SIMPLE FUSION SYSTEMS OVER p-GROUPS WITH ABELIAN SUBGROUP OF INDEX p : I

BOB OLIVER

Abstract. For an odd prime p, we look at simple fusion systems over a finite nonabelian p-group S which has an abelian subgroup A of index p. When S has more than one such subgroup, we reduce this to a case already studied by Ruiz and Viruel. When A is the unique abelian subgroup of index p in S and is not essential (equivalently, is not radical) in the fusion system, we give a complete list of all possibilities which can occur. This includes several families of exotic fusion systems, including some which have proper strongly closed subgroups.

A saturated fusion system F over a finite p-group S is a category whose objects are the subgroups of S, whose morphisms are injective homomorphisms between subgroups, and whose morphism sets satisfy certain axioms first formulated by Puig and motivated by the properties of conjugacy relations among p-subgroups of a finite group. For example, for each finite group G and each S ∈ Syl_{p}(G), the category F_{S}(G), whose objects are the subgroups of G and whose morphisms are those homomorphisms induced by conjugation in G, is a saturated fusion system over S. We refer to Puig’s paper [Pg], and to [AKO] and [Cr], for more background details on saturated fusion systems.

A saturated fusion system F over a finite p-group S is realizable if F = F_{S}(G) for some finite group G with S ∈ Syl_{p}(G); otherwise it will be called exotic. When p is odd, many examples have been constructed of exotic fusion systems over p-groups. We refer in particular to the classifications by Díaz, Ruiz, and Viruel of saturated fusion systems over extraspecial p-groups of order p^{3} and exponent p [RV] and over p-groups of rank two [DRV]; and to a more general procedure described in [BLO4, § 5] for constructing such examples and checking that they are saturated.

A saturated fusion system F is simple if it contains no nontrivial proper normal subsystems (see [AKO, Definition I.6.1] or [Cr, §§ 5.4 & 8.1] for the precise definition of a normal subsystem). In this paper, we study simple fusion systems over nonabelian p-groups which have an abelian subgroup of index p. When p = 2, by [AOV1, Propositions 4.3–4.4] and [AOV2, Propositions 3.1 & 5.2(a)], each such fusion system is isomorphic to the fusion system of PSL_{2}(q) or of PSL_{3}(q) for some odd q, and the 2-group in question is dihedral, semidihedral, or a wreath product C_{2^{n}} ≀ C_{2}.

When p is odd, it turns out that this class of p-groups supports a much richer collection of simple fusion systems, many of which are exotic. Let F be a simple fusion system over the nonabelian p-group S, with abelian subgroup A of index p. We split this into three different cases:

(1) S has more than one abelian subgroup of index p; (2) A is the unique abelian subgroup of index p in S, and A is not F-essential (Definition

2000 Mathematics Subject Classification. Primary 20D20. Secondary 20E32, 20E45.

Key words and phrases. finite groups, finite simple groups, Sylow subgroups, fusion.

B. Oliver is partially supported by UMR 7539 of the CNRS, and by project ANR BLAN08-2_338236, HGRT.

1.1); and

(3) A is the unique abelian subgroup of index p in S, and A is F-essential.

In case (1), we show that S is extraspecial of order p^{3} and exponent p (Theorem 2.1), and hence that we are in the situation considered by [RV]. In case (2), we give a complete list of all such fusion systems (Theorem 2.8), and determine which of them are exotic (most of them). This list includes many of the examples constructed earlier by Díaz, Ruiz, and Viruel [DRV] in a different context. Among the exotic examples constructed are several which are also noteworthy for having proper strongly closed subgroups (Definition 1.1).

Case (3) is more complicated, since it depends heavily on representation theory (via the action of Aut_{F}(A) on A). This will be handled in a later paper together with David Craven.

1. Background

We first recall some of the terminology used for certain subgroups in a fusion system.

Definition 1.1. Fix a prime p, a finite p-group S, and a saturated fusion system F over S. Let P ≤ S be any subgroup.

• P^{F} denotes the set of subgroups of S which are F-conjugate (isomorphic in F) to P. Also, g^{F} denotes the F-conjugacy class of an element g ∈ S (the set of images of g under morphisms in F).

• P is fully normalized in F ( fully centralized in F) if |N_{S}(P)| ≥ |N_{S}(Q)| (|C_{S}(P)| ≤ |C_{S}(Q)|) for each Q ∈ P^{F}.

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

• P is F-essential if P < S, P is F-centric and fully normalized in F, and Out_{F}(P) =^{def} Aut_{F}(P)/Inn(P) contains a strongly p-embedded subgroup. Here, a proper subgroup H < G of a finite group G is strongly p-embedded if p||H|, and p∤|H ∩ gHg^{−1}| for each g ∈ G∖H. Let E_{F} denote the set of all F-essential subgroups of S.

• P is normal in F if each morphism φ ∈ Hom_{F}(Q, R) in F extends to a morphism φ ∈ Hom_{F}(P Q, P R) such that φ(P) = P. The maximal normal p-subgroup of a saturated fusion system F is denoted O_{p}(F).

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

• 𝔣𝔬𝔠(F) = ⟨gh^{−1} | g ∈ S, h ∈ g^{F}⟩.

Let O^{p}(F) and O^{p′}(F) denote the smallest normal fusion subsystems of p-power index, and of index prime to p, respectively. Such normal subsystems are defined by analogy with finite groups, and we refer to [AKO, § I.7] or [Cr, § 7.5] for precise definitions and references.

Definition 1.2. A saturated fusion system F is reduced if O_{p}(F) = 1, and O^{p}(F) = F = O^{p′}(F). A saturated fusion system is simple if it contains no nontrivial proper normal fusion subsystems, in the sense of [AKO, Definition I.6.1] or [Cr, §§ 5.4 & 8.1].

For any saturated fusion system F over S, O^{p}(F), O^{p′}(F), and F_{O_{p}(F)}(O_{p}(F)) are all normal subsystems. Hence F is reduced if it is simple. If E ⊴ F is any normal subsystem over the subgroup T ⊴ S, then by definition of normality, T is strongly closed in F. Thus a reduced fusion system is simple if it has no proper nontrivial strongly closed subgroups.

There are reduced fusion systems which are not simple; constructed, for example, by taking direct products or wreath products. In this paper, our main interest in reduced fusion systems is as an intermediate step towards showing that they are simple. We refer to [AOV1, Theorems A–C] for the original motivation for defining them.

The next proposition lists some of the standard tools for handling O_{p}(F) and O^{p}(F).

Proposition 1.3. The following hold for any saturated fusion system F over a finite p-group S.

(a) Each morphism in F is a composite of restrictions of elements of Aut_{F}(P) for P ∈ E_{F} ∪ {S}. Moreover, each morphism in F is a composite of restrictions of elements in Aut_{F}(S), and in O^{p′}(Aut_{F}(P)) for P ∈ E_{F}.

(b) Assume Q ⊴ S has the property that for each P ∈ E_{F} ∪ {S}, P ≥ Q and Aut_{F}(P) normalizes Q. Then Q ⊴ F.

(c) 𝔣𝔬𝔠(F) = ⟨[Aut_{F}(P), P] | P ∈ E_{F} ∪ {S}⟩.

(d) O^{p}(F) = F if and only if 𝔣𝔬𝔠(F) = S.

Proof. The first statement in (a) is shown in [Pg, § 5], and also in [OV, Corollary 2.6], while the stronger statement follows from [O1, Proposition 1.10(a,b)]. Point (b) is shown in [AKO, Proposition I.4.5], and point (d) in [AKO, Corollary I.7.5]. Point (c) is an immediate consequence of (a) and the definition of 𝔣𝔬𝔠(F). □

Determining O^{p′}(F) is more difficult, in general, but the following lemma suffices for our purposes.

Lemma 1.4. Let F be a saturated fusion system over a finite p-group S. Assume that each P ∈ E_{F} is minimal among all F-centric subgroups. For each P ∈ E_{F}, set

Aut_{F}^{(P)}(S) = {α ∈ Aut_{F}(S) | α(P) = P, α|_{P} ∈ O^{p′}(Aut_{F}(P))} .

Then O^{p′}(F) = F if and only if Aut_{F}(S) = ⟨Inn(S), Aut_{F}^{(P)}(S) | P ∈ E_{F}⟩.

Proof. Set H_{P} = Aut_{F}^{(P)}(S) and H = ⟨Inn(S), H_{P} | P ∈ E_{F}⟩ for short. Let F^{c} ⊆ F be the full subcategory with objects the F-centric subgroups of S.

