REDUCTIONS TO SIMPLE FUSION SYSTEMS

BOB OLIVER

Abstract. We prove that if E ⊴ F are saturated fusion systems over p-groups T ⊴ S, such that C_{S}(E) ≤ T, and either Aut_{F}(T)/Aut_{E}(T) or Out(E) is psolvable, then F can be “reduced” to E by alternately taking normal subsystems of p-power index or of index prime to p. In particular, this is the case whenever E is simple and “tamely realized” by a known simple group K. This answers a question posed by Michael Aschbacher, and is useful when analyzing involution centralizers in simple fusion systems, in connection with his program for reproving parts of the classification of finite simple groups by classifying certain 2-fusion systems.

When p is a prime and S is a finite p-group, a saturated fusion system over S is a category whose objects are the subgroups of S, whose morphisms are injective group homomorphisms between the subgroups, and which satisfies a certain list of axioms motivated by the Sylow theorems for finite groups (Definition 1.1). For example, when G is a finite group and S ∈ Syl_{p}(G), the p-fusion system of G is the category F_{S}(G) whose objects are the subgroups of S, and where for each P, Q ≤ S, Hom_{F_{S}(G)}(P, Q) is the set of those homomorphisms induced by conjugation in G.

Normal fusion subsystems of a saturated fusion system are defined by analogy with normal subgroups of a group (Definition 1.4). Among the normal subsystems, we look at two particular classes: those of index prime to p (defined over the same Sylow subgroup), and those of p-power index (see the discussions before and after Lemma 1.10). A natural question arises: when E ⊴ F, under what conditions can F be “reduced” to E via a sequence of steps, where one alternates taking normal subsystems of p-power index and normal subsystems of index prime to p?

Our main theorem (Theorem 2.3) says that if E ⊴ F are saturated fusion systems over p-groups T ⊴ S, such that C_{S}(E) ≤ T, and either Aut_{F}(T)/Aut_{E}(T) or Out(E) is p-solvable, then F can be reduced to E in the above sense. In particular, if E is the fusion system of a known finite simple group K, and is “tamely realized” by K in the sense that Out(K) surjects onto Out(E) (see Section 2), then Out(E) is solvable since Out(K) is solvable by the Schreier conjecture, and hence F reduces to E.

This paper was motivated by a question posed by Michael Aschbacher. The above situation arises frequently when analyzing centralizers of involutions in simple fusion systems. If F is the centralizer of an involution and E = F^{∗}(F) denotes the generalized Fitting subsystem of F (see [As, Chapter 9]), then the hypothesis C_{S}(E) ≤ T always holds, and Out(E) is solvable by Schreier’s conjecture if E/Z(E) is tamely realized by a known simple group K. Hence in this situation, Theorem 2.3 together with results in [AOV] imply that F itself is realized by a certain extension of K. (See Corollary 2.5 for a slightly more general situation where this applies.)

2000 Mathematics Subject Classification. Primary 20E25. Secondary 20D20, 20D05, 20D25, 20D45.

Key words and phrases. Fusion systems, Sylow subgroups, finite simple groups, generalized Fitting subgroup, p-solvable groups.

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

Throughout the paper, all p-groups are assumed to be finite. Composition is always taken from right to left.

The author would like to thank the Universitat Autònoma de Barcelona for its hospitality while much of this paper was being written.

1. Normal subsystems with p-solvable quotient

Since the details of the definition of a saturated fusion system play an important role in the proofs here, we begin by recalling some of this terminology. When G is a group and P, Q ≤ G are subgroups, Inj(P, Q) denotes the set of injective homomorphisms P ⟶ Q, and Hom_{G}(P, Q) is the set of homomorphisms of the form (x ↦ gxg^{−1}) for g ∈ G. A fusion system over a p-group S is a category F where Ob(F) is the set of subgroups of S, and where for each P, Q ≤ S,

(i) Hom_{S}(P, Q) ⊆ Hom_{F}(P, Q) ⊆ Inj(P, Q), and

(ii) φ ∈ Hom_{F}(P, Q) implies φ^{−1} ∈ Hom_{F}(φ(P), P).

By analogy with the terminology for groups, we say that P, Q ≤ S are F-conjugate if they are isomorphic in F, and let P^{F} denote the set of subgroups F-conjugate to P.

Definition 1.1 ([RS]). Let F be a fusion system over a p-group S.

(a) A subgroup P ≤ S is fully automized in F if Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)).

(b) A subgroup P ≤ S is receptive in F if for each Q ∈ P^{F} and each φ ∈ Iso_{F}(Q, P), φ extends to a homomorphism φ ∈ Hom_{F}(N_{φ}, S), where

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

(c) The fusion system F is saturated if each F-conjugacy class of subgroups of S contains at least one member which is fully automized and receptive in F.

(d) When H is a set of subgroups of F closed under F-conjugacy, we say that F is H-saturated if each member of H is F-conjugate to a subgroup which is fully automized and receptive in F. We say that F is H-generated if each morphism in F is a composite of restrictions of morphisms between members of H.

(e) When X is a set of injective homomorphisms between subgroups of S, ⟨X⟩ denotes the fusion system over S generated by X: the smallest fusion system which contains X. When F_{0} is a category whose objects are subgroups of S and whose morphisms are injective homomorphisms, we write ⟨F_{0}⟩ = ⟨Mor(F_{0})⟩.

We next recall some of the terminology for subgroups in a fusion system. When F is a fusion system over S and P ≤ S, we write Out_{F}(P) = Aut_{F}(P)/Inn(P).

Definition 1.2. Let F be a fusion system over a p-group S, and let P ≤ S be a subgroup. Then

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

(b) P is F-centric if C_{S}(Q) ≤ Q for each Q ≤ P; (c) P is F-radical if O_{p}(Out_{F}(P)) = 1; and

(d) P is strongly closed in F if φ(P_{0}) ≤ P for each P_{0} ≤ P and each φ ∈ Hom_{F}(P_{0}, S).

We also let F^{cr} ⊆ F^{c} denote the sets of subgroups of S which are F-centric and F-radical, or F-centric, respectively.

The following lemma describes the relations between some of these conditions on subgroups. Point (a) and (b) are due to Roberts and Shpectorov [RS] and are also shown in [AKO, Lemma I.2.6(c)], while (c) is immediate from the definitions.

