ≥ 2, and set H = O_{2′}(G). Let s ∈ S be the (unique) involution. Since sH ∈ Z(G/H) by the Brauer-Suzuki theorem [BS, Theorem 2], [s, H] ≠ 1. For each prime p||H|, the number of Sylow p-subgroups of H is odd, and hence there is at least one subgroup H_{p} ∈ Syl_{p}(H) which is normalized by S. Since H is generated by these H_{p}, at least one of them is not centralized by s. So upon replacing H by some appropriate Sylow subgroup H_{p} and G by H_{p}S, we can assume G = HS where H is a normal p-subgroup.

By a theorem of Thompson [G, Theorem 5.3.13], there is a characteristic subgroup Q ≤ H such that S still acts faithfully on Q, Fr(Q) = [Q, Q] ≤ Z(Q) is elementary abelian, and Q has exponent p. Set Q_{0} = [s, Q] ≠ 1. Then Q_{0} is S-invariant since s ∈ Z(S), and is generated by elements [s, g] (g ∈ Q) which are inverted under conjugation by s. Upon replacing H by Q_{0} and G by Q_{0}S, we are reduced to the case where H has exponent p, Z(H) ≥ Fr(H), and s acts on H/Fr(H) via −Id.

Fix an irreducible 𝔽_{2}[H]-module W ⊆ V with nontrivial H-action. Let K ◁ H be the kernel of the action; thus H/K acts faithfully and irreducibly on W. Then Z(H/K) is cyclic, since otherwise any faithful representation would split, and hence |Z(H/K)| = p since H has exponent p. Set S_{0} = N_{S}(K), and set W^{∗} = ⟨u(W) | u ∈ S_{0}⟩ ⊆ V. For each u ∈ S, u(W^{∗}) is a sum of faithful irreducible H/uKu^{−1}-modules. Hence the |S/S_{0}| distinct submodules u(W^{∗}) are linearly independent, and so rk(V) ≥ |S/S_{0}|· rk(W^{∗}). Also, either S_{0} = 1 and rk([s, V]) ≥ \frac{1}{2}|S|, or s ∈ S_{0} and rk([s, V]) ≥ |S/S_{0}|· rk([s, W^{∗}]). So we are done if S_{0} = 1. Otherwise, upon replacing V by W^{∗} and G by HS_{0}/K, we are reduced to the case where Z(H) ≅ C_{p} and V is a sum of faithful, irreducible 𝔽_{2}[H]-modules.

If H is abelian, then H ≅ C_{p}, sgs^{−1} = g^{−1} for g ∈ H, and thus S permutes freely the nontrivial irreducible 𝔽_{2}[H]-representations. So rk(V) = dim_{𝔽_{2}}(𝔽_{2} ⊗_{𝔽_{2}} V) ≥ |S| in this case, and rk([s, V]) ≥ \frac{1}{2}|S|.

If H is nonabelian, then since Fr(H) ≤ Z(H) ≅ C_{p}, H must be extraspecial of order p^{1+2r} for some r ≥ 1. All faithful irreducible 𝔽_{2}[H]-modules have the same rank ep^{r} for some e ≥ 2 depending on p. By construction, s acts via −Id on H/Z(H). Hence S acts freely on (H/Fr(H))∖1, so |S| = 2^{k}|(p^{2r} − 1), and 2^{k−1} must divide one of the factors p^{r} ± 1. So rk(V) ≥ 2p^{r} ≥ 2(2^{k−1} − 1) ≥ 2^{k} − 2^{k−2} = 3·2^{k−2} when k ≥ 3. When k ≤ 3, rk(V) ≥ 2p ≥ 6 ≥ 3·2^{k−2}, and thus this lower bound on rk(V) holds for all k.

Fix any g ∈ H∖Z(H) such that sgs^{−1} = g^{−1}. The eigenvalues (in 𝔽_{2}) of the action of g on V include all p-th roots of unity with equal multiplicity, and the action of s sends the eigenspace of ζ to that of ζ^{−1}. Thus rk([s, V]) = \frac{p−1}{2p}· rk(V) ≥ \frac{1}{3} rk(V). □

1.3 General results on groups

The following result is useful when listing subgroups of Out(P), for a p-group P, which have a given Sylow p-subgroup. The most important case is that where Q ◁ G and H_{0} ≤ H ≤ G are such that Q = O_{p}(G), G = QH, and H_{0} ∈ Syl_{p}(H). But we also have applications which require the more general setting.

Proposition 1.8. Fix a prime p, a finite group G, and a normal abelian p-subgroup Q ◁ G. Let H ≤ G be such that Q ∩ H = 1, and let H_{0} ≤ H be of index prime to p. Consider the set

H = {H′ ≤ G | H′ ∩ Q = 1, QH′ = QH, H_{0} ≤ H′}.

Then for each H′ ∈ H, there is g ∈ C_{Q}(H_{0}) such that H′ = gHg^{−1}.

Proof. Fix H′ ∈ H, and define χ: H → Q by setting χ(h) = h′h^{−1}, where h′ is the unique element in H′∩hQ. By straightforward calculation, χ(h_{1}h_{2}) = χ(h_{1})·h_{1}χ(h_{2})h_{1}^{−1} for all h_{1}, h_{2} ∈ H, and thus χ ∈ Z^{1}(H; Q) is a 1-cocycle. Also, χ|_{H_{0}} = 1. Since H^{1}(H; Q) injects into H^{1}(H_{0}; Q) (Q is abelian and [H : H_{0}] is prime to p), this means that χ is a coboundary; i.e., χ(h) = ghg^{−1}h^{−1} for some g ∈ Q. Thus H′ = gHg^{−1}. Also, [g, H_{0}] = 1, since ghg^{−1}h^{−1} = χ(h) = 1 for each h ∈ H_{0}. □

As an example of why Q must be assumed abelian in the above proposition, consider the group G = GL_{2}(3) (and p = 2). Set

Q = O_{2}(G) = ⟨( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , ( \begin{matrix}1 & 1 \\ 1 & −1 \end{matrix} )⟩ ≅ Q_{8}.

Consider the subgroups

H = ⟨( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} )⟩ ≅ Σ_{3}, H′ = ⟨( \begin{matrix}1 & 0 \\ −1 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} )⟩, and H_{0} = ⟨( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} )⟩.

Then H and H′ are both splittings of the surjection G → G/Q ≅ Σ_{3} which contain H_{0} as Sylow 2-subgroup, but they are not conjugate in G. Instead, H′ is G-conjugate to the subgroup H′′ = ⟨( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} )⟩. The 1-cocycle H → Q which sends the subgroup of order three to I and its complement to −I is nontrivial in H^{1}(H; Q_{8}), but its restriction is trivial in H^{1}(H_{0}; Q_{8}).

The following very elementary lemma will be used later to list subgroups of a given 2-group which are not normal, and have index two in their normalizers.

Lemma 1.9. Assume S is a 2-group with a normal subgroup S_{0} ◁ S, such that S_{0} and S/S_{0} are both elementary abelian and |S/S_{0}| ≤ 4. Assume P ≤ S is such that P is not normal and |N_{S}(P)/P | = 2. Set P_{0} = P ∩ S_{0}. Let m be the number of cosets xS_{0} ∈ S/S_{0} such that xS_{0} ≠ S_{0} and [x, S_{0}] ≤ P_{0}. Then one of the six cases listed in Table 1.1 holds.

rk(S_{0}/P_{0}) |P/P_{0}| |S/S_{0}| m other properties
(a)  1  1  2, 4 0
(b)  1  2  4  1 P_{0} ◁̸ S; [x, S_{0}] ≤ P_{0} ⇔ x ∈ P S_{0}
(c)  1  2  4  3 P_{0} ◁ S, [S, S_{0}] ≨ [S, S] ≰ P_{0}
(d)  2  2  2, 4 0 P S_{0}/P_{0} ≅ D_{8}
(e)  2  4  4  1
(f)  3, 4  4  4  0 rk(C_{S_{0}/P_{0}}(P/P_{0})) = 1

Table 1.1 Proof. Since S_{0} is abelian, [x, S_{0}] depends only on the class of x in S/S_{0}. So m is well defined, independently of the choice of coset representatives.

By assumption, P is not normal in S. Since S/S_{0} is abelian, this implies P_{0} ≨ S_{0}. For g ∈ S_{0}, g ∈ N_{S}(P) if and only if [g, P] ≤ P_{0}, or equivalently, gP_{0} ∈ C_{S_{0}/P_{0}}(P/P_{0}). Hence since |N_{S}(P)/P | = 2 and C_{S_{0}/P_{0}}(P/P_{0}) ≠ 1,

|C_{S_{0}/P_{0}}(P/P_{0})| = 2 and N_{S}(P) ≤ P S_{0}.  (6)

We next claim that if we regard S_{0}/P_{0} as an 𝔽_{2}[P/P_{0}]-module via the conjugation action, then

S_{0}/P_{0} is isomorphic to a submodule of 𝔽_{2}[P/P_{0}] and rk(S_{0}/P_{0}) ≤ |P/P_{0}|. (7)

Since 𝔽_{2}[P/P_{0}] is injective as a module over itself (since its dual is projective), the (unique) monomorphism from C_{S_{0}/P_{0}}(P/P_{0}) ≅ 𝔽_{2} into the fixed subspace of 𝔽_{2}[P/P_{0}] extends to an 𝔽_{2}[P/P_{0}]-linear homomorphism φ from S_{0}/P_{0} to 𝔽_{2}[P/P_{0}]. Also, φ must be injective, since otherwise its kernel would have to contain the fixed subgroup C_{S_{0}/P_{0}}(P/P_{0}), and this proves (7).

We are now ready to consider the individual cases. Assume first rk(S_{0}/P_{0}) = 1. If |P/P_{0}| = 1, then P ≤ S_{0}, and [x, S_{0}] ≤ P_{0} = P only for x ∈ N_{S}(P) = S_{0}. Thus m = 0, and we are in the situation of (a). If |P/P_{0}| > 1, then |P/P_{0}| < |S/S_{0}| (since otherwise [S : P] = 2 and P is normal), and thus |P/P_{0}| = 2 and |S/S_{0}| = 4. Also, N_{S}(P) = P S_{0} by (6). If P_{0} ◁ S, then S_{0}/P_{0} ≅ C_{2} is central in S/P_{0}, so [S, S_{0}] ≤ P_{0}, and m = 3. Also, [S, S] ≰ P_{0} in this case (otherwise P would be normal in S), so [S, S_{0}] ≨ [S, S], and we are in case (c). If P_{0} ◁̸ S, then P S_{0} = N_{S}(P_{0}), S_{0}/P_{0} is central in P S_{0}/P_{0}, so [x, S_{0}] ≤ P_{0} exactly when x ∈ P S_{0}. Thus m = 1, and we are in case (b).

If rk(S_{0}/P_{0}) = 2 and |P/P_{0}| = 2, then S_{0}/P_{0} is free as an 𝔽_{2}[P/P_{0}]-module by (7), so P S_{0}/P_{0} is nonabelian of order 8 containing C_{2}^{2} and hence isomorphic to D_{8}. Every automorphism of D_{8} which leaves invariant a subgroup isomorphic to C_{2}^{2} is inner: this follows as a special case of Lemma 1.2, but also from the well known description of Out(D_{8}) ≅ C_{2}. So for each y ∈ N_{S}(P_{0}), there is g ∈ P S_{0} such that the conjugation action of yg on P S_{0}/P_{0} ≅ D_{8} is the identity. In other words, c_{yg}|_{P S_{0}} ≡ Id (mod P_{0}), and in particular, yg ∈ N_{S}(P) ≤ P S_{0} by (6). This proves that N_{S}(P_{0}) = P S_{0}, and thus [x, P_{0}] ≰ P_{0} for x ∈ S∖P S_{0}. Also, since S_{0}/P_{0} is not fixed by the action of P/P_{0}, [x, S_{0}] ≰ P_{0} for x ∈ P S_{0}∖S_{0}. Thus m = 0, and we are in the situation of (d).

By (7), it remains only to consider the cases where |P/P_{0}| = |S/S_{0}| = 4 and 2 ≤ rk(S_{0}/P_{0}) ≤ 4. If rk(S_{0}/P_{0}) = 2, then since P/P_{0} acts on it nontrivially by (6), it must be a free 𝔽_{2}[P/P_{1}] module for some P_{1} ≤ P of index two containing P_{0}. Hence [x, S_{0}] ≤ P_{0} for x ∈ P_{1} (x centralizes S_{0}/P_{0}), [x, S_{0}] ≰ P_{0} for x ∈ S∖P_{1}, so m = 1, and we are in case (e). If rk(S_{0}/P_{0}) = 3, 4, then S_{0}/P_{0} is isomorphic to a submodule of index one or two in 𝔽_{2}[P/O_{0}] by (7), hence each x ∈ S∖S_{0} acts nontrivially on S_{0}/P_{0}, and so m = 0. Together with (6), this proves we are in case (f). □

2. Fusion systems

We first recall the definition of an (abstract) saturated fusion system. For any group G and any x ∈ G, c_{x} denotes conjugation by x (c_{x}(g) = xgx^{−1}). For H, K ≤ G, we write

Hom_{G}(H, K) = {φ ∈ Hom(H, K) | φ = c_{x} some x ∈ G}.

We also set Aut_{G}(H) = Hom_{G}(H, H) ≅ N_{G}(H)/C_{G}(H).

Definition 2.1 ([Pg], [BLO2, Definition 1.1]). A fusion system over a finite p-group S is a category F, with Ob(F) the set of all subgroups of S, which satisfies the following two properties for all P, Q ≤ S:

• Hom_{S}(P, Q) ⊆ Hom_{F}(P, Q) ⊆ Inj(P, Q); and

• each φ ∈ Hom_{F}(P, Q) is the composite of an isomorphism in F followed by an inclusion.

When F is a fusion system over S, two subgroups P, Q ≤ S are said to be F-conjugate if they are isomorphic as objects of the category F. A subgroup P ≤ S is called fully centralized in F (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.

Definition 2.2 ([Pg], [BLO2, Definition 1.2]). A fusion system F over a finite p-group S is saturated if the following two conditions hold:

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

(II) (Extension axiom) If P ≤ S and φ ∈ Hom_{F}(P, S) are such that φ(P) is fully centralized, and if we set

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

then there is φ ∈ Hom_{F}(N_{φ}, S) such that φ|_{P} = φ.

For any finite group G and any Sylow subgroup S ∈ Syl_{p}(G), the fusion system of G (at p) is the category F_{S}(G), whose objects are the subgroups of S, and with morphism sets Mor_{F_{S}(G)}(P, Q) = Hom_{G}(P, Q). This is easily shown to be saturated using the Sylow theorems (cf. [BLO2, Proposition 1.3]). A saturated fusion system is exotic if it is not the fusion system of any finite group.

The following definitions play a central role in this paper. In general, when F is a fusion system over S and P ≤ S, we write Out_{F}(P) = Aut_{F}(P)/Inn(P) and Out_{S}(P) = Aut_{S}(P)/Inn(P).

Definition 2.3. Fix a prime p, a p-group S, and a saturated fusion system F over S. Let P ≤ S be any subgroup.

• P is F-centric if C_{S}(P′) = Z(P′) for all P′ which is F-conjugate to P.

• P is F-radical if O_{p}(Out_{F}(P)) = 1; i.e., if Out_{F}(P) contains no nontrivial normal p-subgroup.

• P is F-essential if P is F-centric and fully normalized in F, and Out_{F}(P) contains a strongly embedded subgroup at p.

• P is central in F if every morphism φ ∈ Hom_{F}(Q, R) in F extends to a morphism φ ∈ Hom_{F}(P Q, P R) such that φ|_{P} = Id_{P}.

• P is normal in F if P ◁ S, and every morphism φ ∈ Hom_{F}(Q, R) in F extends to a morphism φ ∈ Hom_{F}(P Q, P R) such that φ(P) = P.

• The fusion system F is nonconstrained if there is no subgroup P ≤ S which is F-centric and normal in F.

• For any φ ∈ Aut(S), φFφ^{−1} denotes the fusion system over S defined by

Hom_{φFφ^{−1}}(P, Q) = φ·Hom_{F}(φ^{−1}(P), φ^{−1}(Q))·φ^{−1}

for all P, Q ≤ S.

When F = F_{S}(G) for a finite group G with S ∈ Syl_{p}(G), then P ≤ S is F-centric if and only if it is p-centric in G: that is, Z(P) ∈ Syl_{p}(C_{G}(P)), or equivalently, C_{G}(P) = Z(P) × C′_{G}(P) for some (unique) subgroup C′_{G}(P) of order prime to p. The subgroup P is F-essential if and only if it is p-centric in G, N_{S}(P) ∈ Syl_{p}(N_{G}(P)), and N_{G}(P)/(P ·C_{G}(P)) has a strongly embedded subgroup at p.

We say that a fusion system is “centerfree” if it contains no nontrivial central subgroup. Our main goal in this paper is to develop techniques for listing, for a given 2-group S, all centerfree nonconstrained saturated fusion systems over S (up to isomorphism). This restriction is motivated in part by the two results stated in the following theorem: they imply that any minimal exotic fusion system is centerfree and nonconstrained.

Theorem 2.4. Let F be a saturated fusion system over a finite p-group S.

(a) If F is constrained, then there is up to isomorphism a unique p′-reduced p-con-strained finite group G (i.e., O_{p′}(G) = 1 and C_{G}(O_{p}(G)) ≤ O_{p}(G)) such that F ≅ F_{S}(G).

(b) If A ◁ S is central in F, then F is exotic if and only if F/A is exotic. Here, F/A is the fusion system over S/A such that for all P, Q ≤ S containing A, Hom_{F/A}(P/A, Q/A) is the image of Hom_{F}(P, Q) under projection.

Proof. See [BCGLO1, Proposition C] and [BCGLO2, Corollary 6.14]. In both cases, much more precise results are shown. In (a), one can choose G with normal psubgroup Q such that Q ≅ O_{p}(F) (the maximal normal p-subgroup of F) and G/Q ≅ Aut_{F}(O_{p}(F)). Under the hypotheses of (b), if F/A is the fusion system of a finite group G, then F is the fusion system of a central extension of G by A. □

One of the key problems when constructing fusion systems over a p-group S is to determine which subgroups of S can contribute automorphisms; i.e., for which P ≤ S the group Aut_{F}(P) need not be generated by restrictions of automorphisms of larger subgroups. This is what motivates the definition of F-essential subgroups. The following proposition and corollary are due to Puig [Pg, Theorem 5.8], and were originally pointed out to us by Grodal.

Proposition 2.5. Let F be a saturated fusion system over a p-group S, and let P ≨ S be an F-centric subgroup which is fully normalized in F. Then P is F-essential if and only if Aut_{F}(P) is not generated by restrictions of morphisms between strictly larger subgroups of S.

Proof. Since P is fully normalized, Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)). Since P ≨ S, we have N_{S}(P) ≩ P, and so Aut_{S}(P) ≩ Inn(P) since P is F-centric.

Let G_{0} ≤ Aut_{F}(P) be the subgroup generated by those φ ∈ Aut_{F}(P) which extend to morphisms between strictly larger subgroups of S. We first claim that

G_{0} = ⟨φ ∈ Aut_{F}(P) | φ^{−1}Aut_{S}(P)φ ∩ Aut_{S}(P) ≩ Inn(P)⟩.  (1)

To see this, fix φ ∈ Aut_{F}(P) such that φ^{−1}Aut_{S}(P)φ∩Aut_{S}(P) ≩ Inn(P), and consider the group

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

Then Aut_{N_{φ}}(P) = φ^{−1}Aut_{S}(P)φ ∩ Aut_{S}(P) ≩ Inn(P), so N_{φ} ≩ P. By the extension axiom, φ extends to a morphism in Hom_{F}(N_{φ}, S), and this proves that φ ∈ G_{0}.

Conversely, if φ ∈ Aut_{F}(P) extends to φ ∈ Hom_{F}(Q, S) for some Q ≩ P, then φAut_{Q}(P)φ^{−1} ≤ Aut_{S}(P), and

φ^{−1}Aut_{S}(P)φ ∩ Aut_{S}(P) ≥ Aut_{Q}(P) ≩ Inn(P).

This proves (1). For all α ∈ Aut_{F}(P)∖G_{0} and all β ∈ G_{0},

α^{−1}β^{−1}Aut_{S}(P)(βα) ∩ Aut_{S}(P) = Inn(P)

by (1), and thus the intersection of each Sylow subgroup of α^{−1}G_{0}α with Aut_{S}(P) is Inn(P). In other words, Inn(P) ∈ Syl_{p}(αG_{0}α^{−1} ∩ G_{0}) for all α ∈ Aut_{F}(P)∖G_{0}, which implies that G_{0}/Inn(P) is strongly embedded in Out_{F}(P) = Aut_{F}(P)/Inn(P). Conversely, if Out_{F}(P) contains any strongly embedded subgroup at p, then there is a strongly embedded subgroup H which contains the Sylow p-subgroup Out_{S}(P), and G_{0}/Inn(P) ≤ H ≨ Out_{F}(P) by (1) and the definition of a strongly embedded subgroup. □

As a corollary, we get Alperin’s fusion theorem stated for restriction to essential subgroups. Roughly, it says that every saturated fusion system is generated by automorphisms of S and of essential subgroups, and their restrictions.

Corollary 2.6. Fix a saturated fusion system F over a p-group S. Then for each P, P′ ≤ S and each φ ∈ Iso_{F}(P, P′), there are subgroups P = P_{0}, P_{1}, …, P_{k} = P′, subgroups Q_{i} ≥ ⟨P_{i−1}, P_{i}⟩ (i = 1, …, k) which are F-essential or equal to S, and automorphisms φ_{i} ∈ Aut_{F}(Q_{i}), such that φ_{i}(P_{i−1}) = P_{i} for all i and φ = (φ_{k}|_{P_{k−1}}) ∘ ⋯ ∘ (φ_{1}|_{P_{0}}).

Proof. By Alperin’s fusion theorem in the form shown in [BLO2, Theorem A.10], this holds if we allow the Q_{i} to be any F-centric F-radical subgroups of S which are fully normalized in F. So the corollary follows immediately from that together with Proposition 2.5. □

3. Semicritical and critical subgroups

The following definition gives necessary conditions for subgroups of a p-group to possibly be F-radical or F-essential in some fusion system.

Definition 3.1. Let S be a finite p-group. A subgroup P ≨ S will be called semicritical if the following two conditions hold:

(a) P is (p-)centric in S; and

(b) Out_{S}(P) ∩ O_{p}(Out(P)) = 1.

A subgroup P ≤ S will be called critical if it is semicritical, and if

(c) there are subgroups

Out_{S}(P) ≤ G_{0} ≨ G ≤ Out(P)

such that G_{0} is strongly embedded in G at p and Out_{S}(P) ∈ Syl_{p}(G).

The importance of (semi)critical subgroups lies in the following proposition.

Proposition 3.2. Fix a p-group S, a saturated fusion system F over S, and a subgroup P ≨ S. If P is F-centric and F-radical, then it is a semicritical subgroup of S. If P is F-essential, then P is a critical subgroup of S.

Proof. Set G = Out_{F}(P). If P is F-centric and F-radical, then

Out_{S}(P) ∩ O_{p}(Out(P)) ≤ G ∩ O_{p}(Out(P)) ≤ O_{p}(G) = 1,

and so P is a semicritical subgroup of S.

If P is F-essential, then by definition, P is F-centric (hence centric in S), fully normalized in F, and G =^{def} Out_{F}(P) contains a strongly embedded subgroup G_{0} ≨ G at p. Since any strongly embedded subgroup at p contains a Sylow p-subgroup, we can assume (after replacing G_{0} by a conjugate subgroup if necessary) that G_{0} ≥ Out_{S}(P) ∈ Syl_{p}(G). Since O_{p}(G) ≤ gG_{0}g^{−1} ∩ G_{0} for all g ∈ G, this shows that O_{p}(G) = 1, hence that P is F-radical, and thus a semicritical subgroup of S. This proves that P is critical in S. □

For the above definition to be useful, simple criteria are needed which allow us to eliminate most subgroups as not being critical. This works best when p = 2. The following proposition gives some criteria for doing this; criteria which are useful mostly when P has index ≥ 4 in its normalizer. For example, point (a) implies that P is not critical in S if Out_{S}(P) contains a subgroup isomorphic to D_{8} — since D_{8} contains noncentral involutions.

Recall that when V is a vector space and α is a linear automorphism of V, we write

[α, V] = Im[V ⟶^{α−Id} V].

Proposition 3.3. Fix a critical subgroup P of a 2-group S, and set S_{0} = N_{S}(P)/P ≅ Out_{S}(P). Then the following hold.

(a) Either S_{0} is cyclic, or Z(S_{0}) = {g ∈ S_{0} | g^{2} = 1}. If rk(Z(S_{0})) > 1, then |S_{0}| = |Z(S_{0})|^{m} for m = 1, 2, or 3.

(b) All involutions in S_{0} are conjugate in Out(P), and hence in Aut(P/Fr(P)). In fact, there is a subgroup R ≤ Out(P) (or R ≤ Aut(P/Fr(P))) of odd order, which normalizes S_{0} and permutes its involutions transitively.

(c) Set |S_{0}| = 2^{k}. Then rk(P/Fr(P)) ≥ 2k. If k ≥ 2, then rk([s, P/Fr(P)]) ≥ 2 for all 1 ≠ s ∈ S_{0}.

(d) Assume Z(S_{0}) ≅ C_{2}^{n} with n ≥ 2, and fix 1 ≠ s ∈ Z(S_{0}). Then rk([s, P/Fr(P)]) ≥ n.

Proof. Fix subgroups

Out_{S}(P) ≤ G_{0} ≨ G ≤ Out(P)

such that G_{0} is strongly embedded in G and Out_{S}(P) ∈ Syl_{2}(G). In particular, O_{2}(G) = 1. Since the kernel of the natural map from Out(P) to Out(P/Fr(P)) is a 2-group by Lemma 1.1, the induced action of G on P/Fr(P) is still faithful.

By Bender’s theorem ([Bd] or Theorem 1.6), either S_{0} ≅ Out_{S}(P) is cyclic or quaternion, or O^{2′}(G/O_{2′}(G)) is isomorphic to one of the simple groups P SL_{2}(2^{n}), P SU_{3}(2^{n}), or Sz(2^{n}) (where n ≥ 2, and n is odd in the last case).

(a) This is clear if S_{0} is cyclic or quaternion. If not, let L be the simple group L = O^{2′}(G/O_{2′}(G)). If L ≅ P SL_{2}(2^{n}), then S_{0} ≅ C_{2}^{n}. If L ≅ Sz(q), where q = 2^{n} for odd n ≥ 3, then by Suzuki’s description of the Sylow 2-subgroups [Sz, §4, Lemma 1] (see also [Sz, §9]), |S_{0}| = q^{2}, Z(S_{0}) ≅ C_{2}^{n}, and all involutions in S_{0} are in Z(S_{0}). So (a) holds in both of these cases.

If L ≅ P SU_{3}(q), where q = 2^{n} for odd n ≥ 3, then we can identify

S_{0} = \{V (r, s) | r, s ∈ 𝔽_{q^{2}} | r + r̄ = ss̄\} where r̄ = r^{q} and V (r, s) = ( \begin{matrix}1 & s & r \\ 0 & 1 & s̄ \\ 0 & 0 & 1 \end{matrix}) .

Also, V (r, s)·V (u, v) = V (r + u + sv̄, s + v). Thus |S_{0}| = 2^{3n}, Z(S_{0}) = {V (r, 0) | r ∈ 𝔽_{q}} ≅ C_{2}^{n}, and V (r, s)^{2} = V (ss̄, 0) = 1 only if s = 0. So (a) holds in this case, also.

(b) By [Sz2, Lemma 6.4.4], all involutions in G_{0} are conjugate to each other. Since the involutions in S_{0} are all in Z(S_{0}), they must be conjugate to each other by elements in N_{G_{0}}(S_{0}); and we can write N_{G_{0}}(S_{0}) = S_{0}⋊R where |R| is odd.

(c,d) These follow immediately from Lemma 1.7, applied with V = P/Fr(P). □

Proposition 3.3 will be our main tool when identifying those critical subgroups which have index ≥ 4 in their normalizer. The following lemma is an easy consequence of Lemma 1.1, and will be useful in many situations in the index two case. It will often be applied with Θ = 1, or with Θ = Z_{2}(P) (the subgroup such that Z_{2}(P)/Z(P) = Z(P/Z(P))).

Lemma 3.4. Fix a prime p, a p-group S, a subgroup P ≤ S, and a subgroup Θ ≤ P characteristic in P. Assume there is g ∈ N_{S}(P)∖P such that

(a) [g, P] ≤ Θ·Fr(P), and

(b) [g, Θ] ≤ Fr(P).

Then c_{g} ∈ O_{p}(Aut(P)), and hence P is not semicritical in S.

Proof. Point (a) implies that c_{g} is the identity on P/Θ·Fr(P), and (b) implies it is the identity on Θ·Fr(P)/Fr(P). Hence c_{g} ∈ O_{p}(Aut(P)) by Lemma 1.1, and so P is not semicritical in S. □

Lemma 3.4 will be our main tool when looking for critical subgroups of index two in their normalizer. But there are two more, closely related, lemmas which will also be useful in certain cases. The following one can be thought of as a refinement of Lemma 3.4, at least when p = 2.

Lemma 3.5. Fix a 2-group S, a semicritical subgroup P ≤ S, and g ∈ N_{S}(P)∖P such that c_{g} has order two in Out_{S}(P). Then there is α ∈ Aut(P) of odd order, and x ∈ [g, P], such that x ∉ Fr(P) and [g, α(x)] ∉ Fr(P).

Proof. Since P is semicritical, c_{g} ∉ O_{2}(Out(P)). Hence by the Baer-Suzuki theorem (cf. [A1, Theorem 39.6]), there is β ∈ Out(P) such that Δ = ⟨c_{g}, βc_{g}β^{−1}⟩ ≤ Out(P) is not a 2-group. Thus Δ contains a dihedral group of order 2m for m odd, and we can assume that β is chosen so that |Δ| = 2m. Let γ̂ ∈ Δ ≤ Out(P) be an element of order m inverted by c_{g}, and let γ ∈ Aut(P) be an automorphism of odd order whose class in Out(P) is γ̂.

Set V = P/Fr(P), regarded as an 𝔽_{2}[Δ]-module. Since γ̂ has odd order, V splits as a product V = C_{V} (γ̂) × V′, where γ̂ acts on V′ = [γ̂, V] without fixed component. Let V_{0} ⊆ V′ be an irreducible 𝔽_{2}[Δ]-submodule. Since γ̂ acts nontrivially on V_{0} (and the subgroup of elements of Δ which act trivially is normal), c_{g} acts nontrivially on V_{0}, and hence [g, V_{0}] ≠ 1.

Fix 1 ≠ x̂ ∈ [g, V_{0}] ≤ C_{V_{0}}(g). The γ̂-orbit of x̂ is Δ-invariant (since c_{g}(x̂) = x̂ and Δ = ⟨γ̂, c_{g}⟩), and hence it generates V_{0} since V_{0} is irreducible. Thus there is i such that γ̂^{i}(x̂) ∉ C_{V_{0}}(g). Now choose x ∈ [g, P] such that xFr(P) = x̂; then [g, γ^{i}(x)] ∉ Fr(P). Set α = γ^{i}; then α and x satisfy the conclusion of the lemma. □

In the special case of Lemma 3.5 where [g, S] has order two, one can take this much farther. Recall Z_{2}(S) ◁ S is the subgroup such that Z_{2}(S)/Z(S) = Z(S/Z(S)).

Lemma 3.6. Let S be a 2-group, and fix elements z ∈ Z(S) and g ∈ Z_{2}(S) such that z^{2} = 1 and [g, S] ≤ ⟨z⟩. Assume P is a critical subgroup of S such that g ∉ P. Then the following hold.

(a) |N_{S}(P)/P | = 2, z ∉ Fr(P), and P = C_{S}(h) for some h ∈ S such that h^{2} = 1 and [g, h] = z.

(b) If y ∈ S∖⟨g, z⟩ is such that [y, S] ≤ ⟨z⟩, then either y ∈ Z(P) and h is not S-conjugate to yh, or gy ∈ Z(P) and h is not S-conjugate to gyh.

Proof. (a) By assumption (and since P is centric), [g, P] ≤ [g, S] = ⟨z⟩ ≤ Z(S) ≤ P. In particular, g ∈ N_{S}(P). By Lemma 3.4 (applied with Θ = 1), [g, P] ≰ Fr(P), and thus [g, P] = ⟨z⟩ and z ∉ Fr(P). Since rk([g, P/Fr(P)]) = 1, |N_{S}(P)/P | = 2 by Proposition 3.3(c).

Set Θ = Ω_{1}(Z(P)): the 2-torsion subgroup of Z(P). Then [g, P] ≤ ⟨z⟩ ≤ Θ, so [g, Θ] ≰ Fr(P) by Lemma 3.4 again, and thus z ∈ [g, Θ]. Fix h ∈ Θ such that [g, h] = z. In particular, h^{2} = 1 and P ≤ C_{S}(h).

Now |N_{S}(P)/P | = 2, g ∈ N_{S}(P), and g ∉ C_{S}(h) imply that N_{C_{S}(h)}(P) = P. Hence P = C_{S}(h) since otherwise its normalizer in C_{S}(h) would be strictly larger.

(b) Since [y, P] ≤ [y, S] ≤ ⟨z⟩, y ∈ N_{S}(P). If neither y nor gy is in P, then yP and gP are distinct nonidentity elements of N(P)/P, which is impossible since |N(P)/P | = 2. Thus one of them is in P; say y ∈ P. If y ∉ Z(P), then z ∈ [y, P] ≤ Fr(P), which again contradicts (a). Thus y ∈ Z(P).

It remains to show h is not S-conjugate to yh. Assume otherwise: let a be such that aha^{−1} = yh. Then aP a^{−1} = C_{S}(aha^{−1}) = C_{S}(yh) ≥ P, so a ∈ N_{S}(P). Thus h, zh, yh ∈ Z(P) are all N_{S}(P)-conjugate to h, so |N_{S}(P)/P | > 2, which again contradicts (a). □

We can now outline the general procedure which will be used to determine all of the critical subgroups of a given 2-group S. We first try to find a normal subgroup S_{0} ◁ S, as large as possible, which we can show is contained in all critical subgroups. For example, in many cases, we do this for S_{0} = Z_{2}(S), using Lemmas 3.5 and 3.6. We then search for critical subgroups P ≤ S such that |N_{S}(P)/P | = 2, by first applying Lemma 1.9 (when possible) to list all subgroups of index 2 in their normalizer, and then applying Lemma 3.4 to eliminate most of them. Afterwards, we search for subgroups P ≤ S such that |N_{S}(P)/P | = 2^{k} ≥ 4, rk(P/Fr(P)) ≥ 2k, and rk([s, P/Fr(P)]) ≥ 2 for all s ∈ N_{S}(P)∖P, and check (using Proposition 3.3) which of them could be critical. In practice, this seems to work surprisingly well on groups of order ≤ 2^{10}, at least on those where we have tested it.

4. Fusion systems over the Sylow 2-subgroup of J_{2} and J_{3}

We are now ready to begin working with some concrete examples. In the next four sections, we list all nonconstrained centerfree saturated fusion systems over each of four different 2-groups S. In each case, this procedure can be broken up into three steps: first determine the critical subgroups of S (or a list of subgroups which includes all critical subgroups), then determine the automorphism group of each critical subgroup, and finally work out all possible combinations of which critical subgroups can be Fessential for any given F and what their F-automorphism groups can be. The last step is carried out only up to isomorphism, in the sense that we make a list of fusion systems over S and show that for each F, there is some φ ∈ Aut(S) such that φFφ^{−1} is in the list (see Definition 2.3). If we did find a candidate for a new exotic fusion system, then there would be the additional step of proving that it is saturated; but otherwise this is done by identifying it (by elimination) with the fusion system of some finite group.

In this section and the next, S_{0} = UT_{3}(4) denotes the group of 3×3 upper triangular matrices over 𝔽_{4} with 1’s in all diagonal entries. Let e_{ij}^{a} ∈ S_{0} (for i < j) be the elementary matrix with entry a ∈ 𝔽_{4} in the (i, j) position, and set E_{ij} = {e_{ij}^{a} | a ∈ 𝔽_{4}}. Thus, for example,

Z(S_{0}) = [S_{0}, S_{0}] = E_{13} = {e_{13}^{a} | a ∈ 𝔽_{4}}.

We note here for reference throughout this section the relations

(e_{12}^{a}e_{23}^{b})^{2} = [e_{12}^{a}, e_{23}^{b}] = e_{13}^{ab} for all a, b ∈ 𝔽_{4}  (1)

We also let c_{ij}^{a} denote conjugation by e_{ij}^{a}, as an automorphism of S_{0} and also as a homomorphism between subgroups of S_{0} or groups containing S_{0}, and write ⟨c_{ij}^{∗}⟩ = {c_{ij}^{a} | a ∈ 𝔽_{4}}.

Let a ↦ a = a^{2} denote the field automorphism on 𝔽_{4}, and let M ↦ M denote the induced field automorphism on S_{0}. Let τ ∈ Aut(S_{0}) be the automorphism “transpose inverse” which sends e_{ij}^{a} to e_{4−j,4−i}^{a}. Consider the semidirect product

S_{ϕθ} = UT_{3}(4) ⋊ ⟨ϕ, θ⟩,

where for M ∈ S_{0} = UT_{3}(4), ϕMϕ^{−1} = M and θMθ^{−1} = τ(M) (and ϕ^{2} = θ^{2} = [ϕ, θ] = 1). Thus

τ(( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & c & b \\ 0 & 1 & a \\ 0 & 0 & 1 \end{matrix} )^{−1} = ( \begin{matrix}1 & c & b+ac \\ 0 & 1 & a \\ 0 & 0 & 1 \end{matrix} )  and c_{θ}(( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & c̄ & b+ac \\ 0 & 1 & ā \\ 0 & 0 & 1 \end{matrix} ) ; (2)

and S_{ϕθ} is a Sylow 2-subgroup of the full automorphism group Aut(P SL_{3}(4)) = P GL_{3}(4) ⋊ ⟨ϕ, θ⟩. In this section, we determine the nonconstrained saturated fusion systems over the group

S_{θ} =^{def} UT_{3}(4) ⋊ ⟨θ⟩,

while in the next section we work with the group S_{ϕ} =^{def} UT_{3}(4) ⋊ ⟨ϕ⟩.