Since no F-centric subgroup is properly contained in an essential subgroup by assumption, two F-centric subgroups are F-conjugate only if they are in the same Aut_{F}(S)-orbit by Proposition 1.3(a). Thus all F-centric subgroups are fully normalized in F. In particular, P ∈ E_{F} and Q ∈ P^{F} imply Q ∈ E_{F}. For α ∈ Aut_{F}(S) and P ∈ E_{F}, αH_{P} α^{−1} = H_{α(P)}; and thus H ⊴ Aut_{F}(S).

If H = Aut_{F}(S), then O^{p′}(F) = F by [BCGLO, Theorem 5.4] or [AKO, Theorem I.7.7], since H ≤ Aut_{F}^{0}(S) in the notation of those references.

Now assume H < Aut_{F}(S). We claim that for each P ≤ S which is F-centric,

Hom_{F}(P, S) = \begin{cases}\{α|_{P} | α ∈ Aut_{F}(S)\} & if P ∉ E_{F} \\ \{α|_{P} | α ∈ Aut_{F}(S)\} ∘ O^{p′}(Aut_{F}(P)) & if P ∈ E_{F} . \end{cases}  (1)

If P ∉ E_{F}, then since no subgroup in P^{F} is contained in an essential subgroup, each α ∈ Hom_{F}(P, S) extends to some α ∈ Aut_{F}(S) by Proposition 1.3(a), and (1) holds in this case. If P ∈ E_{F}, then each morphism in Hom_{F}(P, S) is a composite of restrictions of elements in Aut_{F}(S), and of elements in O^{p′}(Aut_{F}(Q)) for Q ∈ P^{F}. Via the relations (α|_{Q})Aut_{F}(Q)(α|_{Q})^{−1} = Aut_{F}(α(Q)) for α ∈ Aut_{F}(S), these morphisms can be arranged in the above form, proving the second case of (1).

Define a map θ: Mor(F^{c}) ⟶ Aut_{F}(S)/H as follows. For each φ ∈ Hom_{F}(P, Q) in F^{c}, set θ(φ) = αH (for α ∈ Aut_{F}(S)) if φ = α|_{P}, or if P ∈ E_{F} and φ = α|_{P} ∘ β where β ∈ O^{p′}(Aut_{F}(P)). If φ = α|_{P} = α′|_{P}, then α′ = α ∘c_{g} for some g ∈ Z(P) (cf. [AKO, Lemma I.5.6]), so α′H = αH. If P ∈ E_{F} and φ = α|_{P} ∘ β = α′|_{P} ∘ β′ (where β, β′ ∈ O^{p′}(Aut_{F}(P))), then α = α′γ for some γ ∈ Inn(S)H_{P} by the same lemma and the definition of H_{P}. Thus θ is well defined in all cases.

By construction, θ sends composites to products, and sends O^{p′}(Aut_{F}(Q)) to the identity for each Q ∈ Ob(F^{c}) since p∤|Aut_{F}(S)/H|. So O^{p′}(F) ⫋ F by [AKO, Theorem I.7.7(c)]. More precisely, ⟨θ^{−1}(1)⟩ is a proper normal subsystem of index prime to p in F. □

The next two lemmas provide the tools we will need to determine which of the fusion systems we construct are realizable and which are exotic.

Lemma 1.5 ([DRV]). Let F be a reduced fusion system over a p-group S. Assume, for each 1 ≠ P ⊴ S strongly closed in F, that P is centric in S, is not elementary abelian, and does not factor as a product of two or more subgroups which are permuted transitively by Aut_{F}(P). Then if F is realizable, it is the fusion system of a finite simple group.

Proof. By [DRV, Proposition 2.19], F = F_{S}(G) for some almost simple group G. We must show that when F is reduced, G can be chosen to be simple.

Assume G is not simple, and that no proper subgroup of G realizes F. Since the outer automorphism group of every simple group is solvable (cf. [GLS3, Theorem 7.1.1(a)]), there is a proper normal subgroup G_{0} ⊴ G of index p or of index prime to p. If [G:G_{0}] = p, then by the focal subgroup theorem [G, Theorem 7.3.4], 𝔣𝔬𝔠(F) ≤ S ∩ G_{0} < S, which is impossible since O^{p}(F) = F (Proposition 1.3(d)). If [G:G_{0}] is prime to p, then F_{S}(G_{0}) is a normal subsystem of index prime to p in F by [BCGLO, Definition 3.1(b)]: it is saturated, and Aut_{G_{0}}(P) ≥ O^{p′}(Aut_{G}(P)) for each P ≤ S. Since F is reduced, it has no proper normal subsystems of index prime to p, so F = F_{S}(G_{0}), which contradicts the minimality assumption on G. We thus have a contradiction in either case, and so G can be taken to be simple. □

Lemma 1.6. Fix an odd prime p. Assume G is a finite simple group for which S ∈ Syl_{p}(G) is nonabelian and contains a unique abelian subgroup A < S of index p. Assume also that |Z(S)| = p, |S/[S, S]| = p^{2}, and A is not essential in G. Then p = 3, and G is isomorphic to one of the groups PSL_{3}(q) (q ≡ 1 (mod 3)), P SU_{3}(q) (q ≡ −1 (mod 3)), G_{2}(q) (3 ∤ q), or 3D_{4}(q) (3 ∤ q).

Proof. Since A is not essential, Aut_{S}(A) ⊴ Aut_{G}(A). Equivalently, S ⊴ N_{G}(A), and hence N_{G}(S) = N_{G}(A). If B < S is any other abelian subgroup, then either B ≤ A, or B ∩ A ≤ Z(S) and hence |B| ≤ p^{2}.

If G is an alternating group, then G ≅ A_{n} for some n = ap + b where p ≤ a < 2p and 0 ≤ b < p. Then A ≅ (C_{p})^{a}, and Aut_{G}(A) has index two in Aut_{Σ_{n}}(A) ≅ C_{p−1} ≀ Σ_{a}. Since no such group has a normal subgroup of order p, G is not an alternating group.

If G is a sporadic group, then by the tables in [GL, § 1.5] or [GLS3, § 5.3], in almost all cases, either |S| ≤ p^{3}, or S is abelian, or S contains an extraspecial group of type p^{1+2k} for k ≥ 2, or S contains a special group of type 3^{2+4}. The exceptions are (G, p) = (J_{3}, 3), where Z(S) ≅ C_{3}^{2}; (Co_{1}, 5), where J(S) ≅ C_{5}^{3} and N_{G}(J(S))/J(S) ≅ C_{4} × Σ_{5}; and (F_{3}, 3), where G contains a subgroup C_{3}^{4} ⋊ SL_{2}(9). Thus G is not a sporadic group.

If G is a group of Lie type in characteristic p, then since S is nonabelian and |S/[S, S]| = p^{2}, G must be one of the groups L_{3}(p), Sp_{4}(p), U_{3}(p), or G_{2}(p) (see the description in [GLS3, Theorem 3.3.1] of the central series for S). By the commutator relations, any abelian subgroup of index p in S would have to be a parabolic subgroup, and hence essential in G.

Thus by the classification theorem, G is a group of Lie type in characteristic different from p. If G ≅ PSL_{n}(q) for some q and |S| ≥ p^{4}, and k is the order of q in 𝔽_{p}^{×}, then S has a normal abelian subgroup A of order at least p^{3} with Aut_{G}(A) ≅ C_{k} ≀ Σ_{n} for some n, and this has a normal Sylow subgroup of order p only when p = n = 3 and k = 1. Thus G ≅ PSL_{3}(q) for q ≡ 1 (mod 3). A similar argument in the unitary case shows that PSU_{3}(q) for q ≡ 2 (mod 3) is the only possibility. If G is symplectic or orthogonal, then Aut_{G}(A) always contains C_{2} ≀ Σ_{n} or (C_{2})^{n−1} ⋊ Σ_{n}, so A is not essential.

In all other cases, by the description in [GL, 10-1] of the Sylow subgroups, there is a normal abelian p-subgroup P_{H} in S, which must be contained in A by the above remarks, which is maximal abelian, and whose index in S is determined by the tables there. In particular, A has index p in S only in the following cases:

• p = 3, G = G_{2}(q) or ^{3}D_{4}(q) (q ≡ ±1 (mod 3)), or G = ^{2}F_{4}(2^{2k+1});

• p = 5, G = E_{6}(q) (q ≡ 1 (mod 5)), G = ^{2}E_{6}(q) (q ≡ −1 (mod 5)), G = E_{7}(q) (q ≡ ±1 (mod 5)), G = E_{8}(q) (q ≡ ±2 (mod 5));

• p = 7, G = E_{7}(q) or E_{8}(q) (q ≡ ±1 (mod 7)).

