FUSION SYSTEMS OVER A SYLOW p-SUBGROUP OF G_{2}(p)

CHRIS PARKER AND JASON SEMERARO

Abstract. For S a Sylow p-subgroup of the group G_{2}(p) for p odd, up to isomorphism of fusion systems, we determine all saturated fusion systems F on S with O_{p}(F) = 1. For p ≠ 7, all such fusion systems are realized by finite groups whereas for p = 7 there are 29 saturated fusion systems of which 27 are exotic.

1. Introduction

Let p ≥ 3 and S be a Sylow p-subgroup of the group G_{2}(p). The purpose of this paper is to give a complete classification of all saturated fusion systems F over S with O_{p}(F) = 1. This may be viewed as a contribution to a program which aims to classify all saturated fusion systems over maximal unipotent subgroups of finite groups of Lie Type of rank 2 and is thus a natural continuation of work carried out in [6, 13, 22]. In a different direction, when p ≥ 5 our paper contributes to the problem of listing all saturated fusion systems F over a Sylow p-subgroup with an extraspecial p-subgroup of index p, currently under investigation by the first author and Raul Moragues Moncho. An infinite family of such fusion systems was discovered recently by the first author and Stroth [20], and it is the p-group underlying the smallest member of this family on which we focus our attention. It will also form part of the classification of fusion systems of sectional p-rank 4 for odd primes p. All of these contributions add to our knowledge of saturated fusion systems defined on p-groups for odd primes p and so extend our understanding of how exotic fusion systems arise at odd primes [1, Problem 7.4] and [2, Problem 7.6].

When p ≥ 5, the problem naturally breaks into three stages. First in Section 3 we give a presentation for S and provide a concrete description of its action on the unique extraspecial subgroup Q of index p. Using this description, if F is a saturated fusion system on S we whittle down the possibilities for the F-essential subgroups in Section 4 by using results concerning the way in which automorphisms of a p-group act on various subgroups and conditions on the existence of certain lifts of automorphism groups which arise because of the saturation axiom.

Armed with a small list of possibilities for the F-essential subgroups, in Section 5 we proceed to analyse the various combinations of essential subgroups and morphisms for F which have the potential to lead to a saturated fusion system. Here we are especially reliant on a short list of possibilities for the group Aut_{F}(Q) (Theorem 2.11) which follows from some results obtained by the second author together with Craven and Oliver in [8]. One issue that arises during this stage is the question of whether or not a fusion system is uniquely determined by the above data. We develop some techniques to answer this, especially relying on some delicate calculations of automorphism groups carried out at the end of Section 3. Generally, our scheme is as follows: suppose for simplicity that we have just two essential subgroups Q and R in S and suppose that we know that Out_{F}(R) contains a normal subgroup isomorphic to SL_{2}(p). Then the saturation axiom and the existence of this subgroup forces the existence of certain morphisms in Aut_{F}(S) and then in Aut_{F}(Q) by restriction. Now we use just the existence of these automorphisms to uniquely determine Aut_{F}(Q) as a subgroup of Aut(Q) containing Aut_{S}(Q). This involves adjusting the entire fusion system by members of Aut(S) and this is fine as we only determine F up to isomorphism. Now Aut_{F}(Q) determines elements of Aut_{F}(S) and these all restrict to elements of Aut_{F}(R) and so give us the subgroup N_{Aut_{F} (R)}(Aut_{S}(R)). Next we prove that in Aut(R) there is a unique subgroup X containing N_{Aut_{F} (R)}(Aut_{S}(R)) with Aut_{S}(R) ∈ Syl_{p}(X) and O^{p′}(X) ≅ SL_{2}(p). Thus we must have Aut_{F}(R) = X and this is uniquely determined.

2010 Mathematics Subject Classification. 20D20, 20D05.

Key words and phrases. groups of Lie type, fusion systems, exotic fusion systems.

In the final stage, in Section 6, we examine each candidate fusion system F in turn and establish (a) its existence, (b) whether it is saturated and (c) whether it is realizable as the fusion system of a finite (almost simple) group. Here the fact that the fusion systems are uniquely determined by the structure of the automorphism groups of their essential subgroups is used implicitly. In all but finitely many cases, we obtain an affirmative answer to (a) and (b) from an affirmative answer to (c). In the remaining cases, it is always possible to realize F as the fusion system of a free amalgamated product of finite groups and saturation is established using the geometry of the associated coset graph (Theorem 6.1).

When p = 3, the two unipotent radical subgroups of G_{2}(3) are isomorphic so that although the overall strategy of the proof is the same, the individual arguments are somewhat different. In addition, in this case there is only one group to consider and we can support our arguments by computer calculations [3]. This case is treated in the final section.

Our main theorem is as follows:

Theorem 1.1. Suppose that p ⩾ 3, S is a Sylow p-subgroup of G_{2}(p) and F is a saturated fusion system over S with O_{p}(F) = 1. Then either F is isomorphic to the fusion system of G_{2}(p), Aut(G_{2}(3)) or p ∈ {5, 7} and F is isomorphic to one of 32 examples tabulated in Table 5.1. Furthermore, each of the fusion systems given in Table 5.1 is saturated.

The examples described in Table 5.1 include the fusion systems of the sporadic simple groups Ly, HN, B, the almost simple group Aut(HN) (all for p = 5) and M when p = 7. It also includes 27 exotic fusion systems which all occur when p = 7. Two of the exotic systems were discovered by Parker and Stroth [20] and the remainder are new to this article. They all are in some way related to the Monster sporadic simple group, though it is not the case that the Monster is “universal” in the sense that it “contains” all the smaller examples. This is somehow a subtle point. The fact is, and this plays no part in the classification, that in GSp_{4}(7), the subgroup 3 × 2. Sym(7) does not contain GL_{2}(7) but rather only a half of this group and so the fusion system that comes from G_{2}(7) is not contained in the fusion system determined by the Monster when p = 7.

Corollary 1.2. Suppose that p ⩾ 3, S is a Sylow p-subgroup of G_{2}(p) and F is a saturated fusion system over S with O_{p}(F) = 1. Then either F is realized by a finite group or p = 7 and F is one of 27 fusion systems listed in Table 5.1.

We close the introduction with a few words about our notation. We use [1, 7, 10] for standard group theoretic and fusion theoretic conventions. Particularly we use [1, 7] as a sources for the introduction of fusion systems in Section 2. The field of order p is denoted by 𝔽_{p}, the symmetric and alternating groups of degree n are denoted by Alt(n) and Sym(n) respectively and other than that we follow classical nomenclature for the finite simple groups and their near relatives. The Frobenius group of order 20 is written as Frob(20) and cyclic groups are mostly represented just by their order. The notation 2_{−}^{1+4} denotes an extraspecial group of −-type and order 2^{5} and, for p odd, p_{+}^{1+2} is extraspecial of order p^{3} and exponent p. We use G = A ∘ B to indicate that G is a central product of the groups A and B. We follow the atlas conventions for group extensions. This means that an “upper” dot informs the reader that an extension is non-split. When we write G ∼ A.B we read that G has a normal subgroup isomorphic to A and a corresponding quotient isomorphic to B. This provides a handy but inaccurate description of group structures. In our case, each time we use this notation the groups will be determined uniquely up to isomorphism as a subgroup of GSp_{4}(p) or GL_{2}(p). We also point out that the notation SL_{2}(7).2 will denote the unique normal subgroup of GL_{2}(7) of index 3.

2. Preliminaries: fusion systems and group theory

We begin by recalling the definition of a fusion system. For a group G, p-subgroup S of G and P, Q ≤ S define

N_{G}(P, Q) = {g ∈ G | P^{g} ≤ Q} and Hom_{G}(P, Q) = {c_{g} | g ∈ N_{G}(P, Q)},

where c_{g} is the conjugation map induced by g:

c_{g} : x ↦ g^{−1}xg.

Define F_{S}(G) to be the category with objects all the subgroups of S, and for objects P and Q of F_{S}(G), the set of morphisms from P to Q is

Mor_{F_{S}(G)}(P, Q) = Hom_{G}(P, Q).

Then F_{S}(G) is an example of a fusion system on S as defined, for example, in [1, Definition 2.1]. If S is a finite p-group and F is a fusion system on S we say that F is realizable if there exists a finite group G with S ∈ Syl_{p}(G) such that F = F_{S}(G). Otherwise F is said to be exotic. If P ≤ S, then define the set of F-conjugates of P to be

P^{F} = {P α | α ∈ Hom_{F}(P, S)}

and similarly, for g ∈ G, we use

g^{F} = {gα | α ∈ Hom_{F}(⟨g⟩, S)

for the set of images of g under morphisms in F. For P ≤ S, we put Aut_{F}(P) = Mor_{F}(P, P), Aut_{S}(P) = Hom_{S}(P, P), Inn(P) the inner automorphisms of P and Out_{F}(P) = Aut_{F}(P)/ Inn(P). Note that Aut_{F}(Q) ≅ Aut_{F}(P) for each Q ∈ P^{F}. Two fusion systems F and F′ on S are isomorphic if there exists α ∈ Aut(S) such that for all P, Q ≤ S,

Hom_{F′}(P α, Qα) = α^{−1}|_{P α} Hom_{F}(P, Q)α.

A proper subgroup H < G of a finite group G is strongly p-embedded in G if p divides |H| and p does not divide |H ∩ H^{g}| for each g ∈ G \ H. The next definition summarizes the main concepts we will need when dealing with fusion systems:

Definition 2.1. Let F be a fusion system on a finite p-group S and P, Q ≤ S. Then, • P is fully F-normalized provided |N_{S}(P)| ≥ |N_{S}(Q)| for all Q ∈ P^{F};

• P is fully F-centralized provided |C_{S}(P)| ≥ |C_{S}(Q)| for all Q ∈ P^{F};

• P is fully F-automized provided Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P));

• P is F-centric if C_{S}(Q) = Z(Q) for all Q ∈ P^{F};

• P is F-essential if P < S, P is F-centric and fully F-normalized and Out_{F}(P) contains a strongly p-embedded subgroup; write E_{F} (or simply E) to denote the set of F-essential subgroups of F;

• P is strongly F-closed if for each g ∈ P, g^{F} ⊆ P;

• if α ∈ Hom_{F}(P, Q) is an isomorphism,

N_{α} = {g ∈ N_{S}(P) | α^{−1}c_{g}α ∈ Aut_{S}(P α)}

is the α-extension control subgroup of S;

• Q is F-receptive provided for all isomorphisms α ∈ Hom_{F}(P, Q), there exists α̃ ∈ Hom_{F}(N_{α}, S) such that α̃|_{P} = α;

• P is saturated provided there exists Q ∈ P^{F} such that Q is simultaneously – fully F-automized; and

– F-receptive;

• F is saturated if every subgroup of S is saturated.

Saturated fusion systems are the main focus of study. Suppose that F is saturated. Then, by [1, Lemma 2.6 (c)], a subgroup Q is fully F-normalized if and only if it is fully F-automized and F-receptive. In particular, F-essential subgroups are both fully F-automized and F-receptive. We shall exploit the saturation property as follows. Suppose that Q is F-receptive. If α ∈ N_{Aut_{F} (Q)}(Aut_{S}(Q)), then

N_{α} = {x ∈ N_{S}(Q) | α^{−1}c_{x}α ∈ N_{Aut_{F} (Q)}(Aut_{S}(Q))} = N_{S}(Q)

and so there exists α̃ ∈ Hom_{F}(N_{S}(Q), S) extending α. Since

N_{S}(Q)α̃ ≤ N_{S}(Qα̃) = N_{S}(Qα) = N_{S}(Q),

we have α̃ ∈ Aut_{F}(N_{S}(Q)). Therefore every α ∈ N_{Aut_{F} (Q)}(Aut_{S}(Q)) extends to an element of Aut_{F}(N_{S}(Q)).

Recall that when G is a finite group and S ∈ Syl_{p}(G), we have that F_{S}(G) is saturated. If X is a set of injective morphisms between various subgroups of S, then we may define ⟨X⟩ to be the fusion system obtained by intersecting all the fusion systems on S which have the members of X as morphisms.

The next result is commonly referred to in the literature as “Alperin’s Theorem.”

Theorem 2.2. Let F be a saturated fusion system on a finite p-group S. Then

F = ⟨Aut_{F}(E) | E ∈ E_{F} ∪ {S}⟩.

Note that a subgroup Q ≤ S is normal in F if and only if Q ≤ ⋂_{P ∈E_{F}} P and, for P ∈ E_{F} ∪ {S}, Q is Aut_{F}(P)-invariant (see [1, Proposition 4.5].) The subgroup O_{p}(F) of S is the largest normal subgroup of F. Recall the definition of O^{p′}(F) which can be found in [7, Section 7.5], and that a subsystem of F has index prime to p (or p′-index) in F if it contains O^{p′}(F). Define

O_{∗}^{p′}(F) = ⟨O^{p′}(Aut_{F}(R)) | R ≤ S⟩.

Then put

Aut_{F}^{0}(S) = ⟨α ∈ Aut_{F}(S) | α|_{P} ∈ Hom_{O_{∗}^{p′}(F)}(P, S) for some F-centric subgroup P ⟩,

and set

Γ_{p′}(F) = Aut_{F}(S)/ Aut_{F}^{0}(S).

We have the following:

Theorem 2.3. There is a one-to-one correspondence between saturated fusion systems on S of index prime to p (which contain O^{p′}(Aut_{F}(P)) for all P ≤ S) and subgroups of Γ_{p′}(F). In particular, O^{p′}(F) is the unique minimal fusion system of index prime to p.

Proof. See [1, Theorem 7.7]. □

When proving that a fusion system is saturated, the following theorem is a basic tool:

Theorem 2.4. Let F be a fusion system on a finite p-group S and let C denote the set of all F-centric subgroups. Suppose that F = ⟨Aut_{F}(P) | P ∈ C⟩. If P is saturated for each P ∈ C then F is saturated.

Proof. See [5, Theorem A]. □

Sometimes we consider the fusion system determined by G, the universal completion of an amalgam G_{1} ≥ G_{12} ≤ G_{2} of finite groups with S a Sylow p-subgroup of either G_{1} or G_{2} (or both). We define the coset graph of G_{1} and G_{2} in G to be the graph Γ = Γ(G, G_{1}, G_{2}, G_{12}) which has

V (Γ) = {G_{i}g | g ∈ G, i ∈ {1, 2}} and E(Γ) = {{G_{i}g, G_{j}h} | G_{i}g ∩ G_{j}h ≠ ∅, i ≠ j}.

Since G is the universal completion of the amalgam, Γ is a tree [24, Theorem 6]. It is easy to verify that G acts on Γ by right multiplication. We shall always consider amalgams which are “simple” in the sense that no normal subgroup of G is contained in G_{12}. In this case, the action of G on Γ is faithful. Finally, we note that the stabiliser of the vertex G_{i}g is just G_{i}^{g} and that the edge-stabilizers are G-conjugate to G_{12}.

The following result shows that the saturation of F_{S}(G) is determined to some extent by the graph Γ and the action of G on it. The proof of this result, which is taken from [18], requires that we remember that when a finite group acts on a tree without exchanging the vertices of some edge, then it fixes a vertex.

Theorem 2.5. Let A = (G_{1} ≥ G_{12} ≤ G_{2}) be an amalgam of finite groups, assume that Syl_{p}(G_{12}) ⊆ Syl_{p}(G_{2}) and fix S_{i} ∈ Syl_{p}(G_{i}) with S_{2} ≤ S_{1}. Assume that G = G_{1} ∗_{G_{12}} G_{2} is the universal completion of A and write Γ = Γ(G, G_{1}, G_{2}, G_{12}) for the coset graph. Suppose that:

(a) for all F_{S_{1}}(G)-centric subgroups P of S, Γ^{P} is finite; and

(b) each F_{S_{i}}(G_{i})-essential subgroup is F_{S_{1}}(G)-centric. Then F_{S_{1}}(G) is saturated.