The following subgroups will play an important role throughout this section:

A_{1} = ⟨E_{12}, E_{13}⟩ = \{( \begin{matrix}1 & a & b \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) | a, b ∈ 𝔽_{4}\}  Q_{0} = \{( \begin{matrix}1 & a & b \\ 0 & 1 & ā \\ 0 & 0 & 1 \end{matrix} ) | a, b ∈ 𝔽_{4}\}
A_{2} = ⟨E_{13}, E_{23}⟩ = \{( \begin{matrix}1 & 0 & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{matrix} ) | a, b ∈ 𝔽_{4}\}  Q = ⟨Q_{0}, θ⟩ .

Thus A_{1} and A_{2} are the “rectangular subgroups”, both isomorphic to C_{2}^{4}; while Q_{0} ≅ C_{2} × Q_{8} and Q ≅ Q_{8} ×_{C_{2}} D_{8}. Also,

Q_{0}/E_{13} = [θ, S_{0}/E_{13}] = C_{S_{0}/E_{13}}(θ) , Q_{0} = [S_{θ}, S_{θ}] ◁ S_{θ} , and Q ◁ S_{θ} . (3)

We start with some elementary facts about S_{θ} and its subgroups. Throughout this section and the next, ω denotes an element in 𝔽_{4}∖𝔽_{2}, so that 𝔽_{4} = {0, 1, ω, ω̄}.

Lemma 4.1. (a) All involutions in S_{θ}∖S_{0} are S_{0}-conjugate to θ. For each involution g ∈ S_{θ}∖S_{0}, C_{S_{0}}(g) ≅ Q_{8}, C_{S_{0}}(g) ≤ Q_{0}, and e_{13}^{1} ∈ Fr(C_{S_{0}}(g)).

(b) All involutions in S_{0} are in A_{1} or in A_{2}, and all involutions in S_{θ}∖S_{0} are in Q.

(c) A_{1} and A_{2} are the only subgroups of S_{θ} isomorphic to C_{2}^{4}.

(d) The subgroups S_{0} and Q are both characteristic in S_{θ}.

Proof. (a) Conjugation by θ acts on E_{13} with fixed subgroup ⟨e_{13}^{1}⟩, and on S_{0}/E_{13} by exchanging the complementary subgroups A_{1}/E_{13} and A_{2}/E_{13}. The hypotheses of Lemma 1.4 thus hold, and all involutions in the coset θS_{0} are S_{0}-conjugate to θ.

By (2), C_{S_{0}}(θ) is generated by e_{13}^{1}, together with matrices ( \begin{matrix}1 & a & ω \\ 0 & 1 & ā \\ 0 & 0 & 1 \end{matrix} ) for 0 ≠ a ∈ 𝔽_{4}. Thus C_{S_{0}}(θ) ≤ Q_{0} (this also follows from (3)), has order 8, and is isomorphic to Q_{8} since it is not cyclic and its only involution is e_{13}^{1} (by (1)). Since each involution g ∈ S_{θ}∖S_{0} is conjugate to θ, and since Q_{0} ◁ S_{θ}, the same holds for C_{S_{0}}(g).

(b) The first statement holds by (1). Since all involutions in S_{θ}∖S_{0} are S_{0}-conjugate to θ, and since θ ∈ Q and Q ◁ S_{θ}, all such involutions are in Q.

(c) Assume A ≤ S_{θ} and A ≅ C_{2}^{4}. If A ≤ S_{0}, then A ⊆ (A_{1} ∪ A_{2}) by (b), and A = A_{1} or A_{2} since no element of A_{1}∖E_{13} commutes with any element of A_{2}∖E_{13}. If A ≰ S_{0} and g ∈ A∖S_{0}, then A ∩ S_{0} ≅ C_{2}^{3}, A ∩ S_{0} ≤ C_{S_{0}}(g) ≅ Q_{8} by (a), and this is impossible.

(d) The subgroup S_{0} = ⟨A_{1}, A_{2}⟩ is characteristic by (c). By (b), Q is generated by the centralizers of all involutions in S_{θ}∖S_{0}, and so it is also characteristic. □

4.1 Candidates for critical subgroups The following proposition is the main result of this subsection.

Proposition 4.2. If P is a critical subgroup of S_{θ}, then P is one of the subgroups Q, S_{0} = UT_{3}(4), A_{1}, or A_{2}.

Proposition 4.2 follows immediately from Lemmas 4.3 and 4.4. We first deal with the normal critical subgroups.

Lemma 4.3. If P ◁ S_{θ} is a normal critical subgroup of S_{θ}, then P = Q or P = S_{0} = UT_{3}(4).

Proof. By Proposition 3.3(c), rk(P/Fr(P)) ≥ 2k if |S_{θ}/P | = 2^{k}. Thus 2^{7} ≥ |S_{θ}| ≥ 2^{k}·2^{2k} = 2^{3k}, so k ≤ 2 and |S_{θ}/P | ≤ 4. In particular, S_{θ}/P is abelian, and so P ≥ [S_{θ}, S_{θ}] = Q_{0} by (3).

Assume first that |S_{θ}/P | = 4. Then |P | = 2^{5} and rk(P/Fr(P)) ≥ 4, so |Fr(P)| ≤ 2. It follows that Fr(P) = Fr(Q_{0}) = ⟨e_{13}^{1}⟩.

If P ≤ S_{0}, then P = ⟨Q_{0}, e_{12}^{a}⟩ for some a ≠ 0, and hence Fr(P) ≥ E_{13}, which we saw is impossible. Thus P = ⟨Q_{0}, hθ⟩ for some h ∈ S_{0}; and since S_{0} = Q_{0}E_{12}, we can assume h = e_{12}^{a} for some a ∈ 𝔽_{4}. Also, (hθ)^{2} = [e_{12}^{a}, θ] ∈ Fr(P) = ⟨e_{13}^{1}⟩, and this is possible only if a = 0. Thus P = ⟨Q_{0}, θ⟩ = Q in this case.

Now assume |S_{θ}/P | = 2, and fix g ∈ S_{θ}∖P. Since S_{θ}/Fr(S_{θ}) ≅ C_{2}^{3}, there are seven subgroups of index 2 in S_{θ}. Assume P ≠ S_{0}. Then P = ⟨Q_{0}, e_{12}^{a}, e_{12}^{b}θ⟩ for some a, b ∈ 𝔽_{4} where a ≠ 0. Also, [Q_{0}, e_{12}^{a}] = E_{13}. If b ∉ {0, a}, then

Fr(P) = ⟨E_{13}, [e_{12}^{a}, θ], [e_{12}^{b}, θ]⟩ = Q_{0} = Fr(S_{θ}),

so [g, P] ≤ Fr(P) for g ∈ S_{θ}∖P, and P is not critical by Lemma 3.4 (applied with Θ = 1).

We are left with the case b ∈ {0, a}, and thus P = ⟨Q_{0}, e_{12}^{a}, θ⟩. Then Fr(P) = ⟨E_{13}, [e_{12}^{a}, θ]⟩ ≅ C_{2} × C_{4}, and so E_{13} is characteristic in P since it is the 2-torsion subgroup of Fr(P). Thus C_{P} (E_{13}) = P_{0} =^{def} P ∩S_{0} is characteristic in P. For g ∈ S_{0}∖P, [g, P] ≤ P_{0} and [g, P_{0}] ≤ E_{13} ≤ Fr(P); and thus P is not critical by Lemma 3.4 applied with Θ = P_{0}. □

It now remains to show:

Lemma 4.4. If P ≤ S_{θ} is a critical subgroup and not normal, then P = A_{1} or P = A_{2}.

Proof. Fix such a P. Assume first that |N(P)/P | ≥ 4. Since S_{θ} has order 2^{7}, |N(P)| ≤ 2^{6}, and so |P | ≤ 2^{4}. Since P is critical, we must have rk(P/F r(P)) ≥ 4 by Proposition 3.3(c). This can only happen if P ≅ C_{2}^{4}, and by Lemma 4.1(c), P = A_{1} or P = A_{2}.

Now assume |N(P)/P | = 2. Then Q_{0} = Fr(S_{θ}) ≰ P because P not normal in S_{θ}. Also, e_{13}^{1} ∈ Z(S_{θ}) ≤ P since P is centric. If e_{13}^{ω} ∉ P, then by Lemma 3.6, there is some h ∈ S_{θ}∖S_{0} (S_{0} = C_{S_{θ}}(e_{13}^{ω})) such that h^{2} = 1, P = C_{S_{θ}}(h), and e_{13}^{1} ∉ Fr(P). But this contradicts Lemma 4.1(a), and thus E_{13} ≤ P. Also, Q_{0} ≤ N(P) since [Q_{0}, S_{θ}] = E_{13} ≤ P.

Thus [Q_{0} : P ∩ Q_{0}] = 2. Since |Q_{0}| = 2^{4}, E_{13} ≨ P ∩ Q_{0} ≨ Q_{0}, and there is a unique a ∈ 𝔽_{4}∖0 such that e_{12}^{a}eā ; 23 ∈ P. Hence (e_{12}^{a}eā ; 23)^{2} = e_{13}^{1} ∈ Fr(P) by (1). Fix

b ∈ 𝔽_{4}∖{a, 0} and set g = e_{12}^{b}eb̄ ; 23 ∈ N(P)∖P. By (3), [g, P] ≤ [Q_{0}, S_{θ}] = E_{13}.

By Lemma 3.5, there is some α ∈ Aut(P), and elements x ∈ [g, P] ≤ E_{13} and y = α(x), such that x ∉ Fr(P) and [g, y] ∉ Fr(P). Since e_{13}^{1} ∈ Fr(P), this means x = e_{13}^{c} for some c ∈ {ω, ω̄}. Also, y^{2} = α(x^{2}) = 1.

If y ∉ S_{0}, then by Lemma 4.1(a), y = hθh^{−1} for some h ∈ S_{0}, and so

[g, y] = [g, hθh^{−1}] = h[h^{−1}gh, θ]h^{−1} ∈ h[Q_{0}, θ]h^{−1} = ⟨e_{13}^{1}⟩ ≤ Fr(P)

(recall Q_{0} ◁ S_{θ} by (3)). But [g, y] ∉ Fr(P) by assumption, so we conclude y ∈ S_{0}.

Since y^{2} = 1, y ∈ A_{1} or A_{2} by Lemma 4.1(b). Also, y ∉ E_{13} since [g, y] ∈ [S_{0}, y] ≠ 1. We can assume (upon replacing P by θP θ^{−1} if necessary) that y ∈ A_{1}∖E_{13}. Fix d ∈ 𝔽_{4}∖0 such that y ∈ e_{12}^{d}E_{13}. By (1),

[y, e_{12}^{a}eā ; 23] = [e_{12}^{d}, e_{12}^{a}eā ; 23] = e_{13}^{dā} ∈ [y, P] ∩ E_{13} ≤ Fr(P).  (4)

Now, [y, P] = α([x, P]) ≤ α(⟨e_{13}^{1}⟩) has order at most two, while [y, P] ∩ E_{13} ≠ 1 by (4). Since e_{13}^{1} ∈ Fr(P) but x = e_{13}^{c} ∉ Fr(P) (and [y, P] ≤ Fr(P)), this implies [y, P] = ⟨e_{13}⟩. It also implies [x, P] = [e_{13}^{c}, P] ≠ 1, and hence P ≰ S_{0}. Fix r ∈ S_{0} such that rθ ∈ P; then (rθ)y(rθ)^{−1} ∈ A_{2}∖E_{13}, so [y, rθ] ∉ E_{13}, which is impossible since [y, P] = ⟨e_{13}^{1}⟩. Thus there are no critical subgroups of this form. □

4.2 Automorphisms of critical subgroups

Before describing the automorphism group of S_{0}, we need to give names to some automorphisms. For each f ∈ Hom_{𝔽_{2}}(𝔽_{4}, 𝔽_{4}), define ρ_{1}^{f}, ρ_{2}^{f} ∈ Aut(S_{0}) by setting ρ_{i}|_{A_{i}} = Id, and

ρ_{1}^{f}(e_{23}^{x}) = e_{23}^{x}e_{13}^{f(x)} and ρ_{2}^{f}(e_{12}^{x}) = e_{12}^{x}e_{13}^{f(x)}.

Note that ρ_{i}^{f} ∘ ρ_{i}^{f′} = ρ_{i}^{f+f′}, and hence R_{i} =^{def} {ρ_{i}^{f} | f ∈ Hom_{𝔽_{2}}(𝔽_{4}, 𝔽_{4})} is a subgroup of Aut(S_{0}) isomorphic to C_{2}^{4}. One easily sees that R_{1} and R_{2} commute in Aut(S_{0}), and that they generate the group of all automorphisms of S_{0} which induce the identity on E_{13} and on S_{0}/E_{13}. Thus R_{1} × R_{2} is a normal subgroup of Aut(S_{0}), and is contained in O_{2}(Aut(S_{0})).

Next define γ_{0}, γ_{1} ∈ Aut(S_{0}) by letting γ_{0} be conjugation by diag(ω, 1, ω̄), and letting γ_{1} be conjugation by diag(ω, 1, ω). Thus

γ_{0}(( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & ωa & ω̄b \\ 0 & 1 & ωc \\ 0 & 0 & 1 \end{matrix} ) and γ_{1}(( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & ωa & b \\ 0 & 1 & ω̄c \\ 0 & 0 & 1 \end{matrix} ) .

Also, γ_{0} and γ_{1} both have order 3,

Γ_{0} =^{def} ⟨γ_{0}, c_{θ}⟩ ≅ Σ_{3}, Γ_{1} =^{def} ⟨γ_{1}, τ⟩ ≅ Σ_{3},

and [Γ_{0}, Γ_{1}] = 1 in Aut(S_{0}).

Lemma 4.5. (a) Aut(S_{0}) = (R_{1} × R_{2})·(Γ_{0} × Γ_{1}) ≅ C_{2}^{8} ⋊ (Σ_{3} × Σ_{3}), and hence

Out(S_{0}) = ((R_{1}/⟨c_{12}^{∗}⟩) × (R_{2}/⟨c_{23}^{∗}⟩))·(Γ_{0} × Γ_{1}) ≅ C_{2}^{4} ⋊ (Σ_{3} × Σ_{3}).

(b) Restriction induces an isomorphism

Out(S_{θ}) ⟶_{≅}^{Res} C_{Out(S_{0})}(c_{θ})/⟨c_{θ}⟩.

(c) Set H = ⟨Aut_{S_{0}}(A_{1}), γ_{0}|_{A_{1}}⟩ ≅ A_{4}. Then for any pair of subgroups U, U_{0} ≤ C_{Aut(A_{1})}(H) of order three, there is ψ ∈ Aut(S_{θ}) such that ψ|_{S_{0}} commutes with γ_{0} in Aut(S_{0}), and such that (ψ|_{A_{1}})U(ψ|_{A_{1}})^{−1} = U_{0}.

Proof. (a) The elements we have defined clearly generate subgroups of Aut(S_{0}) and Out(S_{0}) of the form described in (a). It remains to show that

Aut(S_{0}) = ⟨R_{1}, R_{2}, Γ_{0}, Γ_{1}⟩.  (5)

Let α ∈ Aut(S_{0}) be arbitrary. By Lemma 4.1(c), α either sends each subgroup A_{i} to itself or switches them. Hence there is α_{1} ∈ {α, τα} such that α_{1}(A_{i}) = A_{i} for i = 1, 2.

Next, we can choose r, s ∈ {0, 1, 2} such that if we set α_{2} = γ_{1}^{r}γ_{2}^{s}α_{1}, then α_{2}(e_{12}^{1}) ≡ e_{12}^{1} and α_{2}(e_{23}^{1}) ≡ e_{23}^{1} (mod E_{13}). Finally, there is α_{3} ∈ {c_{ϕ}α_{2}, α_{2}} such that α_{3}(e_{12}^{a}) ≡ e_{12}^{a} (mod E_{13}) for all a ∈ 𝔽_{4}. By (1) (and since E_{13} ≤ Z(S_{0})), for all a ∈ 𝔽_{4},

α_{3}(e_{13}^{a}) = [α_{3}(e_{12}^{a}), α_{3}(e_{23}^{1})] = [e_{12}^{a}, e_{23}^{1}] = e_{13}^{a}

and hence

[e_{12}^{1}, e_{23}^{a}] = e_{13}^{a} = [α_{3}(e_{12}^{1}), α_{3}(e_{23}^{a})] = [e_{12}^{1}, α_{3}(e_{23}^{a})].

Since α_{3}(e_{23}^{a}) ∈ E_{13}E_{23}, this implies α_{3}(e_{23}^{a}) ≡ e_{23}^{a} (mod E_{13}) for all a. Thus α_{3} induces the identity on S_{0}/E_{13} and on E_{13}.

Let φ: S_{0}/E_{13} ⟶ E_{13} be the function such that for all g ∈ S_{0}, α_{3}(g) = g·φ(gE_{13}). Since E_{13} = Z(S_{0}) and α is a homomorphism, φ is also a homomorphism. So there is a pair of functions f, f′ ∈ Hom_{𝔽_{2}}(𝔽_{4}, 𝔽_{4}) such that φ(e_{12}^{a}e_{23}^{b}E_{13}) = e_{13}^{f(a)+f′(b)}, and α_{3} = ρ_{2}^{f} ∘ ρ_{1}^{f′} ∈ R_{1} × R_{2}. Since α ∈ (Γ_{0} × Γ_{1}) ∘ α_{3}, this proves (5).

(b) By Lemma 4.1(d), S_{0} is characteristic in S_{θ}. Also, Z(S_{0}) = E_{13} is free as an 𝔽_{2}[⟨c_{θ}⟩]-module. So by Corollary 1.3, the map induced by restriction

Out(S_{θ}) ⟶_{≅}^{Res} N_{Out(S_{0})}(Out_{S_{θ}}(S_{0}))/Out_{S_{θ}}(S_{0}) = C_{Out(S_{0})}(⟨c_{θ}⟩)/⟨c_{θ}⟩

is an isomorphism.

(c) For each α ∈ Aut(A_{1}), let M(α) ∈ GL_{4}(2) be the matrix of α with respect to the ordered basis {e_{13}^{1}, e_{13}^{ω}, e_{12}^{1}, e_{12}^{ω}} for A_{1} as a vector space over 𝔽_{2}. Thus M(H) is generated by

M(c_{23}^{1}) = (\begin{matrix}I & I \\ 0 & I \end{matrix})  M(c_{23}^{ω}) = (\begin{matrix}I & Z \\ 0 & I \end{matrix})  and M(γ_{0}|_{A_{1}}) = (\begin{matrix}Z^{−1} & 0 \\ 0 & Z \end{matrix}) ,

where matrices are written in 2 × 2 blocks and Z = ( \begin{matrix}0 & 1 \\ 1 & 1 \end{matrix} ). From this, it follows that

M(C_{Aut(A_{1})}(H)) = \{(\begin{matrix}B & C \\ 0 & B \end{matrix}) | B ∈ ⟨Z⟩, C ∈ M_{2}(𝔽_{2}), CZ = Z^{−1}C\} .

In particular, O_{2}(C_{Aut(A_{1})}(H)) = C_{R_{2}^{∗}}(H), where R_{2}^{∗} = {α|_{A_{1}} | α ∈ R_{2}}; and also C_{Aut(A_{1})}(H)/O_{2}(C_{Aut(A_{1})}(H)) ≅ C_{3}. So for any pair of subgroups U, U_{0} ≤ C_{Aut(A_{1})}(H) of order three, there is η ∈ R_{2} such that η|_{A_{1}} ∈ C_{R_{2}^{∗}}(H) and (η|_{A_{1}})U(η|_{A_{1}})^{−1} = U_{0}.

Now, c_{θ}ηc_{θ}^{−1} ∈ R_{1} commutes with η ∈ R_{2}, so η ∘ (c_{θ}ηc_{θ}^{−1}) commutes with c_{θ} in Aut(S_{0}). Hence this can be extended to ψ ∈ Aut(S_{θ}) by setting ψ(θ) = θ. Also, ψ|_{A_{1}} = η|_{A_{1}} since elements of R_{1} are the identity on A_{1}. Since γ_{0}ηγ_{0}^{−1} is equal to η after restriction to A_{1} and both are the identity on A_{2}, they are equal as elements of R_{2} ≤ Aut(S_{0}). So γ_{0} = c_{θ}γ_{0}^{−1}c_{θ} also commutes with c_{θ}ηc_{θ}^{−1} ∈ R_{1}, and thus commutes with their composite ψ|_{S_{0}}. □

Now set

Γ = ⟨γ_{1}, γ_{0}, c_{θ}⟩ ≤ Aut(S_{0}) so that Γ ≅ C_{3} × Σ_{3}.

To simplify notation, we also write Γ_{0} ≤ Γ to denote their images in Out(S_{0}). Finally, define γ̇_{1} ∈ Aut(S_{θ}) by setting γ̇_{1}|_{S_{0}} = γ_{1} (conjugation by diag(ω, 1, ω)) and γ̇_{1}(θ) = θ.

Lemma 4.6. Let F be any saturated fusion system over S_{θ} for which S_{0} is F-essential. Then there is some φ ∈ Aut(S_{θ}) such that either

• Out_{φFφ^{−1}}(S_{0}) = Γ_{0} ≅ Σ_{3} and Out_{φFφ^{−1}}(S_{θ}) = 1; or

• Out_{φFφ^{−1}}(S_{0}) = Γ ≅ C_{3} × Σ_{3} and Out_{φFφ^{−1}}(S_{θ}) = ⟨[γ̇_{1}]⟩.

Proof. By Lemma 4.5(a,b),

Out(S_{θ})/O_{2}(Out(S_{θ})) ≅ C_{Out(S_{0})}(⟨c_{θ}⟩)/O_{2}(C_{Out(S_{0})}(⟨c_{θ}⟩)) ≅ Σ_{3}

(represented by Γ_{1}). Thus |Out_{F}(S_{θ})| = 1 or 3, since it contains Out_{S_{θ}}(S_{θ}) = 1 as a Sylow 2-subgroup.

Set Q = O_{2}(Out(S_{0})) for short. Then Out_{F}(S_{0}) ∩ Q = 1, and by Lemma 4.5(a),

Out(S_{0})/Q = ⟨γ_{0}, c_{θ}⟩ × ⟨γ_{1}, τ⟩ ≅ Σ_{3} × Σ_{3}.

Also, Out_{F}(S_{0})·Q/Q contains ⟨c_{θ}⟩ as a Sylow 2-subgroup not in the center (since O_{2}(Out_{F}(S_{0})) = 1), and this is possible only if Out_{F}(S_{0})·Q = Γ_{0}·Q or Γ·Q. By Lemma 1.8, there is some φ_{0} ∈ O_{2}(Aut(S_{0})) such that [[φ_{0}], c_{θ}] = 1 in Out(S_{0}) and φ_{0}Out_{F}(S_{0})φ_{0}^{−1} is equal to Γ_{0} or Γ. By Lemma 4.5(b), φ_{0} extends to some φ ∈ Aut(S_{θ}), and thus Out_{φFφ^{−1}}(S_{0}) = Γ_{0} or Γ.

If Out_{φFφ^{−1}}(S_{0}) = Γ, then by the extension axiom, γ_{1} extends to an element of Aut_{φFφ^{−1}}(S_{θ}). Hence Out_{φFφ^{−1}}(S_{θ}) = ⟨[γ̇_{1}]⟩ since this extension is unique (mod Inn(S_{θ})) by Lemma 4.5(b) again.

Conversely, if Out_{φFφ^{−1}}(S_{θ}) has order 3, then a generator of this group restricts to an element of order three in Aut_{F}(S_{0}) (since S_{0} is characteristic in S_{θ}). Since no element of order three in Γ_{0} extends to S_{θ}, we conclude that Out_{φFφ^{−1}}(S_{0}) = Γ. □

We now check the possibilities for Aut_{F}(A_{i}) when the A_{i} are essential. Consider the following subgroups of Aut(A_{i}):

Λ_{i} =^{def} Aut_{GL_{3}(4)}(A_{i}) ≅ GL_{2}(4) and Λ_{i}^{0} =^{def} [Λ_{i}, Λ_{i}] ≅ SL_{2}(4).

Thus Λ_{i} is the group of those automorphisms of A_{i} induced by conjugation by elements of GL_{3}(4) ≥ S_{0}. Note that we can regard A_{i} as a vector space over 𝔽_{4}, where scalar multiplication is given by u·e_{ij}^{x} = e_{ij}^{ux} for u, x ∈ 𝔽_{4}; and then Λ_{i} = Aut_{𝔽_{4}}(A_{i}) is the group of 𝔽_{4}-linear automorphisms. Since A_{1} and A_{2} are S_{θ}-conjugate, Aut_{F}(A_{1}) = Λ_{1} (= Λ_{1}^{0}) if and only if Aut_{F}(A_{2}) = Λ_{2} (= Λ_{2}^{0}).

Lemma 4.7. Let F be any saturated fusion system over S_{θ}, and assume A_{1} and A_{2} are F-essential. Then S_{0} is also F-essential. There is an automorphism φ ∈ Aut(S_{θ}) such that either

• Aut_{φFφ^{−1}}(A_{i}) = Λ_{i}^{0}, Out_{φFφ^{−1}}(S_{0}) = Γ_{0} ≅ Σ_{3}, and Out_{φFφ^{−1}}(S_{θ}) = 1; or

• Aut_{φFφ^{−1}}(A_{i}) = Λ_{i}, Out_{φFφ^{−1}}(S_{0}) = Γ ≅ C_{3} × Σ_{3}, and Out_{φFφ^{−1}}(S_{θ}) = ⟨[γ̇_{1}]⟩.

Proof. Set Δ = Aut_{F}(A_{1}). Thus Δ is a subgroup of Aut(A_{1}) ≅ GL_{4}(2) ≅ A_{8} which has Aut_{S_{θ}}(A_{1}) ≅ C_{2}^{2} as Sylow 2-subgroup, and which contains a strongly embedded subgroup. By Bender’s theorem (Theorem 1.6), O^{2′}(Δ/O_{2′}(Δ)) is isomorphic to SL_{2}(4) ≅ A_{5}. The only nontrivial odd order subgroup of GL_{4}(2) which has A_{5} in its normalizer is C_{3}, with normalizer isomorphic to SL_{2}(4) ⋊ ⟨ϕ⟩ ≅ (C_{3} × A_{5}) ⋊ C_{2}. If H ≤ GL_{4}(2) ≅ A_{8} and H ≅ A_{5}, then since the only proper subgroups of A_{5} of index ≤ 8 have index 5 and 6, each orbit of H acting on {1, …, 8} has length 1, 5, or 6. Thus H is in one of two conjugacy classes: either it acts as SL_{2}(4) with respect to some 𝔽_{4}-vector space structure, or it acts via the permutation action on 𝔽_{2}^{5}/diag. Since the fixed set of Aut_{S_{θ}}(A_{1}) = ⟨c_{23}^{∗}⟩ acting on A_{1} is 2-dimensional, this last action cannot occur. We conclude that Δ must be Aut(A_{1})-conjugate to Λ_{1}^{0} or Λ_{1}.

Let Δ^{0} ◁ Δ be the subgroup isomorphic to Λ_{1}^{0} ≅ SL_{2}(4). Thus Δ^{0} has odd index in Δ, and ⟨c_{23}^{∗}⟩ ∈ Syl_{2}(Δ^{0}) = Syl_{2}(Δ). Hence there is an element of order three in N_{Δ^{0}}(⟨c_{23}^{∗}⟩), which by the extension axiom extends to some ξ ∈ Aut_{F}(S_{0}). Since Δ^{0} is Aut(A_{1})-conjugate to Λ_{1}^{0}, ξ|_{A_{1}} acts without fixed component, and in particular acts nontrivially on E_{13}. Hence [ξ] does not commute with c_{θ} in Out_{F}(S_{0}), and so S_{0} is also F-essential. By Lemma 4.6, we can assume (after replacing F by ψFψ^{−1} for some appropriate ψ ∈ Aut(S_{θ})) that Aut_{F}(S_{0}) = Γ_{0} or Aut_{F}(S_{0}) = Γ. In either case, γ_{0}|_{A_{1}} ∈ Aut_{F}(A_{1}).

Since Δ is Aut(A_{1})-conjugate to Λ_{1}^{0} or Λ_{1}, it is contained in the centralizer of some subgroup U ≤ Aut(A_{1}) of order three which acts on A_{1} without fixed component, thus defining a 𝔽_{4}-vector space structure. Similarly, the “standard” subgroups Λ_{1}^{0} ≤ Λ_{1} are centralized by U_{0} = ⟨γ_{0}γ_{1}|_{A_{1}}⟩; i.e., by conjugation by diag(ω, 1, 1).

If Out_{F}(S_{0}) = Γ, then Δ ≥ ⟨Aut_{S_{0}}(A_{1}), γ_{0}|_{A_{1}}, γ_{1}|_{A_{1}}⟩, and so ⟨γ_{0}|_{A_{1}}, γ_{1}|_{A_{1}}⟩ ∈ Syl_{3}(Δ). Thus Δ ≅ GL_{2}(4), U = Z(Δ) ≤ ⟨γ_{0}|_{A_{1}}, γ_{1}|_{A_{1}}⟩, and hence

U ≤ C_{⟨γ_{0}|_{A_{1}},γ_{1}|_{A_{1}}⟩}(Aut_{S_{0}}(A_{1})) = U_{0}.

This proves that U = U_{0}, and hence Δ = Λ_{1} in this case.

Now assume Aut_{F}(S_{0}) = Γ_{0}. Set H = ⟨Aut_{S_{0}}(A_{1}), γ_{0}|_{A_{1}}⟩ ≅ A_{4}. Then H ≤ Δ, and so U and U_{0} both centralize H. By Lemma 4.5(c), there is φ ∈ Aut(S_{θ}) such that [φ|_{S_{0}}, γ_{0}] = 1 in Out(S_{0}) and (φ|_{A_{1}})U(φ|_{A_{1}})^{−1} = U_{0}. Thus

(φ|_{A_{1}})Δ(φ|_{A_{1}})^{−1} ≤ C_{Aut(A_{1})}(U_{0}) = Λ_{1},

and so Aut_{φFφ^{−1}}(A_{1}) = φΔφ^{−1} = Λ_{1}^{0} or Λ_{1}. Also, since φ commutes with γ_{0}, we still have Aut_{φFφ^{−1}}(S_{0}) = Γ_{0}. Since γ_{1}|_{A_{1}} is not the restriction of an element of Γ_{0}, it cannot be in Aut_{φFφ^{−1}}(A_{1}), and thus Aut_{φFφ^{−1}}(A_{1}) = Λ_{1}^{0}.

In either case, Aut_{φFφ^{−1}}(A_{2}) = c_{θ}Aut_{φFφ^{−1}}(A_{1})c_{θ}^{−1} is equal to Λ_{2}^{0} or Λ_{2}. □

4.3 Fusion systems over S_{θ}

Theorem 4.8. Let F be any nonconstrained saturated fusion system over the group S_{θ} = UT_{3}(4) ⋊ ⟨θ⟩, where θ acts on UT_{3}(4) ≤ P GL_{3}(4) by sending a matrix M to τ(M). Then F is isomorphic to the fusion system of one of the groups P SL_{3}(4) ⋊ ⟨θ⟩, P GL_{3}(4) ⋊ ⟨θ⟩, J_{2}, or J_{3}.

Proof. By Proposition 4.2, the only possible F-essential subgroups are S_{0}, Q, A_{1}, and A_{2}. If S_{0} is not F-essential, then by Lemma 4.7, neither A_{1} nor A_{2} is F-essential. Hence Q is the only F-essential subgroup, and F is generated by automorphisms of Q and S_{θ}. Since Q is characteristic in S_{θ} (Lemma 4.1(d)), this implies Q ◁ F, which contradicts the assumption that F is nonconstrained. Thus S_{0} must be F-essential.

If S_{0} is the only F-essential subgroup, then since it is also characteristic in S_{θ} (Lemma 4.1(d) again), it would be normal in F, again contradicting the assumption that F is nonconstrained. Thus either Q is F-essential, or A_{1} and A_{2} are F-essential, or all of them are.

Since Q ≅ D_{8}×_{C_{2}}Q_{8}, Inn(Q) is the group of automorphisms which induce the identity on Q/Z(Q), and Out(Q) ≅ Σ_{5} is the group which permutes the five involutions in Q/Z(Q) which lift to involutions in Q. Hence if Q is F-essential, then Out_{F}(Q) = A_{5}: this is the only subgroup which contains Out_{S_{θ}}(Q) as Sylow 2-subgroup and which has a strongly embedded subgroup.

Case 1: Assume first that Q is not F-essential, and hence that S_{0} and the A_{i} are F-essential. Let F_{1} and F_{2} be the fusion systems over S_{θ} generated by the following automorphism groups and their restrictions:

Out_{F_{1}}(S_{θ}) = 1  Out_{F_{1}}(S_{0}) = Γ_{0} ≅ Σ_{3}  Aut_{F_{1}}(A_{i}) = Λ_{i}^{0}
Out_{F_{2}}(S_{θ}) = ⟨[γ̇_{1}]⟩  Out_{F_{2}}(S_{0}) = Γ ≅ C_{3} × Σ_{3}  Aut_{F_{2}}(A_{i}) = Λ_{i} .

Here, the subgroups Λ_{i}^{0} ≤ Λ_{i} ≤ Aut(A_{i}) are defined as before: Λ_{i} = Aut_{GL_{3}(4)}(A_{i}) ≅ GL_{2}(4) and Λ_{i}^{0} ≅ SL_{2}(4) is its commutator subgroup.

By Lemma 4.7, we can assume (after replacing F by φFφ^{−1} for appropriate φ) that either Aut_{F}(A_{i}) = Λ_{i}^{0} (for i = 1, 2) and Out_{F}(S_{0}) = Γ_{0}, or Aut_{F}(A_{i}) = Λ_{i} and Out_{F}(S_{0}) = Γ. Furthermore, by Lemma 4.6, Out_{F}(S_{θ}) is determined (exactly) by Out_{F}(S_{0}). Since F is generated by automorphisms of S_{θ}, S_{0}, and the A_{i} and their restrictions, this proves that F = F_{1} or F = F_{2}. In other words, these are the only possible isomorphism classes of saturated fusion systems over S_{θ} satisfying these conditions.

If G is one of the groups P SL_{3}(4)⋊⟨θ⟩ or P GL_{3}(4)⋊⟨θ⟩, then any S ∈ Syl_{2}(G) is isomorphic to S_{θ}, and S_{0} is strongly closed in F_{S}(G) under any identification S = S_{θ}. Hence Q ∩ S_{0} ≅ C_{2} × Q_{8} is invariant under the action of Aut_{G}(Q), which is impossible if Out_{F}(Q) ≅ A_{5}. Thus Q is not F-essential. Since F_{S}(G) is nonconstrained and centerfree, it must be isomorphic to F_{1} or F_{2}; and by comparing automorphism groups of the A_{i}, one sees that F_{S_{θ}}(P SL_{3}(4)⋊⟨θ⟩) ≅ F_{1} and F_{S_{θ}}(P GL_{3}(4)⋊⟨θ⟩) ≅ F_{2}.

Case 2: Now assume Q and S_{0} are both F-essential. Let F_{3} and F_{4} be the fusion systems over S_{θ} generated by the following automorphism groups and their restrictions:

Out_{F_{3}}(S_{θ}) = ⟨[γ̇_{1}]⟩ Out_{F_{3}}(S_{0}) = Γ Out_{F_{3}}(Q) = A_{5}
Out_{F_{4}}(S_{θ}) = ⟨[γ̇_{1}]⟩ Out_{F_{4}}(S_{0}) = Γ Out_{F_{4}}(Q) = A_{5} Aut_{F_{4}}(A_{i}) = Λ_{i} .

For F = F_{3} or F_{4}, all involutions in E_{13}, and all involutions in S_{0}∖E_{13}, are F-conjugate via automorphisms of S_{0}. Also, all noncentral involutions in Q are F-conjugate via automorphisms of Q, and hence (by Lemma 4.1(a)) all involutions in S_{θ}∖S_{0} are Fconjugate to the involutions in E_{13}. Thus there are exactly two F-conjugacy classes of involutions when F = F_{3} (those in S_{0}∖E_{13} and the others); while these form a single class if F = F_{4} (since the A_{i} are F-essential).

For arbitrary F of this type, Out_{F}(Q) = A_{5} has index 2 in Out(Q), and so Aut_{F}(Q) contains all automorphisms of Q of odd order. Since γ̇_{1}|_{Q} has order 3 in Aut_{F}(Q), it must extend (by the extension axiom) to some automorphism in Aut_{F}(S_{θ}). Thus Out_{F}(S_{θ}) has order 3 by Lemma 4.6. By Lemmas 4.6 and 4.7, we can assume (after replacing F by φFφ^{−1} for appropriate φ) that Out_{F}(S_{0}) = Γ and Out_{F}(S_{θ}) = ⟨[γ̇_{1}]⟩, and also that Aut_{F}(A_{i}) = Λ_{i} if the A_{i} are F-essential. Thus F = F_{3} or F = F_{4}.

By Janko’s original characterization of the sporadic simple groups J_{2} and J_{3} [J], both contain involution centralizers of odd index isomorphic to (D_{8} ×_{C_{2}} Q_{8})⋊A_{5}, and J_{2} has two conjugacy classes of involutions while J_{3} has only one class. Also, S_{θ} is isomorphic to the Sylow 2-subgroups of these groups; this is shown explicitly in [GH, p.331], and also follows since S_{θ} is a Sylow 2-subgroup of (D_{8} ×_{C_{2}} Q_{8}) ⋊ A_{5}. Thus F_{S_{θ}}(J_{2}) ≅ F_{3} and F_{S_{θ}}(J_{3}) ≅ F_{4}. □

In fact, the main result of [GH] is that if G is a finite group with Sylow 2-subgroup isomorphic to S_{θ}, then either G/O_{2′}(G) is isomorphic to one of the groups P SL_{3}(4) ⋊ ⟨θ⟩, P GL_{3}(4) ⋊ ⟨θ⟩, J_{2}, or J_{3}, or G/O_{2′}(G) ≅ C_{G}(x) for some involution x.

