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

BOB OLIVER AND ALBERT RUIZ

Abstract. We finish the classification, begun in two earlier papers, of all simple fusion systems over finite nonabelian p-groups with an abelian subgroup of index p. In particular, this gives many new examples illustrating the enormous variety of exotic examples that can arise. In addition, we classify all simple fusion systems over infinite nonabelian discrete p-toral groups with an abelian subgroup of index p. In all of these cases (finite or infinite), we reduce the problem to one of listing all 𝔽_{p}G-modules (for G finite) satisfying certain conditions: a problem which was solved in the earlier paper [CrOS] using the classification of finite simple groups.

A saturated fusion system over a finite p-group S is a category whose objects are the subgroups of S, and whose morphisms are injective homomorphisms between the subgroups, and which satisfy some additional conditions first formulated by Puig (who called them “Frobenius S-categories” in [Pg]) and motivated in part by the Sylow theorems for finite groups. For example, if G is a finite group and S ∈ Syl_{p}(G), then the category F_{S}(G), whose objects are the subgroups of S and whose morphisms are the homomorphisms between subgroups defined via conjugation in G, is a saturated fusion system over S. We refer to [Pg], [AKO, Part I], or [Cr] for the basic definitions and properties of saturated fusion systems.

A saturated fusion system is realizable if it is isomorphic to F_{S}(G) for some finite group G and some S ∈ Syl_{p}(G); it is exotic otherwise. Here, by an isomorphism of fusion systems we mean an isomorphism of categories that is induced by an isomorphism between the underlying p-groups. Exotic fusion systems over finite p-groups seem to be quite rare for p = 2 (the only known examples are those constructed in [LO] and others easily derived from them), but many examples of them are known for odd primes p.

A discrete p-torus is a group of the form (ℤ/p^{∞})^{r} for some r ≥ 0, where ℤ/p^{∞} is the union of the cyclic groups ℤ/p^{k} via the obvious inclusions ℤ/p^{k} < ℤ/p^{k+1}. A discrete p-toral group is a group containing a discrete p-torus as a normal subgroup of p-power index. Saturated fusion systems over discrete p-toral groups were defined and studied in [BLO3], motivated by the special case of fusion systems for compact Lie groups and p-compact groups.

A fusion system is simple if it is saturated and contains no proper nontrivial normal fusion subsystems (see Definition 1.4). As a special case, very rich in exotic examples, we have been looking at simple fusion systems F over finite nonabelian p-groups S with an abelian subgroup A of index p. By [AOV2, Proposition 5.2(a)], if p = 2, then S is dihedral, semidihedral, or a wreath product of the form C_{2^{k}} ≀ C_{2}, and hence F is isomorphic to the fusion system of PSL_{2}(q) or PSL_{3}(q) for some odd q. Fusion systems over extraspecial groups of order p^{3} and exponent p were listed in [RV], and by [Ol, Theorem 2.1], these include the only simple fusion systems over nonabelian p-groups containing more than one abelian subgroup of index p. The other cases where p is odd and A is not essential (equivalently, not radical) in F were handled in [Ol, Theorem 2.8], while those where A is essential and of exponent p was handled in [CrOS]. So it remains to describe those cases where A is essential and not elementary abelian (and the unique abelian subgroup of index p). This, together with analogous results about simple fusion systems over infinite discrete p-toral groups with abelian subgroup of index p, are the main results of this paper.

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

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

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

A. Ruiz is partially supported by MICINN-FEDER project number MTM2016-80439-P.

To simplify the following summary of our results, we use the term “index-p-triple” to denote a triple (F, S, A), where S is a nonabelian discrete p-toral group (finite or infinite) with abelian subgroup A of index p, and F is a simple fusion system over S. Our main results are shown in Sections 4 and 5, where we handle separately the finite and infinite cases. In each of these sections, we first list, in Theorems 4.5 and 5.11, all index-p-triples (F, S, A), for S finite or infinite, in terms of the pair (G, A) where G = Aut_{F}(A) and A is regarded as a ℤ_{p}G-module. Theorem 4.5 is taken directly from [CrOS, Theorem 2.8], while Theorem 5.11 is new. For completeness in the infinite case, we also show that each index-2-triple (F, S, A) with |S| = ∞ is isomorphic to that of SO(3) or PSU(3) (Theorem 5.6), and that for each p there is (up to isomorphism) a unique index-p-triple (F, S, A) where |S| = ∞ and A is not essential (Theorem 5.12).

The main theorems, Theorems A and B, appear at the ends of Sections 4 and 5, respectively. In Theorem A, for p odd, we prove that each index-p-triple (F, S, A), where A is finite, essential in F, and not elementary abelian, is determined by G = Aut_{F}(A), V = Ω_{1}(A) regarded as an 𝔽_{p}G-module, the exponent of A, and some additional information needed when A is not homocyclic. In all cases, rk(A) ≥ p − 1, and A is homocyclic whenever rk(A) ≥ p. Also, A is always isomorphic to some quotient of a ℤ_{p}G-lattice.

Theorem B can be thought of as a “limiting case” of the classification in Theorem A. It says that each index-p-triple (F, S, A) such that A is infinite and essential in F is determined by the pair (G, V), where G = Aut_{F}(A), and V = Ω_{1}(A) is regarded as an 𝔽_{p}G-module. In all such cases, A is a discrete p-torus of rank at least p − 1. We also determine which of the fusion systems we list are realized as fusion systems of compact Lie groups or p-compact groups.

Theorems A and B reduce our classification problems to questions about 𝔽_{p}G-modules with certain properties. These questions were already studied in [CrOS], using the classification of finite simple groups, and the results in that paper that are relevant in this one are summarized in Section 6. Theorems A and B together with Proposition 6.1 and Table 6.1 allow us to completely list all simple fusion systems over nonabelian discrete p-toral groups (finite or infinite) with abelian subgroup of index p that is not elementary abelian. In particular, as in the earlier papers [Ol] and [CrOS], we find a very large, very rich variety of exotic fusion systems over finite p-groups (at least for p ≥ 5).

This work was motivated in part by the following questions and problems, all of which are familiar to people working in this field.

Q1: For a fixed odd prime p, a complete classification of all simple fusion systems over finite p-groups, or even a conjecture as to how they could be classified, seems way out of reach for now. But based on the many examples already known, is there any meaningful way in which one could begin to systematize them; for example, by splitting up the problem into simpler cases? Alternatively, is there a class of simple fusion systems over finite p-groups, much less restrictive than the one we look at here, for which there might be some chance of classifying its members?

Q2: Find some criterion which can be used to prove that some (or at least one!) of the examples constructed here or earlier (over finite p-groups for odd primes p) are exotic, without invoking the classification of finite simple groups.

Q3: A torsion linear group in defining characteristic q is a subgroup Γ ≤ GL_{n}(K), for some n ≥ 1 and some field K of characteristic q, such that all elements of Γ have finite order. If p is a prime and Γ is a torsion linear group in defining characteristic different from p, then by [BLO3, § 8], there is a maximal discrete p-toral subgroup S ≤ Γ, unique up to conjugation, and F_{S}(Γ) is a saturated fusion system. Are there any saturated fusion systems over discrete p-toral groups (for any prime p) which we can prove are not fusion systems of torsion linear groups?

The notation used in this paper is mostly standard. We let A ∘ B denote a central product of A and B. When g and h are in a group G, we set ^{g}h = ghg^{−1} and h^{g} = g^{−1}hg. When A is an abelian group and β ∈ Aut(A), we write [β, A] = ⟨β(x)x^{−1} | x ∈ A⟩. When P is a p-group, we let Fr(P) denote its Frattini subgroup, and for k ≥ 1 set

Ω_{k}(P) = ⟨g ∈ P | g^{p^{k}} = 1⟩ and ℧^{k}(P) = ⟨g^{p^{k}} | g ∈ P ⟩.

We would like to thank the Centre for Symmetry and Deformation at Copenhagen University, and the Universitat Autònoma de Barcelona, for their hospitality in allowing us to get together on several different occasions.

1. Background

We first recall some of the definitions and standard terminology used when working with fusion systems. Recall that a discrete p-toral group is a group that contains a normal subgroup of p-power index isomorphic to (ℤ/p^{∞})^{r} for some r ≥ 0. A fusion system over a discrete p-toral group S is a category F whose objects are the subgroups of S, and where for each P, Q ≤ S, the set Hom_{F}(P, Q) is a set of injective homomorphisms from P to Q that includes all those induced by conjugation in S, and such that for each φ ∈ Hom_{F}(P, Q), we have φ ∈ Hom_{F}(P, φ(P)) and φ^{−1} ∈ Hom_{F}(φ(P), P).

Define the rank rk(S) of a discrete p-torus S by setting rk(S) = r if S ≅ (ℤ/p^{∞})^{r}. If S is a discrete p-toral group with normal discrete p-torus S_{0} ⊴ S of p-power index, then we refer to S_{0} as the identity component of S, and set |S| = (rk(S_{0}), |S/S_{0}|), where such pairs are ordered lexicographically. Thus if T is another discrete p-toral group with identity component T_{0}, then |S| ≤ |T | if rk(S_{0}) < rk(T_{0}), or if rk(S_{0}) = rk(T_{0}) and |S/S_{0}| ≤ |T/T_{0}|. Note that the identity component of S, and hence |S|, are uniquely determined since a discrete p-torus has no proper subgroups of finite index.

Definition 1.1. Fix a prime p, a discrete p-toral group S, and a fusion system F over S.

• For each P ≤ S and each g ∈ S, P^{F} denotes the set of subgroups of S which are Fconjugate (isomorphic in F) to P, and g^{F} denotes the F-conjugacy class of g (the set of images of g under morphisms in F).

• A subgroup P ≤ S 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}.

• A subgroup P ≤ S is fully automized in F if Out_{S}(P) ∈ Syl_{p}(Out_{F}(P)), and is receptive in F if for each Q ∈ P^{F} and each φ ∈ Iso_{F}(Q, P), there is φ ∈ Hom_{F}(N_{φ}, S) such that φ|_{P} = φ, where

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

• The fusion system F is saturated if

– (Sylow axiom) each fully normalized subgroup of S is fully automized and fully centralized;

– (extension axiom) each fully centralized subgroup of S is receptive; and

– (continuity axiom, when |S| = ∞) if P_{1} ≤ P_{2} ≤ P_{3} ≤ ⋯ is an increasing sequence of subgroups of S with P = ⋃_{i=1}^{∞} P_{i}, and φ ∈ Hom(P, S) is such that φ|_{P_{i}} ∈ Hom_{F}(P_{i}, S) for each i ≥ 1, then φ ∈ Hom_{F}(P, S).

The above definition of a saturated fusion system is the one given in [BLO2] and [BLO3, Definition 2.2]. It will not be used directly in this paper (saturation of the fusion systems we construct will be shown using later theorems), but we will frequently refer to the extension axiom as a property of saturated fusion systems.

We now need some additional definitions, to describe certain subgroups in a saturated fusion system. Definition 1.2. Fix a prime p, a discrete p-toral group S, and a saturated fusion system F over S. Let P ≤ S be any subgroup. Recall [BLO3, Proposition 2.3] that Out_{F}(P) = Aut_{F}(P)/Inn(P) is finite.

• P is F-centric if C_{S}(Q) = Z(Q) for each Q ∈ P^{F}, and is F-radical if O_{p}(Out_{F}(P)) = 1.

• 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 (P ⊴ 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. Proposition 1.3. Let F be a saturated fusion system over a discrete p-toral group S.

(a) Each morphism in F is a composite of restrictions of elements in Aut_{F}(P) for P ≤ S that is fully normalized in F, F-centric and F-radical.

(b) Each morphism in F is a composite of restrictions of elements in Aut_{F}(P) for P ∈ E_{F} ∪ {S}.

(c) 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.

Proof. Point (a) is shown in [BLO3, Theorem 3.6].

By [BLO3, Proposition 2.3], Out_{F}(P) = Aut_{F}(P)/Inn(P) is always finite. For each such P < S that is not F-essential, Aut_{F}(P) is generated by automorphisms that can be extended to strictly larger subgroups: this is shown in [AKO, Proposition I.3.3] in the finite case, and the same argument applies when S is infinite. Point (b) now follows from (a) and induction, and (in the infinite case) since there are only finitely many S-conjugacy classes of subgroups of S that are F-centric and F-radical [BLO3, Corollary 3.5].

Point (c) follows easily from (b), just as in the finite case [AKO, Proposition I.4.5]. □ Definition 1.4. Let F be a saturated fusion system over a discrete p-toral group S. A saturated fusion subsystem E over T ≤ S is normal in F (E ⊴ F) if • T is strongly closed in F (in particular, T ⊴ S);

• (invariance condition) each α ∈ Aut_{F}(T) is fusion preserving in the sense that it extends to an automorphism of E;

• (Frattini condition) for each P ≤ T and each φ ∈ Hom_{F}(P, T), there are α ∈ Aut_{F}(T) and φ_{0} ∈ Hom_{E}(P, T) such that φ = α ∘ φ_{0}; and

• (extension condition) each α ∈ Aut_{E}(T) extends to some α ∈ Aut_{F}(T C_{S}(T)) such that [α, C_{S}(T)] ≤ Z(T).

The fusion system F is simple if it contains no proper nontrivial normal subsystems.

For more discussion of the definition and properties of normal fusion subsystems over finite p-groups, we refer to [AKO, § I.6] or [Cr, §§ 5.4 & 8.1]. Note in particular the alternative definition used in [Cr]: a saturated fusion subsystem E ≤ F over a subgroup T that is strongly closed in F is normal if the extension condition holds, and also the strong invariance condition: for each P ≤ Q ≤ T, and each φ ∈ Hom_{E}(P, Q) and ψ ∈ Hom_{F}(Q, T), ψ ∘ φ ∘ (ψ|_{P})^{−1} ∈ Hom_{E}(ψ(P), T). When S is finite, this is equivalent to the above definition by [AKO, Proposition I.6.4], and a similar argument (made more complicated because there are infinitely many subgroups) applies when S and T are discrete p-toral. Since the Frattini condition will be important in Section 5, we work here with the above definition.

Proposition 1.5. Fix a prime p, and let F be a saturated fusion system over an infinite discrete p-toral group S. Let S_{0} be the identity component of S, and assume that each element of S is F-conjugate to an element of S_{0}.

(a) If F is realized by a compact Lie group G with identity connected component G_{0}, then G/G_{0} has order prime to p. If in addition, F is simple, then F is realized by the connected, simple group G_{0}/Z(G_{0}), where Z(G_{0}) is finite of order prime to p.

(b) If F is realized by a p-compact group X, then X is connected. If in addition, F is simple, then so is X.

In either case, if F is simple, then the action of the Weyl group Aut_{F}(S_{0}) on the ℚ_{p}-vector space ℚ ⊗_{ℤ} Hom(S_{0}, ℚ_{p}/ℤ_{p}) is irreducible and generated by pseudoreflections.

Proof. If F = F_{S}(G) where G is a compact Lie group with identity connected component G_{0} and maximal discrete p-toral subgroup S, then S ∩ G_{0} is strongly closed in F since G_{0} ⊴ G, S_{0} ≤ S ∩ G_{0}, and so S ≤ G_{0}. Hence G/G_{0} has order prime to p. Also, F_{S}(G_{0}) ⊴ F_{S}(G): the invariance and extension conditions are easily checked, and the Frattini condition holds since G = G_{0}N_{G}(S) by the Frattini argument.

If in addition, F is simple, then F_{S}(G_{0}) = F_{S}(G) = F, and Z(G_{0}) is finite of order prime to p since Z(F) = 1. Hence F is also realized by G_{0}/Z(G_{0}), which is simple.

If F ≅ F_{S}(X) for some p-compact group X with S ∈ Syl_{p}(X), then X is connected by [GLR, Proposition 4.9(a)]. Then X is a central product of connected, simple p-compact groups, and hence is simple if F is simple.

Whenever F is realized by a connected p-compact group X (possibly a compact connected Lie group), then by [DW, Theorem 9.7(ii)], the action of the Weyl group Aut_{F}(S_{0}) on ℚ ⊗_{ℤ} H^{2}(BS_{0}; ℤ_{p}) is generated by pseudoreflections, where

H^{2}(BS_{0}; ℤ_{p}) ≅ H^{2}(S_{0}; ℤ_{p}) ≅ H^{1}(S_{0}; ℚ_{p}/ℤ_{p}) ≅ Hom(S_{0}, ℚ_{p}/ℤ_{p}).

If this is not irreducible as a group generated by pseudoreflections, then by the classification of connected p-compact groups in [AGMV, Theorem 1.2] (for p odd) and in [AG, Theorem 1.1] or [Mø, Corollary 1.2] (for p = 2), X must be a nontrivial central product of simple factors, and hence F is not simple. □

We also recall the definition of a reduced fusion system, but only for fusion systems over finite p-groups. Recall [AKO, § I.7] that in this setting, O^{p}(F) and O^{p′}(F) are the smallest (normal) fusion systems in F of p-power index and of index prime to p, respectively.

Definition 1.6. A saturated fusion system F over a finite p-group S is reduced if O_{p}(F) = 1, and O^{p}(F) = F = O^{p′}(F).

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

To avoid the question of whether O^{p}(F) is defined when F is a saturated fusion system over an infinite discrete p-toral group, we will not consider reduced fusion systems in this setting.

2. Reduced or simple fusion systems over nonabelian discrete p-toral groups with index p abelian subgroup

In this section, p is an arbitrary prime. We want to study simple fusion systems over nonabelian discrete p-toral groups (possibly finite) which contain an abelian subgroup of index p. Most of the results here were shown in [CrOS], but only in the case where |A| < ∞ and p is odd.

We first fix some notation which will be used throughout the rest of the paper. As usual, for a group S, we define Z_{m}(S) for all m ≥ 1 by setting Z_{1}(S) = Z(S), and setting Z_{m}(S)/Z_{m−1}(S) = Z(S/Z_{m−1}(S)) for m ≥ 2.

Notation 2.1. Fix a nonabelian discrete p-toral group S with abelian subgroup A of index p, and a saturated fusion system F over S. Define

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

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

H = {Z⟨x⟩ | x ∈ S∖A}  G = Aut_{F}(A)
B = {Z_{2}⟨x⟩ | x ∈ S∖A}  U = Aut_{S}(A) ∈ Syl_{p}(G) .

Lemma 2.2. Assume Notation 2.1. Then E_{F} ⊆ {A} ∪ H ∪ B, and |N_{S}(P)/P | = p for each P ∈ E_{F}. If E_{F} ⊈ {A}, then Z_{2} ≤ A and |Z_{2}/Z| = p.

Proof. Fix some P ∈ E_{F} ∖ {A}. Then P ≰ A since P is F-centric. Set P_{0} = P ∩ A, and fix some element x ∈ P ∖P_{0}. Since Out_{F}(P) contains a strongly p-embedded subgroup, O_{p}(Out_{F}(P)) = 1 (cf. [AKO, Proposition A.7(c)]).

We must show that P ∈ H ∪ B, |N_{S}(P)/P | = p, Z_{2} ≤ A, and |Z_{2}/Z| = p. (Clearly, |N_{S}(P)/P | = p if P = A.) Case 1: Assume P is nonabelian. Since Z ≤ P (P is F-centric), Z(P) = C_{P_{0}}(x) = Z. For each g ∈ N_{A}(P)∖P, c_{g} is the identity on P_{0} and on P/P_{0}. If P_{0} is characteristic in P, then c_{g} ∈ O_{p}(Aut_{F}(P)) by Lemma A.1, which is impossible since O_{p}(Out_{F}(P)) = 1. Thus P_{0} is not characteristic in P, and hence is not the unique abelian subgroup of index p in P. So by Lemma A.3, |P_{0}/Z| = p and |[P, P]| = |[x, P_{0}]| = p. Also, P/Z is abelian since |P/Z| = p^{2}, so [x, P_{0}] ≤ Z, and hence P_{0} ≤ Z_{2}. Note that P_{0} > Z, since P is nonabelian.

If P_{0} < Z_{2}, then for y ∈ Z_{2} ∖ P, [y, P] ≤ Z = Z(P), so y ∈ N_{S}(P) ∖ P and c_{y} ∈ O_{p}(Aut(P)), contradicting the assumption that P is F-essential. Thus P_{0} = Z_{2}, so P ∈ B, Z_{2} ≤ A, and |Z_{2}/Z| = p. Finally, |N_{S}(P)/P | = p by Lemma A.6, applied with Out_{F}(P) in the role of G, Out_{S}(P) ≅ N_{S}(P)/P in the role of S, and P/Z_{0} (if P ∈ B) in the role of A. Note that [P, P] = [x, Z_{2}] ≤ Z_{0} and C_{P/Z_{0}}(N_{S}(P)) = Z_{2}/Z_{0}. Case 2: If P ∈ E_{F} is abelian, then P_{0} = Z: it contains Z since P is centric, and cannot be larger since then P would be nonabelian. Hence P = Z⟨x⟩ ∈ H. Also, Aut_{A}(P) = Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)) centralizes P_{0}. The conditions of Lemma A.6 thus hold (with P and Aut_{F}(P) in the roles of A and G), so |N_{S}(P)/P | = |Aut_{S}(P)| = p. Since [S:P] > p (A is the unique abelian subgroup of index p), this implies that S/Z is nonabelian, so [x, A] ≰ Z, and Z_{2} ≤ A.

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

Lemma 2.3. Let S, F, etc. be as in Notation 2.1. If, for some x ∈ S ∖ A, Z_{2}⟨x⟩ ∈ E_{F}, then Z⟨x⟩ is not F-centric, and hence Z⟨x⟩ ∉ E_{F}.

Proof. Assume x ∈ S ∖ A and Z_{2}⟨x⟩ ∈ E_{F}, and set P = Z_{2}⟨x⟩ ∈ B. Then |Z_{2}/Z| = p by Lemma 2.2, so |P/Z| = p^{2}, and Z_{2} is not normalized by Aut_{F}(P) since P is essential.

Let P be the set of all subgroups of index p in P which contain Z. Then P ⫌ {Z_{2}}, so P/Z ≅ C_{p}^{2} (i.e., is not cyclic), and Aut_{S}(P) permutes transitively the p members of P ∖ {Z_{2}}. So Aut_{F}(P) must act transitively on P, hence Z⟨x⟩ is F-conjugate to Z_{2}, and is not F-centric. So Z⟨x⟩ ∉ E_{F} in this case. □

Lemma 2.4. Assume Notation 2.1, and also A ⋬ F ( ⟺ E_{F} ⊈ {A}). Then |Z_{0}| = p, and |Z_{i}(S)/Z_{i−1}(S)| = p for all i > 1 such that Z_{i}(S) < S. If |A| < ∞, then |A/ZS′| = p.

Proof. Fix x ∈ S∖A. Let ψ ∈ End(A) be the homomorphism ψ(g) = [g, x]. Thus Ker(ψ) = Z and Im(ψ) = S′.

By Lemma 2.2 and since E_{F} ⊈ {A}, (H ∪ B) ∩ E_{F} ≠ ∅, Z_{2} ≤ A, and |Z_{2}/Z| = p. Since Z_{2}/Z = C_{A/Z}(x), Z_{2} = ψ^{−1}(Z), and so ψ sends Z_{2} onto Z_{0} = Z ∩ S′ with kernel Z. Thus |Z_{0}| = |Z_{2}/Z| = p.

Set Z_{i} = Z_{i}(S) for each i ≥ 0, and let k > 2 be the smallest index such that Z_{k} = S. Thus S/Z_{k−2} is nonabelian, so Z_{k−2} ≤ A, and Z_{k−1}/Z_{k−2} = Z(S/Z_{k−2}) ≤ A/Z_{k−2}. Hence Z_{i} ≤ A, and Z_{i} = ψ^{−1}(Z_{i−1}), for all i < k. In particular, ψ induces a monomorphism from Z_{i}/Z_{i−1} into Z_{i−1}/Z_{i−2} for each 3 ≤ i < k, so |Z_{i}/Z_{i−1}| ≤ p, with equality since Z_{i}(S) > Z_{i−1}(S) whenever Z_{i−1}(S) < S.

If |A| < ∞, then |ZS′| = |Z|·|S′|/|Z_{0}| = |A|/|Z_{0}|, and hence ZS′ has index p in A. □

Lemma 2.5. Let A ⊴ S, F, H, B, etc., be as in Notation 2.1. Assume P ∈ E_{F} where P ∈ H ∪ B, and set X = 1 if P ∈ H and X = Z_{0} if P ∈ B. Define P_{1}, P_{2} ≤ P by setting

P_{1}/X = C_{P/X}(O^{p′}(Aut_{F}(P))) and P_{2} = [O^{p′}(Aut_{F}(P)), P].