Proof. Since, for i = 1, 2, G_{i} is finite, F_{S_{i}}(G_{i}) is a saturated fusion system on S_{i} and hence F_{S_{i}}(G_{i}) is generated by the set of F_{S_{i}}(G_{i})-essential subgroups by Alperin’s Theorem. Since, by [21, Theorem 1],

F_{S_{1}}(G) = ⟨F_{S_{1}}(G_{1}), F_{S_{2}}(G_{2})⟩,

(b) implies that F_{S_{1}}(G) is generated by Aut_{F_{i}}(X) for X an F_{S_{1}}(G)-centric subgroup of S. Thus F_{S_{1}}(G) is saturated provided each F_{S_{1}}(G)-centric subgroup is saturated by Theorem 2.4.

Put S = S_{1} ≥ S_{2}, F = F_{S}(G) and assume that P ≤ S is an F-centric subgroup of S. Since G_{1}, G_{2} and Γ^{P} are finite, the subgroup K of N_{G}(P) which fixes every vertex of Γ^{P} is finite. Now N_{G}(P)/K embeds into Aut(Γ^{P}) and so is also finite. Thus N_{G}(P) is finite and so N_{G}(P) is contained in Stab_{G}(α) for some α ∈ Γ^{P}. Therefore N_{G}(P) is G-conjugate to a subgroup of either G_{1} or G_{2}. Hence we may choose a conjugate P^{f} of P so that either

N_{G}(P^{f}) ≤ G_{1} and R ∈ Syl_{p}(N_{G}(P^{f})) has R ≤ S,

or

N_{G}(P^{f}) ≤ G_{2} and R ∈ Syl_{p}(N_{G}(P^{f})) has R ≤ S_{2}.

Thus

Aut_{S}(P^{f}) = RC_{G}(P^{f})/C_{G}(P^{f}) ∈ Syl_{p}(Aut_{G}(P^{f}))

and hence P^{f} is fully F-automized.

It remains to prove that every F-centric subgroup P in S is F-receptive. So assume that c_{g} ∈ Hom_{F}(U, P) and define

N = N_{c_{g}} = {h ∈ N_{S}(U) | c_{g^{−1}hg} ∈ Aut_{S}(P)}.

Then

N^{g}C_{G}(P) ≤ N_{S}(P)C_{G}(P).

Since P is F-centric and C_{G}(P) is finite,

C_{G}(P) = Z(P) × O^{p′}(C_{G}(P)).

Thus N_{S}(P) ∈ Syl_{p}(N_{S}(P)C_{G}(P)). So there exists x ∈ C_{G}(P) such that N^{gx} ≤ N_{S}(P). Set y = gx. Then c_{y} ∈ Hom_{G}(N, N_{S}(P)) and c_{y} extends c_{g} ∈ Hom(U, P). We have shown that P is F-receptive. This completes the proof. □

We will also need the following result from [23] which gives conditions under which one can enlarge a saturated fusion system on a p-group S to form a new saturated fusion system, by adding morphisms of certain subgroups.

Theorem 2.6. Let F_{0} be a saturated fusion system on a finite p-group S. For 1 ⩽ i ⩽ m, let W_{i} ≤ S be a fully F_{0}-normalized subgroup with W_{i}φ ≰ W_{j} for each φ ∈ Hom_{F_{0}}(W_{i}, S) and i ≠ j. Set K_{i} = Out_{F_{0}}(W_{i}) and let Δ_{i} ≤ Out(W_{i}) be such that K_{i} is a strongly p-embedded subgroup of Δ_{i}. Also write

F = ⟨F_{0}; Δ_{1}, …, Δ_{m}⟩.

Assume further that for each 1 ⩽ i ⩽ m,

(a) W_{i} is F_{0}-centric (hence F-centric) and minimal under inclusion amongst all F-centric subgroups; and

(b) no proper subgroup of W_{i} is F_{0}-essential. Then F is saturated.

Proof. See [23, Theorem C]. □

We now develop some tools for listing the possible F-essential subgroups of a p-group S when F is a saturated fusion system on S. We need two basic facts concerning the way in which a p-group acts on its subnormal subgroups.

Lemma 2.7. Let E be a finite p-group and A ≤ Aut(E). Suppose there exists a normal chain

1 = E_{0} ⊴ E_{1} ⊴ E_{2} ⊴ ⋯ ⊴ E_{m} = E

of subgroups such that for each α ∈ A, E_{i}α = E_{i} for all 0 ≤ i ≤ m. If for all 1 ≤ i ≤ m, A centralizes E_{i}/E_{i−1}, then A ≤ O_{p}(Aut(E)).

Proof. See [10, 5.3.2]. □

Lemma 2.8 (Burnside). Let S be a finite p-group. Then C_{Aut(S)}(S/Φ(S)) ≤ O_{p}(Aut(S)).

Proof. See [10, 5.1.4]. □

The following result can also be found in [17, Lemma 3.4].

Lemma 2.9. Let S be a finite p-group and F ≤ E ≤ S be such that F is characteristic in E. If there exists g ∈ N_{S}(E)\E such that

(a) [g, E] ≤ F Φ(E) and

(b) [g, F] ≤ Φ(E), then E is not an F-essential subgroup in any saturated fusion system on S.

Proof. Since C_{Aut(E)}(E/Φ(E)) ≤ O_{p}(Aut(E)) by Burnside’s Lemma 2.8 and since F Φ(E) is normal in E, if follows from Lemma 2.7 that g ∈ O_{p}(Aut(E)). But then O_{p}(Aut(E)) ≰ Inn(E) which means that Out_{S}(E) ∩ O_{p}(Out(E)) ≠ 1 and hence E ∉ E_{F} for any saturated fusion system F on S. □

We also need some results about the certain subgroups of PGL_{3}(p) and GSp_{4}(p).

Proposition 2.10. Let G be a subgroup of PGL_{3}(p) which contains a strongly p-embedded subgroup. Then O^{p′}(G) is isomorphic to one of the following groups:

PSL_{2}(p), SL_{2}(p) or 13:3 (p = 3).

Proof. See [12, Proposition]. □

Theorem 2.11. Let G be a group with a non-normal Sylow p-subgroup ⟨x⟩ of order p with the property that | Aut_{G}(U)| = p − 1. Let V be a faithful, 4-dimensional 𝔽_{p}G-module which supports a non-degenerate symplectic form such that the matrix in GL(V) representing x has exactly one non-trivial Jordan block. Then one of the following holds:

(a) the image of G in PGL(V) is not an almost simple group, p = 5, F^{∗}(G) = 4 ∘ 2^{1+4} or 2_{−}^{1+4} and either

(i) G/F^{∗}(G) ≅ Sym(5); or (ii) G/F^{∗}(G) ≅ Frob(20);

(b) the image of G in PGL(V) is an almost simple group, G operates irreducibly on V and (i) p ≥ 5 and O^{p′}(G) ≅ SL_{2}(p); (ii) p = 5 and O^{5′}(G) ≅ 2. Alt(6); (iii) p = 7 and O^{7′}(G) ≅ 2. Alt(7); or

(c) the image of G in PGL(V) is an almost simple group, G operates reducibly on V, p = 5, O^{5′}(G) ≅ SL_{2}(5) and V is an indecomposable module with irreducible submodule and quotient of dimension 2.

Furthermore, all the examples in (a), (b) and (c) and their normalizers in GSp_{4}(p) are uniquely determined up to conjugacy in GSp_{4}(p).

Proof. We consider the problem in GL(V). Suppose first that the projection of G into PGL(V) is not an almost simple group. Then, by [8, Proposition 4.4], p = 5, O_{5′}(G) = F^{∗}(G) is isomorphic to one of 4 ∘ 2^{1+4} or 2_{−}^{1+4} and either

(1) G/F^{∗}(G) = Sym(6);

(2) G/F^{∗}(G) = Sym(5); or

(3) G/F^{∗}(G) = Frob(20). By [4, Tables 8.12 and 8.13], case (1) cannot occur and the remaining examples are listed in part (a). Using [8, Proposition 4.4] again yields candidates for O^{p′}(G) when the projection of G into PGL(V) is an almost simple group and provides the examples in (b) and (c). Using [4, Tables 8.12 and 8.13], we see that all the examples exist in GSp_{4}(p). By [8, Lemma 3.4 (c)] the examples in parts (a) and (b) operate absolutely irreducibly on V. By [4, Lemmas 1.8.9, 1.8.10 (ii)], these examples are uniquely determined up to conjugacy in GSp_{4}(p).

In case (c), because V is indecomposable, the 2-space preserved by G is isotropic. Thus G is contained in a maximal parabolic subgroup P of GSp_{4}(5) which leaves an isotropic 2-space invariant. To see uniqueness here, we note that the 1-cohomology of the 3-dimensional 𝔽_{5} SL_{2}(5)- module has dimension 1 (see [8, Lemma 3.11]). Thus there are five O^{5′}(P) conjugacy classes of subgroups isomorphic to SL_{2}(5) contained in O^{5′}(P). One of these acts completely reducibly on V and the others are all conjugate by an element of order 4 in P. Thus G is uniquely determined up to conjugacy in this case as well. □

We end this section with a result about finite simple groups which will be required when proving that certain fusion systems we construct are exotic.

Theorem 2.12. Suppose that p ≥ 5 and let G be a finite simple group with a Sylow p-subgroup isomorphic to that of G_{2}(p). Then one of the following holds:

(a) p = 5 and G ∈ {G_{2}(5), B, HN, Ly};

(b) p = 7 and G ∈ {G_{2}(7), M};

(c) p > 7 and G = G_{2}(p).

Proof. When p = 7, see [20, Lemma 3.4]. The other cases are dealt with in [16, Theorem]. □

3. Definition and basic properties of a Sylow p-subgroup of G_{2}(p) when p ≥ 5

3.1. Construction of S. Let q = p^{f} with p ≥ 5 a prime and 𝔽 be a field of order q. In what follows, we construct a group S which is isomorphic to a Sylow p-subgroup of G_{2}(q) (see the Appendix). To this end, we start with V the 4-dimensional subspace of homogeneous polynomials of degree 3 in 𝔽[X, Y]. Then L = 𝔽^{×} × GL_{2}(𝔽) acts on V via the 𝔽-linear extension of

X^{a}Y^{b} · (t, ( \begin{matrix}α & β \\ γ & δ \end{matrix} )) = t · (αX + βY)^{a} · (γX + δY)^{b}

where a + b = 3. We define a bilinear function β : V × V → 𝔽 by first defining β on basis vectors by

β(X^{a}Y^{b}, X^{c}Y^{d}) = \{ \begin{matrix}0, & if a ≠ d; \\ \frac{(−1)^{a}}{\binom{3}{a}} , & if a = d \end{matrix}

and extending linearly. Let Q be the group (V × 𝔽^{+}, ∗) where

(v, y) ∗ (w, z) = (v + w, y + z + β(v, w)),

for (v, y), (w, z) ∈ Q. In [20, Lemma 2.2] it is noted that Q is a special group with the property that

Z(Q) = {(0, λ) | λ ∈ 𝔽}.

We now construct the group S by extending the action of L on V to an action on Q defined as follows: For (t, A) ∈ L and (v, z) ∈ Q,

(3.1)  (v, z)^{(t,A)} = (v.(t, A), t^{2}(det A)^{3}z).

A simple check (carried out in the discussion before [20, Lemma 2.3]) shows that this action is a group action (in the sense that ((v, y)(w, z))^{(t,A)} = (v, y)^{(t,A)}(w, z)^{(t,A)}) and that the kernel of the action is

(\{(μ^{−3}, ( \begin{matrix}μ & 0 \\ 0 & μ \end{matrix} )) | μ ∈ 𝔽^{×}\} . (3.2)

As in [20], let

B_{0} = 𝔽^{×} × \{( \begin{matrix}α & 0 \\ γ & β \end{matrix} ) | α, β ∈ 𝔽^{×}, γ ∈ 𝔽\} and S_{0} = \{1\} × \{( \begin{matrix}1 & 0 \\ γ & 1 \end{matrix} ) | γ ∈ 𝔽\}

and set

P = LQ, B = B_{0}Q and S = S_{0}Q. (3.3)

For λ ∈ 𝔽, we define the following elements of Q:

x_{6}(λ) = (0, −2λ), x_{5}(λ) = (−λX^{3}, 0), x_{4}(λ) = (3λX^{2}Y, 0),
x_{3}(λ) = (−3λXY^{2}, 0), x_{2}(λ) = (λY^{3}, 0).

Also write

x_{1}(λ) = (1, ( \begin{matrix}1 & 0 \\ λ & 1 \end{matrix} )) ∈ S_{0}.

Observe that

S = ⟨x_{1}(λ), x_{2}(μ), x_{3}(ν), x_{4}(ξ), x_{5}(o), x_{6}(π) | λ, μ, ν, ξ, o, π ∈ 𝔽⟩.

3.2. Properties of S and some subgroups. We now specialize to the case when f = 1, so that S is a Sylow p-subgroup of G_{2}(p). By the discussion in Section 3.1, S = ⟨x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}⟩ where for each 1 ≤ j ≤ 6 we write x_{j} = x_{j}(1). Note that S has nilpotency class 5 and so S is of maximal class. Thus let

1 < Z = Z_{1} < Z_{2} < Z_{3} < Z_{4} < Z_{5} = S

be the upper (and lower) central series of S where, for ease of notation, we set Z = Z(S) and, for 2 ≤ i ≤ 4, Z_{i} = Z_{i}(S). Of particular importance to us will be the groups

Q = ⟨x_{2}, x_{3}, x_{4}, x_{5}, x_{6}⟩

and

R = ⟨x_{1}, x_{3}, x_{4}, x_{5}, x_{6}⟩.

From the construction of S, we see that

Lemma 3.1. The subgroup Q is extraspecial of order p^{5} and exponent p and x_{1} acts on Q/Z with a single Jordan block. □

Lemma 3.2. The following hold:

(a) Z = ⟨x_{6}⟩ and Z_{2} = ⟨x_{6}, x_{5}⟩;

(b) R = C_{S}(Z_{2});

(c) Z_{3} = ⟨x_{6}, x_{5}, x_{4}⟩ is elementary abelian and

Z_{4} = C_{Q}(Z_{2}) = Q ∩ R = Φ(S) = ⟨x_{6}, x_{5}, x_{4}, x_{3}⟩;

(d) Q and R are characteristic maximal subgroups of S; and

(e) the non-trivial normal subgroups of S of order at most p^{4} are the subgroups Z_{i} with 1 ≤ i ≤ 4 and Z_{4} = Φ(S).

Proof. Parts (a) and (b) follow directly from considering the description of S. Since β(X^{3}, X^{2}Y) = 0, we have [x_{4}, x_{5}] = 1 so that Z_{3} is abelian, and hence elementary abelian. Similarly, x_{3}, x_{4} and x_{5} all centralize x_{2} so that Z_{4} ⊆ C_{Q}(Z_{2}). Since β(X^{3}, Y^{3}) ≠ 0, x_{2} ∉ C_{Q}(Z_{2}) so Z_{4} = C_{Q}(Z_{2}) and the remaining equalities in (c) are clear. To see that Q is characteristic, we note that Q/Z is the unique abelian subgroup of order p^{4} in S/Z. That R is characteristic follows from the fact that R = C_{S}(Z_{2}) and Z_{2} is characteristic in S. Thus (d) is proved. Part (e) follows from the fact that S has maximal class so that the upper central series for S and the lower central series for S coincide. □

Lemma 3.3. Suppose X is a maximal subgroup of S with X ≠ Q. Then (a) Z_{3} = Φ(X) is characteristic in X;

(b) Z_{2} is characteristic in X; and

(c) either Z_{4} is characteristic in X or X = R.