When p = 3 and G = ^{2}F_{4}(2^{2k+1}), then Aut_{G}(A) ≅ GL_{2}(3) [Ma, Proposition 1.2]. When p = 5 and G = E_{8}(q) for q ≡ ±2 (mod 5), then Aut_{G}(A)/O_{2}(Aut_{G}(A)) ≅ Σ_{6} [LSS, Table 5.2]. In all other cases when p = 5, 7, Aut_{G}(A) is the Weyl group of E_{m} for m = 6, 7, 8, and contains a (quasi)simple subgroup of index two.

Thus the only cases where Aut_{G}(A) contains a normal Sylow p-subgroup of order p are those where p = 3 and G = G_{2}(q) or ^{3}D_{4}(q). □

We finish the section with a few elementary group theoretic results.

Lemma 1.7. Fix a prime p, a finite p-group P, and a group G ≤ Aut(P) of automorphisms of P. Let P_{0} ⊴ P_{1} ⊴ ⋯ ⊴ P_{m} = P be a sequence of subgroups, all normal in P and normalized by G, such that P_{0} ≤ Fr(P). Let H ≤ G be the subgroup of those g ∈ G which act via the identity on P_{i}/P_{i−1} for each 1 ≤ i ≤ m. Then H is a normal p-subgroup of G.

Proof. See, e.g., [G, Theorems 5.3.2 & 5.1.4]. □

Lemma 1.8. Assume P is a nonabelian group of order p^{3}, for some odd prime p. Then either P has exponent p and Out(P) ≅ GL_{2}(p), or P has exponent p^{2} and O_{p}(Out(P)) ∈ Syl_{p}(Out(P)).

Proof. If P has exponent p, then each automorphism of P/[P, P] ≅ C_{p}^{2} lifts to an automorphism of P. Also, each automorphism of P which induces the identity on P/[P, P] is inner, so Out(P) ≅ Aut(P/[P, P]) ≅ GL_{2}(p).

If P has exponent p^{2}, then it contains a unique subgroup Q < P with Q ≅ C_{p}^{2}. So by Lemma 1.7, there is a homomorphism from Aut(P) to Aut(P/Q) × Aut(Q/[P, P]) ≅ C_{p−1} × C_{p−1} whose kernel is O_{p}(Aut(P)). □

We will adopt the usual notation, and write p_{+}^{1+2} and p_{−}^{1+2} for nonabelian groups of order p^{3} and of exponent p or p^{2}, respectively.

Lemma 1.9. Let S be a nonabelian p-group, and assume A ⊴ S is an abelian subgroup of index p. Then either

(a) |[S, S]| ≥ p^{2}, |S/Z(S)| ≥ p^{3}, and A is the unique abelian subgroup of index p in S; or

(b) |[S, S]| = p, S/Z(S) ≅ C_{p}^{2}, and S contains exactly p + 1 abelian subgroups of index p.

Proof. Fix any x ∈ S∖A. Then [S, S] = [x, A] is the image of (Id − c_{x}) as a homomorphism from A to itself, and Z(S) = C_{A}(x) is its kernel. Hence |S/Z(S)| = p·|A/Z(S)| = p·|[S, S]|. Since S is nonabelian, S/Z(S) is not cyclic.

If B is a second abelian subgroup of index p in S, then S = AB, so A ∩ B ≤ Z(S), and |S/Z(S)| ≤ p^{2}. Thus |[S, S]| = p in this case. Conversely, if |S/Z(S)| ≤ p^{2}, then S/Z(S) ≅ C_{p}^{2} since it is noncyclic, and each subgroup of index p in S containing Z(S) is abelian (generated by Z(S) and one more element). Since each abelian subgroup of index p in S contains Z(S), S has exactly p + 1 abelian subgroups of index p. □

We finish the section with two lemmas which deal with actions on finite abelian p-groups.

Lemma 1.10. Fix a finite abelian p-group A and a subgroup G ≤ Aut(A). Assume, for S ∈ Syl_{p}(G), that S ⋬ G and |A/C_{A}(S)| = p. Then |S| = p.

Proof. Let S_{1}, S_{2} ∈ Syl_{p}(G) be two distinct Sylow subgroups, and set A_{i} = C_{A}(S_{i}). For i = 1, 2, S_{i} induces the trivial action on A/A_{i} ≅ C_{p}, and hence [S_{i}, A] ≤ A_{i}. Also, A_{1} ≠ A_{2}, since otherwise ⟨S_{1}, S_{2}⟩ would act trivially on A_{1} and on A/A_{1}, and hence would be a p-group by Lemma 1.7.

If |S_{i}| > p, then |[S_{i}, A]| > p, so there are elements 1 ≠ x_{i} ∈ S_{i} such that [x_{i}, A] ≤ A_{1} ∩A_{2}. Set T = ⟨x_{1}, x_{2}⟩. Then T acts trivially on A_{1} ∩ A_{2} and on A/(A_{1} ∩ A_{2}), so T is a p-group by Lemma 1.7, while C_{A}(T) = C_{A}(x_{1}) ∩ C_{A}(x_{2}) = A_{1} ∩ A_{2}. Since this contradicts the assumption that |A/C_{A}(S)| = p for each S ∈ Syl_{p}(G), we conclude that |S_{i}| = p. □

Lemma 1.11. Fix a finite abelian p-group A, and a subgroup G ≤ Aut(A). Assume the following.

(i) Each Sylow p-subgroup of G has order p and is not normal in G.

(ii) For each x ∈ G of order p, [x, A] has order p, and hence C_{A}(x) has index p.

Set H = O^{p′}(G), A_{1} = C_{A}(H), and A_{2} = [H, A]. Then G normalizes A_{1} and A_{2}, A = A_{1} × A_{2}, and H ≅ SL_{2}(p) acts faithfully on A_{2} ≅ C_{p}^{2}. There are groups of automorphisms G_{i} ≤ Aut(A_{i}) (i = 1, 2), such that p ∤ |G_{1}|, G_{2} ≥ Aut_{H}(A_{2}) ≅ SL_{2}(p), and G ⊴ G_{1} × G_{2} (as a subgroup of Aut(A)) with index dividing p − 1.

Proof. For each B ≤ A, let 𝕍_{1}(B) be the set of subgroups of B of order p. Define

θ: Syl_{p}(G) ⟶ 𝕍_{1}(A)

by setting θ(S) = [S, A]. If [S, A] = [T, A] for S, T ∈ Syl_{p}(G), then ⟨S, T ⟩ acts via the identity on [S, A] and on A/[S, A], so ⟨S, T ⟩ is a p-group by Lemma 1.7, and S = T. Thus θ is injective.

Assume S_{1}, S_{2} ∈ Syl_{p}(G) are distinct, set K = ⟨S_{1}, S_{2}⟩, and consider the action of K on [K, A] = θ(S_{1})θ(S_{2}) ≅ C_{p}^{2}. Since K acts trivially on A/[K, A], the subgroup K_{0} = C_{K}([K, A]) ⊴ K is a normal p-subgroup by Lemma 1.7, hence contained in all Sylow psubgroups of K, and hence K_{0} = 1 by assumption. Hence K acts faithfully on [K, A] ≅ C_{p}^{2}. Under an appropriate choice of basis for [K, A], S_{1} and S_{2} are the groups of (strict) upper and lower triangular matrices in GL_{2}(p), and thus generate SL_{2}(p). So K ≅ SL_{2}(p). Also, θ(Syl_{p}(K)) = 𝕍_{1}([K, A]) = 𝕍_{1}(θ(S_{1})θ(S_{2})).

Thus θ(Syl_{p}(G)) = 𝕍_{1}(A_{2}) for some elementary abelian subgroup A_{2} ≤ A. If rk(A_{2}) ≥ 3, then there are distinct subgroups T_{1}, T_{2} ∈ Syl_{p}(G) such that [T_{1}, A] ≤ C_{A}(T_{2}), ⟨T_{1}, T_{2}⟩ induces the identity on [T_{1}, A], C_{A}(T_{2})/[T_{1}, A], and A/C_{A}(T_{2}), so ⟨T_{1}, T_{2}⟩ is a p-group by Lemma 1.7 again, which is impossible. We conclude that rk(A_{2}) = 2, and hence that O^{p′}(G) = ⟨Syl_{p}(G)⟩ ≅ SL_{2}(p).