5. Fusion systems over the Sylow 2-subgroup of M_{22}

Again in this section, S_{0} = UT_{3}(4) denotes the group of 3×3 upper triangular matrices over 𝔽_{4} with 1 in all diagonal entries, x ↦ x = x^{2} denotes the field automorphism on 𝔽_{4}, and M ↦ M denotes the induced field automorphism on S_{0}. Set S_{ϕ} = S_{0} ⋊ ⟨ϕ⟩, where ϕMϕ^{−1} = M for all M ∈ S_{0} and ϕ^{2} = 1. We want to list all nonconstrained centerfree saturated fusion systems over S_{ϕ}, up to isomorphism.

As before, e_{ij}^{a} ∈ S_{0} (for i < j) denotes the elementary matrix with entry a ∈ 𝔽_{4} in the (i, j) position, satisfying the relations

(e_{12}^{a}e_{23}^{b})^{2} = [e_{12}^{a}, e_{23}^{b}] = e_{13}^{ab} for all a, b ∈ 𝔽_{4}.  (1)

Also, E_{ij} = {e_{ij}^{a} | a ∈ 𝔽_{4}}, c_{ij}^{a} denotes conjugation by e_{ij}^{a}, and ω denotes an element in 𝔽_{4}∖𝔽_{2}. Thus 𝔽_{4} = {0, 1, ω, ω}. Finally,

Z(S_{0}) = E_{13} = ⟨e_{13}^{1}, e_{13}^{ω}⟩,  Z(S_{ϕ}) = ⟨e_{13}^{1}⟩,  and [S_{ϕ}, S_{ϕ}] = ⟨E_{13}, e_{12}^{1}, e_{23}^{1}⟩.

The following subgroups will play an important role in this section:

A_{1} = ⟨E_{12}, E_{13}⟩ ≅ C_{2}^{4}  H_{1} = ⟨A_{1}, ϕ⟩  N_{1} = ⟨A_{1}, e_{23}^{1}, ϕ⟩
A_{2} = ⟨E_{13}, E_{23}⟩ ≅ C_{2}^{4}  H_{2} = ⟨A_{2}, ϕ⟩  N_{2} = ⟨A_{2}, e_{12}^{1}, ϕ⟩ .

Note that N_{i} = N_{S_{ϕ}}(H_{i}).

Lemma 5.1. (a) For any involution g ∈ S_{ϕ}∖S_{0}, g is S_{0}-conjugate to ϕ, C_{S_{0}}(g) ≤ ⟨E_{13}, e_{12}^{1}, e_{23}^{1}⟩, C_{S_{0}}(g) ∩ E_{13} = ⟨e_{13}^{1}⟩, and e_{13}^{1} ∈ Fr(C_{S_{0}}(g)).

(b) A_{1} and A_{2} are the only subgroups of S_{ϕ} isomorphic to C_{2}^{4}.

Proof. (a) Since C_{E_{13}}(ϕ) = [ϕ, E_{13}] and C_{S_{0}/E_{13}}(ϕ) = [ϕ, S_{0}/E_{13}], the hypotheses of Lemma 1.4 apply to the pair S_{0} ◁ S_{ϕ}. Hence each involution g ∈ S_{ϕ}∖S_{0} is conjugate to ϕ, and C_{S_{0}}(g) is S_{0}-conjugate to C_{S_{0}}(ϕ) = ⟨e_{13}^{1}, e_{12}^{1}, e_{23}^{1}⟩ ≅ D_{8} for such g. Since the subgroups ⟨E_{13}, e_{12}^{1}, e_{23}^{1}⟩, E_{13}, and ⟨e_{13}^{1}⟩ are all normal in S_{ϕ}, and since C_{S_{0}}(ϕ) satisfies all of the above conditions, so does C_{S_{0}}(g).

(b) By Lemma 4.1, A_{1} and A_{2} are the only subgroups of S_{0} isomorphic to C_{2}^{4}. So assume P ≰ S_{0} and P ≅ C_{2}^{4}. Set P_{0} = P ∩ S_{0}, and fix g ∈ P ∖P_{0}. Then C_{2}^{3} ≅ P_{0} ≤ C_{S_{0}}(g), and we just showed in the proof of (a) that C_{S_{0}}(g) ≅ D_{8}. So this situation is impossible. □

5.1 Candidates for critical subgroups Our main result here is the following:

Proposition 5.2. If P is a critical subgroup of S_{ϕ} then P is equal to one of the subgroups S_{0} = UT_{3}(4), N_{1}, or N_{2}; or is conjugate to H_{1} or H_{2}.

Proof. In Lemma 5.3, we show that if P is normal, then P is one of the subgroups S_{0}, N_{1}, or N_{2}. In Lemma 5.4, we show that if P is not normal and has index 2 in its normalizer, then P is conjugate to H_{1} or H_{2}.

Now assume P is not normal and |N(P)/P | ≥ 4. Since S_{ϕ} has order 2^{7}, |N(P)| ≤ 2^{6}, and so |P | ≤ 2^{4}. Since P is critical, rk(P/Fr(P)) ≥ 4 by Proposition 3.3(c). This implies P ≅ C_{2}^{4}, so P = A_{1} or A_{2} by Lemma 5.1(b), and these subgroups are normal. □

For use in the proofs of Lemmas 5.3 and 5.4, we define the subgroup

Q_{0} = ⟨E_{13}, e_{12}^{1}, e_{23}^{1}⟩ ≅ C_{2} × D_{8}.

Then

Q_{0}/E_{13} = [ϕ, S_{0}/E_{13}] = C_{S_{0}/E_{13}}(ϕ) and Q_{0} = [S_{ϕ}, S_{ϕ}] ◁ S_{ϕ} .  (2)

Lemma 5.3. If P ◁ S_{ϕ} is a normal critical subgroup of S_{ϕ}, then P is one of the three subgroups S_{0}, N_{1}, or N_{2}.

Proof. By Proposition 3.3(c), rk(P/Fr(P)) ≥ 2k if |S_{ϕ}/P | = 2^{k}. Thus |S_{ϕ}| ≥ 2^{3k}, so k ≤ 2, and |S_{ϕ}/P | ≤ 4. Hence S_{ϕ}/P is abelian, so P ≥ Q_{0} = [S_{ϕ}, S_{ϕ}] by (2), and Fr(P) ≥ Fr(Q_{0}) = ⟨e_{13}^{1}⟩.

Case 1 : If |S_{ϕ}/P | = 4, then |P | = 2^{5} and |P/Q_{0}| = 2. Since rk(P/Fr(P)) ≥ 4 and Fr(P) ≠ 1, we have Fr(P) = Fr(Q_{0}) = ⟨e_{13}^{1}⟩.

Set 𝔛 = {e_{12}^{ω}, e_{23}^{ω}, e_{12}^{ω}e_{23}^{ω}} (as a set of elements of S_{0}). If P = ⟨Q_{0}, x⟩ for some x ∈ 𝔛, then [x, Q_{0}] ≰ ⟨e_{13}^{1}⟩ by the relations in (1), so Fr(P) ≰ ⟨e_{13}^{1}⟩. If P = ⟨Q_{0}, xϕ⟩ for some x ∈ 𝔛, then (xϕ)^{2} = [x, ϕ] ∉ ⟨e_{13}^{1}⟩, and again Fr(P) ≰ ⟨e_{13}^{1}⟩. So these cases cannot occur.

This leaves only the possibility

P = ⟨Q_{0}, ϕ⟩ = ⟨e_{13}^{ω}, ϕ⟩ ×_{⟨e_{13}^{1}⟩} ⟨e_{12}^{1}, e_{23}^{1}⟩ ≅ D_{8} ×_{C_{2}} D_{8} ≅ Q_{8} ×_{C_{2}} Q_{8}.

Then Out(P) ≅ Σ_{3} ≀ C_{2}. If P were critical, then by Proposition 3.3(b), there would be an odd order subgroup of Out(P) which normalizes Out_{S_{ϕ}}(P) = ⟨e_{12}^{ω}, e_{23}^{ω}⟩ ≅ C_{2}^{2} and permutes its involutions transitively, and this is not the case. Thus this group P is not critical; and we conclude that S_{ϕ} contains no normal critical subgroups of index 4.

Case 2 : Assume |S_{ϕ}/P | = 2, and fix g ∈ S_{ϕ}∖P. Since S_{ϕ}/Fr(S_{ϕ}) ≅ C_{2}^{3}, there are seven subgroups of index 2 in S_{ϕ}. If Fr(P) = Q_{0} = [S_{ϕ}, S_{ϕ}], then [g, P] ≤ Fr(P), and so P is not critical by Lemma 3.4 (applied with Θ = 1). This is the case for three of the seven subgroups

⟨Q_{0}, e_{12}^{ω}ϕ, e_{23}^{ω}⟩,  ⟨Q_{0}, e_{12}^{ω}, e_{23}^{ω}ϕ⟩,  and ⟨Q_{0}, e_{12}^{ω}ϕ, e_{23}^{ω}ϕ⟩;

which leaves only S_{0}, N_{1}, N_{2}, and N_{3} = ⟨Q_{0}, e_{12}^{ω}e_{23}^{ω}, ϕ⟩ to consider. So it remains to check that N_{3} is not critical.

Now, Fr(N_{3}) = ⟨E_{13}, e_{12}^{1}e_{23}^{1}⟩ ≅ C_{2} × C_{4}, and hence its 2-torsion subgroup E_{13} is characteristic in N_{3}. Let Θ ≤ N_{3} be such that Θ/E_{13} = Z(N_{3}/E_{13}). Then Θ = [S_{ϕ}, S_{ϕ}] is characteristic in N_{3}, and for g ∈ S_{ϕ}∖N_{3} [g, N_{3}] ≤ Θ, and [g, Θ] ≤ E_{13} ≤ Fr(N_{3}). So also in this case, P is not critical by Lemma 3.4. □

It remains to handle the critical subgroups which are not normal.

Lemma 5.4. Let P ≤ S_{ϕ} be a critical subgroup with index 2 in N_{S_{ϕ}}(P) and not normal in S_{ϕ}. Then P is S_{ϕ}-conjugate to H_{1} or H_{2}.

Proof. We have e_{13}^{1} ∈ P since P is centric. If e_{13}^{ω} ∉ P, then by Lemma 3.6, there is h ∈ S_{ϕ}∖S_{0} (S_{0} = C_{S_{ϕ}}(e_{13}^{ω})) such that h^{2} = 1, P = C_{S_{ϕ}}(h), and e_{13}^{1} ∉ Fr(P). This is impossible by Lemma 5.1(a), and hence E_{13} ≤ P.

Now, Q_{0} = ⟨E_{13}, e_{12}^{1}, e_{23}^{1}⟩ ≰ P because P is not normal in S_{ϕ} (see (2)). Also, [P, Q_{0}] ≤ [S_{ϕ}, Q_{0}] = E_{13} ≤ P, so N_{S_{ϕ}}(P) ≥ Q_{0}. Thus [Q_{0} : P ∩ Q_{0}] = 2: it cannot be larger because |N_{S_{ϕ}}(P)/P | = 2. So exactly one of the matrices e_{12}^{1}, e_{23}^{1} or e_{12}^{1}e_{23}^{1} is in

P. By symmetry, we can assume that g =^{def} e_{23}^{1} ∉ P (hence g generates N(P)/P), and that P contains e_{12}^{1} or e_{12}^{1}e_{23}^{1}.

If P ≤ S_{0}, then [P, S_{0}] ≤ E_{13} ≤ P, and S_{0} ≤ N(P). Thus N(P) = S_{0} (since P is not normal in S_{ϕ}), and [S_{0} : P] = 2. It follows that P = ⟨E_{13}, e_{12}^{1}h_{1}, e_{12}^{ω}h_{2}, e_{23}^{ω}h_{3}⟩ for some h_{i} ∈ ⟨g⟩ = ⟨e_{23}^{1}⟩. Then e_{23}^{a} ∈ P for some a ∈ {ω, ω̄}, and Fr(P) contains the elements

[e_{12}^{1}h_{1}, e_{23}^{a}] = e_{13}^{a} and [e_{12}^{ω}h_{2}, e_{23}^{a}] = e_{13}^{aω}

(using (1) again). Thus Fr(P) = E_{13} ≥ [g, P], and so P is not critical by Lemma 3.4 applied with Θ = 1.

Now assume P ≰ S_{0}, and set P_{0} = P ∩ S_{0}. Then |P_{0}| ≤ 2^{4}, since |P | ≤ \frac{1}{4}|S| = 2^{5}. If e_{12}^{1} ∈ P, then since [⟨e_{23}^{1}, e_{12}^{ω}⟩, S_{ϕ}] ≤ ⟨E_{13}, e_{12}^{1}⟩ ≤ P, ⟨e_{23}^{1}, e_{12}^{ω}⟩ ≤ N(P), and so e_{12}^{ω}h ∈ P for some h ∈ ⟨g⟩. Thus P_{0} = ⟨E_{13}, e_{12}^{1}, e_{12}^{ω}h⟩. Furthermore, S_{ϕ}/P_{0} ≅ D_{8}, and D_{8} contains exactly two conjugacy classes of subgroups which are not normal. Since P ≰ S_{0}, this proves that up to conjugacy, P = ⟨E_{13}, e_{12}^{1}, e_{12}^{ω}h, ϕ⟩ for some h ∈ ⟨g⟩. If h = 1, then P = H_{1}. If h = g = e_{23}^{1}, then (e_{12}^{ω}e_{23}^{1})^{2} = e_{13}^{ω} ∈ Fr(P) by (1), so Fr(P) ≥ E_{13} = [g, P], and again P is not critical by Lemma 3.4.

By a similar argument, if e_{12}^{1}e_{23}^{1} ∈ P, then e_{12}^{ω}e_{23}^{ω}h ∈ P for some h ∈ ⟨g⟩, and (again up to conjugacy) P = ⟨E_{13}, e_{12}^{1}e_{23}^{1}, e_{12}^{ω}e_{23}^{ω}h, ϕ⟩. If h = 1, then by (1),

Fr(P) ≥ ⟨(e_{12}^{1}e_{23}^{1})^{2} = e_{13}^{1}, (e_{12}^{ω}e_{23}^{ω})^{2} = eω̄ ; 13⟩ = E_{13},

so Fr(P) = ⟨E_{13}, [ϕ, e_{12}^{ω}e_{23}^{ω}]⟩ = ⟨E_{13}, e_{12}^{1}e_{23}^{1}⟩. Thus [g, P] ≤ Fr(P) in this case, and P is not critical by Lemma 3.4. If h = e_{23}^{1}, then

P = ⟨E_{13}, e_{12}^{1}e_{23}^{1}, e_{12}^{ω}e_{23}^{ω}, ϕ⟩,  Z(P) = ⟨e_{13}^{1}⟩,  Z_{2}(P) = ⟨E_{13}, e_{12}^{1}e_{23}^{1}⟩;

so [g, P] ≤ Z_{2}(P) and [g, Z_{2}(P)] ≤ Fr(P), and P is not critical by Lemma 3.4 applied with Θ = Z_{2}(P). □

5.2 Automorphisms of critical subgroups

By Proposition 5.2, the only critical subgroups of S_{ϕ}, and hence the only essential subgroups in a saturated fusion system over S_{ϕ}, are S_{0}, N_{1}, N_{2}, and subgroups conjugate to H_{1} and H_{2}. The automorphism group of S_{0} was computed in Lemma 4.5(a). In this subsection, we first compute Out(H_{1}) and Out(N_{1}), and then determine all possibilities for Out_{F}(S_{0}), Out_{F}(H_{i}), and Out_{F}(N_{i}) when F is a saturated fusion system over S_{ϕ}.

We first recall some of the notation used for automorphisms of S_{0}. For each f ∈ Hom_{𝔽_{2}}(𝔽_{4}, 𝔽_{4}), we defined ρ_{1}^{f}, ρ_{2}^{f} ∈ Aut(S_{0}) by setting

ρ_{1}^{f}(( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & a & b+f(a) \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} ) and ρ_{2}^{f}(( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & a & b+f(c) \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} );

and set R_{i} = {ρ_{i}^{f} | f ∈ Hom_{𝔽_{2}}(𝔽_{4}, 𝔽_{4})} ≅ C_{2}^{4}. Also, we defined γ_{0}, γ_{1}, τ ∈ Aut(S_{0}) by setting

γ_{0}(( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & ωa & ω̄b \\ 0 & 1 & ωc \\ 0 & 0 & 1 \end{matrix} ) , γ_{1}(( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & ωa & b \\ 0 & 1 & ω̄c \\ 0 & 0 & 1 \end{matrix} ) , τ(( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & c & b \\ 0 & 1 & a \\ 0 & 0 & 1 \end{matrix} )^{−1} ;

and set Γ_{0} = ⟨γ_{0}, c_{ϕ}τ⟩ and Γ_{1} = ⟨γ_{1}, τ⟩. By Lemma 4.5(a),

Out(S_{0}) = ((R_{1}/⟨c_{23}^{∗}⟩) × (R_{2}/⟨c_{12}^{∗}⟩))·(Γ_{0} × Γ_{1}) ≅ C_{2}^{4} ⋊ (Σ_{3} × Σ_{3}).

Lemma 5.5. The group Out(S_{ϕ}) is a 2-group. If α ∈ Aut(S_{0}) commutes with c_{ϕ} as elements of Out(S_{0}), then α extends to an automorphism of S_{ϕ}.

Proof. Since c_{ϕ} acts freely on the basis {e_{13}^{ω}, eω̄ ; 13} of Z(S_{0}), and since S_{0} is a characteristic subgroup of S_{ϕ}, the map induced by restriction

Out(S_{ϕ}) ⟶^{≅} N_{Out(S_{0})}(⟨c_{ϕ}⟩)/⟨c_{ϕ}⟩ = C_{Out(S_{0})}(c_{ϕ})/⟨c_{ϕ}⟩

is an isomorphism by Corollary 1.3. This proves the last statement. Since the centralizer of c_{ϕ} in

Out(S_{0})/O_{2}(Out(S_{0})) ≅ Σ_{3} × Σ_{3}

has order 4, C_{Out(S_{0})}(c_{ϕ}) is a 2-group, and hence Out(S_{ϕ}) is a 2-group. □

We next check the possibilities for Out_{F}(S_{0}) when F is a saturated fusion system.

Lemma 5.6. If F is a saturated fusion system over S_{ϕ}, then there is an automorphism φ ∈ Aut(S_{ϕ}) such that

Aut_{φFφ^{−1}}(S_{0}) ≤ ⟨γ_{0}, γ_{1}, Aut_{S_{ϕ}}(S_{0})⟩.

Proof. Set Δ = Out_{F}(S_{0}) and Q = O_{2}(Out(S_{0})) for short. Then Δ ∩ Q = 1 since Out_{S_{ϕ}}(S_{0}) = ⟨c_{ϕ}⟩ ∈ Syl_{2}(Δ) (S_{0} is fully normalized since it is normal). So there is a unique subgroup Δ′ ≤ ⟨[γ_{0}], [γ_{1}], c_{ϕ}⟩ such that QΔ = QΔ′ in Out(S_{0}).

By Proposition 1.8, there is α ∈ Aut(S_{ϕ}) such that [α] ∈ C_{Q}(c_{ϕ}) and Δ′ = [α]Δ[α]^{−1}. Then α extends to an automorphism φ ∈ Aut(S_{ϕ}) by Lemma 5.5, and

Out_{φFφ^{−1}}(S_{0}) = [α]Δ[α]^{−1} = Δ′ ≤ ⟨[γ_{0}], [γ_{1}], c_{ϕ}⟩.  □

We next describe Out(P) for P = H_{i} and N_{i}, and list the possibilities for Out_{F}(P) when F is a saturated fusion system over S_{ϕ}. When doing this, it will be helpful to translate automorphisms of A_{1} to matrices.

Define ρ_{i}^{∗} ∈ Aut(S_{0}) by setting

ρ_{1}^{∗}(( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & a & b+ā \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} ) and ρ_{2}^{∗}(( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & a & b+c̄ \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix}) .

Thus ρ_{i}^{∗} ∈ R_{i} is the identity on A_{3−i}, and ρ_{2}^{∗} = τρ_{1}^{∗}τ^{−1}. The ρ_{i}^{∗} commute with c_{ϕ} in Aut(S_{0}), and hence extend to automorphisms ρ̇_{i}^{∗} ∈ Aut(S_{ϕ}) by sending ϕ to itself. Similarly, we let τ̇ ∈ Aut(S_{ϕ}) be the extension of τ which sends ϕ to itself.

Let η_{1} ∈ Aut(H_{1}) be the automorphism such that

η_{1}(ϕ) = ϕ, η_{1}(e_{13}^{a}) = e_{12}^{a}, and η_{1}(e_{12}^{a}) = e_{12}^{a}e_{13}^{a} (for all a ∈ 𝔽_{4}).

Define η′_{1} ∈ Aut(H_{1}) by setting η′_{1} = ρ̇_{1}^{∗}η_{1}ρ̇_{1}^{∗}^{−1}. Finally, let η_{2}, η′_{2} ∈ Aut(H_{2}) be the automorphisms η_{2} = τ̇η_{1}τ̇^{−1} and η′_{2} = τ̇η′_{1}τ̇^{−1}.

Lemma 5.7. The following hold for any saturated fusion system F over S_{ϕ}.

(a) If H_{i} is F-essential for i = 1 or 2, then

Out_{F}(H_{i}) = ⟨[η_{i}], Out_{N_{i}}(H_{i})⟩ ≅ Σ_{3}  or  Out_{F}(H_{i}) = ⟨[η′_{i}], Out_{N_{i}}(H_{i})⟩ ≅ Σ_{3} .

(b) If Out_{F}(S_{0}) ≤ ⟨[γ_{0}], c_{ϕ}⟩ and H_{1} is F-essential, then there is φ ∈ Aut(S_{ϕ}) such that Out_{φFφ^{−1}}(S_{0}) = Out_{F}(S_{0}) and Out_{φFφ^{−1}}(H_{1}) = ⟨[η_{1}], c_{23}^{1}⟩. If in addition, H_{2} is Fessential, then φ can be chosen such that we also have Out_{φFφ^{−1}}(H_{2}) = ⟨[η_{2}], c_{12}^{1}⟩.

Proof. Since c_{ϕ} acts freely on the basis {e_{13}^{ω}, eω̄ ; 13, e_{12}^{ω}, eω̄ ; 12} of A_{1}, and since A_{1} is a characteristic subgroup of H_{1}, the map induced by restriction

Out(H_{1}) ⟶_{≅}^{Res_{A_{1}}} N_{Aut(A_{1})}(⟨c_{ϕ}⟩)/⟨c_{ϕ}⟩ = C_{Aut(A_{1})}(c_{ϕ})/⟨c_{ϕ}⟩

is an isomorphism by Corollary 1.3.

For each α ∈ Aut(A_{1}), let M(α) denote the matrix for α with respect to the ordered basis {e_{13}^{1}, e_{12}^{1}, e_{13}^{ω}, e_{12}^{ω}}. Matrices will be written as 2 × 2 blocks, where

I = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) , J = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) , Z = ( \begin{matrix}0 & 1 \\ 1 & 1 \end{matrix} ) , and Y = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) .

For example, M(c_{ϕ}) = ( \begin{matrix}I & I \\ 0 & I \end{matrix}) and M(c_{23}^{1}) = ( \begin{matrix}J & 0 \\ 0 & J \end{matrix}). By direct computation,

C_{GL_{4}(2)}(( \begin{matrix}I & I \\ 0 & I \end{matrix})) = \{( \begin{matrix}B & C \\ 0 & B \end{matrix}) | B ∈ GL_{2}(2), C ∈ M_{2}(𝔽_{2})\} ≅ C_{2}^{4} ⋊ GL_{2}(2).  (3)

Hence Out(H_{1}) ≅ C_{2}^{3} ⋊ GL_{2}(2) ≅ C_{2}^{3} ⋊ Σ_{3}. Also, since M(η_{1}|_{A_{1}}) = ( \begin{matrix}Z & 0 \\ 0 & Z \end{matrix}) and

M(c_{23}^{1}|_{A_{1}}) = ( \begin{matrix}J & 0 \\ 0 & J \end{matrix}) (and ⟨Z, J⟩ = GL_{2}(2)),

Out(H_{1}) = O_{2}(Out(H_{1}))·⟨[η_{1}], c_{23}^{1}⟩.

(a) We prove this for H_{1}; the case H_{2} then follows by symmetry. Assume H_{1} is F-essential for some saturated fusion system F. Set Δ = Out_{F}(H_{1}) and Q = O_{2}(Out(H_{1})) for short. Then Δ ∩ Q = 1, QΔ = Out(H_{1}), and c_{23}^{1} ∈ Δ. By Proposition 1.8, Δ = ⟨[αη_{1}α^{−1}], c_{23}^{1}⟩ for some α ∈ O_{2}(Aut(H_{1})) (thus [α] ∈ Q) which centralizes c_{23}^{1} in Out(H_{1}). Translated to matrices, and since we are working modulo ⟨M(c_{ϕ})⟩ = ⟨( \begin{matrix}I & I \\ 0 & I \end{matrix})⟩, this means that M(α|_{A_{1}}) = ( \begin{matrix}I & C \\ 0 & I \end{matrix}) for some C ∈ M_{2}(𝔽_{2}) (by (3)), and that JCJ^{−1} = C or C + I. Since JZJ^{−1} = I + Z, we get

C ∈ ⟨C_{M_{2}(𝔽_{2})}(J), Z⟩ = ⟨I, Y, Z⟩

(as an additive subgroup of M_{2}(𝔽_{2})). Also, since ( \begin{matrix}I & Z \\ 0 & I \end{matrix}) commutes with ( \begin{matrix}Z & 0 \\ 0 & Z \end{matrix}) (the matrix of η_{1}|_{A_{1}}), and since we are working modulo ⟨M(c_{ϕ})⟩ = ⟨( \begin{matrix}I & I \\ 0 & I \end{matrix})⟩, we can always choose C = 0 or C = Y. Since ( \begin{matrix}I & Y \\ 0 & I \end{matrix} ) = ( \begin{matrix}J & Y \\ 0 & J \end{matrix} ) ( \begin{matrix}J & 0 \\ 0 & J \end{matrix}) where ( \begin{matrix}J & Y \\ 0 & J \end{matrix} ) = M(ρ_{1}^{∗}|_{A_{1}}), this shows that we can take α = Id or α = (ρ̇_{1}^{∗}c_{23}^{1})|_{H_{1}}. Also,

(ρ̇_{1}^{∗}c_{23}^{1})η_{1}(ρ̇_{1}^{∗}c_{23}^{1})^{−1} = ρ̇_{1}^{∗}(c_{23}^{1}η_{1}c_{23}^{1}^{−1})ρ̇_{1}^{∗}^{−1} = ρ̇_{1}^{∗}η_{1}^{−1}ρ̇_{1}^{∗}^{−1} = η′_{1}^{−1},

and thus Δ must be one of the two groups ⟨[η_{1}], c_{23}^{1}⟩ or ⟨[η′_{1}], c_{23}^{1}⟩.

(b) Now assume Out_{F}(S_{0}) ≤ ⟨γ_{0}, c_{ϕ}⟩, and H_{1} is F-essential. Set φ = Id_{S_{ϕ}} if Out_{F}(H_{1}) = ⟨[η_{1}], c_{23}^{1}⟩, and φ = ρ̇_{1}^{∗} ∈ Aut(S_{ϕ}) if Out_{F}(H_{1}) = ⟨[η′_{1}], c_{23}^{1}⟩. In either case, Out_{φFφ^{−1}}(H_{1}) = ⟨[η_{1}], c_{23}^{1}⟩. Also, φ|_{H_{2}} = Id and φ|_{S_{0}} commutes with γ_{0}, and thus Out_{φFφ^{−1}}(P) = Out_{F}(P) for P = S_{0} and H_{2}.

Similarly, if H_{2} is also F-essential, we can set ψ = Id_{S_{ϕ}} if Out_{F}(H_{2}) = ⟨[η_{2}], c_{12}^{1}⟩, and ψ = ρ̇_{2}^{∗} ∈ Aut(S_{ϕ}) if Out_{F}(H_{2}) = ⟨[η′_{2}], c_{23}^{1}⟩. Then Out_{ψFψ^{−1}}(H_{2}) = ⟨[η_{2}], c_{12}^{1}⟩, and Out_{ψFψ^{−1}}(P) = Out_{F}(P) for P = S_{0} and H_{1}. □

We now turn our attention to N_{1} and N_{2}. Consider the basis

b_{1} = {e_{12}^{ω}, e_{12}^{ω}e_{13}^{ω}, eω̄ ; 12, eω̄ ; 12eω̄ ; 13}

of A_{1}, which Aut_{N_{1}}(A_{1}) = ⟨c_{23}^{1}, c_{ϕ}⟩ permutes freely. Let ν_{1} ∈ Aut(N_{1}) be the automorphism such that

ν_{1}(e_{12}^{ω}) = e_{12}^{ω}e_{13}^{ω}, ν_{1}(e_{12}^{ω}e_{13}^{ω}) = eω̄ ; 12, ν_{1}(eω̄ ; 12) = e_{12}^{ω},

ν_{1}(eω̄ ; 12eω̄ ; 13) = eω̄ ; 12eω̄ ; 13, ν_{1}(e_{23}^{1}) = e_{23}^{1}ϕ , and ν_{1}(ϕ) = e_{23}^{1} .

Thus ν_{1} permutes cyclically the first three elements in b_{1} and fixes the fourth, and from this it is easily seen to be an automorphism of N_{1} = A_{1}⋊⟨e_{23}^{1}, ϕ⟩. Set ν_{2} = τ̇ν_{1}τ̇^{−1}.

Lemma 5.8. If F is a saturated fusion system over S_{ϕ}, and N_{i} is F-essential for i = 1 or 2, then Out_{F}(N_{i}) = ⟨[ν_{i}], Out_{S_{ϕ}}(N_{i})⟩ ≅ Σ_{3}. If N_{i} is not F-essential, then Out_{F}(N_{i}) = Out_{S_{ϕ}}(N_{i}).

Proof. We prove this for N_{1}. Since N_{1}/A_{1} acts freely on the basis b_{1}, and since A_{1} is characteristic in N_{1}, the map induced by restriction

Out(N_{1}) ⟶^{≅} N_{Aut(A_{1})}(⟨c_{ϕ}, c_{23}^{1}⟩)/⟨c_{ϕ}, c_{23}^{1}⟩

is an isomorphism by Corollary 1.3.

The action of ⟨c_{ϕ}, c_{23}^{1}⟩ ≅ C_{2}^{2} on A_{1} permutes the elements of ⟨E_{13}, e_{12}^{1}⟩ in orbits of order one or two, and permutes the remaining eight elements in two orbits of order four:

b_{1} = {e_{12}^{ω}, e_{12}^{ω}e_{13}^{ω}, eω̄ ; 12, eω̄ ; 12eω̄ ; 13} and b_{2} = {e_{12}^{ω}e_{13}^{1}, e_{12}^{ω}eω̄ ; 13, eω̄ ; 12e_{13}^{1}, eω̄ ; 12e_{13}^{ω}} ,

each of which is a basis. Hence each element of the normalizer of ⟨c_{ϕ}, c_{23}^{1}⟩ either sends each of these bases to itself or exchanges them. Clearly, each permutation of the basis b_{1} defines an element of N_{Aut(A_{1})}(⟨c_{ϕ}, c_{23}^{1}⟩) (and determines a permutation of b_{2}), and so these define a subgroup isomorphic to Σ_{4} and of index at most two in this normalizer. The automorphism which sends each element of b_{1} to the product of the other three elements centralizes ⟨c_{ϕ}, c_{23}^{1}⟩ and exchanges the two bases.

This proves that N_{Aut(A_{1})}(⟨c_{ϕ}, c_{23}^{1}⟩) ≅ C_{2} × Σ_{4}, and hence

Out(N_{1}) ≅ N_{Aut(A_{1})}(⟨c_{ϕ}, c_{23}^{1}⟩)/⟨c_{ϕ}, c_{23}^{1}⟩ ≅ C_{2} × Σ_{3}.

The class of ν_{1} in Out(N_{1}) thus generates the unique subgroup of order three. So either Out_{F}(N_{1}) = ⟨[ν_{1}], Out_{S_{ϕ}}(N_{1})⟩ ≅ Σ_{3}, in which case N_{1} is F-essential, or Out_{F}(N_{1}) = Out_{S_{ϕ}}(N_{1}) and N_{1} is not F-essential. □

We now describe some restrictions on which combinations of subgroups can be essential in a centerfree nonconstrained saturated fusion system.

Lemma 5.9. Let F be any centerfree nonconstrained saturated fusion system over S_{ϕ}. Then for each of i = 1 and 2, either H_{i} or N_{i} is F-essential, but not both. If N_{1} and N_{2} are both F-essential, then Out_{F}(S_{0}) ≰ ⟨γ_{1}, c_{ϕ}⟩.

Proof. By Proposition 5.2 (and since Out(S_{ϕ}) is a 2-group), F is generated by automorphisms in Inn(S_{ϕ}), Aut_{F}(S_{0}), Aut_{F}(H_{i}), and Aut_{F}(N_{i}) (for i = 1, 2), and their restrictions. Since ⟨c_{ϕ}⟩ ∈ Syl_{2}(Out_{F}(S_{0})), each α ∈ Aut_{F}(S_{0}) must send A_{1} and A_{2} to themselves.

If neither H_{1} nor N_{1} is F-essential, then all morphisms in F are composites of restrictions of automorphisms of S_{ϕ}, S_{0}, H_{2}, and N_{2}, all of which send A_{2} to itself. Hence A_{2} is normal in F, which contradicts the assumption that F is nonconstrained. Similarly, if neither H_{2} nor N_{2} is F-essential, then A_{1} ◁ F, which again contradicts our assumption.

Thus at least one subgroup in each pair (H_{1}, N_{1}) and (H_{2}, N_{2}) must be F-essential. If N_{1} is F-essential, then ν_{1} ∈ Aut_{F}(N_{1}) by Lemma 5.8, and ν_{1}(H_{1}) = ⟨A_{1}, e_{23}^{1}⟩. This last subgroup is normal in S_{ϕ}, while N(H_{1}) = N_{1}. Hence H_{1} is not fully normalized in F, and so cannot be F-essential. Similarly, if N_{2} is F-essential, then H_{2} is not.

It remains to prove the last statement. Assume otherwise: assume N_{1} and N_{2} are F-essential, and Out_{F}(S_{0}) ≤ ⟨γ_{1}, c_{ϕ}⟩. Then neither H_{1} nor H_{2} is F-essential, so F is generated by automorphisms of S_{ϕ}, N_{1}, and N_{2}; together with γ_{1}, c_{ϕ} ∈ Aut(S_{0}). All of these automorphisms fix e_{13}^{1} (since S_{ϕ}, N_{1}, and N_{2} all have center e_{13}^{1}). Thus e_{13}^{1} is in the center of F, and this contradicts the assumption that F is centerfree. □

5.3 Fusion systems over S_{ϕ}

In order to better describe the subgroups generated by certain sets of elements of the Aut(A_{i}), we define an explicit isomorphism from Aut(A_{1}) to the alternating group A_{8}. We first describe this on an abstract 4-dimensional 𝔽_{2}-vector space V with ordered basis {v_{1}, v_{2}, v_{3}, v_{4}}.

Let Λ^{2}(V) = (V ⊗V)/⟨v ⊗ v | v ∈ V ⟩ be the second exterior power of V, let [v ⊗w] ∈ Λ^{2}(V) be the class of v ⊗ w, and set v_{ij} = [v_{i} ⊗ v_{j}]. Thus {v_{ij} | i < j} is a basis for Λ^{2}(V). Define 𝔮: Λ^{2}(V) ⟶ 𝔽_{2} by setting 𝔮(x) = 0 if x = [v ⊗ w] for some v, w ∈ V, and 𝔮(x) = 1 otherwise. Let 𝔟: V × V ⟶ 𝔽_{2} be the associated form 𝔟(x, y) = 𝔮(x + y) + 𝔮(x) + 𝔮(y). Thus 𝔮(v_{ij}) = 0 for all i, j, and 𝔟(v_{ij}, v_{kl}) = 1 if i, j, k, l are distinct and is zero otherwise. One can show that 𝔟 is bilinear and hence 𝔮 is quadratic by comparing them with the bilinear and quadratic forms which take the same values on the v_{ij}. Hence this defines an explicit isomorphism from Aut(V) ≅ GL_{4}(2) to Ω(Λ^{2}(V), 𝔮) ≅ Ω_{6}^{+}(2) (the commutator subgroup of the orthogonal group O(Λ^{2}(V), 𝔮)), by sending α to Λ^{2}(α).

We next construct an explicit isomorphism Ω(Λ^{2}(V), 𝔮) ≅ A_{8}. Let P_{e}(\underline{8}) be the group of subsets of even order in \underline{8} = \{1, 2, …, 8\}, regarded as an 𝔽_{2}-vector space with addition given by symmetric difference X + Y = ((X∖Y) ∪ (Y ∖X)). Let 𝔮 be the quadratic form on P_{e}(\underline{8})/⟨\underline{8}⟩ defined by 𝔮(X) = \frac{1}{2}|X|, associated to the bilinear form 𝔟(X, Y) = |X∩Y |. The symmetric group Σ_{8} acts on P_{e}(\underline{8})/⟨\underline{8}⟩ preserving the form, and this defines isomorphisms Σ_{8} ⟶^{≅} SO(P_{e}(\underline{8})/⟨\underline{8}⟩, 𝔮) and A_{8} ⟶^{≅} Ω(P_{e}(\underline{8})/⟨\underline{8}⟩, 𝔮).

Define κ: Λ^{2}(V) ⟶^{≅} P_{e}(\underline{8})/⟨\underline{8}⟩ by setting

κ(v_{12}) = {1234}  κ(v_{13}) = {1256}  κ(v_{14}) = {1357}
κ(v_{34}) = {1238}  κ(v_{24}) = {2356}  κ(v_{23}) = {1367}

This clearly preserves the quadratic forms on the two spaces. Let

χ_{V} : Aut(V) ⟶_{≅}^{Λ^{2}(−)} Ω(Λ^{2}(V), 𝔮) ⟶_{≅}^{κ_{∗}} Ω(P_{e}(\underline{8})/⟨\underline{8}⟩, 𝔮) ⟵^{≅} A_{8}

denote the isomorphism induced by Λ^{2}(−) and κ.

We apply this here with V = A_{1}, and with the ordered basis {v_{1}, v_{2}, v_{3}, v_{4}} = {e_{13}^{1}, e_{13}^{ω}, e_{12}^{1}, e_{12}^{ω}}. We first give an explicit example of how χ_{A_{1}}(α) can be determined in practice for α ∈ Aut(A_{1}).