Proof. As X ≠ Q, we have S = QX and, as Z_{4} = Φ(S), also Z_{4} < X. Now note that, as [Z_{4}, X] ≥ [Z_{4}, Q] = Z,

[Z_{4}, X] = [Z_{4}, QX] = [Z_{4}, S] = Z_{3}

and so [X, X] ≥ [Z_{4}, X] = Z_{3}. Since |X/Z_{3}| = p^{2}, we deduce that [X, X] = Φ(X) = Z_{3}. In particular, Z_{3} is characteristic in X. This proves (a).

Now

[Z_{3}, X] = [Z_{3}, QX] = [Z_{3}, S] = Z_{2}

and so (b) holds. By Lemma 3.2 (c) Z_{4} centralizes Z_{2}. Let α ∈ Aut(X) and assume that Z_{4}α ≠ Z_{4}. Then X = Z_{4}Z_{4}α. Since Z_{4}α centralizes Z_{2}α, and Z_{2}α = Z_{2} by (b), we have X ≤ R and as X is maximal, X = R. This proves (c).

□

As remarked in the introduction, we need to prove that each of the fusion systems F we construct is uniquely determined by the F-automorphism groups of E_{F} ∪{S}. For this, a detailed description of the automorphism groups of Q, R and S is helpful.

3.3. The structure of Aut(Q). The structure of the automorphism group of an extraspecial p-group of exponent p is well known, and we state it here only for convenience:

Proposition 3.4. Set A = Aut(Q) and \overline{A} = Out(Q). There exists θ ∈ A of order p − 1 such that A = ⟨θ⟩ · C_{A}(Z(Q)) and ⟨θ⟩ ∩ C_{A}(Z(Q)) = 1. Moreover \overline{C_{A}(Z(Q))} ≅ Sp_{4}(p) and \overline{A} ≅ GSp_{4}(p).

Proof. See [26, Theorem 1]. □

3.4. The structure of Aut(R). The next lemma provide us with a rather precise description of Aut(R).

Lemma 3.5. Let A = Aut(R), \overline{A} = Out(R) and put

A = {(x, y) ∈ R × R | R = ⟨x, y⟩}.

Then the following hold:

(a) Inn(R) ≅ p_{+}^{1+2};

(b) | Aut(R)| = p^{7}(p^{2} − 1)(p − 1);

(c) if (x, y), (x_{1}, y_{1}) ∈ A, then there exists θ ∈ Aut(R) such that xθ = x_{1} and yθ = y_{1};

(d) A/O_{p}(A) ≅ GL_{2}(p);

(e) O_{p}(\overline{A}) is elementary abelian of order p^{3} and as an O^{p′}(A/O_{p}(A))-module is isomorphic to the module of 2 × 2-matrices over 𝔽_{p} of trace 0 acted upon by conjugation by SL_{2}(p);

(f) Z(\overline{A}) has order 2; and

(g) there exists a subgroup \overline{X} of \overline{A} with \overline{X} ≅ GL_{2}(p).

Proof. Since S has maximal class and Q∩R = Z_{4}, [R, Q∩R] = [S, Z_{4}] = Z_{3} and so Inn(R) ≅ R/Z_{2} is extraspecial. Since (Q ∩ R)/Z_{2} is elementary abelian and x_{1}Z_{2} has order p we see that Inn(R) has exponent p. This proves (a).

By [19, Lemma 5.2], the map

: C_{Aut(R)}(R/Z(R)) → Hom(R, Z(R))

which sends Ψ ∈ C_{Aut(R)}(R/Z(R)) to the map Ψ̃ ∈ Hom(R, Z(R)) defined, for g ∈ R, by gΨ̃ = g^{−1}(gΨ) is an isomorphism. Moreover is Aut(R)-invariant. Indeed suppose that α ∈ Aut(R) and g ∈ R. Then, for θ ∈ Hom(R, Z(R)), we have

gθ^{α} = gα^{−1}θα (3.4)

and so we calculate

gΨ̃^{α} = g^{−1}(gΨ^{α}) = g^{−1}(gα^{−1}Ψα) = (g^{−1}α^{−1}(gα^{−1}Ψ))α = ((gα^{−1})Ψ̃)α = gΨ̃^{α}.

Since Hom(R, Z(R)) ≅ Hom(R/Φ(R), Z(R)) we see that C_{Aut(R)}(R/Z(R)) is isomorphic to the set of all linear transformations from a 2-space to a 2-space. Thus C_{Aut(R)}(R/Z(R)) is elementary abelian of order p^{4}.

Next we collect some automorphisms of R which can be obtained from a parabolic subgroup P in G = G_{2}(p). After identifying S with a Sylow p-subgroup of G, the relevant parabolic subgroup is P = N_{G}(R) and there we observe P/C_{G}(R) = Aut_{G}(R) ≅ p_{+}^{1+2}: GL_{2}(p).

Also

Aut_{G}(R) ∩ C_{Aut(R)}(R/Z(R)) = Inn(R) ∩ C_{Aut(R)}(R/Z(R)) = Z(Inn(R)) = Φ(R)/Z(R) = Z_{3}/Z_{2} has order p. Hence Aut(R) has order at least

|C_{Aut(R)}(R/Z(R)) Aut_{G}(R)| = \frac{p^{4}.p^{3}.p.(p^{2} − 1)(p − 1)}{p} = p^{7}(p^{2} − 1)(p − 1).

We now establish an upper bound for | Aut(R)| and thus simultaneously prove parts (b) and (c). Since R/Φ(R) = R/Z_{3} has order p^{2}, there exist x, y ∈ R such that R = ⟨x, y⟩. We count the possible number of images of x and y under an automorphism θ of R. Plainly, R = ⟨xθ, yθ⟩, xθ ∉ Φ(R) and yθ ∉ ⟨xθ⟩Φ(R). There are at most

|R| − |Φ(R)| = p^{5} − p^{3}

choices for xθ and then

|R| − |⟨xθ⟩Φ(R)| = p^{5} − p^{4}

choices for yθ. Thus there are at most (p^{5} − p^{3})(p^{5} − p^{4}) = p^{7}(p^{2} − 1)(p − 1) automorphisms of R. Thus (b) and (c) hold.

Furthermore, from the discussion the in proof of (b), we see that \overline{A} contains \overline{Aut_{G}(R)} ≅ GL_{2}(p) (which gives (d) and (g)) and Inn(R) ∩ C_{Aut(R)}(R/Z(R)) has order p. Hence O_{p}(A) = Inn(R)C_{Aut(R)}(R/Z(R)) and

O_{p}(\overline{A}) ≅ C_{Aut(R)}(R/Z(R))/(Inn(R) ∩ C_{Aut(R)}(R/Z(R)))

and this isomorphism is as A-groups. In particular, O_{p}(\overline{A}) is elementary abelian of order p^{3}. Since C_{Aut(R)}(R/Z(R)) is isomorphic to the set of all linear transformations from a 2-space to a 2-space and is also an A-group, we infer that as an O^{p′}(A/O_{p}(A))-module, O_{p}(\overline{A}) is isomorphic to the module of trace zero 2 × 2-matrices over 𝔽_{p} with SL_{2}(p) acting by conjugation. This proves (e).

Because O_{p}(\overline{A}) is a minimal normal subgroup of \overline{A} by (e), Z(\overline{A}) ∩ O_{p}(\overline{A}) = 1 so

Z(\overline{A}) ≅ Z(\overline{A})O_{p}(\overline{A})/O_{p}(\overline{A}) ≤ Z(\overline{A}/O_{p}(\overline{A})) ≅ Z(GL_{2}(p))

by (d). Thus we need to determine the centre of the preimage of Z(\overline{A}/O_{p}(\overline{A})). Since O_{p}(\overline{A}) is abelian, it suffices to determine which elements of Z(\overline{A}/O_{p}(\overline{A})) lift to elements of A which centralize O_{p}(\overline{A}).

Let (x, y) ∈ A with y ∈ Q. Then, by (c), the map x ↦ x^{a}, y ↦ y^{a} for 1 ≤ a ≤ p − 1 extends to a unique automorphism θ of R and θ ∈ Z(\overline{A}/O_{p}(\overline{A})). Then define l = [x, y], m = [l, x] and n = [l, y]. Notice that, as R/Z_{2} ≅ Inn(R) is extraspecial, l ∈ Z_{3} \ Z_{2}. Because C_{Q}(Z_{3}) = Z_{3}, we then see that 1 ≠ n ∈ Z. As x acts on Q/Z with a single Jordan block, we have C_{Q/Z}(x) = Z_{2}/Z and so m ∈ Z_{2} \ Z. This shows that Z_{2} = ⟨m, n⟩. Now we use [10, Lemmas 2.2.1 and 2.2.2] to notice first that

lθ = [xθ, yθ] = [x^{a}, y^{a}] = [x, y]^{a^{2}}z

for some z ∈ Z_{2} and then calculate that

mθ = [ℓθ, xθ] = [[x, y]^{a^{2}}z, x^{a}] = [[x, y]^{a^{2}}, x^{a}] = [[x, y], x]^{a^{3}} = m^{a^{3}}

where the third equality follows from [10, Theorem 2.2.1]. Similarly, we determine

nθ = [[x, y]^{a^{2}}z, y^{a}] = [[x, y]^{a^{2}}, y^{a}] = [[x, y], y]^{a^{3}} = n^{a^{3}}. (3.5)

Now using Equation (3.4) and noting that θ operates as the scalar a on R/Φ(R) and a^{3} or Z_{2}, we calculate, for Ψ ∈ C_{Aut(R)}(R/Z(R)) and g ∈ R/Φ(R),

gΨ̃^{θ} = gθ^{−1}Ψ̃θ = g^{a^{−1}} Ψ̃θ = ((gΨ̃)^{a^{−1}})θ = (gΨ̃)^{a^{−1}a^{3}} = (gΨ̃)^{a^{2}}

Thus we see that θ centralizes C_{Aut(R)}(R/Z(R)) if and only if (gΨ̃)^{a^{2}} = gΨ̃ for all g ∈ R/Φ(R) and Ψ ∈ C_{Aut(R)}(R/Z(R)) which is if and only if a^{2} = 1. As θ induces a scalar action on C_{Aut(R)}(R/Z(R)) and O_{p}(\overline{A}) = C_{Aut(R)}(R/Z(R))/(Inn(R) ∩ C_{Aut(R)}(R/Z(R))), we now deduce that C\overline{A}(O_{p}(\overline{A})) has order 2p^{3} and part (f) follows from this. □

It is perhaps interesting to note that Equation 3.5 implies that Aut(R)/C_{Aut(R)}(R/Φ(R)) ≅ GL_{2}(p) whereas Aut(R)/C_{Aut(R)}(Z(R)) ≅ GL_{2}(p)/X where X is central of order (p − 1, 3).

Lemma 3.6. Let \overline{A} = Out(R) and suppose that \overline{Y} ≤ \overline{A}, \overline{T} ∈ Syl_{p}(\overline{Y}) with |\overline{T} | = p and |C\overline{Y} (\overline{T})| > 2. Assume that \overline{X} ≤ \overline{A} with \overline{X} ≅ SL_{2}(p) and \overline{Y} ≤ N\overline{A}(\overline{X}). Then \overline{XY} ≤ C\overline{A}(C\overline{Y} (\overline{T})) ≅ GL_{2}(p).

Proof. By Lemma 3.5 (d) and (e), \overline{A} has shape p^{3}: GL_{2}(p) and \overline{U} = O_{p}(\overline{A}) is a minimal normal subgroup of \overline{UX}. Since \overline{Y} normalizes \overline{X} and N\overline{U}(\overline{X}) = 1, we deduce that \overline{XY} is isomorphic to a subgroup of \overline{A}/\overline{U} ≅ GL_{2}(p). In particular, |C\overline{Y} (\overline{T})| ≤ Z(\overline{XY}) from the structure of GL_{2}(p). Now, as C\overline{Y} (\overline{T}) has order greater than 2, Lemma 3.5 (f) and the fact that \overline{U} is a minimal normal subgroup of \overline{UX} imply that C\overline{Y} (\overline{T}) ∩ \overline{U} = 1 (note that C\overline{Y} (\overline{T}) ∩ \overline{U} is normalized by \overline{X}.) Thus C\overline{A}(C\overline{Y} (\overline{T})) ≅ GL_{2}(p) and this proves the result. □

3.5. The structure of Aut(S). We conclude this section with description of Aut(S).

Lemma 3.7. Suppose that X is a group and Y is a subgroup of X of index p where p is a prime. Then [X, C_{Aut(X)}(Y)] ≤ Z(Y).

Proof. Select x ∈ X \ Y and notice that since p is prime every element z of X can be written as z = yx^{i} for some y ∈ Y and 1 ≤ i ≤ p. Now for each α ∈ C_{Aut(X)}(Y),

[z, α] = [yx^{i}, α] = [y, α]^{x^{i}}[x^{i}, α] = [x^{i}, α]

and so it suffices to show that [x^{i}, α] ∈ Z(Y).

Let y_{1} ∈ Y. Then (x^{i})^{y_{1}} = y_{2}x^{i} for y_{2} = [y_{1}, x^{−i}] and so

[x^{i}, α]^{y_{1}} = (x^{−i}(x^{i})α)^{y_{1}} = (x^{−i})^{y_{1}}((x^{i})α^{(y_{1})α}) = (x^{−i})^{y_{1}})((x^{i})^{y_{1}})α
= (y_{2}x^{i})^{−1}(y_{2}x^{i})α = x^{−i}y_{2}^{−1}y_{2}(x^{i})α = [x^{i}, α]

Hence [x^{i}, α] ∈ Z(Y) and consequently [X, α] ≤ Z(Y). The result follows. □

We remark that the following result can be deduced from [9, Theorem 6.2].

Proposition 3.8. There is an exact sequence

(3.6)  1 ⟶ Hom(S/Q, Z(S)) ⟶ Aut(S) ⟶^{ρ} N_{Aut(Q)}(Aut_{S}(Q)) ⟶ 1,

where ρ is given by α ↦ α|_{Q}, the restriction of α to Q. In particular, | Aut(S)| = p^{7}(p − 1)^{2}.

Proof. If α ∈ ker ρ, then α|_{Q} is the identity automorphism of Q. Thus Lemma 3.7 implies that α ∈ C_{Aut(S)}(S/Z(Q)) = C_{Aut(S)}(S/Z(S)). Hence by [19, Lemma 5.2] α corresponds to an element of Hom(S/Φ(S), Z(S)) which has Q/Φ(S) in its kernel. But then α corresponds to an element of Hom(S/Q, Z(S)) and this group has order p. Thus in order to show that (3.6) is exact, it suffices to show that ρ is surjective.

Now in Out(Q) ≅ GSp_{4}(p) the normalizer of a subgroup of order p whose non-trivial elements operate with a single Jordan block on the natural module has order p^{2}(p − 1)^{2} (see [14, Table 8.1a] to obtain the centralizer in Sp_{4}(p) and the claim follows from this as such a subgroup is contained in GL_{2}(p) ≤ GSp_{4}(p)). Since N_{Aut(Q)}(Aut_{S}(Q)) fits into an exact sequence

1 ⟶ Inn(Q) ⟶ N_{Aut(Q)}(Aut_{S}(Q)) ⟶ N_{Out(Q)}(Out_{S}(Q)) ⟶ 1,

it follows that every element of N_{Aut(Q)}(Aut_{S}(Q)) lifts to an element of Aut(S), and hence ρ is surjective. Lastly, since N_{Out(Q)}(Out_{S}(Q)) ∼ p^{2}:(p−1)^{2}, we have that | Aut(S)| = p.p^{4}.p^{2}(p−1)^{2} = p^{7}(p − 1)^{2}, as required. □

4. Candidates for the essential subgroups when p ≥ 5

Let S be the p-group defined in Section 3 and adopt all the notation introduced there. We also require the following additional piece of notation:

Notation 4.1. Define:

\begin{aligned}W_{x} & = ⟨Z, x⟩ & x ∈ S \ (Q ∪ R); \\ U_{x} & = ⟨Z_{2}, x⟩ & x ∈ S \ Q. \end{aligned}

Also put

W = {W_{x} | x ∈ S\(Q ∪ R)} and U = {U_{x} | x ∈ S\Q}.