Lemma 1.3. In a saturated fusion system F over a p-group S, for each subgroup P ≤ S,

(a) P is fully centralized if and only if P is receptive;

(b) P is fully normalized if and only if P is fully automized and receptive; and

(c) if P is F-centric, then P is fully centralized and hence receptive.

As described in the introduction, normal fusion subsystems play a central role here.

Definition 1.4. Let F be a saturated fusion system over a p-group S, and let E ≤ F be a saturated fusion subsystem over T ≤ S. The subsystem E 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 with respect to E (i.e., extends to (α, α̂) ∈ Aut(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).

Finally, we will frequently need to use the following version of Alperin’s fusion theorem for fusion systems. This is the version shown in [BLO2, Theorem A.10]. For a stronger version due to Puig, see, e.g., [AKO, Theorems I.3.5–6].

Theorem 1.5. If F is a saturated fusion system over a p-group S, then

F = ⟨Aut_{F}(P) | P ≤ S is F-centric, F-radical, and fully normalized in F⟩.

Thus each morphism in F is a composite of restrictions of F-automorphisms of such subgroups.

If E ⊴ F are saturated fusion systems, then E has index prime to p in F if they are both over the same p-group S, and for each P ≤ S, Aut_{E}(P) ≥ O^{p′}(Aut_{F}(P)). Let O^{p′}(F) ⊴ F be the smallest normal subsystem of index prime to p in F [BCGLO2, Theorem 5.4]. We first fix some tools for constructing normal subsystems of this type. The following lemma is basically the same as Theorem I.7.7 in [AKO], but stated in a slightly more general setting.

Lemma 1.6. Let F be a saturated fusion system over a p-group S. Let H ⊆ F^{c} be a nonempty set of F-centric subgroups of S such that (i) H is closed under F-conjugacy and overgroups; and

(ii) for each P ∈ F^{c} ∖H, there is P^{∗} ∈ P^{F} such that Out_{S}(P^{∗})∩O_{p}(Out_{F}(P^{∗})) ≠ 1.

Let F^{∗} ⊆ F be the full subcategory with object set H, let Δ be a finite group of order prime to p, and let χ: Mor(F^{∗}) ⟶ Δ be such that

(iii) χ(incl_{P}^{S}) = 1 for all P ∈ H,

(iv) χ(ψφ) = χ(ψ)χ(φ) for each composable pair of morphisms ψ and φ in F^{∗}, and

(v) χ(Aut_{F}(S)) = Δ.

Let F_{0}^{∗} ⊆ F^{∗} be the subcategory with the same objects, and with Mor(F_{0}^{∗}) = χ^{−1}(1). Set F_{0} = ⟨F_{0}^{∗}⟩: the fusion system over S generated by F_{0}^{∗}. Then F_{0} ⊴ F is a normal saturated fusion subsystem, O^{p′}(F) ≤ F_{0} ≤ F, and Aut_{F_{0}}(S) = Ker(χ|_{Aut_{F} (S)}).

Proof. By (iv), F_{0}^{∗} is a subcategory of F^{∗}. Since |Δ| is prime to p, χ(Inn(S)) = 1, and hence Inn(S) ≤ Aut_{F_{0}^{∗}}(S). Also, by (iii) and (iv),

χ(φ′) = χ(φ) when φ′ ∈ Mor(F^{∗}) is any restriction of φ ∈ Mor(F^{∗}). (1)

Since H is closed under overgroups by (i), we thus have Hom_{F_{0}}(P, Q) = Hom_{F_{0}^{∗}}(P, Q) for all P, Q ∈ H.

We next show the following:

(2) For each P, Q ∈ H and φ ∈ Hom_{F^{∗}}(P, Q), there are morphisms α ∈ Aut_{F}(S) and φ_{0} ∈ Hom_{F_{0}^{∗}}(P, Q^{∗}), where Q^{∗} = α^{−1}(Q), such that φ = α|_{Q^{∗}} ∘ φ_{0}.

(3) If P, Q ∈ H and Q ∈ P^{F}, then there is α ∈ Aut_{F}(S) such that α(Q) ∈ P^{F_{0}}.

(4) If P ∈ H is fully automized in F, then it is fully automized in F_{0}.

(5) If P ∈ H is receptive in F, then it is receptive in F_{0}.

Point (2) follows from (v) (and (1)): choose α such that χ(α) = χ(φ), and set φ_{0} = α^{−1}φ ∈ Mor(F_{0}^{∗}). Point (3) follows immediately from (2). Point (4) holds since if Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)), then Aut_{S}(P) is also a Sylow p-subgroup in Aut_{F_{0}}(P).

If P ∈ H is receptive in F, then each φ ∈ Hom_{F_{0}}(Q, S) with φ(Q) = P extends to some φ ∈ Hom_{F}(N_{φ}, S), where N_{φ} = {x ∈ N_{S}(Q) | φc_{x}φ^{−1} ∈ Aut_{S}(P)}. Then N_{φ} ∈ H by (i) and χ(φ) = χ(φ) = 1 by (1), so φ ∈ Hom_{F_{0}}(N_{φ}, S). Thus P is receptive in F_{0} in this case, and this proves (5).

Since F is saturated, for each P ∈ H, there is Q ∈ P^{F} which is fully automized and receptive in F. By (3), there is α ∈ Aut_{F}(S) such that α(Q) ∈ P^{F_{0}}. Then α(Q) is also fully automized and receptive in F, hence in F_{0} by (4) and (5). Thus F_{0} is H-saturated, and it is F_{0}-generated by definition. So by (ii) and [BCGLO1, Theorem 2.2], F_{0} is saturated.

We claim that

α ∈ Aut_{F}(S), φ ∈ Mor(F_{0}) ⟹ αφα^{−1} ∈ Mor(F_{0}).  (6)

Here, αφα^{−1} means composition on each side with the appropriate restriction of α or α^{−1}. This holds for φ ∈ Mor(F_{0}^{∗}) by definition (and (1)), and hence holds for all composites of restrictions of such morphisms. Since F_{0} = ⟨F_{0}^{∗}⟩, (6) holds for all φ ∈ Mor(F_{0}).

