FUSION SYSTEMS AND AMALGAMS

KASPER K. S. ANDERSEN, BOB OLIVER, AND JOANA VENTURA

Abstract. We study reduced fusion systems from the point of view of their essential subgroups, using the classification by Goldschmidt and Fan of amalgams of prime index to analyze certain pairs of such subgroups. Our results are applied here to study reduced fusion systems over 2-groups of order at most 64, and also reduced fusion systems over 2-groups having abelian subgroups of index two. More applications are given in later papers.

A saturated fusion system over a finite p-group S is a category whose objects are the subgroups of S, whose morphisms are monomorphisms between subgroups, and which satisfy certain axioms first formulated by Puig [Pg2] and motivated by conjugacy relations among p-subgroups of a given finite group. A saturated fusion system is reduced if it has no proper normal subsystem of p-power index, no proper normal subsystem of index prime to p, and no nontrivial normal p-subgroup. (All three of these concepts are defined by analogy with finite groups.) Reduced fusion systems need not be simple, in that they can have proper nontrivial normal subsystems. They were introduced by us in [AOV] as forming a class of fusion systems which is small enough to be manageable, but still large enough to detect any fusion systems (reduced or not) which are “exotic” (not defined via conjugacy relations in any finite group).

When G is a finite group and S ∈ Syl_{p}(G), the version of Alperin’s fusion theorem shown by Goldschmidt [Gd1] says that all G-conjugacy relations among subgroups of S are generated by Aut_{G}(S) (automorphisms induced by conjugation in G), together with Aut_{G}(P) for certain “essential” proper subgroups of S, and restrictions of such automorphisms. There is a version of this result for abstract fusion systems (see Theorem 1.2), which says that a fusion system F is generated by F-automorphisms of F-essential subgroups (Definition 1.1). Our goal in this and our other papers is to study, and to classify in certain cases, reduced fusion systems from the point of view of their essential subgroups and generating automorphisms.

This point of view was introduced in [OV], where two of us described how fusion systems over a given 2-group S could be classified by first listing the subgroups of S which potentially could be essential, using Bender’s theorem on groups with strongly embedded subgroups. When we try to extend those methods to larger classes of groups, it is useful to search for pairs of essential subgroups via theorems of Goldschmidt and Fan classifying certain types of amalgams.

2010 Mathematics Subject Classification. Primary 20D20. Secondary 20E06, 20E45.

Key words and phrases. Finite groups, fusion, amalgams, fusion systems.

K. K. S. Andersen was partially supported by the Danish National Research Foundation (DNRF)

through the Centre for Symmetry and Deformation.

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

J. Ventura was partially supported by FCT through program POCI 2010/FEDER and project PTDC/MAT/098317/2008.

The situation we want to study is the following. Assume F is a saturated fusion system over a finite 2-group S, and P_{1}, P_{2} < S are F-essential subgroups of index two in their normalizer. In addition, we assume either that P_{1} and P_{2} have index two in S, or that N_{S}(P_{1}) = N_{S}(P_{2}) < S and P_{1}, P_{2} are S-conjugate to each other. Set P = P_{1}P_{2} = N_{S}(P_{i}). Then there are finite groups G_{i} > P such that P_{i} ⊴ G_{i}, G_{i}/P_{i} ≅ D_{2p_{i}} for some odd prime p_{i}, and Aut_{G_{i}}(P_{i}) ≤ Aut_{F}(P_{i}). By applying the classifications of amalgams by Goldschmidt and Fan to the triple (G_{1} > P < G_{2}), we get information about S and the P_{i}. In this paper, we only deal with certain cases (see Theorems 4.5 and 4.6), but these are the cases which occur most often in “small” examples.

Applications of these results are given in Section 5. The reduced fusion systems over 2-groups of order at most 32, and the groups of order 64 which support reduced fusion systems, are all listed in Theorems 5.3 and 5.4, respectively. These are preceded by Propositions 5.1 and 5.2, which list various conditions on a reduced fusion system over a 2-group S which imply that S is dihedral of order at least 8, semidihedral of order at least 16, or a wreath product C_{2^{n}} ≀ C_{2} for n ≥ 2. Furthermore, in these cases, F is isomorphic to the fusion system of PSL_{2}(q) for some q ≡ ±1 (mod 8), or of PSL_{3}(q) for some odd q. For example, by Proposition 5.2(a,b,c,e), these conclusions hold whenever F is a reduced fusion system over a 2-group S, where either

• S contains an abelian subgroup of index two; or

• [S, S] is cyclic; or

• there is a subgroup Q < S such that |N_{S}(Q)/Q| = 2, Out_{S}(Q) ≰ O_{2}(Out(Q)), and either Q is abelian or |Q| ≤ 16.

These results are applied in a later paper by the same authors, where we combine them with a computer search to list reduced fusion systems over 2-groups of order at most 2^{9}. They have also been applied by the second author when classifying reduced fusion systems over 2-groups of sectional rank at most four.

Notation: For any group G, we let G^{ab} = G/[G, G] denote its abelianization. Also, C_{n} denotes a (multiplicative) cyclic group of order n, and D_{2^{m}}, SD_{2^{m}}, and Q_{2^{m}} denote dihedral, semidihedral, and quaternion groups of order 2^{m}. As usual, when P is a finite p-group for some prime p, then Ω_{1}(P) = ⟨g ∈ P | g^{p} = 1⟩, and Fr(P) = ⟨a^{p}, [a, b] | a, b ∈ P ⟩ (the Frattini subgroup). For any finite group G, O_{p}(G) is the largest normal p-subgroup of G, and O^{p}(G) is the smallest normal subgroup of p-power index.

When G acts on a group X, we let C_{X}(G) be the subgroup of elements of X fixed by G. When A ⊆ G and B ⊆ X are subsets, we set [A, B] = ⟨g(x)x^{−1} | g ∈ A, x ∈ B⟩. When g, h are elements of any group G, we write their commutator [g, h] = ghg^{−1}h^{−1}.

1. Saturated fusion systems

When G is a finite group and S ∈ Syl_{p}(G), the fusion system of G over S is the category F_{S}(G) whose objects are the subgroups of S, and where Mor_{F_{S}(G)}(P, Q) = Hom_{G}(P, Q) is the set of monomorphisms from P to Q induced by conjugation in G. An abstract fusion system F over a finite p-group S is a category whose objects are the subgroups of S, whose morphisms are monomorphisms of groups including all those induced by conjugation in S, and where for each φ ∈ Hom_{F}(P, Q) = Mor_{F}(P, Q), φ restricts to an F-isomorphism from P to φ(P) ≤ Q. A fusion system is saturated if it satisfies certain additional conditions. Rather than listing those conditions here, we refer to [AKO, Definition I.2.2] or our earlier paper [AOV].

In particular, for any finite G with S ∈ Syl_{p}(G), F_{S}(G) is a saturated fusion system (cf. [AKO, Theorem I.2.3]). An abstract fusion system F over S is called realizable if F = F_{S}(G) for some finite group G with S ∈ Syl_{p}(G), and is called exotic otherwise.

If G is a finite group and p is a prime, then a proper subgroup H < G is strongly p-embedded in G if p||H|, and for each g ∈ G∖H, p∤|H ∩ gHg^{−1}|. We refer to [AKO, Proposition A.7] for a very brief survey of some of the properties of strongly p-embedded subgroups, and to [A1, § 46] or [Sz2, § 6.4] for more details.

Definition 1.1. Fix a prime p, a finite p-group S, and a saturated fusion system F over S. Let P ≤ S be any subgroup. Set Out_{F}(P) = Aut_{F}(P)/Inn(P).

• P^{F} denotes the set of subgroups of S which are F-conjugate to P; i.e., isomorphic to P in the category F. For each g ∈ S, g^{F} denotes the F-conjugacy class of g.

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

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

• P is F-essential if P < S, P is F-centric and fully normalized in F, and Out_{F}(P) contains a strongly p-embedded subgroup.

• P is normal in F (P ⊴ F) if P ⊴ S and every morphism φ ∈ Hom_{F}(Q, R) in F extends to a morphism φ ∈ Hom_{F}(P Q, P R) such that φ(P) = P.

• O_{p}(F) denotes the largest subgroup of S which is normal in F.

• N_{F}(P) ⊆ F denotes the largest fusion subsystem over N_{S}(P) (i.e., the largest subcategory of F which is a fusion system over N_{S}(P)) which contains P as normal subgroup.

• For each φ ∈ Aut(S), φFφ^{−1} is the fusion system over S defined by

Hom_{φFφ^{−1}}(P, Q) = {(φ|_{φ^{−1}(Q)}) ∘ ψ ∘ (φ|_{φ^{−1}(P)})^{−1} | ψ ∈ Hom_{F}(φ^{−1}(P), φ^{−1}(Q))}

for all P, Q ≤ S.

It follows immediately from the definition of a normal subgroup in F that the maximal subgroup O_{p}(F) ⊴ F is uniquely defined. The notation is, of course, chosen by analogy with that for finite groups.

We now look at essential subgroups of a fusion system.

Theorem 1.2. Let F be a saturated fusion system over a finite p-group S. Then each morphism in F is a composite of restrictions of morphisms in Aut_{F}(S), and of morphisms in O^{p′}(Aut_{F}(P)) for F-essential subgroups P ≤ S.

Proof. See, e.g., [O1, Proposition 1.10(a,b)]. In fact, a proper subgroup P < S fully normalized in F is F-essential exactly when Aut_{F}(P) is not generated by restrictions of morphisms between strictly larger subgroups of S. (See [OV, Proposition 2.5] or [AKO, Proposition I.3.3(b)] for more details.) □

The next proposition follows easily from Theorem 1.2, together with the definition of a normal p-subgroup.

Proposition 1.3. Let F be a saturated fusion system over a finite p-group S, and fix Q ≤ S. Assume, whenever P = S or P is F-essential, that Q ≤ P and each α ∈ Aut_{F}(P) sends Q to itself. Then Q ⊴ F.

Proof. See, e.g., [AKO, Proposition I.4.5] for details. □

The next two lemmas are our main tools for detecting essential subgroups, or rather, for proving that certain subgroups are not essential.

Lemma 1.4 ([OV, Lemma 3.4]). Fix a prime p, a finite p-group S, a subgroup P ≤ S, and a characteristic subgroup Θ ≤ P. Assume there is g ∈ N_{S}(P)∖P such that

(a) [g, P] ≤ Θ·Fr(P), and

(b) [g, Θ] ≤ Fr(P).

Then c_{g} ∈ O_{p}(Aut(P)). Hence P is not F-essential for any saturated fusion system F over S.

In fact, [OV, Lemma 3.4] is stated in terms of “(semi)critical subgroups” of a finite p-group S rather than essential subgroups. We refer to [OV, Definition 3.1] for the definition of critical subgroups, and just note here that by [OV, Proposition 3.2], each F-essential subgroup (for any saturated fusion system F over S) is critical in S. This remark also applies to the next lemma, which is a special case of [OV, Proposition 3.3(c)].

Lemma 1.5. Let S be a finite 2-group. Assume that P ≤ S is F-essential for some saturated fusion system F over S, and also that |N_{S}(P)/P | ≥ 4. Then rk(P/Fr(P)) ≥ 4, and rk([s, P/Fr(P)]) ≥ 2 for all s ∈ N_{S}(P)∖P.

We next recall the definitions of the focal and hyperfocal subgroups of a saturated fusion system, defined by analogy with the finite group case.

Definition 1.6. Let F be a saturated fusion system over a finite p-group S. The focal subgroup of F is the subgroup

𝔣𝔬𝔠(F) =^{def} ⟨g^{−1}h | g, h ∈ S and h ∈ g^{F}⟩
= ⟨g^{−1}α(g) | g ∈ P ≤ S, P = S or P is F-essential, α ∈ Aut_{F}(P)⟩.

The hyperfocal subgroup of F is the subgroup

𝔥𝔶𝔭(F) = ⟨g^{−1}α(g) | g ∈ P ≤ S, α ∈ O^{p}(Aut_{F}(P))⟩.

The two definitions of 𝔣𝔬𝔠(F) are equivalent by Theorem 1.2. In the definition of 𝔥𝔶𝔭(F), we could equivalently restrict to automorphisms of order prime to p. When F = F_{S}(G) for a finite group G and S ∈ Syl_{p}(G), then 𝔣𝔬𝔠(F) = S ∩[G, G] by the focal subgroup theorem (cf. [G, Theorem 7.3.4]), and 𝔥𝔶𝔭(F) = S ∩O^{p}(G) by the hyperfocal theorem of Puig [Pg1, § 1.1].

Next recall the following definitions from [5a2].

Definition 1.7. Let F be a saturated fusion system over a finite p-group S, and let F_{0} ⊆ F be a saturated fusion subsystem over a subgroup S_{0} ≤ S.

(a) F_{0} has p-power index in F if 𝔥𝔶𝔭(F) ≤ S_{0} ≤ S, and Aut_{F_{0}}(P) ≥ O^{p}(Aut_{F}(P)) for all P ≤ S_{0}.

(b) F_{0} has index prime to p in F if S_{0} = S, and Aut_{F_{0}}(P) ≥ O^{p′}(Aut_{F}(P)) for all P ≤ S.

By [5a2, Theorems 4.3 & 5.4], each saturated fusion system F over a finite p-group S contains a unique minimal saturated fusion subsystem O^{p}(F) of p-power index (over 𝔥𝔶𝔭(F)), and a unique minimal saturated fusion subsystem O^{p′}(F) of index prime to p (over S). Furthermore:

Proposition 1.8. For any saturated fusion system F over a finite p-group S,

F = O^{p}(F) ⟺ 𝔥𝔶𝔭(F) = S ⟺ 𝔣𝔬𝔠(F) = S .

Proof. See, e.g., [AKO, Corollary I.7.5]. The second equivalence follows upon checking that the image of 𝔣𝔬𝔠(F) in S/𝔥𝔶𝔭(F) is precisely its commutator subgroup (cf. [AKO, Lemma I.7.2]). □

2. Reduced fusion systems

This paper is centered around the special class of reduced fusion systems, which are defined as follows.

Definition 2.1. A reduced fusion system is a saturated fusion system F such that

• F has no nontrivial normal p-subgroups,

• F has no proper subsystem of p-power index, and

• F has no proper subsystem of index prime to p.

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

Definition 2.1 was originally formulated in [AOV], and was motivated by Theorems A and B in that paper. Very roughly, those theorems describe a way to “detect” exotic fusion systems while looking only at reduced fusion systems.

In this section, we give some conditions on a fusion system which are necessary for it to be reduced (equivalently, conditions which imply that it is not reduced). We begin with two very general results.

Lemma 2.2. If F is a reduced fusion system over a nontrivial finite 2-group S, and E is the set of F-essential subgroups of S, then |E| ≥ 2 and [S:⟨E⟩] ≠ 2.

Proof. By Proposition 1.3, S ⊴ F if E = ∅, while P ⊴ F if E = {P } for some P. So F is not reduced (O_{p}(F) ≠ 1) if |E| ≤ 1.

If [S:⟨E⟩] = 2, then [Aut_{F}(S), S] ≤ ⟨E⟩ since Aut_{F}(S) acts trivially on S/⟨E⟩ ≅ C_{2}. Since 𝔣𝔬𝔠(F) is generated by [Aut_{F}(S), S] and the [Aut_{F}(P), P] for P ∈ E, 𝔣𝔬𝔠(F) ≤ ⟨E⟩ < S. So F is not reduced by Proposition 1.8. □

The next proposition is a simple application of a transfer homomorphism for fusion systems.

Proposition 2.3. Let F be a saturated fusion system over a finite 2-group S.

(a) Assume there is g ∈ Ω_{1}(Z(S))∖[S, S] such that each α ∈ Aut_{F}(S) sends the coset g[S, S] to itself. Then g ∉ 𝔣𝔬𝔠(F).

(b) More generally, let U ⊴ S be such that each element of Aut_{F}(S) sends U to itself, and U ≤ [P, P] for each P < S of index two. Assume there is g ∈ S∖[S, S] such that [g, S] ≤ U, g^{2} ∈ U, and each α ∈ Aut_{F}(S) sends the coset g[S, S] to itself. Then g ∉ 𝔣𝔬𝔠(F).

In either case, F is not reduced.

Proof. We refer to [AKO, § I.8] for some of the properties of the transfer homomorphism trf_{F}: S/𝔣𝔬𝔠(F) ⟶ S^{ab} when F is a saturated fusion system over S. Let [g] ∈ S^{ab} be the class of g. Since (a) is a special case of (b) (the case U = 1), and was shown in [AKO, Corollary I.8.5], we assume g satisfies the conditions of (b).

For P < S, let trf_{P}^{S} : S^{ab} ⟶ P^{ab} be the usual transfer homomorphism (cf. [AKO, Lemma I.8.1(b)]). If [S:P] = 2, then trf_{P}^{S} ([g]) = [g^{2}] if g ∉ P, and trf_{P}^{S} ([g]) = [gxgx^{−1}] if g ∈ P and x ∈ S∖P. This follows from the construction in [AKO] upon taking coset representatives {1, x}. Since g^{2} ∈ U, gxgx^{−1} = g^{2}[g^{−1}, x] ∈ U, and U ≤ [P, P], trf_{P}^{S} ([g]) = 1. Since this holds for each P < S of index two, trf_{P}^{S} ([g]) = 1 for each P < S since transfers compose (cf. [AKO, Lemma I.8.1(d)]).

By assumption, for each α ∈ Aut_{F}(S), α([g]) = [g]. So by [AKO, Proposition I.8.4(a)], trf_{F}([g]) = [g]^{k} ≠ 1 where k = |Out_{F}(S)| is odd. Thus g ∉ 𝔣𝔬𝔠(F) since trf_{F} is well defined, so 𝔣𝔬𝔠(F) < S, O^{2}(F) ≠ F by Proposition 1.8, and F is not reduced. □

The next lemma is an application of Lemma A.8, together with the transfer homomorphism for fusion systems (cf. [AKO, § I.8]). Recall that a finite group G is metacyclic if it has a normal cyclic subgroup H ⊴ G such that G/H is also cyclic.

Lemma 2.4. Let S be a finite 2-group, and let F be a saturated fusion system over S. Let E be the set of F-essential subgroups of S.

(a) Assume P ∈ E is such that [N_{S}(P), P] is cyclic. Then there are decompositions P = P_{0}P_{1} and Out_{F}(P) = Γ_{0}×Γ_{1}, where for i = 0, 1, [P, P] ≤ P_{i} ⊴ P, Γ_{i} sends P_{i} to itself and acts trivially on P_{1−i}/[P, P], Γ_{0} has odd order, and Γ_{1} ≅ Σ_{3}. Either

(i) P is abelian, P_{1} ≅ C_{2^{n}} × C_{2^{n}} for some n ≥ 1, C_{P_{1}}(N_{S}(P)) ≅ C_{2^{n}} and

[N_{S}(P), P] ≅ C_{2^{n}} are both direct factors of P_{1}, and P_{0} ∩ P_{1} = 1; or

(ii) P_{1} ≅ Q_{8}, [P_{0}, P_{1}] = 1, and P_{0} ∩ P_{1} = [P, P] = Z(P_{1}).

(b) If the image of Aut_{F}(S) in Aut(S/Z(S)) is a 2-group, then Aut_{F}(S) = Δ×Inn(S) for some (unique) subgroup Δ ≤ Aut_{F}(S) of odd order.

(c) Assume F is reduced and the image of Aut_{F}(S) in Aut(S/Z(S)) is a 2-group. Assume also, for each P ∈ E, that [N_{S}(P), P] is cyclic and the factor P_{0} of point (a) is contained in Z(S). Then Out_{F}(S) = 1, and Ω_{1}(Z(S)) ≤ [S, S].

Proof. (a) By Lemma 1.5, |Out_{S}(P)| = |N_{S}(P)/P | = 2. If P is abelian, then by Lemma A.8, P and Aut_{F}(P) have decompositions as described in (i). More precisely, Lemma A.8 says that P_{1} ≅ C_{2^{n}} × C_{2^{n}} and [N_{S}(P), P] = [N_{S}(P), P_{1}] ≅ C_{2^{n}} for some n ≥ 1, and hence that [N_{S}(P), P] is a direct factor of P_{1}. Also, for x ∈ N_{S}(P)∖P, C_{P_{1}}(N_{S}(P)) = C_{P_{1}}(x) is the kernel of the map P_{1} ⟶ P_{1} which sends g to [x, g], so P_{1}/C_{P_{1}}(x) ≅ [x, P_{1}] ≅ C_{2^{n}}, and C_{P_{1}}(x) ≅ C_{2^{n}} is also a direct factor of P_{1}.

Assume P is nonabelian. By Lemma A.2 (applied with P_{0} = [P, P]), the kernel of the action of Aut_{F}(P) on P^{ab} is a 2-group. Also, O_{2}(Out_{F}(P)) = 1 (i.e., Out_{S}(P) ⋬ Out_{F}(P)), since Out_{F}(P) has a strongly 2-embedded subgroup. Thus Out_{F}(P) acts faithfully on P^{ab}.