Set H = O^{p′}(G), and set A_{1} = C_{A}(H). Then |A/A_{1}| = p^{2}, A_{1} ∩ A_{2} = 1 since H ≅ SL_{2}(p) acts faitfully on A_{2}, and hence A = A_{1} × A_{2}. Since H ⊴ G, the subgroups A_{1} = C_{A}(H) and A_{2} = [H, A] are both G-invariant. Set G_{i} = Aut_{G}(A_{i}) (i = 1, 2). Thus G_{1} and G_{2} are quotient groups of G, G_{1} has order prime to p, G_{2} ≤ Aut(A_{2}) ≅ GL_{2}(p) and contains SL_{2}(p), and G ⊴ G_{1} × G_{2} with index dividing p−1 = [GL_{2}(p):SL_{2}(p)]. □

2. Reduced fusion systems over nonabelian p-groups with index p abelian subgroup

Throughout this section, p is an odd prime. We want to describe all simple fusion systems over nonabelian p-groups which contain an abelian subgroup of index p. We begin by showing that if S has more than one abelian subgroup of index p, and there is a simple fusion system over S, then S must be extraspecial of order p^{3} and exponent p (Theorem 2.1). This is the case already handled by Ruiz and Viruel [RV]. Afterwards, we develop the tools needed to study simple or reduced fusion systems over a p-group S which contains a unique abelian subgroup A < S of index p. Our main result is Theorem 2.8, which lists simple fusion systems over such S when A is not essential. The more complicated case, that where the unique abelian subgroup of index p is essential, will be handled in a later paper.

Theorem 2.1. Assume p is odd, and let S be a nonabelian p-group containing more than one abelian subgroup of index p. If there is a simple (or reduced) fusion system over S, then S is extraspecial of order p^{3} and exponent p.

Proof. Set Z = Z(S) and S′ = [S, S] for short. By Lemma 1.9, |S′| = p and S/Z ≅ C_{p}^{2}. In particular, S′ ≤ Z. The only proper subgroups centric in S are the abelian subgroups of index p, and hence they are the only possible F-essential subgroups.

Fix some A ∈ E_{F}, and set G = Aut_{F}(A). Then Aut_{S}(A) ≅ S/A has order p, it is a Sylow p-subgroup of G, but is not the only Sylow p-subgroup since A ∈ E_{F}. Also, [S, A] = S′ has order p.

The hypotheses of Lemma 1.11 thus hold. So if we set H = O^{p′}(G), A_{1} = C_{A}(H), and A_{2} = [H, A], then A = A_{1} × A_{2}, A_{1} and A_{2} are normalized by G, A_{2} ≅ C_{p}^{2}, and H ≅ SL_{2}(p) acts trivially on A_{1} and faithfully on A_{2}. In particular, upon restricting to the normalizer in H of Aut_{S}(A), we find a subgroup of order p − 1 in H which acts faithfully on S′ = Z ∩ A_{2} and acts trivially on A_{1} ≤ Z.

Thus Z = S′ × A_{1}, and A_{1} is the unique subgroup of Z which is complementary to S′ and normalized by Aut_{F}(Z). It follows that A_{1} is Aut_{F}(S)-invariant, and (by the same argument) is also normalized by Aut_{F}(P) for each P ∈ E_{F}. By Proposition 1.3(b), A_{1} ⊴ F. Since F is reduced, this implies that A_{1} = 1, and so Z = Z(S) = S′ has order p.

Thus S is a nonabelian group of order p^{3}. If S ≅ p_{−}^{1+2}, then there is a unique subgroup Q ≤ S isomorphic to C_{p}^{2}, this is the only possible F-essential subgroup, so Q ⊴ F, again contradicting the assumption F is reduced. We conclude that S ≅ p_{+}^{1+2}. □

Fusion systems over extraspecial groups of order p^{3} and exponent p have already been classified by Ruiz and Viruel [RV]. In particular, they showed that there are exactly three distinct exotic fusion systems over such groups, all for p = 7, of which two are simple.

We now turn to the case where S has a unique abelian subgroup of index p. We first fix some notation.

Notation 2.2. Fix a nonabelian p-group S with unique abelian subgroup A of index p, and a saturated fusion system F over S. Define

S′ = [S, S] , Z = Z(S) , Z_{0} = Z ∩ S′ , Z_{2} = Z_{2}(S) , A_{0} = ZS′ .

Thus Z_{0} ≤ S′ ≤ A_{0} ≤ A and Z_{0} ≤ Z ≤ Z_{2} ≤ A_{0}. Also, set

H = {Z⟨x⟩ | x ∈ S∖A} and B = {Z_{2}⟨x⟩ | x ∈ S∖A} .

Recall that a p-group P of order p^{n} has maximal class if its lower (or upper) central series has length n − 1.

Lemma 2.3. Assume the notation and hypotheses of 2.2.

(a) For each P ∈ E_{F}, P ∈ {A} ∪ H ∪ B and |N_{S}(P)/P | = p.

(b) If Z_{2}⟨x⟩ ∈ E_{F} for some x ∈ S∖A, then Z⟨x⟩ is not F-centric and Z⟨x⟩ ∉ E_{F}.

(c) If E_{F} ⊈ {A}, then |Z_{0}| = p = |A/A_{0}| = |Z_{2}/Z|. Also, S/Z has maximal class.

(d) There is x ∈ S∖A such that A_{0}⟨x⟩ is normalized by Aut_{F}(S).

(e) For each P ∈ E_{F} and each α ∈ N_{Aut_{F} (P)}(Aut_{S}(P)), α extends to some α ∈ Aut_{F}(S).

Proof. (a) Fix some P ∈ E_{F} where P ≠ A. Then P ≰ A since P is F-centric. Set P_{0} = P ∩A, and fix some element x ∈ P ∖P_{0}. Since Out_{F}(P) contains a strongly p-embedded subgroup, O_{p}(Out_{F}(P)) = 1 (cf. [AKO, Proposition A.7(c)]). (a1) Assume P is nonabelian. Since Z ≤ P (P is F-centric), Z(P) = C_{P_{0}}(x) = Z. For each g ∈ N_{A}(P)∖P, c_{g} is the identity on P_{0} and on P/P_{0}. If P_{0} is characteristic in p, then c_{g} ∈ O_{p}(Aut_{F}(P)) by Lemma 1.7, which is impossible since O_{p}(Out_{F}(P)) = 1. Thus P_{0} is not characteristic in P, and hence is not the unique abelian subgroup of index p in P. By Lemma 1.9, |P_{0}/Z| = p, P/Z(P) ≅ P/Z ≅ C_{p}^{2}, and [P, P] = [x, P_{0}] ≅ C_{p}.

Now, Aut_{F}(P) maps injectively to Aut(Z(P)) × Aut(P/Z(P)) (since the kernel is a pgroup and O_{p}(Aut_{F}(P)) = 1); Aut_{S}(P) is sent trivially to Aut(Z(P)), and so |N_{S}(P)/P | = p since p^{2} does not divide the order of Aut(P/Z(P)) ≅ GL_{2}(p). For g ∈ N_{A}(P)∖P, [g, P] ≰ Z (otherwise c_{g} induces the identity on P/Z(P) and on Z(P) = Z, which would imply c_{g} ∈ O_{p}(Aut_{F}(P)) = Inn(P)), so g ∉ Z_{2}. Thus Z_{2} ≤ P_{0}, and Z_{2} = P_{0} since Z_{2} > Z and |P_{0}/Z| = p. So P = Z_{2}⟨x⟩ ∈ B. (a2) Assume P ∈ E_{F} is abelian (and P ≠ A). Then P_{0} = Z: it contains Z since P is centric, and cannot be larger since then P would be nonabelian. Hence P = Z⟨x⟩ ∈ H. Also, Aut_{A}(P) = Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)) centralizes P_{0}. The conditions of Lemma 1.10 thus hold (with P and Aut_{F}(P) in the roles of A and G), so |N_{S}(P)/P | = |Aut_{S}(P)| = p.

For each g ∈ A, g ∈ N_{S}(P) if and only if [g, x] ∈ P_{0} = Z, if and only if gZ ∈ C_{A/Z}(x) = Z(S/Z) = Z_{2}/Z. Thus N_{A}(P) = Z_{2}, N_{S}(P) = Z_{2}⟨x⟩ = Z_{2}P, and |Z_{2}/Z| = |N_{S}(P)/P | = p.

(b) Assume x ∈ S∖A is such that Z_{2}⟨x⟩ is F-essential. There are p + 1 subgroups of index p in Z_{2}⟨x⟩ which contain Z(Z_{2}⟨x⟩) = Z, and by the proof of (a1), Aut_{S}(Z_{2}⟨x⟩) permutes all of them except Z_{2} transitively. Since Z_{2}⟨x⟩ ∈ E_{F}, Z_{2} is not normalized by Aut_{F}(Z_{2}⟨x⟩), and there is α ∈ Aut_{F}(Z_{2}⟨x⟩) such that α(Z⟨x⟩) = Z_{2} < A. Thus Z⟨x⟩ is not F-centric, and hence cannot be F-essential.