We next check that

∀ φ ∈ Mor(F), ∃ φ_{0} ∈ Mor(F_{0}), α ∈ Aut_{F}(S) such that φ = αφ_{0}. (7)

Since H ⊇ F^{cr} by (ii), F = ⟨F^{∗}⟩ by Alperin’s fusion theorem. Hence F = ⟨F_{0}, Aut_{F}(S)⟩ by (2). By [BCGLO2, Lemma 3.4.c] and (6), this suffices to prove (7).

Since the extension condition for normality holds trivially in this case, this proves that F_{0} ⊴ F (see [AKO, Definition I.6.1]). Hence F_{0} ≥ O^{p′}(F), since F and F_{0} are both saturated fusion systems over S (see [AOV, Lemma 1.26]). □

The next lemma is needed to check that point (ii) holds when we apply Lemma 1.6.

Lemma 1.7. Let E ⊴ F be saturated fusion systems over T ⊴ S. Then for each P ∈ F^{c} such that P ∩T ∉ E^{c}, there is P^{∗} ∈ P^{F} such that Out_{S}(P^{∗})∩O_{p}(Out_{F}(P^{∗})) ≠ 1. In particular, P ∉ F^{cr}.

Proof. By [AKO, Lemma I.2.6.c], there is φ ∈ Hom_{F}(N_{S}(P ∩ T), S) such that φ(P ∩ T) is fully normalized in F. Set P^{∗} = φ(P); then φ(P ∩ T) = P^{∗} ∩ T since T is strongly closed in F, and hence P^{∗} ∩T is fully normalized in F. Since P ∩T ∉ E^{c}, there is Q ∈ (P ∩ T)^{E} ⊆ (P^{∗} ∩ T)^{F} such that C_{T} (Q) ≰ Q, each ψ ∈ Iso_{F}(Q, P^{∗} ∩ T) extends to QC_{S}(Q), and hence C_{T} (P^{∗} ∩ T) ≰ P^{∗}.

Thus P^{∗}C_{T} (P^{∗}∩T) > P^{∗}, so N_{P^{∗}C_{T} (P^{∗}∩T)}(P^{∗}) > P^{∗}, and there is x ∈ N_{T} (P^{∗})∖P^{∗} such that [x, P^{∗} ∩ T] = 1. Conjugation by x induces the identity on P^{∗} ∩ T and on P^{∗}/(P^{∗} ∩ T), so c_{x} ∈ O_{p}(Aut_{F}(P^{∗})). Also, c_{x} ∉ Inn(P^{∗}) since P^{∗} ∈ F^{c}, so 1 ≠ [c_{x}] ∈ Out_{S}(P^{∗}) ∩ O_{p}(Out_{F}(P^{∗})). □

The next proposition provides a more explicit way to construct proper normal subsystems of index prime to p. Note that the existence of a normal subsystem over the strongly closed subgroup T is crucial, as is clearly seen by considering the case where T = S. In fact, the proposition is rather trivial when T = S or T = 1, and is useful only when 1 ≠ T < S.

Proposition 1.8. Let E ⊴ F be saturated fusion systems over p-groups T ⊴ S. Let χ_{0} : Aut_{F}(T) ⟶ Δ be a surjective homomorphism, for some Δ ≠ 1 of order prime to p, such that Aut_{E}(T) ≤ Ker(χ_{0}). Then there is a unique proper normal subsystem F_{0} ⊴ F over S such that

Aut_{F_{0}}(S) = {α ∈ Aut_{F}(S) | α|_{T} ∈ Ker(χ_{0})}  (1)

and F_{0} ≥ E. In particular, O^{p′}(F) ≤ F_{0} < F.

Proof. The uniqueness of F_{0} follows from (1) and [BCGLO2, Theorem 5.4]: a saturated fusion subsystem of index prime to p in F is uniquely determined by the automizer of S.

Let F|_{E^{c}} ⊆ F be the full subcategory with objects in E^{c}. We first show that there is a map

χ: Mor(F|_{E^{c}}) ⟶ Δ

which extends χ_{0}, which sends composites to products, and which sends Mor(E^{c}) to the identity. By the Frattini condition for a normal fusion subsystem [AKO, Definition I.6.1], for each P, Q ∈ E^{c} and each φ ∈ Hom_{F}(P, Q), there are α ∈ Aut_{F}(T) and φ_{0} ∈ Hom_{E}(P, Q_{1}), where Q_{1} = α^{−1}(Q), such that φ = α|_{Q_{1}} ∘ φ_{0}. In this situation, the conditions imposed on χ imply that χ(φ) = χ(α) = χ_{0}(α). It remains to prove that this is independent of the choice of decomposition of φ, and that it sends composites to products.

To see that χ sends composites to products when it is uniquely defined, fix a composable pair of morphisms ψ, φ ∈ Mor(F): a pair such that ψ ∘ φ is defined. Assume φ = αφ_{0} and ψ = βψ_{0} (after appropriate restrictions of α and β), where φ_{0}, ψ_{0} ∈ Mor(E) and α, β ∈ Aut_{F}(T). Thus χ(φ) = χ_{0}(α) and χ(ψ) = χ_{0}(β). Also,

ψ ∘ φ = βψ_{0}αφ_{0} = (βα)(α^{−1}ψ_{0}α)φ_{0}

where α^{−1}ψ_{0}α ∈ Mor(E) since E ⊴ F. So χ(ψ ∘ φ) = χ_{0}(βα).

Again fix P, Q ∈ E^{c} and φ ∈ Hom_{F}(P, Q). Let φ = α|_{Q_{1}} ∘ φ_{0} = β|_{Q_{2}} ∘ ψ_{0} be two decompositions, where α, β ∈ Aut_{F}(T), Q_{1} = α^{−1}(Q), Q_{2} = β^{−1}(Q), and φ_{0}, ψ_{0} ∈ Mor(E). If P = Q = T, then all of these morphisms lie in Aut_{F}(T), and χ_{0}(α) = χ_{0}(β) since χ_{0}(Aut_{E}(T)) = 1. So assume P < T, and also assume inductively that χ is uniquely defined on all morphisms between subgroups in E^{c} strictly larger than P. Then (β^{−1}α|_{Q_{1}})φ_{0} = ψ_{0} ∈ Hom_{E}(P, Q_{2}). Since β^{−1}α|_{Q_{1}} ∈ Hom_{E}(Q_{1}, Q_{2}) extends in F to T, it also extends in E to N_{T} (Q_{1}) > Q_{1} (recall that all E-centric subgroups are receptive by Lemma 1.3(c)). Let γ ∈ Hom_{E}(N_{T} (Q_{1}), T) be such that β^{−1}α|_{Q_{1}} = γ|_{Q_{1}}. Since Q_{1} ∈ E^{c}, β^{−1}α(N_{T} (Q_{1})) = γ(N_{T} (Q_{1})), so α^{−1}βγ ∈ Aut_{F}(N_{T} (Q_{1})). This automorphism has p-power order since it is the identity on Q_{1} and on N_{T} (Q_{1})/Q_{1}, and hence χ(α^{−1}βγ) = 1. Since all of these homomorphisms involve subgroups strictly larger than P, χ(α)^{−1}χ(β)χ(γ) = 1, where χ(γ) = 1 since γ ∈ Mor(E^{c}). So χ_{0}(α) = χ_{0}(β), and the two decompositions of φ give the same value for χ(φ).