Set P′ = [P, P] ≠ 1. By Lemma A.8, applied to the Out_{F}(P)-action on P^{ab} = P/P′, there are decompositions P = P_{0}P_{1} and Out_{F}(P) = Γ_{0} × Γ_{1} such that P_{i} ⊴ P is Aut_{F}(P)-invariant, P_{0} ∩ P_{1} = P′, P_{1}/P′ ≅ C_{2^{n}} × C_{2^{n}} for some n ≥ 1, Γ_{0} has odd order and acts trivially on P_{1}/P′, and Γ_{1} ≅ Σ_{3} acts trivially on P_{0}/P′. Also (by the same lemma), [N_{S}(P), P^{ab}] = [N_{S}(P), P_{1}/P′] ≅ C_{2^{n}}, so

P_{1}/[N_{S}(P), P] ≅ (P_{1}/P′)/[N_{S}(P), P_{1}/P′] ≅ C_{2^{n}}.

Since [N_{S}(P), P] is cyclic by assumption, P_{1} is metacyclic.

Any [α] ∈ Γ_{1} ≅ Σ_{3} of order 3 lifts to some α ∈ Aut_{F}(P), and upon replacing α by α^{k} for appropriate k, we can assume α has order 3. If P_{1} is abelian, then P_{1} ≅ C_{2^{n}} × C_{2^{m}} where m > n (since P′ ≠ 1), which is impossible by Corollary A.3(a). So by Lemma A.7, P_{1} ≅ Q_{8}. Hence P_{1}/P′ ≅ C_{2}^{2}, and P′ = [P_{1}, P_{1}] = Z(P_{1}) ≅ C_{2}.

For x ∈ P_{0} and y ∈ P_{1}, [x, y] = α([x, y]) = [x, α(y)] since |P′| = 2 and α acts trivially on P_{0}/P′. Hence [x, y^{−1}α(y)] = 1, and [P_{0}, P_{1}] = 1 since [α, P_{1}] = P_{1}.

(b) Assume that the image of Aut_{F}(S) in Aut(S/Z(S)) is a 2-group. Then for each α ∈ Aut_{F}(S) of odd order, α induces the identity on S/Z(S). So for each g ∈ S, there is x ∈ Z(S) such that α(g) = xg, and hence αc_{g}α^{−1} = c_{α(g)} = c_{g}.

Thus each element of odd order in Aut_{F}(S) commutes with Inn(S), so Aut_{F}(S) = Inn(S)C_{Aut_{F} (S)}(Inn(S)). Since Inn(S) is 2-centric in Aut_{F}(S), C_{Aut_{F} (S)}(Inn(S)) = Z(Inn(S)) × Δ where Δ has odd order (cf. [BLO1, Lemma A.4]). Thus Aut_{F}(S) = Inn(S) × Δ.

(c) Let Δ be as in (b), and set Q = [Δ, S] ≤ Z(S). We first show that Q ⊴ F. Fix P ∈ E, and let P = P_{0}P_{1} and Out_{F}(P) = Γ_{0} × Γ_{1} be the decompositions of (a).

For each δ ∈ Δ, δ(P) = P since [Δ, S] = Q ≤ Z(S) ≤ P, so δ|_{P} ∈ Aut_{F}(P). Also, [δ|_{P}] ∈ N_{Out_{F} (P)}(Out_{S}(P)) = Γ_{0} × Out_{S}(P) since δ(N_{S}(P)) = N_{S}(P), so [δ|_{P}] ∈ Γ_{0} since it has odd order. Thus [δ, P] ≤ P_{0}. Hence

Q = [Δ, S] = [Δ, [Δ, S]] ≤ [Δ, P] ≤ P_{0},

where the second equality holds by [G, Theorem 5.3.6].

Fix β ∈ Aut_{F}(P) of odd order, and let [β] be its class in Out_{F}(P). If [β] ∈ Γ_{0}, then since P_{0} ≤ Z(S) by assumption, β|_{P_{0}} extends to an element of odd order in Aut_{F}(S) by the extension axiom (i.e., since P_{0} is fully centralized), and thus extends to an element of Δ. So β(Q) = Q in this case. If [β] ∈ Γ_{1}, then β induces the identity on P_{0}/[P, P] and on [P, P] (since |[P, P]| ≤ 2), and hence β|_{P_{0}} = Id_{P_{0}} (and β|_{Q} = Id_{Q}) by Lemma A.2. Since Aut_{S}(P) acts trivially on Z(S), this proves that all elements of Aut_{F}(P) send Q to itself.

Since this holds for each P ∈ E, Q ⊴ F by Proposition 1.3. Hence Q = [Δ, S] = 1 since F is reduced, so Out_{F}(S) ≅ Δ = 1, and Ω_{1}(Z(S)) ≤ [S, S] by Proposition 2.3(a) (and since F is reduced). □

The next proposition will be greatly generalized in Section 5, as a consequence of the results in Section 4 using amalgams.

Proposition 2.5. Let S be any finite nonabelian 2-group such that [S, S] is cyclic and S has an abelian subgroup of index two. Then either S ≅ D_{2^{n}} (n ≥ 3), SD_{2^{n}} (n ≥ 4), or C_{2^{n}} ≀ C_{2} (n ≥ 2); or there is no reduced fusion system over S.

Proof. Let A < S be abelian of index two in S. By Lemma A.6(a), all elements in (S/Z(S))∖(A/Z(S)) have order two, and A/Z(S) ≅ [S, S]. Since [S, S] is cyclic, S/Z(S) is dihedral (or ≅ C_{2}^{2}).

Assume F is a reduced fusion system over S, and let E be the set of F-essential subgroups of S. We first show that the hypotheses of Lemma 2.4(c) hold. For each P ∈ E, [N_{S}(P), P] is cyclic since [S, S] is cyclic, and hence Lemma 2.4(a) applies to P. Let P_{0}, P_{1} ⊴ P be as in that lemma; thus P = P_{0}P_{1} and P_{0} ∩ P_{1} = [P, P].

If P is abelian, it must be maximal abelian since it is centric. So either P = A, in which case Z(S) = C_{P} (S) = C_{P} (N_{S}(P)) and P/Z(S) = A/Z(S) is cyclic; or P A = S, in which case P ∩ A = Z(S), so |P/Z(S)| = 2, Z(S) ≤ C_{P} (N_{S}(P)) < P, and hence Z(S) = C_{P} (N_{S}(P)). In either case, Z(S) = C_{P} (N_{S}(P)) ≥ P_{0} and P/Z(S) is cyclic.

If P is nonabelian, then by Lemma 2.4(a.ii), P_{1} ≅ Q_{8}, [P, P] = Z(P_{1}), and [P_{0}, P_{1}] = 1. In particular, Z(S) ≤ C_{P} (P_{1}) = P_{0}. If Z(S) < P_{0}, then P/Z(S) contains a subgroup isomorphic to Q_{8} (if Z(P_{1}) ≰ Z(S)) or C_{2}^{3}, both of which are impossible since S/Z(S) is dihedral. Hence Z(S) = P_{0}.

Recall that S/Z(S) is dihedral of order ≥ 4. If |S/Z(S)| ≥ 8, then Aut(S/Z(S)) is a 2-group by Corollary A.3(b). If |S/Z(S)| = 4, then subgroups A_{1}, A_{2}, A_{3} of index two in S which contain Z(S) are all abelian, they are the only proper subgroups centric in S, and hence the only subgroups which could be in E. If A_{i} ∈ E (recall E ≠ ∅ by Lemma 2.2), then by Lemma 2.4(a.i), it contains a direct factor A_{i1} ≅ C_{2^{m}} ×C_{2^{m}} (some m ≥ 1), which in turn contains C_{A_{i1}}(S) ≅ C_{2^{m}} and [S, S] = [S, A_{i}] ≅ C_{2^{m}} as direct factors. Thus Z(S) = C_{A_{i}}(S) ≅ A_{i0}×C_{2^{m}} and [S, S] are both direct factors of A_{i}, and so [S, S] ≰ Fr(Z(S)). By Lemma A.6(d), there is no automorphism of S which permutes the A_{i} transitively, and thus the image of Aut_{F}(S) in Aut(S/Z(S)) ≅ Aut(C_{2}^{2}) is a 2-group.

The hypotheses of Lemma 2.4(c) thus hold, and so Ω_{1}(Z(S)) ≤ [S, S]. Since [S, S] is cyclic, this implies that |Ω_{1}(Z(S))| = 2, and hence that Z(S) is cyclic.

If |Z(S)| = 2, then Z(S) = Ω_{1}(Z(S)) ≤ [S, S]. So S^{ab} ≅ (S/Z(S))^{ab} ≅ C_{2}^{2}, which implies S is dihedral, semidihedral, or quaternion (cf. [G, Theorem 5.4.5]). If S ≅ Q_{2^{n}}, then by Lemma 2.4(a), for each P ∈ E, P ≅ Q_{8} and Z(P) = Z(S). Hence Z(S) ⊴ F by Proposition 1.3, which contradicts the assumption that F is reduced.

Now assume |Z(S)| = 2^{m} for m ≥ 2; we will show that S ≅ C_{2^{m}} ≀C_{2}. If P ≤ S is any nonabelian subgroup, then since [S, S] is cyclic, Ω_{1}([S, S]) = Ω_{1}([P, P]) is characteristic in P. So if all F-essential subgroups are nonabelian, then Ω_{1}([S, S]) is characteristic in each of them, and hence is normal in F by Proposition 1.3. Since this contradicts the assumption that F is reduced, there is an abelian subgroup P ∈ E, and we already saw that this implies P/Z(S) is cyclic. Since Z(S) is also cyclic, P has rank two, and P ≅ (C_{2^{n}})^{2} by Lemma 2.4(a.i) (i.e., P_{0} = 1). Then n = m ≥ 2, since Z(S) and P/Z(S) are cyclic of order 2^{m} and 2^{2n−m}, respectively. Also, P/Z(S) ≅ C_{2^{m}} is normal in the dihedral group S/Z(S), and hence P ⊴ S. So [S:P] = 2 by Lemma 1.5 (and since rk(P) = 2). Choose any t ∈ S∖P, fix a ∈ P such that aZ(S) generates P/Z(S), and set b = tat^{−1}. Then a^{2^{m−1}} ∉ Z(S) implies 1 ≠ [t, a^{2^{m−1}}] = (ba^{−1})^{2^{m−1}}, so |ba^{−1}| = 2^{m}, and P = ⟨a, b⟩. Also, tbt^{−1} = a since t^{2} ∈ P, so Z(S) = ⟨ab⟩. Let i be such that t^{2} = (ab)^{i}; then (a^{−i}t)^{2} = 1, and this finishes the proof that S ≅ C_{2^{m}} ≀ C_{2}. □

In Section 5, as applications of our main theorems, we will generalize Proposition 2.5 by giving different (weaker) conditions on a 2-group each of which implies the conclusion of Proposition 2.5. For example, by Proposition 5.2(a,b), the same conclusion holds if S has an abelian subgroup of index 2 or [S, S] is cyclic.

3. Two examples

For use in Section 4, we determine here the essential subgroups of the simple groups PSU_{3}(3) and M_{12}. At the same time, since reduced fusion systems over wreath products C_{2^{n}} ≀ C_{2} play an important role in Section 5 (and in Proposition 2.5), we determine all reduced fusion systems over such groups.

We begin with the wreath products. Let v_{2}(−) denote the 2-adic valuation: v_{2}(n) = k if 2^{k}|n and 2^{k+1}∤n.

Proposition 3.1. Assume S = ⟨a, b, t⟩ ≅ C_{2^{m}} ≀C_{2} for some m ≥ 2, where A =^{def} ⟨a, b⟩ ≅ C_{2^{m}} × C_{2^{m}}, t^{2} = 1, and tat^{−1} = b. Set

Q = ⟨ab, a^{2^{m−1}}, t⟩ ≅ C_{2^{m}} ×_{C_{2}} D_{8} ≅ C_{2^{m}} ×_{C_{2}} Q_{8} .

(a) If F is a saturated fusion system over S, then the only subgroups of S which could be F-essential are A and the subgroups S-conjugate to Q. If O^{2}(F) = F, then all of these subgroups are F-essential.

(b) Up to isomorphism, there is a unique saturated fusion system F over S such that O^{2}(F) = F. Also, F is reduced, and is isomorphic to the fusion system of PSL_{3}(q) for any prime power q such that v_{2}(q−1) = m, and to the fusion system of PSU_{3}(q) for any prime power q such that v_{2}(q + 1) = m.

Proof. Let F be a saturated fusion system over S, and let E be the set of F-essential subgroups of S. If P ∈ E and |N_{S}(P)/P | ≥ 4, then rk(P/Fr(P)) ≥ 4 by Lemma 1.5. Since P ∩ A is abelian of rank ≤ 2 and has index ≤ 2 in P, this is impossible. So |N_{S}(P)/P | = 2.

Assume P ≠ A. Since P is centric in S, Z(S) = ⟨ab⟩ ≤ P. Also, P ≰ A since P is centric, and P ≱ A since P ∉ {A, S}. Thus P ∩ A = ⟨ab, a^{2i}⟩ for some i, and P = ⟨ab, a^{2i}, a^{j}t⟩ for some j. If a^{2i} = 1, then P is abelian with a cyclic subgroup Z(S) of index two and order ≥ 4, and Aut(P) is a 2-group by Corollary A.3(a). If |a^{2i}| ≥ 4, then Z(P) = ⟨ab⟩ = Z(S) is cyclic and P/Z(P) is dihedral of order 2·|a^{2i}| ≥ 8 (all elements in (S/Z(S))∖(A/Z(S)) have order two by Lemma A.6(a)), so Aut(P) is a 2-group by Lemma A.2 and Corollary A.3(b). Thus P can be essential only if P = ⟨ab, a^{2i}, a^{j}t⟩ where |a^{2i}| = 2. If j is odd, then (a^{j}t)^{2} = (ab)^{j} generates Z(S), P^{ab} ≅ C_{2} × C_{2^{m}}, and Aut(P) is a 2-group by Corollary A.3(a,c). This leaves only the possiblity P = ⟨ab, a^{2^{m−1}}, a^{j}t⟩ = a^{j/2}Qa^{−j/2} for even j. Thus E ⊆ {A} ∪ Q, where Q denotes the S-conjugacy class of Q.

Now assume O^{2}(F) = F. Then 𝔣𝔬𝔠(F) = S by Proposition 1.8. By Corollary A.3(a,c) (and since S^{ab} ≅ C_{2^{m}} × C_{2}), Aut(S) is a 2-group, and so Out_{F}(S) = 1. Hence 𝔣𝔬𝔠(F) ≤ ⟨[S, S], E⟩. Since the images of A and of Q are both properly contained in S^{ab}, E = {A} ∪ Q. This proves (a).

Each of the three abelian subgroups of index two in Q is isomorphic to C_{2^{m}} × C_{2} and contains exactly two elements of order 4 not in Z(S). Hence Q contains exactly six such elements, and they generate a subgroup Q_{0} which is the unique subgroup of Q isomorphic to Q_{8}. Since Q = Q_{0}Z(S) and Q_{0} ∩Z(S) = ⟨(ab)^{2^{m−1}}⟩ (and Z(S) = Z(Q)), Out(Q) = Out(Z(S)) × Out(Q_{0}), where Out(Q_{0}) ≅ Σ_{3}, and Out(Z(S)) is a 2-group since Z(S) is cyclic. Hence Out_{F}(Q) ≅ Σ_{3}, and Aut_{F}(Q) acts via the identity on Z(S) and the full automorphism group of Q_{0}. Thus Aut_{F}(Q) is uniquely determined.

By Lemma A.2, Aut(A)/O_{2}(Aut(A)) ≅ Aut(A/Fr(A)) ≅ Σ_{3}. Hence Aut_{F}(A) ≅ Σ_{3}. Set c = (ab)^{−1} ∈ Z(S), and let {a′, b′, c} be its Aut_{F}(A)-orbit. Thus a′, b′, c represent the three involutions in A/Fr(A), and any two of them generate A. Also, a′b′c = 1 since Aut_{F}(A) fixes a′b′c, and hence c = (a′b′)^{−1}.

Since c ∈ Z(S) = C_{A}(t), c_{t} exchanges a′ and b′ and fixes c. We can thus assume the a and b were chosen so that a = a′ and b = b′. So up to an automorphism of S (i.e., a relabelling of its generators), Aut_{F}(A) is uniquely determined. Thus F is uniquely determined up to isomorphism by (a) and Theorem 1.2.

Now, O^{2′}(F) = F since Out_{F}(S) = 1 (F is generated by O^{2′}(F) and Aut_{F}(S) by Theorem 1.2). If P ⊴ F, then P is contained in all F-essential subgroups (cf. [AKO, Proposition I.4.5]), and hence is contained in their intersection ⟨ab, a^{2^{m−1}}⟩. Then P ≤ ⟨ab⟩ since it is Aut_{F}(Q)-invariant, and so P = 1 since it is Aut_{F}(A)-invariant. Thus O_{2}(F) = 1, and F is reduced.

If q is a prime power with v_{2}(q − 1) = m, then GL_{2}(q) ≤ SL_{3}(q) and hence PSL_{3}(q) have Sylow 2-subgroups isomorphic to S; while if v_{2}(q +1) = m, then GU_{2}(q) ≤ SU_{3}(q) and PSU_{3}(q) have Sylow 2-subgroups isomorphic to S (cf. [CF, pp. 142–143]). Set G = PSL_{3}(q) or PSU_{3}(q), as appropriate, and identify S ∈ Syl_{2}(G). Since G is simple, 𝔣𝔬𝔠(F_{S}(G)) = S ∩ [G, G] = S by the focal subgroup theorem (cf. [G, Theorem 7.3.4]), and hence F_{S}(G) ≅ F. □

We now look at 2-groups of type M_{12}.

Proposition 3.2. Consider the group S = C_{4}^{2}_{a,b} ⋊ C_{2}^{2}_{r,t}, where rar^{−1} = a^{−1}, rbr^{−1} = b^{−1},

tat^{−1} = b, and tbt^{−1} = a. Set A = ⟨a, b⟩ ≅ C_{4}^{2}, R = ⟨A, r⟩ ≅ C_{4}^{2} ⋊ C_{2}, and

Q = ⟨a^{2}, ab, r, t⟩ = ⟨ab^{−1}, a^{2}t⟩ ×_{⟨a^{2}b^{2}⟩} ⟨ab, a^{2}rt⟩ ≅ Q_{8} ×_{C_{2}} Q_{8} .

For any saturated fusion system F over S, the set of F-essential subgroups is contained in {Q, R}, with equality if F is reduced.

Proof. Let E be the set of F-essential subgroups of S. If F is reduced, then |E| ≥ 2 by Lemma 2.2. So it suffices to prove that E ⊆ {R, Q} for each saturated F.

Fix P ∈ E. Then Z(S) = ⟨a^{2}b^{2}⟩ ≤ P. Assume first that Fr(S) = ⟨a^{2}, ab⟩ ≰ P, and fix g ∈ Fr(S)∖P. Then [g, P] ≤ [g, S] = ⟨a^{2}b^{2}⟩, so |N_{S}(P)/P | = 2 by Lemma 1.5, and a^{2}b^{2} ∉ Fr(P) by Lemma 1.4 (applied with Θ = 1). In particular, since (ab)^{2} = (ab^{−1})^{2} = a^{2}b^{2}, neither ab nor ab^{−1} is in P. Hence a^{2} ∈ P, since |N_{S}(P)/P | = 2. Also, P ≤ C_{S}(a^{2}) = ⟨a, b, r⟩ since a^{2}b^{2} ∉ Fr(P), N_{S}(P) ≥ ⟨a, b, r⟩ since ⟨a, b, r⟩/⟨a^{2}, b^{2}⟩ is abelian and P ≥ ⟨a^{2}, b^{2}⟩, and hence [⟨a, b, r⟩:P] ≤ 2. So up to S-conjugacy, P = ⟨a, b^{2}, r⟩ or ⟨a, b^{2}, br⟩ (recall ⟨a^{2}, ab⟩ ≰ P). In either case, P ≅ C_{2} × D_{8}, so Aut(P) is a 2-group by Lemma A.2 and since P contains a unique subgroup isomorphic to C_{2} ×C_{4}, and hence P ∉ E.

Thus P ≥ Fr(S), so P ⊴ S. If [S:P] ≥ 4, then |P | ≤ 16, rk(P/Fr(P)) ≥ 4 by Lemma 1.5, and hence P ≅ C_{2}^{4}. This is impossible since P ≥ Fr(S) ≅ C_{4} × C_{2}, and so [S:P] = 2. If P = ⟨a, b, t⟩ or ⟨a, b, rt⟩, then P/[P, P] ≅ C_{4} ×C_{2}, Aut(P) is a 2-group by Corollary A.3(a,c), and hence P ∉ E. So R = ⟨a, b, r⟩ is the only (possible) subgroup in E which contains A.

Now assume P = P_{ij} = ⟨ab, a^{2}, a^{i}r, a^{j}t⟩ for i, j = 0, 1: these are the remaining four subgroups of index 2 in S. Let Z_{2}(P) ⊴ P be the subgroup such that Z_{2}(P)/Z(P) = Z(P/Z(P)). Then Z(P) = ⟨a^{2}b^{2}⟩ and Z_{2}(P) ≥ ⟨a^{2}, ab⟩. If (i, j) ≠ (0, 0), then the relations

[r, at] = a^{2}, [ar, t] = ab^{−1}, [ar, at] = a^{−1}b^{−1}

show that Z_{2}(P) = ⟨a^{2}, ab⟩. So [a, P] ≤ Z_{2}(P), [a, Z_{2}(P)] = 1, and P ∉ E by Lemma 1.4. Thus Q = P_{00} is the only possible subgroup in E which does not contain A. □

With a little more work, one can show that the only reduced fusion systems over S (as above) are those of M_{12} and G_{2}(3). But we leave that for a later paper.