Then O^{p′}(Out_{F}(P)) ≅ SL_{2}(p), and the following hold.

(a) If P ∈ H, then P_{1} < Z, Z = P_{1} × Z_{0}, Z_{0} < P_{2} ≅ C_{p}^{2}, and P = P_{1} × P_{2}. If p is odd, then P_{1} is the unique Aut_{F}(Z)-invariant subgroup of Z such that Z = P_{1} × Z_{0}.

(b) If P ∈ B, then P_{1} = Z, P_{2} is extraspecial of order p^{3}, P_{2} ≅ Q_{8} if p = 2 while P_{2} has exponent p if p is odd, and P_{1} ∩ P_{2} = Z(P_{2}) = Z_{0} = [P, P]. Thus P = P_{1} ×_{Z_{0}} P_{2}.

Proof. To simplify notation, set H = Out_{F}(P), H_{0} = O^{p′}(H), and T = Out_{S}(P) ∈ Syl_{p}(H).

If P ∈ B, then [P, P] ≤ Z(P) ∩ S′ = Z ∩ S′ = Z_{0}, with equality since |Z_{0}| = p by Lemma 2.4. Thus P/X is abelian in both cases. Also, [N_{S}(P), P/X] = Z_{0} (if P ∈ H) or Z_{2}/Z_{0} (if P ∈ B), and thus has order p in both cases. So by Proposition A.7, applied to the H-action on P/X, we have H_{0} ≅ SL_{2}(p), P/X = (P_{1}/X) × (P_{2}/X), and P_{2}/X ≅ C_{p}^{2}.

If P ∈ H (so X = 1), then P_{1} = C_{P} (H_{0}) ≤ C_{P} (T) = Z, and [Z:P_{1}] = p since [P:P_{1}] = p^{2}. Also, P_{2} ≥ [T, P] = Z_{0}, so P_{1} ∩ Z_{0} = 1, and Z = P_{1} × Z_{0}. If p is odd, then N_{H_{0}}(T) is a semidirect product of the form C_{p} ⋊ C_{p−1}. Fix α ∈ N_{H_{0}}(T) of order p − 1; then α acts on Z_{0} = [T, P] with order p − 1 and acts trivially on P_{1}. Thus α|_{Z} ∈ Aut_{F}(Z), and P_{1} is the only subgroup which is a complement to Z_{0} in Z and could be normalized by Aut_{F}(Z). Since Aut_{F}(Z) has order prime to p, there is at least one such subgroup, and hence P_{1} is Aut_{F}(Z)-invariant.

If P ∈ B, then X = Z_{0}, and P_{2}/Z_{0} ≅ C_{p}^{2}. Also, H_{0} ≅ SL_{2}(p) acts faithfully on P_{2}, and this is possible only if Z(P_{2}) = Z_{0}, and P_{2} ≅ Q_{8} (if p = 2) or P_{2} is extraspecial of exponent p (if p is odd). Also, P_{1} has index p^{2} in P since P_{1} ∩ P_{2} = Z_{0}, P_{1} ≤ Z(P) since P = P_{1} ×_{Z_{0}} P_{2} is a central product, and hence P_{1} = Z(P) = Z. □

Corollary 2.6. In the situation of Notation 2.1, if A ⋬ F (i.e., if E_{F} ⊈ {A}) and p is odd, then S splits over A: there is x ∈ S ∖ A of order p.

Proof. Fix P ∈ E_{F} ∖ {A}. By Lemma 2.2, P ∈ H ∪ B. In either case, by Lemma 2.5, there is x ∈ P ∖ A of order p. □

We now restrict to the case where p is odd.

Lemma 2.7. Assume Notation 2.1, and also that p is odd and A ⋬ F. Then O_{p}(F) = 1 if and only if either there are no nontrivial G-invariant subgroups of Z, or E_{F} ∩ H ≠ ∅ and Z_{0} is the only G-invariant subgroup of Z.

Proof. The following proof is essentially the same as the proof in [CrOS, Lemma 2.7(a)] in the finite case.

Assume first that Q =^{def} O_{p}(F) ≠ 1. Since A ⋬ F, there is P ∈ E_{F} ∖ {A} ⊆ B ∪ H. If P ∈ H, then Q ≤ Z: the intersection of the subgroups S-conjugate to P. If P ∈ B, then Q ≤ Z_{2} by a similar argument, and then Q ≤ Z since that is the intersection of the subgroups in the Aut_{F}(P)-orbit of Z_{2}. 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). This proves one implication.

Conversely, assume that 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, and so O_{p}(F) ≠ 1.

Now assume that E_{F} ∩ H ≠ ∅, and also that Q ≠ Z_{0}. By [CrOS, Lemma 2.3(b)] (the argument easily extends to the infinite case), there is a unique Aut_{F}(Z)-invariant factorization Z = Z_{0} × Z̃. Set Q̃ = Q ∩ Z̃. If Q ≥ Z_{0}, then Q = Q̃ × 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 ≤ Z̃ and hence Q = Q̃. Since Q ≠ Z_{0}, we have Q̃ ≠ 1 in either case.

For each φ ∈ Aut_{F}(A) = G, φ(Q̃) ≤ Q ≤ Z, so by the extension axiom, φ|Q̃ extends to some φ ∈ Aut_{F}(S), and φ(Q̃) = φ(Q̃) = Q̃ since Q̃ is Aut_{F}(Z)-invariant. So by the same arguments as those applied above to Q, Q̃ 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(Z̃) = Z̃ by [CrOS, Lemma 2.3(b)], so α|Z̃ extends to an element of Aut_{F}(S) and hence of Aut_{F}(Z), and in particular, α(Q̃) = Q̃. Thus 1 ≠ Q̃ ⊴ F, and hence O_{p}(F) ≠ 1. □

Without the assumption that p be odd in Lemma 2.7, the 2-fusion system F of PΣL_{2}(q^{2}) is a counterexample for each prime power q ≡ ±1 (mod 8). Here, PΣL_{2}(q^{2}) = PSL_{2}(q^{2})⟨θ⟩ where θ acts on PSL_{2}(q^{2}) as a field automorphism of order 2 (and θ^{2} = 1). Then O_{2}(F) = 1, S ≅ D_{2^{m}} × C_{2} for some m ≥ 4 depending on q, A ≅ C_{2^{m−1}} × C_{2}, Z = Z(S) = Ω_{1}(A), and G = Aut_{S}(A) acts trivially on Z (so that all subgroups of Z are G-invariant).

Lemma 2.8. Assume Notation 2.1, and also that p is odd and O_{p}(F) = 1. Let A_{2} ≤ A_{1} ≤ A be G-invariant subgroups such that A_{1} ≤ ZA_{2}. Then either A_{1} = A_{2}, or A_{1} = Z_{0} × A_{2} and Z_{0} is G-invariant.

Proof. Fix a class xA_{2} ∈ A_{1}/A_{2}. By assumption, we can assume x ∈ Z. Since G acts on A_{1}/A_{2} and U acts trivially on this quotient, G_{0} = O^{p′}(G) also acts trivially. Hence α(x) ∈ xA_{2} for each α ∈ G_{0}. Let α_{1}, …, α_{k} ∈ G_{0} be left coset representatives for U (so

p ∤ k = [G_{0}:U]), and set y = (∏_{i=1}^{k} α_{i}(x))^{1/k}. Then y ∈ xA_{2} since α_{i}(x) ∈ xA_{2} for each i, and y ∈ C_{A}(G_{0}). This shows that A_{1} ≤ C_{A}(G_{0})A_{2}.

Now, C_{A}(G_{0}) is a subgroup of Z = C_{A}(U) normalized by G. So by Lemma 2.7 and since O_{p}(F) = 1, C_{A}(G_{0}) ≤ Z_{0}. Thus A_{1} ≤ Z_{0}A_{2}. If A_{1} > A_{2}, then A_{1} = A_{2} × Z_{0} since |Z_{0}| = p, and Z_{0} = C_{A}(G_{0}) is G-invariant. □

The following notation, taken from [CrOS, Notation 2.4], will be used throughout the rest of the paper.

Notation 2.9. Assume Notation 2.1, and also that |Z_{0}| = p. Set

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

Set

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

Define

μ: Aut^{∨}(S) ⟶ Δ and μ_{A}: Aut^{∨}(A) ⟶ Δ

by setting, for α ∈ Aut^{∨}(S),

μ(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}

and μ_{A}(α|_{A}) = μ(α) if α ∈ Aut_{F}^{∨}(S).

3. Minimally active modules

We first recall some notation from [CrOS, Definition 3.1 & 3.3].

Definition 3.1. For each prime p,

• 𝒢_{p} is the class of finite groups G with U ∈ Syl_{p}(G) such that |U| = p and U ⋬ G; and

• 𝒢_{p}^{∧} is the class of those G ∈ 𝒢_{p} such that |Out_{G}(U)| = p − 1 for U ∈ Syl_{p}(G).

For G ∈ 𝒢_{p}, an 𝔽_{p}G-module is minimally active if its restriction to U ∈ Syl_{p}(G) has exactly one Jordan block with nontrivial action.

The next lemma explains the importance of minimally active modules here. In particular, it means that many of the tables and results in [CrOS, § 4–5] can be applied to get information about Aut_{F}(A) and Ω_{1}(A).

Lemma 3.2. Assume Notation 2.1 and 2.9, and also that p is odd, A ∈ E_{F}, and O_{p}(F) = 1. Set V = Ω_{1}(A) and

Aut_{F}^{∨}(V) = {β ∈ N_{Aut_{F} (V)}(Aut_{S}(V)) | [β, Ω_{1}(Z)] ≤ Z_{0}}
= {α|_{V} | α ∈ Aut_{F}(S), [α, Ω_{1}(Z)] ≤ Z_{0}},

and define μ_{V} : Aut_{F}^{∨}(V) ⟶ Δ by setting μ_{V} (α|_{V}) = μ(α). Then

(a) G ∈ 𝒢_{p}^{∧};

(b) V, and A/Fr(A) if |A| < ∞, are both faithful, minimally active, and indecomposable as 𝔽_{p}G-modules; and

(c) μ_{A}(Aut_{F}^{∨}(A)) = \begin{cases}μ_{V} (Aut_{F}^{∨}(V)) & if Z_{0} ≰ Fr(A) \\ μ_{V} (Aut_{F}^{∨}(V)) ∩ Δ_{0} & if Z_{0} ≤ Fr(A). \end{cases}

Proof. (a) By assumption, U = Aut_{S}(A) ∈ Syl_{p}(G) has order p. Since A ∈ E_{F}, U is not normal in G = Aut_{F}(A), and hence G ∈ 𝒢_{p}.

Since O_{p}(F) = 1, there is P ∈ E_{F} ∩(H∪B). By Lemma 2.5(a,b), O^{p′}(Out_{F}(P)) ≅ SL_{2}(p). Choose α ∈ O^{p′}(Aut_{F}(P)) of order p − 1 whose class in Out_{F}(P) normalizes Out_{S}(P) ≅ C_{p}; then α extends to an element of Aut_{F}(N_{S}(P)) and hence (since P is maximal among Fessential subgroups) to some α ∈ Aut_{F}(S). Then α|_{A} normalizes U and its class in Aut_{G}(U) has order p − 1, so G ∈ 𝒢_{p}^{∧}.

(b) Set A = A/Fr(A). If |A| < ∞, then since G acts faithfully on A, [G, Theorems 5.2.4 & 5.3.5] imply that C_{G}(V) and C_{G}(A) are both normal p-subgroups of G. Since U is not normal by (a), G acts faithfully on V and on A in this case. If |A| = ∞, then G acts faithfully on Ω_{m}(A) for m large enough, and hence acts faithfully on V by the above argument.

Since |Z_{0}| = p by Lemma 2.4, where Z_{0} = S′ ∩ Z = [U, V] ∩ C_{V} (U), the 𝔽_{p}U-module V |_{U} has exactly one Jordan block with nontrivial action of U. So V is minimally active. If |A| < ∞, then ZS′ = C_{A}(U)[U, A] has index p in A by Lemma 2.4, so C_{A}(U)[U, A] has index at most p in A, and hence A is minimally active.

If V = V_{1} × V_{2}, where each V_{i} is a nontrivial 𝔽_{p}G-submodule, then by [CrOS, Lemma 3.4(a)], we can assume (after exchanging indices if needed) that V_{1} ≤ Z and (since it is a summand) V_{1} ∩ Z_{0} = 1. But this contradicts Lemma 2.7.

Assume |A| < ∞. If A = X × Y where X, Y ≤ A are 𝔽_{p}G-submodules, then by the Krull-Schmidt theorem, one of the factors, say X, contains a nontrivial Jordan block, while U acts trivially on the other factor. Thus X ≥ [U, A] and X ≰ ZS′/Fr(A). Let X ≤ A be such that Fr(A) ≤ X and X/Fr(A) = X. Then X ≥ S′ and X ≰ ZS′, so XZ = A. By Lemma 2.8, either X = A (and X = A) or A = X × Z_{0} (which is impossible since Z_{0} ≤ S′ ≤ X). Thus X = A, and A is indecomposable.

(c) By definition, the restriction to V of each element in Aut_{F}^{∨}(A) lies in Aut_{F}^{∨}(V), and hence μ_{A}(Aut_{F}^{∨}(A)) ≤ μ_{V} (Aut_{F}^{∨}(V)). If Z_{0} ≤ Fr(Z), then choose z ∈ Z such that 1 ≠ z^{p} ∈ Z_{0}. For each β ∈ Aut_{F}^{∨}(A), β(z) = z^{pk+1} for some k since [β, Z] ≤ Z_{0}, and hence β|_{Z_{0}} = Id and μ_{A}(β) ∈ Δ_{0}. Thus μ_{A}(Aut_{F}^{∨}(A)) is contained in the right hand side in (c).

Now assume that β ∈ Aut_{F}^{∨}(V), where β = α|_{V} for α ∈ Aut_{F}(S). If Z_{0} ≤ Fr(A), then assume also that μ_{V} (β) ≤ Δ_{0}; i.e., that β|_{Z_{0}} = Id. Upon replacing β by β^{p^{k}} and α by α^{p^{k}} for appropriate k, we can also assume that α has order prime to p without changing μ(α). Then Z = C_{Z}(α) × [α, Z] by [G, Theorem 5.2.3] and since Z is the union of the finite abelian p-groups Ω_{i}(Z), and [α, Ω_{1}(Z)] = [β, Ω_{1}(Z)] ≤ Z_{0} since β ∈ Aut_{F}^{∨}(V). Also, Ω_{1}([α, Z]) = [α, Ω_{1}(Z)] ≤ Z_{0} (since it can’t be any larger). If Z_{0} ≰ Fr(Z), then this implies that [α, Z] ≤ Z_{0}, hence that α ∈ Aut_{F}^{∨}(S). If Z_{0} ≤ Fr(Z), then Ω_{1}([α, Z]) ≤ Z_{0} ≤ C_{Z}(α) implies that Ω_{1}([α, Z]) = 1 and hence [α, Z] = 1, so again α ∈ Aut_{F}^{∨}(S). Thus μ_{V} (β) = μ_{A}(α|_{A}) ∈ μ_{A}(Aut_{F}^{∨}(A)), and the right hand side in (c) is contained in μ_{A}(Aut_{F}^{∨}(A)). □

The following basic properties of minimally active indecomposable modules, taken from [CrOS], play an important role in the rest of the paper.

Lemma 3.3 ([CrOS, Proposition 3.7]). Fix an odd prime p, a finite group G ∈ 𝒢_{p}, and U ∈ Syl_{p}(G). Let V be a faithful, minimally active, indecomposable 𝔽_{p}G-module. Then

(a) dim(V) ≤ p implies that V |_{U} is indecomposable and thus contains a unique Jordan block;

(b) dim(V) ≥ p + 1 implies that V |_{U} is the direct sum of a Jordan block of dimension p and a module with trivial action of U; and

(c) dim(C_{V} (U)) = 1 if dim(V) ≤ p, while dim(C_{V} (U)) = dim(V) − p + 1 if dim(V) ≥ p.

The next lemma is closely related to [CrOS, Lemma 1.11].

Lemma 3.4. Fix an odd prime p, let G be a finite group such that U ∈ Syl_{p}(G) has order p, and set N = N_{G}(U). Let V be a faithful, minimally active, indecomposable 𝔽_{p}G-module such that dim(V) ≤ p. Then C_{V} (U) and V/[U, V] are both 1-dimensional, and the following hold.

(a) If dim(V) = p, then V/[U, V] and C_{V} (U) are 1-dimensional, and isomorphic as 𝔽_{p}[N/U]- modules.

(b) The projective cover and the injective envelope of V |_{N} are both p-dimensional.

(c) If dim(V) = p − 1, and there is an 𝔽_{p}G-submodule V_{0} < V with dim(V_{0}) = 1, then there is a projective 𝔽_{p}G-module W such that dim(W) = p and W has a submodule isomorphic to V.

(d) Let W be another 𝔽_{p}G-module such that dim(W) = dim(V), and assume that C_{W} (U) ≅ C_{V} (U) as 𝔽_{p}[N_{G}(U)/U]-modules. Then W ≅ V as 𝔽_{p}[N_{G}(U)]-modules, and as 𝔽_{p}Gmodules if dim(V) < p.

Proof. (a) By Lemma 3.3, U acts on V with only one Jordan block, so dim(C_{V} (U)) = 1 and dim(V/[U, V]) = 1. By [CrOS, Lemma 1.11(b)], if g ∈ N_{G}(U) and t ∈ (ℤ/p)^{×} are such that g acts on V/[U, V] via multiplication by t, then for some r ∈ (ℤ/p)^{×}, g acts on C_{V} (U) via multiplication by tr^{m−1} = tr^{p−1} = t. Thus V/[U, V] and C_{V} (U) are isomorphic as 𝔽_{p}[N_{G}(U)/U]-modules.

(b) By the Schur-Zassenhaus theorem, there is H < N of index p such that N = HU. Set V_{0} = C_{V} (U), regarded as an 𝔽_{p}[N/U]-module, and also as an 𝔽_{p}H-module via the natural isomorphism H ≅ N/U. Set V̂ = Ind_{H}^{N}(V_{0}): a projective and injective p-dimensional 𝔽_{p}Nmodule. Then

V̂/[U, V̂] ≅ 𝔽_{p}[N/U] ⊗_{𝔽_{p}N} 𝔽_{p}N ⊗_{𝔽_{p}H} V_{0} ≅ 𝔽_{p}[N/U] ⊗_{𝔽_{p}H} V_{0} ≅ V_{0},

and so CV̂(U) ≅ V_{0} by (a). Thus V̂ is the injective envelope of V_{0} when regarded as an 𝔽_{p}N-module. In particular, an isomorphism C_{V} (U) ≅ CV̂(U) extends to an 𝔽_{p}N-linear homomorphism V ⟶ V̂ which is injective since it sends the socle C_{V} (U) injectively. Thus V̂ is an injective envelope of V |_{N}. The statements about projective envelopes is shown in a similar way (or by dualizing).

(c) Assume that dim(V) = p − 1, and that there is an 𝔽_{p}G-submodule V_{0} < V with dim(V_{0}) = 1. Then V_{0} = C_{V} (U), since this is the unique 1-dimensional submodule of V as an 𝔽_{p}U-module. By (b), there is an injective (hence projective) 𝔽_{p}N-module Ŵ containing V |_{N} as a submodule. By (a), Ŵ/V ≅ V_{0}|_{N} as 𝔽_{p}N-modules.

Consider the homomorphisms

Ext_{𝔽_{p}G}^{1}(V_{0}, V) ⟶^{Φ_{1}} Ext_{𝔽_{p}N}^{1}(V_{0}, V) ⟶^{Φ_{2}} Ext_{𝔽_{p}U}^{1}(V_{0}, V)

induced by restrictions of rings. Since U has index prime to p in G, Φ_{1} and Φ_{2} are injective, and the images of Φ_{2} and Φ_{2}Φ_{1} are certain subgroups of stable elements (see [Ben, Proposition 3.8.2] for this version of the stable elements theorem). Since Ext_{𝔽_{p}}^{1}(V_{0}, V) = 0, we need only consider stability of elements with respect to automorphisms of U, and hence Im(Φ_{2}) = Im(Φ_{2}Φ_{1}).

Thus Φ_{1} is an isomorphism. Interpreted in terms of extensions, this implies that there is an extension 0 ⟶ V ⟶ W ⟶ V_{0} ⟶ 0 of 𝔽_{p}G-modules such that W |_{N} ≅ Ŵ. In particular, W is projective since Ŵ is projective as an 𝔽_{p}N-module.

(d) By (b), the injective envelope V̂ of V |_{N} is p-dimensional. Hence CV̂(U) ≅ C_{V} (U) ≅ C_{W} (U), so V̂ is also the injective envelope of C_{W} (U), and hence of W |_{N} since C_{W} (U) is its socle. Since dim(V) = dim(W), and V̂ contains a unique 𝔽_{p}N-submodule of each dimension m ≤ p, we conclude that V ≅ W are isomorphic as 𝔽_{p}N-modules.

If dim(V) < p, then U is a vertex of V and of W and they are the Green correspondents of V |_{N} and W |_{N}, respectively (see [Ben, § 3.12]). So V ≅ W as 𝔽_{p}G-modules. □

Point (d) need not hold if dim(V) = p. As an example, fix p ≥ 5, set G = SL_{2}(p) and choose U ∈ Syl_{p}(G), let V be the simple p-dimensional 𝔽_{p}G-module, and let W be the projective cover of the trivial 1-dimensional 𝔽_{p}G-module. Using the fact that V is the (p − 1)-st symmetric power of the natural 2-dimensional 𝔽_{p}G-module, it is not hard to see that C_{V} (U) is 1-dimensional with trivial N_{G}(U)/U-action. The same holds for W by construction, where dim(W) = p. We refer to [Al, pp. 48–52] or the discussion in [CrOS, § 6] for more detail.

A minimally active indecomposable 𝔽_{p}G-module of dimension at least p + 2 is simple by [CrOS, Proposition 3.7(c)]. This is not true for modules of dimension p+1, but the following lemma gives some information about such modules. Lemma 3.5. Fix a finite group G ∈ 𝒢_{p} with U ∈ Syl_{p}(G). Let V be a finite, minimally active, indecomposable 𝔽_{p}G-module of rank p + 1. If 0 ≠ V_{0} < V is a proper nontrivial submodule, then V_{0}|_{U} and (V/V_{0})|_{U} are both indecomposable 𝔽_{p}U-modules with nontrivial action. In particular, 2 ≤ dim(V_{0}) ≤ p − 1.

Proof. Recall |U| = p since G ∈ 𝒢_{p}. By [CrOS, Proposition 3.7(a)], V |_{U} ≅ 𝔽_{p}U ⊕ 𝔽_{p}; i.e., V |_{U} has Jordan blocks of dimension p and 1.

Fix a proper nontrivial submodule 0 ≠ V_{0} < V, and assume that V_{0}|_{U} is decomposable or has trivial action as an 𝔽_{p}U-module. We will first show that V_{0} always has a 1-dimensional 𝔽_{p}G-submodule, and then show that this is impossible. In particular, this shows that dim(V_{0}) ≥ 2. The corresponding results for V/V_{0} then follow by dualizing.

If V_{0}|_{U} is decomposable with nontrivial action, then V_{0}|_{U} is the sum of a 1-dimensional module with trivial action and an indecomposalble module of dimension at most p − 1. By [CrOS, Proposition 3.7(a)], V_{0} is decomposable as an 𝔽_{p}G-module, and thus has a 1dimensional summand.

If U acts trivially on V_{0}, then dim(V_{0}) ≤ 2 and O^{p′}(G) (the normal closure of U in G) acts trivially on V_{0}. If dim(V_{0}) = 2, then V_{0} = C_{V} (U), and [U, V]∩V_{0} is a 1-dimensional subspace normalized by N_{G}(U). Since G = O^{p′}(G)N_{G}(U) by the Frattini argument, [U, V] ∩ V_{0} is a 1-dimensional 𝔽_{p}G-submodule.

We are thus reduced to the case where dim(V_{0}) = 1. If V_{0} ≠ [U, V], then (V/V_{0})|_{U} is indecomposable (consists of one Jordan block), and hence is 𝔽_{p}U-free. So V/V_{0} is projective, contradicting the assumption that V is indecomposable.

