REDUCED FUSION SYSTEMS OVER p-GROUPS WITH ABELIAN SUBGROUP OF INDEX p: II

DAVID A. CRAVEN, BOB OLIVER, AND JASON SEMERARO

Abstract. Let p be an odd prime, and let S be a p-group with a unique elementary abelian subgroup A of index p. We classify the simple fusion systems over all such groups S in which A is essential. The resulting list, which depends on the classification of finite simple groups, includes a large variety of new, exotic simple fusion systems.

A saturated fusion system F over a finite p-group S is a category whose objects are the subgroups of S, whose morphisms are monomorphisms between subgroups, and whose morphism sets satisfy certain axioms, formulated originally by Puig and motivated by properties of conjugacy relations between p-subgroups of a finite group. For example, to each finite group G and each Sylow p-subgroup S of G, one associates the saturated fusion system F_{S}(G) over S whose morphisms are those homomorphisms induced by conjugation in G. We refer to [AKO] or [Cr1] for a detailed introduction to the theory of saturated fusion systems, and to the beginning of Section 1 for a little more detail about these definitions.

A saturated fusion system F is simple if it contains no non-trivial proper normal subsystems (see [AKO, Definition I.6.1] or [Cr1, Sections 5.4 and 8.1] for the precise definition of a normal subsystem). In this paper, we continue the study, started in [Ol], of simple fusion systems F over non-abelian p-groups S which have an abelian subgroup of index p. For p = 2, this was handled in [AOV2, Proposition 5.2(a)]: S must be dihedral, semidihedral or wreathed, and F must be the 2-fusion system of PSL_{2}(q) for q ≡ ±1 (mod 8) or of PSL_{3}(q) for q odd. So assume p is an odd prime. If S has more than one such subgroup, then |S| = p^{3} by [Ol, Theorem 2.1], and this case was dealt with earlier in [RV]. Thus we assume that S has a unique abelian subgroup of index p, which we denote A. If A is not F-essential (see Definition 1.1), then we are in the situation of [Ol], and [Ol, Theorem 2.8] gives a complete characterization of simple fusion systems on S. In contrast to the situation when p = 2, most of the fusion systems found in [Ol] are exotic; i.e., they are not fusion systems of finite groups.

In this paper, we handle the case when A is an F-essential subgroup and has exponent p, and again find a very large variety of exotic fusion systems. Our main tool is Theorem 2.8, which gives precise details concerning the way in which the structure of F is controlled by the action of Aut_{F}(A) on A. Indeed, Theorem 2.8 and its Corollary 2.10 reduce the problem of classifying fusion systems on S to that of determining all pairs (G, A), where G is a finite group (the candidate for Aut_{F}(A)) and A is an 𝔽_{p}G-module that satisfy a certain list of conditions.

Date: June 17, 2016.

2000 Mathematics Subject Classification. Primary 20D20. Secondary 20C20, 20D05, 20E45.

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

D. Craven is financially supported by a Royal Society University Research Fellowship.

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

J. Semeraro was an EPSRC-funded DPhil student while some of this research was carried out.

The bulk of our analysis is thus centred on classifying modules satisfying the required conditions. After some preliminary results in Section 3, the main results are summarized in Theorem 4.1 and Table 4.1. Certain cases are then covered in more detail in Propositions 4.2, 4.3, and 4.4. By combining these results with Theorem 2.8, one can get a complete list of all simple fusion systems of the type described. (A few explicit examples are worked out at the end of Section 4.) Most of the results involving lists of modules depend on the classification of finite simple groups (CFSG), which is thus assumed throughout Sections 4 and 5 and also in Lemma 1.7.

Our strategy for listing modules is based on Aschbacher’s classification [A1] of subgroups G < GL_{n}(p). This splits into two cases, according to whether or not the image of G in PGL_{n}(p) is almost simple. If it is not almost simple, Aschbacher gives a short list of possibilities, which we make more explicit (in our situation) in Section 5 (Propositions 5.3 and 5.4). When G/Z(G) is almost simple, we use CFSG to check each of the possibilities for G in Sections 6–11.

We are still left with the case where S has a unique abelian subgroup of index p which is F-essential and of exponent greater than p. The second author plans to handle this in a later paper with Albert Ruiz.

Our notation is mostly standard. For example, p_{±}^{1+2k} denotes an extraspecial group of order p^{2k+1}, where (for odd p) p_{+}^{1+2k} has exponent p and p_{−}^{1+2k} exponent p^{2}. Also, ^{x}g = xgx^{−1}, g^{x} = x^{−1}gx, G′ is the derived (commutator) subgroup of G, and A ∘ B denotes a central product of groups A and B.

As usual, F (G) denotes the Fitting subgroup of the finite group G: the largest normal nilpotent subgroup of G (i.e., the product of the subgroups O_{p}(G) for all p). Also, E(G) denotes the layer: the central product of the components of G (the subnormal quasisimple subgroups). Thus F^{∗}(G) = F (G)E(G) is the generalized Fitting subgroup.

The following definition will be useful in Sections 6–11.

Definition 0.1. Let H be a finite simple group. A finite group G is of type H if Z(G) is cyclic and F^{∗}(G)/Z(G) ≅ H.

1. Background

We begin by recalling some definitions. When G is a finite group and S ∈ Syl_{p}(G), the p-fusion system of G is the category F_{S}(G) whose objects are the subgroups of S, and where for P, Q ≤ S,

Hom_{F_{S}(G)}(P, Q) = Hom_{G}(P, Q)
=^{def} {φ ∈ Hom(P, Q) | φ = c_{g} for some g ∈ G such that ^{g}P ≤ Q}.

More generally, a saturated fusion system over a p-group S is a category F whose objects are the subgroups of S, where Hom_{F}(P, Q) is a set of injective homomorphisms from P to Q for each P, Q ≤ S, and which satisfies certain axioms, due originally to Puig, and listed (in the form we use them) in [AKO, Definition I.2.2] and [Cr1, § 4.1].

The following terminology is used to describe 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}⟩.

The above definition of F-essential subgroups is motivated by the following version for fusion systems, due to Puig, of the Alperin–Goldschmidt fusion theorem.

Theorem 1.2 ([AKO, Theorem I.3.5], [Cr1, Theorem 4.51]). For each saturated fusion system F over a finite p-group S, each morphism in F is a composite of restrictions of F-automorphisms of S and of F-essential subgroups of S.

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 [Cr1, § 7.5] for their precise definitions and properties.

Definition 1.3. For any saturated fusion system F,

• F is reduced if O_{p}(F) = 1, O^{p}(F) = F, and O^{p′}(F) = F;

• F is simple if it contains no non-trivial proper normal fusion subsystems, in the sense of [AKO, Definition I.6.1] or [Cr1, §§ 5.4 & 8.1]; and

• F is realizable if F = F_{S}(G) for some finite group G with Sylow p-subgroup S.

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 of F. Hence F is reduced if it is simple. If E ⊴ F is a 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 non-trivial strongly closed subgroups.

The next proposition gives some very general conditions for a fusion system to be reduced.

Proposition 1.4. The following hold for a saturated fusion system F over a finite p-group S.

(a) For each Q ⊴ S, Q ⊴ F if and only if for each P ∈ E_{F} ∪ {S}, Q ≤ P and Q is Aut_{F}(P)-invariant.

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

(c) In all cases, O^{p}(F) = F if and only if 𝔣𝔬𝔠(F) = S.

(d) If each P ∈ E_{F} is minimal in the set of all F-centric subgroups, then O^{p′}(F) = F if and only if Aut_{F}(S) = ⟨Inn(S), Aut_{F}^{(P)}(S) | P ∈ E_{F}⟩, where for P ≤ S,

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

Proof. Point (a) is shown in [AKO, Proposition I.4.5], and point (c) in [AKO, Corollary I.7.5]. Point (b) follows from the definition and Alperin’s fusion theorem (Theorem 1.2). Point (d) is shown in [Ol, Lemma 1.4]. □

In order to be able to identify which of the fusion systems we construct are realizable, it will be helpful to know that when realizable, they can be realized by finite simple groups.

Lemma 1.5 ([DRV], [Ol, Lemma 1.5]). Let F be a reduced fusion system over a p-group S. Assume, for each strongly F-closed subgroup 1 ≠ P ⊴ S, that is P 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). Under these conditions, if F is realizable, then it is the fusion system of a finite simple group.

The next lemma gives a very simple, necessary condition for a p-group S to have an abelian subgroup of index p.

Lemma 1.6. Let p be any prime, and let S be a non-abelian p-group which contains an abelian subgroup of index p. Then |Z(S)|·|[S, S]| = \frac{1}{p}|S|.

Proof. Let A ⊴ S be an abelian subgroup of index p, fix x ∈ S ∖ A, and let φ ∈ End(A) be the homomorphism φ(a) = [a, x]. Then Z(S) = C_{A}(x) = Ker(φ), and [S, S] = [x, A] = Im(φ). □

Lemma 1.5 motivates the next lemma: a list of all finite simple groups whose fusion systems are of the type we are studying.

Lemma 1.7. Assume the classification of finite simple groups. Let p be an odd prime, and let F be a reduced fusion system over a finite p-group S which contains a unique abelian subgroup A of index p. If F is realizable, then it is isomorphic to the fusion system of one of the following simple groups:

(a) A_{pn}, where p ≤ n < 2p;

(b) Sp_{4}(p);

(c) PSL_{n}(q), where p|(q − 1) and p ≤ n < 2p;

(d) P Ω_{2n}^{+}(q), where p|(q − 1) and p ≤ n < 2p;

(e) ^{3}D_{4}(q) or ^{2}F_{4}(q), where p = 3 and q is prime to 3;

(f) E_{n}(q), where p|(q − 1), p = 5 if n = 6, 7, and p = 7 if n = 7, 8;

(g) E_{8}(q), where p = 5 and q ≡ ±2 (mod 5); or

(h) Co_{1}, where p = 5.

Proof. Since A is the unique abelian subgroup of index p, |S| ≥ p^{4}. By Lemma 1.5, F is the fusion system of a finite simple group G.

Case 1: If G is an alternating group, then G ≅ A_{n} for some p^{2} ≤ n < 2p^{2}. Since F = F_{S}(G) is reduced, we can choose n to be a multiple of p.

Case 2: Assume that G is of Lie type in characteristic p, and fix S ∈ Syl_{p}(G). We prove that the only possibility is G = PSp_{4}(p), by showing that in all other cases, either S is abelian, or it has more than one abelian subgroup of index p, or it has none at all.

Assume first that G is one of the following groups:

PSL_{4}(p) ≅ Ω_{6}^{+}(p), PSU_{4}(p) ≅ Ω_{6}^{−}(p), PSp_{6}(p), G_{2}(p), ^{2}G_{2}(3^{k}).  (1)

We will show that S has no abelian group of index p in any of these cases. If H < PSL_{4}(p) is the stabilizer of a projective plane and a point in the plane, then O_{p}(H) ≅ p_{+}^{1+4}. If G = PSU_{4}(p) or PSp_{6}(p), and H < G is the stabilizer of an isotropic point or a point, respectively, then O_{p}(H) ≅ p_{+}^{1+4}. Thus if G is one of the groups PSL_{4}(p), PSU_{4}(p), or PSp_{6}(p), then S contains an extraspecial subgroup of order p^{5}, and hence contains no abelian subgroup of index p.

For p ≥ 5, G_{2}(p) also contains an extraspecial p-group of order p^{5} (see [Wi2, p.127]). If G ≅ G_{2}(3), then its parabolic subgroups have the form (C_{3}^{2}×3_{+}^{1+2})⋊GL_{2}(3). If S is contained in this group, then each abelian subgroup of S must intersect the subgroup (C_{3}^{2} × 3_{+}^{1+2}) with index at least 3, so that SL_{2}(3) must act trivially in particular on the C_{3}^{2} factor, while it actually acts as on the natural module (see [Wi2, p.125]). So we eliminate these cases.

Now assume that G ≅ ^{2}G_{2}(3^{k}) for k ≥ 3. By the main theorem in [Wa], |S| = (p^{k})^{3} = |Z(S)|·|[S, S]|. So by Lemma 1.6, S does not contain an abelian subgroup of index p.

Thus S has no abelian subgroup of index p if G is one of the groups in (1), or any group which contains one of them. In this way, we can eliminate all larger classical groups, as well as ^{3}D_{4}(p) > G_{2}(p), E_{n}(p) > F_{4}(p) > Spin_{9}(p) (n = 6, 7, 8), and ^{2}E_{6}(p) > F_{4}(p) (see, e.g., [Wi2, Chapter 4] for descriptions of these inclusions), and also the groups of the same type over larger fields of characteristic p.

Since PSL_{2}(p^{k}) has abelian Sylow p-subgroups, it remains to consider the groups

PSL_{3}(p^{k}), PSU_{3}(p^{k}), PSp_{4}(p^{k}) ≅ Ω_{5}(p^{k}),  (2)

for k ≥ 1. The Sylow p-subgroups of PSL_{3}(p) and PSU_{3}(p) are extraspecial of order p^{3}, hence have more than one abelian subgroup of index p, while those of PSp_{4}(p) do have a unique such subgroup. By [GLS3, Theorem 3.3.1.a], if G is one of the groups in (2), then |S|, |Z(S)|, and |[S, S]| are all powers of p^{k}, and so if k > 1, S contains no abelian subgroup of index p by Lemma 1.6.

Case 3: Assume that G = ^{r}𝔾(q) is a group of Lie type in characteristic different from p. By [BMO2, Lemma 6.9], and since the Sylow p-subgroups of G are non-abelian, G has a p-fusion system isomorphic to that of one of the following groups:

(i) PSL_{n}(q) for some n ≥ p; or

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

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

(iv) G_{2}(q), F_{4}(q), E_{6}(q), E_{7}(q), or E_{8}(q) where p|(q − 1); or

(v) E_{8}(q) where p = 5 and q ≡ ±2 (mod 5).

Assume that G = PSL_{n}(q). If e = ord_{p}(q) > 1, then by [Ru, Theorem B], the p-fusion system of G has a proper normal subsystem of index e, and hence is not reduced. Thus p|(q − 1), and p ≤ n < 2p since the Sylow p-subgroups have abelian subgroups of index p. We are thus in the situation of (c).

Assume that G = P Ω_{2n}^{ε}(q) is as in (ii). Set e = ord_{p}(q). If e is even, then by [BMO1, Proposition A.3], the p-fusion system of GO_{2n}^{ε}(q) is isomorphic to that of SL_{2n}(q). So F is normal of index 2 in the fusion system of SL_{2n}(q); this contains a normal subsystem of index e by [Ru, Theorem B] again, and hence F has a normal subsystem of index e/2. So F is reduced only if e = 2. If e is odd, then by [BMO1, Theorem A(a,b)], we can assume that q is a square, GO_{2n}^{ε}(q) and SL_{2n}(\sqrt{q}) have isomorphic p-fusion systems by [BMO1, Proposition A.3] again, and so F is reduced only if e = 1.

We can thus assume that q ≡ ±1 (mod p), and p|(q − 1) if n is odd (since ε = +1). If n is even and q ≡ −1 (mod p), then by [BMO1, Theorem A(b)], G has the same p-fusion system as P Ω_{2n}^{ε}(q^{∗}) for some q^{∗} ≡ 1 (mod p). So we can assume that p|(q − 1) in all cases, and are in the situation of (d).

Cases (iii) and (v) correspond to (e) and (g), respectively. In case (iv), if G = 𝔾(q) and p|(q − 1), then the order of the Weyl group of 𝔾 must be a multiple of p but not of p^{2}, and so (𝔾, p) is one of the pairs (G_{2}, 3), (E_{6}, 5), (E_{7}, 5), (E_{7}, 7), or (E_{8}, 7). The 3-fusion system of G_{2}(q) is not reduced, since it is the fusion system of PSL_{3}(q):2 or PSU_{3}(q):2 [A2, (16.11)]. So we are in the situation of (f).

Case 4: 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), (Co_{1}, 5), and (Th, 3). When S ∈ Syl_{3}(J_{3}), |S| = 3^{5}, Z(S) ≅ C_{3}^{2} and [S, S] ≅ C_{3}^{3} (see [FR, § 3]), so by Lemma 1.6, S does not contain an abelian subgroup of index 3. Since Th contains a subgroup isomorphic to G_{2}(3) [Pa, (3.12)], whose Sylow 3-subgroups were already shown not to have abelian subgroups of index 3, the same holds for Th. Thus p = 5 and G = Co_{1}. □

We finish the section with some miscellaneous group-theoretic results that will be needed later.

Lemma 1.8. Fix a prime p, a finite p-group P, and a group G ≤ Aut(P) of automorphisms of P. Let Fr(P) = P_{0} ⊴ P_{1} ⊴ ⋯ ⊴ P_{m} = P be a sequence of subgroups, all normal in P and normalized by G. 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, and hence H ≤ O_{p}(G).

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

Lemma 1.9. Fix a finite abelian p-group A and a subgroup G ≤ Aut(A), and choose U ∈ Syl_{p}(G). Then

C_{A}(U) ≤ [G, A] ⟺ C_{A}(G) ≤ [G, A] ⟺ C_{A}(G) ≤ [U, A] .

Proof. Each of the first and third inequalities clearly implies the second, so it suffices to show that the second implies each of the other two. Assume C_{A}(G) ≤ [G, A].

If C_{A}(U) ≰ [G, A], then choose z ∈ C_{A}(U) ∖ [G, A]. Let X be the G-orbit of z and set m = |X|; p∤m since z is fixed by U. Let ẑ be the product of the elements in X. Then ẑ ∈ C_{A}(G), and ẑ ∈ z^{m}·[G, A]. Thus ẑ ∈ C_{A}(G) ∖ [G, A], contradicting our assumption.

Now assume C_{A}(G) ≰ [U, A]. Since ℤ/p^{∞} is injective as an abelian group, there is a homomorphism φ ∈ Hom(A, ℤ/p^{∞}) such that [U, A] ≤ Ker(φ) but C_{A}(G) ≰ Ker(φ). Let X be the G-orbit of φ under the action of G on Hom(A, ℤ/p^{∞}), and set m = |X|. Then p∤m since φ is fixed by U. Let φ̂ be the product of the elements of X. Then φ̂ is G-invariant, so [G, A] ≤ Ker(φ̂). Fix z ∈ C_{A}(G) such that φ(z) ≠ 0; then φ̂(z) = m·φ(z) ≠ 0. Thus z ∈ C_{A}(G) ∖ [G, A], again contradicting our assumption. □

Lemma 1.10 ([Ol, 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 A_{1} = C_{A}(O^{p′}(G)) and A_{2} = [O^{p′}(G), A]. Then G normalizes A_{1} and A_{2}, A = A_{1} × A_{2}, and O^{p′}(G) ≅ SL_{2}(p) acts faithfully on A_{2} ≅ C_{p}^{2}.

Lemma 1.11. Let V be a finite abelian p-group (written additively), and fix a subgroup G ≤ Aut(V) with U ∈ Syl_{p}(G) of order p. Assume also that dim(C_{V} (U) ∩ [U, V]) = 1. Set V_{0} = C_{V} (U)[U, V]. Define inductively W_{1} > W_{2} > ⋯ by setting W_{1} = [U, V], and W_{n+1} = [U, W_{n}] for n ≥ 1. Let m be the smallest integer such that W_{m} = 0. Then the following hold.

(a) For each 1 ≤ i ≤ m − 1, |W_{i}/W_{i+1}| = p = |V/V_{0}|.

(b) Fix g ∈ N_{G}(U). Let r, t ∈ (ℤ/p)^{×} be such that ^{g}u = u^{r} for all u ∈ U, and g induces multiplication by t on V/V_{0}. Then for each 1 ≤ i ≤ m − 1, g induces multiplication by tr^{i} on W_{i}/W_{i+1}.

Proof. (a) Fix a generator u ∈ U, and let φ: V ⟶ V be the homomorphism φ(v) = [u, v]. For each 0 ≠ W ≤ V normalized by U, Ker(φ|_{W}) = C_{W} (U), Im(φ|_{W}) = [U, W], and thus |W/[U, W]| = |C_{W} (U)| > 1. In particular, if 0 ≠ W ≤ W_{1} = [U, V], then 0 ≠ C_{W} (U) ≤ C_{W_{1}}(U), with equality since C_{W_{1}}(U) = C_{V} (U) ∩ [U, V] has order p by assumption. Hence [U, W] has index p in W.

Thus |W_{i}/W_{i+1}| = p for each 1 ≤ i < m. Also, since |C_{V} (U)|·|[U, V]| = |V |, |V/V_{0}| = |C_{V} (U) ∩ [U, V]| = p.

(b) Fix g ∈ N_{G}(U), and let r, t ∈ (ℤ/p)^{×} be as above. For each x = [u, v] ∈ W_{1}, g(x) = [^{g}u, g(v)] ≡ [u^{r}, tv] ≡ rt[u, v] modulo [U, V_{0}] = [U, W_{1}] = W_{2}. This proves the result when i = 1, and the other cases follow inductively. □

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

Throughout this section, p is an odd prime. We want to describe all reduced fusion systems over non-abelian p-groups which contain an abelian subgroup of index p. If S has more than one abelian subgroup of index p, then by [Ol, Theorem 2.1], either S is extraspecial of order p^{3} and exponent p (the case already handled by Ruiz and Viruel in [RV]), or there are no reduced fusion systems over S. In [Ol, Theorem 2.8], the second author handled the case where S contains a unique abelian subgroup of index p and that subgroup is not essential. We now look at the more complicated case: that where S contains a unique abelian subgroup of index p and it is essential (Theorem 2.8 below).

Notation 2.1. Fix a 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} = Z·S′ .

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

H = {Z⟨x⟩ | x ∈ S ∖ A} and (when Z_{2} ≤ A_{0}) B = {Z_{2}⟨x⟩ | x ∈ S ∖ A} .

Lemma 2.2. Assume the notation and hypotheses of 2.1. Then the following hold.

(a) For each P ∈ E_{F}, either P = A, or P is abelian and P ∈ H, or P is non-abelian and P ∈ B. In all cases, |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 A ⋬ F (equivalently, if E_{F} ⊈ {A}), then |Z_{0}| = p.

(d) If |Z_{0}| = p, then |A/A_{0}| = p, |Z_{2}/Z| = p, Z_{2} ≤ A_{0}, and Z_{2} ∩ S′ ≅ C_{p}^{2}. Also, there are elements x ∈ S ∖ A and a ∈ A ∖ A_{0} such that A_{0}⟨x⟩ and S′⟨a⟩ are normalized by Aut_{F}(S). If some element of S ∖ A has order p, then we can choose x to have order p.

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

(f) For each x ∈ S ∖ A and each g ∈ A_{0}, Z⟨x⟩ is S-conjugate to Z⟨gx⟩, and Z_{2}⟨x⟩ is S-conjugate to Z_{2}⟨gx⟩.

Proof. (a,b,c,e) See points (a), (c), (b), and (e), respectively, in [Ol, Lemma 2.3].

(d) Assume that |Z_{0}| = p. Fix a generator α ∈ Aut_{S}(A) ≅ C_{p}. Let f: A ⟶ A be the homomorphism f(x) = x^{−1}α(x). Then Ker(f) = C_{A}(α) = Z and Im(f) = [α, A] = S′, so |Z|·|S′| = |A|, and |A/A_{0}| = |A/ZS′| = |Z ∩ S′| = p. Also, |S′| > p since A is the unique abelian subgroup of index p in S, so S/Z is non-abelian, and Z_{2}/Z = Z(S/Z) = C_{A/Z}(α). Let f: A/Z ⟶ A/Z be the homomorphism induced by f on the quotient; then |Z_{2}/Z| = |Ker(f)| = |(A/Z)/Im(f)| = |A/ZS′| = p.

Since Z_{2}/Z = Z(S/Z) has order p (and |S/Z| > p), Z_{2}/Z must be contained in [S/Z, S/Z] = S′Z/Z = A_{0}/Z. Thus Z_{2} ≤ A_{0}, so Z_{2}S′ = ZS′, and hence |Z_{2} ∩ S′| = |Z_{2}/Z|·|Z ∩ S′| = p^{2}. It remains to show that Z_{2} ∩ S′ is not cyclic.

For each x ∈ S′ = [α, A], x = y^{−1}α(y) for some y ∈ A, so ∏_{i=0}^{p−1} α^{i}(x) = 1. Hence if Z_{2} ∩ S′ ≅ C_{p^{2}} is generated by x, then α(x) = x^{1+kp} for some k such that p ∤ k, and ∑_{i=0}^{p−1}(1 + kp)^{i} = ((1 + kp)^{p} − 1)/kp ≡ 0 (mod p^{2}). Since (1 + kp)^{p} ≡ 1 + kp^{2} (mod p^{3}), this is impossible.

Let B ≤ A be minimal among subgroups which are normalized by Aut_{F}(S) such that B ≥ S′ and BA_{0} = A. The natural surjection B/S′Fr(B) −↠ A/A_{0} of 𝔽_{p}[Out_{F}(S)]- modules is split since p ∤ |Out_{F}(S)|. Hence B/S′Fr(B) ≅ A/A_{0} by the minimality of B, and B/S′ is cyclic since B/S′Fr(B) is cyclic. For any generator aS′ of B/S′, a ∈ A ∖ A_{0}, and B = S′⟨a⟩ is normalized by Aut_{F}(S).

For each x ∈ S ∖ A, x^{p} ∈ C_{A}(x) = Z ≤ A_{0}. Hence S/A_{0} ≅ C_{p}^{2}. Since A/A_{0} is an 𝔽_{p}[Out_{F}(S)]-submodule of S/A_{0} (and since p ∤ |Out_{F}(S)|), S/A_{0} splits as a product S/A_{0} = (A/A_{0}) × (R/A_{0}) where R is Aut_{F}(S)-invariant. Let x be any element of R ∖ A ⊆ S ∖ A; then R = A_{0}⟨x⟩.

Assume that there is y ∈ S ∖ A such that y^{p} = 1. If A_{0}⟨y⟩ is normalized by Aut_{F}(S), we are done. Otherwise, there is y′ ∈ S ∖ A such that y′^{p} = 1 and y^{−1}y′ ∉ A_{0}. Set g = y^{−1}y′; then

1 = y′^{p} = (yg)^{p} = (ygy^{−1})(y^{2}gy^{−2}) ⋯ (y^{p}gy^{−p})y^{p} ,

so ∏_{i=1}^{p}(y^{i}gy^{−i}) = 1. Then (yg^{j})^{p} = ∏_{i=1}^{p}(y^{i}g^{j}y^{−i})y^{p} = 1 for all j ∈ ℤ by a similar computation. Since g ∈ A ∖ A_{0}, there is j such that yg^{j} ∈ xA_{0}. Upon replacing x by yg^{j}, we can arrange that x^{p} = 1.

(f) Fix x ∈ S ∖ A and g ∈ A_{0} = S′Z, and choose z ∈ Z and g′ ∈ S′ such that g = g′z. Then there is h ∈ A such that g′ = h^{−1}(^{x}h), gx = zh^{−1}xh, and so Z⟨gx⟩ = Z⟨x^{h}⟩ = Z⟨x⟩^{h} and Z_{2}⟨gx⟩ = Z_{2}⟨x^{h}⟩ = Z_{2}⟨x⟩^{h}. □

Lemma 2.3. Let A ⊴ S, F, H, B, etc., be as in Notation 2.1. Assume that A ⋬ F.

(a) If P ∈ B ∩E_{F}, then P = ZP^{∗} for some unique Aut_{F}(P)-invariant extraspecial subgroup P^{∗} of order p^{3} and exponent p, with centre Z(P^{∗}) = Z_{0} = Z∩P^{∗}. Also, O^{p′}(Out_{F}(P)) ≅ SL_{2}(p), and this group acts faithfully on P^{∗}/Z_{0} ≅ P/Z ≅ C_{p}^{2} and acts trivially on Z.

(b) If P ∈ H∩E_{F}, then there is a unique subgroup Z^{∗} < Z which is normalized by Aut_{F}(Z) and such that Z = Z_{0} × Z^{∗}. Also, Z^{∗} is normalized by Aut_{F}(P), P = Z^{∗} × P^{∗} for some unique Aut_{F}(P)-invariant subgroup P^{∗} ≅ C_{p}^{2} which contains Z_{0}, O^{p′}(Aut_{F}(P)) ≅ SL_{2}(p), and this group acts faithfully on P^{∗} and acts trivially on Z^{∗}.

(c) There is x ∈ S ∖ A of order p; i.e., S splits over A.

Proof. (a) Assume that P ∈ B ∩ E_{F}. Then |P/Z| = |P/Z_{2}|·|Z_{2}/Z| = p^{2} (Lemma 2.2(c,d)), so P/Z is abelian, and [P, P] ≤ Z ∩ S′ = Z_{0}. Thus [P, P] = Z_{0} since |Z_{0}| = p and P is non-abelian.

By Lemma 2.2(a), Out_{S}(P) ∈ Syl_{p}(Out_{F}(P)) has order p. Also, [N_{S}(P), P] ≤ P ∩ S′ = Z_{2} ∩ S′, and Z_{2} ∩ S′ ≅ C_{p}^{2} by Lemma 2.2(c,d). Thus [Aut_{S}(P), P/Z_{0}] ≤ (Z_{2} ∩ S′)/Z_{0} ≅ C_{p}, with equality since P is essential and hence Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)) cannot act trivially on P/[P, P].

By Lemma 1.10 applied to the Out_{F}(P)-action on P/Z_{0}, P = P_{1}P_{2}, where Aut_{F}(P) normalizes P_{1} and P_{2}, P_{1}/Z_{0} = C_{P/Z_{0}}(O^{p′}(Out_{F}(P))), and P_{1} ∩ P_{2} = Z_{0}. Thus P_{1}/Z_{0} is the intersection of the subgroups in the Aut_{F}(P)-orbit of C_{P/Z_{0}}(Aut_{S}(P)) = Z_{2}/Z_{0}, hence contains Z(P)/Z_{0} = Z/Z_{0}, with equality since |P/Z| = p^{2} = |P_{2}/Z_{0}|. Thus P_{1} = Z, and we set P^{∗} = P_{2}. Since P^{∗}/Z_{0} ≅ C_{p}^{2} by Lemma 1.10 again, P^{∗} is extraspecial of order p^{3}, and has exponent p since otherwise its automorphism group would be a p-group. The last statement follows immediately from Lemma 1.10.