4. Detecting essential subgroups via amalgams

We are now ready to describe how theorems of Goldschmidt and Fan [Gd2, Fn] on amalgams can be used to get information about essential subgroups of index two in their normalizer for saturated fusion systems over 2-groups. Throughout the section, in the statements of lemmas and in the proofs of Theorems 4.5 and 4.6, we will refer repeatedly to the following set of hypotheses.

(∗) Assume P_{1}, P_{2} ≤ P ≤ G_{1}, G_{2} are finite groups such that the following hold: • P is a 2-group, [P:P_{1}] = [P:P_{2}] = 2, and P = P_{1}P_{2}. • For i = 1, 2, P_{i} ⊴ G_{i}, G_{i}/P_{i} ≅ D_{2p_{i}} for some odd prime p_{i}, and C_{G_{i}}(P_{i}) ≤ P_{i}. Set P_{12} = P_{1} ∩ P_{2}, and let T ≤ P_{12} be the largest subgroup which is normal in both G_{1} and G_{2}.

Clearly, hypotheses (∗) imply that [G_{i}:P] = p_{i} and P_{i} = O_{2}(G_{i}) for i = 1, 2. In particular, in the terminology of Goldschmidt [Gd2], the triple (G_{1} > P < G_{2}) is an amalgam of index (p_{1}, p_{2}).

The following lemma helps to explain the motivation for these hypotheses.

Lemma 4.1. Fix a finite 2-group S and a saturated fusion system F over S.

(a) Assume P_{1}, P_{2} ≤ S are distinct F-essential subgroups of index two in S. Then there are groups G_{1} > S < G_{2}, and odd primes p_{1} and p_{2}, such that Out_{G_{i}}(P_{i}) ≤ Out_{F}(P_{i}), and such that hypotheses (∗) hold with P = S.

(b) Let P_{1} ≤ S be an F-essential subgroup which is not normal and has index two in its normalizer. Set P = N_{S}(P_{1}) < S, choose x ∈ N_{S}(P)∖P such that x^{2} ∈ P, and set P_{2} = xP_{1}x^{−1}. Then there are groups G_{1} > P < G_{2} and an odd prime p, such that Out_{F}(P_{i}) ≥ Out_{G_{i}}(P_{i}) and hypotheses (∗) hold with p_{1} = p_{2} = p. Also, x ∈ N_{S}(T), where T is as defined in (∗), and there is an isomorphism

β: G_{1} ⟶^{≅} G_{2} such that β|_{P} = c_{x}|_{P}.

Proof. Assume the hypotheses of (a) or (b), where P = S in (a). Since each P_{i} is Fessential (and |N_{S}(P_{i})/P_{i}| = 2), Out_{S}(P_{i}) ∈ Syl_{2}(Out_{F}(P_{i})) and is not the only Sylow 2-subgroup. Fix 1 ≠ g ∈ Out_{S}(P_{i}), and let h ∈ Out_{F}(P_{i}) be any other involution. Then ⟨g, h⟩ is dihedral since it is generated by two involutions, and it has order 2n for some odd integer n > 1 since 4∤|Out_{F}(P_{i})|. Upon choosing an appropriate subgroup Γ_{i} ≤ ⟨g, h⟩, we can arrange that Out_{S}(P_{i}) ≤ Γ_{i} ≤ Out_{F}(P_{i}) and Γ_{i} ≅ D_{2p_{i}} for some odd prime p_{i}.

Fix i = 1, 2, and set F_{i} = N_{F}(P_{i}) (see Definition 1.1). This is a saturated fusion system over N_{S}(P_{i}) = P (cf. [AKO, Theorem I.5.5]), and is constrained in the sense of [AKO, Definition I.4.8] since P_{i} ⊴ F_{i} and C_{P} (P_{i}) ≤ P_{i}. By the model theorem in the form of [AKO, Theorem III.5.10(a)], there is a finite group G_{i}^{∗} ≥ P such that P_{i} ⊴ G_{i}^{∗}, C_{G_{i}^{∗}}(P_{i}) ≤ P_{i}, P ∈ Syl_{2}(G_{i}^{∗}), and F_{P} (G_{i}^{∗}) = F_{i}. In particular, Out_{G_{i}^{∗}}(P_{i}) = Out_{F}(P_{i}). Let G_{i} ≤ G_{i}^{∗} be the unique subgroup such that P ≤ G_{i} and Out_{G_{i}}(P_{i}) = Γ_{i}. Then these groups satisfy hypotheses (∗).

In the situation of (b), we can assume that Γ_{2} is chosen so that Γ_{2} = [c_{x}]Γ_{1}[c_{x}]^{−1} ≤ Out_{F}(P_{2}). Choose G_{1} as in (a), and then choose G_{2} together with an isomorphism β ∈ Iso(G_{1}, G_{2}) such that β|_{P} = c_{x}|_{P}. Since T is the unique largest subgroup of P_{12} which is normal in G_{1} and G_{2}, x ∈ N_{S}(T) since c_{x} exchanges P_{1} and P_{2} (recall x^{2} ∈ P = N_{S}(P_{1})). □

As usual, we say T is centric in a group X ≥ T if C_{X}(T) ≤ T. In general, our results using hypotheses (∗) split into separate cases, depending on whether or not T is centric in P.

We say that a finite group G is strictly p-constrained for a prime p if O_{p}(G) is centric in G. The question of whether one or both of the groups G_{i}/T (under hypotheses (∗)) is strictly 2-constrained plays an important role in Fan’s classification of amalgams of type (p_{1}, p_{2}) [Fn].

Lemma 4.2. Assume hypotheses (∗). Then the following hold.

(a) If T is centric in P, then T is also centric in G_{1} and G_{2}.

(b) If O_{p_{1}}(G_{1}/T) ≠ 1 or O_{p_{2}}(G_{2}/T) ≠ 1, then T is centric in P.

(c) For i = 1, 2, G_{i}/T is strictly 2-constrained if and only if O_{p_{i}}(G_{i}/T) = 1.

(d) Assume T is centric in P, and let S ≥ P be any finite 2-group such that N_{S}(P_{i}) = P for i = 1, 2. Then T is centric in S.

(e) Assume T is centric in P, and let S ≥ P be as in (d). Set Ĝ_{i} = Out_{G_{i}}(T), Ŝ = Out_{S}(T), and Ĝ = ⟨Ĝ_{1}, Ĝ_{2}, Ŝ⟩ ≤ Out(T), and assume Ŝ ∈ Syl_{2}(Ĝ). Then O_{2}(Ĝ) = 1, and Ĝ acts faithfully on T/Fr(T).

Proof. Let S ≥ P be any finite 2-group as in (d); i.e., such that N_{S}(P_{i}) = P for i = 1, 2. Set

P^{0} = C_{P} (T), G_{i}^{0} = C_{G_{i}}(T) (i = 1, 2), and S^{0} = C_{S}(T) .

Note that P^{0} ⊴ P, G_{i}^{0} ⊴ G_{i}, and P^{0} ∈ Syl_{2}(G_{i}^{0}) since T is normal in P and the G_{i}.

If p_{i}∤|G_{i}^{0}|, then G_{i}^{0} is a 2-group, and G_{i}^{0} ≤ O_{2}(G_{i}) = P_{i} since G_{i}^{0} ⊴ G_{i}. If P^{0} = G_{1}^{0} = G_{2}^{0}, then P^{0} ≤ P_{12} and is normal in G_{1} and G_{2}, so P^{0} ≤ T by the maximality of T. To summarize:

p_{i}∤|G_{i}^{0}| ⟹ P^{0} = G_{i}^{0} ≤ P_{i} and P^{0} = G_{1}^{0} = G_{2}^{0} ⟹ P^{0} ≤ T .  (1)

We next claim that when i = 1 or 2,

O_{p_{i}}(G_{i}/T) ≠ 1 or P^{0} ≤ T ⟹ p_{i}∤|G_{i}^{0}| .  (2)

To see this, assume p_{i}||G_{i}^{0}|, and fix g_{i} ∈ G_{i}^{0} of order p_{i}. Then [g_{i}, T] = 1, while [g_{i}, P_{i}] ≠ 1 by (∗) (C_{G_{i}}(P_{i}) ≤ P_{i}). So by Lemma A.2, g_{i} acts nontrivially on P_{i}/T; i.e., [g_{i}, P_{i}] ≰ T. Thus ⟨T, g_{i}⟩ ⋬ G_{i}, so O_{p_{i}}(G_{i}/T) = 1. Also, G_{i}^{0}T ⊴ G_{i} contains the normal closure of ⟨T, g_{i}⟩ in G_{i}, so G_{i}^{0}T > ⟨T, g_{i}⟩, P^{0}T > T, and thus P^{0} ≰ T.

(a) If T is centric in P (i.e., P^{0} ≤ T), then it is centric in the G_{i} by (2) and (1).

(b) Let H_{i} ⊴ G_{i} be the subgroup such that H_{i}/T = O_{p_{i}}(G_{i}/T). If H_{1}/T ≠ 1 and H_{2}/T ≠ 1, then P^{0} ≤ T by (2) and (1) again. So assume H_{1}/T ≠ 1 and H_{2}/T = 1. By (2) and (1), P^{0} = G_{1}^{0} ≤ P_{1}. Also, P_{12} ⊴ G_{1} = H_{1}P since

[P_{12}, H_{1}] ≤ [P_{1}, H_{1}] ≤ P_{1} ∩ H_{1} = T ≤ P_{12}

(recall P_{1}/T is a 2-group and |H_{1}/T | = p_{1}). Since G_{1}^{0} ∩P = P^{0} = G_{2}^{0} ∩P and G_{1}^{0} ≤ P_{1},

G_{2}^{0} ∩ P_{2} = G_{1}^{0} ∩ P_{2} = G_{1}^{0} ∩ P_{12} ⊴ G_{1} .

Also, G_{2}^{0} ∩ P_{2} ⊴ G_{2}, so G_{2}^{0} ∩ P_{2} ≤ T by the maximality of T, and the induced map G_{2}^{0}T/T ⟶ G_{2}/P_{2} ≅ D_{2p_{2}} is injective. Then O_{p_{2}}(G_{2}^{0}T/T) = 1 since G_{2}^{0}T/T ⊴ G_{2}/T and O_{p_{2}}(G_{2}/T) = 1, and hence G_{2}^{0}T/T = 1 since it is isomorphic to a normal subgroup of D_{2p_{2}}. Thus P^{0} ≤ G_{2}^{0} ≤ T, so T is centric in P.

(c) By assumption, for i = 1, 2, G_{i}/T is solvable of order 2^{n}p_{i} for some n. Hence the Fitting subgroup F (G_{i}/T) = O_{2}(G_{i}/T)O_{p_{i}}(G_{i}/T) is always centric in G_{i}/T (cf. [G, Theorem 6.1.3]). So G_{i}/T is strictly 2-constrained if and only if O_{p_{i}}(G_{i}/T) = 1.

(d) Assume T is centric in P. Then S^{0} ⊴ N_{S}(T) and S^{0} ∩ P = P^{0} ≤ T. Since N_{S}(P_{i}) = P by assumption, N_{S^{0}P_{i}}(P_{i}) = P ∩ S^{0}P_{i} = P_{i}, so S^{0}P_{i} = P_{i} by Lemma A.1. Thus C_{S}(T) = S^{0} ≤ P, so C_{S}(T) = C_{P} (T), and T is centric in S since it is centric in P.

(e) Assume T is centric in P, and hence also centric in G_{1}, G_{2}, and S by (a) and (d). Assume Ŝ ∈ Syl_{2}(Ĝ), and set Q = O_{2}(Ĝ) ≤ Ŝ for short. For i = 1, 2, Q ∩ Ĝ_{i} ≤ O_{2}(Ĝ_{i}) = P̂_{i}, and hence

N_{QP̂_{i}}(P̂_{i}) = NŜ(P̂_{i}) ∩ QP̂_{i} = P̂ ∩ QP̂_{i} = (P̂ ∩ Q)P̂_{i} = P̂_{i} .

By Lemma A.1, this implies QP̂_{i} = P̂_{i}, and hence Q ≤ P̂_{i}.

Thus Q ≤ P̂_{12}. Hence Q = R̂ = Out_{R}(T) for some unique R ≤ P_{12} such that R ≥ T, and R ⊴ G_{i} (i = 1, 2) since Q = O_{2}(Ĝ) ⊴ Ĝ. Thus R = T by definition of T, and Q = 1.

Since Ĝ ≤ Out(T), the kernel of the induced Ĝ-action on T/Fr(T) is a 2-group by Lemma A.2, and is trivial since O_{2}(Ĝ) = 1. So the action is faithful. □

The next lemma will be needed to handle the cases involving amalgams whose maximal normal subgroup is not centric. As usual, when F is a fusion system over a finite p-group S, a subgroup P ≤ S is strongly closed in F if no element of P is F-conjugate to an element of S∖P. For example, if F = F_{S}(G) for a finite group G, and H ⊴ G, then S ∩ H is strongly closed in F.

Lemma 4.3. Assume hypotheses (∗), and also that T is not centric in P. Set P = P/T and G_{i} = G_{i}/T for short, and let F^{∗} be the smallest fusion system over P which contains F_{P} (G_{1}) and F_{P} (G_{2}). Assume

(a) F^{∗} = F_{P} (Γ) for some finite perfect group Γ for which P ∈ Syl_{2}(Γ);

(b) no nontrivial proper subgroup of P is strongly closed in F^{∗}; (c) G_{i}/O^{2}(G_{i}) is abelian for i = 1, 2; and

(d) (P ∩ [G_{1}, G_{1}]) ∩ (P ∩ [G_{2}, G_{2}]) ≤ [P , P].

Then C_{P} (T)·T = P, and C_{G_{i}}(T)·T = G_{i} for i = 1, 2. If, in addition, we define

U_{i} = P ∩ O^{2}(G_{i}) , U = U_{1}U_{2} ⊴ P , and Z = U ∩ T ,

then [U, T] = 1, P = UT, and U ∈ Syl_{2}(Γ̃) for some finite perfect group Γ̃ such that Z ≤ Z(Γ̃) and Γ̃/Z ≅ Γ.

Proof. In general, for X ≤ G_{i} or g ∈ G_{i} (i = 1, 2), we let X = XT/T or g = gT denote the image of X or g, respectively, in G_{i}. Write P^{0} = C_{P} (T) and G_{i}^{0} = C_{G_{i}}(T) for short. Then G_{i}^{0} ⊴ G_{i} since T ⊴ G_{i}, and hence G_{i}^{0} ⊴ G_{i}. So P^{0} = P ∩ G_{i}^{0} is strongly closed in F_{P} (G_{1}) and F_{P} (G_{2}), and hence is strongly closed in F^{∗} = ⟨F_{P} (G_{1}), F_{P} (G_{2})⟩. Since T is not centric in P by assumption, P^{0} = C_{P} (T) ≰ T, so P^{0} ≠ 1, and hence P^{0} = P by (b). This in turn implies that G_{i}^{0} ≥ P for i = 1, 2, and hence G_{i}^{0} = G_{i} since G_{i} is the normal closure of P (recall that G_{i}/P_{i} ≅ D_{2p_{i}}). We have now shown that

C_{P} (T)·T = P^{0}T = P and C_{G_{i}}(T)·T = G_{i}^{0}T = G_{i} (i = 1, 2) .

In particular, O^{2}(G_{i}) = O^{2}(G_{i}^{0}), and hence U_{i} = P ∩ O^{2}(G_{i}^{0}). Set

U_{i}^{•} = P ∩ [G_{i}^{0}, G_{i}^{0}] , U^{•} = U_{1}^{•}U_{2}^{•} ⊴ P^{0} , and Z^{•} = U^{•} ∩ T .

Then U_{i} ≤ U_{i}^{•} (see Lemma A.4), and hence U ≤ U^{•} and Z ≤ Z^{•}.

Now, U_{i} = P ∩ O^{2}(G_{i}) = P ∩ [G_{i}, G_{i}] = U_{i}^{•}, where the first and third equalities hold since P ≥ T, and the second holds by (c) and Lemma A.4. Hence

U = U_{1}U_{2} = ⟨P ∩ [G_{1}, G_{1}], P ∩ [G_{2}, G_{2}]⟩ = 𝔣𝔬𝔠(F^{∗}) = 𝔣𝔬𝔠(F_{P} (Γ)) = P ,

where the third and fifth equalities hold by the focal subgroup theorem (cf. [G, Theorem 7.3.4]) applied to G_{1}, G_{2}, and Γ (and since Γ is perfect by (a)). Thus UT = P. So after taking intersections with P^{0} = C_{P} (T), we get UZ(T) = P^{0}. Since [U, Z(T)] ≤ [U^{•}, T] = 1 by definition,

[U, U] = [P^{0}, P^{0}] .

Assume u = u_{1}u_{2} ∈ T ∩ U^{•} = Z(T) ∩ U^{•}, where u_{i} ∈ U_{i}^{•}. Then u = 1 and u_{i} ∈ P ∩ [G_{i}, G_{i}], so u_{1} = u_{2}^{−1} ∈ [P , P] by (d). Thus u_{i} ∈ P^{0} ∩ [P^{0}T, P^{0}T]T = [P^{0}, P^{0}]Z(T) (recall that [P^{0}, T] = 1). Write u_{i} = g_{i}t_{i} where g_{i} ∈ [P^{0}, P^{0}] ≤ U_{i}^{•} and t_{i} ∈ Z(T) ∩ U_{i}^{•}. Since Z(T) ∩ U_{i}^{•} = Z(T) ∩ [G_{i}^{0}, G_{i}^{0}] = Z(T) ∩ [P^{0}, P^{0}], the last equality by Proposition A.5, we see that g_{1}g_{2}, t_{1}, and t_{2} all lie in Z(T) ∩ [P^{0}, P^{0}]. This proves that

Z^{•} = T ∩ U^{•} = Z(T) ∩ [P^{0}, P^{0}] = Z(T) ∩ [U, U] ≤ T ∩ U = Z .

Hence Z = Z^{•}, since we already saw that Z ≤ Z^{•}.

Let τ ∈ H^{2}(P; Z(T)) and τ_{i} ∈ H^{2}(G_{i}; Z(T)) be the classes of the central extensions P^{0} and G_{i}^{0}, respectively. Thus τ = τ_{i}|_{P}, so τ is stable with respect to fusion in G_{i}, and hence is stable with respect to the fusion system F^{∗}. Since F^{∗} = F_{P} (Γ) by (a), τ is the restriction of a unique element τ_{Γ} ∈ H^{2}(Γ; Z(T)), where Γ acts trivially on Z(T) (cf. [CE, Theorem XII.10.1]). Let Γ̂ be the corresponding central extension of Z(T) by Γ. Thus P^{0} ∈ Syl_{2}(Γ̂).

Set Γ̃ = [Γ̂, Γ̂]. Since Γ̂/Z(T) ≅ Γ is perfect, Γ̂ = Γ̃·Z(T) where [Γ̃, Z(T)] = 1, so Γ̃ = [Γ̂, Γ̂] = [Γ̃, Γ̃]. Thus Γ̃ is perfect. Since P^{0} ∈ Syl_{2}(Γ̂), we have

Z = Z^{•} = Z(T) ∩ [P^{0}, P^{0}] = Z(T) ∩ [Γ̂, Γ̂] = Z(T) ∩ Γ̃ ,

the third equality by Proposition A.5. Recall that P^{0} = UZ(T). Hence P^{0}/U ≅ Z(T)/Z ≅ Γ/Γ̃, and so U ∈ Syl_{2}(Γ̃). □

We recall the terminology which will be used in the statements of our main theorems. By an amalgam is meant here a triple of groups G = (G_{1} > H < G_{2}). A (proper) completion of G is a group G, together with injections ρ_{i}: G_{i} ⟶ G, such that ρ_{1}|_{H} = ρ_{2}|_{H} and G = ⟨ρ_{1}(G_{1}), ρ_{2}(G_{2})⟩. The universal completion is the amalgamated free product G_{1} ∗_{H} G_{2}. When H has prime index in G_{1} and in G_{2}, then the amalgam G is primitive if no nontrivial subgroup of H is normal in G_{1} and in G_{2}. An isomorphism of amalgams from (G_{1} > H < G_{2}) to (G_{1}^{∗} > H^{∗} < G_{2}^{∗}) is a triple of isomorphisms

α: H ⟶^{≅} H^{∗} and β_{i}: G_{i} ⟶^{≅} G_{i}^{∗} such that β_{1}|_{H} = α = β_{2}|_{H}.

Lemma 4.4. For each n = 1, …, 6, let G_{1} = G_{1}^{(n)}, G_{2} = G_{2}^{(n)}, and Γ = Γ^{(n)} be the groups listed in case (n) of Table 4.1. Then there is a primitive amalgam (G_{1} > S < G_{2}) with completion Γ, where S = S^{(n)} ∈ Syl_{2}(G_{i}) and [G_{i}:S] = 3 for i = 1, 2, and this amalgam is uniquely determined up to isomorphism. Set F^{(n)} = ⟨F_{S_{(n)}}(G_{1}^{(n)}), F_{S_{(n)}}(G_{2}^{(n)})⟩.

(a) For each n = 1, 3, 5, Γ^{(n)} can be identified with a subgroup of index two in Γ^{(n+1)} in such a way that the amalgam (G_{1}^{(n)} > S^{(n)} < G_{2}^{(n)}) is contained in (G_{1}^{(n+1)} > S^{(n+1)} < G_{2}^{(n+1)}) with index two, and the normal closure of S^{(n)} in G_{i}^{(n+1)} equals G_{i}^{(n)} for i = 1, 2.