Thus V_{0} = [U, V] is an 𝔽_{p}G-submodule, and V/V_{0} has Jordan blocks of length 1 and p−1. So by [CrOS, Proposition 3.7(a)], it is decomposable: there are submodules W_{1}, W_{2} < V such that V/V_{0} = (W_{1}/V_{0}) ⊕ (W_{2}/V_{0}), where dim(W_{1}) = p, and (W_{1}/V_{0})|_{U} is an (indecomposable) Jordan block. If [U, W_{1}] = 0, then [U, W_{2}] = V_{0}, and V |_{U} contains Jordan blocks of dimension p − 1 and 2, which we saw is impossible. Thus [U, W_{1}] = V_{0}, so W_{1}|_{U} is indecomposable, W_{1} is projective and injective, and this again contradicts the assumption that V is indecomposable.

This proves that U acts nontrivially on V_{0}, and in particular, dim(V_{0}) ≥ 2. A similar argument applied to the dual V^{∗} shows that U acts nontrivially on V/V_{0}, and that dim(V_{0}) ≤ p − 1. □

The following definitions will be useful. Definition 3.6. For a finite group G, a ℤ_{p}G-lattice is a finitely generated ℤ_{p}G-module that is free as a ℤ_{p}-module (hence a lattice in a finitely generated ℚ_{p}G-module). A discrete G-p-torus is a discrete p-torus equipped with an action of G by automorphisms.

Let ℚ_{p}(ζ) ⊇ ℤ_{p}[ζ] denote the extensions of ℚ_{p} ⊇ ℤ_{p} by a primitive p-th root of unity ζ. When U is a group of order p, we regard ℚ_{p}(ζ) and ℤ_{p}[ζ] as ℤ_{p}U-modules under some choice of identification U ≅ ⟨ζ⟩. Lemma 3.7. Fix an odd prime p, a group G ∈ 𝒢_{p}, and U ∈ Syl_{p}(G).

(a) Let Λ be a ℤ_{p}G-lattice such that Λ/pΛ is faithful and minimally active as an 𝔽_{p}Gmodule. Then Λ/C_{Λ}(U) ≅ ℤ_{p}[ζ] as ℤ_{p}U-modules, and [U, Λ] + C_{Λ}(U) has index p in Λ.

(b) Let A be a discrete G-p-torus such that Ω_{1}(A) is faithful and minimally active as an 𝔽_{p}G-module. Then A/C_{A}(U) ≅ ℚ_{p}(ζ)/ℤ_{p}[ζ] as ℤ_{p}U-modules, and |[U, A]∩C_{A}(U)| = p.

(c) Let X be a finite, faithful ℤ_{p}G-module. Then G acts faithfully on Ω_{1}(X) and on X/pX. Among the following conditions:

(1) Ω_{1}(X) is minimally active as an 𝔽_{p}G-module.

(2) X/pX is minimally active as an 𝔽_{p}G-module.

(3) |[U, X] ∩ C_{X}(U)| = p.

(4) [U, X] + C_{X}(U) has index p in X.

we have (1) ⟸ (3) ⟺ (4) ⟹ (2). If X ≅ Λ/Λ_{0} for some ℤ_{p}G-lattice Λ and some submodule Λ_{0} ≤ pΛ, then all four conditions are equivalent.

(d) If X is a finite, faithful ℤ_{p}G-module such that condition (c.4) holds, then for each x ∈ X ∖ ([U, X] + C_{X}(U)), X = C_{X}(U) + ℤ_{p}U·x.

Proof. Fix a generator u ∈ U.

(a) Set M = ℚ_{p} ⊗_{ℤ_{p}} Λ. By Lemma A.5(a), M/C_{M}(U) is isomorphic, as a ℚ_{p}U-module, to a sum of copies of ℚ_{p}(ζ), where ζ is a primitive p-th root of unity. In particular, each Jordan block for the action of U on Λ/(C_{Λ}(U) + pΛ) has length at most p − 1. Since Λ/pΛ is minimally active, it follows that M/C_{M}(U) ≅ ℚ_{p}(ζ), since otherwise Λ/(C_{Λ}(U) + pΛ) would have rank at least 2(p − 1) and hence U would be fixed by a submodule of rank at least p ≥ 3. Hence Λ/C_{Λ}(U) ≅ ℤ_{p}[ζ] by Lemma A.5(c).

Consider the short exact sequence

0 ⟶ C_{Λ}(U) ⟶^{incl} Λ ⟶^{φ} [U, Λ] ⟶ 0,

where φ(x) = u(x) − x for all x ∈ Λ. We just saw that [U, Λ] ≅ ℤ_{p}[ζ]. Under this identification, φ|_{[U,Λ]} is multiplication by 1 − ζ, and so its image has index p in [U, Λ]. Thus C_{Λ}(U) + [U, Λ] has index p in Λ.

(b) Let A be a discrete G-p-torus such that Ω_{1}(A) is faithful and minimally active as an 𝔽_{p}G-module. Set Λ = Hom_{ℤ_{p}}(A, ℚ_{p}/ℤ_{p}), regarded as a ℤ_{p}G-lattice. Then Ω_{1}(A) ≅ Λ/pΛ by Proposition A.4, so Λ/pΛ is minimally active. We just saw, in the proof of (a), that this implies that Λ/C_{Λ}(U) ≅ ℤ_{p}[ζ]. So after taking tensor products with ℚ_{p}/ℤ_{p} and applying Proposition A.4 again, we get that A/(ℚ_{p}/ℤ_{p} ⊗_{ℤ_{p}} C_{Λ}(U)) ≅ ℚ_{p}(ζ)/ℤ_{p}[ζ]. Hence A/C_{A}(U) ≅ ℚ_{p}(ζ)/R for some ℤ_{p}U-lattice R < ℚ_{p}(ζ) that contains ℤ_{p}[ζ] with finite index. Then R ≅ ℤ_{p}[ζ] by Lemma A.5(c), ℚ_{p}(ζ) = ℚ_{p}·R, and so A/C_{A}(U) ≅ ℚ_{p}(ζ)/ℤ_{p}[ζ].

Consider the short exact sequence

0 ⟶ C_{A}(U) ⟶^{incl} A ⟶^{φ} [U, A] ⟶ 0,

where φ(x) = u(x) − x for all x ∈ A. We just saw that [U, A] ≅ ℚ_{p}(ζ)/ℤ_{p}[ζ]. Under this identification, φ|_{[U,A]} is multiplication by 1 − ζ, and so its kernel has order p. Thus |C_{A}(U) ∩ [U, A]| = p.

(c) We have |X| = |C_{X}(U)|·|[U, X]|: X is finite, and [U, X] is the image of the homomorphism X ⟶^{1−u} X while C_{X}(U) is its kernel. Hence (3) and (4) are equivalent.

If (3) holds, then [U, Ω_{1}(X)] ∩ C_{Ω_{1}(X)}(U) also has order p (since it cannot be trivial). Since the rank of this intersection is the number of Jordan blocks in Ω_{1}(X) with nontrivial U-action, we see that Ω_{1}(X) is minimally active in this case. So (3) implies (1); and a similar argument shows that (4) implies (2).

Assume that Λ_{0} < Λ are ℤ_{p}G-lattices such that Λ_{0} ≤ pΛ and Λ/Λ_{0} ≅ X. In particular, Λ/pΛ ≅ X/pX. So if X/pX is minimally active, then [U, Λ] + C_{Λ}(U) has index p in Λ by (a), and hence [U, X] + C_{X}(U) has index p in X (since it cannot be all of X). Thus (2) implies (4) in this case.

We continue to assume that X ≅ Λ/Λ_{0}, and set R_{p} = ℚ_{p}/ℤ_{p} for short. We have an exact sequence

0 ⟶ Tor_{ℤ_{p}}(R_{p}, X) ⟶ R_{p} ⊗_{ℤ_{p}} Λ_{0} ⟶ R_{p} ⊗_{ℤ_{p}} Λ ⟶ R_{p} ⊗_{ℤ_{p}} X ⟶ 0.

Also, by tensoring the short exact sequence 0 → ℤ_{p} → ℚ_{p} → R_{p} → 0 by X and using the fact that ℚ_{p} is flat over Z_{p}, we see that R_{p} ⊗_{ℤ_{p}} X = 0, and Tor_{ℤ_{p}}(R_{p}, X) ≅ X as ℤ_{p}Gmodules. Thus X is isomorphic to a subgroup of the discrete G-p-torus A = R_{p} ⊗_{ℤ_{p}} Λ_{0} (see Proposition A.4), where Ω_{1}(X) ≅ Ω_{1}(A). With the help of (b), we now see that (1) implies (3).

(d) Set X = X/pX for short. By (c), X is minimally active as an 𝔽_{p}G-module. Hence there is y ∈ X such that X = C_{X}(U) + 𝔽_{p}U·y. Thus [U, X] = (1 − u)𝔽_{p}U·y. Choose y ∈ X whose class modulo p is y; then [U, X] ≤ (1 − u)ℤ_{p}U·y + pX.

By assumption, X = C_{X}(U) + [U, X] + ℤ_{p}·x. So there are ξ ∈ ℤ_{p}U and r ∈ ℤ_{p} such that y ∈ rx + (1 − u)ξ·y + C_{X}(U) + pX. Then (1 − (1 − u)ξ)y ∈ rx + C_{X}(U) + pX, where 1 − (1 − u)ξ is invertible in ℤ_{p}U since (1 − u)^{p} ∈ pℤ_{p}U. Thus y ∈ ℤ_{p}U·x + C_{X}(U) + pX, and hence X = ℤ_{p}U·x + C_{X}(U) + pX. Since pX is the Frattini subgroup of X, it now follows that X = ℤ_{p}U·x + C_{X}(U). □

In the rest of the section, we look at questions of existence and uniqueness of finite ℤ_{p}Gmodules or discrete G-p-tori A for which Ω_{1}(A) is isomorphic to a given minimally active, indecomposable 𝔽_{p}G-module.

Proposition 3.8. Fix an odd prime p and a group G ∈ 𝒢_{p}, and let V be a faithful, minimally active, indecomposable 𝔽_{p}G-module.

(a) If dim(V) ≥ p − 1, then there are a ℚ_{p}G-module M and a ℤ_{p}G-lattice Λ ≤ M such that V ≅ Λ/pΛ.

(b) If dim(V) = p, and there is an 𝔽_{p}G-submodule V_{1} < V of dimension 1, then M and Λ can be chosen as in (a) such that M contains a 1-dimensional ℚ_{p}G-submodule.

Proof. Fix U ∈ Syl_{p}(G), and let u ∈ U be a generator. Let ζ be a p-th root of unity, and regard ℚ_{p}(ζ) as a ℚ_{p}U-module where u acts by multiplication by ζ. Thus ℤ_{p}[ζ] is a ℤ_{p}Ulattice in ℚ_{p}(ζ). We also write 𝔽_{p}[ζ] = ℤ_{p}[ζ]/pℤ_{p}[ζ]. Thus 𝔽_{p}[ζ] ≅ 𝔽_{p}[u]/⟨(1 − u)^{p−1}⟩, and V |_{U} ≅ 𝔽_{p}[ζ] since V is minimally active and indecomposable [CrOS, Proposition 3.7(a)].

(a) If dim(V) ≥ p, then V is a trivial source module by [CrOS, Proposition 3.7(b)], and hence V is the mod p reduction of some ℤ_{p}G-lattice (see [Ben, Corollary 3.11.4.i]). So for the rest of the proof, we assume that dim(V) = p − 1.

Set Λ̂ = Ind_{U}^{G}(ℤ_{p}[ζ]) and V̂ = Ind_{U}^{G}(V |_{U}). Then V̂ ≅ Λ̂/pΛ̂. Since induction of representations is adjoint to restriction, the identity on V extends to a surjective 𝔽_{p}G-linear homomorphism α: V̂ ⟶ V. This is split by an 𝔽_{p}U-linear map, and hence (by averaging over cosets of U) by an 𝔽_{p}G-linear homomorphism β: V ⟶ V̂. Set e_{0} = βα ∈ End_{𝔽_{p}G}(V̂). Thus e_{0} is an idempotent in this endomorphism ring, and e_{0}V̂ ≅ V.

We want to lift e_{0} to an idempotent in End_{ℤ_{p}G}(Λ̂). By the Mackey double coset formula,

Λ̂|_{U} = (Ind_{U}^{G}(ℤ_{p}[ζ]))|_{U} ≅ (ℤ_{p}[ζ])^{m} × (ℤ_{p}U)^{n}

as ℤ_{p}U-modules for some m, n ≥ 0. Hence as ℤ_{p}-modules,

End_{ℤ_{p}G}(Λ̂) ≅ Hom_{ℤ_{p}U}(ℤ_{p}[ζ], Λ̂) ≅ (End_{ℤ_{p}U}(ℤ_{p}[ζ]))^{m} × (Hom_{ℤ_{p}U}(ℤ_{p}[ζ], ℤ_{p}U))^{n}
≅ (ℤ_{p}[ζ])^{m} × ((1 − u)ℤ_{p}U)^{n}

where the last isomorphism follows upon sending a homomorphism φ to φ(1). Since

End_{𝔽_{p}G}(Λ̂) ≅ (𝔽_{p}[ζ])^{m} × ((1 − u)𝔽_{p}U)^{n}

by a similar argument, the natural homomorphism from End_{ℤ_{p}G}(Λ̂) to End_{𝔽_{p}G}(V̂) is surjective (and reduction mod p). So e_{0} lifts to an idempotent e ∈ End_{ℤ_{p}U}(Λ̂) (see [Ben, Proposition 1.9.4]).

Now set Λ = eΛ̂. Then Λ/pΛ ≅ V, and Λ is a ℤ_{p}G-lattice in the ℚ_{p}G-module M = Q_{p} ⊗_{ℤ_{p}} Λ.

(b) We repeat the proof of (a), but keeping control of the submodule as well as V. Set V_{2} = V and V_{3} = V_{2}/V_{1}, and set V̂_{i} = Ind_{U}^{G}(V_{i}|_{U}) for i = 1, 2, 3. We thus have short exact sequences

0 ⟶ V_{1} ⟶^{f} V_{2} ⟶^{g} V_{3} ⟶ 0 and 0 ⟶ V̂_{1} \underrightarrow{f̂} V̂_{2} \underrightarrow{ĝ} V̂_{3} ⟶ 0.

Let α_{i}: V̂_{i} ⟶ V_{i} be the natural map, and let β_{i}: V_{i} ⟶ V̂_{i} be the 𝔽_{p}G-linear splitting of α_{i} obtained by taking the natural 𝔽_{p}U-linear inclusion and then averaging over cosets of U in G. Thus α_{i} ∘ β_{i} = Id_{V_{i}} (upon composing from right to left), while e_{i} =^{def} β_{i} ∘ α_{i} is an idempotent in End_{𝔽_{p}G}(V̂_{i}). All of these commute with the natural homomorphisms f, f̂, g, and ĝ, and so we get a commutative diagram with exact rows:

\xymatrix{0 \ar[r] & V̂_{1} \ar[r]^{f̂} \ar[d]_{e_{1}} & V̂_{2} \ar[r]^{ĝ} \ar[d]_{e_{2}} & V̂_{3} \ar[r] \ar[d]_{e_{3}} & 0 \\ 0 \ar[r] & V̂_{1} \ar[r]^{f̂} & V̂_{2} \ar[r]^{ĝ} & V̂_{3} \ar[r] & 0 .}  (1)

Now set Λ_{1}^{0} = ℤ_{p}, Λ_{2}^{0} = ℤ_{p}U, Λ_{3}^{0} = ℤ_{p}[ζ], and let φ and ψ be such that

0 ⟶ Λ_{1}^{0} ⟶^{φ} Λ_{2}^{0} ⟶^{ψ} Λ_{3}^{0} ⟶ 0

is a short exact sequence of ℤ_{p}U-modules. We identify V_{i}|_{U} = Λ_{i}^{0}/pΛ_{i}^{0} in such a way that f and g are the reductions modulo p of φ and ψ, respectively. Set Λ̂_{i} = Ind_{U}^{G}(Λ_{i}^{0}), so that V̂_{i} = Λ̂_{i}/pΛ̂_{i} as 𝔽_{p}G-modules, and let φ̂ and ψ̂ be the homomorphisms induced by φ and ψ. We claim that the e_{i} can be lifted to elements ε_{i} ∈ End_{ℤ_{p}G}(Λ̂_{i}) that make the following diagram commute:

\xymatrix{0 \ar[r] & Λ̂_{1} \ar[r]^{φ̂} \ar[d]_{ε_{1}} & Λ̂_{2} \ar[r]^{ψ̂} \ar[d]_{ε_{2}} & Λ̂_{3} \ar[r] \ar[d]_{ε_{3}} & 0 \\ 0 \ar[r] & Λ̂_{1} \ar[r]^{φ̂} & Λ̂_{2} \ar[r]^{ψ̂} & Λ̂_{3} \ar[r] & 0 .}  (2)

To see this, we identify

End_{ℤ_{p}G}(Λ̂_{i}) ≅ Hom_{ℤ_{p}U}(Λ_{i}^{0}, (Ind_{U}^{G}(Λ_{i}^{0}))|_{U}),

where by the Mackey double coset formula, for some indexing sets J and K independent of i ∈ {1, 2, 3},

(Ind_{U}^{G}(Λ_{i}^{0}))|_{U} ≅ ⨁_{j∈J} Λ_{i}^{0} ⊕ ⨁_{k∈K}(ℤ_{p}U ⊗_{ℤ_{p}} Λ_{i}^{0}).

Fix ε_{2} ∈ End_{ℤ_{p}G}(Λ̂_{2}), and set ε_{2}(1) = ((u_{j})_{j∈J}, (v_{k})_{k∈K}) with respect to the above decomposition (and where 1 is the identity in Λ_{2}^{0} = ℤ_{p}U). Then ε_{2} induces endomorphisms ε_{1} and ε_{3} such that (2) commutes if and only if ∑_{i=0}^{p−1} u^{i}(v_{k}) ∈ Ker(ψ̂) for each k ∈ K. Note that ψ̂, after restriction to the summands for some k ∈ K, is a surjection of one free ℤ_{p}U-module onto another. Hence ε_{2} can always be chosen (as a lifting of e_{2}) to induce ε_{1} and ε_{3} since the above condition holds modulo p by the commutativity of (1).

Since ε_{i} is a lifting of the idempotent e_{i}, we have ε_{i}^{2} ≡ ε_{i} (mod p). Hence (ε_{i})^{2p^{k}} ≡ (ε_{i})^{p^{k}} (mod p^{k+1}) for each k ≥ 1. Upon replacing ε_{i} by the limit of the (ε_{i})^{p^{k}}, we can arrange that each ε_{i} is an idempotent in End_{ℤ_{p}G}(Λ̂_{i}) (and that the above diagram still commutes). Set Λ = ε_{2}Λ̂_{2} and M = ℚ_{p} ⊗_{ℤ_{p}} Λ. Thus M is a ℚ_{p}G-module with a 1-dimensional submodule, and has a ℤ_{p}G-lattice Λ such that Λ/pΛ ≅ V. □

We now turn to questions of uniqueness, looking first at the finite case. When R is a ring and M is an R-module, we let Ann_{R}(x) denote the annihilator of an element x ∈ M.

Proposition 3.9. Fix an odd prime p, a finite group G ∈ 𝒢_{p}, and U ∈ Syl_{p}(G). Let A_{1} and A_{2} be finite ℤ_{p}G-modules, and assume that |C_{A_{i}}(U) ∩ [U, A_{i}]| = p for i = 1, 2. Set σ = ∑_{u∈U} u ∈ ℤ_{p}U.

Assume that there is a ℤ_{p}U-linear isomorphism φ: A_{1} ⟶ A_{2} whose reduction modulo p is 𝔽_{p}G-linear. Then A_{1} ≅ A_{2} as ℤ_{p}G-modules. In particular, this happens if A_{1}/pA_{1} ≅ A_{2}/pA_{2} and either

(a) A_{1} and A_{2} are homocyclic of the same exponent and σ·A_{i} ≰ pA_{i} for i = 1, 2; or

(b) there are elements a_{1} ∈ A_{1} and a_{2} ∈ A_{2} such that A_{1} = ℤ_{p}U·a_{1}, A_{2} = ℤ_{p}U·a_{2}, and Ann_{ℤ_{p}U}(a_{1}) = Ann_{ℤ_{p}U}(a_{2}).

Proof. Set A_{i} = A_{i}/pA_{i} for i = 1, 2. For each X ≤ A_{i} and g ∈ A_{i}, set X = (X + pA_{i})/pA_{i} and g = g + pA_{i}. Set Z_{i} = C_{A_{i}}(U) and S′_{i} = [U, A_{i}]. By assumption, |Z_{i} ∩ S′_{i}| = p, and hence |A_{i}/(Z_{i} + S′_{i})| = p by Lemma 3.7(c).

Assume φ: A_{1} ⟶ A_{2} is a ℤ_{p}U-linear isomorphism whose reduction φ: A_{1} ⟶ A_{2} modulo p is 𝔽_{p}G-linear. Let g_{1}, …, g_{k} be a set of representatives for the left cosets gU in G (where k = |G/U| is prime to p), and define ψ: A_{1} ⟶ A_{2} by setting ψ(λ) = \frac{1}{k}(∑_{j=1}^{k} g_{j}φ(g_{j}^{−1}λ)). Then ψ is ℤ_{p}G-linear, its reduction modulo p is equal to φ since φ is 𝔽_{p}G-linear, it is surjective since the reduction mod p is surjective, and is an isomorphism since |A_{1}| = |A_{2}|.

It remains to prove that each of (a) and (b) implies the existence of the homomorphism φ. Fix an 𝔽_{p}G-linear isomorphism φ: A_{1} ⟶ A_{2}.

(a) Assume A_{1} and A_{2} are homocyclic of the same exponent p^{k}, and for i = 1, 2, σ·A_{i} ≰ pA_{i}. Choose a_{1} ∈ A_{1} such that σ·a_{1} ∉ pA_{1}, and let a_{2} ∈ A_{2} be such that φ(a_{1}) = a_{2}. Thus for i = 1, 2, σ·a_{i} ≠ 0, and so {u(a_{i}) | u ∈ U} is a basis for 𝔽_{p}U·a_{i}, and Ann_{𝔽_{p}U}(a_{i}) = 0. Note that a_{i} ∉ Z_{i} + S′_{i}, since σ·Z_{i} ≤ pZ_{i} and σ·S′_{i} = 0.

We claim that each element of C_{A_{i}}(U) lifts to an element of C_{A_{i}}(U); i.e., that

C_{A_{i}}(U) = Z_{i}.  (1)

To see this, let g ∈ A_{i} be such that g ∈ C_{A_{i}}(U). By Lemma 3.7(d), g = ξ·a_{i} + z for some

ξ ∈ ℤ_{p}U and z ∈ Z_{i}. Then ξ·a_{i} is fixed by U since g is, and C_{ℤ_{p}U·a_{i}}(U) = ⟨σ·a_{i}⟩ since

Ann_{𝔽_{p}U}(a_{i}) = 0. Hence there is k ∈ ℤ such that kσ·a_{i} = ξ·a_{i}; and g = kσ·a_{i} + z where kσ·a_{i} + z ∈ Z_{i}. This proves (1).

Set m = rk(A_{1}) − p = rk(A_{2}) − p (possibly m = 0). By (1), we can choose elements x_{1}, …, x_{m} ∈ Z_{1} such that A_{1} = 𝔽_{p}U·a_{1} ⊕ ⟨x_{1}, …, x_{m}⟩. By (1) again, there are elements y_{1}, …, y_{m} ∈ Z_{2} such that φ(x_{i}) = y_{i} for each i. Then

{u(a_{1}) | u ∈ U} ∪ {x_{1}, …, x_{m}} and {u(a_{2}) | u ∈ U} ∪ {y_{1}, …, y_{m}}

are bases for A_{1} and A_{2}, respectively, and φ sends the first basis to the second. Since A_{1} and A_{2} are both homocyclic of exponent p^{k}, the sets {u(a_{1}), x_{1}, …, x_{m} | u ∈ U} and {u(a_{2}), y_{1}, …, y_{m} | u ∈ U} are bases for A_{1} and A_{2}, respectively, as ℤ/p^{k}-modules. Thus φ lifts to a ℤ_{p}U-linear isomorphism φ: A_{1} ⟶ A_{2}, defined by setting φ(u(a_{1})) = u(a_{2}) for u ∈ U and φ(x_{i}) = y_{i} for each i.