(c) Since E_{F} ⊈ {A}, (H ∪ B) ∩ E_{F} ≠ ∅ by (a). Hence |Z_{2}/Z| = p by the proofs of (a1) and (a2). Also, S′ = [x, A] = Im(c_{x} − Id_{A}) and Z = C_{A}(x) = Ker(c_{x} − Id_{A}), so |S′|·|Z| = |A|. Since Z_{2}/Z = C_{A/Z}(x), Z_{2} = (c_{x} − Id_{A})^{−1}(Z), so (c_{x} − Id_{A}) sends Z_{2}/Z isomorphically to Z_{0} = Z ∩ S′. Hence |Z_{0}| = p, and A_{0} = ZS′ has index p in A.

Define inductively A_{1} = [x, A] = S′, and A_{n} = [x, A_{n−1}] = [S, A_{n−1}] for n ≥ 2. If n ≥ 1 and A_{n} ≠ 1, then A_{n+1} = (c_{x} − Id)(A_{n}), Ker((c_{x} − Id)|_{A_{n}}) = C_{A_{n}}(x) = Z_{0} since C_{A_{n}}(x) ≠ 1 and |Z_{0}| = p, and so |A_{n}/A_{n+1}| = |Z_{0}| = p. Since A_{n}Z/Z is the n-th term in the lower central series for S/Z, and since S/A_{1}Z = S/A_{0} ≅ C_{p}^{2}, this proves that S/Z has maximal class.

(d) For each x ∈ S∖A, x^{p} ∈ C_{A}(x) = Z ≤ A_{0}. Hence S/A_{0} ≅ C_{p}^{2} by (c). Also, the Out_{F}(S)-action on S/A_{0} normalizes A/A_{0} since A is characterisitic in S. Since Out_{F}(S) has order prime to p, there is an Out_{F}(S)-invariant splitting of A/A_{0} < S/A_{0}. If xA_{0} generates such a splitting (where x ∈ S∖A), then A_{0}⟨x⟩ is normalized by Aut_{F}(S).

(e) Fix P ∈ E_{F} and α ∈ N_{Aut_{F} (P)}(Aut_{S}(P)). By the extension axiom, α extends to some α̂ ∈ Aut_{F}(N_{S}(P)). By (a,b), P is maximal among all F-essential subgroups: either P = A, or P ∈ B, or P = Z⟨x⟩ ∈ H for some x ∈ S∖A and Z_{2}⟨x⟩ ∉ E_{F}. So α̂ extends to an element α ∈ Aut_{F}(S) by Proposition 1.3(a). □

Lemma 2.4. Assume the notation and hypotheses of 2.2, and also that O_{p}(F) = 1 and A ∉ E_{F}. Then H ∩ E_{F} ≠ ∅, Z = Z_{0}, S′ = A_{0}, and S has maximal class.

Proof. Since A ∉ E_{F}, E_{F} ⊆ H ∪ B by Lemma 2.3(a). If E_{F} ⊆ B, then Z = Z(P) for each P ∈ E_{F}, so Z ⊴ F by Proposition 1.3(b), which contradicts our assumption. Thus there is Q ∈ H ∩ E_{F}.

By Lemma 2.3(a), Aut_{S}(Q) ∈ Syl_{p}(Aut_{F}(Q)) has order p. Also, for each g ∈ N_{S}(Q)∖Q, [g, Q] ≤ Q ∩ S′ = Z ∩ S′ = Z_{0}, where |Z_{0}| = p by Lemma 2.3(c). Hence by Lemma 1.11, if we set Γ_{Q} = O^{p′}(Aut_{F}(Q)), Q_{1} = C_{Q}(Γ_{Q}) ≤ Z(S), and Q_{2} = [Γ_{Q}, Q] ≥ Z_{0}, then Q_{1} and Q_{2} are both normalized by Aut_{F}(Q) and Q = Q_{1} × Q_{2}. Also, Γ_{Q} ≅ SL_{2}(p) acts faithfully on Q_{2} ≅ C_{p}^{2} and trivially on Q_{1}.

In particular, there is a subgroup H ≤ N_{Γ_{Q}}(Aut_{S}(Q)) of order p − 1 which acts as the full group of automorphisms of Z_{0} and of Q_{2}/Z_{0}, and acts trivially on Q_{1}. Since H restricts to a subgroup of Aut_{F}(Z), this shows that Z = Q_{1} × Z_{0}, and that Q_{1} is the unique complement to Z_{0} in Z which is normalized by Aut_{F}(Z).

By a similar argument, Q_{1} is normalized by Aut_{F}(P) for each P ∈ H ∩ E_{F}. If P ∈ B ∩ E_{F} or P = S, then for each α ∈ Aut_{F}(P), α(Z) = Z since Z = Z(P), so α|_{Z} = Aut_{F}(Z), and α(Q_{1}) = Q_{1}. Thus Q_{1} ⊴ F by Proposition 1.3(b), and Q_{1} = 1 (hence Z = Z_{0}) since O_{p}(F) = 1. Also, A_{0} = ZS′ = S′ since Z = Z_{0} ≤ S′.

Since |Z(S)| = p and S/Z(S) has maximal class, S also has maximal class. □

We now need some more notation.

Notation 2.5. Assume the notation and hypotheses of 2.2, and also that |Z_{0}| = |A/A_{0}| = |Z_{2}/Z| = p. Fix a ∈ A∖A_{0} and x ∈ S∖A, where x is chosen so that ⟨A_{0}, x⟩ is normalized by Aut_{F}(S) (Lemma 2.3(d)). For each i = 0, 1, …, p − 1, define

H_{i} = ⟨Z, xa^{i}⟩ ∈ H and B_{i} = ⟨Z_{2}, xa^{i}⟩ ∈ B .

Let H_{i} and B_{i} denote the S-conjugacy classes of H_{i} and B_{i}, respectively, and set

H_{∗} = H_{1} ∪ ⋯ ∪ H_{p−1} and B_{∗} = B_{1} ∪ ⋯ ∪ B_{p−1}.

Thus H = H_{0} ∪ H_{∗} and B = B_{0} ∪ B_{∗}, since |A/A_{0}| = p. Also, for P ≤ S, set

Aut_{F}^{(P)}(S) = {α ∈ Aut_{F}(S) | α(P) = P, α|_{P} ∈ O^{p′}(Aut_{F}(P))} ≤ Aut_{F}(S) .

Set

Δ = (ℤ/p)^{×} × (ℤ/p)^{×} , and Δ_{i} = {(r, r^{i}) | r ∈ (ℤ/p)^{×}} ≤ Δ (for i ∈ ℤ).

Define

μ: Aut(S) ⟶ Δ and μ̂: Out(S) ⟶ Δ

by setting, for α ∈ Aut(S),

μ(α) = μ̂([α]) = (r, s) if \begin{cases}α(x) ∈ x^{r}A & for x ∈ S∖A \\ α(g) = g^{s} & for g ∈ Z_{0} . \end{cases}

The next two lemmas describe the role played by μ and Δ in controlling these fusion systems.

Lemma 2.6. Assume the notation and hypotheses of 2.2 and 2.5, and let m ≥ 3 be such that |A/Z| = p^{m−1}. Then the following hold for each α ∈ Aut(S).

(a) Set (r, s) = μ(α), and let u be such that α(g) ∈ g^{u}A_{0} for each g ∈ A∖A_{0}. Then s ≡ ur^{m−1} (mod p − 1).

(b) Either μ(α) ∈ Δ_{m}, and α normalizes each of the S-conjugacy classes H_{i} and B_{i} (0 ≤ i ≤ p − 1); or μ(α) ∉ Δ_{m}, and α normalizes only the classes H_{0} and B_{0}. Also, α acts via the identity on A/A_{0} if and only if μ(α) ∈ Δ_{m−1}.

Proof. (a) Define inductively A_{1} = [x, A] = S′, and A_{n} = [x, A_{n−1}] for n ≥ 2. If n ≥ 1 and A_{n} ≠ 1, then A_{n+1} = (c_{x} − Id)(A_{n}), Ker((c_{x} − Id)|_{A_{n}}) = Z_{0}, and so |A_{n}/A_{n+1}| = |Z_{0}| = p. Since |A_{1}| = |A|/|Z| = p^{m−1}, this shows that A_{m−1} = Z_{0} and A_{m} = 1.

Fix α ∈ Aut(S), set μ(α) = (r, s), and let u ∈ (ℤ/p)^{×} be such that α(g) ∈ g^{u}A_{0} for each g ∈ A∖A_{0}. Then for each k, α acts on A_{k}/A_{k−1} via g ↦ g^{ur^{k}}. In particular, α acts on Z_{0} via g ↦ g^{ur^{m−1}}, and hence s ≡ ur^{m−1} (mod p − 1).