(b) For each 1 ≤ n ≤ 6, if Q ≤ S^{(n)} is strongly closed in F^{(n)}, then either Q = 1 or Q = S^{(n)}, or n is even and Q = S^{(n−1)}.

(c) For n = 1, 3, 5, F^{(n)} = F_{S_{(n)}}(Γ^{(n)}). (This also holds when n = 2, 4, 6, but we will not need that.)

Proof. These are the amalgams denoted G_{i} and G_{i}^{1} for i = 3, 4, 5 in [Gd2, Table 1]. In all but the first case, Goldschmidt’s choice of completion is the same as the one listed here in Table 4.1. (Note that Aut(U_{3}(3)) ≅ G_{2}(2).) In case (1), Goldschmidt lists L_{3}(2) as a completion, but A_{6} is easily seen to be a completion for the same amalgam. The uniqueness of the amalgams (for given G_{i} and Γ) follows from the classification in [Gd2, Theorem A].

Point (a) follows from Goldschmidt’s construction of the amalgams [Gd2, 3.5, 3.7, 3.8], and also by a direct inspection of the groups in question.

Point (b) can be checked case-by-case. When n is even, it follows from point (a), point (b) for n−1, and the observation that no central subgroup of order two in S^{(n)} is strongly closed. So assume n is odd. Since Z(S^{(n)}) is cyclic in all three cases (n = 1, 3, 5), each nontrivial normal subgroup of S = S^{(n)} contains Ω_{1}(Z(S)).

Consider, for example, the case n = 5, where S = ⟨a, b, r, t⟩ in the notation of Proposition 3.2. Set A = ⟨a, b⟩ ≅ C_{4}^{2}, A_{0} = Ω_{1}(A) = ⟨a^{2}, b^{2}⟩, P_{1} = O_{2}(G_{1}) =

1The groups G_{1} in cases (4) and (5) are not isomorphic. See [Gd2, Table 1] for more details.

(G_{1}, G_{2}) ≅  Γ  U ≅  |Z| [S:UT]
(1)  (Σ_{4}, Σ_{4})  A_{6}  D_{8}  1  1
Q_{16}  2  1
(2)  (C_{2} × Σ_{4}, C_{2} × Σ_{4})  Σ_{6}  D_{8}  1  2
Q_{16}  2  2
(3) ((Q_{8} ×_{C_{2}} C_{4}).Σ_{3}, C_{4}^{2} ⋊ Σ_{3})  U_{3}(3)  C_{4} ≀ C_{2}  1  1
(4) ((Q_{8} ×_{C_{2}} Q_{8}) ⋊ Σ_{3}, C_{4}^{2} ⋊ D_{12})^{1} Aut(U_{3}(3)) C_{4} ≀ C_{2}  1  2
(5) ((Q_{8} ×_{C_{2}} Q_{8}) ⋊ Σ_{3}, C_{4}^{2} ⋊ D_{12})^{1}  M_{12}  Syl_{2}(M_{12}) 1  1
Syl_{2}(2M_{12}) 2  1
(6) ((Q_{8} ×_{C_{2}} Q_{8}).D_{12},  Aut(M_{12}) Syl_{2}(M_{12}) 1  2
C_{4}^{2} ⋊ ((C_{2}^{2} × C_{3}) ⋊ C_{2}))  Syl_{2}(2M_{12}) 2  2

Table 4.1

⟨a^{2}, ab, r, t⟩ ≅ Q_{8} ×_{C_{2}} Q_{8}, and P_{2} = O_{2}(G_{2}) = ⟨A, r⟩ ≅ C_{4}^{2} ⋊ C_{2}. If 1 ≠ P_{∗} ≤ S is strongly closed, then P_{∗} ≥ Z(S) = ⟨a^{2}b^{2}⟩ and P_{∗} ∩P_{2} ⊴ G_{2} imply P_{∗} ≥ A_{0} (since A_{0} = ⟨a^{2}, b^{2}⟩ is the normal closure in G_{2} of ⟨a^{2}b^{2}⟩). If P_{∗}/A_{0} ≤ Z(S/A_{0}) = ⟨A_{0}, ab, r⟩/A_{0}, then P_{∗} ≤ P_{1} ∩ P_{2} and P_{∗} ⊴ G_{i} for i = 1, 2, which contradicts the assumption that the amalgam is primitive. Thus P_{∗}/A_{0} ≰ Z(S/A_{0}) and P_{∗} ⊴ S. Every normal subgroup of S/A_{0} ≅ D_{8} × C_{2} is either contained in its center or contains its commutator subgroup, so P_{∗} ≥ [S, S] = ⟨A_{0}, ab⟩. But then P_{∗} ≥ A since P_{∗} ∩ P_{2} ⊴ G_{2}, and so P_{∗} = S since P_{∗} ∩ P_{1} ⊴ G_{1} (and by [A2, Lemma 5.3(2)]).

Point (c) is clear for n = 1. When n = 3 or 5, then F^{(n)} ⊆ F_{S}(Γ) since Γ is a proper completion of (G_{1} > S < G_{2}), and P_{i} =^{def} O_{2}(G_{i}) for i = 1, 2 are the only (possible) F_{S}(Γ)-essential subgroups of Γ by Propositions 3.1 or 3.2, respectively. So to show that F^{(n)} = F_{S}(Γ), it suffices to show that Out_{G_{i}}(P_{i}) ≅ Out_{Γ}(P_{i}) for i = 1, 2. When n = 5 and i = 1, this is shown in [A2, Lemma 5.3(2)], and in all of the other cases, it follows since Aut(P_{i})/O_{2}(Aut(P_{i})) ≅ Σ_{3} by Lemma A.2. □

Theorem 4.5. Let F be a saturated fusion system over a finite 2-group S, with distinct F-essential subgroups P_{1} and P_{2} of index two in S. Then there are finite groups G_{1} > S < G_{2} such that for i = 1, 2, P_{i} ⊴ G_{i}, G_{i}/P_{i} ≅ Out_{G_{i}}(P_{i}) ≅ D_{2p_{i}} for some odd prime p_{i}, and Aut_{G_{i}}(P_{i}) ≤ Aut_{F}(P_{i}). Let T ≤ P_{1} ∩ P_{2} be the largest subgroup normal in both G_{1} and G_{2}. Then these groups satisfy hypotheses (∗) with P = S.

If T is not centric in S, then p_{1} = p_{2} = 3, and (G_{1}/T > S/T < G_{2}/T) is one of the amalgams listed in Table 4.1. Set

U_{i} = S ∩ O^{2}(G_{i}) (i = 1, 2),  U = U_{1}U_{2},  and  Z = U ∩ T.

Then U ⊴ S, [U, T] = 1; and G_{i} = G_{i}/T, U, Z, and [S:UT] are as listed in Table 4.1.

Proof. Set P = S. By Lemma 4.1(a), there are groups G_{1} > P < G_{2} as described such that the P_{i} < P < G_{i} satisfy hypotheses (∗). Let T be as in (∗), and set G_{i} = G_{i}/T and S = S/T.

Assume T is not centric in S. By Lemma 4.2(b,c), O_{p_{i}}(G_{i}) = 1, and G_{i} is strictly 2constrained, for i = 1, 2. So by a theorem of Fan [Fn, Theorem 1], either (G_{1} > S < G_{2}) is one of the amalgams listed by Goldschmidt in [Gd2, Table 1], or it is the ^{2}F_{4}(2)′- or ^{2}F_{4}(2)-amalgam (points (2) and (3) in [Fn, Theorem 1]). It cannot be either of the last two, since that would require that G_{i}/O_{2}(G_{i}) ≅ Sz(2) ≅ C_{5} ⋊ C_{4} for i = 1 or 2 (see, e.g., [Wi, Theorem 1], or the discussion in [Car, § 8.5] of the Levi decomposition of parabolic subgroups of groups of Lie type). Hence it is one of the six amalgams listed in Table 4.1, since the others listed by Goldschmidt involve groups which are not strictly 2-constrained.

Assume we are in case (n) in Table 4.1. When n is odd, we apply Lemma 4.3. Conditions (a) and (b) in the lemma were shown in Lemma 4.4(c,b), respectively, and (c) and (d) are easily checked case-by-case. So by that lemma, [U, T] = 1, UT = S, and U ∈ Syl_{2}(Γ̃) for some finite perfect group Γ̃ such that Z ≤ Z(Γ̃) and Γ̃/Z ≅ Γ. Here, Γ ≅ A_{6}, U_{3}(3), or M_{12}, when n = 1, 3, or 5. These groups have Schur multiplier C_{6}, 1, and C_{2}, respectively (see [A1, (33.15)], [Gr, Theorem 2], and [Mz]). Hence |Z| ≤ 2, with equality possible only when n = 1 or 5.

When n is even, then by Lemma 4.4(a), there are subgroups G̃_{i} < G_{i} and S̃ < S of index two, all containing T, such that S̃ = S∩G̃_{i}, and such that (G̃_{1}/T > S̃/T < G̃_{2}/T ) is an amalgam of type (n − 1). Hence U_{i} = S ∩ O^{2}(G_{i}) = S̃∩ O^{2}(G̃_{i}), and so U = U_{1}U_{2} plays the same role for the new amalgam as for the original one. Also, (∗) holds for (G̃_{1} > S̃ < G̃_{2}), and T is the largest subgroup normal in G̃_{1} and G̃_{2} since the quotient amalgam is primitive. If T were centric in S̃, then it would be centric in the G̃_{i} by Lemma 4.2(a), and since T is not centric in S, T ·C_{S}(T) = T ·C_{G_{i}}(T) > T would be a strictly larger subgroup normal in the G_{i}. Since this contradicts the choice of T, we conclude that T is not centric in S̃, and hence that the result follows by the argument in the last paragraph applied to (G̃_{1} > S̃ < G̃_{2}). □

Theorem 4.5 will be used when looking for pairs of essential subgroups of index two in S. We next turn to the problem of identifying essential subgroups which have index two in their normalizer but are not normal. The idea is to apply the classification of amalgams by Goldschmidt and Fan to a pair of essential subgroups which are conjugate in S, and have index two in their common normalizer.

Theorem 4.6. Let F be a saturated fusion system over a finite 2-group S. Let P_{1} ≤ S be an F-essential subgroup which is not normal in S, and such that |N_{S}(P_{1})/P_{1}| = 2. Set P = N_{S}(P_{1}) < S. Choose x ∈ N_{S}(P)∖P such that x^{2} ∈ P, and set P_{2} = xP_{1}x^{−1}. Then there are finite groups G_{1} > P < G_{2}, an odd prime p, and an isomorphism β: G_{1} ⟶^{≅} G_{2} such that β|_{P} = c_{x} ∈ Aut_{S}(P), and for i = 1, 2, Out_{G_{i}}(P_{i}) ≤ Out_{F}(P_{i}), P_{i} ⊴ G_{i}, and Out_{G_{i}}(P_{i}) ≅ G_{i}/P_{i} ≅ D_{2p}. Let T ≤ P_{1} ∩ P_{2} be the largest subgroup which is normal in G_{1} and in G_{2}.

(a) If T is not centric in S, then p = 3. Set U_{i} = P ∩ O^{2}(G_{i}) ⊴ P, U = U_{1}U_{2}, and Z = T ∩ U. Set W = Fr(U), S_{∗} = N_{S}(W), and let Δ be the normal closure of U in S_{∗}. Then the following hold.

(i) [T, U] = 1; and either U_{i} ≅ C_{2}^{2}, U ≅ D_{8}, and Z = 1; or U_{i} ≅ Q_{8}, U ≅ Q_{16},

and Z = Z(U).

(ii) There is a subgroup T^{•} ⊴ P such that T ≤ T^{•} < P_{1} ∩ P_{2}, [T^{•}:T] ≤ 2;

(ii.1) P_{i} = T^{•}U_{i} and T^{•} ∩ U = T ∩ U = Z; and

(ii.2) [S_{∗}:T^{•}Δ] = 2, C_{S_{∗}}(U_{1}) = T^{•}U_{1} if U ≅ D_{8}, C_{S_{∗}}(U_{1}) = T^{•} if U ≅ Q_{16}.

Also, G_{i}/T, |Z|, U_{i}, U, Δ, [T^{•}:T], and [T^{•}, U] are as described in Table 4.2.

(b) If T is centric in P or (equivalently) centric in S, then [S, S] is nonabelian, O_{2}(Out_{F}(T)) = 1, and Out_{F}(T) acts faithfully on T/Fr(T).

(c) In the situation of (a), if [S, S] is abelian, or if T^{•}/Z is abelian, or more generally if T^{•} contains no quaternion subgroup of order 16 and T^{•}/Z contains no dihedral subgroup of order 8, then S_{∗} = S.

\begin{matrix}G_{i}/T ≅ & |Z| & U_{i} ≅ & U ≅ & Δ ≅ & |T^{•}/T | & [T^{•}, U] \\ Σ_{4} & 1 & C_{2}^{2} & D_{8} & D_{2^{n}}, n ≥ 3 & 1 & 1 \\ Σ_{4} & 2 & Q_{8} & Q_{16} & Q_{2^{n}}, n ≥ 4 & 1 & 1 \\ Σ_{4} × C_{2} & 1 & C_{2}^{2} & D_{8} & D_{2^{n}}, n ≥ 3 & 2 & 1 \\ Σ_{4} × C_{2} & 2 & Q_{8} & Q_{16} & Q_{2^{n}}, n ≥ 4 & 2 & Z \end{matrix}

Table 4.2

Proof. By Lemma 4.1(b), there are finite groups G_{1} > P < G_{2} such that P_{i} ⊴ G_{i}, G_{i}/P_{i} ≅ D_{2p} for some odd prime p, Out_{S}(P_{i}) ≤ Out_{G_{i}}(P_{i}) ≤ Out_{F}(P_{i}), and β|_{P} = c_{x}|_{P} for some β ∈ Iso(G_{1}, G_{2}). Thus (∗) holds with p_{1} = p_{2} = p. Also, x ∈ N_{S}(T), where T ≤ P_{1} ∩ P_{2} is the largest subgroup normal in G_{1} and G_{2}.

(a) If T is not centric in S, then it is not centric in P by Lemma 4.2(d). Hence the G_{i}/T are strictly 2-constrained by Lemma 4.2(b,c). By [Fn, Theorem 1], they are among the groups listed in [Gd2, Table 1], and hence are isomorphic to Σ_{4} or C_{2} × Σ_{4} (and p = 3).

Case 1: G_{i}/T ≅ Σ_{4}. We apply Lemma 4.3 with Γ = A_{6} (see Table 4.1). Conditions (a) and (b) follow from Lemma 4.4(c,b), and conditions (c) and (d) are easily checked. Recall that U_{i} = P ∩O^{2}(G_{i}), U = U_{1}U_{2}, and Z = T ∩U. By the lemma, [T, U] = 1 and P = UT; and U ∈ Syl_{2}(Γ̃), where Γ̃ is a finite perfect group such that Z ≤ Z(Γ̃) and Γ̃/Z ≅ A_{6}. Thus Γ̃ is isomorphic to A_{6} ≅ PSL_{2}(9) or its 2-fold central extension SL_{2}(9) (cf. [A1, 33.15]), and U ≅ D_{8} or Q_{16}. The image of U_{i} in P/T ≅ U/Z is a Sylow 2-subgroup of O^{2}(G_{i}/T) ≅ A_{4}, so P_{i}/T = O_{2}(G_{i}/T) = U_{i}T/T ≅ C_{2}^{2}, and thus P_{i} = U_{i}T. Since each nontrivial subgroup of Q_{16} contains its center,

either U ≅ D_{8}, U_{i} ≅ C_{2}^{2}, and Z = 1; or U ≅ Q_{16}, U_{i} ≅ Q_{8}, and |Z| = 2.

Set T^{•} = T. We have now proven (i) and (ii.1) in this case, and proven the information on the first two lines in Table 4.2, except for that about Δ.

Case 2: G_{i}/T ≅ C_{2} × Σ_{4}. By Lemma 4.4(a), there are subgroups G̃_{i} < G_{i}, P̃_{i} < P_{i}, and P̃ < P of index two, all containing T and satisfying (∗), such that G̃_{1} > P̃ < G̃_{2}, P̃_{i} = O_{2}(G̃_{i}), and (G̃_{1}/T > P̃/T < G̃_{2}/T ) is an amalgam of the type handled in Case 1. Also, U_{i} = O^{2}(G_{i}) ∩ P = O^{2}(G̃_{i}) ∩ P̃_{i} (i.e., U_{i} and U play the same role for the smaller amalgam as for the original one), and T is not centric in P̃ by a similar argument to the one used at the end of the proof of Theorem 4.5. So the conclusions in Case 1 hold after replacing P and P_{i} by P̃ and P̃_{i}. In particular, (i) holds.

Let T^{•}/T ≤ P_{1}/T ≅ C_{2}^{3} be the subgroup of order two which is fixed by the conjugation action of G_{1}/P_{1} ≅ Σ_{3}. Then P_{1}/T = (T^{•}/T)·(T U_{1}/T) and P_{2}/T = (T^{•}/T)·(T U_{2}/T) (but note that T^{•}/T is not fixed by the action of Aut_{G_{2}}(P_{2}) on P_{2}/T). So P_{1} = T^{•}U_{1}, P_{2} = T^{•}U_{2}, and T^{•} ∩ U = T ∩ U = Z. This finishes the proof of (ii.1).

Now, [T^{•}, U] ≤ T^{•} ∩ U = Z since T^{•} and U are both normal in P. Hence [T^{•}, U] = 1 if |Z| = 1. Now assume |Z| = 2, U ≅ Q_{16}, and W = Fr(U) ≅ C_{4}, and fix g ∈ T^{•}∖T. Since Z is central in U, [g, −] is a homomorphism from U to Z, and so [g, W] = 1. Also, gT ∈ Z(G_{1}/T) by definition of T^{•}. Choose a ∈ G_{1} such that aT has order three in G_{1}/T and W ≠ aW a^{−1}. Then aga^{−1} ∈ gT, aW a^{−1} ≤ P ∩ O^{2}(G_{1}) = U_{1}, and [g, aW a^{−1}] = [aga^{−1}, aW a^{−1}] = 1 since [T, U] = 1 by (i) and [g, W] = 1. Since U_{1} = ⟨W, aW a^{−1}⟩, we conclude that [T^{•}, U_{1}] = [g, U_{1}] = 1.

By a similar argument, there is g′ ∈ P_{2}∖T such that g′T ∈ P_{2}/T ≅ C_{2}^{3} is the involution fixed by the action of G_{2}/P_{2} ≅ Σ_{3}, and such that [g′, U_{2}] = 1. Then gT and g′T are both in the center of P/T ≅ C_{2} × D_{8}, not in [P/T, P/T] = W T/T, and gT ≠ g′T since otherwise ⟨T, g⟩ would be normal in G_{1} and in G_{2} (contradicting the maximality of T). So g′ ∈ gwT for some w ∈ W ∖Z. Thus [gw, U_{2}] = 1, and since [w, U_{2}] = Z (w has order 4 in U_{2} ≅ Q_{8}), [T^{•}, U] = [g, U_{2}] = Z.

This finishes the proof of the information in Table 4.2, except for that about Δ.

Both cases. If U ≅ D_{8}, then by Lemma B.4, Δ ⊴ S_{∗} is dihedral, T^{•} ∩ Δ = 1, [S_{∗}:T^{•}Δ] = 2, and all noncentral involutions in Δ are S_{∗}-conjugate. Fix y ∈ U_{1}∖Z(U). For g ∈ S_{∗}, c_{g}(y) is Δ-conjugate to y if g ∈ T^{•}Δ (recall [T^{•}, U_{1}] = 1), so c_{g}(y) lies in the other Δ-conjugacy class of noncentral involutions if g ∉ T^{•}Δ. Thus C_{S_{∗}}(U_{1}) = C_{T^{•}Δ}(U_{1}) = T^{•}U_{1}.

If U ≅ Q_{16}, then by Lemma B.4 applied to U/Z ≤ S_{∗}/Z, Δ/Z ⊴ S_{∗}/Z is dihedral, T^{•} ∩ Δ = Z, [S_{∗}:T^{•}Δ] = 2, and all noncentral involutions in Δ/Z are S_{∗}/Z-conjugate. Since all involutions in Δ/Z are S_{∗}/Z-conjugate to elements of U_{1}/Z where U_{1} ≅ Q_{8}, there are no involutions in Δ∖Z, and Δ is quaternion. Since [T^{•}, U_{1}] = 1, and since the non-normal subgroups of order four in Δ are S_{∗}-conjugate, a similar argument to that used in the last paragraph (applied with y ∈ U_{1}∖Fr(U)) shows that C_{S_{∗}}(U_{1}) = C_{T^{•}Δ}(U_{1}) = T^{•}.

This proves (ii.2), and the description of Δ in Table 4.2.