(b) Let a_{i} ∈ A_{i} (for i = 1, 2) be such that ℤ_{p}U·a_{i} = A_{i} and Ann_{ℤ_{p}U}(a_{1}) = Ann_{ℤ_{p}U}(a_{2}). Let ξ ∈ ℤ_{p}U be such that φ(a_{1}) = ξ·a_{2}. Thus ξ·a_{2} generates A_{2} as an 𝔽_{p}U-module, and since (1 − u)A_{2} ≤ S′_{2} for 1 ≠ u ∈ U, ξ is not in the ideal (1 − u)ℤ_{p}U + pℤ_{p}U of index p in ℤ_{p}U. Since this is the unique maximal ideal in ℤ_{p}U, ξ is invertible, and we can replace a_{2} by ξ·a_{2} without changing Ann_{ℤ_{p}U}(a_{2}).

Let φ: A_{1} ⟶ A_{2} be the unique ℤ_{p}U-linear homomorphism such that φ(a_{1}) = a_{2}. Its reduction modulo p is φ, since φ(a_{1}) = a_{2} and φ is 𝔽_{p}U-linear. □

It remains to prove the analogous uniqueness result for discrete p-tori.

Lemma 3.10. Fix an odd prime p, a finite group G ∈ 𝒢_{p}, and U ∈ Syl_{p}(G). Let A_{1} and A_{2} be discrete, G-p-tori, and assume that

(i) Ω_{1}(A_{1}) and Ω_{1}(A_{2}) are faithful, minimally active, and indecomposable as 𝔽_{p}G-modules; and

(ii) Ω_{1}(A_{1}) ≅ Ω_{1}(A_{2}) as 𝔽_{p}G-modules.

Then A_{1} ≅ A_{2} as ℤ_{p}G-modules.

Proof. By Lemma 3.7(b), we also have that

(a) [U, A_{i}] ∩ C_{A_{i}}(U) has order p for i = 1, 2.

Assume, for each k ≥ 1, that Ω_{k}(A_{1}) ≅ Ω_{k}(A_{2}) as ℤ_{p}G-modules, and let X_{k} be the set of ℤ_{p}G-linear isomorphisms Ω_{k}(A_{1}) ⟶^{≅} Ω_{k}(A_{2}). Then X_{k} is finite since the Ω_{k}(A_{i}) are finite, X_{k} ≠ ∅ by assumption, and if k ≥ 2, restriction to Ω_{k−1}(A_{i}) defines a map X_{k} ⟶ X_{k−1}. So the inverse limit of the X_{k} is nonempty, and each element in the inverse limit determines a ℤ_{p}G-linear isomorphism A_{1} ≅ A_{2}.

It remains to show that Ω_{k}(A_{1}) ≅ Ω_{k}(A_{2}) for each k. Since Ω_{k}(A_{i})/pΩ_{k}(A_{i}) ≅ Ω_{1}(A_{i}) as 𝔽_{p}G-modules (multiplication by p^{k−1} defines an isomorphism), we have Ω_{k}(A_{1})/pΩ_{k}(A_{1}) ≅ Ω_{k}(A_{2})/pΩ_{k}(A_{2}), and both are faithful, minimally active, and indecomposable.

Set σ = ∑_{u∈U} u ∈ ℤ_{p}U, as usual. If rk(A_{i}) ≥ p, then by Lemma 3.3(a,b), σ·Ω_{k}(A_{i}) ≰ pΩ_{k}(A_{i}). Since Ω_{k}(A_{1}) and Ω_{k}(A_{2}) are both homocyclic of exponent p^{k}, they are isomorphic as ℤ_{p}G-modules by Proposition 3.9(a).

If rk(A_{i}) = p − 1 for i = 1, 2, then by Proposition A.4, A_{i} ≅ (ℚ_{p}/ℤ_{p}) ⊗_{ℤ_{p}} Λ_{i} for some (p − 1)-dimensional ℤ_{p}G-lattice Λ_{i}. Since G acts faithfully on the lattices, Λ_{1} ≅ Λ_{2} ≅ ℤ_{p}[ζ] as ℤ_{p}U-modules by Lemma A.5(a,c) (where ζ is a primitive p-th root of unity). Hence for i = 1, 2, A_{i} ≅ ℚ_{p}(ζ)/ℤ_{p}[ζ], and Ω_{k}(A_{i}) ≅ ℤ_{p}[ζ]/p^{k}ℤ_{p}[ζ], as ℤ_{p}U-modules. So there is a_{i} ∈ Ω_{k}(A_{i}) such that ℤ_{p}U·a_{i} = Ω_{k}(A_{i}) and Ann_{ℤ_{p}U}(a_{i}) is the ideal generated by p^{k} and σ. Proposition 3.9(b) now applies to conclude that Ω_{k}(A_{1}) ≅ Ω_{k}(A_{2}) as ℤ_{p}G-modules. □

4. Reduced fusion systems over finite nonabelian p-groups with index p abelian subgroup (p odd)

Throughout this section, p is an odd prime, and A is finite. As noted in the introduction, the corresponding questionfor finite 2-groups was answered in [AOV2, Proposition 5.2(a)].

Lemma 4.1 ([CrOS, Lemma 2.2(d,e,f)]). Assume the notation and hypotheses of 2.1, and also that p is odd and |A| < ∞. Set A_{0} = ZS′. Then the following hold.

(a) If A ⋬ F, then 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.

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

(c) 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⟩.

We now fix some more notation, based on Lemma 4.1.

Notation 4.2. Assume Notation 2.1. Assume also that p is odd, S is finite, and A ⋬ F, and hence that |Z_{0}| = |A/ZS′| = p by Lemma 2.4. Fix a ∈ A ∖ ZS′ and x ∈ S ∖ A, chosen such that ZS′⟨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 4.1(a)). 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}.

For each P ≤ S, set

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

When |Z_{0}| = p, then by Lemma 4.1(c), H = H_{0} ∪ H_{∗} and B = B_{0} ∪ B_{∗}. Note that for x, x′ ∈ S ∖ A, Z⟨x⟩ is S-conjugate to Z⟨x′⟩ or Z_{2}⟨x⟩ is S-conjugate to Z_{2}⟨x′⟩ only if x′x^{−1} ∈ ZS′. So in fact, each of the sets H and B is a union of p distinct S-conjugacy classes: the classes H_{i} and B_{i} for 0 ≤ i ≤ p − 1.

Lemma 4.3 ([CrOS, Lemma 2.5(a,b)]). Let p be an odd prime, let S be a finite nonabelian p-group with a unique abelian subgroup A ⊴ S of index p, and let F be a saturated fusion system over S such that A ⋬ F. We use the notation of Notation 2.1 and 2.9, set A_{0} = ZS′, and 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).

Lemma 4.4 ([CrOS, Lemma 2.6(a)]). Let p be an odd prime, let S be a finite nonabelian pgroup with a unique abelian subgroup A ⊴ S of index p, and let F be a saturated fusion system over S. We use the notation of Notation 2.1 and 2.9. Let m be such that |A/Z| = p^{m−1}. Fix P ∈ H ∪ B, and set

H_{P} = N_{Aut_{F} (S)}(P) ,  Ĥ_{P} = \{α|_{P} | α ∈ H_{P} \} ,  and t = \begin{cases}−1 & if P ∈ H \\ 0 & if P ∈ B. \end{cases}

If P ∈ E_{F}, then Aut_{F}^{(P)}(S) ≤ Aut_{F}^{∨}(S) and μ(Aut_{F}^{(P)}(S)) = Δ_{t}. If P ∈ H_{∗} or P ∈ B_{∗}, then m ≡ t (mod p − 1).

Theorem 4.5 ([CrOS, Theorem 2.8]). Fix an odd prime p, and a finite nonabelian 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, 2.9, and 4.2, and also set A_{0} = ZS′, E_{0} = E_{F}∖{A}, and G = Aut_{F}(A). Thus U = Aut_{S}(A) ∈ Syl_{p}(G). Let m ≥ 3 be such that |A/Z| = p^{m−1}. 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 nontrivial 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 4.1, where σ = ∑_{u∈U} u ∈ ℤ_{p}U.

μ_{A}(Aut_{F}^{∨}(A)) G = O^{p′}(G)X where m (mod p − 1) σ·A  E_{0}
(i)  Δ  X = Aut_{F}^{∨}(A)  ≡ 0  ≤ Fr(Z) H_{0} ∪ B_{∗}
(ii)  Δ  X = Aut_{F}^{∨}(A)  ≡ −1  ≤ Fr(Z) B_{0} ∪ H_{∗}
(iii′)  ≡ −1  ≤ Fr(Z) ⋃_{i∈I} H_{i}
(iii′′)  ≥ Δ_{−1}  X = μ_{A}^{−1}(Δ_{−1})  −  −  H_{0}
(iv′)  X = μ_{A}^{−1}(Δ_{0})  ≡ 0  ≤ Fr(Z) ⋃_{i∈I} B_{i}

(iv′′)

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

Table 4.1

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 simple fusion system F over A ⋊ U with Aut_{F}(A) = G and E_{F} = E_{0} ∪ {A}, unique up to isomorphism. When A is not elementary abelian, all such fusion systems are exotic, except for the fusion systems of the simple groups listed in Table 4.2. 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.

We now look for a more precise description of the group A when it is finite but not elementary abelian. The following notation will be useful when describing more precisely elements and subgroups of A.

Notation 4.6. Assume Notation 2.1 and 4.2, and set u = c_{x} ∈ U = Aut_{S}(A). Set σ = 1 + u + u^{2} + … + u^{p−1} ∈ ℤ_{p}U. Regard A as a ℤ_{p}U-module, and define

Ψ: ℤ_{p}U ⟶ A

Γ  p  conditions  rk(A)  e  m  G = Aut_{Γ}(A)  E_{0}
PSL_{p}(q) p p^{2}|(q−1), p > 3 p−1 v_{p}(q−1) e(p−1) − 1  Σ_{p}  H_{0} ∪ H_{∗}
PSL_{n}(q) p p^{2}|(q−1), p<n<2p n−1 v_{p}(q−1) e(p−1) + 1  Σ_{n}  B_{0}
P Ω_{2n}^{+}(q) p p^{2}|(q−1), p≤n<2p n  v_{p}(q−1) e(p−1) + 1 C_{2}^{n−1} ⋊ Σ_{n}  B_{0}
2F_{4}(q) 3  2  v_{3}(q+1)  2e  GL_{2}(3)  B_{0} ∪ B_{∗}
E_{n}(q) 5 n = 6, 7, p^{2}|(q−1) n  v_{5}(q−1)  4e + 1  W (E_{n})  B_{0}
E_{n}(q) 7 n = 7, 8, p^{2}|(q−1) n  v_{7}(q−1)  6e + 1  W (E_{n})  B_{0}
E_{8}(q) 5 v_{5}(q^{2} + 1) ≥ 2  4 v_{5}(q^{4}−1)  4e  (4 ∘ 2^{1+4}).Σ_{6} B_{0} ∪ B_{∗}

Table 4.2. In this table, e is such that p^{e} is the exponent of A, and we restrict to the cases where e ≥ 2. In all cases except when Γ ≅ PSL_{p}(q), A is homocyclic.

by setting Ψ(ξ) = ξ·a. Thus Ψ(∑_{i=0}^{p−1} n_{i}u^{i}) = ∏_{i=0}^{p−1} u^{i}(a)^{n_{i}} for n_{i} ∈ ℤ_{p}.

Set ζ = e^{2πi/p}, R = ℤ_{p}[ζ], and 𝔭 = (1−ζ)R. Thus 𝔭 is the unique maximal ideal in R. We identify R = ℤ_{p}U/σℤ_{p}U, by sending ζ ∈ R to the class of u modulo ⟨σ⟩.

The basic properties of Ψ, and the role of Ψ(σ), are described in the following lemma. Recall that ℧^{k}(P) = ⟨g^{p^{k}} | g ∈ P ⟩, when P is a p-group and k ≥ 1.

Lemma 4.7. Assume Notation 2.1 and 4.6, where A ∈ E_{F}, and A ⋬ F is finite and not elementary abelian. Let m be such that |A/Z| = p^{m−1}. Then

(a) Im(Ψ) ∩ Z = Z_{0}⟨Ψ(σ)⟩;

(b) Ψ induces an isomorphism A/Z ≅ R/𝔭^{m−1} via the identification R = ℤ_{p}U/⟨σ⟩; and

(c) Ψ((1 − u)^{m}) = 1 and Z_{0} = ⟨Ψ((u − 1)^{m−1})⟩.

Furthermore, the following all hold.

(d) The homomorphism Ψ is surjective if and only if rk(A) ≤ p, if and only if Z is cyclic. If Ψ(σ) ∈ Fr(A), then rk(A) < p and Ψ is surjective.

(e) Either

• Ψ(σ) = 1, in which case rk(A) = p − 1, Z = Z_{0}, and Ψ induces an isomorphism A ≅ R/𝔭^{m} via the identification R = ℤ_{p}U/⟨σ⟩; or

• Ψ(σ) ∉ Fr(Z), in which case E_{F} ⊆ {A} ∪ H_{0} or E_{F} ⊆ {A} ∪ B_{0}.

(f) If Ψ(σ) ≠ 1 and Ψ(σ) ∈ Z_{0}, then m ≡ 1 (mod p − 1). If Ψ(σ) ∉ Z_{0}, then μ(Aut_{F}^{∨}(S)) = Δ_{m−1}; and either m ≡ 1 (mod p − 1) and E_{F} ∖ {A} = B_{0}, or m ≡ 0 (mod p − 1) and E_{F} ∖ {A} = H_{0}.

(g) If Ψ is not surjective, then A is homocyclic.

Proof. Set A = A/Fr(A) for short. For B ≤ A or g ∈ A, let B ≤ A or g ∈ A denote their images in A under projection. Let Ψ: ℤ_{p}U ⟶ A be the composite of Ψ followed by projection to A.

(a) Since (1 − u)ℤ_{p}U + σℤ has index p in ℤ_{p}U,

Im(Ψ) = Ψ((1 − u)ℤ_{p}U)⟨Ψ(σ)⟩⟨Ψ(1)⟩ = S′⟨Ψ(σ)⟩⟨a⟩,

where a^{p} ∈ S′⟨Ψ(σ)⟩. Since a ∉ Z and Ψ(σ) ∈ Z, we have

Im(Ψ) ∩ Z = C_{Im(Ψ)}(U) = C_{S′⟨Ψ(σ)⟩}(U) = (S′ ∩ Z)·⟨Ψ(σ)⟩ = Z_{0}⟨Ψ(σ)⟩.

(b,c) Since Ψ(σ) ∈ Z, Ψ induces a homomorphism from ℤ_{p}U/⟨σ⟩ ≅ R to A/Z, which is onto since A = Z·Im(Ψ) by Lemma 3.7(d). Since |A/Z| = p^{m−1} by assumption (and since 𝔭 is the unique maximal ideal in R that contains p), we have A/Z ≅ R/𝔭^{m−1}. Hence Ψ((u − 1)^{m−1}) ∈ Z and Ψ((u − 1)^{m−2}) ∉ Z, and the latter implies that Ψ((u − 1)^{m−1}) ≠ 1. Thus in all cases (and since |Z_{0}| = p), Ψ((u − 1)^{m}) = 1 and ⟨Ψ((u − 1)^{m−1})⟩ = Z_{0}.

(d) If rk(A) ≤ p, then rk(A) ≤ p, and by [CrOS, Proposition 3.7(a)], A|_{U} is indecomposable. Hence Ψ is onto in this case, and so Ψ is also onto. Conversely, if rk(A) > p = rk(ℤ_{p}U), then Ψ is clearly not surjective.

By Lemmas 3.2(b) and 3.3(c), Ω_{1}(Z) = C_{Ω_{1}(A)}(U) has rank 1 if and only if rk(Ω_{1}(A)) ≤ p. Hence Z is cyclic if and only if rk(A) ≤ p.

If Ψ(σ) ∈ Fr(A), then rk(Im(Ψ)) ≤ p − 1. Hence A has no nontrivial Jordan block of rank p, and by [CrOS, Proposition 3.7(a)] again, A is indecomposable as an 𝔽_{p}U-module. So rk(A) = rk(A) < p, and Ψ is onto.

(e) If Ψ(σ) ∈ Fr(Z) ≤ Fr(A), then Ψ is surjective by (a), so Z = Z_{0}⟨Ψ(σ)⟩ ≤ Z_{0}·Fr(Z), and hence Z = Z_{0} and Ψ(σ) ∈ Fr(Z_{0}) = 1. Thus Ψ factors through a surjection Ψ^{∗}: ℤ_{p}U/⟨σ⟩ ≅ R ⟶ A, and induces an isomorphism A ≅ R/I for some ideal I in R. Since 𝔭 is the only prime ideal in R of p-power index (and |R/𝔭| = p), and since |A| = p^{m−1}|Z| = p^{m} (recall Z = Z_{0} by (a)), we have I = 𝔭^{m}.

Since E_{F} ⊈ {A} (Notation 4.6), x^{p} = 1 by Notation 4.6 and Lemma 4.1. For each b ∈ A,

(bx)^{p} = (bx)^{p}x^{−p} = b·^{x}b·^{x^{2}}b ⋯^{x^{p−1}}b = ∏_{i=0}^{p−1} u^{i}(b).

If Ψ(σ) = ∏_{i=0}^{p−1} u^{i}(a) ∉ Fr(Z), then ∏_{i=0}^{p−1} u^{i}(b) ≠ 1 for each b ∈ A ∖ ZS′ = ⋃_{i=1}^{p−1} a^{i}ZS′. So by Lemma 2.5, no member of H_{∗} ∪ B_{∗} can be essential, and E_{F} ⊆ {A} ∪ H_{0} ∪ B_{0}.

(f) Assume A ⋬ F, and thus E_{F} ⊈ {A}. Fix P ∈ E_{F} ∩ (H ∪ B) and α ∈ Aut_{F}^{(P)}(S) ≤ Aut_{F}^{∨}(S) (Lemma 4.4).

Set μ(α) = (r, s), and let t be as in Lemma 4.3(b). Thus s ≡ tr^{m−1} (mod p) and α(a) ≡ a^{t} (mod ZS′), so α(a) = Ψ(ξ) for some ξ ≡ t (mod ⟨1 − u, p⟩). Also, α(x) ∈ x^{r}A, so α(u^{i}(g)) = u^{ri}(α(g)) for all i and g ∈ A. Thus

α(Ψ(σ)) = ∏_{i=0}^{p−1} α(u^{i}(a)) = ∏_{i=0}^{p−1} u^{ri}(α(a)) = Ψ(∑_{i=0}^{p−1} ξu^{ri})

= Ψ(ξσ) ≡ Ψ(tσ) . (mod Ψ(pσ)) (1)

In other words, α(Ψ(σ)) ≡ Ψ(σ)^{t} (mod ⟨Ψ(σ)^{p}⟩).

If Ψ(σ) ≠ 1 and Ψ(σ) ∈ Z_{0}, then t ≡ s (mod p) by (1) (and by definition of μ), and hence r^{m−1} ≡ 1 (mod p). Since this holds for arbitary α and hence for arbitary r prime to p by [CrOS, Lemma 2.6(a)] and since P ∈ E_{F} ∩ (H ∪ B), it follows that m ≡ 1 (mod p − 1).

Now assume Ψ(σ) ∉ Z_{0}. By (1) and since [α, Z] ≤ Z_{0}, we have t ≡ 1 and s ≡ r^{m−1}. Since this holds for arbitrary α ∈ Aut_{F}^{(P)}(S) (in particular, for arbitrary r prime to p), it follows that μ(Aut_{F}^{(P)}(S)) ≤ μ(Aut_{F}^{∨}(S)) ≤ Δ_{m−1}, with equality by Lemma 4.4. So by Lemma 4.4, P ∉ H_{∗} ∪ B_{∗}, and either P ∈ H_{0} and Δ_{m−1} = Δ_{−1} (so m ≡ 0 (mod p − 1)); or P ∈ B_{0} and Δ_{m−1} = Δ_{0} (so m ≡ 1 (mod p − 1)).

(g) Assume that Ψ is not onto, and hence by (d) that rk(A) ≥ p + 1 and Ψ(σ) ∉ Fr(A). Let k ≥ 2 be such that A has exponent p^{k}. If A/Z has strictly smaller exponent, then 1 ≠ ℧^{k−1}(A) ≤ Z, and thus ℧^{k−1}(A) is an 𝔽_{p}G-submodule of the minimally active, indecomposable module Ω_{1}(A) upon which U acts trivially. If rk(A) = p + 1, this contradicts Lemma 3.5, while if rk(A) ≥ p + 2, this is impossible since Ω_{1}(A) is simple by [CrOS, Proposition 3.7(c)]. Thus A/Z ≅ R/𝔭^{m−1} also has exponent p^{k} ≥ p^{2}, and hence m − 1 ≥ p. So by (c), and since (u − 1)^{p} ∈ pℤ_{p}U, we have Z_{0} = ⟨Ψ((u − 1)^{m−1})⟩ ≤ Fr(A).

Now, rk(A/Z) = rk(R/𝔭^{m−1}) = p − 1 since m ≥ p, and rk(Z) = rk(C_{Ω_{1}(A)}(U)) = rk(A) − (p − 1) by Lemma 3.3(c). If Z_{0} is a direct factor in Z, then rk(Z/Z_{0}) = rk(Z) − 1, so rk(A/Z_{0}) ≤ rk(A/Z) + rk(Z/Z_{0}) = rk(A) − 1. Thus no minimal generating set for A/Z_{0} lifts to a generating set for A, so Z_{0} ≰ Fr(A), which contradicts what we just showed. Thus Z_{0} is not a direct factor in Z, and so E_{F} ∩ H = ∅ by Lemma 2.5(a).

In particular, m ≡ 1 (mod p − 1) by (f), and hence A/Z ≅ R/𝔭^{m−1} is homocyclic of rank p − 1 and exponent p^{k}. Thus A and A/Z are both ℤ/p^{k}-modules and A/Z is free, so A ≅ Z × (A/Z) as abelian groups. Since ℧^{k−1}(A) ∩ Z = C_{℧^{k−1}(A)}(U) ≠ 1, this shows that rk(℧^{k−1}(A)) ≥ p.

If A is not homocyclic, then ℧^{k−1}(A) < Ω_{1}(A) is a nontrivial proper 𝔽_{p}G-submodule, where Ω_{1}(A) is faithful, minimally active, and indecomposable by Lemma 3.2(b). Hence rk(A) = dim(Ω_{1}(A)) = p + 1, since Ω_{1}(A) is simple if dim(Ω_{1}(A)) ≥ p + 2 by [CrOS, Proposition 3.7(c)]. So dim(℧^{k−1}(A)) ≤ p − 1 by Lemma 3.5. This contradicts what was shown in the last paragraph, and we conclude that A is homocyclic. □

Lemma 4.8. Let p be an odd prime, let U be a group of order p, and set σ = ∑_{u∈U} u ∈ ℤU. Then for each 1 ≠ u ∈ U and each k ≥ 1,

(u − 1)^{k(p−1)} ≡ (−1)^{k−1}(p^{k−1}σ − p^{k}) (mod p^{k}(u − 1)ℤU).

Proof. Since \binom{p−1}{k} ≡ (−1)^{k} (mod p) for each 0 ≤ k ≤ p − 1, we have (u − 1)^{p−1} ≡ σ (mod pℤU). Hence

(u − 1)^{p−1} ≡ σ − p (mod p(u − 1)ℤU)  (1)

since they are congruent modulo p and modulo u − 1. This proves the lemma when k = 1. When k > 1, (1) together with the congruence for (u − 1)^{(k−1)(p−1)} give

(u − 1)^{k(p−1)} = (u − 1)^{p−1}·(u − 1)^{(k−1)(p−1)}
≡ (u − 1)^{p−1}·(−1)^{k−2}(p^{k−2}σ − p^{k−1})  (mod (u − 1)^{p−1}·p^{k−1}(u − 1))
≡ (σ − p)·(−1)^{k−2}(p^{k−2}σ − p^{k−1})  (mod p(u − 1)(p^{k−2}σ − p^{k−1}))
= (−1)^{k−1}(p^{k−1}σ − p^{k});

and the congruences hold modulo p^{k}(u−1) since p(u−1) divides (u−1)^{p} and (u−1)σ = 0. □

Proposition 4.9. Assume the notation of 2.1, 2.9, and 4.6. Assume also that A is finite and not elementary abelian, that A ∈ E_{F}, and that O_{p}(F) = 1. Let m ≥ 3 be such that |A/Z| = p^{m−1}, and let k ≥ 2 be such that A has exponent p^{k}. Then one of the following holds, as summarized in Table 4.3, where G = Aut_{F}(A).