Thus χ is uniquely defined. Set

H^{∗} = {P ∈ F^{c} | P ∩ T ∈ E^{c}},

and let χ̂ be the composite

χ̂: Mor(F|_{H^{∗}}) ⟶^{R} Mor(F|_{E^{c}}) ⟶^{χ} Δ,

where R sends φ ∈ Hom_{F}(P, Q) to φ|_{P ∩T} ∈ Hom_{F}(P ∩ T, Q ∩ T). Since T ⊴ S, T is fully normalized, and hence is fully automized and receptive (Lemma 1.3(b)). Hence Aut_{S}(T) lies in Syl_{p}(Aut_{F}(T)) = Syl_{p}(O^{p′}(Aut_{F}(T))), and so

Aut_{F}(T) = N_{Aut_{F} (T)}(Aut_{S}(T))·O^{p′}(Aut_{F}(T))

by the Frattini argument. Since T is receptive, each α ∈ N_{Aut_{F} (T)}(Aut_{S}(T)) extends to some α ∈ Aut_{F}(S), and χ̂(α) = χ̂(α)χ̂(incl_{T}^{S}) = χ̂(α). Since χ̂(O^{p′}(Aut_{F}(T))) = 1, χ̂(Aut_{F}(S)) = χ̂(Aut_{F}(T)) = Δ.

Condition (ii) in Lemma 1.6 holds by Lemma 1.7. We just checked condition (v), condition (iv) holds for χ̂ since it holds for χ, and the other two are clear. So by that lemma, there is F_{0} < F which is normal in F and contains O^{p′}(F), and such that Aut_{F_{0}}(S) is as required.

It remains to show that F_{0} ≥ E. If P ∈ E^{c} and P is fully centralized in F, then each φ ∈ Aut_{E}(P) extends to some φ ∈ Aut_{F}(P C_{S}(P)), where P C_{S}(P) ∈ F^{c} and χ̂(φ) = χ(φ) = 1, so φ and hence φ are in Mor(F_{0}). If P ∈ E^{c} is arbitary, then ψ(P) is fully centralized in F for some ψ ∈ Hom_{F}(P, T), and Aut_{E}(P) = (Aut_{E}(φ(P)))^{φ} and Aut_{F_{0}}(P) = (Aut_{F_{0}}(φ(P)))^{φ} since E and F_{0} are normal in F (see [AKO, Proposition I.6.4(d)], applied with Q = P). Hence Aut_{E}(P) ≤ Aut_{F_{0}}(P) for all P ∈ E^{c}, and E ≤ F_{0} by Alperin’s fusion theorem. □

Corollary 1.9. If E ⊴ F are saturated fusion systems over T ⊴ S, and

Aut_{E}(T)O^{p′}(Aut_{F}(T)) < Aut_{F}(T),

then O^{p′}(F) < F.

We now turn to constructions of normal subsystems of p-power index. Recall first the definition of the hyperfocal subgroup 𝔥𝔶𝔭(F) for a saturated fusion system F over a p-group S:

𝔥𝔶𝔭(F) = ⟨[O^{p}(Aut_{F}(P)), P] | P ≤ S⟩ ⊴ S.

Lemma 1.10. Let F be a saturated fusion system over a p-group S, and assume T ⊴ S is strongly closed in F. Then

(a) 𝔥𝔶𝔭(F/T) = T ·𝔥𝔶𝔭(F)/T;

(b) T C_{S}(T) is strongly closed in F; and

(c) the natural isomorphism S/T C_{S}(T) ≅ Out_{S}(T) extends to an isomorphism of fusion systems F/T C_{S}(T) ≅ F_{Out_{S}(T)}(Out_{F}(T)).

Proof. By [AKO, Theorem II.5.12], and since T is strongly closed in F, there is a morphism of fusion systems (Ψ, Ψ̂): F ⟶ F/T which sends P to P T/T and sends φ ∈ Hom_{F}(P, Q) to the induced homomorphism between quotient groups.

Set T̂ = T C_{S}(T) for short.

(a) If P ≤ S, and φ ∈ Aut_{F}(P) has order prime to p, then Ψ̂(φ) ∈ Aut_{F/T} (P T/T) also has order prime to p. Hence T [φ, P]/T ≤ 𝔥𝔶𝔭(F/T). Since 𝔥𝔶𝔭(F) is generated by such commutators [φ, P] by definition, this proves that T ·𝔥𝔶𝔭(F)/T ≤ 𝔥𝔶𝔭(F/T).

Conversely, for each P/T ≤ S/T, and each ψ ∈ Aut_{F/T} (P/T) of order prime to p, ψ lifts to some ψ̂ ∈ Aut_{F}(P) by definition of F/T, and [ψ, P/T] = T [ψ̂, P]/T where [ψ̂, P] ≤ 𝔥𝔶𝔭(F). Since 𝔥𝔶𝔭(F/T) is generated by such commutators [ψ, P/T], this proves that 𝔥𝔶𝔭(F/T) ≤ T ·𝔥𝔶𝔭(F)/T.

(b) Fix P ≤ T̂ and φ ∈ Hom_{F}(P, S). Choose φ̂ ∈ Hom_{F}(P T, S) such that Ψ̂(φ̂) = Ψ̂(φ) ∈ Hom_{F/T} (P T/T, S/T). Then φ̂(T) = T, P T ≤ T̂ = T C_{S}(T), so P T = T C_{P T} (T), and φ(P) ≤ φ̂(P T) = T C_{φ̂(P T)}(T) ≤ T̂. Thus T̂ is strongly closed.