(c) Assume S > S_{∗}, so N_{S}(S_{∗}) > S_{∗} by Lemma A.1. Fix y ∈ N_{S}(S_{∗})∖S_{∗} such that y^{2} ∈ S_{∗}. Set Δ^{∗} = yΔy^{−1}, Z^{∗} = yZy^{−1}, W^{∗} = yW y^{−1}, and N = Δ ∩ Δ^{∗}. The subgroups Δ and Δ^{∗} are both normal in S_{∗}, so N ⊴ S_{∗} is normal in each of them, and [Δ, Δ^{∗}] ≤ N. If Z = 1 (if Δ is dihedral), then W^{∗} ≠ W, W^{∗} ≤ Z(S_{∗}) (since W ≤ Z(S_{∗})) and hence W^{∗} ≰ Δ, and N = 1 since each nontrivial normal subgroup of Δ^{∗} contains W^{∗}. Thus [Δ^{∗}, Δ] = 1, Δ^{∗} ≤ C_{S_{∗}}(Δ) ≤ T^{•}W, and thus T^{•} ≅ T^{•}W/W contains a dihedral subgroup of order 8. Also, since the commutator subgroup of ⟨Δ, y⟩ contains all elements of the form g^{−1}(ygy^{−1}) for g ∈ Δ, it surjects onto Δ under the projection from Δ·Δ^{∗}, and thus is nonabelian. So [S, S] is nonabelian in this case.

If |Z| = 2 (if Δ is quaternion) and Z^{∗} ≠ Z, then a similar argument shows that Δ^{∗} ≤ T^{•} and hence T^{•} contains a quaternion subgroup of order 16. If Z^{∗} = Z, we apply the argument in the last paragraph to the conjugation action of y on S_{∗}/Z, where y(W/Z)y^{−1} ≠ (W/Z) by assumption. Thus [Δ^{∗}, Δ] ≤ Z, so Δ^{∗} ≤ C_{S_{∗}}(U_{1})·W = T^{•}W since Ker[Aut(Δ) → Aut(Δ/Z)] ≅ C_{2}^{2} is generated by Aut_{W} (Δ) and an automorphism which is the identity on U_{1}. So in this case, T^{•}W/W ≅ T^{•}/Z contains a dihedral subgroup of order 8. In both cases, [S, S] is nonabelian by the argument used in the last paragraph.

(b) If T is centric in P, then it is centric in S by Lemma 4.2(d), while the converse is immediate. Assume both of these hold.

If T is not fully normalized, then there is some φ ∈ Hom_{F}(N_{S}(T), S) such that φ(T) is fully normalized (cf. [AKO, Lemma I.2.6(c)]). Upon replacing T by φ(T), P_{i} by φ(P_{i}), P by φ(P), etc., we can assume T is fully normalized (and (∗) still holds and T is still centric in P).

Set Ŝ = Out_{S}(T), Ĝ_{i} = Out_{G_{i}}(T), and similarly for subgroups of S, and set Ĝ = ⟨Ĝ_{1}, Ĝ_{2}, Ŝ⟩ ≤ Out_{F}(T). Then Ŝ ∈ Syl_{2}(Ĝ) by the Sylow axiom. Set V = T/Fr(T), written additively, and regarded as an 𝔽_{2}[Out(T)]-module. By Lemma 4.2(e), O_{2}(Ĝ) = 1 and Ĝ acts faithfully on V. Hence O_{2}(Out_{F}(T)) = 1, and by Lemma A.2, Out_{F}(T) acts faithfully on V.

It remains to show that [S, S] is nonabelian. Set V_{0} = [Ŝ, V]. Since every element of Ŝ lifts to an element of S, and every element of V_{0} lifts to an element of [S, S], it will suffice to show that [Ŝ, Ŝ] acts nontrivially on V_{0}. Assume otherwise: assume [Ŝ, Ŝ] acts trivially on V_{0}. If g ∈ [Ŝ, Ŝ], then for each v ∈ V, g acts trivially on [g, v] = gv−v ∈ V_{0}, so (g − 1)^{2}v = (g^{2} − 1)v = 0, and g^{2}v = v. Thus g^{2} = 1 since Ŝ acts faithfully on V. It follows that [Ŝ, Ŝ] is elementary abelian.

If Ĝ_{1} ≅ Ĝ_{2} are strictly 2-constrained, then by the argument used in the second paragraph in the proof of Theorem 4.5, (Ĝ_{1} > P̂ < Ĝ_{2}) must be one of the amalgams listed in Table 4.1. Hence P̂ ≅ D_{8} or D_{8} × C_{2} (the only cases in the table with G_{1} ≅ G_{2}), and P̂_{i} = O_{2}(Ĝ_{i}) ≅ C_{2}^{2} or C_{2}^{3}, respectively. Note that P̂_{12} = Z(P̂) in either case. Choose x_{1} ∈ P̂_{1}∖P̂_{12}. Then x_{1} is Ŝ-conjugate to some x_{2} ∈ P̂_{2}∖P̂_{12}, and x_{1}x_{2}^{−1} ∈ [Ŝ, Ŝ] has order 4. So this case is impossible.

Thus Ĝ_{1} ≅ Ĝ_{2} are not strictly 2-constrained. By Lemma 4.2(c), O_{p}(Ĝ_{i}) ≠ 1 for i = 1, 2. Thus [P̂_{i}, O_{p}(Ĝ_{i})] = 1 since both are normal in Ĝ_{i} (and have relatively prime order), so P̂_{12} ⊴ Ĝ_{i} for i = 1, 2. Hence P_{12} ⊴ G_{i}, and P_{12} = T by definition of T. In other words, P̂_{12} = 1, P̂_{i} ≅ C_{2}, and so P̂ ≅ C_{2}^{2}.

Let a_{i} ∈ P̂_{i} ≅ C_{2} be the generator, and let g ∈ O_{p}(Ĝ_{1}) be an element of order p. Thus Ĝ_{1} = ⟨a_{1}, a_{2}, g⟩ ≅ D_{4p}, ⟨g, a_{2}⟩ ≅ D_{2p}, and [g, a_{1}] = 1. Choose t ∈ Ŝ such that ta_{i}t^{−1} = a_{3−i}. By assumption, and since a_{1}a_{2} ∈ [Ŝ, Ŝ], [a_{1}a_{2}, [Ŝ, V]] = 0.

Set W = [g, V]; W ≠ 0 since Ĝ_{1} acts faithfully on V. We claim that

C_{W} (a_{2}) = [a_{2}, W] and [a_{1}, W] = 0 .  (1)

If w ∈ C_{W} (a_{2}), then a_{2}(g^{i}w) = g^{−i}a_{2}w = g^{−i}w for each i, and ∑_{i=0}^{p−1} g^{i}w = 0 since w ∈ W = [g, V]. Thus w = (1 + a_{2})(∑_{i=1}^{(p−1)/2} g^{i}w) ∈ [a_{2}, W], proving the first statement, and in fact proving that C_{W_{0}}(a_{2}) = [a_{2}, W_{0}] for each Ĝ_{1}-invariant subgroup W_{0} ≤ W.

So if we set W_{0} = [a_{1}, W], then C_{W_{0}}(a_{2}) = [a_{2}, W_{0}] = [a_{1}a_{2}, W_{0}] ≤ [a_{1}a_{2}, [Ŝ, V]], where the last group is zero by assumption. Hence W_{0} = 0, finishing the proof of (1).

Since W = [g, V] is a Ĝ_{1}-invariant direct summand of V, W ∩ [a_{1}, V] = [a_{1}, W] = 0 by (1). So if v ∈ W ∩ t(W), then a_{1}(v) = v since v ∈ W, v ∈ C_{t(W)}(a_{1}) = [a_{1}, t(W)] by (1), and so v ∈ W ∩ [a_{1}, V] = 0. Thus W ∩ t(W) = 0. So if we choose any w ∈ W such that a_{2}(w) ≠ w, then t(w) − w ∈ [Ŝ, V], and hence

a_{1}a_{2}(t(w) − w) = t(w) − w ⟹ w − a_{1}a_{2}(w) = t(w) − ta_{1}a_{2}(w) ∈ W ∩ t(W) = 0 .

Thus a_{1}(w) ≠ a_{1}a_{2}(w) = w, which contradicts (1). □

5. Some applications

We finish the paper with some applications of Theorems 4.5 and 4.6. Following the terminology of [G, § 16.7], we say that a 2-group S is wreathed if S ≅ C_{2^{m}} ≀ C_{2} for some m ≥ 2. These groups arise as Sylow 2-subgroups of GL_{2}(q) and (P)SL_{3}(q) when q ≡ 1 (mod 4).

Proposition 5.1. Fix a finite 2-group S containing a normal subgroup Δ ⊴ S which is dihedral or quaternion of order ≥ 8. Assume

(a) for some dihedral or quaternion subgroup Δ_{0} ≤ Δ of order 8, C_{S}(Δ_{0}) is abelian; and

(b) two of the three subgroups of index two in Δ are S-conjugate.

Then either S is dihedral, semidihedral, or wreathed, or there is no reduced fusion system over S.

Proof. We first fix some notation for elements and subgroups of S. Let A ⊴ Δ be the unique cyclic subgroup of index two in Δ which is normal in S (A is characteristic in Δ unless Δ ≅ Q_{8}). Set Z = Z(Δ), and let W ≤ A be the subgroup of order 4. Fix a generator a ∈ A, and fix b ∈ Δ_{0}∖A. Thus Δ = ⟨a, b⟩ and Δ_{0} = ⟨W, b⟩. Set A_{0} = ⟨a^{2}⟩, and set T = C_{S}(Δ_{0}). To summarize these definitions for later reference,

Z_{=Z(Δ)} ≤ A_{0}_{=⟨a^{2}⟩} < A_{=⟨a⟩} < Δ_{=⟨a,b⟩} ⊴ S , W_{≅C_{4}} ≤ A , Δ_{0} = ⟨W, b⟩, T = C_{S}(Δ_{0}).

By (a) and (b), we are in the situation of Lemma B.3. In particular,

[S:T Δ] = 2 and ∀ g ∈ S, gbg^{−1} = a^{j}b where \begin{cases}j is even & if g ∈ T Δ \\ j is odd & if g ∉ T Δ \end{cases} (1)

by Lemma B.3(b). Note also that

∀ g ∈ S, gag^{−1} = a^{i} where \begin{cases}i ∈ 4ℤ + 1 & if g ∈ C_{S}(W) \\ i ∈ 4ℤ − 1 & if g ∉ C_{S}(W) . \end{cases}  (2)

We claim that the following hold.

T A_{0} ⊴ S, S/T A_{0} ≅ D_{8}, and Z(S/T A_{0}) = T A/T A_{0} .  (3)
[S, S] is abelian, and [S, S] = ⟨A, [y, T]⟩ for each y ∈ S∖T Δ .  (4)
[S, S] = A ⟹ ∀ g ∈ S∖C_{S}(W), C_{T A}(g)·A = T A and |C_{T A}(g)| = |T |. (5)

Point (3) and the first statement in (4) are shown in Lemma B.3(c,d). The last statement in (4) holds since [S, S] ≥ A by Lemma B.3(a), S = ⟨Δ, T, y⟩, and hence [S, S] = ⟨[Δ, S], [y, T]⟩ = ⟨A, [y, T]⟩.

To see (5), fix g ∈ S∖C_{S}(W) and t ∈ T. By (2), gag^{−1} = a^{4k−1} for some k ∈ ℤ. Also, t^{−1}gt = a^{j}g for some j since [S, S] = A, and j is even (a^{j} ∈ A_{0}) since T A_{0} ⊴ S by (3). Choose i such that i(1 − 2k) ≡ (j/2) (mod |A|). Then a^{i}ga^{−i} = a^{j}g = t^{−1}gt, so ta^{i} ∈ C_{T A}(g). This proves the first statement in (5), and the second then follows since C_{T A}(g) ∩ A = C_{A}(g) = Z = T ∩ A.

Set S = S/T A_{0} (T A_{0} ⊴ S by (3)). We write g = gT A_{0} ∈ S for g ∈ S, and Q = QT A_{0}/T A_{0} for Q ≤ S: the images of g and Q in S.

Fix x ∈ S∖T Δ. Upon replacing x by bx if necessary, we can assume x ∈ C_{S}(W)∖T Δ. Also, xbx^{−1} = a^{j}b for some odd j by (1), and upon replacing x by an appropriate element of xA, we can assume xbx^{−1} = ab. By (2), xax^{−1} = a^{i} for some i ∈ 1 + 4ℤ. Then x^{2}bx^{−2} = a^{i+1}b where i + 1 ∈ 2 + 4ℤ; x^{2} ∈ T A since S/T A ≅ S/Z(S) ≅ C_{2}^{2} by (3); and hence x^{2} ∈ T A_{0}a. To summarize,

S = ⟨T A, b, x⟩, C_{S}(W) = ⟨T A, x⟩, xbx^{−1} = ab, x^{2} = a ∈ S.  (6)

Assume F is a reduced fusion system over S, and let E be the set of F-essential subgroups of S. Define

b_{0} = x, b_{1} = b, b_{2} = bx, H_{i} = ⟨T A, b_{i}⟩, E_{i} = {P ∈ E | P ≤ H_{i}}

for i = 0, 1, 2. Thus H_{0} = C_{S}(W) and H_{1} = T Δ. We will prove the following statements.

E = E_{0} ∪ E_{1} ∪ E_{2}, and P ∈ E_{i} ⟹ the normal closure of P is H_{i}.  (7)
E_{0} ≠ ∅ ⟹ S is wreathed.  (8)
E_{2} ≠ ∅ ⟹ [S, S] = A, ∃ b^{•} ∈ T Ab_{2} s.t. ⟨A, b^{•}⟩ dihedral or quaternion. (9)
E_{0} = ∅ ⟹ E_{1} ≠ ∅, E_{2} ≠ ∅, and S is dihedral or semidihedral.  (10)

The proposition then follows from (8) and (10).

Points (7), (8), and (9) will be shown in Steps 1 and 2: essential subgroups of index two in S will be handled in Step 1, and the others in Step 2. It is in Step 2 that Theorem 4.6 plays a crucial role. Point (10) will be shown in Step 3.

Step 1: Fix P ∈ E such that [S:P] = 2. Thus P ≥ [S, S] ≥ A.

We consider seven cases. In the first four, we show P cannot be F-essential using Lemma 1.4 applied with Θ one of the following characteristic subgroups Θ_{i} ≤ P:

Θ_{1} = Z(P) , Θ_{2} = ⟨g ∈ P | [P:C_{P} (g)] ≤ 2⟩ , or Θ_{3} = ⟨β(A) | β ∈ Aut(P)⟩ .

By Lemma 1.4, P ∉ E if for some i = 1, 2, 3 and some g ∈ S∖P, [g, P] ≤ Θ_{i}Fr(P) and [g, Θ_{i}] ≤ Fr(P).

Since P ≥ A, either P ≱ T, or P is one of the H_{i}. In the first two cases below, P ≱ T, and we set T_{0} = P ∩ T, choose t^{•} ∈ T ∖T_{0} ⊆ S∖P, and let t_{1}, t_{2} ∈ {1, t^{•}} be such that P = ⟨T_{0}A, bt_{1}, xt_{2}⟩. Since [xt_{2}, bt_{1}] = [x, b] = a, [xt_{2}, bt_{1}] ∈ [S, S] ∩ T A_{0}a.

• P ≱ T, [t^{•}, P] ≤ AZ(S), and Θ = Θ_{1} = Z(P). Then P = S, so Z(P) ≤ Z(S) = T A by (3), and Z(P) ≤ T A. Thus [t^{•}, Z(P)] ≤ [t^{•}, T A] ≤ ⟨a^{4}⟩ ≤ Fr(P). Also, [t^{•}, P] ≤ [t^{•}, S] ≤ T A_{0} since t^{•} ∈ T A_{0} ⊴ S by (3). So by assumption (and since Z(S) ≤ C_{S}(Δ_{0}) = T), [t^{•}, P] ≤ T A_{0} ∩ AZ(S) = Z(S)A_{0} ≤ Θ_{1}Fr(P). Thus P ∉ E.

• P ≱ T, [t^{•}, P] ≰ AZ(S), and Θ = Θ_{2}. Then [t^{•}, P] ≤ T A_{0} ∩ [S, S] since t^{•} ∈ T A_{0} ⊴ S. Since [t^{•}, T Δ] = [t^{•}, A] ≤ A_{0} (recall ⟨T, b⟩ is abelian) and [t^{•}, P] ≰ AZ(S), [t^{•}, x] = [t^{•}, xt_{2}] = sa^{2i} for some s ∈ T_{0}∖AZ(S). Then [s, A] = 1 since sa^{2i} ∈ [S, S], A ≤ [S, S], and [S, S] is abelian by (4). Hence [s, T Δ] = 1, and [s, g] ≠ 1 for g ∈ S∖T Δ since s ∉ Z(S).

Now, s ∈ Θ since C_{P} (s) = T Δ ∩ P. Also, [t^{•}, T Δ] ≤ A_{0} ⊴ S, so [t^{•}, P] ≤ ⟨A_{0}, [t^{•}, x]⟩ = ⟨A_{0}, s⟩ ≤ ΘFr(P). For all g ∈ P ∖T Δ, C_{P} (g) ≤ ⟨T_{0}A, g⟩ since C_{S}(g) = ⟨a, g⟩, and this inclusion is strict since [s, g] ≠ 1. Thus g ∉ Θ. So Θ ≤ T Δ, and hence [t^{•}, Θ] ≤ [T, T Δ] ≤ ⟨a^{4}⟩ ≤ Fr(P). Thus P ∉ E.

• P ∈ {H_{0}, H_{2}} and Θ = Θ_{3} ≤ T A. Then b ∈ S∖P, [b, P] ≤ A ≤ Θ since bA ∈ Z(S/A), and [b, Θ] ≤ [b, T A] ≤ A_{0} ≤ Fr(P). Thus P ∉ E.

• P = H_{1}, [S, S] = A, and Θ = Θ_{3} ≤ T A. Then x ∈ S∖P and [x, P] ≤ [S, S] = A ≤ Θ. Also, since T A = Z(S) and hence [S, T A] ≤ T A_{0}, [x, Θ] ≤ [S, T A] ≤ A ∩ T A_{0} = A_{0} ≤ Fr(P). Thus P ∉ E.

• P = H_{1}, [S, S] > A, and Θ_{3} ≤ T A. We will see later that we don’t need to consider this case separately.

• P = H_{0}, and Θ_{3} ≰ T A. Since Θ_{3} ≰ T A, there is β ∈ Aut(P) such that β(A) ≰ T A. Set u = β(a) ∈ P ∖T A = T Ax and U = ⟨u⟩ = β(A) ⊴ P. Then u ∈ {x, ax}, so u^{2} = a by (6), and u^{2} = ta^{i} where t ∈ T and i is odd. Set C = A ∩ U and 2^{m} = [A:C] = [U:C], so [A, U] ≤ C, C = ⟨a^{2^{m}}⟩ = ⟨u^{2^{m}}⟩, and AU/C ≅ (C_{2^{m}})^{2}. Then t^{2^{m−1}} = (u^{2}a^{−i})^{2^{m−1}} ≡ u^{2^{m}}a^{−i·2^{m−1}} ≡ a^{2^{m−1}} (mod C) since i is odd, so t^{2^{m−1}} = a^{k·2^{m−1}} for some odd k, a^{k·2^{m−1}} ∈ A ∩ T = Z has order ≤ 2, and hence |a| = 2^{m}. Thus C = 1, and AU ≅ A × A ≅ (C_{2^{m}})^{2}.

Now, [T, u^{2}] ≤ [T, T A] ≤ A, and [T, u^{2}] ≤ U since U ⊴ P. Hence [T, u^{2}] = 1, and [T, A] = [T, a^{i}] = 1 since u^{2} = ta^{i} for i odd. Thus [A, P] = [A, T AU] = 1, [U, P] = β([A, P]) = 1, and so P = T AU is abelian. Also, T ≤ Z(S) since S = ⟨P, b⟩ and [T, b] = 1. Hence [S, S] = A by (4).

Thus [S, S] = A is cyclic and H_{0} = P is abelian of index two in S, so we are in the situation of Proposition 2.5. Since S ≥ AU ≅ A×A, S is not dihedral or semidihedral, and hence is wreathed.

• P ∈ {H_{1}, H_{2}} and Θ_{3} ≰ T A. Let i = 1, 2 be such that P = H_{i} = ⟨T A, b_{i}⟩. Since Θ_{3} ≰ T A, there is β ∈ Aut(P) such that β(a) ∉ T A. Set b^{•} = β(a); thus b^{•} ∈ T Ab_{i} ⊆ S∖C_{S}(W), so b^{•}ab^{•}^{−1} = a^{j} for some j ∈ −1 + 4ℤ by (2). Then

β(A) ≥ β([a, P]) = [b^{•}, P] ≥ [b^{•}, A] = ⟨a^{2}⟩ = A_{0}

since A ≥ [A, P], [β(A):A_{0}] = 2 since |β(A)| = |A|, and so b^{•}^{2} = a^{2ℓ} for some odd ℓ. Thus a^{2} = b^{•}a^{2}b^{•}^{−1} = a^{2j}, so j ≡ 1 (mod \frac{1}{2}|a|). Since j ≡ 3 (mod 4), this proves that |a| = |b^{•}| = 4, [b^{•}, a] = a^{2}, and thus ⟨A, b^{•}⟩ = ⟨a, b^{•}⟩ ≅ Q_{8}. Also, A = W, so [A, T] = 1, and T A is abelian.

If P = H_{2} = ⟨T A, bx⟩, then [b^{•}, T] ≤ (⟨b^{•}⟩ ∩ T A) = Z since ⟨b^{•}⟩ = β(A) ⊴ P. Hence [S, S] = ⟨A, [b^{•}, T]⟩ = A by (4).