(a) If Ψ(σ) = 1, then Ψ is onto, Ker(Ψ) = ⟨σ, (u − 1)^{m}⟩, rk(A) = p − 1, Z = Z_{0} = ⟨Ψ((u − 1)^{m−1})⟩, and A ≅ R/𝔭^{m} as ℤ_{p}U-modules.

(b) If Ψ(σ) ∉ Fr(Z) and A is homocyclic, then rk(A) ≥ rk(Im(Ψ)) = p, rk(Z) = rk(A) − p + 1, and Im(Ψ) and Z are both direct factors in A and homocyclic of exponent p^{k}. Also, E_{F} = {A} ∪ B_{0}. Either Ψ is onto and rk(A) = p, or Ψ is not onto and rk(A) > p. If rk(A) ≥ p + 2, then A/Fr(A) ≅ Ω_{1}(A) are irreducible 𝔽_{p}[Aut_{F}(A)]-modules.

(c) If Ψ(σ) ∉ Fr(Z) and A is not homocyclic, then Ψ is onto, rk(A) = p − 1, m ≡ 1 (mod p − 1), Ker(Ψ) = ⟨p^{k}, p^{k−1} − ℓσ⟩ for some ℓ prime to p, and E_{F} = {A} ∪ H_{0}. Also, A ≅ (C_{p^{k−1}})^{p−2} × C_{p^{k}}, where ℧^{k−1}(A) = Z = Z_{0} = ⟨Ψ(σ)⟩. If k = 2, then ℓ ≢ 1 (mod p).

Case  (a)  (b)  (c)
Ψ(σ)  Ψ(σ) = 1  Ψ(σ) ∉ Fr(Z)

A homocyclic?

yes if (p − 1) | m
no if (p − 1) ∤ m

yes no

Ψ onto? yes

yes if rk(A) = p
no if rk(A) > p

yes

rk(A)  p − 1  r ≥ p  p − 1
Ker(Ψ)  ⟨(u − 1)^{m}, σ⟩  p^{k}ℤ_{p}U  ⟨p^{k}, p^{k−1} − ℓσ⟩ (p ∤ ℓ)
A  ≅ R/𝔭^{m}  ≅ (C_{p^{k}})^{r}  ≅ (C_{p^{k−1}})^{p−2} × C_{p^{k}}
Z  = Z_{0}  ≅ (C_{p^{k}})^{r−p+1}  = Z_{0}
Z_{0}  ⟨Ψ((u − 1)^{m−1})⟩ ⟨Ψ(p^{k−1}σ)⟩  ⟨Ψ(p^{k−1})⟩ = ⟨Ψ(σ)⟩
m  k(p − 1) + 1  (k − 1)(p − 1) + 1
E_{F}∖{A}  (see Table 4.1)  B_{0}  H_{0}

Table 4.3

Proof. If E_{F} = {A}, then A ⊴ F by Proposition 1.3(c), contradicting the assumption that O_{p}(F) = 1. Thus E_{F} ⫌ {A}.

Case 1: Ψ(σ) ∈ Fr(Z). In this case, Ψ is surjective by Lemma 4.7(d) and since Fr(Z) ≤ Fr(A). By Lemma 4.7(e), Ψ(σ) = 1, rk(A) = p − 1, Z = Z_{0}, and A ≅ R/𝔭^{m}. In particular, Ker(Ψ) = ⟨σ, (u − 1)^{m}⟩, while Z_{0} = ⟨Ψ((u − 1)^{m−1})⟩ by Lemma 4.7(c). We are thus in the situation of (a).

Case 2: Ψ(σ) ∉ Fr(Z) and A is homocyclic. Recall that k ≥ 2 is such that A has exponent p^{k}.

If rk(A) < p, then Ψ is onto by Lemma 4.7(d), and Ω_{1}(Z) = C_{Ω_{1}(A)}(U) has rank 1 by Lemma 3.3(c) and since Ω_{1}(A) is minimally active and indecomposable by Lemma 3.2(b). Thus Z is cyclic, and since A is homocyclic of rank at least 2, A/Z ≅ R/𝔭^{m−1} also has exponent p^{k} ≥ p^{2}. Hence rk(A) = rk(A/Z) = p − 1. Also, (A/Z)/℧^{k−1}(A/Z) ≅ (C_{p^{k−1}})^{p−1}, so |Z| = p, and Z = Z_{0}. Thus |A| = |A/Z|·|Z| = p^{m}, and m ≡ 0 (mod p − 1) since A is homocyclic of rank p − 1. But then Ψ(σ) ∉ Z_{0} by Lemma 4.7(f), a contradiction.

Thus rk(A) ≥ p, and Ψ(σ) ∉ Fr(A) by Lemma 4.7(d). So the homomorphism Ψ: 𝔽_{p}U ⟶ A/Fr(A) is injective, and A/Fr(A) contains a Jordan block Im(Ψ) of rank p. Since A is homocyclic of exponent p^{k}, |Im(Ψ)| ≥ p^{pk}, and thus Ker(Ψ) = p^{k}ℤ_{p}U. So Im(Ψ) ≅ ℤ/p^{k}U. Also,

p^{m−1} = |A/Z| = |Im(Ψ)/⟨Ψ(σ)⟩| = p^{k(p−1)},

and so m = k(p − 1) + 1 ≡ 1 (mod p − 1).

Now, Z_{0} = ⟨Ψ(p^{k−1}σ)⟩, and Ψ(σ) ∉ Z_{0} since k ≥ 2. Hence E_{F} ∖ {A} = B_{0} by Lemma 4.7(f). Also, Ψ is surjective if and only if rk(A) = p (Lemma 4.7(d)), and we are in the situation of case (b).

Case 3: Ψ(σ) ∉ Fr(Z) and A is not homocyclic. Set k′ = [m/(p − 1)]. Since m ≡ 0, 1 (mod p − 1) by Lemma 4.7(f), A/Z ≅ R/𝔭^{m−1} has exponent p^{k′}, and hence ℧^{k′}(A) ≤ Z. So p^{k′}(u − 1) ∈ Ker(Ψ).

Now, Ψ is onto by Lemma 4.7(g), and hence rk(A) ≤ p and Z is cyclic by Lemma 4.7(d). If Ψ(σ) ∉ Z_{0}, then Z_{0} < Z and is not a direct factor, so E_{F} ∩ H = ∅ by Lemma 2.5(a). Thus

Ψ onto ⟹ Ψ(σ) ∈ Z_{0} or E_{F} ∩ H = ∅.  (1)

Case 3.1: ℧^{k′}(A) ≠ 1. Since ℧^{k′}(A) ≤ Z is invariant under the action of G = Aut_{F}(A), Lemma 2.7 implies that ℧^{k′}(A) = Z_{0} and E_{F} ∩ H ≠ ∅. In particular, ℧^{k′}(A) = ⟨Ψ(p^{k′})⟩ has order p, and Ψ(σ) ∈ Z_{0} by (1). Thus ⟨Ψ(σ)⟩ = Z_{0} = ⟨Ψ(p^{k′})⟩, so p^{k′} − ℓσ ∈ Ker(σ) for some ℓ prime to p.

Now, rk(A) < p by Lemma 4.7(d) and since Ψ(σ) ∈ Z_{0} ≤ Fr(A). Also, Z = Z_{0}⟨Ψ(σ)⟩ = Z_{0} by Lemma 4.7(a), and m ≡ 1 (mod p − 1) by Lemma 4.7(f) and since Ψ(σ) ∈ Z_{0}. So |A| = |A/Z|·|Z| = p^{m} = p^{k′(p−1)+1}, and A/℧^{k′}(A) has exponent k′, rank at most p − 1, and order p^{k′(p−1)}. This proves that A/℧^{k′}(A) is homocyclic of rank p − 1 and exponent p^{k′}, and hence that A ≅ (C_{p^{k′}})^{p−2} × C_{p^{k′+1}}. Thus k′ = k − 1 (recall A has exponent p^{k}). This also shows that ⟨p^{k−1}, σ⟩ has index p in Ker(Ψ), and hence (since p^{k−1} − ℓσ is in the kernel) that Ker(Ψ) = ⟨p^{k}, p^{k−1} − ℓσ⟩.

If k = 2 and ℓ ≡ 1 (mod p), then Ker(Ψ) = ⟨p^{2}, p − σ⟩, and (u − 1)^{p−1} ∈ Ker(Ψ) by Lemma 4.8. But then Ψ((u − 1)^{p−2}) ∈ Z where Z = Z_{0} = ℧^{1}(A), and this is impossible since A/℧^{1}(A) has rank p − 1.

Finally, E_{F} = {A} ∪ H_{0} by Lemma 4.7(e) and since E_{F} ∩ H ≠ ∅. We are thus in the situation of case (c).

Case 3.2: ℧^{k′}(A) = 1. Since Ψ(σ) ∈ Z_{0} ∖ 1 or E_{F} ∩ H = ∅ by (1), we have m ≡ 1 (mod p−1) by Lemma 4.7(f). Thus m−1 = k′(p−1), so 1 ≠ Z_{0} = ⟨Ψ((1 − u)^{m−1})⟩ = ⟨Ψ(p^{k′−1}σ)⟩ by Lemma 4.8. Since p^{k′−1}σ ∉ Ker(Ψ), we have |Z| ≥ |Ψ(σ)| = p^{k′}, and |A| = |A/Z|·|Z| ≥ p^{m−1+k′} = p^{k′p}. Hence A = Im(Ψ) is homocyclic of rank p and exponent p^{k′}, contradicting our assumption. □

We now give some examples to show that all cases listed in Proposition 4.9 can occur.

Example 4.10. We list here some examples of pairs (G, A) satisfying the hypotheses of Theorem 4.5. In all cases, we assume that G ∈ U and U ∈ Syl_{p}(G). By Lemma 3.2(b), Ω_{1}(A) and A/Fr(A) must be minimally active and indecomposable.

(a): Each ℚ_{p}G-module of dimension p − 1 whose restriction to U is isomorphic to the canonical action on ℚ_{p}(ζ) can be used to construct homocyclic examples of arbitrary exponent, by adding scalars as needed to meet one of the conditions in Table 4.1. More interesting are examples where A is not homocyclic. By Proposition 4.9, Ω_{1}(A) and A/Fr(A) must be not only minimally active and indecomposable of dimension p − 1, but also not simple. By Table 6.1, this occurs only when A_{p} ≤ G ≤ Σ_{p}×C_{p−1}, SL_{2}(p) ≤ G ≤ GL_{2}(p), or PSL_{2}(p) ≤ G ≤ PGL_{2}(p)×C_{p−1}. By Proposition 3.8(a), for each minimally active, indecomposable 𝔽_{p}G-module V of dimension p − 1, there is a ℤ_{p}G-lattice Λ such that Λ/pΛ ≅ V. If 0 ≠ V_{0} < V is a nontrivial proper 𝔽_{p}G-submodule, and Λ_{0} < Λ is such that pΛ < Λ_{0} and Λ_{0}/pΛ ≅ V_{0}, then we can take A ≅ Λ_{0}/p^{k}Λ for arbitrary k ≥ 2.

(b): These are homocyclic, and there are many such examples, obtained from the 𝔽_{p}Gmodules in Table 6.1 of dimension at least p (all of them are reductions of lattices in ℚ_{p}G-modules). Lemma 3.2(c) together with Theorem 4.5 imply, very roughly, that each 𝔽_{p}G-module that yields simple fusion systems with elementary abelian A and with E_{F} ⊆ {A} ∪ B will also give simple fusion systems with A of exponent p^{k} for arbitary k > 1. Some of the resulting fusion systems are realizable (see Table 4.2), while “most” are exotic.

(c): Fix an odd prime p, k ≥ 2, and ℓ prime to p such that ℓ ≢ 1 (mod p) if k = 2. Set G = Σ_{p} × C_{p−1} and G_{0} = O^{p′}(G) ≅ A_{p}, and set U = ⟨(1 2 ⋯ p)⟩ ∈ Syl_{p}(G). Let Λ ≅ (ℤ_{p})^{p} be the ℤ_{p}G-lattice upon which Σ_{p} acts by permuting a ℤ_{p}-basis {e_{1}, …, e_{p}}, and where the factor C_{p−1} acts via multiplication by (p − 1)-st roots of unity in ℤ_{p}^{×}. Now define

A = Λ/⟨p^{k}Λ, p^{k−1}e_{i} − ℓ(e_{1} + ⋯ + e_{p}) | 1 ≤ i ≤ p⟩ ,

and let e_{i} ∈ A be the class of e_{i} ∈ Λ. This defines a finite ℤ_{p}G-module of rank p − 1 and exponent p^{k}, as described in the last column in Table 4.3, where Ψ: ℤ_{p}U ⟶ A is defined by setting Ψ(ξ) = ξ·e_{1}.

Set Z = C_{A}(U). Note that |p^{k−1}A| = p, and A/p^{k−1}A ≅ Λ/⟨p^{k−1}Λ, e_{1} + ⋯ + e_{p}⟩ ≅ ℤ_{p}[ζ]/(p^{k−1}). Hence Z ≥ p^{k−1}A, and |Z| > p only if p^{k−2}(e_{1} + 2e_{2} + ⋯ + pe_{p}) ∈ Z. But this last is the case only if p^{k−2}((e_{1} + ⋯ + e_{p}) − pe_{1}) = 0, which is not possible since we assumed that either k ≥ 3 or ℓ ≢ 1 (mod p). Thus |Z| = p in all cases.

Now set S = A ⋊ U, and set G^{∨} = N_{G}(U). (Note that Z = Z_{0} in the notation of 2.1.) Define μ_{A}: G^{∨} ⟶ Δ as in Notation 2.9. One easily checks that μ_{A}(G^{∨}) = Δ. Set G = G_{0}μ_{A}^{−1}(Δ_{−1}). This now defines an action which satisfies the conditions in Theorem 4.5, including condition (d.iii′′).

We now combine Theorem 4.5 with Proposition 4.9 to prove our main result on simple fusion systems over finite p-groups with an abelian subgroup of index p and exponent at least p^{2}. Recall that E_{F} is the set of essential subgroups in a fusion system F (Definition 1.2), and that 𝒢_{p}^{∧} is a certain class of finite groups (Definition 3.1).

Theorem A. Fix an odd prime p.

(a) Let F be a simple fusion system over a finite nonabelian p-group S with an abelian subgroup A < S of index p such that A ∈ E_{F}. Let k be such that A has exponent p^{k}, and assume k ≥ 2 (A is not elementary abelian). Set G = Aut_{F}(A), U = Aut_{S}(A) ∈ Syl_{p}(G), V = Ω_{1}(A), and V_{0} = ℧^{k−1}(A) ≤ V. Let G_{(V)}^{∨} = {α ∈ N_{G}(U) | [α, C_{V} (U)] ≤ [U, V]} and μ_{V} : G_{(V)}^{∨} ⟶ Δ be as in Notation 2.9. Let Ψ: ℤ_{p}U ⟶ A be as in Notation 4.6. Then G ∈ 𝒢_{p}^{∧}, restriction defines an isomorphism G ≅ Aut_{F}(V), and for some t ∈ {0, −1},

V is a faithful, minimally active, indecomposable 𝔽_{p}G-module, and [G, V_{0}] = V_{0}. Also, one of the cases in Table 4.4 holds, where V has no 1-dimensional submodule in cases (iv′) and (iv′′).

(∗_{fin})

Tbl.4.1 Tbl.4.3 dim(V) r = dim(V_{0})  Ker(Ψ)  μ_{V} (G_{(V)}^{∨})  G =  E_{F}∖{A}
(b)  ≥ p  V_{0} = V  ⟨p^{k}⟩
(iv′′)  1 ≤ r ≤ p − 1  ≥ Δ_{0} O^{p′}(G)μ_{V}^{−1}(Δ_{0})  B_{0}
(iv′)  ⋃_{i∈I} B_{i}
(i)  V_{0} = V ⟨p^{k}, σ,  Δ  O^{p′}(G)·G_{(V)}^{∨} H_{0} ∪ B_{∗}
(ii)  (a)  p − 1  p^{k−1}(u − 1)^{r}⟩  Δ  O^{p′}(G)·G_{(V)}^{∨} B_{0} ∪ H_{∗}
(iii′)  r = p − 2  ⋃_{i∈I} H_{i}
1 ≤ r ≤ p − 1  ≥ Δ_{−1} O^{p′}(G)μ_{V}^{−1}(Δ_{−1})
(iii′′) (c)  p − 1  r = 1  ⟨p^{k}, p^{k−1} − ℓσ⟩  H_{0}

Table 4.4

(b) Conversely, assume that G ∈ 𝒢_{p}^{∧} and U ∈ Syl_{p}(G), and that V is a faithful, minimally active, indecomposable 𝔽_{p}G-module satisfying (∗_{fin}) for some submodule 0 ≠ V_{0} ≤ V, where G^{∨} is the subgroup of all elements g ∈ N_{G}(U) such that [g, C_{V} (U)] ≤ [U, V]. Then for each k ≥ 2, there is a simple fusion system F over a finite p-group S containing an abelian subgroup A of index p, and such that G ≅ Aut_{F}(A), A has exponent p^{k}, Ω_{1}(A) ≅ V and ℧^{k−1}(A) ≅ V_{0} as 𝔽_{p}G-modules, and with E_{F} ∖ {A} as described in Table 4.1. Furthermore, any other simple fusion system with these properties, and with the same essential subgroups, is isomorphic to F.

Proof. (a) Set U = Aut_{S}(A) ∈ Syl_{p}(G) and Z = C_{A}(U).

Under the above assumptions, G ∈ 𝒢_{p}^{∧} by Lemma 3.2(a), V is faithful, minimally active, and indecomposable by Lemma 3.2(b), and dim(V) = rk(A) ≥ p − 1 by Proposition 4.9. In particular, restriction induces a monomorphism G = Aut_{F}(A) ⟶ Aut_{F}(V), and this is an isomorphism by the extension axiom (Definition 1.1) and since A = C_{S}(V). From now on, we identify G = Aut_{F}(V).

Set G_{(A)}^{∨} = {α ∈ N_{G}(U) | [α, C_{A}(U)] ≤ [U, A]}. Thus G_{(A)}^{∨} ≤ G_{(V)}^{∨}, and μ_{A}: G_{(A)}^{∨} ⟶ Δ is as in Notation 2.9. For some Δ_{x} ∈ {Δ, Δ_{0}, Δ_{−1}}, μ_{A}(G_{(A)}^{∨}) ≥ Δ_{x} and G = O^{p′}(G)μ_{A}^{−1}(Δ_{x}) by Theorem 4.5(d). So the same holds if we replace G_{(A)}^{∨} by G_{(V)}^{∨} and μ_{A} by μ_{V}.

Assume we are in case (a) of Table 4.3. Then Z = Z_{0} has order p, so |A| = p^{m}, and m ≡ 0 (mod p − 1) if and only if A is homocyclic. Also, Ψ(σ) = 1. Thus we are in case (i), (iii′′), (iv′), or (iv′′) of Table 4.1 if A is homocyclic, or in case (ii), (iii′), (iii′′), or (iv′′) if A is not homocyclic. Since m ≡ 0 (mod p − 1) and Ψ(σ) = 1 when A is homocyclic, (iv′′) is a special case of (iv′). The other information follows from the two earlier tables.

Now assume we are in case (b) or (c) in Table 4.3. Then Ψ(σ) ∉ Fr(Z), so this corresponds to case (iii′′) or (iv′′) in Table 4.1. Since E_{F} ∖ {A} = H_{0} in cases (c) and (iii′′), and E_{F} ∖ {A} = B_{0} in cases (b) and (iv′′), these are the only possible correspondences. In case (b), A is homocyclic, and so we have V_{0} = V in Table 4.4. In case (c), A ≅ (C_{p^{k−1}})^{p−2} × C_{p^{k}}, so V = Ω_{1}(A) has a submodule V_{0} = ℧^{k}(A) of rank 1.

There is a surjective homomorphism ψ: A ⟶ V_{0} of ℤ_{p}G-modules, defined by setting ψ(a) = a^{p^{k−1}}. Since [G, A] = A by Theorem 4.5(c), we have [G, V_{0}] = V_{0}.

If V = Ω_{1}(A) has a 1-dimensional 𝔽_{p}G-submodule W, then W ∈ C_{V} (U) ≤ C_{A}(U), so W = Z_{0} = C_{A}(U) ∩ [U, A] by Theorem 4.5(b), which is impossible in cases (iv′) and (iv′′) of Theorem 4.5(d).

(b) Fix G ∈ 𝒢_{p}^{∧}, U ∈ Syl_{p}(G), and t ∈ {0, −1}, and let V be an 𝔽_{p}G-module that satisfies (∗_{fin}), where G^{∨} is the subgroup of all elements g ∈ N_{G}(U) such that [g, C_{V} (U)] ≤ [U, V]. Fix k ≥ 2.

Assume we have chosen a finite ℤ_{p}G-module A such that Ω_{1}(A) ≅ V as 𝔽_{p}G-modules, and such that either dim(V) = p − 1 and |C_{A}(U)| = p, or dim(V) ≥ p and A is homocyclic. Let G_{(A)}^{∨} ≤ G_{(V)}^{∨} be as in the proof of (a). If rk(V) = rk(A) = p − 1, then since C_{A}(U) ≅ C_{V} (U) has order p, G_{(A)}^{∨} = G_{(V)}^{∨} = N_{G}(U) and μ_{A} = μ_{V}. So the properties of G_{(V)}^{∨} and μ_{V} in Table 4.4 also hold for G_{(A)}^{∨} and μ_{A}.

If rk(V) = rk(A) ≥ p and A is homocyclic, then μ_{V} (G_{(V)}^{∨}) ≥ Δ_{0}, and G = O^{p′}(G)μ_{V}^{−1}(Δ_{0}). In such cases, we could have G_{(A)}^{∨} < G_{(V)}^{∨}, but for each α ∈ μ_{V}^{−1}(Δ_{0}) of order prime to p, α acts trivially on C_{V} (U) ≅ Ω_{1}(C_{A}(U)) by definition of μ_{V} (and Lemma A.1), and hence also acts trivially on C_{A}(U) (see [G, Theorem 5.2.4]). Thus μ_{V}^{−1}(Δ_{0}) = μ_{A}^{−1}(Δ_{0}), and so the information in Table 4.4 still holds if we replace G_{(V)}^{∨} and μ_{V} by G_{(A)}^{∨} and μ_{A}.

Cases 4.9(a,b): Assume that we are in Case (a) or (b) in Proposition 4.9. By Proposition 3.8(a), there is a ℤ_{p}G-lattice Λ such that Λ/pΛ ≅ V as 𝔽_{p}G-modules. Let Λ_{0} ≤ Λ be such that Λ_{0} ≥ pΛ and Λ_{0}/pΛ ≅ V_{0}, and set A = Λ_{0}/p^{k}Λ. Set S = A ⋊ U, Z = Z(S) = C_{A}(U), S′ = [S, S] = [U, A], and Z_{0} = Z ∩ S′.

We first check that conditions (a)–(d) in Theorem 4.5 all hold. Condition (d) follows immediately from (∗_{fin}). Condition (a) (|Z_{0}| = p) follows from Lemma 3.7(c) and since A is defined to be a quotient group of a ℤ_{p}G-lattice.

Assume 1 ≠ B ≤ Z is G-invariant; we claim that B = Z_{0}. If rk(A) = p − 1, then Z = Z_{0}, and there is nothing to prove. If not, then rk(A) ≥ p, we are in case (iv′′), and so V has no 1-dimensional 𝔽_{p}G-submodule. Thus rk(Z) ≥ rk(B) ≥ 2, and hence dim(V) ≥ p + 1. If dim(V) = p + 1, then by Lemma 3.5, every nontrivial 𝔽_{p}G-submodule has nontrivial action of U, contradicting the assumption B ≤ Z. If dim(V) ≥ p + 2, then V is simple by [CrOS, Proposition 3.7(c)]. So Condition 4.5(b) holds in all cases.

If rk(A) ≥ p, then V_{0} = V, A/Fr(A) ≅ V_{0}, and so [G, A] = A since [G, V_{0}] = V_{0}. If rk(A) = p − 1, then by Lemma A.5(a,b,c), Λ_{0}|_{U} ≅ ℤ_{p}[ζ] as ℤ_{p}U-modules, and the radical of Λ_{0}|_{U} has index p. Thus pΛ is contained in the radical, and in [G, A] = A since V_{0} ≅ Λ_{0}/pΛ and [G, V_{0}] = V_{0}. This proves 4.5(c).