The goal of this section is to prove the following result:

Theorem 4.2. Let F be a saturated fusion system on S and denote by E the set of F-essential subgroups. Then

E ⊆ {Q, R} ∪ W.

Moreover, if W ∩ E ≠ ∅, then p = 7.

Thus our hypotheses are that F is a saturated fusion system on S with O_{p}(F) = 1 and E = E_{F} is the set of F-essential subgroups of S. The proof of Theorem 4.2 will proceed in a series of steps.

Lemma 4.3. If E ≤ Q is F-essential, then E = Q.

Proof. Suppose that E is F-essential with E ≤ Q but that E ≠ Q. Then N_{Q}(E) > E and [E, N_{Q}(E)] ≤ Q′ = Φ(Q). If Φ(E) ≠ 1, then we have Φ(E) = Φ(Q), [E, N_{Q}(E)] ≤ Φ(E) and [Φ(E), N_{Q}(E)] = 1. Thus Lemma 2.9 implies that E is not essential, a contradiction. Therefore Φ(E) = 1 and E is elementary abelian. Since E ≥ C_{Q}(E), we deduce that E is a maximal abelian subgroup of Q. Hence |E| = p^{3}. Now Q/E embeds into Aut_{F}(E) and so Proposition 2.10 provides a contradiction as |Q/E| = p^{2}. Hence, if E ≤ Q and E ∈ E, then E = Q.

□

Lemma 4.4. If E ≰ Q is F-essential, then either E = R or E ∈ U ∪ W.

Proof. Since E is F-centric, Z ≤ C_{S}(E) ≤ E, so we may assume that |E| = p^{t} for some 2 ≤ t ≤ 5. If t = 2 then E = Z⟨x⟩ and as E must be centric, C_{S}(E) ≱ Z_{2}. Hence, as E ≰ Q, we have x ∈ S\(Q ∪ R). Thus E ∈ W in this case.

Suppose that t ⩾ 3. Then Z ≤ Q ∩ E and, as Q/Z, is abelian, we have Q ∩ E ⊴ Q. As Q is normal in S, Q ∩ E is normal in E and so Q ∩ E is normal in S = ⟨Q, E⟩. Therefore, by Lemma 3.2(e) we have that E ∩ Q = Z_{t−1}.

If t = 3 (so that E ∩ Q = Z_{2}), then E ∈ U.

It remains to show that if t > 3 then E = R. Suppose that t = 4. Then E = ⟨Z_{3}, x⟩ for some x ∈ S\Q. We have

Z_{2} = [E ∩ Q, S] = [E ∩ Q, EQ] = [E ∩ Q, E] ≤ [E, E]

and so we infer that Z_{2} = [E, E] = Φ(E). Notice that Z_{3} is elementary abelian. If Z_{3} is normalized by Aut_{F}(E), then using Lemma 2.7 together with [E, N_{S}(E)] ≤ Z_{3} and [Z_{3}, N_{S}(E)] ≤ Z_{2} = Φ(E) yields that Aut_{S}(E) ≤ O_{p}(Aut_{F}(E)) contrary to E being F-essential. Hence there exists α ∈ Aut_{F}(E) such that Z_{3} ≠ Z_{3}α. As |E| = p^{4}, we have E = Z_{3}Z_{3}α and Z_{3} ∩ Z_{3}α = Z_{2}. Since Z_{3} is elementary abelian, this means that Z_{2} = Φ(E) = Z(E) and we remark that this group is elementary abelian. Let x ∈ Z_{3} \ Z(E) and y ∈ Z_{3}α \ Z(X). Then E = ⟨x, y⟩ and x and y have order p. Set N = ⟨[x, y]⟩. Then N ≤ Φ(E) = Z(E) and N has order p. But then E/N is generated by xN and yN and these elements commute and have order p. It follows that E/N has order both p^{3} and p^{2}, a contradiction.

Finally, suppose that t = 5 so E is a maximal subgroup of S which is not equal to Q. If E ≠ R, we claim that the hypotheses of Lemma 2.9 are satisfied with F = Z_{4}. Indeed, Z_{4} is characteristic in E by Lemma 3.3 (c). Moreover Φ(E) = Z_{3}, so that for any x ∈ S\E,

[x, E] ≤ [S, S] = Z_{4} and [x, F] = [x, Z_{4}] ≤ Z_{3} = Φ(E).

Hence by Lemma 2.9 E is not F-essential. Thus, if E ≠ Q is an F-essential subgroup of order p^{5}, then E = R and this completes the proof. □

We also observe the following fact which can also be deduced from the remark after Lemma 3.5.

Lemma 4.5. If R ∈ E, then Out_{F}(R) is isomorphic to a subgroup of GL_{2}(p) and O^{p′}(Out_{F}(R)) ≅ SL_{2}(p). Furthermore, O^{p′}(Out_{F}(R)) acts faithfully on R/Φ(R) and on Z_{2} = Z(R).

Proof. By Lemma 2.8, Out_{F}(R) acts faithfully on R/Φ(R) which is elementary abelian of order p^{2}. Since R ∈ E and any two distinct cyclic subgroups of order p in GL_{2}(p) generate SL_{2}(p), the main statement follows and the action of O^{p′}(Out_{F}(R)) on R/Φ(R) is of course faithful. To see that the action on Z(R) is faithful, it suffices to show that the central involution t of O^{p′}(Out_{F}(R)) acts non-trivially on Z(R). Notice that O^{p′}(Out_{F}(R)) centralizes the cyclic group Z_{3}/Z_{2}. Hence, as t inverts Z_{4}/Z_{3}, t also inverts [Z_{4}, Z_{3}] = Z. This proves the claim. □

Lemma 4.6. Out_{F}(S) is conjugate to a subgroup of diagonal matrices in GL_{2}(p). In particular, Out_{F}(S) can be assumed to be a subgroup of the group generated by the images of the morphisms

d = (t, ( \begin{matrix}λ & 0 \\ 0 & μ \end{matrix} ))

where t, λ, μ ∈ 𝔽_{p}^{×}. The element d centralizes Z if and only if t^{2}λ^{3}μ^{3} = 1.

Proof. By Lemma 3.2 (d), Q and R are characteristic subgroups of S. Thus Out_{F}(S) is conjugate to a subgroup of diagonal matrices in GL_{2}(p). We can describe the action of Out_{F}(S) on S by explicitly by calculation using Equation (3.1) and this gives the second part of the result. □

Lemma 4.7. Suppose that D ≤ Out_{F}(S) normalizes a non-trivial proper subgroup of S/Z_{4} which is not equal to Q/Z_{4} or R/Z_{4}. If D centralizes Z, then D has order dividing 5.

Proof. Let d ∈ D^{#}. Then, by Lemma 4.6, d = (t, ( \begin{matrix}λ & 0 \\ 0 & μ \end{matrix} )) with t^{2}λ^{3}μ^{3} = 1. We calculate that on Q/Z_{4} (which is generated by Y^{3}, d acts by scaling Q/Z_{4} by tμ^{3} and on R/Z_{4} we calculate ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ) to ( \begin{matrix}1 & 0 \\ λμ^{−1} & 1 \end{matrix} ). Thus for a diagonal subgroup to remain fixed by d, we require tμ^{3} = λμ^{−1}. This gives t^{2}μ^{6} = λ^{2}μ^{−2} and then, substituting t^{2}μ^{3} = λ^{−3} and rearranging, we obtain μ^{5} = λ^{5}. Now d^{5} = (t^{5}, ( \begin{matrix}λ^{5} & 0 \\ 0 & λ^{5} \end{matrix} )) where t^{5} = λ^{−15} (as tμ^{3} = λμ^{−1}). But then, by considering Equation 3.2, d^{5} centralizes S and thus d has order dividing 5. Since C_{Out_{F} (S)}(Z) is cyclic, it follows that D is cyclic of order dividing 5. □

We use Lemmas 4.6 and 4.7 to help eliminate the possibility that F contains an essential subgroup in U. We achieve this in the next three lemmas.

Lemma 4.8. If U_{x} ∈ E for some x ∈ S\Q, then U_{x} is abelian (equivalently U_{x} ≤ R).

Proof. Write E = U_{x} for some x ∈ S\Q. If E is non-abelian, then x ∉ R and E ≅ p_{+}^{1+2} with [E, E] = Z. Since Out_{F}(E) acts faithfully on E/Φ(E), we have Out_{F}(E) is isomorphic to a subgroup of GL_{2}(p) containing SL_{2}(p) just as in Lemma 4.5. Let C = C_{Aut_{F} (E)}(Z). Then C/ Inn(E) ≅ SL_{2}(p) and N_{C}(Aut_{S}(E)) is cyclic of order p − 1. Since F is saturated, the elements of C extend to a maps in Aut_{F}(N_{S}(E)). Now using x ∉ R, we see that each α ∈ C is the restriction of an element \overline{α} ∈ Aut_{F}(S) by Lemmas 4.3 and 4.4. But then Out_{F}(S) contains a subgroup C_{0} of order p − 1 which centralizes Z and whose elements restrict to elements in C. We have that C_{0} normalizes Z_{4}E and so, as p − 1 does not divide 5, Lemma 4.7 implies that EZ_{4} = R or EZ_{4} = Q. Since x ∉ Q ∪ R, we have a contradiction. □

We have the following observation:

Lemma 4.9. Suppose that U_{x} ∈ E. Then R ∉ E. In particular, O_{p}(Aut_{F}(R)) = Aut_{S}(R) = S/Z_{2}.

Proof. Suppose that R ∈ E. Then Lemma 4.5 implies that O^{p′}(Out_{F}(R)) ≅ SL_{2}(p). By Lemma 4.8, U_{x} ≤ R and U_{x}Φ(R) is a maximal subgroup of R. Since O^{p′}(Aut_{F}(R)) ≅ SL_{2}(p) acts transitively on the maximal subgroups of R, we see that U_{x}Φ(R) is conjugate to Q∩R by some α ∈ O^{p′}(Aut_{F}(R)). Thus U_{0} = U_{x}α ≥ Z_{2} and N_{S}(U_{0}) ≥ Q. Since N_{S}(U_{x}) = U_{x}Φ(R), we see that U_{x} is not fully Fnormalized and thus U_{x} ∉ E. This proves the claim. □

Lemma 4.10. Suppose U_{x} ∈ E for some x ∈ S \ Q. Then O^{p′}(Aut_{F}(U_{x})) ≅ SL_{2}(p) and the following statements hold:

(a) As an 𝔽_{p}O^{p′}(Aut_{F}(U_{x}))-module, U_{x} is the direct sum of a 2-dimensional module and a 1-dimensional trivial module.

(b) Z = C_{U_{x}}(O^{p′}(Aut_{F}(U_{x}))).

Proof. By Lemma 4.8, U_{x} is elementary abelian and so, as U_{x} is centric we may regard U_{x} as a faithful 𝔽_{p} Aut_{F}(U_{x})-module. In particular, Aut_{F}(U_{x}) is isomorphic to a subgroup of GL_{3}(p). Since Aut_{F}(U_{x}) has a strongly p-embedded subgroup and p ≥ 5, Proposition 2.10 yields O^{p′}(Aut_{F}(U_{x})) is isomorphic to either of PSL_{2}(p) or SL_{2}(p). Now Aut_{S}(U_{x}) = Aut_{Z_{3}}(U_{x}) and Z_{3} is abelian by Lemma 3.2, we have

[U_{x}, Aut_{S}(U_{x}), Aut_{S}(U_{x})] = [U_{x}, Z_{3}, Z_{3}] ≤ [Z_{3}, Z_{3}] = 1.

In particular, as a subgroup of GL_{3}(p), the Jordan form of the elements of Aut_{S}(U_{x}) have one block of size 2 and a trivial block. It follows that O^{p′}(Aut_{F}(U_{x})) ≅ SL_{2}(p) because the p-elements of PSL_{2}(p) have Jordan block of size 3. Let τ ∈ O^{p′}(Aut_{F}(U_{x})) be an involution contained in the centre of SL_{2}(p). Then τ ∈ Z(Aut_{F}(U_{x})) and U_{x} = [U_{x}, τ] ⊕ C_{U_{x}}(τ) is an O^{p′}(Aut_{F}(U_{x})) decomposition of U_{x} as the direct sum of a 2-dimensional module and a 1-dimensional trivial module.

We now prove (b). We have [U_{x}, Aut_{S}(U_{x})] = [U_{x}, Z_{3}] ≤ [S, Z_{3}] = Z_{2} and, by (a), [U_{x}, Aut_{S}(U_{x})] has order p. Since Aut_{S}(U_{x}) = Aut_{Z_{3}}(U_{x}) and Z_{3} is abelian, we have C_{U_{x}}(Aut_{S}(U_{x})) = C_{U_{x}}(Z_{3}) = Z_{2}. If [U_{x}, Aut_{S}(U_{x})] ≤ Z, then [S, Z_{3}] = [U_{x}Q, Z_{3}] ≤ Z, which is impossible. Thus [U_{x}, Aut_{S}(U_{x})] is a subgroup of Z_{2} of order p which is not contained in Z. Let τ ∈ Z(Aut_{F}(U_{x})) have order 2. Then [U_{x}, Aut_{S}(U_{x})] is inverted by τ and to prove the result it suffices to show that Z is normalized by τ for then Z_{2} = [U_{x}, Aut_{S}(U_{x})]Z with Z centralized by τ. Suppose that τ does not normalize Z. Since τ normalizes Aut_{S}(U_{x}) and F is saturated, τ lifts to \overline{τ} ∈ Aut_{F}(N_{S}(U_{x})). Now using Theorem 2.2 and Lemma 4.9, we see that \overline{τ} is the restriction of some τ^{∗} ∈ Aut_{F}(S). But then

Zτ = Z\overline{τ} = Zτ^{∗} = Z,

which is a contradiction. This proves (b). □

Lemma 4.11. U_{x} ∉ E for all x ∈ S \ Q.

Proof. We have O^{p′}(Aut_{F}(U_{x})) ≅ SL_{2}(p) by Lemma 4.10. Let t be an involution which projects into Z(O^{p′}(Aut_{F}(U_{x}))). Then t ∈ N_{Aut_{F} (U_{x})}(Aut_{S}(U_{x})) and so t = τ̂|_{U_{x}} for some τ̂ ∈ Aut_{F}(N_{S}(U_{x})). Since R ∉ E and U_{x} ≰ Q, τ̂ must extend to a map τ ∈ Aut_{F}(S) by Lemmas 4.3 and 4.4. Now, by Lemma 4.10 (a) and (b), τ centralizes Z, inverts Z_{2}/Z and inverts R/Z_{4} = U_{x}Z_{4}/Z_{4} ≅ U_{x}/Z_{2}. Furthermore, as Q is characteristic in S, τ acts on Q/Z_{4}. Since τ has even order, τ does not invert Q/Z_{4} by Lemma 4.7. Let a ∈ Q \ Z_{4} and b ∈ Z_{2} \ Z with bτ = b^{−1}. Then, for some 1 ≤ e ≤ p − 1, aZ_{4}τ = a^{e}Z_{4} and so, as [a, b] ∈ Z^{#}, we obtain

[a, b] = [aZ_{4}, b]τ = [aZ_{4}τ, b^{−1}] = [a^{e}, b^{−1}] = [a, b]^{−e}.

