AUTOMORPHISMS OF FUSION SYSTEMS OF FINITE SIMPLE GROUPS OF LIE TYPE

CARLES BROTO, JESPER M. MØLLER, AND BOB OLIVER

Abstract. For a finite group G of Lie type and a prime p, we compare the automorphism groups of the fusion and linking systems of G at p with the automorphism group of G itself. When p is the defining characteristic of G, they are all isomorphic, with a very short list of exceptions. When p is different from the defining characteristic, the situation is much more complex, but can always be reduced to a case where the natural map from Out(G) to outer automorphisms of the fusion or linking system is split surjective. This work is motivated in part by questions involving extending the local structure of a group by a group of automorphisms, and in part by wanting to describe self homotopy equivalences of BG_{p}^{∧} in terms of Out(G).

Contents

Introduction 1 1. Tame and reduced fusion systems 10 2. Background on finite groups of Lie type 18 3. Automorphisms of groups of Lie type 25 4. The equicharacteristic case 30 5. The cross characteristic case: I 43 6. The cross characteristic case: II 68 Appendix A. Injectivity of μ_{G} by Bob Oliver 81 References 99

Introduction

When p is a prime, G is a finite group, and S ∈ Syl_{p}(G), the fusion system of G at S is the category F_{S}(G) whose objects are the subgroups of S, and whose morphisms are those homomorphisms between subgroups induced by conjugation in G. In this paper, we are

2000 Mathematics Subject Classification. Primary 20D06. Secondary 20D20, 20D45, 20E42, 55R35. Key words and phrases. groups of Lie type, fusion systems, automorphisms, classifying spaces. C. Broto is partially supported by MICINN grant MTM2010-20692 and MINECO grant MTM2013-42293-

P.

J. Møller is partially supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92) and by Villum Fonden through the project Experimental Mathematics in Number Theory, Operator Algebras, and Topology.

B. Oliver is partially supported by UMR 7539 of the CNRS, and by project ANR BLAN08-2 338236, HGRT.

All three authors wish to thank Københavns Universitet, the Universitat Autònoma de Barcelona, and especially the Centre for Symmetry and Deformation in Copenhagen, for their hospitality while much of this work was carried out.

interested in comparing automorphisms of G, when G is a simple group of Lie type, with those of the fusion system of G at a Sylow p-subgroup of G (for different primes p).

Rather than work with automorphisms of F_{S}(G) itself, it turns out to be more natural in many situations to study the group Out(L_{S}^{c}(G)) of outer automorphisms of the centric linking system of G. We refer to Section 1 for the definition of L_{S}^{c}(G), and to Definition 1.2 for precise definitions of Out(F_{S}(G)) and Out(L_{S}^{c}(G)). These are defined in such a way that there are natural homomorphisms

Out(G) ⟶^{κ_{G}} Out(L_{S}^{c}(G)) ⟶^{μ_{G}} Out(F_{S}(G)) and κ_{G} = μ_{G} ∘ κ_{G} .

For example, if S controls fusion in G (i.e., if S has a normal complement), then Out(F_{S}(G)) = Out(S), and κ_{G} is induced by projection to S. The fusion system F_{S}(G) is tamely realized by G if κ_{G} is split surjective, and is tame if it is tamely realized by some finite group G^{∗} where S ∈ Syl_{p}(G^{∗}) and F_{S}(G) = F_{S}(G^{∗}). Tameness plays an important role in Aschbacher’s program for shortening parts of the proof of the classification of finite simple groups by classifying simple fusion systems over finite 2-groups. We say more about this later in the introduction, just before the statement of Theorem C.

By [BLO1, Theorem B], Out(L_{S}^{c}(G)) ≅ Out(BG_{p}^{∧}): the group of homotopy classes of self homotopy equivalences of the p-completed classifying space of G. Thus one of the motivations for this paper is to compute Out(BG_{p}^{∧}) when G is a finite simple group of Lie type (in characteristic p or in characteristic different from p), and compare it with Out(G).

Following the notation used in [GLS3], for each prime p, we let 𝔏𝔦𝔢(p) denote the class of finite groups of Lie type in characteristic p, and let 𝔏𝔦𝔢 denote the union of the classes 𝔏𝔦𝔢(p) for all primes p. (See Definition 2.1 for the precise definition.) We say that G ∈ 𝔏𝔦𝔢(p) is of adjoint type if Z(G) = 1, and is of universal type if it has no nontrivial central extensions which are in 𝔏𝔦𝔢(p). For example, for n ≥ 2 and q a power of p, PSL_{n}(q) is of adjoint type and SL_{n}(q) of universal type.

Our results can be most simply stated in the “equi-characteristic case”: when working with p-fusion of G ∈ 𝔏𝔦𝔢(p).

Theorem A. Let p be a prime. Assume that G ∈ 𝔏𝔦𝔢(p) and is of universal or adjoint type, and also that (G, p) ≅ (Sz(2), 2). Fix S ∈ Syl_{p}(G). Then the composite homomorphism

κ_{G}: Out(G) ⟶^{κ_{G}} Out(L_{S}^{c}(G)) ⟶^{μ_{G}} Out(F_{S}(G))

is an isomorphism, and κ_{G} and μ_{G} are isomorphisms except when G ≅ PSL_{3}(2).

Proof. Assume G is of adjoint type. When G ≅ G̸L_{3}(2), μ_{G} is an isomorphism by [O1, Proposition 4.3]^{1} or [O2, Theorems C & 6.2]. The injectivity of κ_{G} = μ_{G} ∘ κ_{G} (in all cases) is shown in Lemma 4.3. The surjectivity of κ_{G} is shown in Proposition 4.5 when G has Lie rank at least three, and in Proposition 4.8 when G has Lie rank 1 and G ≅ S̸z(2). When G has Lie rank 2, κ_{G} is onto (when G ≅ S̸L_{3}(2)) by Proposition 4.12, 4.14, 4.15, 4.16, or 4.17. (See Notation 4.1(H) for the definition of Lie rank used here.)

If G is of universal type, then by Proposition 3.8, G/Z(G) ∈ 𝔏𝔦𝔢(p) is of adjoint type where Z(G) has order prime to p. Also, Out(G) ≅ Out(G/Z(G)) by [GLS3, Theorem 2.5.14(d)]. Hence F_{S}(G) ≅ F_{S}(G/Z(G)) and L_{S}^{c}(G) ≅ L_{S}^{c}(G/Z(G)); and κ_{G} and/or κ_{G} is an isomorphism if κ_{G/Z(G)} and/or κ_{G/Z(G)}, respectively, is an isomorphism. □

1Steve Smith recently pointed out to the third author an error in the proof of this proposition. One can get around this problem either via a more direct case-by-case argument (see the remark in the middle of page 345 in [O1]), or by applying [O3, Theorem C]. The proof of the latter result uses the classification of finite simple groups, but as described by Glauberman and Lynd [GLn, § 3], the proof in [O3] (for odd p)

When G = PSL_{3}(2) and p = 2, Out(G) ≅ Out(F_{S}(G)) ≅ C_{2} (and κ_{G} is an isomorphism, while Out(L_{S}^{c}(G)) ≅ C_{2}^{2}. When G = Sz(2) ≅ C_{5} ⋊ C_{4} and p = 2, Out(G) = 1, while Out(L_{S}^{c}(G)) ≅ Aut(C_{4}) ≅ C_{2}. Thus these groups are exceptions to Theorem A.

To simplify the statement of the next theorem, for finite groups G and H, we write G ∼_{p} H to mean that there are Sylow subgroups S ∈ Syl_{p}(G) and T ∈ Syl_{p}(H), together with an isomorphism φ: S ⟶^{≅} T which induces an isomorphism of categories F_{S}(G) ≅ F_{T} (H) (i.e., φ is fusion preserving in the sense of Definition 1.2).

Theorem B. Fix a pair of distinct primes p and q_{0}, and a group G ∈ 𝔏𝔦𝔢(q_{0}) of universal or adjoint type. Assume that the Sylow p-subgroups of G are nonabelian. Then there is a prime q_{0}^{∗} ≠ p, and a group G^{∗} ∈ 𝔏𝔦𝔢(q_{0}^{∗}) of universal or adjoint type, respectively, as described in Tables 0.1–0.3, such that G^{∗} ∼_{p} G and κ_{G^{∗}} is split surjective. If, furthermore, p is odd or G^{∗} has universal type, then μ_{G^{∗}} is an isomorphism, and hence κ_{G^{∗}} is also split surjective.

Proof. Case 1: Assume p is odd and G is of universal type. Since μ_{G} is an isomorphism by [O1, Theorem C], κ_{G} or κ_{G^{∗}} is (split) surjective if and only if κ_{G} or κ_{G^{∗}} is.

By Proposition 6.8, we can choose a prime q_{0}^{∗} and a group G^{∗} ∈ 𝔏𝔦𝔢(q_{0}^{∗}) such that either

(1.a) G^{∗} ≅ 𝔾(q^{∗}) or ^{2}𝔾(q^{∗}), for some 𝔾 with Weyl group W and q^{∗} a power of q_{0}^{∗}, and has a σ-setup which satisfies the conditions in Hypotheses 5.1 and 5.11, and

(1.a.1) −Id ∉ W and G^{∗} is a Chevalley group, or (1.a.2) −Id ∈ W and q^{∗} has even order in 𝔽_{p}^{×}; or (1.b) p = 3, q_{0}^{∗} = 2, G ≅ ^{3}D_{4}(q) or ^{2}F_{4}(q) for q some power of q_{0}, and G^{∗} ≅ ^{3}D_{4}(q^{∗}) or

2F_{4}(q^{∗}) for q^{∗} some power of 2.

Also (by the same proposition), if p = 3 and G^{∗} = F_{4}(q^{∗}), then we can assume q_{0}^{∗} = 2.

In case (1.b), κ_{G^{∗}} is split surjective by Proposition 6.9. In case (1.a), it is surjective by Proposition 5.15. In case (1.a.1), κ_{G^{∗}} is split by Proposition 5.16(b,c). In case (1.a.2), if G^{∗} is a Chevalley group, then κ_{G^{∗}} is split by Proposition 5.16(c).

This leaves only case (1.a.2) when G^{∗} is a twisted group. The only irreducible root systems which have nontrivial graph automorphisms and for which −Id ∈ W are those of type D_{n} for even n. Hence G^{∗} = Spin_{2n}^{−}(q^{∗}) for some even n ≥ 4. By the last statement in Proposition 6.8, G^{∗} is one of the groups listed in Proposition 1.10, and so q^{n} ≡ −1 (mod p). Hence κ_{G^{∗}} is split surjective by Example 6.6(a), and we are done also in this case.

Case 2: Now assume p = 2 and G is of universal type. By Proposition 6.2, there is an odd prime q_{0}^{∗}, a group G^{∗} ∈ 𝔏𝔦𝔢(q_{0}^{∗}), and S^{∗} ∈ Syl_{p}(G^{∗}), such that F_{S}(G) ≅ F_{S^{∗}}(G^{∗}) and G^{∗} has a σ-setup which satisfies Hypotheses 5.1 and 5.11. By the same proposition, if G^{∗} ≅ G_{2}(q^{∗}), then we can arrange that q^{∗} = 5 or q_{0}^{∗} = 3. If G^{∗} ≅ G_{2}(5), then by Propositions 6.3 and A.6, G^{∗} ∼_{2} G_{2}(3), κ_{G_{2}(3)} is split surjective, and μ_{G_{2}(3)} is injective.

In all remaining cases (i.e., G^{∗} ≅ G̸_{2}(q^{∗}) or q_{0}^{∗} = 3), κ_{G^{∗}} is split surjective by Proposition 5.16(a). If G^{∗} is a linear, symplectic, or orthogonal group, or an exceptional Chevalley group, then μ_{G^{∗}} is injective by Proposition A.3 or A.12, respectively. If G^{∗} ≅ SU_{n}(q^{∗}) or ^{2}E_{6}(q^{∗}), then by Theorem 1.8(d), F_{S^{∗}}(G^{∗}) is isomorphic to the fusion system of SL_{n}(q^{∨}) or E_{6}(q^{∨}) for some odd prime power q^{∨}, and so μ_{G^{∗}} is injective by A.3 or A.12 again. Since neither the triality groups ^{3}D_{4}(q) nor the Suzuki or Ree groups satisfy Hypotheses 5.1, this shows that μ_{G^{∗}} is injective in all cases, and hence that κ_{G^{∗}} is also split surjective.

can be modified to use an earlier result of Glauberman [Gl2, Theorem A1.4], and through that avoiding the classification.

Case 3: Now assume G is of adjoint type. Then G ≅ G_{u}/Z for some G_{u} ∈ 𝔏𝔦𝔢(q_{0}) of universal type and Z ≤ Z(G_{u}). By Proposition 3.8, Z = Z(G_{u}) and has order prime to q_{0}.

By Case 1 or 2, there is a prime q_{0}^{∗} ≠ p and a group G_{u}^{∗} ∈ 𝔏𝔦𝔢(q_{0}^{∗}) of universal type such that G_{u}^{∗} ∼_{p} G_{u} and κ_{G_{u}^{∗}} is split surjective. Also, G_{u}^{∗} is p-perfect by definition of 𝔏𝔦𝔢(q_{0}^{∗}) (and since q_{0}^{∗} ≠ p), and H^{2}(G_{u}^{∗}; ℤ/p) = 0 by Proposition 3.8. Set G^{∗} = G_{u}^{∗}/Z(G_{u}^{∗}). By Proposition 1.7, with G_{u}^{∗}/O_{p′}(G_{u}^{∗}) in the role of G, κ_{G^{∗}} is also split surjective.

It remains to check that G ∼_{p} G^{∗}. Assume first that G_{u} and G_{u}^{∗} have σ-setups which satisfy Hypotheses 5.1. Fix S ∈ Syl_{p}(G_{u}) and S^{∗} ∈ Syl_{p}(G_{u}^{∗}), and a fusion preserving isomorphism φ: S ⟶ S^{∗} (Definition 1.2(a)). By Corollary 5.10, Z(F_{S}(G_{u})) = O_{p}(Z(G_{u})) and Z(F_{S^{∗}}(G_{u}^{∗})) = O_{p}(Z(G_{u}^{∗})). Since φ is fusion preserving, it sends Z(F_{S}(G_{u})) onto Z(F_{S^{∗}}(G_{u}^{∗})), and thus sends O_{p}(Z(G_{u})) onto O_{p}(Z(G_{u}^{∗})). Hence φ induces a fusion preserving isomorphism between Sylow subgroups of G = G_{u}/Z(G_{u}) and G^{∗} = G_{u}^{∗}/Z(G_{u}^{∗}).

The only cases we considered where G or G^{∗} does not satisfy Hypotheses 5.1 were those in case (1.b) above. In those cases, G ≅ ^{2}F_{4}(q) or ^{3}D_{4}(q) and G^{∗} ≅ ^{2}F_{4}(q^{∗}) or ^{3}D_{4}(q^{∗}) for some q and q^{∗}, hence G and G^{∗} are also of universal type (d = 1 in the notation of [Ca, Lemma 14.1.2(iii)]), and so there is nothing more to prove. □

Since the strategy for replacing G by G^{∗} is quite elaborate, we summarize these replacements in Tables 0.1, 0.2, and 0.3 at the end of the introduction.

The last statement in Theorem B is not true in general when G^{∗} is of adjoint type. For example, if G^{∗} ≅ PSL_{2}(9), p = 2, and S^{∗} ∈ Syl_{2}(G^{∗}), then Out(G^{∗}) ≅ Out(L_{S^{∗}}^{c}(G^{∗})) ≅ C_{2}^{2}, while Out(S^{∗}, F_{S^{∗}}(G^{∗})) ≅ C_{2}. By comparison, if G̃^{∗} ≅ SL_{2}(9) is the universal group, then Out(S̃^{∗}, FS̃^{∗}(G̃^{∗})) ≅ C_{2}^{2}, and κG̃^{∗} and μG̃^{∗} are isomorphisms.

As noted briefly above, a fusion system F_{S}(G) is called tame if there is a finite group G^{∗} such that G^{∗} ∼_{p} G and κ_{G^{∗}} is split surjective. In this situation, we say that G^{∗} tamely realizes the fusion system F_{S}(G). By [AOV, Theorem B], if F_{S}(G) is not tame, then some extension of it is an “exotic” fusion system; i.e., an abstract fusion system not induced by any finite group. (See Section 1 for more details.) The original goal of this paper was to determine whether all fusion systems of simple groups of Lie type (at all primes) are tame, and this follows as an immediate consequence of Theorems A and B. Hence this approach cannot be used to construct new, exotic fusion systems.

Determining which simple fusion systems over finite 2-groups are tame, and tamely realizable by finite simple groups, plays an important role in Aschbacher’s program for classifying simple fusion systems over 2-groups (see [AKO, Part II] or [A3]). Given such a fusion system F over a 2-group S, and an involution x ∈ S, assume that the centralizer fusion system C_{F}(x) contains a normal quasisimple subsystem E ⊴ C_{F}(x). If E is tamely realized by a finite quasisimple group K, then under certain additional assumptions, one can show that the entire centralizer C_{F}(x) is the fusion system of some finite extension of K. (See, e.g., [O6, Corollaries 2.4 & 2.5].) This is part of our motivation for looking at this question, and is also part of the reason why we try to give as much information as possible as to which groups tamely realize which fusion systems.

Theorem C. For any prime p and any G ∈ 𝔏𝔦𝔢 of universal or adjoint type, the p-fusion system of G is tame. If the Sylow p-subgroups of G are nonabelian, or if p is the defining characteristic and G ≅ S̸z(2), then its fusion system is tamely realized by some other group in 𝔏𝔦𝔢.

Proof. If S ∈ Syl_{p}(G) is abelian, then the p-fusion in G is controlled by N_{G}(S), and F_{S}(G) is tame by Proposition 1.6. If p = 2 and G ≅ SL_{3}(2), then the fusion system of G is tamely realized by PSL_{2}(9). In all other cases, the claims follow from Theorems A and B. □

We have stated the above three theorems only for groups of Lie type, but in fact, we proved at the same time the corresponding results for the Tits group:

Theorem D. Set G = ^{2}F_{4}(2)′ (the Tits group). Then for each prime p, the p-fusion system of G is tame. If p = 2 or p = 3, then κ_{G} is an isomorphism.

Proof. The second statement is shown in Proposition 4.17 when p = 2, and in Proposition 6.9 when p = 3. When p > 3, the Sylow p-subgroups of G are abelian (|G| = 2^{11} · 3^{3} · 5^{2} · 13), so G is tame by Proposition 1.6(b). □

As one example, if p = 2 and G = PSL_{2}(17), then κ_{G} is not surjective, but G^{∗} = PSL_{2}(81) (of adjoint type) has the same 2-fusion system and κ_{G^{∗}} is an isomorphism [BLO1, Proposition 7.9]. Also, κ_{G^{∗}} is non-split surjective with kernel generated by the field automorphism of order two by [BLO1, Lemma 7.8]. However, if we consider the universal group G̃^{∗} = SL_{2}(81), then κG̃^{∗} and κG̃^{∗} are both isomorphisms by [BL, Proposition 5.5] (note that Out(F) = Out(S) in this situation).

As another, more complicated example, consider the case where p = 41 and G = Spin_{4k}^{−}(9). By [St1, (3.2)–(3.6)], Outdiag(G) ≅ C_{2}, and Out(G) ≅ C_{2} × C_{4} is generated by a diagonal element of order 2 and a field automorphism of order 4 (whose square is a graph automorphism of order 2). Also, μ_{G} is an isomorphism by Proposition A.3, so κ_{G} is surjective, or split surjective, if and only if κ_{G} is. We refer to the proof of Lemma 6.5, and to Table 6.1 in that proof, for details of a σ-setup for G in which the normalizer of a maximal torus contains a Sylow p-subgroup S. In particular, S is nonabelian if k ≥ 41. By Proposition 5.16(d) and Example 6.6(a,b), when k ≥ 41, κ_{G} is surjective, κ_{G} is split (with Ker(κ_{G}) = Outdiag(G)) when k is odd, and κ_{G} is not split (Ker(κ_{G}) ≅ C_{2} × C_{2}) when k is even. By Proposition 1.9(c), when k is even, G ∼_{41} G^{∗} for G^{∗} = Spin_{4k−1}(9), and κ_{G^{∗}} is split surjective (with Ker(κ_{G^{∗}}) = Outdiag(G^{∗})) by Proposition 5.16(c). Thus F_{S}(G) is tame in all cases: tamely realized by G itself when k is odd and by Spin_{4k−1}(9) when k is even. Note that when k is odd, since the graph automorphism does not act trivially on any Sylow p-subgroup, the pfusion system of G (equivalently, of SO_{4k}^{−}(9)) is not isomorphic to that of the full orthogonal group O_{4k}^{−}(9), so by [BMO, Proposition A.3(b)], it is not isomorphic to that of Spin_{4k+1}(9) either (nor to that of Spin_{4k−1}(9) since its Sylow p-subgroups are smaller).

Other examples are given in Examples 5.17 and 6.6. For more details, in the situation of Theorem B, about for which groups G the homomorphism κ_{G} is surjective or split surjective, see Propositions 5.15 and 5.16.

The following theorem was shown while proving Theorem B, and could be of independent interest. It is closely related to [Ma2, Theorem 5.19]. The case where p is odd was handled by Gorenstein and Lyons [GL, 10-2(1,2)].

Theorem E. Assume G ∈ 𝔏𝔦𝔢(q_{0}) is of universal type for some odd prime q_{0}. Fix S ∈ Syl_{2}(G). Then S contains a unique abelian subgroup of maximal order, except when G ≅ Sp_{2n}(q) for some n ≥ 1 and some q ≡ ±3 (mod 8).

Proof. Assume S is nonabelian; otherwise there is nothing to prove. Since q_{0} is odd, and since the Sylow 2-subgroups of ^{2}G_{2}(3^{2k+1}) are abelian for all k ≥ 1 [Ree, Theorem 8.5], G must be a Chevalley or Steinberg group. If G ≅ ^{3}D_{4}(q), then (up to isomorphism) S ∈ Syl_{2}(G_{2}(q)) by [BMO, Example 4.4]. So we can assume that G ≅ ^{r}𝔾(q) for some odd prime power q, some 𝔾, and r = 1 or 2.

If q ≡ 3 (mod 4), then choose another prime power q^{∗} ≡ 1 (mod 4) such that v_{2}(q^{∗} − 1) = v_{2}(q + 1) (where v_{2}(m) = k if 2^{k}|n and 2^{k+1} ∤ n). Then ⟨q^{∗}⟩ = ⟨ − q⟩ and ⟨ − q^{∗}⟩ = ⟨q⟩ as closed subgroups of (ℤ_{2})^{×}. By [BMO, Theorem A] (see also Theorem 1.8), there is a group G^{∗} ≅ ^{t}𝔾(q^{∗}) (where t ≤ 2) whose 2-fusion system is equivalent to that of G. We can thus assume that q ≡ 1 (mod 4). So by Lemma 6.1, G has a σ-setup which satisfies Hypotheses 5.1. By Proposition 5.13(a), S contains a unique abelian subgroup of maximal order, unless q ≡ 5 (mod 8) and G ≅ Sp_{2n}(q) for some n ≥ 1. □

In fact, when G ≅ Sp_{2n}(q) for q ≡ ±3 (mod 8), then S ∈ Syl_{2}(G) is isomorphic to (Q_{8})^{n} ⋊ P for P ∈ Syl_{2}(Σ_{n}), S contains 3^{n} abelian subgroups of maximal order 2^{2n}, and all of them are conjugate to each other in N_{G}(S).

The main definitions and results about tame and reduced fusion systems are given in Section 1. We then set up our general notation for finite groups of Lie type in Sections 2 and 3, deal with the equicharacteristic case in Section 4, and with the cross characteristic case in Sections 5 and 6. The kernel of μ_{G}, and thus the relation between automorphism groups of the fusion and linking systems, is handled in an appendix.

The third author would like to thank Richard Weiss for explaining how to apply the Delgado-Stellmacher paper [DS] to simplify some of our arguments (see Section 4), and also thank Andy Chermak and Sergei Shpectorov for first pointing out this connection. All three authors would especially like to thank the referee for reading the paper very thoroughly and for the many suggestions for improvements.

Notation: In general, when C is a category and x ∈ Ob(C), we let Aut_{C}(x) denote the group of automorphisms of x in C. When F is a fusion system and P ∈ Ob(F), we set Out_{F}(P) = Aut_{F}(P)/Inn(P).

For any group G and g ∈ G, c_{g} ∈ Aut(G) denotes the automorphism c_{g}(h) = ghg^{−1}. Thus for H ≤ G, ^{g}H = c_{g}(H) and H^{g} = c_{g}^{−1}(H). When G, H, K are all subgroups of a group Γ, we define

T_{G}(H, K) = {g ∈ G |^{g}H ≤ K}
Hom_{G}(H, K) = {c_{g} ∈ Hom(H, K) | g ∈ T_{G}(H, K)} .

We let Aut_{G}(H) be the group Aut_{G}(H) = Hom_{G}(H, H). When H ≤ G (so Aut_{G}(H) ≥ Inn(H)), we also write Out_{G}(H) = Aut_{G}(H)/Inn(H).

Tables of substitutions for Theorem B. We now present tables which describe the strategy for replacing G by G^{∗} in the context of Theorem B. In all three tables, an entry within the column G^{∗} means that the given group is p-locally equivalent to G and tamely realizes its fusion system, while an entry “G ∼_{p} X” carried over two columns means that the group X is p-locally equivalent to G but does not tamely realize its fusion system. In other words, in the latter case, X is one step towards finding the appropriate group G^{∗}, but one must continue, following the information in the tables for G = X.

Whenever G^{∗} is listed as satisfying (III.1), (III.2), or (III.3), this holds by Lemma 6.1, Lemma 6.4, or Lemma 6.5 or 6.7, respectively.

AUTOMORPHISMS OF FUSION SYSTEMS OF FINITE SIMPLE GROUPS OF LIE TYPE 7

The following notation is used in Table 0.1 (the case p = 2):

• q_{0}^{∗} = 3 or 5, and q_{0}^{∗} = 3 when 𝔾 = G_{2};

• q^{∗} = (q_{0}^{∗})^{2^{k}} is such that ⟨q⟩ = ⟨q^{∗}⟩; and

• q^{∨} = (q_{0}^{∗})^{2^{ℓ}} is such that ⟨ − q⟩ = ⟨q^{∨}⟩ (equivalently, ⟨q⟩ = ⟨ − q^{∨}⟩). In all cases except when G^{∗} ≅ G_{2}(3), G^{∗} satisfies case (III.1) of Hypotheses 5.1 by Lemma 6.1, and κ_{G^{∗}} is split surjective by Proposition 5.16(a). When G^{∗} ≅ G_{2}(3), κ_{G^{∗}} is an isomorphism by Proposition 6.3. For all odd q, ^{3}D_{4}(q) ∼_{2} G_{2}(q) by [BMO, Example 4.4(a)].

𝔾  G  q  G^{∗}  G ∼_{2} G^{∗}
A_{n}, E_{6}  1 (mod 4) 𝔾(q^{∗}) Thm. 1.8(a)
D_{n} (n odd) 𝔾(q) 3 (mod 4) ^{2}𝔾(q^{∨}) Thm. 1.8(d)
B_{n}, C_{n}, D_{2m},  1 (mod 4) 𝔾(q^{∗})
F_{4}, E_{7}, E_{8}  𝔾(q) 3 (mod 4) 𝔾(q^{∨}) Thm. 1.8(c)
1 (mod 8) 𝔾(q^{∗})
G_{2}(q) 7 (mod 8) 𝔾(q^{∨}) Thm. 1.8(c)
3D_{4}(q) 3,5 (mod 8) G_{2}(3)
A_{n}, E_{6}  1 (mod 4) ^{2}𝔾(q^{∗}) Thm. 1.8(b)
D_{n} (n odd) ^{2}𝔾(q) 3 (mod 4) 𝔾(q^{∨}) Thm. 1.8(d)
D_{n}  1 (mod 4) ^{2}D_{n}(q^{∗})
(n even) ^{2}D_{n}(q) 3 (mod 4) ^{2}D_{n}(q^{∨}) Thm. 1.8(c)
2G_{2}(q)  S abelian

Table 0.1. Substitutions in cross-characteristic for p = 2

8 CARLES BROTO, JESPER M. MØLLER, AND BOB OLIVER

The following notation is used in Tables 0.2 and 0.3, where p is always an odd prime:

• q_{0}^{∗} is any given odd prime whose class generates (ℤ/p^{2})^{×}.

• q^{∗} = (q_{0}^{∗})^{b} is such that ⟨q⟩ = ⟨q^{∗}⟩ and b|(p − 1)p^{ℓ} for some ℓ ≥ 0.

• q^{∨} = (q_{0}^{∗})^{c} is such that ⟨ − q⟩ = ⟨q^{∨}⟩ and c|(p − 1)p^{ℓ} for some ℓ ≥ 0.

G  q, p  G^{∗}  Hyp 5.1 κ_{G^{∗}} split surj.  G ∼_{p} G^{∗}
SL_{n}(q)  all cases  SL_{n}(q^{∗}) (III.1,3)  Th. 1.8(a)
SU_{n}(q)  all cases  SL_{n}(q^{∨}) (III.1,3)  Th. 1.8(d)
Sp_{2n}(q) or  ord_{p}(q) even  SL_{2n}(q^{∗}) (III.3) Prop. 5.16(b,c) Prop. 1.9(a,b)

Spin_{2n+1}(q) ord_{p}(q) odd SL_{2n}(q^{∨}) (III.3)

Prop. 1.9(a,b) Th. 1.8(c)

q^{n} ≢ ε (mod p) G ∼_{p} Spin_{2n−1}(q) — Prop. 1.9(c)
\begin{matrix}q^{n} ≡ ε (mod p) \\ n odd, ε = 1 \end{matrix} Spin_{2n}^{+}(q^{∗}) (III.1,3)  Th. 1.8(a)
Spin_{2n}^{ε}(q) \begin{matrix}q^{n} ≡ ε (mod p) \\ n odd, ε = −1 \end{matrix} Spin_{2n}^{+}(q^{∨}) (III.1,3) Prop. 5.16(b,c) Th. 1.8(d)

q^{n} ≡ ε (mod p) n, ord_{p}(q) even

Spin_{2n}^{ε}(q^{∗}) (III.3)

Prop. 5.16(c) Ex. 6.6(a)

Th. 1.8(a,b)

q^{n} ≡ ε (mod p) n even, ε = 1 ord_{p}(q) odd

Spin_{2n}^{+}(q^{∨}) (III.3) Prop. 5.16(c) Th. 1.8(c)

Table 0.2. Substitutions in cross-characteristic for p odd: classical groups

G  p  ord_{p}(q)G^{∗}  Hyp 5.1 κ_{G^{∗}} split surj.  G ∼_{p} G^{∗}

2B_{2}(q)  all cases  S abelian
3  1 ^{3}D_{4}(q^{∗})
3D_{4}(q)  3  2 ^{3}D_{4}(q^{∗})

(q_{0}^{∗} = 2)

Prop. 6.9

Theorem 1.8(b)

≥ 5 —  S abelian
3  1 G_{2}(q^{∨})
G_{2}(q)  3  2 G_{2}(q^{∗})

(III.2)

Prop. 5.16(c)

Theorem 1.8(c)

≥ 5 —  S abelian
2G_{2}(q)  all cases  S abelian
3  1 F_{4}(q^{∨}) (III.2)
F_{4}(q)  3  2 F_{4}(q^{∗}) (q_{0}^{∗} = 2)

Prop. 5.16(c)

Theorem 1.8(c)

≥ 5 —  S abelian
2F_{4}(q) or 3  2 ^{2}F_{4}(q^{∗})  —  Prop. 6.9  Prop. 6.8(b)
2F_{4}(2)′ ≥ 5  S abelian
3,5 1 E_{6}(q^{∗}) (III.1) Prop. 5.16(b) Theorem 1.8(a)
E_{6}(q)  3  2  G ∼_{p} F_{4}(q^{2})  —

Theorem 1.8(d) [BMO, Ex. 4.4]

other cases  S abelian
2E_{6}(q)  all cases  G ∼_{p} E_{6}(q^{∨})  —  Theorem 1.8(d)
3,5,7 1 E_{7}(q^{∨})

E_{7}(q) 3,5,7 2 E_{7}(q^{∗})

(III.2)

Prop. 5.16(c)

Theorem 1.8(c)

other cases  S abelian
3,5,7 1 E_{8}(q^{∨})

E_{8}(q) 3,5,7 2 E_{8}(q^{∗})

(III.2)

Prop. 5.16(c)

Theorem 1.8(c)

5  4 E_{8}(q^{∗}) (III.3) Prop. 5.16(c) Theorem 1.8(a)
other cases  S abelian

Table 0.3. Substitutions in cross-characteristic for p odd: exceptional groups

1. Tame and reduced fusion systems

Throughout this section, p always denotes a fixed prime. Before defining tameness of fusion systems more precisely, we first recall the definitions of fusion and linking systems of finite groups, and of automorphism groups of fusion and linking systems.

Definition 1.1. Fix a finite group G and a Sylow p-subgroup S ≤ G.

(a) The fusion system of G is the category F_{S}(G) whose objects are the subgroups of S, and where Mor_{F_{S}(G)}(P, Q) = Hom_{G}(P, Q) for each P, Q ≤ S.

(b) A subgroup P ≤ S is p-centric in G if Z(P) ∈ Syl_{p}(C_{G}(P)); equivalently, if C_{G}(P) = Z(P) × C′_{G}(P) for a (unique) subgroup C′_{G}(P) of order prime to p.

(c) The centric linking system of G is the category L_{S}^{c}(G) whose objects are the p-centric subgroups of G, and where Mor_{L_{S}^{c}(G)}(P, Q) = T_{G}(P, Q)/C′_{G}(P) for each pair of objects P, Q. Let π: L_{S}^{c}(G) ⟶ F_{S}(G) denote the natural functor: π is the inclusion on objects, and sends the class of g ∈ T_{G}(P, Q) to c_{g} ∈ Mor_{F_{S}(G)}(P, Q).

(d) For P, Q ≤ S p-centric in G and g ∈ T_{G}(P, Q), we let [[g]]_{P,Q} ∈ Mor_{L_{S}^{c}(G)}(P, Q) denote the class of g, and set [[g]]_{P} = [[g]]_{P,P} if g ∈ N_{G}(P). For each subgroup H ≤ N_{G}(P), [[H]]_{P} denotes the image of H in Aut_{L}(P) = N_{G}(P)/C′_{G}(P).

The following definitions of automorphism groups are taken from [AOV, Definition 1.13 & Lemma 1.14], where they are formulated more generally for abstract fusion and linking systems.

Definition 1.2. Let G be a finite group with S ∈ Syl_{p}(G), and set F = F_{S}(G) and L = L_{S}^{c}(G).

(a) If H is another finite group with T ∈ Syl_{p}(H), then an isomorphism φ: S ⟶^{≅} T is called fusion preserving (with respect to G and H) if for each P, Q ≤ S,

Hom_{H}(φ(P), φ(Q)) = φ ∘ Hom_{G}(P, Q) ∘ φ^{−1} .

(Composition is from right to left.) Equivalently, φ is fusion preserving if it induces an isomorphism of categories F_{S}(G) ⟶^{≅} F_{T} (H).

(b) Let Aut(F) ≤ Aut(S) be the group of fusion preserving automorphisms of S. Set Out(F) = Aut(F)/Aut_{F}(S).

(c) For each pair of objects P ≤ Q in L, set ι_{P,Q} = [[1]]_{P,Q} ∈ Mor_{L}(P, Q), which we call the inclusion in L of P in Q. For each P, we call [[P]] = [[P]]_{P} ≤ Aut_{L}(P) the distinguished subgroup of Aut_{L}(P).

(d) Let Aut(L) be the group of automorphisms α of the category L such that α sends inclusions to inclusions and distinguished subgroups to distinguished subgroups. For γ ∈ Aut_{L}(S), let c_{γ} ∈ Aut(L) be the automorphism which sends an object P to π(γ)(P), and sends ψ ∈ Mor_{L}(P, Q) to γ′ψ(γ′′)^{−1} where γ′ and γ′′ are appropriate restrictions of γ. Set

Out(L) = Aut(L)/{c_{γ} | γ ∈ Aut_{L}(S)} .

(e) Let κ_{G}: Out(G) ⟶ Out(L) be the homomorphism which sends the class [α], for α ∈ Aut(G) such that α(S) = S, to the class of β ∈ Aut(L_{S}^{c}(G)), where β(P) = α(P) for an object P, and β([[g]]_{P,Q}) = [[α(g)]]_{α(P),α(Q)} for g ∈ T_{G}(P, Q).

(f) Define μ_{G}: Out(L) ⟶ Out(F) by restriction: μ_{G}([β]) = [β_{S}|_{S}] for β ∈ Aut(L_{S}^{c}(G)), where β_{S} is the induced automorphism of Aut_{L}(S), and β_{S}|_{S} ∈ Aut(S) is its restriction to S when we identify S with its image in Aut_{L}(S) = N_{G}(S)/C′_{G}(S).

(g) Set κ_{G} = μ_{G} ∘ κ_{G}: Out(G) ⟶ Out(F): the homomorphism which sends the class of α ∈ N_{Aut(G)}(S) to the class of α|_{S}.

By [AOV, Lemma 1.14], the above definition of Out(L) is equivalent to that in [BLO2], and by [BLO2, Lemma 8.2], both are equivalent to that in [BLO1]. So by [BLO1, Theorem 4.5(a)], Out(L_{S}^{c}(G)) ≅ Out(BG_{p}^{∧}): the group of homotopy classes of self homotopy equivalences of the space BG_{p}^{∧}.

We refer to [AOV, § 2.2] and [AOV, § 1.3] for more details about the definitions of κ_{G} and μ_{G} and the proofs that they are well defined. Note that μ is defined there for an arbitrary linking system, not necessarily one realized by a group.

We are now ready to define tameness. Again, we restrict attention to fusion systems of finite groups, and refer to [AOV, § 2.2] for the definition in the more abstract setting.

Definition 1.3. For a finite group G and S ∈ Syl_{p}(G), the fusion system F_{S}(G) is tame if there is a finite group G^{∗} which satisfies:

• there is a fusion preserving isomorphism S ⟶^{≅} S^{∗} for some S^{∗} ∈ Syl_{p}(G^{∗}); and

• the homomorphism κ_{G^{∗}}: Out(G^{∗}) ⟶ Out_{typ}(L_{S}^{c}(G^{∗})) ≅ Out(BG^{∗}_{p}^{∧}) is split surjective.

In this situation, we say that G^{∗} tamely realizes the fusion system F_{S}(G).

The above definition is complicated by the fact that two finite groups can have isomorphic fusion systems but different outer automorphism groups. For example, set G = P SL_{2}(9) ≅ A_{6} and H = P SL_{2}(7) ≅ GL_{3}(2). The Sylow subgroups of both groups are dihedral of order 8, and it is not hard to see that any isomorphism between Sylow subgroups is fusion preserving. But Out(G) ≅ C_{2}^{2} while Out(H) ≅ C_{2} (see Theorem 3.4 below). Also, κ_{G} is an isomorphism, while κ_{H} fails to be onto (see [BLO1, Proposition 7.9]). In conclusion, the 2-fusion system of both groups is tame, even though κ_{H} is not split surjective.

This definition of tameness was motivated in part in [AOV] by an attempt to construct new, “exotic” fusion systems (abstract fusion systems not realized by any finite group) as extensions of a known fusion system by an automorphism. Very roughly, if α ∈ Aut(L_{S}^{c}(G)) is not in the image of κ_{G}, and not in the image of κ_{G^{∗}} for any other finite group G^{∗} which has the same fusion and linking systems, then one can construct and extension of F_{S}(G) by α which is not isomorphic to the fusion system of any finite group. This shows why we are interested in the surjectivity of κ_{G}; to see the importance of its being split, we refer to the proof of [AOV, Theorem B].

It is usually simpler to work with automorphisms of a p-group which preserve fusion than with automorphisms of a linking system. So in most cases, we prove tameness for the fusion system of a group G by first showing that κ_{G} = μ_{G} ∘ κ_{G} is split surjective, and then showing that μ_{G} is injective. The following elementary lemma will be useful.

Lemma 1.4. Fix a finite group G and S ∈ Syl_{p}(G), and set F = F_{S}(G). Then

(a) κ_{G} is surjective if and only if each φ ∈ Aut(F) extends to some φ ∈ Aut(G), and

(b) Ker(κ_{G}) ≅ C_{Aut(G)}(S)/Aut_{C_{G}(S)}(G).

Proof. This follows from the following diagram

\xymatrix{0 \ar[r] & Aut_{N_{G}(S)}(G) \ar[r] \ar@{->>}[d] & N_{Aut(G)}(S) \ar[r] \ar[d]^{restr} & Out(G) \ar[r] \ar[d]^{_{κ_{G}}} & 0 \\ 0 \ar[r] & Aut_{N_{G}(S)}(S) \ar[r] & Aut(F) \ar[r] & Out(F) \ar[r] & 0}

with exact rows. □

The next lemma can be useful when κ_{G} or κ_{G} is surjective but not split.

Lemma 1.5. Fix a prime p, a finite group G, and S ∈ Syl_{p}(G).

(a) Assume Ĝ ≥ G is such that G ⊴ Ĝ, p ∤ |Ĝ/G|, and OutĜ(G) ≤ Ker(κ_{G}). Then F_{S}(Ĝ) = F_{S}(G) and L_{S}^{c}(Ĝ) ≅ L_{S}^{c}(G).

(b) If κ_{G} is surjective and Ker(κ_{G}) has order prime to p, then there is Ĝ ≥ G/O_{p′}(Z(G)) such that F_{S}(Ĝ) = F_{S}(G) (where we identify S with its image in G/O_{p′}(G)) and κĜ is split surjective. In particular, F_{S}(G) is tame, and is tamely realized by Ĝ.

Proof. (a) Since OutĜ(G) ≤ Ker(κ_{G}), each coset of G in Ĝ contains an element which centralizes S. (Recall that κ_{G} is induced by the restriction homomorphism from N_{Aut(G)}(S) to Aut(F).) Thus F_{S}(Ĝ) = F_{S}(G) and L_{S}^{c}(Ĝ) = L_{S}^{c}(G).

(b) Since G and G/O_{p′}(Z(G)) have isomorphic fusion systems at p, we can assume that Z(G) is a p-group. Set K = Ker(κ_{G}) ≤ Out(G). Since H^{i}(K; Z(G)) = 0 for i = 2, 3, by the obstruction theory for group extensions [McL, Theorems IV.8.7–8], there is an extension Ĝ of G by K such that G ⊴ Ĝ and Ĝ/G ≅ K = OutĜ(G). In particular, CĜ(G) ≤ G. Since K = Ker(κ_{G}) ≤ Ker(κ_{G}), F_{S}(Ĝ) = F_{S}(G), and L_{S}^{c}(Ĝ) = L_{S}^{c}(G) by (a).

By [OV, Lemma 1.2], and since K ⊴ Out(G) and H^{i}(K; Z(G)) = 0 for i = 1, 2, each automorphism of G extends to an automorphism of Ĝ which is unique modulo inner automorphisms. Thus Out(Ĝ) contains a subgroup isomorphic to Out(G)/K, and κĜ sends this subgroup isomorphically onto Out(L_{S}^{c}(Ĝ)). So κĜ is split surjective, and F_{S}(G) is tame. □

The next proposition is really a result about constrained fusion systems (cf. [AKO, Definition I.4.8]): it says that every constrained fusion system is tame. Since we are dealing here only with fusion systems of finite groups, we state it instead in terms of p-constrained groups.

Proposition 1.6. Fix a finite group G and a Sylow subgroup S ∈ Syl_{p}(G).

(a) If C_{G}(O_{p}(G)) ≤ O_{p}(G), then κ_{G} and μ_{G} are both isomorphisms:

Out(G) ⟶_{≅}^{κ_{G}} Out(L_{S}^{c}(G)) ⟶_{≅}^{μ_{G}} Out(F_{S}(G)) .

(b) If S is abelian, or more generally if N_{G}(S) controls p-fusion in G, then F_{S}(G) is tame, and is tamely realized by N_{G}(S)/O_{p′}(C_{G}(S)).

Proof. (a) Set Q = O_{p}(G), F = F_{S}(G), and L = L_{S}^{c}(G). Then Aut_{L}(Q) = G, so (α ↦ α_{Q}) defines a homomorphism

Φ: Aut(L) ⟶ Aut(Aut_{L}(Q)) = Aut(G)

whose image lies in N_{Aut(G)}(S). For each α ∈ Ker(Φ), α_{Q} = Id_{G} and hence α = Id_{L}. (Here, it is important that α sends inclusions to inclusions.) Thus Φ is a monomorphism. Also, α = c_{γ} for some γ ∈ Aut_{L}(S) if and only if α_{Q} = c_{g} for some g ∈ N_{G}(S), so Φ factors through a monomorphism Φ from Out(L) to N_{Aut(G)}(S)/Aut_{G}(S) ≅ Out(G), and Φ_{G} ∘ κ_{G} = Id_{Out(G)}. Thus κ_{G} is an isomorphism.

In the terminology in [AKO, § I.4], G is a model for F = F_{S}(G). By the uniqueness of models (cf. [AKO, Theorem III.5.10(c)]), each β ∈ Aut(F) extends to some χ ∈ Aut(G), and χ is unique modulo Aut_{Z(S)}(G). Hence κ_{G} is an isomorphism, and so is μ_{G}.

(b) If N_{G}(S) controls p-fusion in G, then N_{G}(S) ∼_{p} G. Also, N_{G}(S) ∼_{p} G^{∗} where G^{∗} = N_{G}(S)/O_{p′}(C_{G}(S)), G^{∗} satisfies the hypotheses of (a), and hence tamely realizes F_{S}(G). In particular, this holds whenever S is abelian by Burnside’s theorem. □

When working with groups of Lie type when p is not the defining characteristic, it is easier to work with the universal groups rather than those in adjoint form (μ_{G} is better behaved in such cases). The next proposition is needed to show that tameness for fusion systems of groups of universal type implies the corresponding result for groups of adjoint type.

Proposition 1.7. Let G be a finite p-perfect group such that O_{p′}(G) = 1 and H_{2}(G; ℤ/p) = 0 (i.e., such that each central extension of G by a finite p-group splits). Choose S ∈ Syl_{p}(G), and set Z = Z(G) ≤ S. If F_{S}(G) is tamely realized by G, then F_{S/Z}(G/Z) is tamely realized by G/Z.

Proof. Let H be the set of all P ≤ S such that P ≥ Z and P/Z is p-centric in G/Z, and let L_{S}^{H}(G) ⊆ L_{S}^{c}(G) be the full subcategory with object set H. By [AOV, Lemma 2.17], L_{S}^{H}(G) is a linking system associated to F_{S}(G) in the sense of [AOV, Definition 1.9]. Hence the homomorphism

R: Out(L_{S}^{c}(G)) ⟶^{≅} Out(L_{S}^{H}(G))

induced by restriction is an isomorphism by [AOV, Lemma 1.17].

Set F = F_{S}(G), L = L_{S}^{H}(G), G = G/Z, S = S/Z, F = F_{S}(G), and L = L_{S}^{c}(G) for short. Consider the following square:

\xymatrix{Out(G) \ar[rr]^{κ_{G}} \ar[d]^{μ} &  & Out(L) ≅ Out(L_{S}^{c}(G)) \\ Out(G) \ar[r]^{κ_{G}} & Out(L) . \ar[ur]_{ν}^{1−1}}  (1)

Here, μ sends the class of an automorphism of G to the class of the induced automorphism of G = G/Z(G).

Assume that ν has been defined so that (1) commutes and ν is injective. If κ_{G} is onto, then ν is onto and hence an isomorphism, so κ_{G} is also onto. Similarly, if κ_{G} is split surjective, then κ_{G} is also split surjective. Thus F is tamely realized by G if F is tamely realized by G, which is what we needed to show.

It thus remains to construct the monomorphism ν, by sending the class of α ∈ Aut(L) to the class of a lifting of α to L. So in the rest of the proof, we show the existence and uniqueness of such a lifting.

Let pr: L ⟶ L denote the projection. Let End(L) be the monoid of functors from L to itself which send inclusions to inclusions and distinguished subgroups into distinguished subgroups. (Thus Aut(L) is the group of elements of End(L) which are invertible.) We will prove the following two statements:

(2) For each α ∈ Aut(L), there is a functor α̃ ∈ End(L) such that pr ∘ α̃ = α ∘ pr.

(3) If β ∈ End(L) is such that pr ∘ β = pr, then β = Id_{L}.

Assume that (2) and (3) hold; we call α̃ a “lifting” of α in the situation of (2). For each α ∈ Aut(L), there are liftings α̃ of α and α̃^{∗} of α^{−1} in End(L), and these are inverses to each other by (3). Hence α̃ ∈ Aut(L), and is the unique such lifting of α by (3) again.

Define ν: Out(L) ⟶ Out(L) by setting ν([α]) = [α̃] when α̃ is the unique lifting of α. This is well defined as a homomorphism on Aut(L) by the existence and uniqueness of the lifting; and it factors through Out(L) since conjugation by γ ∈ Aut_{L}(S) lifts to conjugation by γ ∈ Aut_{L}(S) for any γ ∈ pr_{S}^{−1}(γ).

Thus ν is a well defined homomorphism, and is clearly injective. The square (1) commutes since for each β ∈ Aut(G) such that β(S) = S, κ_{G}([β]) and νκ_{G}μ([β]) are the classes of liftings of the same automorphism of L.

It remains to prove (2) and (3).

Proof of (2): For each α ∈ Aut(L), consider the pullback diagram

\xymatrix{L̃ \ar[rr]^{ρ_{1}} \ar[d]^{ρ_{2}} &  & L \ar[d]^{pr} \\ L \ar[r]^{pr} \ar@{-->}[urr]^{α̃} & L \ar[r]^{α}_{≅} & _{L.}}  (4)

Each functor in (4) is bijective on objects, and the diagram restricts to a pullback square of morphism sets for each pair of objects in L (and their inverse images in L and L̃).

Since the natural projection G ⟶ G is a central extension with kernel Z, the projection functor pr: L ⟶ L is also a central extension of linking systems in the sense of [5a2, Definition 6.9] with kernel Z. Since ρ_{2} is the pullback of a central extension, it is also a central extension of linking systems by [5a2, Proposition 6.10], applied with ω = pr^{∗}α^{∗}(ω_{0}) ∈ Z^{2}(L; Z), where ω_{0} is a 2-cocycle on L which determines the extension pr. By [BLO1, Proposition 1.1], H^{2}(|L|; 𝔽_{p}) ≅ H^{2}(G; 𝔽_{p}), where the last group is zero by assumption. Hence H^{2}(|L|; Z) = 0, so ω is a coboundary, and ρ_{2} is the product extension by [5a2, Theorem 6.13]. In other words, L̃ ≅ L_{Z}^{c}(Z)×L, where L_{Z}^{c}(Z) has one object and automorphism group Z, and there is a subcategory L_{0} ⊆ L̃ (with the same objects) which is sent isomorphically to L by ρ_{2}. Set α̃ = ρ_{1} ∘ (ρ_{2}|_{L_{0}})^{−1}.

We first check that α̃ sends distinguished subgroups to distinguished subgroups. Let pr_{S}: S ⟶ S = S/Z be the projection. Fix an object P in L, and set Q = α̃(P). Then Q/Z = α(P/Z), and α_{P/Z}([[P/Z]]) = [[Q/Z]], so α̃_{P} ([[P]]) ≤ pr_{S}^{−1}([[Q/Z]]) = [[Q]].

For each subgroup P ∈ Ob(L), there is a unique element z_{P} ∈ Z such that α̃(ι_{P,S}) = ι_{α̃(P),S} ∘[[z_{P}]]_{α̃(P)}. Note that z_{S} = 1. Define a new functor β: L ⟶ L by setting β(P) = α̃(P) on objects and for each φ ∈ Mor_{L}(P, Q), β(φ) = [[z_{Q}]]_{α̃(Q)} ∘ α̃(φ) ∘ [[z_{P}]]_{α̃(P)}^{−1}. Then β is still a lifting of α, and for each P:

β(ι_{P,S}) = [[z_{S}]]_{S} ∘ α̃(ι_{P,S}) ∘ [[z_{P}]]_{α̃(P)}^{−1} = ι_{α̃(P),S} ∘ [[z_{P}]]_{α̃(P)} ∘ [[z_{P}]]_{α̃(P)}^{−1} = ι_{α̃(P),S} .

For arbitrary P ≤ Q, since ι_{α̃(P),α̃(Q)} is the unique morphism whose composite with ι_{α̃(Q),S} is ι_{α̃(P),S} (see [BLO2, Lemma 1.10(a)]), β sends ι_{P,Q} to ι_{α̃(P),α̃(Q)}.

Thus, upon replacing α̃ by β, we can assume that α̃ sends inclusions to inclusions. This finishes the proof of (2).

Proof of (3): Assume that β ∈ End(L) is a lift of the identity on L. Let B(Z) be the category with one object * and with morphism group Z. Define a functor χ: L ⟶ B(Z) by sending all objects in L to * , and by sending a morphism [[g]] ∈ Mor_{L}(P, Q) to the unique element z ∈ Z such that β_{P,Q}([[g]]) = [[gz]] = [[zg]]. (Recall that Z ≤ Z(G).)

Now,

H^{1}(|L|; 𝔽_{p}) ≅ H^{1}(|L_{S}^{c}(G)|; 𝔽_{p}) ≅ H^{1}(BG; 𝔽_{p}) ≅ H^{1}(G; 𝔽_{p}) = 0 ,

where the first isomorphism holds by [5a1, Theorem B] and the second by [BLO1, Proposition 1.1]. Hence Hom(π_{1}(|L|), 𝔽_{p}) ≅ Hom(H_{1}(|L|), 𝔽_{p}) ≅ H^{1}(|L|; 𝔽_{p}) = 0, where the second isomorphism holds by the universal coefficient theorem (cf. [McL, Theorem III.4.1]), and so Hom(π_{1}(|L|), Z) = 0. In particular, the homomorphism χ̂: π_{1}(|L|) ⟶ π_{1}(|B(Z)|) ≅ Z induced by χ is trivial.

Thus for each ψ ∈ Mor_{L}(P, Q), the loop in |L| formed by ψ and the inclusions ι_{P,S} and ι_{Q,S} is sent to 1 ∈ Z. Since β sends inclusions to inclusions, this proves that χ_{P,Q}(ψ) = 1, and hence that β_{P,Q}(ψ) = ψ. Thus β = Id_{L}. □

By Proposition 1.7, when proving tameness for fusion systems of simple groups of Lie type, it suffices to look at the universal groups (such as SL_{n}(q), SU_{n}(q)) rather than the simple groups (PSL_{n}(q), PSU_{n}(q)). However, it is important to note that the proposition is false if we replace automorphisms of the linking systems by those of the fusion system. For example, set G = SL_{2}(3^{4}) and G = P SL_{2}(3^{4}). Then S ≅ Q_{32} and S ≅ D_{16}, Out(F_{S}(G)) = Out(S) ≅ Out(G) ≅ C_{4} × C_{2} (and κ_{G} is an isomorphism), while Out(G) ≅ C_{4} × C_{2} and Out(S, F_{S}(G)) = Out(S) ≅ C_{2} × C_{2}.

We already gave one example of two groups which have the same fusion system but different outer automorphism groups. That is a special case of the main theorem in our earlier paper, where we construct many examples of different groups of Lie type with isomorphic fusion systems. Since this plays a crucial role in Section 6, where we handle the cross characteristic case, we restate the theorem here.

As in the introduction, we write G ∼_{p} H to mean that there is a fusion preserving isomorphism from a Sylow p-subgroup of G to one of H.

Theorem 1.8 ([BMO, Theorem A]). Fix a prime p, a connected reductive group scheme 𝔾 over ℤ, and a pair of prime powers q and q^{∗} both prime to p. Then the following hold.

(a) 𝔾(q) ∼_{p} 𝔾(q^{∗}) if ⟨q⟩ = ⟨q^{∗}⟩ as subgroups of ℤ_{p}^{×}.

(b) If 𝔾 is of type A_{n}, D_{n}, or E_{6}, and τ is a graph automorphism of 𝔾, then ^{τ}𝔾(q) ∼_{p} ^{τ}𝔾(q^{∗}) if ⟨q⟩ = ⟨q^{∗}⟩ as subgroups of ℤ_{p}^{×}.

(c) If the Weyl group of 𝔾 contains an element which acts on the maximal torus by inverting all elements, then 𝔾(q) ∼_{p} 𝔾(q^{∗}) (or ^{τ}𝔾(q) ∼_{p} ^{τ}𝔾(q^{∗}) for τ as in (b)) if ⟨ − 1, q⟩ = ⟨ − 1, q^{∗}⟩ as subgroups of ℤ_{p}^{×}.

(d) If 𝔾 is of type A_{n}, D_{n} for n odd, or E_{6}, and τ is a graph automorphism of 𝔾 of order two, then ^{τ}𝔾(q) ∼_{p} 𝔾(q^{∗}) if ⟨ − q⟩ = ⟨q^{∗}⟩ as subgroups of ℤ_{p}^{×}.

The next proposition is of similar type, but much more elementary.

Proposition 1.9. Fix an odd prime p, a prime power q prime to p, n ≥ 2, and ε ∈ {±1}. Then

(a) Sp_{2n}(q) ∼_{p} SL_{2n}(q) if ord_{p}(q) is even; (b) Sp_{2n}(q) ∼_{p} Spin_{2n+1}(q); and

(c) Spin_{2n}^{ε}(q) ∼_{p} Spin_{2n−1}(q) if q is odd and q^{n} ≢ ε (mod p).

Proof. If we replace Spin_{m}^{±}(q) by SO_{m}^{±}(q) in (b) and (c), then these three points are shown in [BMO, Proposition A.3] as points (d), (a), and (c), respectively. When q is a power of 2, (b) holds because the groups are isomorphic (see [Ta, Theorem 11.9]). So it remains to show that

Spin_{m}^{ε}(q) ∼_{p} Ω_{m}^{ε}(q) ∼_{p} SO_{m}^{ε}(q)

for all m ≥ 3 (even or odd) and q odd. The first equivalence holds since p is odd and Ω_{m}^{ε}(q) ≅ Spin_{m}^{ε}(q)/K where |K| = 2. The second holds by Lemma 1.5(a), and since Out_{SO_{m}^{ε}(q)}(Ω_{m}^{ε}(q)) is generated by the class of a diagonal automorphism of order 2 (see, e.g., [GLS3, § 2.7]) and hence can be chosen to commute with a Sylow p-subgroup. This last statement is shown in Lemma 5.9 below, and holds since for appropriate choices of algebraic group G containing the given group G, and of maximal torus T ≤ G, a Sylow p-subgroup of G is contained in N_{G}(T) (see [GLS3, Theorem 4.10.2]) and the diagonal automorphisms of G are induced by conjugation by elements in N_{T} (G) (see Proposition 3.5(c)). □

Theorem 1.8 and Proposition 1.9, together with some other, similar relations in [BMO], lead to the following proposition, which when p is odd provides a relatively short list of “p-local equivalence class representatives” for groups of Lie type in characteristic different from p.