(c) Fix P, Q ≤ S which contain T̂, and φ ∈ Hom_{F}(P, Q). Let φ̂ ∈ Hom_{F/T̂}(P/T̂, Q/T̂) be the induced homomorphism, and let [φ|_{T}] ∈ Out_{F}(T) be the class of φ|_{T} ∈ Aut_{F}(T). Then for x ∈ P and c_{x} ∈ Aut_{P} (T), (φ|_{T})c_{x}(φ|_{T})^{−1} = c_{φ(x)} ∈ Aut_{Q}(T). So if we identify Out_{P} (T) ≅ P/T̂ and Out_{Q}(T) ≅ Q/T̂, then φ̂ is conjugation by [φ|_{T}], and hence a morphism in F_{Out_{S}(T)}(Out_{F}(T)).

Conversely, if conjugation by the class of ψ ∈ Aut_{F}(T) sends Out_{P} (T) into Out_{Q}(T) for P, Q ≥ T̂, then ψ extends to some ψ^{∗} ∈ Hom_{F}(P, Q) since T ⊴ S is receptive (Lemma 1.3(b)), and hence c_{[ψ]} ∈ Hom_{Out_{F} (T)}(Out_{P} (T), Out_{Q}(T)) is identified with ψ̂^{∗} ∈ Hom_{F/T̂}(P/T̂, Q/T̂). □

In [BCGLO2, § 3], a fusion subsystem F_{0} ≤ F over U ≤ S is defined to have p-power index if U ≥ 𝔥𝔶𝔭(F) and Aut_{F_{0}}(P) ≥ O^{p}(Aut_{F}(P)) for each P ≤ U. By [AKO, Theorem I.7.4], if F is saturated, then for each U ≤ S containing 𝔥𝔶𝔭(F), there is a unique saturated fusion subsystem F_{U} ≤ F over U of p-power index in F, and F_{U} ⊴ F if U ⊴ S.

Proposition 1.11. If F is a saturated fusion system over a p-group S, and T ⊴ S is strongly closed in F, then

𝔥𝔶𝔭(F) ≤ {x ∈ S | c_{x} ∈ O^{p}(Aut_{F}(T))Inn(T)}.

If, in addition, E ⊴ F and F_{0} ⊴ F are normal subsystems over T and U, respectively, where U ≥ T ·𝔥𝔶𝔭(F) and F_{0} has p-power index in F, then E ≤ F_{0}.

Proof. Set T̂ = T C_{S}(T) and Q = {x ∈ S | c_{x} ∈ O^{p}(Aut_{F}(T))Inn(T)} ≥ T̂, for short, and let ω: S/T̂ ⟶^{≅} Out_{S}(T) be the natural isomorphism. By Lemma 1.10(b), T̂ is strongly closed in F, and by Lemma 1.10(c), ω induces an isomorphism of fusion systems F/T̂ ≅ F_{Out_{S}(T)}(Out_{F}(T)). Puig’s hyperfocal theorem for groups now implies that

ω(𝔥𝔶𝔭(F/T̂)) = 𝔥𝔶𝔭(F_{Out_{S}(T)}(Out_{F}(T))) = O^{p}(Out_{F}(T)) ∩ Out_{S}(T)
= (O^{p}(Aut_{F}(T))Inn(T) ∩ Aut_{S}(T))/Inn(T)
= Aut_{Q}(T)/Inn(T) = ω(Q/T̂).

Hence 𝔥𝔶𝔭(F/T̂) = Q/T̂, so Q = T̂·𝔥𝔶𝔭(F) by Lemma 1.10(a).

Now let U ⊴ S be any normal subgroup containing T ·𝔥𝔶𝔭(F), and assume that E and F_{0} are normal subsystems in F over T and U, respectively, where F_{0} has p-power index in F. If P ≤ T is fully normalized in E, then Aut_{E}(P) = Aut_{T} (P)O^{p}(Aut_{E}(P)) since Aut_{T} (P) ∈ Syl_{p}(Aut_{E}(P)), Aut_{T} (P) ≤ Aut_{F_{0}}(P) since F_{0} is a fusion system over U ≥ T, and O^{p}(Aut_{E}(P)) ≤ O^{p}(Aut_{F}(P)) ≤ Aut_{F_{0}}(P) where the second inclusion holds since F_{0} ⊴ F has p-power index. Hence Aut_{E}(P) ≤ Aut_{F_{0}}(P) for all such P, and E ≤ F_{0} since E is generated by such automorphisms by Alperin’s fusion theorem. □

The next lemma will be needed to show that certain subnormal systems are normal, when iterating inductively Propositions 1.8 and 1.11.

Lemma 1.12 ([As, 7.4]). Let F_{2} ⊴ F_{1} ⊴ F be saturated fusion systems over pgroups S_{2} ⊴ S_{1} ⊴ S. Assume, for each α ∈ Aut_{F}(S_{1}), that ^{α}F_{2} = F_{2}. Assume also that C_{S}(S_{2}) ≤ S_{2}. Then F_{2} ⊴ F.

For an arbitrary saturated fusion system F, let F^{∞} ⊴ F be the limit after applying O^{p}(−) and O^{p′}(−) until the sequence stabilizes. More precisely, set F^{∞} = ⋂_{i=0}^{∞} F^{(i)}, where the sequence {F^{(i)}} is defined by setting F^{(0)} = F and F^{(i+1)} = O^{p′}(O^{p}(F^{(i)})).

Lemma 1.13. Let F = F_{0} > F_{1} > ⋯ > F_{m} = E be any sequence of saturated fusion systems, each normal in F and each of p-power index or of index prime to p in the preceeding one. Then F^{∞} = E^{∞}.