By Theorem 4.5, there is a unique simple fusion system F over A ⋊ U such that G = Aut_{F}(A), A ∈ E_{F}, and E_{F} ∖ {A} is as described in Table 4.4. Since A is unique (up to isomorphism of ℤ_{p}G-modules) by Proposition 3.9(a) (in case 4.9(b)) or 3.9(b) (in case 4.9(a)), this shows that F is uniquely determined by V. Case 4.9(c): Assume that we are in Case (c) in Proposition 4.9. In particular, dim(V) = p−1 and dim(V_{0}) = 1. By Lemma 3.4(c), there is a projective, minimally active 𝔽_{p}G-module W > V such that dim(W) = p and thus dim(W/V) = 1.

By Proposition 3.8(b), there is a ℤ_{p}G-lattice Λ such that Λ/pΛ ≅ W, and such that Λ has a ℤ_{p}G-submodule Λ_{0} = C_{Λ}(G_{0}) of rank 1. In particular, Λ is free as a ℤ_{p}U-module since W is free as an 𝔽_{p}U-module. Let Λ_{V} < Λ be the ℤ_{p}G-sublattice of index p such that Λ_{V} /pΛ ≅ V. Define

Â = Λ/(p^{k}Λ + p^{k−1}Λ_{V} + pΛ_{0}) ≅ C_{p^{k}} × (C_{p^{k−1}})^{p−2} × C_{p} .

Then Ω_{1}(Â) is a p-dimensional 𝔽_{p}G-module, and contains a 2-dimensional submodule

Â_{0} = (p^{k−1}Λ + Λ_{0})/(p^{k}Λ + p^{k−1}Λ_{V} + pΛ_{0})
≅ (p^{k−1}Λ/(p^{k}Λ + p^{k−1}Λ_{V})) ⊕ (Λ_{0}/pΛ_{0}) ≅ (W/V) ⊕ V_{0} .

Now, V_{0} ≅ W/V as 𝔽_{p}[N_{G}(U)]-modules by Lemma 3.4(a) and since dim(W) = p. Also, O^{p′}(G) acts trivially on each of them and G = O^{p′}(G)N_{G}(U) by the Frattini argument, so V_{0} and W/V are isomorphic as 𝔽_{p}G-modules, and any 𝔽_{p}-linear isomorphism is 𝔽_{p}G-linear. Hence for fixed a ∈ Λ ∖ Λ_{V} and fixed ℓ prime to p such that ℓ ≢ 1 (mod p) if k = 2, the quotient group

A = Â/Â_{1} where Â_{1} = ⟨[p^{k−1}a − ℓσ·a]⟩ ≤ Â_{0}

is a quotient group of Â where the two summands of Â_{0} have been identified, and hence is a ℤ_{p}G-module. Here, as usual, σ = ∑_{u∈U} u ∈ ℤ_{p}U. Note that Â_{1} is independent of the choice of a, and depends on ℓ only modulo p.

Since Â/Â_{0} ≅ Λ/(p^{k−1}Λ + Λ_{0}), where Λ/Λ_{0} ≅ ℤ_{p}[ζ] as ℤ_{p}U-modules by Lemma A.5(c), we have that |C_{Â/Â_{0}}(U)| = p, and is generated by the class of p^{k−2}(u−1)^{p−2}a for 1 ≠ u ∈ U. The class of p^{k−2}(u − 1)^{p−2}a in A is fixed by U if and only if as classes in Â,

[p^{k−2}(u − 1)^{p−1}a] = [p^{k−2}(σ − p)a] = [p^{k−2}σ·a] − [p^{k−1}a] ∈ Â_{1}

(where the second equality holds by Lemma 4.8). But this fails to hold under our hypotheses: either k > 2, in which case [p^{k−2}σ·a] = 0 and [p^{k−1}a] ≠ 0 in Â; or k = 2 and ℓ ≢ 1 (mod p), in which case [σ·a] − [pa] ∉ Â_{1}. Thus in all such cases, C_{A}(U) = Â_{0}/Â_{1} and has order p.

Since V_{0} ≅ Λ_{0}/pΛ_{0} ≅ Â_{0}/Â_{1} = C_{A}(U) as 𝔽_{p}G-modules, and since V and Ω_{1}(A) are both (p − 1)-dimensional minimally active 𝔽_{p}G-modules, Lemma 3.4(d) applies to show that V ≅ Ω_{1}(A) as 𝔽_{p}G-modules.

By construction, (G, A) satisfies all of the conditions in case (c) of Proposition 4.9, as well as condition (d.iii′′) of Theorem 4.5. Since |Z| = |C_{A}(U)| = p, conditions (a) and (b) in 4.5 also hold: |Z_{0}| = p, and no nontrivial subgroup of Z is G-invariant except possibly Z_{0} = Z. Finally, 4.5(c) holds ([G, A] = A) since A/[U, A] = A/Ω_{k−1}(A) has order p, and has nontrivial action of G since G acts nontrivially on Z_{0} = ℧^{k−1}(A) (since μ_{A}(G^{∨}) ≥ Δ_{−1}).

By Theorem 4.5, there is a unique simple fusion system F over A ⋊ U such that G = Aut_{F}(A), A ∈ E_{F}, and E_{F} = {A}∪H. Since A is unique up to isomorphism of ℤ_{p}G-modules by Proposition 3.9(b), this shows that F is uniquely determined by V. □

5. Simple fusion systems over nonabelian discrete p-toral groups with abelian subgroup of index p

We now focus on the case where A and S are infinite.

Lemma 5.1. Assume Notation 2.1. If |A| = ∞ and O_{p}(F) = 1, then |A/Z| = ∞.

Proof. Let S_{0} ⊴ S denote the identity component of S. If |A/Z| < ∞, then S_{0} ≤ Z, so S_{0} is contained in (and is characterisitic in) each P ∈ E_{F}. Thus S_{0} ⊴ F by Proposition 1.3(c), so S_{0} ≤ O_{p}(F) = 1, contradicting the assumption that |A| = ∞. □

Corollary 5.2. Assume Notation 2.1. If |A| = ∞ and O_{p}(F) = 1, then A is the only abelian subgroup of index p in S.

Proof. Since |A/Z| = ∞ by Lemma 5.1, this follows from Lemma A.3. □

Lemma 5.3. Assume Notation 2.1. If |A| = ∞ and O_{p}(F) = 1, then A/Z and S′ are both discrete p-tori of rank p − 1.

Proof. Since |A/Z| = ∞ by Lemma 5.1 and C_{A/Z}(U) = Z_{2}/Z has order p by Lemma 2.4, A/Z ≅ (ℤ/p^{∞})^{p−1} by Lemma A.5(d). Also, A/Z ≅ S′ by Lemma A.3. □

Lemma 5.4. Assume Notation 2.1. If |A| = ∞ and O_{p}(F) = 1, then A = ZS′. As one consequence, each of the sets B and H consists of one S-conjugacy class.

Proof. Fix a generator u ∈ U, and define χ: A/Z ⟶ A/Z by setting χ(aZ) = [a, u]Z. Then Im(χ) = ZS′/Z, and Ker(χ) = Z_{2}/Z has order p by Lemma 2.4. Since A/Z is a discrete p-torus by Lemma 5.3, χ must be onto, and hence ZS′ = A.

Thus if P = Z⟨x⟩ and Q = Z⟨y⟩ are two members of H, where yx^{−1} ∈ A, then there are z ∈ Z and a ∈ A such that yx^{−1} = axa^{−1}x^{−1}z. Then y ∈ ^{a}xZ, so Q = ^{a}P, and P and Q are S-conjugate. A similar argument shows that all members of B are S-conjugate. □

Part of the next lemma follows from Lemma 2.7 when p is odd. But since we also need it here when p = 2, we prove it independently of the earlier lemma.

Lemma 5.5. Assume Notation 2.1, and also that |A| = ∞, O_{p}(F) = 1, and A ∉ E_{F}. Then Z = Z_{0}, E_{F} = H, and A = S′ is a discrete p-torus of rank p − 1.

Proof. Since A ∉ E_{F}, Lemmas 2.2 and 2.3 imply that E_{F} = B or E_{F} = H. If E_{F} = B, then since Z = Z(S) is normalized by Aut_{F}(S) and by Aut_{F}(P) for each P ∈ B (Z is characteristic in P by Lemma 2.5(b)), Z ⊴ F by Proposition 1.3(c), contradicting the assumption that O_{p}(F) = 1. Thus E_{F} = H.

For P = Z⟨x⟩ ∈ H, by Lemma 2.5(a), P = P_{1} × P_{2}, where P_{1} = C_{P} (O^{p′}(Aut_{F}(P))) < Z, Z = P_{1} × Z_{0}, and Z_{0} < P_{2} ≅ C_{p} × C_{p}, and where each factor P_{i} is normalized by Aut_{F}(P). If P^{∗} ∈ H is another member, then P^{∗} = ^{g}P for some g ∈ S by Lemma 5.4, and P_{1} = gP_{1} = C_{^{g}P} (O^{p′}(Aut_{F}(^{g}P))) is also normalized by Aut_{F}(^{g}P). Finally, P_{1} is normalized by Aut_{F}(S) since Aut_{F}(S) = Inn(S)·N_{Aut_{F} (S)}(P) by the Frattini argument, and thus P_{1} ⊴ F by Proposition 1.3(c). So P_{1} ≤ O_{p}(F) = 1, and hence Z = Z_{0}.

Since Z = Z_{0}, we have A = S′ by Lemma 5.4, and so A is a discrete p-torus of rank p − 1 by Lemma 5.3. □

When A is finite and p is odd, it was shown in [Ol, Lemma 2.4] that O_{p}(F) = 1 and A ∉ E_{F} imply Z = Z_{0}. When A is finite and p = 2, this is not true: for each odd prime p, the 2-fusion system of PSL_{2}(p^{2})⟨ϕ⟩, where ϕ is a field automorphism of order 2, is a counterexample. Note that in this case, S ≅ D × C_{2} for some dihedral 2-group D (whose order depends on p).

The case p = 2 is now very easy to handle.

Theorem 5.6. Let S be an infinite nonabelian discrete 2-toral group with abelian subgroup A < S of index 2. Let F be a saturated fusion system over S such that O_{2}(F) = 1. Then F is isomorphic to the 2-fusion system of SO(3) (if A is not F-essential) or of PSU(3) (if A is F-essential).

Proof. We use Notation 2.1. Then |A/Z| = ∞ by Lemma 5.1, and A is the unique abelian subgroup of index 2 in S by Corollary 5.2. Also, |Z_{0}| = 2 by Lemma 2.4, and E_{F} ∖ {A} = H or B by Lemmas 2.2 and 5.4 (and since A ⊴ F if E_{F} ⊆ {A}).

Case 1: Assume first that A ∉ E_{F}. Then by Lemma 5.5, E_{F} = H, Z = Z_{0}, and A = S′ is a discrete p-torus of rank 1. Thus A ≅ ℤ/2^{∞}, where this group is inverted by the action of S/A. Also, for each P ∈ E_{F} = H, P ≅ C_{2} × C_{2} and hence Aut_{F}(P) = Aut(P) ≅ Σ_{3}. Thus there is a unique choice of fusion system F on S, and it must be isomorphic to the fusion system of SO(3).

Case 2: Now assume that A is F-essential, and set G = Aut_{F}(A). Then Aut_{S}(A) ∈ Syl_{2}(G) has order 2, so |G| = 2m for some odd m. By Proposition A.7, we can write G = G_{1} × G_{2} and A = A_{1} × A_{2}, where G_{i} acts faithfully on A_{i} and trivially on A_{3−i} for i = 1, 2, where |G_{1}| is odd, G_{2} ≅ Σ_{3}, and A_{2} ≅ C_{2^{k}} × C_{2^{k}} for some 1 ≤ k ≤ ∞.

Now, |A_{2}| = ∞ since A_{1} ≤ Z and |A/Z| = ∞. Hence Z_{0} is not a direct factor in Z, so H ∩ E_{F} = ∅ by Lemma 2.5(a), and E_{F} = {A} ∪ B. Each α ∈ Aut_{F}(S) normalizes A and hence normalizes A_{1}. For each P ∈ B and each α ∈ Aut_{F}(P), α(Z) = Z since Z = Z(P), α|_{Z} ∈ Aut_{F}(Z) extends to α ∈ Aut_{F}(S), and thus α(A_{1}) = A_{1}. So A_{1} ⊴ F, and A_{1} ≤ O_{2}(F) = 1.

Thus A ≅ (ℤ/2^{∞})^{2} and Aut_{F}(A) = G ≅ Σ_{3}. For each P ∈ B = E_{F} ∖ {A}, P ∈ (ℤ/2^{∞}) ×_{C_{2}} Q_{8}, and the subgroup isomorphic to Q_{8} is unique. Hence Out_{F}(P) ≅ Σ_{3} is uniquely determined, and F is determined uniquely by Aut_{F}(A). So F is the 2-fusion system of PSU(3). □

We now focus on the cases where p is odd.

Proposition 5.7. Assume Notation 2.1. Assume also that p is odd, |A| = ∞, and O_{p}(F) = 1. Then A is a discrete p-torus. If rk(A) ≥ p, then Z is also a discrete p-torus, and has rank rk(A) − p + 1.

Proof. We first apply Lemma 2.8, with A_{1} = A and A_{2} the identity component of A. Since S′ is a discrete p-torus by Lemma 5.3, we have A_{2}Z ≥ ZS′ = A = A_{1} by Lemma 5.4. So by Lemma 2.8, A ≤ A_{2}Z_{0} = A_{2}, the last equality since Z_{0} ≤ S′ ≤ A_{2}. Thus A is a discrete p-torus.

Set G = Aut_{F}(A) and V = Ω_{1}(A), and choose 1 ≠ u ∈ U ∈ Syl_{p}(G). By Lemma 3.2(a,b), G ∈ 𝒢_{p}^{∧} (so |U| = p), and V is faithful, minimally active, and indecomposable as an 𝔽_{p}Gmodule. So if dim(V) = rk(A) ≥ p, then by Lemma 3.3(a,b), the action of u on V has a Jordan block of length p, and hence dim(Ω_{1}(Z)) = dim(C_{V} (U)) = rk(A) − p + 1.

Let Z_{1}, Z_{2} ≤ Z be such that Z_{1} is a discrete p-torus, Z_{2} is a finite abelian p-group, and Z = Z_{1} × Z_{2}. Since rk(A/Z) = p − 1 by Lemma 5.3, rk(Z_{1}) = rk(A) − p + 1 = dim(Ω_{1}(Z)). So Z_{2} = 1, and Z = Z_{1} is a discrete p-torus. □

Lemma 5.8. Assume Notation 2.1, and also that p is odd, |A| = ∞, and O_{p}(F) = 1. Then no proper nontrivial subgroup of S is strongly closed in F. Thus F is simple if and only if it contains no proper normal subsystem over S.

Proof. 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 1.3(c), contradicting the assumption that 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.

Fix P ∈ E_{F} ∖ {A} ⊆ B ∪ H (recall A ⋬ F). If P ∈ B, then the Aut_{F}(P)-orbit of g ∈ (Q ∩ Z_{2}) ∖ Z is not contained in A. If P ∈ H, then the Aut_{F}(P)-orbit of Z_{0} ≤ Q 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, and contains [U, A] where G_{0} is the normal closure of U in G, Q ≥ [G_{0}, A]. Since C_{A}(G_{0}) ≤ C_{A}(U) = Z and C_{A}(G_{0}) is normalized by G, C_{A}(G_{0}) ≤ Z_{0} ≤ [U, A] by Lemma 2.7. So by Lemma A.2, [G_{0}, A] ≥ C_{A}(U) = Z. Thus Q ≥ ZS′ = A, and so Q = S since Q ≰ A.

The last statement is immediate. □

Lemma 5.9. Assume we are in the situation of Notation 2.1 and 2.9, and also that |A| = ∞ and S splits over A.

(a) The kernel of μ: Aut^{∨}(S) ⟶ Δ does not contain any elements of finite order prime to p.

(b) Fix Q ∈ B∪H, and set t = 0 if Q ∈ B, t = −1 if Q ∈ H. Assume that μ(Aut_{F}^{∨}(S)) ≥ Δ_{t}. Then there are unique subgroups Z̃ ≤ Z and Q̃ ≥ Q ∩ S′ which are normalized by N_{Aut_{F} (S)}(Q), and are such that the following hold.

(i) If Q ∈ H, then Q = Z̃ × Q̃ and Q̃ ≅ C_{p} × C_{p}.

(ii) If Q ∈ B, then Z̃ = Z, Q = ZQ̃, Z ∩ Q̃ = Z_{0} = Z(Q̃), and Q̃ is extraspecial of order p^{3} and exponent p.

(iii) Thus in all cases, Out(Q̃) ≅ GL_{2}(p). If α ∈ Aut(Q) is such that α|Z̃ = Id, α(Q̃) = Q̃, α(Q ∩ A) = Q ∩ A, and α|Q̃ ∈ O^{p′}(Aut(Q̃)), then α extends to some α ∈ N_{Aut_{F}^{∨} (S)}(Q).

Proof. (a) Fix α ∈ Ker(μ) of finite order prime to p. Then α induces the identity on Z/Z_{0} since α ∈ Aut_{F}^{∨}(S), and on Z_{0} and S/A since μ(α) = (1, 1). In particular, α|_{Z} = Id by Lemma A.1, and α|_{A} is ℤ_{p}U-linear.

Fix x ∈ S ∖ A, and let ψ ∈ End(A) be the homomorphism ψ(g) = [g, x]. Then ψ commutes with α|_{A} since α(x) ∈ xA, and ψ induces an injection from Z_{i}(S)/Z_{i−1}(S) into Z_{i−1}(S)/Z_{i−2}(S) for each i ≥ 2. Since α|_{Z} = Id, this shows that α induces the identity on Z_{i}(S)/Z_{i−1}(S) for each i, and hence that α|_{Z_{i}(S)} = Id for each i by Lemma A.1 again. Thus α|_{Ω_{1}(A)} = Id since Ω_{1}(A) ≤ Z_{i}(S) for some i, so α|_{Ω_{m}(A)} = Id for each m ≥ 1 by [G, Theorem 5.2.4]. So α|_{A} = Id, and α = Id_{S} by Lemma A.1 again.

(b) This proof is essentially the same as that of [CrOS, Lemma 2.6(b)] (a similar result but with |A| < ∞). We sketch an alternative argument here.

Set K = N_{Aut_{F}^{∨} (S)}(Q) for short. By the Frattini argument and since all members of the Aut(S)-orbit of Q are S-conjugate to Q (Lemma 5.4), Aut_{F}^{∨}(S) = Inn(S)·K. Hence μ(K) ≥ Δ_{t}. Also, |N_{Inn(S)}(Q)| ≤ |N_{S}(Q)/Z| < ∞ since |N_{S}(Q)/Q| = p by Lemma 2.2, and N_{Inn(S)}(Q) = Aut_{N_{S}(Q)}(S) is normal of index prime to p in K since Inn(S) ⊴ Aut_{F}(S) has finite index prime to p by the Sylow axiom.

By the Schur-Zassenhaus theorem, there is K_{0} < K of order prime to p such that K = K_{0}·Aut_{N_{S}(Q)}(S). Set Z̃ = C_{Q}(K_{0}) and Q̃ = [K_{0}, Q].

• If Q ∈ H and μ(K_{0}) = μ(K) ≥ Δ_{−1}, then K_{0} acts nontrivially on Q/Z and on Z_{0}, and trivially on Z/Z_{0}. Hence Z̃ ≤ Z and Q̃ ∩ A = Z_{0}. Also, Q = Z̃ × Q̃ and Z = Z̃ × Z_{0} by [G, Theorem 5.2.3] (applied to the subgroups Ω_{m}(Q) for m ≥ 1), and Q̃ ≅ C_{p} ×C_{p} since S splits over A. In particular, Out(Q̃) ≅ GL_{2}(p).

If β ∈ K_{0} is such that μ(β) generates Δ_{−1}, then μ(β) = (r, r^{−1}) for some generator r of (ℤ/p)^{×}, so β|Q̃ acts on Q̃ ≅ C_{p} × C_{p} as the matrix ( \begin{matrix}r & 0 \\ 0 & r^{−1} \end{matrix} ) for an appropriate choice of basis. Thus Aut_{S}(Q̃)⟨β|Q̃⟩ = N_{O^{p′}(Aut(Q̃))}(Z_{0}) where O^{p′}(Aut(Q̃)) ≅ SL_{2}(p), proving (iii) in this case.

• If Q ∈ B and μ(K_{0}) = μ(K) ≥ Δ_{0}, then α acts nontrivially on Q/Z_{2}, and trivially on Z_{0} and Z/Z_{0}. Hence α acts trivially on Z by Lemma A.1, nontrivially on Z_{2}/Z, and so Z̃ = Z and Q̃ ∩ A = Z_{2} ∩ S′. Also, Q̃ is extraspecial of order p^{3} and (since S splits over A) exponent p. In particular, Out(Q̃) ≅ GL_{2}(p). By [G, Theorem 5.2.3], applied to the abelian p-groups Ω_{m}(Q/Z_{0}) for m ≥ 1, Q = Z̃Q̃ and Z̃ ∩ Q̃ = Z_{0}.

If β ∈ K_{0} is such that μ(β) generates Δ_{0}, then μ(β) = (r, 1) for some generator r of (ℤ/p)^{×}, so [β|Q̃] has order (p−1) in O^{p′}(Aut(Q̃))/Inn(Q̃) ≅ SL_{2}(p). So Aut_{S}(Q̃)⟨β|Q̃⟩ = N_{O^{p′}(Aut(Q̃))}(Q ∩ A) in this case, again proving (iii).

Since Z̃ ≤ Z ≤ C_{Q}(N_{S}(Q)) and Q̃ ≥ Q∩S′ ≥ [N_{S}(Q), Q] in all cases, we have Z̃ = C_{Q}(K) and Q̃ = [K, Q]. Thus Z̃ and Q̃ are independent of the choice of K_{0}, and are normalized by K = N_{Aut_{F}^{∨} (S)}(Q). Since Aut_{F}^{∨}(S) is normal in Aut_{F}(S) (the kernel of a homomorphism to Aut(Z/Z_{0})), we see that K is normal in N_{Aut_{F} (S)}(Q), and hence Z̃ and Q̃ are also normalized by N_{Aut_{F} (S)}(Q). These are easily seen to be the unique subgroups that satisfy the required conditions. □

When F is a fusion system over a discrete p-toral group S, then for each Q ≤ S, we define another subgroup Q^{•} ≤ S as follows. Let T be the identity component of S. If m ≥ 0 is the smallest integer such that g^{p^{m}} ∈ T for each g ∈ S, and Q^{[m]} = ℧^{m}(Q) = ⟨g^{p^{m}} | g ∈ Q⟩, then Q^{•} =^{def} Q·I(Q^{[m]})_{0}, where I(Q^{[m]}) = C_{T} (C_{Aut_{F} (T)}(Q^{[m]})) and I(Q^{[m]})_{0} is its identity component. Thus Q ≤ Q^{•} ≤ QT for each Q. See [BLO3, Definition 3.1] or [BLO6, Definition 3.1] for more detail, as well as the motivation for this construction.

Lemma 5.10. Assume the notation in 2.1, and also that |A| = ∞. For each Q ∈ B ∪ H ∪ {A, S}, Q^{•} = Q.

Proof. By Proposition 5.7, A is the identity component of S. Thus A and G play the role of T and W = Aut_{F}(T) in [BLO6, Definition 3.1]. Since Q ≤ Q^{•} ≤ QA for each Q ≤ S, we have A^{•} = A and S^{•} = S.

Now assume Q ∈ H∪B. By assumption, S/A has exponent p = p^{1}. Since Q/Z ≅ C_{p} or C_{p}^{2}

(Lemma 2.5), Q^{[1]} =^{def} ⟨g^{p} | g ∈ Q⟩ ≤ Z. Hence C_{G}(Q^{[1]}) ≥ U, and I(Q^{[1]}) =^{def} C_{A}(C_{G}(Q^{[1]})) ≤

C_{A}(U) = Z. It follows that Q^{•} ≤ Q·I(Q^{[1]}) = Q. □

We are now ready to prove our main theorem used to construct simple fusion systems over infinite discrete p-toral groups with abelian subgroup of index p.

Theorem 5.11. Fix an odd prime p. Let S be an infinite nonabelian discret p-toral group which contains an abelian subgroup A ⊴ S of index p, and let F be a simple fusion system over S. We use the notation of 2.1 and 2.9. Then the following hold:

(a) U ∈ Syl_{p}(G) and S splits over A.

(b) A is a discrete p-torus of rank at least p − 1, Z_{0} = C_{A}(U) ∩ [U, A] has order p, and A = C_{A}(U)·[U, A].