Proposition 1.10. Fix an odd prime p, and assume G ∈ 𝔏𝔦𝔢(q_{0}) is of universal type for some prime q_{0} ≠ p. Assume also that the Sylow p-subgroups of G are nonabelian. Then there is a group G^{∗} ∈ 𝔏𝔦𝔢(q_{0}^{∗}) of universal type for some q_{0}^{∗} ≠ p, such that G^{∗} ∼_{p} G and G^{∗} is one of the groups in the following list:

(a) SL_{n}(q^{∗}) for some n ≥ p; or

(b) Spin_{2n}^{ε}(q^{∗}), where n ≥ p, ε = ±1, (q^{∗})^{n} ≡ ε (mod p), and ε = +1 if n is odd; or

(c) ^{3}D_{4}(q^{∗}) or ^{2}F_{4}(q^{∗}), where p = 3 and q^{∗} is a power of 2; or

(d) 𝔾(q^{∗}), where 𝔾 = G_{2}, F_{4}, E_{6}, E_{7}, or E_{8}, p | |W (𝔾)|, and q^{∗} ≡ 1 (mod p); or

(e) E_{8}(q^{∗}), where p = 5 and q^{∗} ≡ ±2 (mod 5).

Furthermore, in all cases except (c), we can take q_{0}^{∗} to be any given prime whose class generates (ℤ/p^{2})^{×}, and choose G^{∗} so that q^{∗} = (q_{0}^{∗})^{b} where b|(p − 1)p^{ℓ} for some ℓ.

Proof. Let q be such that G ≅ ^{τ}𝔾(q) for some τ and some 𝔾. Thus q is a power of q_{0}. Fix a prime q_{0}^{∗} as specified above. By Lemma 1.11(a), there are positive integers b, c, and powers q^{∗} = (q_{0}^{∗})^{b} and q^{∨} = (q_{0}^{∗})^{c} such that ⟨q⟩ = ⟨q^{∗}⟩, ⟨ − q⟩ = ⟨q^{∨}⟩, and b, c|(p − 1)p^{ℓ} for some ℓ ≥ 0.

(i) Assume G ≅ Sz(q), ^{2}G_{2}(q), ^{2}F_{4}(q), or G ≅ ^{3}D_{4}(q). Since p ≠ q_{0}, and since S ∈ Syl_{p}(G) is nonabelian, p divides the order of the Weyl group W of 𝔾 by [GL, 10-1(3)]. The Weyl group of B_{2} is a 2-group, and 2 and 3 are the only primes which divide the orders of the Weyl groups of G_{2}, F_{4}, and D_{4}. Hence p = 3, G ≅ ^{2̸}G_{2}(q) since that is defined only in characteristic 3, and so G ≅ ^{2}F_{4}(q) or ^{3}D_{4}(q). Set G^{∗} = ^{2}F_{4}(q^{∗}) or ^{3}D_{4}(q^{∗}), respectively, where q_{0}^{∗} = 2. Then G^{∗} ∼_{p} G, and we are in case (c).

(ii) If G = SU_{n}(q) or ^{2}E_{6}(q), then by Theorem 1.8(d), G ∼_{p} G^{∗} where G^{∗} ≅ SL_{n}(q^{∨}) or E_{6}(q^{∨}), respectively. So we can replace G by a Chevalley group in these cases.

(iii) Assume G = Sp_{2n}(q) for some n and q. If ord_{p}(q) is even, then by Proposition 1.9(a), G ∼_{p} SL_{2n}(q). If ord_{p}(q) is odd, then ord_{p}(q^{∨}) is even since ⟨q^{∨}⟩ = ⟨ − q⟩ in 𝔽_{p}^{×}, and G ∼_{p} Sp_{2n}(q^{∨}) by Theorem 1.8(c). So G is always p-locally equivalent to a linear group in this case.

(iv) Assume G = Spin_{2n+1}(q) for some n and q. Then G ∼_{p} Sp_{2n}(q) by Proposition 1.9(b). So G is p-locally equivalent to a linear group by (iii).

(v) If G = SL_{n}(q), set G^{∗} = SL_{n}(q^{∗}). Then G^{∗} ∼_{p} G by Theorem 1.8(a), n ≥ p since the Sylow p-subgroups of G are nonabelian, and we are in the situation of (a).

(vi) Assume G = Spin_{2n}^{ε}(q) for some n and q, and ε = ±1. If q is a power of 2, then by using point (a) or (b) of Theorem 1.8, we can arrange that q be odd. If q^{n} ≢ ε (mod p), then G ∼_{p} Spin_{2n−1}(q) by Proposition 1.9(c), and this is p-equivalent to a linear group by (iv). So we are left with the case where q^{n} ≡ ε (mod p). If n is odd and ε = −1, set G^{∗} = Spin_{2n}^{+}(q^{∨}) ∼_{p} G (Theorem 1.8(d)). Otherwise, set G^{∗} = Spin_{2n}^{ε}(q^{∗}) ∼_{p} G (Theorem 1.8(a,b)). In either case, we are in the situation of (b).

We are left with the cases where G = 𝔾(q) for some exceptional Lie group 𝔾. By [GL, 10-1(3)] and since the Sylow p-subgroups of G are nonabelian, p | |W (𝔾)|. If ord_{p}(q) = 1, then G^{∗} = 𝔾(q^{∗}) ∼_{p} G by Theorem 1.8(a). If ord_{p}(q) = 2 and 𝔾 ≠ E_{6}, then G^{∗} = 𝔾(q^{∨}) ∼_{p} G by Theorem 1.8(c), where q^{∨} ≡ 1 (mod p). In either case, we are in the situation of (d).

If ord_{p}(q) = 2 and G = E_{6}(q), then ⟨q⟩ = ⟨ − q^{2}⟩ as closed subgroups of ℤ_{p}^{×} (note that v_{p}(q^{2} − 1) = v_{p}((−q^{2})^{2} − 1)). So by Theorem 1.8(d) and Example 4.4 in [BMO], G = E_{6}(q) ∼_{p} ^{2}E_{6}(q^{2}) ∼_{p} F_{4}(q^{2}). So we can choose G^{∗} satisfying (d) as in the last paragraph.

Assume ord_{p}(q) > 2. By [GL, 10-1(3)], for S ∈ Syl_{p}(G) to be nonabelian, there must be some n ≥ 1 such that p · ord_{p}(q) | n, and such that q^{n} − 1 appears as a factor in the formula for |𝔾(q)| (see, e.g., [GL, Table 4-2] or [Ca, Theorem 9.4.10 & Proposition 10.2.5]). Since ord_{p}(q)|(p − 1), this shows that the case ord_{p}(q) > 2 appears only for the group E_{8}(q), and only when p = 5 and ord_{p}(q) = 4. In particular, q, q^{∗} ≡ ±2 (mod 5). Set G^{∗} = E_{8}(q^{∗}); then G^{∗} ∼_{p} G by Theorem 1.8(a), and we are in the situation of (e). □

The following lemma was needed in the proof of Proposition 1.10 to reduce still further the prime powers under consideration.

Lemma 1.11. Fix a prime p, and an integer q prime to p such that q ≠ ±1.

(a) If p is odd, then for any prime r_{0} whose class generates (ℤ/p^{2})^{×}, there is b ≥ 1 such that ⟨q⟩ = ⟨(r_{0})^{b}⟩, and b|(p − 1)p^{ℓ} for some ℓ.

(b) If p = 2, then either ⟨q⟩ = ⟨3⟩, or ⟨q⟩ = ⟨5⟩, or there are ε = ±1 and k ≥ 1 such that

ε ≡ q (mod 8) and ⟨q⟩ = ⟨ε · 3^{2^{k}}⟩.

Proof. Since q ∈ ℤ and |q| > 1, ⟨q⟩ is infinite.

(a) If p is odd, then for each n ≥ 1, (ℤ/p^{n})^{×} ≅ (ℤ/p)^{×} × (ℤ/p^{n−1}) is cyclic and generated by the class of r_{0}. Hence ℤ_{p}^{×} ≅ (ℤ/p)^{×} × (ℤ_{p}, +), and ⟨r_{0}⟩ = ℤ_{p}^{×}. Also, ⟨q⟩ ≥ 1 + p^{ℓ}ℤ_{p} for some ℓ ≥ 1, since each infinite, closed subgroup of (ℤ_{p}, +) contains p^{k}ℤ_{p} for some k.

Set b = [ℤ_{p}^{×} : ⟨q⟩] = [(ℤ/p^{ℓ})^{×} : ⟨q+p^{ℓ}ℤ⟩]|(p − 1)p^{ℓ−1}. Then ⟨q⟩ = ⟨(r_{0})^{b}⟩.

(b) If p = 2, then ℤ_{2}^{×} = {±1} × ⟨3⟩, where ⟨3⟩ ≅ (ℤ_{2}, +). Hence the only infinite closed

subgroups of ⟨3⟩ are those of the form ⟨3^{2^{k}}⟩ for some k ≥ 0. So ⟨q⟩ = ⟨ε · 3^{2^{k}}⟩ for some k ≥ 0 and some ε = ±1, and the result follows since ⟨5⟩ = ⟨ − 3⟩. □

We also note, for use in Section 4, the following more technical result.

Lemma 1.12. Let G be a finite group, fix S ∈ Syl_{p}(G), and set F = F_{S}(G). Let P ≤ S be such that C_{G}(P) ≤ P and N_{S}(P) ∈ Syl_{p}(N_{G}(P)). Then for each φ ∈ Aut(F) such that φ(P) = P, φ|_{N_{S}(P)} extends to an automorphism φ of N_{G}(P).

Proof. Since C_{G}(P) ≤ P and N_{S}(P) ∈ Syl_{p}(N_{G}(P)), N_{G}(P) is a model for the fusion system E = F_{N_{S}(P)}(N_{G}(P)) in the sense of [AKO, Definition I.4.8]. By the strong uniqueness property for models [AKO, Theorem I.4.9(b)], and since φ|_{N_{S}(P)} preserves fusion in E, φ|_{N_{S}(P)} extends to an automorphism of the model. □

The following elementary lemma will be useful in Sections 5 and 6; for example, when computing orders of Sylow subgroups of groups of Lie type.

Lemma 1.13. Fix a prime p. Assume q ≡ 1 (mod p), and q ≡ 1 (mod 4) if p = 2. Then for each n ≥ 1, v_{p}(q^{n} − 1) = v_{p}(q − 1) + v_{p}(n).

Proof. Set r = v_{p}(q − 1), and let k be such that q = 1 + p^{r}k. Then q^{n} = 1 + np^{r}k + ξ, where v_{p}(np^{r}k) = v_{p}(n) + r, and where each term in ξ has strictly larger valuation. □

2. Background on finite groups of Lie type

In this section and the next, we fix the notation to be used for finite groups of Lie type, and list some of the (mostly standard) results which will be needed later. We begin by recalling the following concepts used in [GLS3]. We do not repeat the definitions of maximal tori and Borel subgroups in algebraic groups, but refer instead to [GLS3, §§ 1.4–1.6].

Definition 2.1 ([GLS3, Definitions 1.7.1, 1.15.1, 2.2.1]). Fix a prime q_{0}.

(a) A connected algebraic group G over 𝔽_{q_{0}} is simple if [G, G] ≠ 1, and all proper closed normal subgroups of G are finite and central. If G is simple, then it is of universal type if it is simply connected, and of adjoint type if Z(G) = 1.

(b) A Steinberg endomorphism of a connected simple algebraic group G is a surjective algebraic endomorphism σ ∈ End(G) whose fixed subgroup is finite.

(c) A σ-setup for a finite group G is a pair (G, σ), where G is a simple algebraic group over 𝔽_{q_{0}}, and where σ is a Steinberg endomorphism of G such that G = O^{q_{0}′}(C_{G}(σ)).

(d) Let 𝔏𝔦𝔢(q_{0}) denote the class of finite groups with σ-setup (G, σ) where G is simple and is defined in characteristic q_{0}, and let 𝔏𝔦𝔢 be the union of the classes 𝔏𝔦𝔢(q_{0}) for all primes q_{0}. We say that G is of universal (adjoint) type if G is of universal (adjoint) type.

If G is universal, then C_{G}(σ) is generated by elements of q_{0}-power order (see [St3, Theorem 12.4]), and hence G = C_{G}(σ) in (c) above. In general, C_{G}(σ) = G · C_{T} (σ) (cf. [GLS3, Theorem 2.2.6]).

A root group in a connected algebraic group G over 𝔽_{q_{0}} with a given maximal torus T is a one-parameter closed subgroup (thus isomorphic to 𝔽_{q_{0}}) which is normalized by T.

The roots of G are the characters for the T -actions on the root groups, and lie in the ℤlattice X(T) = Hom(T , 𝔽_{q_{0}}^{×}) of characters of T. (Note that this is the group of algebraic homomorphisms, and that Hom(𝔽_{q_{0}}^{×}, 𝔽_{q_{0}}^{×}) ≅ ℤ.) The roots are regarded as lying in the ℝ-vector space V = ℝ ⊗_{ℤ} X(T). We refer to [GLS3, § 1.9] for details about roots and root subgroups of algebraic groups, and to [Brb, Chapitre VI] for a detailed survey of root systems.

The following notation and hypotheses will be used throughout this paper, when working with a finite group of Lie type defined via a σ-setup.

Notation 2.2. Let (G, σ) be a σ-setup for the finite group G, where G is a connected, simple algebraic group over 𝔽_{q_{0}} for a prime q_{0}. When convenient, we also write G = 𝔾(𝔽_{q_{0}}), where 𝔾 is a group scheme over ℤ.

(A) The maximal torus and Weyl group of G. Fix a maximal torus T in G such that σ(T) = T. Let W = N_{G}(T)/T be the Weyl group of G (and of 𝔾).

(B) The root system of G. Let Σ be the set of all roots of G with respect to T, and let X_{α} < G denote the root group for the root α ∈ Σ. Thus X_{α} = {x_{α}(u) | u ∈ 𝔽_{q_{0}}} with respect to some fixed Chevalley parametrization of G. Set V = ℝ⊗_{ℤ} X(T): a real vector space with inner product (−, −) upon which the Weyl group W acts orthogonally. Let Π ⊆ Σ be a fundamental system of roots, and let Σ_{+} ⊆ Σ be the set of positive roots with respect to Π. For each α ∈ Σ_{+}, let ht(α) denote the height of α: the number of summands in the decomposition of α as a sum of fundamental roots.

For each α ∈ Σ, let w_{α} ∈ W be the reflection in the hyperplane α^{⊥} ⊆ V.

For α ∈ Σ and λ ∈ 𝔽_{q_{0}}^{×}, let n_{α}(λ) ∈ ⟨X_{α}, X_{−α}⟩ and h_{α}(λ) ∈ T ∩ ⟨X_{α}, X_{−α}⟩ be as defined in [Ca, § 6.4] or [GLS3, Theorem 1.12.1]: the images of ( \begin{matrix}0 & λ \\ −λ^{−1} & 0 \end{matrix} ) and ( \begin{matrix}λ & 0 \\ 0 & λ^{−1} \end{matrix} ), respectively, under the homomorphism SL_{2}(𝔽_{q_{0}}) ⟶ G that sends ( \begin{matrix}1 & u \\ 0 & 1 \end{matrix} ) to x_{α}(u) and ( \begin{matrix}1 & 0 \\ v & 1 \end{matrix} ) to x_{−α}(v). Equivalently, n_{α}(λ) = x_{α}(λ)x_{−α}(−λ^{−1})x_{α}(λ) and h_{α}(λ) = n_{α}(λ)n_{α}(1)^{−1}.

(C) The maximal torus, root system and Weyl group of G. Set T = T ∩ G. Let τ ∈ Aut(V) and ρ ∈ Aut(Σ) be the orthogonal automorphism and permutation, respectively, such that for each α ∈ Σ, σ(X_{α}) = X_{ρ(α)} and ρ(α) is a positive multiple of τ(α). Set W_{0} = C_{W} (τ). If ρ(Π) = Π, then set V_{0} = C_{V} (τ), and let pr_{V_{0}}^{⊥} be the orthogonal projection of V onto V_{0}. Let Σ̂ be the set of equivalence classes in Σ determined by τ, where α, β ∈ Σ are equivalent if pr_{V_{0}}^{⊥}(α) is a positive scalar multiple of pr_{V_{0}}^{⊥}(β) (see [GLS3, Definition 2.3.1] or [Ca, § 13.2]). Let Π̂ ⊆ Σ̂_{+} denote the images in Σ̂ of Π ⊆ Σ_{+}. For each α̂ ∈ Σ̂, set Xα̂ = ⟨X_{α} | α ∈ α̂⟩ and Xα̂ = C_{Xα̂}(σ). When α ∈ Σ is of minimal height in its class α̂ ∈ Σ̂, and q′ = |Xab ; α̂|, then for u ∈ 𝔽_{q′}, let x̂_{α}(u) ∈ Xα̂ be an element whose image under projection to X_{α} is x_{α}(u) (uniquely determined modulo [Xα̂, Xα̂]). For α ∈ Π and λ ∈ 𝔽_{q_{0}}^{×}, let ĥ_{α}(λ) ∈ T be an element in G ∩ ⟨h_{β}(𝔽_{q_{0}}^{×}) | β ∈ α̂⟩ whose component in h_{α}(𝔽_{q_{0}}^{×}) is h_{α}(λ) (if there is such an element).

To see that τ and ρ exist as defined in point (C), recall that the root groups X_{α} for α ∈ Σ are the unique closed subgroups of G which are isomorphic to (𝔽_{q_{0}}, +) and normalized by T (see, e.g., [GLS3, Theorem 1.9.5(a,b)]). Since σ is algebraic (hence continuous) and bijective, σ^{−1} sends root subgroups to root subgroups, and σ permutes the root subgroups (hence the roots) since there are only finitely many of them. Using Chevalley’s commutator formula, one sees that this permutation ρ of Σ preserves angles between roots, and hence (up to positive scalar multiple) extends to an orthogonal automorphism of V.

These definitions of x̂_{α}(u) ∈ Xα̂ and ĥ_{α}(λ) ∈ T are slightly different from the definitions in [GLS3, § 2.4] of elements xα̂(u) and hα̂(λ). We choose this notation to emphasize that these elements depend on the choice of α ∈ Σ, not only on its class α̂ ∈ Σ̂. This will be important in some of the relations we need to use in Section 5.

Lemma 2.3. Under the assumptions of Notation 2.2, the action of W on T restricts to an action of W_{0} on T, and the natural isomorphism N_{G}(T)/T ≅ W restricts to an isomorphism

(N_{G}(T) ∩ N_{G}(T))/T ≅ C_{W} (τ) = W_{0} .

Proof. For each α ∈ Σ, n_{α}(1) = x_{α}(1)x_{−α}(−1)x_{α}(1) represents the reflection w_{α} ∈ W, and hence σ(n_{α}) ∈ ⟨X_{ρ(α)}, X_{−ρ(α)}⟩ ∩ N_{G}(T) represents the reflection w_{ρ(α)} = ^{τ}(w_{α}). Since W is generated by the w_{α} for α ∈ Σ, we conclude that σ and τ have the same action on W.

Thus the identification N_{G}(T)/T ≅ W restricts to the following inclusions:

(N_{G}(T) ∩ N_{G}(T))/T ≤ C_{N_{G}(T)}(σ)/C_{T} (σ) ≤ C_{N_{G}(T)/T} (σ) ≅ C_{W} (τ) = W_{0} .

If w ∈ W_{0} represents the coset xT ⊆ N_{G}(T), then x^{−1}σ(x) ∈ T. By the Lang-Steinberg theorem, each element of T has the form t^{−1}σ(t) for some t ∈ T, and hence we can choose x such that σ(x) = x. Then x ∈ C_{G}(σ), and hence x normalizes G = O^{q′_{0}}(C_{G}(σ)) and T = G ∩ T. Since C_{G}(σ) = GC_{T} (σ) (see [GLS3, Theorem 2.2.6(g)] or [St3, Corollary 12.3(a)]), some element of xT lies in N_{G}(T). So the above inclusions are equalities. □

The roots in G are defined formally as characters of its maximal torus T. But it will be useful to distinguish the (abstract) root α ∈ Σ from the character θ_{α} ∈ Hom(T , 𝔽_{q_{0}}^{×}) ⊆ V.

For each root α ∈ Σ ⊆ V, let α^{∨} ∈ V^{∗} be the corresponding co-root (dual root): the unique element such that (α^{∨}, α) = 2 and w_{α} is reflection in the hyperplane Ker(α^{∨}). Since we identify V = V^{∗} via a W -invariant inner product, α^{∨} = 2α/(α, α). Point (c) of the next lemma says that α^{∨} = h_{α}, when we regard h_{α} ∈ Hom(𝔽_{q_{0}}^{×}, T) as an element in V^{∗}.

Lemma 2.4. Assume we are in the situation of (A) and (B) in Notation 2.2.

(a) We have C_{G}(T) = T. In particular, Z(G) ≤ T, and is finite of order prime to the defining characteristic q_{0}.

(b) The maximal torus T in G is generated by the elements h_{α}(λ) for α ∈ Π and λ ∈ 𝔽_{q_{0}}^{×}. If G is universal, and λ_{α} ∈ 𝔽_{q_{0}} are such that ∏_{α∈Π} h_{α}(λ_{α}) = 1, then λ_{α} = 1 for each α ∈ Π. Thus

T = ∏_{α∈Π} h_{α}(𝔽_{q_{0}}^{×}),

and h_{α} is injective for each α.

(c) For each β ∈ Σ, let θ_{β} ∈ X(T) = Hom(T , 𝔽_{q_{0}}^{×}) be the character such that

tx_{β}(u) = x_{β}(θ_{β}(t)·u)

for t ∈ T and u ∈ 𝔽_{q_{0}}. Then

θ_{β}(h_{α}(λ)) = λ^{(α^{∨},β)} for β, α ∈ Σ, λ ∈ 𝔽_{q_{0}}^{×}.

The product homomorphism θ_{Π} = ∏ θ_{β}: T ⟶ ∏_{β∈Π} 𝔽_{q_{0}}^{×} is surjective, and Ker(θ_{Π}) = Z(G).

(d) If α, β_{1}, …, β_{k} ∈ Σ and n_{1}, …, n_{k} ∈ ℤ are such that α^{∨} = n_{1}β_{1}^{∨} + … + n_{k}β_{k}^{∨}, then for each λ ∈ 𝔽_{q_{0}}^{×}, h_{α}(λ) = h_{β_{1}}(λ^{n_{1}}) ⋯ h_{β_{k}}(λ^{n_{k}}).

(e) For each w ∈ W, α ∈ Σ, and λ ∈ 𝔽_{q_{0}}^{×}, and each n ∈ N_{G}(T) such that nT = w ∈ N_{G}(T)/T ≅ W, ^{n}(X_{α}) = X_{w(α)} and ^{n}(h_{α}(λ)) = h_{w(α)}(λ). For each α, β ∈ Σ and each λ ∈ 𝔽_{q_{0}}^{×},

w_{α}(h_{β}(λ)) = h_{w_{α}(β)}(λ) = h_{β}(λ)h_{α}(λ^{−(β^{∨},α)}) .

Hence w_{α}(t) = t · h_{α}(θ_{α}(t))^{−1} for each t ∈ T.

Proof. (a) By [Hu, Proposition 24.1.A], the maximal torus T is regular (i.e., contained in only finitely many Borel subgroups). So C_{G}(T) = T by [Hu, Corollary 26.2.A]. Hence Z(G) ≤ T, it is finite since G is assumed simple, and so it has order prime to the defining characteristic q_{0}.

We claim that it suffices to prove the relations in (c)–(e) in the adjoint group G/Z(G), and hence that we can use the results in [Ca, §§ 7.1–2]. For relations in T, this holds since T is infinitely divisible and Z(G) is finite (thus each homomorphism to T /Z(G) has at most one lifting to T). For relations in a root group X_{α}, this holds since each element of X_{α}Z(G) of order q_{0} lies in X_{α}, since |Z(G)| is prime to q_{0} by (a).

(b) This is stated without proof in [GLS3, Theorem 1.12.5(b)], and with a brief sketch of a proof in [St4, p. 122]. We show here how it follows from the classification of reductive algebraic groups in terms of root data (see, e.g., [Sp, § 10]).

Consider the homomorphism

h_{Π} : T̃ =^{def} ∏_{α∈Π} 𝔽_{q_{0}}^{×} ⟶ T

which sends (λ_{α})_{α∈Π} to ∏_{α} h_{α}(λ_{α}). Then h_{Π} is surjective with finite kernel (see [Ca, § 7.1]). It remains to show that it is an isomorphism when G is of universal type.

We recall some of the notation used in [Sp, § 7]. To G is associated the root datum (X(T), Σ, X^{∨}(T), Σ^{∨}), where

X(T) = Hom(T , 𝔽_{q_{0}}^{×}), X^{∨}(T) = Hom(𝔽_{q_{0}}^{×}, T), Σ^{∨} = {α^{∨} = h_{α} | α ∈ Σ} ⊆ X^{∨}(T) .

As noted before, X(T) and X^{∨}(T) are groups of algebraic homomorphisms, and are free abelian groups of finite rank dual to each other. Recall that Σ ⊆ X(T), since we identify a root α with the character θ_{α}.

Set Y^{∨} = ℤΣ^{∨} ⊆ X^{∨}(T), and let Y ⊇ X(T) be its dual. Then (Y, Σ, Y^{∨}, Σ^{∨}) is still a root datum as defined in [Sp, § 7.4]. By [Sp, Proposition 10.1.3] and its proof, it is realized by a connected algebraic group G̃ with maximal torus T̃, which lies in a central extension f: G̃ ⟶ G which extends h_{Π}. Since G is of universal type, f and hence h_{Π} are isomorphisms.

22 CARLES BROTO, JESPER M. MØLLER, AND BOB OLIVER

(c) Let ℤΣ ≤ V be the additive subgroup generated by Σ. In the notation of [Ca, pp. 97–98], for each α ∈ Σ and λ ∈ 𝔽_{q_{0}}^{×}, h_{α}(λ) = h(χ_{α,λ}) where

χ_{α,λ} ∈ Hom(ℤΣ, 𝔽_{q_{0}}^{×}) is defined by χ_{α,λ}(v) = λ^{2(α,v)/(α,α)} = λ^{(α^{∨},v)}.

Also, by [Ca, p. 100], for each χ ∈ Hom(ℤΣ, 𝔽_{q_{0}}^{×}), β ∈ Σ, and u ∈ 𝔽_{q_{0}}, ^{h(χ)}x_{β}(u) = x_{β}(χ(β)·u). Thus there are homomorphisms θ_{β} ∈ Hom(T , 𝔽_{q_{0}}^{×}), for each β ∈ Σ, such that tx_{β}(u) = x_{β}(θ_{β}(t)·u), and θ_{β}(h(χ)) = χ(β) for each χ. For each α ∈ Σ and λ ∈ 𝔽_{q_{0}}^{×},

θ_{β}(h_{α}(λ)) = θ_{β}(h(χ_{α,λ})) = χ_{α,λ}(β) = λ^{(α^{∨},β)} .  (1)

Assume t ∈ Ker(θ_{Π}). Thus t ∈ Ker(θ_{α}) for all α ∈ Π, and hence for all α ∈ Σ ⊆ ℤΠ. So [t, X_{α}] = 1 for all α ∈ Σ, these root subgroups generate G (see [Sp, Corollary 8.2.10]), and this proves that t ∈ Z(G). The converse is clear: t ∈ Z(G) implies t ∈ T by (a), and hence θ_{β}(t) = 1 for all β ∈ Π by definition of θ_{β}.

It remains to show that θ_{Π} sends T onto ∏_{β∈Π} 𝔽_{q_{0}}. Consider the homomorphisms

T̃ =^{def} ∏_{α∈Π} 𝔽_{q_{0}}^{×} ⟶^{h_{Π}} T ⟶^{θ_{Π}} ∏_{β∈Π} 𝔽_{q_{0}}^{×} ,  (2)

where h_{Π} was defined in the proof of (b). We just saw that θ_{Π} ∘h_{Π} has matrix ((α^{∨}, β))_{α,β∈Π}, which has nonzero determinant since Π ⊆ V and Π^{∨} ⊆ V^{∗} are bases. Since 𝔽_{q_{0}}^{×} is divisible and its finite subgroups are cyclic, this implies that θ_{Π} ∘ h_{Π} is onto, and hence θ_{Π} is onto.

(d) This follows immediately from (c), where we showed, for α ∈ Σ, that α^{∨} can be identified with h_{α} in Hom(𝔽_{q_{0}}^{×}, T) ⊆ V^{∗}.

(e) The first statement (^{n}(X_{α}) = X_{w(α)} and ^{n}(h_{α}(λ)) = h_{w(α)}(λ)) is shown in [Ca, Lemma 7.2.1(ii) & Theorem 7.2.2]. By the usual formula for an orthogonal reflection, w_{α}(β) = β − \frac{2(α,β)}{(α,α)}α = β − (α^{∨}, β)α. Here, we regard w_{α} as an automorphism of V (not of T). Since w_{α}(β) and β have the same norm,

w_{α}(β)^{∨} = \frac{2w_{α}(β)}{(β, β)} = \frac{2β}{(β, β)} − \frac{2(α, β)}{(β, β)} · \frac{2α}{(α, α)} = β^{∨} − (β^{∨}, α) · α^{∨} ,

and by (d),

w_{α}(h_{β}(λ)) = h_{w_{α}(β)}(λ) = h_{β}(λ)h_{α}(λ^{−(β^{∨},α)}) = h_{β}(λ)h_{α}(θ_{α}(h_{β}(λ))^{−1})

where the last equality follows from (c). Since T is generated by the h_{β}(λ) by (b), this implies that w_{α}(t) = t · h_{α}(θ_{α}(t))^{−1} for all t ∈ T. □

For any algebraic group H, H^{0} denotes its identity connected component. The following proposition holds for any connected, reductive group, but we state it only in the context of Notation 2.2. Recall the homomorphisms θ_{β} ∈ Hom(T , 𝔽_{q_{0}}^{×}), defined for β ∈ Σ in Lemma 2.4(c).

Proposition 2.5. Assume Notation 2.2. For any subgroup H ≤ T, C_{G}(H) is an algebraic group, C_{G}(H)^{0} is reductive, and

\begin{aligned}C_{G}(H)^{0} & = ⟨T , X_{α} | α ∈ Σ, H ≤ Ker(θ_{α})⟩ \\ C_{G}(H) & = C_{G}(H)^{0} · \{g ∈ N_{G}(T) | [g, H] = 1\} . \end{aligned}  (3)

If, furthermore, G is of universal type, then Z(G) = C_{T} (W).

Proof. The description of C_{G}(H)^{0} is shown in [Ca2, Theorem 3.5.3] when H is finite and cyclic, and the proof given there also applies in the more general case. For each g ∈ C_{G}(H), c_{g}(T) is another maximal torus in C_{G}(H)^{0}, so gh ∈ C_{N_{G}(T)}(H) for some h ∈ C_{G}(H)^{0}, and thus C_{G}(H) = C_{G}(H)^{0} · C_{N_{G}(T)}(H).

Assume G is of universal type. Since Z(G) ≤ T by Lemma 2.4(a), we have Z(G) ≤ C_{T} (W). Conversely, by Lemma 2.4(b), for each t ∈ T and each α ∈ Σ, ^{t}(x_{α}(u)) = x_{α}(θ_{α}(t)u), and θ_{−α}(t) = θ_{α}(t)^{−1}. Hence also ^{t}(n_{α}(1)) = n_{α}(θ_{α}(t)) (see the formula for n_{α}(λ) in Notation 2.2(B)). If t ∈ C_{T} (W), then [t, n_{α}(1)] = 1 for each α, and since G is of universal type, ⟨X_{α}, X_{−α}⟩ ≅ SL_{2}(𝔽_{q_{0}}). Thus θ_{α}(t) = 1 for all α ∈ Σ, t acts trivially on all root subgroups, and so t ∈ Z(G). □

We now look more closely at the lattice ℤΣ^{∨} generated by the dual roots.

Lemma 2.6. Assume Notation 2.2(A,B), and also that 𝔾 (and hence G) is of universal type.

(a) There is an isomorphism

Φ: ℤΣ^{∨} ⊗_{ℤ} 𝔽_{q_{0}}^{×} ⟶ T

with the property that Φ(α^{∨} ⊗ λ) = h_{α}(λ) for each α ∈ Σ and each λ ∈ 𝔽_{q_{0}}^{×}.

Fix some λ ∈ 𝔽_{q_{0}}^{×}, and set m = |λ|. Set Φ_{λ} = Φ(−, λ): ℤΣ^{∨} ⟶ T.

(b) The map Φ_{λ} is ℤ[W]-linear, Ker(Φ_{λ}) = mℤΣ^{∨}, and Im(Φ_{λ}) = {t ∈ T | t^{m} = 1}.

(c) Fix t ∈ T and x ∈ ℤΣ^{∨} such that Φ_{λ}(x) = t, and also such that

∥x∥ < \frac{1}{2}m · min\{∥α^{∨}∥ | α ∈ Π\}.

Then C_{W} (t) = C_{W} (x).

(d) If m = |λ| ≥ 4, then for each α ∈ Σ, C_{W} (h_{α}(λ)) = C_{W} (α).

Proof. (a,b) Identify ℤΣ^{∨} as a subgroup of Hom(𝔽_{q_{0}}^{×}, T), and let

Φ: ℤΣ^{∨} × 𝔽_{q_{0}}^{×} ⟶ T

be the evaluation pairing. This is bilinear, hence induces a homomorphism on the tensor product, and Φ(α^{∨}, λ) = h_{α}(λ) by Lemma 2.4(c). Since {α^{∨} | α ∈ Π} is a ℤ-basis for ℤΣ^{∨} (since Σ^{∨} is a root system by [Brb, § VI.1, Proposition 2]), and since G is of universal type, Φ is an isomorphism by Lemma 2.4(b).

In particular, for fixed λ ∈ 𝔽_{q_{0}}^{×} of order m, Φ(−, λ) induces an isomorphism from the quotient group ℤΣ^{∨}/mℤΣ^{∨} onto the m-torsion subgroup of T.

(c) Clearly, C_{W} (x) ≤ C_{W} (t); it remains to prove the opposite inclusion. Fix w ∈ C_{W} (t). By (a), w(x) ≡ x (mod mℤΣ^{∨}).

Set r = min\{∥α^{∨}∥ | α ∈ Π\}. For each α ∈ Σ, ∥α^{∨}∥ = \sqrt{k} · r for some k = 1, 2, 3, and hence (α^{∨}, α^{∨}) ∈ r^{2}ℤ. For each α, β ∈ Σ, 2(α^{∨}, β^{∨})/(α^{∨}, α^{∨}) ∈ ℤ (cf. [Ca, Definition 2.1.1]), and hence (α^{∨}, β^{∨}) ∈ \frac{1}{2}r^{2}ℤ. Thus (x, x) ∈ r^{2}ℤ for each x ∈ ℤΣ^{∨}, and in particular, min{∥x∥ | 0 ≠ x ∈ ℤΣ^{∨}} = r.

By assumption, ∥w(x)∥ = ∥x∥ < mr/2, so ∥w(x) − x∥ < mr. Since each nonzero element in mℤΣ^{∨} has norm at least mr, this proves that w(x) − x = 0, and hence that w ∈ C_{W} (x).

(d) This is the special case of (c), where x = α^{∨} and t = h_{α}(λ). □

Lemma 2.7. Assume Notation 2.2, and assume also that 𝔾 is of universal type. Let Γ < Aut(V) be any finite group of isometries of (V, Σ). Then there is an action of Γ on T, where g(h_{α}(u)) = h_{g(α)}(u) for each g ∈ Γ, α ∈ Σ, and u ∈ 𝔽_{q_{0}}^{×}. Fix m ≥ 3 such that q_{0} ∤ m, and set T_{m} = {t ∈ T | t^{m} = 1}. Then Γ acts faithfully on T_{m}. If 1 ≠ g ∈ Γ and ℓ ∈ ℤ are such that g(t) = t^{ℓ} for each t ∈ T_{m}, then ℓ ≡ −1 (mod m).

Proof. The action of Γ on T is well defined by the relations in Lemma 2.4(d,b).

Now fix m ≥ 3 prime to q_{0}, and let T_{m} < T be the m-torsion subgroup. It suffices to prove the rest of the lemma when m = p is an odd prime, or when m = 4 and p = 2. Fix λ ∈ 𝔽_{q_{0}}^{×} of order m, and let Φ_{λ}: ℤΣ^{∨} ⟶ T be the homomorphism of Lemma 2.6(b). By definition of Φ_{λ}, it commutes with the actions of Γ on ℤΣ^{∨} < V and on T_{m}.

Assume 1 ≠ g ∈ Γ and ℓ ∈ ℤ are such that g(t) = t^{ℓ} for each t ∈ T_{m}. Set r = dim(V), and let B ∈ GL_{r}(ℤ) be the matrix for the action of g on ℤΣ^{∨}, with respect to some ℤ-basis of ℤΣ^{∨}. Then |g| = |B|, and B ≡ ℓI (mod mM_{r}(ℤ)). If p = 2 (m = 4), let μ ∈ {±1} be such that ℓ ≡ μ (mod 4). If p is odd (so m = p), then let μ ∈ (ℤ_{p})^{×} be such that μ ≡ ℓ (mod p) and μ^{p−1} = 1. Set B′ = μ^{−1}B ∈ GL_{r}(ℤ_{p}). Thus B′ also has finite order, and B′ ≡ I (mod mM_{r}(ℤ_{p})).

The logarithm and exponential maps define inverse bijections

I + mM_{r}(ℤ_{p}) \overrightarrow{⟵}_{exp}^{ln} mM_{r}(ℤ_{p}) .

They are not homomorphisms, but they do have the property that ln(M^{k}) = k ln(M) for each M ∈ I + mM_{r}(ℤ_{p}) and each k ≥ 1. In particular, the only element of finite order in I + mM_{r}(ℤ_{p}) is the identity. Thus B′ = I, so B = μI. Since μ ∈ ℤ and B ≠ I, we have μ = −1 and B = −I. □

The following lemma about the lattice ℤΣ^{∨} will also be useful when working with the Weyl group action on certain subgroups of T.

Lemma 2.8. Assume Notation 2.2(A,B). Set Λ = ℤΣ^{∨}: the lattice in V generated by the dual roots. Assume that there are b ∈ W of order 2, and a splitting Λ = Λ_{+} × Λ_{−}, such that Λ_{+}, Λ_{−} ≠ 0 and b acts on Λ_{±} via ±Id. Then 𝔾 ≅ C_{n} (= Sp_{2n}) for some n ≥ 2.

Proof. Fix b ∈ W and a splitting Λ = Λ_{+} ×Λ_{−} as above. When considering individual cases, we use the notation of Bourbaki [Brb, Planches I–IX] to describe the (dual) roots, lattice, and Weyl group.

• If 𝔾 = A_{n} (n ≥ 2), then Λ = {(a_{0}, …, a_{n}) ∈ ℤ^{n+1} | a_{0} + … + a_{n} = 0}, and b exchanges certain coordinates pairwise. Choose v ∈ Λ with coordinates 1, −1, and otherwise 0; where the two nonzero entries are in separate orbits of b of which at least one is nonfixed. Then v ∉ Λ_{+} × Λ_{−}, a contradiction.

• If 𝔾 = G_{2}, then as described in [Brb, Planche IX], Λ is generated by the dual fundamental roots (1, −1, 0) and (\frac{2}{3}, −\frac{1}{3}, −\frac{1}{3}), and does not have an orthogonal basis.

• If 𝔾 = B_{n} (n ≥ 3), D_{n} (n ≥ 4), or F_{4}, then Λ < ℤ^{n} is the sublattice of n-tuples the sum of whose coordinates is even. Also, b acts by permuting the coordinates and changing sign (or we can assume it acts this way in the F_{4} case). Choose v with two 1’s and the rest 0, where the 1’s are in separate b-orbits, of which either at least one is nonfixed, or both are fixed and exactly one is negated. Then v ∉ Λ_{+} × Λ_{−}, a contradiction.

• If 𝔾 = E_{8}, then Λ = Λ(E_{8}) < ℝ^{8} is generated by \frac{1}{2}(1, 1, …, 1) and the n-tuples of integers whose sum is even. We can assume (up to conjugation) that b acts as a signed permutation. Choose v as in the last case.

• If 𝔾 = E_{7}, then Λ < ℝ^{8} is the lattice of all x = (x_{1}, …, x_{8}) ∈ Λ(E_{8}) such that x_{7} = −x_{8}. Up to conjugation, b can be again be assumed to act on A via a signed permutation (permuting only the first six coordinates), and v can be chosen as in the last case.

• If 𝔾 = E_{6}, then Λ < ℝ^{8} is the lattice of all x = (x_{1}, …, x_{8}) ∈ Λ(E_{8}) such that x_{6} = x_{7} = −x_{8}. Also, W contains a subgroup isomorphic to 2^{4} : S_{5} with odd index which acts on the remaining five coordinates via signed permutations. So b and v can be taken as in the last three cases. □

We finish the section with a very elementary lemma.

It will be useful to know, in certain situations, that each coset of T in N_{G}(T) contains elements of G.

Lemma 2.9. Assume that we are in the situation of Notation 2.2(A,B). Assume also that σ acts on T via (t ↦ t^{m}) for some 1 ≠ m ∈ ℤ. Then for each g ∈ N_{G}(T), gT ∩ C_{G}(σ) ≠ ∅.

Proof. Since σ|_{T} ∈ Z(Aut(T)), we have g^{−1}σ(g) ∈ C_{G}(T) = T, the last equality by Lemma 2.4(a). So for each t ∈ T, σ(gt) = gt if and only if g^{−1}σ(g) = t^{1−m}. Since T ≅ (𝔽_{q_{0}}^{×})^{r} for some r, and 𝔽_{q_{0}} is algebraically complete (and 1 − m ≠ 0), this always has solutions. □

3. Automorphisms of groups of Lie type

Since automorphisms of G play a central role in this paper, we need to fix our notation (mostly taken from [GLS3]) for certain subgroups and elements of Aut(G). We begin with automorphisms of the algebraic group G.

Definition 3.1. Let G and its root system Σ be as in Notation 2.2(A,B).

(a) When q is any power of q_{0} (the defining characteristic of G), let ψ_{q} ∈ End(G) be the field endomorphism defined by ψ_{q}(x_{α}(u)) = x_{α}(u^{q}) for each α ∈ Σ and each u ∈ 𝔽_{q_{0}}. Set Φ_{G} = {ψ_{q_{0}^{b}} | b ≥ 1}: the monoid of all field endomorphisms of G.

(b) Let Γ_{G} be the group or set of graph automorphisms of G as defined in [GLS3, Definition 1.15.5(e)]. Thus when (𝔾, q_{0}) ≠ (B_{2}, 2), (G_{2}, 3), nor (F_{4}, 2), Γ_{G} is the group of all γ ∈ Aut(G) of the form γ(x_{α}(u)) = x_{ρ(α)}(u) (all α ∈ ±Π and u ∈ 𝔽_{q_{0}}) for some isometry ρ of Σ such that ρ(Π) = Π. If (𝔾, q_{0}) = (B_{2}, 2), (G_{2}, 3), or (F_{4}, 2), then Γ_{G} = {1, ψ}, where for the angle-preserving permutation ρ of Σ which exchanges long and short roots and sends Π to itself, ψ(x_{α}(u)) = x_{ρ(α)}(u) when α is a long root and ψ(x_{α}(u)) = x_{ρ(α)}(u^{q_{0}}) when α is short.

(c) A Steinberg endomorphism σ of G is “standard” if σ = ψ_{q} ∘ γ = γ ∘ ψ_{q}, where q is a power of q_{0} and γ ∈ Γ_{G}. A σ-setup (G, σ) for a finite subgroup G < G is standard if σ is standard.

By [GLS3, Theorem 2.2.3], for any G with σ-setup (G, σ) as in Notation 2.2, G is Gconjugate to a subgroup G^{∗} which has a standard σ-setup. This will be made more precise in Proposition 3.6(a).

Most of the time in this paper, we will be working with standard σ-setups. But there are a few cases where we will need to work with setups which are not standard, which is why this condition is not included in Notation 2.2.

Following the usual terminology, we call G a “Chevalley group” if it has a standard σsetup where γ = Id in the notation of Definition 3.1; i.e., if G ≅ 𝔾(q) where q is some power of q_{0}. In this case, the root groups Xα̂ are all abelian and isomorphic to 𝔽_{q}. When G has a standard σ-setup with γ ≠ Id, we refer to G as a “twisted group”, and the different possible structures of its root groups are described in [GLS3, Table 2.4]. We also refer to G as a “Steinberg group” if γ ≠ Id and is an algebraic automorphism of G; i.e., if G is a twisted group and not a Suzuki or Ree group.

The following lemma will be useful in Sections 5 and 6.

Lemma 3.2. Assume G is as in Notation 2.2(A,B). Then for each algebraic automorphism γ of G which normalizes T, there is an orthogonal automorphism τ of V such that τ(Σ) = Σ, and

γ(X_{α}) = X_{τ(α)}  and  γ(h_{α}(λ)) = h_{τ(α)}(λ)

for each α ∈ Σ and each λ ∈ 𝔽_{q_{0}}^{×}. In particular, |γ|_{T} | = |τ| < ∞. If, in addition, γ normalizes each of the root groups X_{α} (i.e., τ = Id), then γ ∈ Aut_{T} (G).

Proof. By [GLS3, Theorem 1.15.2(b)], and since γ is an algebraic automorphism of G, γ = c_{g} ∘γ_{0} for some g ∈ G and some γ_{0} ∈ Γ_{G}. Furthermore, γ_{0} has the form: γ_{0}(x_{α}(u)) = x_{χ(α)}(u) for all α ∈ Σ and u ∈ 𝔽_{q_{0}}, and some isometry χ ∈ Aut(V) such that χ(Π) = Π. Since γ and γ_{0} both normalize T, we have g ∈ N_{G}(T).

Thus by Lemma 2.4(e), there is τ ∈ Aut(V) such that τ(Σ) = Σ, and γ(X_{α}) = X_{τ(α)} and γ(h_{α}(λ)) = h_{τ(α)}(λ) for each α ∈ Σ and λ ∈ 𝔽_{q_{0}}^{×}. In particular, |γ|_{T} | = |τ|.

If τ = Id, then γ_{0} = Id and g ∈ T. Thus γ ∈ Aut_{T} (G). □

We next fix notation for automorphisms of G.

Definition 3.3. Let G and G be as in Notation 2.2(A,B,C), where in addition, we assume the σ-setup is standard.

(a) Set

Inndiag(G) = Aut_{T} (G)Inn(G)  and  Outdiag(G) = Inndiag(G)/Inn(G) .

(b) Set Φ_{G} = {ψ_{q}|_{G} | q = q_{0}^{b}, b ≥ 1}, the group of field automorphisms of G.

(c) If G is a Chevalley group, set Γ_{G} = {γ|_{G} | γ ∈ Γ_{G}}, the group of graph automorphisms of G. Set Γ_{G} = 1 if G is a twisted group (a Steinberg, Suzuki, or Ree group).

Note that in [GLS3, Definition 2.5.13], when G has a standard σ-setup (G, σ), Inndiag(G) is defined to be the group of automorphisms induced by conjugation by elements of C_{G/Z(G)}(σ)

(lifted to G). By [GLS3, Lemma 2.5.8], this is equal to Inndiag(G) as defined above when G is of adjoint form, and hence also in the general case (since Z(G) ≤ T).

AUTOMORPHISMS OF FUSION SYSTEMS OF FINITE SIMPLE GROUPS OF LIE TYPE 27

Steinberg’s theorem on automorphisms of groups of Lie type can now be stated.

Theorem 3.4 ([St1, § 3]). Let G be a finite group of Lie type. Assume that (G, σ) is a standard σ-setup for G, where G is in adjoint or universal form. Then

Aut(G) = Inndiag(G)Φ_{G}Γ_{G} ,

where Inndiag(G) ⊴ Aut(G) and Inndiag(G) ∩ (Φ_{G}Γ_{G}) = 1.

Proof. See, e.g., [GLS3, Theorem 2.5.12(a)] (together with [GLS3, Theorem 2.5.14(d)]). Most of this follows from the main result in [St1], and from [St2, Theorems 30 & 36]. □

We also need the following characterizations of Inndiag(G) which are independent of the choice of σ-setup.

Proposition 3.5. Assume the hypotheses and notation in 2.2. Then

(a) C_{G}(G) = Z(G);

(b) N_{G}(G) = GN_{T} (G); and

(c) Inndiag(G) = Aut_{T} (G)Inn(G) = Aut_{G}(G) and hence Outdiag(G) = Out_{T} (G).

In fact, (b) and (c) hold if we replace T by any σ-invariant maximal torus in G.

Proof. (a) Since the statement is independent of the choice of σ-setup, we can assume that σ is standard. Set U = ∏_{α∈Σ_{+}} X_{α} and U^{∗} = ∏_{α∈Σ_{+}} X_{−α}.

Fix g ∈ C_{G}(G). Since G has a BN-pair (see [Ca, Proposition 8.2.1]), it has a Bruhat decomposition G = BNB = UNU [Ca, Proposition 8.2.2(i)], where B = T U and N = N_{G}(T). Write g = unv, where u, v ∈ U and n ∈ N. For each x ∈ U ∩ G, ^{g}x = ^{u}(^{nv}x) ∈ U implies that ^{nv}x = ^{n}(^{v}x) ∈ U.

Since n ∈ N_{G}(T), conjugation by n permutes the root groups of G, in a way determined by the class w = nT ∈ W = N_{G}(T)/T. Thus w sends each (positive) root in the decomposition of ^{v}x to a positive root. For each α ∈ Σ_{+}, x̂_{α}(1) ∈ G, ^{v}(x̂_{α}(1)) has α in its decomposition, and hence w(α) ∈ Σ_{+}.

Thus w sends all positive roots to positive roots, so w(Π) = Π, and w = 1 by [Ca, Corollary 2.2.3]. So n ∈ T, and g = unv ∈ T U.

By the same argument applied to the negative root groups, g ∈ T U^{∗}. Hence g ∈ T.

For each α ∈ Σ, g ∈ T commutes with x̂_{α}(1) ∈ G, and hence g centralizes X_{β} for each β ∈ α̂ (Lemma 2.4(c)). Thus g centralizes all root groups in G, so g ∈ Z(G).

(b) Let T^{∗} be any σ-invariant maximal torus in G. Fix g ∈ N_{G}(G). Then g^{−1} · σ(g) ∈ C_{G}(G) = Z(G) ≤ T^{∗} by (a). By Lang’s theorem [GLS3, Theorem 2.1.1], there is t ∈ T^{∗} such that g^{−1} · σ(g) = t^{−1} · σ(t). Hence gt^{−1} ∈ C_{G}(σ) = G · C_{T_{∗}}(σ), where the last equality holds by [GLS3, Theorem 2.2.6(g)]. So g ∈ GT^{∗}, and g ∈ GN_{T_{∗}}(G) since g normalizes G.

(c) By (b), Aut_{G}(G) = Aut_{T_{∗}}(G)Inn(G) for each σ-invariant maximal torus T^{∗}. By definition, Inndiag(G) = Aut_{T_{∗}}(G)Inn(G) when T^{∗} is the maximal torus in a standard σ-setup for G. Hence Inndiag(G) = Aut_{G}(G) = Aut_{T_{∗}}(G)Inn(G) for all such T^{∗}. □

We refer to [GLS3, Definitions 1.15.5(a,e) & 2.5.10] for more details about the definitions of Φ_{G} and Γ_{G}. The next proposition describes how to identify these subgroups when working in a nonstandard setup.

Proposition 3.6. Assume G, T, and the root system of G, are as in Notation 2.2(A,B). Let σ be any Steinberg endomorphism of G, and set G = O^{q′_{0}}(C_{G}(σ)).

(a) There is a standard Steinberg endomorphism σ^{∗} of G such that if we set G^{∗} = O^{q′_{0}}(C_{G}(σ^{∗})), then there is x ∈ G such that G = ^{x}(G^{∗}).

Fix G^{∗}, σ^{∗}, and x as in (a). Let Inndiag(G^{∗}), Φ_{G^{∗}}, and Γ_{G^{∗}} be as in Definition 3.3 (with respect to the σ-setup (G, σ^{∗})). Set Inndiag(G) = c_{x}Inndiag(G^{∗})c_{x}^{−1}, Φ_{G} = c_{x}Φ_{G^{∗}}c_{x}^{−1}, and Γ_{G} = c_{x}Γ_{G^{∗}}c_{x}^{−1}, all as subgroups of Aut(G). Then the following hold.

(b) Inndiag(G) = Aut_{G}(G).

(c) For each α ∈ Φ_{G}Γ_{G} such that α|_{G^{∗}} ∈ Φ_{G^{∗}}Γ_{G^{∗}}, and each β ∈ α · Inn(G) such that β(G) = G, β|_{G} ≡ c_{x}(α)c_{x}^{−1} (mod Inndiag(G)).

(d) If ψ_{q_{0}} normalizes G, then Inndiag(G)Φ_{G} = Inndiag(G)⟨ψ_{q_{0}}|_{G}⟩.

Thus the subgroups Φ_{G} and Γ_{G} are well defined modulo Inndiag(G), independently of the choice of standard σ-setup for G.

Proof. (a) See, e.g., [GLS3, Theorem 2.2.3]: for any given choice of maximal torus, positive roots, and parametrizations of the root groups, each Steinberg automorphism of G is conjugate, by an element of Inn(G), to a Steinberg automorphism of standard type.

(b) This follows immediately from Proposition 3.5(c).

(c) By assumption, β ≡ α ≡ c_{x}αc_{x}^{−1} (mod Inn(G)). Since β and c_{x}αc_{x}^{−1} both normalize G, β|_{G} ≡ c_{x}α^{∗}c_{x}^{−1} modulo Aut_{G}(G) = Inndiag(G).

(d) If ψ_{q_{0}} normalizes G, then (c), applied with α = β = ψ_{q_{0}}, implies that as elements of Aut(G)/Inndiag(G), [ψ_{q_{0}}|_{G}] = [c_{x}(ψ_{q_{0}}|_{G^{∗}})c_{x}^{−1}] generates the image of Φ_{G}. □

Lemma 3.7. Assume G, T, σ, G = O^{q′_{0}}(C_{G}(σ)), and the root system of G, are as in Notation 2.2(A,B). Assume that φ ∈ Aut(T) is the restriction of an algebraic automorphism of G such that [φ, σ|_{T}] = 1. Then there is an algebraic automorphism φ ∈ Aut(G) such that φ|_{T} = φ, [φ, σ] = 1, and φ(G) = G.

Proof. By assumption, there is φ ∈ Aut(G) such that φ|_{T} = φ. Also, [φ, σ] is an algebraic automorphism of G by [GLS3, Theorem 1.15.7(a)], it is the identity on T, and hence [φ, σ] = c_{t} for some t ∈ T by Lemma 3.2. Using the Lang-Steinberg theorem, upon replacing φ by c_{u}φ for appropriate u ∈ T, we can arrange that [φ, σ] = 1. In particular, φ(G) = G. □

The following proposition is well known, but it seems to be difficult to find references where it is proven.

Proposition 3.8. Fix a prime q_{0}, and a group G ∈ 𝔏𝔦𝔢(q_{0}) of universal type. Then Z(G) has order prime to q_{0}, G/Z(G) ∈ 𝔏𝔦𝔢(q_{0}) and is of adjoint type, and Z(G/Z(G)) = 1. If G/Z(G) is simple, then each central extension of G by a group of order prime to q_{0} splits (equivalently, H^{2}(G; ℤ/p) = 0 for all primes p ≠ q_{0}).

Proof. Let (G, σ) be a σ-setup for G, and choose a maximal torus and positive roots in G. We can thus assume Notation 2.2. By Lemma 2.4(a), Z(G) is finite of order prime to q_{0}. Since Z(G) ≤ C_{G}(G) = Z(G) by Proposition 3.5(a), Z(G) also has order prime to q_{0}.

Set G_{a} = G/Z(G) and let G_{a} < G_{a} be the image of G under projection. Thus G_{a} is an algebraic group of adjoint type, and G_{a} = O^{q_{0}′}(C_{G_{a}}(σ_{a})) ∈ 𝔏𝔦𝔢(q_{0}) where σ_{a} ∈ End(G_{a}) is induced by σ. Also, Z(G_{a}) ≤ Z(G_{a}) = 1 by Proposition 3.5(a) again.

It remains to prove the statement about central extensions. When G is a Chevalley group, this was shown in [St4, Théorème 4.5]. It was shown in [St6, Corollary 6.2] when G ≅ ^{2}A_{n}(q) for n even, and in [AG] when G ≅ ^{2}G_{2}(q) or Sz(q). The remaining cases follow by similar arguments (see [St5, 9.4 & 12.4]). (See also [Cu, § 1], as well as Theorem 6.1.4 and Tables 6.1.2 and 6.1.3 in [GLS3].) □

The next proposition shows that in most cases, C_{G}(T) = T. In Section 5, we will see some conditions which imply that C_{G}(O_{p}(T)) = T when p is a prime different from the defining characteristic.

Proposition 3.9. Let (G, σ) be a σ-setup for G, where G and G are of universal type. Assume Notation 2.2, and in particular, that we have fixed a maximal torus T and a root system Σ in G.

(a) Assume that C_{G}(T)^{0} ≩ T, where (−)^{0} denotes the connected component of the identity. Then there is α ∈ Σ_{+} such that θ_{α}(T) = 1. Also, there is β ∈ Hom(T , 𝔽_{q_{0}}^{×}) such that θ_{α} = β^{−1}σ^{∗}(β); i.e., θ_{α}(t) = β(t^{−1}σ(t)) for each t ∈ T.

(b) If the σ-setup is standard, then C_{G}(T)^{0} = T except possibly when G ≅ ^{r}𝔾(2) for some 𝔾 and some r ≤ 3, or when G ≅ A_{1}(3), C_{n}(3) for n ≥ 2, or ^{2}G_{2}(3).