(b) Assume that P ∈ H ∩ E_{F}. Thus P is abelian. By Lemma 2.2(a), Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)) has order p. Also, for each g ∈ N_{S}(P) ∖ P, 1 ≠ [g, P] ≤ P ∩ S′ = Z_{0}, so |[g, P]| = p by Lemma 2.2(c). Hence by Lemma 1.10, P = Z^{∗} × P^{∗}, where Z^{∗} ≤ Z and P^{∗} ≥ Z(P) ∩ S′ = Z_{0} are both Aut_{F}(P)-invariant. Also, by the same lemma, O^{p′}(Aut_{F}(P)) ≅ SL_{2}(p), and this subgroup acts faithfully on P^{∗} ≅ C_{p}^{2} and trivially on Z^{∗}.

In particular, there is a subgroup H ≤ N_{O^{p′}(Aut_{F} (P))}(Aut_{S}(P)) of order p − 1 which acts as the full group of automorphisms of Z_{0} and of P^{∗}/Z_{0}, and acts trivially on Z^{∗}. Since H restricts to a subgroup of Aut_{F}(Z), this shows that Z = Z^{∗} × Z_{0} is the unique Aut_{F}(Z)- invariant splitting of Z with one factor Z_{0}.

(c) Since A ⋬ F, there must be P ∈ E_{F} in H ∪ B. So there is x ∈ S ∖ A of order p by the descriptions of P in (a) and (b). □

We now need to fix some more notation.

Notation 2.4. Assume Notation 2.1. Assume also that |Z_{0}| = p, and hence that |A/A_{0}| = p. Fix a ∈ A ∖ A_{0} and x ∈ S ∖ A, chosen such that A_{0}⟨x⟩ and S′⟨a⟩ are each normalized by Aut_{F}(S), and such that x^{p} = 1 if any element of S ∖ A has order p (Lemma 2.2(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_{∗} by Lemma 2.2(f) and since |A/A_{0}| = p.

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}

Finally, set

Aut_{F}^{∨}(S) = {α ∈ Aut_{F}(S) | [α, Z] ≤ Z_{0}},
Out_{F}^{∨}(S) = Aut_{F}^{∨}(S)/Inn(S),
Aut_{F}^{∨}(A) = {α|_{A} | α ∈ Aut_{F}^{∨}(S)} = {β ∈ N_{Aut_{F} (A)}(Aut_{S}(A)) | [β, Z] ≤ Z_{0}},
Aut_{F}^{(P)}(S) = {α ∈ Aut_{F}(S) | α(P) = P, α|_{P} ∈ O^{p′}(Aut_{F}(P))}  (all P ≤ S).

Lemma 2.5. Let S be a finite p-group with a unique abelian subgroup A ⊴ S of index p, and let F be a saturated fusion system over S. Assume that |Z_{0}| = p, and use Notation 2.1 and 2.4. Let m ≥ 3 be such that |A/Z| = p^{m−1}. Then the following hold.

(a) μ̂|_{Out_{F}^{∨} (S)} is injective.

(b) Fix α ∈ Aut(S), set (r, s) = μ(α), and let t be such that α(g) ∈ g^{t}A_{0} for each g ∈ A∖A_{0}. Then s ≡ tr^{m−1} (mod p).

(c) For α ∈ Aut_{F}(S), 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}.

(d) Assume that x^{p} = 1, and set σ = ∏_{i=0}^{p−1} ^{x^{i}}(a) = (ax)^{p}x^{−p}. For each P ∈ H_{0} ∪ B_{0}, P splits over P ∩ A. For each P ∈ H_{∗} ∪ B_{∗}, P splits over P ∩ A if and only if σ ∈ Fr(Z).

Proof. (b) This follows from Lemma 1.11(b), applied with A, A_{0}, S′, and Z_{0} in the role of V, V_{0}, W_{1}, and W_{m−1}. Note that p^{m−1} = |A/Z| = |S′|, |W_{1}| = p^{m−1} by Lemma 1.11(a), and thus m plays the same role here as in Lemma 1.11.

(a) Assume that α ∈ Aut_{F}^{∨}(S) has order prime to p, and μ(α) = 1. Then α induces the identity on S/A, on A/A_{0} by (b), and on A_{0}/S′ since [α, ZS′] ≤ Z_{0}S′ = S′ by definition of Aut_{F}^{∨}(S). So α = Id_{S} by Lemma 1.8. Thus Aut_{F}^{∨}(S) ∩ Ker(μ) is a p-group, and hence equal to Inn(S).

(c) Fix α ∈ Aut_{F}(S), and let r, s, t be as in (b). Then α acts via the identity on A/A_{0} if and only if t = 1; equivalently (by (b)) if and only if μ̂([α]) = (r, s) ∈ Δ_{m−1}.

Since α(A_{0}⟨x⟩) = A_{0}⟨x⟩ by assumption (see Notation 2.4), α(x) ∈ A_{0}x^{r}. Hence for each 0 ≤ i < p, α(A_{0}⟨xa^{i}⟩) = A_{0}⟨x^{r}a^{it}⟩. Thus α(H_{0}) = H_{0} and α(B_{0}) = B_{0} in all cases, while for 0 < j < p, α(H_{j}) = H_{j} or α(B_{j}) = B_{j} only if r ≡ t (mod p − 1). By (b), this holds if and only if s ≡ r^{m} (mod p − 1); i.e., if and only if μ(α) ∈ Δ_{m}.

(d) Since x^{p} = 1, P splits over P ∩ A for all P ∈ H_{0} ∪ B_{0}. For each 1 ≤ i ≤ p − 1, (xa^{i})^{p} = σ^{i}, so there is z ∈ Z with xa^{i}z of order p exactly when σ ∈ Fr(Z). This proves the claim for P ∈ H_{∗}.

Assume that P ∈ B_{i} for 1 ≤ i ≤ p − 1. If P splits over P ∩ A, then Z_{2}⟨xa^{i}⟩ splits (it is Sconjugate to P), and hence (xa^{i}y)^{p} = 1 for some y ∈ Z_{2}. Also, [xa^{i}, y] = [x, y] ∈ [S, Z_{2}] = Z_{0}, so 1 = (xa^{i}y)^{p} = (xa^{i})^{p}y^{p} = σ^{i}y^{p}, and hence σ ∈ Fr(Z_{2}). Also, since Z_{2} = Z(Z_{2} ∩ S′) where Z_{2} ∩ S′ ≅ C_{p}^{2}, Z_{2} ≅ Z × C_{p}, and hence Fr(Z_{2}) = Fr(Z). The converse is clear: if σ = z^{p} for some z ∈ Z, then Z_{2}⟨xa^{i}⟩ is split over Z_{2} by ⟨xa^{i}z^{−i}⟩. □

Lemma 2.6. Let S be a finite p-group with a unique abelian subgroup A ⊴ S of index p, and let F be a saturated fusion system over S. We use Notation 2.1, and let m be such that |A/Z| = p^{m−1}. Fix P ∈ H ∪ B. Set t = −1 if P ∈ H or set t = 0 if P ∈ B.

(a) If P ∈ E_{F}, then in the notation of 2.4, Aut_{F}^{(P)}(S) ≤ Aut_{F}^{∨}(S) and μ(Aut_{F}^{(P)}(S)) = Δ_{t}. If in addition, P ∈ H_{∗} ∪ B_{∗}, then m ≡ t (mod p − 1).

(b) Conversely, assume that μ(N_{Aut_{F}^{∨} (S)}(P)) ≥ Δ_{t}, and also that P splits over P ∩ A. Then there is a unique subgroup Θ ≤ Aut(P) such that

(i) Aut_{S}(P) ∈ Syl_{p}(Θ),

(ii) Θ ≥ Inn(P) and O^{p′}(Θ)/Inn(P) ≅ SL_{2}(p),

(iii) [α, Z] ≤ Z_{0} for each α ∈ O^{p′}(Θ), and

(iv) N_{Θ}(Aut_{S}(P)) = {α|_{P} | α ∈ N_{Aut_{F} (S)}(P)}.

Proof. (a) Assume that P ∈ E_{F}. By Lemma 2.3(a,b), whether P ∈ H or P ∈ B, O^{p′}(Out_{F}(P)) ≅ SL_{2}(p), and acts trivially on Z/Z_{0}. Thus Aut_{F}^{(P)}(S) ≤ Aut_{F}^{∨}(S).

If P ∈ H, then by Lemma 2.3(b), there is P^{∗} ≤ P such that P^{∗} ∩ A = Z_{0}, and O^{p′}(Aut_{F}(P)) ≅ SL_{2}(p) acts faithfully on P^{∗} ≅ C_{p}^{2}. Each element of the normalizer N_{O^{p′}(Aut_{F} (P))}(Aut_{S}(P)) ≅ C_{p} ⋊ C_{p−1} extends to an element of Aut_{F}^{(P)}(S) by Lemma 2.2(e); and since this normalizer contains all diagonal matrices ( \begin{matrix}u & 0 \\ 0 & u^{−1} \end{matrix} ) for u ∈ (ℤ/p)^{×}, μ(Aut_{F}^{(P)}(S)) is the set {(u, u^{−1}) | u ∈ (ℤ/p)^{×}} = Δ_{−1}.

If P ∈ B, then by Lemma 2.3(a), there is P^{∗} ≤ P such that Z_{0} ≤ P^{∗} ∩ A ≤ Z_{2}, P^{∗} is extraspecial of order p^{3} and exponent p, and O^{p′}(Out_{F}(P)) ≅ SL_{2}(p) acts faithfully on P^{∗}/Z_{0} ≅ C_{p}^{2}. Each element of N_{O^{p′}(Out_{F} (P))}(Out_{S}(P)) ≅ C_{p} ⋊ C_{p−1} extends to an element of Aut_{F}^{(P)}(S) by Lemma 2.2(e); this normalizer acts trivially on Z_{0}, and hence μ(Aut_{F}^{(P)}) = {(u, 1) | u ∈ (ℤ/p)^{×}} = Δ_{0}.

If P ∈ H_{∗}∪B_{∗}, then μ(Aut_{F}^{(P)}(S)) ≤ Δ_{m} by Lemma 2.5(c), since each element in Aut_{F}^{(P)}(S) normalizes P. So Δ_{m} = Δ_{t}, and m ≡ t (mod p − 1).

(b) Set

Λ_{P} = \{α|_{P} | α ∈ N_{Aut_{F} (S)}(P)\} and P̂ = P ∩ S′ = \begin{cases}Z ∩ S′ = Z_{0} & if P ∈ H \\ Z_{2} ∩ S′ ≅ C_{p}^{2} & if P ∈ B. \end{cases}

Then Λ_{P} normalizes P̂, and the induced action of Λ_{P} on P/P̂ factors through the quotient group Λ_{P} /Aut_{S}(P) of order prime to p and normalizes (A ∩ P)/P̂. Since P splits over P ∩ A, there is a subgroup P^{∗}/P̂ ≤ P/P̂ of order p, also normalized by Λ_{P}, such that P/P̂ = ((P ∩ A)/P̂) × (P^{∗}/P̂). Since Z_{2} ≤ A_{0} = ZS′ by Lemma 2.2(d), P = P^{∗}Z and P^{∗} ∩ Z = Z_{0}. Also, |P^{∗}| = p^{2} if P ∈ H, and P^{∗} is extraspecial of order p^{3} if P ∈ B.

Assume first that P ∈ H. Choose α ∈ N_{Aut_{F}^{∨} (S)}(P) such that μ(α) generates Δ_{−1}. Then α(P^{∗}) = P^{∗} since P^{∗} is normalized by Λ_{P}, α acts non-trivially on Z_{0} and on P^{∗}/Z_{0}, and induces the identity on Z/Z_{0}. Set Z^{∗} = C_{Z}(α); thus Z = Z_{0} × Z^{∗}. Also, P^{∗} = [α, P], and P^{∗} ≅ C_{p}^{2} since P = Z^{∗} × P^{∗} splits over Z. Since Aut_{F}(Z) has order prime to p, and normalizes Z_{0} which is a direct factor in Z, there is an Aut_{F}(Z)-invariant subgroup of Z complementary to Z_{0}, and this can only be Z^{∗} = C_{Z}(α). In particular, Z^{∗} is normalized by Aut_{F}(S), and hence by Λ_{P}.

Now assume that P ∈ B. Choose α ∈ N_{Aut_{F}^{∨} (S)}(P) such that μ(α) generates Δ_{0}. Then α(P^{∗}) = P^{∗} since P^{∗} is normalized by Λ_{P}, α acts non-trivially on P^{∗}/(P^{∗} ∩ A) and on (P^{∗}∩A)/Z_{0} and trivially on Z_{0}, and hence P^{∗} has exponent p. Also, since P^{∗}/Z_{0} = [α, P/Z_{0}], the choice of P^{∗} was unique.

We have now shown that O^{p′}(Out(P^{∗})) ≅ SL_{2}(p) in both cases (P ∈ H or P ∈ B). Define

Θ_{0} = \begin{cases}\{α ∈ Aut(P) | α|_{P^{∗}} ∈ O^{p′}(Aut(P^{∗})), α|_{Z^{∗}} = Id\} & if P ∈ H \\ \{α ∈ Aut(P) | α|_{P^{∗}} ∈ O^{p′}(Aut(P^{∗})), α|_{Z} = Id\} & if P ∈ B \end{cases}

and set Θ = Λ_{P} Θ_{0}. Since Λ_{P} normalizes Θ_{0}, Θ is a subgroup of Aut(P). Also, Θ ≥ Θ_{0} ≥ Inn(P), and Θ_{0}/Inn(P) ≅ SL_{2}(p). \underline{Proof of (i)–(iii)} Since Θ = Λ_{P} Θ_{0} where Λ_{P} normalizes Θ_{0}, |Θ/Θ_{0}| = |Λ_{P} /(Λ_{P} ∩ Θ_{0})|, and this is prime to p since Λ_{P} ∩ Θ_{0} ≥ Aut_{S}(P) ∈ Syl_{p}(Λ_{P}). Thus Θ_{0} = O^{p′}(Θ), and Aut_{S}(P) ∈ Syl_{p}(Θ). So (i) and (ii) hold. Also, (iii) holds since [α, Z] ≤ Z_{0} for all α ∈ Θ_{0} = O^{p′}(Θ). \underline{Proof of (iv)} By construction, Λ_{P} ≤ N_{Θ}(Aut_{S}(P)). So to prove (iv), it suffices to show that Λ_{P} ≥ N_{Θ_{0}}(Aut_{S}(P)). Fix a generator u ∈ (ℤ/p)^{×}. By assumption, there is α ∈ N_{Aut_{F}^{∨} (S)}(P) such that μ(α) = (u, u^{t}). Thus α|_{P} ∈ Λ_{P}, and [α, Z] ≤ Z_{0}. Upon replacing α by an appropriate power, if necessary, we can assume that |α| is prime to p.

If P ∈ H, then α|_{Z^{∗}} = Id (recall that α|_{Z} ∈ Aut_{F}(Z) normalizes Z^{∗}), t = −1, so α acts on P^{∗} ≅ C_{p}^{2} via diag(u, u^{−1}). If P ∈ B, then t = 0, α|_{Z} = Id since α induces the identity on Z_{0} and on Z/Z_{0}, and α acts on P^{∗}/Z_{0} ≅ C_{p}^{2} via diag(u, u^{−1}). Thus α|_{P} ∈ Θ_{0}, and N_{Θ_{0}}(Aut_{S}(P)) = Aut_{S}(P)⟨α⟩ ≤ Λ_{P}. This finishes the proof of (iv). \underline{Proof of uniqueness} Let Θ^{∗} ≤ Aut(P) be another subgroup which satisfies (i)–(iv). Since Aut_{S}(P) ∈ Syl_{p}(Θ^{∗}) by (i), Θ^{∗} = N_{Θ}(Aut_{S}(P))O^{p′}(Θ^{∗}) by the Frattini argument, and so Θ^{∗} = Λ_{P} O^{p′}(Θ^{∗}) by (iv).

It remains to prove that O^{p′}(Θ^{∗}) = Θ_{0}. Since O^{p′}(Θ^{∗})/Inn(P) ≅ SL_{2}(p) ≅ Θ_{0}/Inn(P) by (ii), it suffices to show that O^{p′}(Θ^{∗}) ≤ Θ_{0}. If P ∈ B, then Aut_{S}(P) ∈ Syl_{p}(Θ^{∗}) acts trivially on Z and Z = Z(P) is characteristic in P, so O^{p′}(Θ^{∗}) also acts trivially on Z, and hence O^{p′}(Θ^{∗}) ≤ Θ_{0} by definition and the uniqueness of P^{∗}.

If P ∈ H, then by Lemma 1.10, P = Z^{∨} ×P^{∨} where O^{p′}(Θ^{∗}) ≅ SL_{2}(p) acts trivially on Z^{∨} and acts faithfully on P^{∨} ≅ C_{p}^{2}. We showed above that there is α ∈ N_{Aut_{F}^{∨} (S)}(P) such that Z^{∗} = C_{P} (α) and P^{∗} = [α, P], and α|_{P} ∈ Λ_{P} ≤ Θ^{∗} by (iv). Hence Z^{∗} = Z^{∨} and P^{∗} = P^{∨}, and so O^{p′}(Θ^{∗}) ≤ Θ_{0}. □

In the next lemma, we describe the conditions for a saturated fusion system F over S to be reduced. This requires some more precise information about the F-essential subgroups and their automorphisms in this situation.

In the proofs of the next lemma and theorem, we refer several times to the extension axiom for saturated fusion systems. This axiom states that in a saturated fusion system F over a p-group S, if φ ∈ Hom_{F}(P, Q) is such that Q = φ(P) is fully centralized in F, and if P ≥ P is such that P ⊴ P and φAut_{P} (P)φ^{−1} ≤ Aut_{S}(Q), then there is φ ∈ Hom_{F}(P , S) which extends φ. We refer to [AKO, Proposition I.2.5] for more detail, including a description of how this axiom can be used to characterize saturated fusion systems.

Lemma 2.7. Fix an odd prime p, and a p-group S which contains a unique abelian subgroup A ⊴ S of index p. Let F be a saturated fusion system over S, and assume that A ⋬ F. We use the notation of 2.1 and 2.4. Assume also that A ∈ E_{F}, and set G = Aut_{F}(A) and U = Aut_{S}(A) ∈ Syl_{p}(G). Then the following hold.

(a) O_{p}(F) = 1 if and only if either (i) there are no non-trivial G-invariant subgroups of Z; or

(ii) E_{F} ∩ H ≠ ∅ and Z_{0} is the only non-trivial G-invariant subgroup of Z.

(b) If O_{p}(F) = 1, then O^{p}(F) = F if and only if [G, A] = A.

(c) O^{p′}(F) = F if and only if one of the following holds: either

(i) E_{F} ∖ {A} = H_{0} ∪ B_{∗}, and Aut_{F}(S) = ⟨Aut_{F}^{∨}(S), Aut_{F}^{(A)}(S)⟩; or

(ii) E_{F} ∖ {A} = B_{0} ∪ H_{∗}, and Aut_{F}(S) = ⟨Aut_{F}^{∨}(S), Aut_{F}^{(A)}(S)⟩; or

(iii) E_{F} ∖ {A} ⊆ H and Aut_{F}(S) = ⟨(Aut_{F}^{∨}(S) ∩ μ^{−1}(Δ_{−1})), Aut_{F}^{(A)}(S)⟩; or

(iv) E_{F} ∖ {A} ⊆ B and Aut_{F}(S) = ⟨(Aut_{F}^{∨}(S) ∩ μ^{−1}(Δ_{0})), Aut_{F}^{(A)}(S)⟩.

Proof. (a) Upon replacing the statements by their negatives, we must show that O_{p}(F) ≠ 1 if and only if

there is a non-trivial subgroup 1 ≠ Q ≤ Z normalized by G, such that either

Q ≠ Z_{0} or E_{F} ∩ H = ∅. (1)

Assume first that O_{p}(F) ≠ 1, and set Q = O_{p}(F). Since A ∈ E_{F}, Q ≤ A and is Ginvariant by Proposition 1.4(a). Since A ⋬ F, there is some P ∈ E_{F} ∖ {A} ⊆ B ∪ H. By Proposition 1.4(a) again, if P ∈ H, then Q ≤ P ∩ A = Z, while if P ∈ B, then Q ≤ ⋂_{α∈Aut_{F} (P)} α(P ∩ A) = Z. Thus Q is a non-trivial G-invariant subgroup of Z. If Q = Z_{0}, then E_{F} ∩ H = ∅, since for P ∈ E_{F} ∩ H, Z_{0} is not normalized by Aut_{F}(P) (Z_{0} < P^{∗} ≅ C_{p}^{2} in the notation of Lemma 2.3(b)). Thus (1) holds in this case.

Conversely, assume that (1) holds. In particular, 1 ≠ Q ≤ Z is G-invariant. For each α ∈ Aut_{F}(S), α(A) = A since A is the unique abelian subgroup of index p, so α|_{A} ∈ G, and thus α(Q) = Q. Since each element of Aut_{F}(Z) extends to S by the extension axiom, Q is also normalized by Aut_{F}(Z). Also, for each P ∈ E_{F} ∩ B, Z = Z(P) is characteristic in P and so Q is also normalized by Aut_{F}(P). In particular, if E_{F} ∩ H = ∅, then Q ⊴ F by Proposition 1.4(a), so O_{p}(F) ≠ 1.

Now assume that E_{F} ∩ H ≠ ∅, and hence by assumption that Q ≠ Z_{0}. By Lemma 2.3(d), there is a unique Aut_{F}(Z)-invariant splitting Z = Z_{0}×Z^{∗}. Set Q^{∗} = Q∩Z^{∗}. If Q ≥ Z_{0}, then Q = Q^{∗} × Z_{0}. Otherwise, Q ∩ Z_{0} = 1 (recall |Z_{0}| = p), and since Q is Aut_{F}(Z)-invariant, the uniqueness of the splitting implies that Q ≤ Z^{∗} and hence Q = Q^{∗}. Since Q ≠ Z_{0}, we have Q^{∗} ≠ 1 in either case.

For each φ ∈ Aut_{F}(A) = G, φ(Q^{∗}) ≤ Q ≤ Z, so by the extension axiom, φ|_{Q^{∗}} extends to some φ ∈ Aut_{F}(S), and φ(Q^{∗}) = φ(Q^{∗}) = Q^{∗} since Q^{∗} is Aut_{F}(Z)-invariant. So by the same arguments as those applied above to Q, Q^{∗} is normalized by Aut_{F}(P) for each P ∈ ({S} ∪ E_{F}) ∖ H. If P ∈ E_{F} ∩ H, then for each α ∈ Aut_{F}(P), α(Z^{∗}) = Z^{∗} by Lemma 2.3(b), so α|_{Z^{∗}} extends to an element of Aut_{F}(S) and hence of Aut_{F}(Z), and in particular, α(Q^{∗}) = Q^{∗}. Thus 1 ≠ Q^{∗} ⊴ F, and hence O_{p}(F) ≠ 1.

(b) Assume that O_{p}(F) = 1 and [G, A] < A. Since C_{A}(G) ≤ Z_{0} ≤ [U, A] by (a), [G, A] ≥ C_{A}(U) = Z by Lemma 1.9. Hence [G, A] ≥ ZS′ = A_{0}, with equality since |A/A_{0}| = p. The G-action on A/A_{0} is thus trivial, and hence μ̂(Out_{F}(S)) ≤ Δ_{m−1} by Lemma 2.5(c), where p^{m−1} = |A/Z|.

By Lemma 2.6(a), for each 1 ≤ i ≤ p − 1, H_{i} ⊆ E_{F} implies μ̂(Out_{F}(S)) ≥ Δ_{−1} = Δ_{m}, while B_{i} ⊆ E_{F} implies μ̂(Out_{F}(S)) ≥ Δ_{0} = Δ_{m}. Thus E_{F} ∖ {A} ⊆ H_{0} ∪ B_{0}, so 𝔣𝔬𝔠(F) ≤ ⟨[G, A], H_{0}, B_{0}⟩ = A_{0}⟨x⟩ < S by Proposition 1.4(b), and O^{p}(F) ≠ F by Proposition 1.4(c).

Conversely, if [G, A] = A, then 𝔣𝔬𝔠(F) ≥ A. Since A ⋬ F, E_{F} ⫌ {A}, and hence 𝔣𝔬𝔠(F) = S. So O^{p}(F) = F by Proposition 1.4(c).

(c) By Proposition 1.4(d), and since Aut_{F}^{(A)}(S) ≥ Inn(S), O^{p′}(F) = F if and only if Aut_{F}(S) is generated by the subgroups Aut_{F}^{(P)}(S) for all P ∈ E_{F}. By Lemma 2.6(a), if P ∈ E_{F} ∖{A}, then Aut_{F}^{(P)}(S) = Aut_{F}^{∨}(S) ∩ μ^{−1}(Δ_{t}), where t = −1 if P ∈ H and t = 0 if P ∈ B. Hence the conditions in (c.iii) are necessary and sufficient if E_{F} ∖ {A} ⊆ H, and those in (c.iv) are necessary and sufficient if E_{F} ∖ {A} ⊆ B.

If E_{F} ∖ {A} contains subgroups in both H and B, then μ(Aut_{F}^{∨}(S)) ≥ Δ_{0}Δ_{−1} = Δ by Lemma 2.6(a), and Aut_{F}(S) = ⟨Aut_{F}^{(A)}(S), Aut_{F}^{∨}(S)⟩ by the above remarks. Also, by Lemma 2.2(b), some H_{i} or B_{i} must be essential for 0 < i < p, and by Lemma 2.6(a), m ≡ −1 (mod p − 1) if H_{i} ∈ E_{F}, while m ≡ 0 (mod p − 1) if B_{i} ∈ E_{F}. Also, all subgroups in H_{∗} and in B_{∗} are F-conjugate by Lemma 2.5(c) and since μ(Aut_{F}^{∨}(S)) = Δ. Hence E_{F} ∖ {A} = H_{0} ∪ B_{∗} or B_{0} ∪ H_{∗}, and we are in the situation of (c.i) or (c.ii). □

We are now ready to describe the reduced fusion systems over non-abelian p-groups S which contain a unique abelian subgroup of index p which is essential. Recall that we defined

Aut_{F}^{∨}(A) = {α ∈ N_{G}(U) | [α, Z] ≤ Z_{0}} = {α|_{A} | α ∈ Aut_{F}^{∨}(S)} .

Let

μ_{A}: Aut_{F}^{∨}(A) ⟶ Δ

be the homomorphism defined by setting μ_{A}(α) = μ(α) for some α ∈ Aut_{F}(S) which extends α. Since α ∈ N_{G}(U) = N_{Aut_{F} (A)}(Aut_{S}(A)), it does extend to some α ∈ Aut_{F}(S) by the extension axiom. If β is another extension, then β^{−1}α ∈ Aut_{F}(S) induces the identity on A and (since C_{S}(A) = A) on S/A, and hence by definition of μ lies in Ker(μ). Thus μ_{A}(α) = μ(α) is independent of the choice of α.

Theorem 2.8. Fix an odd prime p, and a p-group S which contains a unique abelian subgroup A ⊴ S of index p. Let F be a reduced fusion system over S for which A is F-essential. We use the notation of 2.1 and 2.4, and also set E_{0} = E_{F} ∖ {A} and G = Aut_{F}(A), so that U = Aut_{S}(A) ∈ Syl_{p}(G). Let m ≥ 3 be such that |A/Z| = p^{m−1}. Set σ = ∏_{i=0}^{p−1} ^{x^{i}}(a) = (ax)^{p}x^{−p} ∈ Z. Then the following hold:

(a) Z_{0} = C_{A}(U) ∩ [U, A] has order p, and hence A_{0} = C_{A}(U)[U, A] has index p in A.

(b) There are no non-trivial G-invariant subgroups of Z = C_{A}(U), aside (possibly) from Z_{0}.

(c) [G, A] = A.

(d) One of the conditions (i)–(iv) holds, described in Table 2.1.

Conversely, for each G, A, U ∈ Syl_{p}(G), and E_{0} ⊆ H∪B which satisfy conditions (a)–(d), where |U| = p and U ⋬ G, there is a reduced fusion system F over A⋊U with Aut_{F}(A) = G and E_{F} = E_{0} ∪ {A}, unique up to isomorphism. All such fusion systems are simple. All such fusion systems are exotic, except for the fusion systems of the simple groups listed in Table 2.2.

μ_{A}(Aut_{F}^{∨}(A)) G = O^{p′}(G)X where m (mod p − 1)  σ  E_{0}
(i)  Δ  X = Aut_{F}^{∨}(A)  ≡ 0  σ ∈ Fr(Z) H_{0} ∪ B_{∗}
(ii)  Δ  X = Aut_{F}^{∨}(A)  ≡ −1  σ ∈ Fr(Z) B_{0} ∪ H_{∗}
≡ −1  σ ∈ Fr(Z) union of H_{i}’s
(iii)  ≥ Δ_{−1}  X = μ_{A}^{−1}(Δ_{−1})  −  −  H_{0}
X = μ_{A}^{−1}(Δ_{0})  ≡ 0  σ ∈ Fr(Z) union of B_{i}’s

(iv)

≥ Δ_{0}
Z_{0} not G-invariant  −  −  B_{0}

Table 2.1

Such a fusion system F has a proper strongly closed subgroup if and only if A_{0} = C_{A}(U)[U, A] is G-invariant, and E_{0} = H_{i} or B_{i} for some i, in which case A_{0}H_{i} = A_{0}B_{i} is strongly closed.

In Table 2.2, e is such that p^{e} is the exponent of A. In all cases except when Γ ≅ PSL_{p}(q) and e > 1, A is homocyclic.