Hence e = p − 1. As we have argued that Q/Z_{4} is not inverted by τ, this is a contradiction. □

Lemma 4.12. If W_{x} ∈ E for some x ∈ S\(Q ∪ R), then p = 7.

Proof. This follows from the argument in [20, Theorem 2.9]. □

We can now prove Theorem 4.2.

Proof of Theorem 4.2. This follows from Lemmas 4.3, 4.4, 4.11 and 4.12. □

5. Determining the fusion systems up to isomorphism when p ≥ 5

Our hypotheses for this section are that F is a saturated fusion system on S with O_{p}(F) = 1 and E is the set of F-essential subgroups of S. Here is the result we shall prove:

Theorem 5.1. Let p ⩾ 5, S be a Sylow p-subgroup of G_{2}(p) and F be a saturated fusion system on S with O_{p}(F) = 1. Then either F is isomorphic to the fusion system of G_{2}(p) or else p ≤ 7 and F is isomorphic to a subsystem of p′-index in one of the fusion systems listed in Table 5.1.

Table 5.1. Exceptional fusion systems F on S with O_{p}(F) = 1

p Out_{F}(W) Out_{F}(R)  Out_{F}(Q)  Out_{F}(S) Example Γ_{p′}(F)
F_{5}^{0} 5  –  GL_{2}(5)  2. Alt(6).4  4 × 4  Ly  1
F_{5}^{1} 5  –  GL_{2}(5) 4 ∘ 2_{−}^{1+4}.Frob(20) 4 × 4 Aut(HN)  2
F_{5}^{2} 5  –  GL_{2}(5)  2_{−}^{1+4}. Alt(5).4  4 × 4  B  1
F_{7}^{0} 7  –  GL_{2}(7) 3 × 2. Sym(7)  6 × 6  –  1
F_{7}^{1}(1_{1}) 7 SL_{2}(7)  –  –  6  –  1
F_{7}^{1}(2_{1}) 7 SL_{2}(7)  –  –  6  –  1
F_{7}^{1}(2_{2}) 7 SL_{2}(7)  –  –  6  –  1
F_{7}^{1}(2_{3}) 7 SL_{2}(7)  –  –  6 × 2  –  2
F_{7}^{1}(3_{1}) 7 SL_{2}(7)  –  –  6  –  1
F_{7}^{1}(3_{2}) 7 SL_{2}(7)  –  –  6  –  1
F_{7}^{1}(3_{3}) 7 SL_{2}(7)  –  –  6  –  1
F_{7}^{1}(3_{4}) 7 SL_{2}(7)  –  –  6 × 3  –  3
F_{7}^{1}(4_{1}) 7 SL_{2}(7)  –  –  6  –  1
F_{7}^{1}(4_{2}) 7 SL_{2}(7)  –  –  6  –  1
F_{7}^{1}(4_{3}) 7 SL_{2}(7)  –  –  6 × 2  –  2
F_{7}^{1}(5) 7 SL_{2}(7)  –  –  6  –  1
F_{7}^{1}(6) 7 SL_{2}(7)  –  –  6 × 6  –  6
F_{7}^{2}(1) 7 SL_{2}(7) SL_{2}(7).2  –  6 × 2  –  1
F_{7}^{2}(2) 7 SL_{2}(7) SL_{2}(7).2  –  6 × 2  –  1
F_{7}^{2}(3) 7 SL_{2}(7) GL_{2}(7)  –  6 × 6  –  3
F_{7}^{3} 7 SL_{2}(7)  –  GL_{2}(7)  6 × 6  –  1
F_{7}^{4} 7 SL_{2}(7)  –  3 × 2. Sym(7)  6 × 6  –  1
F_{7}^{5} 7 SL_{2}(7) GL_{2}(7)  GL_{2}(7)  6 × 6  –  1
F_{7}^{6} 7 SL_{2}(7) GL_{2}(7) 3 × 2. Sym(7)  6 × 6  M  1

A description of the fusion systems in Table 5.1 is developed throughout this section. Especially for the fusion systems F_{7}^{1}(j_{i}) see the discussion surrounding Notation 5.16.

We first present a preliminary result for the case when Q ∈ E.

Lemma 5.2. Suppose Q ∈ E and | Aut_{F}(Q) : C_{Aut_{F} (Q)}(Aut_{S}(Q))| = | Aut_{Aut_{F} (Q)}(Aut_{S}(Q))| = p − 1. Then either Out_{F}(Q) ≅ GL_{2}(p) or one of the following holds:

(a) p = 5 and Out_{F}(Q) ∼ 2. Alt(6).4; (b) p = 5 and Out_{F}(Q) ∼ 4 ∘ 2_{−}^{1+4}.Frob(20);

(c) p = 5 and Out_{F}(Q) ∼ 2_{−}^{1+4}.Frob(20);

(d) p = 5 and Out_{F}(Q) ∼ 2_{−}^{1+4}. Alt(5).4;

(e) p = 5 and Out_{F}(Q) ≅ GL_{2}(5) with Out_{F}(Q) having reducible action on Q/Z; or

(f) p = 7 and Out_{F}(Q) ∼ 3 × 2. Sym(7). Furthermore, in each case Aut_{F}(Q) is uniquely determined up to conjugacy as a subgroup of Aut(Q).

Proof. The possibilities for the isomorphism type of Out_{F}(Q) are an immediate consequence of Theorem 2.11. □

5.1. The case E ∩ W = ∅. By Theorem 4.2, E ∩ W ≠ ∅ implies that p = 7 and so the typical case occurs when E ⊆ {Q, R}. We consider this scenario in this subsection.

Lemma 5.3. If E ⊆ {Q, R}, then E = {Q, R}.

Proof. Since S ≠ O_{p}(X), we may suppose that E ⊂ {Q, R} has a unique element X ∈ {Q, R}. Then

F = ⟨Aut_{F}(X), Aut_{F}(S)⟩

by Theorem 2.2. By Lemma 3.2 (d), X is a characteristic subgroup of S and thus we see that X = O_{p}(F) = 1, which is a contradiction. □

In our proof of uniqueness of the fusion systems, we remark that the next result only demonstrates the existence of certain morphisms.

Lemma 5.4. If R ∈ E, then | Aut_{F}(Q) : C_{Aut_{F} (Q)}(Z)| = p−1 and | Aut_{Aut_{F} (Q)}(Aut_{S}(Q))| = p−1. In particular, if Q ∈ E, then Aut_{F}(Q) is determined uniquely as a subgroup of Aut(Q) up to conjugacy and is as tabulated in Table 5.1.

Proof. By Lemma 4.5, O^{p′}(Out_{F}(R)) = SL_{2}(p) and this group acts faithfully on Z(R). Thus there exists θ ∈ N_{Aut_{F} (R)}(Aut_{S}(R)) such that θ induces an automorphism of Z of order p − 1. Since F is saturated, θ extends to an element of Aut_{F}(S) and then by restriction we obtain an element θ_{0} of Aut_{F}(Q) which acts on Z with order p − 1 = | Aut(Z)|. Hence θ ∉ C_{Aut_{F} (Q)}(Z) and then, by Proposition 3.4, | Aut_{F}(Q) : C_{Aut_{F} (Q)}(Z)| = p − 1. Now we note that θ acts on R/(Q ∩ R) ≅ RQ/Q = S/Q faithfully and so we also have | Aut_{Aut_{F} (Q)}(Aut_{S}(Q))| = p − 1. The final statement follows from Lemma 5.2. □

The relevance of the final statement in the previous lemma comes from the following observation. If H and K are subgroups of Aut(Q) which contain Aut_{S}(Q) as a Sylow p-subgroup then H and K are Aut(Q)-conjugate if and only if they are N_{Aut(Q)}(Aut_{S}(Q))-conjugate. By Proposition 3.8, conjugation by elements of N_{Aut(Q)}(Aut_{S}(Q)) lift to automorphisms of S and therefore we may assume when constructing our fusion systems that Aut_{F}(Q) = H. After this, no further choices up to conjugacy in Aut_{F}(X) for X ∈ E can be made in such an arbitrary fashion. We also have the following consequence of Lemma 5.4:

Corollary 5.5. If {Q, R} ⊆ E, then either Out_{F}(S) has order (p − 1)^{2} or p = 5, Lemma 5.2 case (b) holds and Out_{F}(S) has order 8. In any case, |C_{Out_{F} (R)}(Out_{S}(R))| > 2.

Proof. Since elements of N_{Aut_{F} (Q)}(Aut_{S}(Q)) extend to Aut_{F}(S), the first part of the corollary follows by scrutinizing the groups listed in Table 5.1. As the elements of Aut_{F}(S) restrict to members of N_{Aut_{F} (R)}(Aut_{S}(R)) and Out_{S}(R) has order p, if | Out_{F}(S)| has order (p − 1)^{2} then |C_{Out_{F} (R)}(Out_{S}(R))| = p − 1 > 2. So suppose that Lemma 5.2 case (b) occurs. Recall that O^{p′}(Out_{F}(R)) ≅ SL_{2}(5). Hence

Out_{F}(R) = N_{Out_{F} (R)}(Out_{S}(R))O^{p′}(Out_{F}(R)) ∼ SL_{2}(5).2

and this group is a subgroup of GL_{2}(5). From these observations we conclude that Out_{F}(R) ≅ 4 ∘ SL_{2}(5). This proves the claim. □

Lemma 5.6. If {Q, R} ⊆ E, then either Out_{F}(R) ≅ GL_{2}(p) or p = 5 and Out_{F}(R) ≅ 4 ∘ SL_{2}(5) with Aut_{F}(Q) as described in 5.2 (c). Furthermore Aut_{F}(R) is uniquely determined as a subgroup of Aut(R).

Proof. Since R ∈ E, Out_{F}(R) is isomorphic to a subgroup of GL_{2}(p) and O^{p′}(Out_{F}(R)) ≅ SL_{2}(p) by Lemma 4.5. By Corollary 5.5 we have Out_{F}(R) is isomorphic to GL_{2}(p) or Lemma 5.2 case (b) holds with Out_{F}(R) ≅ 4 ∘ SL_{2}(5). Set

\overline{Y} = N_{Out_{F} (R)}(Out_{S}(R)) and \overline{T} = Out_{S}(R).

Then C\overline{Y} (\overline{T}) has order greater than 2 by Corollary 5.5. Thus Lemma 3.6 implies that Aut_{F}(R) is uniquely determined as a subgroup of Aut(R). □

Corollary 5.7. Suppose that {Q, R} ⊆ E. Then ⟨Aut_{F}(Q), Aut_{F}(R), Aut_{F}(S)⟩ is uniquely determined as a subsystem of F. In particular, if E = {Q, R}, then F is uniquely determined by Aut_{F}(Q), Aut_{F}(R) and Aut_{F}(S).

Proof. This follows from Lemmas 5.3, 5.4 and 5.6. □

Lemma 5.8. Suppose that E ⊆ {Q, R}. Then Γ_{p′}(F) = 1 or p = 5, Lemma 5.2 (c) holds and Γ_{5′}(F) = 2.

Proof. We have O^{p′}(Out_{F}(R)) ≅ SL_{2}(p) and so this group contributes a cyclic group of order p − 1 which acts faithfully on Z to Out_{F}^{0}(S) = Aut_{F}^{0}(S)/ Inn(S). If N_{O^{p′}(Out_{F} (Q))}(Aut_{S}(Q)) has order p − 1, then, as O^{p′}(Aut_{F}(Q)) centralizes Z, we have | Out_{F}^{0}(S)| = (p − 1)^{2} and we obtain Γ_{p′}(F) = 1. Now examining the groups in listed in Lemma 5.2, yields that the only possibility for Γ_{p′}(F) to be non-trivial arises when p = 5 and Lemma 5.2 (b) holds. In this case Out_{F}(S) ≅ 4 × 4 and Out_{F}^{0}(S) has index 2. □

Theorem 5.9. Suppose that E ⊆ {Q, R} and F is a saturated fusion system on S with O_{p}(F) = 1. Then E = {Q, R} and F is isomorphic to the fusion system of G_{2}(p) or to F_{5}^{0}, F_{5}^{1}, F_{5}^{2} or F_{7}^{0} or to a subsystem of index 2 in F_{5}^{1} as listed Table 5.1.

Proof. From Corollary 5.7, F is uniquely determined. Thus we only need to check that O_{p}(F) = 1. If Aut_{F}(Q) acts irreducibly on Q/Z, then the only candidates for O_{p}(F) are Z and Q. Since O_{p}(F) is contained in all the F-essential subgroups and Aut_{F}(R) does not normalize Z, we are done. The only possibility which arises with Aut_{F}(Q) acting reducibly on Q/Z, occurs in Lemma 5.2 (e). In this case, p = 5 and using the detail in Lemma 2.11 (c), we see that Aut_{F}(Q) leaves invariant the unique normal subgroup of S of order 5^{3}. That is Aut_{F}(Q) leaves Z_{3} invariant. Since Z_{3} = Φ(R) is also invariant under the action of Aut_{F}(R), we have O_{5}(F) = Z_{3} in this case, a contradiction. □

5.2. The case E ∩ W ≠ ∅. In this subsection, we have E ∩ W ≠ ∅ and consequently p = 7 by Lemma 4.12. Since G_{2}(7) has a 7-dimensional representation over 𝔽_{7}, S has exponent 7. In fact, as S is now a fixed group, we may use Magma [3] to perform calculations in S and also to calculate in the automorphism group of subgroups of S.

Lemma 5.10. Suppose that W ∈ E ∩ W. Then (a) Aut_{F}(W) ≅ SL_{2}(7) is uniquely determined;

(b) |N_{Aut_{F} (S)}(W) Inn(S)/ Inn(S)| = 6;

(c) | Aut_{F}(Q) : C_{Aut_{F} (Q)}(Z)| = 6; and

(d) | Aut_{Aut_{F} (Q)}(Aut_{S}(Q))| = 6.

Furthermore, the subgroup N_{Aut_{F} (S)}(W) Inn(S)/ Inn(S) acts as scalar matrices on S/[S, S] and is independent of the choice of W ∈ E ∩ W.

Proof. Suppose that W ∈ W. Then O^{p′}(Aut_{F}(W)) ≅ SL_{2}(7). Since F is saturated, any element of N_{Aut_{F} (W)}(Aut_{S}(W)) extends to an automorphism of N_{S}(W) and then to an automorphism of S. Let D ≤ Out_{F}(S) denote the subgroup generated by these extended automorphisms. Since Q[S, S] ≠ W [S, S] ≠ R[S, S], we see that D normalizes three distinct cyclic groups of order 7. It follows that D acts as scalars on S/[S, S] and so |D| ≤ 6. On the other hand, |N_{Aut_{F} (W)}(Aut_{S}(W))/ Aut_{S}(W)| ≥ 6 with equality if and only if Aut_{F}(W) = O^{p′}(Aut_{F}(W)). This proves (a) and part (b) follows as D = N_{Aut_{F} (S)}(W) Inn(S)/ Inn(S).

Moving on to (c) and (d). By (a) there exists θ ∈ N_{Aut_{F} (W)}(Aut_{S}(W)) such that θ induces an automorphism of Z of order 6. Since F is saturated and Z is fully F-centralized, θ extends to an element of Aut_{F}(S) and then by restriction we obtain an element of Aut_{F}(Q) which acts on Z with order 6 = | Aut(Z)|. This proves (c). Since θ acts on R/(Q ∩ R) ≅ S/Q faithfully, we also have | Aut_{Aut_{F} (Q)}(Aut_{S}(Q))| = 6 which proves (d). □

Motivated by Lemma 5.10 (b), for an arbitrary subgroup W ∈ W ∩ E we define

Δ = N_{Aut_{F} (S)}(W).