(c) If C_{G}(T)^{0} = T, then N_{G}(T)/T ≅ W_{0}.

Proof. (a) By Proposition 2.5, and since C_{G}(T)^{0} > T, there is α ∈ Σ such that T ≤ Ker(θ_{α}) (equivalently, [T, X_{α}] = 1). Since Ker(θ_{−α}) = Ker(θ_{α}), we can assume that α ∈ Σ_{+}.

Since G is of universal type, G = C_{G}(σ) and T = C_{T} (σ). Hence there is a short exact sequence

1 ⟶ T ⟶ T ⟶^{t↦t^{−1}σ(t)} T ,

where the last map is onto by the Lang-Steinberg theorem. Upon dualizing, and regarding Hom(T , 𝔽_{q_{0}}^{×}) additively, we get an exact sequence

0 ⟶ Hom(T , 𝔽_{q_{0}}^{×}) ⟶^{σ^{∗}−Id} Hom(T , 𝔽_{q_{0}}^{×}) ⟶^{restr} Hom(T, 𝔽_{q_{0}}^{×})

(see also [Ca2, Proposition 3.2.3]), where Hom(T , 𝔽_{q_{0}}^{×}) is the group of algebraic homomorphisms. Since θ_{α} is in the kernel of the restriction map, by assumption, it has the form β^{−1}σ^{∗}(β) for some β ∈ Hom(T , 𝔽_{q_{0}}^{×}).

(b) Let P (Σ) and Q(Σ) be as in [Brb, § VI.1.9] (but with Σ in place of R to denote the root system). Thus Q(Σ) = ℤΣ, the integral lattice generated by Σ, and

P (Σ) = {v ∈ V | (v, α^{∨}) ∈ ℤ for all α ∈ Σ} ≥ Q(Σ) .

For each v ∈ P (Σ), define θ_{v} ∈ X(T) = Hom(T , 𝔽_{q_{0}}^{×}) by setting θ_{v}(h_{α}(λ)) = λ^{(v,α^{∨})} for α ∈ Π and λ ∈ 𝔽_{q_{0}}. Since G is of universal type, this is a well defined homomorphism by Lemma 2.4(b), and the same formula holds for all α ∈ Σ by Lemma 2.4(d). By Lemma 2.4(c), this extends our definition of θ_{β} for β ∈ Σ ⊆ P (Σ).

Recall that Hom(𝔽_{q_{0}}^{×}, 𝔽_{q_{0}}^{×}) ≅ ℤ. For each θ ∈ X(T) and each α ∈ Σ, let n_{θ,α} ∈ ℤ be such that θ(h_{α}(λ)) = λ^{n_{θ,α}} for all λ ∈ 𝔽_{q_{0}}^{×}. For given θ, there is v ∈ P (Σ) such that (v, α^{∨}) = n_{θ,α} for all α ∈ Π, and hence (by Lemma 2.4(d)) for all α ∈ Σ. Then θ = θ_{v} as defined above. In this way, we identify P (Σ) with the lattice X(T) of characters for T, while identifying Q(Σ) with ℤΣ.

From the appendix to Chapter VI in [Brb] (Planches I–IX), we obtain the following table:

root system Σ  A_{n}  C_{n}  B_{n}, D_{n}  G_{2} F_{4} E_{6} E_{7} E_{8}
min\{∥v∥ | v ∈ P (Σ)\} \sqrt{n/(n+1)} 1 min\{\sqrt{n/4}, 1\} \sqrt{2} 1 \sqrt{4/3} \sqrt{2} \sqrt{2}
max\{∥α∥ | α ∈ Σ\}  \sqrt{2}  2  \sqrt{2}  \sqrt{6} \sqrt{2} \sqrt{2} \sqrt{2} \sqrt{2}

Here, the norms are given with respect to the descriptions of these lattices in [Brb] as subgroups of Euclidean spaces.

Assume C_{G}(T)^{0} ≩ T. By (a), there are α ∈ Σ_{+} and β ∈ Hom(T , 𝔽_{q_{0}}^{×}) such that α = β^{−1}σ^{∗}(β). If we regard α and β as elements in the normed vector space V, then ∥α∥ = ∥σ^{∗}(β)−β∥ ≥ ∥σ^{∗}(β)∥−∥β∥. If G = ^{r}𝔾(q) (and σ is a standard setup), then ∥σ^{∗}(β)∥ = q∥β∥, except when G is a Suzuki or Ree group in which case ∥σ^{∗}(β)∥ = \sqrt{q}∥β∥. Thus

\frac{∥α∥}{∥β∥} + 1 ≥ \begin{cases}q & if G is a Chevalley or Steinberg group \\ \sqrt{q} & if G is a Suzuki or Ree group. \end{cases}

By the above table, this is possible only if q = 2, or if G is isomorphic to one of the groups A_{1}(3), B_{2}(3), C_{n}(3) (n ≥ 3), ^{2}G_{2}(3), or ^{2}B_{2}(8).

Assume G ≅ ^{2}B_{2}(8) ≅ Sz(8). It is most convenient to use the root system for C_{2} constructed in [Brb]: P (Σ) = ℤ^{2}, and Σ = {(±2, 0), (0, ±2), (±1, ±1)}. Then α and β satisfy the above inequality only if ∥α∥ = 2, ∥β∥ = 1, and ∥α + β∥ = \sqrt{8}. So (α, β) = \frac{3}{2}, which is impossible for α, β ∈ ℤ^{2}. Hence C_{G}(T)^{0} = T in this case.

(c) If C_{G}(T)^{0} = T, then N_{G}(T) ≤ N_{G}(T), and so N_{G}(T)/T ≅ W_{0} by Lemma 2.3. □

The following, more technical lemma will be needed in Section 6.

Lemma 3.10. Assume the hypotheses and notation in 2.2, and also that the σ-setup (G, σ) is standard. Then under the action of W_{0} on Σ̂, each orbit contains elements of Π̂.

Proof. When ρ = Id, this is [Ca, Proposition 2.1.8]. When ρ ≠ Id, it follows from the descriptions of W_{0} and Σ̂ in [Ca, §§ 13.2–13.3]. □

4. The equicharacteristic case

The following notation will be used in this section.

Notation 4.1. Assume the notation in 2.2, and also that ρ(Π) = Π, q_{0} = p, and Z(G) = 1. Thus G = 𝔾(𝔽_{p}) is a connected, simple group over 𝔽_{p} in adjoint form, σ is a Steinberg endomorphism of G of standard form, and G = O^{p′}(C_{G}(σ)).

(D) Set U = ⟨X_{α} | α ∈ Σ_{+}⟩ and B =^{def} N_{G}(U) = UT (the Borel subgroup of G). Set

U = C_{U}(σ) = ⟨Xα̂ | α̂ ∈ Σ̂_{+}⟩ ,  B = N_{G}(U) ,  and  T = T ∩ G.

Thus U = ∏_{α̂∈Σ̂_{+}} Xα̂ ∈ Syl_{p}(G), and B = UT. (See, e.g., [GLS3, Theorems 2.3.4(d) & 2.3.7], or [Ca, Theorems 5.3.3(ii) & 9.4.10] in the case of Chevalley groups.) When Ĵ ⫋ Π̂ is the image in Σ̂_{+} of a τ-invariant subset J ⫋ Π, let UĴ ≤ U be the subgroup generated by root groups for positive roots in Σ_{+}∖⟨J⟩ (the unipotent radical subgroup associated to Ĵ), and set 𝔓Ĵ = N_{G}(UĴ) = B⟨X_{−α̂} | α ∈ ⟨J⟩⟩ (the parabolic subgroup associated to Ĵ). Thus U = U_{∅} and B = 𝔓_{∅}. We also write Uα̂ = U_{{α̂}} and 𝔓α̂ = 𝔓_{{α̂}} for each α̂ ∈ Π̂.

(E) The height of a positive root α = ∑_{γ∈Π} n_{γ}γ ∈ Σ_{+} (n_{γ} ≥ 0) is defined by ht(α) = ∑_{γ∈Π} n_{γ}. The height ht(α̂) of a class of roots α̂ ∈ Σ̂_{+} is the minimum of the heights of roots in the class α̂.

(F) Set F = F_{U}(G) and L = L_{U}^{c}(G).

(G) Set U_{0} = ⟨Xα̂ | α̂ ∈ Σ̂_{+}, α̂ ∩ Π = ∅⟩ = ⟨Xα̂ | ht(α̂) ≥ 2⟩.

(H) The Lie rank of G is equal to |Π̂|; equivalently, to the number of maximal parabolic subgroups containing B.

For example, assume σ = ψ_{q} ∘ γ, where γ ∈ Aut(G) is a graph automorphism which induces ρ ∈ Aut(Σ_{+}), and ψ_{q} is the field automorphism induced by t ↦ t^{q}. Then for α̂ ∈ Σ̂, Xα̂ ≅ 𝔽_{q} when α̂ = {α} contains only one root, Xα̂ ≅ 𝔽_{q^{a}} if α̂ = {ρ^{i}(α)} is the ρ-orbit of α with length a, and Xα̂ is nonabelian if α̂ contains a root α and sums of roots in its ρ-orbit.

We need the following, stronger version of Theorem 3.4.

Theorem 4.2 ([St1, § 3]). Assume G is as in Notation 2.2 and 4.1. If α ∈ Aut(G) is such that α(U) = U, then α = c_{u}dfg for unique automorphisms c_{u} ∈ Aut_{U}(G), d ∈ Inndiag(G) = Aut_{T} (G), f ∈ Φ_{G}, and g ∈ Γ_{G}.

Proof. Let N_{Aut(G)}(U) ≤ Aut(G) and N_{Inndiag(G)}(U) ≤ Inndiag(G) be the subgroups of those automorphisms that send U to itself. Since Φ_{G}Γ_{G} ≤ N_{Aut(G)}(U) by definition, Theorem 3.4 implies that N_{Aut(G)}(U) = N_{Inndiag(G)}(U) · (Φ_{G}Γ_{G}), a semidirect product. Since Φ_{G} ∩ Γ_{G} = 1, it remains to show that N_{Inndiag(G)}(U) = Aut_{U}(G)Aut_{T} (G) and Aut_{U}(G) ∩ Aut_{T} (G) = 1. The first is immediate: since Aut_{T} (G) ≤ N_{Aut(G)}(U) and N_{G}(U) = T U,

N_{Inndiag(G)}(U) = (Inn(G)Aut_{T} (G)) ∩ N_{Aut(G)}(U)
= Aut_{N_{G}(U)}(G)Aut_{T} (G) = Aut_{U}(G)Aut_{T} (G) .

Finally, if c_{u} = c_{t} ∈ Aut(G) where u ∈ U and t ∈ T, then c_{u} = Id_{G}, since u has p-power order and t has order prime to p. □

Lemma 4.3. Assume G ∈ 𝔏𝔦𝔢(p). Then for U ∈ Syl_{p}(G), κ_{G} sends Out(G) injectively into Out(F).

Proof. Assume that κ_{G/Z(G)} is injective. We claim that Aut(G) injects into Aut(G/Z(G)), and hence that κ_{G} is also injective. To see this, fix α ∈ Aut(G) such that [α, G] ≤ Z(G). Recall that Z(G) has order prime to p (Proposition 3.8). For each g ∈ G of p-power order, α(g) = gz for some z ∈ Z(G), and z = 1 since otherwise |zg| > |g|. Since G is generated by such elements by definition of 𝔏𝔦𝔢(p), α = Id_{G}, proving the claim. It thus suffices to prove the lemma when G is in adjoint form.

We can thus assume Notation 4.1. By Lemma 1.4, it will suffice to prove that C_{Aut(G)}(U) ≤ Inn(G). Fix β ∈ Aut(G) such that β|_{U} = Id_{U}. By Theorem 4.2, there are unique automorphisms c_{u} ∈ Aut_{U}(G), d ∈ Aut_{T} (G), f ∈ Φ_{G}, and g ∈ Γ_{G} such that β = c_{u}dfg.

If g ≠ Id, then it permutes the fundamental root groups nontrivially, while c_{u}df|_{U} sends each such group to itself modulo higher root groups and commutators. Hence g = Id. Similarly, f = Id, since otherwise β would act on the fundamental root groups (modulo higher root groups) via some automorphism other than a translation.

Thus β = c_{u}d, where d = c_{t} for some t ∈ N_{T} (G). Then u has p-power order while t has order prime to p, so d|_{U} = c_{t}|_{U} = Id. By Lemma 2.4(c), c_{t} sends each root group in U to itself via x_{α}(u) ↦ x_{α}(θ_{α}(t)·u) for some character θ_{α} ∈ Hom(T , 𝔽_{p}^{×}) which is linear in α. For each α̂ ∈ Σ̂_{+}, c_{t}|_{Xα̂} = Id implies that θ_{α}(t) = 1 for all α ∈ α̂. Thus θ_{α}(t) = 1 for all α ∈ Σ_{+}, so c_{t} = Id_{G}, and β = c_{u} ∈ Inn(G). □

It now remains, when proving Theorem A, to show the surjectivity of κ_{G}. This will be done case-by-case. We first handle groups of Lie rank at least three, then those of rank one, and finally those of rank two.

For simplicity, we state the next two propositions only for groups of adjoint type, but they also hold without this restriction. The first implies that each element of Aut(F) permutes the subgroups UĴ (as defined in Notation 4.1), and that each element of Aut(L_{S}^{c}(G)) induces an automorphism of the amalgam of parabolics 𝔓Ĵ for Ĵ ⫋ Π̂.

Proposition 4.4. Assume Notation 4.1. For 1 ≠ P ≤ U, the following are equivalent:

(i) P = UĴ for some Ĵ ⫋ Π̂;

(ii) P ⊴ B, C_{U}(P) ≤ P, and O_{p}(Out_{F}(P)) = 1; and

(iii) P ⊴ B, C_{G}(P) ≤ P, and O_{p}(N_{G}(P)) = P.

Hence for each φ ∈ Aut(F), φ permutes the subgroups UĴ, and in particular permutes the subgroups Uα̂ for α̂ ∈ Π̂.

Proof. (i) ⟹ (iii): For each Ĵ ⫋ Π̂, C_{G}(UĴ) = Z(UĴ) by [GLS3, Theorem 2.6.5(e)] (recall that G is of adjoint type). Also, O_{p}(N_{G}(UĴ)) = O_{p}(𝔓Ĵ) = UĴ, and UĴ is normal in B since N_{G}(UĴ) = 𝔓Ĵ ≥ B. (iii) ⟹ (ii): This holds since Out_{F}(P) ≅ N_{G}(P)/P C_{G}(P). (ii) ⟹ (i): In this case, P ⊴ B, so N_{G}(P) ≥ B, and N_{G}(P) = 𝔓Ĵ for some Ĵ ⫋ Π̂ (cf. [Ca, Theorem 8.3.2]). Then P ≤ O_{p}(𝔓Ĵ) = UĴ. Also, UĴC_{G}(P)/P C_{G}(P) ≤ O_{p}(N_{G}(P)/P C_{G}(P)) = 1, so UĴ ≤ P C_{G}(P). Since UĴ ≤ U, this implies that UĴ ≤ P C_{U}(P) = P; i.e., that P = UĴ. So (i) holds.

The last statement follows from the equivalence of (i) and (ii). □

When G has large Lie rank, Theorem A now follows from properties of Tits buildings.

Proposition 4.5. Assume G ∈ 𝔏𝔦𝔢(p) is of adjoint type and has Lie rank at least 3. Fix U ∈ Syl_{p}(G). Then κ_{G} is split surjective.

Proof. Set L = L_{U}^{c}(G). By Proposition 4.4, for each α ∈ Aut(L), α permutes the subgroups UĴ for Ĵ ⫋ Π̂. For each such Ĵ, C_{G}(UĴ) = Z(UĴ), so Aut_{L}(UĴ) = N_{G}(UĴ) = 𝔓Ĵ. Thus α induces an automorphism of the amalgam of parabolic subgroups 𝔓Ĵ. Since G is the amalgamated sum of these subgroups by a theorem of Tits (see [Ti, Theorem 13.5] or [Se, p. 95, Corollary 3]), α extends to a unique automorphism α of G.

Thus α ↦ α defines a homomorphism ŝ: Aut(L) ⟶ Aut(G). If α = c_{γ} for γ ∈ Aut_{L}(U) = N_{G}(U), then α is conjugation by γ ∈ G and hence lies in Inn(G). Hence ŝ factors through s: Out(L) ⟶ Out(G), κ_{G} ∘ s = Id_{Out(L)}, and thus κ_{G} is split surjective. □

Before we can handle the rank 1 case, two elementary lemmas are needed.

Lemma 4.6. Let G be a finite group with normal Sylow p-subgroup S ⊴ G such that C_{G}(S) ≤ S. Fix subgroups 1 = S_{0} < S_{1} < ⋯ < S_{k} = S normal in G such that

(i) S_{k−1} ≤ Fr(S); and

(ii) for each 1 ≤ i ≤ k − 1, S_{i} is characteristic in G, [S, S_{i}] ≤ S_{i−1}, S_{i}/S_{i−1} has exponent p, and Hom_{𝔽_{p}[G/S]}(S/Fr(S), S_{i}/S_{i−1}) = 0 (i.e., no irreducible 𝔽_{p}[G/S]-submodule of S_{i}/S_{i−1} appears as a submodule of S/Fr(S)).

Let α ∈ Aut(G) be such that [α, S] ≤ S_{k−1}. Then α ∈ Aut_{S}(G).

Proof. For 1 ≠ g ∈ G of order prime to p, the conjugation action of g on S is nontrivial since C_{G}(S) ≤ S, and hence the conjugation action on S/Fr(S) is also nontrivial (see [G, Theorem 5.3.5]). Thus G/S acts faithfully on S/Fr(S). Since α induces the identity on S/Fr(S), α also induces the identity on G/S.

Assume first that α|_{S} = Id. Since S is a p-group and G/S has order prime to p, H^{1}(G/S; Z(S)) = 0. So by [OV, Lemma 1.2], α ∈ Inn(G). If g ∈ G is such that α = c_{g}, then [g, S] = 1 since α|_{S} = Id, and g ∈ S since C_{G}(S) ≤ S. Thus α ∈ Aut_{S}(G) in this case.

In particular, this proves the lemma when k = 1. So assume k ≥ 2. We can assume inductively that the lemma holds for G/S_{1}, and hence can arrange (after composing by an appropriate element of Aut_{S}(G)) that α induces the identity on G/S_{1}.

Let φ ∈ Hom(S, S_{1}) be such that α(x) = xφ(x) for each x ∈ S (a homomorphism since S_{1} ≤ Z(S)). Then φ factors through φ ∈ Hom(S/Fr(S), S_{1}) since S_{1} is elementary abelian, and φ is a homomorphism of 𝔽_{p}[G/S]-modules since α(g) ≡ g (mod S_{1}) for each g ∈ G (and S_{1} ≤ Z(S)). Thus φ = 1 since Hom_{G/S}(S_{k}/S_{k−1}, S_{1}) = 0 by (ii), so α|_{S} = Id, and we already showed that this implies α ∈ Aut_{S}(G). □

The next lemma will be useful when checking the hypotheses of Lemma 4.6.

Lemma 4.7. Fix a prime p and e ≥ 1, and set q = p^{e} and Γ = 𝔽_{q}^{×}. For each a ∈ ℤ, set V_{a} = 𝔽_{q}, regarded as an 𝔽_{p}Γ-module with action λ(x) = λ^{a}x for λ ∈ Γ and x ∈ 𝔽_{q}.

(a) For each a, V_{a} is 𝔽_{p}Γ-irreducible if and only if a/ gcd(a, q − 1) does not divide p^{t} − 1 for any t|e, t < e.

(b) For each a, b ∈ ℤ, V_{a} ≅ V_{b} as 𝔽_{p}Γ-modules if and only if a ≡ bp^{i} (mod q − 1) for some i ∈ ℤ.

Proof. (a) Set d = gcd(a, q − 1), and let t be the order of p in (ℤ/\frac{q−1}{d})^{×}. Thus t|e since \frac{q−1}{d} |(p^{e} − 1). If t < e, then λ^{a} ∈ 𝔽_{p^{t}} for each λ ∈ 𝔽_{q}, so 0 ≠ 𝔽_{p^{t}} ⫋ V_{a} is a proper 𝔽_{p}Γ-submodule, and V_{a} is reducible.

Conversely, if V_{a} is reducible, then it contains a proper submodule 0 ≠ W ⫋ V_{a} of dimension i, some 0 < i < e. All Γ-orbits in V_{a}∖0, hence in W ∖0, have length \frac{q−1}{d}, so \frac{q−1}{d} |(p^{i} − 1), and t ≤ i < e.

(b) For each a ∈ ℤ, let V_{a} ≅ 𝔽_{q} be the 𝔽_{q}Γ-module where Γ acts via λ(x) = λ^{a}x. Then 𝔽_{q} ⊗_{𝔽_{p}} V_{a} ≅ V_{a} ⊕ V_{ap} ⊕ ⋯ ⊕ V_{ap^{e−1}} as 𝔽_{q}Γ-modules. Since V_{b} ≅ V_{a} if and only if b ≡ a (mod q − 1), V_{b} ≅ V_{a} if and only if b ≡ ap^{i} (mod q − 1) for some i. □

In principle, we don’t need to look at the fusion systems of the simple groups of Lie rank 1 if we only want to prove tameness. Their fusion is controlled by the Borel subgroup, so their fusion systems are tame by Proposition 1.6. But the following proposition is needed when proving Theorem A in its stronger form, and will also be used when working with groups of larger Lie rank.

Proposition 4.8. Fix a prime p, and a group G ∈ 𝔏𝔦𝔢(p) of Lie rank 1. Assume (G, p) ≅ (Sz(2), 2). Then each φ ∈ Aut(F) extends to an automorphism of G. Also, if [φ, U] ≤ [U, U], then φ ∈ Inn(U).

Proof. If G is of universal form, then Z(G) is cyclic of order prime to p by Proposition 3.8. For each Z ≤ Z(G), Out(G/Z) ≅ Out(G) by [GLS3, Theorem 2.5.14(d)], and Out(F_{U}(G/Z)) ≅ Out(F_{U}(G)) since G and G/Z have the same p-fusion systems. It thus suffices to prove the proposition when G has adjoint form.

Assume first G = PSL_{2}(q). Thus U ≅ 𝔽_{q} (as an additive group), T ≅ C_{(q−1)/ε} where ε = gcd(q − 1, 2), and Γ =^{def} Aut_{T} (U) is the subgroup of index ε in 𝔽_{q}^{×}. If φ ∈ Aut(U) is fusion preserving, then under these identifications, there is α ∈ Aut(Γ) such that α(u)φ(v) = φ(uv) for each u ∈ Γ ≤ 𝔽_{q}^{×} and v ∈ 𝔽_{q}. After composing with an appropriate diagonal automorphism (conjugation by a diagonal element of P GL_{2}(q)), we can assume that φ(1) = 1. Hence the above formula (with v = 1) implies that α = φ|_{Γ}, and thus that φ(uv) = φ(u)φ(v) for each u, v ∈ 𝔽_{q} with u ∈ Γ. If ε = 1, then φ acts as a field automorphism on U, hence is the restriction of a field automorphism of G, and we are done. Otherwise, there is u ∈ Γ such that 𝔽_{q} = 𝔽_{p}(u), u and φ(u) have the same minimal polynomial over 𝔽_{p}, and there is ψ ∈ Aut(𝔽_{q}) (a field automorphism) such that ψ(u) = φ(u). Thus ψ(u^{i}) = φ(u^{i}) for each i, so ψ = φ since both are additive homomorphisms, and hence φ extends to a field automorphism of G. (Note that this argument also holds when q = 3 and Γ = 1.)

Next assume G = PSU_{3}(q). Following the conventions in [H, Satz II.10.12(b)], we identify

U = \{[[a, b]] | a, b ∈ 𝔽_{q^{2}}, b + b^{q} = −a^{q+1}\}  where  [[a, b]] = ( \begin{matrix}1 & a & b \\ 0 & 1 & −a^{q} \\ 0 & 0 & 1 \end{matrix} );
T = {d(λ) | λ ∈ 𝔽_{q^{2}}^{×}}  where  d(λ) = diag(λ^{−q}, λ^{q−1}, λ).

Here, whenever we write a matrix, we mean its class in PSU_{3}(q). Then B = UT = N_{G}(U) ≤ G (see [H, Satz II.10.12(b)]), and

[[a, b]] · [[c, d]] = [[a + c, b + d − ac^{q}]] and ^{d(λ)}[[a, b]] = [[λ^{1−2q}a, λ^{−1−q}b]] .

Set ε = gcd(2q − 1, q^{2} − 1) = gcd(2q − 1, q^{2} − 2q) = gcd(q + 1, 3). Then d(λ) = 1 exactly when λ^{ε} = 1, C_{T} (U) = 1, and hence |T | = |Aut_{B}(U/Z(U))| = (q^{2} − 1)/ε. If q > 2, then |T | does not divide p^{i} − 1 for any power 1 < p^{i} < q^{2}, and by Lemma 4.7(a), U/Z(U) and Z(U) are both irreducible as 𝔽_{p}[T]-modules. (Note, in particular, the cases q = 5 and q = 8, where (U/Z(U), T) is isomorphic to (𝔽_{25}, C_{8}) and (𝔽_{64}, C_{21}), respectively.)

Fix φ ∈ Aut(F), and extend it to α ∈ Aut(B) (Lemma 1.12). Via the same argument as that used when G = PSL_{2}(q), we can arrange (without changing the class of φ modulo Im(κ_{G})) that φ ≡ Id (mod [U, U]). If q > 2, then the hypotheses of Lemma 4.6 hold (with [U, U] < U < B in the role of S_{1} < S_{2} = S < G), so α ∈ Aut_{U}(B) and φ ∈ Inn(U).

If G ≅ PSU_{3}(2) ≅ C_{3}^{2} ⋊ Q_{8} (cf. [Ta, p. 123–124]), then U ≅ Q_{8} and T = 1, so Out(F) = Out(U) ≅ Σ_{3}. By Theorem 3.4 (or by direct computation), Out(G) = Outdiag(G)Φ_{G} has order six, since |Outdiag(G)| = gcd(3, q + 1) = 3 and |Φ_{G}| = 2. Thus κ_{G} is an isomorphism, since it is injective by Lemma 4.3.

The proof when G = Sz(q) is similar. Set θ = \sqrt{2q}. We follow the notation in [HB, § XI.3], and identify U as the group of all S(a, b) for a, b ∈ 𝔽_{q} and T < B = N_{G}(U) as the group of all d(λ) for λ ∈ 𝔽_{q}^{×}, with relations

S(a, b) · S(c, d) = S(a + c, b + d + a^{θ}c) and ^{d(λ)}S(a, b) = S(λa, λ^{1+θ}b) .

As in the last case, we can arrange that φ ∈ Aut(F) is the identity modulo [U, U]. Since q ≥ 8 (q ≠ 2 by hypothesis), Z(U) and U/Z(U) are nonisomorphic, irreducible 𝔽_{2}T -modules by Lemma 4.7(a,b) (and since Z(U) ≅ V_{1+θ} and U/Z(U) ≅ V_{1} in the notation of that lemma). We can thus apply Lemma 4.6 to show that φ ∈ Inn(U).

It remains to handle the Ree groups ^{2}G_{2}(q), where q = 3^{m} for some odd m ≥ 1. Set θ = \sqrt{3q}. We use the notation in [HB, Theorem XI.13.2], and identify U = (𝔽_{q})^{3} with multiplication given by

(x_{1}, y_{1}, z_{1})·(x_{2}, y_{2}, z_{2}) = (x_{1} + x_{2}, y_{1} + y_{2} + x_{1}·x_{2}^{θ}, z_{1} + z_{2} − x_{1}·y_{2} + y_{1}·x_{2} − x_{1}·x_{1}^{θ}·x_{2}) .

Note that x^{θ^{2}} = x^{3}. Let T ≤ B = N_{G}(U) be the set of all d(λ) for λ ∈ 𝔽_{q}^{×}, acting on U via

d(λ)(x, y, z) = (λx, λ^{θ+1}y, λ^{θ+2}z).

Again, we first reduce to the case where φ ∈ Aut(F) is such that [φ, U] ≤ [U, U], and extend φ to α ∈ Aut(B). If q > 3, then U/[U, U] ≅ V_{1}, [U, U]/Z(U) ≅ V_{θ+1}, and Z(U) ≅ V_{θ+2} are irreducible and pairwise nonisomorphic as 𝔽_{3}T -modules by Lemma 4.7 (for V_{a} as defined in that lemma), since neither θ + 1 nor θ + 2 is a power of 3. So φ ∈ Inn(U) by Lemma 4.6.

If q = 3, then U = ⟨a, b⟩, where |a| = 9, |b| = 3, and [a, b] = a^{3}. Set Q_{i} = ⟨ab^{i}⟩ ≅ C_{9} (i = 0, 1, 2): the three subgroups of U isomorphic to C_{9}. Let Aut^{0}(U) ≤ Aut(U) be the group of those α ∈ Aut(U) which send each Q_{i} to itself. For each such α, the induced action on U/Z(U) sends each subgroup of order three to itself, hence is the identity or (g ↦ g^{−1}), and the latter is seen to be impossible using the relation [a, b] = a^{3}. Thus each α ∈ Aut^{0}(U) induces the identity on U/Z(U) and on Z(U), and has the form α(g) = gφ(g) for some φ ∈ Hom(U/Z(U), Z(U)). So Aut^{0}(U) = Inn(U) since they both have order 9 (and clearly Inn(U) ≤ Aut^{0}(U)). The action of Aut(U) on {Q_{0}, Q_{1}, Q_{2}} thus defines an embedding of Out(U) into Σ_{3}, and the automorphisms (a, b) ↦ (ab, b) and (a, b) ↦ (a^{−1}, b) show that Out(U) ≅ Σ_{3}. Since |Out_{F}(U)| = 2 and Aut_{F}(U) ⊴ Aut(F), it follows that Out(F) = 1 = Out(G). (See also [BC, Theorem 2] for more discussion about Aut(U).) □

It remains to show that κ_{G} (at the prime p) is surjective when G ∈ 𝔏𝔦𝔢(p) has Lie rank 2, with the one exception when G ≅ SL_{3}(2). Our proof is based on ideas taken from the article of Delgado and Stellmacher [DS], even though in the end, we do not actually need to refer to any of their results in our argument. The third author would like to thank Richard Weiss for explaining many of the details of how to apply the results in [DS], and also to Andy Chermak and Sergey Shpectorov for first pointing out the connection.

Fix a prime p, and a finite group G ∈ 𝔏𝔦𝔢(p) of Lie rank two. We assume Notation 2.2 and 4.1. In particular, (G, σ) is a σ-setup for G, T ≤ G is a maximal torus, U ∈ Syl_{p}(G) is generated by the positive root subgroups, and B = N_{G}(U) is a Borel subgroup. Set Π̂ = {α̂_{1}, α̂_{2}}, and set 𝔓_{1} = 𝔓α̂_{1} = ⟨B, X_{−α̂_{1}}⟩ and 𝔓_{2} = 𝔓α̂_{2} = ⟨B, X_{−α̂_{2}}⟩: the two maximal parabolic subgroups of G containing B. Our proofs are based on the following observation:

Lemma 4.9. Assume, for G ∈ 𝔏𝔦𝔢(p) of rank 2 and its amalgam of parabolics as above, that each automorphism of the amalgam (𝔓_{1} > B < 𝔓_{2}) extends to an automorphism

of G. (∗)

Then κ_{G} is surjective.

Here, by an automorphism of the amalgam, we mean a pair (χ_{1}, χ_{2}), where either χ_{i} ∈ Aut(𝔓_{i}) for i = 1, 2 or χ_{i} ∈ Iso(𝔓_{i}, 𝔓_{3−i}) for i = 1, 2, and also χ_{1}|_{B} = χ_{2}|_{B}.

Proof. Set L = L_{U}^{c}(G) and U_{i} = O_{p}(𝔓_{i}). By Proposition 4.4, each χ ∈ Aut(L) either sends U_{1} and U_{2} to themselves or exchanges them. For each i = 1, 2, C_{G}(U_{i}) ≤ U_{i}, so Aut_{L}(U_{i}) = N_{G}(U_{i}) = 𝔓_{i}. Thus χ induces an automorphism of the amalgam (𝔓_{1} > B < 𝔓_{2}). By assumption, this extends to an automorphism χ of G, and κ_{G}(χ) = ξ. □

Set 𝔊 = 𝔓_{1} *_{B}𝔓_{2}: the amalgamated free product over B. Let ρ: 𝔊 ⟶ G be the natural surjective homomorphism. Since each automorphism of the amalgam induces an automorphism of 𝔊, (∗) holds if for each automorphism of (𝔓_{1} > B < 𝔓_{2}), the induced automorphism of 𝔊 sends Ker(ρ) to itself.

Let Δ be the tree corresponding to the amalgam (𝔓_{1} > B < 𝔓_{2}). Thus Δ has a vertex [g𝔓_{i}] for each coset g𝔓_{i} (for all g ∈ 𝔊 and i = 1, 2), and an edge g(e_{B}) connecting [g𝔓_{1}] to [g𝔓_{2}] for each coset gB in 𝔊. Also, 𝔊 acts on Δ via its canonical action on the cosets, and in particular, it acts on g(e_{B}) with stabilizer subgroup ^{g}B.

Similarly, let Δ_{G} be the graph of G with respect to the same amalgam: the graph with vertex set (G/𝔓_{1}) ∪ (G/𝔓_{2}) and edge set G/B. Equivalently, since 𝔓_{1}, 𝔓_{2}, and B are self-normalizing, Δ_{G} is the graph whose vertices are the maximal parabolics in G and whose edges are the Borel subgroups. Let ρ̂: Δ ⟶ Δ_{G} be the canonical map which sends a vertex [g𝔓_{i}] in Δ to the vertex in Δ_{G} corresponding to the image of g𝔓_{i} in G.

Fix a subgroup N ≤ G such that (B, N) is a BN-pair for G, and such that B ∩N = T and N/T ≅ W_{0} (where T and W_{0} are as defined in Notation 2.2). We refer to [Ca, §§ 8.2, 13.5] for the definition of BN-pairs, and the proof that G has a BN-pair (B, N) which satisfies these conditions. For i = 1, 2, choose t_{i} ∈ (N ∩ 𝔓_{i})∖B = (N ∩ 𝔓_{i})∖T. Since (N ∩ 𝔓_{i})/T ≅ C_{2} and N = ⟨N ∩ 𝔓_{1}, N ∩ 𝔓_{2}⟩, we have N = T ⟨t_{1}, t_{2}⟩, consistent with the notation in [DS]. Note that T can be the trivial subgroup. We also regard the t_{i} ∈ 𝔓_{i} as elements of 𝔊, and T ≤ B as a subgroup of 𝔊, when appropriate.

Let 𝒯 be the union of the edges in the T ⟨t_{1}, t_{2}⟩-orbit of e_{B}. Thus 𝒯 is a path of infinite length in Δ of the following form:

⋯  •  •  •  •  •_{[t_{1}t_{2}𝔓_{1}]}  _{[t_{1}𝔓_{2}]}  _{[𝔓_{1}]}  _{[𝔓_{2}]}  _{[t_{2}𝔓_{1}]}^{t_{1}t_{2}t_{1}(e_{B})}  ^{t_{1}t_{2}(e_{B})}  ^{t_{1}(e_{B})}  ^{e_{B}}  ^{t_{2}(e_{B})}  ^{t_{2}t_{1}(e_{B})} ⋯

Thus ρ̂(𝒯 ) is an apartment in the building Δ_{G} under Tits’s definition and construction of these structures in [Ti, 3.2.6].

A path in Δ is always understood not to double back on itself.

Lemma 4.10. Let G, Δ, (T, t_{1}, t_{2}), and 𝒯 be as above, and let n ∈ {3, 4, 6, 8} be such that W_{0} ≅ D_{2n}. Then each path in Δ of length at most n + 1 is contained in g(𝒯 ) for some g ∈ 𝔊.

Proof. A path of length 1 is an edge, and is in the 𝔊-orbit of e_{B} which has stabilizer group B. If e_{B} is extended to a path of length 2 with the edge t_{i}(e_{B}) (i = 1 or 2), then this path has stabilizer group

B ∩^{t_{i}}B = ∏_{α̂∈Σ̂_{+}∖{α̂_{i}}} Xα̂ · T .

(Recall that ^{t_{i}}Xα̂_{i} = X_{−α̂_{i}}, and X_{−α̂_{i}} ∩ B = 1 by [Ca, Lemma 7.1.2].) Thus the stabilizer subgroup has index p^{j} in B, where p^{j} = |Xα̂_{i}|. Furthermore, |𝔓_{i}/B| = 1 + p^{j}, since by [Ca, Proposition 8.2.2(ii)],

𝔓_{i} = B ∪ (Bt_{i}B) where |Bt_{i}B| = |B| · |B/(B ∩^{t_{i}}B)| = |B| · p^{j} .

Hence there are exactly p^{j} extensions of e_{B} to a path of length 2 containing the vertex [𝔓_{i}] in the interior, and these are permuted transitively by B.

Upon continuing this argument, we see inductively that for all 2 ≤ k ≤ n + 1, the paths of length k starting at e_{B} with endpoint [𝔓_{3−i}] are permuted transitively by B, and of them, the one contained in 𝒯 has stabilizer subgroup the product of T with (n + 1 − k) root subgroups in U. (Recall that B = T U, and U is the product of n root subgroups.) Since 𝔊 acts transitively on the set of edges in Δ, each path of length k is in the 𝔊-orbit of one which begins with e_{B} (and with endpoint [𝔓_{1}] or [𝔓_{2}]), and hence in the 𝔊-orbit of a subpath of 𝒯 . □

Proposition 4.11. Let G, 𝔊, and (T, t_{1}, t_{2}) be as above, and let n ∈ {3, 4, 6, 8} be such that W_{0} ≅ D_{2n}. Assume that

for each (χ_{1}, χ_{2}) ∈ Aut(𝔓_{1} > B < 𝔓_{2}), where χ_{i} ∈ Aut(𝔓_{i}) or χ_{i} ∈
Iso(𝔓_{i}, 𝔓_{3−i}) for i = 1, 2, we have (χ_{1}(t_{1})χ_{2}(t_{2}))^{n} ∈ χ_{1}(T) ≤ G.

(†)

Then (∗) holds (each automorphism of (𝔓_{1} > B < 𝔓_{2}) extends to an automorphism of G), and hence κ_{G} is onto.

Proof. Let ≈ be the equivalence relation on the set of vertices in Δ generated by setting x ≈ y if x and y are of distance 2n apart in some path in the 𝔊-orbit of 𝒯 . Since T ⟨t_{1}, t_{2}⟩/T ≅ D_{2n} as a subgroup of N_{G}(T)/T, the natural map ρ̂: Δ ⟶ Δ_{G} sends 𝒯 to a loop of length 2n, and hence sends all apartments in the 𝔊-orbit of 𝒯 to loops of length 2n. Hence ρ̂: Δ ⟶ Δ_{G} factors through Δ/≈. We will show that ρ̂ induces an isomorphism (Δ/≈) ≅ Δ_{G} of graphs, and then use that and (†) to prove the proposition.

We claim that

Δ_{G} contains no loops of length strictly less than 2n; and  (1)
each pair of points in Δ/≈ is connected by a path of length at most n.  (2)

Assume (1) does not hold: let L be a loop of minimal length 2k (k < n). Fix edges σ_{i} = [x_{i}, y_{i}] in L (i = 1, 2) such that the shortest path from x_{i} to y_{3−i} in L has length k − 1, and let L_{0} ⊆ L be the path of length k + 1 from x_{1} through y_{1} and x_{2} to y_{2}. Then L_{0} lifts to a path of length k + 1 ≤ n in Δ, this is contained in some apartment in the 𝔊-orbit of 𝒯 by Lemma 4.10, and hence L_{0} ⊆ Σ, where Σ ⊆ Δ_{G} is an apartment in the G-orbit of ρ̂(𝒯 ). By [Ti, Theorem 3.3] or [Br, p. 86], there is a retraction of Δ_{G} onto Σ. Hence the path from y_{2} to x_{1} in Σ has length at most k − 1, which is impossible since Σ is a loop of length 2n and L_{0} is a path of length k + 1 ≤ n in Σ. (See also [Br, § IV.3, Exercise 1]. Point (1) also follows since Δ_{G} is a generalized n-gon in the sense of Tits [Br, p. 117], and hence any two vertices are joined by at most one path of length less than n.)

Now assume (2) does not hold: let x, y be vertices in Δ such that the shortest path between their classes in Δ/≈ has length k ≥ n + 1. Upon replacing x and y by other vertices in their equivalence classes, if needed, we can assume that the path [x, y] in Δ has length k. Let z be the vertex in the path [x, y] of distance n + 1 from x. By Lemma 4.10, [x, z] is contained in g(𝒯 ) for some g ∈ 𝔊; let x′ be the vertex in g(𝒯 ) of distance 2n from x and distance n − 1 from z. Then x′ ≈ x, and [x′, y] has length at most (n − 1) + (k − n − 1) = k − 2, a contradiction. This proves (2).

Assume the map (Δ/≈) ⟶ Δ_{G} induced by ρ̂ is not an isomorphism of graphs, and let x and y be distinct vertices in Δ/≈ whose images are equal in Δ_{G}. By (2), there is a path from x to y of length at most n, and of even length since the graph is bipartite. This path cannot have length 2 since ρ̂: Δ ⟶ Δ_{G} preserves valence, so its image in Δ_{G} is a loop of length at most n, and this contradicts (1). We conclude that Δ_{G} ≅ Δ/≈.

Let (χ_{1}, χ_{2}) be an automorphism of the amalgam (𝔓_{1} > B < 𝔓_{2}), let χ ∈ Aut(𝔊) be the induced automorphism of the amalgamated free product, and let χ̂ ∈ Aut(Δ) be the automorphism which sends a vertex [g𝔓_{i}] to [χ(g𝔓_{i})]. By (†),

(χ_{1}(t_{1})χ_{2}(t_{2}))^{n} ∈ χ_{1}(T) = χ_{2}(T) ≤ C_{G}(ρ̂(χ̂(𝒯 )))

where χ_{1}(t_{1})χ_{2}(t_{2}) acts on χ̂(𝒯 ) by translating it by distance 2. Hence ρ̂(χ̂(𝒯 )) is a loop of length 2n in Δ_{G}. So ρ̂ ∘ χ̂ factors through (Δ/≈) ≅ Δ_{G}, and since Δ_{G} is a finite graph, the induced map Δ_{G} ⟶ Δ_{G} is an automorphism of Δ_{G}. So χ sends Ker[𝔊 ⟶^{ρ} G] to itself, and thus induces an automorphism of G. The last statement (κ_{G} is onto) now follows from Lemma 4.9. □

It remains to find conditions under which (†) holds. The following proposition handles all but a small number of cases.

Proposition 4.12. Assume N = N_{G}(T) (and hence N_{G}(T)/T is dihedral of order 2n). Then (†) holds, and hence each automorphism of the amalgam (𝔓_{1} > B < 𝔓_{2}) extends to an automorphism of G. In particular, (†) and (∗) hold, and hence κ_{G} is onto, whenever G = ^{r}X_{n}(q) ∈ 𝔏𝔦𝔢(p) has Lie rank 2 for q > 2 and G ≅ S̸p_{4}(3).

Proof. Assume that N_{G}(T) = N = T ⟨t_{1}, t_{2}⟩. Then the choices of the t_{i} are unique modulo T. Also, any two choices of T are B-conjugate, so each automorphism of the amalgam is Bconjugate to one which sends 𝒯 to itself. Thus (†) holds, and so (∗) follows from Proposition 4.11.

The last statement now follows from Proposition 3.9. Note that if (†) holds for G of universal type, then it also holds for G/Z(G) of adjoint type. □

What can go wrong, and what does go wrong when G = SL_{3}(2), is that an automorphism of the amalgam can send t_{1}, t_{2} to another pair of elements whose product (modulo T) has order strictly greater than 2n. This happens when 𝒯 is sent to another path not in the 𝔊-orbit of 𝒯 : one whose image in Δ_{G} is a loop of a different length.

Example 4.13. Assume G = SL_{3}(2). In particular, T = 1. Let B be the group of upper triangular matrices, let t_{1} and t_{2} be the permutation matrices for (1 2) and (2 3), respectively, and set 𝔓_{i} = ⟨B, t_{i}⟩.

Consider the automorphism α of the amalgam which is the identity on 𝔓_{1} (hence on B), and which is conjugation by e_{13} (the involution in Z(B)) on 𝔓_{2}. Set t′_{i} = α(t_{i}). Thus

t′_{1} = ( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} )  and  t′_{2} = ( \begin{matrix}1 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ) .

One checks that t′_{1}t′_{2} has order 4, so that ⟨t′_{1}, t′_{2}⟩ ≅ D_{8} while ⟨t_{1}, t_{2}⟩ ≅ D_{6}. In other words, α sends the lifting (from Δ_{G} to Δ) of a loop of length 6 to the lifting of a loop of length 8, hence is not compatible with the relation ≈, hence does not extend to an automorphism of G.

We are left with seven cases: four cases with n = 4, two with n = 6, and one with n = 8. Those with n = 4 are relatively easy to handle.

Proposition 4.14. Assume G is one of the groups Sp_{4}(2), PSp_{4}(3), PSU_{4}(2), or PSU_{5}(2). Then (†) holds, and hence (∗) also holds and κ_{G} is onto.

Proof. In all cases, we work in the universal groups Sp_{4}(q) and SU_{n}(2), but the arguments are unchanged if we replace the subgroups described below by their images in the adjoint group. Recall that p is always the defining characteristic, so the second and third cases are distinct, even though PSp_{4}(3) ≅ SU_{4}(2) (see [Wi, § 3.12.4] or [Ta, Corollary 10.19]).

Let (χ_{1}, χ_{2}) be an automorphism of (𝔓_{1} > B < 𝔓_{2}). Since all subgroups of B isomorphic to T are conjugate to T by the Schur-Zassenhaus theorem, we can also assume that χ_{i}(T) = T. Set χ_{0} = χ_{1}|_{B} = χ_{2}|_{B} and t_{i}^{∗} = χ_{i}(t_{i}) for short; we must show that |t_{1}^{∗}t_{2}^{∗}| = n = 4. Note that t_{1}^{∗}t_{2}^{∗} has order at least 4, since otherwise Δ_{G} would contain a loop of length strictly less than 8 = 2n, which is impossible by point (1) in the proof of Proposition 4.11.

G = Sp_{4}(2) ≅ Σ_{6} : Set G′ = [G, G]: the subgroup of index 2. The elements x_{γ}(1) for γ ∈ Σ are all Aut(G)-conjugate: the long roots and the short roots are all W -conjugate and a graph automorphism exchanges them. Since these elements generate G, none of them are in G′. Hence for i = 1, 2, all involutions in

⟨x_{α_{i}}(1), x_{−α_{i}}(1)⟩ ≅ GL_{2}(2) ≅ Σ_{3}

lie in G∖G′, and in particular, t_{i} ∈ G∖G′.

Each automorphism of the amalgam sends the focal subgroup to itself (as a subgroup of B), and hence also sends the intersections 𝔓_{i} ∩ G′ to themselves. So t_{1}^{∗}, t_{2}^{∗} ∈ G∖G′, and t_{1}^{∗}t_{2}^{∗} ∈ G′ ≅ A_{6}. It follows that |t_{1}^{∗}t_{2}^{∗}| ≤ 5, and |t_{1}^{∗}t_{2}^{∗}| = 4 since every dihedral subgroup of order 10 in Σ_{6} is contained in A_{6}.

G = Sp_{4}(3) : In this case, T ≅ C_{2}^{2}, and N_{G}(T) ≅ SL_{2}(3) ≀ C_{2}. Hence N_{G}(T)/T ≅ A_{4} ≀ C_{2} contains elements of order 2, 3, 4, and 6, but no dihedral subgroups of order 12. Since t_{1}^{∗}t_{2}^{∗} has order at least 4, |t_{1}^{∗}t_{2}^{∗}| = 4, and condition (†) holds.

G = SU_{n}(2) for n = 4 or 5 : We regard these as matrix groups via

SU_{n}(2) = {M ∈ SL_{n}(4) | M^{t} = M^{−1}} where (a_{ij})^{t} = (a_{n+1−j,n+1−i}) ,

and where x = x^{2} for x ∈ 𝔽_{4}. We can then take B to be the group of upper triangular matrices in SU_{n}(2), U the group of strict upper triangular matrices, and T the group of diagonal matrices. We thus have

T = {diag(x, x^{−1}, x^{−1}, x) | x ∈ 𝔽_{4}} ≅ C_{3}  if n = 4
T = {diag(x, y, xy, y, x) | x, y ∈ 𝔽_{4}} ≅ C_{3}^{2}  if n = 5.

Since N_{G}(T) must permute the eigenspaces of the action of T on 𝔽_{4}^{n}, we have N_{GU_{n}(2)}(T) ≅ GU_{2}(2) ≀ C_{2} (if n = 4) or (GU_{2}(2) ≀ C_{2}) × 𝔽_{4}^{×} (if n = 5). So in both cases,

N_{G}(T)/T ≅ PGU_{2}(2) ≀ C_{2} ≅ Σ_{3} ≀ C_{2} ≅ C_{3}^{2} ⋊ D_{8} .

Set Q = N_{G}(T)/O_{3}(N_{G}(T)) ≅ D_{8}, and let ψ: N_{G}(T) ⟶ Q be the natural projection. Set Q_{0} = ψ(C_{G}(T)). Since C_{G}(T)/T ≅ Σ_{3} × Σ_{3} (the subgroup of elements which send each eigenspace to itself), Q_{0} ≅ C_{2}^{2} and C_{G}(T) = ψ^{−1}(Q_{0}).

Choose the indexing of the parabolics such that 𝔓_{1} is the subgroup of elements which fix an isotropic point and 𝔓_{2} of those which fix an isotropic line. Thus

𝔓_{1} = \{( \begin{matrix}u & v & x \\ 0 & A & w \\ 0 & 0 & u \end{matrix} )| A ∈ GU_{n−2}(2)\}

and

𝔓_{2} = \begin{cases}\{( \begin{matrix}A & X \\ 0 & (A^{t})^{−1} \end{matrix} )| A ∈ SL_{2}(4)\} & if n = 4 \\ \{( \begin{matrix}A & v & X \\ 0 & u & w \\ 0 & 0 & (A^{t})^{−1} \end{matrix} )| A ∈ GL_{2}(4)\} & if n = 5. \end{cases}

Then ψ(N_{𝔓_{1}}(T)) ≤ Q_{0}: no matrix in 𝔓_{1} can normalize T and exchange its eigenspaces. Also, N_{B}(T) contains C_{U}(T) = ⟨e_{1,n}(1), e_{2,n−1}(1)⟩, where e_{i,j}(u) denotes the elementary matrix with unique off-diagonal entry u in position (i, j). Thus Q_{0} ≥ ψ(N_{𝔓_{1}}(T)) ≥ ψ(N_{B}(T)) ≅ C_{2}^{2}, so these inclusions are all equalities. Also, 𝔓_{2} contains the permutation matrix for the permutation (1 2)(n−1 n), this element exchanges the eigenspaces of rank 2 for T, and so ψ(N_{𝔓_{2}}(T)) = Q.

Since T ⟨t_{1}, t_{2}⟩/T ≅ D_{8}, ⟨ψ(t_{1}), ψ(t_{2})⟩ = Q, and so ψ(t_{1}) ∈ Q_{0}∖Z(Q) and ψ(t_{2}) ∈ Q∖Q_{0}. Since (χ_{1}, χ_{2}) induces an automorphism of the amalgam (Q > Q_{0} = Q_{0}), this implies that ψ(t_{1}^{∗}) ∈ Q_{0}∖Z(Q) and ψ(t_{2}^{∗}) ∈ Q∖Q_{0}. But then ⟨ψ(t_{1}^{∗}), ψ(t_{2}^{∗})⟩ = Q since these elements generate modulo Z(Q), so |t_{1}^{∗}t_{2}^{∗}| ∈ 4ℤ, and |t_{1}^{∗}t_{2}^{∗}| = 4 since N_{G}(T)/T ≅ Σ_{3} ≀ C_{2} contains no elements of order 12. □

It remains to handle the groups G_{2}(2), ^{3}D_{4}(2), and ^{2}F_{4}(2). In the first two cases, if t_{i}^{∗} is an arbitrary involution in N_{𝔓_{i}}(T)∖N_{B}(T) for i = 1, 2, then t_{1}^{∗}t_{2}^{∗} can have order 6, 7, 8, or 12 when G = G_{2}(2), or order 6 or 8 when G = ^{3}D_{4}(2), and there does not seem to be any way to prove condition (†) short of analyzing automorphisms of the amalgam sufficiently to prove (∗) directly.

Let {α, β} be a fundamental system in the root system of G_{2} where α is the long root. Let α, α′, α′′ be the three long positive roots, and β, β′, β′′ the three short positive roots, as described in (3) below.

Let γ_{0}, γ_{1}, γ_{2}, γ_{3} denote the four fundamental roots in the D_{4} root system, where γ_{0} is in the center of the Dynkin diagram, and the other three are permuted cyclically by the triality automorphism. Set γ_{ij} = γ_{i} + γ_{j} (when it is a root), etc. We identify the six classes of positive roots in ^{3}D_{4} with the roots in G_{2} by identifying the following two diagrams:

[figure: β; α′; β′′; β′; α; α′′; −β; −α; G_{2}; \{γ_{1},γ_{2},γ_{3}\}; γ_{0123}; ^{\{ \begin{matrix}γ_{012}, \\ γ_{023}, \\ γ_{013} \end{matrix} \}}; ^{\{ \begin{matrix}γ_{01}, \\ γ_{02}, \\ γ_{03} \end{matrix} \}}; γ_{0}; γ_{00123}; −γ̂_{1}; −γ_{0}; ^{3}D_{4}]  (3)

The following list gives all nontrivial commutator relations among root subgroups of G_{2}(q) or ^{3}D_{4}(q) (see [GLS3, Theorems 1.12.1(b) & 2.4.5(b)]):

[x_{α}(u), x_{β}(v)] ≡ x_{β′}(±uv)x_{β′′}(±uv^{1+q})  (mod X_{α′}X_{α′′})  (4)
[x_{β′}(u), x_{β}(v)] ≡ x_{β′′}(±(uv^{q} + u^{q}v))  (mod X_{α′}X_{α′′})  (5)
[x_{α}(u), x_{α′}(v)] = x_{α′′}(±uv)  (6)
[x_{β′}(u), x_{β′′}(v)] = x_{α′′}(±Tr(uv^{q}))  (7)
[x_{β′′}(u), x_{β}(v)] = x_{α′}(±Tr(u^{q}v)) .  (8)

Again, Tr: 𝔽_{q^{3}} ⟶ 𝔽_{q} denotes the trace. Note that when G = G_{2}(q), then u, v ∈ 𝔽_{q} in all cases, and hence u^{q} = u^{q^{2}} = u, u^{q+q^{2}} = u^{2}, and Tr(u) = 3u. When G = ^{3}D_{4}(q), the notation x_{β}(−), x_{β′}(−), and x_{β′′}(−) is somewhat ambiguous (and formula (5) depends on making the right choice), but this doesn’t affect the arguments given below.

Proposition 4.15. Assume p = 2 and G = G_{2}(2). Then (∗) holds: each automorphism of the amalgam (𝔓_{α} > B < 𝔓_{β}) extends to an automorphism of G. (In fact, each automorphism of the amalgam is conjugation by some element of B.) In particular, κ_{G} is onto.

Proof. In this case, T = 1, and

𝔓_{α} ≅ (C_{4} × C_{4}) ⋊ D_{12} and 𝔓_{β} ≅ (Q_{8} ×_{C_{2}} Q_{8}) ⋊ Σ_{3} .

Also, B = U has presentation U = A ⋊ ⟨r, t⟩, where

A = ⟨a, b⟩ ≅ C_{4} × C_{4}, ⟨r, t⟩ ≅ C_{2}^{2}, ^{r}a = a^{−1}, ^{r}b = b^{−1}, ^{t}a = b, ^{t}b = a .

In terms of the generators x_{γ} = x_{γ}(1) for γ ∈ Σ_{+}, we have A = ⟨x_{β′}x_{β}, x_{β′′}x_{β}⟩ and Ω_{1}(A) = ⟨x_{α′}, x_{α′′}⟩, and we can take r = x_{β′′}, t = x_{α}, and a = x_{β}x_{β′′} (and then b = ^{t}a). Note that (5) takes the more precise form [x_{β′}, x_{β}] = x_{α′}x_{α′′} in this case. Also,

U_{α} = A⟨r⟩ ≅ (C_{4} × C_{4}) ⋊ C_{2}
U_{β} = ⟨ab^{−1}, a^{2}t⟩ ×_{⟨a^{2}b^{2}⟩} ⟨ab, a^{2}rt⟩ ≅ Q_{8} ×_{C_{2}} Q_{8}
U ∩ G′ = A⟨t⟩ ≅ C_{4} ≀ C_{2} .

The last formula holds since G′ = [G, G] ≅ SU_{3}(3) has index two in G (see [Wi, § 4.4.4] or [Di, pp. 146–150]), since x_{α}, x_{α′}, x_{α′′} ∈ G′ (note that x_{α} = [x_{−β}, x_{β′}]), and since x_{β}, x_{β′}, and x_{β′′} are all G-conjugate and hence none of them lies in G′.

Fix an automorphism (χ_{α}, χ_{β}) of the amalgam (𝔓_{α} > B < 𝔓_{β}), and set χ_{0} = χ_{α}|_{B} = χ_{β}|_{B} ∈ Aut(B). Then χ_{0} normalizes each of the subgroups U_{α}, U_{β}, and U ∩ G′. Also, χ_{0} normalizes U_{α} ∩ G′ = A, and since U_{β} ∩ G′ = ⟨ab, ab^{−1}, t⟩ ≅ Q_{8} ×_{C_{2}} C_{4} contains a unique quaternion subgroup, χ_{0} normalizes each of the two quaternion subgroups in U_{β}. After composing by an appropriate element of Aut_{U}(𝔓_{β}), we can arrange that χ_{0}(ab) = ab and χ_{0}(ab^{−1}) = ab^{−1}. In particular, χ_{0} induces the identity on Ω_{1}(A) and hence also on A/Ω_{1}(A).