To summarize, if P ∈ E has index 2 in S, then either P = H_{0} and S is wreathed, or P = H_{2} and [S, S] = A, or P = H_{1}. Also, if P ∈ {H_{1}, H_{2}} and [S, S] = A, then P = T Δ^{•} for some Q_{8} ≅ Δ^{•} ⊴ P, and P = C_{T A}(Δ^{•})Δ^{•} by (5). So this proves (7), (8), and (9) for essential subgroups of index two in S.

Step 2: Now assume P ∈ E where [S:P] > 2. If |N_{S}(P)/P | > 2, then by Lemma 1.5, rk([g, P/Fr(P)]) ≥ 2 for all g ∈ N_{S}(P)∖P. For each i ∈ ℤ, [a^{i}, S] = ⟨a^{2i}⟩, so a^{i} ∈ N_{S}(P) if a^{2i} ∈ P, which implies a^{i} ∈ P since [a^{i}, S] is cyclic. Thus A ≤ P by induction on |A ∩ P |, so a^{2} ∈ Fr(P). For t ∈ T, [t, S] ≤ ⟨a^{4}, [t, x]⟩, and thus P ≥ T A. Hence P = T A since [S:P] ≥ 4, so [b, P] = ⟨a^{2}⟩ ≤ Fr(P) (and b ∈ N_{S}(P)), which contradicts Lemma 1.5. We conclude that |N_{S}(P)/P | = 2.

In particular, P is not normal in S. By Theorem 4.6(b,c), and since [S, S] is abelian by (4), P is of the type described in Theorem 4.6(a) with S_{∗} = S. Thus P = C_{S}(U^{•})U^{•}, where U^{•} ≅ C_{2}^{2} or Q_{8}, and the normal closure Δ^{•} of U^{•} in S is dihedral of order ≥ 8 or quaternion of order ≥ 16. Let A^{•} ⊴ Δ^{•} be the unique cyclic subgroup of index two, and fix b^{•} ∈ U^{•}∖A^{•}. Thus Δ^{•} = ⟨A^{•}, b^{•}⟩.

Since A^{•} ≤ Δ^{•} are both normal in S, the coset A^{•}b^{•} = Δ^{•}∖A^{•} is a union of Sconjugacy classes, and its elements are all conjugate since otherwise [b^{•}, S] = [b^{•}, A^{•}] has index two in A^{•} and the normal closure of U^{•} is strictly contained in Δ^{•}. Hence

A^{•} = [b^{•}, S] ≤ [S, S] ≤ T A

(recall that S/T A ≅ C_{2}^{2} is abelian by (3)). We claim that Δ^{•} ≰ T A, and hence b^{•} ∉ T A. Assume otherwise: then Z(Δ^{•}) ≤ [Δ^{•}, Δ^{•}] ≤ [T A, T A] ≤ A, so Z(Δ^{•}) = Z, and the dihedral group Δ^{•}/Z(Δ^{•}) is generated by elements of order two in T A/Z. Since each element of order two in T A/Z ≅ (A/Z) ⋊ (T/Z) lies in T W/Z, Δ^{•} is contained in the abelian group T W, which is impossible.

Thus b^{•} ∈ H_{i}∖T A = T Ab_{i} for some unique i = 0, 1, 2. Since b^{•} ∉ T A, b^{•} is not central in S ≅ D_{8} (see (3)), so C_{S}(b^{•}) = ⟨b^{•}, a⟩ = H_{i}. Hence C_{S}(b^{•}) ≤ H_{i} = ⟨T A, b^{•}⟩, so P = C_{S}(U^{•})·U^{•} ≤ C_{S}(b^{•})A^{•} ≤ H_{i}, and P ∈ E_{i}. Since the normal closure of P has index two in S = S_{∗} by Theorem 4.6(a) again, it must be equal to H_{i}. This proves (7).

Assume i = 0 (so b^{•} ∈ T Ax), or i = 2 (so b^{•} ∈ T Abx). Since [b, T A] = A_{0} = ⟨a^{2}⟩ and [x, b] = a, we get ⟨[b^{•}, b]⟩ = A in all cases. Thus A^{•} = [b^{•}, S] ≥ A, A^{•} = [S, S] = A, and hence A^{•} = A. (If A^{•} > A, then since both are cyclic, they could not have the same nontrivial image in any quotient group.) So Δ^{•} = ⟨A, b^{•}⟩. Also, since S = ⟨T Δ, b^{•}⟩ in these cases, (4) implies that

[S, S] = ⟨A, [b^{•}, T]⟩ = A .

Together with Step 1, this proves (9).

The subgroup W ≤ A^{•} = A of order 4 is central in H_{0} and not central in Δ^{•}. Thus P ∉ E_{0}. Together with Step 1, this finishes the proof of (8): E_{0} = ∅ or E_{0} = {H_{0}}, and in the latter case, S is wreathed.

Step 3: We have now proven (7), (8), and (9), and it remains to prove (10). So assume E_{0} = ∅. Then E_{1} ≠ ∅ and E_{2} ≠ ∅ by (7) and Lemma 2.2, and [S, S] = A by (9). It remains to prove that S is dihedral or semidihedral.

Fix P_{2} = C_{S}(U^{•})U^{•} ∈ E_{2} with normal closure H_{2}, as described in Step 2. Choose b^{•} ∈ U^{•}∖A, so that ⟨A, b^{•}⟩ is the normal closure of U^{•} and is dihedral or quaternion. Thus b^{•} ∈ T Abx, and Δ̂ =^{def} ⟨A, b, b^{•}⟩ ⊴ S is dihedral, semidihedral, or quaternion by Lemma B.3(e). Also, T Δ̂ = ⟨T A, b, b^{•}⟩ = S. Let Â < Δ̂ be the cyclic subgroup of index two (so Δ̂ = ⟨Â, b⟩, [Â:A] = 2, and Â ⊴ S).

If S contains an abelian subgroup of index two, then since [S, S] = A is cyclic, S is dihedral, semidihedral, or wreathed by Proposition 2.5. Since [S, S] = A is strictly contained in the larger cyclic subgroup Â, S cannot be wreathed.

Assume S does not contain an abelian subgroup of index two. Thus T Â is nonabelian, so [T, Â] ≠ 1 (T and Â are both abelian), and the homomorphism c: T ⟶ Aut(Â) induced by conjugation is nontrivial. Also, Im(c) is cyclic (each t ∈ T acts via (a ↦ a^{i}) for i ∈ 1 + 4ℤ by (2)), and C_{T} (Â) = Ker(c) = Z(S) since S = ⟨T Â, b⟩, [T, T] = [T, b] = 1, and T =^{def} C_{S}(Δ_{0}) ≥ Z(S). Also, C_{T} (A) = {t ∈ T | |c(t)| ≤ 2}, since for i ∈ 4ℤ + 1, i ≡ 1 (mod |A|) if and only if i^{2} ≡ 1 (mod |Â|). So [C_{T} (A):C_{T} (Â)] = 2. Thus

C_{S}(A)/Z(S) = C_{T} (A)Â/C_{T} (Â) ≅ (C_{T} (A)/C_{T} (Â)) × (Â/Z) ≅ C_{2} × (Â/Z)

since [C_{T} (A), Â] = Z, so Aut(C_{S}(A)/Z(S)) is a 2-group by Corollary A.3(a). Hence each element of odd order in Aut_{F}(S) induces the identity on C_{S}(A)/Z(S), and so the action of the odd order group Out_{F}(S) on S^{ab} = S/A induces the identity on C_{S}(A)/Z(S)A.

We want to apply Proposition 2.3(b), with U = Z = [S, S] ∩ Z(S). For each P < S of index two, 1 ≠ [P, P] ≤ [S, S] = A, so [P, P] ≥ Z since A is cyclic. Choose g_{0} ∈ C_{T} (A)∖C_{T} (Â), and let g_{1} be such that g_{1}A = ∏_{α∈Out_{F} (S)} α(g_{0}A). Since Out_{F}(S) acts trivially on C_{S}(A)/Z(S)A, g_{1} ≡ g_{0}^{k} (mod Z(S)A), where k = |Out_{F}(S)| is odd, and we can assume g_{1} ∈ T and g_{1} ≡ g_{0}^{k} (mod Z(S)). Thus |c(g_{1})| = |c(g_{0}^{k})| = 2 since k is odd and [g_{0}, Â] = Z, so g_{1} ∉ Z. Fix g ∈ ⟨g_{1}⟩ such that g ∉ Z but g^{2} ∈ Z. Then |c(g)| ≤ 2, so [g, S] = [g, Â] ≤ Z. By construction, every element of Aut_{F}(S) sends gA to itself.

Thus F is not reduced by Proposition 2.3, applied with U = Z and g as above. □

We now prove some other versions of Proposition 5.1, by listing different hypotheses which give the same conclusion.

Proposition 5.2. Let S be a finite nonabelian 2-group which satisfies at least one of the following conditions.

(a) There is an abelian subgroup A ⊴ S of index two.

(b) [S, S] is cyclic.

(c) There is an abelian subgroup Q < S such that |N_{S}(Q)/Q| = 2 and Out_{S}(Q) ≰ O_{2}(Out(Q)).

(d) There is a subgroup Q = Z(Q)U in S, where U ≅ Q_{8}, |N_{S}(Q)/Q| = 2, and Out_{S}(Q) exchanges two of the three abelian subgroups of index two in Q.

(e) There is a subgroup Q < S such that |Q| ≤ 16, |N_{S}(Q)/Q| = 2, and Out_{S}(Q) ≰ O_{2}(Out(Q)).

Then either S is dihedral, semidihedral, or wreathed, or there is no reduced fusion system over S.

Proof. Assume there is a reduced fusion system F over S, and let E be the set of all Fessential subgroups. Let DSW be the class of all dihedral, semidihedral, and wreathed 2-groups; we must show S ∈ DSW.

(a) Let A ≤ S be an abelian subgroup of index two. If R ∈ E and R ≠ A, then R ≰ A since R is centric in S, and so AR = S. So there is g ∈ A ∩ N_{S}(R)∖R, and [g, R] is cyclic, generated by [g, h] for any h ∈ R∖A. Hence by Lemma 1.5, |N_{S}(R)/R| = 2.

For each x ∈ S∖A, C_{S}(x) = ⟨C_{A}(x), x⟩ is abelian. Hence

R ≤ S, R ≰ A ⟹ C_{S}(R) is abelian.  (1)

By Theorem 4.6(b,c), and since [S, S] ≤ A is abelian, each R ∈ E either has index two in S, or has the form described in Theorem 4.6(a) with S_{∗} = S. In the latter case, there is Δ ⊴ S which is dihedral of order ≥ 8 or quaternion of order ≥ 16, and the noncyclic subgroups of index two in Δ are S-conjugate since Δ is the normal closure of a subgroup U_{1} ≅ C_{2}^{2} or Q_{8}. Also, for Δ_{0} ≤ Δ dihedral or quaternion of order 8, C_{S}(Δ_{0}) is abelian by (1). The hypotheses of Proposition 5.1 thus hold, and so S ∈ DSW.

We are left with the case where each R ∈ E has index two in S. If all F-essential subgroups are abelian, then S has at least two abelian subgroups of index two (Lemma 2.2), so |[S, S]| = 2 by Lemma A.6(c), and S ∈ DSW by Proposition 2.5.

Assume R ∈ E is nonabelian. If A ∩ R is the only abelian subgroup of index two in R, then it is characteristic in R. For g ∈ A∖R, [g, R] ≤ R ∩ A, [g, R ∩ A] = 1, and this contradicts Lemma 1.4 (applied with Θ = R ∩ A).

Thus by Lemma A.6(b,c), R has three abelian subgroups of index two, [R:Z(R)] = 4, and |[R, R]| = 2. Set Z = [R, R] ≤ Z(R), and fix a generator z ∈ Z. Fix g ∈ A∖R and h ∈ R∖A, and set y = [h, g]. Thus [g, R] = ⟨y⟩, and

hgh^{−1} = gy ⟹ g = h^{2}gh^{−2} = gy(hyh^{−1}) ⟹ [h, y] = hyh^{−1}y^{−1} = y^{−2}.

If y^{2} = 1, then [h, y] = 1, so [g, R] = ⟨y⟩ ≤ Z(R), [g, Z(R)] = 1 since Z(R) ≤ A, and this again contradicts Lemma 1.4 (applied with Θ = Z(R)). Thus y^{2} = [h, y]^{−1} ≠ 1, so y^{2} = z ∈ [R, R], and [S, R] = ⟨y, [R, R]⟩ = ⟨y⟩ is cyclic.

By Lemma 2.4(a.ii) (and since R is nonabelian), there are R_{1} ⊴ R and Out_{S}(R) ≤ Γ_{1} ⊴ Out_{F}(R) such that R_{1} ≅ Q_{8}, Aut_{F}(R) sends R_{1} to itself, and Γ_{1} ≅ Σ_{3} acts faithfully on R_{1}/Fr(R_{1}). Thus R_{1} ⊴ S, Aut_{S}(R) exchanges two of the three subgroups of index two in R_{1}, and C_{S}(R_{1}) is abelian by (1). Hence S ∈ DSW by Proposition 5.1.

(b) Assume [S, S] is cyclic, and let Z ≤ [S, S] be the subgroup of order 2. If some P ∈ E has index ≥ 4 in its normalizer, then rk([s, P/Fr(P)]) ≥ 2 for each s ∈ N_{S}(P)∖P by Lemma 1.5, which is impossible since [S, S] is cyclic. If P ∈ E is nonabelian, then 1 ≠ [P, P] ≤ [S, S], and Z = Ω_{1}([P, P]) is characteristic in P. So if each P ∈ E is nonabelian, then Z is characteristic in S and in each P ∈ E, hence Z ⊴ F by Proposition 1.3, and F is not reduced.

Thus some R ∈ E is abelian and has index 2 in its normalizer. If [S:R] = 2, then the result follows from (a). Otherwise, by Theorem 4.6, there is Δ ⊴ S which is dihedral of order ≥ 8 (R ≱ Q_{8} since it is abelian), and the noncyclic subgroups of index two in Δ are S-conjugate since Δ is the normal closure of a subgroup U_{1} ≅ C_{2}^{2} in R. Also, for Δ_{0} ≤ Δ dihedral of order 8 containing U_{1}, C_{S}(Δ_{0}) ≤ C_{S}(U_{1}) ≤ R is abelian. The hypotheses of Proposition 5.1 thus hold, and so S ∈ DSW.

(c) If S contains an abelian subgroup of index two, then S ∈ DSW by (a). If there is an abelian subgroup Q < S which satisfies the assumptions in (c) and is not normal, then by Lemma B.5(a), there are subgroups Δ_{0} ≤ Δ ⊴ S such that Δ is dihedral, Δ_{0} ≅ D_{8}, C_{S}(Δ_{0}) ≤ Q is abelian, and the noncentral involutions in Δ are all S-conjugate. Then S ∈ DSW by Proposition 5.1.

(d) Fix Q = Z(Q)U ≤ S, where U ≅ Q_{8}, |N_{S}(Q)/Q| = 2, and Out_{S}(Q) exchanges two abelian subgroups A_{1} and A_{2} of index two in Q. Set Z = Z(U) = [Q, Q] ≅ C_{2}, let z ∈ Z be the generator, choose u_{1} ∈ A_{1} ∩ U of order 4, and set u_{2} = xu_{1}x^{−1} ∈ A_{2} for some x ∈ N_{S}(Q)∖Q. Then u_{2}^{2} = u_{1}^{2} = z since [x, z] = 1, u_{2} ∉ Z(Q), and so [u_{1}, u_{2}] = z and ⟨u_{1}, u_{2}⟩ ≅ Q_{8}. So upon replacing U by ⟨u_{1}, u_{2}⟩, we can assume U ⊴ N_{S}(Q).

Set S_{∗} = C_{S}(Z), and let Δ be the normal closure of U in S_{∗}. Then Q/Z is abelian, and [N_{S_{∗}/Z}(Q/Z):Q/Z] = 2 since [N_{S}(Q):Q] = 2 and N_{S_{∗}}(Q) = N_{S}(Q). Also, U/Z ≅ C_{2}^{2} is a direct factor in Q/Z, and [N_{S}(Q)/Z, U/Z] ≠ 1 since Out_{S}(Q) exchanges two of the abelian subgroups of U.

If Δ > U, then Δ/Z is dihedral by Lemma B.5(b), and all involutions in Δ/Z are S_{∗}-conjugate to elements of U/Z. Hence there are no involutions in Δ∖Z, and Δ is quaternion.

Set R = N_{C_{S}(U)}(Z(Q)). Since conjugation by each y ∈ N_{S}(Q)∖Q exchanges A_{1} and A_{2}, C_{S}(U)∩N_{S}(Q) ≤ Q. Also, R normalizes Z(Q)U = Q, so R ≤ C_{S}(U)∩N_{S}(Q) ≤ Q. Hence R ≤ C_{Q}(U) = Z(Q), so C_{S}(U) = Z(Q) by Lemma A.1.

If S_{∗} < S, then choose y ∈ N_{S}(S_{∗})∖S_{∗}, and set β = c_{y} ∈ Aut(S_{∗}). Thus β(Z) ≠ Z, and Δ ∩ β(Δ) = 1 since Z and β(Z) contain the only elements of order two in Δ and β(Δ), respectively. So [Δ, β(Δ)] ≤ Δ ∩ β(Δ) = 1 since both subgroups are normal, which is impossible since C_{S}(Δ) ≤ C_{S}(U) = Z(Q) is abelian. Thus S_{∗} = S.

If Δ = U, then by assumption, Aut_{S}(Q) exchanges two of the abelian subgroups of index two in Q = Z(Q)U, and hence exchanges two of the index two subgroups of U = Δ. If Δ > U, the two noncyclic subgroups of index two in Δ are S-conjugate since Δ is generated by the S-conjugates of U < Δ. Since C_{S}(U) = Z(Q) is abelian, S ∈ DSW by Proposition 5.1, applied with Δ_{0} = U.

(e) Assume Q < S is such that |Q| ≤ 16, |N_{S}(Q)/Q| = 2, and Out_{S}(Q) ≰ O_{2}(Out(Q)). In particular, Out(Q) is not a 2-group. By Corollary A.3(a,c), Q^{ab} ≅ C̸_{4} ×C_{2}, so either Q is abelian or Q^{ab} is elementary abelian. If Q is abelian, then S ∈ DSW by (c). If Q is nonabelian, then by Corollary A.3(b) and the list of groups of order 16 (cf. [Bu, § 74]), Q is isomorphic to Q_{8}, C_{2} × Q_{8}, or C_{4} ×_{C_{2}} Q_{8}. (Note that Aut(C_{2} × D_{8}) is a 2-group by Lemma A.2, since it contains a unique subgroup isomorphic to C_{2} × C_{4}.) In each of these cases, by Lemma A.2 applied to the chain 1 < Fr(Q) ≤ Z(Q) < Q, O_{2}(Aut(Q)) contains all α ∈ Aut(Q) which act trivially on Q/Z(Q). Since Aut_{S}(Q) ≰ O_{2}(Aut(Q)), Aut_{S}(Q) acts nontrivially on Q/Z(Q) and hence contains elements which exchange two of the three abelian subgroups of index two in Q. So S ∈ DSW by (d). □

Each reduced fusion system over a dihedral, semidihedral, or wreathed 2-group is isomorphic to the fusion system of PSL_{2}(q) for q ≡ ±1 (mod 8), or of PSU_{3}(q) or PSL_{3}(q) for q ≡ 1 (mod 4). Fusion systems over dihedral and semidihedral 2-groups have been listed by several people; cf. [AOV, § 4.1] for the reduced case. For wreathed 2-groups, this was shown in Proposition 3.1. The fusion systems (at the prime 2) of PSU_{3}(q) and PSL_{3}(q) for q ≡ 3 (mod 4) also have this form (see, e.g., [BMO, Theorem A(d)]).

Proposition 5.2(e) has as an easy consequence:

Theorem 5.3. Let F be a reduced fusion system over a nontrivial finite 2-group S of order at most 32. Then S is dihedral, semidihedral, or wreathed, and F is isomorphic to the fusion system of PSL_{2}(q) for q ≡ ±1 (mod 8), or that of PSL_{3}(q) for q odd.

Proof. By Lemma 2.2, there exists an F-essential subgroup Q < S. Then |Q| ≤ 16, and |N_{S}(Q)/Q| = 2, since otherwise |S| ≥ |N_{S}(Q)| ≥ 64 by Lemma 1.5. Since Out_{F}(Q) contains a strongly 2-embedded subgroup, O_{2}(Out_{F}(Q)) = 1 (cf. [AKO, Proposition A.7(c)]). Hence Out_{S}(Q) ≰ O_{2}(Out(Q)), so S is dihedral, semidihedral or wreathed by Theorem 5.2(e). The description of F follows from the above remarks. □

We finish with a slightly less easy consequence of the results in this section and in Section 4: a list of those groups of order 64 which support reduced fusion systems. For all n and q, UT_{n}(q) denotes the group of strictly upper triangular matrices over 𝔽_{q} (i.e., those with 1’s on the diagonal). A fusion system over a p-group S is indecomposable if it is not isomorphic to a product of fusion systems over nontrivial subgroups of S.

Theorem 5.4. Let F be a reduced, indecomposable fusion system over S, where |S| = 64. Then S is isomorphic to one of the groups D_{64}, SD_{64}, UT_{4}(2), UT_{3}(4), or to a Sylow 2-subgroup of M_{12}.