Proof. We prove that E^{∞} = F^{∞} by induction on |Mor(F)|. In particular, it suffices to do this when m = 1. If E ⊴ F has p-power index, then O^{p}(E) = O^{p}(F) [Cr, Theorem 7.53(ii)], and E^{∞} = F^{∞} by definition. So assume E has index prime to p in F. Set T = 𝔥𝔶𝔭(F). Since 𝔥𝔶𝔭(E) ≤ T by definition, there is a unique normal subsystem E_{T} ⊴ E of p-power index over T, and by [AKO, Theorem I.7.4], E_{T} = ⟨Inn(T), O^{p}(Aut_{E}(P)) | P ≤ T ⟩ ≤ O^{p}(F). Also, E_{T} ⊴ F by Lemma 1.12 and the uniqueness of E_{T} (and since T ⊴ S). Hence E_{T} ⊴ O^{p}(F), and E_{T} has index prime to p in O^{p}(F) since they are both fusion systems over T [AOV, Lemma 1.26]. Hence

E^{∞} = (E_{T})^{∞} = (O^{p′}(O^{p}(F)))^{∞} = F^{∞},

where the second equality holds by the induction hypothesis. □

We are now ready to combine Propositions 1.8 and 1.11 to get the following:

Theorem 1.14. Let E ⊴ F be saturated fusion systems over p-groups T ⊴ S. Assume that Aut_{F}(T)/Aut_{E}(T) is p-solvable (equivalently, that Out_{F}(T) is p-solvable). Then there is a normal saturated subsystem F_{0} ⊴ F over T C_{S}(T), such that F_{0} ≥ E, F_{0} ≥ F^{∞}, (F_{0})^{∞} = F^{∞}, and Aut_{F_{0}}(T) = Aut_{E}(T).

Proof. Choose a sequence of subgroups Aut_{F}(T) = G_{m} > G_{m−1} > ⋯ > G_{0} = Aut_{E}(T), all normal in G_{m}, and such that for each i, G_{i}/G_{i−1} is a p-group or a p′-group. For each i, set S_{i} = N_{S}^{G_{i}}(T) = {x ∈ S | c_{x} ∈ G_{i}}. Thus S_{i} ⊴ S and Aut_{S_{i}}(T) ∈ Syl_{p}(G_{i}) for each i, S_{m} = S, and S_{0} = T C_{S}(T).

We claim that there are subsystems F = F_{m} > F_{m−1} > ⋯ > F_{0} = E in F, each normal in F and of p-power index or of index prime to p in the preceeding one, where F_{i} is over S_{i}, and such that Aut_{F_{i}}(T) = G_{i} for each i. If this holds, then (F_{0})^{∞} = F^{∞} by Lemma 1.13, and F_{0} satisfies all of the other conditions listed above.

Assume F_{i} ⊴ F has been constructed as claimed, with i ≥ 1 and F_{i} ≥ E. If G_{i}/G_{i−1} is a p′-group, then S_{i−1} = S_{i}. Let F_{i−1} ⊴ F_{i} be as in Proposition 1.8. In particular, F_{i−1} has index prime to p in F_{i} and contains E. For each α ∈ Aut_{F}(S_{i}), α|_{T} ∈ Aut_{F}(T) = G_{m} normalizes G_{i−1} = Aut_{F_{i−1}}(T), and hence α normalizes F_{i−1} by the uniqueness in Proposition 1.8. So F_{i−1} ⊴ F by Lemma 1.12.

If G_{i}/G_{i−1} is a p-group, then 𝔥𝔶𝔭(F_{i}) ≤ S_{i−1} by Proposition 1.11, so there is a unique normal subsystem F_{i−1} ⊴ F_{i} over S_{i−1} of p-power index (see [AKO, Theorem I.7.4]), and F_{i−1} ≥ E by Proposition 1.11 again. For each α ∈ Aut_{F}(S_{i}), α(S_{i−1}) = S_{i−1} since S_{i−1} = N_{S}^{G_{i−1}}(T) and α|_{T} ∈ G_{m} normalizes G_{i−1}, so α normalizes F_{i−1} by its uniqueness. Thus F_{i−1} ⊴ F by Lemma 1.12. □

2. Reductions to centric normal subsystems

In order to get further results, we must also work with the linking system associated to a fusion system. When G is a group and P, Q ≤ G, we set T_{G}(P, Q) = {g ∈ G |^{g}P ≤ Q} (the transporter set). When F is a saturated fusion system over a pgroup S, a centric linking system associated to F is a category L with Ob(L) = F^{c}, the set of F-centric subgroups of S, together with a pair of functors

T_{F^{c}}(S) ⟶^{δ} L ⟶^{π} F

which satisfy certain axioms listed in [AKO, Definition III.4.1] and [BLO2, Definition 1.7]. Here, T_{F^{c}}(S) is the transporter category of S: the category with object set F^{c}, and with Mor_{T_{F^{c}}(S)}(P, Q) = T_{S}(P, Q) (where composition is given by multiplication in S). Also, δ is the identity on objects and injective on morphism sets, π is the inclusion on objects and surjective on morphism sets, and π ∘ δ sends g ∈ T_{S}(P, Q) to c_{g} ∈ Hom_{F}(P, Q). The motivating example is the category L_{S}^{c}(G), when G is a finite group and S ∈ Syl_{p}(G), where

Ob(L_{S}^{c}(G)) = F_{S}(G)^{c} = {P ≤ S | C_{G}(P) = Z(P) × O_{p′}(C_{G}(P))}
Mor_{L_{S}^{c}(G)}(P, Q) = T_{G}(P, Q)/O_{p′}(C_{G}(P)).

By comparison, note that Hom_{F_{S}(G)}(P, Q) ≅ T_{G}(P, Q)/C_{G}(P).

Since we are working with extensions of fusion and linking systems, we also need to handle their automorphism groups. Automorphisms of fusion systems are straightforward. When F is a saturated fusion system over a p-group S, an automorphism α ∈ Aut(S) is said to be fusion preserving if it induces an automorphism of the category F, and we set

Aut(S, F) = {α ∈ Aut(S) | α is fusion preserving}
Out(S, F) = Aut(S, F)/Aut_{F}(S).

Let L be a centric linking system associated to F, and let δ be the functor described above. For each P ∈ F^{c} = Ob(L), set ι_{P} = δ_{P,S}(1) ∈ Hom_{L}(P, S) (the “inclusion” of P in S in the category L), and set [[P]] = δ_{P} (P) ≤ Aut_{L}(P). Define

Aut_{typ}^{I}(L) = {β ∈ Aut(L) | β(ι_{P}) = ι_{β(P)}, β([[P]]) = [[β(P)]], ∀ P ∈ F^{c}}
Out_{typ}(L) = Aut_{typ}^{I}(L)/⟨c_{γ} | γ ∈ Aut_{L}(S)⟩.