Γ  p  conditions  rk(A)  e  m  G = Aut_{Γ}(A)  E_{0}
A_{pn} p  p ≤ n < 2p  n  1  p  \frac{1}{2}C_{p−1} ≀ S_{n}  H_{0}
Sp_{4}(p) p  3  1  3  GL_{2}(p)/{±I}  B_{0}
PSL_{p}(q) p v_{p}(q−1) = 1, p > 3 p−2  1  p − 2  S_{p}  H_{0} ∪ H_{∗}
PSL_{p}(q) p p^{2}|(q−1), p > 3  p−1 v_{p}(q−1) e(p−1) − 1  S_{p}  H_{0} ∪ H_{∗}
PSL_{n}(q) p p|(q−1), p<n<2p n−1 v_{p}(q−1) e(p−1) + 1  S_{n}  B_{0}
P Ω_{2n}^{+}(q) p p|(q−1), p≤n<2p  n  v_{p}(q−1) e(p−1) + 1 C_{2}^{n−1} ⋊ S_{n}  B_{0}
2F_{4}(q) 3  q ≥ 8  2  v_{3}(q+1)  2e  GL_{2}(3)  B_{0} ∪ B_{∗}
E_{n}(q) 5 n = 6, 7, p|(q−1)  n  v_{p}(q−1)  4e + 1  W (E_{n})  B_{0}
E_{n}(q) 7 n = 7, 8, p|(q−1)  n  v_{p}(q−1)  6e + 1  W (E_{n})  B_{0}
E_{8}(q) 5 q ≡ ±2 (mod 5)  4 v_{5}(q^{4}−1)  4e  (C_{4} ∘ 2^{1+4}).S_{6} H_{0} ∪ B_{∗}
Co^{1} 5  3  1  3  4 × S_{5}  B_{0} ∪ H_{∗}

Table 2.2

Proof. We prove in Step 1 that conditions (a)–(d) are necessary, and in Step 2 that they are sufficient for the existence of a reduced fusion system. In Step 3, we list all strongly closed subgroups for the fusion systems constructed in Step 2, and then prove in Step 4 that they are all simple. In Step 5, we handle the question of which of these fusion systems are realizable.

Step 1: Assume that F is a reduced fusion system over S. We must show that conditions (a)–(d) hold.

(a) By Lemma 2.2(c,d), |Z_{0}| = p = |A/A_{0}|.

(b,c) Since F is reduced, O_{p}(F) = 1 and O^{p}(F) = F. So these claims follow from points (a) and (b), respectively, in Lemma 2.7.

(d) Cases (i)–(iv) here correspond exactly to cases (i)–(iv) of Lemma 2.7(c). The conditions on E_{0} follow immediately from that lemma, while the conditions on μ_{A}(Aut_{F}^{∨}(A)) = μ(Aut_{F}^{∨}(S)) and those on m (mod p − 1) follow from Lemma 2.6(a). By Lemma 2.5(d), for P ∈ H_{∗} ∪ B_{∗}, σ ∈ Fr(Z) if and only if P splits over P ∩ A, and this is a necessary condition to have P ∈ E_{F} by Lemma 2.3(a,b).

To see why the conditions on G hold, let

R: Aut_{F}(S) ⟶ N_{Aut_{F} (A)}(Aut_{S}(A)) = N_{G}(U)

be the homomorphism induced by restriction. By the extension axiom and by definition, R sends subgroups of Aut_{F}(S) as follows:

X R^{−1}(Y) Aut_{F}(S) Aut_{F}^{∨}(S) Aut_{F}^{(A)}(S) Aut_{F}^{∨}(S) ∩ μ^{−1}(Δ_{r})
R(X)  Y  N_{G}(U) Aut_{F}^{∨}(A) N_{O^{p′}(G)}(U)  μ_{A}^{−1}(Δ_{r})

(1)

In other words, the groups in the second row are the images of those in the first, and those in the first are the inverse images of those in the second. By the Frattini argument, G = O^{p′}(G)·N_{G}(U) = O^{p′}(G)·R(Aut_{F}(S)). The claims in (d) about generators for G now follow from the formulas for Aut_{F}(S) in Lemma 2.7(c) (and since R(Aut_{F}^{(A)}(S)) ≤ O^{p′}(G)).

Step 2: Now assume that A, G, and E_{0} are as above and satisfy (a)–(d). We will show that they are realized by a unique reduced fusion system F.

Set Γ = A ⋊ G, and identify S = A ⋊ U ∈ Syl_{p}(Γ). Choose a generator x ∈ U < S. Choose a ∈ A ∖ A_{0} so that S′⟨a⟩ is normalized by N_{G}(U). Set Z = Z(S), Z_{2} = Z_{2}(S), H_{i} = Z⟨xa^{i}⟩, and B_{i} = Z_{2}⟨xa^{i}⟩, as in Notation 2.1 and 2.4.

Let Q_{1}, …, Q_{k} ∈ E_{0} ∩{B_{i}, H_{i} | 0 ≤ i ≤ p−1} be a set of Γ-conjugacy class representatives for the subgroups in E_{0}. For each i ≤ k, set K_{i} = Aut_{Γ}(Q_{i}). Set t_{i} = −1 if Q_{i} ∈ H, and t_{i} = 0 if Q_{i} ∈ B. Since σ ∈ Fr(Z) whenever Q_{i} ∈ H_{∗} ∪ B_{∗}, Q_{i} splits over Q_{i} ∩ A in all cases by Lemma 2.5(d). By the assumptions in (d), there is α ∈ Aut_{F}^{∨}(S) such that μ(α) generates Δ_{t}. By Lemma 2.5(c) and the assumptions on m, α(Q_{i}) is S-conjugate to Q_{i}. So upon composing α with an inner automorphism (hence in Ker(μ)), we can arrange that α ∈ N_{Aut_{F}^{∨} (S)}(Q_{i}).

We want to apply Lemma 2.6(b), with F_{S}(Γ) in the role of F and Q_{i} in the role of P. Thus Aut_{F}^{∨}(A) = Aut_{F_{S}(Γ)}^{∨}(S). Each element of K_{i} = Aut_{Γ}(Q_{i}) extends to an element of Aut_{Γ}(S) since N_{Γ}(Q_{i}) ≤ N_{Γ}(AQ_{i}), and so K_{i} = {α|_{Q_{i}} | α ∈ N_{Aut_{F} (S)}(Q_{i})}. By Lemma 2.6(b), there is a unique subgroup Θ_{i} ≤ Aut(Q_{i}) such that Θ_{i} ≥ Inn(Q_{i}), Aut_{S}(Q_{i}) ∈ Syl_{p}(Θ_{i}), N_{Θ_{i}}(Aut_{S}(Q_{i})) = K_{i}, [α, Z] ≤ Z_{0} for α ∈ O^{p′}(Θ_{i}), and O^{p′}(Θ_{i})/Inn(Q_{i}) ≅ SL_{2}(p).

Set F = ⟨F_{S}(Γ), Θ_{1}, …, Θ_{k}⟩: the smallest fusion system over S which contains F_{S}(Γ), and such that Aut_{F}(Q_{i}) ≥ Θ_{i} for each i. By [BLO4, Proposition 5.1], to see that F is saturated, it suffices to check that the following conditions hold.

(1) For i ≠ j, Q_{i} is not Γ-conjugate to a subgroup of Q_{j}. By assumption, Q_{i} and Q_{j} are not Γ-conjugate, and hence are not F-conjugate since there is no larger group in the generating set which could conjugate the one into the other. If Q_{i} = H_{k} and Q_{j} = B_{ℓ} for some k, ℓ ∈ {0, 1, …, p − 1}, then by (d), either k = 0 and ℓ ≠ 0 or vice versa, so B_{k} and B_{ℓ} are not Aut_{F}(S)-conjugate (hence not Γ-conjugate, hence not F-conjugate) by Lemma 2.5(c).

(2) For each i, Q_{i} is p-centric in Γ, but no proper subgroup P < Q_{i} is F-centric nor an essential p-subgroup of Γ. In all cases, C_{Γ}(Q_{i}) ∩ A = Z = Z(Q_{i}) ∩ A. Also, Q_{i} ∈ H implies C_{Γ}(Q_{i}) ≤ A ⋊ C_{G}(Z) where U ∈ Syl_{p}(C_{G}(Z)), while Q_{i} ∈ B implies C_{Γ}(Q_{i}) ≤ A ⋊ C_{G}(Z_{2}) where C_{G}(Z_{2}) has order prime to p. Thus Z(Q_{i}) ∈ Syl_{p}(C_{Γ}(Q_{i})) in both cases, so Q_{i} is p-centric in Γ by definition.

Each proper subgroup of Q_{i} either does not contain Z or is in the Θ_{i}-orbit of (hence F-conjugate to) a proper subgroup of A, and in either case, is not F-centric. Since A ⊴ Γ (so A ⊴ F_{S}(Γ)), each essential p-subgroup of Γ contains A by Proposition 1.4(a).

(3) For each i, p ∤ [Θ_{i}:K_{i}] and K_{i}/Inn(Q_{i}) is strongly p-embedded in Θ_{i}/Inn(Q_{i}). By assumption, Aut_{S}(Q_{i}) ∈ Syl_{p}(Θ_{i}), and K_{i} = N_{Θ_{i}}(Aut_{S}(Q_{i})). Thus Out_{S}(Q_{i}) ∈ Syl_{p}(Θ_{i}/Inn(Q_{i})) has order p (and is not normal), and so its normalizer has index prime to p and is strongly p-embedded in Θ_{i}/Inn(Q_{i}).

It remains to prove that F is reduced. By (b), there are no non-trivial G-invariant subgroups of Z except possibly for Z_{0}, E_{F} ∩H ≠ ∅ in cases (d.i)–(d.iii), and Z_{0} is not G-invariant in case (d.iv). Hence O_{p}(F) = 1 by Lemma 2.7(a). Also, O^{p}(F) = F by Lemma 2.7(b), and since [G, A] = A by (c).

As for showing that O^{p′}(F) = F, we claim that the necessary conditions in Lemma 2.7(c.i–c.iv) follow from (d.i–d.iv). The conditions on E_{0} = E_{F} ∖ {A} are clear. As for the conditions on Aut_{F}(S), these follow since if G = O^{p}(G)X by (d) for X ≤ N_{Aut(A)}(U), and R: Aut(S) ⟶ N_{Aut(A)}(U) is as in Step 1, then

Aut_{F}(S) = R^{−1}(N_{G}(U)) = R^{−1}(N_{O^{p′}(G)}(U))·R^{−1}(X) = Aut_{F}^{(A)}(S)·R^{−1}(X),

where R^{−1}(X) is described in (1).

The uniqueness of F follows from the uniqueness of the Θ_{i} (Lemma 2.6(b)).

Step 3: We next list the proper non-trivial strongly closed subgroups in the reduced fusion system F constructed in Step 2.

Assume that 1 ≠ Q < S is strongly closed in F. If Q ≤ Z, then Q is contained in all F-essential subgroups, so Q ⊴ F by Proposition 1.4(a), which is impossible since O_{p}(F) = 1. Thus Q ≰ Z.

Now, (QZ/Z) ∩ Z(S/Z) ≠ 1 since Q ⊴ S, so Q ∩ Z_{2} ≰ Z. Fix g ∈ (Q ∩ Z_{2}) ∖ Z. Then Q ≥ [g, S] = Z_{0} since Q ⊴ S.

Since A ⋬ F, there is P ∈ E_{0}. If P ∈ B, then the Aut_{F}(P)-orbit of g ∈ Z_{2} ≤ P is not contained in A. If P ∈ H, then the Aut_{F}(P)-orbit of Z_{0} ≤ P is not contained in A. So in either case, Q ≰ A. Hence Q ≥ [Q, S] ≥ [U, A] = S′.

Set G_{0} = O^{p′}(G). Since Q ∩ A is normalized by the action of G = Aut_{F}(A), and contains [U, A] where U ∈ Syl_{p}(G), Q ≥ [G_{0}, A]. Since [G, A] = A by (c), the group G/G_{0} of order prime to p acts on A/[G_{0}, A] with C_{A/[G_{0},A]}(G/G_{0}) = 1. Also, G = G_{0}Aut_{F}^{∨}(A) by (d), so C_{A/[G_{0},A]}(Aut_{F}^{∨}(A)) = 1. Since Aut_{F}^{∨}(A) acts trivially on Z/Z_{0} (and Z_{0} ≤ S′ ≤ [G_{0}, A]), the natural homomorphism Z/Z_{0} ⟶ A/[G_{0}, A] has trivial image. Hence A_{0} = ZS′ ≤ [G_{0}, A] ≤ Q.

Since Q < S by assumption, this proves that Q = A_{0}⟨xa^{i}⟩ = A_{0}H_{i} for some i, and has index p in S. If A_{0}H_{i} is strongly closed, then neither H_{j} nor B_{j} can be F-essential for any j ≠ i (0 ≤ j ≤ p − 1), so E_{0} = H_{i} or B_{i}.

Conversely, if E_{0} = H_{i} or B_{i} for some 0 ≤ i ≤ p − 1 and A_{0} is normalized by G, then A_{0}H_{i} is normalized by Aut_{F}(S) and Aut_{F}(A) and contains all F-essential subgroups other than A, hence is strongly closed in F.

Step 4: We now prove that F is simple. Assume otherwise. Then there is a normal subsystem E ⊴ F over a proper strongly closed subgroup 1 ≠ T < S. We just saw that this implies T = A_{0}H_{i} for some i, and E_{0} = H_{i} or B_{i}.

Assume first that m ≥ 4. Thus |S′| = |A/Z| = p^{m−1} ≥ p^{3}. For P ∈ E_{0}, we have P < T, so N_{S}(P) ≤ A_{0}P = T. Thus Aut_{S}(P) ≤ Aut_{E}(P) and Aut_{E}(P) ⊴ Aut_{F}(P), which imply that P ∈ E_{E}. (Since Aut_{E}(P) has Sylow p-subgroups which are non-normal and have order p, it has strongly p-embedded subgroups.) Also, A_{0} = A ∩ T is the unique abelian subgroup of index p in T since |[T, T]| = \frac{1}{p}|S′| ≥ p^{2}, so by Steps 1 and 2 applied to E, we have E_{E} = {A_{0}, H_{i}} or {A_{0}, B_{i}}. Here, H_{i} or B_{i} plays the role for E that H or B plays for F. So by Lemma 2.6(a) (or by Table 2.1 case (iii) or (iv)), m − 1 ≡ t (mod p − 1), where as usual, t = −1 if E_{0} = H_{i} and t = 0 if E_{0} = B_{i}. Also, G = O^{p′}(G)μ_{A}^{−1}(Δ_{t}), and by Lemma 2.5(c), elements in μ_{A}^{−1}(Δ_{t}) = μ_{A}^{−1}(Δ_{m−1}) act trivially on A/A_{0}. Since O^{p′}(G) acts trivially on A/A_{0} ≅ C_{p}, this proves that [G, A] ≤ A_{0}, which contradicts (c). We conclude that F is simple.

Now assume that m = 3. Thus |[U, A_{0}]| = |A_{0}/Z| = p^{m−2} = p, and A_{0} is G-invariant. By Lemma 1.10 applied to the G-action on A_{0}, A_{0} = C_{A_{0}}(O^{p′}(G)) × [O^{p′}(G), A_{0}], the first factor is G-invariant and contained in Z, and hence is trivial by (b). So by the same lemma, A_{0} ≅ C_{p}^{2} and O^{p′}(G) ≅ SL_{2}(p). If α ∈ N_{O^{p′}(G)}(U) has order p − 1, then for some generator u ∈ (ℤ/p)^{×}, α(z) = z^{u} for z ∈ Z_{0}, and α(z) ∈ z^{u^{−1}}Z_{0} for z ∈ Z_{2} ∖ Z_{0} = A_{0} ∖ Z_{0}. In the notation of Lemma 1.11(b) (applied with A in the role of V), tr = u^{−1} and tr^{2} = u. So t = u^{−3}, and thus α(a) ∈ a^{u^{−3}}A_{0} for a ∈ A ∖ A_{0}. Since O^{p′}(G) acts trivially on A/A_{0}, this implies that u^{−3} ≡ 1 (mod p), so (p − 1)|3, which is impossible. Thus there are no strongly closed subgroups in this case, and so F is simple.

Step 5: By Lemma 1.5, if F is realizable, it is isomorphic to the fusion system of a finite simple group. Hence by Lemma 1.7, it is isomorphic to the fusion system of one of the simple groups listed in Table 2.2. □

Recall that when U ≅ C_{p}, then 𝔽_{p}U ≅ 𝔽_{p}[X]/(X^{p}), and hence the indecomposable 𝔽_{p}Umodules are those of the form 𝔽_{p}[X]/(X^{i}) for 1 ≤ i ≤ p. These are the “Jordan blocks” of an 𝔽_{p}U-module.

Notation 2.9. Assume that G is a finite group, with U ∈ Syl_{p}(G) of order p. Let M be an 𝔽_{p}G-module, and set Z = C_{M}(U) and Z_{0} = Z ∩ [U, M]. Assume that dim(Z_{0}) = 1; equivalently, that M|_{U} has just one non-trivial Jordan block. In this situation, we set

G^{∨} = {α ∈ N_{G}(U) | [α, Z] ≤ Z_{0}} ,

and define μ_{A}: G^{∨} ⟶ Δ by setting μ_{A}(g) = (r, s) if ^{g}u = u^{r} and g(z) = z^{s} for all u ∈ U and z ∈ Z_{0}.

Using this notation, we now get as an immediate consequence of Theorem 2.8 the following list of necessary and sufficient conditions for an 𝔽_{p}G-representation A to give rise to a simple fusion system in the way described by the theorem.

Corollary 2.10. Fix an odd prime p, a finite group G, and a finite dimensional, faithful 𝔽_{p}G-module A. Fix U ∈ Syl_{p}(G), and assume that |U| = p and U ⋬ G. Set Z = C_{A}(U), Z_{0} = Z ∩ [U, A], and m = dim(A/Z) + 1, and assume that m ≥ 3. Then there is a simple fusion system F over a finite p-group S which contains A as its unique abelian subgroup of index p, where G = Aut_{F}(A), if and only if the following conditions hold:

(a) |Z_{0}| = p; (b) there are no non-trivial G-invariant subgroups of Z, aside (possibly) from Z_{0};

(c) [G, A] = A; and

(d) one of the following holds: either

(d.1) μ_{A}(G^{∨}) = Δ, G = O^{p′}(G)G^{∨}, m ≡ 0, −1 (mod p − 1), and dim(A) ≤ p − 1; or

(d.1′) μ_{A}(G^{∨}) = Δ, G = O^{p′}(G)G^{∨}, m = p − 1 or p − 2 or m = p = 3, and dim(A) ≤

p − 1; or

(d.2) μ_{A}(G^{∨}) ≥ Δ_{−1} and G = O^{p′}(G)·μ_{A}^{−1}(Δ_{−1}); or

(d.3) μ_{A}(G^{∨}) ≥ Δ_{0}, G = O^{p′}(G)·μ_{A}^{−1}(Δ_{0}), and Z_{0} is not G-invariant.

We will see in Section 3 (Propositions 3.5 and 3.7) that m = min(rk(A), p) in the situation of Corollary 2.10.

We finish the section with some examples which show that proper strongly closed subgroups can be found in simple fusion systems of the type constructed in Theorem 2.8.

Example 2.11. Fix a prime p ≥ 5. Set Γ = S_{p} × 𝔽_{p}^{×}, and let M ≅ 𝔽_{p}^{p} be the 𝔽_{p}Γmodule where S_{p} acts by permuting the coordinates, and where a ∈ 𝔽_{p}^{×} acts via a·Id_{M}. Fix U ∈ Syl_{p}(Γ).

(a) Set A = M and S = A ⋊ U. Let μ_{A}: N_{Γ}(U) ⟶ Δ be as in Notation 2.9. Set G = O^{p}(Γ)μ^{−1}(Δ_{−1}). By Theorem 2.8 case (d.iii), there is a simple fusion system F over S such that Aut_{F}(A) = Aut_{G}(A), and such that E_{F} = {A} ∪ H_{0} in the notation of 2.4. Furthermore, the subgroup A_{0}U, where A_{0} = [U, A], is strongly closed in F.

(b) Set A = M/C_{M}(O^{p′}(Γ)) and S = A ⋊ U. Thus |A| = p^{p−1} and |S| = p^{p}. Let μ_{A}: N_{Γ}(U) ⟶ Δ be as in Notation 2.9. Set G = O^{p′}(Γ)μ^{−1}(Δ_{0}). By Theorem 2.8 case (d.iv), there is a simple fusion system F over S such that Aut_{F}(A) = Aut_{G}(A), and such that E_{F} = {A} ∪ B_{0} in the notation of 2.4. Furthermore, the subgroup A_{0}U, where A_{0} = [U, A], is strongly closed in F.

(c) Let A, S, μ_{A}, and G be as in (b). Fix I ⊆ {0, 1, …, p − 1} with |I| ≥ 2. By Theorem 2.8 case (d.iv), there is a simple fusion system F over S such that Aut_{F}(A) = Aut_{G}(A) and E_{F} = {A} ∪ ⋃_{i∈I} B_{i}, and no proper non-trivial subgroup of S is strongly closed in F.

In all of these cases, G has index 2 in Γ, and F is exotic by Table 2.2 and the theorem.

3. Representation-theoretic preliminaries

From now on, we restrict attention to the case where the abelian group A in Section 2 has exponent p. In other words, we are looking at 𝔽_{p}G-modules, for certain finite groups G, for which the conditions in Corollary 2.10 are satisfied. We begin with some representation theory that we will need, in particular the representation theory of groups with a Sylow p-subgroup of order p, which is very well understood. Throughout this section, p is an odd prime.

Definition 3.1. Let 𝒢_{p} be the class of finite groups whose Sylow p-subgroups are not normal and have order p. Let 𝒢_{p}^{∧} be the class of all G ∈ 𝒢_{p} such that |Aut_{G}(U)| = p − 1 for U ∈ Syl_{p}(G).

The following notation will be used throughout this section, and in much of the rest of the paper.

Notation 3.2. When p is odd and G ∈ 𝒢_{p}, we

• fix an element x of order p in G and set U = ⟨x⟩ ∈ Syl_{p}(G); and

• set N = N_{G}(U), C = C_{G}(U), and C′ = C′_{G}(U) = O_{p′}(C_{G}(U)).

For background to the following discussion about vertices, sources, and Green correspondents of 𝔽_{p}G-modules, we refer to [Be, Chapter 3], and especially to Sections 3.10–3.12.

Let V be an indecomposable 𝔽_{p}G-module. A vertex for V is a minimal subgroup P ≤ G such that V is relatively P -projective; i.e., such that each surjection W → V which splits 𝔽_{p}P -linearly is also 𝔽_{p}G-linearly split. This is always a p-subgroup of G, and is uniquely determined up to conjugacy. If G ∈ 𝒢_{p}, then since U ∈ Syl_{p}(G) has order p, either V is 𝔽_{p}G-projective and has trivial vertex, or V is non-projective and U is a vertex of V. Note that V is projective if and only if V |_{U} is projective, equivalently, if U (or x) acts on V with Jordan blocks all of size p.

In general, if P is a vertex of V then the restriction V |_{N_{G}(P)} of V to N_{G}(P) is the direct sum of an indecomposable 𝔽_{p}N_{G}(P)-module W with vertex P, the Green correspondent of V, and other indecomposable modules with vertices that are contained in intersections P^{g} ∩ N_{G}(P) for g ∉ N_{G}(P), so in particular not equal to P and of order at most |P |. Thus when U ∈ Syl_{p}(G) has order p and V is not projective, V |_{N} ≅ W ⊕ X (recall N = N_{G}(U)), where W (the Green correspondent) is indecomposable and non-projective, and where X is projective.

The restriction of any non-projective, indecomposable 𝔽_{p}G-module V to U is a sum of free modules and of copies of a fixed 𝔽_{p}U-module T, called the source of V. (In general, U is a sum of conjugates of the source, but for cyclic groups conjugate modules are isomorphic.) If the source of V is the trivial module (i.e., V |_{U} is a sum of free modules and trivial modules) then V is said to be trivial source. The source of an indecomposable 𝔽_{p}G-module and the source of its Green correspondent are the same.

As C is normal in N, the simple 𝔽_{p}N-modules restrict to C as a sum of simple modules. However, as C = U × C′, we see that the simple 𝔽_{p}C-modules are just the simple 𝔽_{p}C′- modules, which are irreducible ordinary characters as C′ is a p′-group. In particular, all simple 𝔽_{p}N-modules are trivial source, and indeed these are the only trivial-source 𝔽_{p}Nmodules. Hence if V is a simple, trivial-source 𝔽_{p}G-module then its Green correspondent is also simple.

The following definition will be useful in our discussion.

Definition 3.3. When G ∈ 𝒢_{p}, an 𝔽_{p}G-module M is minimally active if the action of x on M has at most one non-trivial Jordan block (i.e., at most one Jordan block with non-trivial action).

Lemma 3.4. Fix a field k of characteristic p, assume that G ∈ 𝒢_{p}, and let M be a minimally active kG-module upon which U acts non-trivially.

(a) If M = M_{1} ⊕M_{2}, where the M_{i} are kG-submodules, then O^{p′}(G) acts trivially on exactly one of the M_{i}.

(b) M is indecomposable if and only if C_{M}(O^{p′}(G)) ≤ [O^{p′}(G), M].

(c) If M is indecomposable, then M is absolutely indecomposable. If M is simple then M is absolutely simple.

(d) We can decompose M|_{N} = M_{0} ⊕ M_{1}, where U acts trivially on M_{0} and M_{1}|_{U} is indecomposable.

Proof. (a) Since M is minimally active, a generator x ∈ U acts with at most one nontrivial Jordan block. So U must act trivially on at least one of the M_{i}, and hence O^{p′}(G) acts trivially on it.

(b) Set H = O^{p′}(G) for short. If M is decomposable, then C_{M}(H) ≰ [H, M] by (a). Assume, conversely, that C_{M}(H) ≰ [H, M]. Since k[G/H] is semisimple, there is a kG-

submodule V_{0} ≤ C_{M}(H) such that V_{0} ≠ 0 and C_{M}(H) = V_{0} ⊕ (C_{M}(H) ∩ [H, M]). For the same reason, there is V_{1} ≤ M such that V_{1} ≥ [H, M] and M/[H, M] = (V_{0}+[H, M])/[H, M]⊕ (V_{1}/[H, M]). Thus M = V_{0} ⊕ V_{1}, V_{1} ≠ 0 since [H, M] ≠ 0 (since U acts non-trivially), and thus M is decomposable.

(c) Set M = k ⊗_{k} M for short. If M is indecomposable, then C_{M}(O^{p′}(G)) ≤ [O^{p′}(G), M] by (b), so C_{M}(O^{p′}(G)) ≤ [O^{p′}(G), M], and M is indecomposable by (b) again.

If M is simple, then End_{kG}(M) ≅ 𝔽_{p^{k}} for some k ≥ 1. Then k | dim(C_{M}(U)∩[U, M]) = 1, so k = 1, and M is absolutely simple.

(d) We apply (a) and (b) with N and U in the role of G and O^{p′}(G). If C_{M}(U) ≤ [U, M], then M|_{U} is indecomposable, and we take (M_{0}, M_{1}) = (0, M). Otherwise, M|_{N} is decomposable by (b), and U acts trivially on all but one of its indecomposable summands by (a). □

Minimally active modules are what is needed in the situation of Theorem 2.8 and Corollary 2.10. This is made more precise in the next proposition.

Proposition 3.5. If A is an 𝔽_{p}G-module that satisfies the hypotheses of Corollary 2.10, then G ∈ 𝒢_{p}^{∧}, and A is minimally active and indecomposable.

Proof. By Corollary 2.10(a), dim(C_{A}(U) ∩ [U, A]) = 1, so the action of x has only one non-trivial Jordan block. Thus A is minimally active. Also, U ∈ Syl_{p}(G) has order p and |Aut_{G}(U)| = p − 1 by the conditions in Corollary 2.10(d), so G ∈ 𝒢_{p}^{∧}.

If A = A_{1} ⊕ A_{2}, where the A_{i} are 𝔽_{p}G-submodules, then by Lemma 3.4, at least one of its direct factors lies in Z = C_{A}(U) and intersects trivially with [U, A]. In the terminology of Corollary 2.10, this is a non-trivial G-invariant subgroup of Z which is not equal to Z_{0}, contradicting point (b) in the corollary. Thus A is indecomposable. □

We note the following easy lemma.

Lemma 3.6. The property of being minimally active is preserved under taking submodules, quotients, dual, tensoring by a 1-dimensional module, and restricting to subgroups in 𝒢_{p}.

We will show that of the almost simple groups which lie in 𝒢_{p}^{∧}, very few possess minimally active modules. The previous lemma shows that if M is a minimally active 𝔽_{p}G-module and H ≤ G, then M|_{H} is also minimally active. This means that we can use induction and embed, say, SL_{n−1}(q) into SL_{n}(q). Here the automizers of cyclic subgroups are the same, and so if SL_{n−1}(q) is not in 𝒢_{p}^{∧} then neither is SL_{n}(q).

The next result describes some of the basic properties of minimally active modules.

Proposition 3.7. Let G ∈ 𝒢_{p}, and assume Notation 3.2. Then the following hold for each indecomposable, minimally active 𝔽_{p}G-module M on which U acts faithfully.

(a) Either

• dim(M) < p and M|_{U} is indecomposable, or • dim(M) = p, M|_{U} is indecomposable and free, and M is projective, or

• dim(M) > p, M is a trivial-source module, and M|_{U} is the sum of one copy of 𝔽_{p}U and a module with fixed action.

(b) M is a trivial-source module if and only if dim(M) ≥ p.

(c) If dim(M) ≥ p + 2 then M is simple and absolutely simple.

(d) If M is simple and has trivial source, then M is the reduction modulo p of a ℤ_{p}-lattice in a simple ℚ_{p}G-module. Moreover, χ_{M} extends to an absolutely irreducible ordinary character.

(e) If dim(M) > p, then the Green correspondent V of M is simple.

Proof. (a) By Lemma 3.4(d), we can write M|_{N} = V_{0} ⊕ V_{1}, where V_{1}|_{U} is a non-trivial Jordan block and V_{0} has fixed action of U. (Recall that N = N_{G}(U).) If V_{0} = 0, then dim(M) ≤ p, as p is the largest size of a Jordan block, and M|_{U} ≅ V_{1}|_{U} is indecomposable. If in addition, dim(V_{1}) = p, then V_{1} ≅ M|_{U} is projective, and hence M is projective since U ∈ Syl_{p}(G) (cf. [Be, Corollary 3.6.9]).