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

(d) Either

(i) E_{F} ∖ {A} = H, rk(A) = p − 1, μ_{A}(Aut_{F}^{∨}(A)) ≥ Δ_{−1}, and G = O^{p′}(G)·μ_{A}^{−1}(Δ_{−1}); or (ii) E_{F} ∖ {A} = B, rk(A) ≥ p − 1, μ_{A}(Aut_{F}^{∨}(A)) ≥ Δ_{0}, μ_{A}(Aut_{F}^{∨}(A)) = Δ_{0} if

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

Here, we regard μ_{A} as a homomorphism defined on Aut_{F}^{∨}(A).

Conversely, let S be an infinite discrete p-toral group containing an abelian subgroup A ⊴ S of index p, let G ≤ Aut(A) be such that Aut_{S}(A) ∈ Syl_{p}(G), adopt the notation in 2.1 and 2.9. Assume that (a)–(d) hold, with Aut_{F}^{∨}(A) replaced by G∩Aut^{∨}(A) and E_{F} ∖{A} replaced by E_{0} = H or B in (d). Then there is a unique simple fusion system F over S such that G = Aut_{F}(A) and E_{F} ∖ {A} = E_{0}.

Proof. We prove in Steps 1 and 3 that conditions (a)–(d) are necessary, and prove the converse in Step 2.

Step 1: Assume that F is a simple fusion system over S. We must show that conditions (a)–(d) hold. By Corollary 5.2, A is the unique abelian subgroup of index p in S.

(a,b,c) Point (a) holds by Corollary 2.6 and since A is fully automized. The last two statements in (b) hold by Lemmas 2.4 and 5.4, and (c) holds by Lemma 2.7 and since O_{p}(F) = 1. Finally, A is a discrete p-torus by Proposition 5.7, and rk(A) ≥ p − 1 by Lemma 5.3.

(d) Since A ⋬ F, there is P ∈ E_{F} ∩ (H ∪ B). Set t = 0 if P ∈ B, and t = −1 if P ∈ H.

Set H = Aut_{F}(P) and H_{0} = O^{p′}(H). By Lemma 2.5, H_{0}/Inn(P) ≅ SL_{2}(p), and acts trivially on Z/Z_{0}. Since Aut_{S}(P) ∈ Syl_{p}(H) = Syl_{p}(H_{0}), we can choose α ∈ N_{H_{0}}(Aut_{S}(P)) of order p−1 in H/Inn(P). By the extension axiom, α extends to an element of Aut_{F}(N_{S}(P)), and since P is maximal among F-essential subgroups by Lemmas 2.2 and 2.3, α = α̂|_{P} for some α̂ ∈ Aut_{F}(S). Set α_{0} = α̂|_{A} ∈ G.

Now, α_{0} induces the identity on Z/Z_{0} since α does, and α_{0} ∈ N_{G}(U) since it extends to S. Thus α_{0} ∈ Aut_{F}^{∨}(A). By Lemma 2.5, α acts as an element of SL_{2}(p) on P/Z ≅ C_{p}^{2} (if P ∈ B) or on P/P_{1} ≅ C_{p}^{2} where Z = P_{1} ×Z_{0} (if P ∈ H). Hence for some s ∈ (ℤ/p)^{×} of order p − 1, μ_{A}(α_{0}) = (s, s^{−1}) if P ∈ H, or μ_{A}(α_{0}) = (s, 1) if P ∈ B. Thus μ_{A}(Aut_{F}^{∨}(A)) ≥ Δ_{t} in either case.

Assume rk(A) ≥ p. By Proposition 5.7, Z is a discrete p-torus. Hence each element of Aut_{F}^{∨}(A) induces the identity on Z_{0} since it induces the identity on Z/Z_{0}, and μ_{A}(Aut_{F}^{∨}(A)) ≤ Δ_{0}.

Set G_{0} = O^{p′}(G)·μ_{A}^{−1}(Δ_{t}). Since Ker(μ_{A}|_{Aut_{F}^{∨} (A)}) = U by Lemma 5.9(a), and since μ_{A}(Aut_{F}^{∨}(A)) ≥ Δ_{t} by assumption, we have G ≥ μ_{A}^{−1}(Δ_{t}), and hence G ≥ G_{0}. We will show in Step 3 (with the help of the constructions in Step 2) that G = G_{0}, thus finishing the proof of (d).

Step 2: Now assume that S, A, and G are as above, and set G^{∨} = G ∩ Aut^{∨}(A). Assume that (a) and (b) hold, and also that μ_{A}(G^{∨}) ≥ Δ_{t} for some t ∈ {0, −1}. (Note that G^{∨} ≤ N_{G}(U) since each element is the restriction of an automorphism of S.) We must show that these are realized by a unique saturated fusion system F, which is simple if (c) and (d) hold. Set E_{0} = H if t = −1, or E_{0} = B if t = 0.

Set Γ = A ⋊ G, and identify S = A ⋊ U ∈ Syl_{p}(Γ). Choose a generator x ∈ U < S. Set Z = Z(S), Z_{2} = Z_{2}(S), as in Notation 2.1.

Set Q = Z⟨x⟩ if E_{0} = H, or Q = Z_{2}⟨x⟩ if E_{0} = B. Thus Q ∈ E_{0}, and each member of E_{0} is S-conjugate to Q by Lemma 5.4. Set K = Aut_{Γ}(Q). By assumption, there is α ∈ N_{G^{∨}}(U) such that μ_{A}(α) generates Δ_{t}, and α extends to some α ∈ Aut^{∨}(S) such that α(x) ∈ U. In particular, α(Q) = Q, and α ∈ Aut_{Γ}(S).

By Lemma 5.9(b), applied with F_{S}(Γ) in the role of F, there are unique subgroups Z̃ ≤ Z and Q̃ ≥ Q∩S′, both normalized by N_{Aut_{Γ}(S)}(Q), and such that Q = Z̃×Q̃ and Q̃ ≅ C_{p} ×C_{p} if Q ∈ H; or Z̃ = Z, Q = ZQ̃, Z ∩ Q̃ = Z_{0}, and Q̃ is extraspecial of order p^{3} and exponent p if Q ∈ B. Let Θ ≤ Aut(Q) be the unique subgroup containing Inn(Q) that acts trivially on Z̃, normalizes Q̃, and is such that Θ/Inn(Q) ≅ SL_{2}(p).

We next claim that

(1) each α ∈ Aut_{Γ}(Q) extends to some α ∈ N_{Aut_{Γ}(S)}(Q);

(2) Aut_{Γ}(Q) normalizes Θ;

(3) Aut_{S}(Q) ∈ Syl_{p}(Θ) = Syl_{p}(ΘAut_{Γ}(Q)); and

(4) N_{Θ}(Aut_{S}(Q)) ≤ Aut_{Γ}(Q).

Point (1) holds since S = QA where A ⊴ Γ, and hence N_{Γ}(Q) ≤ N_{Γ}(S). By assumption, each element of N_{Aut_{Γ}(S)}(Q) normalizes Z̃ and Q̃, and hence normalizes Θ. Thus (2) follows from (1). Since |Out_{S}(Q)| = |N_{S}(Q)/Q| = p by Lemma 2.2, this acts trivially on Z ≥ Z̃ and normalizes Q̃, and Out(Q̃) ≅ GL_{2}(p) where Syl_{p}(GL_{2}(p)) = Syl_{p}(SL_{2}(p)), we see that Out_{S}(Q) ∈ Syl_{p}(Θ/Inn(Q)) and hence that Aut_{S}(Q) ∈ Syl_{p}(Θ). Also, Θ has index prime to p in ΘAut_{Γ}(Q) since Aut_{S}(Q) ∈ Syl_{p}(Aut_{Γ}(Q)), and this proves (3). Finally, (4) follows from Lemma 5.9(b.iii).

Set F = ⟨F_{S}(Γ), Θ⟩: the smallest fusion system over S which contains F_{S}(Γ) and such that Aut_{F}(Q) ≥ Θ. Set K = {S, A} ∪ E_{0}. Then K is invariant under F-conjugacy, and is closed in the space of all subgroups of S [BLO6, Definition 1.11]. Thus condition (i) in [BLO6, Theorem 4.2] holds for K; and condition (iii) holds (P ∈ K and P ≤ Q ≤ P^{•} imply Q ∈ K) since P = P^{•} for each P ∈ K (Lemma 5.10).

By Lemma 2.3, if E_{0} = B, then the members of H are not F-centric. So in all cases, if P ≤ S is F-centric and P ∉ K, then P is not contained in any member of E_{F} = E_{0} ∪ {A}, and hence Out_{S}(P) ⊴ Out_{F}(P). This proves condition (iv) in [BLO6, Theorem 4.2]:

O_{p}(Out_{F}(P)) ∩ Out_{S}(P) ≠ 1

whenever P ≤ S is F-centric and not in K.

We refer to [BLO6, Definition 1.11] for the definitions of “K-generated” and “K-saturated”. By construction, F is K-generated. To show that F is K-saturated, we must prove that each P ∈ K is fully automized and receptive in F (Definition 1.1). If P = A or P = S, then Aut_{F}(P) = Aut_{Γ}(P), and this is easily checked. So it remains to show this when P = Q. By (2), Aut_{F}(Q) = Θ·Aut_{Γ}(Q). So Q is fully automized by (3). If α ∈ N_{Aut_{F} (Q)}(Aut_{S}(Q)), then α ∈ Aut_{Γ}(Q) by (4), and hence extends to some α ∈ Aut_{Γ}(S) by (1). So Q is also receptive. This finishes the proof of condition (ii) in [BLO6, Theorem 4.2], and hence F is saturated by that theorem.

Now assume (c) and (d) hold; we must prove that F is simple. By (c), there are no non-trivial G-invariant subgroups of Z except possibly for Z_{0}. Also, E_{F} ⊇ H in case (d.i), and Z_{0} is not G-invariant in case (d.ii). Hence O_{p}(F) = 1 by Lemma 2.7. By Lemma 5.8, F is simple if there are no proper normal fusion subsystems in F over S.

Assume F_{0} ⊴ F is a normal fusion subsystem over S, and set G_{0} = Aut_{F_{0}}(A). Then G_{0} ⊴ G, and G_{0} ≥ O^{p′}(G) since it is the normal closure of U = Aut_{S}(A). Also, μ_{A}(Aut_{F_{0}}^{∨}(A)) ≥ Δ_{t} by Step 1, applied with F_{0} in the role of F. Since μ_{A} is injective on G^{∨}/Aut_{S}(A) by Lemma 5.9(a), we have G_{0} ≥ O^{p′}(G)·μ_{A}^{−1}(Δ_{t}) = G. Thus G_{0} = G, and Aut_{F_{0}}(S) = Aut_{F}(S) by the extension axiom, so F_{0} = F by the Frattini condition on a normal subsystem (see Definition 1.4). This finishes the proof that F is simple.

The uniqueness of F follows from the uniqueness of Z̃ and Q̃ in Lemma 5.9(b).

Step 3: We return to the situation of Step 1, where it remains only to prove that G_{0} = G. By Step 2, there is a unique saturated fusion subsystem F_{0} ≤ F over S such that E_{F_{0}} = E_{F} and Aut_{F_{0}}(A) = G_{0}. The invariance condition on F_{0} ≤ F (Definition 1.4) holds by the uniqueness of F_{0}, and the Frattini condition holds since G = O^{p′}(G)N_{G}(U) ≤ G_{0}N_{G}(U) (where each element of N_{G}(U) extends to an element of Aut_{F}(S) by the extension axiom). Thus F_{0} ⊴ F, so F_{0} = F since F is simple, and hence G_{0} = G. □

As a special case, we next show that for each prime p, there is (up to isomorphism) a unique simple fusion system over an infinite discrete p-toral group with abelian subgroup of index p which is not essential.

Theorem 5.12. For each odd prime p, there is, up to isomorphism, a unique simple fusion system F over an infinite nonabelian discrete p-toral group S which contains an abelian subgroup A < S of index p that is not F-essential. The following hold for each such p, F, S, and A:

(a) The group A is a discrete p-torus of rank p − 1, and S splits over A. Also, Aut_{F}(A) ≅ U ⋊ C_{p−1}, Out_{F}(S) ≅ C_{p−1}, and E_{F} = H.

(b) Fix a prime q ≠ p, set Γ = PSL_{p}(𝔽_{q}), let Ã < Γ be the subgroup of classes of diagonal matrices of p-power order, and set S̃ = Ã⟨x̃⟩ ∈ Syl_{p}(Γ) for some permutation matrix x̃ of order p. Then there is an isomorphism S ≅ S̃ that restricts to an isomorphism A ≅ Ã, and induces isomorphisms Aut_{F}(S) ≅ Aut_{Γ}(S̃) and Aut_{F}(A) ≅ Aut_{N_{Γ}(S̃)}(Ã).

(c) If p = 3, then F is isomorphic to the 3-fusion system of PSU(3), and also to the 3-fusion system of PSL_{3}(𝔽_{q}) for each prime q ≠ 3. For p ≥ 5, F is not realized by any compact Lie group, nor by any p-compact group.

Proof. We use the notation of 2.1 and 2.9. In particular, G = Aut_{F}(A).

(a,b) Assume F is a simple fusion system over an infinite discrete p-toral group S with an abelian subgroup A < S of index p such that A ∉ E_{F}. By Lemma 5.5, Z = Z_{0}, E_{F} = H, and A = S′ is a discrete p-torus of rank p − 1. In particular, |Z| = |Z_{0}| = p. Also, S splits over A by Corollary 2.6.

It remains to describe G = Aut_{F}(A) and Aut_{F}(S) and prove (b). Since A ∉ E_{F}, U ⊴ G, and Out_{F}(S) ≅ G/U. (Each α ∈ Aut_{F}(A) extends to Aut_{F}(S) by the extension axiom.)

Since S splits over A, each α ∈ N_{Aut(A)}(U) extends to an automorphism of S. Since Z = Z_{0}, this implies that N_{Aut(A)}(U) = Aut^{∨}(A). Also, μ_{A}(G) = Δ_{−1} by Theorem 5.11(d) and since O^{p′}(G) = U ≤ Ker(μ_{A}).

Let R = ℤ_{p}[ζ] and 𝔭 = (1 − ζ)R be as in Notation 4.6, regarded as ℤ_{p}U-modules. By Proposition A.4, A ≅ (ℚ_{p}/ℤ_{p})⊗_{ℤ_{p}} Λ and Λ ≅ Hom_{ℤ_{p}}(ℚ_{p}/ℤ_{p}, A) for some (p−1)-dimensional ℤ_{p}G-lattice Λ, and Λ|_{U} ≅ R as ℤ_{p}U-modules by Lemma A.5(c). These isomorphisms induce isomorphisms of automorphism groups

Aut^{∨}(A) = N_{Aut(A)}(U) ≅ N_{Aut(R)}(U) ≅ C_{Aut(R)}(U) ⋊ Gal(ℚ_{p}(ζ)/ℚ_{p})
≅ R^{×} ⋊ Gal(ℚ_{p}(ζ)/ℚ_{p}) ≅ ((1 + 𝔭) × 𝔽_{p}^{×}) ⋊ Gal(ℚ_{p}(ζ)/ℚ_{p}),

and these send Ker(μ_{A}) ≤ Aut^{∨}(A) (the group of automorphisms of A that commute with U and are the identity on Z = Z_{0}) onto 1 + 𝔭, and send the subgroup sc(𝔽_{p}^{×}) ≤ Aut^{∨}(A) of scalar multiplication by (p − 1)-st roots of unity onto 𝔽_{p}^{×}. Thus

Aut^{∨}(A) = N_{Aut(A)}(U) = (Ker(μ_{A}) × sc(𝔽_{p}^{×})) ⋊ W

for a certain subgroup W ≅ C_{p−1}. Set G = (U×sc(𝔽_{p}^{×}))·W < Aut^{∨}(A): a subgroup of order p(p − 1)^{2}.

Since Aut^{∨}(A)/Ker(μ_{A}) has order (p − 1)^{2}, and since Ker(μ_{A}) ≅ (1 + 𝔭) is an abelian prop-group and hence uniquely m-divisible for each m prime to p, we have H^{i}(H; Ker(μ_{A})) = 0 for each H ≤ Aut^{∨}(A)/Ker(μ_{A}) and each i > 0. Hence for each subgroup K < Aut^{∨}(A) of order prime to p, K ∩ Ker(μ_{A}) = 1 since Ker(μ_{A}) is a pro-p-group, Ker(μ_{A})·K splits over Ker(μ_{A}) with a splitting unique up to conjugacy, and thus K is conjugate by an element of Ker(μ_{A}) to a subgroup of sc(𝔽_{p}^{×})·W. In particular, G is conjugate to a subgroup of G, and we can assume (without changing the isomorphism type of F) that G ≤ G. Finally, one easily sees that μ_{A} sends G onto Δ with kernel U, and hence that G = (μ_{A}|_{G})^{−1}(Δ_{−1}) ≅ C_{p} ⋊C_{p−1} is uniquely determined.

A natural isomorphism A ≅ Ã is most easily seen by identifying Γ = PGL_{p}(𝔽_{q}), so that Ã is the quotient of (ℤ/p^{∞})^{p} by the diagonal ℤ/p^{∞} ≅ O_{p}(Z(Γ)), with the permutation action of Aut_{Γ}(Ã) ≅ Σ_{p}. This then extends to an isomorphism of S ≅ A ⋊ U with S̃ = Ã⟨x̃⟩, and of Aut_{F}(A) with Aut_{N_{Γ}(S̃)}(Ã).

Existence and uniqueness of F: Let A, U ⊴ G, and S = A ⋊ ⟨x⟩ be as described in the proof of (b). Since μ_{A}(G) = Δ_{−1}, conditions (a)–(d) in Theorem 5.11 all hold with E_{F} = H. So such an F exists by that theorem. It is unique up to isomorphism by the uniqueness in the theorem and by the restrictions shown in the proof of (b).

(c) If F is realized by a compact Lie group or a p-compact group, then by Proposition 1.5 and since all elements in S are F-conjugate to elements in A, F is realized by a connected, simple p-compact group, and the action of the Weyl group Aut_{F}(A) on ℚ⊗_{ℤ} Hom(A, ℚ_{p}/ℤ_{p}) is generated by pseudoreflections. But if p ≥ 5, then Aut_{F}(A) ≅ C_{p} ⋊ C_{p−1} contains no pseudoreflections other than the identity. So p = 3, and we easily check that F is realized by PSU(3), or by PSL_{3}(𝔽_{q}) for q ≠ 3. □

We can now describe the simple fusion systems over discrete p-toral groups with discrete p-torus of index p in terms of the classification of certain faithful, minimally active, indecomposable modules carried out in [CrOS].

Theorem B. Fix an odd prime p.

(a) Let F be a simple fusion system over an infinite nonabelian discrete p-toral group S with an abelian subgroup A < S of index p. Assume also that A is F-essential. Set G = Aut_{F}(A) and V = Ω_{1}(A), and let G^{∨} = Aut_{F}^{∨}(A) and μ_{A}: G^{∨} ⟶ Δ be as in Notation 2.9. Then A is a discrete p-torus, S splits over A, G ∈ 𝒢_{p}^{∧}, and for some t ∈ {0, −1},

V is a faithful, minimally active, indecomposable 𝔽_{p}G-module. Either dim(V) = p − 1, μ_{A}(G^{∨}) ≥ Δ_{t} and G = O^{p′}(G)μ_{A}^{−1}(Δ_{t}); or dim(V) ≥ p, t = 0, μ_{A}(G^{∨}) = Δ_{0}, and G = O^{p′}(G)·G^{∨}. Also, E_{F} = {A} ∪ H if t = −1, while E_{F} = {A} ∪ B if t = 0. If t = 0, then V contains no 1-dimensional 𝔽_{p}G-submodule.

(∗_{∞})

(b) Conversely, assume that G ∈ 𝒢_{p}^{∧}, U ∈ Syl_{p}(G), and t ∈ {0, −1}, and that V is an 𝔽_{p}Gmodule that satisfies (∗_{∞}), where G^{∨} is the subgroup of all elements α ∈ N_{G}(U) such that [α, C_{V} (U)] ≤ [U, V]. Then there are a discrete G-p-torus A and a simple fusion system F over S = A ⋊ U such that Aut_{F}(A) = Aut_{G}(A) ≅ G, such that Ω_{1}(A) ≅ V as 𝔽_{p}G-modules, and such that E_{F} = {A} ∪ H if t = −1, or E_{F} = {A} ∪ B if t = 0. Furthermore, any other simple fusion system with these properties is isomorphic to F.

(c) Among the fusion systems specified in (b), the only ones that are realized as fusion systems of compact Lie groups or of p-compact groups are those listed in Table 5.1.

p conditions

rk(A) G = Aut_{F}(A) E_{0}

p-cpct. gp.

tors. lin. gp.

p  p ≥ 5  p−1  Σ_{p}  H PSU(p)  PSL_{p}(𝔽_{q})
p p < n < 2p  n−1  Σ_{n}  B  PSU(n)  PSL_{n}(𝔽_{q})
p p ≤ n < 2p, n ≥ 4 n  C_{2}^{n−1} ⋊ Σ_{n} B PSO(2n)  P Ω_{2n}(𝔽_{q})
p \begin{matrix}2 < m | (p − 1) \\ p ≤ n < 2p, n ≥ 4 \end{matrix} n (C_{m})^{n−1} ⋊ Σ_{n} B X(m, m, n)
5  n = 6, 7  n  W (E_{n})  B  E_{n}  E_{n}(𝔽_{q})
7  n = 7, 8  n  W (E_{n})  B  E_{n}  E_{n}(𝔽_{q})
3  2  GL_{2}(3)  B  X_{12}  C_{F_{4}(K)}(γ)
5  4  (4 ∘ 2^{1+4}).Σ_{5} B  X_{29}
5  4  (4 ∘ 2^{1+4}).Σ_{6} B  X_{31}  E_{8}(K)
7  6 6_{1}·PSU_{4}(3).2_{2} B  X_{34}

Table 5.1. The sixth column lists a compact Lie group or a p-compact group that realizes the fusion system F described in the first five columns. Here, X(m, m, n) denotes the p-compact group with Weyl group G(m, m, n) in the notation of [ST, § 2], and X_{k} the one with Weyl group number k in [ST, Table VII]. In the last column, we give, in some cases, a torsion linear group that realizes F: q ≠ p is prime, K ⊆ \overline{𝔽}_{2} is the union of the odd degree extensions of 𝔽_{2}, and γ ∈ Aut(F_{4}(K)) is a graph automorphism of order 2. In the fourth column, B.C means an extension of B by C, and the subscripts in the entry 6_{1}·PSU_{4}(3).2_{2} are Atlas notation [At, p. 52].

Proof. (a) Set U = Aut_{S}(A) ∈ Syl_{p}(G) and Z = C_{A}(U).

Under the above assumptions, A is a discrete p-torus by Proposition 5.7, G ∈ 𝒢_{p}^{∧} by Lemma 3.2(a), V is faithful, minimally active, and indecomposable by Lemma 3.2(b), and rk(V) = rk(A) ≥ p − 1 by Lemma 5.3. Also, for some t ∈ {0, −1}, μ_{A}(G^{∨}) ≥ Δ_{t} and G = O^{p′}(G)μ_{A}^{−1}(Δ_{t}), and E_{F} is as described in (∗_{∞}), by Theorem 5.11(d). Since A ⋬ F, S splits over A by Corollary 2.6. If t = 0 and V_{0} < V is a 1-dimensional 𝔽_{p}G-submodule, then V_{0} ≤ C_{V} (U) ≤ Z, which is impossible by Theorem 5.11(c,d.ii).

Assume rk(V) = rk(A) ≥ p. Since V is minimally active and indecomposable, V |_{U} is the direct sum of a free module 𝔽_{p}U and an 𝔽_{p}-vector space with trivial U-action by Lemma 3.3, and hence Ω_{1}(Z) = C_{V} (U) has rank rk(A) − p + 1. Also, Z is a discrete p-torus by Proposition 5.7, so for each α ∈ G^{∨} = Aut_{F}^{∨}(A), α acts via the identity on Z_{0} since it acts via the identity on Z/Z_{0} (see Notation 2.9). Thus μ_{A}(α) ∈ Δ_{0} by definition of μ_{A}, so t = 0 and μ_{A}(G^{∨}) = Δ_{0} in this case, finishing the proof of (∗_{∞}).