There is a natural homomorphism μ_{L}: Out_{typ}(L) ⟶ Out(S, F), defined by restriction to [[S]] ≤ Aut_{L}(S). We refer to [AKO, § III.4.3] or [AOV, § 1.3] for more details on these definitions.

When F is a saturated fusion system over a p-group S, it is straightforward to define the centralizer fusion system C_{F}(P) of a subgroup P ≤ S (see [AKO, Definition I.5.3]): this is a fusion subsystem over C_{S}(P) which is saturated if P is fully centralized in F (i.e., receptive). One can also define C_{F}(E) when E is a normal subsystem in F (see [As, Chapter 6]), but this is much more complicated. For our purposes here, it will suffice to work with the following somewhat simpler definition. If E ⊴ F are saturated fusion systems over T ⊴ S, then by [As, 6.7], there is a (unique) subgroup C_{S}(E) ≤ C_{S}(T) with the property that for P ≤ C_{S}(T), P ≤ C_{S}(E) if and only if E ≤ C_{F}(P).

Proposition 2.1. Let E ⊴ F be a pair of saturated fusion systems over the pgroups T ⊴ S such that C_{S}(E) ≤ T. Let L be a centric linking system associated to E. Then the natural homomorphism S ⟶ Aut(T, E) lifts to a homomorphism ω̃: S ⟶ Aut_{typ}^{I}(L), and this factors through an injective homomorphism ω: S/T ⟶ Out_{typ}(L).

Proof. Since the conclusion of the proposition involves only S and E, we can assume that F = SE as defined in [As, Theorem 5, Chapter 8]. In particular, E ≥ O^{p}(F). So by [AOV, Proposition 1.31(a)], there is a pair of linking systems L ⊴ L^{∗} associated to E ⊴ F, where Ob(L) = E^{c}, and Ob(L^{∗}) is the set of all P ≤ S such that P ∩ T ∈ Ob(L). (This was shown in [AOV] only when E = O^{p}(F), but the same argument applies in this situation.)

This inclusion L ⊴ L^{∗} induces a natural homomorphism ω̃ from S to Aut_{typ}^{I}(L), defined by conjugation in L^{∗}, and which factors through a homomorphism

ω: S/T ⟶ Out_{typ}(L) = Aut_{typ}^{I}(L)/{c_{γ} | γ ∈ Aut_{L}(T)}.

Assume ω is not injective, and let x ∈ S ∖ T be such that xT ∈ Ker(ω). Thus ω̃(x) = c_{δ_{T} (x)} is conjugation by some element γ ∈ Aut_{L}(T). Since ω̃(x) has p-power order, we can assume that γ has p-power order, and hence γ = δ_{T} (y) for some y ∈ T. Upon replacing x by xy^{−1}, we can arrange that c_{δ_{T} (x)} = Id_{L}, and hence that δ_{T} (x) and its restrictions commute with all morphisms in L. In particular, x ∈ C_{S}(T).

Fix P, Q ≤ T and ψ ∈ Iso_{L}(P, Q). We just showed that δ_{Q}(x)ψ = ψδ_{P} (x). So by [Ol, Proposition 4.e], ψ extends to an isomorphism ψ ∈ Iso_{L^{∗}}(P ⟨x⟩, Q⟨x⟩). Set y = π(ψ)(x), where π(ψ) ∈ Hom_{F}(P ⟨x⟩, Q⟨x⟩). By axiom (C) for a linking system [AKO, Definition III.4.1],

ψ ∘ δ_{P ⟨x⟩}(x) = δ_{Q⟨x⟩}(y) ∘ ψ.

But this is also equal to δ_{Q⟨x⟩}(x) ∘ ψ since extensions are unique in a linking system [Ol, Proposition 4.e or 4.f], and so δ_{Q⟨x⟩}(x) = δ_{Q⟨x⟩}(y). Since δ is injective by [Ol, Proposition 4.c], we have x = y = π(ψ)(x).

Thus all isomorphisms in E between objects in L extend to morphisms in F which send x to itself. Since Ob(L) = E^{c}, this statement holds for all morphisms in E by Alperin’s fusion theorem. So E ≤ C_{F}(x), hence x ∈ C_{S}(E), which contradicts the assumption that C_{S}(E) ≤ T. We conclude that ω is injective. □

We saw in Theorem 1.14 the importance of getting control of the quotient group T C_{S}(T)/T, when E ⊴ F are saturated fusion systems over T ⊴ S.

Corollary 2.2. Let E ⊴ F be saturated fusion systems over T ⊴ S such that C_{S}(E) ≤ T. Then T C_{S}(T)/T is abelian, and C_{S}(T) ≤ T if p is odd.

Proof. Let L be a centric linking system associated to E. Consider the homomorphisms

S/T ⟶^{ω} Out_{typ}(L) ⟶^{μ=μ_{L}} Out(T, E) ≤ Aut(T)/Aut_{E}(T)

where ω is injective by Proposition 2.1 and μ(ω(xT)) = [c_{x}] for x ∈ S. Thus T C_{S}(T)/T = Ker(μ ∘ ω) injects into Ker(μ). In particular, C_{S}(T) ≤ T if μ is injective, and this always holds if p is odd by [O2, Theorem C] and [GL]. Otherwise, T C_{S}(T)/T is abelian since Ker(μ) is abelian (see [AKO, Proposition III.5.12]). □

We are now ready to prove our main result, which says that under appropriate conditions on E ⊴ F, F reduces down to E in the sense which we have been studying.

Theorem 2.3. Let E ⊴ F be saturated fusion systems over T ⊴ S such that C_{S}(E) ≤ T. Assume either

(a) Aut_{F}(T)/Aut_{E}(T) is p-solvable; or

(b) Out(T, E) is p-solvable.

Then F^{∞} = E^{∞}.

Proof. Since

Aut_{F}(T)/Aut_{E}(T) ≤ Aut(T, E)/Aut_{E}(T) = Out(T, E),

Aut_{F}(T)/Aut_{E}(T) is p-solvable if Out(T, E) is p-solvable, and thus (b) implies (a). So from now on, we assume Aut_{F}(T)/Aut_{E}(T) is p-solvable. By Theorem 1.14, it suffices to prove this when S = T C_{S}(T) and Aut_{F}(T) = Aut_{E}(T). So by Corollary 2.2, S/T is abelian.