If V_{0} ≠ 0, then M is not projective, so M|_{N} is the sum of a projective module and the Green correspondent of M. Thus V_{1} is projective, hence of rank p, and so dim(M) > p. Also, M is a trivial source module since U acts trivially on V_{0}.

(b) If dim(M) > p, then we are done by (a). If dim(M) = p, then M is projective, and hence has trivial vertex and trivial source. If dim(M) < p, then M|_{U} is indecomposable, hence is the source of M, and has non-trivial action since M|_{U} is assumed to be faithful.

(c) Assume that dim(M) ≥ p + 2. By Lemma 3.4(d), M|_{N} = V_{0} ⊕ V_{1}, where V_{1} is 𝔽_{p}N-projective (and V_{1}|_{U} ≅ 𝔽_{p}U), and V_{0} is the Green correspondent to M and is an indecomposable (hence simple) 𝔽_{p}[N/U]-module. Note that by the Frattini argument, G = O^{p′}(G)N.

Assume first that there is a non-trivial submodule 0 ≠ M_{0} < M on which U acts trivially. Then O^{p′}(G) acts trivially on M_{0}, and M_{0} ≤ C_{M}(U) = V_{0} ⊕ C_{V_{1}}(U). Since V_{0} is 𝔽_{p}Nirreducible, dim(V_{0}) = dim(M) − p ≥ 2, and dim(C_{V_{1}}(U)) = 1, either M_{0} ≥ V_{0} or M_{0} = C_{V_{1}}(U). If M_{0} ≥ V_{0}, then V_{0} is an 𝔽_{p}G-submodule of M (recall G = O^{p′}(G)N), hence a direct summand of M since M/V_{0} is 𝔽_{p}G-projective, which contradicts the indecomposability of M.

Thus M_{0} = C_{V_{1}}(U). As M/M_{0} does not satisfy any of the conditions in (a), (since U acts faithfully) it must be decomposable. By the Krull–Schmidt theorem, each proper direct sum decomposition of (M/M_{0})|_{N} has a summand isomorphic to V_{0}, so M/M_{0} contains a direct summand whose restriction to N is isomorphic to V_{0}, and which (by an argument similar to that in the last paragraph) must be equal to the image of V_{0} in M/M_{0}. Hence V_{0} ⊕ C_{V_{1}}(U) = C_{M}(U) is an 𝔽_{p}G-submodule, and we just showed that this is impossible.

Now assume that 1 ≠ M_{0} < M is an arbitrary non-trivial proper submodule. We just showed that U acts non-trivially on M_{0}, and by a similar argument applied to the dual M^{∗}, U also acts non-trivially on M/M_{0}. If either of M_{0} or M/M_{0} is decomposable, then it has a direct factor on which U acts trivially (Lemma 3.4), which contradicts the fact that M has no submodules or quotients on which U acts trivially. So each of M_{0} and M/M_{0} has one of the forms listed in (a). Since M is minimally active, M|_{U} and (M/M_{0})|_{U} must both be indecomposable, so dim(C_{M}(U)) ≤ dim(C_{M_{0}}(U)) + dim(C_{M/M_{0}}(U)) = 2. But we already saw that dim(C_{M}(U)) = dim(V_{0}) + 1 = dim(M) − p + 1 ≥ 3, so this is impossible. Absolute simplicity now comes from Lemma 3.4(c).

(d) That M is the mod p reduction of a ℤ_{p}-lattice M̂ is a general property of all trivialsource modules (see [Be, Corollary 3.11.4(i)]). If ℚ_{p}M̂ is not simple, then it contains a non-trivial proper submodule 0 ≠ W < ℚ_{p}M̂, and the mod p reduction of W ∩ M̂ is a proper 𝔽_{p}G-submodule of M, contradicting the assumption that M is simple. Since this also holds for 𝔽_{p^{n}} for n ≥ 1, we get absolute irreducibility since M is absolutely simple by Lemma 3.4(c).

By Lemma 3.4(c), 𝔽_{p} ⊗_{𝔽_{p}} M is a simple 𝔽_{p}G-module. Hence by a similar argument, K ⊗_{ℚ_{p}} (ℚ_{p}M̂) is a simple KG-module for each finite extension K ⊃ ℚ_{p} by roots of unity. So ℚ_{p} ⊗_{ℚ_{p}} (ℚ_{p}M̂) is simple, and the character of ℚ_{p}M̂ is irreducible when regarded as a complex character of G.

(e) By (a), M has trivial source, so its Green correspondent is an indecomposable 𝔽_{p}[N/U]- module, hence irreducible since N/U has order prime to p. □

The next lemma will be useful when showing that certain extensions of minimally active modules are again minimally active.

Lemma 3.8. Fix G ∈ 𝒢_{p}. Assume that V is an indecomposable 𝔽_{p}G-module of dimension at most p + 1, and that V has a nonzero submodule or quotient module which is minimally active. Then V is also minimally active.

Proof. Assume V is not minimally active. Then V |_{N} is the Green correspondent of V, and is thus indecomposable. By [Al, p. 42] and since U is normal and cyclic in N, V |_{N} is uniserial in the sense of [Al, p. 26]. (Alperin always assumes we are working over an algebraically closed field, but this proof does not use that.) In particular, the socle C_{V} (U) and the top V/[U, V] are both irreducible N/U-modules, and they have rank at least 2 since V is not minimally active. But this is impossible: if W < V is a submodule, then C_{W} (U) < C_{V} (U) has rank 1 if W is minimally active, while the image of W in V/[U, V] has corank 1 if V/W is minimally active. □

As a consequence of Proposition 3.7, if M is a simple, minimally active module, then either dim(M) ≤ p, or M has as Green correspondent the reduction modulo p of an irreducible ordinary character of G, whose minimal degrees are known in the case where G is quasisimple. The next result will help us to classify such modules.

Proposition 3.9. The following hold for each faithful, indecomposable, minimally active 𝔽_{p}G-module M.

(a) Suppose that dim(M) > p, and set a = dim(M) − p. Then • a divides |N/U|;

• if N/U is abelian, then a = 1;

• if C is abelian, then a divides |N/C|; and

• if C > U, then a ≤ |C/U| − 1.

(b) If O^{p′}(G) = ⟨x, y⟩ for some x, y ∈ G, where |x| = |y| = p or |x| = 2 and |y| = p, then for each central extension G̃ of G of degree prime to p, the dimension of each minimally active faithful indecomposable 𝔽_{p}G̃-module is at most 2p − 2.

Proof. (a) If dim(M) = p + 1, then all four statements hold. So we may assume that dim(M) > p+1. In particular, by Proposition 3.7(c), M is absolutely simple, and the Green correspondent W of M is an absolutely simple module for N, and for the p′-group N/U since U acts trivially on W. Thus χ_{W} is an ordinary irreducible character for N/U. Also, dim(W) = dim(M) − p = a.

In particular, dim(W) divides |N/U| (see, e.g., [Is, Theorem 3.11]), and dim(W) = 1 if N/U is abelian. If C is abelian, then we apply a theorem of Ito (see [Is, Theorem 6.15]) to get that dim(W) divides |N/C|. This proves the first three statements.

Set W = 𝔽_{p} ⊗_{𝔽_{p}} W, where 𝔽_{p} ⊇ 𝔽_{p} is the algebraic closure. By Clifford theory (see [Fe, Theorem III.2.12]), W |_{C} ≅ e·[⨁_{i=1}^{k} W_{i}], where e ≥ 1, and where W_{1}, …, W_{k} are pairwise distinct irreducible 𝔽_{p}[C/U]-modules which form one orbit under the N/C-action on the set of all irreducible representations. Also, e = 1 since N/C is cyclic (see [Fe, Theorem III.2.14]), so W |_{C} is a sum of distinct irreducible representations. Since C/U ≠ 1, N/C cannot act transitively on the set Irr(C/U), and hence dim_{𝔽_{p}}(W) < dim_{𝔽_{p}}(𝔽_{p}[C/U]) = |C/U|.

(b) By Lemma 3.4(b), M is 𝔽_{p}O^{p′}(G)-indecomposable if it is 𝔽_{p}G-indecomposable. So we can assume G = O^{p′}(G).

If G = ⟨x, y⟩ where x and y have order p, then since a minimally active module M has at most one non-trivial Jordan block, C_{M}(x) and C_{M}(y) have codimension at most p − 1. This means that C_{M}(x) ∩ C_{M}(y) = C_{M}(O^{p′}(G)) has codimension at most 2p − 2. Since M is simple, dim(M) ≤ 2p − 2, as needed.

If G = ⟨t, y⟩ where |t| = 2 and |y| = p, then ⟨^{t}y, y⟩ is normal of index at most 2 in G, hence equal to G since G = O^{p′}(G). If G̃ is a central extension of G of degree prime to p, where G = ⟨x, y⟩ and |x| = |y| = p, then x and y lift to x̃, ỹ ∈ G̃ of order p, and ⟨x̃, ỹ⟩ ≥ O^{p′}(G̃). So in both cases, we are back in the first situation. □

Proposition 3.9(b) is useful because both sporadic and alternating groups are known to be generated by two elements of order p whenever the Sylow p-subgroup is cyclic (see [Cr2] for sporadic groups), and this can be checked for any individual groups we might encounter. In general, it appears that with few exceptions this is always true for finite simple groups, and such a statement is currently under investigation by the first author.

As a generalization of part of Proposition 3.9, we get the following condition, which is useful for bounding the size of minimally active modules for a group in terms of minimally active modules for a subgroup.

Proposition 3.10. Let H be a subgroup of G such that O^{p′}(H) = H. Suppose that O^{p′}(G) ≤ ⟨H_{1}, …, H_{n}⟩ for some set H_{1}, …, H_{n} of n conjugates of H. Let s be the maximal dimension of an indecomposable, faithful, minimally active 𝔽_{p}H-module. If M is an indecomposable, faithful, minimally active 𝔽_{p}G-module such that C_{M}(O^{p′}(G)) = 0, then dim(M) ≤ ns. In particular, if C_{H}(U) is abelian, then dim(M) < 2np.

Proof. The proof is similar to that of Proposition 3.9(b). If M is an indecomposable, faithful, minimally active 𝔽_{p}G-module, then by Lemma 3.4(a), for each 1 ≤ i ≤ n, the restriction of M to H_{i} must have a summand N_{i} of codimension at most s on which H_{i} acts trivially. The intersection of the N_{i} has codimension at most ns, and is contained in the proper submodule C_{M}(O^{p′}(G)). Since C_{M}(O^{p′}(G)) = 0 we have dim(M) ≤ ns, as claimed. □

Having found minimally active, simple modules, we would like to know whether there are minimally active, indecomposable modules built from them. In almost every case we will see that if V is minimally active then dim(V) > (p + 1)/2, so that Proposition 3.7(c) eliminates any extensions between non-trivial modules. However, this leaves open the possibility that V is minimally active for G and H^{1}(G, V) ≠ 0, so that V has a minimally active extension with the trivial module; for example, the permutation module for the symmetric group S_{p} is minimally active and has a trivial submodule and a trivial quotient. Of course, by Proposition 3.7(c) again, dim(V) must be at most p for this to work. The next lemma deals with this case, when V is self dual.

Lemma 3.11. Let G be a group in 𝒢_{p}^{∧}, and let V be a self-dual, simple, minimally active module with dim(V) ≤ p. If H^{1}(G, V) ≠ 0 then dim(V) = p − 2.

Proof. Set m = dim(V). If m = p, then V is projective and H^{1}(G, V) = 0, so m < p. Since 0 ≠ H^{1}(G; V) ≅ Ext_{𝔽_{p}G}^{1}(𝔽_{p}, V), there is an indecomposable extension M of 𝔽_{p} by V (with V as submodule). Since restriction sends H^{1}(G; V) injectively into H^{1}(U; V), M|_{U} is indecomposable and consists of one Jordan block. Set M_{0} = [U, M] = V, and M_{i} = [U, M_{i−1}] for each i ≥ 1; then dim(M_{i}) = m − i for each 0 ≤ i ≤ m.

Fix g ∈ N such that ^{g}u = u^{r} where r generates (ℤ/p)^{×}. By Lemma 1.11(b), and since g acts trivially on M/M_{0} = M/V, g acts on each M_{i−1}/M_{i} by multiplication by r^{i}. In particular, it acts on M_{0}/M_{1} by multiplication by r and on M_{m−1} by multiplication by r^{m}, and since V is self-dual, r^{m} ≡ r^{−1} (mod p). Thus m+1 ≡ 0 (mod p−1), and since 0 < m < p, we have m = p − 2. □

In the next lemma, we give some tools for handling some of the other properties listed in Corollary 2.10, especially those involving the homomorphism μ_{V} : G^{∨} ⟶ Δ of Notation 2.9. For any finite abelian group M, let Aut_{sc}(M) be the group of scalar automorphisms: those of the form (x ↦ kx) for k prime to |M|.

Lemma 3.12. Fix G ∈ 𝒢_{p}, and let V be a faithful, minimally active 𝔽_{p}G-module. Let U ∈ Syl_{p}(G), G^{∨} ≤ N_{G}(U), and μ_{V} : G^{∨} ⟶ Δ be as in Notation 2.9.

(a) The homomorphism μ_{V} sends G^{∨}/U injectively into Δ.

(b) If G ∈ 𝒢_{p}^{∧}, dim(V) ≤ p, and G ≥ Aut_{sc}(V), then G^{∨} = N_{G}(U) and μ_{V} (G^{∨}) = Δ.

(c) Assume that dim(V) ≥ p, and set Z = C_{V} (U). Then for each g ∈ N_{G}(U)∖C_{G}(U), χ_{V} (g) = χ_{Z}(g).

Proof. Let V_{0} = C_{V} (U)[U, V] and [U, V] = W_{1} > W_{2} > ⋯ > W_{m} = 0 be as in Lemma 1.11.

(a) This is essentially Lemma 2.5(a), but we give another proof here. Assume that g ∈ G^{∨} has order prime to p, and μ_{V} (g) = (1, 1). By Lemma 1.11(b), g acts via the identity on V/V_{0}, and on W_{i}/W_{i+1} for each 1 ≤ i ≤ m − 1 (r = t = 1 in the notation of the lemma). By definition of G^{∨}, g acts via the identity on C_{V} (U)/W_{m−1}, and hence also acts via the identity on C_{V} (U)[U, V]/[U, V] = V_{0}/W_{1}. So by Lemma 1.8, g acts trivially on V, and hence g = 1 since G acts faithfully.

Thus Ker(μ_{V} |_{G^{∨}}) ≤ U, and the opposite inclusion is clear.

(b) Since dim(V) ≤ p, V |_{U} contains only one Jordan block. So dim(C_{V} (U)) = 1, Z = Z_{0} in the notation of Corollary 2.10, and G^{∨} = N_{G}(U).

Let r ∈ (ℤ/p)^{×} be a generator, and let ψ_{r} ∈ Aut_{sc}(V) ≤ N_{G}(U) be the automorphism (a ↦ a^{r}). Then μ_{V} (ψ_{r}) = (1, r).

By assumption, there is g ∈ N_{G}(U) such that ^{g}u = u^{r} for each u ∈ U. Then μ_{V} (g) = (r, s) for some s, and thus μ_{V} (G^{∨}) ≥ ⟨μ_{V} (g), μ_{V} (ψ_{r})⟩ = ⟨(1, r), (r, s)⟩ = Δ.

(c) Let W_{0} ≤ V and V_{0} ≤ Z be such that W_{0} is a non-trivial Jordan block for the action of U and V = W_{0} ⊕ V_{0}. Since dim(V) ≥ p, dim(W_{0}) = p by Proposition 3.7(a). Hence dim(V/Z) = dim(W_{0}/C_{W_{0}}(U)) = p − 1.

Let [U, V] = [U, W_{0}] = W_{1} > W_{2} > ⋯ > W_{p} = 0 be as defined above. By Lemma 1.11(a), |W_{i}/W_{i+1}| = p for each 1 ≤ i ≤ p − 1, and also for i = 0 since W_{0}/W_{1} ≅ W_{0}Z/W_{1}Z = V/V_{0}.

Fix g ∈ N_{G}(U). Let r, t ∈ (ℤ/p)^{×} be such that ^{g}u = u^{r} for each u ∈ U, and g acts on W_{0}/W_{1} ≅ V/V_{0} via multiplication by t. Then by Lemma 1.11(b), for each 1 ≤ i ≤ p − 1, g acts on W_{i}/W_{i+1} via multiplication by tr^{i}. The action of g on W_{0}/W_{p−1} ≅ V/Z thus has eigenvalues t, tr, tr^{2}, …, tr^{p−2}. Hence χ_{V} (g) − χ_{Z}(g) = ∑_{i=0}^{p−2} ψ(tr^{i}) for some embedding ψ: (ℤ/p)^{×} ⟶ ℂ^{×}. If g ∉ C_{G}(U), then ψ(r) ≠ 1, the sum of this geometric series is zero since ψ(r)^{p−1} = 1, and hence χ_{V} (g) = χ_{Z}(g). □

We now summarize the tools which will be used to compute μ_{V} (G^{∨}) in later sections.

Proposition 3.13. Fix G ∈ 𝒢_{p}^{∧}, and let V be a faithful, minimally active 𝔽_{p}G-module such that G ≥ Aut_{sc}(V). Let U ∈ Syl_{p}(G), G^{∨}, and μ_{V} be as in Notation 2.9.

(a) If dim(V) ≤ p, then μ_{V} (G^{∨}) = Δ and C_{G}(U) = U × Aut_{sc}(V).

(b) If dim(V) > p, then |μ_{V} (G^{∨})| = |G^{∨}/U| ≤ \frac{|N_{G}(U)/U|}{p−1}, with equality if dim(V) = p + 1.

(c) If dim(V) = p + 1, and g ∈ N_{G}(U) has order prime to p and is such that c_{g} generates Aut(U), then the following hold.

(i) If |χ_{V} (g)| = 2, then μ_{V} (G^{∨}) ≥ Δ_{0}.

(ii) If χ_{V} (g) = 0, then μ_{V} (G^{∨}) ≥ Δ_{(p−1)/2}.

(iii) If |χ_{V} (g)| = 1, then μ_{V} (G^{∨}) ≥ Δ_{ε(p−1)/3} for some ε = ±1.

Proof. (a) The first statement was shown in Lemma 3.12(b). Hence |N_{G}(U)/U| = |G^{∨}/U| = |Δ| = (p−1)^{2} by Lemma 3.12(b,a), so |C_{G}(U)| = p(p−1), and the centralizer is as described.

(b) The first equality holds since Ker(μ_{V}) = U by Lemma 3.12(a). Also, Aut_{sc}(V) ≤ C_{G}(U) and intersects trivially with G^{∨}, so G^{∨} has index at least p − 1 in N_{G}(U). If dim(V) = p + 1, then dim(Z/Z_{0}) = 1 (in the notation of 2.9), so for each g ∈ N_{G}(U), the coset gAut_{sc}(V) contains a unique element in G^{∨}. Hence G^{∨} has index exactly p − 1 in N_{G}(U) in this case.

(c) Now assume that dim(V) = p + 1, and fix g ∈ N_{G}(U) of order p − 1 such that c_{g} generates Aut(U). Let g′ ∈ gAut_{sc}(V) be the unique element in G^{∨}. Let r ∈ (ℤ/p)^{×} be such that ^{g}u = ^{g′}u = u^{r} for each u ∈ U.

Set Z = C_{V} (U) and Z_{0} = Z ∩ [U, V]. Thus dim(Z) = 2, dim(Z_{0}) = 1, and χ_{V} (g) = χ_{Z}(g) by Lemma 3.12(c). Also, for some choice of monomorphism ψ: (ℤ/p)^{×} ⟶ ℂ^{×}, χ_{Z}(g) = ψ(s) + ψ(t), where g acts on Z_{0} via multiplication by s and on Z/Z_{0} via multiplication by t (s, t ∈ (ℤ/p)^{×}). Since g′ acts on Z/Z_{0} via the identity by definition of G^{∨}, it acts on Z_{0} via multiplication by st^{−1}, and hence μ_{V} (g′) = (r, st^{−1}).

Recall that r generates (ℤ/p)^{×} by the assumption on g. If |χ_{V} (g)| = |ψ(s) + ψ(t)| = 2, then s = t, so μ_{V} (g′) = (r, 1) generates Δ_{0}. If χ_{V} (g) = 0, then s = −t, so μ_{V} (g′) = (r, −1) generates Δ_{(p−1)/2}. If |χ_{V} (g)| = 1, then ψ(st^{−1}) = ψ(s)/ψ(t) must be a primitive cube root of unity, and hence μ_{V} (g′) = (r, st^{−1}) generates Δ_{(p−1)/3} or Δ_{−(p−1)/3}. □

4. Representations which occur in simple fusion systems: a summary

In this section, we present a summary of the rest of paper, by outlining our classification of all possible pairs (G, A) satisfying parts (a) to (d) in Theorem 2.8 or Corollary 2.10. From now on, these will be regarded as 𝔽_{p}G-modules with additive structure (as opposed to the multiplicative group structure on A), and will be denoted by V to emphasize this. Throughout this section and the next, we assume the classification of finite simple groups.

Table 4.1 is an attempt at tabulating this information, but its notation requires explanation. When the image of G in PGL(V) is almost simple, the group G_{0} = F^{∗}(G) is listed in the second column, and a group G ≤ N_{GL(V)}(G_{0}) such that G/G_{0} is a p′-subgroup of maximal order in N_{GL(V)}(G_{0})/G_{0} is listed in the fourth column. In all cases, N_{GL(V)}(G_{0})/G_{0} is solvable, so the choice of G is unique up to conjugacy, and we can assume that G_{0} ≤ G ≤ G. (In almost all cases, G = N_{GL(V)}(G_{0}).)

When the image of G in PGL(V) is not almost simple, the second column is left blank, G is contained in the group G in the fourth column, and we provide more information on the possibilities for G later in this section. In all cases, the third column lists the possible dimensions of the minimally active module V that becomes the subgroup A in the saturated fusion system. The fifth and sixth columns list the images μ_{V} (G_{0}^{∨}) ≤ μ_{V} (G^{∨}) ≤ Δ, and the final column gives the information as to whether this representation leads to a realizable fusion system (R) and/or an exotic fusion system (E) with superscripts indicating the number of such systems. In some cases we are necessarily vague when considering whole collections of possible groups G.

Theorem 4.1. Assume that G ∈ 𝒢_{p}^{∧}, and let V be a minimally active, faithful, indecomposable 𝔽_{p}G-module which satisfies the hypotheses of Corollary 2.10. Then either

(a) the image of G in PGL(V) is not almost simple, and G ≤ G with the given action on V for one of the pairs (G, V) listed in Table 4.1 with no entry G_{0}; or

(b) the image of G in PGL(V) is almost simple, and G_{0} ≤ G ≤ G for one of the triples (G_{0}, G, V) listed in Table 4.1.

When G_{0} ≅ SL_{2}(p) or A_{p} for p ≥ 5, more precise descriptions of the modules are given in Propositions 4.2 and 4.3, respectively.

Proof. Let V be a minimally active, faithful, indecomposable 𝔽_{p}G-module with dim(V) ≥ 3. When the image of G in PGL(V) is not almost simple, the possibilities for G and V are listed in cases (b)–(e) of Proposition 5.4. When the image of G in PGL(V) is almost simple, and G_{0} = F^{∗}(G), the possible pairs (G_{0}, V) are determined in Sections 6–11: SL_{2}(p) is handled in Proposition 6.1 (the only groups of Lie type in defining characteristic p which appear), sporadic groups in Proposition 7.1, alternating groups in Proposition 8.1, linear, unitary and symplectic groups in Propositions 10.1, 10.2 and 10.3, and orthogonal groups in Proposition 10.4. Finally, the exceptional groups are dealt with in Proposition 11.1. Together, these show that in all cases, (G, V) appears as an entry in one of the two Tables 4.1 or 4.2, as described in (a) or (b).

If V is a simple 𝔽_{p}G_{0}-module, then it is absolutely simple by Lemma 3.4(c), and hence C_{GL(V)}(G_{0}) = Aut_{sc}(V) (the group of scalar automorphisms). So G = O^{p}(N_{GL(V)}(G_{0})) is an extension of G_{0}Aut_{sc}(V) by outer automorphisms of G_{0} and hence can be determined from the tables of modular characters. When V is indecomposable but not simple, G has a similar form but is not in general the full normalizer of G_{0} in GL(V) (see, e.g., the second-to-last sentence in Proposition 6.1.

In both tables, μ_{V} (G^{∨}) = Δ by Proposition 3.13(a) whenever dim(V) ≤ p. When G_{0} ≅ A_{n} (n ≥ p+2) or G ≤ C_{p−1}≀S_{n} (n > p), μ_{V} (G^{∨}) is easily calculated using the definition, and when G_{0} ≅ SL_{2}(p) and dim(V) = p + 1, it is calculated in Section 6. When G ≅ (C_{3} × 2_{+}^{1+6}).S_{8}, μ_{V} (G^{∨}) is determined in Lemma 5.5. In all other cases where dim(V) = p+1, μ_{V} (G^{∨}) can be determined using points (b) and (c) in Proposition 3.13. This leaves only the representation

p  G_{0}  dim(V)  G  μ_{V} (G^{∨}) μ_{V} (G_{0}^{∨}) E, R
SL_{2}(p) or PSL_{2}(p)

3 ≤ n ≤ p ^{(4.2)} socle of dim. i

GL_{2}(p) or  Δ  {(u^{2}, u^{i−1})} ER
p  (p ≥ 5)  2/(p−1) ^{(4.2)} PGL_{2}(p) × C_{p−1} Δ_{−1}  \frac{1}{2}Δ_{−1}  E
p  A_{p} (p ≥ 5)  [1]/p−2/[1] ^{(4.3)} S_{p} × (p−1)  Δ \frac{1}{2}Δ_{0} or \frac{1}{2}Δ_{−1} ER
p  A_{p+1} (p ≥ 5)  p  S_{p+1} × (p−1)  Δ  \frac{1}{2}Δ_{0}  ER
p A_{n} (p+2 ≤ n ≤ 2p−1)  n − 1  S_{n} × (p−1)  Δ_{0}  \frac{1}{2}Δ_{0}  R
p  —  n ^{(4.4(b))}  C_{p−1} ≀ S_{n} (n ≥ p) Δ  —  ER
3  —  2/2 ^{(4.4(c))}  GL_{2}(3)  Δ_{1}  —  E
5  2·A_{6}  4  4 ∘ 2·S_{6}  Δ  Δ_{1/2}  E
5  —  4 ^{(4.4(d))}  (C_{4} ∘ 2^{1+4}).S_{6}  Δ  —  ER
5 PSp_{4}(3) = W (E_{6})′  6  W (E_{6}) × 4  Δ_{0}.2  \frac{1}{2}Δ_{0}  R
5 Sp_{6}(2) = W (E_{7})′  7  G_{0} × 4  Δ_{0}  \frac{1}{2}Δ_{0}  R
7  2·A_{7}  4  2·S_{7} × 3  Δ  Δ_{3/2}  E
7  6·PSL_{3}(4)  6  G_{0}.2_{1}  Δ  𝔽_{p}^{×2} × 𝔽_{p}^{×} E
7  6_{1}·PSU_{4}(3)  6  G_{0}.2_{2}  Δ  𝔽_{p}^{×2} × 𝔽_{p}^{×} E
7  PSU_{3}(3)  6  G_{0}.2 × 6  Δ  \frac{1}{2}Δ_{1}  E
7  PSU_{3}(3)  7  G_{0}.2 × 6  Δ  \frac{1}{2}Δ_{0}  E
7  SL_{2}(8)  7  G_{0}:3 × 6  Δ  \frac{1}{3}Δ_{1}  E
7 Sp_{6}(2) = W (E_{7})′  7  G_{0} × 6  Δ  Δ_{3}  ER
7 2·Ω_{8}^{+}(2) = W (E_{8})′  8  W (E_{8}) × 3  Δ_{0}.2  Δ_{3}  R
11  J_{1}  7  G_{0} × 10  Δ  Δ_{3}  E
11  PSU_{5}(2)  10  G_{0}.2 × 10  Δ  \frac{1}{2}Δ_{2}  E
11  2·M_{12}  10^{[2]}  G_{0}.2 × 5  Δ  Δ_{1/2} , Δ_{7/2} E
11  2·M_{22}  10^{[2]}  G_{0}.2 × 5  Δ  Δ_{1/2} , Δ_{7/2} E
13  PSU_{3}(4)  12  G_{0}.4 × 12  Δ  \frac{1}{3}Δ_{1}  E

In the third column, a superscript in brackets shows the number of distinct representations of G_{0} of the given dimension (not counting them as distinct if they differ by an automorphism of G_{0}). A superscript ^{(4.x)} means that these representations (and/or the groups) are described more precisely in Proposition 4.x. When k, ℓ are relatively prime, we set Δ_{k/ℓ} = {(u^{ℓ}, u^{k}) | u ∈ 𝔽_{p}^{×}} (always cyclic of order p − 1).

Table 4.1. Groups in 𝒢_{p}^{∧} with minimally active modules of dimension at least 3 which appear in reduced fusion systems

where p = 5, G_{0} ≅ Sp_{6}(2), and dim(V) = 7: |μ_{V} (G^{∨})| = p − 1 by Proposition 3.13(b), and the structure of fusion in E_{7}(q) (when v_{5}(q − 1) = 1) together with Table 2.1 show that it must be Δ_{0}.

The determination of μ_{V} (G_{0}^{∨}) is slightly more delicate than that of μ_{V} (G^{∨}). But in almost all cases, this can either be done either directly using the definitions, or with the help of character tables and Green correspondence, or by examining V |_{H} for some H < G_{0} isomorphic to SL_{2}(p) or PSL_{2}(p).