Let g ∈ 𝔓_{α} be an element of order 3, chosen so that ^{g}(a^{2}) = b^{2} and ^{g}(b^{2}) = a^{2}b^{2}. The image of ⟨g⟩ in 𝔓_{α}/A ≅ D_{12} is normal, so χ_{α}(g) ∈ Ag. Let x ∈ Ω_{1}(A) be such that χ_{α}(b) = χ_{0}(b) = ax. Then ^{g}b ∈ ⟨ab, b^{2}⟩ ≤ C_{A}(χ_{0}), so ^{g}b = χ_{α}(^{g}b) = ^{g}(bx) implies that ^{g}x = 1 and hence x = 1. Thus χ_{0}|_{A} = Id. Also, χ_{α}(⟨g⟩) ∈ Syl_{3}(𝔓_{α}) is conjugate to ⟨g⟩ by an element of A, so we can arrange that χ_{α}(⟨g⟩) = ⟨g⟩ and hence that χ_{α}|_{A⟨g⟩} = Id. But then χ_{α} is the identity modulo C_{𝔓_{α}}(A⟨g⟩) = Z(A⟨g⟩) = 1, so χ_{α} = Id_{𝔓_{α}}.

Since χ_{β}|_{U_{β}} = Id, χ_{β} induces the identity modulo C_{𝔓_{β}}(U_{β}) = Z(U_{β}) ≅ C_{2}. It thus has the form χ_{β}(x) = xψ(x) for some ψ ∈ Hom(𝔓_{β}, Z(U_{β})). Hence χ_{β} = Id, since it is the identity on U ∈ Syl_{2}(𝔓_{β}). □

Proposition 4.16. Assume p = 2 and G = ^{3}D_{4}(2). Then (∗) holds, and κ_{G} is onto.

Proof. In this case, T ≅ 𝔽_{8}^{×} ≅ C_{7}, 𝔓_{α}/U_{α} ≅ C_{7} × Σ_{3}, and 𝔓_{β}/U_{β} ≅ SL_{2}(8). Also, by (6) and (7), U_{β} is extraspecial with center X_{α′′}. Fix an automorphism (χ_{α}, χ_{β}) of the amalgam (𝔓_{α} > B < 𝔓_{β}), and set χ_{0} = χ_{α}|_{B} = χ_{β}|_{B}. We must show that χ_{α} and χ_{β} are the restrictions of some automorphism of G.

By Theorem 3.4, and since Outdiag(SL_{2}(8)) = 1 = Γ_{SL_{2}(8)}, Out(𝔓_{β}/U_{β}) ≅ Out(SL_{2}(8)) is generated by field automorphisms, and hence automorphisms which are restrictions of field automorphisms of G. So we can compose χ_{β} and χ_{α} by restrictions of elements of Aut_{B}(G)Φ_{G} = N_{Aut_{𝔓_{β}} (G)}(U)Φ_{G}, to arrange that χ_{β} induces the identity on 𝔓_{β}/U_{β}. Then, upon composing them by some element of Aut_{U}(G), we can also arrange that χ_{0}(T) = T. Since X_{β′} and X_{β′′} are dual to each other by (7) and hence nonisomorphic as 𝔽_{2}[T]-modules, χ_{0} sends each of them to itself.

Since χ_{0}(T) = T, χ_{0} sends C_{U}(T) = X_{α}X_{α′}X_{α′′} ≅ D_{8} to itself. It cannot exchange the two subgroups X_{α}X_{α′′} and X_{α′}X_{α′′} (the first is not contained in U_{α} and the second is), so χ_{0}|_{C_{U} (T)} ∈ Inn(C_{U}(T)). Hence after composing by an element of Aut_{C_{U} (T)}(G), we can arrange that χ_{0} is the identity on this subgroup. Also, by applying (4) with u = 1, and since χ_{0}|_{X_{β}} ≡ Id (mod U_{β}) and [X_{α}, U_{β}] ≤ X_{α′′}, we see that χ_{0} is the identity on X_{β′}X_{β′′}. We conclude that χ_{0} is the identity on U_{β}.

Since χ_{β} induces the identity on U_{β} and on 𝔓_{β}/U_{β}, it has the form χ_{β}(x) = xψ(x) (all x ∈ 𝔓_{β}) for some

ψ ∈ Hom(𝔓_{β}/U_{β}; Z(U_{β})) ≅ Hom(SL_{2}(8), C_{2}) = 1 .

So χ_{β} = Id_{𝔓_{β}}.

Now, C_{𝔓_{α}}(T) ≅ Σ_{4} ×C_{7}, and Out(Σ_{4}) = 1. Hence χ_{α}|_{C_{𝔓α}(T)} must be conjugation by some element z ∈ Z(C_{U}(T)) = X_{α′′} = Z(𝔓_{β}). After composing χ_{α} and χ_{β} by restrictions of c_{z}, we can thus assume that χ_{α} is the identity on C_{𝔓_{α}}(T) (and still χ_{β} = Id_{𝔓_{β}}). Since χ_{α}|_{U} = Id and 𝔓_{α} = ⟨U, C_{𝔓_{α}}(T)⟩, we have χ_{α} = Id_{𝔓_{α}}. □

It remains only to handle ^{2}F_{4}(2) and the Tits group.

Proposition 4.17. Assume G = ^{2}F_{4}(2)′ or ^{2}F_{4}(2). Then κ_{G} is an isomorphism.

Proof. By the pullback square in [AOV, Lemma 2.15] (and since Out(L) is independent of the choice of objects in L by [AOV, Lemma 1.17]), κ_{G} is an isomorphism when G = ^{2}F_{4}(2) if it is an isomorphism when G is the Tits group. So from now on, we assume G = ^{2}F_{4}(2)′.

We adopt the notation for subgroups of G used by Parrott [Pa]. Fix T ∈ Syl_{2}(G), and set Z = Z(T) ≅ C_{2}, H = C_{G}(Z), and J = O_{2}(H). Let z ∈ Z be a generator. Then H is the parabolic subgroup of order 2^{11} · 5, |J| = 2^{9}, and H/J ≅ C_{5} ⋊ C_{4}. Set E = [J, J]. By [Pa, Lemma 1], E = Z_{2}(J) = Fr(J) ≅ C_{2}^{5}, and by the proof of that lemma, the Sylow 5-subgroups of H act irreducibly on J/E ≅ C_{2}^{4} and on E/Z ≅ C_{2}^{4}. Since each element of Aut_{H/J}(J/E) sends C_{J/E}(T/J) ≅ C_{2} to itself,

Aut_{H/J}(J/E) = {Id_{J/E}} and

|Hom_{H/J}(J/E, E/Z)| ≤ |Hom_{H/J}(J/E, J/E)| = 2 . (9)

Let N > T be the other parabolic, and set K = O_{2}(N). Thus N/K ≅ Σ_{3}, and [T : K] = 2. Fix P ∈ Syl_{5}(H) ⊆ Syl_{5}(G) (so P ≅ C_{5}). By [Pa, p. 674], H/E = (J/E) · (N_{G}(P)/Z),

where N_{G}(P)/Z ≅ H/J ≅ C_{5} ⋊ C_{4}. For each β ∈ Aut(H) such that β(T) = T, there is β_{1} ≡ β (mod Aut_{J}(H)) such that β_{1}(P) = P. Since each automorphism of H/J which sends T/J ≅ C_{4} to itself is conjugation by an element of T/J, there is β_{2} ≡ β_{1} (mod Aut_{N_{T} (P)}(H)) such that β_{2} induces the identity on H/J. By (9), β_{2} also induces the identity on J/E, and hence on H/E = (J/E) · (N_{G}(P)/Z). Thus

N_{Aut(H)}(T) = Aut_{T} (H) · {β ∈ Aut(H) | β(P) = P, [β, H] ≤ E} .  (10)

Now set L = L_{T}^{c} (G) for short, and identify N = Aut_{L}(K) and H = Aut_{L}(J). For each α ∈ Aut(L), let α_{H} ∈ Aut(H) and α_{N} ∈ Aut(N) be the induced automorphisms, and set α_{T} = α_{H}|_{T} = α_{N}|_{T}. Set

A_{0} = {α ∈ Aut(L) | [α_{H}, H] ≤ E and α_{H}|_{P} = Id_{P} } .

By (10), each class in Out(L) contains at least one automorphism in A_{0}.

Fix α ∈ A_{0}. Since [α_{H}, H] must be normal in H, we have [α_{H}, H] ∈ {E, Z, 1}. If [α_{H}, H] = Z, then α_{H}|_{JP} = Id, so [α_{H}, K] = [α_{N}, K] = Z, which is impossible since Z is not normal in N by [Pa, Lemma 6] (or since z ∉ Z(G) and G = ⟨H, N⟩). Thus either α_{H} = Id, or [α_{H}, H] = E.

If α_{H} = Id_{H}, then α_{N}|_{T} = Id. In this case, α_{N} determines an element of H^{1}(N/K; Z(K)) whose restriction to H^{1}(T/K; Z(K)) is trivial, and since this restriction map for H^{1}(−; Z(K)) is injective (since T/K ∈ Syl_{2}(N/K)), α_{N} ∈ Inn(N) (see, e.g., [OV, Lemma 1.2]). Hence α_{N} ∈ Aut_{Z}(N) since α_{N}|_{T} = Id (and Z = Z(T)). So α ∈ Aut_{Z}(L) in this case, and [α] = 1 ∈ Out(L).

Set H = H/Z, and similarly for subgroups of H. Let α_{H} ∈ Aut(H) and α_{T} ∈ Aut(T) be the automorphisms induced by α_{H} and α_{T}, and set β = α_{T} |\overline{J}. Then E = Z(J) since E = Z_{2}(J), so β(g) = gφ(ĝ) for some φ ∈ Hom_{H/J}(J/E, E). If φ = 1, so that [α, J] ≤ Z, then since α|_{P} = Id, we have [α_{H}, H] < E and so α_{H} = Id.

We have now constructed a homomorphism from A_{0} to Hom_{H/J}(J/E, E) with kernel Aut_{Z}(L). Thus

|Out(L)| ≤ |A_{0}/Aut_{Z}(L)| ≤ |Hom_{H/J}(J/E, E)| ≤ 2 .

where the last inequality holds by (9). Since |Out(G)| = 2 by [GrL, Theorem 2], and since κ_{G} is injective by Lemma 4.3, this proves that κ_{G} is an isomorphism.

Alternatively, this can be shown using results in [Fn]. Since T/[T, T] ≅ C_{2} × C_{4} by the above description of T/E (where E ≤ [T, T]), Aut(T) and hence Out(L) are 2-groups. So each automorphism of the amalgam H > T < N determines a larger amalgam. Since the only extension of this amalgam is to that of ^{2}F_{4}(2) by [Fn, Theorem 1], |Out(L)| = 2. □

5. The cross characteristic case: I

Throughout this section, we will work with groups G = C_{G}(σ) which satisfy the conditions in Hypotheses 5.1 below. In particular, 5.1(I) implies that G is not a Suzuki or Ree group. We will see in Section 6 (Proposition 6.8) that while these hypotheses are far from including all finite Chevalley and Steinberg groups, their fusion systems at the prime p do include almost all of those we need to consider.

For any finite abelian group B, we denote its “scalar automorphisms” by

ψ_{k}^{B} ∈ Aut(B),  ψ_{k}^{B}(g) = g^{k}  for all k such that (k, |B|) = 1

and define the group of its scalar automorphisms

Aut_{sc}(B) = {ψ_{k}^{B} | (k, |B|) = 1} ≤ Z(Aut(B)) .

Hypotheses 5.1. Assume we are in the situation of Notation 2.2(A,B,C).

(I) Let p be a prime distinct from q_{0} such that p||W_{0}|. Assume also that σ = ψ_{q} ∘ γ = γ ∘ ψ_{q} ∈ End(G), where

• q is a power of the prime q_{0};

• ψ_{q} ∈ Φ_{G} is the field automorphism (see Definition 3.1(a)); and

• γ ∈ Aut(G) is an algebraic automorphism of finite order which sends T to itself and commutes with ψ_{q_{0}} (so that ψ_{q_{0}}(G) = G).

Also, there is a free ⟨τ⟩-orbit of the form

{α_{1}, α_{2}, …, α_{s}}  or  {±α_{1}, ±α_{2}, …, ±α_{s}}

in Σ such that the set {α_{1}, α_{2}, …, α_{s}} is linearly independent in V.

(II) The algebraic group G is of universal type, and N_{G}(T) contains a Sylow p-subgroup of G. Set A = O_{p}(T), and fix S ∈ Syl_{p}(N_{G}(T)) ⊆ Syl_{p}(G); thus A ≤ S.

(III) Assume one of the following holds: either

(III.1) q ≡ 1 (mod p), q ≡ 1 (mod 4) if p = 2, |γ| ≤ 2, and γ ∈ Γ_{G} (thus ρ(Π) = Π); or

(III.2) p is odd, q ≡ −1 (mod p), G is a Chevalley group (i.e., γ ∈ Inn(G)), and γ(t) = t^{−1} for each t ∈ T; or

(III.3) p is odd, |τ| = ord_{p}(q) ≥ 2, C_{A}(O_{p′}(W_{0})) = 1, C_{S}(Ω_{1}(A)) = A, Aut_{G}(A) = Aut_{W_{0}}(A),

N_{Aut(A)}(Aut_{W_{0}}(A)) ≤ Aut_{sc}(A)Aut_{Aut(G)}(A)

where Aut_{Aut(G)}(A) = {δ|_{A} | δ ∈ Aut(G), δ(A) = A}, and

Aut_{W_{0}}(A) ∩ Aut_{sc}(A) ≤ \begin{cases}⟨γ|_{A}⟩ & if 2|ord_{p}(q) or −Id ∉ W \\ ⟨γ|_{A}, ψ_{−1}^{A}⟩ & otherwise, \end{cases}

Since W_{0} acts on T by Lemma 2.3, it also acts on A = O_{p}(T).

We will see in Lemma 5.3 that the conditions C_{S}(Ω_{1}(A)) = A (or C_{S}(A) = A when p = 2) and Aut_{G}(A) = Aut_{W_{0}}(A), both assumed here in (III.3), also hold in cases (III.1) and (III.2).

Recall, in the situation of (III.3), that |τ| = |γ|_{T} | by Lemma 3.2.

Note that the above hypotheses eliminate the possibility that G be a Suzuki or Ree group. Since we always assume the Sylow p-subgroups are nonabelian, the only such case which needs to be considered here (when q_{0} ≠ p) is that of ^{2}F_{4}(q) when p = 3, and this will be handled separately.

By Lemma 3.2, whenever σ = ψ_{q} ∘ γ, and γ is an algebraic automorphism of G which normalizes T, there is τ ∈ Aut(V) such that τ(Σ) = Σ and σ(X_{α}) = X_{τ(α)} for each α ∈ Σ. So under Hypotheses 5.1, the condition at the beginning of Notation 2.2(C) holds automatically, and with ρ = τ|_{Σ}. To simplify the notation, throughout this section and the next, we write τ = ρ to denote this induced permutation of Σ.

The following notation will be used throughout this section, in addition to that in Notation 2.2. Note that Π̂ and Σ̂ are defined in Notation 2.2(C) only when ρ(Π) = Π, and hence only in case (III.1) of Hypotheses 5.1. It will be convenient, in some of the proofs in this section, to extend this definition to case (III.2).

Recall (Notation 2.2) that for α ∈ Σ, w_{α} ∈ W denotes the reflection in the hyperplane α^{⊥} ⊆ V.

Notation 5.2. Assume we are in the situation of Notation 2.2 and Hypotheses 5.1.

(D) If (III.2) holds, then set Σ̂ = Σ, Π̂ = Π, and V_{0} = V. Note that W_{0} = W in this case.

(E) If (III.1) holds, then for each α̂ ∈ Σ̂, let wα̂ ∈ W_{0} be the element in ⟨w_{α} | α ∈ α̂⟩ which acts on V_{0} as the reflection across the hyperplane ⟨α̂⟩^{⊥}, and which exchanges the positive and negative roots in the set ⟨α̂⟩ ∩ Σ. (Such an element exists and lies in W_{0} by [Ca, Proposition 13.1.2].)

(F) If (III.1) or (III.2) holds, then for each α ∈ Σ and each α̂ ∈ Σ̂, set

K_{α} = ⟨X_{α}, X_{−α}⟩  T_{α} = h_{α}(𝔽_{q_{0}}^{×})
Kα̂ = ⟨K_{α} | α ∈ α̂⟩  T α̂ = ⟨T_{α} | α ∈ α̂⟩ .

(G) Set N = N_{G}(T)/O_{p′}(T), and identify A = O_{p}(T) with T/O_{p′}(T) ⊴ N. If (III.1) or (III.2) holds, then for α̂ ∈ Σ̂, set Aα̂ = A ∩ T α̂.

(H) Set F = F_{S}(G), and

Aut(A, F) = {β ∈ Aut(A) | β = β|_{A}, some β ∈ Aut(F)} .

Set Aut_{diag}(F) = C_{Aut(F)}(A) = {β ∈ Aut(F) | β|_{A} = Id}, and let Out_{diag}(F) be the image of Aut_{diag}(F) in Out(F).

Note that when (G, σ) is a standard setup (i.e., in case (III.1)), W_{0} acts faithfully on V_{0} (see [Ca, Lemma 13.1.1]).

Recall that N = N_{G}(T)/O_{p′}(T). We identify A = O_{p}(T) with T/O_{p′}(T) ⊴ N.

Lemma 5.3. Assume Hypotheses 5.1 and Notation 5.2.

(a) If condition (III.1) or (III.2) holds, then C_{W} (A) = 1, C_{G}(A) = C_{G}(T) = T, C_{G}(A) = T, and C_{S}(A) = A. If p is odd, then C_{W} (Ω_{1}(A)) = 1 and C_{S}(Ω_{1}(A)) = A.

(b) If C_{G}(A)^{0} = T (in particular, if (III.1) or (III.2) holds), then N_{G}(A) = N_{G}(T) ≤ N_{G}(T), and the inclusion of N_{G}(T) in N_{G}(T) induces isomorphisms W_{0} ≅ N_{G}(T)/T ≅ N/A. Thus Aut_{G}(A) = Aut_{W_{0}}(A).

Proof. (a) Assume condition (III.1) or (III.2) holds. We first prove that C_{W} (A) = 1, and also that C_{W} (Ω_{1}(A)) = 1 when p is odd.

If p is odd, set A_{0} = Ω_{1}(A) and p̂ = p. If p = 2, set A_{0} = Ω_{2}(A) and p̂ = 4. Thus in all cases, A_{0} is the p̂-torsion subgroup of A. Set ε = 1 if we are in case (III.1), or ε = −1 in case (III.2). By assumption, p̂|(q − ε). Choose λ ∈ 𝔽_{q}^{×} (or λ ∈ 𝔽_{q^{2}}^{×} if ε = −1) of order p̂. Set Π = {α_{1}, …, α_{r}}. Fix w ∈ C_{W} (A_{0}).

Assume first G = 𝔾(q), a Chevalley group. Then T = {t ∈ T | t^{q−ε} = 1}, and A_{0} contains all elements of order p̂ in T. So w = 1 by Lemma 2.7.

Now assume that Id ≠ γ ∈ Γ_{G}; i.e., G is one of the Steinberg groups ^{2}A_{n}(q), ^{2}D_{n}(q), or 2E_{6}(q). Then C_{G}(γ) is a simple algebraic group of type B_{m}, C_{m}, or F_{4} (cf. [Ca, § 13.1–3]) with root system Σ̂ ⊆ V_{0} = C_{V} (τ), and A_{0} contains all p̂-torsion in C_{T} (γ). By Lemma 2.7 again, w|_{V_{0}} = Id. Since w and τ are both orthogonal, w also sends the (−1)-eigenspace for the action of τ to itself, and thus w ∈ C_{W} (τ) = W_{0}. But W_{0} acts faithfully on V_{0} (see, e.g., [Ca, 13.1.1]), so w = 1.

Thus C_{W} (A_{0}) = 1. Hence C_{G}(A_{0}) = T by Proposition 2.5, and the other statements follow immediately.

(b) If C_{G}(A)^{0} = T, then N_{G}(T) ≤ N_{G}(A) ≤ N_{G}(T) (recall that A is the p-power torsion in T). If g ∈ N_{G}(T) and σ(g) = g, then g also normalizes T = C_{T} (σ). Thus N_{G}(T) = N_{G}(A) ≤ N_{G}(T), and hence N_{G}(T)/T ≅ W_{0} by Lemma 2.3. The identification N/A ≅ N_{G}(T)/T is immediate from the definition of N. □

We next look at the centralizer of the Weyl group acting on T or T.

Lemma 5.4. Assume Hypotheses 5.1, case (III.1), and Notation 5.2.

(a) Assume that all classes in Σ̂ have order 1 or 2. (Equivalently, τ(α) = α or τ(α) ⊥ α for each α ∈ Σ.) Then C_{T} (W_{0}) = C_{T} (W) = Z(G), and Z(G) = C_{T} (W_{0}).

(b) Assume that Σ̂ contains classes of order 3. Then G ≅ SL_{2n−1}(𝔽_{q_{0}}) and G ≅ SU_{2n−1}(q) for some n ≥ 2. Also, C_{T} (W_{0}) ≅ 𝔽_{q_{0}}^{×}, and σ(t) = t^{−q} for all t ∈ C_{T} (W_{0}).

Proof. (a) Assume that τ(α) = α or τ(α) ⊥ α for each α ∈ Σ. We first show, for each α̂ = {α, τ(α)} ∈ Π̂, that C_{T} (wα̂) = C_{T} (w_{α}, w_{τ(α)}). This is clear if α = τ(α). If α ⊥ τ(α), then wα̂ = w_{α}w_{τ(α)}, so if t ∈ C_{T} (wα̂), then w_{α}(t) = w_{τ(α)}(t) and t^{−1}w_{α}(t) = t^{−1}w_{τ(α)}(t). Also, t^{−1}w_{α}(t) ∈ T_{α} and t^{−1}w_{τ(α)}(t) ∈ T_{τ(α)} by Lemma 2.4(e). Since T_{α} ∩ T_{τ(α)} = 1 by Lemma 2.4(b), t^{−1}w_{α}(t) = 1, and hence t ∈ C_{T} (w_{α}, w_{τ(α)}).

Since W = ⟨w_{α} | α ∈ Π⟩, this proves that C_{T} (W_{0}) = C_{T} (W). Since G is universal, C_{T} (W) = Z(G) by Proposition 2.5. In particular, C_{T} (W_{0}) ≤ G ∩ Z(G) ≤ Z(G); while Z(G) ≤ C_{T} (W_{0}) since C_{G}(T) = T by Lemma 5.3(a).

(b) Assume Σ̂ contains a class of order 3. Then by [GLS3, (2.3.2)], γ ≠ Id, 𝔾 ≅ SL_{2n−1}, and G ≅ SU_{2n−1}(q) (some n ≥ 2). Also, if we identify

T = {diag(λ_{1}, …, λ_{2n−1}) | λ_{i} ∈ 𝔽_{q_{0}}^{×}, λ_{1}λ_{2} ⋯ λ_{2n−1} = 1} ,

and identify W = Σ_{2n−1} with its action on T permuting the coordinates, then

γ(diag(λ_{1}, …, λ_{2n−1})) = diag(λ_{2n−1}^{−1}, …, λ_{1}^{−1}),

and W_{0} ≅ C_{2} ≀ Σ_{n−1} is generated by the permutations (i 2n−i) and (i j)(2n−i 2n−j) for i, j < n. So C_{T} (W_{0}) is the group of all matrices diag(λ_{1}, …, λ_{2n−1}) such that λ_{i} = λ_{1} for all i ≠ n and λ_{n} = λ_{1}^{−(2n−2)}, and C_{T} (W_{0}) ≅ 𝔽_{q_{0}}^{×}. Also, γ inverts C_{T} (W_{0}), so σ(t) = t^{−q} for t ∈ C_{T} (W_{0}). □

Recall (Notation 5.2(F)) that when case (III.1) of Hypotheses 5.1 holds (in particular, when p = 2), we set Kα̂ = ⟨K_{α} | α ∈ α̂⟩ for α̂ ∈ Σ̂, where K_{α} = ⟨X_{α}, X_{−α}⟩. The conditions in (III.1) imply that each class in Σ̂ is of the form {α}, {α, τ(α)}, or {α, τ(α), α + τ(α)} for some α. This last case occurs only when G ≅ SU_{n}(q) for some odd n ≥ 3 and some q ≡ 1 (mod p or mod 4).

Lemma 5.5. Assume Hypotheses 5.1, case (III.1), and Notation 5.2. For each α ∈ Σ, K_{α} ≅ SL_{2}(𝔽_{q_{0}}). For each α̂ ∈ Σ̂, Kα̂ ≅ SL_{2}(𝔽_{q_{0}}), SL_{2}(𝔽_{q_{0}})×SL_{2}(𝔽_{q_{0}}), or SL_{3}(𝔽_{q_{0}}) whenever the class α̂ has order 1, 2, or 3, respectively. Also, G ∩ Kα̂ is isomorphic to SL_{2}(q), SL_{2}(q^{2}), or SU_{3}(q), respectively, in these three cases.

Proof. By Lemma 3.10, each class in Σ̂ is in the W_{0}-orbit of a class in Π̂. So it suffices to prove the statements about K_{α} and Kα̂ when α ∈ Π, and when α̂ ∈ Π̂ is its equivalence class.

By Lemma 2.4(b) (and since G is universal), K_{α} ≅ SL_{2}(𝔽_{q_{0}}) for each α ∈ Π. So when α = τ(α) (when |α̂| = 1), Kα̂ = K_{α} ≅ SL_{2}(𝔽_{q_{0}}).

When α ≠ τ(α) and they are not orthogonal, then G ≅ SL_{2n+1}(𝔽_{q_{0}}) for some n, and the inclusion of SL_{3}(𝔽_{q_{0}}) is clear. When α ⊥ τ(α), then [K_{α}, K_{τ(α)}] = 1, and K_{α} ∩ K_{τ(α)} = 1 by Lemma 2.4(b) and since G is universal, and since the intersection is contained in the centers of the two factors and hence in the maximal tori. Hence K̄α̂ = ⟨X̄_{±α}, X̄_{±τ(α)}⟩ ≅ K_{α} × K_{τ(α)} ≅ SL_{2}(𝔽_{q_{0}}) × SL_{2}(𝔽_{q_{0}}).

In all cases, since G is universal, G ∩ Kα̂ = C_{G}(σ) ∩ Kα̂ = C_{Kα̂}(σ). If α = τ(α), then γ

acts trivially on Kα̂, and C_{Kα̂}(σ) ≅ SL_{2}(q). If α ⊥ τ(α) then γ exchanges the two factors and C_{Kα̂}(σ) ≅ SL_{2}(q^{2}). Finally, if α ≠ τ(α) and they are not orthogonal, then γ is the

graph automorphism of SL_{3}(𝔽_{q_{0}}), so C_{Kα̂}(σ) ≅ SU_{3}(q). □

Recall, for α̂ ∈ Σ̂, that Aα̂ = A ∩ T α̂, where T α̂ = ⟨h_{α}(𝔽_{q_{0}}^{×}) | α ∈ α̂⟩. These subgroups are described in the next lemma.

Lemma 5.6. Assume that G and (G, σ) satisfy Hypotheses 5.1, case (III.1) or (III.2). Assume also Notation 5.2.

(a) If τ ≠ Id (hence we are in Case (III.1)), then for each α̂ ∈ Σ̂,

wα̂ = \begin{cases}w_{α} & if α̂ = \{α\} \\ w_{α}w_{τ(α)} & if α̂ = \{α, τ(α)\}, α ⊥ τ(α) \\ w_{α+τ(α)} = w_{α}w_{τ(α)}w_{α} & if α̂ = \{α, τ(α), α + τ(α)\}, \end{cases}

T α̂ = \begin{cases}T_{α} & if α̂ = \{α\} \\ T_{α} × T_{τ(α)} & if α, τ(α) ∈ α̂, α ≠ τ(α), \end{cases}

and

ĥ_{α}(λ) = \begin{cases}h_{α}(λ) & if α̂ = \{α\}, λ ∈ 𝔽_{q}^{×} \\ h_{α}(λ)h_{τ(α)}(λ^{q}) & if α, τ(α) ∈ α̂, α ≠ τ(α), λ ∈ 𝔽_{q^{2}}^{×}. \end{cases}

(b) In all cases, T = C_{T} (ψ_{q}γ) = ∏_{α̂∈Π̂} C_{T α̂}(ψ_{q}γ) and hence A = ∏_{α̂∈Π̂} Aα̂ (direct products).

(c) Set ε = 1 if we are in case (III.1), or ε = −1 if we are in case (III.2). Set m = v_{p}(q−ε). For each α̂ ∈ Σ̂,

G ∩ T α̂ = \begin{cases}\{h_{α}(λ) | λ ∈ 𝔽_{q^{2}}^{×}, λ^{εq} = λ\} ≅ C_{q−ε} & if α̂ = \{α\} \\ ĥ_{α}(𝔽_{q^{2}}^{×}) ≅ C_{q^{2}−1} & if α, τ(α) ∈ α̂, α ≠ τ(α). \end{cases}

In particular,

Aα̂ ≅ C_{p^{m}} if p is odd; Aα̂ ≅ \begin{cases}C_{2^{m}} & if p = 2 and |α̂| = 1 \\ C_{2^{m+1}} & if p = 2 and |α̂| ≥ 2. \end{cases}

Proof. Recall that Σ̂ and Π̂ are defined in Notation 2.2(C) only when ρ(Π) = Π; i.e., in case (III.1) of Hypotheses 5.1. In case (III.2), they were defined in Notation 5.2(D) by setting Σ̂ = Σ and Π̂ = Π (and also W_{0} = W in this case).

(a,c) If we are in case (III.1) of Hypotheses 5.1 (where the σ-setup is standard), then by Lemma 3.10, each orbit of W_{0} under its action on Σ̂ contains an element of Π̂. If we are in case (III.2), then since W_{0} = W, Σ̂ = Σ, and Π̂ = Π, the same statement follows from [Ca, Proposition 2.1.8]. So it suffices to prove these two points when α̂ ∈ Π̂.

The formulas for wα̂, T α̂, and ĥ_{α}(λ), and the description of G ∩ T α̂ = C_{T α̂}(σ), are clear when α̂ = {α}. So assume now that α̂ = {α, τ(α)} or {α, τ(α), α + τ(α)}, where α ≠ τ(α).

By the definition in Notation 5.2(E), wα̂ ∈ ⟨w_{α}, w_{τ(α)}⟩ acts on V_{0} = C_{V} (τ) as the reflection across the hyperplane ⟨α̂⟩^{⊥}, and exchanges the positive and negative roots in ⟨α̂⟩ ∩ Σ. If α ⊥ τ(α), then [w_{α}, w_{τ(α)}] = 1, and hence wα̂ is the product of these reflections. If |α̂| = 3, then ⟨w_{α}, w_{τ(α)}⟩ ≅ Σ_{3}, and one sees by inspection that w_{α+τ(α)} = w_{α}w_{τ(α)}w_{α} is the only element which satisfies the above conditions.

If |α̂| = 3, then T_{α+τ(α)} ≤ T_{α}T_{τ(α)} by Lemma 2.4(d). Hence T α̂ = T_{α}T_{τ(α)} whenever α ≠ τ(α) ∈ α̂. We can assume α, τ(α) ∈ Π, and so T_{α} ∩ T_{τ(α)} = 1 by Lemma 2.4(b).

By definition (see Notation 2.2(C)), for λ ∈ 𝔽_{q_{0}}, if ĥ_{α}(λ) is defined, it has the form h_{α}(λ)h_{τ(α)}(μ) for some μ ∈ 𝔽_{q_{0}}. Since

σ(h_{α}(λ)h_{τ(α)}(μ)) = h_{α}(μ^{q})h_{τ(α)}(λ^{q}) ,

this element lies in G if and only if μ = λ^{q} and λ^{q^{2}} = λ; i.e., λ ∈ 𝔽_{q^{2}}^{×}.

This proves the formulas for ĥ_{α}(λ) in (a), and also the description of G ∩ T α̂ in (c). The last statement in (c) is now immediate, since v_{p}(q^{2} − 1) = m + v_{p}(q + 1) = m (if p is odd) or m + 1 (if p = 2).

(b) By Lemma 2.4(b), T = ∏_{α∈Π} T_{α} = ∏_{α̂∈Π̂} T α̂ (a direct product), the last equality by (a). The direct product decompositions for T and A = O_{p}(T) follow immediately. □

We would like to know that fusion preserving automorphisms of S (i.e., elements of Aut(F_{S}(G))) permute the subgroups Aα̂ ≤ S. We next characterize (when possible) these subgroups in terms of fusion in S. Recall the definition of the focal subgroup of a saturated fusion system F over a finite p-group S:

𝔣𝔬𝔠(F) = ⟨xy^{−1} | x, y ∈ S, x is F-conjugate to y⟩.

By the focal subgroup theorem for groups (cf. [G, Theorem 7.3.4]), if F = F_{S}(G) for some finite group G with S ∈ Syl_{p}(G), then 𝔣𝔬𝔠(F) = S ∩ [G, G].

Lemma 5.7. Assume Hypotheses 5.1, case (III.1) or (III.2), and Notation 5.2.

(a) If p is odd, then [wα̂, A] = Aα̂ for each α̂ ∈ Σ̂. If p = 2, then for each α̂ ∈ Σ̂, [wα̂, A] ≤ Aα̂ with index at most 2, and [wα̂, A] = Aα̂ with the following exceptions:

• τ = Id, 𝔾 ≅ A_{1}, and α̂ = {α}; or

• τ = Id, 𝔾 ≅ C_{n} for n ≥ 2 (or B_{2}), and α̂ = {α} where α is a long root; or

• |τ| = 2, 𝔾 ≅ D_{n} for n ≥ 3 (or A_{3}), and α̂ = {α, τ(α)} where α ⊥ τ(α); or

• |τ| = 2, 𝔾 ≅ A_{2n} for n ≥ 1, and |α̂| = 3.

(b) For each w ∈ W_{0} of order 2, w = wα̂ for some α̂ ∈ Σ̂ if and only if [w, A] is cyclic.

(c) If p = 2, then for each α̂ ∈ Σ̂,

C_{G}(C_{A}(wα̂)) = \begin{cases}T Kα̂ & if |α̂| ≤ 2 \\ T K_{α+τ(α)} & if α̂ = \{α, τ(α), α + τ(α)\}. \end{cases}  (1)

If in addition, |α̂| ≤ 2, then

Aα̂ = A ∩ [C_{G}(C_{A}(wα̂)), C_{G}(C_{A}(wα̂))] = A ∩ 𝔣𝔬𝔠(C_{F}(C_{A}(wα̂))).

Proof. As in the proof of Lemma 5.6, we can assume in the proofs of (a) and (c) that α̂ ∈ Π̂.

(a) Fix α ∈ Π, and let α̂ ∈ Π̂ be its class. By Lemma 2.4(e) and since wα̂ ∈ ⟨w_{α}, w_{τ(α)}⟩, we have [wα̂, A] ≤ A ∩ T α̂ = Aα̂ in all cases. By the same lemma, wα̂(ĥ_{α}(λ)) = ĥ_{α}(λ^{−1}) for all λ ∈ 𝔽_{q_{0}}^{×} if |α̂| ≤ 2; and wα̂(ĥ_{α}(λ)) = ĥ_{α}(λ^{−q}) for λ ∈ 𝔽_{q^{2}}^{×} if |α̂| = 3. So [wα̂, A] = Aα̂ if p is odd, and (since Aα̂ is cyclic by Lemma 5.6(c)) [wα̂, A] has index at most 2 in Aα̂ if p = 2.

Assume now that p = 2, and hence that q ≡ 1 (mod 4). If τ = Id (and hence α̂ = {α}), then for each β ∈ Π and each λ ∈ 𝔽_{q}^{×}, Lemma 2.4(e) implies that

w_{α}(h_{β}(λ)) = \begin{cases}h_{β}(λ) & if β ⊥ α \\ h_{β}(λ)h_{α}(λ) & if β ⊥̸ α, ∥β∥ ≥ ∥α∥ \\ h_{β}(λ)h_{α}(λ^{k}) & if β ⊥̸ α, ∥α∥ = \sqrt{k} · ∥β∥, k = 1, 2, 3. \end{cases}

(Note that w_{α}(β^{∨}) = β^{∨}, β^{∨} + α^{∨}, or β^{∨} + kα^{∨}, respectively, in these three cases.) Since T is generated by the h_{β}(λ) for β ∈ Π and λ ∈ 𝔽_{q}^{×}, it follows that [w_{α}, A] has index 2 in A_{α} exactly when |Π| = 1, or there are roots with two lengths and ratio \sqrt{2}, α is a long root, and is orthogonal to all other long roots in Π. This happens only when 𝔾 ≅ A_{1} or C_{n}.

Now assume |τ| = 2. In particular, all roots in Σ have the same length. By Lemmas 2.4(e) and 5.6(a), for each β ∈ Π∖α̂ such that β ⊥̸ α and with class β̂ ∈ Π̂, we have

wα̂(ĥ_{β}(λ)) = \begin{cases}ĥ_{β}(λ)ĥ_{α}(λ) & if |β̂| = 1 and λ ∈ 𝔽_{q}^{×} \\ ĥ_{β}(λ)ĥ_{α}(λ) & if |β̂| ≥ 2, |α̂| = 2, and λ ∈ 𝔽_{q^{2}}^{×} \\ ĥ_{β}(λ)ĥ_{α}(λ^{q+1}) & if |β̂| ≥ 2, |α̂| = 1 or 3, and λ ∈ 𝔽_{q^{2}}^{×} \end{cases}

By these formulas and Lemma 5.6(c), [wα̂, A] = Aα̂ exactly when |α̂| = 1, or |α̂| = 2 and there is some β ∈ Π such that β ⊥̸ α and β ≠ τ(β). The only cases where this does not happen are when 𝔾 ≅ D_{n} or A_{3} and |α̂| = 2, and when 𝔾 ≅ A_{2n} and |α̂| ≥ 3.

(b) For each α̂ ∈ Σ̂, [wα̂, A] ≤ Aα̂ by (a), and hence is cyclic. It remains to prove the converse.

Recall (Notation 5.2(D)) that when we are in case (III.2) (and hence the setup is not standard), we define V_{0} = V. By assumption, G is always a Chevalley group in this case.

Let w ∈ W_{0} be an element of order 2 which is not equal to wα̂ for any α̂. If G is a Chevalley group (if W_{0} = W and V_{0} = V), then C_{V} (w) contains no points in the interior of any Weyl chamber, since W permutes freely the Weyl chambers (see [Brb, § V.3.2, Théorème 1(iii)]). Since w is not the reflection in a root hyperplane, it follows that dim(V/C_{V} (w)) ≥ 2. If G is a Steinberg group (thus in case (III.1) with a standard setup), then W_{0} acts on V_{0} as the Weyl group of a certain root system on V_{0} (see [Ca, § 13.3]), so dim(V_{0}/C_{V_{0}}(w)) ≥ 2 by a similar argument.

Set ε = +1 if we are in case (III.1), or ε = −1 if we are in case (III.2). Set m = v_{p}(q − ε), and choose λ ∈ (𝔽_{q^{2}})^{×} of order p^{m}. Set Λ = ℤΣ^{∨}, regarded as the lattice in V with ℤ-basis Π^{∨} = {α^{∨} | α ∈ Π}. Let

Φ_{λ}: Λ/p^{m}Λ ⟶ T

be the ℤ[W]-linear monomorphism of Lemma 2.6(b) with image the p^{m}-torsion in T. Thus Φ_{λ}(α^{∨}) = h_{α}(λ) for each α ∈ Σ. Also, σ(h_{α}(λ)) = h_{τ(α)}(λ) for each α ∈ Σ (λ^{q} = λ by assumption), and thus Φ_{λ} commutes with the actions of τ on Λ < V and of σ on T.

Set Λ_{0} = C_{Λ}(τ) in case (III.1), or Λ_{0} = Λ in case (III.2). Then C_{Λ/p^{m}Λ}(τ) = Λ_{0}/p^{m}Λ_{0} in case (III.1), since τ permutes the basis Π^{∨} of Λ. We claim that Φ_{λ} restricts to a ℤ[W_{0}]-linear isomorphism

Φ_{0} : Λ_{0}/p^{m}Λ_{0} ⟶^{≅} Ω_{m}(A) ,

where Ω_{m}(A) is the p^{m}-torsion subgroup of A and hence of T = C_{T} (σ). If G is a Chevalley group (in either case (III.1) or (III.2)), then Λ_{0} = Λ, so Im(Φ_{0}) is the p^{m}-torsion subgroup of T and equal to Ω_{m}(A). If G is a Steinberg group, then ε = +1, each element of order dividing p^{m} in T is fixed by ψ^{q}, and hence lies in Ω_{m}(A) if and only if it is fixed by γ (thus in Φ_{λ}(C_{Λ/p^{m}Λ}(τ))).

Thus [w, A] ≥ [w, Ω_{m}(A)] ≅ [w, Λ_{0}/p^{m}Λ_{0}]. Set B = Λ_{0}/p^{m}Λ_{0} for short; we will show that [w, B] is noncyclic. Set

r = rk(Λ_{0}) = dim(V_{0}) and s = rk(C_{Λ_{0}}(w)) = dim_{ℝ}(C_{V_{0}}(w)) ≤ r − 2 .

For each b ∈ C_{B}(w), and each v ∈ Λ_{0} such that b = v + p^{m}Λ_{0}, v + w(v) ∈ C_{Λ_{0}}(w) maps to 2b ∈ C_{B}(w). Thus B ≅ (ℤ/p^{m})^{r}, while {2b | b ∈ C_{B}(w)} is contained in C_{Λ_{0}}(w)/p^{m}C_{Λ_{0}}(w) ≅ (ℤ/p^{m})^{s}. Since p^{m} > 2 by assumption (and r − s ≥ 2), it follows that B/C_{B}(w) ≅ [w, B] is not cyclic.

(c) Fix α̂ ∈ Σ̂. We set up our notation as follows.

Case (1): |α̂| = 1 or 3. Set α^{∗} = α if α̂ = {α} (where τ(α) = α), or α^{∗} = α + τ(α) if

α̂ = {α, τ(α), α + τ(α)}. Set wα̂ = w_{α^{∗}}, Wα̂ = ⟨wα̂⟩, and Δ = {±α^{∗}} ⊆ Σ.

Case (2): |α̂| = 2. Thus α̂ = {α, τ(α)}, where α ⊥ τ(α). Set wα̂ = w_{α}w_{τ(α)}, Wα̂ = ⟨w_{α}, w_{τ(α)}⟩, and Δ = {±α, ±τ(α)} ⊆ Σ.

In case (1), by Lemma 2.4(c,e),

C_{T} (wα̂) = C_{T} (w_{α^{∗}}) = Ker(θ_{α^{∗}}) = C_{T} (X_{α^{∗}}) = C_{T} (X_{−α^{∗}}) .

Hence C_{G}(C_{A}(wα̂)) ≥ C_{G}(C_{T} (wα̂)) ≥ T ⟨X_{α^{∗}}, X_{−α^{∗}}⟩ = T K_{α^{∗}}. In case (2), by the same lemma,

C_{T} (wα̂) = C_{T} (⟨w_{α}, w_{τ(α)}⟩) = C_{T} (⟨X_{α}, X_{−α}, X_{τ(α)}, X_{−τ(α)}⟩) = C_{T} (K_{α}K_{τ(α)})

so that C_{G}(C_{A}(wα̂)) ≥ T Kα̂. This proves one of the inclusions in (1). By Proposition 2.5, the opposite inclusion will follow once we show that

C_{W} (C_{A}(wα̂)) ≤ Wα̂ .  (2)

Fix w ∈ C_{W} (C_{A}(wα̂)).

• Let β ∈ Σ ∩ Δ^{⊥} be such that β = τ(β). Then h_{β}(λ) ∈ C_{A}(wα̂) for λ ∈ 𝔽_{q_{0}}^{×} of order 4, so w(h_{β}(λ)) = h_{β}(λ), and β ∈ C_{V} (w) by Lemma 2.6(d).

• Let β ∈ Σ ∩ Δ^{⊥} be such that β ≠ τ(β), and set β′ = τ(β) for short. Let r ≥ 2 be such that q ≡ 1 + 2^{r} (mod 2^{r+1}), and choose λ ∈ 𝔽_{q_{0}}^{×} of order 2^{r+1}. Set a = 1 − 2^{r}, so λ^{a} = λ^{q}. Then

h_{β}(λ)h_{β′}(λ^{a}), h_{β}(λ^{a})h_{β′}(λ) ∈ C_{A}(wα̂) ≤ C_{T} (w) .

Also, ∥β + aβ′∥ = ∥aβ + β′∥ < (1 − a)∥β∥ = \frac{1}{2}|λ|∥β∥ since a < 0 and β′ ≠ −β (since τ(Σ_{+}) = Σ_{+}). Thus β + aβ′, aβ + β′ ∈ C_{V} (w) by Lemma 2.6(c), so β, β′ ∈ C_{V} (w).

• Let β ∈ Σ be such that β = τ(β) and β ∉ Δ^{⊥}, and set η = β +wα̂(β). Since wα̂τ = τwα̂ in Aut(V), τ(η) = η. Since β ∉ Δ^{⊥} = C_{V} (wα̂), we have wα̂(β) ≠ β, and hence ∥η∥ < 2∥β∥. For λ ∈ 𝔽_{q_{0}}^{×} of order 4, t = h_{β}(λ)h_{wα̂(β)}(λ) ∈ C_{A}(wα̂), so w(t) = t, and η = β + wα̂(β) ∈ C_{V} (w) by Lemma 2.6(c).

Consider the set

Σ^{∗} = (Σ ∩ Δ^{⊥}) ∪ {β + wα̂(β) | β ∈ Σ, τ(β) = β, β ⊥̸ Δ} ⊆ V .

We have just shown that w(η) = η for each η ∈ Σ^{∗}, and hence that w|_{⟨Σ^{∗}⟩} = Id. We next claim that

Σ ∩ (Σ^{∗})^{⊥} = Δ except when 𝔾 ≅ A_{2} and |τ| = 2.  (3)

From the description of the root systems in [Brb, Planches I–IX], whenever 𝔾 is not of type A_{n}, we get that ⟨Σ ∩ β^{⊥}⟩ is a hyperplane in V for each β ∈ Σ. (It suffices to check this for one root in Σ, or for one short root and one long root.) In particular, (3) holds whenever |α̂| = 1 or 3 and 𝔾 is of one of these types. If |α̂| = 2, so |τ| = 2 and α̂ = {α, τ(α)} where α ⊥ τ(α), and 𝔾 = D_{n} or E_{6}, then a similar check shows that ⟨Σ ∩ {α, τ(α)}^{⊥}⟩ has codimension 2 in V, and hence that (3) holds. For example, when 𝔾 = E_{6}, it suffices to check this with the roots α = α_{3} = ε_{2} − ε_{1} and τ(α) = α_{5} = ε_{4} − ε_{3} in the notation of [Brb, Planche V].

Now assume 𝔾 ≅ A_{n} for some n. If n ≥ 3, then ⟨Σ ∩ β^{⊥}⟩ has codimension 2 for β ∈ Σ, but the only roots in the orthogonal complement of this space are ±β. Thus (3) holds for A_{n} when n ≥ 3 and |α̂| = 1 or 3, and the cases n = 1, 2 are easily checked. If α̂ = {α, τ(α)} where α ⊥ τ(α), then n ≥ 3, and we can take α = ε_{1} − ε_{2} and τ(α) = ε_{n} − ε_{n+1} in the notation of [Brb, Planche I], where τ(x_{1}, …, x_{n+1}) = (−x_{n+1}, …, −x_{1}). In this case, Σ^{∗} contains all roots ε_{i} − ε_{j} for 3 ≤ i < j ≤ n − 1 as well as (ε_{1} − ε_{n+1}) + (ε_{2} − ε_{n}), and these elements suffice to show that ±α and ±τ(α) are the only roots in (Σ^{∗})^{⊥}. This finishes the proof of (3).

By (3), when G ≅ S̸U_{3}(q), the only reflection hyperplanes which contain ⟨Σ^{∗}⟩ are those in the set {β^{⊥} | β ∈ Δ}. Fix a “generic” element v ∈ ⟨Σ^{∗}⟩; i.e., one which is not contained in any other hyperplane. In case (1), v is contained in only the one reflection hyperplane α^{∗}^{⊥}, and hence is in the closure of exactly two Weyl chambers for (Σ, W): chambers which are exchanged by wα̂. In case (2), v is contained in the two reflection hyperplanes α^{⊥} and τ(α)^{⊥}, and hence in the closure of four Weyl chambers which are permuted freely and transitively by Wα̂ = ⟨w_{α}, w_{τ(α)}⟩. Since W permutes the Weyl chambers freely and transitively (see [Brb, § V.3.2, Théorème 1(iii)]), and since ⟨w, Wα̂⟩ permutes the chambers whose closures contain v, we have w ∈ Wα̂.

This proves (2) when G ≅ S̸U_{3}(q). If G ≅ SU_{3}(q), then h_{α^{∗}}(−1) ∈ C_{A}(wα̂). But no element of order 2 in T < SL_{3}(𝔽_{q_{0}}) centralizes the full Weyl group W ≅ Σ_{3}, so (2) also holds in this case. This finishes the proof of (1).

If |α̂| ≤ 2, then

C_{G}(C_{A}(wα̂)) = G ∩ C_{G}(C_{A}(wα̂)) = T (G ∩ Kα̂)

where by Lemma 5.5, G∩Kα̂ ≅ SL_{2}(q) or SL_{2}(q^{2}). Hence C_{G}(C_{A}(wα̂)) has commutator subgroup G∩Kα̂, and focal subgroup Aα̂. Since C_{F}(C_{A}(wα̂)) is the fusion system of C_{G}(C_{A}(wα̂)) (cf. [AKO, Proposition I.5.4]), this proves the last statement. □

Recall (Notation 5.2(H)) that Aut(A, F) is the group of automorphisms of A which extend to elements of Aut(F). The next result describes the structure of Aut(A, F) for a group G in the situation of case (III.1) or (III.2) of Hypotheses 5.1. Recall that W_{0} acts faithfully on A by Lemma 5.3(a), and hence that W_{0} ≅ Aut_{N}(A) = Aut_{N_{G}(T)}(A) by Lemma 5.3(b). It will be convenient to identify W_{0} with this subgroup of Aut(A). Since each element of Aut(A, F) is fusion preserving, this group normalizes and hence acts on W_{0}, and W_{0}Aut(A, F) is a subgroup of Aut(A).

For convenience, we set Aut_{Aut(G)}(A) = {δ|_{A} | δ ∈ Aut(G), δ(A) = A}.

Lemma 5.8. Assume that G and (G, σ) satisfy Hypotheses 5.1, case (III.1) or (III.2). Assume also Notation 5.2.

(a) C_{W_{0}Aut(A,F)}(W_{0}) ≤ W_{0}Aut_{sc}(A).

(b) Aut(A, F) ≤ Aut_{sc}(A)Aut_{Aut(G)}(A), with the exceptions • (G, p) ≅ (^{2}E_{6}(q), 3), or

• (G, p) ≅ (G_{2}(q), 2) and q_{0} ≠ 3, or

• (G, p) ≅ (F_{4}(q), 3) and q_{0} ≠ 2.

(c) In all cases, the index of Aut(A, F) ∩ Aut_{sc}(A)Aut_{Aut(G)}(A) in Aut(A, F) is at most 2 .

Proof. Recall that in Notation 2.2(C), V_{0}, Σ̂, and Π̂ are defined when ρ(Π) = Π, and hence in case (III.1) of Hypotheses 5.1. In case (III.2), we defined V_{0} = V, Σ̂ = Σ, and Π̂ = Π in Notation 5.2(D). So under the hypotheses of the lemma (and since G is always a Chevalley group in case (III.2)), we have V_{0} = V and Π̂ = Π if and only if G is a Chevalley group.

Set ε = 1 if we are in case (III.1), ε = −1 if we are in case (III.2), and m = v_{p}(q − ε).

Step 1: We first prove that

φ ∈ C_{W_{0}Aut(A,F)}(W_{0}) ⟹ φ(Aα̂) = Aα̂ for all α̂ ∈ Σ̂.  (4)

If p is odd, then Aα̂ = [wα̂, A] by Lemma 5.7(a), so (4) is immediate.

Next assume that p = 2, and also that |α̂| ≤ 2. Write φ = w ∘ φ_{0}, where w ∈ W_{0} and φ_{0} ∈ Aut(A, F). Then φ_{0}(C_{A}(wα̂)) = w^{−1}(C_{A}(wα̂)) = C_{A}(wβ̂), where β̂ = w^{−1}(α̂). By definition of Aut(A, F) (Notation 5.2), φ_{0} = φ_{0}|_{A} for some φ_{0} ∈ Aut(F). Since φ_{0} is fusion preserving, it sends 𝔣𝔬𝔠(C_{F}(C_{A}(wα̂))) onto 𝔣𝔬𝔠(C_{F}(C_{A}(wβ̂))). Since these focal subgroups are Aα̂ and Aβ̂, respectively, by Lemma 5.7(c), φ(Aα̂) = w(Aβ̂) = A_{w(β̂)} = Aα̂ also in this case (the second equality by Lemma 2.4(e)).

It remains to consider the case where p = 2 and |α̂| = 3, and thus where G ≅ SU_{2n+1}(q) for some n ≥ 1. There is a subgroup (H_{1} × ⋯ × H_{n}) ⋊ Σ_{n} < G of odd index, where H_{i} ≅ GU_{2}(q). Fix S_{i} ∈ Syl_{2}(H_{i}); then S_{i} ≅ SD_{2^{k}} where k = v_{2}(q^{2} − 1) + 1 ≥ 4. Let A_{i}, Q_{i} < S_{i} denote the cyclic and quaternion subgroups of index 2 in S_{i}. Then we can take A = A_{1} × ⋯ × A_{n} ≅ (C_{2^{k−1}})^{n}, N = (S_{1} × ⋯ × S_{n}) ⋊ Σ_{n}, and S ∈ Syl_{2}(N).

There are exactly n classes α̂_{1}, …, α̂_{n} ∈ Σ̂_{+} of order 3, which we label so that [wα̂_{i}, A] ≤ A_{i} ([wα̂_{i}, A] = A ∩ Q_{i}). Equivalently, these are chosen so that wα̂_{i} acts on A via conjugation by an element of S_{i}∖A_{i}. Let α_{i}^{∗} ∈ Σ_{+} be the root in the class α̂_{i} which is the sum of the other two.

Write φ = w ∘ φ_{0}, where w ∈ W_{0} and φ_{0} ∈ Aut(A, F), and let φ_{0} ∈ Aut(F) be such that φ_{0} = φ_{0}|_{A}. For each 1 ≤ i ≤ n, φ_{0}(C_{A}(wα̂_{i})) = w^{−1}(C_{A}(wα̂_{i})) = C_{A}(wα̂_{f(i)}), where f ∈ Σ_{n} is such that α̂_{f(i)} = w^{−1}(α̂_{i}). Since φ_{0} is fusion preserving, it sends 𝔣𝔬𝔠(C_{F}(C_{A}(wα̂_{i}))) onto 𝔣𝔬𝔠(C_{F}(C_{A}(wα̂_{f(i)}))). By Lemma 5.7(c), C_{G}(C_{A}(wα̂_{i})) = G ∩ (T K_{α_{i}^{∗}}), its commutator subgroup is G ∩ K_{α_{i}^{∗}} ≅ SL_{2}(q), and hence 𝔣𝔬𝔠(C_{F}(C_{A}(wα̂_{i}))) = Q_{i}. Thus φ_{0}(Q_{i}) = Q_{f(i)}.

For each i, set Q_{i}^{∗} = ⟨Q_{j} | j ≠ i⟩. Then C_{G}(Q_{i}^{∗}) is the product of G ∩ Kα̂_{i} ≅ SL_{3}(q) (Lemma 5.5) with Z(Q_{i}^{∗}). Thus φ_{0} sends 𝔣𝔬𝔠(C_{F}(Q_{i}^{∗})) = S_{i} to 𝔣𝔬𝔠(C_{F}(Q_{f(i)}^{∗})) = S_{f(i)}, and hence φ_{0}(A_{i}) = A_{f(i)}. So φ(A_{i}) = w(A_{f(i)}) = A_{i} for each i where A_{i} = Aα̂_{i}, and this finishes the proof of (4).

Step 2: We next prove point (a): that C_{W_{0}Aut(A,F)}(W_{0}) ≤ W_{0}Aut_{sc}(A). Let φ ∈ W_{0}Aut(A, F) be an element which centralizes Aut_{N}(A) ≅ N/A ≅ W_{0}. By (4), φ(Aα̂) = Aα̂ for each α̂ ∈ Σ̂. Since Aα̂ is cyclic for each α̂ ∈ Σ̂_{+} by Lemma 5.6(c), φ|_{Aα̂} is multiplication by some unique uα̂ ∈ (ℤ/qα̂)^{×}, where qα̂ = |Aα̂|. We must show that uα̂ is independent of α̂.

Assume first that τ = Id. By Lemma 5.6(c), |A_{α}| = p^{m} for each α ∈ Π. Fix α_{1}, α_{2} ∈ Π and β ∈ Σ_{+} such that \frac{1}{k}β = \frac{1}{k}α_{1} + α_{2}, where either • k = 1 and all three roots have the same length; or

• k ∈ \{2, 3\} and ∥β∥ = ∥α_{1}∥ = \sqrt{k} · ∥α_{2}∥.

The relation between the three roots is chosen so that h_{β}(λ) = h_{α_{1}}(λ)h_{α_{2}}(λ) for all λ ∈ 𝔽_{q_{0}}^{×} by Lemma 2.4(d). Hence u_{α_{1}} ≡ u_{β} ≡ u_{α_{2}} (mod p^{m}) by Lemma 5.6(b). By the connectivity of the Dynkin diagram, the u_{α} for α ∈ Π are all equal, and φ ∈ Aut_{sc}(A).

Now assume |τ| = 2; the argument is similar but slightly more complicated. By assumption, 𝔾 is of type A_{n}, D_{n}, or E_{n}; i.e., all roots have the same length. Set m′ = v_{p}(q^{2} − 1); then m′ = m if p is odd, and m′ = m + 1 if p = 2. Fix α_{1}, α_{2} ∈ Π such that α_{1} ≠ τ(α_{2}) and β =^{def} α_{1} + α_{2} ∈ Σ_{+}. Choose λ ∈ 𝔽_{q_{0}}^{×} of order p^{m′}.