Consider the case α = c_{23}^{1}. By (1), c_{23}^{1}(e_{13}^{a}) = e_{13}^{a} and c_{23}^{1}(e_{12}^{a}) = e_{12}^{a}e_{13}^{a}, so that (upon writing elements additively)

c_{23}^{1}(v_{1}) = v_{1}, c_{23}^{1}(v_{2}) = v_{2}, c_{23}^{1}(v_{3}) = v_{1} + v_{3}, c_{23}^{1}(v_{4}) = v_{2} + v_{4}.

Hence Λ^{2}(c_{ϕ}) and κ_{∗}(Λ^{2}(c_{ϕ})) make the following assignments:

v_{12} ↦ v_{12}  v_{13} ↦ v_{13}  v_{14} ↦ v_{12} + v_{14}
{1234} ↦ {1234}  {1256} ↦ {1256}  {1357} ↦ {2457}

and

v_{12} + v_{34} ↦ v_{14} + v_{23} + v_{34} v_{13} + v_{24} ↦ v_{13} + v_{24} v_{14} + v_{23} ↦ v_{14} + v_{23}
{48} ↦ {47}  {13} ↦ {13}  {56} ↦ {56}.

Note that by taking sums of complementary pairs in the second row, we got information on how κ_{∗}(Λ^{2}(c_{ϕ})) acts on certain sets of order two. Recall that in the quotient group P_{e}(\underline{8})/⟨\underline{8}⟩, each subset of \underline{8} is identified with its complement. So we also get that {57} = {13} + {1357} is sent to {13} + {2457} = {68}. Since {56} is left invariant, the permutation which induces κ_{∗}(Λ^{2}(c_{ϕ})) must exchange 5 and 6 and send 7 to 8. Upon continuing with arguments of this type, we eventually show that κ_{∗}(Λ^{2}(c_{ϕ})) is induced by the permutation (5 6)(7 8), and hence that χ_{A_{1}}(c_{23}^{1}) = (5 6)(7 8). In fact, if one just wants to check that (5 6)(7 8) is indeed the right answer, the procedure is much simpler: it suffices to check that this permutation does indeed induce κ_{∗}(Λ^{2}(c_{ϕ})) on the six basis elements as listed above.

We now list images under χ_{A_{1}} of several of the automorphisms we need to consider. In each case, M(α) denotes the matrix of α with respect to the ordered basis {e_{13}^{1}, e_{13}^{ω}, e_{12}^{1}, e_{12}^{ω}}:

\begin{matrix}α = & c_{ϕ} & c_{23}^{1} & c_{23}^{ω} & ρ_{1}^{∗}|_{A_{1}} \\ M(α) = & (\begin{matrix}J & 0 \\ 0 & J \end{matrix}) & (\begin{matrix}I & I \\ 0 & I \end{matrix}) & (\begin{matrix}I & Z \\ 0 & I \end{matrix}) & (\begin{matrix}I & J \\ 0 & I \end{matrix}) \\ χ_{A_{1}}(α) = & (1 2)(5 6) & (5 6)(7 8) & (5 8)(6 7) & (1 2)(3 4) . \end{matrix}  (4)

Here, J = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) and Z = ( \begin{matrix}0 & 1 \\ 1 & 1 \end{matrix} ). We also get the following values for χ(α|_{A_{1}}), for certain automorphisms α ∈ Aut(P) of order 3 which can occur in Aut_{F}(P):

\begin{matrix}(α, P) = & (γ_{0}, S_{0}) & (γ_{1}, S_{0}) & (ν_{1}, N_{1}) & (η_{1}, H_{1}) & (η′_{1}, H_{1}) \\ M(α|_{A_{1}}) = & (\begin{matrix}Z^{−1} & 0 \\ 0 & Z \end{matrix}) & (\begin{matrix}I & 0 \\ 0 & Z \end{matrix}) & ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) & (\begin{matrix}0 & I \\ I & I \end{matrix}) & (\begin{matrix}J & J \\ I & I+J \end{matrix}) \\ χ_{A_{1}}(α|_{A_{1}}) = & (5 6 7) & (1 3 2)(5 7 6) & (2 5 8)(1 6 7) & (4 8 7) & (3 8 7) . \end{matrix} (5)

This is now applied in the following lemma, which identifies certain groups of automorphisms of A_{1}.

Lemma 5.10. (a) ⟨Aut_{S_{ϕ}}(A_{1}), η_{1}⟩ ≅ ⟨Aut_{S_{ϕ}}(A_{1}), η′_{1}⟩ ≅ Σ_{5} and γ_{0}|_{A_{1}} belongs to both of these groups of automorphisms;

(b) ⟨Aut_{S_{ϕ}}(A_{1}), η_{1}, γ_{1}⟩ = ⟨Aut_{S_{ϕ}}(A_{1}), η_{1}, γ_{0}, γ_{1}⟩ ≅ (C_{3} × A_{5}) ⋊ C_{2} ≅ ΓL_{2}(4);

(c) ⟨Aut_{S_{ϕ}}(A_{1}), η′_{1}, γ_{1}⟩ = ⟨Aut_{S_{ϕ}}(A_{1}), ν_{1}, η′_{1}, γ_{0}, γ_{1}⟩ ≅ A_{7};

(d) ⟨Aut_{S_{ϕ}}(A_{1}), ν_{1}, γ_{0}⟩ ≅ A_{6};

(e) ⟨Aut_{S_{ϕ}}(A_{1}), ν_{1}, γ_{0}γ_{1}⟩ = ⟨Aut_{S_{ϕ}}(A_{1}), ν_{1}, γ_{0}, γ_{1}⟩ ≅ A_{7}. Here, we write ν_{1}, η_{1}, η′_{1}, and γ_{i}, but mean their restrictions to A_{1}.

Proof. The proof will be based on the isomorphism χ = χ_{A_{1}} : Aut(A_{1}) ⟶^{≅} A_{8} constructed above. To simplify notation, we identify these two groups, and omit “χ(−)” where it would be appropriate.

Whenever I and J are disjoint subsets of \underline{8} = \{1, …, 8\} (m ≥ 1), we let A_{I,J} ≤ A_{8} (A_{I} ≤ A_{8}) denote the subgroups of permutations which leave I and J invariant (leave I invariant), and fix all other elements in \underline{8}. Elements of the subsets are listed without brackets or commas. Thus, for example, A_{125678} (≅ A_{6}) is the subgroup of even permutations which fix 3 and 4, while A_{12;5678} contains those permutations which fix 3 and 4 and leave the subset {1, 2} invariant.

We refer to (4) and (5) for the images in A_{8} of certain elements of Aut(A_{1}).

(a) Consider first

H_{a} =^{def} ⟨Aut_{S_{ϕ}}(A_{1}), η_{1}⟩ = ⟨c_{23}^{ω}, c_{23}^{1}, c_{ϕ}, η_{1}⟩ = ⟨(5 8)(6 7), (5 6)(7 8), (1 2)(5 6), (4 8 7)⟩ .

Then H_{a} ≤ A_{12;45678}. Also, the image of H_{a} under projection to Σ_{5} (permutations of {4, 5, 6, 7, 8}) contains the 2-cycle (5 6) and the 5-cycle c_{23}^{ω}η_{1} = (5 8 6 7 4) (where we compose from right to left). Thus the projection is surjective, and this proves that

H_{a} = A_{12;45678} ≅ Σ_{5}.  (6)

In particular, γ_{0} = (5 6 7) ∈ H_{a}.

Simiarly, if we set

H′_{a} =^{def} ⟨Aut_{S_{ϕ}}(A_{1}), η′_{1}⟩ = ⟨c_{23}^{ω}, c_{23}^{1}, c_{ϕ}, η′_{1}⟩ = ⟨(5 8)(6 7), (5 6)(7 8), (1 2)(5 6), (3 8 7)⟩ ,

then

H′_{a} = A_{12;35678} ≅ Σ_{5} and hence γ_{0} = (5 6 7) ∈ H′_{a}.  (7)

(b) By (6),

H_{b} =^{def} ⟨Aut_{S_{ϕ}}(A_{1}), η_{1}, γ_{1}⟩ = ⟨H_{a}, γ_{1}⟩ = ⟨A_{12;45678}, (1 3 2)(5 7 6)⟩ = A_{123;45678}.

Thus H_{b} ≅ (C_{3} × A_{5}) ⋊ C_{2} ≅ ΓL_{2}(4) and γ_{0} = (5 6 7) ∈ H_{b}.

(c) By (7),

H_{c} =^{def} ⟨Aut_{S_{ϕ}}(A_{1}), η′_{1}, γ_{1}⟩ = ⟨A_{12;35678}, (1 3 2)(5 7 6)⟩ = A_{1235678} ≅ A_{7}.

In particular, ν_{1} = (2 5 8)(1 6 7) and γ_{0} = (5 6 7) are both in H_{c}.

(d) We have

H_{d} =^{def} ⟨Aut_{S_{ϕ}}(A_{1}), γ_{0}, ν_{1}⟩ = ⟨(1 2)(5 6), (5 8)(6 7), (5 6)(7 8), (5 6 7), (2 5 8)(1 6 7)⟩

= ⟨A_{12;5678}, (2 5 8)(1 6 7)⟩ = A_{125678} ≅ A_{6}.

(e) Consider the subgroup

H_{e} =^{def} ⟨Aut_{S_{ϕ}}(A_{1}), ν_{1}, γ_{0}γ_{1}⟩ = ⟨Aut_{S_{ϕ}}(A_{1}), (2 5 8)(1 6 7), (1 3 2)⟩.

Then ν_{1}^{−1}(1 3 2)ν_{1} = (7 3 8) = η′_{1} ∈ H_{e}, and so

H_{e} = ⟨H′_{a}, (2 5 8)(1 6 7), (1 3 2)⟩ = ⟨A_{12;35678}, (2 5 8)(1 6 7), (1 3 2)⟩ = A_{1235678}.

Thus H_{e} ≅ A_{7}, and γ_{0}, γ_{1} ∈ H_{e}. □

We are now ready to list fusion systems over S_{ϕ}. In the statement and the proof of the following theorem, we follow the usual notation by writing P ΓL_{n}(q) = P GL_{n}(q)⋊⟨ϕ⟩ and P ΣL_{n}(q) = P SL_{n}(q)⋊⟨ϕ⟩, where ϕ is a generator of Aut(𝔽_{q}) (extended to an automorphism on matrix groups).

Theorem 5.11. If F is a nonconstrained centerfree saturated fusion system over S_{ϕ}, then it is isomorphic to the fusion system of one of the following groups: M_{22}, M_{23}, McL, P ΣL_{3}(4), P ΓL_{3}(4), or P SL_{4}(5) ≅ P Ω_{6}^{+}(5).

Proof. Let F be a saturated fusion system over S_{ϕ}. Assume F is nonconstrained and centerfree. By Lemma 5.6, upon replacing F by φFφ^{−1} for some φ ∈ Aut(S_{ϕ}), we can assume that

Out_{F}(S_{0}) ≤ ⟨[γ_{0}], [γ_{1}], c_{ϕ}⟩.  (8)

We first list the different choices for the set of F-essential subgroups, then we list the different combinations for Aut_{F}(P) (or Out_{F}(P)) for each F-essential subgroup P. Using that, we show that F is isomorphic to one of a list of six explicitly defined fusion systems over S_{ϕ}, which we then compare with those in the statement of the theorem.

The following are some conditions which must hold for F:

(a) Out_{F}(S_{ϕ}) = 1. This holds since Out(S_{ϕ}) is a 2-group (Lemma 5.5).

(b) The only possible F-essential subgroups are S_{0}, N_{1}, N_{2}, H_{1}, H_{2}, and their conjugates (Proposition 5.2).

(c) Exactly one of the subgroups H_{1} or N_{1} is essential, and exactly one of the subgroups H_{2} or N_{2} is essential (Lemma 5.9).

(d) If H_{i} is F-essential (i = 1 or 2), then γ_{0} ∈ Aut_{F}(S_{0}). If H_{1} is F-essential, then by Lemma 5.7(a), η_{1} or η′_{1} is in Aut_{F}(H_{1}), so γ_{0}|_{A_{1}} ∈ Aut_{F}(A_{1}) by Lemma 5.10(a). This is the restriction of an automorphism in Aut_{F}(S_{0}) by the extension axiom, and thus γ_{0} ∈ Aut_{F}(S_{0}) by (8). Similarly, if H_{2} is F-essential, then τγ_{0}τ^{−1} = γ_{0} ∈ Aut_{F}(S_{0}).

(e) If γ_{0}, γ_{1} ∈ Aut_{F}(S_{0}), and H_{i} is essential for i = 1 or 2, then Out_{F}(H_{i}) = ⟨[η_{i}], Out_{S_{ϕ}}(H_{i})⟩. To see this when i = 1, assume otherwise: thus η′_{1} ∈ Aut_{F}(H_{1}) by Lemma 5.7(a). Then ν_{1}|_{A_{1}} ∈ Aut_{F}(A_{1}) by Lemma 5.10(c), which implies by the extension axiom that ν_{1}|_{A_{1}} extends to an automorphism in Aut_{F}(N_{1}). Thus N_{1} is F-essential by Lemma 5.8, so H_{1} is not F-essential by Lemma 5.9, which is a contradiction.

(f) If N_{1} and N_{2} are both F-essential, then at least one of the automorphisms γ_{0}, γ_{0}γ_{1}, or γ_{0}γ_{1}^{−1} must be in Aut_{F}(S_{0}). To see this, note first that by (8), Out_{F}(S_{0}) = ⟨Δ, c_{ϕ}⟩ for some Δ ≤ ⟨[γ_{0}], [γ_{1}]⟩ ≅ C_{3}^{2}. Also, by Lemma 5.9, Δ ≰ ⟨[γ_{1}]⟩. Thus for some i, [γ_{0}γ_{1}^{i}] ∈ Δ ≤ Out_{F}(S_{0}).

If H_{1} or H_{2} is F-essential, then by (d), γ_{0} ∈ Aut_{F}(S_{0}). If neither of these groups is F-essential, then N_{1} and N_{2} are both F-essential by (c), and so γ_{0}γ_{1}^{i} ∈ Aut_{F}(S_{0}) (some i = 0, ±1) by (f). Since γ_{0}, γ_{1} ∈ Aut(S_{0}) are both inverted by c_{ϕ}, Out_{F}(S_{0}) contains a subgroup Σ_{3} in all cases, and hence S_{0} is F-essential for any F.

Together with points (b) and (c), this shows that the choices for the set of F-essential subgroups (up to conjugacy) are among the following:

{H_{1}, H_{2}, S_{0}} , {H_{1}, N_{2}, S_{0}} , {N_{1}, H_{2}, S_{0}} and {N_{1}, N_{2}, S_{0}} .  (9)

If N_{1} and H_{2} are F-essential, then H_{1} and N_{2} are essential in the fusion system τ̇Fτ̇^{−1} which is isomorphic to F. We claim that upon combining (9) with the restrictions on the automorphism groups Out_{F}(−) imposed by points (a) and (d)–(f), we are reduced to the following list of eleven candidates for fusion systems F, up to isomorphism:

\begin{matrix}\{H_{1}; H_{2}; S_{0}\} : & \{η_{1}; η_{2}; γ_{0}\} , \{η_{1}; η_{2}; γ_{0}, γ_{1}\} , \{η′_{1}; η_{2}; γ_{0}\} , \{η′_{1}; η′_{2}; γ_{0}\} ; \\ \{H_{1}; N_{2}; S_{0}\} : & \{η_{1}; ν_{2}; γ_{0}\} , \{η_{1}; ν_{2}; γ_{0}, γ_{1}\} , \{η′_{1}; ν_{2}; γ_{0}\} ; \\ \{N_{1}; N_{2}; S_{0}\} : & \{ν_{1}; ν_{2}; γ_{0}\} , \{ν_{1}; ν_{2}; γ_{0}, γ_{1}\} , \{ν_{1}; ν_{2}; γ_{0}γ_{1}\} , \{ν_{1}; ν_{2}; γ_{0}γ_{1}^{−1}\} . \end{matrix} (10)

The first entry in each row of (10) gives the F-essential subgroups. The later entries list, for each F-essential subgroup P, generators of Aut_{F}(P) in addition to Aut_{S_{ϕ}}(P). In each case, F is the fusion system generated by the given automorphism groups of the given essential subgroups and Inn(S_{ϕ}); i.e., the fusion system generated by Inn(S_{ϕ}), γ_{k} ∈ Aut(S_{0}), η_{i} or η′_{i} in Aut(H_{i}), and ν_{j} ∈ Aut(N_{j}), where k, i, and j are as listed. Thus, for example, the last entry in the first row describes the fusion system generated by Inn(S_{ϕ}), γ_{0} ∈ Aut(S_{0}), η′_{1} ∈ Aut(H_{1}), and η′_{2} ∈ Aut(H_{2}).

We next justify the claim. When H_{1} or H_{2} is F-essential, then Out_{F}(S_{0}) = ⟨[γ_{0}], c_{ϕ}⟩ or ⟨[γ_{0}], [γ_{1}], c_{ϕ}⟩ by (d). By (e), the second is possible only if η_{i} ∈ Aut_{F}(H_{i}) for all H_{i} which are F-essential. Thus the seven fusion systems listed in the first two rows are the only possible ones for which H_{1} or H_{2} is F-essential (up to replacing F by τ̇Fτ̇^{−1}). If N_{1} and N_{2} are F-essential, then by (f), F must be one of the four fusion systems listed in the third row of (10).

By Lemma 5.10(e), if N_{1} is F-essential (so ν_{1} ∈ Aut_{F}(N_{1}) by Lemma 5.8) and γ_{0}γ_{1} ∈ Aut_{F}(S_{0}), then γ_{0}|_{A_{1}}, γ_{1}|_{A_{1}} ∈ Aut_{F}(A_{1}). So by the extension axiom (and (8)), γ_{0}, γ_{1} ∈ Aut_{F}(S_{0}) in this case. Likewise, if γ_{0}γ_{1}^{−1} = τ(γ_{0}γ_{1})τ^{−1} ∈ Aut_{F}(S_{0}) and N_{2} is F-essential, then γ_{0}, γ_{1} ∈ Aut_{F}(S_{0}). In other words, F cannot have the form corresponding to either of the last two entries in the last row of (10).

By Lemma 5.7(b), if Out_{F}(S_{0}) = ⟨[γ_{0}], c_{ϕ}⟩, H_{1} is F-essential, and Out_{F}(H_{1}) = ⟨[η′_{1}], c_{23}^{1}⟩, then there is an automorphism φ ∈ Aut(S_{ϕ}) such that Out_{φFφ^{−1}}(S_{0}) = Out_{F}(S_{0}) and Out_{φFφ^{−1}}(H_{1}) = ⟨[η_{1}], c_{23}^{1}⟩. If, furthermore, H_{2} is also F-essential, then φ can be chosen such that Out_{φFφ^{−1}}(H_{2}) = ⟨[η_{2}], c_{12}^{1}⟩. In other words, we can eliminate all of the cases in the first two rows of (10) which involve η′_{1} or η′_{2}, since the corresponding fusion systems are isomorphic to others in the list.

Out_{F}(S_{0}) F-essential  Aut_{F}(A_{1})  Aut_{F}(A_{2})  G
⟨[γ_{0}], c_{ϕ}⟩ H_{1}, H_{2}, S_{0}  Σ_{5}  Σ_{5}  P ΣL_{3}(4)
⟨[γ_{0}], [γ_{1}], c_{ϕ}⟩ H_{1}, H_{2}, S_{0} (C_{3} × A_{5}) ⋊ C_{2} (C_{3} × A_{5}) ⋊ C_{2}  P ΓL_{3}(4)
⟨[γ_{0}], c_{ϕ}⟩  H_{1}, N_{2}, S_{0}  Σ_{5}  A_{6}  M_{22}
⟨[γ_{0}], [γ_{1}], c_{ϕ}⟩ H_{1}, N_{2}, S_{0} (C_{3} × A_{5}) ⋊ C_{2}  A_{7}  M_{23}
⟨[γ_{0}], c_{ϕ}⟩  N_{1}, N_{2}, S_{0}  A_{6}  A_{6}  P SL_{4}(5) ≅ P Ω_{6}^{+}(5)
⟨[γ_{0}], [γ_{1}], c_{ϕ}⟩ N_{1}, N_{2}, S_{0}  A_{7}  A_{7}  McL

Table 5.2

We have now shown that F is isomorphic to one of six fusion systems: the first two in each row of (10). These six are described in more detail in Table 5.2, where in all cases, Out_{F}(H_{i}) = ⟨[η_{i}], Out_{S_{ϕ}}(H_{i})⟩ if H_{i} is F-essential. If H_{i} is F-essential, then Aut_{F}(A_{i}) ≅ Σ_{5} if Out_{F}(S_{0}) = ⟨[γ_{0}], c_{ϕ}⟩ by Lemma 5.10(a), while Aut_{F}(A_{i}) ≅ (C_{3} × A_{5}) ⋊ C_{2} if Out_{F}(S_{0}) = ⟨[γ_{0}], c_{ϕ}⟩ by Lemma 5.10(b). The descriptions of Aut_{F}(A_{i}) when N_{i} is F-essential follow in a similar way from Lemma 5.10(d,e). By inspection, these six fusion systems are distinguished by the groups Aut_{F}(A_{1}) and Aut_{F}(A_{2}) as described in the table.

It remains to prove that the groups G listed in the table do realize these fusion systems: that they all have Sylow 2-subgroups isomorphic to S_{ϕ}, and have automorphism groups Aut_{G}(A_{i}) as described. This is clear for the groups P ΣL_{3}(4) and P ΓL_{3}(4) using the well-known isomorphisms ΣL_{2}(4) ≅ Σ_{5} and ΓL_{2}(4) ≅ (C_{3}×A_{5})⋊C_{2} (or by directly determining Aut_{G}(H_{i}) and Aut_{G}(S_{0})).

Since A_{6} has no subgroup of index 7 or 8, and A_{7} no subgroup of index 8, the group GL_{4}(2) ≅ A_{8} contains unique conjugacy classes of subgroups isomorphic to A_{6} and A_{7}. Since A_{6} and A_{7} are simple, this implies that up to isomorphism, there are unique semidirect products C_{2}^{4} ⋊ A_{6} and C_{2}^{4} ⋊ A_{7} which are not direct products. By Lemma 5.10, Aut_{S_{ϕ}}(A_{i}) is contained in a subgroup isomorphic to A_{7}, and hence S_{ϕ} is a Sylow 2-subgroup of the (of any) semidirect product C_{2}^{4}⋊A_{6} or C_{2}^{4}⋊A_{7} which is not a direct product.

When q ≡ ±5 (mod 8), then P Ω_{6}^{±}(q) is the commutator subgroup of the projective orthogonal group of a quadratic form on V = 𝔽_{q}^{6} with orthonormal basis {v_{1}, …, v_{6}}. This group contains two conjugacy classes of subgroups C_{2}^{4}⋊A_{6}: the groups of automorphisms which preserve up to sign one of the two bases {v_{i}} or {v_{1} ± v_{2}, v_{3} ± v_{4}, v_{5} ± v_{6}}. (These two orthogonal bases are inequivalent, since 2 is always a nonsquare for such q.) Since these are subgroups of odd index, P Ω_{6}^{±}(q) has Sylow 2-subgroups isomorphic to S_{ϕ}, and its fusion system is the one with these automorphism groups (and is independent of q).

As for the other groups, M_{22} contains subgroups C_{2}^{4} ⋊ Σ_{5} (the quintet subgroup) and C_{2}^{4} ⋊ A_{6} (the hexad subgroup); while M_{23} contains (C_{2}^{4} ⋊ C_{3}) ⋊ Σ_{5} (the quintet subgroup) and C_{2}^{4} ⋊ A_{7} (the heptad subgroup). See [Co, Table 3] for more detail. By [Fi, Theorem 1], McLaughlin’s group McL contains two conjugacy classes of subgroups C_{2}^{4}⋊A_{7}. So all three of these groups have the fusion systems described in Table 5.2. □

Note also that McL contains M_{22}, P Ω_{6}^{−}(3), and P ΣL_{3}(4) as subgroups of odd index, while M_{23} contains M_{22} and P ΣL_{3}(4) as subgroups of odd index.

6. Fusion systems over UT_{5}(2)

Throughout this section, T = UT_{5}(2) denotes the group of 5 × 5 upper triangular matrices over 𝔽_{2}. We let e_{ij} ∈ T (for i < j) be the elementary matrix with nontrivial entry in the (i, j) position. Also, c_{ij} denotes conjugation by e_{ij}, regarded as an automorphism of T or as a homomorphism between subgroups of T. For later reference, we note here the following relations among the e_{ij}:

(e_{ij}e_{kℓ})^{2} = [e_{ij}, e_{kℓ}] = \begin{cases}e_{iℓ} & if j = k \\ e_{kj} & if i = ℓ \\ 1 & if i ≠ ℓ and j ≠ k. \end{cases}  (1)

For any pair of sets of indices I, J ⊆ {1, 2, 3, 4, 5}, let E_{I;J} ≤ T denote the subgroup generated by all e_{ij} for i ∈ I and j ∈ J (and i < j). In particular, we focus attention on the “rectangular” subgroups A_{1} = E_{12;345}, A_{2} = E_{123;45}, U_{1} = E_{1;2345}, and U_{2} = E_{1234;5}. These can be described pictorally as follows:

A_{1} =  A_{2} =  U_{1} =  U_{2} =

We also need to consider the following index two subgroups Q_{i}:

Q_{1} =_{=A_{1}A_{2}U_{2}}  Q_{2} =_{=A_{2}U_{1}U_{2}}  Q_{3} =_{=A_{1}U_{1}U_{2}}  Q_{4} =_{=A_{1}A_{2}U_{1}}  .

We will show in Proposition 6.5 that the Q_{i} are the only critical subgroups of T.

The following lemma is very elementary and well known; we include it here for the sake of completeness.

Lemma 6.1. The only elementary abelian subgroups of rank 6 in T are A_{1} and A_{2}.

Proof. Set R = E_{123;345} = A_{1}A_{2} and R_{0} = Z(R) = E_{12;45} for short. By (1), all involutions in R are in A_{1} ∪ A_{2}, and no element of A_{1}∖R_{0} commutes with any element of A_{2}∖R_{0}. Hence each elementary abelian subgroup of R is contained in A_{1} or in A_{2}.

Assume A ≤ T is elementary abelian of rank six, and set B = A ∩ R. We just saw that B is contained in A_{1} or A_{2}; it suffices to handle the case B ≤ A_{1}. Since T/R ≅ C_{2}^{2}, rk(B) ≥ 4. If rk(B) = 4, then AR = T, so there are elements g, h ∈ A such that g ∈ e_{12}R and h ∈ e_{45}R. Then

B ∩ R_{0} ≤ C_{R_{0}}(⟨g, h⟩) = C_{R_{0}}(⟨e_{12}, e_{45}⟩) = ⟨e_{15}⟩

(since R = C_{T} (R_{0})), so rk(B) ≤ 1 + rk(A_{1}/R_{0}) = 3, a contradiction. Thus rk(B) = 5, A = ⟨B, g⟩ for some g ∈ T ∖R, B ∩ R_{0} ≤ C_{R_{0}}(g) has rank at least three, and this is impossible since rk(C_{R_{0}}(a)) = 2 for a = e_{12}, e_{45}, or e_{12}e_{45}. □

6.1 Determining the critical subgroups Throughout this subsection, we write

T′ = [T, T] = and Z_{2} = ⟨e_{15}, e_{14}, e_{25}⟩ =

for short. These subgroups will appear repeatedly. Using (1), they are seen to be terms in the upper and lower central series for T:

Z_{2} = [T, T′] = Z_{2}(T) and T′ = Z_{3}(T).  (2)

Also, τ ∈ Aut(T) is the automorphism τ(e_{ij}) = e_{6−j,6−i}. We first show:

Lemma 6.2. All critical subgroups of T contain Z_{2}.

Proof. Fix a critical subgroup P ≤ T, and assume first that e_{14} ∉ P. We apply Lemma 3.6 with z = e_{15}, g = e_{14}, and y = e_{25}. By the proposition, P = C_{T} (h) for some h such that [e_{14}, h] = e_{15}. Also, either e_{25} ∈ Z(P) and h is not T -conjugate to e_{25}h, or e_{14}e_{25} ∈ Z(P) and h is not T -conjugate to e_{14}e_{25}h.

If e_{25} ∈ Z(P), then since [h, e_{14}] = e_{15} ≠ 1,

h ∈ C_{T} (e_{25}) ∖ C_{T} (e_{14}) = ⟨e_{45}, A_{1}A_{2}⟩ ∖ ⟨e_{12}, A_{1}A_{2}⟩ = e_{45}·A_{1}A_{2}.

Since [A_{1}A_{2}, e_{24}] = 1, [h, e_{24}] = [e_{45}, e_{24}] = e_{25}, contradicting the condition that h not be T -conjugate to e_{25}h. Similarly, if e_{14}e_{25} ∈ Z(P), then

h ∈ C_{T} (e_{14}e_{25}) ∖ C_{T} (e_{14}) = ⟨e_{12}e_{45}, A_{1}A_{2}⟩ ∖ ⟨e_{12}, A_{1}A_{2}⟩ = e_{12}e_{45}·A_{1}A_{2},

so [h, e_{24}] = [e_{12}e_{45}, e_{24}] = e_{14}e_{25}, contradicting the condition that h not be T -conjugate to e_{14}e_{25}h.

This proves that e_{14} ∈ P, and a similar argument shows that e_{25} ∈ P. □

We next reduce to the case of subgroups having index 2 in their normalizers.

Lemma 6.3. If P is a critical subgroup of T, then |N_{T} (P)/P | = 2.

Proof. Assume otherwise: let P be a critical subgroup of T with |N(P)/P | ≥ 4. By Proposition 3.3(c),

rk([g, P/Fr(P)]) ≥ 2 for each g ∈ N(P)∖P  (3)

and

|N(P)/P | = 2^{k} ⟹ rk(P/Fr(P)) ≥ 2k.  (4)

By Lemma 6.2, P ≥ Z_{2}. Hence [T′, P] ≤ [T′, T] = Z_{2} ≤ P by (2), so N(P) ≥ T′. We now consider separately the cases where e_{15} ∈ Fr(P) or e_{15} ∉ Fr(P). We will frequently be using (1) for commutator and squaring relations, without referring to it each time.

Case 1: Assume first that e_{15} ∈ Fr(P). Since [e_{13}, P] ≤ [e_{13}, T] = ⟨e_{14}, e_{15}⟩, this implies rk([e_{13}, P/Fr(P)]) ≤ 1. Hence e_{13} ∈ P by (3), and e_{35} ∈ P by symmetry.

We claim that

(a) e_{14} ∉ Fr(P) implies P ≤ ⟨T′, e_{12}, e_{23}, e_{45}⟩; and

(b) e_{25} ∉ Fr(P) implies P ≤ ⟨T′, e_{12}, e_{34}, e_{45}⟩.

If e_{14} ∉ Fr(P), then since e_{13} ∈ P and e_{15} ∈ Fr(P), this implies e_{14}, e_{14}e_{15} ∉ [e_{13}, P]. Also, [e_{13}, T] = ⟨e_{14}, e_{15}⟩, and hence

P ≤ {g ∈ T | [e_{13}, g] ∈ ⟨e_{15}⟩} = ⟨e_{35}, C_{T} (e_{13})⟩ = ⟨T′, e_{12}, e_{23}, e_{45}⟩.

Point (b) follows by symmetry with respect to τ ∈ Aut(T).

Case 1a: Assume e_{24} ∉ P. Thus e_{24} ∈ N_{T} (P)∖P, and rk([e_{24}, P/Fr(P)]) ≥ 2 by (3). Since [e_{24}, T] = ⟨e_{14}, e_{25}⟩, this implies that

e_{14}, e_{25} ∈ [e_{24}, P] and Fr(P) ∩ Z_{2} = ⟨e_{15}⟩.  (5)

The second point, together with (a) and (b), implies P ≤ ⟨T′, e_{12}, e_{45}⟩.

Set P_{0} = ⟨Z_{2}, e_{13}, e_{15}⟩. We have now shown that P_{0} ≤ P ≤ ⟨P_{0}, e_{24}, e_{12}, e_{45}⟩, and that e_{24} ∉ P. Also, since [e_{24}, P_{0}] = 1 and |[e_{24}, P]| ≥ 4 by (5), [P:P_{0}] ≥ 4. We conclude that P = ⟨P_{0}, e_{12}x, e_{45}y⟩ for some x, y ∈ ⟨e_{24}⟩.

By (5) again, (e_{12}e_{24})^{2} = e_{14} ∉ Fr(P) and (e_{45}e_{24})^{2} = e_{25} ∉ Fr(P). Hence x = y = 1, and P = ⟨Z_{2}, e_{13}, e_{35}, e_{12}, e_{45}⟩. But then Out_{T} (P) ≅ T/P ≅ D_{8} has noncentral involutions, while by Proposition 3.3(a), this is impossible for a critical subgroup P ≤ T.

Case 1b: Now assume e_{24} ∈ P. Thus T′ ≤ P, and so P is normal in T. Also, [P:T′] = 16/[T:P] ≤ 4 since [T:P] ≥ 4.

Assume first e_{12}, e_{45} ∈ P. Then P = ⟨T′, e_{12}, e_{45}⟩ (since it cannot be larger), and

Fr(P) = ⟨Fr(T′), [e_{12}, T′], [e_{45}, T′]⟩ = ⟨e_{15}, [e_{12}, e_{24}], [e_{45}, e_{24}]⟩ = Z_{2}.

So [e_{23}, P/Fr(P)] = ⟨e_{13}⟩ has rank one, which contradicts (3).

Thus either e_{12} ∉ P or e_{45} ∉ P. By symmetry (with respect to τ ∈ Aut(T)), we can assume e_{12} ∉ P. Then rk([e_{12}, P/Fr(P)]) ≥ 2 by (3). Since [e_{12}, P] ≤ [e_{12}, T] = ⟨e_{13}, e_{14}, e_{15}⟩ and e_{15} ∈ Fr(P), this implies e_{13}, e_{14} ∉ Fr(P). Hence by (a), P ≤ ⟨T′, e_{12}, e_{23}, e_{45}⟩. If [P:T′] ≤ 2, then T/P ≅ C_{2}^{k} for k ≥ 3, hence rk([e_{12}, P/Fr(P)]) ≥ 3 by Proposition 3.3(d), and we just saw this is impossible.

Thus [P:T′] = 4, and e_{12} ∉ P ≤ ⟨T′, e_{12}, e_{23}, e_{45}⟩. It follows that P = ⟨T′, e_{23}x, e_{45}y⟩ for some x, y ∈ ⟨e_{12}⟩. If y ≠ 1, then e_{45} ∈ N_{T} (P)∖P, rk([e_{45}, P/Fr(P)]) ≥ 2, which is impossible since [e_{45}, P] ≤ ⟨e_{15}, e_{25}, e_{35}⟩ and e_{25} = [e_{23}x, e_{35}] ∈ Fr(P). Thus y = 1. If x ≠ 1, then (e_{12}e_{23})^{2} = e_{13} ∈ Fr(P), while we already showed that e_{13} ∉ Fr(P).

We are thus left with the case P = ⟨T′, e_{23}, e_{45}⟩. Then Fr(P) = ⟨e_{15}, [e_{23}, e_{35}]⟩ = ⟨e_{15}, e_{25}⟩. So

rk([e_{12}, P/Fr(P)]) = rk(⟨e_{13}, e_{14}, e_{15}⟩/⟨e_{15}⟩) = 2
rk([e_{34}, P/Fr(P)]) = rk(⟨e_{14}, e_{24}, e_{35}⟩) = 3.

But this contradicts Proposition 3.3(b), which says that all involutions in Out_{T} (P) are conjugate in Out(P), and hence that [e_{12}, P/Fr(P)] and [e_{34}, P/Fr(P)] have the same rank. So this subgroup is not critical.

Case 2: Now assume e_{15} ∉ Fr(P). Since e_{14} ∈ P and [e_{14}, T] = ⟨e_{15}⟩, this implies e_{14} ∈ Z(P), and similarly e_{25} ∈ Z(P). So P ≤ C_{T} (Z_{2}) = A_{1}A_{2}. Since P is centric in T, this also implies P ≥ Z(A_{1}A_{2}) = A_{1} ∩ A_{2} = ⟨Z_{2}, e_{24}⟩. Set R_{0} = ⟨Z_{2}, e_{24}⟩ for short.

If |P/R_{0}| ≤ 2, then |P | ≤ 2^{5}, so rk(P/Fr(P)) ≤ 5 and |N(P)/P | ≥ |A_{1}A_{2}/P | ≥ 2^{3}. This contradicts (4), and we conclude |P/R_{0}| ≥ 4.

Now, e_{13}e_{35} ∉ P since (e_{13}e_{35})^{2} = e_{15} ∉ Fr(P). Assume neither e_{13} nor e_{35} is in P. Since |P/R_{0}| ≥ 4, this implies P = ⟨R_{0}, e_{23}x, e_{34}y⟩ for some x, y ∈ ⟨e_{13}, e_{35}⟩. Also,

e_{12}P e_{12}^{−1} = ⟨R_{0}, e_{23}e_{13}x, e_{34}y⟩ and e_{45}P e_{45}^{−1} = ⟨R_{0}, e_{23}x, e_{34}e_{35}y⟩,

so up to conjugacy, we can assume x ∈ ⟨e_{35}⟩ and y ∈ ⟨e_{13}⟩. Since rk([e_{13}, P/Fr(P)]) ≥ 2 by (3), e_{14} = (e_{13}e_{34})^{2} ∉ Fr(P), so y = 1. By a similar argument, x = 1, and thus P = ⟨R_{0}, e_{23}, e_{34}⟩. But then [e_{13}, P] = ⟨e_{14}⟩, contradicting (3) again.

Thus either e_{13} ∈ P or e_{35} ∈ P, and they cannot both be in P since [e_{13}, e_{35}] = e_{15} ∉ Fr(P). By symmetry (with respect to τ ∈ Aut(T)), it suffices to consider the case e_{35} ∈ P and e_{13} ∉ P. Then e_{45} ∈ N(P) since [e_{45}, T] ≤ P, and thus N(P) ≥ ⟨A_{1}A_{2}, e_{45}⟩ has order ≥ 2^{9}. If |P | ≤ 2^{6}, then |N(P)/P | ≥ 2^{3}, so rk(P/Fr(P)) ≥ 6 by (4), P is elementary abelian of rank 6, and P = A_{2} by Lemma 6.1. But since N(A_{2})/A_{2} = T/A_{2} has order 16, A_{2} is not critical by (4).