Finally, with the help of Table 2.1, we determine which of these representations appear in simple fusion systems (i.e., satisfy the conditions in Corollary 2.10 for some choice of G ≤ G containing G_{0}), and in those cases we determine μ_{V} (G^{∨}) and give dim(V). Among those fusion systems, Table 2.2 tells us which are realizable. □

p  G_{0}  dim(V)  G  μ_{V} (G^{∨})
p (P)SL_{2}(p) type j/i ^{(4.2)}_{(i+j = p+1, j ≠ 2)} \begin{matrix}GL_{2}(p) or \\ PGL_{2}(p) × C_{p−1} \end{matrix} Δ_{i−1}
7  —  8 ^{(4.4(e))}  (C_{3} × 2_{+}^{1+6}).S_{8}  Δ_{3}
7 SL_{2}(8)  8  G_{0}:3 × 6  Δ_{3}
7 2·Sp_{6}(2)  8  G_{0} × 3  Δ_{3}
7  2·A_{7}  4/4  2·S_{7} × 3  Δ_{3}
7  2·A_{8}  8  2·S_{8} × 3  Δ_{3}
7  2·A_{9}  8  G_{0} × 3  Δ_{3}
11 2·M_{12}  12  G_{0}.2 × 5  Δ_{5}
13 ^{2}B_{2}(8)  14^{[2]}  G_{0}:3 × 12  Δ_{±4}
13 G_{2}(3)  14  G_{0}:2 × 12  Δ_{6}
17 Sp_{4}(4)  18  G_{0}.4 × 16  Δ_{8}

Table 4.2. Groups in 𝒢_{p}^{∧} with minimally active modules of dimension at least 3 which do not satisfy the hypotheses of Corollary 2.10

The first few rows of Tables 4.1 and 4.2 require the most explanation. The next proposition concerns the first row, and is basically a restatement of Proposition 6.1.

Proposition 4.2. Let G_{0}, V, and G be as in Theorem 4.1, and assume that G_{0} = SL_{2}(p). The indecomposable minimally active 𝔽_{p}G_{0}-modules with faithful action of G_{0} or of G_{0}/Z(G_{0}) are described as follows (where unique always means up to isomorphism).

(a) A unique simple 𝔽_{p}G_{0}-module V_{i} of dimension i for each 2 ≤ i ≤ p. Each simple 𝔽_{p}G_{0}- module is isomorphic to V_{i} for some 1 ≤ i ≤ p, where V_{1} = 𝔽_{p} is the trivial module, V_{2} is the natural module for G_{0} = SL_{2}(p), and V_{i} = Sym^{i−1}(V_{2}) for 3 ≤ i ≤ p (the (i−1)-st symmetric power).

(b) A unique (p − 1)-dimensional indecomposable module V_{j,i} of type V_{j}/V_{i} for each 1 ≤ i, j ≤ p − 2 such that i + j = p − 1.

(c) A unique p-dimensional indecomposable module V_{1,p−2,1} of type V_{1}/V_{p−2}/V_{1}.

(d) A unique (p + 1)-dimensional indecomposable module V_{j,i} of type V_{j}/V_{i} for each 2 ≤ i, j ≤ p − 1 such that i + j = p + 1.

If the simple composition factors of V are even dimensional, then G_{0} = SL_{2}(p) acts faithfully on V, and G ≅ GL_{2}(p). If the simple composition factors of V are odd dimensional, then the action of G_{0} factors through PSL_{2}(p), and G ≅ PGL_{2}(p) × C_{p−1}. Also, G = N_{Aut(V)}(G_{0}) except when V ≅ V_{i,i} for i = (p ± 1)/2 or V ≅ V_{1,p−2,1}, in which cases G has index p in the normalizer.

We now consider the case where G_{0} is A_{p}.

Proposition 4.3. Let G_{0}, V, and G be as in Theorem 4.1, and assume that G_{0} = A_{p}. Then there is a unique faithful, simple, minimally active 𝔽_{p}G_{0}-module W, of dimension p − 2. There are unique non-simple indecomposable minimally active modules of each of the types 𝔽_{p}/W, W/𝔽_{p}, and 𝔽_{p}/W/𝔽_{p}, where the last is the permutation module. Also, G ≅ S_{p} × C_{p−1} in all cases, and is equal to N_{Aut(V)}(G_{0}) except when V is the permutation module, in which case G has index p in the normalizer. All of these representations give rise to simple fusion systems via Theorem 2.8 (for some choice of G), but only W itself gives rise to simple, realizable fusion systems.

Proof. See Proposition 8.1 for a determination of the minimally active modules. □

It remains to describe the minimally active, indecomposable 𝔽_{p}G-modules, when G ∈ 𝒢_{p}^{∧} and the image of G in the projective group is not almost simple.

Proposition 4.4. Assume that G ∈ 𝒢_{p}^{∧}. Let V be a minimally active, faithful, indecomposable 𝔽_{p}G-module, and set n = dim(V). Then one of the following holds.

(a) The image of G in PGL(V) is almost simple, and p divides the order of its socle.

(b) G ≤ C_{p−1} ≀ S_{n} (n ≥ p) acts as a group of monomial matrices on V ≅ (𝔽_{p})^{n}. More precisely, if we set K = O_{p′}(G), then

K = {(a_{1}, …, a_{n}) ∈ (C_{p−1})^{n} | a_{1}^{t} = a_{2}^{t} = ⋯ = a_{n}^{t}, (a_{1} ⋯ a_{n}, a_{1}^{t}) ∈ R}

for some 1 ≠ t|(p − 1) and some R ≤ C_{p−1} × C_{p−1}; and one of the following holds:

• n = p and G/K ≅ C_{p} ⋊ C_{p−1};

• n = p + 1 and G/K ≅ PGL_{2}(p);

• p ≤ n ≤ 2p − 1 and G/K ≅ S_{n};

• p + 2 ≤ n ≤ 2p − 1 and G/K ≅ A_{n}.

Also, V |_{K} splits as a direct sum of pairwise non-isomorphic 1-dimensional 𝔽_{p}K-modules which are permuted 2-transitively by G/K.

(c) p = 3, G ≅ 2_{−}^{1+2}.S_{3} ≅ Q_{8}.S_{3} ≅ GL_{2}(3), and either n = 2 and V is simple, or n = 4 and V is non-simple of type 2/2.

(d) p = 5, n = 4, O_{5′}(G) ≅ C_{4} ∘ 2^{1+4} or 2_{−}^{1+4}, and G/O_{5′}(G) ≅ S_{6} ≅ Sp_{4}(2), S_{5} ≅ SO_{4}^{−}(2), or C_{5} ⋊ C_{4}. (Note that C_{4} ∘ 2_{+}^{1+4} ≅ C_{4} ∘ 2_{−}^{1+4}. If O_{5′}(G) ≅ 2_{−}^{1+4} then G/O_{5′}(G) is not S_{6}.)

(e) p = 7, n = 8, O_{7′}(G) ≅ C_{3} × 2_{+}^{1+6} or 2_{+}^{1+6}, and G/O_{7′}(G) ≅ S_{8} ≅ SO_{6}^{+}(2), S_{7}, PGL_{2}(7), or C_{7} ⋊ C_{6}.

Proof. This will be proved in Section 5, as Propositions 5.3 and 5.4. □

In order to describe more explicitly how to get from Table 4.1 to actual fusion systems, we list a few cases in more detail in Table 4.3. The first four columns in the table correspond to information given in Table 4.1, while the last four columns consist of separate rows for the different cases (corresponding to cases (i)–(iv) in Table 2.1). In the last column, E (or E^{+}) means that there is one (or more than one) exotic fusion system of this type; otherwise, a group is given which realizes it.

For example, when p ≥ 5, G_{0} ≅ PSL_{2}(p), and V is the simple (p−2)-dimensional 𝔽_{p}G_{0}- module, we have m = dim(V) ≡ −1 (mod p − 1) and μ_{V} (G^{∨}) = Δ. Hence there are three families of simple fusion systems which arise in this way, corresponding to the three cases (ii), (iii), (iv) in Table 2.1. The fusion systems of types (ii) and (iv) are unique by Theorem 2.8, while in case (iii), E_{F} ∖ {A} can be any union of H_{i}’s. In all cases, Aut_{F}(A) = G ≥ G_{0} is determined (as a subgroup of G) by the third column in Table 2.1. For P ∈ E_{F} ∩ (H ∪ B), Aut_{F}(P) is determined by Lemma 2.6. By Table 2.2, all of these fusion systems are exotic if p ≥ 7. If p = 5, then the fusion systems of type (iii) are exotic, that of type (iv) is realized by Sp_{4}(5), and that of type (ii) is realized by Co_{1}.

In contrast, when p = 7, G_{0} ≅ 2·Ω_{8}^{+}(2) ≅ W (E_{8})′, and V is the simple 8-dimensional 𝔽_{p}G_{0}- representation, then m = min(dim(V), p) = 7, μ_{V} (G^{∨}) = Δ_{0}.2 = Δ_{0}Δ_{3}, and μ_{V} (G_{0}^{∨}) = Δ_{3}. By Table 2.1, and since m ≢ 0, −1 (mod p), any simple fusion system which realizes (G, V) must be in case (iii) or (iv), and it cannot be in case (iii) since μ_{V} (G^{∨}) ≱ Δ_{−1}. So there is exactly one simple fusion system F of this type, of type (iv) and hence with E_{F} = {A} ∪ B_{0}. Also, Aut_{F}(A) = G must contain G_{0} with index 2 by the condition in the third column of Table 2.1, so G ≅ W (E_{8}). By Table 2.2, F is realized by E_{8}(q) for any q such that v_{7}(q − 1) = 1.

5. Reduction to almost simple groups

In this section, we analyse the possibilities for (G, V) as in Theorem 4.1 when the image of G in PGL(V) is not almost simple, by using Aschbacher’s classification of the maximal subgroups of GL_{n}(p). The almost simple cases will be handled in the later sections.

Before proving a general result, we look at representations of G ≤ C_{p−1} ≀ S_{n} on 𝔽_{p}^{n}, acting via monomial matrices. Two lemmas are first needed.

Lemma 5.1. Assume that p is an odd prime and H is a finite group with a Sylow p-subgroup U of order p such that |N_{H}(U)/C_{H}(U)| = p − 1. Assume also that H acts faithfully and transitively on a set Ω in such a way that each x ∈ H of order p acts via a p-cycle. Then H acts primitively and 2-transitively on Ω, and one of the following holds:

(a) |Ω| = p and H ≅ C_{p} ⋊ C_{p−1};

(b) |Ω| = p + 1 and H ≅ PGL_{2}(p);

(c) p ≤ |Ω| ≤ 2p − 1 and H = S_{Ω};

(d) p + 2 ≤ |Ω| ≤ 2p − 1 and H = A_{Ω}; or

(e) p = 7, |Ω| = 9, and H ≅ ΣL_{2}(8).

Proof. Fix U ∈ Syl_{p}(H) and 1 ≠ x ∈ U. We first show that H is primitive on Ω. If Σ is a block of a system of imprimitivity for the action of H on Ω (thus |Σ| > 1), then x must stabilize Σ, as otherwise x must move at least p|Σ| > p points, contradicting our assumption. Choose Σ so that x acts non-trivially on Σ and trivially on Ω ∖ Σ. Since H acts transitively on any system of imprimitivity, there exists h ∈ H such that x^{h} acts non-trivially on a different block Σ^{h} but trivially on Σ, ⟨x, x^{h}⟩ is a subgroup of order p^{2} in H, contradicting the assumption that U ∈ Syl_{p}(G). Thus H acts primitively on Ω.

We now appeal to the classification of primitive permutation groups containing a p-cycle, as listed, for example, in [Zi]. By that theorem, one of the following holds, where n = |Ω|.

32 DAVID A. CRAVEN, BOB OLIVER, AND JASON SEMERARO

p  G_{0}  dim(V) μ_{V} (G_{0}^{∨})  G  μ_{V} (G^{∨})  E_{0}  Group/E
4 ≤ n ≤ p−3  \begin{matrix}GL_{2}(p)/Z \\ |Z| = (p−1, n−1) \end{matrix} Δ_{0}Δ_{\frac{n−1}{2}}  B_{0}  E
p  SL_{2}(p)  n even  Δ_{\frac{n−1}{2}}  \begin{matrix}\frac{1}{ε}GL_{2}(p) \\ ε = (p−1, n+1) \end{matrix} Δ_{−1}Δ_{\frac{n−1}{2}} H_{0}  E
G_{0}.2 × (p−1)  Δ  B_{0} ∪ H_{∗} \begin{matrix}Co_{1} (p=5) \\ E (p≥7) \end{matrix}
p  PSL_{2}(p)  p − 2  \frac{1}{2}Δ_{−1}  PGL_{2}(p)  Δ_{−1}  ⋃ H_{i}  E
(p ≥ 5)  G_{0}.2 × \frac{p−1}{2} Δ_{0}.(\frac{p−1}{2}) B_{0}  \begin{matrix}Sp_{4}(5) (p=5) \\ E (p≥7) \end{matrix}
p \begin{matrix}A_{n} \\ (p+2 ≤ n ≤ 2p−1) \end{matrix} n − 1  \frac{1}{2}Δ_{0}  S_{n}  Δ_{0}  B_{0}  PSL_{n}(q)
2·S_{7}  Δ_{0}Δ_{3}  B_{0}  E
7  2·A_{7}  4  Δ_{3/2}  2·S_{7} × 3  Δ  H_{0}  E
G_{0}.2_{1}  Δ  B_{0} ∪ H_{∗}  E
7  6·PSL_{3}(4)  6  \frac{1}{2}Δ  G_{0}.2_{1}  Δ  ⋃ H_{i}  E
G_{0}.2_{1}  Δ  B_{0}  E
G_{0} × 2  Δ_{0}Δ_{3}  B_{0}  E_{7}(q)
7 Sp_{6}(2) = W (E_{7})′  7  Δ_{3}  G_{0} × 3  Δ_{−1}Δ_{3}  H_{0}  E
7 2·Ω_{8}^{+}(2) ≅ W (E_{8})′  8  Δ_{3}  G.2 ≅ W (E_{8})  Δ_{0}.2  B_{0}  E_{8}(q)
10  G_{0}.2 × 5  Δ  H_{0} ∪ B_{∗}  E
11  2·M_{12}  (V |_{SL_{2}(11)}  Δ_{1/2}  G_{0}.2 × 5  Δ  H_{0}  E
type 8/2)  G_{0}.2 × 5  Δ  ⋃ B_{i}  E
G_{0}.4 × 12  Δ  H_{0} ∪ B_{∗}  E
13  PSU_{3}(4)  12  \frac{1}{3}Δ_{1}  G_{0}.4 × 3  Δ_{−1}Δ_{4}  H_{0}  E
G_{0}.4 × 3  Δ_{0}Δ_{4}  ⋃ B_{i}  E

In all cases, q is a prime power such that v_{p}(q − 1) = 1, and Δ_{k/ℓ} is as in Table 4.1.

Table 4.3. Some examples

(a) n = p, and H is a subgroup of AGL_{1}(p) = C_{p} ⋊ C_{p−1};

(b) n = p, p = \frac{q^{d}−1}{q−1} for some prime power q and d ≥ 2, and PSL_{d}(q) ≤ H ≤ PΓ L_{d}(q);

(c) n = p + 1, p = 2^{d} − 1 is a Mersenne prime, and AGL_{1}(2^{d}) ≤ H ≤ AGL_{d}(2);

(d) n = p + 1, H ≅ PSL_{2}(p) or H ≅ PGL_{2}(p);

(e) n = p = 11, and H = PSL_{2}(11) or H = M_{11}, or n = p = 23 and H = M_{23};

(f) n = p + 1 = 12, and H = M_{11} or H = M_{12}, or n = p + 1 = 24 and H = M_{24};

(g) n = p + 2, p = 2^{d} − 1 a Mersenne prime, and H ≅ PSL_{2}(2^{d}) or PΣL_{2}(2^{d}); or

(h) H is A_{n} or S_{n}, and p ≤ n ≤ 2p − 1.

Thus the structure of H is very tightly controlled. Since |Aut_{H}(U)| = p − 1, we can be even more restrictive: (a) can occur only with H = C_{p} ⋊ C_{p−1}, and (d) can occur only with H ≅ PGL_{2}(p). In case (b), if q = r^{k} where r is prime, then p − 1 = |Aut_{H}(U)| ≤ kd, so (q^{d}−1)/(q−1) ≤ kd+1, which gives (p, q, d) = (3, 2, 2) or (5, 4, 2), and H ≅ S_{3} or S_{5}. In case (c), where |Aut_{H}(U)| ≤ d, we get 2^{d} − 1 = p = d + 1 and hence d = 2, p = 3, and H ≅ S_{4}. For (e) and (f), |Aut_{H}(U)| = (p − 1)/2, so these do not occur. For (g), |Aut_{H}(U)| = 2 or 2d, so this occurs only when (p, d) = (3, 2) or (7, 3), and H ≅ PSL_{2}(4) ≅ A_{5}, PΣL_{2}(4) ≅ S_{5}, or PΣL_{2}(8). Of course (h) can occur, but not when H = A_{p} or H = A_{p+1}. By inspection, all of these actions are 2-transitive. □

Lemma 5.2. Fix a prime p, and a subgroup H ≤ S_{p} such that H ∈ 𝒢_{p}^{∧}. Fix m > 1 prime to p, and regard (ℤ/m)^{p} as a ℤ/m[H]-module via the inclusion H ≤ S_{p}. Let M ≤ (ℤ/m)^{p} be a submodule, and assume that U acts non-trivially on M/qM for each prime q | m. Then M ≥ I, where

I = {(x_{1}, …, x_{p}) | x_{i} ∈ ℤ/m, x_{1} + ⋯ + x_{p} = 0} .

Proof. Fix U ∈ Syl_{p}(H). Since (m, p) = 1, M|_{U} = C_{M}(U)⊕(M ∩I), so U acts non-trivially on (M ∩ I)/q(M ∩ I) for each prime q | m. So upon replacing M by M ∩ I, we can assume that M ≤ I.

Assume first that m is a prime; thus ℤ/m ≅ 𝔽_{m}. As an 𝔽_{m}U-module, (𝔽_{m})^{p} factors as a product of irreducible modules of which one is 1-dimensional with trivial action, and the others are permuted transitively by the action of N_{H}(U)/U (since |Aut_{H}(U)| = p − 1). Hence each 𝔽_{m}H-submodule of 𝔽_{m}^{p} either has trivial action of U, or contains all of the factors which have non-trivial action and thus contains their sum I. Since the action of U on M is non-trivial by assumption, we have M ≥ I.

Now assume that m = q^{a} where q is prime and a > 1. By assumption, M/qM has nontrivial action of U, and we just showed that this implies that I ≤ M +(qℤ/mℤ)^{p}. Hence there is x = (x_{1}, …, x_{p}) ∈ M such that x_{1} ≡ 1 (mod q), x_{2} ≡ −1 (mod q), and x_{i} ≡ 0 (mod q) for each i ≥ 2. Then q^{a−1}x and its U-translates generate q^{a−1}I, so q^{a−1}I ≤ M. Since q^{a−2}x ≡ (q^{a−2}, −q^{a−2}, 0, …, 0) (mod q^{a−1}I) (recall that M ≤ I), we have (q^{a−2}, −q^{a−2}, 0, …, 0) ∈ M, and hence q^{a−2}I ≤ M. Upon continuing in this way, we get I ≤ M.

If m is not a prime power, the result now follows upon splitting M and I as products of their Sylow subgroups. □

We are now ready to describe the groups and modules which appear in case (b) of Proposition 4.4.

Proposition 5.3. Set G = C_{p−1} ≀ S_{n} (n ≥ p), and let V ≅ 𝔽_{p}^{n} be the natural 𝔽_{p}G-module where G acts via monomial matrices. Let G ≤ G be such that G ∈ 𝒢_{p}^{∧} and V |_{G} is a simple, minimally active module. Assume also that the image of G in PGL(V) is not almost simple. Let K ⊴ G be the subgroup of G acting via diagonal matrices (K ≅ C_{p−1}^{n}), and set K = G ∩ K. Then for some 1 < t | (p − 1) and some R ≤ C_{p−1} × C_{p−1},

K = {(a_{1}, …, a_{n}) ∈ (C_{p−1})^{n} | a_{1}^{t} = a_{2}^{t} = ⋯ = a_{n}^{t}, (a_{1} ⋯ a_{n}, a_{1}^{t}) ∈ R};

and one of the following holds:

(a) n = p and G/K ≅ C_{p} ⋊ C_{p−1};

(b) n = p + 1 and G/K ≅ PGL_{2}(p);

(c) p ≤ n ≤ 2p − 1 and G/K ≅ S_{n}; (d) p + 2 ≤ n ≤ 2p − 1 and G/K ≅ A_{n}.

(e) p = 7, n = 9, and G/K ≅ ΣL_{2}(8).

Proof. Set H = G/K for short, with its natural action on Ω = {1, 2, …, n}. If H is intransitive on Ω, then Ω = Ω_{1} ∪ Ω_{2} with Ω_{i} being H-invariant. Then V = 𝔽_{p}Ω_{1} ⊕ 𝔽_{p}Ω_{2}, a sum of 𝔽_{p}G-submodules, which is impossible since V is assumed to be indecomposable. Thus the action of H on Ω is transitive.

Fix U ∈ Syl_{p}(G). By assumption, |U| = p, U ⋬ G, and |Aut_{G}(U)| = p − 1. Each 1 ≠ x ∈ U acts as a single p-cycle on Ω, since x has a single non-trivial Jordan block on M. So by Lemma 5.1, H (as a subgroup of S_{n}) is one of the groups listed in (a)–(d). In particular, H acts 2-transitively on Ω.

Set B = {a_{1}a_{2}^{−1} | (a_{1}, …, a_{n}) ∈ K} ≤ C_{p−1}. Since H acts 2-transitively on Ω, any other pair of distinct elements of Ω defines the same subgroup. Set

K_{B} = {(a_{1}, …, a_{n}) ∈ (C_{p−1})^{n} | a_{1} ≡ ⋯ ≡ a_{n} (mod B)} .

Thus K ≤ K_{B}. Define

Φ: K_{B} ⟶ C_{p−1} × (C_{p−1}/B) by setting Φ(a_{1}, …, a_{n}) = (a_{1} ⋯ a_{n}, a_{1}B) .

If B = 1, then each element of K acts via multiplication by scalars, so either U ⊴ G or the image of G in PGL(V) is almost simple. Since we assume that neither of these holds, B ≠ 1.

Fix a generator u ∈ U. Without loss of generality, we can assume that elements in Ω are arranged so that u = (1 2 … p). Let H_{0} be the image of N_{H}(U) in S_{p} via restriction of its action. Choose a = (a_{1}, …, a_{n}) such that a_{1}a_{2}^{−1} generates B, and set b = a^{−1}u(a). Thus b = (b_{1}, …, b_{p}, 1, …, 1), where b_{i} ∈ B for 1 ≤ i ≤ p, b_{1}b_{2} … b_{p} = 1, and b_{2} generates B. Since |B| is prime to p, the sequence b_{1}, …, b_{p} is not constant modulo q for any q | |B|. We regard b as an element of B^{p} ∩ K, which is a subgroup invariant under the action of H_{0}. By Lemma 5.2, K contains the group of all (b_{1}, …, b_{p}, 1, …, 1) such that b_{i} ∈ B and b_{1} ⋯ b_{p} = 1. Since the action of H on Ω is 2-transitive, it now follows that

K ≥ {(a_{1}, …, a_{n}) ∈ B^{n} | a_{1} ⋯ a_{n} = 1} = Ker(Φ) .

Set R = Im(Φ). Thus K = {a ∈ K_{B} | Φ(a) ∈ R}, and this translates to the description of K given above (where t = |B|). □

The possibilities in Table 4.1 where there is no G_{0}, corresponding to point (a) in Theorem 4.1, arise in an analysis of subgroups of GL_{n}(p) which lie in 𝒢_{p}^{∧} and whose action on the associated module is minimally active. Using Aschbacher’s classification of the maximal subgroups of GL_{n}(p), we restrict the options for G, leaving either almost simple groups or a few other possibilities. The next proposition performs that reduction.

Proposition 5.4. Assume that G ∈ 𝒢_{p}^{∧}, let V be a minimally active, faithful, indecomposable 𝔽_{p}G-module, and set n = dim(V). Then one of the following holds.

(a) The image of G in PGL(V) is almost simple, and p | |F^{∗}(G)|.

(b) G ≤ C_{p−1} ≀ S_{n} (n ≥ p) and acts as a group of monomial matrices.

(c) p = 3, G ≅ 2_{−}^{1+2}.S_{3} ≅ Q_{8}.S_{3} ≅ GL_{2}(3), and either n = 2 and V is simple, or n = 4 and V is non-simple of type 2/2.

(d) p = 5, n = 4, O_{5′}(G) ≅ C_{4} ∘ 2^{1+4} or 2_{−}^{1+4}, and G/O_{5′}(G) ≅ S_{6} ≅ Sp_{4}(2), S_{5} ≅ SO_{4}^{−}(2), or C_{5} ⋊ C_{4}. (Note that C_{4} ∘ 2_{+}^{1+4} ≅ C_{4} ∘ 2_{−}^{1+4}. If O_{5′}(G) ≅ 2_{−}^{1+4} then G/O_{5′}(G) is not S_{6}.)

(e) p = 7, n = 8, O_{7′}(G) ≅ C_{3} × 2_{+}^{1+6} or 2_{+}^{1+6}, and G/O_{7′}(G) ≅ S_{8} ≅ SO_{6}^{+}(2), S_{7}, PGL_{2}(7), or C_{7} ⋊ C_{6}.

Proof. Fix U ∈ Syl_{p}(G). By assumption, U ⋬ G, |U| = p, and |Aut_{G}(U)| = p − 1.

Case 1: Assume first that the image of G in PGL(V) is almost simple, and set G_{0} = F^{∗}(G) and Γ = G_{0}/Z(G_{0}). Thus G_{0} is quasisimple, Γ is simple, and we must show that p | |Γ|.

Assume otherwise; then p | |Out(G_{0})|, and hence p | |Out(Γ)|. Since p is an odd prime (and clearly Γ ≅ A̸_{6}), this is impossible when Γ is an alternating or sporadic group [GL, § I.5]. Hence Γ is of Lie type. From the tables in [GL, § I.7] it follows that p ∤ |Outdiag(Γ)|, and hence that Out(Γ) possesses a normal p-complement. But then Aut_{G}(U) = 1, which contradicts our assumptions on G.

Case 2: Assume that V is a simple 𝔽_{p}G-module, and that the image of G in PGL(V) is not almost simple. By the main theorem in [A1], G is contained in one of an explicit list of geometrically defined subgroups of GL_{n}(p), which fall into eight classes C_{i} for 1 ≤ i ≤ 8. Of these, C_{8} consists of the subgroups Sp_{n}(p) (if n is even) and GO_{n}(p) (for all choices of quadratic form). If G ≤ GL(𝔽_{p}^{n}, 𝔮) for some symplectic or quadratic form 𝔮, then G is contained in a subgroup in one of the classes C_{i} for this classical group.

Assume that G ≤ G ∈ C_{i} for i ≤ 7. Since V is simple, i ≠ 1. Since p is a prime, C_{5} = ∅. If i = 3, then G ≤ GL_{m}(p^{k}) where mk = n and k > 1, which is impossible since each Jordan block over 𝔽_{p^{k}} splits as a sum of k Jordan blocks over 𝔽_{p}. If i = 4 or 7, then G is contained in a tensor product or wreath tensor product of representations, which again implies that no element of order p acts with exactly one non-trivial Jordan block.

Now assume that i = 2, so that G ≤ G ≅ GL_{m}(p) ≀ S_{k} for some m, k such that mk = n and k > 1. If m = 1, then we are in the situation of (b). So assume that m ≥ 2, set H = G ∩ (GL_{m}(p))^{k}, and let H_{i} ≤ GL_{m}(p) be the image of H under projection to the i-th factor (1 ≤ i ≤ k). The H_{i} are all isomorphic, since otherwise G/H would not permute them transitively, contradicting the assumption that V is simple. If p||H_{i}| for all i, then since p^{2} ∤ |G|, the elements of order p in G act with m ≥ 2 non-trivial Jordan blocks, a contradiction. Thus there is an element of order p which non-trivially permutes some set of the H_{i}, this acts on V with k ≥ 2 non-trivial Jordan blocks, which again is a contradiction.

We are left with the case where G ≤ G ∈ C_{6}. Then for some prime r|(p − 1) and some k ≥ 1, n = r^{k} and G = N_{GL_{n}(p)}(K) where K ≅ C_{p−1} ∘ r_{±}^{1+2k} acts as a group of monomial matrices whose non-zero entries are r-th roots of unity. Also, G/K ≅ C_{Out(K)}(C_{p−1}) is isomorphic to Sp_{2k}(r), or (if r = 2 and 4 ∤ (p − 1)) to GO_{2k}^{±}(r). In particular, p | |Sp_{2k}(r)|.

Set k = 𝔽_{p}, or k = 𝔽_{p^{2}} if r = 2. The action of the subgroup (k^{×} ∘r^{1+2k}).(Sp_{2k−2}(r)×Sp_{2}(r)) on k⊗_{𝔽_{p}} V factors as a tensor product V ⊗_{k}W, where (k^{×} ∘r^{1+2(k−1)}).Sp_{2k−2}(r) acts on V with dim_{k}(V) = r^{k−1}, and (k^{×} ∘ r^{1+2}).Sp_{2}(r) acts on W with dim_{k}(W) = r. So if p||Sp_{2k−2}(r)|, then the Jordan blocks for elements of order p occur in multiples of r, which is impossible since there is a unique non-trivial block. We conclude that p|(r^{2k} − 1), so p|(r^{k} ± 1), and dim(V) = r^{k} ≡ ±1 (mod p).

For each non-central element x ∈ K, C_{Sp_{2k}(r)}(x) has index r^{2k} − 1 in the symplectic group and hence has order prime to p. Thus C_{K}(U) = Z(K) = Aut_{sc}(V), and N_{KU}(U) = UAut_{sc}(V) ≅ C_{p(p−1)}. Also, V is irreducible as an 𝔽_{p}K-module and hence as an 𝔽_{p}[KU]- module, and the KU-Green correspondent of V is indecomposable as an 𝔽_{p}[UAut_{sc}(V)]- module and hence of dimension at most p. Since dim(V) ≡ ±1 (mod p), the Green correspondent either has dimension 1, in which case dim(V) = p + 1 since V is minimally active, or it has dimension p − 1, in which case dim(V) = p − 1.