If α_{1} ≠ τ(α_{1}) and α_{2} ≠ τ(α_{2}), then |Aα̂_{1}| = |Aα̂_{2}| = p^{m′} by Lemma 5.6(c), and

ĥ_{α_{1}}(λ)ĥ_{α_{2}}(λ) = h_{α_{1}}(λ)h_{τ(α_{1})}(λ^{q})h_{α_{2}}(λ)h_{τ(α_{2})}(λ^{q})
= h_{β}(λ)h_{τ(β)}(λ^{q}) = ĥ_{β}(λ) ∈ Aβ̂ .

Hence

(ĥ_{α_{1}}(λ)ĥ_{α_{2}}(λ))^{uβ̂} = φ(ĥ_{α_{1}}(λ)ĥ_{α_{2}}(λ)) = ĥ_{α_{1}}(λ)^{uα̂_{1}} · ĥ_{α_{2}}(λ)^{uα̂_{2}} ,

and together with Lemma 5.6(b), this proves that uα̂_{1} ≡ uβ̂ ≡ uα̂_{2} (mod p^{m′}).

If τ(α_{i}) = α_{i} for i = 1, 2, then a similar argument shows that uα̂_{1} ≡ uβ̂ ≡ uα̂_{2} (mod p^{m}). It remains to handle the case where α_{1} ≠ τ(α_{1}) and α_{2} = τ(α_{2}). In this case, |Aα̂_{1}| = p^{m′} and |Aα̂_{2}| = p^{m} by Lemma 5.6(c), and these groups are generated by ĥ_{α_{1}}(λ) = h_{α_{1}}(λ)h_{τ(α_{1})}(λ^{q}) and h_{α_{2}}(λ^{q+1}), respectively. Then

ĥ_{α_{1}}(λ)ĥ_{α_{2}}(λ^{q+1}) = h_{α_{1}}(λ)h_{τ(α_{1})}(λ^{q})h_{α_{2}}(λ^{q+1}) = h_{β}(λ)h_{τ(β)}(λ^{q}) = ĥ_{β}(λ) ∈ Aβ̂ ,

so

(ĥ_{α_{1}}(λ)ĥ_{α_{2}}(λ^{q+1}))^{uβ̂} = φ(ĥ_{α_{1}}(λ)ĥ_{α_{2}}(λ^{q+1})) = ĥ_{α_{1}}(λ)^{uα̂_{1}} · h_{α_{2}}(λ^{q+1})^{uα̂_{2}} ,

and uα̂_{1} ≡ uβ̂ ≡ uα̂_{2} (mod p^{m}) by Lemma 5.6(b) again.

Since the Dynkin diagram is connected, and since the subdiagram of nodes in free orbits in the quotient diagram is also connected, this shows that the uα̂ are all congruent for α̂ ∈ Π̂ (modulo p^{m} or p^{m′}, depending on where they are defined), and hence that φ ∈ Aut_{sc}(A).

Step 3: Consider the subset WΠ̂ = {wα̂ | α̂ ∈ Π̂}. We need to study the subgroup N_{W_{0}Aut(A,F)}(WΠ̂): the group of elements of W_{0}Aut(A, F) which permute the set WΠ̂. Note that W_{0} = ⟨WΠ̂⟩ (see, e.g., [Ca, Proposition 13.1.2], and recall that W_{0} = W and Π̂ = Π in case (III.2)). We first show that

Aut(A, F) ≤ W_{0}N_{W_{0}Aut(A,F)}(WΠ̂) .  (5)

Write Π̂ = {α̂_{1}, …, α̂_{k}}, ordered so that for each 2 ≤ i ≤ k, α̂_{i} is orthogonal to all but one of the α̂_{j} for j < i. Here, α̂_{i} ⊥ α̂_{j} means orthogonal as vectors in V_{0}. Thus wα̂_{i} commutes with all but one of the wα̂_{j} for j < i. By inspection of the Dynkin diagram of 𝔾 (or the quotient of that diagram by τ), this is always possible.

Fix φ ∈ Aut(A, F). In particular, φ normalizes W_{0} (recall that we identify W_{0} = Aut_{W_{0}}(A)) since φ is fusion preserving. (Recall that Aut_{G}(A) = Aut_{W_{0}}(A) by Lemma 5.3(b).) We must show that some element of φW_{0} normalizes the set WΠ̂.

By definition of Aut(A, F) (Notation 5.2), φ = φ|_{A} for some φ ∈ Aut(F). Since φ is fusion preserving, φ normalizes Aut_{F}(A) = Aut_{G}(A), where Aut_{G}(A) ≅ N/A ≅ W_{0} since C_{N}(A) = A by Lemma 5.3(a). Thus there is a unique automorphism φ̂ ∈ Aut(W_{0}) such that φ̂(w) = φ ∘ w ∘ φ^{−1} for each w ∈ W_{0}.

For each i, since |φ̂(wα̂_{i})| = 2 and [φ̂(wα̂_{i}), A] ≅ [wα̂_{i}, A] is cyclic, φ̂(wα̂_{i}) = w_{α′_{i}} for some α̂′_{i} ∈ Σ̂ by Lemma 5.7(b), where α̂′_{i} is uniquely determined only up to sign. For i ≠ j,

α̂_{i} ⊥ α̂_{j} ⟺ [wα̂_{i}, wα̂_{j}] = 1 ⟺ [φ̂(wα̂_{i}), φ̂(wα̂_{j})] = 1 ⟺ α̂′_{i} ⊥ α̂′_{j} .

So via the assumption about orthogonality, we can choose successively elements α̂′_{1}, α̂′_{2}, …, α̂′_{k} such that φ̂(wα̂_{i}) = w_{α′_{i}} for each i, and ⟨α̂′_{i}, α̂′_{j}⟩ ≤ 0 for i ≠ j.

For each i ≠ j, since |wα̂_{i}wα̂_{j}| = |w_{α′_{i}}w_{α′_{j}}|, the angle (in V_{0}) between α̂_{i} and α̂_{j} is equal to that between α̂′_{i} and α̂′_{j} (by assumption, both angles are between π/2 and π). The roots α̂′_{i} for 1 ≤ i ≤ k thus generate Σ̂ as a root system on V_{0} with Weyl group W_{0}, and hence are the fundamental roots for another Weyl chamber for Σ̂. (Recall that Σ̂ = Σ, V_{0} = V, and W_{0} = W in case (III.2).) Since W_{0} permutes the Weyl chambers transitively [Brb, § VI.1.5, Theorem 2(i)], there is w ∈ W_{0} which sends the set {wα̂_{i}} onto {φ̂(wα̂_{i})}. Thus c_{w}^{−1} ∘ φ ∈ N_{W_{0}Aut(A,F)}(WΠ̂), so φ ∈ W_{0}N_{W_{0}Aut(A,F)}(WΠ̂), and this proves (5).

Step 4: Set Aut_{W_{0}Aut(A,F)}(WΠ̂) = N_{W_{0}Aut(A,F)}(WΠ̂)/C_{W_{0}Aut(A,F)}(WΠ̂): the group of permutations of the set WΠ̂ which are induced by elements of W_{0}Aut(A, F). By (a) (Step 2) and (5), and since W_{0} = ⟨WΠ̂⟩, there is a surjection

Aut_{W_{0}Aut(A,F)}(WΠ̂) −↠^{onto} \frac{W_{0}N_{W_{0}Aut(A,F)}(WΠ̂)}{W_{0}C_{W_{0}Aut(A,F)}(WΠ̂)} = \frac{W_{0}Aut(A, F)}{W_{0}Aut_{sc}(A)} .  (6)

To finish the proof of the lemma, we must show that each element of the group Aut_{W_{0}Aut(A,F)}(WΠ̂) is represented by an element of Aut_{Aut(G)}(A) (i.e., the restriction of an automorphism of G), with the exceptions listed in point (b).

In the proof of Step 3, we saw that each element of Aut_{W_{0}Aut(A,F)}(WΠ̂) preserves angles between the corresponding elements of Π̂, and hence induces an automorphism of the Coxeter diagram for (V_{0}, Σ̂) (i.e., the Dynkin diagram without orientation on the edges).

Case 1: Assume G = 𝔾(q) is a Chevalley group. The automorphisms of the Coxeter diagram of 𝔾 are well known, and we have

|Aut_{W_{0}Aut(A,F)}(WΠ̂)| ≤ \begin{cases}6 & if 𝔾 ≅ D_{4} \\ 2 & if 𝔾 ≅ A_{n} (n ≥ 2), D_{n} (n ≥ 5), E_{6}, B_{2}, G_{2}, or F_{4} \\ 1 & otherwise. \end{cases} (7)

In case (III.1) (i.e., when the setup is standard), all of these automorphisms are realized by restrictions of graph automorphisms in Γ_{G} (see [Ca, §§ 12.2–4]), except possibly when G ≅ B_{2}(q), G_{2}(q), or F_{4}(q). In case (III.2), with the same three exceptions, each such automorphism is realized by some graph automorphism φ ∈ Γ_{G}, and φ|_{T} commutes with σ|_{T} ∈ Z(Aut(T)). Hence by Lemma 3.7, φ|_{T} extends to an automorphism of G whose restriction to A induces the given symmetry of the Coxeter diagram. Together with (6), this proves the lemma for Chevalley groups, with the above exceptions.

If G ≅ B_{2}(q) or F_{4}(q) and p ≠ 2, then |Aut_{W_{0}Aut(A,F)}(WΠ̂)| = 2, and the nontrivial element is represented by an element of Aut_{Γ_{G}}(A) exactly when q_{0} = 2. This proves the lemma in these cases, and a similar argument holds when G ≅ G_{2}(q) and p ≠ 3.

It remains to check the cases where (G, p) ≅ (B_{2}(q), 2), (G_{2}(q), 3), or (F_{4}(q), 2). We claim that Aut_{W_{0}Aut(A,F)}(WΠ̂) = 1 in these three cases; then the three groups in (6) are trivial, and so Aut(A, F) ≤ W_{0}Aut_{sc}(A). If (𝔾, p) = (B_{2}, 2) or (G_{2}, 3), then with the help of Lemma 2.4(d,b), one shows that the subgroups Ω_{1}(A_{α}) are all equal for α a short root, and are all distinct for the distinct (positive) long roots. More precisely, of the p + 1 subgroups of order p in Ω_{1}(A) ≅ C_{p}^{2}, one is equal to A_{α} when α is any of the short roots in Σ_{+}, while each of the other p is equal to A_{α} for one distinct long root α. Since Ω_{1}(A_{α}) = Ω_{1}([w_{α}, A]) for each α, no element of N_{W_{0}Aut(A,F)}(WΠ̂) can exchange the long and short roots, so Aut_{W_{0}Aut(A,F)}(WΠ̂) = 1.

Now assume (𝔾, p) = (F_{4}, 2). Let α, β ∈ Π be such that α is long, β is short, and α ⊥̸ β. Then α and β generate a root system of type B_{2}, and by the argument in the last paragraph, no element of N_{W_{0}Aut(A,F)}(WΠ̂) can exchange them. Thus no element in N_{W_{0}Aut(A,F)}(WΠ̂) can exchange the long and short roots in 𝔾, so again Aut_{W_{0}Aut(A,F)}(WΠ̂) = 1.

Case 2: Assume G is a Steinberg group. In particular, we are in case (III.1). The Coxeter diagram for the root system (V_{0}, Σ̂) has type B_{n}, C_{n}, or F_{4} (recall that we excluded the triality groups ^{3}D_{4}(q) in Hypotheses 5.1), and hence has a nontrivial automorphism only when it has type B_{2} or F_{4}. It thus suffices to consider the groups G = ^{2}A_{3}(q), ^{2}A_{4}(q), and 2E_{6}(q).

For these groups, the elements ĥ_{α}(λ) for λ ∈ 𝔽_{q}^{×}, and hence the (q − 1)-torsion in the subgroups Tα̂ for α̂ ∈ Σ̂_{+}, have relations similar to those among the corresponding subgroups of T when G = B_{2}(q) or F_{4}(q). This follows from Lemma 2.6(b): if λ ∈ 𝔽_{q}^{×} is a generator, then Φ_{λ} restricts to an isomorphism from C_{ℤΣ^{∨}}(τ)/(q − 1) to the (q − 1)-torsion in T, and the elements in Π̂ can be identified in a natural way with a basis for C_{ℤΣ^{∨}}(τ). Hence when p = 2, certain subgroups Ω_{1}(Aα̂) are equal for distinct α̂ ∈ Σ̂_{+}, proving that no element in N_{W_{0}Aut(A,F)}(WΠ̂) can exchange the two classes of roots. Thus the same argument as that used in Case 1 when (G, p) = (B_{2}(q), 2) or (F_{4}(q), 2) applies to prove that N_{W_{0}Aut(A,F)}(WΠ̂) = Aut_{sc}(A) in these cases.

Since p||W_{0}| by Hypotheses 5.1(I), we are left only with the case where p = 3 and G = 2E_{6}(q) for some q ≡ 1 (mod 3). Then (V_{0}, Σ̂) is the root system of F_{4}, so Aut(A, F) ∩ W_{0}Aut_{sc}(A) has index at most 2 in Aut(A, F) by (6) and (7). Thus (c) holds in this case. (In fact, the fusion system of G is isomorphic to that of F_{4}(q) by [BMO, Example 4.4], and does have an “exotic” graph automorphism.) □

We now look at groups which satisfy any of the cases (III.1), (III.2), or (III.3) in Hypotheses 5.1. Recall that κ_{G} = μ_{G} ∘ κ_{G}: Out(G) ⟶ Out(F).

Lemma 5.9. Assume Hypotheses 5.1 and Notation 5.2. Then each element φ ∈ Aut_{diag}(F) is the restriction of a diagonal automorphism of G. More precisely, κ_{G} restricts to an epimorphism from Outdiag(G) onto Out_{diag}(F) whose kernel is the p′-torsion subgroup. Also, C_{A}(W_{0}) = O_{p}(Z(G)).

Proof. In general, whenever H is a group and B ⊴ H is a normal abelian subgroup, we let Aut_{diag}(H, B) be the group of all φ ∈ Aut(H) such that φ|_{B} = Id_{B} and [φ, H] ≤ B, and let Out_{diag}(H, B) be the image of Aut_{diag}(H, B) in Out(H). There is a natural isomorphism Aut_{diag}(H, B)/Aut_{B}(H) ⟶_{≅}^{η_{H,B}} H^{1}(H/B; B) (cf. [Sz1, 2.8.7]), and hence H^{1}(H/B; B) sur-

jects onto Out_{diag}(H, B). If B is centric in H (if C_{H}(B) = B), then Out_{diag}(H, B) ≅ H^{1}(H/B; B) since Aut_{B}(H) = Inn(H) ∩ Aut_{diag}(H, B).

In particular, Out_{diag}(S, A) is a p-group since H^{1}(S/A; A) is a p-group. Also, C_{S}(A) = A by Lemma 5.3(a) (or by assumption in case (III.3)), and hence we have Out_{diag}(S, A) ≅ Aut_{diag}(S, A)/Aut_{A}(S). So Aut_{diag}(S, A) is a p-group, and its subgroup Aut_{diag}(F) is a p-group. It follows that

Aut_{diag}(F) ∩ Aut_{G}(S) = Aut_{diag}(F) ∩ Inn(S) = Aut_{A}(S) ,

and thus Out_{diag}(F) ≅ Aut_{diag}(F)/Aut_{A}(S).

Since Outdiag(G) = Out_{T} (G) by Proposition 3.5(c), we get κ_{G}(Outdiag(G)) ≤ Out_{diag}(F). In particular, κ_{G} sends all torsion prime to p in Outdiag(G) to the identity. It remains to show that it sends O_{p}(Outdiag(G)) isomorphically to Out_{diag}(F).

Consider the following commutative diagram of automorphism groups and cohomology groups:

\xymatrix{Out_{diag}(F) ≅ Aut_{diag}(F)/Aut_{A}(S) \ar[r]^{χ} \ar[d]^{incl} & H^{1}(Aut_{G}(A); A) \ar[d]^{ρ_{2}} \\ Out_{diag}(S, A) ≅ Aut_{diag}(S, A)/Aut_{A}(S) \ar[r]^{η_{S,A}}_{≅} & H^{1}(Aut_{S}(A); A) .}  (8)

Here, ρ_{2} is induced by restriction, and is injective by [CE, Theorem XII.10.1] and since Aut_{S}(A) ∈ Syl_{p}(Aut_{G}(A)) (since A ⊴ S ∈ Syl_{p}(G)). For each ω ∈ Aut_{diag}(F), since ω is fusion preserving, η_{S,A}([ω]) ∈ H^{1}(Aut_{S}(A); A) is stable with respect to Aut_{G}(A)- fusion, and hence by [CE, Theorem XII.10.1] is the restriction of a unique element χ([ω]) ∈ H^{1}(Aut_{G}(A); A).

The rest of the proof splits into two parts, depending on which of cases (III.1), (III.2), or (III.3) in Hypotheses 5.1 holds. Recall that Aut_{F}(A) = Aut_{G}(A) = Aut_{W_{0}}(A): the second equality by Lemma 5.3(b) in cases (III.1) or (III.2), or by assumption in case (III.3).

Cases (III.2) and (III.3): We show that in these cases, Outdiag(G), Out_{diag}(F), Z(G), and C_{A}(W_{0}) all have order prime to p. Recall that p is odd in both cases. By hypothesis in case (III.3), and since γ|_{T} ∈ O_{p′}(W_{0}) inverts T in case (III.2), C_{A}(O_{p′}(W_{0})) = 1. In particular, C_{A}(W_{0}) = 1. Since Z(G) ≤ Z(G) by Proposition 3.5(a), and Z(G) ≤ T by Lemma 2.4(a), Z(G) ≤ G ∩ C_{T} (W) ≤ C_{T} (W_{0}), so O_{p}(Z(G)) ≤ C_{A}(W_{0}) = 1. This proves the last statement.

Now, O_{p}(Outdiag(G)) = 1 since Outdiag(G) ≅ Z(G) (see [GLS3, Theorem 2.5.12(c)]) and O_{p}(Z(G)) = 1. Also,

H^{1}(Aut_{G}(A); A) = H^{1}(Aut_{W_{0}}(A); A)
≅ H^{1}(Aut_{W_{0}}(A)/Aut_{O_{p′}(W_{0})}(A); C_{A}(O_{p′}(W_{0}))) = 0

since A is a p-group and C_{A}(O_{p′}(W_{0})) = 1. Hence Out_{diag}(F) = 1 by diagram (8).

Case (III.1): Since C_{W} (A) = 1 by Lemma 5.3(a) (and since Aut_{G}(A) = Aut_{W_{0}}(A)), we can identify H^{1}(Aut_{G}(A); A) = H^{1}(W_{0}; A). Consider the following commutative diagram of automorphism groups and cohomology groups

\xymatrix{O_{p}(Outdiag(G)) \ar[r]^{R} \ar[dd]_{_{κ_{G}}} & O_{p}(Out_{diag}(N_{G}(T), T)) \ar[r]^{η_{N(T),T}}_{≅} \ar[d]_{≅}^{σ_{1}} & H^{1}(W_{0}; T)_{(p)} \ar[d]_{≅}^{σ_{2}} \\  & Out_{diag}(N, A) \ar[r]^{η_{N,A}}_{≅} \ar[d]^{ρ_{1}} & H^{1}(W_{0}; A) \ar[d]^{ρ_{2}} \\ Out_{diag}(F) \ar[r]^{incl} \ar[ur]^{χ_{0}} \ar[urr]^{χ} & Out_{diag}(S, A) \ar[r]^{η_{S,A}}_{≅} & H^{1}(S/A; A)}  (9)

where R is induced by restriction to N_{G}(T). By Lemma 5.3(a), T is centric in N_{G}(T) and A is centric in N, so the three η’s are well defined and isomorphisms (i.e., Out_{diag}(N, A) = Aut_{diag}(N, A)/Aut_{A}(N), etc.). The maps σ_{i} are induced by dividing out by O_{p′}(T), and are isomorphisms since A = O_{p}(T). The maps ρ_{i} are induced by restriction, and are injective since S/A ∈ Syl_{p}(W_{0}) (see [CE, Theorem XII.10.1]).

Consider the short exact sequence

1 ⟶ T ⟶ T ⟶^{Ψ} T ⟶ 1,

where Ψ(t) = t^{−1} · γψ_{q}(t) = t^{−1}γ(t^{q}) for t ∈ T. Let

1 ⟶ C_{T} (W_{0}) ⟶ C_{T} (W_{0}) ⟶^{Ψ_{∗}} C_{T} (W_{0}) ⟶^{δ} H^{1}(W_{0}; T) ⟶^{θ} H^{1}(W_{0}; T)  (10)

be the induced cohomology exact sequence for the W_{0}-action, and recall that H^{1}(W_{0}; A) ≅ H^{1}(W_{0}; T)_{(p)} by (9). We claim that

(11) |O_{p}(Outdiag(G))| = |Im(δ)_{(p)}| = |O_{p}(Z(G))| = |C_{A}(W_{0})|;

(12) R is injective; and

(13) χ(Out_{diag}(F)) ≤ Ker(θ).

These three points will be shown below. It then follows from the commutativity of diagram (9) (and since Im(δ) = Ker(θ)) that κ_{G} sends O_{p}(Outdiag(G)) isomorphically onto Out_{diag}(F).

Proof of (11) and (12): Assume first that γ ≠ Id and 𝔾 ≅ SL_{2n−1} (some n ≥ 1). Thus G ≅ SU_{2n−1}(q). By [St1, 3.4], Outdiag(G) and Z(G) are cyclic of order (q + 1, 2n − 1), and hence have no p-torsion (recall p|(q − 1)). By Lemma 5.4(b), C_{T} (W_{0}) ≅ 𝔽_{q_{0}}^{×}, and σ(u) = u^{−q} for u ∈ C_{T} (W_{0}). Thus Ψ_{∗}(u) = u^{−1}σ(u) = u^{−1−q} for u ∈ C_{T} (W_{0}), so Ψ_{∗} is onto, and Im(δ) = 1 ≅ O_{p}(Outdiag(G)) in this case. Also, C_{T} (W_{0}) = Ker(Ψ_{∗}) has order q + 1, so C_{A}(W_{0}) = O_{p}(C_{T} (W_{0})) = 1.

Now assume γ = Id or 𝔾 ≠ SL_{2n−1}. By Lemma 5.4, in all such cases,

C_{T} (W_{0}) = C_{T} (W) = Z(G) and C_{T} (W_{0}) = Z(G) .  (14)

In particular, these groups are all finite, and hence |Im(δ)| = |Z(G)| by the exactness of (10). By [GLS3, Theorem 2.5.12(c)], Outdiag(G) ≅ Z(G) in all cases, and hence |Outdiag(G)| = |Im(δ)|.

If [φ] ∈ Ker(R), then we can assume that it is the class of φ ∈ Aut_{T} (G). Thus φ = c_{x} for some x ∈ N_{T} (G), and φ|_{N_{G}(T)} = c_{y} for some y ∈ N_{G}(T) which centralizes A. Then y ∈ C_{G}(A) = T by Lemma 5.3(a), and upon replacing φ by c_{y}^{−1} ∘ φ and x by y^{−1}x (without changing the class [φ]), we can arrange that φ|_{N_{G}(T)} = Id. Then x ∈ C_{T} (W_{0}) since it centralizes N_{G}(T) (and since N_{G}(T)/T ≅ W_{0} by Lemma 5.3(b)), so x ∈ Z(G) by (14), and hence φ = Id_{G}. Thus R is injective.

Proof of (13): Fix φ ∈ Aut_{diag}(F). Choose φ ∈ Aut_{diag}(N, A) such that φ|_{S} = φ (i.e., such that [φ] = χ_{0}([φ]) in diagram (9)). Recall that W_{0} ≅ N/A by Lemma 5.3(b). Let \underline{c}: W_{0} ≅ N/A ⟶ A be such that φ(g) = \underline{c}(gA)·g for each g ∈ N; thus η_{N,A}([φ]) = [\underline{c}]. We must show that θ([\underline{c}]) = 1: that this is a consequence of φ being fusion preserving.

For each α̂ ∈ Π̂, set uα̂ = \underline{c}(wα̂). Thus for g ∈ N, φ(g) = uα̂g if g ∈ wα̂ (as a coset of A in N). Since w2 ; α̂ = 1, g^{2} = φ(g^{2}) = (uα̂g)^{2}, and hence wα̂(uα̂) = u−1 ; α̂. We claim that uα̂ ∈ Aα̂ = A ∩ Kα̂ for each α̂ ∈ Π̂.

• If p is odd, then uα̂ ∈ Aα̂, since Aα̂ = {a ∈ A | wα̂(a) = a^{−1}} by Lemma 2.4(e).

• If p = 2, wα̂ ∈ S/A, and |α̂| ≤ 2, choose gα̂ ∈ S ∩ Kα̂ such that wα̂ = gα̂A. (For example, if we set g = ∏_{α∈α̂} n_{α}(1) (see Notation 2.2(B)), then g ∈ N_{G}(T) represents the class wα̂ ∈ W_{0}, and is T -conjugate to an element of S ∩ Kα̂.) By Lemma 5.7(c), C_{G}(C_{A}(wα̂)) = G ∩ T Kα̂, where G ∩ Kα̂ ≅ SL_{2}(q) or SL_{2}(q^{2}) by Lemma 5.5. Hence

𝔣𝔬𝔠(C_{F}(C_{A}(wα̂))) = 𝔣𝔬𝔠(C_{G}(C_{A}(wα̂))) = S ∩ [G ∩ T Kα̂, G ∩ T Kα̂] = S ∩ Kα̂

(see the remarks before Lemma 5.7), and gα̂ lies in this subgroup. Since φ is fusion preserving, φ(gα̂) ∈ 𝔣𝔬𝔠(C_{F}(C_{A}(wα̂))). By Lemma 5.7(c) again,

uα̂ = φ(gα̂) · g−1 ; α̂ ∈ A ∩ 𝔣𝔬𝔠(C_{F}(C_{A}(wα̂))) = Aα̂ .

• If p = 2, wα̂ ∈ S/A, and α̂ = {α, τ(α), α^{∗}} where α^{∗} = α + τ(α), then wα̂ = w_{α^{∗}}. Choose gα̂ ∈ S ∩ K_{α^{∗}} such that gα̂A = wα̂ ∈ N/A. (For example, there is such a gα̂ which is T - conjugate to n_{α^{∗}}(1).) By Lemma 5.7(c), C_{G}(C_{A}(wα̂)) = G∩T K_{α^{∗}}, G∩K_{α^{∗}} ≅ SL_{2}(q), and hence gα̂ ∈ 𝔣𝔬𝔠(C_{F}(C_{A}(wα̂))). So φ(gα̂) ∈ 𝔣𝔬𝔠(C_{F}(C_{A}(wα̂))) since φ|_{S} is fusion preserving. By Lemma 5.7(c),

uα̂ = φ(gα̂) · g−1 ; α̂ ∈ A ∩ 𝔣𝔬𝔠(C_{F}(C_{A}(wα̂))) = A ∩ K_{α^{∗}} ≤ Aα̂ .

• If p = 2 and wα̂ ∉ S/A ∈ Syl_{2}(W_{0}), then it is W_{0}-conjugate to some other reflection wβ̂ ∈ S/A (for β̂ ∈ Σ̂_{+}), \underline{c}(wβ̂) ∈ Aβ̂ by the above argument, and hence uα̂ = \underline{c}(wα̂) ∈ Aα̂.

Consider the homomorphism

Φ = (Φ_{α})_{α∈Π} : T ⟶ ∏_{α∈Π} T_{α} where Φ_{α}(t) = t^{−1}w_{α}(t) ∀ t ∈ T , α ∈ Π.

Since W = ⟨w_{α} | α ∈ Π⟩, we have Ker(Φ) = C_{T} (W) = Z(G) is finite (Proposition 2.5). Thus Φ is (isomorphic to) a homomorphism from (𝔽_{q_{0}}^{×})^{r} to itself with finite kernel (where r = |Π|), and any such homomorphism is surjective since 𝔽_{q_{0}}^{×} has no subgroups of finite index.

Choose elements v_{α} ∈ T_{α} for α ∈ Π as follows.

• If α̂ = {α} where τ(α) = α, we set v_{α} = uα̂.

• If α̂ = {α, τ(α)}, where α ⊥ τ(α), then T α̂ = T_{α} × T_{τ(α)}, and we let v_{α}, v_{τ(α)} be such that v_{α}v_{τ(α)} = uα̂.

• If α̂ = {α, τ(α), α^{∗}} where α^{∗} = α + τ(α), then uα̂ = h_{α}(λ)h_{τ(α)}(λ′) for some λ, λ′ ∈ 𝔽_{q_{0}}^{×},

wα̂(h_{α}(λ)h_{τ(α)}(λ′)) = h_{α}(λ′^{−1})h_{τ(α)}(λ^{−1})

by Lemma 2.4(e), and λ = λ′ since wα̂(uα̂) = u−1 ; α̂. Set v_{α} = h_{α}(λ) and v_{τ(α)} = 1. (This depends on the choice of α ∈ α̂ ∩ Π.)

Let t ∈ T be such that Φ(t) = (v_{α})_{α∈Π}. We claim that t^{−1}wα̂(t) = uα̂ for each α̂ ∈ Π̂. This is clear when |α̂| ≤ 2. If α̂ = {α, τ(α), α^{∗}} and λ are as above, then

wα̂(t) = w_{α^{∗}}(t) = w_{τ(α)}w_{α}w_{τ(α)}(t) = w_{τ(α)}(w_{α}(t)) = w_{τ(α)}(t · h_{α}(λ))
= t · w_{τ(α)}(h_{α}(λ)) = t · h_{α^{∗}}(λ) = t · uα̂.

Thus \underline{c}(wα̂) = dt(wα̂) for each α̂ ∈ Π̂. Since W_{0} = ⟨wα̂ | α̂ ∈ Π̂⟩ (and since \underline{c} and dt are both cocycles), this implies that \underline{c} = dt, and hence that [\underline{c}] = 0 in H^{1}(W_{0}; T). □

As one consequence of Lemma 5.9, the Z^{∗}-theorem holds for these groups. This is known to hold for all finite groups (see [GLS3, § 7.8]), but its proof for odd p depends on the classification of finite simple groups, which we prefer not to assume here.

Corollary 5.10. Assume that G ∈ 𝔏𝔦𝔢(q_{0}), p ≠ q_{0}, and S ∈ Syl_{p}(G) satisfy Hypotheses 5.1. Then Z(F_{S}(G)) = O_{p}(Z(G)).

Proof. By Lemma 5.9, O_{p}(Z(G)) = C_{A}(W_{0}). By Lemma 5.3(a,b), or by hypothesis in Case 5.1(III.3), C_{S}(A) = A and Aut_{G}(A) = Aut_{W_{0}}(A). Hence Z(F_{S}(G)) ≤ O_{p}(Z(G)), while the other inclusion is clear. □

We now need the following additional hypotheses, in order to be able to compare Aut_{sc}(A) with the group of field automorphisms of G. With the help of Lemma 1.11, we will see in Section 6 that we can always arrange for them to hold.

Hypotheses 5.11. Fix a prime p and a prime power q. Assume that q = q_{0}^{b} where q_{0} is prime, b ≥ 1, q_{0} ≠ p, and

(i) q_{0} ≡ ±3 (mod 8) if p = 2;

(ii) the class of q_{0} generates (ℤ/p^{2})^{×} if p is odd; and

(iii) b|(p − 1)p^{ℓ} for some ℓ ≥ 0.

We will also say that “G satisfies Hypotheses 5.11” (for a given prime p) if G ≅ ^{t}𝔾(q) for some t and 𝔾, and some q which satisfies the above conditions.

By Hypothesis 5.1(I), ψ_{q_{0}}(G) = G, and thus all field endomorphisms of G normalize G. When G has a standard σ-setup, Φ_{G} was defined to be the group of restrictions of such endomorphisms ψ_{q_{0}^{a}} ∈ Φ_{G} for a ≥ 0. Under our Hypotheses 5.1, this applies only when we are in case (III.1) (although Proposition 3.6 describes the relation between Φ_{G} and ψ_{q_{0}} in the other cases). In what follows, it will be useful to set

Φ̂_{G} = ⟨ψ_{q_{0}}|_{G}⟩ ≤ Aut(G).

By Proposition 3.6(d), Inndiag(G)Φ̂_{G} = Inndiag(G)Φ_{G}. However, Φ̂_{G} can be strictly larger than Φ_{G}, and Φ̂_{G} ∩ Inndiag(G) need not be trivial. For example, if G = SL_{n}(q) where p does not divide q − 1, then there is a σ-setup with σ = c_{x}ψ_{q} for some x ∈ N_{G}(T) that satisfies Hypotheses 5.1 (see Lemma 6.5), and ψ_{q}|_{G} = c_{x}^{−1}|_{G} ∈ Inndiag(G). Note that since each element of Φ̂_{G} acts on T via (t ↦ t^{r}) for some r, Φ̂_{G} normalizes T and each of its subgroups.

Recall that τ ∈ Aut(V) is the automorphism induced by σ, and also denotes the induced permutation of Σ.

Lemma 5.12. Assume Hypotheses 5.1 and 5.11 and Notation 5.2. Let

χ_{0} : Φ̂_{G} ⟶ Aut(A, F)

be the homomorphism induced by restriction from G to A. Set m = |τ| = |γ|_{T} |. Then the following hold.

(a) Either T has exponent q^{m} − 1; or p is odd, m = ord_{p}(q), m is even, and (q^{m/2} + 1)|expt(T)|(q^{m} − 1).

(b) If p is odd, then χ_{0}(Φ̂_{G}) = Aut_{sc}(A). If p = 2, then χ_{0}(Φ̂_{G}) has index 2 in Aut_{sc}(A), and Aut_{sc}(A) = Im(χ_{0})⟨ψ_{−1}^{A}⟩.

(c) If p = 2, then χ_{0} is injective. If p is odd, then

Ker(χ_{0}) = \begin{cases}⟨ψ_{q}|_{G}⟩ = ⟨γ|_{G}⟩ & in case (III.1) \\ ⟨(ψ_{q}|_{G})^{m}⟩ = ⟨γ^{m}|_{G}⟩ = Φ̂_{G} ∩ Aut_{T} (G) & in cases (III.2) and (III.3). \end{cases}

Proof. We first recall some of the assumptions in cases (III.1–3) of Hypotheses 5.1:

case (III.1) ord_{p}(q) = 1, m = |γ|, and m ≤ 2
case (III.2) ord_{p}(q) = m = 2  p is odd
case (III.3) ord_{p}(q) = m  p is odd

(15)

(Recall that γ is a graph automorphsm in case (III.1), so |γ| = |τ| = m.) In all of these cases, p|(q^{m} − 1) since ord_{p}(q)|m.

(a) For each t ∈ T = C_{T} (ψ_{q} ∘ γ), t^{q} = ψ_{q}(t) = γ^{−1}(t). Hence t = γ^{−m}(t) = (ψ_{q})^{m}(t) = t^{q^{m}}, and t^{q^{m}−1} = 1. Thus expt(T)|(q^{m} − 1).

By Hypotheses 5.1(I), there is a linearly independent subset Ω = {α_{1}, …, α_{s}} ⊆ Σ such that either Ω or ±Ω = {±α_{1}, …, ±α_{s}} is a free ⟨τ⟩-orbit in Σ. Assume Ω is a free orbit (this always happens in case (III.1)). In particular, m = |τ| = s. For each 1 ≠ λ ∈ 𝔽_{q_{0}}^{×} such that λ^{q^{s}−1} = 1, the element

t(λ) = ∏_{i=0}^{m−1} h_{τ^{i}(α_{1})}(λ^{q^{i}})

is fixed by σ = ψ_{q} ∘ γ (recall σ(h_{β}(λ)) = h_{τ(β)}(λ^{q}) for each β ∈ Σ by Lemma 3.2). Hence t(λ) ∈ T, and t(λ) ≠ 1 when λ ≠ 1 by Lemma 2.4(d,b). Thus T contains the subgroup {t(λ) | λ^{q^{m}−1} = 1} of order q^{m} − 1, this subgroup is cyclic (isomorphic to a subgroup of 𝔽_{q_{0}}^{×}), and hence expt(T) = q^{m} − 1.

Assume now that ±Ω is a free ⟨τ⟩-orbit (thus m = |τ| = 2s). In particular, we are not in case (III.1), so p is odd and m = ord_{p}(q). Then τ^{i}(α_{1}) = −α_{1} for some 0 < i < 2s, and i = s since τ^{2i}(α_{1}) = α_{1}. For each 1 ≠ λ ∈ 𝔽_{q_{0}}^{×} such that λ^{q^{s}+1} = 1,

t(λ) = ∏_{i=0}^{s−1} h_{τ^{i}(α_{1})}(λ^{q^{i}})

is fixed by σ = ψ_{q} ∘ γ by Lemma 3.2 and since h_{τ^{s}(α_{1})}(λ^{q^{s}}) = h_{−α_{1}}(λ^{−1}) = h_{α_{1}}(λ). Hence t(λ) ∈ T, and t(λ) ≠ 1 when λ ≠ 1 by Lemma 2.4 again. Thus {t(λ) | λ^{q^{e}+1} = 1} ≤ T is cyclic of order q^{s} + 1, and so (q^{s} + 1)|expt(T).

(b) By definition, Im(χ_{0}) = χ_{0}(Φ̂_{G}) is generated by χ_{0}(ψ_{q_{0}}) = ψ_{q_{0}}|_{A}, which acts on A via (a ↦ a^{q_{0}}). If p is odd, then by Hypotheses 5.11(ii), the class of q_{0} generates (ℤ/p^{2})^{×}, and hence generates (ℤ/p^{k})^{×} for each k > 0. So Im(χ_{0}) = Aut_{sc}(A) in this case.

If p = 2, then q_{0} ≡ ±3 (mod 8) by Hypotheses 5.11(i). So for each k ≥ 2, ⟨q_{0}⟩ has index 2 in (ℤ/2^{k})^{×} = ⟨q_{0}, −1⟩. Hence Im(χ_{0}) = ⟨ψ_{q_{0}}|_{A}⟩ has index 2 in Aut_{sc}(A) = ⟨ψ_{q_{0}}|_{A}, ψ_{−1}^{A}⟩.

(c) Set ϕ_{0} = ψ_{q_{0}}|_{G}, a generator of Φ̂_{G}. Then (ϕ_{0})^{b} = ψ_{q}|_{G} = (γ|_{G})^{−1} since G = C_{G}(ψ_{q} ∘ γ), and so |ϕ_{0}|_{T} | divides b|γ|_{T} | = bm. Also, (ϕ_{0})^{bm} = (γ|_{G})^{−m} ∈ Aut_{T} (G) by Lemma 3.2.

By (a), either expt(T) = q^{m} − 1; or m is even, p is odd, ord_{p}(q) = m, and (q^{m/2} + 1)|expt(T)|(q^{m} − 1). In the latter case, v_{p}(q^{m/2} + 1) = v_{p}(q^{m} − 1) > 0 since p ∤ (q^{m/2} − 1). Thus

expt(A) = p^{e} where e = v_{p}(q^{m} − 1) = v_{p}(q_{0}^{bm} − 1) > 0 .  (16)

If p = 2, then we are in case (III.1). In particular, q = q_{0}^{b} ≡ 1 (mod 4), and m ≤ 2. Also, b (and hence bm) is a power of 2 by Hypotheses 5.11(iii). If bm = 1, then q = q_{0} ≡ 5 (mod 8), so e = v_{2}(q−1) = 2. If bm is even, then e = v_{2}(q_{0}^{bm}−1) = v_{2}(q_{0}^{2}−1)+v_{2}(bm/2) = 3+v_{2}(bm/2) by Lemma 1.13. Thus in all cases, e = 2 + v_{2}(bm). So Im(χ_{0}) ≤ Aut_{sc}(A) ≅ (ℤ/2^{e})^{×} has order 2^{e−2} = bm. Since (ψ_{q_{0}}|_{G})^{bm} = (ψ_{q}|_{G})^{m} = (γ^{−1}|_{G})^{m} = Id_{G} (recall m = |γ| in case (III.1)), χ_{0} is injective.

Now assume p is odd, and set m_{0} = ord_{p}(q). Then b|(p−1)p^{ℓ} for some ℓ ≥ 0 by Hypotheses 5.11(iii), and q = q_{0}^{b} where the class of q_{0} generates (ℤ/p^{k})^{×} for each k ≥ 1. For r ∈ ℤ, q^{r} = q_{0}^{br} ≡ 1 (mod p) if and only if (p − 1)|br. Hence bm_{0} = b · ord_{p}(q) = (p − 1)p^{ℓ} for some ℓ ≥ 0. Since v_{p}(q_{0}^{p−1} − 1) = 1, and since m = m_{0} or 2m_{0}, Lemma 1.13 implies that

e = v_{p}(q^{m} − 1) = v_{p}(q_{0}^{bm} − 1) = v_{p}(q_{0}^{bm_{0}} − 1) = 1 + v_{p}(p^{ℓ}) = 1 + ℓ .

Thus ℓ = e − 1, where p^{e} = expt(A) by (16), so |Aut_{sc}(A)| = (p − 1)p^{e−1} = bm_{0}. Since χ_{0} sends the generator ϕ_{0} of Φ̂_{G} to the generator χ_{0}(ϕ_{0}) of Aut_{sc}(A), this proves that Ker(χ_{0}) = ⟨ψ_{q}^{m_{0}}|_{G}⟩ = ⟨γ^{m_{0}}|_{G}⟩. The descriptions in the different cases now follow immediately. Note that in cases (III.2) and (III.3) (where m = m_{0}), ϕ_{0}^{bm} = γ^{−m}|_{G} ∈ Aut_{T} (G) by Lemma 3.2. The converse is immediate: Φ̂_{G} ∩ Aut_{T} (G) ≤ Ker(χ_{0}). □

Before applying these results to describe Out(F) and the homomorphism κ_{G}, we need to know in which cases the subgroup A is characteristic in S.

Proposition 5.13. Assume Hypotheses 5.1 and Notation 5.2.

(a) If p = 2, then A is characteristic in S, and is the unique abelian subgroup of S of order |A|, except when q ≡ 5 (mod 8) and G ≅ Sp_{2n}(q) for some n ≥ 1.

(b) If p is odd, then A is characteristic in S, and Ω_{1}(A) is the unique elementary abelian subgroup of S of maximal rank, except when p = 3, q ≡ 1 (mod 3), v_{3}(q − 1) = 1, and G ≅ SU_{3}(q) or G_{2}(q).

In all cases, each normal subgroup of S isomorphic to A is N_{G}(S)-conjugate to A.

Proof. If p is odd, then by [GL, 10-2(1,2)], there is a unique elementary p-subgroup E ≤ S of rank equal to that of A (denoted r_{m_{0}} in [GL]), except when p = 3 and G is isomorphic to one of the groups SL_{3}(q) (q ≡ 1 (mod 3)), SU_{3}(q) (q ≡ −1 (mod 3)), or G_{2}(q), ^{3}D_{4}(q), or ^{2}F_{4}(q) (q ≡ ±1 (mod 3)). When there is a unique such subgroup E, then A = C_{S}(E) by Lemma 5.3(a) (or by assumption in case (III.3)), and hence A is characteristic in S.

Among the exceptions, SL_{3}(q) and G_{2}(q) are the only ones which satisfy Hypotheses 5.1. In both cases, S is an extension of A ≅ (C_{3^{ℓ}})^{2} by C_{3}, where ℓ = v_{3}(q − 1), and where Z(S) = C_{A}(S) has order 3. If ℓ > 1, then A is the unique abelian subgroup of index p in S. If ℓ = 1, then S is extraspecial of order 3^{3} and exponent 3. By Theorem 1.8(a), we can assume q = 4 without changing the isomorphism type of the fusion system, so G contains SU_{3}(2). This is a semidirect product S ⋊ Q_{8} (cf. [Ta, p. 123–124]), and hence the four subgroups of S of order 9 are N_{G}(S)-conjugate.

It remains to prove the proposition when p = 2. We use [O3, § 2] as a reference for information about best offenders, since this contains what we need in a brief presentation. Assume A is not the unique abelian subgroup of S of order |A|. Then there is an abelian subgroup 1 ≠ B ≤ W_{0} such that |B| · |C_{A}(B)| ≥ |A|. In other words, the action of the Weyl group W_{0} on A has a nontrivial best offender [O3, Definition 2.1(b)]. Hence by Timmesfeld’s replacement theorem [O3, Theorem 2.5], there is a quadratic best offender 1 ≠ B ≤ W_{0}: an offender such that [B, [B, A]] = 1.

We consider three different cases.

Case 1: G ≅ 𝔾(q) is a Chevalley group, where either q ≡ 1 (mod 8), or G ≅ Sp_{2n}(q) for any n ≥ 1. Set n = rk(A) = rk(T): the Lie rank of G (or of 𝔾). Set ℓ = v_{2}(q − 1) ≥ 2. Then A ≅ (C_{2^{ℓ}})^{n} is the group of all 2^{ℓ}-torsion elements in T (or in T). Since the result is clear when n = 1 (G ≅ SL_{2}(q) ≅ Sp_{2}(q), A ≅ C_{2^{ℓ}}, and S ≅ Q_{2^{ℓ+1}}), we assume n ≥ 2.

Let Λ = ℤΣ^{∨} be the lattice in V generated by the dual roots. By Lemma 2.6(b), there are ℤ[W]-linear isomorphisms A ≅ Λ/2^{ℓ}Λ and Ω_{1}(A) ≅ Λ/2Λ.

Assume first that B acts faithfully on Ω_{1}(A). Since B has quadratic action, it is elementary abelian [O3, Lemma 2.4]. Set k = rk(B); thus B ≅ C_{2}^{k} and |A/C_{A}(B)| ≤ 2^{k}.

Since the B-action on V is faithful, the characters χ ∈ Hom(B, {±1}) which have nontrivial eigenspace on V generate the dual group B^{∗}. So we can choose a basis χ_{1}, …, χ_{k} for B^{∗} such that each χ_{i} has nontrivial eigenspace. Let b ∈ B be the unique element such that χ_{i}(b) = −1 for each i = 1, …, k. Let V_{+}, V_{−} be the ±1-eigenspaces for the b-action on V, and set Λ_{±} = Λ ∩ V_{±}. By construction, dim(V_{−}) ≥ k.

Let v ∈ Λ be an element whose class modulo 2^{ℓ}Λ is fixed by b, and write v = v_{+} + v_{−} where v_{±} ∈ V_{±}. Then 2v_{−} = v − b(v) ∈ 2^{ℓ}Λ ∩ V_{−} = 2^{ℓ}Λ_{−}, so v_{−} ∈ 2^{ℓ−1}Λ_{−} and v_{+} = v − v_{−} ∈ Λ ∩ V_{+} = Λ_{+}. Thus C_{Λ/2^{ℓ}Λ}(b) = (Λ_{+} × 2^{ℓ−1}Λ_{−})/2^{ℓ}Λ. Set r = rk(Λ_{−}) = dim(V_{−}) ≥ k; then

2^{k} ≥ |A/C_{A}(B)| ≥ |A/C_{A}(b)| = |Λ/(Λ_{+} × 2^{ℓ−1}Λ_{−})| = 2^{r(ℓ−1)} · |Λ/(Λ_{+} × Λ_{−})|
≥ 2^{k(ℓ−1)} · |Λ/(Λ_{+} × Λ_{−})|.

In particular, Λ = Λ_{+} × Λ_{−}. But then b acts trivially on Λ/2Λ, hence on Ω_{1}(A), which contradicts our assumption.

Thus B does not act faithfully on Ω_{1}(A). Set B_{0} = C_{B}(Ω_{1}(A)) ≅ C_{B}(Λ/2Λ) ≠ 1. If −Id_{V} ∈ B_{0}, then it inverts A, [B, Ω_{1}(A)] ≤ [B, [B_{0}, A]] = 1 since B acts quadratically, so B = B_{0}, and |B_{0}| ≥ |A/C_{A}(B)| ≥ |A/Ω_{1}(A)| = 2^{(ℓ−1)n}. If b ∈ B_{0} is such that b^{2} = −Id_{V}, then b defines a ℂ-vector space structure on V, and hence does not induce the identity on Λ/2Λ, a contradiction.

Thus there is b ∈ B_{0} which does not act on V via ±Id. Let V_{±} ≠ 0 be the ±1-eigenspaces for the b-action on V, and set Λ_{±} = Λ∩V_{±}. For each v ∈ Λ, v−b(v) ∈ 2Λ since b acts trivially on Ω_{1}(A) ≅ Λ/2Λ. Set v = v_{+} + v_{−}, where v_{±} ∈ V_{±}. Then 2v_{−} = v − b(v) ∈ 2Λ ∩ V_{−} = 2Λ_{−} implies that v_{−} ∈ Λ_{−}, and hence v_{+} ∈ Λ_{+}. Thus v ∈ Λ_{+} × Λ_{−}, so by Lemma 2.8, 𝔾 ≅ C_{n}. By assumption, q ≡ 1 (mod 8), so ℓ ≥ 3, and [b, [b, Λ/2^{ℓ}Λ]] ≥ 4Λ_{−}/2^{ℓ}Λ_{−} ≠ 1, contradicting the assumption that B acts quadratically on A.

Case 2: G ≅ Sp_{2n}(q) for some n ≥ 1 and some q ≡ 5 (mod 8). Fix subgroups H_{i} ≤ G (1 ≤ i ≤ n) and K < G such that H_{i} ≅ Sp_{2}(q) for each i, K ≅ Σ_{n} is the group of permutation matrices (in 2 × 2 blocks), and K normalizes H = H_{1} × ⋯ × H_{n} and permutes the factors in the obvious way. We can also fix isomorphisms χ_{i}: H_{i} ⟶^{≅} Sp_{2}(q) such that the action of K on the H_{i} commutes with the χ_{i}.

Fix subgroups Â < Q̂ < Sp_{2}(q), where Q̂ ≅ Q_{8} (a Sylow 2-subgroup), and Â ≅ C_{4} is contained in the maximal torus. Set Q_{i} = χ_{i}^{−1}(Q̂) and A_{i} = χ_{i}^{−1}(Â), and set Q = Q_{1}Q_{2} ⋯ Q_{n} and A = A_{1}A_{2} ⋯ A_{n}. Thus A = O_{2}(T) is as in Hypotheses 5.1(III): the 2power torsion in the maximal torus of G. By [CF, § I], S = QR for some R ∈ Syl_{2}(K). Also, W ≅ QK/A ≅ C_{2} ≀ Σ_{n} acts on A via signed permutations of the coordinates.

Let B be any nontrivial best offender in W on A. Consider the action of B on the set {1, 2, …, n}, let X_{1}, …, X_{k} be the set of orbits, and set d_{i} = |X_{i}|. For 1 ≤ i ≤ k, let A_{i} ≤ A be the subgroup of elements whose coordinates vanish except for those in positions in X_{i}; thus A_{i} ≅ (C_{4})^{d_{i}} and A = A_{1} × ⋯ × A_{k}. Set B_{i} = B/C_{B}(A_{i}); then |B| ≤ ∏_{i=1}^{k} |B_{i}|. Since B is abelian, either |B_{i}| = d_{i} and B_{i} permutes the coordinates freely, or |B_{i}| = 2d_{i} and there is a unique involution in B_{i} which inverts all coordinates in A_{i}. In the first case, |C_{A_{i}}(B_{i})| = 4, and so |B_{i}| · |C_{A_{i}}(B_{i})| = d_{i} · 4 ≤ 4^{d_{i}} = |A_{i}| with equality only if d_{i} = 1. In the second case, |C_{A_{i}}(B_{i})| = 2, and again |B_{i}| · |C_{A_{i}}(B_{i})| = 2d_{i} · 2 ≤ 4^{d_{i}} = |A_{i}| with equality only if d_{i} = 1. Since

∏_{i=1}^{k} |A_{i}| = |A| ≤ |B| · |C_{A}(B)| = |B| · ∏_{i=1}^{k} |C_{A_{i}}(B_{i})| ≤ ∏_{i=1}^{k}(|B_{i}| · |C_{A_{i}}(B_{i})|) ,

we conclude that d_{i} = 1 for all i, and hence that B acts only by changing signs in certain coordinates.

For each 1 ≤ i ≤ n, let pr_{i}: Q ⟶ Q_{i} be the projection onto the i-th factor. If A^{∗} ≤ S is abelian of order 4^{n}, then A^{∗}A/A is a best offender in W on A, and hence A^{∗} ≤ Q by the last paragraph. Also, pr_{i}(A^{∗}) is cyclic of order at most 4 for each i, and since |A^{∗}| = 4^{n}, pr_{i}(A^{∗}) ≅ C_{4} for each i and A^{∗} = ∏_{i=1}^{n} pr_{i}(A^{∗}). Thus there are exactly 3^{n} such subgroups.

Now assume A^{∗} ⊴ S, and set A_{i}^{∗} = pr_{i}(A^{∗}) ≤ Q_{i} for short. Since A^{∗} is normal, the subgroups χ_{i}(A_{i}^{∗}) ≤ Q̂ < Sp_{2}(q) are equal for all i lying in any R-orbit of the set {1, 2, …, n}. Hence we can choose elements x_{1}, x_{2}, …, x_{n}, where x_{i} ∈ N_{H_{i}}(Q_{i}) ≅ SL_{2}(3) and ^{x_{i}}(A_{i}) = A_{i}^{∗} for each i, and such that χ_{i}(x_{i}) ∈ Sp_{2}(q) is constant on each R-orbit. Set x = x_{1}x_{2} ⋯ x_{n}; then ^{x}A = A^{∗}, and x ∈ N_{G}(S).

Case 3: G is a Steinberg group. Assume γ ∈ Γ_{G} is a graph automorphism of order 2, and that G = C_{G}(σ) where σ = γψ_{q}. Set G_{0} = C_{G}(γ, ψ_{q}); thus G_{0} ≤ G. Set ℓ = v_{2}(q−1) ≥ 2. We must again show that the action of W_{0} on A has no nontrivial best offenders.

If G ≅ ^{2}E_{6}(q) or Spin_{2n}^{−}(q) (n ≥ 4), then G_{0} ≅ F_{4}(q) or Spin_{2n−1}(q), respectively, and W_{0} is the Weyl group of G_{0}. If 1 ≠ B ≤ W_{0} is a best offender in W_{0} on A, then it is also a best offender on Ω_{ℓ}(A) ≤ G_{0} (see [O3, Lemma 2.2(a)]), which is impossible by Case 1.

If G ≅ SU_{2n+1}(q) ≅ ^{2}A_{2n}(q), then S ≅ (SD_{2^{ℓ+2}})^{n} ⋊R for some R ∈ Syl_{2}(Σ_{n}) [CF, pp. 143– 144]. Thus A ≅ (C_{2^{ℓ+1}})^{n}, W_{0} ≅ C_{2} ≀ Σ_{n}, Σ_{n} < W_{0} acts on A by permuting the coordinates, and the subgroup W_{1} ≅ (C_{2})^{n} in W_{0} has a basis each element of which acts on one coordinate by (a ↦ a^{2^{ℓ}−1}). If B ≤ W_{0} is a nontrivial quadratic best offender on A, then it is also a best offender on Ω_{ℓ}(A) [O3, Lemma 2.2(a)], hence is contained in W_{1} by the argument in Case 2, which is impossible since no nontrivial element in this subgroup acts quadratically. Thus A is characteristic in this case.

It remains to consider the case where G ≅ SU_{2n}(q) ≅ ^{2}A_{2n−1}(q). Since the case SU_{2}(q) ≅ Sp_{2}(q) has already been handled, we can assume n ≥ 2. Set Ĝ = GU_{2n}(q) > G, set G_{0} = GU_{2}(q) × ⋯ × GU_{2}(q) ≤ Ĝ, and set G_{1} = NĜ(G_{0}) ≅ GU_{2}(q) ≀ Σ_{n}. Then G_{1} has odd index in Ĝ [CF, pp. 143–144], so we can assume S ≤ G_{1} ∩ G. Fix H_{0} ∈ Syl_{2}(G_{0}); thus H_{0} ≅ (SD_{2^{ℓ+2}})^{n}. Since v_{2}(q + 1) = 1, and since the Sylow 2-subgroups of SU_{2}(q) are quaternion,

G ∩ H_{0} = Ker[H_{0} ≅ (SD_{2^{ℓ+2}})^{n} ⟶^{χ^{n}} C_{2}^{n} ⟶^{sum} C_{2}] ,

where χ: SD_{2^{ℓ+2}} ⟶ C_{2} is the surjection with quaternion kernel. As in the last case, W_{0} ≅ C_{2} ≀ Σ_{n} with normal subgroup W_{1} ≅ C_{2}^{n}. If B ≤ W_{0} is a nontrivial quadratic best offender on A, then it is also a best offender on Ω_{ℓ}(A) [O3, Lemma 2.2(a)], so B ≤ W_{1} by the argument used in Case 2. Since no nontrivial element in W_{1} acts quadratically on A, we conclude that A is characteristic in this case. □

The next lemma is needed to deal with the fact that not all restrictions to A of automorphisms of G lie in Aut(A, F) (since they need not normalize S).

Lemma 5.14. Let G be any finite group, fix S ∈ Syl_{p}(G), and let S_{0} ⊴ S be a normal subgroup. Let φ ∈ Aut(G) be such that φ(S_{0}) = S_{0} and φ|_{S_{0}} ∈ N_{Aut(S_{0})}(Aut_{S}(S_{0})). Then there is φ′ ∈ Aut(G) such that φ′|_{S_{0}} = φ|_{S_{0}}, φ′(S) = S, and φ′ ≡ φ (mod Inn(G)).

Proof. Since φ|_{S_{0}} normalizes Aut_{S}(S_{0}), and c_{φ(g)} = φc_{g}φ^{−1} for each g ∈ G, we have Aut_{φ(S)}(S_{0}) = ^{φ}Aut_{S}(S_{0}) = Aut_{S}(S_{0}). Hence φ(S) ≤ C_{G}(S_{0})S. Since S normalizes C_{G}(S_{0}) and S ∈ Syl_{p}(C_{G}(S_{0})S), we have φ(S) = ^{x}S for some x ∈ C_{G}(S_{0}). Set φ′ = c_{x}^{−1} ∘φ ∈ Aut(G); then φ′(S) = S and φ′|_{S_{0}} = φ|_{S_{0}}. □

In the next two propositions, we will be referring to the short exact sequence

1 ⟶ Aut_{diag}(F) ⟶ N_{Aut(F)}(A) ⟶^{R} Aut(A, F) ⟶ 1 .  (17)

Here, R is induced by restriction, and Aut(A, F) = Im(R) and Aut_{diag}(F) = Ker(R) by definition of these two groups (Notation 5.2(H)). By the last statement in Proposition 5.13, in all cases, each class in Out(F) is represented by elements of N_{Aut(F)}(A).