Thus |P | = 2^{7}, P has index 2 in A_{1}A_{2}, and hence P = ⟨R_{0}, e_{23}x, e_{34}y, e_{35}⟩ for some x, y ∈ ⟨e_{13}⟩. Also, since rk([e_{13}, P/Fr(P)]) ≥ 2 by (3) again, [e_{13}, P] = [e_{13}, T] = ⟨e_{14}, e_{15}⟩ and ⟨e_{14}, e_{15}⟩ ∩ Fr(P) = 1. Thus (e_{13}e_{34})^{2} = e_{14} ∉ Fr(P), implying y = 1. Since e_{12}P e_{12}^{−1} = ⟨R_{0}, e_{23}e_{13}x, e_{34}, e_{35}⟩, we can now assume up to conjugacy that P = ⟨R_{0}, e_{23}, e_{34}, e_{35}⟩. In this case, Z(P) = R_{0}, [e_{13}, P] ≤ Z(P) and [e_{13}, Z(P)] = 1, and P is not critical by Lemma 3.4 applied with Θ = Z(P). □

The following lemma will be used when determining the normal critical subgroups of index two in T. We formulate it here in a more general form, so it can also be applied in the next section.

Lemma 6.4. Assume S = ⟨g_{1}, g_{2}, g_{3}, g_{4}⟩ is a group of order 2^{7}, with center Z =^{def} Z(S) = Fr(S) = ⟨z_{1}, z_{2}, z_{3}⟩ ≅ C_{2}^{3}, satisfying the relations g_{i}^{2} = 1 (i = 1, 2, 3, 4), [g_{i}, g_{i+1}] = z_{i} (i = 1, 2, 3), and [g_{i}, g_{j}] = 1 when |i − j| ≥ 2. Consider the subgroups

U_{i} = ⟨Z, g_{j} | j ≠ i⟩ (1 ≤ i ≤ 4), U_{13} = ⟨Z, g_{1}g_{3}, g_{2}, g_{4}⟩, U_{24} = ⟨Z, g_{1}, g_{3}, g_{2}g_{4}⟩.

Let P ◁ S be a subgroup of index two, not equal to U_{i} for any i = 1, 2, 3, 4. Then either Fr(P) = Z; or P = U_{13} or U_{24}, Fr(P) = ⟨z_{1}x, z_{3}y⟩ for some x, y ∈ ⟨z_{2}⟩, and Z(P) = Z.

Proof. If g_{1} ∈ P, then g_{2}a ∈ P for some a ∈ ⟨g_{3}, g_{4}⟩ (since P ≠ U_{2}), and [g_{1}, g_{2}a] = z_{1} ∈ Fr(P). If g_{2} ∈ P, then g_{1}a ∈ P for some a ∈ ⟨g_{3}, g_{4}⟩ (P ≠ U_{1}), and [g_{2}, g_{1}a] ∈ {z_{1}, z_{1}z_{2}} is in Fr(P). If g_{1}g_{2} ∈ P, then (g_{1}g_{2})^{2} = z_{1} ∈ Fr(P). Since [S:P] = 2, one of the elements g_{1}, g_{2}, g_{1}g_{2} is in P, so in all cases, z_{1}x ∈ Fr(P) for some x ∈ ⟨z_{2}⟩. By a similar argument, z_{3}y ∈ Fr(P) for some y ∈ ⟨z_{2}⟩.

Thus Fr(P) = Z whenever z_{2} ∈ Fr(P). If neither g_{2} nor g_{3} is in P, then g_{2}g_{3} ∈ P, and z_{2} = (g_{2}g_{3})^{2} ∈ Fr(P). If g_{2} ∈ P and g_{3}x ∈ P for x ∈ ⟨g_{4}⟩, then z_{2} = [g_{2}, g_{3}x] ∈ Fr(P). If g_{2} ∈ P and neither g_{3} nor g_{3}g_{4} is in P, then g_{4} ∈ P, and hence P = ⟨Z, g_{2}, g_{4}, g_{1}g_{3}⟩ = U_{13} (since P ≠ U_{3}). By a similar argument, if g_{3} ∈ P, then either z_{2} ∈ Fr(P) or P = U_{24}. Thus Fr(P) = Z with these two exceptions.

If P = U_{13} = ⟨Z, g_{2}, g_{4}, g_{1}g_{3}⟩, then clearly Z(P) ≥ Z. If g = g_{1}^{i}g_{2}^{j}g_{3}^{k}g_{4}^{ℓ}x ∈ Z(P), where i, j, k, ℓ = 0, 1 and x ∈ Z, then i = k = 0 since [g, g_{2}] = 1, and j = ℓ = 0 since [g, g_{1}g_{3}] = 1. Thus g = x ∈ Z, and so Z(P) = Z. The proof that Z(U_{24}) = Z is similar. □

We are now ready to handle the subgroups of T which contain Z_{2} and have index 2 in their normalizer. This requires some detailed case-by-case checks.

Proposition 6.5. The only possible critical subgroups of T = UT_{5}(2) are the subgroups Q_{i} (i = 1, 2, 3, 4) of index 2.

Proof. Let P be a critical subgroup of T. By Lemma 6.3, |N(P)/P | = 2. By Lemma 3.4,

g ∈ N(P)∖P, Θ char P ⟹ [g, P] ≰ Θ·Fr(P) or [g, Θ] ≰ Fr(P). (6)

Case 1: Assume P ◁ T. Thus P has index 2 in T, and P ≥ [T, T] = T′. Also, e_{15} = [e_{13}, e_{35}] ∈ Fr(P),

e_{14} = [e_{34}, e_{13}] = [e_{12}, e_{24}] = [e_{12}e_{34}, e_{24}] ∈ Fr(P)

since one of the elements e_{12}, e_{34}, or e_{12}e_{34} is in P, and similarly e_{25} ∈ Fr(P). Thus Fr(P) ≥ Z_{2}. For any g ∈ T ∖P, [g, P] ≤ T′ and [g, T′] ≤ [T, T′] = Z_{2} ≤ Fr(P) (2). Hence by (6), T′ is not characteristic in P.

We must show P = Q_{i} for some i = 1, 2, 3, 4. Assume otherwise: assume P is not one of the Q_{i}. Consider the group S of Lemma 6.4. There is an epimorphism φ: T ⟶ S, defined by φ(e_{i,i+1}) = g_{i} and φ(e_{i,i+2}) = z_{i}, with Ker(φ) = Z_{2}. Since P ≠ Q_{i} for each i, φ(P) satisfies the hypotheses of the lemma. So either Fr(P) = φ^{−1}(Z(S)) = T′ and hence T′ is characteristic in P; or P = ⟨e_{12}, e_{34}, e_{23}e_{45}⟩ or ⟨e_{12}e_{34}, e_{23}, e_{45}⟩.

By Lemma 6.4 again, in both of these last two cases, Fr(P) = ⟨Z_{2}, e_{13}x, e_{35}y⟩ for some x, y ∈ ⟨e_{24}⟩, and Z(P/Z_{2}) = φ^{−1}(Z(S))/Z_{2} = T′/Z_{2}. Thus Z(Fr(P)) = Z_{2}, and hence Z_{2} and T′ are both characteristic in P. But we have already seen that this implies P cannot be critical.

Case 2: Now assume P ◁̸ T. Thus P ≱ T′ = [T, T], while P ≥ Z_{2} by Lemma 6.2. Since [T, T′] = Z_{2} ≤ P by (2), T′ ≤ N_{T} (P). So we can always choose g ∈ N_{T′}(P)∖P, in which case [g, P] ≤ [T′, T] = Z_{2}. By (6), applied with Θ = 1 or Θ = Z_{2}(P) ≥ Z_{2},

Fr(P) ≱ Z_{2},  and [g, Z_{2}(P)] ≰ Fr(P) for g ∈ N_{T′}(P)∖P  (7)

We next claim that

{e_{13}, e_{35}} ∩ P ≠ ∅.  (8)

Assume otherwise: assume e_{13}, e_{35} ∉ P. Then both of these are in N(P), and since |N(P)/P | = 2, e_{13}e_{35} ∈ P. By Lemma 3.5, there is α ∈ Aut(P) of odd order, and x ∈ [e_{13}, P], such that x ∉ Fr(P) and [e_{13}, α(x)] ∉ Fr(P). Set y = α(x). Since x ∈ {e_{14}, e_{14}e_{15}} has order two, y^{2} = 1. Also, [e_{13}, y] ∈ {e_{14}, e_{14}e_{15}}, and [e_{13}e_{35}, y] ∉ {e_{14}, e_{14}e_{15}} since e_{14} ∉ Fr(P).

Set Q = U_{1}U_{2} = ⟨e_{12}, e_{13}, e_{14}, e_{15}, e_{25}, e_{35}, e_{45}⟩. By the commutator relations (1), [e_{13}, y] ∈ {e_{14}, e_{14}e_{15}} implies y ≡ e_{34} (mod ⟨Q, e_{23}, e_{24}⟩). Combined with the condition [e_{13}e_{35}, y] ∉ {e_{14}, e_{14}e_{15}}, we have y ≡ e_{23}e_{34} (mod ⟨Q, e_{24}⟩. But then the class yQ has order four in T/Q ≅ D_{8}, which contradicts the assumption y^{2} = 1. This finishes the proof of (8).

Set T_{0} = ⟨T′, e_{12}, e_{45}⟩. We want to apply Lemma 1.9 to identify subgroups of S = T/Z_{2} of index two in their normalizer. To do this, we regard T/Z_{2} as an extension

1 ⟶ T_{0}/Z_{2}_{=⟨e_{13},e_{24},e_{35},e_{12},e_{45}⟩} ⟶ T/Z_{2} ⟶ T/T_{0}_{=⟨e_{23},e_{34}⟩} ⟶ 1,

where S_{0} = T_{0}/Z_{2} ≅ C_{2}^{5} and S/S_{0} ≅ C_{2}^{2}. Using the notation of Lemma 1.9 (but with P a subgroup of T and not of S = T/Z_{2}), we set P_{0} = P ∩ T_{0}.

Recall, in the notation of Lemma 1.9, that m is the number of classes xT_{0} ∈ T/T_{0} such that xT_{0} ≠ T_{0} and [x, T_{0}] ≤ P_{0}. Since [e_{23}, T_{0}/Z_{2}] = ⟨e_{13}Z_{2}⟩, [e_{34}, T_{0}/Z_{2}] = ⟨e_{35}Z_{2}⟩, and [e_{23}e_{34}, T_{0}/Z_{2}] = ⟨e_{13}Z_{2}, e_{35}Z_{2}⟩, we see that

m = 2^{k} − 1 where k = |{e_{13}, e_{35}} ∩ P |.  (9)

Thus (8) implies m ≥ 1.

By Lemma 1.9, we must consider the following cases, where we omit those where m = 0. In all cases, since N_{T} (P) ≥ T′ and P ≱ T′, [T′:P ∩ T′] = 2. Recall that we always choose g ∈ N_{T′}(P)∖P.

(b) rk(T_{0}/P_{0}) = 1, |P/P_{0}| = 2, m = 1, and P_{0} ◁̸ S. Then {e_{13}, e_{35}} ⊈ P by (9), and we can choose g ∈ {e_{13}, e_{35}} in N_{T} (P)∖P. Since P_{0} has index two in T_{0} and does not contain g, there are elements x, y, z ∈ ⟨g⟩ such that e_{24}x, e_{12}y, e_{45}z ∈ P. Thus e_{14} = [e_{12}y, e_{24}x] and e_{25} = [e_{24}x, e_{45}z] are both in Fr(P), so Z_{2} ≤ Fr(P), which contradicts (7).

(c) rk(T_{0}/P_{0}) = 1, |P/P_{0}| = 2, m = 3, and P_{0} ◁ S. Then P_{0} ≥ ⟨Z_{2}, e_{13}, e_{35}⟩ by (9). Hence e_{24} ∉ P (P ≱ T′), and we take g = e_{24}. For x ∈ T_{0}, (e_{23}e_{34}x)^{2} ≡ (e_{23}e_{34})^{2} = e_{24} (mod [T, T_{0}] ≤ P_{0}), so (e_{23}e_{34}x)^{2} ∉ P_{0}, and e_{23}e_{34}x ∉ P. So up to symmetry, we can assume P T_{0} = ⟨T_{0}, e_{23}⟩. Thus P = ⟨Z_{2}, e_{13}, e_{35}, e_{12}x, e_{45}y, e_{23}z⟩ for some x, y, z ∈ ⟨g⟩. In all cases, Z(P) = ⟨e_{15}⟩, Z_{2}(P) ≤ ⟨T′, e_{45}⟩, and [e_{24}, Z_{2}(P)] ≤ ⟨e_{15}, e_{25}⟩ ≤ Fr(P). So this case is impossible by (7).

(e) rk(T_{0}/P_{0}) = 2, |P/P_{0}| = 4, and m = 1 . By (9), exactly one of the elements e_{13} or e_{35} is in P_{0}. Up to symmetry, we can assume e_{13} ∈ P_{0} while e_{35} ∉ P_{0}. Set g = e_{35}. Since [e_{34}, T_{0}/Z_{2}] = ⟨e_{35}Z_{2}⟩ is not in P/Z_{2}, and since P_{0}/Z_{2} is invariant under the conjugation action of e_{34} on T_{0}/Z_{2} (since |P/P_{0}| = 4), P_{0}/Z_{2} ≤ C_{T_{0}/Z_{2}}(e_{34}) = ⟨e_{12}Z_{2}, T′/Z_{2}⟩. Also, |P_{0}/Z_{2}| = 8 since |T_{0}/P_{0}| = 4 and |T_{0}/Z_{2}| = 32, and so P_{0} = ⟨Z_{2}, e_{13}, e_{24}x, e_{12}y⟩ for some x, y ∈ ⟨g⟩.

Now, e_{15} = [e_{12}y, e_{25}] ∈ Fr(P). By Lemma 3.5, there is α ∈ Aut(P) and r ∈ [e_{35}, P] such that r ∉ Fr(P) and [e_{35}, α(r)] ∉ Fr(P). Set s = α(r). Since [e_{35}, T] = ⟨e_{25}, e_{15}⟩, these conditions imply r, [e_{35}, s] ∈ {e_{25}, e_{25}e_{15}}. Also, s^{2} = 1 since r^{2} = 1.

Set H = E_{1234;45} = ⟨A_{2}, e_{45}⟩ ≤ C_{T} (e_{35}). The condition on [e_{35}, s] (together with (1)) implies s = e_{23}v for v ∈ ⟨H, e_{12}, e_{13}⟩; and v ∈ ⟨H, e_{13}⟩ since s^{2} = 1.

Set K = ⟨Z_{2}, e_{35}⟩. Since |P/P_{0}| = 4, and since ⟨P_{0}, e_{35}, e_{45}⟩ = T_{0}, there is w ∈ ⟨e_{35}, e_{45}⟩ such that e_{34}w ∈ P. Then [s, P] contains the elements

[s, e_{12}y] ∈ [e_{23}·⟨e_{13}, H⟩, e_{12}H] ∈ e_{13}H and [s, e_{34}w] = [e_{23}v, e_{34}w] ∈ e_{24}K,

where the last inclusion holds since [v, e_{34}w] ∈ [⟨e_{13}, H⟩, H] ≤ K and [T, w] ≤ K. Thus |[s, P]| ≥ 4, which is impossible since [s, P] = α([r, P]) = ⟨α(e_{15})⟩.

This finishes the proof that P is not critical when it is not normal. □

6.2 Automorphisms of critical subgroups

Recall that τ ∈ Aut(T) denotes the transpose along the “back” diagonal composed with A ↦ A^{−1}; i.e., the automorphism τ(e_{ij}) = e_{6−j,6−i}. We claim there are also automorphisms θ_{1}, ψ_{1} ∈ Aut(T) such that

θ_{1}(e_{23}) = e_{23}e_{15} , ψ_{1}(e_{12}) = e_{12}e_{35} , ψ_{1}(e_{13}) = e_{13}e_{15}e_{25} ;

and which send all other generators e_{ij} to themselves. This is clear for θ_{1}, since it has the form θ_{1}(g) = g·φ(g) for some φ ∈ Hom(T, Z(T)). By a similar argument, ψ_{1}|_{Q_{1}} is an automorphism of Q_{1}, and it extends to an automorphism of T if ψ_{1}([e_{12}, g]) = [e_{12}e_{35}, ψ_{1}(g)] for all g ∈ Q_{1}. This is clear when g = e_{ij} for j ≥ 4 (g = ψ_{1}(g) commutes with e_{12} and e_{35}), and holds for the other two generators by direct calculation:

ψ_{1}([e_{12}, e_{13}]) = 1 = [e_{12}e_{35}, e_{13}e_{15}e_{25}] and ψ_{1}([e_{12}, e_{23}]) = e_{13}e_{15}e_{25} = [e_{12}e_{35}, e_{23}].

We also define θ_{2}, ψ_{2} ∈ Aut(T) by setting θ_{2} = τθ_{1}τ^{−1} and ψ_{2} = τψ_{1}τ^{−1}. It is helpful to visualize these automorphisms pictorially as follows:

θ_{1} =  [figure] ,  ψ_{1} = [figure] ,  θ_{2} =  [figure] ,  ψ_{2} =  [figure] .

For each φ ∈ {θ_{i}, ψ_{i}} and each i < j, the arrows in the diagram for φ starting in position (i, j) point to the positions of the basis elements which occur in e_{ij}^{−1}φ(e_{ij}).

Let Aut^{0}(T) ≤ Aut(T) be the subgroup of automorphisms which send A_{1} to itself, and set Out^{0}(T) = Aut^{0}(T)/Inn(T). By Lemma 6.1, each automorphism of T either sends the A_{i} to themselves or exchanges them, so Aut(T) = Aut^{0}(T) ⋊ ⟨τ⟩.

For each i = 1, 2, 3, 4, N_{GL_{5}(2)}(Q_{i}) is the group of all A = (a_{jk}) ∈ GL_{5}(2) such that a_{jk} = 0 for all j > k such that (j, k) ≠ (i + 1, i). Thus N_{GL_{5}(2)}(Q_{i})/Q_{i} ≅ GL_{2}(2) ≅ Σ_{3}, and is generated by the classes (mod Q_{i}) of e_{i,i+1} and the permutation matrix for the transposition (i i+1). We now define

Δ_{i} = Out_{GL_{5}(2)}(Q_{i}) = ⟨c_{i,i+1}, [σ_{i,i+1}]⟩ ≅ Σ_{3} ,  (10)

where σ_{i,i+1} ∈ Aut(Q_{i}) is conjugation by that permutation matrix; i.e., the automorphism which exchanges the i-th and (i+1)-st rows and columns.

Proposition 6.6. (a) Out^{0}(T) = ⟨[θ_{1}], [θ_{2}], [ψ_{1}], [ψ_{2}]⟩ ≅ C_{2}^{4}. Hence |Out(T)| = 2^{5}.

(b) For each i = 1, 2, 3, 4, Out(Q_{i}) = O_{2}(Out(Q_{i}))·Δ_{i}, and O_{2}(Out(Q_{i})) is elementary abelian. If F is a saturated fusion system over T and Q_{i} is F-essential, then Out_{F}(Q_{i}) = [φ]Δ_{i}[φ]^{−1} for some φ ∈ O_{2}(Aut(Q_{i})) which extends to an automorphism of T.

Proof. Set

R = E_{123;345} = A_{1}A_{2} and R_{0} = E_{12;45} = A_{1} ∩ A_{2} = Z(R) = Fr(R).

Let Out^{0}(R) ≤ Out(R) be the subgroup of classes of automorphisms which send A_{1} to itself. In Steps 1 and 2, we describe Out^{0}(T), Out(Q_{1}), and Out(Q_{4}) by comparison with Out^{0}(R). Then in Step 3, we prove (b) for Q_{2} and Q_{3}.

In Steps 2 and 3, it will be helpful to represent automorphisms of A_{1} by matrices. So for each α ∈ A_{1}, M(α) will denote the matrix for α|_{A_{1}} with respect to the ordered basis

b = {e_{15}, e_{25}, e_{14}, e_{24}, e_{13}, e_{23}}.

Step 1: Let κ̃ be the homomorphism from Aut^{0}(R) to Aut(R/A_{1}) × Aut(R/A_{2}) induced by the projections of R onto R/A_{2} and R/A_{1}. Then Inn(R) ≤ Ker(κ̃), so κ̃ induces a homomorphism κ on Out^{0}(R). We claim the sequence

1 ⟶ O_{2}(Out(R)) ⟶^{incl} Out^{0}(R) ⟶^{κ} Aut(R/A_{1}) × Aut(R/A_{2}) ⟶ 1 (11)

is exact. Here, Aut(R/A_{i}) ≅ Σ_{3} since R/A_{i} ≅ C_{2}^{2}, and κ is onto since its restriction to the subgroup Out_{GL_{5}(2)}(R) ≅ Σ_{3}×Σ_{3} is onto. So O_{2}(Out(R)) ≤ Ker(κ). Conversely, for each α ∈ Aut(R) such that [α] ∈ Ker(κ), α induces the identity on R/(A_{1}∩A_{2}) = R/R_{0} where R_{0} = Fr(R), and hence α ∈ O_{2}(Aut(R)) by Lemma 1.1. Thus (11) is exact.

If α ∈ Aut(R) induces the identity on R/R_{0}, then it also restricts to the identity on R_{0} — since [e_{i3}, e_{3j}] = e_{ij} for i = 1, 2 and j = 4, 5 by (1). Hence each such α has the form α(g) = g·α̂(gR_{0}) for some map α̂ from R/R_{0} to R_{0}, and α̂ is a homomorphism since R_{0} = Z(R). Thus O_{2}(Aut(R)) ≅ Hom(R/R_{0}, R_{0}) ≅ C_{2}^{16}. Since Inn(R) ≅ R/R_{0} has rank 4, O_{2}(Out(R)) ≅ C_{2}^{12}.

Now, R_{0} = Z(R) is free as a module over 𝔽_{2}[T/R] = 𝔽_{2}[⟨c_{12}, c_{45}⟩]. Also, R is generated by the only subgroups of T isomorphic to C_{2}^{6} (Lemma 6.1), and hence is characteristic in any subgroup of T which contains it. So if P is any of the groups T, Q_{1}, or Q_{4}, then restriction to R induces an isomorphism

Out(P) ⟶_{≅}^{Res_{R}} N_{Out(R)}(Out_{P} (R))/Out_{P} (R).  (12)

by Corollary 1.3. When P = Q_{i} for i = 1 or 4, then κ sends Out_{P} (R) ≅ C_{2} nontrivially to one of the factors Aut(R/A_{1}) or Aut(R/A_{2}) in the extension (11), and sends Δ_{i} isomorphically to the other factor. Thus

κ(N_{Out(R)}(Out_{P} (R))) = κ(Out_{P} (R)·Δ_{i}) ≅ C_{2} × Σ_{3},

and hence

Out(Q_{i}) = O_{2}(Out(Q_{i}))·Δ_{i} where O_{2}(Out(Q_{i})) ≅ C_{O_{2}(Out(R))}(Out_{P} (R)).

In particular, O_{2}(Out(Q_{i})) is elementary abelian.

Assume Q_{i} is F-essential, and set Δ′_{i} = Out_{F}(Q_{i}). Then Δ′_{i} ∩ O_{2}(Out(Q_{i})) = 1 since O_{2}(Δ′_{i}) = 1, and O_{2}(Out(Q_{i}))·Δ′_{i} = Out(Q_{i}) since otherwise Δ′_{i} would have order two. Hence by Proposition 1.8, Δ′_{i} = φΔ_{i}φ^{−1} for some φ ∈ Out(Q_{i}) which centralizes Out_{T} (Q_{i}). Since Z(Q_{i}) ≅ C_{2}^{2} is a free 𝔽_{2}[T/Q_{i}]-module, H^{2}(T/Q_{i}; Z(Q_{i})) = 0, and Lemma 1.2 implies that φ extends to an automorphism of T.

Step 2: When P = T, (12) restricts to an isomorphism

Out^{0}(T) ⟶_{≅}^{Res_{R}} N_{Out^{0}(R)}(⟨c_{12}, c_{45}⟩)/⟨c_{12}, c_{45}⟩ ≅ C_{O_{2}(Out(R))}(⟨c_{12}, c_{45}⟩),

where the last isomorphism follows using (11). We now prove point (a) by describing this centralizer explicitly. Write O_{2}(Aut(R)) = A_{1} × A_{2}, where A_{1} ≅ Hom(R/A_{2}, R_{0}) is the subgroup of automorphisms which are the identity on A_{2} and on R/A_{2}, and A_{2} ≅ Hom(R/A_{1}, R_{0}) is the subgroup of automorphisms which are the identity on A_{1} and on R/A_{1}. Set

Â_{1} = A_{1}/⟨c_{34}, c_{35}⟩ and Â_{2} = A_{2}/⟨c_{13}, c_{23}⟩;

thus O_{2}(Out(R)) = Â_{1} × Â_{2}. The actions of c_{12} and c_{45} clearly preserve this decomposition, and hence Res_{R} induces an isomorphism

Out^{0}(T) ≅ C_{O_{2}(Out(R))}(⟨c_{12}, c_{45}⟩) = CÂ_{1}(⟨c_{12}, c_{45}⟩) × CÂ_{2}(⟨c_{12}, c_{45}⟩).  (13)

Recall that M(−) is the matrix for an automorphism of A_{1} with respect to the basis b defined above. Thus

\{M(α|_{A_{1}}) | α ∈ A_{1}\} = \{( \begin{matrix}I & 0 & B \\ 0 & I & C \\ 0 & 0 & I \end{matrix} ) | B, C ∈ M_{2}(𝔽_{2})\}.

Write λ(B, C) = ( \begin{matrix}I & 0 & B \\ 0 & I & C \\ 0 & 0 & I \end{matrix} ) for short; then M(c_{34}) = λ(0, I) and M(c_{35}) = λ(I, 0).

Now, c_{45} and c_{12} act on these matrices via conjugation by ( \begin{matrix}I & I & 0 \\ 0 & I & 0 \\ 0 & 0 & I \end{matrix} ) and by ( \begin{matrix}J & 0 & 0 \\ 0 & J & 0 \\ 0 & 0 & J \end{matrix} ), respectively, where J = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ). Hence

[c_{45}, λ(B, C)] = λ(C, 0) and [c_{12}, λ(B, C)] = λ(JBJ^{−1}+B, JCJ^{−1}+C).

From this it follows that M induces an isomorphism

CÂ_{1}(⟨c_{12}, c_{45}⟩) ⟶^{≅} {λ(B, 0) | JBJ^{−1}+B ∈ {0, I}}/⟨λ(I, 0)⟩.

Also, J ( \begin{matrix}a & b \\ c & d \end{matrix}) J^{−1}+ ( \begin{matrix}a & b \\ c & d \end{matrix}) = ( \begin{matrix}c & a+c+d \\ 0 & c \end{matrix}) ∈ \{0, I\} if and only if a + c + d = 0. Since M(θ_{1}|_{R}) = λ(( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) , 0) and M(ψ_{1}|_{R}) = λ(( \begin{matrix}1 & 0 \\ 1 & 0 \end{matrix} ) , 0), this proves that

CÂ_{1}(⟨c_{12}, c_{45}⟩) = ⟨θ_{1}|_{R}, ψ_{1}|_{R}⟩ ≅ C_{2}^{2}.

After combining this with the corresponding argument for Â_{2}, and with (13), we have now proven that Out^{0}(T) ≅ C_{2}^{4} with basis {[θ_{1}], [ψ_{1}], [θ_{2}], [ψ_{2}]}.

Step 3: It remains to prove (b) for Q_{2} and Q_{3}. We do this for Q_{3}, and the result for Q_{2} then follows via conjugation by τ. Recall that σ_{34} ∈ Aut(Q_{3}) is the automorphism which switches the third and fourth rows and columns. We define automorphisms β_{1}, …, β_{5} ∈ Aut(Q_{3}) as follows:

β_{1} = θ_{1}|_{Q_{3}}, β_{2} = σ_{34}β_{1}σ_{34}, β_{3} = ψ_{1}|_{Q_{3}}, β_{4} = σ_{34}β_{3}σ_{34}, and β_{5} = ψ_{2}|_{Q_{3}}.

These can be described pictorially as follows:

β_{1} = [figure]  β_{2} =  [figure]  β_{3} = [figure]  β_{4} = [figure]  β_{5} = [figure] .

Thus, for example, β_{4}(e_{12}) = e_{12}e_{45}, β_{4}(e_{14}) = e_{14}e_{15}e_{25}, and β_{4} sends all of the other generators e_{ij} to themselves. We will show that O_{2}(Out(Q_{3})) ≅ C_{2}^{5} with the classes of these elements as basis.

By Lemma 1.2, there is a short exact sequence

1 → H^{1}(Q_{3}/A_{1}; A_{1}) ⟶ Out(Q_{3}) ⟶^{Res_{A_{1}}} N_{Aut(A_{1})}(Aut_{Q_{3}}(A_{1}))/Aut_{Q_{3}}(A_{1}) → 1.

(14)

In terms of matrices, we are looking for the centralizer in GL_{6}(2) of

M(c_{12}) = ( \begin{matrix}J & 0 & 0 \\ 0 & J & 0 \\ 0 & 0 & J \end{matrix} ) , M(c_{35}) = ( \begin{matrix}I & 0 & I \\ 0 & I & 0 \\ 0 & 0 & I \end{matrix} ) , and M(c_{45}) = ( \begin{matrix}I & I & 0 \\ 0 & I & 0 \\ 0 & 0 & I \end{matrix} ) ,

where J = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) as before. The centralizer in Aut(A_{1}) of ⟨c_{35}, c_{45}⟩ is the group of those α such that M(α) = ( \begin{matrix}A & B & C \\ 0 & A & 0 \\ 0 & 0 & A \end{matrix} ) for some A, B, C ∈ M_{2}(𝔽_{2}) with A invertible; and such a matrix commutes with M(c_{12}) exactly when A, B, and C all commute with J. Since a matrix in M_{2}(𝔽_{2}) commutes with J if and only if it has the form ( \begin{matrix}a & b \\ 0 & a \end{matrix}) for some a, b ∈ 𝔽_{2}, this proves that

M(C_{Aut(A_{1})}(Aut_{Q_{3}}(A_{1}))) = ⟨M(c_{12}), M(c_{35}), M(c_{45}), ( \begin{matrix}I & Y & 0 \\ 0 & I & 0 \\ 0 & 0 & I \end{matrix} ) , ( \begin{matrix}I & 0 & Y \\ 0 & I & 0 \\ 0 & 0 & I \end{matrix} )⟩

where Y = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ). Hence

C_{Aut(A_{1})}(Aut_{Q_{3}}(A_{1})) = ⟨c_{12}, c_{35}, c_{45}, β_{1}|_{A_{1}}, β_{2}|_{A_{1}}⟩ ≅ C_{2}^{5}.  (15)

So C_{Aut(A_{1})}(Aut_{Q_{3}}(A_{1}))/Aut_{Q_{3}}(A_{1}) is a group of order 4 generated by the classes of β_{1}|_{A_{1}} and β_{2}|_{A_{1}}.

By (1), for g ∈ Q_{3}∖A_{1}, [g, A_{1}] = ⟨e_{15}, e_{25}⟩ ≅ C_{2}^{2} if g ∈ ⟨A_{1}, e_{35}, e_{45}⟩, while [g, A_{1}] ≅ C_{2}^{3} otherwise. Hence each β ∈ Aut(Q_{3}) leaves the subgroup ⟨A_{1}, e_{35}, e_{45}⟩ invariant.

The group of automorphisms of Q_{3}/A_{1} ≅ C_{2}^{3} which leave ⟨e_{35}A_{1}, e_{45}A_{1}⟩ invariant is isomorphic to Σ_{4}, and is generated by the actions of β_{4}, β_{3}, and Δ_{3} on Q_{3}/A_{1}. This, together with (15) shows that

N_{Aut(A_{1})}(Aut_{Q_{3}}(A_{1}))/Aut_{Q_{3}}(A_{1}) = Res_{A_{1}}(⟨[β_{1}], [β_{2}], [β_{3}], [β_{4}], Δ_{3}⟩) ≅ C_{2}^{4}⋊Σ_{3}, (16)

where the β_{i}|_{A_{1}} generate an elementary abelian subgroup since their matrices all have

the form ( \begin{matrix}I & B & C \\ 0 & I & 0 \\ 0 & 0 & I \end{matrix} ) for some B, C ∈ M_{2}(𝔽_{2}).

We next claim that H^{1}(Q_{3}/A_{1}; A_{1}) ≅ C_{2}. To see this, set V = A_{1} × ⟨ê_{1}, ê_{2}⟩ ≅ C_{2}^{8}, regarded as an 𝔽_{2}[Q_{3}/A_{1}]-module (with A_{1} as submodule) by setting e_{35}(ê_{i}) = ê_{i}·e_{i4}, e_{45}(ê_{i}) = ê_{i}·e_{i3}, e_{12}(ê_{1}) = ê_{1}, and e_{12}(ê_{2}) = ê_{1}ê_{2}. This module is free (the Q_{3}/A_{1}- orbit of ê_{2} is a basis), and hence is cohomologically trivial. So the exact sequence in cohomology for the extension 1 → A_{1} → V → V/A_{1} → 1 takes the form

H^{0}(Q_{3}/A_{1}; V)_{=⟨e_{15}⟩≅C_{2}} ⟶ H^{0}(Q_{3}/A_{1}; V/A_{1})_{=⟨ê_{1}A_{1}⟩≅C_{2}} ⟶^{δ} H^{1}(Q_{3}/A_{1}; A_{1}) ⟶ H^{1}(Q_{3}/A_{1}; V)_{=0}.

Thus H^{1}(Q_{3}/A_{1}; A_{1}) ≅ C_{2} is generated by δ(ê_{1}A_{1}), which is represented by the cocycle which sends g ∈ Q_{3}/A_{1} to g(ê_{1})ê_{1}^{−1}. This induces an automorphism β ∈ Aut(Q_{3}) such that β|_{⟨A_{1},e_{12}⟩} = Id, β(e_{35}) = e_{35}e_{14}, and β(e_{45}) = e_{45}e_{13}. By inspection, β = β_{5}c_{13}, and this finishes the proof that Ker(Res_{A_{1}}) = ⟨[β_{5}]⟩.

Upon combining this with (14) and (16), we have now shown that Out(Q_{3}) is generated by the [β_{i}] and Δ_{3}. Also, ⟨[β_{5}]⟩ = Ker(Res_{A_{1}}) is normal in Out(Q_{3}), and hence central. The subgroup of elements in Out(Q_{3}) which leave invariant the subgroup U_{2} contains [β_{1}], [β_{2}], [β_{4}], [β_{3}], and Δ_{3}, but not [β_{5}]. Thus ⟨[β_{5}]⟩ splits off as a direct factor in Out(Q_{3}). So by (16), O_{2}(Out(Q_{3})) ≅ C_{2}^{5} with the [β_{i}] as basis.

Assume Q_{3} is F-essential, and set Δ′_{3} = Out_{F}(Q_{3}). Then |Δ′_{3}| = 2n for some odd n > 1 by the Sylow axiom, and n = 3 since it divides |Out(Q_{3})|. Also, Δ′_{3} ∩ O_{2}(Out(Q_{3})) = 1, and hence O_{2}(Out(Q_{3}))·Δ′_{3} = Out(Q_{3}) = O_{2}(Out(Q_{3}))·Δ_{3}. By Proposition 1.8, Δ′_{3} = [β]Δ_{3}[β]^{−1} for some β ∈ Aut_{F}(Q_{3}) which commutes with c_{34} in Out_{F}(Q_{3}). Since c_{34}β_{2}c_{34}^{−1} ≡ β_{1}β_{2} and c_{34}β_{4}c_{34}^{−1} ≡ β_{4}β_{3} (mod Inn(Q_{3})), we have

[β] ∈ C_{Out(Q_{3})}(⟨c_{34}⟩) = ⟨[β_{1}], [β_{3}], [β_{5}], c_{34}⟩.

All of these extend to automorphisms of T by the definitions at the beginning of Step 3, and this finishes the proof of (b) for Q_{3}. □

The following computations will also be needed later.

Lemma 6.7. The following commutativity relations hold:

[ψ_{1}|_{Q_{2}}, Δ_{2}] = 1 in Out(Q_{2})  [(θ_{1}ψ_{1})|_{Q_{1}}, Δ_{1}] = 1 in Out(Q_{1})
[ψ_{2}|_{Q_{3}}, Δ_{3}] = 1 in Out(Q_{3})  [(θ_{2}ψ_{2})|_{Q_{4}}, Δ_{4}] = 1 in Out(Q_{4}) .

Proof. When φ ∈ {θ_{1}, θ_{2}, ψ_{1}, ψ_{2}}, φc_{g}φ^{−1} = c_{φ(g)} and φ(g)g^{−1} ∈ Q_{i} for each g ∈ T and each i = 1, 2, 3, 4, and thus [φ|_{Q_{i}}, c_{i,i+1}] = 1 in Out(Q_{i}). So we need only check the commutators with σ_{i,i+1} (see (10)). This can be done by direct computation, but can also be seen using the pictorial description of these automorphisms. For example,

ψ_{1}|_{Q_{2}} = [figure] = [figure] ∘ c_{35} , σ_{23}(ψ_{1}|_{Q_{2}})σ_{23}^{−1} = [figure] ∘ c_{25} ,

and so [ψ_{1}, σ_{23}] = c_{35}c_{25} ∈ Inn(Q_{2}). Similarly, in Aut(Q_{1}),

θ_{1}ψ_{1}|_{Q_{1}} =  [figure] ,  σ_{12}(θ_{1}ψ_{1}|_{Q_{1}})σ_{12}^{−1} =  [figure] ,  c_{35} =  [figure] ,

and so [θ_{1}ψ_{1}|_{Q_{1}}, σ_{12}] = c_{35}. The remaining cases follow via conjugation with the automorphism τ. □

6.3 Fusion systems over T = UT_{5}(2)

We are now ready to describe the nonconstrained saturated fusion systems over T. We begin by looking at automorphisms of Q_{2} and Q_{3} in such a fusion system.

Proposition 6.8. Let F be a nonconstrained saturated fusion system over T. Then Q_{2} and Q_{3} are both F-essential. Also, F is isomorphic to a fusion system F^{∗} over T such that Out_{F^{∗}}(Q_{2}) = Δ_{2} and Out_{F^{∗}}(Q_{3}) = Δ_{3}.