Thus p = dim(V) ± 1 = r^{k} ± 1, which is possible only if r = 2 and p is a Fermat or Mersenne prime. The action of U ≅ C_{p} on the symplectic space K/Aut_{sc}(V) ≅ (𝔽_{2})^{2k} has at most 2k eigenvalues (in the algebraic closure 𝔽_{2}), and since |Aut_{G}(U)| = p − 1, they must include all (p − 1)-th roots of unity other than 1. Thus p − 1 ≤ 2k, so (p, k) = (3, 1), (5, 2), or (7, 3), which correspond to cases (c)–(e) listed above. Note that 7 ∤ |O_{6}^{−}(2)|, so this case cannot occur.

Set G_{0} = G ∩ K, and regard K/Z(K) ≅ 𝔽_{2}^{2k} as an 𝔽_{2}[G/G_{0}]-module. Since G contains U ≅ C_{p} and |Aut_{G}(U)| = p − 1 (and since 2k = p − 1 in each case), K/Z(K) is a simple module. Hence either G_{0} ≤ Z(K), or G_{0}Z(K) = G. By Lemma 5.1 (and since |Aut_{G}(U)| = p−1), we have G/G_{0} ≅ S_{3} in case (c); G/G_{0} ≅ S_{6}, S_{5}, or C_{5} ⋊C_{4} in case (d); or G/G_{0} ≅ S_{8}, S_{7}, PGL_{2}(7), or C_{7} ⋊ C_{6} in case (e). So G_{0} ≰ Z(K), since otherwise, either the image of G in PGL(V) would be almost simple or U would be normal in G. We are left with the possibilities listed in the proposition.

Case 3: Now assume that V is not simple. Let 0 = V_{0} < V_{1} < ⋯ < V_{k} = V be 𝔽_{p}Gsubmodules such that W_{i} = V_{i}/V_{i−1} is simple for each 1 ≤ i ≤ k. Set H_{i} = C_{G}(W_{i}) for each i. Thus G/H_{i} acts faithfully on W_{i}. Since O_{p}(G) = 1 by assumption, G acts faithfully on W_{1} ⊕ ⋯ ⊕ W_{k} (cf. [Go, Theorem 5.3.2]), so ⋂_{i=1}^{k} H_{i} = 1.

If H ⊴ G and p | |H|, then since p^{2} ∤ |G|, H ≥ O^{p′}(G). Hence there is some 1 ≤ ℓ ≤ k such that p ∤ |H_{ℓ}|. Then V = C_{V} (H_{ℓ}) ⊕ [H_{ℓ}, V] as 𝔽_{p}G-modules, dim(C_{V} (H_{ℓ})) ≥ dim(W_{ℓ}) > 0, and V is indecomposable, so [H_{ℓ}, V] = 0 and hence H_{ℓ} = 1. Thus G acts faithfully on W_{ℓ}, and (G, W_{ℓ}) is one of the pairs listed in cases (a)–(e).

Set K = O_{p′}(G). Let Irr_{V} (K) be the set of irreducible 𝔽_{p}K-characters which appear as summands of V |_{K}, and similarly for Irr_{W_{i}}(K). For each χ ∈ Irr_{V} (K), let V_{χ} ≤ V |_{K} be the submodule generated by all irreducible submodules with character χ. Thus V |_{K} is the direct sum (as 𝔽_{p}K-modules) of the V_{χ} for χ ∈ Irr_{V} (K). Since V is 𝔽_{p}G-indecomposable, the action of G on Irr_{V} (K) induced by conjugation must be transitive. In particular, since the subsets Irr_{W_{i}}(K) are non-empty and G-invariant, we have Irr_{W_{i}}(K) = Irr_{V} (K) for each 1 ≤ i ≤ k.

If the image of G in PGL(W_{ℓ}) is almost simple, then K is cyclic of order dividing p − 1, and Irr_{W_{ℓ}}(K) = Irr_{V} (K) contains just one character. Hence K acts on V via multiplication by scalars, G/K is almost simple, and (G, V) is as in case (a).

Now assume that (G, W_{ℓ}) is in one of the cases (b)–(e). Let G be the maximal group listed in that case (thus G ≤ G), and set K = O_{p′}(G). In case (b), G/K acts 2-transitively on the set Irr_{W_{ℓ}}(K) by Proposition 5.3, so after restriction to K, its elements are all distinct. (By the description of K in Proposition 5.3, they cannot be pairwise isomorphic.) Hence dim(V) ≥ 2 dim(W_{ℓ}) ≥ 2p, which contradicts Proposition 3.7(c) (dim(V) ≤ p + 1).

In each of cases (c)–(e), W_{ℓ}|_{K} is 𝔽_{p}K-simple, so Irr_{V} (K) contains only its character. Hence dim(V) = m dim(W_{ℓ}) = m(p − 1) for some m ≥ 2. Since dim(V) ≤ p + 1 by Proposition 3.7(c), this is possible only when p = 3 and m = 2, as described in case (c). □

We finish the section with a computation which was used in the proof of Theorem 4.1 to show that the representations in case (e) of Proposition 5.4 cannot be used to construct simple fusion systems.

Lemma 5.5. Set G_{0} = 2_{+}^{1+6}, let V be the unique faithful, irreducible 𝔽_{7}[G_{0}]-module, and set G = N_{GL(V)}(G_{0}). Fix U ∈ Syl_{7}(G). Then G/G_{0} ≅ C_{3}×SO_{6}^{+}(2) ≅ C_{3}×S_{8}, |N_{G}(U)/U| = 6^{2}, and μ_{V} (G^{∨}) = Δ_{3} (see Notation 2.9).

Proof. Since V is the unique faithful, irreducible 𝔽_{7}[G_{0}]-module, Out_{G}(G_{0}) = Out(G_{0}) ≅ SO_{6}^{+}(2) ≅ S_{8}, where G_{0}/Z(G_{0}) is the natural module for S_{8}. Since C_{GL(V)}(G_{0}) = Aut_{sc}(V) ≅ C_{6}, this proves that G/G_{0} ≅ C_{3} × S_{8}. Set Z = Z(G_{0}), Ẑ = Aut_{sc}(V) ≅ C_{3} × Z, and Ĝ_{0} = G_{0}Ẑ ≅ C_{3} × G_{0}.

Now, Out_{G}(G_{0}) is contained in the symplectic group Sp_{6}(2), which acts transitively on the 63 involutions in Ĝ_{0}/Ẑ ≅ G_{0}/Z. Since 7^{2} ∤ |Sp_{6}(2)|, this means that C_{G}(x) has order prime to 7 for each x ∈ Ĝ_{0}∖Ẑ, and hence that C_{G}(U) = U × Ẑ and |N_{G}(U)/U| = 6^{2}. So by Proposition 3.13(b), |G^{∨}/U| = |μ_{V} (G^{∨})| = 6, and hence μ_{V} (G^{∨}) = Δ_{m} for some m.

Fix g ∈ G^{∨} such that c_{g} has order 6 in Aut_{G}(U). Then g^{6} = 1 since |G^{∨}/U| = 6.

Identify G_{0} = H ∘ H ∘ H (central product) and V = W ⊗ W ⊗ W, where H ≅ D_{8}, W is a faithful, irreducible 𝔽_{7}H-module (dim(W) = 2), and where G_{0} acts on V as the tensor power of the H-action on W. Choose h_{0} ∈ N_{Aut(W)}(H)∖H of order 2 (since GL_{2}(7) contains an extension of 𝔽_{49}^{×} by a field automorphism, N_{Aut(W)}(H) contains a dihedral subgroup of order 16). Define h ∈ G = N_{GL(V)}(G_{0}) by setting h(w_{1}⊗w_{2}⊗w_{3}) = h_{0}(w_{2})⊗h_{0}(w_{3})⊗h_{0}(w_{1}). Since the action of h on G_{0}/Z ≅ C_{2}^{6} permutes a basis transitively, the image of h in G/Ĝ_{0} ≅ S_{8} is a 6-cycle, thus conjugate to the image of g. Also, |h| = 6, and χ_{V} (h) = 0 since V has a basis permuted by h with cycles of length 6 and 2.

Since C_{G_{0}/Z}(h^{3}) = [h^{3}, G_{0}/Z] (both of rank 3), Ĝ_{0}/Ẑ ≅ G_{0}/Z ≅ C_{2}^{6} is projective and hence cohomologically trivial as an 𝔽_{2}[⟨h⟩]-module. Hence all subgroups of Ĝ_{0}⟨h⟩/Ẑ complementary to Ĝ_{0}/Ẑ are conjugate to each other. We already saw that g is G-conjugate to some element of order 6 in the coset hĜ_{0}, and hence g is conjugate to hz for some z ∈ Ẑ = Aut_{sc}(V). Thus for some 6th root of unity ζ, χ_{V} (g) = χ_{V} (hz) = ζ·χ_{V} (h) = 0. Proposition 3.13(c.ii) now implies that μ_{V} (G^{∨}) = Δ_{3}. □

6. PSL_{2}(p)

Recall Definition 0.1: for a finite simple group L, we say that a finite group G is of type L if Z(G) is cyclic and F^{∗}(G)/Z(G) ≅ L. This concept provides a convenient way to organize the search for all pairs (G, V), where G ∈ 𝒢_{p}^{∧}, V is minimally active, and the image of G in PGL(V) is almost simple. So in the remaining sections, we go systematically through the list of non-abelian simple groups L, and list for each prime p the groups G ∈ 𝒢_{p}^{∧} of type L and their indecomposable, minimally active modules.

In this section, we handle the case where L is a simple group of Lie type in defining characteristic p. If L ∈ 𝒢_{p}, then L is isomorphic to PSL_{2}(p), as all other groups of Lie type have Sylow p-subgroups of order greater than p. We are thus reduced to the cases where G_{0} is isomorphic to PSL_{2}(p) or SL_{2}(p), and consider the simple and indecomposable modules for SL_{2}(p).

There are p different simple modules V_{1}, …, V_{p} for SL_{2}(p), where dim(V_{i}) = i. (One way to construct them is to let V_{2} be the natural module, and set V_{i} = S^{i−1}(V_{2}) for 3 ≤ i ≤ p.) Hence by Proposition 3.7(c), the dimension of each indecomposable minimally active 𝔽_{p}Gmodule is at most p + 1. To determine all such modules, we describe the projective covers of the V_{i}, referring to [Al, pp. 75–79] for more detail. (Although Alperin’s descriptions are only for an algebraically closed field, since all simple modules are defined over 𝔽_{p} the projectives must be so as well.)

The module V_{p} is projective of dimension p. For 2 ≤ i ≤ p − 2, the projective cover of V_{i} is of shape

V_{i}/(V_{p−1−i} ⊕ V_{p+1−i})/V_{i},

while the projective covers of V_{1} and V_{p−1} are of shape V_{1}/V_{p−2}/V_{1} and V_{p−1}/V_{2}/V_{p−1}, respectively. Here ‘/’ delineates the radical layers, with the socle appearing on the right, so that B is the submodule and A the quotient in A/B.

This yields indecomposable modules V_{i}/V_{p−1−i} (for each 1 ≤ i ≤ p − 2) and V_{i}/V_{p+1−i} (each 2 ≤ i ≤ p − 1), of dimension p − 1 and p + 1 respectively (and they are all minimally active by Lemma 3.8). Any minimally active indecomposable modules not yet found must have dimension p or p + 1. If dim(V) = p, then V is projective by Proposition 3.7(a), so V ≅ V_{p} or V is the projective cover of the trivial module V_{1}/V_{p−2}/V_{1} as described above.

If V has dimension p + 1 and has at least three composition factors, then there are either three factors including a copy of V_{2}, or four factors including two copies of V_{1}. In the former case, either V or V^{∗} has a simple socle, and so is a quotient module of one of the projectives above, which by inspection cannot occur. If there are four composition factors, then two are trivial, so at least one of the other two must have non-trivial 1-cohomology. By Lemma 3.11, V_{p−2} is the only such module, so V would have this and three trivial modules as composition factors, which is impossible by the above discussion.

We have now shown that each minimally active indecomposable module V is of one of the following types:

• V ≅ V_{i} for i > 1;

• V ≅ V_{i}/V_{p±1−i}; or

• V ≅ V_{1}/V_{p−2}/V_{1} the projective cover of the trivial module V_{1}.

Also, the action of SL_{2}(p) on each V_{i} extends to one of GL_{2}(p) (being a symmetric power of the natural module).

We claim that the action of G_{0} = SL_{2}(p) on each of the non-simple indecomposable modules listed above also extends to an action of G = GL_{2}(p). To see this, fix i, and let U_{1}, …, U_{p−1} be the distinct simple G-modules whose restriction to G_{0} is isomorphic to V_{i}. (These are obtained by taking one such module, and tensoring it by each of the 1-dimensional G/G_{0}-modules.) By Frobenius reciprocity, the induced module V_{i}|^{G} is isomorphic to the direct sum of the U_{j}. Hence the natural projection of P (V_{i})|^{G} onto V_{i}|^{G} (where P (−) denotes projective cover) lifts to a homomorphism Φ = ⨁_{j=1}^{p−1} Φ_{j} from P (V_{i})|^{G} to ⨁_{j=1}^{p−1} P (U_{j}). Since by definition, the kernel of the natural projection P (U_{j}) ⟶ U_{j} is contained in the radical of P (U_{j}) and hence in all maximal submodules, Im(Φ_{1}) = P (U_{1}), Im(Φ_{2}|_{Ker(Φ_{1})}) = P (U_{2}), etc. Thus Φ is onto. Also, dim(P (U_{j})) ≥ dim(P (V_{i})) for each j since P (U_{j})|_{G_{0}} contains P (V_{i}) as a direct summand. So by comparing dimensions, we see that Φ is an isomorphism and P (U_{j})|_{G_{0}} ≅ P (V_{i}). Thus for each i, the action of G_{0} on P (V_{i}) extends to G, and hence the same holds for the quotient modules of these projective covers listed above.

We next determine the normalizer N_{GL(V)}(G_{0}), when V is one of the G_{0}-modules just listed. We first consider the centralizer C_{GL(V)}(G_{0}), which obviously contains the scalar matrices Aut_{sc}(V). If V is a simple 𝔽_{p}G_{0}-module, then it is absolutely simple by Lemma 3.4(c), and hence C_{GL(V)}(G_{0}) = Aut_{sc}(V) by Schur’s lemma. If V acts indecomposably with socle V_{i} and quotient V_{p±1−i} (both simple), and g ∈ C_{GL(V)}(G_{0}), then g stabilizes V_{i}, and g|_{V_{i}} = u·Id and g ≡ u′·Id (mod V_{i}) for some u, u′ ∈ 𝔽_{p}^{×}. Also, u = u′ since V is indecomposable, and g has the form g(x) = ux + ψ(x + V_{i}) for some ψ ∈ Hom_{G_{0}}(V/V_{i}, V_{i}). Thus either C_{GL(V)}(G_{0}) = Aut_{sc}(V), or V/V_{i} ≅ V_{i} (so i = (p ± 1)/2) and C_{GL(V)}(G_{0}) ≅ Aut_{sc}(V) × C_{p}. A similar argument proves that C_{GL(V)}(G_{0}) = Aut_{sc}(V) × C_{p} when V is the projective cover of the trivial module.

Thus C_{GL(V)}(G_{0})·G_{0} = G_{0} ∘ Aut_{sc}(V) or (G_{0} ∘ Aut_{sc}(V)) × C_{p} in all cases. Also, since N_{GL(V)}(G_{0})/G_{0}·C_{GL(V)}(G_{0}) is a subgroup of Out(G_{0}) ≅ C_{2}, the normalizer N_{GL(V)}(G_{0}) contains C_{GL(V)}(G_{0})·G_{0} with index at most 2. Thus G ≤ G, where G (as defined in Section 4) has index 1 or p in N_{GL(V)}(G_{0}), and in all cases has the form (G_{0} ∘ Aut_{sc}(V)).2. (As noted above, the action of SL_{2}(p) or PSL_{2}(p) on V always extends to the outer automorphism.)

By Proposition 3.13(a), it is only the (p + 1)-dimensional modules that might not produce reduced fusion systems. To know whether they do or not, we need to understand the two modules in the socle of V |_{N} (recall N = N_{G}(U)).

Assume that V is an extension of V_{i} (the submodule) by V_{j}, where i + j = p + 1. Let ζ ∈ 𝔽_{p}^{×} be a generator, and fix the following elements in GL_{2}(p):

g = (\begin{matrix}ζ & 0 \\ 0 & 1 \end{matrix}) ,  h = (\begin{matrix}1 & 0 \\ 0 & ζ \end{matrix}) ,  u = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) .

Thus U = ⟨u⟩, ^{g}u = u^{ζ}, and ^{h}u = u^{ζ^{−1}}. On the natural module V_{2}, g acts with eigenvalues {ζ, 1} and h with eigenvalues {1, ζ}, beginning with those of the socle.

Identify V_{i} = Sym^{i−1}(V_{2}) and V_{j} = Sym^{j−1}(V_{2}). Then g, h have eigenvalues on V_{i} and V_{j} as follows:

V_{i}  V_{j}
g ζ^{i−1}, ζ^{i−2}, …, ζ, 1 ζ^{j−1}, ζ^{j−2}, …, ζ, 1
h 1, ζ, ζ^{2}, …, ζ^{i−1} 1, ζ, ζ^{2}, …, ζ^{j−1}

(in each case from socle to top). In general, these actions of g and h don’t extend to an action on V (while the action of gh^{−1} ∈ SL_{2}(p) does extend). So let z = ζ·Id_{V_{j}}, let ĝ, ĥ have the actions of g, h on V_{i}, but the actions of gz^{i−1} and hz^{i−1} on V_{j}. Since (i − 1) + (j − 1) ≡ 0 (mod p − 1), we get the following eigenvalues:

V_{i}  V_{j}
ĝ ζ^{i−1}, ζ^{i−2}, …, ζ, 1 1, ζ^{−1}, ζ^{−2}, …, ζ^{−j+1}
ĥ 1, ζ, ζ^{2}, …, ζ^{i−1} ζ^{i−1}, ζ^{i}, ζ^{i+1}, …, 1

Then ĝ and ĥ have the same action on the top of V_{i} and the socle of V_{j}, so they do extend to actions on V. In particular, ĝ ∈ G^{∨} and μ_{V} (ĝ) = (ζ, ζ^{i−1}), while h^{∗} =^{def} ĥ ∘ ζ^{1−i}·Id_{V} ∈ G^{∨} and μ_{V} (h^{∗}) = (ζ^{−1}, ζ^{1−i}).

We conclude that μ_{V} (G^{∨}) = Δ_{i−1}. Thus each of the Δ_{k} except Δ_{0} can appear as μ_{V} (G^{∨}) for some choice of V. When i = p − 1 and j = 2, we get μ_{V} (G^{∨}) = Δ_{−1}.

This yields the following proposition.

Proposition 6.1. Let G ∈ 𝒢_{p}^{∧} be a group of type PSL_{2}(p), and let V be a non-trivial, minimally active module for G. Then one of the following holds:

(i) V |_{G_{0}} ≅ V_{i} is simple for i ≥ 2;

(ii) V |_{G_{0}} ≅ V_{p−1−i}/V_{i} is indecomposable of dimension p − 1;

(iii) V |_{G_{0}} is projective of dimension p, of the form V_{1}/V_{p−2}/V_{1};

(iv) V |_{G_{0}} ≅ V_{p+1−i}/V_{i} is indecomposable of dimension p + 1.

Furthermore, G is contained in (G_{0} ∘ Z(GL(V))) .2. In case (iv), μ_{V} (G) ≤ Δ_{i}, and hence μ_{V} (G) = Δ_{−1} only when i = p − 1.

7. Sporadic groups

In this short section, we determine all minimally active modules for those sporadic groups (and their extensions) which lie in 𝒢_{p}^{∧}. For the reader’s convenience, we include a table listing, for each simple sporadic group G_{0}, all primes p for which the Sylow p-subgroup has order p. Those primes for which G_{0} ∈ 𝒢_{p}^{∧} are in bold, and the other primes for which Aut(G_{0}) ∈ 𝒢_{p}^{∧} are in italics.

Group  Primes  Group Primes Group  Primes
M_{11}  5; 11  HS  7; 11  Ru  7, 13; 29
M_{12}  5; 11  McL 7 ; 11  ON  5, 11; 31 ; 19
M_{22}  5; 11 ; 7  Suz 7, 11; 13 F i_{22}  7; 11,13
M_{23}  5; 7,11,23  Co_{3} 7; 11,23 F i_{23}  7, 11, 17; 13,23
M_{24}  5, 11; 7,23  Co_{2} 7, 11; 23 F i′_{24}  11, 13, 17; 29 ; 23
J_{1}  3, 7, 11; 5,19  Co_{1} 11, 13; 23 Ly  7; 11,31,37,67
J_{2}  7  He  17  B  11, 13, 17, 19; 23,31,47
J_{3}  5,19 ; 17  HN 7, 11; 19 M 17, 19, 29, 41;23,31,47,59,71
J_{4} 5, 23;7,29,31,37,43 Th 13, 19; 31

By [Cr2], if G is a sporadic simple group and p | |G|, then with a few exceptions when p = 3 (none of which are in 𝒢_{3}), G = ⟨t, y⟩ where |t| = 2 and |y| = p. So by Proposition 3.9(b), for each central extension G̃ of G of degree prime to p, each minimally active 𝔽_{p}G̃-module has dimension at most 2p − 2.

The table [Js, Table 1] provides a helpful list of the minimal degrees for sporadic groups for each prime. This allows us to eliminate almost all cases from the above table, just by applying the bound dim(V) ≤ 2p − 2. We are left with the following possibilities for G_{0} (or for G_{0}.2 when G_{0} ∉ 𝒢_{p}^{∧}):

Prime  Possibilities for G_{0} or G_{0}.2
7  2·J_{2}, 6·Suz
11 M_{12}.2, 2·M_{12}.2, J_{1}, M_{22}.2, 2·M_{22}.2, [6·Suz]
13  [6·Suz.2], 2·Co_{1}
19  [3·J_{3}.2]

Since our modules are defined over 𝔽_{p} and are minimally active, Z(G_{0}) must act via multiplication by scalars, and hence |Z(G_{0})| | (p − 1) and Z(G_{0}) is central in G in all cases. This criterion allows us to eliminate the three entries in brackets in the above table. Note that the outer automorphisms of 6·Suz and 3·J_{3} invert the centres: as described, e.g., in the tables in [GL, § I.1.5].

When p = 7 and G_{0} = 2·J_{2}, the two 6-dimensional modules over the algebraically closed field amalgamate into a single 12-dimensional module over 𝔽_{7}: this can be seen either by computer or from the Brauer character table on [JLPW, p.105], where we see that there are irrationalities in the Brauer character of this representation, which from [JLPW, p.289] we see require 𝔽_{49}.

By Proposition 3.9, if an 𝔽_{p}G-module V is minimally active (for G ∈ 𝒢_{p}), and dim(V) > p, then dim(V) − p divides |N_{G}(U)/U|, and dim(V) = p + 1 if N_{G}(U)/U is abelian. We can thus eliminate the 𝔽_{p}G-modules in the following dimensions:

p Group dim(V) dim(V) − p N_{G}(U)/U
7 6·Suz  12  5  6·((3 × A_{4}).2)
11 J_{1}  14  3  10
11 M_{12}.2  16  5  10
11 M_{22}.2  20  9  10
13 2·Co_{1}  24  11  2·((6 × A_{4}).2)

We are left with the following cases, all for p = 11. For J_{1}, there is one 7-dimensional module. For 2·M_{12}.2, there are two pairs of modules of dimension 10 and one pair of dimension 12 (where the modules in each pair are isomorphic after restriction to 2·M_{12}). For 2·M_{22}.2, there are two pairs of modules of dimension 10. All of these modules are minimally active.

We have nearly proved the following proposition. Proposition 7.1. Let G ∈ 𝒢_{p}^{∧} be a group of sporadic type, and let V be a non-trivial, minimally active indecomposable module for G. Let G_{0} = E(G) be the unique quasisimple normal subgroup of G. Then p = 11, and one of the following holds:

(i) G_{0} ≅ 2·M_{12}, G/Z(G) ≅ M_{12}:2, and dim(V) = 10 (two possible F_{11}G_{0}-modules) or 12 (one such module); or

(ii) G_{0} ≅ 2·M_{22}, G/Z(G) ≅ M_{22}:2, and dim(V) = 10 (two possible 𝔽_{11}G_{0}-modules); or

(iii) G_{0} ≅ G/Z(G) ≅ J_{1}, and dim(V) = 7 (one module).

Proof. If V is simple, then we are done. So assume that V is indecomposable and not simple. In particular, dim(V) ≤ p + 1 by Proposition 3.7(c). Hence V has one non-trivial composition factor W of dimension at most p and the others are trivial. Then Z(G_{0}) = 1, so G_{0} ≅ J_{1}, dim(W) = 7, W is self-dual since it is the only module in that dimension, and hence H^{1}(G_{0}; W) = 0 by Lemma 3.11. So this case is impossible. □

8. Alternating groups

In this section we determine all minimally active modules for almost quasisimple groups associated to the alternating groups.

Proposition 8.1. Let G ∈ 𝒢_{p}^{∧} be a group of type A_{n} for n ≥ 5, and set G_{0} = E(G). Let V be a non-trivial, minimally active indecomposable module for G. Then one of the following holds:

(i) G_{0} ≅ A_{p}, G/Z(G) ≅ S_{p}, and V |_{G_{0}} is a subquotient of the permutation module, which has structure 1/W/1 for W of dimension p − 2;

(ii) G_{0} ≅ A_{p+1}, G/Z(G) ≅ S_{p+1}, and V |_{G_{0}} is the p-dimensional non-trivial summand of the permutation module;

(iii) G_{0} ≅ A_{n}, G/Z(G) ≅ A_{n} or S_{n} for p + 2 ≤ n ≤ 2p − 1, and V |_{G_{0}} is the (n − 1)- dimensional summand of the permutation module;

(iv) p = 5, G_{0} ≅ A_{5} ≅ PSL_{2}(5), G/Z(G) ≅ S_{5} ≅ PGL_{2}(5), and V is one of the modules described in Proposition 6.1 other than those in (i);

(v) p = 5, G_{0} ≅ 2·A_{5} ≅ SL_{2}(5), G/Z(G) ≅ S_{5}, and V is as in Proposition 6.1;

(vi) p = 5, G_{0} ≅ 2·A_{6}, G/Z(G) ≅ S_{6}, and dim(V) = 4 (two modules);

(vii) p = 7, G_{0} ≅ 2·A_{7}, G/Z(G) ≅ S_{7}, and dim(V) = 4;

(viii) p = 7, G_{0} ≅ 2·A_{8} or 2·A_{9}, G/Z(G) ≅ S_{8} or A_{9}, and dim(V) = 8 (one or two 𝔽_{7}G_{0}-modules, respectively); or

(ix) p = 7, G_{0} ≅ 2·A_{7}, G/Z(G) ≅ S_{7}, dim(V) = 8, and V has the form W/W where W is as in (vii).

Proof. By a result of Miller [Mi, pp.29–30], for each 3 < p ≤ n, A_{n} is generated by an element t of order 2 and an element x of order p (and this is easily seen to hold when (p, n) = (3, 5)). Thus by Proposition 3.9(b), each minimally active 𝔽_{p}A_{n}- or 𝔽_{p}[2·A_{n}]-module has dimension at most 2p−2. Our knowledge of small-dimensional representations of these groups is rather extensive, which makes these cases relatively easy to handle.

The smallest (faithful) simple module for S_{n} is the module arising from the permutation module, having dimension n − 1 − κ_{n}, where κ_{n} = 0 if p ∤ n and κ_{n} = 1 if p | n. By [Jam, Theorem 7 and Table 1] or [BK, Lemma 1.18], if p ≥ 7 and n ≥ 9, or p = 5 and n ≥ 11, the dimension of each larger 𝔽_{p}S_{n}-module is strictly greater than n(n − 5)/2, and hence that of each larger 𝔽_{p}A_{n}-module is greater than n(n − 5)/4. For each pair (p, n) such that n ≥ 10 and p ≤ n < 2p, n(n − 5)/4 > 2p − 2 except when p = n = 11, and the smallest faithful 𝔽_{11}A_{11}-module other than those in point (i) has dimension 36 (see [JLPW]). We are thus reduced to checking the cases 5 ≤ n ≤ 9.

For n ≥ 12, the smallest faithful representation of 2·A_{n} is of dimension 2^{⌊(n−2−κ_{n})/2⌋}, where κ_{n} is as above (see, e.g., [KT]). If V is minimally active, we have dim(V) ≤ 2p − 2 ≤ 2n − 2. Hence

2n − 2 ≥ 2^{⌊(n−2−κ_{n})/2⌋},

which yields n ≤ 11. (Note that since n < 2p, κ_{n} = 0 whenever n ≠ p, in particular, when n is not prime.)

We can get yet more restrictions based on Green correspondence. Assume that G is a central extension of A_{n} for p ≤ n ≤ 2p − 1, and let V be an indecomposable minimally active 𝔽_{p}G-module. Recall (Proposition 3.7(c,e)) that the Green correspondent of V is an absolutely simple N_{G}(U)/U-module.

(1) If p ≤ n ≤ p + 2, then N_{A_{n}}(U)/U is cyclic of order p − 1 or (p − 1)/2, so N_{G}(U)/U is abelian, and dim(V) ≤ p + 1 by Proposition 3.9(a).

(2) If n = p+3, then N_{A_{n}}(U)/U ≅ C_{3} ⋊C_{p−1}. Hence N_{G}(U)/U has an abelian subgroup of index 2, so its irreducible representations have dimension at most 2, and dim(V) ≤ p+2.

(3) If n = p + 4 and G = 2·A_{n} (and acts faithfully on V), then N_{G}(U)/U contains a subgroup H ≅ B ∘ 2·A_{4} with index 2, where B is abelian. Since the absolutely irreducible representations of H on which the central involution acts non-trivially are all 2-dimensional, either dim(V) ≤ p, or dim(V) = p + 2 or p + 4.