(b) Fix α ∈ Aut(S), and let r, s, u be as in (a). Then α acts via the identity on A/A_{0} if and only if u = 1, which by (b) holds if and only if μ(α) ∈ Δ_{m−1}. Similarly, α normalizes each subgroup ⟨A_{0}, xa^{j}⟩ if and only if r ≡ u (mod p − 1); i.e., s ≡ r^{m}, and μ(α) ∈ Δ_{m}. □

Lemma 2.7. Assume the notation and hypotheses of 2.2 and 2.5. Assume also that Z = Z_{0}. Let m be such that |A/Z| = p^{m−1} (hence |A| = p^{m}). Fix P ∈ H ∪ B, and set t = −1 if P ∈ H, t = 0 if P ∈ B.

(a) If P ∈ E_{F}, then P ≅ C_{p}^{2} or p_{+}^{1+2}, and μ(Aut_{F}^{(P)}(S)) = Δ_{t}. If P ∈ H_{∗} or P ∈ B_{∗}, then m ≡ t (mod p − 1).

(b) Conversely, assume that P ≅ C_{p}^{2} or p_{+}^{1+2}, and also that μ(H_{P}) ≥ Δ_{t} where

H_{P} = {α ∈ Aut_{F}(S) | α(P) = P } and Ĥ_{P} = {α|_{P} | α ∈ H_{P} } .

Then there is a subgroup Θ ≤ Out(P) such that O_{p}(Θ) = 1, Out_{S}(P) ∈ Syl_{p}(Θ), and N_{Θ}(Out_{S}(P)) = Ĥ_{P} /Inn(P).

(c) In the situation of (b), or in the situation of (a) when Θ = Out_{F}(P), O^{p′}(Θ) ≅ SL_{2}(p), and

Θ ≅ \begin{cases}SL_{2}(p) & if μ(H_{P}) = Δ_{t} \\ GL_{2}(p) & if μ(H_{P}) = Δ . \end{cases}

Proof. Set P_{0} = P ∩ A, and P_{1} = [S, P_{0}] = [x, P_{0}]. Thus |P/P_{0}| = p and |P/P_{1}| = p^{2} in all cases. Set Γ_{P} = O^{p′}(Out(P)).

(a) Assume P ∈ E_{F}. If P ∈ H, then |P | = p^{2} since |Z| = p, and if P ∈ B, then P is nonabelian of order p^{3}. In either case, by Lemma 1.11, applied to the action of Out_{F}(P) on P/[P, P], P/[P, P] ≅ C_{p}^{2} and the action contains that of SL_{2}(p). Hence by Lemma 1.8, P ≅ C_{p}^{2} or p_{+}^{1+2}, Out(P) ≅ GL_{2}(p), and Out_{F}(P) contains Γ_{P} ≅ SL_{2}(p).

Now, N_{Γ_{P}} (Out_{S}(P)) = Out_{S}(P)⋊⟨α⟩ ≅ C_{p} ⋊C_{p−1}, where for some generator r ∈ (ℤ/p)^{×}, α acts on P/P_{0} via g ↦ g^{r}, and on P_{0}/P_{1} via g ↦ g^{1/r}. By Lemma 2.3(e), α extends to α ∈ Aut_{F}(S). Hence Aut_{F}^{(P)}(S) ≥ Aut_{N_{S}(P)}(S)⟨α⟩, with equality since restrictions of elements in Aut_{F}^{(P)}(S) must be contained in the normalizer of a Sylow p-subgroup in SL_{2}(p). If P ≅ C_{p}^{2}, then μ(α) = (r, r^{−1}), while if P ≅ p_{+}^{1+2}, then α|_{Z} = Id and hence μ(α) = (r, 1). Thus in either case, μ(α) generates Δ_{t}, so μ(Aut_{F}^{(P)}(S)) = Δ_{t}.

If P ∈ H_{∗} ∪ B_{∗}, then μ(α) ∈ Δ_{m} by Lemma 2.6(b), since α normalizes P ∈ H_{i}. So Δ_{m} = Δ_{t}, and m ≡ t (mod p − 1).

(b) By assumption, P ≅ C_{p}^{2} or p_{+}^{1+2}, so Γ_{P} = O^{p′}(Out(P)) ≅ SL_{2}(p). Choose α ∈ H_{P} such that μ(α) generates Δ_{t}. Then for some generator r ∈ (ℤ/p)^{×}, α induces x ↦ x^{r} on P/P_{0} and induces x ↦ x^{1/r} on P_{0}/P_{1}. Thus α|_{P} ∈ Γ_{P} and Aut_{S}(P)⟨α|_{P} ⟩ = N_{Γ_{P}} (Aut_{S}(P)).

Set Θ = Γ_{P} ·(Ĥ_{P} /Inn(P)). Then O_{p}(Θ) = 1 since O_{p}(SL_{2}(p)) = 1 and p ∤ |Ĥ_{P} /Inn(P)|, Out_{S}(P) ∈ Syl_{p}(Θ) since p ∤ [Θ:Γ_{P}], and N_{Θ}(Aut_{S}(P)) = Ĥ_{P} since Aut_{S}(P) ⊴ Ĥ_{P}.

(c) In either case (a) or (b), Θ acts faithfully on P/P_{1} ≅ C_{p}^{2}, and contains O^{p′}(Out(P)) ≅ SL_{2}(p) by Lemma 1.11. Also, N_{Θ}(Out_{S}(P)) = Ĥ_{P} /Inn(P): by assumption in case (b), and by the extension axiom in case (a). The last statement now follows since [Θ:O^{p′}(Out(P))] = [μ(H_{P}):Δ_{t}]. □

We are now ready to state and prove our main theorem: a description of all simple fusion systems in the situation of Notation 2.2 for which A is not essential. In the statement of the theorem, we set

ζ = ζ_{p} = e^{2πi/p}, R = ℤ[ζ], and 𝔭 = (1 − ζ)R .

We also set U = Aut_{S}(A), u = c_{x} ∈ U, and σ = ∑_{i=0}^{p−1} u^{i} ∈ ℤ[U]. We regard A as a ℤ[U]-module, and also when possible as an R-module by setting ζ·a = u(a) for a ∈ A.

Theorem 2.8. Fix an odd prime p, a finite nonabelian p-group S with a unique abelian subgroup A ≤ S of index p, and a simple fusion system F over S for which A is not essential. Assume the notation and hypotheses of 2.2 and 2.5. Let m ≥ 3 be such that |A| = p^{m}. Then |Z| = p, μ̂|_{Out_{F} (S)} is injective, x ∈ S∖A can be chosen so that x^{p} = 1 and A_{0}⟨x⟩ is normalized by Aut_{F}(S), and one of the following holds.

(a) S ≅ (R/𝔭^{m}) ⋊ ⟨x⟩, and either (i) m ≡ −1 (mod p−1), μ̂(Out_{F}(S)) = Δ_{−1}, and E_{F} is the union of between 1 and

p of the S-conjugacy classes H_{i}, with Aut_{F}(H_{i}) ≅ SL_{2}(p) when H_{i} ∈ E_{F}; or

(ii) m ≡ −1 (mod p−1), μ̂(Out_{F}(S)) = Δ, and E_{F} = B_{0} ∪ H_{∗}, where Out_{F}(B_{0}) ≅ GL_{2}(p), the subgroups in H_{∗} are all F-conjugate, and Out_{F}(H_{i}) ≅ SL_{2}(p); or

(iii) m ≡ 0 (mod p−1), μ̂(Out_{F}(S)) = Δ, and E_{F} = H_{0} ∪ B_{∗}, where Out_{F}(H_{0}) ≅

GL_{2}(p), the subgroups in B_{∗} are all F-conjugate, and Out_{F}(B_{i}) ≅ SL_{2}(p); or

(iv) m ≢ 0, −1 (mod p−1), μ̂(Out_{F}(S)) = Δ_{−1}, and E_{F} = H_{0} with Out_{F}(H_{0}) ≅

SL_{2}(p).

(b) As a ℤ[U]-module,

A ≅ ℤ[U]/⟨pσ, p^{k} + ℓσ⟩ where k ≥ 1, ℓ ∈ ℤ, and \begin{cases}p ∤ ℓ & if k ≥ 2 \\ p ∤ ℓ+1 & if k = 1. \end{cases}

Also, m = k(p − 1) + 1 ≡ 1 (mod p−1), S ≅ A ⋊ ⟨x⟩, μ̂(Out_{F}(S)) = Δ_{−1}, E_{F} = H_{0},

and Out_{F}(H_{0}) ≅ SL_{2}(p).

Conversely, in each of these cases, there is up to isomorphism a unique simple fusion system F which satisfies the given description of S, μ̂(Out_{F}(S)), E_{F}, and the groups Out_{F}(P) for P ∈ E_{F}. Furthermore,

• A_{0}H_{0} < S is strongly closed in F in cases (a.iv) and (b), A_{0}H_{i} is strongly closed in F in case (a.i) if E_{F} = H_{i}, and these are the only occurrences of proper strongly closed subgroups in these fusion systems; and

• all of these fusion systems are exotic, with the following exceptions when p = 3.