Then Δ/ Inn(S) is cyclic or order 6.

Lemma 5.11. The following hold:

(a) Inn(S) has six orbits on W;

(b) for W, X ∈ W, W [S, S] = X[S, S] if and only if W and X are in the same S-orbit;

(c) Aut(S) acts transitively on W;

(d) if W ∩ E ≠ ∅, then W is the union of \frac{36}{| Out_{F} (S)|} F-conjugacy classes of subgroups of order 49.

Proof. As S has exponent 7, the number of subgroups of S of order 49 which are not contained in Q or R and contain Z is

|W| = \frac{|S| − |Q| − (|R| − |R ∩ Q|)}{(49 − 7)} = \frac{7^{6} − 7^{5} − (7^{5} − 7^{4})}{42} = 7^{3}.6.

Here we use the fact that W_{x} = W_{x′} if and only if x′ ∈ W_{x}\Z, where x ∈ S\Q ∪ R and W_{x} is as defined in Notation 4.1. Let W ∈ W. Then, as W ≰ Q and W ≰ R, N_{S}(W) = W Z_{2}. Thus |W^{S}| = |S : W Z_{2}| = 7^{3} and so W is the union of six S-orbits. This proves (a). Now W_{x}, W_{y} ∈ W are in the same S-orbit if and only if W_{x}[S, S] = W_{y}[S, S] which is (b). Since Aut(S)/C_{Aut(S)}(S/[S, S]) acts as diagonal matrices on S/[S, S] by Proposition 3.8, we see that Aut(S) acts transitively on X = {W [S, S] | W ∈ W} and hence also on W. Thus (c) holds.

Assume that W ∩ E ≠ ∅. Then, for W ∈ W, |N_{Aut_{F} (S)}(W) Inn(S)/ Inn(S)| = 6 by Lemma 5.10 (b). Therefore Out_{F}(S) has 36/| Out_{F}(S)| orbits on X and so there are 36/| Out_{F}(S)| F-conjugacy classes. This proves (d). □

Lemma 5.12. If W ∩ E ≠ ∅ and Q ∈ E, then Aut_{F}(Q) is uniquely determined up to conjugacy in Aut(Q) and either Out_{F}(Q) ≅ GL_{2}(7) or 3 × 2. Sym(7). Furthermore, Aut_{F}(S) acts transitively on W.

Proof. Combining Lemmas 5.2 and 5.10 we obtain that Aut_{F}(Q) is unique up to conjugacy in Aut(Q) and the described structure of Out_{F}(Q). From the structure of Out_{F}(Q), we see that Out_{F}(S) has order 36. Hence Lemma 5.11 shows that Aut_{F}(S) acts transitively on W. □

Lemma 5.13. If W ∩ E ≠ ∅ and {Q, R} ⊂ E, then Aut_{F}(R) is uniquely determined, Out_{F}(R) ≅ GL_{2}(7) and Out_{F}(S) ≅ 6 × 6.

Proof. The proof of the uniqueness of Aut_{F}(R) follows the same steps as in Lemma 5.6. □

Theorem 5.14. If W ∩ E ≠ ∅ and Q ∈ E, then F is isomorphic to either F_{7}^{3}, F_{7}^{4}, F_{7}^{5} or F_{7}^{6}.

Proof. This follows by combining Lemmas 5.10, 5.12 and 5.13. □

We now move on to the case where E ⊆ W ∪ {R}. This time we take a slightly different route to demonstrate uniqueness of the fusion systems which arise. We start by fixing Aut_{F}(S) in a certain subgroup of Aut(S). Thus we identify Aut_{F}(S) as a subgroup Aut_{B}(S) where B is as defined in (3.3). This identification is justified by Proposition 3.8. A complement to Inn(S) in Aut_{F}(S) can be generated by the conjugation action of elements of the form

d = (t, ( \begin{matrix}λ & 0 \\ 0 & μ \end{matrix} ))

where t, λ, μ ∈ 𝔽_{7}^{×}. Here we recall that by (3.2), the kernel of this action consists of the elements

(μ^{3}, ( \begin{matrix}μ & 0 \\ 0 & μ \end{matrix} )) .

The next lemma is similar to [20, Lemma 2.7].

Lemma 5.15. Suppose that W ∩ E ≠ ∅. Then Δ is generated by the conjugation action of the elements

d = (λμ^{−4}, ( \begin{matrix}λ & 0 \\ 0 & μ \end{matrix} ))

where λ, μ ∈ 𝔽_{7}^{×} and Inn(S).

Proof. Let W ∈ W ∩ E. Then W [S, S]/[S, S] is a subgroup of S/[S, S] of order 7 and Q[S, S] ≠ W [S, S] ≠ R[S, S]. Hence Δ normalizes three subgroups of order 7 in S/[S, S] and therefore acts as scalar matrices. Since Y^{3}. (t, ( \begin{matrix}λ & 0 \\ 0 & μ \end{matrix} )) = tμ^{3}Y^{3} and (1, ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} )) goes to (1, ( \begin{matrix}1 & 0 \\ λμ^{−1} & 1 \end{matrix} )) we require tμ^{3} = μ^{−1}λ and so t = λμ^{−4} as claimed. □

Notation 5.16. Suppose that I = 𝔽_{7}^{×} = {1, 2, 3, 4, 5, 6}. Then the orbits of 𝔽_{7}^{×} on non-empty subsets of I are denoted as follows.

1_{1} = {1},
2_{1} = {1, 2}, 2_{2} = {1, 3}, 2_{3} = {1, 6},
3_{1} = {1, 2, 3}, 3_{2} = {1, 2, 5}, 3_{3} = {1, 2, 6}, 3_{4} = {1, 2, 4},
4_{1} = {1, 2, 3, 4}, 4_{2} = {1, 2, 3, 5}, 4_{3} = {1, 2, 5, 6},
5_{1} = {1, 2, 3, 4, 5},
6_{1} = {1, 2, 3, 4, 5, 6}.

We note that these orbits are regular other than 2_{3}, 4_{3} (both of which have length 3) and 3_{4} (which has length 2.)

By Lemma 5.11

X = {W [S, S]/[S, S] | W ∈ W}

consists of the six diagonal subgroups to Q/[S, S] and R/[S, S] in S/[S, S] and if Y is an S-orbit on W, then {W [S, S]/[S, S] | W ∈ Y} ⊆ X. Since the elements of X correspond to S-orbits on W, we may sensibly denote the S-orbits on W by W_{i} where i ∈ 𝔽_{7}^{×} = I. Now the Aut_{B}(S)-orbits on the non-empty subsets of the set of S-orbits {W_{1}, …, W_{6}} on W have representatives as described in Notation 5.16. Let

B(j_{i}) = Stab_{Aut_{B}(S)}(j_{i}).

For an orbit representative j_{i}, define the fusion systems

G(j_{i}) = ⟨Aut_{F}(W), Δ | W ∈ W_{k}, k ∈ j_{i}⟩

and then put

F_{7}^{1}(j_{i}) = ⟨G(j_{i}), B(j_{i})⟩.

Theorem 5.17. Suppose that E ⊆ W. Then F is isomorphic to a subsystem of 7′-index of F_{7}^{1}(j_{i}) containing G(j_{i}) where j_{i} is an Aut_{B}(S)-orbit on the non-empty subsets of {W_{1}, …, W_{6}}.

Proof. This follows from the previous discussion. □

Theorem 5.18. Suppose W ∩E ≠ ∅ and R ∈ E and Q ∉ E. Then Aut_{F}(R) is uniquely determined and either

(1) Out_{F}(R) ≅ GL_{2}(7), Out_{F}(S) ≅ 6 × 6 and Aut_{F}(S) acts transitively on W; or

(2) Out_{F}(R) ∼ SL_{2}(7).2, Out_{F}(S) ≅ 6 × 2 and Aut_{F}(S) has three orbits each of length two on

W. In particular, F is isomorphic to either F_{7}^{2}(1), F_{7}^{2}(2), F_{7}^{2}(3) or O^{7′}(F_{7}^{2}(3)).

Proof. Let W ∈ W ∩ E and let Δ̃ represent the subgroup of Aut_{F}(R) obtained by lifting the morphisms in N_{Aut_{F} (W)}(Aut_{S}(W)) to automorphisms of S and then restricting them to automorphisms of R. Thus Δ̃ is just the restriction of Δ to R. By Lemma 5.15, Δ̃ consists of restrictions of the elements c_{d} where

d = (λμ^{−4}, ( \begin{matrix}λ & 0 \\ 0 & μ \end{matrix} ))

where λ, μ ∈ 𝔽_{7}^{×}. We calculate that Δ̃ is cyclic of order 6 and that on R/Φ(R) we can select a basis so that such elements act as diagonal matrices diag(λ^{2}μ^{−2}, λμ^{−1}) and so have determinant λ^{3}μ^{−3} which is a cube. Recall from Lemma 5.10 (b) that Δ̃ is independent of the choice of W ∈ W. Thus Out_{F}(R) ≥ ⟨O^{7′}(Out_{F}(R)), Δ̃⟩ ≅ SL_{2}(7).2, the unique subgroup of GL_{2}(7) of index 3. In addition, as Δ̃ acts as scalars on S/[S, S], Out_{S}(R) admits Δ̃ faithfully. Now calculating in Aut(R) using Magma [3] for example, we see that there is a unique subgroup X of Aut(R) containing Inn(R) with X/ Inn(R) ≅ SL_{2}(7) which is normalized by Aut_{S}(R)Δ̃. Furthermore, N_{Aut(R)}(X)/ Inn(R) ≅ GL_{2}(7). This means that Out_{F}(R) ≅ SL_{2}(7).2 or GL_{2}(7) and Aut_{F}(R) is uniquely determined as a subgroup of Aut(R). Since Δ̃ ∩ N_{O^{7′}(Aut_{F} (R)}(Aut_{S}(R)) has index 2 in Δ̃, N_{O^{7′}(Aut_{F} (R)}(Aut_{S}(R)) has 3 orbits of length 2 on X with representative of the first orbit being given by 2_{3} as in Notation 5.16. Let W_{1,6}, W_{3,4} and W_{2,5} be the orbits of N_{O^{7′}(Aut_{F} (R)}(Aut_{S}(R))D̃ on W. If Out_{F}(R) ≅ GL_{2}(7), then Aut_{F}(S) = Aut_{B}(S) is transitive on W and we have no choices to make. Thus in this case

F = ⟨Aut_{F}(R), Aut_{F}(W), Aut_{F}(S)⟩

and this is the fusion system F_{7}^{2}(3). So suppose that Aut_{F}(S) is generated by the lifts of the elements from N_{O^{7′}(Aut_{F} (R))}(Aut_{S}(R)) to S and Δ. Then Out_{F}(S) ≅ 6×2 and Aut_{F}(R) ≅ SL_{2}(7).2. In this case, up to conjugacy by Aut_{B}(S), we may suppose that one of the following holds

F = ⟨Aut_{F}(R), Aut_{F}(S), Aut_{F}(W) | W ∈ W_{1,6}⟩;
F = ⟨Aut_{F}(R), Aut_{F}(S), Aut_{F}(W) | W ∈ W_{1,6} ∪ W_{3,4}⟩; or
F = ⟨Aut_{F}(R), Aut_{F}(S), Aut_{F}(W) | W ∈ W⟩.

These fusion systems are F_{7}^{2}(1), F_{7}^{2}(2) and O^{7′}(F_{7}^{2}(3)) respectively. □

We can now complete the proof of Theorem 5.1:

Proof of Theorem 5.1. This follows from Theorem 5.9 in the case W ∩ E = ∅ and from Theorems 5.14, 5.18 and 5.17 in the case W ∩ E ≠ ∅. □

6. Completing the proof of Theorem 1.1 for p ≥ 5

In order to complete the proof of Theorem 1.1, it remains to show that each of the fusion systems described in Table 5.1 exists and is saturated, and to establish which ones are realizable as fusion systems of finite groups.

Theorem 6.1. Each of the fusion systems listed in Theorem 5.1 exists and is saturated.

Proof. Examining the list of maximal subgroups of G_{2}(p), Ly, Aut(HN), B and M yields that the fusion systems F_{G_{2}(p)}(S), F_{5}^{0}, F_{5}^{1}, F_{5}^{2} and F_{7}^{6} are respectively isomorphic to the fusion systems of G_{2}(p), Ly, Aut(HN), B and M on their Sylow p-subgroups. In particular, each of these fusion systems is saturated. Let F = F_{7}^{0}. Then E = {Q, R} and N_{F}(Q) and N_{F}(R) are saturated fusion systems on S with O_{7}(N_{F}(Q)) = Q and O_{7}(N_{F}(R)) = R. Hence by [1, Theorem 4.9] there exist finite groups G_{1} and G_{2} which realize N_{F}(Q) and N_{F}(R) respectively. Moreover O_{p}(G_{1}) = Q and O_{p}(G_{2}) = R and Q and R are self-centralizing in these groups. In addition, we may realize N_{F}(S) by G_{12} which may be embedded into both G_{1} and G_{2}. Note that this configuration appears in the Monster sporadic simple group and so exists. Let G^{∗} be the free amalgamated product G_{1} ∗_{G_{12}} G_{2} and let Γ be the coset graph Γ(G^{∗}, G_{1}, G_{2}). Since the only F_{S}(G_{1})-essential subgroup is Q and the only F_{S}(G_{2})-essential subgroup is R, to invoke Theorem 2.5, we only have to demonstrate that for any F_{S}(S)-centric subgroup A, the fixed vertex set Γ^{A} is finite.

For adjacent vertices α, β ∈ Γ with α a coset of G_{1} and β a coset of G_{2}, we set Q_{α} = O_{7}(G_{α}) and R_{β} = O_{7}(G_{β}). Thus Q_{α} is G^{∗}-conjugate to Q and R_{β} is G^{∗} conjugate to R. We also set S_{αβ} = O_{7}(G_{αβ}) = G_{α} ∩ G_{β}. Notice that G_{αβ} = N_{G_{α}}(S_{αβ}) = N_{G_{β}}(S_{αβ}) and G_{αβ} is a maximal subgroup of G_{α} and G_{β}.

Assume that A ≤ S_{αβ} is S_{αβ}-centric. Seeking a contradiction we further assume that Γ^{A} is infinite. Notice first that any p-group which stabilizes an arc γ, δ, ε of length 2 is contained in S_{γδ} ∩ S_{δε} = O_{7}(G_{δ}) which is one of R_{γ} or Q_{γ}. Since Γ^{A} is infinite, we may consider a path emanating from the arc α, β of infinite length. We choose notation so that G_{α} = G_{2} and G_{β} = G_{1} and consider a path

α, β, γ, δ, ε, ζ, η

which is fixed by A. Since A stabilizes the arc α, β, γ, δ, A is contained in Q_{β} ∩ R_{γ}. In particular, A ≤ Q_{β} and, as A is S_{αβ}-centric, Z(Q_{β}) = Z(S_{αβ}) ≤ A. Thus

Q_{β} normalizes A.

Notice that Z(Q_{δ}) ≤ Z(R_{γ}) ≤ Q_{β} ≤ S_{αβ}. Therefore, using the fact that A fixes the arc α, β, γ, δ, ε, we deduce first that A ≤ Q_{δ} and second that Z(Q_{δ}) ≤ A. Now we have Z(R_{γ}) = Z(Q_{β})Z(Q_{γ}) ≤ A ≤ Q_{β} ∩ R_{γ} ∩ Q_{δ} = Φ(R_{γ}). Since Φ(R_{γ}) is abelian and Φ(R_{γ}) ≤ Q_{β} ≤ S_{αβ}, we now see that

A = Φ(R_{γ})

because A is S_{αβ}-centric. In particular, A is normalized by G_{γ}. Since A fixes the arc γ, δ, ε, ζ, η we have

A ≤ Q_{δ} ∩ R_{ε} ∩ Q_{ζ} = Φ(R_{ε}).