G_{0} p dimen. p dimen.  G_{0} p dimen. p dimen.
A_{5} 3 3^{∗}, 3^{∗}  5 Prop.6.1  2·A_{5} 3 2^{∗}, 2^{∗}  5 Prop.6.1
A_{6} 5  8^{(1)}  2·A_{6} 5  4,4
3·A_{6} 5 3 ∤ (p − 1)  6·A_{6} 5 6 ∤ (p − 1)
A_{7} 5 8^{(1)} 7 10^{(1)} 2·A_{7} 5 4^{∗}, 4^{∗} 7 4
3·A_{7} 5 3 ∤ (p − 1) 7 6^{†}, 9^{†}  6·A_{7} 5 6 ∤ (p − 1) 7 6^{†}, 6^{†}
A_{8} 5 none  7 none  2·A_{8} 5  8^{(2)}  7  8
A_{9} 5 none  7 none  2·A_{9} 5 8^{(3)}, 8^{(3)} 7  8,8
2·A_{10} 7 none  2·A_{11} 7  8  11 16^{(1)}

Table 8.1. Modules of dimension ≤ 2p − 2 for quasisimple alternating groups

Thus we can restrict attention to faithful representations of 2·A_{n} for n ≤ 11 and of A_{n} for n ≤ 9, as well as those of 3·A_{n} and 6·A_{n} for n = 6, 7. All simple modules for all primes are known for these alternating groups, and we can simply check them one by one and prime by prime. This is done in Table 8.1, where (based on [JLPW]) we list dimensions of all irreducible 𝔽_{p}G_{0}-modules of dimension at most 2p − 2 when G_{0} is a central extension of A_{n} for p ≤ n ≤ 2p − 1, except for the natural modules for A_{n} described in points (i)–(iii).

For the modules listed in the table, an asterisk (−)^{∗} means that it is not realized over 𝔽_{p} (hence does not give rise to any minimally active 𝔽_{p}G_{0}-module); a dagger (−)^{†} means that the module does not extend to S_{n} (when n = p or p + 1), and a superscript (−)^{(i)} for i = 1, 2, 3 means that it is not minimally active by point (i) above. Note that the groups 3·S_{7} and 6·S_{7} are “twisted” in the sense that their outer automorphisms invert their centres, so there are no indecomposable modules for these groups over 𝔽_{7}. The remaining modules (aside from the case (n, p) = (5, 5) of points (iv) and (v)) are shown in boldface, and are, in fact, minimally active, as can be seen by using the Magma command

IndecomposableSummands(Restriction(V,U)) to check the block sizes. These are precisely

the modules listed in points (vi)–(viii).

Now assume that V is indecomposable but not simple. By Proposition 3.7(c), dim(V) ≤ p + 1. If one or more of the simple components of V is 1-dimensional, then G_{0} ≅ A_{n}, and V contains a simple composition factor V_{0} of dimension p − 2 by Lemma 3.11. Hence we are in the situation of (i).

The only remaining possibility is the case where p = 7 and G_{0} ≅ 2·A_{7}, and V is an indecomposable extension of the 4-dimensional module in Table 8.1 by itself. By [Al, Proposition 21.7], there is at most one such extension. From the tables in [JLPW], we see that the restriction to 2·A_{7} of the simple 8-dimensional 2·A_{8}-module of case (viii) has this form. Hence there is a module of this type, and it is the restriction of a 2·S_{7}-module. □

9. Groups of Lie type: notation and preliminaries

We continue our notation that G is a finite group, U is a Sylow p-subgroup of G, and x ∈ U has order p, writing N = N_{G}(U) and C = C_{G}(U), with C = U × C′.

In this section we consider groups of Lie type, and use induction to reduce the problem of classifying minimally active modules to a small set of situations, essentially where the centralizer is abelian. We start with a brief overview of the orders of finite simple groups of Lie type, using [GL] or [MT] as a reference. We assume a passing familiarity with basic concepts from algebraic groups, such as simple connectivity (see [GLS3] for a brief outline of the background assumed, and [MT] for more details). It will be useful, in many cases, to write SL_{n}^{±}(q) or E_{6}^{±}(q), where

SL_{n}^{+}(q) = SL_{n}(q), SL_{n}^{−}(q) = SU_{n}(q), E_{6}^{+}(q) = E_{6}(q), E_{6}^{−}(q) = ^{2}E_{6}(q).

If G = ^{r}𝔾(q) is a finite group of Lie type of universal type, then its order is

|G| = q^{N} ∏_{d} Φ_{d}(q)^{a_{d}},

a power of q times a product of cyclotomic polynomials. However, to get the order of the associated simple group (if there is one), we must divide this order by z = |Z(G)|, an integer. The polynomials in q are given in Tables 9.1 and 9.2. We have dealt with the case where p | q in Section 6, where we saw that the only possibility is SL_{2}(p). So we assume for the rest of the paper that gcd(p, q) = 1.

G  |G|_{q′}  |Z(G)|  d with a_{d} = 1
SL_{2}(q)  (q − 1)(q + 1)  (2, q − 1)  1, 2
\begin{matrix}SL_{n}^{ε}(q) \\ (n ≥ 3) \end{matrix}  ∏_{i=2}^{n}((εq)^{i} − 1)  (n, q − ε)  ⌊n/2⌋+1,...,n−2, n − 1, n

\begin{matrix}Sp_{2n}(q), \\ Spin_{2n+1}(q) \end{matrix} ∏_{i=1}^{n}(q^{2i} − 1) (2, q − 1) ⌊n/2⌋+1,...,n−1, n (odd only)

n+1,...,2n−2, 2n (even only)

Spin_{2n}^{+}(q) (q^{n} − 1) ∏_{i=1}^{n−1}(q^{2i} − 1) \begin{matrix}(4, q^{n} − 1) or \\ (2, q − 1)^{2} \end{matrix} ⌊n/2⌋+1,...,n − 1, n (odd only) n+1,...,2n−4, 2n − 2 (even only)

Spin_{2n}^{−}(q) (q^{n} + 1) ∏_{i=1}^{n−1}(q^{2i} − 1) (4, q^{n} + 1)

⌊n/2⌋+1,...,n − 1 (odd only)
n,...,2n−4, 2n − 2, 2n (even only)

Table 9.1. Orders of classical groups of Lie type (regular d in bold)

G  |G|_{q′}  |Z(G)|  d with a_{d} = 1
2B_{2}(q)  Φ_{1}Φ_{4}  1  1, 4′, 4′′
2G_{2}(q)  Φ_{1}Φ_{2}Φ_{6}  1  1, 2, 6′, 6′′
2F_{4}(q)  Φ_{1}^{2}Φ_{2}^{2}Φ_{4}^{2}Φ_{6}Φ_{12}  1  6, 12′, 12′′
G_{2}(q)  Φ_{1}^{2}Φ_{2}^{2}Φ_{3}Φ_{6}  1  3, 6
3D_{4}(q)  Φ_{1}^{2}Φ_{2}^{2}Φ_{3}^{2}Φ_{6}^{2}Φ_{12}  1  12
F_{4}(q)  Φ_{1}^{4}Φ_{2}^{4}Φ_{3}^{2}Φ_{4}^{2}Φ_{6}^{2}Φ_{8}Φ_{12}  1  8, 12
E_{6}(q)  Φ_{1}^{6}Φ_{2}^{4}Φ_{3}^{3}Φ_{4}^{2}Φ_{5}Φ_{6}^{2}Φ_{8}Φ_{9}Φ_{12}  (3, q − 1)  5, 8, 9, 12
2E_{6}(q)  Φ_{1}^{4}Φ_{2}^{6}Φ_{3}^{2}Φ_{4}^{2}Φ_{6}^{3}Φ_{8}Φ_{10}Φ_{12}Φ_{18}  (3, q + 1)  8, 10, 12, 18

E_{7}(q) Φ_{1}^{7}Φ_{2}^{7}Φ_{3}^{3}Φ_{4}^{2}Φ_{5}Φ_{6}^{3}Φ_{7} Φ_{8}Φ_{9}Φ_{10}Φ_{12}Φ_{14}Φ_{18} (2, q − 1) 5, 7, 8, 9, 10,12, 14, 18

E_{8}(q) Φ_{1}^{8}Φ_{2}^{8}Φ_{3}^{4}Φ_{4}^{4}Φ_{5}^{2}Φ_{6}^{4}Φ_{7}Φ_{8}^{2}Φ_{9} Φ_{10}^{2}Φ_{12}^{2}Φ_{14}Φ_{15}Φ_{18}Φ_{20}Φ_{24}Φ_{30} 1 7,9,14, 15, 18, 20, 24, 30

Table 9.2. Orders of exceptional groups of Lie type (regular d in bold)

In Table 9.2, the values d = 4′, 4′′, etc. represent the factorizations of the cyclotomic polynomials Φ_{d}(q):

G = ^{2}B_{2}(q) q = 2^{2m+1} d = 4  Φ_{4}(q) = (q + \sqrt{2q} + 1)(q − \sqrt{2q} + 1)
G = ^{2}G_{2}(q) q = 3^{2m+1} d = 6  Φ_{6}(q) = (q + \sqrt{3q} + 1)(q − \sqrt{3q} + 1)
G = ^{2}F_{4}(q)  q = 2^{2m+1} d = 12 Φ_{12}(q) = (q^{2} + q\sqrt{2q} + q + \sqrt{2q} + 1)
·(q^{2} − q\sqrt{2q} + q − \sqrt{2q} + 1).

These factors are the orders of the largest cyclic subgroups in G of order dividing Φ_{d}(q).

We claim that

G/Z(G) ∈ 𝒢_{p} ⟹ p ∤ z and hence G ∈ 𝒢_{p}.  (1)

Assume otherwise: let G = ^{r}𝔾(q) be a counterexample. Since p is odd and p | z, z is not a power of 2, and hence 𝔾 = A_{n} or E_{6}. If 𝔾 = E_{6}, then z = (3, q ± 1), so p = 3, which is impossible since 3^{4} | (q^{2} − 1)^{4} | |G| for all q prime to 3 (Table 9.2). Thus G = SL_{n}^{ε}(q) and p | z = (n, q − ε). So n ≥ 3 and (q − ε)^{3} ∤ |G|, which by Table 9.1 implies n = p = 3. But then 3 | (q − ε) implies 3^{2} | (q^{3} − ε), so 3^{3} | |G| and 3^{2} | |G/Z(G)|. This proves (1). In particular, since G ∈ 𝒢_{p},

there exists a unique d such that p | Φ_{d}(q), and for this d, p^{2} ∤ Φ_{d}(q), and a_{d} = 1. (2)

We now want to understand the subgroups N = N_{G}(U) and C = C_{G}(U), which are closely related to the integer d, and not really dependent on the prime p.

We start by assuming that 𝔾 is simply connected, e.g., SL_{n}. In this case, a theorem of Steinberg (see [MT, Theorem 14.16]) states that, for each semisimple element s ∈ 𝔾, the centralizer C_{𝔾}(s) is connected. A semisimple element s ∈ G is regular if, inside the corresponding algebraic group 𝔾, dim(C_{𝔾}(s)) is minimal among all semisimple elements, so equal to the rank of 𝔾. By [MT, Corollary 14.10], if s is regular then C_{𝔾}(s)^{0} = 𝕋 for any maximal torus 𝕋 containing s, and hence s is contained inside a unique maximal torus 𝕋 and C_{𝔾}(s) = 𝕋.

Let F be a Frobenius endomorphism such that G = 𝔾^{F}, and assume for the moment that G is not a Ree or Suzuki group. To continue we need one fact from the theory of d-tori.

Rather than giving a formal definition of d-tori here, we instead refer to [MT, Section 25], and in particular to [MT, Definition 25.6]. The property we need is that by [MT, Theorems 25.14 and 25.19], if a_{d} = 1, then for the finite group G = 𝔾^{F}, there exists a cyclic torus T_{d} = 𝕋_{d}^{F} (called a Sylow d-torus), where 𝕋_{d} is an F -stable torus in 𝔾, such that U ≤ T_{d} and C_{G}(U) = C_{G}(T_{d}).

Let T_{d} = ⟨s⟩. If s is regular then we know that C_{𝔾}(s) is a maximal torus of 𝔾, and so C_{G}(U) = C_{G}(s) ≤ C_{𝔾}(s) is abelian. Note that in general, if s is regular then, although C_{G}(s) is a torus, we have C_{G}(s) ≥ T_{d}, with equality if and only if T_{d} is a maximal torus. This last statement is true if and only if ϕ(d) = rank(G).

If G is a Suzuki or Ree group then the maximal subgroups of G are known (see for example [Wi2]), so we can deduce the same result for those groups and for our particular primes p, and get C_{G}(U) abelian in all cases for Table 9.2. (The d-torus theory can be extended to these groups with some complications coming from the fact that cyclotomic polynomials split, but it is easier for us to use the lists of maximal subgroups directly to prove that C_{G}(U) is abelian.)

We have now shown the following:

Proposition 9.1. Let G = ^{r}𝔾(q) be the universal form of a group of Lie type (so a quasisimple group as given in Tables 9.1 and 9.2), and suppose that G ∈ 𝒢_{p} and p ∤ q. Let d be the multiplicative order of q modulo p. If d is a regular number for G then C_{G}(U) is abelian.

The numbers d that yield regular elements s for a given group 𝔾 and Frobenius endomorphism F were computed by Springer in [Sp, Section 5, 6.9–6.11]. They are given in Tables 9.1 and 9.2, where we list all d such that a_{d} = 1 (i.e., such that the Sylow Φ_{d}-subgroup has order p), and among them list in bold the d that are regular.

In view of Proposition 9.1, we would like to always be in the situation that d is a regular number, but from Tables 9.1 and 9.2 we see that this is not true. However, as we will see, there is always a subgroup H of G, also a group of Lie type, such that U ≤ H and d is regular for H.

For classical groups at least, we also need to understand the structure of N_{G}(U), not just of C_{G}(U), although this is easy and again depends only on d. For d a positive integer, let d be defined by d = 2d if d is odd, d = d/2 if d is even but 4 ∤ d, and d = d if 4 | d. Note that if d is the multiplicative order of q modulo p, then d is the order of −q modulo p.

For SL_{2}(q), the automizer is always of order 2. For SL_{n}(q) and SU_{n}(q) for n ≥ 3, by [FS1, Proposition (3D)], |N/C| = d or d, respectively. For the other classical groups, by [FS2, p.128], |N/C| equals 2d if d is odd or d if d is even, i.e., lcm(d, 2) in all cases.

The order of N/C, when p | Φ_{d}(q), is summarized for the classical groups in Table 9.3.

Group  |N/C| when p | Φ_{d}(q)
PSL_{2}(q)  2
PSL_{n}(q) (n ≥ 3)  d
PSU_{n}(q) d
PSp_{2n}(q) or P Ω_{m}^{±}(q)  lcm(2, d)

Table 9.3

When we work with cyclotomic polynomials, the following relations are useful:

(i) for all n > 1, Φ_{n}(q) | (q^{n} − 1)/(q − 1);

(ii) if n is even but 4 ∤ n, then Φ_{n}(q) | (q^{n/2} + 1)/(q + 1); and

(iii) if 4 | n then Φ_{n}(q) | q^{n/2} + 1.

We are now ready to work with the individual groups: first the classical groups and then the exceptional groups.

10. Determination for classical groups

In this section, we classify minimally active modules for classical groups and their extensions in 𝒢_{p}^{∧} when p is not the defining characteristic. Throughout this section, G = G(q) is a group of Lie type, p ∤ q is a prime dividing |G|, U is a Sylow p-subgroup of G with generator x, and

p has order d modulo q, so that p | Φ_{d}(q).

Let V be a minimally active module for G. In Table 10.1, we list the minimal possible dimension l(G) for a faithful 𝔽_{p}G-module, as determined in [LS], [SZ], or [GT].

G Lower bound for dimension Ref.

Exceptions

SL_{2}(q)  (q − 1)/z (z = (2, q − 1))  [LS] \begin{matrix}l(SL_{2}(4)) = 2 \\ l(SL_{2}(9)) = 3 \end{matrix}
SL_{n}(q)  q^{n−1} − 1  [LS] \begin{matrix}l(SL_{3}(2)) = 2 \\ l(SL_{3}(4)) = 4 \end{matrix}
(n ≥ 3)  q(q^{n−1} − 1)/(q − 1) − 1  [GT] \begin{matrix}l(SL_{4}(2)) = 7 \\ l(SL_{4}(3)) = 26 \end{matrix}
\begin{matrix}SU_{n}(q) \\ (n ≥ 3) \end{matrix}  \begin{matrix}(q^{n} − q)/(q + 1) & (n odd) \\ (q^{n} − 1)/(q + 1) & (n even) \end{matrix}  [LS] \begin{matrix}l(SU_{4}(2)) = 4 \\ l(SU_{4}(3)) = 6 \end{matrix}

Sp_{2n}(q)  \begin{matrix}(q^{n} − 1)/2 & (2 ∤ q) \\ q(q^{n} − 1)(q^{n−1} − 1)/2(q + 1) & (2 | q) \end{matrix} [SZ] l(Sp_{4}(2)′) = 2
\begin{matrix}Spin_{2n+1}(q) \\ (n ≥ 3, 2 ∤ q) \end{matrix}  q^{n−1}(q^{n−1} − 1)  [LS] l(Spin_{7}(3)) = 27
\begin{matrix}Spin_{2n}^{+}(q) \\ (n ≥ 4) \end{matrix}  q^{n−2}(q^{n−1} − 1)  [LS] l(Spin_{8}^{+}(2)) = 8
\begin{matrix}Spin_{2n}^{−}(q) \\ (n ≥ 4) \end{matrix}  (q^{n−1} + 1)(q^{n−2} − 1)  [LS]  —

Table 10.1. Minimal representation dimensions of classical groups

We begin with the linear groups.

Proposition 10.1. Let G ∈ 𝒢_{p}^{∧} be a group of type PSL_{n}(q), where n ≥ 2 and p ∤ q, and set G_{0} = E(G). Let V be a non-trivial, minimally active 𝔽_{p}G-module. Then one of the following holds:

(i) G_{0} is a central extension of PSL_{2}(4) ≅ A_{5}, PSL_{2}(5) ≅ A_{5}, PSL_{2}(9) ≅ A_{6} or PSL_{4}(2) ≅ A_{8}, and the modules are as in Proposition 8.1;

(ii) G_{0} is a central extension of PSL_{2}(7) ≅ PSL_{3}(2), p = 7, and V is one of the modules in Proposition 6.1; (iii) G_{0} ≅ SL_{2}(8), G/Z(G) ≅ SL_{2}(8):3, p = 7, and dim(V) = 7, 8;

(iv) G_{0} ≅ 6·PSL_{3}(4), G/Z(G) ≅ PSL_{3}(4).2, p = 7, and dim(V) = 6.

Proof. In all cases, we fix n and q = r^{t}, where r is prime and t ≥ 1, such that G is of type PSL_{n}(q). If (n, q) = (2, 4), (2, 5), (2, 9), or (4, 2), then PSL_{n}(q) is an alternating group, and we are in the situation of (i). So we assume from now on that (n, q) is not one of these pairs. Since PSL_{3}(2) ≅ PSL_{2}(7), we can also assume that (n, q) ≠ (3, 2).

Among the remaining cases, SL_{n}(q) is the universal central extension of PSL_{n}(q) with only one exception: PSL_{3}(4) has an “exceptional cover”: a central extension of the form 4^{2}·SL_{3}(4) [Wi2, § 3.12]. In all other cases, if p | Φ_{d}(q) for regular d, then C_{G_{0}}(U) is abelian by Proposition 9.1, and hence by Proposition 3.9(a), dim(V) ≤ 2p − 1 for each indecomposable minimally active 𝔽_{p}G_{0}-module V.

Case 1: We start with the groups of type PSL_{2}(q), where q = r^{t}. Thus p | Φ_{d}(q) where d = 1, 2, and the normalizer N_{G_{0}}(U) is dihedral of order Φ_{d}(q) or quaternion of order 2·Φ_{d}(q), containing C_{G_{0}}(U) as a cyclic subgroup of index 2. Hence if V is minimally active, then dim(V) ≤ p + 2 by Proposition 3.9(a).

In the group Aut(SL_{2}(r^{t})), the automizer of U has order at most 2t, so that if G ∈ 𝒢_{p}^{∧} then p ≤ 2t + 1. On the other hand, the smallest dimension for V is (r^{t} − 1)/2 for r odd and r^{t} − 1 for r even, so that dim(V) ≤ p + 2 becomes

r^{t} ≤ dim(V) + 1 ≤ p + 3 ≤ 2t + 4  if r = 2
r^{t} ≤ 2 dim(V) + 1 ≤ 2(p + 2) + 1 ≤ 4t + 7  if r > 2.

If r = 2, then t ≤ 3; while if r is odd, then either t = 2 and r = 3, or t = 1 and r ≤ 11.

When G_{0} ≅ PSL_{2}(11) and p = 3, there is a 5-dimensional 𝔽_{3}G_{0}-module V, but N_{G_{0}}(U)/U ≅ C_{2}^{2} is abelian in this case, so V is not minimally active by Proposition 3.9(a). Since we have already dealt with the alternating groups PSL_{2}(4) ≅ PSL_{2}(5) ≅ A_{5} and PSL_{2}(9) ≅ A_{6}, and do not need to deal with solvable groups, we are left with the groups PSL_{2}(7) with p = 3, and PSL_{2}(8) with p = 7.

For G ≅ SL_{2}(7) and p = 3, the modules of dimension 3 are not defined over 𝔽_{3}, but only over 𝔽_{9}. For G ≅ SL_{2}(8) and p = 7, there are four 7-dimensional modules of which only one extends to SL_{2}(8):3 ∈ 𝒢_{7}^{∧}, and one 8-dimensional module which also extends to SL_{2}(8):3. Both of these are minimally active.

Case 2: We next consider the groups of type PSL_{n}(q) for n ≥ 3, where p | Φ_{d}(q) for some ⌊\frac{n}{2}⌋ + 1 ≤ d ≤ n − 1. By Table 10.1, dim(V) ≥ q(q^{n−1} − 1)/(q − 1) − 1 with the exceptions in Table 10.1, which (aside from cases that we already eliminated) are

(n, q, p) ∈ {(3, 4, 5), (4, 3, 13)}.

(Here we consider those PSL_{n}(q) in the table, together with p such that p | Φ_{d}(q) with ⌊\frac{n}{2}⌋ + 1 ≤ d ≤ n − 1 and p^{2} ∤ |PSL_{n}(q)|.) Furthermore, if V is the reduction modulo p of a complex character (this is true if dim(V) ≥ p + 1 by Proposition 3.7(d)) then dim(V) ≥ q(q^{n−1} − 1)/(q − 1) with the exceptions above.

Suppose firstly that d = n − 1. If (n, q, p) = (3, 4, 5), then U ≅ C_{5} is self-centralizing in PSL_{3}(4), so that C_{G_{0}}(U) is abelian when G_{0} is any central extension. Hence C_{G}(U) is abelian and dim(V) ≤ 2p − 1 in all cases. With the exceptions above, we get

q(q^{n−1} − 1)/(q − 1) − 1 ≤ dim(V) ≤ 2p − 1 ≤ 2Φ_{n−1}(q) − 1 ≤ 2(q^{n−1} − 1)/(q − 1) − 1.

Thus q = 2 and dim(V) = 2p−1, whence dim(V) > p and so V is the reduction of a complex character. So dim(V) ≥ q(q^{n−1} − 1)/(q − 1) by Table 10.1, which is a contradiction.

If G_{0}/Z(G_{0}) ≅ PSL_{3}(4) and p = 5, then the smallest projective representation has dimension 6 (see [JLPW]), but this is for 6.PSL_{3}(4), and as the centre has size 6 the two 6-dimensional modules are not realizable over 𝔽_{5}, so there are no minimally active modules. For G_{0} a central extension of PSL_{4}(3) and p = 13 = Φ_{3}(3), dim(V) ≥ 2p and so there are no minimally active modules (since d = 3 is regular).

If ⌊n/2⌋ + 1 ≤ d < n − 1 and V is minimally active for SL_{n}(q), then n ≥ 5, and there is H ≅ SL_{n−1}(q) such that U ≤ H, where V |_{H} is also minimally active. So by what was just shown, H ≅ SL_{4}(2) and p = 7, and G_{0} ≅ SL_{5}(2). In this case, N_{G_{0}}(U)/U ≅ 3 × S_{3}, so the Green correspondent of V has dimension at most 2, and dim(V) ≤ p + 2 = 9, which contradicts both bounds in Table 10.1.

Case 3: We now assume G is of type PSL_{n}(q), where n ≥ 3 and p | Φ_{n}(q). If G ∈ 𝒢_{p}^{∧}, then by allowing for a possible graph automorphism and a field automorphism of order t, we have that |Aut_{G}(U)| ≤ 2nt, so p ≤ 2nt + 1. On the other hand, by Table 10.1, with the exceptions of SL_{3}(2) and SL_{3}(4), the dimension of any minimally active module V is at least r^{t(n−1)} − 1 (as the group is SL_{n}(r^{t}).2.t), whence dim(V) < 2p (which also holds when G_{0} is a central extension of PSL_{3}(4) and p = 7) becomes

r^{2nt/3} ≤ r^{t(n−1)} ≤ dim(V) + 1 < 2p + 1 ≤ 4nt + 3.

Set ℓ = nt ≥ 3. Then r^{2ℓ/3} ≤ 4ℓ + 2, and from this we see that r = 2 implies ℓ < 9 (hence t ≤ 2), r = 3 implies ℓ < 6 (hence t = 1), and r ≥ 5 is impossible. Upon returning to the inequality r^{t(n−1)} ≤ 4nt + 3, we have q = r^{t} = 4 implies n = 3, q = 3 implies n = 3, and q = 2 implies n ≤ 5. Since we are assuming that (n, q) ≠ (3, 2) or (4, 2), we are left with the following possibilities:

\begin{matrix}(n, q) & (5, 2) & (3, 3) & (3, 4) \\ Φ_{n}(q) & 31 & 13 & 21 \end{matrix}

Note that this list includes the remaining exception to the Landazuri–Seitz bounds.

We can eliminate groups of type PSL_{5}(2) for p = 31 since Aut(PSL_{2}(5)) ∉ 𝒢_{31}^{∧}, and those of type PSL_{3}(3) for p = 13 since Aut(PSL_{3}(3)) ∉ 𝒢_{13}^{∧}. It remains to consider PSL_{3}(4) and its covers when p = 7. In this case, since N_{PSL_{3}(4)}(U)/U has order 3, N_{G}(U)/U is abelian in all cases, and hence dim(V) ≤ p + 1 = 8 by Proposition 3.9(a). We have modules of dimension 8 for 4_{1}·PSL_{3}(4) (but these cannot be defined over 𝔽_{7} as the centre has order 4), and of dimension 6 for 6·PSL_{3}(4), which are indecomposable on restriction to U. □

We now turn to unitary groups.

Proposition 10.2. Let G ∈ 𝒢_{p}^{∧} be a group of type PSU_{n}(q), where n ≥ 3 and p ∤ q, and set G_{0} = E(G). Let V be a non-trivial, minimally active 𝔽_{p}G-module. Then one of the following holds: either

(i) G_{0} ≅ PSU_{3}(3), G/Z(G) ≅ PSU_{3}(3).2, p = 7, and dim(V) = 6, 7; or

(ii) G_{0} ≅ PSU_{3}(4), G/Z(G) ≅ PSU_{3}(4):4, p = 13, and dim(V) = 12; or

(iii) G_{0} ≅ G/Z(G) ≅ PSU_{4}(2), p = 5, and dim(V) = 6; or

(iv) G_{0} ≅ 6_{1}·PSU_{4}(3), G/Z(G) ≅ PSU_{4}(3).2_{2} (G contains the complex reflection group G_{34}), p = 7, and dim(V) = 6; or

(v) G_{0} ≅ PSU_{5}(2), G/Z(G) ≅ PSU_{5}(2).2, p = 11, and dim(V) = 10.

Proof. Let G be of type PSU_{n}(q), where q = r^{t} and r is prime. Since SU_{3}(2) is solvable, we assume (n, q) ≠ (3, 2). Among the other cases, SU_{n}(q) is the universal central extension of PSU_{n}(q) with exactly three exceptions: PSU_{4}(2), PSU_{6}(2), and PSU_{4}(3) [Wi2, § 3.12]. In all other cases, if p | Φ_{d}(q) for regular d, then C_{G_{0}}(U) is abelian, and dim(V) ≤ 2p − 1 when V is indecomposable and minimally active, by Propositions 9.1 and 3.9(a).

By Table 10.1, dim(V) ≥ (q^{n} − q)/(q + 1) (whether n is even or odd), with just two exceptions.

Case 1: Assume that p | Φ_{d}(−q) = Φ_{d}(q) for some ⌊n/2⌋ + 1 ≤ d ≤ n − 1. If d = n − 1, then by the above remarks, either G_{0}/Z(G_{0}) ≅ PSU_{4}(2), PSU_{4}(3), or PSU_{6}(2), or

q(q^{n−1} − 1)/(q + 1) ≤ dim(V) < 2p ≤ 2(q^{n−1} + 1)/(q + 1).

From this, it follows immediately that q = 2 and dim(V) = 2p−1. Since V is minimally active and C_{G_{0}}(U) is abelian, this last condition implies that |Aut_{G_{0}}(U)| = p − 1 by Proposition 3.9(a). Hence n − 1 ≥ p − 1, so n ≥ p > (2^{n−1} − 1)/3, and n ≤ 4. But the group SU_{3}(2) is solvable, and 3^{2} | |PSU_{4}(2)| (3 = Φ_{3}(−2)), so both of these are eliminated.

This leaves the two other exceptional cases. If G_{0}/Z(G_{0}) ≅ PSU_{6}(2) and p | Φ_{5}(−2) = 11, then U is self-centralizing in PSU_{6}(2), and so C_{G_{0}}(U) is abelian for each cover U. Thus dim(V) ≤ 2p − 1 = 21, which contradicts Table 10.1. If G_{0}/Z(G_{0}) ≅ PSU_{4}(3) and p | Φ_{3}(−3) = 7, then since |N_{G_{0}}(U)/U| = 3, dim(V) ≤ p + 1 = 8 when V is minimally active (Proposition 3.9(a)). By [JLPW, p.137], there is a single 6-dimensional module for G_{0} ≅ 6_{1}·PSU_{4}(3) over 𝔽_{7} (using ATLAS notation for the central extension), it extends to 6_{1}·PSU_{4}(3).2_{2} ∈ 𝒢_{7}^{∧}, and all other modules are of dimension larger than 8.