Proof. By Proposition 6.6(a), Out(T) is a 2-group, and hence Out_{F}(T) = 1. So if Q_{3} is not F-essential, then by Proposition 6.5, F is generated by restrictions of automorphisms of Q_{1}, Q_{2}, Q_{4}, all of which send A_{2} to itself. Hence each morphism in F extends to a morphism between subgroups containing A_{2} which sends A_{2} to itself, and so A_{2} is normal in F. But A_{2} is centric in T, and so this contradicts the assumption that F is nonconstrained. Thus Q_{3} is F-essential; and by a similar argument, Q_{2} is also F-essential.

By Proposition 6.6(b), Out_{F}(Q_{3}) = (φ|_{Q_{3}})Δ_{3}(φ|_{Q_{3}})^{−1} for some φ ∈ Aut(T), and φ ∈ Aut^{0}(T) since it leaves Q_{3} invariant. So upon replacing F by φ^{−1}Fφ, we can assume Out_{F}(Q_{3}) = Δ_{3}. Then, by Proposition 6.6(b) again, Out_{F}(Q_{2}) = (ψ|_{Q_{2}})Δ_{2}(ψ|_{Q_{2}})^{−1} for some ψ ∈ Aut^{0}(T). Since θ_{1}|_{Q_{2}} = Id and ψ_{1}|_{Q_{2}} centralizes Δ_{2} (Lemma 6.7), we can assume ψ ∈ ⟨θ_{2}, ψ_{2}⟩. In particular, (ψ|_{Q_{3}})Δ_{3}(ψ|_{Q_{3}})^{−1} = Δ_{3} by Lemma 6.7 again (and since θ_{2}|_{Q_{3}} = Id). So if we set F^{∗} = ψ^{−1}Fψ, then Out_{F^{∗}}(Q_{2}) = Δ_{2} and Out_{F^{∗}}(Q_{3}) = Δ_{3}. □

We now study how the automorphisms of Q_{1} and Q_{4} fit with those of Q_{2} and Q_{3}. In the following proposition, 3Σ_{6} denotes a nonsplit extension with kernel of order 3 and quotient group Σ_{6}.

Proposition 6.9. Fix a nonconstrained saturated fusion system F over T, and assume Out_{F}(Q_{2}) = Δ_{2} and Out_{F}(Q_{3}) = Δ_{3}. Then Q_{1} and Q_{4} are both F-essential. Also, for each pair (i, j) = (1, 2) or (4, 1), either

• Out_{F}(Q_{i}) = Δ_{i} and Aut_{F}(A_{j}) ≅ Σ_{3} × GL_{3}(2); or

• Out_{F}(Q_{i}) = (θ_{j}ψ_{j})Δ_{i}(θ_{j}ψ_{j})^{−1} and Aut_{F}(A_{j}) ≅ 3Σ_{6}.

Proof. By Proposition 6.6(a), Out(T) is a 2-group, and hence Out_{F}(T) = 1. So by Proposition 6.5, F is generated by Inn(T) together with Aut_{F}(Q_{i}) for i = 1, 2, 3, 4.

If neither Q_{1} nor Q_{4} is F-essential, then F is generated by Inn(T) together with Δ_{2}, and Δ_{3}, all of which leave U_{1} and U_{2} invariant. Thus U_{1} and U_{2} would both be normal in F, which contradicts our assumption that F is nonconstrained. So Q_{1} or Q_{4} is F-essential.

For each i = 1, 4, if Q_{i} is F-essential, then by Proposition 6.6(b), Aut_{F}(Q_{i}) = φ_{i}Δ_{i}φ_{i}^{−1} for some φ_{i} ∈ Aut^{0}(T). (We drop “restricted to Q_{i}” to simplify the notation.) Since θ_{1}ψ_{1} commutes with Δ_{1} in Out(Q_{1}) by Lemma 6.7, we can assume φ_{1} ∈ ⟨θ_{1}, θ_{2}, ψ_{2}⟩. Similarly, we can assume φ_{4} ∈ ⟨θ_{1}, θ_{2}, ψ_{1}⟩. Set

σ_{12}^{∗} = φ_{1}σ_{12}φ_{1}^{−1} and σ_{45}^{∗} = φ_{4}σ_{45}φ_{4}^{−1} ,

so that Out_{F}(Q_{1}) = ⟨c_{12}, σ_{12}^{∗}⟩ and Out_{F}(Q_{4}) = ⟨c_{45}, σ_{45}^{∗}⟩.

In Steps 1 and 2, when α ∈ Aut(A_{1}), we again let M(α) ∈ GL_{6}(2) be its matrix with respect to the ordered basis {e_{15}, e_{25}, e_{14}, e_{24}, e_{13}, e_{23}}.

Step 1 We first prove that if Q_{1} is F-essential, then φ_{1} ∈ ⟨θ_{2}, ψ_{2}⟩; while if Q_{4} is F-essential, then φ_{4} ∈ ⟨θ_{1}, ψ_{1}⟩.

Assume Q_{1} is F-essential. Fix X ∈ M_{2}(𝔽_{2}) such that M(φ_{1}|_{A_{1}}) = ( \begin{matrix}I & 0 & X \\ 0 & I & 0 \\ 0 & 0 & I \end{matrix} ). Thus X = 0 if φ_{1} ∈ ⟨θ_{2}, ψ_{2}⟩, and X = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) otherwise. Set W = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ). Then

M(σ_{34}|_{A_{1}}) = ( \begin{matrix}I & 0 & 0 \\ 0 & 0 & I \\ 0 & I & 0 \end{matrix} ) and M(σ_{12}^{∗}|_{A_{1}}) = ( \begin{matrix}I & 0 & X \\ 0 & I & 0 \\ 0 & 0 & I \end{matrix} ) ( \begin{matrix}W & 0 & 0 \\ 0 & W & 0 \\ 0 & 0 & W \end{matrix} ) ( \begin{matrix}I & 0 & X \\ 0 & I & 0 \\ 0 & 0 & I \end{matrix} ) = ( \begin{matrix}W & 0 & Y \\ 0 & W & 0 \\ 0 & 0 & W \end{matrix} ) ,

where σ_{34} ∈ Aut_{F}(Q_{3}) and σ_{12}^{∗} ∈ Aut_{F}(Q_{1}), and Y = XW + W X = 0 or I. So if we set Q_{13} = Q_{1} ∩ Q_{3} and α = [σ_{12}^{∗}|_{Q_{13}}, σ_{34}|_{Q_{13}}] ∈ Aut_{F}(Q_{13}), then

M(α|_{A_{1}}) = [( \begin{matrix}W & 0 & Y \\ 0 & W & 0 \\ 0 & 0 & W \end{matrix} ) , ( \begin{matrix}I & 0 & 0 \\ 0 & 0 & I \\ 0 & I & 0 \end{matrix} )] = ( \begin{matrix}I & Y W & Y W \\ 0 & I & 0 \\ 0 & 0 & I \end{matrix} ) .

Thus α induces the identity on Fr(Q_{13}) = ⟨e_{15}, e_{25}⟩ and on A_{1}/Fr(Q_{13}), and induces the identity on Q_{13}/A_{1} since φ_{1} (and hence σ_{12}^{∗}) does. Since these are characteristic subgroups of Q_{13}, α ∈ O_{2}(Aut_{F}(Q_{13})) ≤ Out_{T} (Q_{13}) by Lemma 1.1. Hence Y W = 0 or Y W = I. Since Y ∈ \{0, I\} and W = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ), we conclude that Y = 0, and thus X = 0. This proves that φ_{1} ∈ ⟨θ_{2}, ψ_{2}⟩ if Q_{1} is F-essential; and also (via conjugation by τ) that φ_{4} ∈ ⟨θ_{1}, ψ_{1}⟩ if Q_{4} is F-essential.

Step 2 We now strengthen the conclusion of Step 1, by proving that Q_{1} F-essential implies φ_{1} ∈ ⟨θ_{2}ψ_{2}⟩, and Q_{4} F-essential implies φ_{4} ∈ ⟨θ_{1}ψ_{1}⟩.

Assume first Q_{1} and Q_{4} are both F-essential; we show φ_{4} ∈ ⟨θ_{1}ψ_{1}⟩. Set Q_{14} =

Q_{1} ∩Q_{4} = A_{1}A_{2}. Let X ∈ M_{2}(𝔽_{2}) be such that M(φ_{4}|_{A_{1}}) = ( \begin{matrix}I & 0 & X \\ 0 & I & 0 \\ 0 & 0 & I \end{matrix} ). Thus X = ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ),

( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ), ( \begin{matrix}1 & 0 \\ 1 & 0 \end{matrix} ), or ( \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} ), depending on whether φ_{4} = Id, θ_{1}, ψ_{1}, or θ_{1}ψ_{1}. Set W = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) as before. Since φ_{1} ∈ ⟨θ_{2}, ψ_{2}⟩ and θ_{2}|_{A_{1}} = ψ_{2}|_{A_{1}} = Id, σ_{12}^{∗}|_{A_{1}} = σ_{12}|_{A_{1}}. Hence

M(σ_{12}^{∗}|_{A_{1}}) = ( \begin{matrix}W & 0 & 0 \\ 0 & W & 0 \\ 0 & 0 & W \end{matrix} ) and M(σ_{45}^{∗}|_{A_{1}}) = ( \begin{matrix}I & 0 & X \\ 0 & I & 0 \\ 0 & 0 & I \end{matrix} ) ( \begin{matrix}0 & I & 0 \\ I & 0 & 0 \\ 0 & 0 & I \end{matrix} ) ( \begin{matrix}I & 0 & X \\ 0 & I & 0 \\ 0 & 0 & I \end{matrix} ) = ( \begin{matrix}0 & I & X \\ I & 0 & X \\ 0 & 0 & I \end{matrix} ) ,

where σ_{12}^{∗} ∈ Aut_{F}(Q_{1}) and σ_{45}^{∗} ∈ Aut_{F}(Q_{4}). Set β = [σ_{12}^{∗}, σ_{45}^{∗}] ∈ Aut_{F}(Q_{14}). Then

M(β|_{A_{1}}) = [( \begin{matrix}W & 0 & 0 \\ 0 & W & 0 \\ 0 & 0 & W \end{matrix} ) , ( \begin{matrix}0 & I & X \\ I & 0 & X \\ 0 & 0 & I \end{matrix} )] = ( \begin{matrix}I & 0 & X+W XW^{−1} \\ 0 & I & X+W XW^{−1} \\ 0 & 0 & I \end{matrix} ) .

Set R_{0} = A_{1} ∩ A_{2}. Thus β induces the identity on A_{1}/R_{0}, and also on A_{2}/R_{0} since φ_{1} (and hence σ_{12}^{∗}) induces the identity on A_{2}/R_{0}. Since R_{0} = Fr(Q_{14}), β ∈ O_{2}(Aut_{F}(Q_{14})) by Lemma 1.1, so β ∈ Aut_{T} (Q_{14}) by the Sylow axiom, and thus

X + W XW^{−1} ∈ \{I, 0\}. If X = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) or ( \begin{matrix}1 & 0 \\ 1 & 0 \end{matrix} ), then X + W XW^{−1} = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) or ( \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} ),

which is impossible. It follows that φ_{4} ∈ ⟨θ_{1}ψ_{1}⟩ in this situation.

Now assume Q_{4} is F-essential and Q_{1} is not. If φ_{4} = θ_{1}, then F is generated by Inn(T) and Aut_{F}(Q_{i}) for i = 2, 3, 4, and all of these automorphism groups leave U_{1} invariant. Thus U_{1} is normal in F in this case, which contradicts the assumption that F is nonconstrained.

Assume φ_{4} = ψ_{1}. Set V = ⟨e_{23}, e_{24}, e_{25}⟩ ≤ A_{1}, and let A ≤ Aut_{F}(A_{1}) be the subgroup of those elements which leave V invariant. Consider the homomorphism

Ψ: A ⟶^{(res,proj)} Aut(V) × Aut(A_{1}/V) \underrightarrow{M̂} GL_{3}(2) × GL_{3}(2)

where M̂ sends a pair of automorphisms to their matrices with respect to the bases {e_{i5}, e_{i4}, e_{i3}} for i = 2 or 1, respectively. Then Ψ(Aut_{Q_{1}}(A_{1})) = {(X, X)} for X ∈ GL_{3}(2) upper triangular, while Ψ(σ_{34}|_{A_{1}}) = (( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} )). Thus Im(Ψ) contains all matrices (M, M) for M ∈ H, where H ≤ GL_{3}(2) is the subgroup of matrices with first column ( \begin{matrix}1 \\ 0 \\ 0 \end{matrix} ). The above computation of M(σ_{45}^{∗}|_{A_{1}}) when φ_{4} = ψ_{1} (hence

X = ( \begin{matrix}1 & 0 \\ 1 & 0 \end{matrix} )) shows that Ψ(σ_{45}^{∗}) = (( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ) , ( \begin{matrix}0 & 1 & 1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{matrix} )).

Now, Ψ is not onto, since otherwise 2^{6}||Aut_{F}(A_{1})|, contradicting the Sylow axiom. Since H is a maximal subgroup in GL_{3}(2) (it has prime index), the above computations show that Im(Ψ) surjects onto each factor GL_{3}(2). Hence the subgroup K of all g ∈ GL_{3}(2) such that (1, g) ∈ Im(Ψ) is normal, and K ≠ GL_{3}(2) since Ψ is not onto. Thus K = 1 since GL_{3}(2) is simple, and Im(Ψ) has the form {(g, α(g))} for some α ∈ Aut(GL_{3}(2)). By the above computations, α|_{H} = Id, and

α(( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}0 & 1 & 1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{matrix} ) implies α(( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} )) = ( \begin{matrix}0 & 1 & 1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ) = ( \begin{matrix}0 & 1 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 0 \end{matrix} ) .

Thus α sends an element of order three to one of order four, which is impossible.

This finishes the proof that φ_{4} ∈ ⟨θ_{1}ψ_{1}⟩ in both cases (Q_{1} F-essential or not). As usual, it then follows by symmetry that φ_{1} ∈ ⟨θ_{2}ψ_{2}⟩ if Q_{1} is F-essential.

Step 3 Assume Q_{4} is F-essential, and Aut_{F}(Q_{4}) = Δ_{4}. If Q_{1} is not F-essential, then F is generated by automorphisms of Q_{2}, Q_{3}, and Q_{4}, all of which leave U_{1} invariant. Hence U_{1} is normal in F, which contradicts the assumption that F is nonconstrained.

Thus Q_{1} is F-essential. Since φ_{1}|_{A_{1}} = Id, the restriction to A_{1} of Aut_{F}(Q_{1}) is equal to that of Δ_{1}. So Aut_{F}(A_{1}) is generated by restrictions of automorphisms of Δ_{i} for i = 1, 3, 4. This is the product of the actions of Δ_{1} on each of the three columns ⟨e_{1i}, e_{2i}⟩ in A_{1} (i = 3, 4, 5) with the actions of ⟨Δ_{3}, Δ_{4}⟩ on each of the two rows. The actions of Δ_{3} and Δ_{4} generate the full GL_{3}(2)-action on each row (any 3 × 3 matrix can be diagonalized by row and column operations on its first two and last two rows and columns), and thus Aut_{F}(A_{1}) ≅ Σ_{3} × GL_{3}(2).

Similarly, if Q_{1} is F-essential and Aut_{F}(Q_{1}) = Δ_{1}, then Q_{4} is also F-essential and Aut_{F}(A_{2}) ≅ Σ_{3} × GL_{3}(2).

Step 4 Now assume Q_{4} is F-essential and Aut_{F}(Q_{4}) = (θ_{1}ψ_{1})Δ_{4}(θ_{1}ψ_{1})^{−1}. We will show that Aut_{F}(A_{1}) ≅ 3Σ_{6}, and that Q_{1} is also F-essential. The corresponding result when Aut_{F}(Q_{1}) = (θ_{2}ψ_{2})Δ_{1}(θ_{2}ψ_{2})^{−1} then follows by symmetry.

Consider the subgroup

Aut_{F}^{0}(A_{1}) =^{def} ⟨Aut_{T} (A_{1}), σ_{34}|_{A_{1}}, σ_{45}^{∗}|_{A_{1}}⟩ ≤ Aut_{F}(A_{1}) :

the subgroup generated by restrictions of elements in Aut_{F}(Q_{i}) for i = 2, 3, 4. This time, we identify A_{1} with 𝔽_{4}^{3}. Fix ω ∈ 𝔽_{4}∖𝔽_{2}, and give A_{1} the structure of a 𝔽_{4}-vector space by setting ωe_{1j} = e_{2j} and ωe_{2j} = e_{1j}e_{2j}. For α ∈ Aut_{𝔽_{4}}(A_{1}), let M^{∗}(α) ∈ GL_{3}(4) be the matrix for α with respect to the 𝔽_{4}-basis {e_{15}, e_{14}, e_{13}}.

Write ω = ω^{2} = ω + 1 ∈ 𝔽_{4}. Then

M^{∗}(c_{34}) = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ) ,  M^{∗}(σ_{34}) = ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ) ,  M^{∗}(c_{45}) = ( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} );

and

M^{∗}(σ_{45}^{∗}) = M^{∗}((θ_{1}ψ_{1})σ_{45}(θ_{1}ψ_{1})^{−1}) = ( \begin{matrix}1 & 0 & ω \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) ( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 & ω \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) = ( \begin{matrix}0 & 1 & ω \\ 1 & 0 & ω \\ 0 & 0 & 1 \end{matrix} ) .

Also, c_{12} acts on A_{1} as the field automorphism ϕ_{2} : (a, b, c) ↦ (ā, b̄, c̄), with respect to the given basis.

Consider the following six points in the projective plane P (𝔽_{4}^{3}): λ_{1} = ⟨(1, ω, 0)⟩, λ_{2} = ⟨(1, ω, 0)⟩, λ_{3} = ⟨(ω, 0, 1)⟩, λ_{4} = ⟨(ω, 1, 1)⟩, λ_{5} = ⟨(ω, 0, 1)⟩, λ_{6} = ⟨(ω, 1, 1)⟩. These form an “oval”, in the sense that no three of them lie in a projective line. By a direct check, the above generators permute these points in the following way:

\begin{matrix}ϕ_{2} & ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ) & ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ) & ( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) & ( \begin{matrix}0 & 1 & ω \\ 1 & 0 & ω \\ 0 & 0 & 1 \end{matrix} ) \\ (1 2)(3 5)(4 6) & (3 4)(5 6) & (1 5)(2 3) & (1 2)(4 6) & (1 2)(3 6) \end{matrix} .

The first two and last two of these permutations generate the subgroup of elements of Σ_{6} which leave {1, 2} invariant, and hence this set of five permutations generates Σ_{6}. Since this extension of C_{3} by Σ_{6} is not split, this proves that Aut_{F}^{0}(A_{1}) ≅ 3Σ_{6}.

Let ζ ∈ Aut(A_{1}) be such that M^{∗}(ζ) = ω·I = diag(ω, ω, ω). Then ζ ∈ Aut_{F}^{0}(A_{1}) by the above computation. Also, ζ commutes with all elements of Aut_{Q_{1}}(A_{1}), so it extends to an element ζ̄ ∈ Aut_{F}(Q_{1}) by the extension axiom. Thus Q_{1} is F-essential since Aut_{F}(Q_{1}) is not a 2-group. Also, the restriction to A_{1} of each element of Aut_{F}(Q_{1}) = ⟨Aut_{T} (Q_{1}), ζ̄⟩ lies in Aut_{F}^{0}(A_{1}), and hence Aut_{F}(A_{1}) = Aut_{F}^{0}(A_{1}) ≅ 3Σ_{6}. □

We can now summarize these results in the following theorem. The much more difficult classification of simple groups with Sylow 2-subgroup UT_{5}(2) is due to Held [He], and is also shown in [A2, Chapter 14].

Theorem 6.10. Every nonconstrained saturated fusion system over T = UT_{5}(2) is isomorphic to the fusion system of one of the simple groups GL_{5}(2), M_{24}, or He.

Proof. By Proposition 6.5, F is generated by Aut_{F}(T) and the Aut_{F}(Q_{i}) for i = 1, 2, 3, 4. Also, Aut_{F}(T) = Inn(T) since Aut(T) is a 2-group (Proposition 6.6(a)). By Proposition 6.8, we can assume Out_{F}(Q_{i}) = Δ_{i} for i = 2, 3. Then by Proposition 6.9, there are at most four possibilities for F, of which two are isomorphic via τ.

We refer to [He], and to [A2, §40], for a description of the groups Aut_{G}(A_{i}) when G = GL_{5}(2), M_{24}, or Held’s group. Each of these groups contains Sylow 2-subgroups S ≅ UT_{5}(2). Also, F_{S}(G) is nonconstrained and centerfree in each case, and hence must be isomorphic to one of the three distinct fusion systems which we found. Thus F is isomorphic to the fusion system of GL_{5}(2) if Aut_{F}(A_{1}) ≅ Aut_{F}(A_{2}) ≅ Σ_{3} × GL_{3}(2) (if Aut_{F}(Q_{i}) = Δ_{i} for i = 1, 4); F is isomorphic to the fusion system of M_{24} if Aut_{F}(A_{1}) ≅ Σ_{3} × GL_{3}(2) and Aut_{F}(A_{2}) ≅ 3Σ_{6} or vice versa (if Aut_{F}(Q_{i}) = Δ_{i} for i = 1 or i = 4 but not both); and F is isomorphic to the fusion system of Held’s group if Aut_{F}(A_{1}) ≅ Aut_{F}(A_{2}) ≅ 3Σ_{6} (if Aut_{F}(Q_{i}) = (θ_{j}ψ_{j})Δ_{i}(θ_{j}ψ_{j})^{−1} for (i, j) = (1, 2) and (4, 1)). □

7. Fusion systems over the Sylow subgroup of Co_{3}

Our notation here for elements in a Sylow 2-subgroup of Spin_{7}(3) is based on that used in [LO]. Fix Y, B ∈ SL_{2}(9) such that Y has order 8 and ⟨Y, B⟩ ≅ Q_{16}, and set A = Y^{2}. In particular, Y^{4} = B^{2} = −I, and ⟨A, B⟩ ≅ Q_{8}. Consider the groups

𝕊_{0} =^{def} ⟨Y, B⟩^{3}/⟨(−I, −I, −I)⟩ and 𝕊 =^{def} 𝕊_{0} ⋊^{(1 2)} C_{2}_{τ},

and let [[X_{1}, X_{2}, X_{3}]] denote the class of (X_{1}, X_{2}, X_{3}) in 𝕊_{0}. Thus

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

Write a_{1} = [[A, I, I]], a_{2} = [[I, A, I]], a_{3} = [[I, I, A]], b_{1} = [[B, I, I]], b_{2} = [[I, B, I]], b_{3} = [[I, I, B]], c = [[Y, Y, Y]], and z_{i} = a_{i}^{2}. Finally, set

T^{∗} = ⟨a_{1}, a_{2}, a_{3}, c, b_{1}, b_{2}, b_{3}, τ ⟩ ≤ 𝕊 :

a group of order 2^{10}. For later reference, we list the following relations in T^{∗} (for all i ≠ j), which in fact form a complete presentation for this group:

a_{i}^{2} = b_{i}^{2} = [a_{i}, b_{i}] = z_{i},  z_{i}^{2} = 1 = z_{1}z_{2}z_{3},  [a_{i}, b_{j}] = 1 = [a_{i}, a_{j}] = [b_{i}, b_{j}];
c^{2} = a_{1}a_{2}a_{3},  [c, a_{i}] = 1,  cb_{i}c^{−1} = a_{i}b_{i},  b_{i}cb_{i}^{−1} = a_{i}^{−1}c; (1)

τ^{2} = 1, τ cτ^{−1} = c, τ a_{i}τ^{−1} = a_{σ(i)}, τ b_{i}τ^{−1} = b_{σ(i)} (where σ = (1 2) ∈ Σ_{3}).

The embedding of T^{∗} as a Sylow 2-subgroup of Spin_{7}(3) is described in detail in [LO, § 2]. For example, the subgroup ⟨a_{1}, b_{1}, a_{2}, b_{2}, a_{3}, b_{3}⟩ is a Sylow subgroup of Spin_{3}(3)×_{C_{2}}Spin_{4}^{+}(3) ≤ Spin_{7}(3), via the identifications Spin_{3}(3) ≅ SL_{2}(3), Spin_{4}^{+}(3) ≅ SL_{2}(3) × SL_{2}(3), and Q_{8} ∈ Syl_{2}(SL_{2}(3)). Instead of repeating that argument here, we give an explicit homomorphism ρ: T^{∗} ⟶ Ω_{7}(3) to help motivate some of our constructions. Let δ_{i} be the diagonal matrix with entry −1 in i-th position and 1 elsewhere, and set δ_{ij} = δ_{i}δ_{j}, etc. Let π_{σ} be the permutation matrix for σ ∈ Σ_{7}. Thus π_{σ}δ_{i}π_{σ}^{−1} = δ_{σ(i)}. Define ρ by setting

ρ(a_{1}) = δ_{14}π_{(12)(34)}  ρ(a_{2}) = δ_{24}π_{(12)(34)}  ρ(a_{3}) = δ_{56}
ρ(b_{1}) = δ_{12}π_{(13)(24)}  ρ(b_{2}) = δ_{23}π_{(13)(24)}  ρ(b_{3}) = δ_{57}
ρ(c) = δ_{46}π_{(34)(56)}  ρ(τ) = δ_{1567}.

It is straightforward to check that the relations in T^{∗} listed above all hold, and hence that this defines a homomorphism with kernel ⟨z_{3}⟩.

Two families of subgroups of T^{∗} will play an important role in what follows. First define

R_{0} = ⟨a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}⟩ R_{1} = ⟨R_{0}, c⟩ R_{2} = ⟨R_{0}, τ ⟩ R_{3} = ⟨R_{0}, cτ ⟩ .

Thus T^{∗}/R_{0} ≅ C_{2}^{2}, and R_{1}, R_{2}, and R_{3} are the three subgroups of index two in T^{∗} which contain R_{0}. Also, Z(R_{0}) = Z(R_{1}) = ⟨z_{1}, z_{2}⟩, while Z(R_{i}) = ⟨z_{3}⟩ = Z(T^{∗}) for i = 2, 3.

Next consider the following subgroups:

Q = ⟨z_{1}, a_{1}a_{2}, a_{3}, b_{1}b_{2}, b_{3}, τ ⟩ = ⟨a_{1}a_{2}, b_{1}b_{2}⟩ ×_{C_{2}} ⟨a_{3}, b_{3}⟩ ×_{C_{2}} ⟨z_{1}, τ ⟩
R_{4} = ⟨Q, a_{1}, c⟩ ;  H_{1} = ⟨Q, c⟩ H_{2} = ⟨Q, a_{1}c⟩ H_{3} = ⟨Q, a_{1}⟩ .

Thus Q is extraspecial of order 2^{7} (a central product of three D_{8}’s), R_{4}/Q ≅ C_{2}^{2}, and H_{1}, H_{2}, and H_{3} are the three subgroups of index two in R_{4} which contain Q. Also, H_{3} ◁ T^{∗}, while H_{2} = b_{1}H_{1}b_{1}^{−1} and N_{T^{∗}}(H_{1}) = N_{T^{∗}}(H_{2}) = R_{4}. These three subgroups will be seen to be permuted transitively by Out(R_{4}).

Consider again the homomorphism ρ: T^{∗} ⟶ Ω_{7}(3) defined above, and also the induced action of T^{∗} on V ≅ 𝔽_{2}^{7} with canonical (orthonormal) basis {e_{1}, …, e_{7}}. By inspection, R_{1} is the subgroup of those elements which act on each of the factors ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ and ⟨e_{5}, e_{6}, e_{7}⟩ with determinant one, and R_{0} is the subgroup of elements whose action on each factor lies in the spinor group. Also, R_{4} is the subgroup of elements which leave invariant each of the summands ⟨e_{1}, e_{2}⟩, ⟨e_{3}, e_{4}⟩, and ⟨e_{5}, e_{6}⟩, while Q is the group of elements which sends each of the ⟨e_{i}⟩ to itself.

Before we begin looking at the critical subgroups of T^{∗}, we prove the following lemma about Q, and about another subgroup A ≅ C_{4}^{3} which we will need to work with.

Lemma 7.1. (a) Set A = ⟨a_{1}, a_{2}, a_{3}, c⟩. Then A ≅ C_{4}^{3}, A ◁ T^{∗}, and T^{∗}/A ≅ C_{2} × D_{8}.

(b) If P ≤ T^{∗} is such that |P | = 2^{7} and |Fr(P)| = 2, then P = Q.

Proof. (a) Since c^{2} = a_{1}a_{2}a_{3} and (a_{1}a_{2}a_{3})^{2} = 1, A = ⟨a_{1}⟩ × ⟨a_{2}⟩ × ⟨c⟩ ≅ C_{4}^{3}. By the relations (1), A is normal in T^{∗}, and T^{∗}/A = ⟨b_{1}A, b_{2}A, b_{3}A, τ A⟩ ≅ D_{8} × C_{2}.

(b) Let A_{0} = ⟨z_{1}, z_{2}, a_{1}a_{2}a_{3}⟩ be the 2-torsion subgroup of A. Since |T^{∗}/A| = 2^{4}, P ∩A is a normal subgroup of P of order at least 2^{3}. If |P ∩A| = 2^{3}, then P/(P ∩A) ≅ T^{∗}/A ≅ C_{2} × D_{8}, so Fr(P) ∩ A = 1, and P ∩ A = A_{0} since it cannot have 4-torsion. Since Fr(T^{∗}/A) = ⟨b_{1}b_{2}A⟩, Fr(P) = ⟨b_{1}b_{2}g⟩ for some g ∈ A, which is impossible since [b_{1}b_{2}g, a_{1}a_{2}a_{3}] = z_{3} ≠ 1.

It follows that |P ∩ A| ≥ 2^{4}. In particular, Fr(P) ≤ A since P ∩ A is not elementary abelian, and P ≥ A_{0} and |P ∩ A| = 2^{4} since P contains no subgroup C_{4}^{2}. So P A/A is an elementary abelian subgroup of order 2^{3} in T^{∗}/A ≅ D_{8} × C_{2}. Hence either P A/A = ⟨b_{1}, b_{2}, b_{3}⟩ and thus P A = R_{1}, or P A/A = ⟨b_{1}b_{2}, b_{3}, τ ⟩ and P A = R_{4}. In either case, b_{3}g ∈ P for some g ∈ A, and so [b_{3}g, a_{1}a_{2}a_{3}] = z_{3} ∈ Fr(P). Thus Fr(P) = ⟨z_{3}⟩.

If P A = R_{1}, then b_{1}g ∈ P for some g ∈ A, so [b_{1}g, a_{1}a_{2}a_{3}] = z_{1} ∈ Fr(P), and we just saw this is impossible. Hence P A = R_{4}.

Consider the quotient group

R_{4}/A_{0} = R_{4}/⟨z_{1}, z_{2}, a_{1}a_{2}a_{3}⟩ = ⟨a_{1}, a_{2}, τ ⟩ ×_{⟨a_{3}⟩} ⟨c, b_{3}⟩ × ⟨b_{1}b_{2}b_{3}⟩ ≅ D_{8} ×_{C_{2}} D_{8} × C_{2}.

Since Fr(P) ≤ A_{0}, P/A_{0} ≅ C_{2}^{4}. Hence Z(R_{4}/A_{0}) ≤ P/A_{0}, since every abelian subgroup of rank four in R_{4}/A_{0} contains the center. In particular, b_{1}b_{2}b_{3} ∈ P, so P/⟨z_{3}⟩ ≤ C_{R_{4}/⟨z_{3}⟩}(b_{1}b_{2}b_{3}) = Q/⟨z_{3}⟩, and hence P = Q. □

In fact, A is the unique abelian subgroup of order 2^{6} in T^{∗}, but we will not need to use that.

7.1 Determining the critical subgroups We start as usual by reducing to the case of subgroups of index 2 in their normalizers.

Lemma 7.2. If P is a critical subgroup of T^{∗}, then |N_{T^{∗}}(P)/P | = 2.

Proof. Assume otherwise: let P be a critical subgroup of T^{∗} with |N_{T^{∗}}(P)/P | ≥ 4. By Proposition 3.3(c),

g ∉ P, [g, P] ≤ P ⟹ rk([g, P/Fr(P)]) ≥ 2.  (2)

Since P is centric in T^{∗}, Z(T^{∗}) = ⟨z_{3}⟩ ≤ P. Since [x, P] ≤ [x, T^{∗}] ≤ ⟨z_{3}⟩ for x ∈ ⟨z_{1}, a_{3}⟩, z_{1}, a_{3} ∈ P by (2). In particular, z_{3} = a_{3}^{2} ∈ Fr(P).

Since [a_{1}a_{2}, P] ≤ [a_{1}a_{2}, T^{∗}] = ⟨z_{1}, z_{3}⟩, rk([a_{1}a_{2}, P/Fr(P)]) ≤ 1, and hence a_{1}a_{2} ∈ P by (2). Similarly, [b_{3}, P] ≤ [b_{3}, T^{∗}] = ⟨a_{3}⟩ implies b_{3} ∈ P. Set T_{0} = ⟨z_{1}, a_{1}a_{2}, a_{3}, b_{3}⟩ ≤ P. Then |T_{0}| = 2^{5}, T_{0} ◁ T^{∗}, and

T^{∗}/T_{0} = R_{4}/T_{0} ⋊ ⟨b_{1}⟩ ≅ C_{2}^{4}_{⟨a_{1},b_{1}b_{2},c,τ⟩} ⋊ C_{2}_{b_{1}}.

Now, [a_{1}, T^{∗}] = ⟨z_{1}, z_{3}, a_{1}a_{2}⟩ = [b_{1}b_{2}, T^{∗}]. So by (2), either Fr(P)∩⟨z_{1}, a_{1}a_{2}⟩ = ⟨z_{3}⟩, or a_{1}, b_{1}b_{2} ∈ P. If a_{1} ∈ P, then z_{1} = a_{1}^{2} ∈ Fr(P), and so b_{1}b_{2} ∈ P, Fr(T^{∗}) ≤ P, and thus P ◁ T^{∗}. Set T_{1} = ⟨T_{0}, a_{1}, b_{1}b_{2}⟩; thus |T_{1}| = 2^{7} and so [P:T_{1}] ≤ 2. If b_{1} ∉ P, then since [T_{1}, b_{1}] ≤ ⟨z_{1}, z_{3}⟩ ≤ Fr(P) and |P/T_{1}| ≤ 2, rk([b_{1}, P/Fr(P)]) ≤ 1, contradicting (2) again. Thus P = ⟨T_{1}, b_{1}⟩ = R_{0}, Fr(P) = ⟨z_{1}, z_{3}⟩, rk([τ , P/Fr(P)]) = 2, and rk([c, P/Fr(P)]) = 3. This contradicts Proposition 3.3(b) (all involutions in Out_{T^{∗}}(P) are conjugate in Out(P)). We now conclude that

a_{1} ∉ P,  and Fr(P) ∩ ⟨z_{1}, a_{1}a_{2}⟩ = ⟨z_{3}⟩.  (3)

Since [a_{1}a_{2}, R_{4}] = ⟨z_{3}⟩ ≤ Fr(P) and [a_{1}a_{2}, b_{1}] = z_{1}, (3) implies P ≤ R_{4}. Also, P ≥ Fr(R_{4}), and so N_{T^{∗}}(P) ≥ R_{4}. By Proposition 3.3(a), all involutions in Out_{T^{∗}}(P) ≅ N_{T^{∗}}(P)/P are central. Hence if P ◁ T^{∗}, then all elements of R_{4}/P are central in T^{∗}/P, which is impossible since a_{1} ∉ P and [c, b_{1}] = a_{1} (c ∉ P since P is normal, and hence cP ∉ Z(T^{∗}/P)). Thus N_{T^{∗}}(P) = R_{4}. Also, R_{4}/P ≅ C_{2}^{k} for k ≥ 2. If k ≥ 3, then rk(P/Fr(P)) ≥ 6 by Proposition 3.3(c), so 2^{7} ≤ |P | = 2^{9−k}, a contradiction. Thus

N_{T^{∗}}(P) = R_{4},  [P : T_{0}] = 4,  and [R_{4} : P] = 4.  (4)

If xb_{1}b_{2} ∈ P for some x ∈ ⟨a_{1}⟩, then since [xb_{1}b_{2}, T^{∗}] ≤ ⟨z_{1}, a_{1}a_{2}⟩ in both cases, [xb_{1}b_{2}, P] = ⟨z_{3}⟩ by (3). Hence

P ≤ \{g ∈ R_{4} | [xb_{1}b_{2}, g] ∈ ⟨z_{3}⟩\} = \begin{cases}⟨T_{0}, b_{1}b_{2}, τ ⟩ & if x = 1 \\ ⟨T_{0}, a_{1}b_{1}b_{2}, a_{2}cτ ⟩ & if x = a_{1}, \end{cases}

and P is equal to one of these groups (the inclusion is an equality) by (4). But both of these groups are normal in T^{∗} — note that b_{1}(a_{2}cτ)b_{1}^{−1} ≡ (a_{1}b_{1}b_{2})(a_{2}cτ) (mod T_{0})

— which contradicts (4). So this case is impossible.

Thus P ∩ ⟨T_{0}, a_{1}, b_{1}b_{2}⟩ = T_{0}. Since [P : T_{0}] = 4 by (4) again,

P = ⟨T_{0}, cx, τ y⟩ = ⟨z_{1}, a_{1}a_{2}, a_{3}, b_{3}, cx, τ y⟩

for some x, y ∈ ⟨a_{1}, b_{1}b_{2}⟩. Since b_{1}cb_{1}^{−1} = a_{1}^{−1}c, it suffices to consider the case where x ∈ ⟨b_{1}b_{2}⟩. Then one of the following happens:

• y ∈ ⟨b_{1}b_{2}⟩, [b_{1}b_{2}, P] = ⟨a_{1}a_{2}, z_{3}⟩, so rk([b_{1}b_{2}, P/Fr(P)]) ≤ 1 contradicting (2);

• y = a_{1}, (τ a_{1})^{2} = a_{1}a_{2} ∈ Fr(P), contradicting (3); or

• y = a_{1}b_{1}b_{2}, (τ a_{1}b_{1}b_{2})^{2} = a_{1}a_{2}z_{2} ∈ Fr(P), contradicting (3).