Since |A| = 7^{3}, we must have A = Φ(R_{ε}). Hence Φ(R_{ε}) = Φ(R_{γ}) and this subgroup is normalized by ⟨G_{γδ}, G_{δε}⟩ = G_{δ}. But then A = Φ(R_{γ}) is normalized by ⟨G_{γ}, G_{δ}⟩ = G^{∗} which is absurd. We conclude that Γ^{A} is finite and thus that F_{7}^{0} is saturated.

We are left only with the cases where E ∩ W ≠ ∅ and F is isomorphic to F_{7}^{1}(j_{i}), F_{7}^{2}(j) or F_{7}^{i} with 3 ≤ i ≤ 5. Let E_{0} = E\(E ∩ W) (so E_{0} ⊆ {Q, R}) and define F_{0} = ⟨Aut_{F}(P) | P ∈ E_{0}⟩. Then F_{0} is saturated because in each case it is the fusion system of a finite group. We intend to apply Lemma 2.6 with {W_{1}, W_{2}, …, W_{m}} a set of representatives for the set of F_{0}-conjugacy classes of subgroups in E ∩ W. Observe that:

- W_{i} is F_{0}-centric and minimal under inclusion amongst all F-centric subgroups (note that any F-centric subgroup must properly contain Z(S));

- no proper subgroup of W_{i} is F_{0}-essential. Hence the hypotheses of Theorem 2.6 are satisfied and F is saturated and exists as the fusion system of a tree of groups. Finally, we note that by Theorem 2.3, the fusion systems corresponding to subgroups of Γ_{7′}(F) are also saturated. □

Theorem 6.2. Let p ⩾ 5, S be a Sylow p-subgroup of G_{2}(p) and F be a saturated fusion system on S with O_{p}(F) = 1. Then either F is isomorphic to the fusion system of G_{2}(p) or p ≤ 7 and one of the following holds:

(a) F is exotic;

(b) F is isomorphic to the fusion system of one of the simple groups listed in Theorem 2.12 parts (a) and (b);

(c) F = F_{5}^{1} and F is isomorphic to the fusion system of Aut(HN).

Proof. Suppose that F = F_{S}(G) for some finite group G with S ∈ Syl_{p}(G). Since F_{S}(G) = F\overline{S}(\overline{G}) where \overline{G} = G/O_{p′}(G), we may suppose that O_{p′}(G) = 1. Let N be a minimal normal subgroup of G. Then 1 ≠ S ∩ N is a normal subgroup of S and so Z(S) ≤ N.

If W ∈ E, then p = 7 and W = ⟨Z(S)^{Aut_{F} (W)}⟩ ≤ N and hence

W Z_{4} = W [W, Q] ≤ W [W, S] = ⟨W^{S}⟩ ≤ N.

Let M = N_{G}(W), then (N ∩ M)C_{G}(W)/C_{G}(W) contains a Sylow p-subgroup of M/C_{G}(W) and so, as N ∩ M is normal in M, and M/C_{G}(W) ≅ SL_{2}(7), (N ∩ M)C_{G}(W) = M. So F_{S}(NS) must contain one of the fusion systems O^{7′}(F_{7}^{i}) for 1 ≤ i ≤ 6. But by Lemma 5.15 Out_{F_{S}(NS)}(S) satisfies [S, Out_{NS}(S)] = S and this means that S ≤ N.

If W ∉ E, then E = {Q, R} by Lemma 5.3. Thus Z_{2} = Z(R) ≤ ⟨Z(S)^{Aut_{F} (R)}⟩ ≤ N, and then Q = ⟨Z(R)^{Aut_{F} (Q)}⟩ ≤ N. Now Z_{4} ≤ N and so R = ⟨Z_{4}^{Aut_{F} (R)}⟩ ≤ N. Thus S = QR ≤ N.

It follows that for all the fusion systems under investigation we have S ∈ Syl_{p}(N). Plainly N is non-abelian and so N is a direct product of isomorphic non-abelian simple groups. Therefore, as Z(S) has order p, we have that N is simple and that G is almost simple. Since S ∈ Syl_{p}(N), Theorem 2.12 shows that either N ≅ G_{2}(p) or p ≤ 7 and N is one of the sporadic simple groups Ly, HN, B or M. Furthermore, in all cases except for N ≅ HN we have Out(N) = 1 and so either G = N or G = Aut(HN). It is now straight forward to match fusion systems to groups and this proves the theorem. □

We now complete the proof of Theorem 1.1:

Proof of Theorem 1.1. This follows on combining Theorems 5.1, 6.1 and 6.2. □

7. Fusion systems on a Sylow 3-subgroup of G_{2}(3)

We classify all saturated fusion systems on S where, in this section, S is the group U constructed in the appendix in the case 𝔽 = 𝔽_{3}. Set

Q_{1} = ⟨x_{β}, x_{α+β}, x_{α+2β}, x_{α+3β}, x_{2α+3β}⟩ and Q_{2} = ⟨x_{α}, x_{α+β}, x_{α+2β}, x_{α+3β}, x_{2α+3β}⟩.

Since S is isomorphic to a Sylow 3-subgroup of G_{2}(3), we can check all the claims that we make about S by computer calculations using [3]. For example, we see that Q_{1} ≅ Q_{2} ≅ 3^{2} × 3_{+}^{1+2} and that every element of order 3 in S is contained in either Q_{1} or Q_{2}. Whenever we use the word calculate in the coming arguments, we mean calculate using Magma [3].

In this section we will prove the following result.

Theorem 7.1. Let S be a Sylow 3-subgroup of G_{2}(3) and F be a saturated fusion system on S with O_{3}(F) = 1. Then F is isomorphic to the fusion system of G_{2}(3) or Aut(G_{2}(3)).

Assume that F is a saturated fusion system on S. To prove Theorem 7.1 it suffices to demonstrate that up to isomorphism there are exactly two possible fusions systems on S with O_{3}(F) = 1.

Lemma 7.2. Suppose that E ≤ S is an F-essential subgroup of F. Then E ≤ Q_{1} or E ≤ Q_{2}.

Proof. Suppose that the claim is false. We first examine the possibility that E ∩ Q_{1} = E ∩ Q_{2}. In this case |E/(E ∩ Q_{1} ∩ Q_{2})| and every element of E \ Q_{1} has order 9. Thus

E ∩ Q_{1} = E ∩ Q_{2} = E ∩ Q_{1} ∩ Q_{2} = Ω_{1}(E).

Because E is centric, E ∩ Q_{1} ≥ Z(S). Since E ≥ Z(S) and [S, Q_{1} ∩ Q_{2}] = Z(S), E is normalized by Q_{1} ∩ Q_{2} and so

[E, Q_{1} ∩ Q_{2}] ≤ E ∩ Q_{1} ∩ Q_{2} = Ω_{1}(E)

(as Q_{1} ∩ Q_{2} is elementary abelian). Furthermore, as Ω_{1}(E) ≤ Q_{1} ∩ Q_{2} and Q_{1} ∩ Q_{2} is abelian, [Ω_{1}(E), Q_{1} ∩ Q_{2}] = 1. Hence

1 ⊴ Ω_{1}(E) ⊴ E

is an Aut_{Q_{1}∩Q_{2}}(E)-invariant chain and we conclude from Lemma 2.7 that Aut_{Q_{1}∩Q_{2}}(E) ≤ O_{p}(Aut_{F}(E)) = Inn(E). Thus Q_{1} ∩ Q_{2} ≤ E and

E is normalized by S = Q_{1}Q_{2}.

Since Z(Q_{1}) ≤ Q_{1} ∩ Q_{2}, we now have Z(E) ≤ C_{S}(Z(Q_{i})) = Q_{i} and so Z(S) ≤ Z(E) ≤ Q_{1} ∩ Q_{2}. Suppose that Z(E) > Z(S). Since, for i = 1, 2, E does not centralizes Z(Q_{i}), and |Z(Q_{i})| = 3^{3}, we have Q_{1} ∩ Q_{2} = Z(Q_{1})Z(E) = Z(Q_{2})Z(E). But then

[Q_{1} ∩ Q_{2}, E] = [Z(Q_{1})Z(E), E] = [Z(Q_{1}), E] = [Z(Q_{1}), Q_{1}E]
= [Z(Q_{1}), Q_{1}Q_{2}] = [Z(Q_{1}), Q_{2}] = Φ(Q_{2})

and

[Q_{1} ∩ Q_{2}, E] = [Z(Q_{2})Z(E), E] = [Z(Q_{2}), E] = [Z(Q_{2}), Q_{2}E]
= [Z(Q_{2}), Q_{1}Q_{2}] = [Z(Q_{2}), Q_{1}] = Φ(Q_{1})

whereas we know Φ(Q_{1}) ≠ Φ(Q_{2}). This contradiction shows that Z(E) = Z(S). Now, recalling that Ω_{1}(E) = Q_{1} ∩ E = E ∩ Q_{1} ∩ Q_{2}, we have

[E, Q_{1}] ≤ Ω_{1}(E)
[Ω_{1}(E), Q_{1}] ≤ [Q_{1} ∩ Q_{2}, Q_{1}] = Φ(Q_{1}) ≤ Z(E) and
[Z(E), Q_{1}] = 1.

Hence

1 ⊴ Z(E) ⊴ Ω_{1}(E) ⊴ E

is an Aut_{Q_{1}}(E)-invariant chain so that Aut_{Q_{1}}(E) ≤ O_{p}(Aut_{F}(E)) = Inn(E) which means that Q_{1} ≤ E and S = E, a contradiction. Hence E ∩ Q_{1} ≠ E ∩ Q_{2}.

Suppose E ∩ Q_{1} ≰ Q_{2}. Then (E ∩ Q_{1})(Q_{1} ∩ Q_{2}) = Q_{1}. So Q_{1} normalizes E ∩ Q_{1} and therefore EQ_{1} = S normalizes E ∩ Q_{1}. It follows that

(E ∩ Q_{1})Z(Q_{1})/Z(Q_{1}) ≥ C_{Q_{1}/Z(Q_{1})}(S) = (Q_{1} ∩ Q_{2})/Z(Q_{1})

and this implies that

(E ∩ Q_{1})Z(Q_{1}) = Q_{1}

and EZ(Q_{1}) = S. Now noting that E ≥ Z(S) and |Z(Q_{1}) : Z(S)| = 3 yields |E| = 3^{5}. Thus E is a maximal subgroup of S. Calculation shows that E′ ≥ Z(S). Thus

[E, S] = [E, EZ(Q_{1})] = E′[E, Z(Q_{1})]
≤ E′[S, Z(Q_{1})] ≤ E′Z(S) = E′.

Hence Aut_{S}(E) centralizes E/E′ and this means that Aut_{S}(E) ≤ O_{p}(Aut_{F}(E)), a contradiction.

□

Lemma 7.3. We have Q_{1} ∩ Q_{2} is not F-essential.

Proof. Suppose to the contrary that E = Q_{1} ∩ Q_{2} is F-essential. Then Out_{F}(E) has a strongly 3-embedded subgroup. Since E is normalized by S, Out_{S}(E) is elementary abelian of order 9. It follows from [11, Theorem 7.6.1] that, setting X = Out_{F}(E), O^{3′}(X/Z(X)) is isomorphic to one of PSL_{2}(9), PSL_{3}(4), or Mat(11). Because the latter two groups have order which does not divide | GL_{4}(3)|, we conclude that O^{3′}(X/Z(X)) ≅ PSL_{2}(9). Since C_{E}(S) = Z(S) has order 9, we deduce that O^{3′}(X) ≅ SL_{2}(9). But then, by [1, Theorem 4.9], N_{F}(E) is realized by a group which contains 3^{4} : SL_{2}(9) and this group has Sylow 3-subgroups of exponent 3, a contradiction. □

Lemma 7.4. The following hold.

(a) Out_{F}(S) is a subgroup of Dih(8);

(b) N_{Aut_{F} (S)}(Q_{1}) = N_{Aut_{F} (S)}(Q_{2}) has index at most 2 in Aut_{F}(S) and N_{Aut_{F} (S)}(Q_{1})/ Inn(S) is elementary abelian; and

(c) if t ∈ N_{Aut_{F} (S)}(Q_{1}) is an involution, then C_{Q_{i}}(t) has order either 3 or 3^{2}.

Proof. This can be checked using [3]. □

Lemma 7.5. Suppose that for i = 1 or 2, E < Q_{i} is F-essential. Then |E| = 3^{4}, E is elementary abelian, O^{3′}(Aut_{F}(E)) ≅ SL_{2}(3) and |[E, t]| = 9 for t ∈ Z(O^{3′}(Aut_{F}(E))) of order 2.

Proof. Without loss of generality, assume that E ≤ Q_{1}. Then, by Lemma 7.3, E ≠ Q_{1} ∩ Q_{2}. Since E ≤ Q_{1}, Z(Q_{1}) ≤ E and, as E is centric, E > Z(Q_{1}). Since, by assumption, E ≠ Q_{1} and using Q_{1} has exponent 3, we obtain E is elementary abelian of order 3^{4}. As E ≠ Q_{1} ∩ Q_{2}, we have N_{S}(E) = Q_{1} and so Aut_{S}(E) = Aut_{Q_{1}}(E) has order 3. Since [E, Q_{1}] = Φ(Q_{1}) has order 3 we may apply the main result of [15] to see that O^{3′}(Aut_{F}(E)) ≅ SL_{2}(3) and that |[E, t]| = 9 for t ∈ Z(O^{3′}(Aut_{F}(E))) of order 2. □

Lemma 7.6. Suppose that for i = 1 or 2, E < Q_{i} is F-essential. Then Q_{i} is F-essential.

Proof. Assume that Q_{i} is not F-essential. Let t ∈ Z(O^{3′}(Aut_{F}(E))) have order 2. Then t normalizes Aut_{S}(E) = Q_{i}/Z(E) and hence lifts to τ ∈ Aut_{F}(Q_{i}) by saturation. Since Q_{i} is not F-essential, τ = σ|_{Q_{i}} for some σ ∈ Aut_{F}(S). Now |C_{Q_{i}}(τ)| = |Q_{i}/E||C_{E}(t)| = 27 by Lemma 7.5. On the other hand, by Lemma 7.4 we have |C_{Q_{i}}(σ|_{Q_{i}})| ≤ 9, a contradiction. Hence Q_{i} is F-essential. □

Lemma 7.7. Suppose that Q_{i} is F-essential for some 1 ≤ i ≤ 2. Then Out_{F}(Q_{i}) acts faithfully on Q_{i}/Z(Q_{i}). In particular, O^{3′}(Out_{F}(Q_{i})) ≅ SL_{2}(3) and Out_{F}(Q) embeds into GL_{2}(3).