Set H = {P ≤ S | P ≥ C_{S}(T)}. If P ∈ F^{c} and P ∉ H, then P C_{S}(T) > P, so N_{P C_{S}(T)}(P) > P, and there is x ∈ N_{C_{S}(T)}(P) ∖ P. Then c_{x} ∈ Aut_{S}(P) induces the identity on P ∩ T and on P/(P ∩ T), so c_{x} ∈ O_{p}(Aut_{F}(P)). Also, c_{x} ∉ Inn(P) since x ∉ P and P ∈ F^{c}, so 1 ≠ [c_{x}] ∈ Out_{S}(P) ∩ O_{p}(Out_{F}(P)).

Let (Ψ, Ψ̂): F ⟶ F/T be the morphism of fusion systems which sends P to P T/T and φ ∈ Mor(F) to the induced homomorphism between quotient groups [AKO, Theorem II.5.12]. Let F^{∗} ⊆ F^{c} be the full subcategory with objects in H∩F^{c}. By definition, for each P ∈ H, P T = T C_{S}(T) = S since P ≥ C_{S}(T). Define χ: Mor(F^{∗}) ⟶ Aut_{F/T} (S/T) by sending φ ∈ Mor(F^{∗}) to Ψ̂(φ) ∈ Aut_{F/T} (S/T). This clearly satisfies conditions (iii)–(v) in Lemma 1.6. Since we just checked condition (ii) in the lemma, and (i) is clear, we conclude that there is a normal fusion subsystem F_{0} ⊴ F containing O^{p′}(F) such that Aut_{F_{0}}(S) = Ker(χ|_{Aut_{F} (S)}) and hence Aut_{F_{0}/T} (S/T) = 1.

Since S/T is abelian and Aut_{F_{0}/T} (S/T) = 1, we have F_{0}/T = F_{S/T} (S/T). So 𝔥𝔶𝔭(F_{0}) ≤ T by Lemma 1.10(a). By [AKO, Theorem I.7.4], there is a unique fusion subsystem F_{1} ⊴ F_{0} over T of p-power index. Also, E ⊴ F_{1} by Proposition 1.11, so E has index prime to p in F_{1} by [AOV, Lemma 1.26]. Thue F^{∞} = (F_{0})^{∞} = (F_{1})^{∞} = E^{∞} by Lemma 1.13. □

To explain the motivation for Theorem 2.3, we recall some definitions in [AOV]. When F = F_{S}(G) and L = L_{S}^{c}(G) for some finite group G with S ∈ Syl_{p}(G), there is a natural homomorphism

Out(G) ≅ N_{Aut(G)}(S)/Aut_{N_{G}(S)}(G) ⟶^{κ_{G}} Out_{typ}(L),

defined by sending α ∈ N_{Aut(G)}(S) to the automorphism of L induced by α. See [AOV, § 2.2] for more details. The fusion system F is tamely realized by G if F ≅ F_{S}(G) and κ_{G} is split surjective, and F is tame if it is tamely realized by some finite group.

Finally, a saturated fusion system F is reduced if O_{p}(F) = 1 and O^{p}(F) = F = O^{p′}(F). If F is any fusion system, and Q = O_{p}(F), set 𝔯𝔢𝔡(F) = (C_{F}(Q)/Z(Q))^{∞}: the reduction of F. By [AOV, Proposition 2.2], 𝔯𝔢𝔡(F) is reduced for any saturated fusion system F. By [AOV, Theorem A], F is tame if its reduction is tame.

Corollary 2.4. Let E ⊴ F be saturated fusion systems over T ⊴ S, where C_{S}(E) ≤ T, and where E is simple and is tamely realized by a known simple group K. Then F^{∞} = E, and F is tamely realized by an extension of K.

Proof. By the Schreier conjecture (see [GLS3, Theorem 7.1.1]), Out(K) is solvable. Set L = L_{T}^{c} (K). Then Out_{typ}(L) is solvable since κ_{K} is surjective, and Out(T, E) is solvable since μ_{L} is surjective by [O2, Theorem C] and [GL]. Hence F^{∞} = E^{∞} by Theorem 2.3, 𝔯𝔢𝔡(F) = E^{∞} = E since E is simple, and so F is tame by [AOV, Theorem 2.20]. More precisely, F is tamely realized by an extension of K by successive applications of [AOV, Proposition 2.16], together with the existence of “compatible” linking systems as made precise in the proof of [AOV, Theorem 2.20]. □

One can take this further by stating it in terms of the generalized Fitting subsystem F^{∗}(F) of a saturated fusion system F [As, 9.9].

Corollary 2.5. Let F be a saturated fusion system. Assume that F^{∗}(F) = O_{p}(F)E (a central product), where E ⊴ F is quasisimple, and where E/Z(E) is tamely realized by a known simple group K. Then 𝔯𝔢𝔡(F) ≅ E/Z(E), and F is tamely realized by a finite group G such that F^{∗}(G) = O_{p}(G)H, where H is quasisimple and H/Z(H) ≅ K.

Proof. Set Q = O_{p}(F). Then C_{F^{∗}(F)}(Q)/Z(Q) ≅ E/Z(E) is simple, and hence the pair C_{F^{∗}(F)}(Q)/Z(Q) ⊴ C_{F}(Q)/Z(Q) satisfies the hypotheses of Corollary 2.4. So by that corollary, 𝔯𝔢𝔡(F) ≅ E/Z(E), and C_{F}(Q)/Z(Q) is tamely realized by an extension of K. Together with [AOV, Theorem 2.20] (and its proof), this implies that F itself is tamely realized by a finite group G of the form described above. □

References

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[GLS3] D. Gorenstein, R. Lyons, & R. Solomon, The classification of the finite simple groups, nr. 3, Amer. Math. Soc. surveys and monogr. 40 #3 (1997)

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[O2] B. Oliver, Existence and uniqueness of linking systems: Chermak’s proof via obstruction theory, Acta Math. 211 (2013), 141–175

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Université Paris 13, Sorbonne Paris Cité, LAGA, UMR 7539 du CNRS, 99, Av. J.-B. Clément, 93430 Villetaneuse, France.

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