– Case (a.i), when E_{F} = H = H_{0} ∪ H_{∗}: F is the 3-fusion system of PSL_{3}(q) for appropriate q ≡ 1 (mod 3), and also of PSU_{3}(q) for appropriate q ≡ −1 (mod 3).

– Case (a.ii): F is the 3-fusion system of ^{3}D_{4}(q) for appropriate q prime to 3.

Proof. Let F be any reduced fusion system over S such that A ∉ E_{F}. By Lemmas 2.3(c) and 2.4, Z = Z_{0}, S′ = A_{0}, and |Z| = |A/A_{0}| = p.

For each α ∈ Ker(μ), α induces the identity on S/A, and induces the identity on A/A_{0} by Lemma 2.6(a). Thus Ker(μ) is a p-group by Lemma 1.7, and Ker(μ) = O_{p}(Aut(S)) since Im(μ) has order prime to p. In particular, μ̂|_{Out_{F} (S)} is injective since p ∤ |Out_{F}(S)|.

By Proposition 1.3(c,d), and since [Aut_{F}(P), P] = P for P ∈ E_{F}, O^{p}(F) = F if and only if ⟨[Aut_{F}(S), S], E_{F}⟩ = S. Since S′ = A_{0} has index p^{2} in S, [Aut_{F}(S), S] = A_{0}⟨x⟩ if μ(Aut_{F}(S)) ≤ Δ_{m−1} and [Aut_{F}(S), S] = S otherwise (Lemma 2.6(b)). Thus

O^{p}(F) = F ⟺ μ(Aut_{F}(S)) ≰ Δ_{m−1} or (H_{∗} ∪ B_{∗}) ∩ E_{F} ≠ ∅ .  (1)

By Lemma 1.4, O^{p′}(F) = F if and only if μ(Aut_{F}(S)) is generated by the subgroups μ(Aut_{F}^{(P)}(S)) for P ∈ E_{F}. Together with Lemma 2.7(a) (and since H ∩ E_{F} ≠ ∅ in all cases by Lemma 2.4), this implies that

O^{p′}(F) = F ⟺ B ∩ E_{F} ≠ ∅ or μ̂(Out_{F}(S)) = Δ_{−1} .  (2)

Step 1: We first check that each of the above choices of S = A ⋊ ⟨x⟩, μ(Aut_{F}(S)), E_{F}, and Out_{F}(P) for P ∈ E_{F}, determines a reduced fusion system which is unique up to isomorphism. In each case, we fix an identification of A with a quotient group of R or of ℤ[U], as described, and let a ∈ A be the element identified with 1 ∈ R or 1 ∈ ℤ[U].

For each P ∈ E_{F}, either P ∈ H and P ≅ C_{p}^{2} or P ∈ B and P ≅ p_{+}^{1+2}, so Out(P) ≅ GL_{2}(p) in either case, and there is a unique subgroup of Out(P) isomorphic to SL_{2}(p). Also, Out(S) is isomorphic to a semidirect product of Ker(μ̂) = O_{p}(Out(S)) with Im(μ̂) ≤ Δ, so by the Schur-Zassenhaus theorem (see [G, Theorem 6.2.1]), the statements about μ(Aut_{F}(S)) determine Aut_{F}(S) up to conjugacy in Aut(S) (hence up to an isomorphism of fusion systems). Since F is generated by Aut_{F}(S), Aut_{F}(P) for P ∈ E_{F}, and restrictions of those automorphisms, it is uniquely determined by such data.

For each i ∈ ℤ prime to p, define λ_{i} ∈ Aut(S) by setting λ_{i}(a) = a^{i} for a ∈ A and λ_{i}(x) = x. For each j ∈ (ℤ/p)^{×}, define ν_{j} ∈ Aut(S) by setting ν_{j}(x) = x^{j} and ν_{j}(a) = a. Then ν_{j}(u^{k}(a)) = u^{jk}(a) for each k (recall u = c_{x}), so ν_{j} acts on A via a Galois automorphism on R in case (a), or via an automorphism of the group U in case (b). In all cases, ν_{j} is well defined by the description of A as a quotient of R or of ℤ[U]. Let Λ < Aut(S) be the subgroup generated by those λ_{i} of order prime to p (i.e., such that i^{p−1} ≡ 1 modulo the exponent of A), and by all ν_{j} for j ∈ (ℤ/p)^{×}. Thus Λ ≅ C_{p−1} × C_{p−1}. By Lemma 2.6(a), μ(Λ) = Δ. By construction, Λ(H_{0}) = H_{0}, Λ(B_{0}) = B_{0}, and Λ permutes each of the sets {H_{1}, …, H_{p−1}} and {B_{1}, …, B_{p−1}}.

Assume we are in the situation of one of cases (a.i)–(a.iv) or (b). Let Λ_{0} ≤ Λ be such that μ(Λ_{0}) is the given subgroup, and set G = S ⋊ Λ_{0}. Let Q_{1}, …, Q_{k} be representatives for the G-conjugacy classes in E_{F} as listed, chosen among the H_{i} and B_{i} for 0 ≤ i ≤ p − 1. By Lemma 2.6(b), Λ_{0} sends the S-conjugacy class of each Q_{i} to itself, except in (a.ii) and (a.iii) when Q ∈ H_{∗} ∪ B_{∗}, in which cases Λ_{0} = Λ contains a subgroup of order p − 1 which sends each class to itself. Thus Λ_{0}, or a subgroup of order p − 1 in Λ_{0}, normalizes each Q_{i} by the above remarks. By Lemma 2.7(b), for each i = 1, …, k, there is a subgroup Θ_{i} ≤ Out(Q_{i}) ≅ GL_{2}(p) such that Out_{S}(Q_{i}) ∈ Syl_{p}(Θ_{i}), Out_{G}(Q_{i}) = N_{Θ_{i}}(Out_{S}(Q_{i})), and O_{p}(Θ_{i}) = 1; and Θ_{i} is uniquely determined because it contains SL_{2}(p) and the normalizer of its Sylow p-subgroup is given.

Set F = ⟨F_{S}(G), Θ_{1}, …, Θ_{k}⟩, and set K_{i} = Out_{G}(Q_{i}). No Q_{i} is G-conjugate to a subgroup of Q_{j} for j ≠ i. For each i,

(1) p∤[Θ_{i}:K_{i}],

(2) Q_{i} is p-centric in G but no proper subgroup of Q_{i} is F-centric or essential in G, and

(3) K_{i} is strongly p-embedded in Θ_{i}.

Hence F is saturated by [BLO4, Proposition 5.1].

Any normal p-subgroup of F must be contained in all essential subgroups, hence is contained in Z ≅ C_{p}; and Z ⋬ F since some P ∈ E_{F} is abelian in each case. Thus O_{p}(F) = 1. By inspection, the conditions on the right-hand side of (1) and (2) hold, so O^{p}(F) = F = O^{p′}(F).

Step 2: We next list the proper strongly closed subgroups in F, and use that to determine whether F is realizable and to prove that F is simple in all cases.

If 1 ≠ Q ⊴ S is strongly closed in F, then it contains Z_{0} = Z(S) ≅ C_{p} (each nontrivial normal subgroup intersects nontrivially with the center), and hence contains each abelian subgroup in E_{F}. Thus H_{i} ⊆ E_{F} implies that Q ≥ ⟨H_{i}⟩ = A_{0}⟨xa^{i}⟩. In particular, Q ≥ A_{0} ≥ Z_{2}, so Q also contains all nonabelian subgroups in E_{F}. Hence if Q < S, then E_{F} = H_{i} and Q = A_{0}H_{i} for some i. By inspection, the only cases where this occurs are (a.iv) and (b), with i = 0, and sometimes in case (a.i). Conversely, A_{0}H_{i} is strongly closed in each of these cases, since it contains ⟨E_{F}⟩ and is normalized by Aut_{F}(S) (by Lemma 2.6(b) and the assumptions in (a.i) when i ≠ 0).

If A_{0}H_{i} is strongly closed, then it is centric in S, is nonabelian, and does not split as a product. So in all cases, by Lemma 1.5, if F is realizable, it is realizable by a finite simple group. Hence by Lemma 1.6, p = 3, and F is the fusion system of one of the simple groups G ≅ PSL_{3}(q), PSU_{3}(q), G_{2}(q), or ^{3}D_{4}(q) for some q prime to 3. When G is a Chevalley group 𝔾(q), then all of the classes H_{i} are conjugate in the Sylow 3-subgroup of 𝔾(q^{3}), so all of them are essential in F since at least one of them is. Thus we must be in case (a.i), and since m ≡ 1 (mod 2), we have G ≅ PSL_{3}(q) for some q ≡ 1 (mod 3). The groups PSU_{3}(q) are handled in a similar way. The 3-fusion system of ^{3}D_{4}(q) has type (a.ii) by the description of its maximal subgroups in [Kl].