Proof. We may as well suppose that i = 1. Let X = C_{Aut_{F} (Q_{1})}(Q_{1}/Z(Q_{1})), we will show that X = Inn(Q_{1}). Since Out_{S}(Q_{1}) acts faithfully on Q_{1}/Z(Q_{1}) and Out_{S}(Q_{1}) has order 3, we have X/ Inn(Q_{1}) has 3′-order. Thus

Q_{1}/Φ(Q_{1}) = [Q_{1}/Φ(Q_{1}), X] × C_{Q_{1}/Φ(Q_{1})}(X)

by coprime action. Since X is normal in Aut_{F}(Q_{1}), this is a non-trivial decomposition which is Aut_{S}(Q_{1}) invariant. Suppose that C_{Q_{1}/Φ(Q_{1})}(X) ∩ Z(Q_{1})/Φ(Q_{1}) ≠ 1. Then

Z(Q_{1})/Φ(Q_{1}) = (C_{Q_{1}/Φ(Q_{1})}(X) ∩ Z(Q_{1})/Φ(Q_{1})) × [Q_{1}/Φ(Q_{1}), X]

as [Q_{1}/Φ(Q_{1}), X] = [Z(Q_{1})/Φ(Q_{1}), X]. This decomposition is also non-trivial and we deduce that Aut_{S}(Q_{1}) centralizes Z(Q_{1})/Φ(Q_{1}) which we calculate is not the case. Hence C_{Q_{1}/Φ(Q_{1})}(X) ∩ Z(Q_{1})/Φ(Q_{1}) = 1 and so [Q_{1}/Φ(Q_{1}), X] = Z(Q_{1})/Φ(Q_{1}). Now

|C_{Q_{1}/Φ(Q_{1})}(X)| = |[Q_{1}/Φ(Q_{1}), X]| = 9.

Thus [Q_{1}, S, S] ≤ Φ(Q_{1}). A calculation in S shows that this is not the case. Hence X = Inn(Q_{1}). Therefore, Out_{F}(Q_{1}) is isomorphic to a subgroup of GL_{2}(3) and, as Out_{S}(Q_{1}) is not normal in Out_{F}(Q_{1}), this proves the lemma. □

Proposition 7.8. If E ≤ S is F-essential, then E ∈ {Q_{1}, Q_{2}}.

Proof. Suppose that E is F-essential and that E ∉ {Q_{1}, Q_{2}}. Then by Lemma 7.2, without loss of generality we may assume Z(Q_{1}) < E < Q_{1}. Then by Lemma 7.6, Q_{1} is F-essential and by Lemma 7.7 Aut_{F}(Q_{1}) acts transitively on the maximal subgroups of Q_{1} containing Z(Q_{1}). In particular, E is F-conjugate to Q_{1} ∩ Q_{2} and this contradicts E being fully F-normalized. This contradiction shows that if E is F-essential, then E ∈ {Q_{1}, Q_{2}}. □

Lemma 7.9. We have Aut_{F}(Q_{1}) ≅ GL_{2}(3) ≅ Aut_{F}(Q_{2}) and Out_{F}(S) is either elementary abelian of order 4 or dihedral of order 8.

Proof. By Proposition 7.8, we may assume that Q_{1} is F-essential. If Q_{1}^{F} is the only class of essential subgroups, then, as O_{3}(F) = 1, we must have that Aut_{F}(S) has an element α which does not normalize Q_{1}. But then Q_{1}α = Q_{2}. It follows that either Q_{1} and Q_{2} are not F-conjugate and are both F-essential or that they are F-conjugate. Thus, by Lemma 7.7,

O^{3′}(Out_{F}(Q_{1})) ≅ O^{3′}(Out_{F}(Q_{2})) ≅ SL_{2}(3).

For 1 ≤ i ≤ 2, let τ_{i} ∈ Aut_{F}(Q_{i}) project to an involution in Z(O^{3′}(Out_{F}(Q_{i}))). Then τ_{i} lifts to τ̂_{i} ∈ Aut_{F}(S_{i}) of order 2. Furthermore, on S/(Q_{1} ∩ Q_{2}), these maps normalize both Q_{1}/(Q_{1} ∩ Q_{2}) and Q_{2}/(Q_{1} ∩ Q_{2}) with τ̂_{1} centralizing S/Q_{1} and inverting Q_{1}/(Q_{1} ∩ Q_{2}) whereas τ̂_{2} centralizes S/Q_{2} and inverts Q_{2}/(Q_{1} ∩ Q_{2}) by Lemma 7.7. It follows that Out_{F}(S) ≥ ⟨τ̂_{1}, τ̂_{2}⟩ Inn(S)/ Inn(S) which is elementary abelian of order 4. Thus

Out_{F}(Q_{1}) = ⟨τ̂_{2}|_{Q_{1}}, O^{3′}(Aut_{F}(Q_{1}))⟩/ Inn(Q_{1}) ≅ GL_{2}(3), and
Out_{F}(Q_{2}) = ⟨τ̂_{1}|_{Q_{2}}, O^{3′}(Aut_{F}(Q_{2}))⟩/ Inn(Q_{2}) ≅ GL_{2}(3).

Finally, either Out_{F}(S) = ⟨τ̂_{1}, τ̂_{2}⟩ Inn(S)/ Inn(S) or Out_{F}(S) ≅ Dih(8) by Lemma 7.4.

□

Proof of Theorem 7.1. By Lemmas 7.4 and 7.9 Aut_{F}(S) has a subgroup Aut_{F}^{0}(S) of index at most 2 which has order 2^{2}3^{4} with elementary abelian Sylow 2-subgroups. This subgroup normalizes both Q_{1} and Q_{2} and is uniquely determined up to conjugacy in Aut(S). We fix it once and for all. Let N = N_{Aut(S)}(Aut_{F}^{0}(S)). Then N has a subgroup N^{0} of index 2 which normalizes both Q_{1} and Q_{2}. Now we calculate using Magma that, for i = 1, 2, the restriction K of Aut_{F}^{0}(S) to Q_{i} is contained in exactly three subgroups X of Aut(Q_{i}) containing Inn(Q_{i}) which have X/ Inn(Q_{i}) ≅ GL_{2}(3). Since K must coincide with N_{Aut_{F} (Q_{i})}(Aut_{S}(Q_{i})) ∼ 3_{+}^{1+2}.2^{2} we see, using Lemma 7.9, that there are exactly three candidates for the subgroup Aut_{F}(Q_{1}) of Aut(Q_{1}) and also three candidates for the subgroup Aut_{F}(Q_{2}) of Aut(Q_{2}). Next we calculate that N_{0} restricted to Q_{1} conjugates these three candidates for Aut_{F}(Q_{1}) together and thus we have a subgroup N^{1} of index 3 in N_{0} which normalizes Aut_{F}(Q_{1}), Aut_{F}^{0}(S). We calculate that the restriction of N^{1} to Q_{2} acts transitively on the three candidates for Aut_{F}(Q_{2}). Thus the triple Aut_{F}^{0}(S), Aut_{F}(Q_{1}), Aut_{F}(Q_{2}) is uniquely determine up to Aut(S) conjugacy. If Out_{F}(S) has order 4, then Aut_{F}^{0}(S) = Aut_{F}(S) and F is uniquely determined up to isomorphism. If Aut_{F}(S) > Aut_{F}(S), we check that Out_{F}^{0}(S) is contained in a unique subgroup of order 8 which conjugates Aut_{F}(Q_{1}) to Aut_{F}(Q_{2}). This proves that there are exactly two saturated fusion systems on S up to isomorphism. Since F_{S}(G_{2}(3)) and F_{S}(Aut(G_{2}(3))) provide examples of fusion systems, we have completed the proof of the theorem. □

8. Appendix

8.1. Construction of a Sylow p-subgroup of G_{2}(q). Following Wilson [25], we construct the Sylow p-subgroup U of G_{2}(q). Let 𝔽 denote a finite field of order q = p^{f} for some prime p ≥ 3 and let 𝕆 = 𝔽[i_{0}, i_{1}, i_{2}, i_{3}, i_{4}, i_{5}, i_{6}] be the octonion algebra where, taking subscripts modulo 7, i_{t}, i_{t+1}, i_{t+3} satisfy same multiplication rules as i, j, k in the quaternions. For completeness, we give the multiplication table below:

i_{0} i_{1} i_{2} i_{3} i_{4} i_{5} i_{6}
i_{0} −1 i_{3} i_{6} −i_{1} i_{5} −i_{4} −i_{2}
i_{1} −i_{3} −1 i_{4} i_{0} −i_{2} i_{6} −i_{5}
i_{2} −i_{6} −i_{4} −1 i_{5} i_{1} −i_{3} i_{0}
i_{3} i_{1} −i_{0} −i_{5} −1 i_{6} i_{2} −i_{4}
i_{4} −i_{5} i_{2} −i_{1} −i_{6} −1 i_{0} i_{3}
i_{5} i_{4} −i_{6} i_{3} −i_{2} −i_{0} −1 i_{1}
i_{6} i_{2} i_{5} −i_{0} i_{4} −i_{3} −i_{1} −1

Wilson defines a new basis for 𝕆 as follows: first choose a, b ∈ 𝔽 such that a^{2} + b^{2} = −1, b ≠ 0 and define a new basis {y_{1}, y_{2}, …, y_{8}} of 𝕆 by setting:

\begin{matrix}2y_{1} = i_{4} + ai_{6} + bi_{0}, & 2y_{8} = i_{4} − ai_{6} − bi_{0} \\ 2y_{2} = i_{2} + bi_{3} + ai_{5}, & 2y_{7} = i_{2} − bi_{3} − ai_{5}, \\ 2y_{3} = i_{1} − bi_{6} + ai_{0}, & 2y_{6} = i_{1} + bi_{6} − ai_{0}, \\ 2y_{4} = 1 + ai_{3} − bi_{5}, & 2y_{5} = 1 − ai_{3} + bi_{5}. \end{matrix}

The new multiplication table is given on [25, p. 123] and involves coefficients ±1. Moreover, on [25, p. 124] Wilson gives a maximal unipotent subgroup U of G_{2}(q) in terms of its action with respect to this basis. Precisely, for λ ∈ 𝔽 it is generated by the matrices

\begin{matrix}x_{2α+3β}(λ) = ( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ −λ & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & λ & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ) , & x_{α+3β}(λ) = ( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ −λ & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & λ & 0 & 0 & 0 & 0 & 1 \end{matrix} ) , & x_{α+2β}(λ) = ( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ −λ & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ λ & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & −λ & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & λ & 0 & 0 & 0 & 1 & 0 \\ λ^{2} & 0 & 0 & −λ & λ & 0 & 0 & 1 \end{matrix} ) , \\ x_{α+β}(λ) = ( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ −λ & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & −λ & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & λ & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & λ^{2} & 0 & −λ & λ & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & λ & 0 & 1 \end{matrix} ) , & x_{β}(λ) = ( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & −λ & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & λ & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ) , & x_{α}(λ) = ( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ −λ & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & λ & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & −λ & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & λ^{2} & λ & −λ & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & λ & 1 \end{matrix} ) , \end{matrix}

which generate the root groups corresponding to the positive roots R = {α, β, α + β, α + 2β, α + 3β, 2α + 3β} of the root system of type G_{2}. It is easy to verify that the following relations hold for all λ, μ ∈ 𝔽:

[x_{β}(λ), x_{α}(μ)] = x_{2α+3β}(2μ^{3}λ^{2})x_{α+3β}(−μ^{3}λ)x_{α+2β}(μ^{2}λ)x_{α+β}(−μλ)
[x_{α+β}(λ), x_{α}(μ)] = x_{2α+3β}(−3μλ^{2})x_{α+3β}(3μ^{2}λ)x_{α+2β}(−2μλ)
[x_{α+2β}(λ), x_{α}(μ)] = x_{α+3β}(−3μλ)
[x_{α+3β}(λ), x_{β}(μ)] = x_{2α+3β}(3μλ)
[x_{α+2β}(λ), x_{α+β}(μ)] = x_{2α+3β}(−μλ)
[x_{r}(λ), x_{s}(μ)] = 1 for all other {r, s} ⊂ R.

Recall the definitions of the elements x_{1}(λ), x_{2}(λ), …, x_{6}(λ) in Section 3.1.

Proposition 8.1. Let ϕ : R ⟶ {1, 2, 3, 4, 5, 6} be the mapping which sends the tuple of elements (α, β, α + β, α + 2β, α + 3β, 2α + 3β) to (1, 2, 3, 4, 5, 6). Then the induced map U ⟶ S given for each λ ∈ 𝔽 by x_{r}(λ) ↦ x_{rϕ}(λ) is an isomorphism.

Proof. This is easily checked. □

References

[1] M. Aschbacher, R. Kessar, and B. Oliver, Fusion systems in algebra and topology, London Mathematical Society Lecture Note Series, 391. Cambridge University Press, Cambridge, (2011).

[2] M. Aschbacher and B. Oliver, Fusion systems, Bull. Amer. Math. Soc. online 2016.

[3] Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 235–265, 1997.

[4] J. Bray, D. Holt and C. Roney-Dougal, The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, LMS Lecture Note Series 407, 2013.

[5] C. Broto, N. Castellana, J. Grodal, R. Levi and B. Oliver, Subgroup families controlling p-local finite groups, Proc. London Math. Soc. (3) 91: 325 – 354, 2005.

[6] M. Clelland, Saturated fusion systems and finite groups, PhD, The University of Birmingham 2007.

[7] D. Craven, The theory of fusion systems, Cambridge Univ. Press 2011.

[8] D. Craven, B. Oliver and J. Semeraro, Reduced fusion systems over p-groups with abelian subgroup of index

    p:II, preprint 2016 https://arxiv.org/pdf/1606.05133v1.pdf.

[9] J. A. Gibbs Automorphisms of certain unipotent groups, J. Alg., 14 2:203–228, 1970.

[10] D. Gorenstein, Finite Groups, Harper & Row, 1968.

[11] D. Gorenstein, R. N. Lyons, R. M. Solomon, The Classification of the Finite Simple Groups, Number 3, Mathematical Surveys and Monographs Volume 40, 1998.

[12] Valentina Grazian, Fusion systems on p-groups of sectional rank 3 for p odd, PhD Thesis, University of Birmingham, expected 2017.

[13] E. Henke and S. Shpectorov, Fusion systems on Sylow subgroups of Sp_{4}(q), in preparation, 2016.

[14] M. Liebeck and G. Seitz, Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras, AMS Mathematical surveys and monographs 180, 2012.

[15] Jack McLaughlin, Some groups generated by transvections. Arch. Math. (Basel) 18 1967 364–368.

[16] Raul Moragues Moncho, Fusion systems on p-groups with an extraspecial subgroup of index p, PhD, University of Birmingham, expected 2018.

[17] B. Oliver and J. Ventura, Saturated fusion systems over 2-groups, Trans. AMS, 361:6661–6728, 2009.

[18] C. Parker, Fusion systems and amalgams, lecture notes, <http://www.math.nsc.ru/conference/isc/2014/lectures/Parker2014.pdf>.

[19] C. Parker and P. Rowley, Local characteristic p-completions of weak BN-pairs, Proc. London Math. Soc., (3)

93:325–394, 2006.

[20] C. Parker and G. Stroth, A family of fusion systems related to the groups Sp_{4}(p^{a}) and G_{2}(p^{a}), Arch. Math.

104, 104:311–323, 2015.

[21] Geoffrey R. Robinson, Amalgams, blocks, weights, fusion systems and nite simple groups, J. Algebra 314 (2007), no. 2, 912–923.

[22] A. Ruiz and Antonio Viruel, The classification of p-local finite groups over the extraspecial group of order p^{3} and exponent p, Math. Z. 248 (2004), no. 1, 45–65.

[23] J. Semeraro, Trees of fusion systems, J. Alg. , 399:1051–1072, 2014.

[24] J. P. Serre, Trees, Springer Verlag, Berlin 1980.

[25] R. A. Wilson The finite simple groups, Springer 2009.

[26] D. L. Winter The automorphism group of an extraspecial p-group, Rocky Mountain J. Math., 2 2:159-168, 1972.

School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom

E-mail address: c.w.parker@bham.ac.uk

Heilbronn Institute for Mathematical Research, Department of Mathematics, University of Bristol, United Kingdom

E-mail address: js13525@bristol.ac.uk