Proposition 5.15. Assume Hypotheses 5.1 and 5.11 and Notation 5.2. Then κ_{G} is surjective, except in the following cases:

• (G, p) ≅ (^{2}E_{6}(q), 3), or

• (G, p) ≅ (G_{2}(q), 2) and q_{0} ≠ 3, or

• (G, p) ≅ (F_{4}(q), 3) and q_{0} ≠ 2.

In the exceptional cases, |Coker(κ_{G})| ≤ 2.

Proof. We first claim that for φ ∈ Aut(F),

φ(A) = A and φ|_{A} ∈ Aut_{sc}(A)Aut_{Aut(G)}(A)  ⟹  [φ] ∈ Im(κ_{G}) .  (18)

To see this, fix such a φ. By Lemma 5.12(b), each element of Aut_{sc}(A), or of Aut_{sc}(A)/⟨ψ_{−1}^{A}⟩ if p = 2, is the restriction of an element of Φ̂_{G}. If p = 2, then we are in case (III.1), the σ-setup is standard, and hence the inversion automorphism ψ_{−1}^{A} is the restriction of an inner automorphism of G (if −Id_{V} ∈ W) or an element of Inn(G)Γ_{G}. Thus φ|_{A} extends to an automorphism of G.

Now, φ|_{A} normalizes Aut_{S}(A) since φ(S) = S. So by Lemma 5.14, φ|_{A} is the restriction of an automorphism of G which normalizes S, and hence is the restriction of an element ψ ∈ Aut(F) such that [ψ] ∈ Im(κ_{G}). Then φψ^{−1} ∈ Ker(R) = Aut_{diag}(F) by the exactness of (17), and [φψ^{−1}] ∈ Im(κ_{G}) by Lemma 5.9. So [φ] ∈ Im(κ_{G}), which proves (18).

By Proposition 5.13, each class in Out(F) is represented by an element of N_{Aut(F)}(A). Hence by (18), |Coker(κ_{G})| is at most the index of Aut(A, F) ∩ Aut_{sc}(A)Aut_{Aut(G)}(A) in Aut(A, F). So by Lemma 5.8, |Coker(κ_{G})| ≤ 2, and κ_{G} is surjective with the exceptions listed above. □

We now want to refine Proposition 5.15, and finish the proof of Theorem B, by determining Ker(κ_{G}) in each case where 5.1 and 5.11 hold and checking whether it is split. In particular, we still want to show that each of these fusion systems is tamely realized by some finite group of Lie type (and not just an extension of such a group by outer automorphisms).

Since O_{p′}(Outdiag(G)) ≤ Ker(κ_{G}) in all cases by Lemma 5.9, κ_{G} induces a quotient homomorphism

κ^{∘}_{G}: Out(G)/O_{p′}(Outdiag(G)) ⟶ Out(F) ,

and it is simpler to describe Ker(κ^{∘}_{G}) than Ker(κ_{G}). The projection of Out(G) onto the quotient Out(G)/O_{p′}(Outdiag(G)) is always split: by Steinberg’s theorem (Theorem 3.4), it splits back to O_{p}(Outdiag(G))Φ_{G}Γ_{G} as defined with respect to some choice of standard setup. (Recall that Outdiag(G) is independent of the σ-setup by Propositions 3.5(c) and 3.6(a).) Hence κ^{∘}_{G} is split surjective if and only if κ_{G} is split surjective.

Proposition 5.16. Assume Hypotheses 5.1 and 5.11 and Notation 5.2. Assume also that none of the following hold: neither

• (G, p) ≅ (^{2}E_{6}(q), 3), nor

• (G, p) ≅ (G_{2}(q), 2) and q_{0} ≠ 3, nor

• (G, p) ≅ (F_{4}(q), 3) and q_{0} ≠ 2.

(a) If p = 2, then κ^{∘}_{G} is an isomorphism, and κ_{G} is split surjective.

(b) Assume that p is odd, and that we are in the situation of case (III.1) of Hypotheses 5.1. Then \sqrt{q} ∈ ℕ, and

Ker(κ^{∘}_{G}) = \begin{cases}⟨[ψ\sqrt{q}]⟩ ≅ C_{2} & if γ = Id and −Id ∈ W \\ ⟨[γ_{0}ψ\sqrt{q}]⟩ ≅ C_{2} & if γ = Id and −Id ∉ W \\ ⟨[ψ\sqrt{q}]⟩ ≅ C_{4} & if γ ≠ Id (G is a Steinberg group) \end{cases}

where in the second case, γ_{0} ∈ Γ_{G} is a graph automorphism of order 2. Hence κ_{G} and κ^{∘}_{G} are split surjective if and only if either γ = Id and −Id ∉ W, or p ≡ 3 (mod 4) and 𝔾 is not F_{4}.

(c) Assume that p is odd, and that we are in the situation of case (III.2) or (III.3) of Hypotheses 5.1. Assume also that G is a Chevalley group (γ ∈ Inn(G)), and that ord_{p}(q) is even or −Id ∉ W_{0}. Let Φ_{G}, Γ_{G} ≤ Aut(G) be as in Proposition 3.6. Then Φ_{G} ∩ Ker(κ^{∘}_{G}) = 1, so |Ker(κ^{∘}_{G})| ≤ |Γ_{G}|, and κ_{G} and κ^{∘}_{G} are split surjective.

(d) Assume that p is odd, and that we are in the situation of case (III.3) of Hypotheses 5.1. Assume also that G is a Steinberg group (γ ∉ Inn(G)), and that ord_{p}(q) is even. Then

Ker(κ^{∘}_{G}) = \begin{cases}⟨[γ|_{G}]⟩ ≅ C_{2} & if γ|_{A} ∈ Aut_{W_{0}}(A) \\ 1 & otherwise. \end{cases}

Hence κ_{G} and κ^{∘}_{G} are split surjective if and only if q is an odd power of q_{0} or Ker(κ_{G}) = O_{p′}(Outdiag(G)). If κ_{G} is not split surjective, then its kernel contains a graph automorphism of order 2 in Out(G)/Outdiag(G).

Proof. In all cases, κ_{G} is surjective by Proposition 5.15 (with the three exceptions listed above).

By definition and Proposition 5.13,

Out(F) = Aut(F)/Aut_{F}(S) ≅ N_{Aut(F)}(A)/N_{Aut_{F} (S)}(A) .

Also, Out_{diag}(F) is the image in Out(F) of Aut_{diag}(F). Since N_{Aut_{F} (S)}(A) is the group of automorphisms of S induced by conjugation by elements in N_{G}(S) ∩ N_{G}(A), the short exact sequence (17) induces a quotient exact sequence

1 ⟶ Out_{diag}(F) ⟶ Out(F) ⟶^{R} Aut(A, F)/Aut_{N_{G}(S)}(A) ⟶ 1 . (19)

We claim that

Aut_{N_{G}(S)}(A) = Aut(A, F) ∩ Aut_{G}(A) .  (20)

That Aut_{N_{G}(S)}(A) is contained in the two other groups is clear. Conversely, assume α ∈ Aut(A, F) ∩ Aut_{G}(A). Then α = c_{g}|_{A} for some g ∈ N_{G}(A), and α ∈ N_{Aut(A)}(Aut_{S}(A)) since it is the restriction of an element of Aut(F). Hence g normalizes SC_{G}(A), and since S ∈ Syl_{p}(SC_{G}(A)), there is h ∈ C_{G}(A) such that hg ∈ N_{G}(S). Thus α = c_{g}|_{A} = c_{hg}|_{A} ∈ Aut_{N_{G}(S)}(A), and this finishes the proof of (20).

By Lemma 5.9, κ_{G} sends Outdiag(G) onto Out_{diag}(F) with kernel the subgroup O_{p′}(Outdiag(G)). Hence by the exactness of (19), restriction to A induces an isomorphism

Ker(κ^{∘}_{G}) ⟶_{≅}^{R_{0}} Ker[Out(G)/Outdiag(G) ⟶ Aut(A, F)/Aut_{N_{G}(S)}(A)]

= Ker[Out(G)/Outdiag(G) ⟶ N_{Aut(A)}(Aut_{G}(A))/Aut_{G}(A)] , (21)

where the equality holds by (20).

Recall that for each ℓ prime to p, ψ_{ℓ}^{A} ∈ Aut_{sc}(A) denotes the automorphism (a ↦ a^{ℓ}).

(a,b) Under either assumption (a) or (b), we are in case (III.1) of Hypotheses 5.1. In particular, (G, σ) is a standard σ-setup for G. Set k = v_{p}(q − 1); then k ≥ 1, and k ≥ 2 if p = 2.

If p is odd, then by Hypotheses 5.11(ii), the class of q_{0} generates (ℤ/p)^{×}. Since q = q_{0}^{b} ≡ 1 (mod p), this implies that (p − 1)|b. In particular, b is even and \sqrt{q} = q_{0}^{b/2} ∈ ℕ in this case. Also, for arbitrary p, Hypotheses 5.11(iii) implies that

b = (p − 1)p^{ℓ} for some ℓ ≥ 0.  (22)

Since Out(G)/Outdiag(G) ≅ Φ_{G}Γ_{G} by Theorem 3.4, where Φ_{G}Γ_{G} normalizes T and hence A, and since Aut_{G}(A) = Aut_{W_{0}}(A) by Lemma 5.3(b), (21) takes the form

Ker(κ^{∘}_{G}) ≅ {φ ∈ Φ_{G}Γ_{G} | φ|_{A} ∈ Aut_{W_{0}}(A)} .  (23)

In fact, when Ker(κ^{∘}_{G}) has order prime to p (which is the case for all examples considered here), the isomorphism in (23) is an equality since the quotient Outdiag(G)/O_{p′}(Outdiag(G)) is a p-group.

Assume first that G = 𝔾(q) is a Chevalley group. Thus σ = ψ_{q} where q ≡ 1 (mod p), and A = {t ∈ T | t^{p^{k}} = 1}. Set

Γ_{G}^{0} = \begin{cases}Γ_{G} & if 𝔾 is not one of B_{2}, F_{4}, or G_{2} \\ 1 & if 𝔾 ≅ B_{2}, F_{4}, or G_{2} \end{cases}

and similarly for Γ_{G}^{0}. By Lemma 2.7 (applied with m = p^{k} ≥ 3), we have Aut_{W} (T) ∩

Aut_{Γ_{G}^{0}}(T) = 1, the group Aut_{W} (T)Aut_{Γ_{G}^{0}}(T) acts faithfully on A, and its action intersects

Aut_{sc}(A) only in ⟨ψ_{−1}^{A}⟩. By Lemma 5.12(b,c), restriction to A sends Φ̂_{G} isomorphically onto Aut_{sc}(A) if p is odd, and with index 2 and ψ_{−1}^{A} not in the image if p = 2. So Φ_{G}Γ_{G}^{0} acts faithfully on A, and

\{φ ∈ Φ_{G}Γ_{G}^{0} | φ|_{A} ∈ Aut_{W_{0}}(A)\} = \begin{cases}1 & if p = 2 \\ ⟨ψ\sqrt{q}⟩ & if p is odd and −Id ∈ W \\ ⟨γ_{0}ψ\sqrt{q}⟩ & if p is odd and −Id ∉ W \end{cases}  (24)

where in the last case, γ_{0} ∈ Γ_{G} is a graph automorphism such that the coset γ_{0}W contains −Id. (Recall that b = (p−1)p^{ℓ} for some ℓ ≥ 0 by (22). Hence \sqrt{q} ≡ −1 modulo p^{k} = expt(A), and ψ\sqrt{q}|_{A} = ψ_{−1}^{A}.)

Thus by (23) and (24), if 𝔾 is not B_{2}, F_{4}, or G_{2}, then κ^{∘}_{G} is injective if p = 2, and |Ker(κ^{∘}_{G})| = 2 if p is odd. When p is odd, since Ker(κ^{∘}_{G}) is normal of order prime to p in Out(G) (hence of order prime to |O_{p}(Outdiag(G))|), Ker(κ^{∘}_{G}) is generated by [ψ\sqrt{q}] if −Id ∈ W (i.e., if there is an inner automorphism which inverts T and hence A), or by [γ_{0}ψ\sqrt{q}] if −Id ∉ W and γ_{0} is as above. In the latter case, κ^{∘}_{G} is split since it sends O_{p}(Outdiag(G))Φ_{G} isomorphically onto Out(F) (note that in this case, 𝔾 ≅ A_{n}, D_{n} for n odd, or E_{6}, and hence Γ_{G} ≅ C_{2}). When Ker(κ^{∘}_{G}) = ⟨[ψ\sqrt{q}]⟩, the map is split if and only if 4 ∤ |Φ_{G}| = b, and since b = (p − 1)p^{m} for some m by (22), this holds exactly when p ≡ 3 (mod 4).

If (G, p) ≅ (B_{2}(q), 2), (F_{4}(q), 2), or (G_{2}(q), 3), then since q_{0} ≠ p, Γ_{G} = 1 = Γ_{G}^{0}. So (23) and (24) again imply that Ker(κ^{∘}_{G}) = 1, 1, or ⟨[ψ\sqrt{q}]⟩ ≅ C_{2}, respectively, and that κ^{∘}_{G} is split in all cases.

Next assume G = G_{2}(q), where p = 2, q = 3^{b}, and b is a power of 2. Then b ≥ 2 since q ≡ 1 (mod 4). By (23) and (24) again, Φ_{G} injects into Out(F). Since Out(G) is cyclic of order 2b, generated by a graph automorphism whose square generates Φ_{G} (and since 2|b), Out(G) injects into Out(F).

If G = F_{4}(q), where p = 3, q = 2^{b}, and b = 2 · 3^{ℓ} for some ℓ ≥ 0, then by (23) and (24), Ker(κ^{∘}_{G}|_{Φ(G)}) = ⟨ψ\sqrt{q}⟩ ≅ C_{2}. Fix 1 ≠ γ ∈ Γ_{G}. If |Ker(κ^{∘}_{G})| > 2, then since Aut_{sc}(Ω_{1}(A)) = {±Id} ≤ Aut_{W} (Ω_{1}(A)), we have wγ|_{Ω_{1}(A)} = Id for some w ∈ W. Since W acts faithfully on Ω_{1}(A) (Lemma 2.7), this would imply that [wγ, W] = 1 in W ⟨γ⟩, and hence that γ acts on W as an inner automorphism, which is impossible since the action of γ exchanges reflections in W for long and short roots, unlike any inner automorphism. Thus Ker(κ^{∘}_{G}) = ⟨ψ\sqrt{q}⟩ ≅ C_{2}. Since Out(G) is cyclic of order 2b = 4 · 3^{ℓ}, neither κ^{∘}_{G} nor κ_{G} splits.

It remains to handle the Steinberg groups. Let ℍ be such that C_{G}(γ) = ℍ(𝔽_{q_{0}}): a simple algebraic group by [GLS3, Theorem 1.15.2(d)]. In particular, G ≥ H = ℍ(q). Also, W_{0} is the Weyl group of ℍ by [GLS3, Theorem 1.15.2(d)] (or by the proof of [St3, Theorem 8.2]). For a ∈ A,

a ∈ H ⟺ γ(a) = a ⟺ ψ_{q}(a) = a^{q} = a ⟺ a ∈ Ω_{k}(A).

Thus Ω_{k}(A) = A ∩ H. So by Lemma 2.7 applied to ℍ(𝔽_{q_{0}}), W_{0} acts faithfully on Ω_{k}(A), and intersects Aut_{sc}(A) at most in ⟨ψ_{−1}^{A}⟩.

If p = 2, then by Lemma 5.12(b), ψ_{−1}^{A} is not the restriction of an element in Φ_{G}. Also, Φ_{G} ≅ C_{2b} is sent injectively into Aut_{sc}(A) by Lemma 5.12(c), so κ^{∘}_{G} is injective by (23).

If p is odd, then A = Ω_{k}(A) ≤ H since v_{p}(q^{2} − 1) = v_{p}(q − 1) = k, and W_{0} acts on A as the Weyl group of B_{m} or C_{m} (some m) or of F_{4} (see [GLS3, Proposition 2.3.2(d)] or [Ca, § 13.3]). Also, ψ_{q_{0}}|_{A} has order b in Aut_{sc}(A) by Lemma 5.12(c). Since (ψ_{q_{0}})^{b/2} = ψ\sqrt{q} where \sqrt{q} ≡ −1 (mod p) (recall b = (p − 1)p^{ℓ} for some ℓ by (22)), and since −Id_{V_{0}} ∈ W_{0} by the above remarks, ψ_{q_{0}}|_{A} has order b/2 modulo Aut_{W_{0}}(A). So by (23) and the remark afterwards, and since Φ_{G} is cyclic of order 2b, Ker(κ^{∘}_{G}) = ⟨[ψ\sqrt{q}]⟩ ≅ C_{4}. In particular, κ^{∘}_{G} is split only if b/2 is odd; equivalently, p ≡ 3 (mod 4).

(c,d) In both of these cases, p is odd, either ord_{p}(q) is even or −Id ∉ W_{0}, and we are in the situation of case (III.2) or (III.3) in Hypothesis 5.1. Then γ|_{G} = (ψ_{q}|_{G})^{−1} since G ≤ C_{G}(γψ_{q}). Also, ψ_{q_{0}}(G) = G by 5.1(I), and hence γ(G) = G. Since ψ_{q_{0}} and γ both normalize T by assumption or by construction, they also normalize T = G ∩ T and A = O_{p}(T). By Proposition 3.6(d), [ψ_{q_{0}}] generates the image of Φ_{G} in Out(G)/Outdiag(G).

We claim that in all cases,

Aut_{G}(A) = Aut_{W_{0}}(A) and Aut_{G}(A) ∩ Aut_{sc}(A) ≤ ⟨γ|_{A}⟩ .  (25)

This holds by assumption in case (III.3), and since ord_{p}(q) is even or −Id ∉ W_{0}. In case (III.2), the first statement holds by Lemma 5.3(b), and the second by Lemma 2.7 (and since W_{0} = W and A contains all p^{k}-torsion in T).

(c) Assume in addition that G is a Chevalley group. Thus γ ∈ Inn(G), so γ|_{G} ∈ Inndiag(G) = Inn(G)Aut_{T} (G) by Proposition 3.6(b), and hence γ|_{A} ∈ Aut_{G}(A). Also, γ|_{A} = (ψ_{q}|_{A})^{−1} = (ψ_{q_{0}}|_{A})^{−b} since σ = γψ_{q} centralizes G ≥ A. Since ψ_{q_{0}}|_{A} has order b · ord_{p}(q) in Aut_{sc}(A) by Lemma 5.12(c), its class in N_{Aut(A)}(Aut_{G}(A))/Aut_{G}(A) has order b by (25).

Thus by (21), κ^{∘}_{G} sends O_{p}(Outdiag(G))Φ_{G} injectively into Out(F). Since Γ_{G} is isomorphic to 1, C_{2}, or Σ_{3} (and since κ^{∘}_{G} is onto by Proposition 5.15), κ^{∘}_{G} and κ_{G} are split.

(d) Assume G is a Steinberg group and ord_{p}(q) is even. In this case, γ ∉ Inn(G), and Out(G)/Outdiag(G) ≅ Φ_{G} is cyclic of order 2b, generated by the class of ψ_{q_{0}}|_{G}. Hence by (21), Ker(κ^{∘}_{G}) is isomorphic to the subgroup of those ψ ∈ Φ_{G} such that ψ|_{A} ∈ Aut_{G}(A). By (25) and since ψ_{q}|_{A} = γ^{−1}|_{A}, Aut_{G}(A) ∩ Aut_{sc}(A) ≤ ⟨ψ_{q}^{A}⟩. Thus |Ker(κ^{∘}_{G})| ≤ 2, and

|Ker(κ^{∘}_{G})| = 2 ⟺ γ|_{A} ∈ Aut_{G}(A) = Aut_{W_{0}}(A) .

When Ker(κ^{∘}_{G}) ≠ 1, κ^{∘}_{G} is split if and only if 4 ∤ |Φ_{G}| = 2b; i.e., when b is odd. □

In the situation of Proposition 5.16(c), if −Id ∉ W, then Ker(κ^{∘}_{G}) = ⟨[γ_{0}ψ\sqrt{q}]⟩ where γ_{0} is a nontrivial graph automorphism. If −Id ∈ W (hence ord_{p}(q) is even), then κ^{∘}_{G} is always injective: either because Γ_{G} = 1, or by the explicit descriptions in the next section of the setups when ord_{p}(q) = 2 (Lemma 6.4), or when ord_{p}(q) > 2 and 𝔾 ≅ D_{2n} (Lemma 6.5).

The following examples help to illustrate some of the complications in the statement of Proposition 5.16.

Example 5.17. Set p = 5. If G = Spin_{4k}^{−}(3^{4}), Sp_{2k}(3^{4}), or SU_{k}(3^{4}) (k ≥ 5), then by Proposition 5.16(b), κ_{G} is surjective but not split. (These groups satisfy case (III.1) of Hypotheses 5.1 by Lemma 6.1.) The fusion systems of the last two are tamely realized by Sp_{2ℓ}(3^{2}) and SL_{n}(3^{2}), respectively (these groups satisfy case (III.2) by Lemma 6.4, hence Proposition 5.16(c) applies). The fusion system of Spin_{4k}^{−}(3^{4}) is also realized by Spin_{4k}^{−}(3^{2}), but not tamely (Example 6.6(b)). It is tamely realized by Spin_{4k−1}(3^{2}) (see Propositions 1.9(c) and 5.16(c)).

6. The cross characteristic case: II

In Section 5, we established certain conditions on a finite group G of Lie type in characteristic q_{0}, on a σ-setup for G, and on a prime p ≠ q_{0}, and then proved that the p-fusion system of G is tame whenever those conditions hold. It remains to prove that for each G of Lie type and each p different from the characteristic, there is another group G^{∗} whose p-fusion system is tame by the results of Section 5, and is isomorphic to that of G.

We first list the groups which satisfy case (III.1) of Hypotheses 5.1.

Lemma 6.1. Fix a prime p and a prime power q ≡ 1 (mod p), where q ≡ 1 (mod 4) if p = 2. Assume G ≅ 𝔾(q) for some simple group scheme 𝔾 over ℤ of universal type, or G ≅ ^{2}𝔾(q) for 𝔾 ≅ A_{n}, D_{n}, or E_{6} of universal type. Then G has a σ-setup (G, σ) such that Hypotheses 5.1, case (III.1) holds.

Proof. Set G = 𝔾(𝔽_{q}), and let ψ_{q} ∈ Φ_{G} be the field automorphism. Set σ = γψ_{q} ∈ End(G), where γ = Id if G ≅ 𝔾(q), and γ ∈ Γ_{G} has order 2 if G ≅ ^{2}𝔾(q).

N_{G}(T) contains a Sylow p-subgroup of G. If γ = Id, then by [Ca, Theorem 9.4.10] (and since G is in universal form), |G| = q^{N} ∏_{i=1}^{r}(q^{d_{i}} −1) for some integers N, d_{1}, …, d_{r} (r = rk(𝔾)), where d_{1}d_{2} ⋯ d_{r} = |W | by [Ca, Theorem 9.3.4]. Also, |T | = (q−1)^{r}, N_{G}(T)/T ≅ W, and so

v_{p}(|G|) = ∑_{i=1}^{r} v_{p}(q^{d_{i}} − 1) = ∑_{i=1}^{r}(v_{p}(q − 1) + v_{p}(d_{i}))
= v_{p}(|T |) + v_{p}(|W |) = v_{p}(N_{G}(T)) ,

where the second equality holds by Lemma 1.13.

If |γ| = 2, then by [Ca, §§ 14.2–3], for N and d_{i} as above, there are ε_{i}, η_{i} ∈ {±1} for 1 ≤ i ≤ r such that |G| = q^{N} ∏_{i=1}^{r}(q^{d_{i}}−ε_{i}) and |T | = ∏_{i=1}^{r}(q−η_{i}). (More precisely, the η_{i} are the eigenvalues of the γ-action on V, and polynomial generators I_{1}, …, I_{r} ∈ ℝ[x_{1}, …, x_{r}]^{W} can be chosen such that deg(I_{i}) = d_{i} and τ(I_{i}) = ε_{i}I_{i}.) By [Ca, Proposition 14.2.1],

|W_{0}| = lim_{t→1} ∏_{i=1}^{r}(\frac{1 − ε_{i}t^{d_{i}}}{1 − η_{i}t} ) ⟹ \begin{matrix}|\{1 ≤ i ≤ r | ε_{i} = 1\}| = |\{1 ≤ i ≤ r | η_{i} = 1\}| \\ and |W_{0}| = ∏\{d_{i} | ε_{i} = +1\}. \end{matrix}

Also, v_{p}(q^{d} + 1) = v_{p}(q + 1) for all d ≥ 1: they are both 0 if p is odd, and both 1 if p = 2. Hence

v_{p}(|G|) − v_{p}(|T |) = ∑_{\begin{matrix}i=1 \\ ε_{i}=+1 \end{matrix}}^{r} v_{p}(\frac{q^{d_{i}} − 1}{q − 1})

= ∑_{\begin{matrix}i=1 \\ ε_{i}=+1 \end{matrix}}^{r} v_{p}(d_{i}) = v_{p}(|W_{0}|) = v_{p}(|N_{G}(T)|) − v_{p}(|T |)

by Lemma 1.13 again, and so N_{G}(T) contains a Sylow p-subgroup of G.

The free ⟨γ⟩-orbit {α} (if γ = Id) or {α, τ(α)} (if |γ| = 2 and α ≠ τ(α)), for any α ∈ Σ, satisfies the hypotheses of this condition.

[γ, ψ_{q_{0}}] = Id since γ ∈ Γ_{G}. □

We are now ready to describe the reduction, when p = 2, to groups with σ-setups satisfying Hypotheses 5.1.

Proposition 6.2. Assume G ∈ 𝔏𝔦𝔢(q_{0}) is of universal type for some odd prime q_{0}. Fix S ∈ Syl_{2}(G), and assume S is nonabelian. Then there is an odd prime q_{0}^{∗}, a group G^{∗} ∈ 𝔏𝔦𝔢(q_{0}^{∗}) of universal type, and S^{∗} ∈ Syl_{2}(G^{∗}), such that F_{S}(G) ≅ F_{S^{∗}}(G^{∗}), and G^{∗} has a σ-setup which satisfies case (III.1) of Hypotheses 5.1 and also Hypotheses 5.11. Moreover, if G^{∗} ≅ G_{2}(q^{∗}) where q^{∗} is a power of q_{0}^{∗}, then we can arrange that either q^{∗} = 5 or q_{0}^{∗} = 3.

Proof. Since q_{0} is odd, and since the Sylow 2-subgroups of ^{2}G_{2}(3^{2k+1}) are abelian for all k ≥ 1 [Ree, Theorem 8.5], G must be a Chevalley or Steinberg group. If G ≅ ^{3}D_{4}(q), then F is also the fusion system of G_{2}(q) by [BMO, Example 4.5]. So we can assume that G ≅ ^{τ}𝔾(q) for some odd prime power q, some 𝔾, and some graph automorphism τ of order 1 or 2.

Let ε ∈ {±1} be such that q ≡ ε (mod 4). By Lemma 1.11, there is a prime q_{0}^{∗} and k ≥ 0

such that ⟨q⟩ = ⟨ε · (q_{0}^{∗})^{2^{k}}⟩, where either q_{0}^{∗} = 5 and k = 0, or q_{0}^{∗} = 3 and k ≥ 1.

If ε = 1, then set G^{∗} = ^{τ}𝔾((q_{0}^{∗})^{2^{k}}), and fix S^{∗} ∈ Syl_{2}(G^{∗}). Then F_{S^{∗}}(G^{∗}) ≅ F_{S}(G) by Theorem 1.8(a), G^{∗} satisfies case (III.1) of Hypotheses 5.1 by Lemma 6.1 (and since (q_{0}^{∗})^{2^{k}} ≡ 1 (mod 4)), and G^{∗} also satisfies Hypotheses 5.11.

Now assume ε = −1. If −Id is in the Weyl group of G, then set G^{∗} = ^{τ}𝔾((q_{0}^{∗})^{2^{k}}). If −Id is not in the Weyl group, then 𝔾 ≅ A_{n}, D_{n} for n odd, or E_{6}, and we set G^{∗} = 𝔾((q_{0}^{∗})^{2^{k}}) if τ ≠ Id, and G^{∗} = ^{2}𝔾((q_{0}^{∗})^{2^{k}}) if G = 𝔾(q). In all cases, for S^{∗} ∈ Syl_{p}(G^{∗}), F_{S^{∗}}(G^{∗}) ≅ F_{S}(G) by Theorem 1.8(c,d), G^{∗} satisfies case (III.1) of Hypotheses 5.1 by Lemma 6.1 again, and also satisfies Hypotheses 5.11.

By construction, if 𝔾 ≅ G_{2}, then either q_{0}^{∗} = 3 or (q_{0}^{∗})^{2^{k}} = 5. □

When G ≅ G_{2}(5) and p = 2, G satisfies Hypotheses 5.1 and 5.11, but κ_{G} is not shown to be surjective in Proposition 5.15 (and in fact, it is not surjective). Hence this case must be handled separately.

Proposition 6.3. Assume p = 2. Set G = G_{2}(5) and G^{∗} = G_{2}(3), and fix S ∈ Syl_{2}(G) and S^{∗} ∈ Syl_{2}(G^{∗}). Then F_{S^{∗}}(G^{∗}) ≅ F_{S}(G) as fusion systems, and κ_{G^{∗}} = μ_{G^{∗}} ∘ κ_{G^{∗}} is an isomorphism from Out(G^{∗}) ≅ C_{2} onto Out(S^{∗}, F_{S^{∗}}(G^{∗})).

Proof. The first statement follows from Theorem 1.8(c). Also, |Out(G)| = 1 and |Out(G^{∗})| = 2 by Theorem 3.4, and since G and G^{∗} have no field automorphisms and all diagonal automorphisms are inner (cf. [St1, 3.4]), and G^{∗} = G_{2}(3) has a nontrivial graph automorphism while G = G_{2}(5) does not [St1, 3.6]. Since G satisfies Hypotheses 5.1 and 5.11, |Coker(κ_{G})| ≤ 2 by Proposition 5.15, so |Out(F_{S}(G))| ≤ 2.

By [O5, Proposition 4.2], S^{∗} contains a unique subgroup Q ≅ Q_{8} ×_{C_{2}} Q_{8} of index 2. Let x ∈ Z(Q) = Z(S^{∗}) be the central involution. Set G = G_{2}(𝔽_{3}) > G^{∗}. Then C_{G}(x) is connected since G is of universal type [St3, Theorem 8.1], so C_{G}(x) ≅ SL_{2}(𝔽_{3}) ×_{C_{2}} SL_{2}(𝔽_{3}) by Proposition 2.5. Furthermore, any outer (graph) automorphism which centralizes x exchanges the two central factors SL_{2}(𝔽_{3}). Hence for each α ∈ Aut(G^{∗})∖Inn(G^{∗}) which normalizes S^{∗}, α exchanges the two factors Q_{8}, and in particular, does not centralize S^{∗}. Thus κ_{G^{∗}} is injective, and hence an isomorphism since |Out(G^{∗})| = 2 and |Out(S^{∗}, F_{S^{∗}}(G^{∗}))| = |Out(F_{S}(G))| ≤ 2. □

We now turn to case (III.2) of Hypotheses 5.1.

Lemma 6.4. Fix an odd prime p, and an odd prime power q prime to p such that q ≡ −1 (mod p). Let G be one of the groups Sp_{2n}(q), Spin_{2n+1}(q), Spin_{4n}^{+}(q) (n ≥ 2), G_{2}(q), F_{4}(q), E_{7}(q), or E_{8}(q) (i.e., G = 𝔾(q) for some 𝔾 whose Weyl group contains −Id), and assume that the Sylow p-subgroups of G are nonabelian. Then G has a σ-setup (G, σ) such that Hypotheses 5.1, case (III.2), hold.

Proof. Assume q = q_{0}^{b} where q_{0} is prime and b ≥ 1. Set G = 𝔾(𝔽_{q_{0}}), and let T < G be a maximal torus. Set r = rk(T) and k = v_{p}(q + 1).

In all of these cases, −Id ∈ W, so there is a coset w_{0} ∈ N_{G}(T)/T which inverts T. Fix g_{0} ∈ N_{G}(T) such that g_{0}T = w_{0} and ψ_{q_{0}}(g_{0}) = g_{0} (Lemma 2.9). Set γ = c_{g_{0}} and σ = γ ∘ ψ_{q}.

We identify G = O^{q′_{0}}(C_{G}(σ)), T = G ∩ T, and A = O_{p}(T). Since σ(t) = t^{−q} for each t ∈ T, T ≅ (C_{q+1})^{r} is the (q + 1)-torsion subgroup of T, and A ≅ (C_{p^{k}})^{r}.

N_{G}(T) contains a Sylow p-subgroup of G. In all cases, by [Ca, Theorem 9.4.10] (and since G is in universal form), |G| = q^{N} ∏_{i=1}^{r}(q^{d_{i}} − 1), where d_{1}d_{2} ⋯ d_{r} = |W | by [Ca, Theorem 9.3.4]. Also, the d_{i} are all even in the cases considered here (see [St2, Theorem 25] or [Ca, Corollary 10.2.4 & Proposition 10.2.5]). Hence by Lemma 1.13 and since p is odd,

v_{p}(|G|) = ∑_{i=1}^{r} v_{p}(q^{d_{i}} − 1) = ∑_{i=1}^{r} v_{p}((q^{2})^{d_{i}/2} − 1) = ∑_{i=1}^{r}(v_{p}(q^{2} − 1) + v_{p}(d_{i}/2))

= r · v_{p}(q + 1) + ∑_{i=1}^{r} v_{p}(d_{i}) = v_{p}(|T |) + v_{p}(|W |) = v_{p}(|N_{G}(T)|) .

[γ, ψ_{q_{0}}] = Id since γ = c_{g_{0}} and ψ_{q_{0}}(g_{0}) = g_{0}.

A free ⟨γ⟩-orbit in Σ. For each α ∈ Σ, {±α} is a free ⟨γ⟩-orbit. □

We now consider case (III.3) of Hypotheses 5.1. By [GL, 10-1,2], when p is odd, each finite group of Lie type has a σ-setup for which N_{G}(T) contains a Sylow p-subgroup of G. Here, we need to construct such setups explicitly enough to be able to check that the other conditions in Hypotheses 5.1 hold.

When p is a prime, A is a finite abelian p-group, and Id ≠ ξ ∈ Aut(A) has order prime to p, we say that ξ is a reflection in A if [A, ξ] is cyclic. In this case, there is a direct product decomposition A = [A, ξ] × C_{A}(ξ), and we call [A, ξ] the reflection subgroup of ξ. This terminology will be used in the proofs of the next two lemmas.

Lemma 6.5. Fix an odd prime p, and an odd prime power q prime to p such that q ≢ 1 (mod p). Let G be one of the classical groups SL_{n}(q), Sp_{2n}(q), Spin_{2n+1}(q), or Spin_{2n}^{±}(q), and assume that the Sylow p-subgroups of G are nonabelian. Then G has a σ-setup (G, σ) such that case (III.3) of Hypotheses 5.1 holds.

Proof. Set m = ord_{p}(q); m > 1 by assumption. We follow Notation 2.2, except that we have yet to fix the σ-setup for G. Thus, for example, q_{0} is the prime of which q is a power.

When defining and working with the σ-setups for the spinor groups, it is sometimes easier to work with orthogonal groups than with their 2-fold covers. For this reason, throughout the proof, we set 𝔾_{c} = SO_{ℓ} when 𝔾 = Spin_{ℓ}, set G_{c} = SO_{ℓ}(𝔽_{q_{0}}) when G = Spin_{ℓ}(𝔽_{q_{0}}), and let χ: G ⟶ G_{c} be the natural surjection. We then set G_{c} = C_{G_{c}}(σ) ≅ SO_{ℓ}^{±}(q), once σ has been chosen so that G = C_{G}(σ) ≅ Spin_{ℓ}^{±}(q), and set T_{c} = χ(T) and T_{c} = C_{T_{c}}(σ). Also, in order to prove the lemma without constantly considering these groups as a separate case, we set G_{c} = G, G_{c} = G, χ = Id, etc. when G is linear or symplectic. Thus G_{c} and G_{c} are classical groups in all cases.

Regard G_{c} as a subgroup of Aut(V , 𝔟), where V is an 𝔽_{q_{0}}-vector space of dimension n, 2n, or 2n + 1, and 𝔟 is a bilinear form. Explicitly, 𝔟 = 0 if 𝔾 = SL_{n}, and 𝔟 has matrix ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )^{⊕n} if 𝔾 = Sp_{2n}, ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )^{⊕n} if 𝔾 = Spin_{2n}, or ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )^{⊕n} ⊕ (1) if 𝔾 = Spin_{2n+1}. Let T_{c} be the group of diagonal matrices in G_{c}, and set

[λ_{1}, …, λ_{n}] = \begin{cases}diag(λ_{1}, …, λ_{n}) & if 𝔾 = SL_{n} \\ diag(λ_{1}, λ_{1}^{−1}, …, λ_{n}, λ_{n}^{−1}) & if 𝔾 = Sp_{2n} or Spin_{2n} \\ diag(λ_{1}, λ_{1}^{−1}, …, λ_{n}, λ_{n}^{−1}, 1) & if 𝔾 = Spin_{2n+1}. \end{cases}

In this way, we identify the maximal torus T_{c} < G_{c} with (𝔽_{q_{0}}^{×})^{n} in the symplectic and orthogonal cases, and with a subgroup of (𝔽_{q_{0}}^{×})^{n} in the linear case.

Set W^{∗} = W (the Weyl group of 𝔾 and of 𝔾_{c}), except when 𝔾 = Spin_{2n}, in which case we let W^{∗} < Aut(T_{c}) be the group of all automorphisms which permute and invert the coordinates. Thus in this last case, W^{∗} ≅ {±1} ≀ Σ_{n}, while W is the group of signed permutations which invert an even number of coordinates (so [W^{∗} : W] = 2). Since W^{∗} induces a group of isometries of the root system for Spin_{2n} and contains W with index 2, it is generated by W and the restriction to T_{c} of a graph automorphism of order 2 (see, e.g., [Brb, § VI.1.5, Proposition 16]).

We next introduce some notation in order to identify certain elements in W^{∗}. For each r, s such that rs ≤ n, let τ _{r}^{s} ∈ Aut(T_{c}) be the Weyl group element induced by the permutation (1 ⋯ r)(r+1 ⋯ 2r) ⋯ ((s−1)r + 1 ⋯ sr); i.e.,

τ _{r}^{s}([λ_{1}, …, λ_{n}]) =

[λ_{r}, λ_{1}, …, λ_{r−1}, λ_{2r}, λ_{r+1}, …, λ_{sr}, λ_{(s−1)r+1}, …, λ_{sr−1}, λ_{sr+1}, …].

For 1 ≤ i ≤ n, let ξ_{i} ∈ Aut(T) be the automorphism which inverts the i-th coordinate. Set τ _{r,+1}^{s} = τ _{r}^{s} and τ _{r,−1}^{s} = τ _{r}^{s}ξ_{r}ξ_{2r} ⋯ ξ_{sr}. Thus for θ = ±1,

τ _{r,θ}^{s}([λ_{1}, …, λ_{n}]) =

[λ_{r}^{θ}, λ_{1}, …, λ_{r−1}, λ_{2r}^{θ}, λ_{r+1}, … λ_{sr}^{θ}, λ_{(s−1)r+1}, …, λ_{sr−1}, λ_{sr+1}, …].

Recall that m = ord_{p}(q). Define parameters μ, θ, k, and κ as follows:

if m is odd : μ = m  θ = 1  κ = [n/μ] = [n/m]
if m is even : μ = m/2 θ = −1 k = [n/m] κ = [n/μ] = [2n/m] .

We can now define our σ-setups for G and G_{c}. Recall that we assume m > 1. In Table 6.1, we define an element w_{0} ∈ W^{∗}, and then describe T_{c} = C_{T_{c}}(w_{0} ∘ ψ_{q}) and W_{0}^{∗} = C_{W^{∗}}(w_{0}) (where W_{0} = C_{W} (w_{0}) has index at most 2 in W_{0}^{∗}). In all cases, we choose γ ∈ Aut(G_{c}) as

G_{c}  conditions w_{0} = γ|_{T_{c}}  T_{c}  W_{0}^{∗}
SL_{n}(q)  τ _{m}^{k}  (C_{q^{m}−1})^{k} × C_{q−1}^{n−mk−1}  (C_{m} ≀ Σ_{k}) × H
Sp_{2n}(q)
SO_{2n+1}(q)  τ _{μ,θ}^{κ}  (C_{q^{μ}−θ})^{κ} × C_{q−1}^{n−κμ}  (C_{2μ} ≀ Σ_{κ}) × H
ε ≠ θ^{κ}, μ∤n τ _{μ,θ}^{κ} ξ_{n} (C_{q^{μ}−θ})^{κ} × C_{q−1}^{n−κμ−1} × C_{q+1} (C_{2μ} ≀ Σ_{κ}) × H
SO_{2n}^{ε}(q) \begin{matrix}ε ≠ θ^{κ}, μ|n \\ θ = −1 \end{matrix} τ _{μ,θ}^{κ−1}  (C_{q^{μ}−θ})^{κ−1} × C_{q−1}^{μ}
\begin{matrix}ε ≠ θ^{κ}, μ|n \\ θ = +1 \end{matrix} τ _{μ,θ}^{κ−1} ξ_{n} (C_{q^{μ}−θ})^{κ−1} × C_{q−1}^{μ−1} × C_{q+1} (C_{2μ} ≀ Σ_{κ−1}) × H

Table 6.1

follows. Write w_{0} = w′_{0} ∘ γ_{0}|_{T_{c}} for some w′_{0} ∈ W and γ_{0} ∈ Γ_{G_{c}} (possibly γ_{0} = Id). Choose g_{0} ∈ N_{G_{c}}(T_{c}) such that g_{0}T_{c} = w′_{0} and ψ_{q_{0}}(g_{0}) = g_{0} (Lemma 2.9), and set γ = c_{g_{0}} ∘ γ_{0}. Then [γ, ψ_{q_{0}}] = Id_{G_{c}}, since c_{g_{0}} and γ_{0} both commute with ψ_{q_{0}}, and we set σ = γ ∘ ψ_{q} = ψ_{q} ∘ γ. When 𝔾 = Spin_{2n} or Spin_{2n+1}, since G is a perfect group and Ker(χ) ≤ Z(G), γ and σ lift to unique endomorphisms of G which we also denote γ and σ (and still [γ, ψ_{q_{0}}] = 1 in Aut(G)).

ε = θ^{κ} In all cases, T ⟶^{χ} T_{c} has kernel and cokernel of order ≤ 2, and so A = O_{p}(T) ≅ O_{p}(T_{c}).

Thus G ≅ C_{G}(σ) and G_{c} ≅ C_{G_{c}}(σ) in all cases, and we identify these groups. Set T = C_{T} (σ), T_{c} = C_{T_{c}}(σ), W_{0}^{∗} = C_{W^{∗}}(γ), and W_{0} = C_{W} (γ). If 𝔾 = Spin_{2n+1} or Spin_{2n}, then χ(T) is the kernel of the homomorphism T_{c} → Ker(χ) which sends χ(t) to t^{−1}σ(t), and thus has index at most 2 in T_{c}. Since p is odd, this proves the statement in the last line of Table 6.1.

In the description of W_{0}^{∗} in Table 6.1, H always denotes a direct factor of order prime to p. The first factor in the description of W_{0}^{∗} acts on the first factor in that of T, and H acts on the other factors.

When G_{c} = SL_{n}(q) and m|n, the second factor C_{q−1}^{−1} in the description of T doesn’t make sense. It should be interpreted to mean that T is “a subgroup of index q − 1 in the first factor (C_{q^{m}−1})^{k}”.

Recall that T_{c} = C_{T_{c}}(γ ∘ ψ_{q}). When U = (𝔽_{q_{0}}^{×})^{μ}, then

C_{U}(τ _{μ,θ}^{1} ∘ ψ_{q}) = {(λ, λ^{q}, λ^{q^{2}}, …, λ^{q^{μ−1}}) | (λ^{q^{μ−1}})^{qθ} = λ}
= {(λ, λ^{q}, λ^{q^{2}}, …, λ^{q^{μ−1}}) | λ^{q^{μ}−θ} = 1} ≅ C_{q^{μ}−θ} .

This explains the description of T_{c} in the symplectic and orthogonal cases: it is always the direct product of (C_{q^{μ}−θ})^{κ} or (C_{q^{μ}−θ})^{κ−1} with a group of order prime to p. (Note that p|(q+1) only when m = 2; i.e., when θ = −1 and 1 = μ|n.)

Since the cyclic permutation (1 2 ⋯ μ) generates its own centralizer in Σ_{μ}, the centralizer of τ _{μ,θ}^{1} in {±1} ≀ Σ_{μ} < Aut((𝔽_{q_{0}}^{×})^{μ}) is generated by τ _{μ,θ}^{1} and ψ_{−1}^{T}. If θ = −1, then (τ _{μ,θ}^{1})^{μ} = ψ_{−1}^{T}, while if θ = 1, then τ _{μ,θ}^{1} has order μ. Since m = μ is odd in the latter case, the centralizer is cyclic of order 2μ in both cases. This is why, in the symplectic and orthogonal cases, the first factor in W_{0}^{∗} is always a wreath product of C_{2μ} with a symmetric group.

We are now ready to check the conditions in case (III.3) of Hypotheses 5.1. N_{G}(T) contains a Sylow p-subgroup of G. Set

e = v_{p}(q^{m} − 1) = v_{p}(q^{μ} − θ) .

The second equality holds since if 2|m, then p ∤ (q^{μ} − 1) and hence e = v_{p}(q^{μ} + 1). Recall also that m|(p − 1), so v_{p}(m) = 0. Consider the information listed in Table 6.2, where the formulas for v_{p}(|T |) = v_{p}(|T_{c}|) and v_{p}(|W_{0}|) follow from Table 6.1, and those for |G| are shown in [St2, Theorems 25 & 35] and also in [Ca, Corollary 10.2.4, Proposition 10.2.5 & Theorem 14.3.2].

For all i > 0, we have

v_{p}(q^{i} − 1) = \begin{cases}e + v_{p}(i/m) & if m|i \\ 0 & if m∤i. \end{cases}

The first case follows from Lemma 1.13, and the second case since m = ord_{p}(q). Using this, we check that v_{p}(q^{2i} − 1) = v_{p}(q^{i} − 1) for all i whenever m is odd, and that

v_{p}(q^{n} − ε) = \begin{cases}e + v_{p}(2n/m) & if m|2n and ε = (−1)^{2n/m} \\ 0 & otherwise. \end{cases}

G  cond.  v_{p}(|G|)  v_{p}(|T |) v_{p}(|W_{0}|)
SL_{n}(q)  ∑_{i=2}^{n} v_{p}(q^{i} − 1)  ke  v_{p}(k!)
Sp_{2n}(q)
Spin_{2n+1}(q)  ∑_{i=1}^{n} v_{p}(q^{2i} − 1)
ε = θ^{κ}  κe  v_{p}(κ!)
Spin_{2n}^{ε}(q) ε ≠ θ^{κ}, μ∤n v_{p}(q^{n} − ε)
ε ≠ θ^{κ}, μ|n + ∑_{i=1}^{n−1} v_{p}(q^{2i} − 1) (κ−1)e v_{p}((κ−1)!)

Table 6.2

So in all cases, v_{p}(|G|) = v_{p}(|T |) + v_{p}(|W_{0}|) by the above relations and the formulas in Table 6.2. Since N_{G}(T)/T ≅ W_{0} by Lemma 5.3(b), this proves that v_{p}(|G|) = v_{p}(|N_{G}(T)|), and hence that N_{G}(T) contains a Sylow p-subgroup of G.

|γ|_{T} | = |τ | = ord_{p}(q) ≥ 2 and [γ, ψ_{q_{0}}] = Id by construction. Note, when G is a spinor group, that these relations hold in G if and only if they hold in G_{c}.

C_{S}(Ω_{1}(A)) = A by Table 6.1 and since p ∤ |H|.

C_{A}(O_{p′}(W_{0})) = 1. By Table 6.1, in all cases, there are r, t ≥ 1 and 1 ≠ s|(p − 1) such that A ≅ (C_{p^{t}})^{r}, and Aut_{W_{0}^{∗}}(A) ≅ C_{s} ≀ Σ_{r} acts on A by acting on and permuting the cyclic factors. In particular, Aut_{O_{p′}(W_{0})}(A) contains a subgroup of index at most 2 in (C_{s})^{r}, this subgroup acts nontrivially on each of the cyclic factors in A, and hence C_{A}(O_{p′}(W_{0})) = 1.

A free ⟨γ⟩-orbit in Σ. This can be defined as described in Table 6.3. In each case, we use the notation of Bourbaki [Brb, pp. 250–258] for the roots of 𝔾. Thus, for example, the roots of SL_{n} are the ±(ε_{i} − ε_{j}) for 1 ≤ i < j ≤ n, and the roots of SO_{2n} the ±ε_{i} ± ε_{j}. Note that since S is assumed nonabelian, p||W_{0}|, and hence n ≥ pm in the linear case, and n ≥ pμ in the other cases.

G  θ = 1  θ = −1
SL_{n}(q)  {ε_{i} − ε_{m+i} | 1 ≤ i ≤ m}
Sp_{2n}(q)  {2ε_{i} | 1 ≤ i ≤ μ}  {±2ε_{i} | 1 ≤ i ≤ μ}
Spin_{2n+1}(q)  {ε_{i} | 1 ≤ i ≤ μ}  {±ε_{i} | 1 ≤ i ≤ μ}
Spin_{2n}^{ε}(q) {ε_{i} − ε_{μ+i} | 1 ≤ i ≤ μ} {±(ε_{i} − ε_{μ+i}) | 1 ≤ i ≤ μ}

Table 6.3

Aut_{W_{0}}(A) ∩ Aut_{sc}(A) ≤ \begin{cases}⟨γ|_{A}⟩ & if ord_{p}(q) even or −Id ∉ W_{0} \\ ⟨γ|_{A}, ψ_{−1}^{A}⟩ & otherwise. \end{cases}

Set K^{∗} = Aut_{W_{0}^{∗}}(A) ∩ Aut_{sc}(A) and K = Aut_{W_{0}}(A) ∩ Aut_{sc}(A) for short. By Table 6.1, |K^{∗}| = m if G ≅ SL_{n}(q), and |K^{∗}| = 2μ otherwise. Also, ⟨γ|_{A}⟩ = ⟨ψ_{q}^{−1}|_{A}⟩ has order ord_{p}(q). Thus K ≤ K^{∗} = ⟨γ|_{A}⟩ except when G is symplectic or orthogonal and m = ord_{p}(q) is odd.

In this last case, K = K^{∗} (so |K| = 2μ = 2m) if W_{0} contains an element which inverts A (hence which inverts T and T); and |K^{∗}/K| = 2 (|K| = m) otherwise.

Aut_{G}(A) = Aut_{W_{0}}(A). Since A = O_{p}(T) ≅ O_{p}(T_{c}) by Table 6.1, it suffices to prove this for G_{c}. Fix g ∈ N_{G_{c}}(A). Since ^{g}T_{c} is a maximal torus in the algebraic group C_{G_{c}}(A) (Proposition 2.5), there is b ∈ C_{G_{c}}(A) such that ^{b}T_{c} = ^{g}T_{c}. Set a = b^{−1}g ∈ N_{G_{c}}(T_{c}); thus c_{a} = c_{g} ∈ Aut(A). Set w = aT_{c} ∈ W = N_{G_{c}}(T_{c})/T_{c}; thus w ∈ N_{W} (A), and w|_{A} = c_{g}|_{A}.

By the descriptions in Table 6.1, we can factor T_{c} = T_{1} ×T_{2}, where γ and each element of N_{W} (A) send each factor to itself, γ|_{T_{2}} = Id, A ≤ T_{1}, and [C_{W} (A), T_{1}] = 1. Since σ(g) = g, σ(a) ≡ a (mod C_{G_{c}}(A)), and so τ(w) ≡ w (mod C_{W} (A)). Thus τ(w)|_{T_{1}} = w|_{T_{1}} since C_{W} (A) acts trivially on this factor, τ(w)|_{T_{2}} = w|_{T_{2}} since γ|_{T_{2}} = Id, and so w ∈ W_{0} = C_{W} (τ).

N_{Aut(A)}(Aut_{W_{0}}(A)) ≤ Aut_{sc}(A)Aut_{Aut(G)}(A). By Table 6.1, for some r, t ≥ 1, A = A_{1}×⋯×A_{r}, where A_{i} ≅ C_{p^{t}} for each i. Also, for some 1 ≠ s|(p−1), Aut_{W_{0}^{∗}}(A) ≅ C_{s}≀Σ_{r} acts on A via faithful actions of C_{s} on each A_{i} and permutations of the A_{i}.

Let Aut_{W_{0}^{∗}}^{0}(A) ⊴ Aut_{W_{0}^{∗}}(A) and Aut_{W_{0}}^{0}(A) ⊴ Aut_{W_{0}}(A) be the subgroups of elements which normalize each cyclic subgroup A_{i}. Thus Aut_{W_{0}^{∗}}^{0}(A) ≅ (C_{s})^{r}, and contains Aut_{W_{0}}^{0}(A) with index at most 2. Case 1: Assume first that Aut_{W_{0}}^{0}(A) is characteristic in Aut_{W_{0}}(A). Fix some α ∈ N_{Aut(A)}(Aut_{W_{0}}(A)). We first show that α ∈ Aut_{W_{0}^{∗}}(A)Aut_{sc}(A).

Since α normalizes Aut_{W_{0}}(A), it also normalizes Aut_{W_{0}}^{0}(A). For each β ∈ Aut_{W_{0}}^{0}(A), [β, A] is a product of A_{i}’s. Hence the factors A_{i} are characterized as the minimal nontrivial intersections of such [β, A], and are permuted by α. So after composing with an appropriate element of Aut_{W_{0}^{∗}}(A), we can assume that α(A_{i}) = A_{i} for each i.

After composing α by an element of Aut_{sc}(A), we can assume that α|_{A_{1}} = Id. Fix i ≠ 1 (2 ≤ i ≤ r), let u ∈ ℤ be such that α|_{A_{i}} = ψ_{u}^{A_{i}} = (a ↦ a^{u}), and choose w ∈ Aut_{W_{0}}(A) such that w(A_{1}) = A_{i}. Then w^{−1}αwα^{−1} ∈ Aut_{W_{0}}(A) since α normalizes Aut_{W_{0}}(A), and (w^{−1}αwα^{−1})|_{A_{1}} = ψ_{u}^{A_{1}}. Hence u^{s} ≡ 1 (mod p^{t} = |A_{1}|), and since this holds for each i, α ∈ Aut_{W_{0}^{∗}}(A).

Thus N_{Aut(A)}(Aut_{W_{0}}(A)) ≤ Aut_{W_{0}^{∗}}(A)Aut_{sc}(A). By Table 6.1, each element of Aut_{W_{0}^{∗}}(A) extends to some φ ∈ Aut_{W^{∗}}(T) which commutes with σ|_{T}. So Aut_{W_{0}^{∗}}(A) ≤ Aut_{Aut(G)}(A) by Lemma 3.7, and this finishes the proof of the claim. Case 2: Now assume that Aut_{W_{0}}^{0}(A) is not characteristic in Aut_{W_{0}}(A). Then r ≤ 4, and since p ≤ r, we have p = 3 and r = 3, 4. Also, s = 2 since s|(p − 1) and s ≠ 1. Thus r = 4, since Aut_{W_{0}}^{0}(A) = O_{2}(Aut_{W_{0}}(A)) if r = 3. Thus Aut_{W_{0}}(A) ≅ C_{2}^{3} ⋊ Σ_{4}: the Weyl group of D_{4}. Also, m = 2 since p = 3, so (in the notation used in the tables) μ = 1, θ = −1, and κ = n. By Table 6.1, G ≅ SO_{8}(q) for some q ≡ 2 (mod 3) (and W_{0} = W).

Now, O_{2}(W) ≅ Q_{8}×_{C_{2}}Q_{8}, and so Out(O_{2}(W)) ≅ Σ_{3}≀C_{2}. Under the action of W/O_{2}(W) ≅ Σ_{3}, the elements of order 3 act on both central factors and those of order 2 exchange the factors. (This is seen by computing their centralizers in O_{2}(W).) It follows that N_{Out(O_{2}(W))}(Out_{W} (O_{2}(W)))/Out_{W} (O_{2}(W)) ≅ Σ_{3} ≅ Γ_{G}, and all classes in this quotient extend to graph automorphisms of G ≅ Spin_{8}(q). So for each α ∈ N_{Aut(A)}(Aut_{W} (A)), after composing with a graph automorphism of G we can arrange that α commutes with O_{2}(W), and in particular, normalizes Aut_{W}^{0} (A). Hence by the same argument as used in Case 1, α ∈ Aut_{sc}(A)Aut_{Aut(G)}(A).

This finishes the proof that this σ-setup for G satisfies case (III.3) of Hypotheses 5.1. □

Example 6.6. Fix distinct odd primes p and q_{0}, and a prime power q = q_{0}^{b} where b is even and ord_{p}q is even. Set G = Spin_{4k}^{−}(q) for some k ≥ 2. Let (G, σ) be the setup for G of Lemma 6.5, where σ = ψ_{q}γ for γ ∈ Aut(G). In the notation of Table 6.1, m = ord_{p}(q), μ = m/2, θ = −1 = ε, n = 2k, and κ = [2k/μ] = [4k/m]. There are three cases to consider:

(a) If q^{2k} ≡ −1 (mod p); equivalently, if m|4k and κ = 4k/m is odd, then ε = θ^{κ}, w_{0} = γ|_{T_{c}} = τ _{μ,θ}^{κ}, rk(A) = κ, and W_{0}^{∗} ≅ C_{m} ≀ Σ_{κ}. Then W_{0}^{∗} acts faithfully on A while w_{0} ∈ W_{0}^{∗}∖W_{0}, and so γ|_{A} ∉ Aut_{W_{0}}(A). Hence by Proposition 5.16(d), κ_{G} is split.