This finishes the proof. □

It remains to handle the subgroups of T^{∗} of index two in their normalizer.

Proposition 7.3. The only critical subgroups in T^{∗} are R_{1}, R_{2}, R_{3}, R_{4}, H_{1}, and H_{2}.

Proof. Fix a critical subgroup P ≤ T^{∗} of index two in its normalizer. By Lemma 3.4,

g ∈ N(P)∖P, Θ char P ⟹ [g, P] ≰ Θ·Fr(P) or [g, Θ] ≰ Fr(P). (5)

In Step 1, we show that ⟨z_{1}, z_{3}, a_{3}⟩ ≤ P, and that ⟨a_{1}, a_{2}, a_{3}⟩ ≤ P if P ≤ R_{1}. In Step 2, we show a_{1}a_{2} ∈ P. We then handle the cases where P is not normal in T^{∗} in Step 3, and those where P ◁ T^{∗} (hence [T^{∗}:P] = 2) in Step 4.

Step 1: Since P is centric, z_{3} ∈ Z(T^{∗}) ≤ P. Since [z_{1}, P] ≤ [z_{1}, T^{∗}] = ⟨z_{3}⟩ ≤ P, and similarly for a_{3}, ⟨z_{1}, a_{3}⟩ ≤ N(P). So if z_{1} ∉ P, then a_{3} or z_{1}a_{3} must be in P, since otherwise |N(P)/P | ≥ 4. Then z_{3} = a_{3}^{2} = (z_{1}a_{3})^{2} ∈ Fr(P), so [z_{1}, P] ≤ Fr(P), and this contradicts (5). This proves that z_{1} ∈ P.

Now assume a_{3} ∉ P. By Lemma 3.6, applied with z = z_{3}, g = a_{3}, and y = z_{1}, there is h ∈ T^{∗} such that [h, a_{3}] = z_{3} ∉ Fr(P), h^{2} = 1, and P = C_{T^{∗}}(h). Also, z_{1} ∈ Z(P) (since z_{1}a_{3} ∉ P), and h is not T^{∗}-conjugate to z_{1}h. Thus h ∈ P ≤ R_{1}, since z_{1} ∈ Z(P). We return to the notation used at the beginning of the section, and write h = [[X_{1}, X_{2}, X_{3}]] for some X_{i} ∈ ⟨Y, B⟩ ≅ Q_{16}. Recall A = Y^{2} and ⟨A, B⟩ ≅ Q_{8}.

The condition [h, a_{3}] = z_{3} implies [X_{3}, A] = −I, and hence X_{3} ∈ ⟨Y ⟩·B. Thus X_{3}^{2} = −I, and hence X_{1}^{2} = X_{2}^{2} = −I since h^{2} = 1. Since h is not T^{∗}-conjugate to z_{1}h, [X_{1}, A] ≠ −I and [X_{1}, B] ≠ −I imply X_{1} = ±I, and thus X_{1}^{2} ≠ −I. Hence this situation is impossible, and we conclude that a_{3} ∈ P.

Now assume P ≤ R_{1} = ⟨a_{i}, b_{i}, c | i = 1, 2, 3⟩; we claim that a_{1}, a_{2} ∈ P. This is clear if P = R_{1}, so we assume P ≨ R_{1}. Then N_{R_{1}}(P)/P ≠ 1, so N(P) ≤ R_{1} since we are assuming |N(P)/P | = 2. Thus P is also critical in R_{1}. In this situation, the same argument we just used to show a_{3} ∈ P also applies to prove that a_{1}, a_{2} ∈ P.

Step 2: We next prove that a_{1}a_{2} ∈ P. Assume otherwise; then a_{1}a_{2} ∈ N_{T^{∗}}(P)∖P. Since [a_{1}a_{2}, P] ≤ [a_{1}a_{2}, T^{∗}] = ⟨z_{1}, z_{2}⟩, and z_{3} = a_{3}^{2} ∈ Fr(P), z_{1} ∉ Fr(P) by (5). By Step 1, P ≰ R_{1}; let g ∈ R_{1} be such that gτ ∈ P.

By Lemma 3.5, there is α ∈ Aut(P) of odd order, and x ∈ [a_{1}a_{2}, P], such that x ∉ Fr(P) and [a_{1}a_{2}, α(x)] ∉ Fr(P). Thus x ∈ {z_{1}, z_{2}}, and [a_{1}a_{2}, α(x)] ∈ {z_{1}, z_{2}}. Set y = α(x) = [[X_{1}, X_{2}, X_{3}]]τ^{k}, where X_{i} ∈ ⟨Y, B⟩ ≅ Q_{16} and k = 0, 1. Then y^{2} = α(x^{2}) = 1. The condition [a_{1}a_{2}, y] ∉ ⟨z_{3}⟩ means that [A, X_{1}] ≠ [A, X_{2}] (recall a_{1}a_{2} = [[A, A, I]] and A = Y^{2}), and thus X_{1} ∈ ⟨Y ⟩ or X_{2} ∈ ⟨Y ⟩ but not both. If k = 1, then y^{2} = [[X_{1}X_{2}, X_{2}X_{1}, X_{3}^{2}]] = 1, which is impossible since X_{1}X_{2} ∉ ⟨Y ⟩. Thus k = 0, and y^{2} = [[X_{1}^{2}, X_{2}^{2}, X_{3}^{2}]] = 1. Since X_{1} or X_{2} has order ≥ 4, this implies X_{i}^{2} = −I for each i = 1, 2, 3. Also, X_{j} ∈ ⟨Y ⟩ for j = 1 or 2, so X_{j} ∈ ⟨A⟩, and thus X_{i} ∈ ⟨A, B⟩ ≅ Q_{8} for each i = 1, 2, 3. We have now shown that y = y_{1}y_{2}y_{3}, where y_{i} ∈ ⟨a_{i}, b_{i}⟩∖⟨z_{i}⟩, and where y_{1} ∈ ⟨a_{1}⟩ or y_{2} ∈ ⟨a_{2}⟩ but not both.

Thus [y, gτ] ≡ [y, τ] ≡ b_{1}b_{2} (mod ⟨a_{1}, a_{2}, a_{3}⟩). Since [y, P] = α([x, P]) = ⟨α(z_{3})⟩ has order 2 (recall x ∈ {z_{1}, z_{2}}), this implies z_{3} ∉ [y, P], so [y, a_{3}] = 1, and y_{3} ∈ ⟨a_{3}⟩. But then [y, gτ] ≡ b_{1}b_{2} (mod ⟨a_{1}, a_{2}⟩), hence it has order four, which again is impossible since [y, P] has order two. We conclude that a_{1}a_{2} ∈ P.

Step 3: Assume P ◁̸ T^{∗}. Set T_{1} = ⟨z_{1}, z_{3}, a_{1}a_{2}, a_{3}⟩ ≤ P, and consider the extension

1 ⟶ R_{0}/T_{1}_{=⟨a_{1},b_{1},b_{2},b_{3}⟩≅C_{2}^{4}} ⟶ T^{∗}/T_{1} ⟶ T^{∗}/R_{0}_{=⟨c,τ⟩≅C_{2}^{2}} ⟶ 1.

We want to apply Lemma 1.9, with S = T^{∗}/T_{1} and S_{0} = R_{0}/T_{1} ≅ C_{2}^{4}, where we recall R_{0} = ⟨a_{i}, b_{i} | i = 1, 2, 3⟩. Set P_{0} = P ∩ R_{0}. Since [⟨a_{1}, b_{1}b_{2}, b_{3}⟩, T^{∗}] ≤ T_{1} ≤ P,

N_{T^{∗}}(P) ≥ T_{2} =^{def} ⟨T_{1}, a_{1}, b_{1}b_{2}, b_{3}⟩ and hence [T_{2} : P ∩ T_{2}] ≤ 2.  (6)

Recall, in the notation of Lemma 1.9, that m is the number of classes xR_{0} ∈ T^{∗}/R_{0} such that xR_{0} ≠ R_{0} and [x, R_{0}] ≤ P_{0}. Since [τ , R_{0}/T_{1}] = ⟨b_{1}b_{2}T_{1}⟩, [c, R_{0}/T_{1}] = ⟨a_{1}T_{1}⟩, and [cτ , R_{0}/T_{1}] = ⟨a_{1}b_{1}b_{2}T_{1}⟩,

m = |{a_{1}, b_{1}b_{2}, a_{1}b_{1}b_{2}} ∩ P |.  (7)

At least one of the elements a_{1}, b_{1}b_{2}, or a_{1}b_{1}b_{2} is in P_{0} by (6), so m ≥ 1. By Lemma 1.9, we are left with the following cases, where g ∈ N_{R_{0}}(P)∖P:

(b) rk(R_{0}/P_{0}) = 1, |P/P_{0}| = 2, m = 1 , and P_{0} ◁̸ T^{∗}. By (7), P_{0} contains exactly one of the elements a_{1}, b_{1}b_{2}, or a_{1}b_{1}b_{2}. Fix g ∈ {a_{1}, b_{1}b_{2}} such that g ∉ P. Let h ∈ {c, τ , cτ } be such that h ∈ P R_{0} (|P R_{0}/R_{0}| = |P/P_{0}| = 2).

Since |R_{0}/P_{0}| = 2, at least one of the elements a_{1}, b_{1}, or a_{1}b_{1} is in P_{0}, and so z_{1} = a_{1}^{2} = b_{1}^{2} = (a_{1}b_{1})^{2} ∈ Fr(P). Since z_{3} = a_{3}^{2} ∈ Fr(P), Fr(P) ≥ ⟨z_{1}, z_{2}⟩.

By Lemma 3.5, there are elements r, s ∈ P such that s = α(r) for some α ∈ Aut(P), r ∈ [g, P], r ∉ Fr(P), and [g, s] ∉ Fr(P). Since [g, P] ≤ [g, T^{∗}] = ⟨z_{1}, a_{1}a_{2}⟩ (recall g ∈ {a_{1}, b_{1}b_{2}}), this means that r, [g, s] ∈ {a_{1}^{i}a_{2}^{j} | i, j = ±1}. In particular, [r, P] ≤ ⟨z_{1}, z_{2}⟩ has exponent two, so [s, P] = α([r, P]) also has exponent two.

Now, s ∈ P ∖R_{0} since [g, R_{0}] ≤ [R_{0}, R_{0}] ≤ ⟨z_{1}, z_{2}⟩, and thus s = hs_{0} for some s_{0} ∈ R_{0}. Since |R_{0}/P_{0}| = 2 and g ∈ R_{0}∖P_{0}, b_{1}x ∈ P_{0} for some x ∈ ⟨g⟩. By the previous paragraph, [hs_{0}, b_{1}x] ∈ [s, P] has order at most 2. Set K = ⟨z_{1}, a_{1}a_{2}⟩ ◁ T^{∗}. Then [s_{0}, b_{1}x] ∈ [R_{0}, R_{0}] ≤ K, [h, x] ∈ [T^{∗}, ⟨a_{1}, b_{1}b_{2}⟩] ≤ K, and

[τ , b_{1}] = b_{2}b_{1}^{−1},  [c, b_{1}] = (a_{1}b_{1})b_{1}^{−1} = a_{1},  and [cτ , b_{1}] = (a_{2}b_{2})b_{1}^{−1}.

Thus [s, b_{1}x] is in one of the cosets b_{1}b_{2}K, a_{1}K, or b_{1}a_{2}b_{2}K. All of the elements in these cosets have order four, in contradiction with what was already shown. So this case is impossible.

(c) rk(R_{0}/P_{0}) = 1, |P/P_{0}| = 2, m = 3 , and P_{0} ◁ T^{∗}. Since [c, τ] = 1, this would imply [T^{∗}, T^{∗}] = [R_{0}, T^{∗}] ≤ P, and hence P ◁ T^{∗}.

(e) rk(R_{0}/P_{0}) = 2, |P/P_{0}| = 4, and m = 1 . By (6), [T_{2}:P_{0} ∩ T_{2}] ≤ 2, where T_{2} = ⟨T_{1}, a_{1}, b_{1}b_{2}, b_{3}⟩ has index two in R_{0}. Since [R_{0}:P_{0}] = 4, this implies that P_{0} ≤ T_{2} with index two. Also, by (7), exactly one of the elements a_{1}, b_{1}b_{2}, or a_{1}b_{1}b_{2} is in P_{0}. This leaves the following possibilities for P:

• (a_{1} ∈ P) P = ⟨T_{1}, a_{1}, b_{3}x, cy, τ z⟩ for some x ∈ ⟨b_{1}b_{2}⟩ and y, z ∈ ⟨b_{1}, b_{2}⟩. We take g = b_{1}b_{2}. In all of these cases, z_{1} = a_{1}^{2} and a_{1}a_{2} ≡ [a_{1}, τ z] (mod ⟨z_{1}, z_{2}⟩) are both in Fr(P), and so [g, P] ≤ ⟨z_{1}, a_{1}a_{2}⟩ ≤ Fr(P).

• (b_{1}b_{2} ∈ P) P = ⟨T_{1}, b_{1}b_{2}, b_{3}x, cy, τ z⟩ for some x ∈ ⟨a_{1}⟩ and y, z ∈ ⟨a_{1}, b_{1}⟩. We take g = a_{1}. Then (cy)^{2} ∈ P_{0} implies y ∈ ⟨a_{1}⟩, and [cy, τ z] ∈ P_{0} implies z ∈ ⟨a_{1}⟩. We can also arrange that y = 1 by replacing P by b_{1}P b_{1}^{−1} if necessary. Then Fr(P) always contains [c, b_{1}b_{2}] = a_{1}a_{2}. Since (b_{3}a_{1})^{2} = z_{2} and [τ a_{1}, b_{1}b_{2}] = z_{2}, [g, P] ≤ ⟨z_{1}, a_{1}a_{2}⟩ ≤ Fr(P) if either x = a_{1} or z = a_{1}. If x = z = 1, then P = H_{1}.

• (a_{1}b_{1}b_{2} ∈ P) P = ⟨T_{1}, a_{1}b_{1}b_{2}, b_{3}x, cy, τ z⟩ for some x ∈ ⟨a_{1}⟩ and y, z ∈ ⟨a_{1}, b_{1}⟩. We take g = a_{1}. Then (cy)^{2} ∈ P_{0} implies y ∈ ⟨a_{1}⟩, and [cy, τ z] ∈ P_{0} implies z ∈ ⟨a_{1}⟩. In all cases, Z(P/⟨z_{1}, z_{2}⟩) ≤ T_{1}/⟨z_{1}, z_{2}⟩. Hence Z(P) ≤ C_{T_{1}}(a_{1}b_{1}b_{2}, b_{3}x) = ⟨z_{1}, z_{2}⟩, and Z_{2}(P) ≤ T_{1}. Thus [g, P] ≤ ⟨z_{1}, a_{1}a_{2}⟩ ≤ Z_{2}(T^{∗}) ≤ Z_{2}(P) and [g, Z_{2}(P)] = 1.

Thus in all cases except when P is conjugate to H_{1}, P fails to be critical by (5).

This finishes Step 3: the only critical subgroups of T^{∗} which are not normal are H_{1} and H_{2}.

Step 4: It remains to handle the case where P ◁ T^{∗}; i.e., where P has index 2 in T^{∗}. Thus P contains

T^{∗}′ = [T^{∗}, T^{∗}] = ⟨a_{1}, a_{2}, a_{3}, b_{1}b_{2}⟩

(recall [c, b_{i}] = a_{i} by (1)). Also, Fr(P) ≥ L_{3}(T^{∗}) = [[T^{∗}, T^{∗}], T^{∗}] = ⟨z_{1}, a_{1}a_{2}⟩: z_{1} = a_{1}^{2} ∈ Fr(P), and a_{1}a_{2} ≡ [a_{1}, τ] ≡ [a_{1}, τ c] ≡ [b_{1}b_{2}, c] (mod ⟨z_{1}, z_{3}⟩) (and one of the elements c, τ, or τ c must be in P). For any g ∈ N_{T^{∗}}(P)∖P, [g, P] ≤ T^{∗}′ and [g, T^{∗}′] ≤ L_{3}(T^{∗}) ≤ Fr(P). Hence T^{∗}′ is not characteristic in P by (5).

Consider the group S of Lemma 6.4. Set x_{1} = b_{3}, x_{2} = b_{3}c, x_{3} = b_{1}, and x_{4} = τ. These have the property that x_{i}^{2} ∈ L_{3}(T^{∗}), the three commutators [x_{i}, x_{i+1}] form a basis for T^{∗}′/L_{3}(T^{∗}) ≅ C_{2}^{3}, and [x_{i}, x_{j}] = 1 when |i − j| ≥ 2. Hence there is an epimorphism φ: T^{∗} ⟶ S, defined by φ(x_{i}) = g_{i}, with Ker(φ) = L_{3}(T^{∗}). By Lemma 6.4, either Fr(P) = T^{∗}′, in which case T^{∗}′ is characteristic and P is not critical; or P is one of the groups

U_{i} = ⟨T^{∗}′, x_{j} | j ≠ i⟩ (1 ≤ i ≤ 4), U_{13} = ⟨T^{∗}′, x_{1}x_{3}, x_{2}, x_{4}⟩, U_{24} = ⟨T^{∗}′, x_{1}, x_{3}, x_{2}x_{4}⟩.

Of these six cases, U_{2} = R_{2}, U_{3} = R_{4}, U_{4} = R_{1}, and U_{24} = R_{3}. So it remains to show that U_{1} and U_{13} are not critical.

If P = U_{13} = ⟨a_{1}, a_{2}, a_{3}, b_{1}b_{2}, b_{1}b_{3}, b_{3}c, τ ⟩, then Fr(P) = ⟨z_{1}, a_{1}a_{2}, a_{1}a_{3}, b_{1}b_{2}⟩. Set H = ⟨a_{1}a_{2}, a_{1}a_{3}⟩ ≅ C_{4}^{2}. Thus is the unique abelian subgroup of index two in Fr(P), since any abelian subgroup not in H is contained in C_{Fr(P)}(g) = ⟨z_{1}, z_{2}, g⟩ for some g ∈ Fr(P)∖H. Hence H is characteristic in P, and so is the subgroup L_{3}(T^{∗}) = ⟨z_{1}, a_{1}a_{2}⟩, since it is the subgroup of elements in H which are inverted under conjugation by b_{1}b_{2}. Also, Z(P/L_{3}(T^{∗})) = T^{∗}′/L_{3}(T^{∗}) by Lemma 6.4 again, so T^{∗}′ is also characteristic, and P is not critical.

If P = U_{1} = ⟨a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}c, τ ⟩, then Fr(P) = ⟨a_{1}, a_{2}, b_{1}b_{2}⟩ again contains a unique abelian subgroup H = ⟨a_{1}, a_{2}⟩ ≅ C_{4}^{2} of index two. So H and Ω_{1}(H) = ⟨z_{1}, z_{2}⟩ are characteristic in P. Also, C_{P} (H) = ⟨a_{1}, a_{2}, a_{3}, b_{3}c⟩ is characteristic, and since (b_{3}c)^{2} = a_{1}a_{2}z_{3}, ⟨a_{1}, a_{2}, a_{3}⟩/⟨z_{1}, z_{2}⟩ is the 2-torsion subgroup of C_{P} (H)/⟨z_{1}, z_{2}⟩. So ⟨a_{1}, a_{2}, a_{3}⟩ is characteristic, T^{∗}′ = ⟨Fr(P), a_{1}, a_{2}, a_{3}⟩ is characteristic, and again P is not critical. □

7.2 Automorphisms of critical subgroups We first define automorphisms β_{1}^{∗}, β_{2}^{∗}, β_{3}^{∗}, β_{4}^{∗} ∈ Aut(T^{∗}) via the following table:

g  a_{1} a_{2} a_{3}  b_{1}  b_{2}  b_{3}  c  τ
β_{1}^{∗}(g) a_{1} a_{2} a_{3}  b_{1}  b_{2}  b_{3}  z_{3}c  τ
β_{2}^{∗}(g) a_{1} a_{2} a_{3}  b_{1}  b_{2}  a_{3}b_{3}  c  τ
β_{3}^{∗}(g) a_{1} a_{2} a_{3} z_{3}b_{1} z_{3}b_{2}  b_{3}  c  τ
β_{4}^{∗}(g) a_{1} a_{2} a_{3} z_{3}b_{1} z_{1}b_{2} z_{2}b_{3}  c  a_{1}a_{2}a_{3}τ

Here, β_{1}^{∗} and β_{3}^{∗} are automorphisms since they have the form β_{i}^{∗}(g) = g·φ_{i}(g) for some φ_{i} ∈ Hom(T^{∗}, Z(T^{∗})); and β_{2}^{∗}|_{R_{2}} and β_{4}^{∗}|_{R_{1}} are automorphisms for similar reasons. One easily checks that β_{2}^{∗}([c, g]) = [c, β_{2}^{∗}(g)] for all g ∈ R_{2}, and hence that β_{2}^{∗} ∈ Aut(T^{∗}). As for β_{4}^{∗}, since (a_{1}a_{2}a_{3}τ)^{2} = τ^{2} = 1, the only tricky point to check is that for i = 1, 2, 3 (and taking indices modulo 3, so that τ b_{i}τ^{−1} = b_{−i}):

[a_{1}a_{2}a_{3}τ , β_{4}^{∗}(b_{i})] = [a_{1}a_{2}a_{3}τ , z_{i−1}b_{i}] = c_{a_{1}a_{2}a_{3}}(z_{−i+1}b_{−i})·(z_{i−1}b_{i})^{−1}

= (z_{−i+1}z_{−i}b_{−i})(z_{i−1}b_{i})^{−1} = z_{−i−1}z_{i−1}b_{−i}b_{i}^{−1} = β_{4}^{∗}(b_{−i}b_{i}^{−1}) = β_{4}^{∗}([τ , b_{i}]).

We will show in Lemma 7.4 that Out(T^{∗}) ≅ C_{2}^{4} with the elements [β_{1}^{∗}], …, [β_{4}^{∗}] as generators.

Next let γ ∈ Aut(R_{1}) be the automorphism of order 3 where γ([[R, S, T]]) = [[T, R, S]] for R, S, T ∈ Q_{16}. Thus γ(a_{i}) = a_{i+1} and γ(b_{i}) = b_{i+1}, with indices taken modulo 3, and γ(c) = c. For all i = 1, 2, 3, 4, set

β_{i} = β_{i}^{∗}|_{R_{1}} and β′_{i} = γβ_{i}γ^{−1}.

Thus β′_{4} = β_{4}. We will see in the next lemma that as subgroups of Out(R_{1}), ⟨[β_{i}], [β′_{i}]⟩ ≅ C_{2}^{2} for i = 1, 2, 3, ⟨[β_{4}]⟩ = ⟨[β′_{4}]⟩ ≅ C_{2}, and each of these is normalized by γ.

For use in the following lemma, we define Q_{i} = ⟨a_{i}, b_{i}⟩ ≤ T^{∗} for i = 1, 2, 3. Thus Q_{1} ≅ Q_{2} ≅ Q_{3} ≅ Q_{8}, R_{0} = Q_{1}Q_{2}Q_{3}, the Q_{i} commute pairwise with each other, and the inclusions Q_{i} ≤ R_{0} define an isomorphism R_{0} ≅ (Q_{8})^{3}/C_{2}. Also, we set

Aut^{0}(R_{0}) = {α ∈ Aut(R_{0}) | α|_{Z(R_{0})} = Id} and Out^{0}(R_{0}) = Aut^{0}(R_{0})/Inn(R_{0}).

Lemma 7.4. (a) If P ≤ T^{∗} and P ≅ R_{0}, then P = R_{0}.

(b) Each α ∈ Aut^{0}(R_{0}) sends each of the subgroups Q_{i}Z(R_{0}) (i = 1, 2, 3) to itself.

(c) Out^{0}(R_{0}) ≅ (Σ_{4})^{3}, and Out(R_{0}) = Out^{0}(R_{0}) ⋊ ⟨[γ], c_{τ}⟩ ≅ Σ_{4} ≀ Σ_{3}. The identification of Σ_{4} with the group of automorphisms of Q_{i}Z(R_{0}) ≅ Q_{8} × C_{2} which are the identity on its center is induced by the action of this automorphism group on the set of the four subgroups of Q_{i}Z(R_{0}) isomorphic to Q_{8}.

(d) Out(R_{1}) = ⟨[β_{1}], [β_{2}], [β_{3}], [β_{4}], [β′_{1}], [β′_{2}], [β′_{3}]⟩⋊⟨[γ], c_{τ}⟩ ≅ C_{2}^{7}⋊Σ_{3}. The conjugation action in Out(R_{1}) of ⟨[γ], c_{τ}⟩ ≅ Σ_{3} on ⟨[β_{i}], [β′_{i}]⟩ ≅ C_{2}^{7} is the following: [c_{τ}, [β_{i}]] = 1 for all i; [[γ], [β_{4}]] = 1; and for i = 1, 2, 3, γβ_{i}γ^{−1} = β′_{i} and γβ′_{i}γ^{−1} ≡ c_{τ}β′_{i}c_{τ}^{−1} ≡ β_{i}β′_{i} (mod Inn(R_{1})).

(e) For k = 2, 3, restriction to R_{0} induces an isomorphism

Out(R_{k}) ⟶^{≅} C_{Out^{0}(R_{0})}(Out_{R_{k}}(R_{0})) ≅ \begin{cases}Σ_{4} × Σ_{4} & if k = 2 \\ Σ_{4} × C_{2}^{2} & if k = 3. \end{cases}

(f) Out(T^{∗}) = ⟨[β_{1}^{∗}], [β_{2}^{∗}], [β_{3}^{∗}], [β_{4}^{∗}]⟩ ≅ C_{2}^{4}. Every automorphism of R_{1} which commutes with c_{τ} in Out(R_{1}) extends to an automorphism of T^{∗}, and every automorphism in Aut^{0}(R_{0}) which commutes with c_{τ} and c_{c} in Out(R_{0}) extends to an automorphism of T^{∗}.

Proof. (a) Let P ≤ T^{∗} be any subgroup isomorphic to R_{0}. Set A = ⟨a_{1}, a_{2}, a_{3}, c⟩ ≅ C_{4}^{3}. For each H ◁ R_{0}, either ⟨z_{1}, z_{2}⟩ ≤ H and R_{0}/H is elementary abelian; or z_{i} ∉ H for some i, Q_{i} ∩ H = 1, and so R_{0}/H contains a subgroup ≅ Q_{8}. Hence for each H ◁ P, either H ≥ Z(P) and P/H is elementary abelian, or P/H contains a subgroup ≅ Q_{8}. When H = P ∩A, then P/H ≅ P A/A is contained in T^{∗}/A ≅ C_{2}×D_{8} (Lemma 7.1(a)), which contains no subgroup isomorphic to Q_{8}. We conclude that Z(P) ≤ H ≤ A, and P A/A is elementary abelian. Also, P ≱ A, since R_{0} contains no subgroup C_{4}^{3}. Thus P A/A ≅ C_{2}^{k}, k ≤ 3 since T^{∗}/A ≅ C_{2} × D_{8}, |P ∩ A| = 2^{8−k} ≤ 2^{5}; and hence k = 3 and P ∩ A ≅ C_{4}^{2} × C_{2}.

Thus either P A/A = ⟨b_{1}, b_{2}, b_{3}⟩ or P A/A = ⟨b_{1}b_{2}, b_{3}, τ ⟩. In either case, b_{3}g ∈ P for some g ∈ A. Hence Z(P) ≤ C_{A}(b_{3}g) = C_{A}(b_{3}) = ⟨a_{1}, a_{2}⟩, and so Z(P) = ⟨z_{1}, z_{2}⟩ = Z(R_{0}) since it is 2-torsion. Thus P ≤ C_{T^{∗}}(⟨z_{1}, z_{2}⟩) = R_{1}. Also,

R_{1}/Z(R_{0}) = ⟨a_{1}, a_{2}, a_{3}, c, b_{1}, b_{2}, b_{3}⟩/Z(R_{0}) = (R_{0}/Z(R_{0}))·⟨c⟩ ≅ C_{2}^{6}⋊C_{2},

and c acts on R_{0}/Z(R_{0}) centralizing only ⟨a_{1}, a_{2}, a_{3}⟩. Hence R_{0}/Z(R_{0}) is the unique elementary abelian subgroup of rank six in R_{1}/Z(R_{0}), so P = R_{0}, and this proves (a).

(b) Fix α ∈ Aut^{0}(R_{0}). Since (α(a_{1}))^{2} = α(a_{1}^{2}) = z_{1}, either α(a_{1}) ∈ Q_{1}Z(R_{0}), or α(a_{1}) = x_{2}x_{3} for some x_{i} ∈ Q_{i} (i = 2, 3) of order 4. In this last case, [x_{2}x_{3}, Q_{i}] = [x_{i}, Q_{i}] = ⟨z_{i}⟩ for i = 2, 3, so [α(a_{1}), R_{0}] = Z(R_{0}). This is impossible, since [a_{1}, R_{0}] = ⟨z_{1}⟩ has order two, and we conclude that α(a_{1}) ∈ Q_{1}Z(R_{0}). Similar arguments show that α(a_{i}), α(b_{i}) ∈ Q_{i}Z(R_{0}) for each i = 1, 2, 3, and thus α(Q_{i}) ≤ Q_{i}Z(R_{0}).

(c) By (b), each α ∈ Aut^{0}(R_{0}) leaves invariant each of the subgroups Q_{i}Z(R_{0}) for i = 1, 2, 3. The image of Out^{0}(R_{0}) under the projection to Aut(R_{0}/Z(R_{0})) is thus the group of automorphisms of C_{2}^{2} × C_{2}^{2} × C_{2}^{2} which send each factor to itself, and hence is isomorphic to (Σ_{3})^{3}. The group of automorphisms of R_{0} which induce the identity on R_{0}/Z(R_{0}) (and hence also on Z(R_{0})) is isomorphic to Hom(R_{0}/Z(R_{0}), Z(R_{0})) ≅ C_{2}^{12}, and this group contains Inn(R_{0}) ≅ R_{0}/Z(R_{0}) ≅ C_{2}^{6}. We thus have an extension

1 ⟶ C_{2}^{6} ⟶ Out^{0}(R_{0}) ⟶ Σ_{3} × Σ_{3} × Σ_{3} ⟶ 1.

In particular, |Out^{0}(R_{0})| = 2^{6}·6^{3} = 24^{3}.

We now make this more explicit. For each i = 1, 2, 3, define

Q_{i1} = Q_{i} = ⟨a_{i}, b_{i}⟩,  Q_{i2} = ⟨a_{i}z_{j}, b_{i}⟩,  Q_{i3} = ⟨a_{i}, b_{i}z_{j}⟩,  Q_{i4} = ⟨a_{i}z_{j}, b_{i}z_{j}⟩

for any j ≠ i; these are the four subgroups of Q_{i}Z(R_{0}) isomorphic to Q_{8}. Let

ω: Out^{0}(R_{0}) ⟶^{≅} Σ_{4} × Σ_{4} × Σ_{4}

be the isomorphism which sends [α], for α ∈ Aut^{0}(R_{0}), to the triple of permutations of the Q_{ik} induced by α. Thus, for example,

ω(β_{2}|_{R_{0}}) = (I, I, (2 4))  ω(β′_{2}|_{R_{0}}) = ((2 4), I, I)
ω(β_{3}|_{R_{0}}) = ((1 3)(2 4), (1 3)(2 4), I)  ω(β′_{3}|_{R_{0}}) = (I, (1 3)(2 4), (1 3)(2 4))
ω(β_{4}|_{R_{0}}) = ((1 3)(2 4), (1 3)(2 4), (1 3)(2 4))  ω(c_{c}) = ((2 4), (2 4), (2 4)) .

If α is such that [α] ∈ Ker(ω), then α sends each Q_{i} to itself via the identity modulo Z(Q_{i}) = ⟨z_{i}⟩. Thus α|_{Q_{i}} ∈ Inn(Q_{i}) for each i, and α ∈ Inn(R_{0}). We conclude that ω is injective, and hence an isomorphism since the source and target both have order 24^{3}.

Since ⟨γ|_{Z(R_{0})}, c_{τ}|_{Z(R_{0})}⟩ = Aut(Z(R_{0})), Aut(R_{0}) = Aut^{0}(R_{0}) ⋊ ⟨γ, c_{τ}⟩, and similarly for Out(R_{0}). Hence ω extends to an isomorphism Out(R_{0}) ⟶^{≅} Σ_{4} ≀ Σ_{3}; for example, by regarding Σ_{4} ≀ Σ_{3} as a group of permutations of the twelve subgroups Q_{ik}.

(d) By Lemma 1.2 (and by (a)), there is an exact sequence

1 ⟶ H^{1}(R_{1}/R_{0}; Z(R_{0})) ⟶^{η} Out(R_{1}) ⟶^{Res_{R_{0}}} C_{Out(R_{0})}(⟨c_{c}⟩)/⟨c_{c}⟩. (8)

Since [c, Z(R_{0})] = 1, H^{1}(R_{1}/R_{0}; Z(R_{0})) = Hom(⟨c⟩, Z(R_{0})) ≅ (ℤ/2)^{2}. Hence Im(η) = ⟨[β_{1}], [β′_{1}]⟩: since [β_{1}] = η(c ↦ z_{3}) (recall β_{1}(c) = z_{3}c and β_{1}|_{R_{0}} = Id), and similarly [β′_{1}] = η(c ↦ z_{1}). From the above table of values of ω(−), we see that

C_{Out^{0}(R_{0})}(c_{c}) = ω^{−1}({(σ_{1}, σ_{2}, σ_{3}) | σ_{i} ∈ ⟨(1 3), (2 4)⟩})
= ⟨β_{2}|_{R_{0}}, β′_{2}|_{R_{0}}, β_{3}|_{R_{0}}, β′_{3}|_{R_{0}}, β_{4}|_{R_{0}}, c_{c}⟩ ≅ C_{2}^{6}.

Hence C_{Out(R_{0})}(c_{c})/⟨c_{c}⟩ ≅ C_{2}^{5}⋊Σ_{3} is generated by the classes of the restrictions of the β_{i} (i = 2, 3, 4), β′_{i} (i = 2, 3), γ, and c_{τ} (recall ⟨γ, c_{τ}⟩ ≅ Σ_{3}). So by (8), Out(R_{1}) is generated by these elements together with [β_{1}] and [β′_{1}].

In particular, this shows that the subgroup A = ⟨a_{1}, a_{2}, a_{3}, c⟩ is characteristic in R_{1} (in fact, it is the only subgroup of T^{∗} isomorphic to C_{4}^{3}). So by Lemma 1.2 again, there is an exact sequence

1 ⟶ H^{1}(R_{1}/A; A) ⟶ Out(R_{1}) ⟶^{Res_{A}} N_{Aut(A)}(Aut_{R_{1}}(A))/Aut_{R_{1}}(A) ,

where Res_{A} is induced by restriction to A. Hence Ker(Res_{A}) = ⟨β_{2}, β_{3}, β_{4}, β′_{2}, β′_{3}⟩ is abelian and normal in Out(R_{1}). Since ⟨β_{1}, β′_{1}⟩ is also normal and abelian, this proves that the subgroup of Out(R_{1}) generated by all seven automorphisms β_{i} and β′_{i} is abelian and normal. Also, this subgroup has exponent two: (β_{2})^{2} = c_{a_{3}}, (β′_{2})^{2} = c_{a_{1}}, and the others have order 2 in Aut(R_{1}).

Thus Out(R_{1}) ≅ C_{2}^{7}⋊Σ_{3}. The description of the action of ⟨γ, c_{τ}⟩ ≅ Σ_{3} on the normal subgroup ⟨β_{i}, β′_{i} | 1 ≤ i ≤ 4⟩ ≅ C_{2}^{7} follows from the splitting of this group as a product of two normal subgroups, together with the fact that the factor ⟨β_{2}, β_{3}, β_{4}, β′_{2}, β′_{3}⟩ is sent injectively to (Σ_{4})^{3} via ω.

(e) Fix k = 2, 3. Set x_{2} = τ and x_{3} = cτ, so R_{k} = ⟨R_{0}, x_{k}⟩. Then R_{0} is characteristic in R_{k} by (a), and Z(R_{0}) = ⟨z_{1}, z_{2}⟩ is free as an module over Out_{R_{k}}(R_{0}) = ⟨c_{x_{k}}⟩. By Corollary 1.3, restriction to R_{0} induces an isomorphism

Out(R_{k}) ⟶_{≅}^{Res_{R_{0}}} C_{Out(R_{0})}(⟨c_{x_{k}}⟩)/⟨c_{x_{k}}⟩.

When k = 2 and x_{k} = τ, then c_{τ} ∈ Out(R_{0}) is the element which exchanges two of the factors Σ_{4}. Hence

Out(R_{2}) ≅ C_{Out(R_{0})}(c_{τ})/⟨c_{τ}⟩ ≅ C_{Out^{0}(R_{0})}(c_{τ}) ≅ {(α, α, β) | α, β ∈ Σ_{4}} ≅ Σ_{4} × Σ_{4} .

When k = 3 and x_{k} = cτ, then the image of c_{cτ} ∈ Out(R_{0}) in Σ_{4} ≀ Σ_{3} is the product of the triple ω(c) = ((2 4), (2 4), (2 4)) with the transposition of the first two factors Σ_{4}. Thus

Out(R_{3}) ≅ C_{Out(R_{0})}(c_{cτ})/⟨c_{cτ}⟩ ≅ C_{Out^{0}(R_{0})}(c_{cτ})
≅ {(α, (2 4)α(2 4), β) | α ∈ Σ_{4}, β ∈ ⟨(1 3), (2 4)⟩} ≅ Σ_{4} × C_{2}^{2} .

(f) Now, T^{∗}/R_{1} = ⟨τ ⟩ ≅ C_{2}, and c_{τ} acts on Z(R_{1}) = ⟨z_{1}, z_{2}⟩ by exchanging z_{1} and z_{2}. So by Corollary 1.3, restriction to R_{1} induces an isomorphism

Out(T^{∗}) ≅ N_{Out(R_{1})}(Out_{T^{∗}}(R_{1}))/Out_{T^{∗}}(R_{1}) = C_{Out(R_{1})}(c_{τ})/⟨c_{τ}⟩ .

By (d), this centralizer is generated by the β_{i} and c_{τ}, and so Out(T^{∗}) ≅ C_{2}^{4} is generated by the β_{i}^{∗}. This also proves that every automorphism of R_{1} which commutes with c_{τ} in Out(R_{1}) extends to T^{∗}.