Proof. Fix F and S, and let E be the set of F-essential subgroups of S. Assume S is neither dihedral nor semidihedral; it cannot be wreathed since |S| is a power of 4.

Case 1: Assume there is P ∈ E such that P ⋬ S and |N_{S}(P)/P | = 2. Then |P | ≤ 16, Out_{S}(P) ≰ O_{2}(Out(P)) since P is essential, and this is impossible by Proposition 5.2(e).

Case 2: Assume that we are not in the situation of Case 1, and that there is P ∈ E such that |N_{S}(P)/P | ≥ 4. By Lemma 1.5, rk(P/Fr(P)) ≥ 4. Thus P ≅ C_{2}^{4} and P ⊴ S. Using Bender’s classification of groups with strongly 2-embedded subgroups [Bd, Satz 1], we see that either Aut_{F}(P) ≥ A_{5}, or Aut_{F}(P) ≅ C_{5}⋊C_{4}, C_{15}⋊C_{4}, or C_{3}^{2}⋊C_{4}. Also, there are exactly two conjugacy classes of subgroups isomorphic to A_{5} in GL_{4}(2) ≅ A_{8} (cf. [Ta, Corollary 6.7]), corresponding to the A_{5}-orbits A_{5}/A_{4} and A_{5}/D_{10}. Case 2A: Assume A_{5} ≤ Aut_{F}(P), acting via the reduced permutation action (i.e., the permutation action on 𝔽_{2}^{5} modulo its fixed subspace). Then Aut_{S}(P) ≅ C_{2}^{2} permutes freely a basis for P ≅ C_{2}^{4}, so the extension of P by S/P splits, and S ≅ UT_{4}(2). Case 2B: Assume A_{5} ≤ Aut_{F}(P), acting via the canonical action of A_{5} ≅ SL_{2}(4) on 𝔽_{4}^{2}. With respect to a suitable basis, each g ∈ S∖P acts via multiplication by a triangular matrix ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) for some x ∈ 𝔽_{4}∖\{0\}, so C_{P} (g) = [g, P] = Z(S) ≅ C_{2}^{2}. Thus g^{2} ∈ C_{P} (g) = Z(S) for each g ∈ S, so S/Z(S) has exponent two and hence is abelian. It follows that [S, S] = Z(S), and in particular, that each R ∈ E is normal in S.

Assume R ∈ E with [S:R] = 2. If P ≤ R, then [g, P] = [S, S] for any g ∈ R∖P. If P ≰ R, then S = P R, rk(P ∩ R) = 3 > rk(Z(S)), and for g ∈ (P ∩ R)∖Z(S), [R, R] ≥ [R, g] = [P R, g] = [Aut_{S}(P), g] = [S, S]. Thus [R, R] = [S, S] in both cases. Hence [S, R] = [R, R] ≤ Fr(R), which contradicts Lemma 1.4 (applied with Θ = 1).

By Lemma 2.2, there is R ∈ E with R ≠ P, and we just showed that [S:R] > 2. Hence |N_{S}(R)/R| ≥ 4, so R ≅ C_{2}^{4} by Lemma 1.5. Also, Z(RP) = Z(S) ≅ C_{2}^{2} since C_{P} (g) = Z(S) for each g ∈ S∖P, so |R ∩ P | ≤ |Z(RP)| = 4, and RP = S. In particular, the extension of P by S/P ≅ C_{2}^{2} is split, and so S ≅ UT_{3}(4). Case 2C: Assume Aut_{S}(P) ≅ S/P ≅ C_{4}, and thus Aut_{F}(P) ≅ C_{5} ⋊ C_{4}, C_{15} ⋊ C_{4}, or C_{3}^{2} ⋊ C_{4}. In either case, there is a basis of P ≅ C_{2}^{4} permuted freely by S/P ≅ C_{4}, and so C_{P} (S) ≤ [S, P] ≅ C_{2}^{3}. If x ∈ S is such that S = ⟨P, x⟩, then x^{4} ∈ C_{P} (S) ≤ [S, P], and so [S, S] = [S, P] and S/[S, S] ≅ C_{2} × C_{4}.

Let Q = ⟨P, x^{2}⟩ < S be the (unique) subgroup of index two which contains P. Thus Q/[S, S] ≅ C_{2}^{2}, and in particular, each elementary abelian subgroup of S is contained in Q.

For each R ∈ E, we have shown that either [S:R] = 2 or R ≅ C_{2}^{4}. If R ≅ C_{2}^{4}, then R ≤ Q, and RP = Q if R ≠ P. (In fact, there is no such subgroup.) If [S:R] = 2 and R ≠ Q, then R ∩ P ≅ C_{2}^{3}, so R ∩ P = [S, P] since it is normal in S, RP = S, and [R, R] = [S, R∩P] has index two in R∩P. So R/[R, R] ≅ C_{2} ×C_{4}, Aut(R) is a 2-group by Corollary A.3(a,c), and R ∉ E. Thus RP = Q for each R ∈ E∖{P }. So E = {P } or ⟨E⟩ = Q, either of which contradicts Lemma 2.2. Hence this case is impossible.

Case 3: We are left with the case where each F-essential subgroup has index two in S. Since |E| ≥ 2 by Lemma 2.2, we can choose distinct P_{1}, P_{2} ∈ E. By Theorem 4.5, we can choose G_{1} > S < G_{2} and T ≤ P_{12} = P_{1} ∩ P_{2} which satisfy conditions (∗) at the beginning of Section 4, with P = S and Aut_{G_{i}}(P_{i}) ≤ Aut_{F}(P_{i}). Thus Aut_{G_{i}}(T) ≤ Aut_{F}(T). Case 3A: Assume T is not centric in S. Let U and Z = T ∩ U be as in Theorem 4.5; [T, U] = 1 in all cases. There are six cases to consider, listed in Table 4.1.

(1) |T | = 8, and S ≅ T ×D_{8} or T ×_{C_{2}} Q_{16}. The first is impossible by [O1, Theorem B] (and since F is indecomposable), and the second by Proposition 5.2(b) ([S, S] = [U, U] is cyclic).

(2) |T | = 4, [S:UT] = 2, U ⊴ S, (U, Z) ≅ (D_{8}, 1) or (Q_{16}, C_{2}), and S/T ≅ C_{2} × D_{8}. Then S = ⟨UT, x⟩ for some x such that xT ∈ Z(S/T). If U ≅ D_{8}, then S = U × ⟨T, x⟩, which is impossible by [O1, Theorem B] again. Hence U ≅ Q_{16}, [x, U] ≤ T ∩ U = Z since [x, S] ≤ T and U ⊴ S, and [x, T] ≤ Z since [T:Z] = 2 and Z ⊴ S. Thus [S, S] = [U, U] is cyclic, and this is impossible by Proposition 5.2(b).

(3) U ≅ C_{4} ≀ C_{2}, Z = 1, and S = UT. Then S ≅ U × C_{2} has an abelian subgroup of index two, which is impossible by Proposition 5.2(a).

(4) U ≅ C_{4} ≀ C_{2}, Z = 1, and [S:UT] = 2. Thus T = 1, and S is of type Aut(U_{3}(3)). By the descriptions of this amalgam in [Gd2, Table 1 & (3.7)], S is isomorphic to a Sylow 2-subgroup of M_{12}.

(5) S = U is a Sylow subgroup of M_{12}.

(6) |U| ≥ 2^{7}, so this is impossible.

In the remaining cases, we assume T is centric in S. Set Ŝ = Out_{S}(T), Ĝ_{i} = Out_{G_{i}}(T), Ĝ = ⟨Ĝ_{1}, Ĝ_{2}⟩ ≤ Out_{F}(T), and P̂_{i} = Out_{P_{i}}(T). Thus Ŝ ∈ Syl_{2}(Ĝ) by the Sylow axiom since T ⊴ S (hence T is fully normalized). Let Γ be a model for T ⊴ N_{F}(T) (cf. [AKO, Theorem III.5.10(a)]), and let G ≤ Γ be such that T ≤ G and Out_{G}(T) = Ĝ. Thus G is a finite group such that

S ∈ Syl_{2}(G), T ⊴ G, C_{G}(T) ≤ T, and Ĝ = Out_{G}(T).

Case 3B: Assume [S:T] = 4. Thus Ŝ ≅ S/T ≅ C_{2}^{2} by (∗), T = P_{12}, P_{i}/T ≅ C_{2} is normal in Ĝ_{i} ≅ G_{i}/T with quotient group G_{i}/P_{i} ≅ D_{2p_{i}}, and hence Ĝ_{i} ≅ C_{2} × D_{2p_{i}} contains an element of order 2p_{i}. Also, Ĝ acts faithfully on T/Fr(T) by Lemma 4.2(e), GL_{4}(2) ≅ A_{8} contains no element of order 2p_{i} for primes p_{i} ≥ 5, and GL_{3}(2) contains no element of order 6. It follows that rk(T/Fr(T)) = 4, so T ≅ C_{2}^{4}, and p_{1} = p_{2} = 3.

Set Ĥ_{i} = O^{2}(Ĝ_{i}) ≅ C_{3}. It is not hard to see, by examining normalizers of subgroups of order three in GL_{4}(2) ≅ A_{8}, that there are exactly two conjugacy classes of subgroups C_{2} × Σ_{3} in GL_{4}(2). Hence for some decomposition T = V_{1} × V_{2}, where V_{i} ≅ C_{2}^{2}, the action of Ĝ_{1} ≅ C_{2} × Σ_{3} on T has the following form: either (i) Σ_{3} acts faithfully on each V_{i}, and C_{2} switches the two factors; or

(ii) Σ_{3} acts faithfully on V_{1} and trivially on V_{2}, and C_{2} acts faithfully on V_{2} and trivially on V_{1}.

In case (i), Ŝ acts by permuting freely a basis of T, and hence S ≅ UT_{4}(2) as seen earlier.

In case (ii), there is a basis of T permuted by Ŝ in two orbits of length 2. Hence the three involutions in Ŝ cannot be conjugate in Ĝ, so FŜ(Ĝ) is the fusion system of Ŝ ≅ C_{2}^{2}. Thus [Ĝ, Ĝ]∩Ŝ = 1 by the focal subgroup theorem (cf. [G, Theorem 7.3.4]), so there is Ĥ ⊴ Ĝ of odd order and index four. Also, P̂_{i} ⊴ Ĝ_{i} for i = 1, 2, where P̂_{i} ≅ C_{2} and Ĝ_{i} ≅ C_{2} × Σ_{3}, so Ĝ_{1} ≠ Ĝ_{2}. Thus Ĝ = ⟨Ĝ_{1}, Ĝ_{2}⟩ contains two distinct subgroups of order 3, they are contained in Ĥ, and hence |Ĥ| > 3. Also, Ĥ cannot contain a normal subgroup K̂ ⊴ Ĥ of order 7, since that would imply Ĝ ≤ N_{Aut(T)}(K̂) ≅ C_{7} ⋊ C_{3}. Since |GL_{4}(2)| = 2^{6}·3^{2}·5·7, Ĥ contains C_{3}^{2} or C_{5}, and in either case, C_{T} (Ĥ) = 1. Let H < G be such that H ≥ T and Aut_{H}(T) = Ĥ. By Lemma A.9, applied to the triple T ⊴ H ⊴ G, G (and hence S) splits as a semidirect product over T. Thus S ≅ D_{8}×D_{8}, which is impossible by [O1, Theorem B] again (F is indecomposable).

Case 3C: Now assume [S:T] ≥ 8. Since Ŝ acts faithfully on T/Fr(T), rk(T/Fr(T)) ≥ 3, so T ≅ C_{2}^{3}, |Ŝ| = |S/T | = 8, and Ŝ ≅ D_{8} since this is a Sylow 2-subgroup of GL_{3}(2). Neither P̂_{1} nor P̂_{2} can be cyclic of order four, since the normalizer in Aut(T) ≅ GL_{3}(2) of such a subgroup has order 8. Hence P̂_{i} ≅ C_{2}^{2} for i = 1, 2, Ĝ_{i} ≅ Σ_{4} (the normalizer in Aut(T) ≅ GL_{3}(2) of P̂_{i}), and Ĝ = ⟨Ĝ_{1}, Ĝ_{2}⟩ = Aut(T). Thus G is an extension of C_{2}^{3} by GL_{3}(2). By [GH, Lemma II.3.4], either the extension is split and S ≅ UT_{4}(2), or it is not split and S is a Sylow 2-subgroup of M_{12}. □

Appendix A. Background on groups

We list here some elementary results about finite groups which are needed throughout the paper, beginning with a well known property of p-groups.

Lemma A.1. If Q < P are finite p-groups for some prime p, then Q < N_{P} (Q).

Proof. See, e.g., [Sz1, Theorem 2.1.6]. □

We next look at automorphisms of finite p-groups.

Lemma A.2. Fix a prime p, a finite p-group P, a subgroup P_{0} ≤ Fr(P), and a sequence of subgroups

P_{0} ⊴ P_{1} ⊴ ⋯ ⊴ P_{k} = P.

Set

A = {α ∈ Aut(P) | x^{−1}α(x) ∈ P_{i−1}, all x ∈ P_{i}, all i = 1, …, k} ≤ Aut(P):

the group of automorphisms which leave each P_{i} invariant, and which induce the identity on each quotient group P_{i}/P_{i−1}. Then A is a p-group. If the P_{i} are all characteristic in P, then A ⊴ Aut(P), and hence A ≤ O_{p}(Aut(P)).

Proof. See, for example, [G, Theorems 5.1.4 & 5.3.2]. □

As an easy exercise, Lemma A.2 implies the following corollary, which contains a list of 2-groups whose automorphism groups are 2-groups. Recall that G^{ab} = G/[G, G] denotes the abelianization of a group G.

Corollary A.3. For a finite 2-group P, Aut(P) is a 2-group if at least one of the following hold:

(a) P ≅ C_{2^{k_{1}}} × C_{2^{k_{2}}} × ⋯ × C_{2^{kr}}, where k_{1}, …, k_{r} are pairwise distinct.

(b) P is dihedral of order ≥ 8, or semidihedral or quaternion of order ≥ 16.

(c) Aut(P^{ab}) is a 2-group.

The next two results involve the intersection of a Sylow subgroup with the commutator subgroup.

Lemma A.4. For any finite group G with S ∈ Syl_{p}(G), S ∩ O^{p}(G) ≤ S ∩ [G, G], with equality if G/O^{p}(G) is abelian.

Proof. Set G′ = [G, G]. Since G/G′ is abelian, its largest p-group quotient G/O^{p}(G)G′ is isomorphic to its Sylow p-subgroup SG′/G′ ≅ S/(S ∩ G′) (and the isomorphism is induced by the inclusion S ≤ G). Hence S ∩ G′ = S ∩ O^{p}(G)G′, and so S ∩ O^{p}(G) ≤ S ∩ G′ with equality if G′ ≤ O^{p}(G). □

The following proposition goes essentially back to Schur [Sch, IX–X].

Proposition A.5. Fix a finite group G with S ∈ Syl_{p}(G). Let Z ≤ Z(G) be a psubgroup. Then Z ∩ [G, G] = Z ∩ [S, S].

Proof. This follows as an application of the transfer in (co)homology. See, e.g., [Hu, Satz IV.2.2]. □

The next lemma describes nonabelian 2-groups with abelian subgroup of index two.

Lemma A.6. Let S be a finite nonabelian 2-group containing an abelian subgroup A ⊴ S of index two. Then the following hold.

(a) [S, S] ≅ A/Z(S), and all elements of (S/Z(S))∖(A/Z(S)) have order two.

(b) If |[S, S]| = 2, then S/Z(S) ≅ C_{2}^{2}, and S contains exactly three abelian subgroups of index two.

(c) If |[S, S]| ≥ 4, then |S/Z(S)| ≥ 8, and A is the unique abelian subgroup of index two in S.

(d) If S contains three abelian subgroups of index two which are permuted transitively by some automorphism of S, then either Z(S) is not a direct factor of A, or [S, S] ≤ Fr(Z(S)).

Proof. (a) For each x ∈ S∖A, x^{2} ∈ C_{A}(x) = Z(S), and thus xZ(S) has order two in S/Z(S). Also, [S, S] = [x, A] is the image of Id−c_{x} as a homomorphism from A to itself, and Z(S) is its kernel. Hence [S, S] ≅ A/Z(S).

(b) Assume |[S, S]| = 2. Then by (a), |S/Z(S)| = 4 and S/Z(S) ≅ C_{2}^{2}. Each abelian subgroup of index two contains Z(S), and each subgroup of index two which contains Z(S) is abelian. So there are exactly three abelian subgroups of index two in S.

(c) Now assume |[S, S]| ≥ 4, so |S/Z(S)| ≥ 8 by (a). If B ≠ A is another abelian subgroup of index two in S, then AB = S, so Z(S) ≥ A ∩ B, and |S/Z(S)| ≤ 4, a contradiction. Thus A is the only abelian subgroup of index two.

(d) Assume S contains three abelian subgroups A = A_{1}, A_{2}, A_{3} of index two which are permuted transitively by some automorphism of S. Thus |[S, S]| = 2 and S/Z(S) ≅ C_{2}^{2} by (b,c). Fix a generator z ∈ [S, S]. If Z(S) is a direct factor of A = A_{1}, then it is a direct factor of each A_{i}, and there are elements a_{i} ∈ A_{i}∖Z(S) of order 2. Then a_{1}a_{2}a_{3} ∈ Z(S) since the a_{i} represent the three nonidentity elements in S/Z(S) ≅ C_{2}^{2}, [a_{i}, a_{j}] = z for distinct i, j ∈ {1, 2, 3} since S = ⟨a_{i}, a_{j}, Z(S)⟩ is nonabelian, and so (a_{1}a_{2}a_{3})^{2} = z^{3} = z. Thus z ∈ Fr(Z(S)). □

Recall that a group G is metacyclic if it contains a normal cyclic subgroup H ⊴ G such that G/H is also cyclic.

Lemma A.7. Let S be a finite metacyclic 2-group with an automorphism γ ≠ Id_{S} of odd order. Then either S ≅ C_{2^{n}} × C_{2^{n}} for some n ≥ 1 and C_{S}(γ) = 1, or S ≅ Q_{8}.

Proof. If S is abelian, then rk(S) ≤ 2, and S ≅ C_{2^{n}} × C_{2^{n}} by Corollary A.3(a) (and since Aut(S) is not a 2-group). Also, C_{S}(γ) = 1 since γ acts nontrivially on Ω_{1}(S) ≅ C_{2}^{2} (see [G, Theorem 5.2.4]).

Now assume S is nonabelian. Fix A = ⟨a⟩ ⊴ S such that S/A is cyclic, and let b ∈ S be such that S = ⟨a, b⟩. Set S′ = [S, S] ≤ A. Then γ|_{S′} = Id since S′ ≤ A is cyclic. Let γ ∈ Aut(S/S′) be the automorphism induced by γ. Then γ ≠ Id_{S/S′} by Lemma A.2, so γ fixes only the identity element by the first paragraph, and C_{S}(γ) = S′. Also, Z(S) is abelian of rank ≤ 2 and γ|_{Z(S)} has nonidentity fixed points (since S′ ∩ Z(S) ≠ 1), so γ|_{Z(S)} = Id by the first paragraph, and Z(S) ≤ S′. Since γ does not induce the identity on S/Fr(S) ≅ C_{2}^{2} (and a ∉ Fr(S)), S = ⟨a, γ(a), Fr(S)⟩, so S = ⟨a, γ(a)⟩ since Fr(S) is the intersection of all maximal subgroups. Thus [S′, S] = 1, and S′ = Z(S). Also, [a^{i}, b^{j}] = [a, b]^{ij} for all i, j ≥ 1 since [a, b] ∈ Z(S).

Set S_{0} = Fr(S) = ⟨a^{2}, b^{2}⟩ ≥ S′ (since [a, b] ∈ ⟨a^{2}⟩). If S_{0} > S′, then γ|_{S_{0}} ≠ Id and C_{S_{0}}(γ) ≠ 1. Hence S_{0} is nonabelian, and we just showed this implies Z(S_{0}) = [S_{0}, S_{0}]. But [S_{0}, S_{0}] = ⟨[a^{2}, b^{2}]⟩ = ⟨[a, b]^{4}⟩ < S′ ≤ Z(S_{0}), so this is impossible. We conclude that S_{0} = S′, and hence is generated by [a, b]. Also, b^{2} ∈ A implies [a, b]^{2} = [a, b^{2}] = 1, so |S′| = 2, |S| = 8, and S ≅ Q_{8}. □

The following result about actions on abelian 2-groups is very useful in certain situations.

Lemma A.8. Fix a finite abelian 2-group A and a subgroup G ≤ Aut(A) with Sylow subgroup S ∈ Syl_{2}(G) of order two. Assume S ≰ Z(G), and [S, A] ≅ C_{2^{n}} for some n ≥ 1. Then there are unique decompositions A = A_{0} × A_{1} and G = G_{0} × G_{1}, such that the G-action on A splits as a product of G_{i}-actions on A_{i}; and such that |G_{0}| is odd, G_{1} ≅ Σ_{3}, and A_{1} ≅ C_{2^{n}} × C_{2^{n}}.

Proof. See, e.g., [O1, Proposition 2.3]. □

Lemma A.9. Fix a prime p, and a finite group G with subgroups A ⊴ B ⊴ G, both normal in G, such that A is an abelian p-group, B/A has order prime to p, and C_{A}(B) = 1. Then G splits as a semidirect product G = A ⋊ H, where H ≅ G/A.