(b) If q^{2k} ≡ 1 (mod p); equivalently, if m|4k and κ = 4k/m is even, then ε ≠ θ^{κ}, γ|_{T_{c}} = τ _{μ,θ}^{κ−1}, rk(A) = κ − 1, and W_{0}^{∗} ≅ (C_{m} ≀ Σ_{κ−1}) × H where H ≅ (C_{2} ≀ Σ_{μ}). Then H acts trivially on A and contains elements in W_{0}^{∗}∖W_{0}, so γ|_{A} ∈ Aut_{W_{0}}(A). Hence κ_{G} is not split.

(c) If q^{2k} ≢ ±1 (mod p); equivalently, if m ∤ 4k, then in either case (κ even or odd), the factor H in the last column of Table 6.1 is nontrivial, acts trivially on A, and contains elements in W_{0}^{∗}∖W_{0}. Hence γ|_{A} ∈ Aut_{W_{0}}(A) in this case, and κ_{G} is not split.

We also need the following lemma, which handles the only case of a Chevalley group of exceptional type which we must show satisfies case (III.3) of Hypotheses 5.1.

Lemma 6.7. Set p = 5, let q be an odd prime power such that q ≡ ±2 (mod 5), and set G = E_{8}(q). Then G has a σ-setup which satisfies Hypotheses 5.1 (case (III.3)).

Proof. We use the notation in 2.2, where q is a power of the odd prime q_{0}, and G = E_{8}(𝔽_{q_{0}}).

By [Brb, Planche VII], the of roots of E_{8} can be given the following form, where {ε_{1}, …, ε_{8}} denotes the standard orthonormal basis of ℝ^{8}:

Σ = \{±ε_{i} ± ε_{j} | 1 ≤ i < j ≤ 8\} ∪ \{\frac{1}{2} ∑_{i=1}^{8}(−1)^{m_{i}}ε_{i} | ∑_{i=1}^{8} m_{i} even\} ⊆ ℝ^{8} .

By the same reference, the Weyl group W is the group of all automorphisms of ℝ^{8} which permute Σ (A(R) = W (R) in the notation of [Brb]). Give ℝ^{8} a complex structure by setting iε_{2k−1} = ε_{2k} and iε_{2k} = −ε_{2k−1}, and set ε_{k}^{∗} = ε_{2k−1} for 1 ≤ k ≤ 4. Multiplication by i permutes Σ, and hence is the action of an element w_{0} ∈ W. Upon writing the elements of Σ with complex coordinates, we get the following equivalent subset Σ^{∗} ⊆ ℂ^{4}:

Σ^{∗} = {(±1 ± i)ε_{k}^{∗} | 1 ≤ k ≤ 4} ∪ {i^{m}ε_{k}^{∗} + i^{n}ε_{ℓ}^{∗} | 1 ≤ k < ℓ ≤ 4, m, n ∈ ℤ}

∪ \{\frac{1 + i}{2} ∑_{k=1}^{4} i^{m_{k}}ε_{k}^{∗} | ∑ m_{k} even\} .

Let ℤΣ ⊆ ℝ^{8} be the lattice generated by Σ. By Lemma 2.4(d) (and since (α, α) = 2 for all α ∈ Σ), we can identify T ≅ ℤΣ ⊗_{ℤ} 𝔽_{q_{0}}^{×} by sending h_{α}(λ) to α ⊗ λ for α ∈ Σ and λ ∈ 𝔽_{q_{0}}^{×}. Set Λ_{0} = ℤΣ ∩ ℤ^{8}, a lattice in ℝ^{8} of index 2 in ℤΣ and in ℤ^{8}. The inclusions of lattices induce homomorphisms

T ≅ ℤΣ ⊗_{ℤ} 𝔽_{q_{0}}^{×} ⟵^{χ_{1}} Λ_{0} ⊗_{ℤ} 𝔽_{q_{0}}^{×} ⟶^{χ_{2}} ℤ^{8} ⊗_{ℤ} 𝔽_{q_{0}}^{×} ≅ (𝔽_{q_{0}}^{×})^{8}

each of which is surjective with kernel of order 2 (since Tor_{ℤ}^{1}(ℤ/2, 𝔽_{q_{0}}^{×}) ≅ ℤ/2). We can thus identify T = (𝔽_{q_{0}}^{×})^{8}, modulo 2-power torsion, in a way so that

α = ∑_{i=1}^{8} k_{i}ε_{i} ∈ Σ, λ ∈ 𝔽_{q_{0}}^{×} ⟹ h_{α}(λ) = (λ^{k_{1}}, …, λ^{k_{8}}) .

Under this identification, by the formula in Lemma 2.4(c),

β = ∑_{i=1}^{8} ℓ_{i}ε_{i} ∈ Σ ⟹ θ_{β}(λ_{1}, …, λ_{8}) = λ_{1}^{ℓ_{1}} ⋯ λ_{8}^{ℓ_{8}}  (1)

for λ_{1}, …, λ_{8} ∈ 𝔽_{q_{0}}^{×}. Also,

w_{0}(λ_{1}, …, λ_{8}) = (λ_{2}^{−1}, λ_{1}, λ_{4}^{−1}, λ_{3}, …, λ_{8}^{−1}, λ_{7})

for each (λ_{1}, …, λ_{8}).

Choose g_{0} ∈ N_{G}(T) such that g_{0}T = w_{0} and ψ_{q_{0}}(g_{0}) = g_{0} (Lemma 2.9), and set γ = c_{g_{0}} ∈ Inn(G). Thus σ = ψ_{q} ∘ γ = γ ∘ ψ_{q}, G = C_{G}(σ), and T = C_{T} (σ). By the Lang-Steinberg theorem [St3, Theorem 10.1], there is h ∈ G such that g = hψ_{q}(h^{−1}); then σ = c_{h}ψ_{q}c_{h}^{−1} and G ≅ C_{G}(ψ_{q}) = E_{8}(q). It remains to check that the setup (G, σ) satisfies the list of conditions in Hypotheses 5.1.

We identify W_{0} = C_{W} (w_{0}) with the group of ℂ-linear automorphisms of ℂ^{4} which permute Σ^{∗}. The order of W_{0} is computed in [Ca3, Table 11] (the entry Γ = D_{4}(a_{1})^{2}), but since we need to know more about its structure, we describe it more precisely here. Let W_{2} ≤ GL_{4}(ℂ) be the group of monomial matrices with nonzero entries ±1 or ±i, and with determinant ±1. Then W_{2} ≤ W_{0}, |W_{2}| = \frac{1}{2} ·4^{4} ·4! = 2^{10} ·3, and W_{2} acts on Σ^{∗} with three orbits corresponding to the three subsets in the above description of Σ^{∗}. The (complex) reflection of order 2 in the hyperplane orthogonal to \frac{1+i}{2}(ε_{1}^{∗} + ε_{2}^{∗} + ε_{3}^{∗} + ε_{4}^{∗}) sends (1 + i)ε_{1}^{∗} to \frac{1+i}{2}(ε_{1}^{∗} − ε_{2}^{∗} − ε_{3}^{∗} − ε_{4}^{∗}), and it sends (ε_{1}^{∗} + iε_{2}^{∗}) to \frac{1+i}{2}(i^{3}ε_{1}^{∗} + iε_{2}^{∗} − ε_{3}^{∗} − ε_{4}^{∗}). Thus W_{0} acts transitively on Σ^{∗}.

Let Σ ⊆ P (ℂ^{4}) be the set of projective points representing elements of Σ^{∗}, and let [α] ∈ Σ denote the class of α ∈ Σ^{∗}. To simplify notation, we also write [x] = [α] for x ∈ ℂ^{4} representing the same point, also when x ∉ Σ^{∗}. Let ∼ denote the relation on Σ: [α] ∼ [β] if α = β, or if α ⊥ β and the projective line ⟨[α], [β]⟩ ⊆ P (ℂ^{4}) contains four other points in Σ. By inspection, [ε_{j}^{∗}] ∼ [ε_{k}^{∗}] for all j, k ∈ {1, 2, 3, 4}, and these are the only elements [α] such that [α] ∼ [ε_{j}^{∗}] for some j. Since this relation is preserved by W_{0}, and W_{0} acts transitively on Σ, we see that ∼ is an equivalence relation on Σ with 15 classes of four elements each. Set Δ = Σ/∼, and let [α]_{Δ} denote the class of [α] in Δ. Thus |Σ| = \frac{1}{4}|Σ| = 60 and |Δ| = 15. Since W_{2} is the stabilizer subgroup of [ε_{1}^{∗}]_{Δ} under the transitive W_{0}-action on Δ, we have |W_{0}| = |W_{2}| · 15 = 2^{10} · 3^{2} · 5.

Let W_{1} ⊴ W_{0} be the subgroup of elements which act trivially on Δ. By inspection, W_{1} ≤ W_{2}, |W_{1}| = 2^{6}, and W_{1} is generated by w_{0} = diag(i, i, i, i), diag(1, 1, −1, −1), diag(1, −1, 1, −1), and the permutation matrices for the permutations (1 2)(3 4) and (1 3)(2 4). Thus W_{1} ≅ C_{4} ×_{C_{2}} D_{8} ×_{C_{2}} D_{8}.

By the above computations, |W_{0}/W_{1}| = 2^{4} · 3^{2} · 5 = |Sp_{4}(2)|. There is a bijection from Δ to the set of maximal isotropic subspaces in W_{1}/Z(W_{1}) which sends a class [α]_{Δ} to the subgroup of those elements in W_{1} which send each of the four projective points in [α]_{Δ} to itself. Hence for each w ∈ C_{W_{0}}(W_{1}), w acts via the identity on Δ, and so w ∈ W_{1} by definition. Thus W_{0}/W_{1} injects into Out(W_{1}) ≅ Σ_{6} × C_{2}, and injects into the first factor since Z(W_{1}) = Z(W_{0}) (≅ C_{4}). So by counting, W_{0}/W_{1} ≅ Σ_{6}. Also, W_{1} = O_{2}(W_{0}).

Set a = v_{5}(q^{4} − 1) = v_{5}(q^{2} + 1), and fix u ∈ 𝔽_{q_{0}}^{×} of order 5^{a}. Let A be as in Notation 5.2(G): the subgroup of elements in T of 5-power order. Thus

A = {(u_{1}, u_{1}^{q}, u_{2}, u_{2}^{q}, u_{3}, u_{3}^{q}, u_{4}, u_{4}^{q}) | u_{1}, u_{2}, u_{3}, u_{4} ∈ ⟨u⟩} ≅ (C_{5^{a}})^{4} .  (2)

By (2) and (1), there is no β ∈ Σ such that A ≤ Ker(θ_{β}). Hence C_{G}(A)^{0} = T by Proposition 2.5. So by Lemma 5.3(b),

N_{G}(A) = N_{G}(T) and N_{G}(T)/T = W_{0} .  (3)

We are now ready to check the conditions in Case (III.3) of Hypotheses 5.1.

N_{G}(T) contains a Sylow p-subgroup of G. Let S be a Sylow p-subgroup of N_{G}(T) which contains A. Since N_{G}(T)/T = W_{0} by (3), A ≅ (C_{5^{a}})^{4}, and W_{0}/O_{2}(W_{0}) ≅ Σ_{6}, |S| = 5^{4a+1}. By [St2, Theorem 25] or [Ca, Corollary 10.2.4 & Proposition 10.2.5], and since v_{5}(q^{k} − 1) = 0 when 4 ∤ k and v_{5}(q^{4ℓ} − 1) = a + v_{5}(ℓ) (Lemma 1.13),

v_{5}(|G|) = v_{5}((q^{24} − 1)(q^{20} − 1)(q^{12} − 1)(q^{8} − 1)) = 4a + 1 .

Thus S ∈ Syl_{p}(G).

|γ|_{T} | = ord_{p}(q) ≥ 2 and [γ, ψ_{q_{0}}] = Id. The first is clear, and the second holds since γ = c_{g_{0}} where ψ_{q_{0}}(g_{0}) = g_{0}. C_{S}(Ω_{1}(A)) = A by the above description of the action of W_{0} on A.

C_{A}(O_{p′}(W_{0})) = 1 since w_{0} ∈ O_{5′}(W_{0}) and C_{A}(w_{0}) = 1.

A free ⟨γ⟩-orbit in Σ. The subset {±(ε_{1} + ε_{3}), ±(ε_{2} + ε_{4})} ⊆ Σ is a free ⟨γ⟩-orbit.

Aut_{W_{0}}(A) ∩ Aut_{sc}(A) ≤ ⟨γ|_{A}⟩. Recall that |γ|_{T} | = 4 and |Aut_{sc}(A)| = 4 · 5^{k} for some k, and W_{0} acts faithfully on A. So if this is not true, then there is an element of order 5 in Z(W_{0}), which is impossible by the above description of W_{0}.

Aut_{G}(A) = Aut_{W_{0}}(A) by (3).

N_{Aut(A)}(Aut_{W_{0}}(A)) ≤ Aut_{sc}(A)Aut_{W_{0}}(A). For j = 1, 2, 3, 4, let A_{j} < A be the cyclic subgroup of all elements as in (2) where u_{k} = 1 for k ≠ j. The group W_{0} contains as subgroup C_{2} ≀ Σ_{4}: the group which permutes pairs of coordinates up to sign. So each of the four subgroups A_{j} is the reflection subgroup of some reflection in W_{0}.

For each φ ∈ C_{Aut(A)}(Aut_{W_{0}}(A)), φ(A_{j}) = A_{j} for each j, and φ(a) = a^{n_{j}} for some n_{j} ∈ (ℤ/5^{a})^{×}. Also, n_{1} = n_{2} = n_{3} = n_{4} since the A_{j} are permuted transitively by elements of W_{0}, and hence φ ∈ Aut_{sc}(A).

Now assume φ ∈ N_{Aut(A)}(Aut_{W_{0}}(A)). Since φ centralizes Z(W_{1}) = ⟨w_{0}⟩ = ⟨diag(i, i, i, i)⟩ (since diag(i, i, i, i) ∈ Z(Aut(A))), c_{φ}|_{W_{1}} ∈ Inn(W_{1}), and we can assume (after composing by an appropriate element of W_{1}) that [φ, W_{1}] = 1. So c_{φ} ∈ Aut(W_{0}) has the form c_{φ}(g) = gχ(g), where g ∈ W_{0}/W_{1} ≅ Σ_{6} is the class of g ∈ W_{0}, and where χ ∈ Hom(W_{0}/W_{1}, Z(W_{1})) ≅ Hom(Σ_{6}, C_{4}) ≅ C_{2} is some homomorphism. Since (w_{0})^{2} inverts the torus T, composition with (w_{0})^{2} does not send reflections (in A) to reflections, and so we must have c_{φ} = Id_{W_{0}}. Thus φ ∈ C_{Aut(A)}(Aut_{W_{0}}(A)) = Aut_{sc}(A) (modulo Aut_{W_{0}}(A)). □

The following lemma now reduces the proof of Theorem B to the cases considered in Section 5, together with certain small cases handled at the end of this section. As before, when p is a prime and p ∤ n, ord_{p}(n) denotes the multiplicative order of n in 𝔽_{p}^{×}.

Proposition 6.8. Fix an odd prime p, and assume G ∈ 𝔏𝔦𝔢(q_{0}) is of universal type for some prime q_{0} ≠ p. Fix S ∈ Syl_{p}(G), and assume S is nonabelian. Then there is a prime q_{0}^{∗} ≠ p, a group G^{∗} ∈ 𝔏𝔦𝔢(q_{0}^{∗}) of universal type, and S^{∗} ∈ Syl_{p}(G^{∗}), such that F_{S}(G) ≅ F_{S^{∗}}(G^{∗}), and one of the following holds: either

(a) G^{∗} has a σ-setup which satisfies Hypotheses 5.1 and 5.11, G^{∗} ≅ 𝔾(q^{∗}) or ^{2}𝔾(q^{∗}) where q^{∗} is a power of q_{0}^{∗}, and

(a.1) −Id ∉ W and G^{∗} is a Chevalley group, or (a.2) −Id ∈ W and ord_{p}(q^{∗}) is even where W is the Weyl group of 𝔾; or

(b) p = 3, q_{0}^{∗} = 2, G ≅ ^{3}D_{4}(q) or ^{2}F_{4}(q) for q some power of q_{0}, and G^{∗} ≅ ^{3}D_{4}(q^{∗}) or ^{2}F_{4}(q^{∗}) for q^{∗} some power of 2.

Moreover, if p = 3 and G^{∗} = F_{4}(q^{∗}) where q^{∗} is a power of q_{0}^{∗}, then we can assume q_{0}^{∗} = 2. In all cases, we can choose G^{∗} to be either one of the groups listed in Proposition 1.10(a–e), or one of E_{7}(q^{∗}) or E_{8}(q^{∗}) for some q^{∗} ≡ −1 (mod p).

Proof. We can assume that G = 𝔾(q) is one of the groups listed in one of the five cases (a)–(e) of Proposition 1.10. In all cases except 1.10(c), we can also assume that G satisfies Hypotheses 5.11, with q_{0} = 2 if p = 3 and 𝔾 = F_{4}, and with q_{0} odd in cases (a) and (b) of 1.10. If G = SL_{n}(q) or Spin_{2n}^{±}(q) where p|(q − 1), or G is in case (d), then G satisfies Hypotheses 5.1 by Lemma 6.1. If G ≅ SL_{n}(q) or Spin_{2n}^{±}(q) where p ∤ (q − 1), then G satisfies Hypotheses 5.1 by Lemma 6.5. This leaves only case (c) in Proposition 1.10, which corresponds to case (b) here, and case (e) (p = 5, G = E_{8}(q), q ≡ ±2 (mod 5)) where G^{∗} satisfies Hypotheses 5.1 by Lemma 6.7.

We next show, in cases (a,b,d,e) of Proposition 1.10, that we can arrange for one of the conditions (a.1) or (a.2) to hold. If −Id ∉ W, then 𝔾 = A_{n}, D_{n} for n odd, or E_{6}, and G is a Chevalley group by the assumptions in cases (a,b,d) of Proposition 1.10. So (a.1) holds. If −Id ∈ W and ord_{p}(q) is even, then (a.2) holds. If −Id ∈ W, ord_{p}(q) is odd, and G = 𝔾(q) is a Chevalley group, then by Theorem 1.8(c), G ∼_{p} 𝔾(q^{∗}) for some q^{∗} = q_{0}^{c} such that ⟨q^{∗}⟩ = ⟨ − q⟩, and ord_{p}(q^{∗}) is even. So we can replace G by 𝔾(q^{∗}) in this last case, and (a.2) holds.

This leaves the case where −Id ∈ W, ord_{p}(q) is odd, and G is not a Chevalley group. By inspection, the first and third conditions both hold only when G = ^{2}D_{n}(q) for n even. So we are in the situation of Proposition 1.10(b), where we also assume q^{n} ≡ −1 (mod p). But then ord_{p}(q) is even, so this case cannot occur. □

We now consider the two families of groups which appear in Proposition 6.8(b): those not covered by Hypotheses 5.1.

Proposition 6.9. Let G be one of the groups ^{3}D_{4}(q) where q is a prime power prime to 3, 2F_{4}(2^{2m+1}) for m ≥ 0, or ^{2}F_{4}(2)′. Then the 3-fusion system of G is tame. If G ≅ ^{3}D_{4}(2^{n}) (n ≥ 1), ^{2}F_{4}(2^{2m+1}) (m ≥ 0), or ^{2}F_{4}(2)′, then κ_{G} is split surjective, and Ker(κ_{G}) is the subgroup of field automorphisms of order prime to 3.

Proof. Fix S ∈ Syl_{3}(G), and set F = F_{S}(G).

If G is the Tits group ^{2}F_{4}(2)′, then S is extraspecial of order 3^{3} and exponent 3, so Out(S) ≅ GL_{2}(3). Also, Out_{G}(S) ≅ D_{8} and Out_{Aut(G)}(S) ≅ SD_{16}, since the normalizer in 2F_{4}(2) of an element of order 3 (the element t_{4} in [Sh]) has the form SU_{3}(2) : 2 ≅ 3_{+}^{1+2} : SD_{16} by [Sh, Table IV] or [Ma1, Proposition 1.2]. Hence Out(F) ≤ N_{Out(S)}(Out_{G}(S))/Out_{G}(S) has order at most 2, and κ_{G} sends Out(G) ≅ C_{2} ([GrL, Theorem 2]) isomorphically to Out(F). If G = ^{2}F_{4}(2), then Out_{G}(S) ≅ SD_{16}, so Out(F) = 1 by a similar argument, and κ_{G} is an isomorphism between trivial groups.

Assume now that G ≅ ^{2}F_{4}(2^{n}) for odd n ≥ 3 or G ≅ ^{3}D_{4}(q) where 3 ∤ q. In order to describe the Sylow 3-subgroups of these groups, set ζ = e^{2πi/3}, R = ℤ[ζ], and 𝔭 = (1 − ζ)R. Let S_{k} be the semidirect product R/𝔭^{k} ⋊ C_{3}, where the quotient acts via multiplication by ζ. Explicitly, set

S_{k} = {(x, i) | x ∈ R/𝔭^{k}, i ∈ ℤ/3} and A_{k} = R/𝔭^{k} × {0},

where (x, i)(y, j) = (x + ζ^{i}y, i + j). Thus |S_{k}| = 3^{k+1}. Set s = (0, 1), so that s(x, 0)s^{−1} = (ζx, 0) for each x ∈ R/𝔭^{k}.

Assume k ≥ 3, so that A_{k} is the unique abelian subgroup of index three in S_{k}. Set S = S_{k} and A = A_{k} for short. We want to describe Out(S). Define automorphisms ξ_{a} (a ∈ (R/𝔭^{k})^{×}), ω, η, and ρ by setting

ξ_{a}(x, i) = (xa, i), η = ξ_{−1}, ω(x, i) = (−x, −i), ρ(x, i) = (x + λ(i), i).  (4)

Here, x ↦ x means complex conjugation, and λ(i) = 1 + ζ + … + ζ^{i−1}. Note, when checking that ρ is an automorphism, that λ(i) + ζ^{i}λ(j) = λ(i + j). Note that ρ^{3} ∈ Inn(S): it is (left) conjugation by (1 − ζ^{2}, 0).

Let Aut^{0}(S) ⊴ Aut(S) be the subgroup of automorphisms which induce the identity on S/[S, S] = S/[s, A], and set Out^{0}(S) = Aut^{0}(S)/Inn(S). Each element in s·[s, A] is conjugate to s, and thus each class in Out^{0}(S) is represented by an automorphism which sends s to itself, which is unique modulo ⟨c_{s}⟩. If φ ∈ Aut(S) and φ(s) = s, then φ|_{A} commutes with c_{s}, thus is R-linear under the identification A ≅ R/𝔭^{k}, and hence φ = ξ_{a} for some a ∈ 1 + 𝔭/𝔭^{k}. Moreover, since

(1 + 𝔭/𝔭^{k})^{×} = (1 + 𝔭^{2}/𝔭^{k})^{×} × ⟨ζ⟩ = (1 + 3R/𝔭^{k})^{×} × ⟨ζ⟩

as multiplicative groups (just compare orders, noting that the groups on the right have trivial intersection), each class in Out^{0}(S) is represented by ξ_{a} for some unique a ∈ 1 + 3R/𝔭^{k}.

Since the images of η, ω, and ρ generate Aut(S)/Aut^{0}(S) (the group of automorphisms of S/[s, A] ≅ C_{3}^{2} which normalize A/[s, A] ≅ C_{3}), this shows that Out(S) is generated by the classes of the automorphisms in (4). In fact, a straightforward check of the relations among them shows that

Out(S) ≅ (Out^{0}(S) ⋊ C_{2}_{[ω]}) × Σ_{3}_{[ρ],[η]} where Out^{0}(S) = {[ξ_{a}] | a ∈ (1 + 3R/𝔭^{k})^{×}}.

Also, ωξ_{a}ω^{−1} = ξā for a ∈ (1 + 3R/𝔭^{k})^{×}.

For each x ∈ 1 + 3R such that x ≡ x (mod 𝔭^{k}), we can write x = r + sζ with r, s ∈ ℤ, and then s(ζ − \overline{ζ}) ∈ 𝔭^{k}, so s ∈ 𝔭^{k−1}, and x ∈ r + s + 𝔭^{k} ⊆ 1 + 3ℤ + 𝔭^{k}. This proves that

C_{Out(S)}(ω) = {[ξ_{a}] | a ∈ ℤ} × ⟨[ω]⟩ × ⟨[ρ], [η]⟩.

For any group G with S ∈ Syl_{3}(G) and S ≅ S_{k}, Out_{G}(S) has order prime to 3, and hence is a 2-group and conjugate to a subgroup of ⟨ω, η⟩ ∈ Syl_{2}(Out(S)). If |Out_{G}(S)| = 4, then we can identify S with S_{k} in a way so that Out_{G}(S) = ⟨[ω], [η]⟩. Then

Out(F) ≤ N_{Out(S)}(⟨[ω], [η]⟩)/⟨[ω], [η]⟩
= C_{Out(S)}(⟨[ω], [η]⟩)/⟨[ω], [η]⟩ = {[ξ_{a}] | a ∈ ℤ} = ⟨[ξ_{2}]⟩ ,

where the first equality holds since O_{3}(Out(S)) has index four in Out(S).

We are now ready to look at the individual groups. Assume G = ^{2}F_{4}(q), where q = 2^{n} and n ≥ 3 is odd. By [St1, 3.2–3.6], Out(G) is cyclic of order n, generated by the field automorphism ψ_{2}. By the main theorem in [Ma1], there is a subgroup N_{G}(T_{8}) ≅ (C_{q+1})^{2} ⋊ GL_{2}(3), the normalizer of a maximal torus, which contains a Sylow 3-subgroup. Hence if we set k = v_{3}(q +1) = v_{3}(4^{n} −1) = 1+v_{3}(n) (Lemma 1.13), we have S ≅ S_{2k} ≅ (C_{3^{k}})^{2} ⋊C_{3}, and Out_{G}(S) = ⟨ω, η⟩ up to conjugacy. So Out(F) is cyclic, generated by ξ_{2} = κ_{G}(ψ_{2}). Since A ≅ (C_{3^{k}})^{2}, and since ξ_{−1} ∈ Out_{G}(S), |Out(F)| = |[ξ_{2}]| = 3^{k−1} where k−1 = v_{3}(n). Thus κ_{G} is surjective, and is split since the Sylow 3-subgroup of Out(G) ≅ C_{n} is sent isomorphically to Out(F).

Next assume G = ^{3}D_{4}(q), where q = 2^{n} for n ≥ 1. By [St1, 3.2–3.6], Out(G) is cyclic of order 3n, generated by the field automorphism ψ_{2} (and where the field automorphism ψ_{2^{n}} of order three is also a graph automorphism). Set k = v_{3}(q^{2} − 1) = v_{3}(2^{2n} − 1) = 1 + v_{3}(n) (Lemma 1.13). Then S ≅ S_{2k+1}: this follows from the description of the Sylow structure in G in [GL, 10-1(4)], and also from the description (based on [Kl]) of its fusion system in [O4, Theorem 2.8] (case (a.ii) of the theorem). Also, Out_{G}(S) = ⟨ω, η⟩ up to conjugacy. So Out(F) is cyclic, generated by ξ_{2} = κ_{G}(ψ_{2}). Since A ≅ C_{3^{k}} ×C_{3^{k+1}}, and since ξ_{−1} ∈ Out_{G}(S), |Out(F)| = |[ξ_{2}]| = 3^{k}. Thus κ_{G} is surjective, and is split since the Sylow 3-subgroup of Out(G) ≅ C_{3n} is sent isomorphically to Out(F).

By Theorem 1.8(b) and Lemma 1.11(a), for each prime power q with 3 ∤ q, the 3-fusion system of ^{3}D_{4}(q) is isomorphic to that of ^{3}D_{4}(2^{n}) for some n. By [O1, Theorem C], μ_{G} is injective in all cases. Thus the 3-fusion systems of all of these groups are tame. □

Appendix A. Injectivity of μ_{G} by Bob Oliver

Recall that for any finite group G and any S ∈ Syl_{p}(G),

μ_{G}: Out(L_{S}^{c}(G)) ⟶ Out(F_{S}(G))

is the homomorphism which sends the class of β ∈ Aut(L_{S}^{c}(G)) to the class of β_{S}|_{S}, where β_{S} is the induced automorphism of Aut_{L_{S}^{c}(G)}(S) = N_{G}(S)/O_{p′}(C_{G}(S)). We need to develop tools for computing Ker(μ_{G}), taking as starting point [AOV, Proposition 4.2].

As usual, for a finite group G and a prime p, a proper subgroup H < G is strongly pembedded in G if p||H|, and p ∤ |H ∩^{g}H| for g ∈ G∖H. The following properties of groups with strongly embedded subgroups will be needed.

Lemma A.1. Fix a prime p and a finite group G.

(a) If G contains a strongly p-embedded subgroup, then O_{p}(G) = 1.

(b) If H < G is strongly p-embedded, and K ⊴ G is a normal subgroup of order prime to p such that KH < G, then HK/K is strongly p-embedded in G/K.

Proof. (a) See, e.g., [AKO, Proposition A.7(c)].

(b) Assume otherwise. Thus there is g ∈ G∖HK such that p||(^{g}HK/K) ∩ (HK/K)|, and hence x ∈ ^{g}HK ∩ HK of order p. Then H ∩ K⟨x⟩ and ^{g}H ∩ K⟨x⟩ have order a multiple of p, so there are elements y ∈ H and z ∈ ^{g}H of order p such that y ≡ x ≡ z (mod K).

Since ⟨y⟩, ⟨z⟩ ∈ Syl_{p}(K⟨x⟩), there is k ∈ K such that ⟨y⟩ = ^{k}⟨z⟩. Then y ∈ H ∩ ^{kg}H, and kg ∉ H since k ∈ K and g ∉ HK. But this is impossible, since H is strongly pembedded. □

For the sake of possible future applications, we state the next proposition in terms of abstract fusion and linking systems. We refer to [AOV], and also to Chapters I.2 and III.4 in [AKO], for the basic definitions. Recall that if F is a fusion system over a finite p-group S, and P ≤ S, then

• P is F-centric if C_{S}(Q) ≤ Q for each Q which is F-conjugate to P;

• P is fully normalized in F if |N_{S}(P)| ≥ |N_{S}(Q)| whenever Q is F-conjugate to P; and

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

For any saturated fusion system F over a finite p-group S, set

Ẑ(F) = {E ≤ S | E elementary abelian, fully normalized in F,

E = Ω_{1}(Z(C_{S}(E))), Aut_{F}(E) has a strongly p-embedded subgroup} .

The following proposition is our main tool for proving that μ_{L} is injective in certain cases. Point (a) will be used to handle the groups Spin_{n}^{±}(q), point (c) the linear and symplectic groups, and point (b) the exceptional Chevalley groups.

Proposition A.2. Fix a saturated fusion system F over a p-group S and an associated centric linking system L. Let E_{1}, …, E_{k} ∈ Ẑ(F) be such that each E ∈ Ẑ(F) is F-conjugate to E_{i} for some unique i. For each i, set P_{i} = C_{S}(E_{i}) and Z_{i} = Z(P_{i}). Then the following hold.

(a) If k = 0 (Ẑ(F) = ∅), then Ker(μ_{L}) = 1.

(b) If k = 1, E_{1} ⊴ S, and Aut_{F}(Ω_{1}(Z(S))) = 1, then Ker(μ_{L}) = 1.

(c) Assume, for each (g_{i})_{i=1}^{k} ∈ ∏_{i=1}^{k} C_{Z_{i}}(Aut_{S}(P_{i})), that there is an element g ∈ C_{Z(S)}(Aut_{F}(S)) such that g_{i} ∈ g · C_{Z_{i}}(Aut_{F}(P_{i})) for each i. Then Ker(μ_{L}) = 1.

(d) If α ∈ Aut(L) is the identity on Aut_{L}(S), and on Aut_{L}(P_{i}) for each 1 ≤ i ≤ k, then α = Id_{L}.

Proof. We first prove point (d). The other three points then follow quickly from that together with [AOV, Proposition 4.2].

We will need to refer a few times to the extension axiom for fusion systems, as stated, e.g., in [AKO, Proposition I.2.5]. As one special case, this says that for P ≤ S and P C_{S}(P) ≤ Q ≤ N_{S}(P), each automorphism in N_{Aut_{F} (P)}(Aut_{Q}(P)) extends to one in Aut_{F}(Q) (a consequence of the Sylow theorems when F = F_{S}(G) for S ∈ Syl_{p}(G)).

(d) Fix α ∈ Aut(L) such that α_{S} = Id_{Aut_{L}(S)}. By [AOV, Proposition 4.2], there are elements g_{P} ∈ C_{Z(P)}(Aut_{S}(P)), defined for each P ∈ Ob(L) which is fully normalized, such that

(i) α_{P} ∈ Aut(Aut_{L}(P)) is conjugation by [[g_{P}]]_{P}; and

(ii) α_{P} = Id if and only if g_{P} ∈ C_{Z(P)}(Aut_{F}(P)).

Note that if we are in an abstract linking system, [[g_{P}]]_{P} ∈ Aut_{L}(P) should be replaced by δ_{P} (g_{P}). Furthermore, for each such P and each ψ ∈ Aut_{L}(P),

α_{P} (ψ) = ψ ⟺ π(ψ)(g_{P}) = g_{P} ,  (1)

where π: L ⟶ F denotes the canonical functor (so π([[g]]) = c_{g} if L = L_{S}^{c}(G) and F = F_{S}(G)). By (i) above, α_{P} (ψ) = ψ if and only if ψ commutes with [[g_{P}]]_{P} in Aut_{L}(P), and this is equivalent to π(ψ)(g_{P}) = g_{P} by axiom (C) in the definition of a linking system (see, e.g., [AKO, Definition III.4.1]) and since (g ↦ [[g]]_{P}) is injective. We leave it as an easy exercise to check this when L = L_{S}^{c}(G) and ψ = [[h]]_{P} for some h ∈ N_{G}(P) (note that [h, g_{P}] ∈ Z(P) since g_{P} ∈ Z(P)).

Now assume α_{P_{i}} is the identity on Aut_{L}(P_{i}) for each 1 ≤ i ≤ k. If α ≠ Id_{L}, then by Alperin’s fusion theorem for linking systems (see [AOV, Theorem 4.1]), there is Q < S such that α_{Q} ≠ Id, while α is the identity on Mor_{L}(P, P^{∗}) for all P, P^{∗} ∈ Ob(L) such that |P |, |P^{∗}| > |Q|. Also, for each Q^{∗} ∈ Q^{F}, there is (by Alperin’s fusion theorem again) an isomorphism χ ∈ Iso_{L}(Q, Q^{∗}) which is a composite of isomorphisms each of which extends to an isomorphism between strictly larger subgroups, and hence is such that α(χ) = χ. Thus

Q^{∗} ∈ Q^{F}  ⟹  α_{Q^{∗}} ≠ Id .  (2)

Set E = Ω_{1}(Z(Q)). Let φ ∈ Hom_{F}(N_{S}(E), S) be such that φ(E) is fully normalized (cf. [AKO, Lemma I.2.6(c)]). Then N_{S}(Q) ≤ N_{S}(E), so |N_{S}(φ(Q))| ≥ |N_{S}(Q)|, and φ(Q) is fully normalized since Q is. Since α_{Q^{∗}} ≠ Id by (2), we can replace Q by Q^{∗} and E by E^{∗}, and arrange that Q and E are both fully normalized in F (and Q is still F-essential).

We will show that Q = C_{S}(E) and E ∈ Ẑ(F). Then E ∈ (E_{i})^{F} for some unique 1 ≤ i ≤ k, and Q ∈ (P_{i})^{F} by the extension axiom (and since E and E_{i} are both fully centralized). But then α_{P_{i}} ≠ Id by (2), contradicting the original assumption about α. We conclude that α = Id, finishing the proof of (d).

Set Γ = Aut_{F}(Q), and set

Γ_{0} = C_{Γ}(E) = {φ ∈ Aut_{F}(Q) | φ|_{E} = Id_{E}} ⊴ Γ
Γ_{1} = ⟨φ ∈ Γ | φ = φ|_{Q} for some φ ∈ Hom_{F}(R, S), R > Q⟩ .

Then Aut_{S}(Q) ≤ Γ_{1}, since each element of Aut_{S}(Q) extends to N_{S}(Q) and N_{S}(Q) > Q (see [Sz1, Theorem 2.1.6]). Hence

Γ_{0}Γ_{1} = O^{p}(Γ_{0}) · Aut_{S}(Q) · Γ_{1} = O^{p}(Γ_{0})Γ_{1} .

For each φ ∈ Γ_{0} of order prime to p, φ|_{Z(Q)} = Id_{Z(Q)} since φ is the identity on E = Ω_{1}(Z(Q)) (cf. [G, Theorem 5.2.4]). Thus g_{Q} ∈ C_{Z(Q)}(O^{p}(Γ_{0})). If φ ∈ Aut_{F}(Q) extends to φ ∈ Hom_{F}(R, S) for some R > Q, then by the maximality of Q, α(ψ) = ψ for each ψ ∈ Mor_{L}(R, S) such that π(ψ) = φ, and since α commutes with restriction (it sends inclusions to themselves), α_{Q} is the identity on ψ|_{Q,Q} ∈ π_{Q}^{−1}(φ). So by (1), φ(g_{Q}) = g_{Q}. Thus φ(g_{Q}) = g_{Q} for all φ ∈ Γ_{1}. Since α_{Q} ≠ Id by assumption, there is some φ ∈ Aut_{F}(Q) such that φ(g_{Q}) ≠ g_{Q} (by (1) again), and we conclude that

g_{Q} ∈ C_{Z(Q)}(Γ_{0}Γ_{1}) and Γ_{0}Γ_{1} < Γ = Aut_{F}(Q) .  (3)

Set Q^{∗} = N_{C_{S}(E)}(Q) ≥ Q. Then Aut_{Q^{∗}}(Q) = Γ_{0} ∩ Aut_{S}(Q) ∈ Syl_{p}(Γ_{0}) since Aut_{S}(Q) ∈ Syl_{p}(Γ), and by the Frattini argument, Γ = N_{Γ}(Aut_{Q^{∗}}(Q))Γ_{0}. If Q^{∗} > Q, then for each φ ∈ N_{Γ}(Aut_{Q^{∗}}(Q)), φ extends to φ ∈ Aut_{F}(Q^{∗}) by the extension axiom. Thus N_{Γ}(Aut_{Q^{∗}}(Q)) ≤ Γ_{1} in this case, so Γ = Γ_{1}Γ_{0}, contradicting (3). We conclude that Q^{∗} = N_{C_{S}(E)}(Q) = Q, and hence that C_{S}(E) = Q (cf. [Sz1, Theorem 2.1.6]).

The homomorphism Γ = Aut_{F}(Q) ⟶ Aut_{F}(E) induced by restriction is surjective by the extension axiom, so Aut_{F}(E) ≅ Γ/Γ_{0}. By [AKO, Proposition I.3.3(b)], Γ_{1}/Inn(Q) is strongly p-embedded in Γ/Inn(Q) = Out_{F}(Q); and Γ_{0}Γ_{1} < Γ by (3). Also, p ∤ |Γ_{0}/Inn(Q)|, since otherwise we would have Γ_{1} ≥ N_{Γ}(T) for some T ∈ Syl_{p}(Γ_{0}), in which case Γ_{1}Γ_{0} ≥ N_{Γ}(T)Γ_{0} = Γ by the Frattini argument. Thus Γ_{1}Γ_{0}/Γ_{0} is strongly p-embedded in Γ/Γ_{0} ≅ Aut_{F}(E) by Lemma A.1(b).

Now, Ω_{1}(Z(C_{S}(E))) = Ω_{1}(Z(Q)) = E, and thus E ∈ Ê(F). We already showed that this implies (d).

(c) Now assume that the hypothesis in (c) holds, and fix [α] ∈ Ker(μ_{L}). By [AOV, Proposition 4.2], there is α ∈ Aut(L) in the class [α] such that α_{S} = Id. For each 1 ≤ i ≤ k, let g_{P_{i}} ∈ C_{Z(P_{i})}(Aut_{S}(P_{i})) be as in the proof of (d). By assumption, there is g ∈ C_{Z(S)}(Aut_{F}(S)) such that g_{P_{i}} ≡ g (mod C_{Z(P_{i})}(Aut_{F}(P_{i})) for each i.

Let β ∈ Aut(L) be conjugation by [[g]]_{S} ∈ Aut_{S}(L) and its restrictions (or by δ_{S}(g) if L is an abstract linking system). Upon replacing α by β^{−1} ∘ α and hence g_{P_{i}} by g^{−1}g_{P_{i}} for each i, we can arrange that g_{P_{i}} ∈ C_{Z(P_{i})}(Aut_{F}(P_{i})) for each i, and hence by (ii) that α_{P_{i}} = Id for each i. Then α = Id by (d), so [α] = 1. Thus Ker(μ_{L}) = 1, proving (c).

(a) This is a special case of (c).

(b) If k = 1, E_{1} ⊴ S, and Aut_{F}(Ω_{1}(Z(S))) = 1, then the group Out_{F}(S) of order prime to p acts trivially on Ω_{1}(Z(S)), and hence acts trivially on Z(S) (cf. [G, Theorem 5.2.4]). Also, P_{1} = C_{S}(E_{1}) ⊴ S, so C_{Z_{1}}(Aut_{S}(P_{1})) = Z(S) = C_{Z(S)}(Aut_{F}(S)), and Ker(μ_{L}) = 1 by (c). □

A.1. Classical groups of Lie type in odd characteristic. Throughout this subsection, we fix an odd prime power q and an integer n ≥ 1. We want to show Ker(μ_{G}) = 1 when G is one of the quasisimple classical groups of universal type over 𝔽_{q}. By Theorem 1.8(d), we need not consider the unitary groups.

Proposition A.3. Fix an odd prime power q. Let G be isomorphic to one of the quasisimple groups SL_{n}(q), Sp_{n}(q) (n = 2m), or Spin_{n}^{±}(q) (n ≥ 3). Then Ker(μ_{G}) = 1.

Proof. Let V, 𝔟, and Ĝ = Aut(V, 𝔟) be such that G = [Ĝ, Ĝ] if G ≅ Sp_{n}(q) or G ≅ SL_{n}(q), and G/⟨z⟩ = [Ĝ, Ĝ] for some z ∈ Z(Ĝ) if G ≅ Spin_{n}^{±}(q) (where z ∈ Z(G)). Thus V is a vector space of dimension n over the field K = 𝔽_{q}, 𝔟 is a trivial, symplectic, or quadratic form, and Ĝ is one of the groups GL_{n}(q), Sp_{2n}(q), or O_{n}^{±}(q).

Fix S ∈ Syl_{2}(G), and set F = F_{S}(G). Set Ẑ = Ẑ(F) for short.

Case 1: Assume G = Spin(V, 𝔟), where 𝔟 is nondegenerate and symmetric. Set Z = Z(G), and let z ∈ Z be such that G/⟨z⟩ = Ω(V, 𝔟). We claim that Ẑ = ∅ in this case, and hence that Ker(μ_{G}) = 1 by Proposition A.2(a).

Fix an elementary abelian 2-subgroup E ≤ G where E ≥ Z. Let V = ⨁_{i=1}^{m} V_{i} be the decomposition as a sum of eigenspaces for the action of E on V. Fix indices j, k ∈ {1, …, m} such that either dim(V_{j}) ≥ 2, or the subspaces have the same discriminant (modulo squares). (Since dim(V) ≥ 3, this can always be done.) Then there is an involution γ ∈ SO(V, 𝔟) such that γ(V_{i}) = V_{i} for all i, γ|_{V_{i}} = Id for i ≠ j, k, det(γ|_{V_{j}}) = det(γ|_{V_{k}}) = −1, and such that the (−1)-eigenspace of γ has discriminant a square. This last condition ensures that γ ∈ Ω(V, 𝔟) (cf. [LO, Lemma A.4(a)]), so we can lift it to g ∈ G. Then for each x ∈ E, c_{g}(x) = x if x has the same eigenvalues on V_{j} and V_{k}, and c_{g}(x) = zx otherwise (see, e.g., [LO, Lemma A.4(c)]). Since z is fixed by all elements of Aut_{F}(E), c_{g} ∈ O_{2}(Aut_{F}(E)), and hence Aut_{F}(E) has no strongly 2-embedded subgroups by Lemma A.1(a). Thus E ∉ Ẑ.

Case 2: Now assume G is linear or symplectic, and fix S ∈ Syl_{2}(G). For each V = {V_{1}, …, V_{k}} such that V = ⨁_{i=1}^{k} V_{i}, and such that V_{i} ⊥ V_{j} for i ≠ j if G is symplectic, set

E(V) = {φ ∈ G | φ|_{V_{i}} = ±Id for each i}.

We claim that each subgroup in Ẑ has this form. To see this, fix E ∈ Ẑ, and let V = {V_{1}, …, V_{k}} be the eigenspaces for the nonzero characters of E. Then E ≤ E(V), V = ⨁_{i=1}^{k} V_{i}, and this is an orthogonal decomposition if G is symplectic. Also, CĜ(E) is the product of the groups Aut(V_{i}, 𝔟|_{V_{i}}). Since E = Ω_{1}(Z(P)) where P = C_{S}(E), E contains the 2-torsion in the center of C_{G}(E), and thus E = E(V). Furthermore, the action of P on each V_{i} must be irreducible (otherwise Ω_{1}(Z(P)) > E), so dim(V_{i}) is a power of 2 for each i.

Again assume E = E(V) ∈ Ẑ for some V. Then AutĜ(E) is a product of symmetric groups: if V contains n_{i} subspaces of dimension i for each i ≥ 1, then AutĜ(E(V)) ≅ ∏_{i≥1} Σ_{n_{i}}. Each such permutation can be realized by a self map of determinant one (if G is linear), so Aut_{G}(E) = AutĜ(E). Since Aut_{G}(E) contains a strongly 2-embedded subgroup by definition of Ẑ (and since a direct product of groups of even order contains no strongly 2-embedded subgroup), Aut_{G}(E) = AutĜ(E) ≅ Σ_{3}.

Write n = dim(V) = 2^{k_{0}} + 2^{k_{1}} + … + 2^{k_{m}}, where 0 ≤ k_{0} < k_{1} < ⋯ < k_{m}. There is an (orthogonal) decomposition V = ⨁_{i=0}^{m} V_{i}, where S acts irreducibly on each V_{i}, and where dim(V_{i}) = 2^{k_{i}} (see [CF, Theorem 1]). For each 1 ≤ i ≤ m, fix an (orthogonal) decomposition W_{i} of V_{i} whose components have dimensions 2^{k_{i−1}}, 2^{k_{i−1}}, 2^{k_{i−1}+1}, …, 2^{k_{i}−1}, and set

V_{i} = {V_{j} | j ≠ i} ∪ W_{i}

and E_{i} = E(V_{i}). Thus V_{i} contains exactly three subspaces of dimension 2^{k_{i−1}}, and the dimensions of the other subspaces are distinct. Hence Aut_{G}(E_{i}) ≅ Σ_{3}, and E_{i} ∈ Ẑ. Conversely, by the above analysis (and since the conjugacy class of E ∈ Ẑ is determined by the dimensions of its eigenspaces), each subgroup in Ẑ is G-conjugate to one of the E_{i}.

For each 1 ≤ i ≤ m, set P_{i} = C_{S}(E_{i}) and Z_{i} = Z(P_{i}) (so E_{i} = Ω_{1}(Z_{i})). Since each element of N_{G}(P_{i}) ≤ N_{G}(E_{i}) permutes members of V_{i} of equal dimension, and the elements of N_{S}(P_{i}) do so only within each of the V_{j}, we have

Z_{i} = {z ∈ G | z|_{X} = λ_{X}^{(z)}Id_{X} for all X ∈ V_{i}, some λ_{X}^{(z)} ∈ O_{2}(𝔽_{q}^{×})}
C_{Z_{i}}(Aut_{S}(P_{i})) = {z ∈ Z_{i} | λ_{X_{i}}^{(z)} = λ_{X′_{i}}^{(z)}}
C_{Z_{i}}(Aut_{G}(P_{i})) = {z ∈ Z_{i} | λ_{X_{i}}^{(z)} = λ_{X′_{i}}^{(z)} = λ_{V_{i−1}}^{(z)}} ,

(4)

where X_{i}, X′_{i}, and V_{i−1} are the three members of the decomposition V_{i} of dimension 2^{k_{i−1}} (and X_{i}, X′_{i} ∈ W_{i}).

Fix (g_{i})_{i=1}^{m} ∈ ∏_{i=1}^{m} C_{Z_{i}}(Aut_{S}(P_{i})). Then g_{i} ∈ C_{Z_{i}}(Aut_{G}(P_{i})) if and only if λ_{V_{i−1}}^{(g_{i})} = λ_{X_{i}}^{(g_{i})}. Choose g ∈ Ĝ such that g|_{V_{i}} = η_{i} · Id for each i, where the η_{i} ∈ O_{2}(𝔽_{q}^{×}) are chosen so that η_{i}/η_{i−1} = λ_{X_{i}}^{(g_{i})}/λ_{V_{i−1}}^{(g_{i})} for each 1 ≤ i ≤ m. If G is linear, then det(g) = θ^{2^{k_{0}}} for some θ ∈ O_{2}(𝔽_{q}^{×}), and upon replacing g by g ∘ θ^{−2^{k_{0}}/n}Id_{V} (recall k_{0} = v_{2}(n)) we can assume g ∈ G. Then g ∈ C_{Z(S)}(Aut_{G}(S)) since it is a multiple of the identity on each V_{i} and has 2-power order. By construction and (4), g ≡ g_{i} (mod C_{Z_{i}}(Aut_{G}(P_{i}))) for each i; so Ker(μ_{G}) = 1 by Proposition A.2(c). □

A.2. Exceptional groups of Lie type in odd characteristic. Throughout this subsection, q_{0} is an odd prime, and q is a power of q_{0}. We show that Ker(μ_{G}) = 1 when G is one of the groups G_{2}(q), F_{4}(q), E_{6}(q), E_{7}(q), or E_{8}(q) and is of universal type.

The following proposition is a special case of [GLS3, Theorem 2.1.5], and is stated and proven explicitly in [O2, Proposition 8.5]. It describes, in many cases, the relationship between conjugacy classes and normalizers in a connected algebraic group and those in the subgroup fixed by a Steinberg endomorphism.

Proposition A.4. Let G be a connected algebraic group over 𝔽_{q_{0}}, let σ be a Steinberg endomorphism of G, and set G = C_{G}(σ). Let H ≤ G be any subgroup, and let H be the set of G-conjugacy classes of subgroups G-conjugate to H. Let N_{G}(H) act on π_{0}(C_{G}(H)) by sending g to xgσ(x)^{−1} (for x ∈ N_{G}(H)). Then there is a bijection

ω: H ⟶^{≅} π_{0}(C_{G}(H))/N_{G}(H),

defined by setting ω([^{x}H]) = [x^{−1}σ(x)] whenever ^{x}H ≤ C_{G}(σ). Also, for each x ∈ G such that xH ≤ G, Aut_{G}(^{x}H) is isomorphic to the stabilizer of [x^{−1}σ(x)] ∈ π_{0}(C_{G}(H))/C_{G}(H) under the action of Aut_{G}(H) on this set.

Since we always assume G is of universal type in this section, the group G = C_{G}(σ) of Proposition A.4 is equal to the group G = O^{q′_{0}}(C_{G}(σ)) of Definition 2.1 and Notation 2.2.

The following definitions will be useful when applying Proposition A.4. For any finite group G, set

SE(G) = {H ≤ G | H has a strongly 2-embedded subgroup}
δ(G) = \begin{cases}min\{[G : H] | H ∈ SE(G)\} & if SE(G) ≠ ∅ \\ ∞ & if SE(G) = ∅. \end{cases}

Thus by Proposition A.4, if H < G is such that |π_{0}(C_{G}(H))| > δ(Out_{G}(H)), then no subgroup H^{∗} ≤ C_{G}(σ) which is G-conjugate to H has the property that Aut_{C_{G}(σ)}(H^{∗}) has a strongly 2-embedded subgroup. The next lemma provides some tools for finding lower bounds for δ(G).

Lemma A.5. (a) For any finite group G, δ(G) ≥ |O_{2}(G)| · δ(G/O_{2}(G)).

(b) If G = G_{1} × G_{2} is finite, and δ(G_{i}) < ∞ for i = 1, 2, then

δ(G) = min{δ(G_{1}) · η(G_{2}) , δ(G_{2}) · η(G_{1})} ,

where η(G_{i}) is the smallest index of any odd order subgroup of G_{i}.

(c) If δ(G) < ∞, and there is a faithful 𝔽_{2}[G]-module V of rank n, then

2^{v_{2}(|G|)−[n/2]} | δ(G).

(d) More concretely, δ(GL_{3}(2)) = 28, δ(GL_{4}(2)) = 112, δ(GL_{5}(2)) = 2^{8} · 7 · 31, and δ(SO_{4}^{+}(2)) = 2 = δ(SO_{4}^{−}(2)). Also, 2^{4} ≤ δ(SO_{6}^{+}(2)) < ∞ and 2^{6} ≤ δ(SO_{7}(2)) < ∞.

Proof. (a) If H ∈ SE(G), then H ∩O_{2}(G) = 1 by Lemma A.1(a). Hence there is a subgroup H^{∗} ≤ G/O_{2}(G) isomorphic to H, and

[G : H] = |O_{2}(G)| · [G/O_{2}(G) : H^{∗}] ≥ |O_{2}(G)| · δ(G/O_{2}(G)) .

(b) If a finite group H has a strongly 2-embedded subgroup, then so does its direct product with any odd order group. Hence δ(G) ≤ δ(G_{i})η(G_{3−i}) for i = 1, 2.

Assume H ≤ G has a strongly 2-embedded subgroup K < H. Set H_{i} = H ∩ G_{i} for i = 1, 2. Since all involutions in H are H-conjugate (see [Sz2, 6.4.4]), H_{1} and H_{2} cannot both have even order. Assume |H_{2}| is odd. Let pr_{1} be projection onto the first factor. If pr_{1}(K) = pr_{1}(H), then there is x ∈ (H∖K) ∩ H_{2}, and this commutes with all Sylow 2-subgroups of H since they lie in G_{1}, contradicting the assumption that K is strongly 2embedded in H. Thus pr_{1}(K) < pr_{1}(H). Then pr_{1}(H) has a strongly 2-embedded subgroup by Lemma A.1(b), and hence

[G : H] = [G_{1} : pr_{1}(H)] · [G_{2} : H_{2}] ≥ δ(G_{1}) · η(G_{2}) .

So δ(G) ≥ δ(G_{i})η(G_{3−i}) for i = 1 or 2.

(c) This follows from [OV, Lemma 1.7(a)]: if H < G has a strongly 2-embedded subgroup, T ∈ Syl_{2}(H), and |T | = 2^{k}, then dim(V) ≥ 2k.

(d) The formulas for δ(SO_{4}^{±}(2)) hold since SO_{4}^{+}(2) ≅ Σ_{3}≀C_{2} contains a subgroup isomorphic to C_{3}^{2} ⋊ C_{4} and SO_{4}^{−}(2) ≅ Σ_{5} a subgroup isomorphic to A_{5}. Since 4|δ(GL_{3}(2)) by (c), and since 7|δ(GL_{3}(2)) (there are no subgroups of order 14 or 42), we have 28|δ(GL_{3}(2)), with equality since Σ_{3} has index 28. The last two (very coarse) estimates follow from (c), and the 6- and 7-dimensional representations of these groups.

Fix n = 4, 5, and set G_{n} = GL_{n}(2). Assume H ≤ G_{n} has a strongly embedded subgroup, where 7||H| or 31||H|. By (c), 2^{4}|δ(G_{4}) and 2^{8}|δ(G_{5}), and thus 8 ∤ |H|. If H is almost simple, then H ≅ A_{5} by Bender’s theorem (see [Sz2, Theorem 6.4.2]), contradicting the assumption about |H|. So by the main theorem in [A1], H must be contained in a member of one of the classes C_{i} (1 ≤ i ≤ 8) defined in that paper. One quickly checks that since (7 · 31, |H|) ≠ 1, H is contained in a member of C_{1}. Thus H is reducible, and since O_{2}(H) = 1, either H is isomorphic to a subgroup of GL_{3}(2) × GL_{n−3}(2), or n = 5 and H < GL_{4}(2). By (b) and since 7||δ(GL_{3}(2)), we must have H ≅ Σ_{3} × (C_{7} ⋊ C_{3}), in which case |H| < 180 = |GL_{2}(4)|. Thus 7|δ(G_{n}) for n = 4, 5, and 31|δ(G_{5}). Since GL_{4}(2) contains a subgroup isomorphic to GL_{2}(4) ≅ C_{3} × A_{5}, we get δ(G_{4}) = 2^{4} · 7 and δ(G_{5}) = 2^{8} · 7 · 31. □

We illustrate the use of the above proposition and lemma by proving the injectivity of μ_{G} when G = G_{2}(q).

Proposition A.6. If G = G_{2}(q) for some odd prime power q, then Ker(μ_{G}) = 1.

Proof. Assume q is a power of the prime q_{0}, set G = G_{2}(𝔽_{q_{0}}), and fix a maximal torus T. We identify G = C_{G}(ψ_{q}), where ψ_{q} is the field automorphism, and acts via (t ↦ t^{q}) on T. Fix S ∈ Syl_{2}(G), and set Ẑ = Ẑ(F_{S}(G)).

Let E ≅ C_{2}^{2} be the 2-torsion subgroup of T. By Proposition 2.5, C_{G}(E) = T ⟨θ⟩ where θ ∈ N_{G}(T) inverts the torus. Thus by Proposition A.4, there are two G-conjugacy classes of subgroups G-conjugate to E, represented by E^{±} (E^{+} = E), where Aut_{G}(E^{±}) = Aut(E^{±}) ≅ Σ_{3} and C_{G}(E^{±}) = (C_{q∓1})^{2} ⋊ C_{2}. The subgroups in one of these classes have centralizer in S isomorphic to C_{2}^{3}, hence are not in Ẑ, while those in the other class do lie in Ẑ. The latter also have normalizer of order 12(q ± 1)^{2} and hence of odd index in G, and thus are normal in some choice of Sylow 2-subgroup.

By [Gr, Table I], for each nontoral elementary abelian 2-subgroup E ≤ G, rk(E) = 3, C_{G}(E) = E, and Aut_{G}(E) ≅ GL_{3}(2). By Proposition A.4, and since δ(Aut_{G}(E)) = 28 > |C_{G}(E)| by Lemma A.5, Aut_{G}(E) contains no strongly 2-embedded subgroup, and thus E ∉ Ẑ.