Finally, if β ∈ Aut^{0}(R_{0}) is such that [β] commutes with c_{τ} and c_{c}, then by the computations of ω(−) in the proof of (c), ω([β]) = (σ_{1}, σ_{2}, σ_{3}) where σ_{1} = σ_{2} and σ_{i} ∈ ⟨(1 3), (2 4)⟩. Hence [β] is in the subgroup generated by the classes of β_{2}|_{R_{0}}, β_{3}|_{R_{3}}, β_{4}|_{R_{0}}, and c_{c}, and so β extends to an automorphism of T^{∗}. □

For i = 1, 2, 3, let η_{i} ∈ Aut^{0}(R_{0}) denote the automorphism of order 3: η_{i}(a_{i}) = b_{i}, η_{i}(b_{i}) = a_{i}b_{i}, and η_{i} fixes a_{j} and b_{j} for j ≠ i. Also, set γ_{0} = γ|_{R_{0}}: the automorphism which permutes the subgroups ⟨a_{i}, b_{i}⟩ cyclically. Thus ⟨[η_{1}], [η_{2}], [η_{3}]⟩ ≅ C_{3}^{3} is a Sylow 3-subgroup of Out^{0}(R_{0}), and ⟨[η_{1}], [η_{2}], [η_{3}], [γ_{0}]⟩ ≅ C_{3} ≀ C_{3} is a Sylow 3-subgroup of Out(R_{0}).

Define η_{12}^{(2)}, η_{3}^{(2)} ∈ Aut(R_{2}) by setting

η_{12}^{(2)}|_{R_{0}} = η_{1}η_{2},  η_{3}^{(2)}|_{R_{0}} = η_{3},  and η_{12}^{(2)}(τ) = η_{3}^{(2)}(τ) = τ .

These are well defined automorphisms since η_{1}η_{2} and η_{3} both commute with c_{τ} in Aut(R_{0}).

Let η^{(3)} ∈ Aut(R_{3}) be any automorphism such that η^{(3)}|_{R_{0}} = η_{1}η_{2}^{−1}. The existence of such an automorphism, and its uniqueness modulo Inn(R_{3}), follows from Lemma 7.4(e) once we check that c_{cτ} commutes with η_{1}η_{2}^{−1} in Out(R_{0}). Since each of the automorphisms η_{1}η_{2}^{−1} and c_{cτ}η_{1}η_{2}^{−1}c_{cτ}^{−1} sends each subgroup Q_{i} to itself, it suffices to show that they induce the same maps on each group Q_{i}/⟨z_{i}⟩, and this is easily checked. Alternatively, under the explicit isomorphism ω: Out^{0}(R_{0}) ⟶^{≅} (Σ_{4})^{3} defined in the proof of Lemma 7.4, they are both sent to ((4 3 2), (2 3 4), I). In fact, by a direct (and long) computation, one can show that η^{(3)} can be chosen such that η^{(3)}(cτ) = ca_{1}^{−1}b_{2}τ, but that will not be needed here.

Proposition 7.5. Let F be a saturated fusion system over T^{∗} for which R_{1} is Fessential. Then |Out_{F}(R_{0})| = 4·3^{n} for some 1 ≤ n ≤ 4, and we say F has “Type n”. Also, F is isomorphic to a fusion system F′ over T^{∗} for which the automorphism groups Out_{F′}(R_{i}) (i = 0, 1, 2, 3) are as described in the following table:

Out_{F′}(R_{0})  Out_{F′}(R_{1}) Out_{F′}(R_{2}) Out_{F′}(R_{3})
Type 1  ⟨c_{c}, [γ_{0}], c_{τ}⟩  ⟨[γ], c_{τ}⟩  ⟨c_{c}⟩  ⟨c_{c}⟩
Type 2  ⟨[η_{1}η_{2}η_{3}], c_{c}, [γ_{0}], c_{τ}⟩  ⟨[γ], c_{τ}⟩ ⟨[η_{12}^{(2)}η_{3}^{(2)}], c_{c}⟩  ⟨c_{c}⟩
Type 3 ⟨[η_{1}η_{2}^{−1}], [η_{2}η_{3}^{−1}], c_{c}, [γ_{0}], c_{τ}⟩ ⟨[γ], c_{τ}⟩ ⟨[η_{12}^{(2)}η_{3}^{(2)}], c_{c}⟩ ⟨[η^{(3)}], c_{c}⟩
Type 4 ⟨[η_{1}], [η_{2}], [η_{3}], c_{c}, [γ_{0}], c_{τ}⟩ ⟨[γ], c_{τ}⟩ ⟨[η_{12}^{(2)}], [η_{3}^{(2)}], c_{c}⟩ ⟨[η^{(3)}], c_{c}⟩

If F has type 1 or 2, then V =^{def} ⟨z_{1}, z_{2}, a_{1}a_{2}a_{3}, b_{1}b_{2}b_{3}⟩ is Out_{F′}(R_{0})-invariant.

Proof. Since Out_{T^{∗}}(R_{1}) = ⟨c_{τ}⟩ ∈ Syl_{2}(Out_{F}(R_{1})) intersects trivially with O_{2}(Out(R_{1})) by Lemma 7.4(d), Out_{F}(R_{1}) is sent injectively under projection to the quotient group Out(R_{1})/O_{2}(Out(R_{1})) ≅ Σ_{3}; and since R_{1} is F-essential, it is sent isomorphically. Thus

Out(R_{1}) = O_{2}(Out(R_{1}))·Out_{F}(R_{1}) = O_{2}(Out(R_{1}))·⟨[γ], c_{τ}⟩.

So by Proposition 1.8, and since O_{2}(Out(R_{1})) is abelian by Proposition 7.4(d), there is β_{1} ∈ O_{2}(Aut(R_{1})) which commutes in Out(R_{1}) with c_{τ} and such that Out_{F}(R_{1}) = ⟨[β_{1}γβ_{1}^{−1}], c_{τ}⟩. Any such β_{1} extends to an automorphism β of T^{∗} by Lemma 7.4(f). So upon replacing F by β^{−1}Fβ, we can assume Out_{F}(R_{1}) = ⟨[γ], c_{τ}⟩. In particular, γ_{0} ∈ Aut_{F}(R_{0}).

Consider the following subgroups of Out(R_{0}):

Q = O_{2}(Out(R_{0})),  H_{0} = ⟨c_{c}, c_{τ}, [γ_{0}]⟩,  Ĥ = ⟨[η_{1}], [η_{2}], [η_{3}], H_{0}⟩;
H^{∗} = Out_{F}(R_{0}) ≥ H_{0}, and H = H^{∗}Q ∩ Ĥ.

By Lemma 7.4(c), Out(R_{0})/Q ≅ Σ_{3} ≀ Σ_{3}, and hence ⟨Q, [η_{1}], [η_{2}], [η_{3}], [γ_{0}]⟩/Q is its only Sylow 3-subgroup which contains the class of γ_{0}. Since [γ_{0}] ∈ Out_{F}(R_{0}) = H^{∗}, H^{∗} is generated by H_{0} together with its Sylow 3-subgroup, and hence is contained in ĤQ. Thus H^{∗}Q ≤ ĤQ, and so H^{∗}Q = HQ. We are now in the situation of Proposition 1.8: there is φ_{0} ∈ Aut(R_{0}) such that [φ_{0}] ∈ C_{Q}(H_{0}) and φ_{0}H^{∗}φ_{0}^{−1} = H. By Lemma 7.4(f), φ_{0} extends to some φ ∈ Aut(T^{∗}). So upon replacing F by φFφ^{−1}, we can arrange that

⟨c_{c}, c_{τ}, [γ_{0}]⟩ ≤ Out_{F}(R_{0}) = H ≤ Ĥ = ⟨c_{c}, c_{τ}, [η_{1}], [η_{2}], [η_{3}], [γ_{0}]⟩.

The only proper γ_{0}-invariant subgroups of ⟨[η_{1}], [η_{2}], [η_{3}]⟩ ≅ C_{3}^{3} are ⟨[η_{1}η_{2}^{−1}], [η_{2}η_{3}^{−1}]⟩ and ⟨[η_{1}η_{2}η_{3}]⟩. Hence Out_{F}(R_{0}) must now be one of the four groups listed in the above table. Also, for i = 2, 3, Out_{F}(R_{i}) is determined by Out_{F}(R_{0}) as described in that table: for each β ∈ Aut_{F}(R_{0}) such that [β] is Out_{R_{i}}(R_{0})-invariant, β extends to an element β^{∗} ∈ Aut(R_{i}) (unique modulo Inn(R_{i})) by Proposition 7.4(e), and β^{∗} ∈ Aut_{F}(R_{i}) by the extension axiom. The last statement (about the subgroup V) follows since V is normal in T^{∗}, and is left invariant by γ_{0} and by η_{1}η_{2}η_{3}. □

We next look at automorphisms of Q, and of the subgroups R_{4} and H_{i} which contain Q. Set Q = Q/Z(Q) for short. Let 𝔮: Q ⟶ 𝔽_{2} be the quadratic form where for any x̄ = xZ(Q) ∈ Q, 𝔮(x̄) = 0 if x^{2} = 1 and 𝔮(x̄) = 1 if x^{2} = z_{3}. Since Inn(Q) is the group of all automorphisms of Q which are the identity modulo Z(Q), Out(Q) ≅ GO(Q, 𝔮).

We want to choose an explicit isomorphism Out(Q) ≅ GO_{6}^{+}(2) ≅ Σ_{8}. Let P_{e}(\underline{8}) be the group of subsets of even order in \underline{8} = \{1, 2, …, 8\}, regarded as an 𝔽_{2}-vector space with addition given by symmetric difference X + Y = ((X∖Y) ∪ (Y ∖X)). Let 𝔮 be the quadratic form on P_{e}(\underline{8})/⟨\underline{8}⟩ defined by 𝔮(X) = \frac{1}{2}|X|, associated to the bilinear form 𝔟(X, Y) = |X ∩ Y |. The induced action of Σ_{8} on P_{e}(\underline{8})/⟨\underline{8}⟩ preserves the form, and thus defines a homomorphism Σ_{8} ⟶ SO(P_{e}(\underline{8})/⟨\underline{8}⟩, 𝔮) which is injective by the simplicity of A_{8} and hence an isomorphism by counting.

Define κ: Q ⟶ P_{e}(\underline{8})/\underline{8} by setting

κ(a_{1}a_{2}) = {34},  κ(a_{3}) = {56},  κ(z_{1}) = {1234} = {5678},
κ(b_{1}b_{2}) = {24},  κ(b_{3}) = {57}  κ(τ) = {1567} = {2348} .

This is motivated by the homomorphism ρ: T^{∗} ⟶ Ω_{7}(3) defined at the beginning of the section: ρ(Q) is the group of diagonal matrices, and κ sends the class of g ∈ Q to the set of positions where ρ(g) has (−1) on the diagonal. So assuming ρ lifts to a homomorphism T^{∗} ⟶ Spin_{7}(3), κ preserves the quadratic forms by standard commutator and squaring relations in the spinor groups (cf. [LO, Lemma A.4]). However, it is much easier to check this directly, by comparing values of the quadratic forms and associated bilinear forms on the basis used above to define κ.

Thus κ induces an isomorphism

χ_{Q} : Out(Q) ≅ GO(Q, 𝔮) ⟶_{≅}^{κ_{∗}} GO(P_{e}(\underline{8})/\underline{8}; 𝔮) ⟵^{≅} Σ_{8}.

To simplify notation, we also regard χ_{Q} as a homomorphism on Aut(Q).

The images under χ_{Q} of automorphisms in Out_{T^{∗}}(Q), and also of the restrictions of η_{12}^{(2)}, η_{3}^{(2)} ∈ Aut(R_{2}), are given in the following table:

\begin{matrix}α & c_{a_{1}} & c_{c} & c_{b_{1}} & η_{12}^{(2)}|_{Q} & η_{3}^{(2)}|_{Q} \\ χ_{Q}(α) & (1 2)(3 4) & (3 4)(5 6) & (1 3)(2 4) & (2 3 4) & (5 7 6) \end{matrix}  (9)

For example, c_{a_{1}} sends b_{1}b_{2} to z_{1}b_{1}b_{2}, sends τ to a_{1}a_{2}^{−1}τ = a_{1}a_{2}z_{2}τ, and sends all of the other generators listed above to themselves. Hence χ_{Q}(c_{a_{1}}) ∈ Σ_{8} sends {24} to {13}, sends {1567} to κ(z_{1}a_{1}a_{2}τ) = {2567} (note z_{2} ≡ z_{1} (mod Z(Q))), and sends each of {34}, {56}, {57}, and {1234} to itself. So χ_{Q}(c_{a_{1}}) = (1 2)(3 4).

We now apply this to describe the automorphisms of R_{4}. In order to “see” better the symmetry of this subgroup, we give it the following, alternative description.

Define

𝕊′ = ⟨z, r_{1}, r_{2}, r_{3}, s_{1}, s_{2}, s_{3} | z^{2} = 1, r_{i}^{4} = z, s_{i}^{2} = z, s_{i}r_{i}s_{i}^{−1} = r_{i}^{−1};

[r_{i}, r_{j}] = 1, [s_{i}, r_{j}] = 1, [s_{i}, s_{j}] = z for all i ≠ j⟩

Thus 𝕊′ is generated by the three subgroups ⟨r_{i}, s_{i}⟩ ≅ Q_{16} (i = 1, 2, 3), which intersect in Z(𝕊′) = ⟨z⟩, and whose cyclic subgroups of order 8 commute with each other. This “twisted” product corresponds to the lifting to Spin_{7}(9) of three copies of GO_{2}^{−}(3) ≅ D_{8} in SO_{7}(3) ≤ Ω_{7}(9).

Define an embedding ψ: R_{4} ⟶ 𝕊′ by setting

ψ(a_{1}) = r_{1}^{−1}r_{2}  ψ(a_{2}) = r_{1}r_{2}  ψ(a_{3}) = r_{3}^{2}  ψ(c) = r_{2}r_{3}
ψ(b_{1}b_{2}) = r_{1}^{2}r_{2}^{2}s_{1}s_{2}  ψ(b_{3}) = s_{3}  ψ(τ) = r_{3}^{2}s_{1} .

Thus

ψ(R_{4}) = ⟨r_{1}^{2}, r_{1}r_{2}, r_{1}r_{3}, s_{1}, s_{2}, s_{3}⟩ and ψ(Q) = ⟨r_{1}^{2}, r_{2}^{2}, r_{3}^{2}, s_{1}, s_{2}, s_{3}⟩.

Also, ψ([[Y^{i}, Y^{j}, Y^{k}]]) = r_{1}^{(j−i)/2}r_{2}^{(j+i)/2}r_{3}^{k} whenever i ≡ j ≡ k (mod 2); and

s_{1} = ψ(a_{3}^{−1}τ),  s_{2} = ψ(z_{1}a_{3}b_{1}b_{2}τ) = ψ(b_{1}(a_{3}^{−1}τ)b_{1}^{−1}),  and s_{3} = ψ(b_{3}).

To simplify notation, we identify elements of R_{4} with their images under ψ. Thus

κ(r_{1}^{2}) = {12},  κ(r_{2}^{2}) = {34},  κ(r_{3}^{2}) = {56},
κ(s_{1}) = {17},  κ(s_{2}) = {37},  κ(s_{3}) = {57}.

Define ε_{1}, ε_{2}, ε_{3}, θ_{1}, θ_{2}, ξ ∈ Aut(R_{4}) to be the restrictions of the automorphisms of 𝕊′ defined in the following table, where, σ, τ ∈ Σ_{3} are the permutations σ = (1 2 3) and τ = (1 2); and δ_{ij} = 1 if i = j and 0 if i ≠ j:

α  ε_{j}(j = 1, 2, 3)  θ_{1}  θ_{2}  ξ  c_{b_{1}a_{1}a_{2}}
α(r_{i})  z^{δ_{ij}}r_{i}  r_{i}  r_{i}  r_{σ(i)}  r_{τ(i)}
α(s_{i})  s_{i}  r_{i}^{2}s_{i}  r_{1}^{2}r_{2}^{2}r_{3}^{2}s_{i}  s_{σ(i)}  s_{τ(i)}
χ_{Q}(α|_{Q})  Id  (1 2)(3 4)(5 6) (7 8) (1 3 5)(2 4 6) (1 3)(2 4)

(10)

Lemma 7.6. Out(R_{4}) = (⟨[ε_{1}], [ε_{2}]⟩⋊⟨[ξ], c_{b_{1}}⟩)×⟨[θ_{1}], [θ_{2}]⟩ ≅ Σ_{4} ×C_{2}^{2}, where ε_{1}ε_{2}ε_{3} = Id_{R_{4}}. If F is a saturated fusion system over T^{∗} and R_{4} is F-essential, then Out_{F}(R_{4}) is one of the groups ⟨[ξ], c_{b_{1}}⟩ or ⟨[ε_{3}ξε_{3}^{−1}], c_{b_{1}}⟩, both isomorphic to Σ_{3}. In either case, the image under χ_{Q} of the restriction to Q of Aut_{F}(R_{4}) is generated by χ_{Q}(Out_{T^{∗}}(Q)) and (1 3 5)(2 4 6).

Proof. By Lemma 7.1(b), each automorphism of R_{4} leaves Q invariant. Hence by Lemma 1.2, there is an exact sequence

1 ⟶ H^{1}(R_{4}/Q; Z(Q)) ⟶^{η} Out(R_{4}) ⟶^{Res_{Q}} N_{Out(Q)}(Out_{R_{4}}(Q))/Out_{R_{4}}(Q).

Also, since R_{4}/Q = ⟨r_{1}r_{2}, r_{2}r_{3}⟩ ≅ C_{2}^{2} and Z(Q) = ⟨z⟩ ≅ C_{2},

H^{1}(R_{4}/Q; Z(Q)) ≅ Hom(R_{4}/Q, Z(Q)) ≅ C_{2}^{2},

and η sends a homomorphism φ to the class of the automorphism (g ↦ g·φ(gQ)). Thus Ker(Res_{Q}) = ⟨[ε_{1}], [ε_{2}]⟩.

By (9), χ_{Q}(Out_{R_{4}}(Q)) = ⟨(1 2)(3 4), (3 4)(5 6)⟩. The normalizer of this subgroup is the group of all permutations which leave {7, 8} invariant, and permute the three subsets {1, 2}, {3, 4}, {5, 6}. Hence

N_{Out(Q)}(Out_{R_{4}}(Q))/Out_{R_{4}}(Q) ≅ Σ_{3} × C_{2}^{2},

where the direct factor C_{2}^{2} is represented by the permutations (1 2)(3 4)(5 6) = χ_{Q}(θ_{1}) and (7 8) = χ_{Q}(θ_{2}) (see (10)). Also, (1 3 5)(2 4 6) = χ_{Q}(ξ) and (1 3)(2 4) = χ_{Q}(c_{b_{1}}) represent generators of the first factor. This proves that Res_{Q} in the above exact sequence is onto, and also shows that Out(R_{4}) is generated by the classes of ε_{1}, ε_{2}, θ_{1}, θ_{2}, ξ, and c_{b_{1}}.

The exact structure of this extension follows from (10) (and the relation ε_{1}ε_{2}ε_{3} = Id). Also, ⟨θ_{1}, θ_{2}⟩ is the subgroup of elements which restrict to the identity on A = ⟨a_{1}, a_{2}, a_{3}, c⟩ = ⟨r_{1}^{2}, r_{1}r_{2}, r_{1}r_{3}⟩, and hence is normal in Out(R_{4}). Thus the subgroup ⟨[ε_{1}], [ε_{2}], [θ_{1}], [θ_{2}]⟩ in Out(R_{4}) is elementary abelian, and ⟨[ξ], c_{b_{1}}⟩ ≅ Σ_{3} permutes the three involutions ε_{i} and acts trivially on ⟨[θ_{1}], [θ_{2}]⟩.

If R_{4} is F-essential for some saturated fusion system F over T^{∗}, then Out_{F}(R_{4}) = ⟨α, c_{b_{1}}⟩ ≅ Σ_{3} for some α ∈ ⟨[ε_{1}], [ε_{2}], [ξ], c_{b_{1}}⟩ ≅ Σ_{4} of order three which is normalized by c_{b_{1}}. Any transposition in Σ_{4} normalizes exactly two subgroups of order 3, and in this case, these are easily seen to be the subgroups ⟨[ξ]⟩ and ⟨[ε_{3}ξε_{3}^{−1}]⟩. □

It remains to examine the outer automorphism groups of the H_{i}.

Lemma 7.7. (a) For i = 1, 2, or 3, let α, α′ ∈ Aut(H_{i}) be two automorphisms of order three such that χ_{Q}(α|_{Q}) = χ_{Q}(α′|_{Q}) in Out(Q). Then [α] = [α′] in Out(H_{i}).

(b) If F is a saturated fusion system over T^{∗} and H_{1} and H_{2} are F-essential, then Out_{F}(H_{i}) = ⟨[α_{i}], c_{c}⟩ ≅ Σ_{3} (i = 1, 2) for some α_{i} ∈ Aut(H_{i}) of order 3 such that χ_{Q}(α_{1}|_{Q}) = (1 2 x) and χ_{Q}(α_{2}|_{Q}) = (3 4 x) for the same x = 7 or 8.

Proof. Set x_{1} = c, x_{2} = a_{1}c, and x_{3} = a_{1}; thus H_{i} = ⟨Q, x_{i}⟩ for i = 1, 2, 3. For each such i, Q is characteristic in H_{i} by Lemma 7.1(b), so there is a well defined homomorphism

Res_{i}: Out(H_{i}) ⟶ C_{Out(Q)}(c_{x_{i}})/⟨c_{x_{i}}⟩

induced by restriction, and Ker(Res_{i}) ≅ H^{1}(H_{i}/Q; Z(Q)) ≅ H^{1}(C_{2}; ℤ/2) has order 2 (Lemma 1.2). In particular, for any [α] ∈ Out(Q) of order three which centralizes c_{x_{i}}, its class in the quotient lifts to at most one element of order three in Out(H_{i}), and this proves (a).

By (9), χ_{Q}(c_{x_{i}}) = (3 4)(5 6) (i = 1), (1 2)(5 6) (i = 2), or (1 2)(3 4) (i = 3). Thus C_{Out(Q)}(c_{x_{i}})/⟨c_{x_{i}}⟩ ≅ C_{2}^{2} × Σ_{4} in all three cases, and |Out(H_{i})| = 2^{k} or 3·2^{k} for some k. By the Sylow axiom, for each i, |Out_{F}(H_{i})| = 2 or 6. So if H_{1} and H_{2} are F-essential, then for i = 1, 2, Out_{F}(H_{i}) = ⟨[α_{i}], c_{x_{3}}⟩ ≅ Σ_{3} for some α_{i} ∈ Aut(H_{i}) of order 3 such that [[α_{i}|_{Q}], c_{x_{i}}] = 1 in Out(Q). Thus χ_{Q}(α_{1}|_{Q}) commutes with χ_{Q}(c_{x_{1}}) = (3 4)(5 6) in Σ_{8}, and is also normalized by χ_{Q}(c_{x_{3}}) = (1 2)(3 4). This is possible only if χ_{Q}(α_{1}|_{Q}) = (1 2 x) for x = 7 or 8. Since H_{2} = b_{1}H_{1}b_{1}^{−1}, and χ_{Q}(c_{b_{1}}) = (1 3)(2 4) by (9) again, we can choose α_{2} = c_{b_{1}}α_{1}c_{b_{1}}^{−1}, in which case χ_{Q}(α_{2}|_{Q}) = (3 4 x). This proves (b). □

In fact, the homomorphisms Res_{i} in the above proof are surjective, and hence |Out(H_{i})| = 3·2^{6}. This can be shown for i = 3 by checking that C_{Out(Q)}(c_{a_{1}})/⟨c_{a_{1}}⟩ is generated by restrictions of automorphisms of R_{2} (those which leave H_{3} invariant). It then follows for i = 1, 2 since ξ ∈ Aut(R_{4}) permutes transitively the H_{i}. However, this will not be needed here.

7.3 Fusion systems over T^{∗} We are now ready to list the saturated fusion systems over T^{∗}.

Theorem 7.8. Every nonconstrained centerfree fusion system over T^{∗} is isomorphic to the fusion system of Sol(3), Co_{3}, or Aut(P Sp_{6}(3)).

Proof. Let F be a nonconstrained fusion system over T^{∗} such that z_{3} is not central in F. By Lemma 7.4(f), Out(T^{∗}) is a 2-group, and hence Out_{F}(T^{∗}) = 1. So by Proposition 7.3, all fusion in F is generated by fusion in R_{1}, R_{2}, R_{3}, R_{4}, H_{1}, and H_{2}. Since each of these except R_{1} has center ⟨z_{3}⟩, R_{1} must be F-essential, since otherwise z_{3} would be central in F.

By Proposition 7.5, there is a fusion system F′ over T^{∗} isomorphic to F, such that that the groups Out_{F′}(R_{i}) for i = 0, 1, 2, 3 are as in one of the four cases listed there. Assume for simplicity F′ = F. Then Out_{F}(R_{1}) = ⟨[γ], c_{τ}⟩, Out_{F}(R_{2}) ≤ ⟨[η_{12}^{(2)}], [η_{3}^{(2)}], c_{c}⟩, and Out_{F}(R_{0}) determines Out_{F}(R_{2}) and Out_{F}(R_{3}).

If all F-essential subgroups contain R_{0}, then R_{0} must be normal in F (Lemma 7.4(a)), which would contradict the assumption that F is nonconstrained. Hence either R_{4}, or H_{1} and H_{2}, are also F-essential. (Recall that H_{1} and H_{2} are conjugate in T^{∗}.) If R_{4} is essential, then H_{1}, H_{2}, and H_{3} are all F-conjugate by Lemma 7.6, since ξ and ε_{3}ξε_{3}^{−1} both permute them transitively. Since H_{3} ◁ T^{∗}, this implies neither H_{1} nor H_{2} is fully normalized in F, and hence neither is F-essential.

The cases where R_{4} is F-essential will be handled in Step 1, and the cases where H_{1} and H_{2} are F-essential in Step 2. Afterwards, the distinct (possible) fusion systems found in those two steps will be identified in Step 3.

Step 1: Assume R_{4} is F-essential. The automorphisms ξ, ε_{3} ∈ Aut(R_{4}) both leave invariant the subgroup

V = ⟨z_{1}, z_{2}, a_{1}a_{2}a_{3}, b_{1}b_{2}b_{3}⟩ = ⟨z, r_{1}^{2}r_{2}^{2}, r_{1}^{2}r_{3}^{3}, s_{1}s_{2}s_{3}⟩ ≅ C_{2}^{4},

and hence Out_{F}(R_{4}) leaves leaves V invariant by Lemma 7.6. Since we are assuming F is not constrained, V is not normal in F, and this implies Out_{F}(R_{0}) does not leave V invariant. So F must be of Type 3 or 4 by Proposition 7.5.

By Lemma 7.6 again, Out_{F}(R_{4}) is one of the two groups ⟨[ξ], c_{b_{1}}⟩ or ⟨[ε_{3}ξε_{3}^{−1}], c_{b_{1}}⟩. So we are reduced to at most four different possibilities for F. We claim that Out_{F}(R_{4}) = ⟨[ε_{3}ξε_{3}^{−1}], c_{b_{1}}⟩ is the only possibility, given our choice of Aut_{F}(R_{1}). This is closely related to [LO2, Lemma 1.2] (and to the error in [LO] which made a correction necessary), but we give a more direct argument here. Consider the subgroup A = ⟨a_{1}, a_{2}, a_{3}, c⟩ ≅ C_{4}^{3} of Lemma 7.1(a). For each α ∈ Aut(A), let M(α) ∈ GL_{3}(ℤ/4) be the matrix for α with respect to the ordered basis {a_{1}, a_{2}, c}. Note that Aut_{F}(A) is generated by restrictions of automorphisms in Aut_{F}(R_{4}) and in Aut_{F}(R_{1}) = ⟨Inn(R_{1}), γ, c_{τ}⟩. Then

M(ξ) = ( \begin{matrix}−1 & 0 & −1 \\ −1 & 0 & 0 \\ 1 & 1 & 1 \end{matrix} ) , M(ε_{3}ξε_{3}^{−1}) = ( \begin{matrix}1 & 2 & −1 \\ 1 & 2 & 0 \\ 1 & 1 & 1 \end{matrix} ) , M(γ) = ( \begin{matrix}0 & −1 & 0 \\ 1 & −1 & 0 \\ 0 & 2 & 1 \end{matrix} ) , M(ξγ) = ( \begin{matrix}0 & −1 & −1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{matrix} )

(where we drop “|_{A}” to simplify the notation); and M((ξγ)^{3}) = ( \begin{matrix}−1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 2 & −1 \end{matrix} ) ≡ Id (mod 2). Since this has order two, (ξγ)^{3} must be conjugate in Aut(A) to some element of the Sylow 2-subgroup Aut_{T^{∗}}(A) = ⟨c_{b_{1}}, c_{b_{2}}, c_{b_{3}}, c_{τ}⟩. But this is impossible, since the only element of Aut_{T^{∗}}(A) which is the identity modulo 2 is c_{b_{1}b_{2}b_{3}}, and M(c_{b_{1}b_{2}b_{3}}) = −Id. So ξ and γ cannot both be in Aut_{F}(A), and thus Out_{F}(R_{4}) = ⟨[ε_{3}ξε_{3}^{−1}], c_{b_{1}}⟩. (In fact, (ε_{3}ξε_{3}^{−1}γ|_{A})^{3} = 1.)

Thus F must be isomorphic to one of two fusion systems, which we denote F_{1} (of Type 3) and F_{2} (of Type 4). The restriction of Aut_{F}(R_{4}) to Q is generated by ξ|_{Q} (since ε_{3}|_{Q} = Id) and Aut_{T^{∗}}(Q). Hence if we let X_{0} ≤ Out_{F}(Q) be the subgroup generated by Out_{T^{∗}}(Q) and classes of restrictions of F-automorphisms of R_{4}, then by (9) and (10),

χ_{Q}(X_{0}) = ⟨(1 2)(3 4)_{χ_{Q}(c_{a_{1}})}, (3 4)(5 6)_{χ_{Q}(c_{c})}⟩ ⋊ ⟨(1 3 5)(2 4 6)_{χ_{Q}(ξ|_{Q})}, (1 3)(2 4)_{χ_{Q}(c_{b_{1}})}⟩.

Since χ_{Q}(Out_{F}(Q)) is generated by χ_{Q}(X_{0}) and restrictions of elements in Out_{F}(R_{2}), Proposition 7.5 and (9) imply

χ_{Q}(Out_{F}(Q)) = \begin{cases}⟨χ_{Q}(X_{0}), (2 3 4)(5 7 6)⟩ & if F = F_{1} \\ ⟨χ_{Q}(X_{0}), (2 3 4), (5 7 6)⟩ & if F = F_{2}. \end{cases}

Since χ_{Q}(X_{0}) contains all even permutations which fix 7 and 8 and permute the three subsets {1, 2}, {3, 4}, {5, 6}, ⟨χ_{Q}(X_{0}), (2 3 4)⟩ contains all even permutations which fix 7 and 8, and so χ_{Q}(Out_{F_{2}}(Q)) ≅ A_{7} is the group of all even permutations which fix 8.

An isomorphism χ_{Q}(Out_{F_{1}}(Q)) ≅ GL_{3}(2) is defined via the bijection \underline{8} ⟶^{≅} 𝔽_{2}^{3} which sends n ∈ \underline{8} to the three digits in the binary expansion of 8 − n. Thus 1 is sent to (1, 1, 1), 2 to (1, 1, 0), 8 to (0, 0, 0), etc.

Step 2: Now assume H_{1} and H_{2} are F-essential (and R_{4} is not). By Lemma 7.7(b), Out_{F}(H_{i}) ≅ Σ_{3} for i = 1, 2, and there are elements α_{i} ∈ Aut_{F}(H_{i}) of order three such that χ_{Q}(α_{1}|_{Q}) = (1 2 x) and χ_{Q}(α_{2}|_{Q}) = (3 4 x) for some fixed x ∈ {7, 8}. Thus χ_{Q}(Out_{F}(Q)) contains ⟨(1 2 x), (3 4 x)⟩, which is the group of all even permutations of the set {1, 2, 3, 4, x}.

Now, since χ_{Q}(η_{12}^{(2)}|_{Q}) = (2 3 4) (where η_{12}^{(2)} ∈ Aut(R_{2})), this implies η_{12}^{(2)}|_{Q} ∈ Aut_{F}(Q), and hence (by the extension axiom) that η_{12}^{(2)} (or some other automorphism with the same restriction) is in Aut_{F}(R_{2}). So F has Type 4 by Proposition 7.5, and χ_{Q}(Out_{F}(Q)) also contains χ_{Q}(η_{3}^{(2)}|_{Q}) = (5 7 6).

If x = 7, then χ_{Q}(Out_{F}(Q)) contains all even permutations which fix the element 8. In particular, it contains χ_{Q}(ξ|_{Q}), where ξ ∈ Aut(R_{4}) has order three (see Step 1). By the extension axiom, ξ|_{Q} ∈ Aut_{F}(Q) extends to an automorphism in Aut_{F}(R_{4}) of order 3, so R_{4} is F-essential, contradicting our original assumption.

Thus x = 8, and

χ_{Q}(Out_{F}(Q)) = ⟨(1 2)(3 4)_{χ_{Q}(c_{a_{1}})}, (1 3)(2 4)_{χ_{Q}(c_{b_{1}})}, (3 4)(5 6)_{χ_{Q}(c_{c})}, (1 2 8)_{χ_{Q}(α_{1})}, (3 4 8)_{χ_{Q}(α_{2})}, (2 3 4)_{χ_{Q}(η_{12}^{(2)})}, (5 7 6)_{χ_{Q}(η_{3}^{(2)})}⟩

is the group of all even permutations which leave the sets {1, 2, 3, 4, 8} and {5, 6, 7} invariant. By Lemma 7.7(a), [α_{i}] ∈ Out(H_{i}) is determined by χ_{Q}(α_{i}|_{Q}) ∈ Out(Q) for i = 1, 2. Hence there is only one fusion system satisfying these conditions, and we denote it by F_{3}.

Step 3: By the construction in [LO], T^{∗} is a Sylow 2-subgroup of Spin_{7}(3), and also of the exotic fusion system Sol(3) constructed there. The sporadic simple group Co_{3} contains 2·Sp_{6}(2) with odd index (cf. [Fi, Theorem 2]). The orthogonal group Ω_{7}(3) contains Sp_{6}(2) with odd index (as a subgroup of index two in the Weyl group of E_{7}), and hence Spin_{7}(3) contains 2·Sp_{6}(2) with odd index. Thus Co_{3} also has Sylow 2-subgroups isomorphic to T^{∗}.

Since Sp_{6}(9) contains the wreath product Sp_{2}(9) ≀ Σ_{3} with odd index, and since Sp_{2}(9) ≅ SL_{2}(9) has Sylow 2-subgroups isomorphic to Q_{16}, the group 𝕊 defined at the beginning of the section is isomorphic to a Sylow 2-subgroup of P Sp_{6}(9), and its subgroup R_{2} = ⟨R_{0}, τ ⟩ is isomorphic to a Sylow 2-subgroup of P Sp_{6}(3). The group Aut(P Sp_{6}(3)) is the extension of P Sp_{6}(3) by its diagonal automorphisms, hence the normalizer of P Sp_{6}(3) in P Sp_{6}(9), and contains P Sp_{6}(3) with index two. Its Sylow 2subgroup is thus isomorphic to a subgroup of index four in 𝕊 of the form ⟨P, τ ⟩ for some R_{0} ≤ P ≤ 𝕊_{0} which is invariant under the action of Σ_{3} permuting the central factors; and this can only be P = ⟨R_{0}, c⟩ = R_{1}. The Sylow 2-subgroups of Aut(P Sp_{6}(3)) are thus isomorphic to ⟨R_{1}, τ ⟩ = T^{∗} ≤ 𝕊.

We showed in Steps 1 and 2 that every nonconstrained centerfree saturated fusion system over T^{∗} is isomorphic to one of the fusion systems F_{1}, F_{2}, or F_{3}. Hence the fusion systems of Co_{3}, Sol(3), and Aut(P Sp_{6}(3)) must be among these, and it remains to identify them.

As shown in Steps 1 and 2, the F_{i} are distinguished by Out_{F_{i}}(Q):

Out_{F_{1}}(Q) ≅ GL_{3}(2), Out_{F_{2}}(Q) ≅ A_{7}, and Out_{F_{3}}(Q) ≅ (A_{5} × C_{3}) ⋊ C_{2}.

By Lemma 7.1(b), Q is the only subgroup of T^{∗} of order 2^{7} with quotient group C_{2}^{6}. Hence to determine Out_{G}(Q) for any finite group G with Sylow 2-subgroups isomorphic to T^{∗}, it suffices to find any subgroup C_{2}^{6} in C_{G}(z)/⟨z⟩ for some involution z in G, and determine its automorphism group.

The centralizer in Co_{3} of a Sylow central involution is isomorphic to 2·Sp_{6}(2) (cf. [Fi, Lemma 4.4]), and Sp_{6}(2) contains a maximal subgroup C_{2}^{6}⋊GL_{3}(2) (the stabilizer subgroup of an isotropic plane). Hence F_{S}(Co_{3}) ≅ F_{1} for S ∈ Syl_{2}(Co_{3}).

The centralizer in Sol(3) of any involution is the fusion system of Spin_{7}(3) [LO, Theorem 2.1], and Ω_{7}(3) contains a maximal subgroup C_{2}^{6}⋊A_{7} (the elements which leave an orthonormal basis invariant up to sign and permutation). Hence F_{T^{∗}}(Sol(3)) ≅ F_{2}.

Finally, the group Aut(P Sp_{6}(3)) has involution centralizer (SL_{2}(3) ×_{C_{2}} Sp_{4}(3))⋊C_{2} (the elements which leave invariant an orthogonal decomposition 𝔽_{3}^{2} × 𝔽_{3}^{4} of the vector space). Since P SL_{2}(3)×P Sp_{4}(3) contains a subgroup (C_{2}^{2}⋊C_{3})×(C_{2}^{4}⋊A_{5}), this shows that F_{S}(Aut(P Sp_{6}(3))) ≃ F_{3} for S ∈ Syl_{2}(Aut(P Sp_{6}(3))). □

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

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

Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049– 001 Lisboa, Portugal

  E-mail address: jventura@math.ist.utl.pt