If ⌊n/2⌋+1 ≤ d < n−1, then n ≥ 5, and there is H < G_{0} such that H/Z(H) ≅ PSU_{n−1}(q) such that U ≤ H and V |_{H} is still indecomposable and minimally active. Hence (n, q) = (5, 3), p = 7, and H ≅ 6_{1}.SU_{4}(3). Since this central extension is not a subgroup of SU_{5}(3) (and this group has no central extensions), there are no minimally active modules for p = 7 and G = SU_{5}(3).

Case 2: Now assume that p | Φ_{n}(−q) = Φ_{n}(q), so that |Aut_{G}(U)| ≤ 2nt by Table 9.3. Thus p ≤ 2nt + 1 and dim(V) < 2p imply (with the three exceptions noted above)

r^{t}(r^{t(n−1)} − 1) ≤ (4nt + 2)(r^{t} + 1)

and hence (since n ≥ 3)

r^{2tn/3} ≤ r^{t(n−1)} ≤ 1 + (4nt + 2)(1 + \frac{1}{r^{t}}) ≤ (4nt + 3)(1 + \frac{1}{r}).

Set ℓ = nt; we thus have r^{2ℓ/3} ≤ \frac{r+1}{r}(4ℓ + 3). When r = 2, this implies ℓ < 9 and hence t ≤ 2; when r = 3 it implies ℓ < 6 and hence t = 1, and there are no solutions for r ≥ 5 and ℓ ≥ 3. If we now go back to the original inequality, we see that the only solutions (including the exceptional cases) are the following ones:

\begin{matrix}(n, q) & (4, 2) & (5, 2) & (6, 2) & (3, 3) & (3, 4) & (4, 3) \\ Φ_{n}(−q) & 5 & 11 & 7 & 7 & 13 & 10 \end{matrix}

Here, we omit the pair (n, q) = (3, 2) since SU_{3}(2) is solvable.

• For PSU_{4}(2) and p = 5, there are two 5-dimensional modules whose irrationalities in their Brauer characters [JLPW, p.62] imply that they need 𝔽_{25} from [JLPW, p.288], and there is a single 6-dimensional module, which is minimally active for PSU_{4}(2) ∈ 𝒢_{5}^{∧}.

• For PSU_{5}(2) and p = 11, we see from [JLPW, p.184] that there is a module of dimension 10 = p − 1 and two of dimension 11, amalgamating over PSU_{5}(2).2 ∈ 𝒢_{11}^{∧}. So the proposition holds in these cases.

• For PSU_{6}(2), the Landazuri–Seitz bound gives dim(V) ≥ 21, so there are no minimally active modules when p = 7.

• For PSU_{3}(3) and p = 7, we need PSU_{3}(3).2 to be in 𝒢_{7}^{∧}. By [JLPW, p.24] there is a single module for G_{0} = PSU_{3}(3) of dimension 6 and one of dimension 7 (the two other dual modules of dimension 7 amalgamate for G).

• For PSU_{3}(4) and p = 13, there is by [JLPW, p.73] a single module of dimension 12, extendible to PSU_{3}(4).4 ∈ 𝒢_{13}^{∧}, and four modules of dimension 13, amalgamating into a single 52-dimensional for PSU_{3}(4).4.

• For PSU_{4}(3) and p = 5, there are many covers, as the Schur multiplier is of order 36. However, for our module to be definable over 𝔽_{5}, the centre of G must have order dividing 4, and so G_{0} is a quotient group of SU_{4}(3), and its smallest simple module has dimension 20 [JLPW, p.128]. Thus dim(V) ≤ 2p − 1 = 9 by Propositions 9.1 and 3.9(a), a contradiction.

This completes the proof. □

We next consider the symplectic groups.

Proposition 10.3. Let G ∈ 𝒢_{p}^{∧} be a group of type PSp_{2n}(q), where n ≥ 2 and p ∤ q, and set G_{0} = E(G). Let V be a non-trivial, minimally active 𝔽_{p}G-module. Then one of the following holds: either

(i) G is of type Sp_{4}(2)′ ≅ A_{6} and p = 5, and V is as in Proposition 8.1; or

(ii) G_{0} ≅ G/Z(G) ≅ PSp_{4}(3) ≅ PSU_{4}(2), p = 5, and dim(V) = 6; or

(iii) G_{0} ≅ Sp_{4}(4), G/Z(G) ≅ Sp_{4}(4).4, p = 17, and dim(V) = 18; or

(iv) G_{0} ≅ G/Z(G) ≅ Sp_{6}(2), p = 5 or p = 7, and dim(V) = 7; or

(v) G_{0} ≅ 2·Sp_{6}(2), G/Z(G) ≅ Sp_{6}(2), p = 7 and dim(V) = 8.

Proof. In the first three cases, we assume that p | Φ_{2n}(q), or (if n is odd) that p | Φ_{n}(q). In particular, in these cases, p ≤ (q^{n} ± 1)/(q ± 1) if n is odd, and p ≤ (q^{n} + 1) if n is even. The remaining possibilities (where d is not regular) are handled inductively in Case 4.

Since the group Sp_{4}(2)′ ≅ A_{6} has already been handled in Proposition 8.1, we assume from now on that (2n, q) ≠ (4, 2). The only (other) case in which Sp_{2n}(q) has a proper central extension is the group Sp_{6}(2). Case 1: Suppose that q is even. By Table 10.1 (and since (2n, q) ≠ (4, 2)), we have dim(V) ≥ q(q^{n} − 1)(q^{n−1} − 1)/2(q + 1). When (2n, q) = (6, 2), we have p = 7 = Φ_{3}(2) (since Φ_{6}(2) = 3 and Sp_{6}(2) ∉ 𝒢_{3}), U is self-centralizing in Sp_{6}(2), so C_{G_{0}}(U) is abelian when G_{0} is any central extension of Sp_{6}(2), and the bound dim(V) < 2p still applies.

Suppose firstly that p | Φ_{n}(q), where n ≥ 3 is odd. The statement dim(V) < 2p yields

q(q^{n} − 1)(q^{n−1} − 1)/2(q + 1) < 2(q^{n} − 1)/(q − 1).

If q ≥ 4 then this can never happen, but if q = 2 then this reduces to (2^{n−1} − 1) < 6, and hence n = 3. Thus (2n, q) = (6, 2), and p = Φ_{3}(2) = 7. In this case, Sp_{6}(2) has a faithful 7-dimensional module and 2·Sp_{6}(2) a faithful 8-dimensional module, and both are minimally active.

If p | Φ_{2n}(q), then since dim(V) < 2p, we get

q(q^{n} − 1)(q^{n−1} − 1)/2(q + 1) < 2(q^{n} + 1),

whence q(q^{n−1} − 1)(q^{n} − 1) < 4(q + 1)(q^{n} + 1), and q ≤ 4. If q = 4, then (4^{n} − 1)(4^{n−1} − 1) < 5(4^{n} + 1), which is satisfied for n = 2 only. If q = 2, we have (2^{n} − 1)(2^{n−1} − 1) < 6(2^{n} + 1), which is satisfied for n ≤ 3 only, but Φ_{6}(2) = 3, so the case (2n, q) = (6, 2) can be eliminated. Since (2n, q) ≠ (4, 2), we are left with the case G_{0} ≅ Sp_{4}(4) and p | Φ_{4}(4) = 17. Then Aut(G_{0}) ≅ G_{0}:4 ∈ 𝒢_{17}^{∧}, and there is a (unique) 18-dimensional simple minimally active 𝔽_{17}G_{0}-module for which the action extends to Aut(G_{0}).

Case 2: If q is odd and n is odd, the smallest cross-characteristic representations are the Weil representations, of dimension (q^{n} − 1)/2 (see Table 10.1 and [GMST]). If p | Φ_{n}(q) or Φ_{2n}(q) then p ≤ (q^{n} ± 1)/(q ± 1), whence the inequality dim(V) < 2p yields

(q^{n} − 1)/2 < 2(q^{n} ± 1)/(q ± 1),

and hence q = 3.

If V is minimally active and G = Sp_{2n}(3) (n odd) lies in 𝒢_{p}^{∧}, and p | Φ_{n}(3), then since there can be no graph or field automorphisms, Aut_{G}(U) has order 2n, so that p = 2n + 1. On the other hand, dim(V) ≥ (q^{n} − 1)/2, and since q and n are both odd, we have that (n, q) = (3, 3), yielding p = 2n + 1 = 7 = Φ_{6}(3) and V has dimension 13, larger than p + 1.

Case 3: Assume that p | Φ_{2n}(q), q is odd, and n is even. Assume that q = r^{t} where r is prime. If G = Sp_{2n}(q^{t}).t, where t is the order of a graph automorphism, then Aut_{G}(U) has order at most 2nt, so that p ≤ 2nt + 1. Thus dim(V) < 2p ≤ 4nt + 2, and using the fact that dim(V) ≥ (q^{n} − 1)/2, we have that

r^{nt} − 1 < 8nt + 4.

As r ≥ 3 and n ≥ 2 is even, we have nt = 2, so t = 1 and (n, q) = (2, 3). But PSp_{4}(3) ≅ PSU_{4}(2), so this case has already been done (Proposition 10.2).

Case 4: Now assume that G ∈ 𝒢_{p}^{∧} is of type PSp_{2n}(q), where p | Φ_{d}(q) for odd d < n or even d < 2n (see Table 9.1), and that there is an indecomposable minimally active 𝔽_{p}G-module V. Then there is H < G of type PSp_{2n−2}(q) with H ∈ 𝒢_{p}^{∧}, and V |_{H} is again minimally active. So we can assume inductively that some indecomposable summand of V |_{H} is already on our list.

Thus we next consider groups of type Sp_{6}(2) (p = 5), PSp_{6}(3) (p = 5), Sp_{6}(4) (p = 17), and Sp_{8}(2) (p = 7). Since Aut(Sp_{6}(4)) ∉ 𝒢_{17}^{∧}, we can eliminate this case.

Assume G is of type Sp_{6}(2) and p = 5. Then N_{G_{0}/Z(G_{0})}(U) ≅ 5:4 × S_{3}, contained in the subgroup Ω_{6}^{+}(2) ≅ S_{8} in Sp_{6}(2). Hence N_{G_{0}/Z(G_{0})}(U)/U contains a cyclic subgroup of index 2, and N_{G_{0}}(U)/U contains an abelian subgroup of index 2. So by Green correspondence, if V is indecomposable and minimally active, then dim(V) ≤ p + 2 = 7. By [JLPW], there is a 7-dimensional module for Sp_{6}(2), and it is minimally active.

If G is of type PSp_{6}(3) and p = 5, the smallest faithful module has dimension at least 13 by Table 10.1, more than twice that of the minimally active module for Sp_{4}(3), so by Proposition 3.10, there are no minimally active 𝔽_{p}G-modules. Similarly, if G is of type Sp_{8}(2) and p = 5 or 7, then dim(V) ≥ 28 by Table 10.1, this is more than twice the dimension of the modules we found for Sp_{6}(2), so this is again impossible by Proposition 3.10. □

It remains to handle the orthogonal groups.

Proposition 10.4. Let G ∈ 𝒢_{p}^{∧} be a group of type Ω_{2n+1}(q) for q odd and n ≥ 3, or of type P Ω_{2n}^{±}(q) for n ≥ 4, where p ∤ q in all cases. Set G_{0} = E(G). Let V be a non-trivial, minimally active 𝔽_{p}G-module. Then p = 7, G_{0} ≅ 2·Ω_{8}^{+}(2), G/Z(G) ≅ Ω_{8}^{+}(2) or Ω_{8}^{+}(2).2, and dim(V) = 8.

Proof. Case 1: Assume that G_{0}/Z(G_{0}) ≅ Ω_{2n+1}(q) (where q is odd). By Table 10.1, if V is a minimally active module for G_{0} or one of its covers, then dim(V) ≥ q^{n−1}(q^{n−1} − 1), except possibly for Ω_{7}(3), which will be handled separately. As with symplectic groups, we use an inductive argument to reduce to the case where p | Φ_{d}(q) for d = n, 2n, starting with the base case of the induction, Ω_{5}(q) = Sp_{4}(q).

If p | Φ_{n}(q), then p ≤ (q^{n} − 1)/(q − 1), and so dim(V) < 2p becomes

q^{n−1}(q^{n−1} − 1) < \frac{2(q^{n} − 1)}{q − 1} ⟹ 2q^{2} ≤ q^{n−1}(q − 1) < \frac{2(q^{n} − 1)}{q^{n−1} − 1} ≤ 2(1 + q),

which has no solutions.

If p | Φ_{2n}(q), then p ≤ (q^{n} + 1), and so dim(V) < 2p becomes

q^{n−1}(q^{n−1} − 1) ≤ 2(q^{n} + 1),

yielding q^{2n−2} ≤ 2q^{n} + q^{n−1} + 2 for n, q ≥ 3, again clearly having no solutions. Thus there are no minimally active modules when p | Φ_{n}(q) or p | Φ_{2n}(q). So by induction, and since the only groups of type Ω_{5}(q) ≅ PSp_{4}(q) with minimally active modules are those for q = 3 (Proposition 10.3(ii)), the only cases left to consider are those where G_{0}/Z(G_{0}) ≅ Ω_{7}(3) and p = 5 | Φ_{4}(3), 7 = Φ_{6}(3), or 13 = Φ_{3}(3).

When p = 5, Proposition 3.10, applied with H < G of type Ω_{5}(q) = PSp_{4}(q), says that the dimension of a minimally active 𝔽_{5}G_{0}-module V is at most twice that of a minimally active 𝔽_{p}H-module (since Ω_{7}(3) is generated by two conjugates of Ω_{5}(3)). Thus dim(V) ≤ 12 by Proposition 10.3(ii), which is impossible by Table 10.1.

When p = 7 or 13, C_{Ω_{7}(3)}(U) is cyclic (of order 14 or 13, respectively), so the centralizer of U is abelian in each cover of Ω_{7}(3). Hence dim(V) < 2p, which again contradicts Table 10.1.

Case 2: We move on to the case where G = Ω_{2n}^{+}(q) (for n ≥ 4). Here we are concerned with p | Φ_{d}(q), where by Table 9.1, d is regular when d = 2n − 2, or d ∈ {n − 1, n} is odd. The Landazuri–Seitz bound from Table 10.1 is q^{n−2}(q^{n−1} − 1), with Ω_{8}^{+}(2) the only exception. This is also the only case which has an exceptional Schur multiplier.

If d = n − 1 or d = n, then p ≤ (q^{n} − 1)/(q − 1), and dim(V) < 2p implies

q^{2} ≤ q^{n−2}(q − 1) < 2(q^{n} − 1)/(q^{n−1} − 1) ≤ 2(1 + q).

This is satisfied only for Ω_{8}^{+}(2), which we need to consider separately in any case. If d = 2n − 2, then p ≤ (q^{n} + 1), and we have

q^{n−2}(q^{n−1} − 1) < 2(q^{n} + 1) ⟹ q ≤ q^{n−3} < 2 + q^{−2} + 2q^{−n},

which again is only satisfied for Ω_{8}^{+}(2).

Now, Ω_{8}^{+}(2) ∈ 𝒢_{p} only for p = 7, C_{Ω_{8}^{+}(2)}(U) = U in this case, and thus C_{G_{0}}(U) is abelian when G_{0} is any central extension of Ω_{8}^{+}(2). Thus we can always assume that dim(V) < 2p = 14. The smallest non-trivial representation of Ω_{8}^{+}(2) itself is of dimension 28, which is too large. There is an 8-dimensional representation for the exceptional cover 2·Ω_{8}^{+}(2), it is minimally active, and it extends to a module for 2·Ω_{8}^{+}(2).2 (the Weyl group of E_{8}).

Since the only minimally active module is for the exceptional cover 2·Ω_{8}^{+}(2), it does not extend to a minimally active module over a group of type Ω_{10}^{+}(2).

Case 3: Finally, consider G of type Ω_{2n}^{−}(q) (again for n ≥ 4). By Table 9.1, d is regular when d = 2n, 2n − 2, or n is even and d = n − 1, and we first consider p | Φ_{d}(q) for such d. The Landazuri–Seitz bound gives dim(V) ≥ (q^{n−1} + 1)(q^{n−2} − 1) (Table 10.1), and we do the same analysis as for the plus-type case.

If d = n − 1, then p ≤ (q^{n−1} − 1)/(q − 1), and since dim(V) < 2p,

(q^{n−1} + 1)(q^{n−2} − 1)(q − 1) < 2(q^{n−1} − 1),

so that (q^{n−2} − 1)(q − 1) < 2, which has no solutions when n ≥ 4. If d = 2n − 2 or 2n, then p ≤ q^{n} + 1, and since dim(V) < 2p, we have

(q^{n−1} + 1)(q^{n−2} − 1) < 2(q^{n} + 1) ⟹ q ≤ q^{n−3} < 2 + 3q^{−n} + q^{−1} − q^{−2}.

Thus (2n, q) = (8, 2). Also, p = Φ_{6}(2) = 3 or p = Φ_{8}(2) = 17, and since 3^{2} | |Ω_{8}^{−}(2)|, we have p = 17. However, upon checking the Brauer character table from [JLPW, p.248], we see that the smallest non-trivial module has dimension 34 = 2p, so this case does not occur.

We now must check the base case of our induction. For G_{0} ≅ Ω_{8}^{−}(q), G_{0} ∈ 𝒢_{p} only when p | Φ_{d}(q) for d = 3, 4, 6, 8. We have already handled d = 3, 6, 8, but we are left with d = 4, for which the centralizer of U might not be abelian. As with the Ω_{7}(q) case, we take two copies of H = SU_{4}(q) ≅ Ω_{6}^{−}(q) inside G_{0} ≅ Ω_{8}^{−}(q) that generate G, and apply Proposition 3.10 to get dim(V) < 4p ≤ 4(q^{2} + 1). Thus the lower bound in Table 10.1 becomes

(q^{3} + 1)(q^{2} − 1) ≤ dim(V) < 4(q^{2} + 1),

which clearly has no solutions, not even for q = 2. This completes the proof. □

11. Exceptional groups

In this section we treat the exceptional groups of Lie type. We maintain the notation of the previous section, so that G is of type 𝔾(q), an exceptional group of Lie type, p ∤ q is a prime dividing G, a Sylow p-subgroup U has order p and is generated by x, and p has order d modulo q, so that p | Φ_{d}(q). In Table 11.1, we list the minimal possible dimensions for V, as determined in [LS] or [SZ].

G Lower bound for dim(V)

Ref.

Exceptions

2B_{2}(q)  (q − 1)\sqrt{q/2}  [LS] l(^{2}B_{2}(8)) = 8
2G_{2}(q)  q(q − 1)  [LS]  —
2F_{4}(q)  q^{4}(q − 1)\sqrt{q/2}  [LS]  —
G_{2}(q)  q(q^{2} − 1)  [SZ] \begin{matrix}l(G_{2}(3)) = 14 \\ l(G_{2}(4)) = 12 \end{matrix}
3D_{4}(q)  q^{3}(q^{2} − 1)  [LS]  —
F_{4}(q) \begin{matrix}q^{6}(q^{2} − 1) & (2 ∤ q) \\ q^{7}(q^{3} − 1)(q − 1) & (2 | q) \end{matrix} [LS] l(F_{4}(2)) ≥ 44
E_{6}^{±}(q)  q^{9}(q^{2} − 1)  [SZ]  —
E_{7}(q)  q^{15}(q^{2} − 1)  [LS]  —
E_{8}(q)  q^{27}(q^{2} − 1)  [LS]  —

Table 11.1. Minimal representation dimensions of exceptional groups

Proposition 11.1. Let G ∈ 𝒢_{p}^{∧} be such that E(G/Z(G)) is an exceptional simple group of Lie type in defining characteristic different from p. Set G_{0} = E(G). Let V be a non-trivial, minimally active 𝔽_{p}G-module. Then one of the following holds: either

(i) G_{0} ≅ ^{2}B_{2}(8), G/Z(G) ≅ ^{2}B_{2}(8):3, p = 13, and dim(V) = 14; or (ii) G_{0} ≅ SL_{2}(8), G/Z(G) ≅ ^{2}G_{2}(3) ≅ SL_{2}(8):3, p = 7, and dim(V) = 7, 8; or

(iii) G_{0} ≅ PSU_{3}(3), G/Z(G) ≅ G_{2}(2) ≅ PSU_{3}(3):2, p = 7, and dim(V) = 6, 7; or

(iv) G_{0} ≅ G_{2}(3), G/Z(G) ≅ G_{2}(3).2, p = 13, and dim(V) = 14.

Proof. In all cases, G_{0}/Z(G_{0}) is one of the exceptional groups listed in Tables 9.2 and 11.1, and unless stated otherwise, p | Φ_{d}(q) for one of the d listed in the first table. Thus U ∈ Syl_{p}(G_{0}) has order p. As usual, V is assumed to be a minimally active 𝔽_{p}G_{0}-module.

Case 1: Assume that G_{0} is a Suzuki or Ree group or G_{0} ≅ ^{3}D_{4}(q). In all of these cases, d is regular by Table 9.2. Aside from ^{2}B_{2}(8), these groups have no exceptional covers [Wi2, § 4.2.4], so C_{G_{0}}(U) is abelian in all other cases, and dim(V) < 2p.

If G_{0}/Z(G_{0}) is a Suzuki group ^{2}B_{2}(q), then p divides one of q − 1, q + \sqrt{2q} + 1 and q − \sqrt{2q} + 1. In particular, p ≤ q + \sqrt{2q} + 1, and since dim(V) ≥ (q − 1)\sqrt{q/2} by Table 11.1,

(q − 1)\sqrt{q/2} ≤ dim(V) < 2p ≤ 2q + 2\sqrt{2q} + 2 ⟹ \sqrt{q} < \sqrt{8}(1 + \frac{1}{q}) + 5/\sqrt{q}.

Hence there are no minimally active modules for q ≥ 32. For q = 8 we can just go through the known character tables [JLPW]. If G_{0} ≅ ^{2}B_{2}(8) (i.e., not an exceptional cover), then of the three primes p = 5, 7, and 13 for which G_{0} ∈ 𝒢_{p}, there is a non-trivial 𝔽_{p}G_{0}-module of dimension less than 2p only for p = 13: the two 14-dimensional simple modules in this case are minimally active and each of them extends to G_{0}:3 ∈ 𝒢_{13}^{∧}. If G_{0} ≅ 2·^{2}B_{2}(8), then Out(G_{0}) = 1 (since Out(^{2}B_{2}(8)) ≅ C_{3} acts faithfully on the Schur multiplier C_{2}^{2}), and G_{0} ∈ 𝒢_{p}^{∧} only for p = 5. In this case, there is an 8-dimensional 𝔽_{5}G_{0}-module, but by Proposition 3.9(a) and since C_{G_{0}}(U) is cyclic of order 10, it cannot be minimally active since (8 − 5) ∤ (p − 1) = 4.

If G_{0} is a small Ree group ^{2}G_{2}(q), then p divides one of q − 1, q + 1, q − \sqrt{3q} + 1 or q + \sqrt{3q} + 1. Together with the lower bound for dim(V) in Table 11.1, this gives

q(q − 1) ≤ dim(V) < 2p ≤ 2q + 2\sqrt{3q} + 2.

For q ≥ 27 we therefore can have no minimally active modules. For q = 3, we have that 2G_{2}(3) ≅ SL_{2}(8):3, and this case was covered in Proposition 10.1.

If G_{0} is a large Ree group ^{2}F_{4}(q), then p divides Φ_{6}(q) or one of the two polynomials into which Φ_{12}(q) splits over ℤ[\sqrt{2}]. The Landazuri–Seitz bound for dim(V) gives

q^{4} ≤ q^{4}(q − 1)\sqrt{q/2} ≤ dim(V) < 2p ≤ 4q^{2} + 4q + 2.

So there are no minimally active modules if q ≥ 8. If q = 2, then G_{0} ∈ 𝒢_{p} only for p = 13, N_{G_{0}}(U)/U is abelian in this case, so dim(V) ≤ p + 1 = 14, while dim(V) ≥ 16 by the above bound.

If G is a triality group G = ^{3}D_{4}(q), then by Table 9.2, p divides Φ_{12}(q) = q^{4} − q^{2} + 1, and together with the bound in Table 11.1, this implies that

q^{3}(q^{2} − 1) ≤ dim(V) < 2p ≤ 2q^{4} − 2q^{2} + 2,

and hence q = 2. But in this case, p = Φ_{12}(2) = 13, and the smallest non-trivial module has dimension 26 by [JLPW]. So there are no minimally active modules in any of these cases.

Case 2: We next consider the small exceptional groups G_{2}(q) and F_{4}(q). Again for these groups, by Table 9.2 and Proposition 9.1, C_{G_{0}}(U) is abelian for all primes p such that G_{0} ∈ 𝒢_{p} (unless possibly G_{0} is one of the exceptional covers 3·G_{2}(3), 2·G_{2}(4), or 2·F_{4}(2)), and hence dim(V) < 2p.

If G_{0} = G_{2}(q), then p | Φ_{3}(q) or p | Φ_{6}(q), so in particular p ≤ q^{2} + q + 1. Together with the Seitz–Zalesskii bound in Table 11.1, this gives

q(q^{2} − 1) ≤ dim(V) < 2p ≤ 2q^{2} + 2q + 2,

for q ≥ 5, yielding no solutions. We have already dealt with the case G_{2}(2) ≅ PSU_{3}(3):2.

If G_{0}/Z(G_{0}) ≅ G_{2}(3), then G_{0} ∈ 𝒢_{p} implies p = 7 or 13, and C_{G_{0}}(U) is abelian since the Sylow 7- and 13-subgroups of G_{2}(3) are self-centralizing. There are no non-trivial 𝔽_{7}G_{0}- modules of degree less than 14, while there are 14-dimensional simple modules for G_{2}(3):2 and p = 13 (see [JLPW]), and they are minimally active.

If G_{0}/Z(G_{0}) ≅ G_{2}(4), then G_{0} ∈ 𝒢_{p} implies p = 7 or 13, and C_{G_{0}}(U) is abelian since the centralizers of the Sylow 7- and 13-subgroups of G_{2}(4) are cyclic of order 21 or 13, respectively [Atlas]. Thus dim(V) < 2p, and by [JLPW], there are 12-dimensional modules for 2·G_{2}(4) over 𝔽_{7} and over 𝔽_{13} to be considered. When p = 7, this module cannot be minimally active by Proposition 3.9(a) and since (12−7) ∤ (p−1) = 6. When p = 13, we have |Aut_{G_{0}}(U)| = 6, so we need to extend to G = 2·G_{2}(4).2. However, from the table on [JLPW, p.277], we see that there are irrationalities in the Brauer character of this representation, and by [JLPW, p.291], it is defined only over 𝔽_{13^{2}}.

For G_{0} = F_{4}(q), the Landazuri–Seitz bounds in Table 11.1 gives the inequalities

q^{6} ≤ dim(V) < 2p ≤ 2· max{Φ_{8}(q), Φ_{12}(q)} = 2q^{4} + 2

when q > 2, and 44 < 2·2^{4} + 2 = 34 when q = 2. Since these have no solutions, G_{0} has no minimally active modules.

Case 3: Assume that G_{0} = E_{6}^{ε}(q) (the universal or adjoint group). If p | Φ_{d}(εq) for d = 8, 9, 12, then C_{G_{0}}(U) is abelian (see Table 9.2), so dim(V) < 2p, while dim(V) ≥ q^{9}(q^{2} − 1) by Table 11.1. Thus

q^{9} ≤ q^{9}(q^{2} − 1) < 2Φ_{d}(εq) ≤ 2Φ_{9}(q) = q^{6} + q^{3} + 1,

which is impossible. This leaves p | Φ_{5}(εq), in which case U ≤ H < G for H of type PSL_{6}^{ε}(q). We already saw in the proofs of Propositions 10.1 (Case 2) and 10.2 (Case 1) that SL_{6}^{ε}(q) has no minimally active projective representations over 𝔽_{p} when p | Φ_{5}(εq), and so G_{0} has no minimally active modules.

If G_{0} is an exceptional cover 2·^{2}E_{6}(2) [Wi2, § 4.11], then G_{0} ∈ 𝒢_{p} only for p = 11, 13, 17, 19. In the last three cases, the Sylow p-subgroup of ^{2}E_{6}(2) is self-centralizing, so C_{G_{0}}(U) is abelian, dim(V) < 2p, and the above argument applies. If p = 11 = Φ_{5}(−2), then the above comparison with U_{6}(2) again applies.

Assume that G_{0} = E_{7}(q) and p | Φ_{d}(q). If d = 7, 9, 14, 18, then C_{G_{0}}(U) is abelian (see Table 9.2), so dim(V) < 2p, while dim(V) ≥ q^{15}(q^{2} − 1) > q^{15} by Table 11.1. Thus q^{15} < 2Φ_{d}(q) ≤ 2Φ_{7}(q) ≤ q^{7} − 1, which is impossible. If d = 5, 8, 10, 12, then U ≤ H < G_{0} for H ≅ E_{6}^{±}(q), and we just showed that these groups have no minimally active modules over 𝔽_{p}. So G_{0} has no minimally active modules.

Finally, assume that G_{0} = E_{8}(q) and p | Φ_{d}(q). If d = 15, 20, 24, 30, then C_{G_{0}}(U) is abelian (see Table 9.2), so dim(V) < 2p, while dim(V) ≥ q^{27}(q^{2} − 1) by Table 11.1. Since Φ_{d} has degree 8 in each of these cases, one easily sees that we cannot have dim(V) < 2p. If d = 7, 9, 14, 18, then U ≤ H < G_{0} for H ≅ E_{7}(q), and we just showed that these groups have no minimally active modules over 𝔽_{p}. So again in this case, G_{0} has no minimally active modules. □

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

  E-mail address: d.a.craven@bham.ac.uk

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

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

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