Thus Ẑ is contained in a unique G-conjugacy class of subgroups of rank 2, and Ker(μ_{G}) = 1 by Proposition A.2(b). □

Throughout the rest of this subsection, fix an odd prime power q, and let 𝔾 be one of the groups F_{4}, E_{6}, E_{7}, or E_{8}.

Hypotheses A.7. Assume G = 𝔾(𝔽_{q_{0}}) and G ≅ 𝔾(q), where q is a power of the odd prime q_{0}, and where 𝔾 = F_{4}, E_{6}, E_{7}, or E_{8} and is of universal type. Fix a maximal torus T < G.

(I) Set T_{(2)} = {t ∈ T | t^{2} = 1}. Let 2A and 2B denote the two G-conjugacy classes of noncentral involutions in G, as defined in [Gr, Table VI], except that when 𝔾 = E_{7}, they denote the classes labelled 2B and 2C, respectively, in that table. For each elementary abelian 2-subgroup E < G, define

𝔮_{E}: E ⟶ 𝔽_{2}

by setting 𝔮(x) = 0 if x ∈ 2B ∪ {1}, and 𝔮(x) = 1 if x ∈ 2A ∪ (Z(G)∖1).

(II) Assume G = C_{G}(ψ_{q}), where ψ_{q} is the field endomorphism with respect to some root structure with maximal torus T. Thus ψ_{q}(t) = t^{q} for all t ∈ T. Fix S ∈ Syl_{2}(G), and set Ẑ = Ẑ(F_{S}(G)).

By [Gr, Lemma 2.16], 𝔮_{T_{(2)}} is a quadratic form on T_{(2)} in all cases, and hence 𝔮_{E} is quadratic for each E ≤ T_{(2)}. In general, 𝔮_{E} need not be quadratic when E is not contained in a maximal torus. In fact, Griess showed in [Gr, Theorems 7.3, 8.2, & 9.2] that in many (but not all) cases, E is contained in a torus if and only if 𝔮_{E} is quadratic (cx(E) ≤ 2 in his terminology).

With the above choices of notation for noncentral involutions, all of the inclusions F_{4} ≤ E_{6} ≤ E_{7} ≤ E_{8} restrict to inclusions of the classes 2A and of the classes 2B. This follows since the forms are quadratic, and also (for E_{7} < E_{8}) from [Gr, Lemma 2.16(iv)].

Lemma A.8. Assume Hypotheses A.7, and let 𝔟 be the bilinear form associated to 𝔮. Define

V_{0} = {v ∈ T_{(2)} | 𝔟(v, T_{(2)}) = 0, 𝔮(v) = 0}
γ_{x} = (v ↦ v + 𝔟(v, x)x) ∈ Aut(T_{(2)}, 𝔮)  for x ∈ T_{(2)} with 𝔮(x) = 1, x ⊥̸ T_{(2)}

Then the following hold.

(a) Aut_{G}(T_{(2)}) = Aut(T_{(2)}, 𝔮).

(b) For each nonisotropic x ∈ T_{(2)}∖T_{(2)}^{⊥}, γ_{x} is the restriction to T_{(2)} of a Weyl reflection on

T. If α ∈ Σ is such that γ_{x} = w_{α}|_{T_{(2)}}, then θ_{α}(v) = (−1)^{𝔟(x,v)} for each v ∈ T_{(2)}.

(c) If 𝔾 = E_{r} (r = 6, 7, 8), then 𝔮 is nondegenerate (V_{0} = 0), and the restriction to T_{(2)} of each Weyl reflection is equal to γ_{x} for some nonisotropic x ∈ T_{(2)}∖T_{(2)}^{⊥}.

(d) If 𝔾 = F_{4}, then dim(V_{0}) = 2, and 𝔮(v) = 1 for all v ∈ T_{(2)}∖V_{0}.

Proof. (a) Since Aut_{G}(T_{(2)}) has to preserve G-conjugacy classes, it is contained in Aut(T_{(2)}, 𝔮). Equality will be shown while proving (c) and (d).

(c) If 𝔾 = E_{r} for r = 6, 7, 8, then 𝔮 is nondegenerate by [Gr, Lemma 2.16]. Hence the only orthogonal transvections are of the form γ_{x} for nonisotropic x, and each Weyl reflection restricts to one of them. By a direct count (using the tables in [Brb]), the number of pairs {±α} of roots in 𝔾 (hence the number of Weyl reflections) is equal to 36, 63, or 120, respectively. This is equal to the number of nonisotropic elements in T_{(2)} ∖ T_{(2)}^{⊥} =

T_{(2)} ∖ Z(G) (see the formula in [Ta, Theorem 11.5] for the number of isotropic elements). So all transvections are restrictions of Weyl reflections, and Aut_{G}(T_{(2)}) = Aut(T_{(2)}, 𝔮).

(d) Assume 𝔾 = F_{4}. Then dim(V_{0}) = 2 and 𝔮^{−1}(1) = T_{(2)}∖V_{0} by [Gr, Lemma 2.16]. Thus |Aut(T_{(2)}, 𝔮)| = 4^{2} · |GL_{2}(2)|^{2} = 2^{6} · 3^{2} = \frac{1}{2}|W | (see [Brb, Planche VIII]), so Aut_{W} (T_{(2)}) = Aut(T_{(2)}, 𝔮) since W also contains −Id.

There are three conjugacy classes of transvections γ ∈ Aut(T_{(2)}, 𝔮): one of order 36 containing those where γ|_{V_{0}} ≠ Id (and hence [γ, T_{(2)}] ≤ V_{0}), and two of order 12 containing those where γ|_{V_{0}} = Id (one where [γ, T_{(2)}] ≤ V_{0} and one where [γ, T_{(2)}] ≰ V_{0}). Since there are two W -orbits of roots (long and short), each containing 12 pairs ±α, the corresponding Weyl reflections must restrict to the last two classes of transvections, of which one is the set of all γ_{x} for x ∈ T_{(2)} ∖ V_{0}.

(b) We showed in the proofs of (c) and (d) that each orthogonal transvection γ_{x} is the restriction of a Weyl reflection. If γ_{x} = w_{α}|_{T_{(2)}} for some root α ∈ Σ, then θ_{α} ∈ Hom(T , 𝔽_{q_{0}}^{×}) (Lemma 2.4(c)), so [T_{(2)} : Ker(θ_{α}|_{T_{(2)}})] ≤ 2. Also, Ker(θ_{α}) ≤ C_{T} (w_{α}) by Lemma 2.4(e), so Ker(θ_{α}|_{T_{(2)}}) ≤ C_{T_{(2)}}(w_{α}) = C_{T_{(2)}}(γ_{x}) = x^{⊥}, with equality since [T_{(2)} : x^{⊥}] = 2. Since θ_{α}(T_{(2)}) ≤ {±1}, it follows that θ_{α}(v) = (−1)^{𝔟(x,v)} for each v ∈ T_{(2)}. □

We are now ready to list the subgroups in Ẑ(𝔾(q)) in all cases. The proof of the following lemma will be given at the end of the section.

Lemma A.9. Let G = 𝔾(𝔽_{q_{0}}) and G = 𝔾(q) be as in Hypotheses A.7. Assume E ∈ Ẑ(G). Then either 𝔾 ≠ E_{7}, rk(E) = 2, and 𝔮_{E} = 0; or 𝔾 = E_{7}, Z = Z(G) ≅ C_{2}, and E = Z × E_{0} where rk(E_{0}) = 2 and 𝔮_{E_{0}} = 0. In all cases, Aut_{G}(E) ≅ Σ_{3}.

Proof. This will be shown in Lemmas A.14 and A.15. □

The next two lemmas will be needed to apply Proposition A.2(b) to these groups. The first is very elementary.

Lemma A.10. Let V be an 𝔽_{2}-vector space of dimension k, and let 𝔮: V ⟶ 𝔽_{2} be a quadratic form on V. For m ≥ 1 such that k > 2m, the number of totally isotropic subspaces of dimension m in V is odd.

Proof. This will be shown by induction on m, starting with the case m = 1. Since k ≥ 3, there is an orthogonal splitting V = V_{1} ⊥ V_{2} where V_{1}, V_{2} ≠ 0. Let k_{i} be the number of isotropic elements in V_{i} (including 0), and set n_{i} = |V_{i}|. The number of isotropic elements in V is then k_{1}k_{2} + (n_{1} − k_{1})(n_{2} − k_{2}), and is even since the n_{i} are even. The number of 1-dimensional isotropic subspaces is thus odd.

Now fix m > 1 (such that k > 2m), and assume the lemma holds for subspaces of dimension m − 1. For each isotropic element x ∈ V, a subspace E ≤ V of dimension m containing x is totally isotropic if and only if E ≤ x^{⊥} and E/⟨x⟩ is isotropic in x^{⊥}/⟨x⟩ with the induced quadratic form. By the induction hypothesis, and since

2 · dim(E/⟨x⟩) = 2(m − 1) < k − 2 ≤ dim(x^{⊥}/⟨x⟩),

the number of isotropic subspaces of dimension m which contain x is odd. Upon taking the sum over all x, and noting that each subspace has been counted 2^{m} − 1 times, we see that the number of isotropic subspaces of dimension m is odd. □

Lemma A.11. Assume Hypotheses A.7(I). Let σ be a Steinberg endomorphism of G such that for some ε = ±1, σ(t) = t^{εq} for each t ∈ T. Set G = C_{G}(σ). Fix E ≤ T_{(2)} of rank 2 such that 𝔮(E) = 0. Then the set of subgroups of G which are G-conjugate to E, and the set of subgroups which are G-conjugate to E, both have odd order and contain all totally isotropic subgroups of rank 2 in T_{(2)}.

Proof. Let 𝔛 ⊇ 𝔛 be the sets of subgroups of G which are G-conjugate to E or G-conjugate to E, respectively. Let 𝔛_{0} be the subset of all totally isotropic subgroups of T_{(2)} of rank 2. If 𝔮 is nondegenerate, then by Witt’s theorem (see [Ta, Theorem 7.4]), Aut_{W} (T_{(2)}) = Aut(T_{(2)}, 𝔮) permutes 𝔛_{0} transitively, and hence all elements in 𝔛_{0} are G-conjugate to E by Lemma 2.9. If in addition, dim(T_{(2)}) ≥ 5, then |𝔛_{0}| is odd by Lemma A.10. Otherwise, by Lemma A.8(c,d), 𝔾 = F_{4} and 𝔛_{0} = {E}. Thus in all cases, 𝔛_{0} ⊆ 𝔛 and |𝔛_{0}| is odd.

Assume 𝔾 = E_{6}. Then C_{G}(T_{(2)}) = T by Proposition 2.5. Consider the conjugation action of T_{(2)} on 𝔛, and let 𝔛_{1} be its fixed point set. Since T_{(2)} ≤ G by the assumptions on σ, this action also normalizes 𝔛. For F ∈ 𝔛_{1}, either the action of T_{(2)} fixes F pointwise, in which case F ∈ 𝔛_{0}, or there are x, y ∈ F such that [x, T_{(2)}] = 1 and [y, T_{(2)}] = ⟨x⟩. In particular, c_{y} ∈ Aut_{G}(T_{(2)}) = SO(T_{(2)}, 𝔮). For each v ∈ T_{(2)} such that [y, v] = x, 𝔮(v) = 𝔮(vx) and 𝔮(x) = 0 imply x ⊥ v, so x ⊥ T_{(2)} since T_{(2)} is generated by those elements. This is impossible since 𝔮 is nondegenerate by Lemma A.8(c), and thus 𝔛_{1} = 𝔛_{0}.

Now assume 𝔾 = F_{4}, E_{7}, or E_{8}. Then −Id ∈ W, so there is θ ∈ N_{G}(T) which inverts T. Then C_{G}(T_{(2)}) = T ⟨θ⟩. By the Lang-Steinberg theorem, there is g ∈ G such that g^{−1}σ(g) ∈ θT; then σ(gtg^{−1}) = gt^{∓q}g^{−1} for t ∈ T, and thus σ acts on gT g^{−1} via t ↦ t^{∓q}. We can thus assume T was chosen so that G ∩ T = C_{T} (σ) contains the 4-torsion subgroup T_{(4)} ≤ T. Let 𝔛_{1} ⊆ 𝔛 be the fixed point set of the conjugation action of T_{(4)} on 𝔛. For F ∈ 𝔛_{1}, either the action of T_{(4)} fixes F pointwise, in which case F ∈ 𝔛_{0}, or there are x, y ∈ F such that [x, T_{(4)}] = 1 and [y, T_{(4)}] = ⟨x⟩. But then [F, T _{(4)}^{∗}] = 1 for some

T _{(4)}^{∗} < T_{(4)} of index two, [F, T_{(2)}] = 1 implies F ≤ T_{(2)}⟨θ⟩; and F ≤ T_{(2)} since no element in

T_{(4)}∖T_{(2)} commutes with any element of T_{(2)}θ. So 𝔛_{1} = 𝔛_{0} in this case.

Thus in both cases, 𝔛_{0} is the fixed point set of an action of a 2-group on 𝔛 which normalizes 𝔛. Since |𝔛_{0}| is odd, so are |𝔛| and |𝔛|. □

We are now ready to prove:

Proposition A.12. Fix an odd prime power q. Assume G is a quasisimple group of universal type isomorphic to G_{2}(q), F_{4}(q), E_{6}(q), E_{7}(q), or E_{8}(q). Then Ker(μ_{G}) = 1.

Proof. This holds when G ≅ G_{2}(q) by Proposition A.6, so we can assume Hypotheses A.7. Let 𝔛 be the set of all elementary abelian 2-subgroups E ≤ G such that either 𝔾 ≠ E_{7}, rk(E) = 2, and 𝔮_{E} = 0; or 𝔾 = E_{7}, rk(E) = 3, and E = Z(G) × E_{0} where 𝔮_{E_{0}} = 0. By Lemma A.11, |𝔛| is odd. In all cases, by Lemma A.9, Ẑ(G) ⊆ 𝔛. By Proposition A.2(a,b), to prove μ_{G} is injective, it remains to show that if Ẑ(G) ≠ ∅, then Ẑ(G) has odd order and is contained in a single G-conjugacy class, and Aut_{G}(Z(S)) = 1.

Fix E ∈ 𝔛 such that E ≤ T_{(2)}. We first claim that if 𝔾 = F_{4}, E_{6}, or E_{7}, then C_{G}(E) is connected, and hence all elements in 𝔛 are G-conjugate to E by Proposition A.4. If 𝔾 = E_{7}, then C_{G}(E) is connected by [Gr, Proposition 9.5(iii)(a)]. If 𝔾 = F_{4} or E_{6}, then for x ∈ E, C_{G}(x) ≅ Spin_{9}(𝔽_{q_{0}}) or 𝔽_{q_{0}} ×_{C_{4}} Spin_{10}(𝔽_{q_{0}}), respectively (see [Gr, Table VI]). Since the centralizer of each element in the simply connected groups Spin_{9}(𝔽_{q_{0}}) and Spin_{10}(𝔽_{q_{0}}) is connected [St3, Theorem 8.1], C_{G}(E) is connected in these cases.

Now assume 𝔾 = E_{8}. We can assume G = C_{G}(ψ_{q}), where ψ_{q} is the field automorphism; in particular, ψ_{q}(t) = t^{q} for t ∈ T. Fix x, y ∈ E such that E = ⟨x, y⟩. By [Gr, Lemma 2.16(ii)], (T_{(2)}, 𝔮) is of positive type (has a 4-dimensional totally isotropic subspace). Hence E^{⊥} = E × V_{1} × V_{2}, where dim(V_{i}) = 2 and 𝔮(V_{i}∖1) = 1 for i = 1, 2, and V_{1} ⊥ V_{2}. Thus (𝔮_{E^{⊥}})^{−1}(1) = ⋃_{i=1}^{2}((V_{i}∖1) × E), and by Lemma A.8(b,c), these are the restrictions to T_{(2)} of Weyl reflections w_{α} for α ∈ Σ such that E ≤ Ker(θ_{α}). Also, C_{W} (E) ≅ W (D_{4}) ≀ C_{2}. By Proposition 2.5, C_{G}(E)^{0} has type D_{4} × D_{4} and |π_{0}(C_{G}(E))| = 2. More precisely, C_{G}(E) = (H_{1} ×_{E} H_{2})⟨δ⟩, where H_{i} ≅ Spin_{8}(𝔽_{q_{0}}) and Z(H_{i}) = E for i = 1, 2, and conjugation by δ ∈ N_{G}(T) exchanges V_{1} and V_{2} and hence exchanges H_{1} and H_{2}.

By Proposition A.4, the two connected components in the centralizer give rise to two G-conjugacy classes of subgroups which are G-conjugate to E, represented by E and gEg^{−1} where g^{−1}σ(g) lies in the nonidentity component of C_{G}(E). Then C_{G}(E) contains a subgroup Spin_{8}^{+}(q)×_{C_{2}^{2}}Spin_{8}^{+}(q) with index 8 (the extension by certain pairs of diagonal automorphisms of the Spin_{8}^{+}(q)-factors, as well as an automorphism which switches the factors). So E = Z(T) for T ∈ Syl_{2}(C_{G}(E)), and E ∈ Ẑ(G). Also, gyg^{−1} ∈ C_{G}(gEg^{−1}) if and only if y ∈ C_{G}(E) and τ̃(y) = y where τ̃ = c_{g^{−1}σ(g)} ∘σ. Then τ̃ switches the central factors in C_{G}(E), and the group C_{C_{G}(E)}(τ̃) splits as a product of E times the group of elements which are

invariant after lifting τ̃ to the 4-fold cover Spin_{8}(𝔽_{q_{0}})≀C_{2}. Since gEg^{−1} intersects trivially with the commutator subgroup of C_{G}(gEg^{−1}), Ω_{1}(Z(T)) > gEg^{−1} for any T ∈ Syl_{2}(C_{G}(gEg^{−1})) (since Z(T) ∩ [T, T] ≠ 1); and thus gEg^{−1} ∉ Ẑ(G). Thus Ẑ(G) is the G-conjugacy class of E, and has odd order by Lemma A.11.

Thus, in all cases, if Ẑ(G) is nonempty, it has odd order and is contained in one Gconjugacy class. Also, Z(S) ≤ C_{E}(Aut_{S}(E)) < E for E ∈ Ẑ(G), so either |Z(S)| = 2, or 𝔾 = E_{7}, Z(S) ≅ C_{2}^{2}, and the three involutions in Z(S) belong to three different G-conjugacy classes. Hence Aut_{G}(Z(S)) = 1. □

It remains to prove Lemma A.9, which is split into the two Lemmas A.14 and A.15. The next proposition will be used to show that certain elementary abelian subgroups are not in Ẑ.

Proposition A.13. Assume Hypotheses A.7. Let E ≤ T_{(2)} and x ∈ T_{(2)}∖E be such that the orbit of x under the C_{W} (E)-action on T_{(2)} has odd order. Then no subgroup of S which is G-conjugate to E is in Ẑ. More generally, if E ≥ E is also elementary abelian, and is such that x is not C_{G}(E)-conjugate to any element of E, then for any L ⊴ G which contains {gxg^{−1} | g ∈ G} ∩ G, no subgroup of S which is G-conjugate to E is in Ẑ.

Proof. In [O2], an elementary abelian p-subgroup E < G is called pivotal if O_{p}(Aut_{G}(E)) = 1, and E = Ω_{1}(Z(P)) for some P ∈ Syl_{p}(C_{G}(E)). In particular, by Lemma A.1(a), the subgroups in Ẑ are all pivotal. Note that T_{(2)} ≤ G by Hypotheses A.7. By [O2, Proposition 8.9], no subgroup satisfying the above conditions can be pivotal, and hence they cannot be in Ẑ. □

In the next two lemmas, we show that in all cases, E ∈ Ẑ implies rk(E) = 2 and 𝔮_{E} = 0 if 𝔾 ≠ E_{7}, with a similar result when 𝔾 = E_{7}. We first handle those subgroups which are toral (contained in a maximal torus in G), and then those which are not toral. By a 2A^{k}-subgroup or subgroup of type 2A^{k} (2B^{k}-subgroup or subgroup of type 2B^{k}) is meant an elementary abelian 2-subgroup of rank k all of whose nonidentity elements are in class 2A (class 2B).

Lemma A.14. Assume Hypotheses A.7. Fix some E ∈ Ẑ which is contained in a maximal torus of G. Then either 𝔾 ≠ E_{7}, rk(E) = 2, and 𝔮_{E} = 0; or 𝔾 = E_{7}, Z = Z(G) ≅ C_{2}, and E = Z × E_{0} where rk(E_{0}) = 2 and 𝔮_{E_{0}} = 0. In all cases, Aut_{G}(E) ≅ Σ_{3}.

Proof. Set Z = O_{2}(Z(G)) ≤ T_{(2)}. Thus |Z| = 2 if 𝔾 = E_{7}, and |Z| = 1 otherwise. Recall that Aut_{G}(T_{(2)}) = Aut_{G}(T_{(2)}) = Aut(T_{(2)}, 𝔮) by Lemmas 2.9 and A.8(a).

The following notation will be used to denote isomorphism types of quadratic forms over 𝔽_{2}. Let [n]^{±} denote the isomorphism class of a nondegenerate form of rank n. When n is even, [n]^{+} denotes the hyperbolic form (with maximal Witt index), and [n]^{−} the form with nonmaximal Witt index. Finally, a subscript “(k)” denotes sum with a k-dimensional trivial form. By [Gr, Lemma 2.16], 𝔮_{T_{(2)}} has type [2]_{(2)}^{−}, [6]^{−}, [7], or [8]^{+} when 𝔾 = F_{4}, E_{6}, E_{7}, or E_{8}, respectively.

Fix E ≤ T_{(2)}; we want to determine whether E can be G-conjugate to an element of Ẑ. Set E_{1} = E ∩ E^{⊥} (the orthogonal complement taken with respect to 𝔮), and set E_{0} = Ker(𝔮_{E_{1}}). Note that E_{1} > E_{0} if 𝔾 = E_{7} (E ≥ Z).

Assume first that E_{0} = 1. If 𝔾 = F_{4}, then T_{(2)} ∩ 2B is a C_{W} (E)-orbit of odd order. If 𝔾 = E_{r} and E_{1} = 1, then E × E^{⊥}, E^{⊥} is C_{W} (E)-invariant, and hence there is 1 ≠ x ∈ E^{⊥} whose C_{W} (E)-orbit has odd order. If 𝔾 = E_{r} and rk(E_{1}) = 1, then E ∩ E^{⊥} = E_{1}, there is an odd number of involutions in E^{⊥}∖E_{1} of each type (isotropic or not), and again there is 1 ≠ x ∈ E^{⊥} whose C_{W} (E)-orbit has odd order. In all cases, x has the property that C_{W} (⟨E, x⟩) has odd index in C_{W} (E). So by Proposition A.13, no subgroup of G which is G-conjugate to E can be in Ẑ.

Thus E_{0} ≠ 1. Set k = rk(E_{0}). Then

|π_{0}(C_{G}(E))| = |C_{W} (E)/⟨w_{α} | α ∈ Σ, E ≤ Ker(θ(α))⟩|  (Proposition 2.5)
≤ |C_{W} (T_{(2)})| · |C_{SO(T_{(2)},𝔮)}(E)/⟨γ_{v} | v ∈ 2A ∩ E^{⊥}⟩|  (Lemma A.8(a,b))
≤ |C_{W} (T_{(2)})| · |C_{SO(T_{(2)},𝔮)}(E_{0}^{⊥})| · |C_{SO(E_{0}^{⊥},𝔮)}(E)/⟨γ_{v} | v ∈ 2A ∩ E^{⊥}⟩| .  (5)

The first factor is easily described:

|C_{W} (T_{(2)})| = 2^{ε} where ε = \begin{cases}1 & if −Id ∈ W (if 𝔾 = F_{4}, E_{7}, E_{8}) \\ 0 & if −Id ∉ W (if 𝔾 = E_{6}). \end{cases}  (6)

We next claim that

|C_{SO(T_{(2)},𝔮)}(E_{0}^{⊥})| ≤ 2^{\binom{k}{2}} ,  (7)

with equality except possibly when 𝔾 = F_{4}. To see this, let F_{1} < T_{(2)} be a subspace complementary to E_{0}^{⊥}. Each α ∈ C_{Aut(T_{(2)})}(E_{0}^{⊥}) has the form α(x) = xψ(x) for some ψ ∈ Hom(F_{1}, E_{0}), and α is orthogonal if and only if x ⊥ ψ(x) for each x. The space of such homomorphisms has dimension at most \binom{k}{2} (corresponding to symmetric k ×k matrices with zeros on the diagonal); with dimension equal to \binom{k}{2} if dim(F_{1}) = dim(E_{0}) (which occurs if 𝔮 is nondegenerate).

Write (E_{0})^{⊥} = E × F_{2}, where E^{⊥} = E_{0} × F_{2} and the form 𝔮_{F_{2}} is nondegenerate. By [Ta, Theorem 11.41], SO(F_{2}, 𝔮_{F_{2}}) is generated by transvections unless 𝔮_{F_{2}} is of type [4]^{+}, in which case the reflections generate a subgroup of SO(F_{2}, 𝔮_{F_{2}}) ≅ Σ_{3} ≀ C_{2} isomorphic to Σ_{3} × Σ_{3}. Also, F_{2} is generated by nonisotropic elements except when 𝔮_{F_{2}} is of type [2]^{+}, and when this is the case, all automorphisms of (E_{0})^{⊥} which induce the identity on E and on (E_{0})^{⊥}/E_{0} are composites of transvections. (Look at the composites γ_{vx} ∘ γ_{v} for v ∈ F_{2} and x ∈ E_{0}.) Hence

|C_{SO(E_{0}^{⊥},𝔮)}(E)/⟨γ_{v} | v ∈ 2A ∩ E^{⊥}⟩| ≤ 2^{η}

where η = 1 if 𝔮_{E^{⊥}} has type [4]_{(k)}^{+}, η = k if 𝔮_{E^{⊥}} has type [2]_{(k)}^{+}, and η = 0 otherwise. Together with (5), (6), and (7), this proves that

|π_{0}(C_{G}(E))| ≤ 2^{\binom{k}{2}+ε+η}  where ε ≤ 1.  (8)

Now, N_{G}(E) ≤ C_{G}(E)^{0}N_{G}(T) by the Frattini argument: each maximal torus which contains E lies in C_{G}(E)^{0} and hence is C_{G}(E)^{0}-conjugate to T. So each element of Aut_{G}(E) is represented by a coset of T in N_{G}(T), and can be chosen to lie in G by Lemma 2.9. Thus the action described in Proposition A.4 which determines the automizers Aut_{G}(E^{∗}) for E^{∗} G-conjugate to E is the conjugation action of Aut_{G}(E) on the set of conjugacy classes in π_{0}(C_{G}(E)). In particular, this action is not transitive, since the identity is fixed.

Set ℓ = rk(E/E_{0}) − 1 if 𝔾 = E_{7} and ℓ = rk(E/E_{0}) otherwise. Every automorphism of E which induces the identity on E_{0}Z and on E/E_{0} is orthogonal, and hence the restriction of an element of O_{2}(C_{W} (E)). Thus |O_{2}(Out_{G}(E))| ≥ 2^{kℓ}. If E^{∗} ∈ Z is G-conjugate to E, then since Aut_{G}(E^{∗}) has a strongly 2-embedded subgroup, 2^{kℓ} ≤ δ(Aut_{G}(E)) < |π_{0}(C_{G}(E))| by Proposition A.4 and Lemma A.5(a), with strict inequality since the action of N_{G}(E) on π_{0}(C_{G}(E)) is not transitive. Together with (8), and since ε ≤ 1, this implies that kℓ ≤ \binom{k}{2} + η ≤ \binom{k}{2} + k. Thus ℓ ≤ \frac{k+1}{2}, and ℓ ≤ \frac{k−1}{2} if η = 0. By definition, η = 0 whenever rk(E_{1}/E_{0}) = 1, which is the case if 𝔾 = E_{7} or ℓ is odd. Since 2k + ℓ ≤ 8, we are thus left with the following possibilities.

• If (k, ℓ) = (3, 2), then 𝔾 = E_{8}, E has form of type [2]_{(3)}^{+}, so E^{⊥} = E_{0} has trivial form, and η = 0. Thus kℓ ≰ \binom{k}{2} + η, so this cannot occur.

• If (k, ℓ) = (3, 1), then 𝔾 = E_{8}, E has form of type ∗ + 3, and rk(E) = rk(E_{1}) = 4. Then Aut_{G}(E) ≅ C_{2}^{3} ⋊ GL_{3}(2), so δ(Aut_{G}(E)) ≥ 2^{3} · 28 by Lemma A.5(a,d). Since |π_{0}(C_{G}(E))| ≤ 16, this case is also impossible.

• If (k, ℓ) = (4, 0), then 𝔾 = E_{8} and E = E_{0} is isotropic of rank 4. By Proposition 2.5 and Lemma A.8(c), C_{G}(E)^{0} = T. By [CG, Proposition 3.8(ii)], π_{0}(C_{G}(E)) is extraspecial of order 2^{7} and Aut_{G}(E) ≅ GL_{4}(2). (This is stated for subgroups of E_{8}(ℂ), but the same argument applies in our situation.) In particular, π_{0}(C_{G}(E)) has just 65 conjugacy classes. Since δ(GL_{4}(2)) = 112 by Lemma A.5(d), Proposition A.4 implies that Aut_{G}(E^{∗}) cannot have a strongly 2-embedded subgroup.

• If (k, ℓ) = (3, 0), then E = Z ×E_{0} where dim(E_{0}) = 3, and Aut_{G}(E) ≅ GL_{3}(2). If 𝔾 = E_{6}

or E_{7}, then E^{⊥} = E, and |π_{0}(C_{G}(E))| ≤ 16 by (8).

If 𝔾 = E_{8}, then (E^{⊥}, 𝔮_{E^{⊥}}) has type [2]_{(3)}^{+}. By the arguments used to prove (8),

|C_{W} (E)| = |C_{W} (T_{(2)})| · |C_{SO(T_{(2)},𝔮)}(E_{0}^{⊥})| · |C_{SO(E_{0}^{⊥},𝔮)}(E)| = 2 · 2^{3} · 2^{7} = 2^{11}.

Also, E^{⊥} contains exactly 8 nonisotropic elements, they are pairwise orthogonal, and hence determine 8 pairwise commuting transvections on T_{(2)}. These extend to 8 Weyl reflections which are pairwise commuting since no two can generate a dihedral subgroup of order 8 (this would imply two roots of different lengths). Hence by Proposition 2.5, C_{G}(E)^{0} has type (A_{1})^{8} and |π_{0}(C_{G}(E))| = 2^{11}/2^{8} = 2^{3}. Since δ(GL_{3}(2)) = 28 by Lemma A.5(c), this case cannot occur.

• If (k, ℓ) = (2, 0), then E = Z × E_{0} where dim(E_{0}) = 2. Then E is as described in the statement of the lemma. □

It remains to handle the nontoral elementary abelian subgroups.

Lemma A.15. Assume Hypotheses A.7. Let E ≤ G be an elementary abelian 2-group which is not contained in a maximal torus of G. Then E ∉ Ẑ.

Proof. To simplify notation, we write 𝕂 = 𝔽_{q_{0}}. Set Z = O_{2}(Z(G)) ≤ T_{(2)}. Thus |Z| = 2 if 𝔾 = E_{7}, and |Z| = 1 otherwise. The maximal nontoral subgroups of G are described in all cases by Griess [Gr].

(A) If 𝔾 = F_{4} or E_{6}, then by [Gr, Theorems 7.3 & 8.2], G contains a unique conjugacy class of maximal nontoral elementary abelian 2-subgroups, represented by W_{5} of rank five. There is a subgroup W_{2} ≤ W_{5} of rank two such that W_{5} ∩ 2A = W_{5}∖W_{2}. Also, Aut_{G}(E_{5}) = Aut(E_{5}, 𝔮_{E_{5}}): the group of all automorphisms of W_{5} which normalize W_{2}. A subgroup E ≤ W_{5} is nontoral if and only if it contains a 2A^{3}-subgroup.

When 𝔾 = F_{4}, we can assume W_{5} = T_{(2)}⟨θ⟩, where θ ∈ N_{G}(T) inverts the torus.

(B) If 𝔾 = E_{7}, then by [Gr, Theorem 9.8(i)], G contains a unique maximal nontoral elementary abelian 2-subgroup W_{6}, of rank six. For any choice of E_{6}(𝕂) < G, W_{5} < E_{6}(𝕂) (as just described) has rank 5, is nontoral since it contains a 2A^{3}-subgroup, and so we can take W_{6} = Z × W_{5}.

Each coset of Z of involutions in G∖Z contains one element of each class 2A and 2B. Together with the above description of E_{5}, this shows that all 2A^{2}-subgroups of W_{6} are contained in W_{5}. Hence for each nontoral subgroup E ≤ W_{6} which contains Z, E ∩ W_{5} is the subgroup generated by 2A^{2}-subgroups of E, thus is normalized by Aut_{G}(E), and so

Aut_{G}(E) ≅ Aut_{G}(E ∩ W_{5}) = Aut(E ∩ W_{5}, 𝔮_{E∩W_{5}}) ≅ Aut(E, 𝔮_{E})
Aut_{G}(W_{6}) ≅ Aut(W_{6}, 𝔮_{W_{6}}) ≅ C_{2}^{6} ⋊ (Σ_{3} × GL_{3}(2))

For Z ≤ E ≤ W_{6}, the subgroup E is nontoral exactly when it contains a 2A^{3}- subgroup. This is immediate from the analogous statement in (A) for E_{6}(𝕂).

(C) If 𝔾 = E_{8}, then by [Gr, Theorem 2.17], G contains two maximal elementary abelian subgroups W_{8} and W_{9}, neither of which is toral [Gr, Theorem 9.2]. An elementary abelian 2-subgroup E ≤ G is nontoral if and only if 𝔮_{E} is not quadratic or E has type 2B^{5} [Gr, Theorem 9.2].

We refer to [Gr, Theorem 2.17] for descriptions of W_{8} and W_{9}. There are subgroups F_{0} ≤ F_{1}, F_{2} ≤ W_{8} such that rk(F_{0}) = 2, rk(F_{1}) = rk(F_{2}) = 5, F_{1} ∩ F_{2} = F_{0}, and W_{8} ∩ 2A = (F_{1}∖F_{0}) ∪ (F_{2}∖F_{0}). Also, Aut_{G}(W_{8}) is the group of those automorphisms of W_{8} which leave F_{0} invariant, and either leave F_{1} and F_{2} invariant or exchange them.

We can assume that W_{9} = T_{(2)}⟨θ⟩, where θ ∈ N_{G}(T) inverts T. Also, W_{9}∖T_{(2)} ⊆ 2B. Hence T_{(2)} = ⟨W_{9} ∩ 2A⟩ is Aut_{G}(W_{9})-invariant. Each automorphism of W_{9} which is the identity on T_{(2)} is induced by conjugation by some element of order 4 in T, and thus Aut_{G}(W_{9}) is the group of all automorphisms whose restriction to T_{(2)} lies in Aut_{G}(T_{(2)}).

We next list other properties of elementary abelian subgroups of G, and of their centralizers and normalizers, which will be needed in the proof.

(D) If 𝔾 = E_{8}, E ≤ G, E ≅ C_{2}^{r}, and |E ∩ 2A| = m, then dim(C_{G}(E)) = 2^{8−r} + 2^{5−r}m − 8.

This follows from character computations: if 𝔤 denotes the Lie algebra of G = E_{8}(𝕂), then dim(C_{G}(E)) = dim(C_{𝔤}(E)) = |E|^{−1} ∑_{x∈E} χ_{𝔤}(x). By [Gr, Table VI], χ_{𝔤}(1) = dim(G) = 248, and χ_{𝔤}(x) = 24 or −8 when x ∈ 2A or 2B, respectively.

(E) If 𝔾 = E_{8}, E ≤ G is an elementary abelian 2-group, and E_{t} < E has index 2 and is

such that E∖E_{t} ⊆ 2B, then there is g ∈ G such that ^{g}E ≤ W_{9} = T_{(2)}⟨θ⟩ and ^{g}E_{t} ≤ T_{(2)}.

It suffices to prove this when E is maximal among such such pairs E_{t} < E. We can assume that E is contained in W_{8} or W_{9}.

If E ≤ W_{8}, then in the notation of (C), F_{0} ≤ E (since E is maximal), and either rk(E ∩ F_{i}) = 3 for i = 1, 2 and rk(E) = 6, or rk(E ∩ F_{i}) = 4 for i = 1, 2 and rk(E) = 7. These imply that |E ∩ 2A| = 8 or 24, respectively, and hence by (D) that dim(C_{G}(E_{t})) = 8 (C_{G}(E_{t})^{0} = T) and dim(C_{G}(E)) = 0. Hence in either case, if g ∈ G is such that ^{g}E_{t} ≤ T_{(2)}, then ^{g}E∖^{g}E_{t} ⊆ θT, and there is t ∈ T such that tgE ≤ T_{(2)}⟨θ⟩ = W_{9}.

If E ≤ W_{9}, set E_{2} = ⟨E ∩ 2A⟩. Then E_{2} ≤ E ∩ T_{(2)} and E_{2} ≤ E_{t}, so there is nothing to prove unless rk(E/E_{2}) ≥ 2. In this case, from the maximality of E, we see that E_{t} = E_{a} × E_{b}, where E_{a} ≅ C_{2}^{2} has type 2ABB, E_{b} is a 2B^{3}-group, and E_{a} ⊥ E_{b} with respect to the form 𝔮. Thus rk(E) = 6, |E ∩ 2A| = 8, and the result follows by the same argument as in the last paragraph.

(F) If 𝔾 = E_{8}, and E ≤ G is a nontoral elementary abelian 2-group, then either E contains

a 2A^{3}-subgroup, or E is G-conjugate to a subgroup of W_{9}.

Assume E ≤ W_{8} is nontoral and contains no 2A^{3}-subgroup. We use the notation F_{0} < F_{1}, F_{2} < W_{8} of (C). Set E_{i} = E ∩ F_{i} for i = 0, 1, 2. Then 𝔮_{E_{1}E_{2}} is quadratic: it is the orthogonal direct sum of 𝔮_{E_{0}}, 𝔮_{E_{1}/E_{0}}, and 𝔮_{E_{2}/E_{0}}, each of which is quadratic since rk(E_{i}/E_{0}) ≤ 2 for i = 1, 2 (E has no 2A^{3}-subgroup). Hence E > E_{1}E_{2} ≥ ⟨E ∩ 2A⟩ since E is nontoral, so E is conjugate to a subgroup of W_{9} by (E).

(G) Let E ≤ G be an elementary abelian 2-subgroup, and let E_{t} ≤ E be maximal among toral subgroups of E. Assume that E_{t}∩E_{t}^{⊥}∩2B = ∅, and that either rk(T)−rk(E_{t}) ≥ 2 or E_{t} ∩ E_{t}^{⊥} = 1. Then E ∉ Ẑ.

To see this, choose F ≥ F_{t} which is G-conjugate to E ≥ E_{t} and such that F_{t} = F ∩ T_{(2)}. By maximality, no element of F ∖F_{t} is C_{G}(F_{t})-conjugate to an element of T. If F_{t} ∩ F_{t}^{⊥} = 1, then some C_{W} (F_{t})-orbit in F_{t}^{⊥}∖1 has odd order. Otherwise, since 𝔮 is linear on F_{t} ∩ F_{t}^{⊥}, we have F_{t} ∩ F_{t}^{⊥} = ⟨y⟩ for some y ∈ 2A, in which case |𝔮_{F_{t}^{⊥}}^{−1}(0)| = |F_{t}^{⊥}|/2 is even since rk(F_{t}^{⊥}) ≥ rk(T) − rk(F_{t}) ≥ 2. So again, some C_{W} (F_{t})- orbit in F_{t}^{⊥}∖1 has odd order in this case. Point (G) now follows from Proposition A.13.

(H) Assume 𝔾 = E_{8}. Let 1 ≠ E_{0} ≤ E ≤ G be elementary abelian 2-subgroups, where rk(E) = 3, and E ∩ 2A = E_{0}∖1. Then

C_{G}(E) ≅ \begin{cases}E × F_{4}(𝕂) & if rk(E_{0}) = 3 \\ E × PSp_{8}(𝕂) & if rk(E_{0}) = 2 \\ E × PSO_{8}(𝕂) & if rk(E_{0}) = 1. \end{cases}

To see this, fix 1 ≠ y ∈ E_{0}, and identify C_{G}(y) ≅ SL_{2}(𝕂) ×_{C_{2}} E_{7}(𝕂). For each x ∈ E∖⟨y⟩, since x and xy are G-conjugate, x ≠ (1, b) for b ∈ E_{7}(𝕂). Thus x = (a, b) for some a ∈ SL_{2}(𝕂) and b ∈ E_{7}(𝕂) both of order 4, and (in the notation of [Gr, Table VI]) b is in class 4A or 4H since b^{2} ∈ Z(E_{7}(𝕂)). By (D) and [Gr, Table VI],

dim(C_{G}(E)) = \begin{cases}80 = dim(C_{E_{7}(𝕂)}(4H)) + 1 & if E has type 2AAA \\ 64 = dim(C_{E_{7}(𝕂)}(4A)) + 1 & if E has type 2ABB, \end{cases}

and thus x ∈ 2A if b ∈ 4H and x ∈ 2B if b ∈ 4A. Thus if E = ⟨y, x_{1}, x_{2}⟩, and x_{i} = (a_{i}, b_{i}), then ⟨a_{1}, a_{2}⟩ ≤ SL_{2}(𝕂) and ⟨b_{1}, b_{2}⟩ ≤ E_{7}(𝕂) are both quaternion of order 8. Point (H) now follows using the description in [Gr, Proposition 9.5(i)] of centralizers of certain quaternion subgroups of E_{7}(𝕂). When combined with the description in [Gr, Table VI] of C_{E_{7}(𝕂)}(4A), this also shows that

F ≅ C_{2}^{2} of type 2ABB ⟹ C_{G}(F)^{0} is of type A_{7}T^{1}  (9)

(i.e., C_{G}(F)^{0} ≅ (SL_{8}(𝕂) × 𝕂^{×})/Z, for some finite subgroup Z ≤ Z(SL_{8}(𝕂)) × 𝕂^{×}).

(I) If U < G is a 2A^{3}-subgroup, then C_{G}(U) = U × H, where H is as follows:

\begin{matrix}𝔾 & F_{4} & E_{6} & E_{7} & E_{8} \\ H & SO_{3}(𝕂) & SL_{3}(𝕂) & Sp_{6}(𝕂) & F_{4}(𝕂) \end{matrix}

When 𝔾 = E_{8}, this is a special case of (H). For x ∈ 2A ∩ F_{4}(𝕂), C_{E_{8}(𝕂)}(x) ≅ SL_{2}(𝕂) ×_{C_{2}} E_{7}(𝕂) by [Gr, 2.14]. Since C_{F_{4}(𝕂)}(x) ≅ SL_{2}(𝕂) ×_{C_{2}} Sp_{6}(𝕂), this shows that C_{E_{7}(𝕂)}(U) ≅ U × Sp_{6}(𝕂).

Similarly, C_{E_{8}(𝕂)}(y) ≅ SL_{3}(𝕂) ×_{C_{3}} E_{6}(𝕂) by [Gr, 2.14] again (where y is in class 3B in his notation). There is only one class of element of order three in F_{4}(𝕂) whose centralizer contains a central factor SL_{3}(𝕂) — C_{F_{4}(𝕂)}(y) ≅ SL_{3}(𝕂) ×_{C_{3}} SL_{3}(𝕂) for y of type 3C in F_{4}(𝕂) — and thus C_{E_{6}(𝕂)}(U) ≅ U × SL_{3}(𝕂).

If 𝔾 = F_{4}, then by [Gr, 2.14], for y ∈ 3C, C_{G}(y) ≅ SL_{3}(𝕂) ×_{C_{3}} SL_{3}(𝕂). Also, the involutions in one factor must all lie in the class 2A and those in the other in 2B. This, together with Proposition 2.5, shows that for U_{2} < U of rank 2, C_{G}(U_{2}) ≅ (T^{2} ×_{C_{3}} SL_{3}(𝕂))⟨θ⟩, where θ inverts a maximal torus. Thus C_{G}(U) = U × C_{SL_{3}(𝕂)}(θ), where by [Gr, Proposition 2.18], C_{SL_{3}(𝕂)}(θ) ≅ SO_{3}(𝕂). This finishes the proof of (I).

For the rest of the proof, we fix a nontoral elementary abelian 2-subgroup E < G. We must show that E ∉ Ẑ. In almost all cases, we do this either by showing that the hypotheses of (G) hold, or by showing that δ(Aut_{G}(E)) > |π_{0}(C_{G}(E))| (where δ(−) is as in Lemma A.5), in which case Aut_{G}(E) has no strongly 2-embedded subgroup by Proposition A.4, and hence E ∉ Ẑ.

By (A), (B), and (F), either E contains a 2A^{3}-subgroup of rank three, or 𝔾 = E_{8} and E is G-conjugate to a subgroup of W_{9}. These two cases will be handled separately.

Case 1: Assume first that E contains a 2A^{3}-subgroup U ≤ E. From the lists in (A,B,C) of maximal nontoral subgroups, there are the following possibilities.

\underline{𝔾 = F_{4}, E_{6}, or E_{7}}: By (A,B), we can write E = U × E_{0} × Z, where E_{0} is a 2B^{k} subgroup (some k ≤ 2) and UE_{0}∖E_{0} ⊆ 2A (and where Z = 1 unless 𝔾 = E_{7}). If k = 0, then E ∉ Ẑ by (G), so assume k ≥ 1. By (I), and since each elementary abelian 2-subgroup of SL_{3}(𝕂) and of Sp_{6}(𝕂) has connected centralizer, π_{0}(C_{G}(E)) ≅ U if 𝔾 = E_{6} or E_{7}. If 𝔾 = F_{4}, then by (I) again, and since the centralizer in SO_{3}(𝕂) ≅ PSL_{2}(𝕂) of any C_{2}^{k} has 2^{k} components, |π_{0}(C_{G}(E))| = 2^{3+k}.

By (A,B) again, Aut_{G}(E) is the group of all automorphisms which normalize E_{0} and UE_{0} and fix Z. Hence

|O_{2}(Aut_{G}(E))| = 2^{3k} and Aut_{G}(E)/O_{2}(Aut_{G}(E)) ≅ GL_{3}(2) × GL_{k}(2) .

So δ(Aut_{G}(E)) ≥ 2^{3k+3} > |π_{0}(C_{G}(E))| by Lemma A.5, and E ∉ Ẑ.

\underline{𝔾 = E_{8}}: By (I), C_{G}(U) = U ×H where H ≅ F_{4}(𝕂). Set E_{2} = E∩H, and let E_{0} = ⟨E_{2}∩2B⟩.

Set k = rk(E_{0}) and ℓ = rk(E_{2}/E_{0}).

If k = 0, then E_{2} has type 2A^{ℓ}, and E∖(U ∪ E_{2}) ⊆ 2B. So each maximal toral subgroup E_{t} < E has the form E_{t} = U_{1} × U_{2}, where rk(U_{1}) = 2, rk(U_{2}) ≤ 2, and E_{t} ∩ 2A = (U_{1} ∪ U_{2})∖1. The hypotheses of (G) thus hold, and so E^{∗} ∉ Ẑ.

Thus k = 1, 2. If ℓ ≤ 2, then E_{2} is toral, and

|π_{0}(C_{G}(E))| = 8 · |π_{0}(C_{H}(E_{2}))| ≤ 2^{3+k}

by formula (8) in the proof of Lemma A.14. (Note that ε = 1 and η = 0 in the notation of that formula.) If ℓ = 3, then |π_{0}(C_{G}(E))| = 2^{6+k} by the argument just given for F_{4}(𝕂). Also, Aut_{G}(E) contains all automorphisms of E which normalize E_{0}, and either normalize UE_{0} and E_{2} or (if ℓ = 3) exchange them: since in the notation of (C), each such automorphism extends to an automorphism of W_{8} which normalizes F_{1} and F_{2}. So |O_{2}(Aut_{G}(E))| ≥ 2^{k(3+ℓ)}, and Aut_{G}(E)/O_{2}(Aut_{G}(E)) ≅ GL_{3}(2) × GL_{k}(2) × GL_{ℓ}(2) or (if ℓ = 3) (GL_{3}(2) ≀ C_{2}) × GL_{k}(2). In all cases, δ(Aut_{G}(E)) ≥ 2^{3k+ℓk+3} > |π_{0}(C_{G}(E))|, so E ∉ Z.

Case 2: Now assume that 𝔾 = E_{8}, and that E is G-conjugate to a subgroup of W_{9}. To simplify the argument, we assume that E ≤ W_{9}, and then prove that no subgroup E^{∗} ∈ Ẑ can be G-conjugate to E. Recall that W_{9} = T_{(2)}⟨θ⟩, where θ ∈ N_{G}(T) inverts the torus and θT_{(2)} ⊆ 2B.

If E ∩ 2A = ∅, then rk(E) = 5. In this case, Aut_{G}(E) ≅ GL_{5}(2) and |C_{G}(E)| = 2^{15} [CG, Proposition 3.8]. (Cohen and Griess work in E_{8}(ℂ), but their argument also holds in our situation.) Since δ(GL_{5}(2)) > 2^{15} by Lemma A.5(d), no E^{∗} ∈ Ẑ can be G-conjugate to E.

Now assume E has 2A-elements, and set E_{2} = ⟨E ∩ 2A⟩. Then E_{2} ≤ T_{(2)} (hence 𝔮_{E_{2}} is quadratic) by the above remarks. Set E_{1} = E_{2}^{⊥} ∩ E_{2} and E_{0} = Ker(𝔮_{E_{1}}). If E_{0} = 1 and rk(E_{2}) ≠ 7, then by (G), no subgroup of S which is G-conjugate to E lies in Ẑ.

It remains to consider the subgroups E for which E_{0} ≠ 1 or rk(E_{2}) = 7. Information about |O_{2}(Aut_{G}(E))| and |π_{0}(C_{G}(E))| for such E is summarized in Table A.1. By the “type of 𝔮_{E}” is meant the type of quadratic form, in the notation used in the proof of Lemma A.14.

We first check that the table includes all cases. If rk(E/E_{2}) = 1, then E_{2} = E ∩ T_{(2)}, and the table lists all types which the form 𝔮_{E_{2}} can have. Note that since E_{2} is generated by nonisotropic vectors, 𝔮_{E_{2}} cannot have type [2]_{(k)}^{+}. If rk(E/E_{2}) = 2, then 𝔮_{E_{2}} is linear, and must be one of the three types listed. Since 𝔮_{E∩T_{(2)}} is quadratic and 𝔮_{E} is not, E_{2} has index at most 2 in E ∩ T_{(2)}.

We claim that

E, F < W_{9}, α ∈ Iso(E, F) such that α(E ∩ T_{(2)}) = F ∩ T_{(2)} and α(E ∩ 2A) = F ∩ 2A ⟹ α = c_{tg} for some t ∈ T and some g ∈ N_{G}(T) = G ∩ N_{G}(T).

(10)

By (C) and Witt’s theorem (see [Ta, Theorem 7.4]), there is g ∈ N_{G}(T) such that α|_{E∩T_{(2)}} = c_{g}, and we can assume g ∈ G by Lemma 2.9. Then ^{g}E∖^{g}(E ∩ T_{(2)}) ≤ θT since θT ∈

Case nr.rk(E/E_{2}) rk(E_{2}/E_{0}) rk(E_{0})\begin{matrix}𝔮_{E_{2}} \\ type \end{matrix} |π_{0}(C_{G}(E^{∗}))| |O_{2}(Aut_{G}(E^{∗}))| δ(Aut_{G}(E^{∗}))

1  1  7  0  [7] ≤ 2^{9}  2^{7}  ≥ 2^{13}
2  1  6^{+}  1 [6]_{(1)}^{+} ≤ 2^{9}  2^{13}  ≥ 2^{17}
3  1  5  1 [5]_{(1)} ≤ 2^{10}  2^{11}  ≥ 3 · 2^{13}
4  1  4^{+}  2 [4]_{(2)}^{+} ≤ 2^{10}  2^{14}  ≥ 2^{16}
5  1  4^{+}  1 [4]_{(1)}^{+} ≤ 2^{8}  2^{9}  ≥ 2^{10}
6  1  4^{−}  1 [4]_{(1)}^{−} ≤ 2^{7}  2^{9}  ≥ 2^{10}
7  1  3  2 [3]_{(2)} ≤ 2^{6}  2^{11}  ≥ 2^{12}
8  1  3  1 [3]_{(1)} ≤ 2^{5}  2^{7}  ≥ 2^{7}
9  1  2^{−}  2 [2]_{(2)}^{−} ≤ 2^{5}  2^{8}  ≥ 2^{9}
10  1  2^{−}  1 [2]_{(1)}^{−} ≤ 2^{4}  2^{5}  ≥ 2^{5}
11  2  1  3 [1]_{(3)} ≤ 2^{12}  2^{11}  ≥ 2^{14}
12  2  1  2 [1]_{(2)} ≤ 2^{8}  2^{8}  ≥ 2^{9}
13  2  1  1 [1]_{(1)} ≤ 2^{5}  2^{5}  ≥ 2^{5}

Table A.1

Z(N_{G}(T))/T, so α = c_{tg} for some t ∈ T. This proves (10). In particular, any two subgroups of W_{9} which have the same data as listed in the first three rows of Table A.1 are G-conjugate.

By (10), together with (E) when rk(E/E_{2}) = 2, we have Aut_{G}(E) = Aut(E, 𝔮_{E}) in all cases. Thus Aut_{G}(E) is the group of all automorphisms of E which normalize E_{0} and E_{2} and preserve the induced quadratic form on E_{2}/E_{0}. This gives the value for |O_{2}(Aut_{G}(E))| in the table, and the lower bounds for δ(Aut_{G}(E)) then follow from Lemma A.5.

In cases 1–6, the upper bounds for |π_{0}(C_{G}(E))| given in the table are proven in [O2, p. 78–79]. In all cases, |π_{0}(C_{G}(E_{2}))| is first computed, using Proposition 2.5 or the upper bound given in formula (8) in the proof of the last lemma, and then [O2, Proposition 8.8] is used to compute an upper bound for |π_{0}(C_{G}(E))|/|π_{0}(C_{G}(E_{2}))|. There is in fact an error in the table on [O2, p. 79] (the group C_{G}(E_{0})_{s}^{0} in the third-to-last column should be SL_{2} × SL_{2} up to finite cover), but correcting this gives in fact a better estimate |π_{0}(C_{G}(E))| ≤ 2^{9}.

Case nr. 11 can be handled in a similar way. Set E_{t} = E ∩ T_{(2)} < E, so that |E/E_{t}| = 2 = |E_{t}/E_{2}|. The form 𝔮_{E_{t}} has type [2]_{(3)}^{+}, while E_{t}^{⊥} has type 2B^{3}. Hence |π_{0}(C_{G}(E_{t}))| ≤ 2^{4} by

(8). By [O2, Proposition 8.8], |π_{0}(C_{G}(E))| ≤ 2^{4+r}, where r = dim(T) = 8.

To handle the remaining cases, fix rank 2 subgroups F_{1}, F_{2} ≤ T_{(2)} < G with involutions of type AAA and ABB, respectively, and consider the information in Table A.2. The description of C_{G}(F_{i}⟨θ⟩) follows from (H). The third through fifth columns give dimensions

dim(C_{G}(F_{i})⟨θ, g⟩) for g as follows:

i  C_{G}(F_{i}⟨θ⟩)  −I_{4} ⊕ I_{4} −I_{2} ⊕ I_{6} order 4 2A 2B
1 F_{1}⟨θ⟩ × PSp_{8}(𝕂)  20  24  16  16 20
2 F_{2}⟨θ⟩ × PSO_{8}(𝕂)  12  16  16  16 12

Table A.2

of centralizers of F_{i}⟨θ⟩⟨g⟩, for g as described after lifting to Sp_{8}(𝕂) or SO_{8}(𝕂). (Here, I_{m} denotes the m × m identity matrix.) The last two columns do this for g ∈ 2A or 2B, respectively, when g ∈ T_{(2)} is orthogonal to F_{i} with respect to the form 𝔮, and the dimensions follow from (D). Thus elements of class 2B lift to involutions in Sp_{8}(𝕂) or SO_{8}(𝕂) with 4-dimensional (−1)-eigenspace, while for i = 1 at least, elements of class 2A lift to elements of order 4 in Sp_{8}(𝕂).

Thus in all of the cases nr. 7–13 in Table A.1, we can identify E = F_{i}⟨θ⟩ × F^{∗}, where i = 1 in nr. 7–10 or i = 2 in nr. 11–13, and where F^{∗} lifts to an abelian subgroup of Sp_{8}(𝕂) or SO_{8}(𝕂) (elementary abelian except for nr. 7–8). This information, together with the following:

H a group, Z ≤ Z(H), |Z| = p, Z ≤ P ≤ H a p-subgroup

⟹ |C_{H/Z}(P)/C_{H}(P)/Z| ≤ |P/Fr(P)|

(applied with H = Sp_{8}(𝕂) or SO_{8}(𝕂)), imply the remaining bounds in the last line of Table A.1.

In all but the last case in Table A.1, δ(Aut_{G}(E)) > |π_{0}(C_{G}(E))|, so no E^{∗} ∈ Ẑ is Gconjugate to E by Proposition A.4. In the last case, by the same proposition, E can be G-conjugate to some E^{∗} ∈ Ẑ only if Aut_{G}(E) acts transitively on π_{0}(C_{G}(E)) ≅ C_{2}^{5} with point stabilizers isomorphic to Σ_{3}. By (10), each class in O_{2}(Aut_{G}(E)) is represented by some element tg ∈ N_{G}(E), where g ∈ N_{G}(T) and t ∈ T. In particular, (tg)σ(tg)^{−1} = tσ(t)^{−1} ∈ T. So each class in the O_{2}(Aut_{G}(E))-orbit of 1 ∈ π_{0}(C_{G}(E)) has nonempty intersection with T. But by (9), C_{G}(F_{2})^{0} ∩ θT = ∅, so θC_{G}(E)^{0} ∩ T = ∅. Thus the action is not transitive on π_{0}(C_{G}(E)), and hence E^{∗} ∉ Ẑ. □

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Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain

  E-mail address: broto@mat.uab.es

Matematisk Institut, Universitetsparken 5, DK–2100 København, Denmark

  E-mail address: moller@math.ku.dk

Université Paris 13, Sorbonne Paris Cité, LAGA, UMR 7539 du CNRS, 99, Av. J.-B. Clément, 93430 Villetaneuse, France.

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