If F is not simple, then since it is reduced, A_{0}H_{i} must be strongly closed for some i, and there is a normal fusion subsystem E ⊴ F over T = A_{0}H_{i}. Then H_{i} splits into p T -conjugacy classes. For each P ∈ H_{i}, N_{S}(P) = Z_{2}P ≤ T since Z_{2} ≤ A_{0}, so Aut_{E}(P) contains the normal closure O^{p′}(Aut_{F}(P)) of Aut_{S}(P), and hence P ∈ E_{E}. So if m ≥ 4 (if |T | ≥ p^{4}), then by Lemma 2.7(a) (and since |A_{0}| = p^{m−1}), m − 1 ≡ −1 (mod p − 1), which does not hold in the cases we are considering. If m = 3 (if T is extraspecial of order p^{3}), then by [RV, Tables 1.1 & 1.2], there is no saturated fusion system over T with exactly p essential (or radical) subgroups of order p^{2}. Thus F is simple.

Step 3: It remains to show that the fusion systems of cases (a) and (b) are the only simple fusion systems satisfying our hypotheses. By Lemma 2.3(a,b), each P ∈ E_{F} lies in exactly one of the classes H_{i} or B_{i} for 0 ≤ i ≤ p − 1, and H_{i} and B_{i} cannot both be in E_{F}.

Recall that U = Aut_{S}(A), u = c_{x} ∈ U, and σ = ∑_{i=0}^{p−1} u^{i} ∈ ℤ[U]. Define

Ψ: ℤ[U] ⟶ A

by setting Ψ(u^{k}) = u^{k}(a) for all k. Then Im(Ψ) is normalized by x, Im(Ψ)⟨x⟩ ≥ ⟨a, x⟩ = S since A_{0} = S′, so Im(Ψ) = A and Ψ is onto.

For each i ∈ ℤ, (xa^{i})^{p} = u(a^{i})u^{2}(a^{i}) ⋯ u^{p}(a^{i})x^{p} = Ψ(iσ)x^{p}. Hence

Ψ(σ) = 1 ⟺ (xa^{i})^{p} = x^{p} for each i ∈ ℤ
⟺ (xa^{i})^{p} = (xa^{j})^{p} for some 0 ≤ i < j ≤ p − 1.

(3)

Since (xa^{i})^{p} = 1 if H_{i} ∈ E_{F} or B_{i} ∈ E_{F}, this shows that x^{p} = 1 if H_{0} ⊆ E_{F}, or if B_{0} ⊆ E_{F}, or if Ψ(σ) = 1. We will see that at least one of these holds in each case.

Note that R ≅ ℤ[U]/⟨σ⟩, and that 𝔭 is the image in R of (1 − u)ℤ[U]. Also, 𝔭^{p−1} = pR: this follows, for example, from the congruence (1 − u)^{p−1} ≡ σ (mod pℤ[U]). Thus each ideal of p-power index in R is a power of 𝔭.

By Lemma 2.4, E_{F} contains at least one abelian subgroup. There are three cases to consider:

Case 1: Assume E_{F} ∩ B ≠ ∅. By Lemma 2.7(a), and since E_{F} ∩ H ≠ ∅, μ(Aut_{F}(S)) ≥ Δ_{0}Δ_{−1} = Δ. Hence all subgroups in H_{∗} and in B_{∗} are F-conjugate (Lemma 2.6(b)), and E_{F} = H_{0} ∪ B_{∗} or B_{0} ∪ H_{∗}.

By Lemma 2.7(a), m ≡ −1 (mod p − 1) if H_{∗} ⊆ E_{F}, while m ≡ 0 (mod p − 1) if B_{∗} ⊆ E_{F}. By Lemma 2.7(c), for P ∈ E_{F}, Out_{F}(P) ≅ GL_{2}(p) if P ∈ H_{0} ∪ B_{0}, while Out_{F}(P) ≅ SL_{2}(p) if P ∈ H_{∗} ∪ B_{∗}.

Since all subgroups in E_{F} have exponent p, (xa^{i})^{p} = 1 for each i. Hence Ψ(σ) = 1 by (3), so Ψ factors through ℤ[U]/⟨σ⟩ ≅ R. We can thus regard A as an R-module, and A ≅ R/𝔭^{m} (recall p^{m} = |A|) by the above remarks. Thus we are in case (a.ii) or (a.iii).

Case 2: Assume E_{F} contains only abelian subgroups, and also that H_{i}, H_{j} ∈ E_{F} for some 0 ≤ i < j ≤ p − 1. Then (xa^{i})^{p} = (xa^{j})^{p} = 1, so σ ∈ Ker(Ψ) by (3), and A ≅ R/𝔭^{m} by the above remarks.

Since H_{j} ∈ E_{F} where j ≠ 0, m ≡ −1 (mod p − 1) by Lemma 2.7(a), and μ(Aut_{F}(S)) ≥ Δ_{m} = Δ_{−1}. Since F is reduced, O^{p′}(F) = F, so μ(Aut_{F}(S)) = Δ_{−1} = Δ_{m} by (2). Thus no two of the H_{i} are F-conjugate (Lemma 2.6(b)), Aut_{F}(H) ≅ SL_{2}(p) for each H ∈ E_{F} by Lemma 2.7(c), and we are in case (a.i).

Case 3: Assume E_{F} = H_{i} for some i. Then μ(Aut_{F}(S)) = Δ_{−1} by (2), and Out_{F}(H_{i}) ≅ SL_{2}(p) by Lemma 2.7(c). Also, by Lemma 2.7(a), either i = 0 or m ≡ −1 (mod p − 1). If i = 0, then by (1), μ(Aut_{F}(S)) ≰ Δ_{m−1}, and hence m ≢ 0 (mod p − 1).

Thus if Ψ(σ) = 1 (so that A ≅ R/𝔭^{m}), then either m ≡ −1 (mod p − 1) and we are in the situation of (a.i); or i = 0, m ≢ 0, −1 (mod p − 1), and we are in the situation of (a.iv).

Now assume Ψ(σ) ≠ 1. By (3), (xa^{j})^{p} = 1 for a unique 0 ≤ j ≤ p − 1, and j = i since subgroups in E_{F} = H_{i} have exponent p. Also, A_{0}⟨xa^{i}⟩ is characteristic in S since it splits over A_{0} while A_{0}⟨xa^{ℓ}⟩ does not split for ℓ ≠ i. So we can assume that x was chosen with x = xa^{i}. Thus i = 0, and E_{F} = H_{0}.

Set I = Ker(Ψ), τ = 1−u and J = σℤ[U]; we identify R = ℤ[U]/J and 𝔭 = (τℤ[U]+J)/J. Then Ψ(J) = ⟨Ψ(σ)⟩ ≠ 1, Ψ(σ) ∈ Z, so Ψ(J) = Z since |Z| = p. Thus J ∩ I = pJ. Also,

A/Z ≅ ℤ[U]/(I + J) ≅ \frac{ℤ[U]/J}{(I + J)/J} ≅ R/𝔭^{m−1} ,

since |A/Z| = p^{m−1} and 𝔭 is the unique maximal ideal in R which contains p. Hence I+J = ⟨σ, τ^{m−1}⟩, Ψ(τ^{m−1}) ∈ Z, and so I = ⟨pσ, τ^{m−1} + tσ⟩ for some t ∈ ℤ. Also, τ^{m} ∈ I since τσ = 0.

Define ε: ℤ[U] ⟶ ℤ by setting ε(∑ n_{i}u^{i}) = ∑ n_{i}. Then Ker(ε) = τℤ[U]. Since |ℤ[U]/(I + τℤ[U])| = |A/A_{0}| = p, we have ε(I) = pℤ, and hence p ∤ t.

Since μ(Aut_{F}(S)) = Δ_{−1}, I is invariant under all automorphisms of U. So (1−u^{r})^{m−1} + tσ ∈ I for each 1 ≤ r ≤ p − 1. Since τ^{m} ∈ I, this implies that

0 ≡ τ^{m−1}(1 + u + … + u^{r−1})^{m−1} + tσ ≡ r^{m−1}τ^{m−1} + tσ (mod I) .

Sice p ∤ t, this implies r^{m−1} ≡ 1 (mod p − 1) for each r, and hence m ≡ 1 (mod p − 1).

Set k = (m − 1)/(p − 1). Then 𝔭^{m−1} = p^{k}R, so I = ⟨pσ, p^{k} + ℓσ⟩ for some ℓ, and the condition ε(I) = pℤ implies that p ∤ ℓ if k ≥ 2, p ∤ (ℓ+1) if k = 1. We are thus in the situation of case (b). □

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