(b) Fix G ∈ 𝒢_{p}^{∧}, U ∈ Syl_{p}(G), and t ∈ {0, −1}, and let V be an 𝔽_{p}G-module that satisfies (∗_{∞}), where G^{∨} is the subgroup of all elements g ∈ N_{G}(U) such that [g, C_{V} (U)] ≤ [U, V]. By Proposition 3.8(a), there is a ℤ_{p}G-lattice Λ such that Λ/pΛ ≅ V as 𝔽_{p}G-modules. Set A = (ℚ_{p}/ℤ_{p})⊗_{ℤ_{p}} Λ: a discrete G-p-torus where Ω_{1}(A) ≅ V as 𝔽_{p}G-modules (see Proposition A.4). To simplify notation, we identify V = Ω_{1}(A). Set S = A⋊U. Set Z = Z(S) = C_{A}(U), S′ = [S, S] = [U, A], and Z_{0} = Z ∩ S′.

We next check that conditions (a)–(d) in Theorem 5.11 all hold. Conditions (a) and (d) follow immediately from (∗_{∞}), and (b) (|Z_{0}| = p) was shown in Lemma 3.7(b).

Assume 1 ≠ B ≤ Z is G-invariant. If rk(A) = p − 1, then Z = Z_{0} has order p, so B = Z_{0}. Otherwise, by (∗_{∞}), t = 0, and V contains no 1-dimensional 𝔽_{p}G-submodule. Thus dim(Ω_{1}(B)) ≥ 2, so dim(V) = rk(A) ≥ p + 1. If dim(V) ≥ p + 2, then V is simple by [CrOS, Proposition 3.7(c)], while if dim(V) = p + 1, then V contains no nontrivial 𝔽_{p}G-submodule with trivial U-action by Lemma 3.5. Thus B = Z_{0}, and this proves condition 5.11(c).

By Theorem 5.11, there is a unique simple fusion system F over S such that G = Aut_{F}(A), and E_{F} ∖ {A} = H (if t = −1) or B (if t = 0). Since A is unique (up to isomorphism of ℤ_{p}G-modules) by Lemma 3.10, this shows that F is uniquely determined by V.

(c) If F is realized by a compact Lie group or a p-compact group, then by Proposition 1.5 and since all elements in S are F-conjugate to elements in A, F is realized by a connected, simple p-compact group, and the action of the Weyl group G = Aut_{F}(A) on ℚ ⊗_{ℤ} Hom(A, ℚ_{p}/ℤ_{p}) is irreducible as a group generated by pseudoreflections. Using the list of pseudoreflection groups and their realizability over ℚ_{p} compiled by Clark & Ewing [CE], as well as the assumption that v_{p}(|G|) = 1, we see that G must be one of the groups listed in Table 5.1, or else one of the other groups G(m, d, n) (of index d in C_{m} ≀ Σ_{n}) for d | m | (p − 1) with d < m. The latter are eliminated by the condition G = O^{p′}(G)·μ_{A}^{−1}(Δ_{0}) in (d.ii) (i.e., the fusion systems of the corresponding p-compact groups are not simple), and so we are left with the groups listed in the table.

Since a p-compact group is determined by its Weyl group by [AGMV, Theorem 1.1], it remains only to check, when rk(A) = p − 1 and based on the constructions of these groups, whether B ⊆ E_{F} or H ⊆ E_{F}. This situation occurs only in the last four cases listed in the table, in which cases the p-compact group was constructed by Aguadé [Ag, §§ 5–7, 10], and the use of SU(p) in his construction shows that extraspecial groups of order p^{3} and exponent p appear as essential subgroups.

In those cases where a torsion linear group is given in Table 5.1, it is a union of a sequence of finite groups that by Table 4.2 realize a sequence of finite fusion subsystems of F. □

The different situations handled in Theorem B are partly summarized in Table 5.2.

6. Examples

Recall Definition 3.1: for a given prime p, 𝒢_{p} is the class of finite groups G with U ∈ Syl_{p}(G) of order p and not normal, and 𝒢_{p}^{∧} is the class of those G ∈ 𝒢_{p} such that Aut_{G}(U) = Aut(U). It remains now to describe explicitly which finite groups G ∈ 𝒢_{p}^{∧} and 𝔽_{p}G-modules V can appear in Theorems A and B. This follows immediately from the work already done in [CrOS], and is stated in Proposition 6.1 and Table 6.1. As in [CrOS], when p is a fixed

dim(V) E_{F} ∖ {A} μ_{A}(G^{∨})  G =  Condition
H  ≥ Δ_{−1} O^{p′}(G)·μ_{A}^{−1}(Δ_{−1})  − p − 1
B ≥ Δ_{0}

V contains no 1-dimensional

≥ p  B  = Δ_{0}

O^{p′}(G)μ_{A}^{−1}(Δ_{0})

𝔽_{p}G-submodule

Table 5.2

prime, we define, for each odd integer i prime to p,

Δ_{i/2} = {(r^{2}, r^{i}) | r ∈ (ℤ/p)^{×}} .

(Compare with the definition of Δ_{i} in Notation 2.9.)

Proposition 6.1. Assume that G ∈ 𝒢_{p}^{∧}, and that V is a faithful, minimally active, indecomposable 𝔽_{p}G-module such that dim(V) ≥ p − 1. If dim(V) ≥ p, then assume also that μ_{V} (G^{∨}) ≥ Δ_{0}; and if dim(V) = p, then assume that V contains no 1-dimensional 𝔽_{p}G-submodule. 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 6.1 with no entry G_{0}; or

(b) the image of G in PGL(V) is almost simple, and G_{0} ≤ G ≤ G with the given action on V for one of the triples (G_{0}, G, V) listed in Table 6.1.

In all cases, the entry under dim(V) gives the dimensions of the composition factors of V; thus a single number means that V is simple.

Proof. We take as starting point the information in [CrOS, Table 4.1]. We drop from that table those cases where dim(V) < p − 1, and also those cases where dim(V) ≥ p and μ_{V} (G^{∨}) ≱ Δ_{0}, or where dim(V) = p and V contains a 1-dimensional 𝔽_{p}G-submodule.

Since the table in [CrOS] is restricted to representations of dimension at least 3, we must add those representations of dimension 2 that appear. Since dim(V) ≥ p − 1, this occurs only for p = 3, and thus G ≤ GL_{2}(3). Since this group is solvable, the image of G in PGL(V) cannot be almost simple, and so this case is covered by the unique row of the table restricted to p = 3. □

We now give two examples, in terms of the pairs (G, V) that appear in Table 6.1, to illustrate how this table can be used to list explicit fusion systems as described by Theorems A and B. When V is an 𝔽_{p}-vector space, we set Aut_{sc}(V) = Z(Aut(V)) ≅ 𝔽_{p}^{×}: the group of automorphisms given by scalar multiplication.

Example 6.2. Fix an odd prime p ≥ 5 and a finite group G ∈ 𝒢_{p}^{∧}, and choose U ∈ Syl_{p}(G). Let V be a simple, (p − 1)-dimensional, minimally active 𝔽_{p}G-module, and assume that Aut_{G}(V) ≥ Aut_{sc}(V). Then μ_{V} (G^{∨}) = Δ by [CrOS, Proposition 3.13(a)]. Let Λ be a ℤ_{p}G-lattice such that Λ/pΛ ≅ V (see Proposition 3.8(a)).

(a) By case (i–a) in Table 4.4, for each k ≥ 2, there is a unique simple fusion systems F over (Λ/p^{k}Λ) ⋊ U, with Aut_{F}(Λ/p^{k}Λ) ≅ G, and such that E_{F} = {A} ∪ H_{0} ∪ B_{∗}.

p  G_{0}  dim(V)  G  μ_{V} (G^{∨}) μ_{V} (G_{0}^{∨})
SL_{2}(p) or PSL_{2}(p) p − 1, p GL_{2}(p) or  Δ Δ_{−1/2}, \frac{1}{2}Δ_{0}
p  (p ≥ 5)  (p−n−1)/n PGL_{2}(p) × C_{p−1} Δ {(u^{2}, u^{n−1})}
(p−2)/1  Δ  \frac{1}{2}Δ_{0}
p  A_{p} (p ≥ 5)  1/(p−2)  Σ_{p} × C_{p−1}  Δ  \frac{1}{2}Δ_{−1}
p  A_{p+1} (p ≥ 5)  p  Σ_{p+1} × C_{p−1}  Δ  \frac{1}{2}Δ_{0}
p A_{n} (p+2 ≤ n ≤ 2p−1) n − 1  Σ_{n} × C_{p−1}  Δ_{0}  \frac{1}{2}Δ_{0}
p  —  n  C_{p−1} ≀ S_{n} (n ≥ p) Δ  —
3  —  2  GL_{2}(3)  Δ  —
5  2·A_{6}  4  4·S_{6}  Δ  Δ_{1/2}
5  —  4  (C_{4} ∘ 2^{1+4}).S_{6}  Δ  —
5 PSp_{4}(3) = W (E_{6})′  6  W (E_{6}) × 4  Δ_{0}.2  \frac{1}{2}Δ_{0}
5 Sp_{6}(2) = W (E_{7})′  7  G_{0} × 4  Δ_{0}  \frac{1}{2}Δ_{0}
7  6·PSL_{3}(4)  6  G_{0}.2_{1}  Δ  𝔽_{p}^{×2} × 𝔽_{p}^{×}
7  6_{1}·PSU_{4}(3)  6  G_{0}.2_{2}  Δ  𝔽_{p}^{×2} × 𝔽_{p}^{×}
7  PSU_{3}(3)  6  G_{0}.2 × 6  Δ  \frac{1}{2}Δ_{1}
7  PSU_{3}(3)  7  G_{0}.2 × 6  Δ  \frac{1}{2}Δ_{0}
7  SL_{2}(8)  7  G_{0}:3 × 6  Δ  \frac{1}{3}Δ_{1}
7 Sp_{6}(2) = W (E_{7})′  7  G_{0} × 6  Δ  Δ_{3}
7 2·Ω_{8}^{+}(2) = W (E_{8})′  8  W (E_{8}) × 3  Δ_{0}.2  Δ_{3}
11  PSU_{5}(2)  10  G_{0}.2 × 10  Δ  \frac{1}{2}Δ_{2}
11  2·M_{12}  10, 10  G_{0}.2 × 5  Δ  Δ_{1/2} , Δ_{7/2}
11  2·M_{22}  10, 10  G_{0}.2 × 5  Δ  Δ_{1/2} , Δ_{7/2}
13  PSU_{3}(4)  12  G_{0}.4 × 12  Δ  \frac{1}{3}Δ_{1}

Table 6.1. Pairs (G, V), where G ∈ 𝒢_{p}^{∧}, G ≤ \overline{G}, G ≥ G_{0} when a quasisimple group G_{0} is given, and where V is a minimally active indecomposable module of dimension at least p − 1, such that μ_{V} (G^{∨}) ≥ Δ_{0} if dim(V) ≥ p, and such that V does not have a 1-dimensional submodule if dim(V) = p. In all cases, dim(V) gives the dimensions of the composition factors in V. Also, 𝔽_{p}^{×2} = {r^{2} | r ∈ 𝔽_{p}^{×}}. The notation B.C, B:C, and B·C for extensions is as in the Atlas [At, p. xx], as well as the subscripts used to make precise certain central extensions or automorphism groups.

(b) By case (iv′–a) in Table 4.4, for each k ≥ 2 and each ∅ ≠ I ⊆ {0, 1, …, p − 1}, there is a unique simple fusion system F_{I} over (Λ/p^{k}Λ) ⋊ U, with Aut_{F_{I}}(Λ/p^{k}Λ) = G_{0}μ_{V}^{−1}(Δ_{0}), and such that E_{F_{I}} = {A} ∪ (⋃_{i∈I} B_{i}).

(c) By case (iii′′–a) in Table 4.4, for each k ≥ 2, there is a unique simple fusion system F over (Λ/p^{k}Λ) ⋊ U, with Aut_{F}(Λ/p^{k}Λ) = G_{0}μ_{V}^{−1}(Δ_{−1}), and such that E_{F} = {A} ∪ H_{0}.

(d) Set A = (ℚ_{p}/ℤ_{p}) ⊗_{ℤ_{p}} Λ, regarded as a discrete G-p-torus. By Theorem B, there are unique simple fusion systems F_{B} and F_{H} over A ⋊ U, with Aut_{F_{B}}(A) ≅ O^{p′}(G)μ_{V}^{−1}(Δ_{0}) and E_{F_{B}} = {A} ∪ B, and Aut_{F_{H}}(A) ≅ O^{p′}(G)μ_{V}^{−1}(Δ_{−1}) and E_{F_{H}} = {A} ∪ H.

Since V is simple (since there is no (p − 2)-dimensional submodule), none of the cases (ii– a), (iii′–a), or (iii′′–c) in Table 4.4 can occur with G_{0} ≤ G ≤ G and V ≅ Ω_{1}(A). Since dim(V) < p, case (iv′′–b) in Table 4.3 cannot occur.

The last column in Table 6.1 can be used to help determine the subgroups O^{p′}(G)μ_{V}^{−1}(Δ_{t}) for i = 0, −1. For example:

• When p = 5, G_{0} ≅ 2·A_{6}, and dim(V) = 4, we have μ_{V} (G_{0}^{∨}) = Δ_{1/2}: the subgroup of order 4 in Δ = (ℤ/5)^{×} × (ℤ/5)^{×} generated by the class of (4, 2). Since Δ_{1/2}Δ_{t} = Δ for t = 0, −1, we have G_{0}μ_{V}^{−1}(Δ_{t}) = G.

• When p = 7, G_{0} ≅ 6·PSL_{3}(4) or 6·PSU_{4}(3), and dim(V) = 6, we have that μ_{V} (G_{0}^{∨}) has index 2 in Δ and does not contain Δ_{t} for any t. So in all cases, G_{0}μ_{V}^{−1}(Δ_{t}) = G: an extension of the form G_{0}.2.

• If p = 7, G_{0} ≅ PSU_{3}(3), and dim(V) = 6, then μ_{V} (G_{0}^{∨}) = \frac{1}{2}Δ_{1}: a subgroup of order 3 that intersects trivially with Δ_{t} for t = 0, −1. So in this case, G_{0}μ_{V}^{−1}(Δ_{t}) has the form G_{0}.2 × 3 (where the precise extension depends on t).

We now look at one case where the 𝔽_{p}G-module V is not simple.

Example 6.3. Let p, G, U, V, and Λ be as in Example 6.2, except that we assume that V is indecomposable but not simple, and contains a (p − 2)-dimensional submodule V_{0} < V. Let Λ_{0} < Λ be a ℤ_{p}G-sublattice of index p such that Λ_{0}/pΛ ≅ V_{0}.

• There are simple fusion systems exactly as described in cases (a), (b), (c), and (d) in Example 6.2. In addition, we have:

(e) By case (ii–a) in Table 4.4, for each k ≥ 2, there is a unique simple fusion systems F over (Λ_{0}/p^{k}Λ) ⋊ U, with Aut_{F}(Λ_{0}/p^{k}Λ) ≅ G, and such that E_{F} = {A} ∪ B_{0} ∪ H_{∗}.

(f) By case (iii′–a) in Table 4.4, for each k ≥ 2 and each ∅ ≠ I ⊆ {0, 1, …, p − 1}, there is a unique simple fusion system F_{I} over (Λ_{0}/p^{k}Λ) ⋊ U, with Aut_{F_{I}}(Λ_{0}/p^{k}Λ) = O^{p′}(G)μ_{V}^{−1}(Δ_{−1}), and such that E_{F_{I}} = {A} ∪ (⋃_{i∈I} H_{i}).

Since there is no 1-dimensional submodule, case (iii′′–c) in Table 4.4 cannot occur with G_{0} ≤ G ≤ G and V ≅ Ω_{1}(A). Since dim(V) < p, case (iv′′–b) in Table 4.3 cannot occur.

If we chose to restrict the above examples to the case dim(V) = p − 1, this is because when dim(V) is larger, there are far fewer possibilities. By Table 4.4, A ≅ Λ/p^{k}Λ for some k ≥ 2, and E_{F} = {A} ∪ B_{0}. Similarly, by Table 5.2, there is only one possibility for F when A is a discrete p-torus with Ω_{1}(A) ≅ V.

Appendix A. Background on groups and representations

We collect here some miscellaneous group theoretic results which were needed earlier. We begin with a few elementary properties of discrete p-toral groups that are easily reduced to the analogous statements about finite p-groups.

Lemma A.1. Fix a prime p, a discrete p-toral group P, and a finite group G ≤ Aut(P) of automorphisms of P. Let 1 = 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., [BLO3, Lemma 1.7(a)]. □

Lemma A.2. Fix an abelian group A each of whose elements has p-power order. Let G ≤ Aut(A) be a finite group of automorphisms, and choose U ∈ Syl_{p}(G). Then

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

Proof. This is shown in [CrOS, Lemma 1.9] when A is a finite abelian p-group, and the proof given there also applies when A is infinite and p-power torsion. □

Lemma A.3. Let S be a nonabelian discrete p-toral group, with abelian subgroup A < S of index p, and set Z = Z(S) = C_{S}(A) and S′ = [S, S] = [S, A]. Then S′ ≅ A/Z. Also, A is the unique abelian subgroup of index p in S if and only if |S′| = |A/Z| > p.

Proof. Choose 1 ≠ u ∈ Aut_{S}(A), and define φ: A ⟶ A by setting φ(a) = a − u(a). Then Z = Ker(φ) and S′ = Im(φ), so A/Z ≅ S′.

If |A/Z| = p, then S/Z ≅ C_{p} × C_{p} (it cannot be cyclic since S is nonabelian), and each subgroup of index p in S containing Z is abelian. Conversely, if B is a second abelian subgroup of index p, then Z = A ∩ B since S = AB, so |A/Z| = p. □

We now turn attention to discrete p-toral groups and discrete G-p-tori. We start with the well known equivalence between discrete G-p-tori and ℤ_{p}G-lattices (see Definition 3.6).

Proposition A.4. Fix a prime p and a finite group G. Then there is a natural bijection

[figure: \left\{{isomorphism classes of \atop discrete G-p-tori}\right\}; ≅; \left\{{isomorphism classes of ℤ_{p}G-lattices \atop in finitely generated ℚ_{p}G-modules}\right\}; A; Hom_{ℤ_{p}}(ℚ_{p}/ℤ_{p}, A); (ℚ_{p}/ℤ_{p}) ⊗_{ℤ_{p}} Λ; Λ]

If A is a discrete G-p-torus and Λ = Hom_{ℤ_{p}}(ℚ_{p}/ℤ_{p}, A), then for each n ≥ 1, evaluation at [1/p^{n}] ∈ ℚ_{p}/ℤ_{p} defines an 𝔽_{p}G-linear isomorphism Λ/p^{n}Λ ⟶^{≅} Ω_{n}(A).

Proof. If Λ is a ℤ_{p}G-lattice, then (ℚ_{p}/ℤ_{p})⊗_{ℤ_{p}} Λ is a discrete G-p-torus, and if A is a discrete G-p-torus, then Hom_{ℤ_{p}}(ℚ_{p}/ℤ_{p}, A) is a ℤ_{p}G-lattice. It is an easy exercise to show that the natural homomorphisms

(ℚ_{p}/ℤ_{p}) ⊗_{ℤ_{p}} Hom_{ℤ_{p}}(ℚ_{p}/ℤ_{p}, A) ⟶^{eval} A

and

Λ ⟶^{λ↦(r↦r⊗λ)} Hom_{ℤ_{p}}(ℚ_{p}/ℤ_{p}, (ℚ_{p}/ℤ_{p}) ⊗_{ℤ_{p}} Λ)

are isomorphisms for each ℤ_{p}G-lattice Λ and each discrete G-p-torus A. The last statement now follows from the short exact sequence

0 ⟶ (p^{−n}ℤ_{p})/ℤ_{p} ⟶^{incl} ℚ_{p}/ℤ_{p} ⟶^{(x↦p^{n}x)} ℚ_{p}/ℤ_{p} ⟶ 0.  □

The next lemma is mostly a well known result in elementary number theory.

Lemma A.5. Fix a prime p, and let U be a group of order p. Let ζ be a primitive p-th root of unity, and regard ℚ_{p}(ζ) and ℤ_{p}[ζ] as ℤ_{p}U-modules via some choice of isomorphism U ≅ ⟨ζ⟩.

(a) There are exactly two irreducible ℚ_{p}U-modules up to isomorphism: a 1-dimensional module with trivial U-action, and a (p − 1)-dimensional module isomorphic to ℚ_{p}(ζ).

(b) The ring ℤ_{p}[ζ] is a local ring with maximal ideal 𝔭 = (1 − ζ)ℤ_{p}[ζ]. Also, pℤ_{p}[ζ] = 𝔭^{p−1}.

(c) Let M be a (p−1)-dimensional irreducible ℚ_{p}U-module, and let Λ < M be a ℤ_{p}U-lattice. Then Λ ≅ ℤ_{p}[ζ] as ℤ_{p}U-modules, and hence Λ/pΛ ≅ ℤ_{p}[ζ]/𝔭^{p−1} is indecomposable as an 𝔽_{p}U-module.

(d) Let B be an infinite abelian discrete p-toral group (written additively), upon which U acts with |C_{B}(U)| = p. Assume also that ∏_{u∈U} u(x) = 1 for each x ∈ B. Then B is a discrete p-torus of rank p − 1, and B ≅ ℚ_{p}(ζ)/ℤ_{p}[ζ] as ℤ_{p}U-modules.

Proof. (a,b) By [Gd, Proposition 6-2-6], (1−ζ)ℤ[ζ] is the only prime ideal in ℤ[ζ] containing pℤ[ζ] = (1 − ζ)^{p−1}ℤ[ζ]. Hence ℚ_{p} ⊗_{ℚ} ℚ(ζ) = ℚ_{p}(ζ), so dim_{ℚ_{p}}(ℚ_{p}(ζ)) = p − 1, and ℤ_{p}[ζ] is a local ring with maximal ideal 𝔭 = (1 − ζ)ℤ_{p}[ζ] where 𝔭^{p−1} = pℤ_{p}[ζ]. This proves (b), and also that ℚ_{p}(ζ) is an irreducible (p − 1)-dimensional ℚ_{p}U-module. So the only other irreducible ℚ_{p}U-module is ℚ_{p} with the trivial action.

(c) By (a), we can assume that M = ℚ_{p}(ζ). Thus Λ is a ℤ_{p}U-lattice in ℚ_{p}(ζ), so p^{m}Λ ≤ ℤ_{p}[ζ] is an ideal for m large enough. Hence p^{m}Λ = 𝔭^{k} = (1 − ζ)^{k}ℤ_{p}[ζ] for some k, and Λ is isomorphic to ℤ_{p}[ζ] as a ℤ_{p}U-module.

(d) Since ∏_{u∈U} u(x) = 1 for each x ∈ B, we can regard B as a ℤ_{p}[ζ]-module. For each n ≥ 1, |Ω_{n}(B)/(1−ζ)Ω_{n}(B)| = |C_{Ω_{n}(B)}(U)| = |C_{B}(U)| = p, and so by (b), there is r_{n} ∈ Ω_{n}(B) that generates Ω_{n}(B) as a ℤ_{p}[ζ]-module. Let R_{n} be the set of all such generators of Ω_{n}(B), and let φ_{n}: R_{n} ⟶ R_{n−1} be the map φ_{n}(r_{n}) = (r_{n})^{p}. The (R_{n}, φ_{n}) thus form an inverse system of nonempty finite sets. An element (r_{n})_{n≥1} in the inverse limit defines an isomorphism ℚ_{p}(ζ)/ℤ_{p}[ζ] ≅ (ℚ_{p}/ℤ_{p}) ⊗_{ℤ_{p}} ℤ_{p}[ζ] ⟶ B of ℤ_{p}[ζ]-modules (hence of ℤ_{p}U-modules), where (1/p^{n}) ⊗ ξ is sent to ξ·r_{n}. □

Lemma A.6. Fix a prime p, an abelian discrete p-toral group A and a finite group of automorphisms G ≤ Aut(A). Assume, for S ∈ Syl_{p}(G), that S ⋬ G and |A/C_{A}(S)| = p. Then |S| = p.

Proof. This is shown in [Ol, Lemma 1.10] when A is finite, and the general case follows since G acts faithfully on Ω_{k}(A) for k large enough. □

Proposition A.7. Fix an abelian discrete p-toral group A, and a subgroup G ≤ Aut(A). Assume the following.

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

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

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

Proof. This is shown in [Ol, Lemma 1.11] when A is finite. The general case then follows by regarding A as the union of the groups Ω_{k}(A) for k ≥ 1. □

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LAGA, Institut Galilée, Av. J-B Clément, 93430 Villetaneuse, France

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

Departament de Mathemàtiques, Edifici C, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain

  E-mail address: albert@mat.uab.cat