Proof. By the spectral sequence for the extension 1 → B/A → G/A → G/B → 1, H^{i}(G/A; A) = 0 for each i ≥ 0 since H^{0}(B/A; A) = C_{A}(B) = 0 (and since (|B/A|, |A|) = 1). In particular, G ≅ A ⋊ (G/A).

Alternatively, since |B/A| is prime to |A|, by the Schur-Zassenhaus theorem [G, Theorem 6.2.1], there is K ≤ B such that KA = B and K ∩ A = 1, and each such subgroup is B-conjugate to K. Hence the B-conjugacy class of K is its G-conjugacy class, so [N_{G}(K):N_{B}(K)] = [G:B]. If a ∈ A ∩ N_{G}(K), then [a, K] ≤ A ∩ K = 1, so a = 1 since C_{A}(K) = C_{A}(B) = 1. Thus N_{G}(K) ∩ A = 1, so N_{B}(K) = K, and G = AN_{G}(K). □

Appendix B. Finite 2-groups with normal dihedral or quaternion subgroups

We prove here some elementary results about certain finite 2-groups which have normal dihedral or quaternion subgroups, and their automorphisms. We begin by stating the following general proposition about automorphisms of products of p-groups.

Proposition B.1 ([O1, Proposition 3.2(a)]). Fix a pair of finite p-groups S_{1} and S_{2}, set S = S_{1} × S_{2}, and let pr_{i} ∈ Hom(S, S_{i}) be the projection. Let α ∈ Aut(S) be such that α(Ω_{1}(Z(S_{1}))) = Ω_{1}(Z(S_{1})). Then for i = 1, 2, pr_{i}(α(S_{i})) = S_{i} and α(S_{i}Z(S_{3−i})) = S_{i}Z(S_{3−i}).

We next look at semidirect products with normal dihedral subgroup.

Lemma B.2. Fix a finite 2-group S, and subgroups Δ ⊴ S and T ≤ S. Assume that S = T Δ, T ∩ Δ = 1, Δ is dihedral of order ≥ 8, and [T, Δ_{0}] = 1 for some dihedral subgroup Δ_{0} ≤ Δ of order 8. Let A, Δ_{1}, Δ_{2} ≤ Δ be the three subgroups of index two where A is cyclic, and let A_{0} = [Δ, Δ] be the subgroup of index two in A. Set Z = Z(Δ) for short. Fix b ∈ Δ_{0}∖A.

(a) Assume φ ∈ Hom(Δ, S) is such that φ(Z) = Z and φ(A) ⊴ S. Then φ(A_{0}) = A_{0}, φ(A) ≤ T A, and φ(b) ∈ T Ab.

(b) Assume α ∈ Aut(S) is such that α(Z) = Z. Then either α sends each of the subgroups T Δ_{1} and T Δ_{2} to itself or it exchanges them.

Proof. Fix a generator a ∈ A; thus A_{0} = ⟨a^{2}⟩. Since Δ ⊴ S and A is characteristic in Δ, A ⊴ S. For all t ∈ T, [t, Δ_{0}] = 1 by assumption, so [t, b] = 1, and [t, a^{k}] = 1 if |a^{k}| ≤ 4. Hence tat^{−1} = a^{4j+1} for some j, so [t, a] ∈ ⟨a^{4}⟩. Thus [T, Δ] ≤ ⟨a^{4}⟩, and so ⟨T, a^{4}⟩ ⊴ S.

(a) Assume φ ∈ Hom(Δ, S) is such that φ(Z) = Z and φ(A) ⊴ S. Since A_{0} = [Δ, Δ], φ(A_{0}) ≤ [S, S] ≤ T A_{0}. If φ(A_{0}) ≤ ⟨T, a^{4}⟩, then there are distinct elements a^{i} ≠ a^{j} in A_{0} such that φ(a^{i}), φ(a^{j}) ∈ T a^{k} for some k. Then 1 ≠ a^{i−j} and φ(a^{i−j}) ∈ T, which is impossible since Z ≤ ⟨a^{i−j}⟩ and φ(Z) = Z ≰ T.

Thus φ(a^{2}) ∉ ⟨T, a^{4}⟩, and the image of φ(a) in S/⟨T, a^{4}⟩ ≅ D_{8} has order 4. Hence φ(a) = ta^{i} for some t ∈ T and some odd i. Also, the image of φ(b) in S/⟨T, a^{4}⟩ must invert that of φ(a), and so φ(b) ∈ T Ab. Since φ(A) ⊴ S, b(ta^{i})b^{−1} = ta^{−i}, and (ta^{−i})^{−1}ta^{i} = a^{2i}, we have a^{2i} ∈ φ(A). Thus A_{0} = ⟨a^{2i}⟩ ≤ φ(A), so φ(A_{0}) = A_{0}. This finishes the proof of (a).

(b) Now assume α ∈ Aut(S) is such that α(Z) = Z. By (a), applied with φ = α|_{Δ}, α(A) ≤ T A and α(A_{0}) = A_{0}. Hence α(⟨a^{4}⟩) = ⟨a^{4}⟩, and α induces an automorphism α of S/⟨a^{4}⟩ ≅ T × D_{8} which sends Z(Δ/⟨a^{4}⟩) = A_{0}/⟨a^{4}⟩ to itself. By Proposition B.1, α(T A_{0}/⟨a^{4}⟩) = T A_{0}/⟨a^{4}⟩, and thus α(T A_{0}) = T A_{0}. Since α(T A) = T A, it now follows that α either sends the two subgroups T Δ_{i}/T A_{0} of S/T A_{0} ≅ C_{2}^{2} to themselves (i = 1, 2) or switches them. □

The next lemma involves a similar situation.

Lemma B.3. Fix a finite 2-group S with a normal dihedral or quaternion subgroup Δ ⊴ S of order ≥ 8. Assume two of the three subgroups of index two in Δ are S-conjugate. Let Δ_{0} ≤ Δ be a dihedral or quaternion subgroup of order 8, and set T = C_{S}(Δ_{0}). Let A ⊴ Δ be the cyclic subgroup of index two: the one which is normal in S if Δ ≅ Q_{8}. Fix a generator a ∈ A, and choose b ∈ Δ_{0}∖A. Let Z ≤ A_{0} < A be the subgroups of order two and index two, respectively. Then the following hold.

(a) A ≤ [S, S].

(b) [S:T Δ] = 2. For each g ∈ S, gbg^{−1} = a^{i}b where i is even if g ∈ T Δ and i is odd if g ∈ S∖T Δ.

(c) T A_{0} ⊴ S, S/T A_{0} ≅ D_{8}, and Z(S/T A_{0}) = T A/T A_{0}.

(d) If T is abelian, then [S, S] is abelian.

(e) Assume b_{1}, b_{2} ∈ S are in distinct cosets of T A, [b_{1}, A] = A_{0} = [b_{2}, A], b_{i}^{2} ∈ A, and ⟨A, b_{i}⟩ ⊴ S. Then ⟨A, b_{1}, b_{2}⟩ is dihedral, semidihedral, or quaternion.

(f) Assume that Δ is dihedral and Z is a direct factor of T. Let α ∈ Aut(S) be such that α^{2} ∈ Inn(S), α(Z) = Z, and α(T Δ) ≠ T Δ. Set Δ̂ = α(Δ)·Δ. Then Δ̂ is dihedral, Δ̂ ⊴ S, [Δ̂:Δ] = 2, T ∩ Δ̂ = Z, and T Δ̂ = S.

Proof. By assumption, Δ ⊴ S. Hence A ⊴ S since A is characteristic in Δ, except when Δ ≅ Q_{8} in which case A was chosen to be the unique subgroup of index two in Δ normal in S.

(a) By assumption, the subgroups ⟨A_{0}, b⟩ and ⟨A_{0}, ba⟩ are S-conjugate. Hence there is x ∈ S such that c_{x}(bA_{0}) = baA_{0}, so xbx^{−1} = ba^{i} for some odd i, and A = ⟨a^{i}⟩ ≤ [S, S].

(b) By definition (and since A = ⟨a⟩ ⊴ S),

T = C_{S}(Δ_{0}) = {g ∈ S | gbg^{−1} = b, gag^{−1} = a^{4j+1} some j}.  (1)

Hence

T Δ = {g ∈ S | c_{g} ∈ Aut_{T} (Δ)Inn(Δ)} = {g ∈ S | gbg^{−1} = a^{i}b some i ≡ 0 (mod 2)}.

Since Δ ⊴ S, this proves that gbg^{−1} = a^{i}b with i odd whenever g ∈ S∖T Δ. Also, T Δ < S since ⟨A_{0}, b⟩ and ⟨A_{0}, ab⟩ are S-conjugate, and hence [S:T Δ] = 2.

(c) If Δ = Δ_{0} has order 8, then T A_{0} = T = C_{S}(Δ) is normal in S since Δ ⊴ S. So assume |Δ| > 8. Since A ⊴ S, 1 ≠ ⟨a^{4}⟩ ⊴ S. Also, [T, A] ≤ ⟨a^{4}⟩ by (1), and hence T Δ/⟨a^{4}⟩ ≅ (T/Z)×D_{8}. For x ∈ S, c_{x}(T Δ) = T Δ by (b), so c_{x}(T A_{0}/⟨a^{4}⟩) = T A_{0}/⟨a^{4}⟩ by Proposition B.1 applied with α = c_{x} ∈ Aut(T Δ/⟨a^{4}⟩). Thus T A_{0} ⊴ S.

Throughout the rest of the proof of the lemma, we set S = S/T A_{0}, and let P ≤ S or g ∈ S be the image of P ≤ S or g ∈ S. Thus T Δ = ⟨a, b⟩ ≅ C_{2}^{2}.

For x ∈ S∖T Δ, c_{x}(a) = a and c_{x}(b) = ab by (b). Thus S ≅ D_{8}, with center ⟨a⟩ = T A.

(d) Assume T is abelian. Since A is cyclic, Aut(A) is abelian, and hence [S, S] is in the kernel of the map S ⟶ Aut(A) induced by conjugation. Thus [S, S] ≤ C_{S}(A). Also, [S, S] ≤ T A since T A is normal of index 4 in S. Since T is abelian, C_{T A}(A) is also abelian, and so is [S, S].

(e) Fix b_{1}, b_{2} as above, and set Δ_{i} = ⟨A, b_{i}⟩ ⊴ S and Δ̂ = Δ_{1}Δ_{2}. Then Δ̂/A ≅ C_{2}^{2} since the Δ_{i} are normal and contain A with index two, so (b_{1}b_{2})^{2} ∈ A. Also, (b_{1}b_{2})^{2} = a in S since b_{1} and b_{2} are in distinct nonidentity cosets of A = Z(S) and have order 2 in S, so (b_{1}b_{2})^{2} = a^{j} for some odd j. Thus Â =^{def} ⟨b_{1}b_{2}⟩ is cyclic of index two in Δ̂, conjugation by b_{1} inverts the subgroup A of index two in Â, and hence Δ̂ is dihedral, semidihedral, or quaternion (cf. [G, Theorem 5.4.4]).

(f) Now assume that Δ is dihedral, and that T = T_{0}Z where T_{0} ∩ Z = 1. Let α ∈ Aut(S) be such that α^{2} ∈ Inn(S), α(Z) = Z, and α(T Δ) ≠ T Δ. Set Δ^{∗} = α(Δ), and Δ̂ = ΔΔ^{∗}. Since Δ and Δ^{∗} are both normal in S and are exchanged by α (since α^{2} ∈ Inn(S)), Δ̂ is normal in S.

Set a_{∗} = α(a) and b_{∗} = α(b). By (c), T A_{0} ⊴ S, and S = S/T A_{0} is dihedral of order 8 with center T A/T A_{0}.

Since A ≤ [S, S] by (a), α(A) ≤ [S, S] ≤ T A. If b_{∗} ∈ T Δ, then α(Δ) ≤ T Δ, and by Lemma B.2(a) (applied with φ = α|_{Δ} and T replaced by T_{0}), b_{∗} ∈ T_{0}Ab. So b_{∗} ∈ {b, ab}. Since [T, b] = 1 (recall b ∈ Δ_{0}), [α(T), b_{∗}] = 1, so α(T) ≤ ⟨a, b⟩ = T Δ. Thus α(T) ≤ T Δ, so α(T Δ) = T Δ, and this contradicts our original assumption about α.

Thus b_{∗} ∉ T Δ. Hence by (b), b_{∗}bb_{∗}^{−1} = a^{i}b for some odd i. Set â = b_{∗}b; then â^{2} = a^{i}, and thus ⟨b_{∗}, b⟩ = ⟨â, b⟩ is a dihedral group which contains Δ with index two. Since α^{2} ∈ Inn(S), α(b_{∗}) = α^{2}(b) = a^{j}b for some j, and so α(â) = a^{j}bb_{∗} = a^{j}(b_{∗}b)^{−1} = a^{j}â^{−1} is in ⟨â⟩. So a_{∗} ∈ ⟨a⟩, ⟨b_{∗}, b⟩ = ΔΔ^{∗} is dihedral, and it contains Δ with index two. Also, T Δ̂ = S since [S:T Δ] = 2 and Δ̂ ≰ T Δ, and T ∩ Δ̂ = T ∩ Δ = Z since T Δ ∩ Δ̂ = Δ. □

This will now be applied to prove the following lemma.

Lemma B.4. Fix a finite 2-group S, and subgroups T, Δ_{0} ≤ S such that Δ_{0} is dihedral of order 8, T ∩ Δ_{0} = 1, and [T, Δ_{0}] = 1. Let U, V ≤ Δ_{0} be the two noncyclic subgroups of order four, and set Z = U ∩ V = Z(Δ_{0}). Assume N_{S}(T U) = T Δ_{0}, N_{S}(Δ_{0}) > T Δ_{0}, and either Z ≤ Z(S) or T contains no subgroup isomorphic to D_{8}. Let Δ ⊴ S be the normal closure of Δ_{0} in S. Then Δ is dihedral, T ∩ Δ = 1, [S:T Δ] = 2, and all noncentral involutions in Δ are S-conjugate.

Proof. Case 1: Assume first that Z ≤ Z(S). Set Δ_{−1} = U. We will construct subgroups Δ_{0} < Δ_{1} < ⋯ < Δ_{m} < S, all normalized by T, such that [S:T Δ_{m}] = 2, and such that for each 0 ≤ i ≤ m,

(i) Δ_{i} is dihedral of order 2^{3+i};

(ii) T ∩ Δ_{i} = 1 and T Δ_{i} = N_{S}(T Δ_{i−1}) = N_{S}(Δ_{i−1}); and

(iii) N_{S}(Δ_{i}) > T Δ_{i}.

To simplify notation, we set S_{i} = T Δ_{i} whenever Δ_{i} has been defined.

When i = 0, the only condition which is not immediate from the hypotheses is that T Δ_{0} = N_{S}(U). One inclusion is clear: N_{S}(U) ≥ T Δ_{0} since [U, T] = 1 and U ⊴ Δ_{0}. If g ∈ N_{N_{S}(U)}(T Δ_{0}), then since g normalizes U and T Δ_{0} and C_{T Δ_{0}}(U) = T U, g ∈ N_{S}(T U) = T Δ_{0}. Thus N_{N_{S}(U)}(T Δ_{0}) = T Δ_{0}, and so N_{S}(U) = T Δ_{0} by Lemma A.1.

Assume, for some i ≥ 0, that we have constructed Δ_{i} which satisfies (i)–(iii). If [S:S_{i}] = 2 (recall S_{i} = T Δ_{i}), then Δ_{i} ⊴ S since N_{S}(Δ_{i}) > S_{i}. For g ∈ S∖S_{i}, g ∉ N_{S}(Δ_{i−1}) = S_{i} implies that g exchanges the two conjugacy classes of noncentral involutions in Δ_{i}, and hence the noncentral involutions in Δ_{i} are all S-conjugate. If i > 0, then the normal closure of Δ_{0} in Δ_{i} is Δ_{i−1} which is not normal in S, and thus Δ_{i} is the normal closure of Δ_{0} in S (this is trivial if i = 0). So the subgroup Δ = Δ_{i} satisfies the conditions in the statement of the lemma.

Now assume [S:S_{i}] ≥ 4. Set S_{i+1} = N_{S}(Δ_{i}) > S_{i}. Let Δ_{i−1} and Δ_{i−1}^{∗} be the two dihedral subgroups of index two in Δ_{i}. Since S_{i} = N_{S}(Δ_{i−1}) and Δ_{i} ⊴ S_{i+1}, conjugation by any element of S_{i+1}∖S_{i} exchanges Δ_{i−1} with Δ_{i−1}^{∗}. The product of any two elements of S_{i+1}∖S_{i} thus lies in S_{i}, so [S_{i+1}:S_{i}] = 2, and S_{i} ⊴ S_{i+1} < S.

For each x ∈ N_{S}(S_{i}), since c_{x}(Z) = Z by assumption (Z ≤ Z(S)), Lemma B.2(b) implies that c_{x} either leaves T Δ_{i−1} and T Δ_{i−1}^{∗} invariant or exchanges them; and leaves them invariant only if x ∈ N_{S}(T Δ_{i−1}) = S_{i}. Thus [N_{S}(S_{i}):S_{i}] = 2. Since S_{i} ⊴ S_{i+1} with index two, this implies N_{S}(S_{i}) = S_{i+1}.

Choose any g ∈ N_{S}(S_{i+1})∖S_{i+1} such that g^{2} ∈ S_{i+1}, and set α = c_{g} ∈ Aut(S_{i+1}). Then α(Z) = Z since Z ≤ Z(S), and α(S_{i}) ≠ S_{i} since g ∉ S_{i+1} = N_{S}(S_{i}). Also, C_{S}(Δ_{0}) = T Z since N_{S}(U) = T Δ_{0} by (ii) when i = 0 (and since C_{S}(Δ_{0}) ≤ C_{S}(U) ≤ N_{S}(U)). The hypotheses of Lemma B.3(f) thus hold (but where T in Lemma B.3 corresponds to T Z here). So if we define Δ_{i+1} = Δ_{i}·α(Δ_{i}), then Δ_{i+1} ⊴ S_{i+1} is dihedral, [Δ_{i+1}:Δ_{i}] = 2, T ∩ Δ_{i+1} = 1, and T Δ_{i+1} = S_{i+1}. Thus (i) and (ii) hold, (iii) holds since g ∈ N_{S}(Δ_{i+1}), and this finishes the induction step in the proof.

Case 2: Now assume Z ≰ Z(S), and set S_{∗} = C_{S}(Z) < S. By Case 1, S_{∗} has the form described in the lemma: it contains a normal dihedral subgroup Δ ⊴ S_{∗} (the normal closure of Δ_{0} in S_{∗}), T ∩ Δ = 1, and [S_{∗}:T Δ] = 2. We prove that T contains a subgroup isomorphic to D_{8}, contradicting the assumptions.

Choose any g ∈ N_{S}(S_{∗})∖S_{∗} such that g^{2} ∈ S_{∗}, and set β = c_{g} ∈ Aut(S_{∗}). Then β(Z) ≠ Z. Assume first that β(Δ) ≰ T Δ, and choose x ∈ β(Δ)∖T Δ. Let A ⊴ Δ be the cyclic subgroup of index two (A ⊴ S_{∗}), and choose b ∈ U∖Z. (Recall C_{2}^{2} ≅ U ≤ Δ_{0}.) If c_{x}(b) ∈ Δ∖A is Δ-conjugate to b, then c_{ax}(U) = U for some a ∈ A, which is impossible since we showed in the proof of Case 1 that N_{S}(U) = T Δ_{0}. Thus b is not Δ-conjugate to xbx^{−1}, so [x, b] generates A, and hence A ≤ β(Δ) since x ∈ β(Δ) ⊴ S_{∗}. Since A is cyclic of order ≥ 4, this is possible only if β(A) = A, which is impossible since β(Z) ≠ Z.

Thus β(Δ) ≤ T Δ. Let ψ ∈ Hom(Δ, T) be the composite Δ ⟶^{β} T Δ −↠ T Δ/Δ ≅ T. Since β(Z) ≠ Z and β(Z) ≤ Z(T Δ), β(Z) ≰ Δ, and hence Z ≰ Ker(ψ). Since any nontrivial normal subgroup of Δ contains Z, this implies that ψ is injective, and thus that T contains a subgroup isomorphic to D_{8}. □

We also need the following corollary to Lemma B.4.

Lemma B.5. Fix a finite 2-group S, and an abelian subgroup P ≤ S such that |N_{S}(P)/P | = 2 and P ⋬ S. Assume either

(a) Aut_{S}(P) ≰ O_{2}(Aut(P)), or

(b) there is a direct factor U ≤ P such that U ≅ C_{2}^{2} and 1 ≠ [N_{S}(P), U] ≤ U.

Then there are subgroups Δ_{0} ≤ Δ ⊴ S such that Δ is dihedral, Δ_{0} ≅ D_{8}, C_{S}(Δ_{0}) ≤ P, and the noncentral involutions in Δ are all S-conjugate. In case (b), Δ can be taken to be the normal closure of U in S.

Proof. Set P̂ = N_{S}(P) < S, fix x ∈ N_{S}(P̂)∖P̂ such that x^{2} ∈ P̂, and set Q = xP x^{−1}. Set Z = [P̂, P̂]. Since P̂ is nonabelian, and P, Q are distinct abelian subgroups of index two, Lemma A.6(c,a) implies that |Z| = 2, and Z(P̂) = P ∩ Q has index 2 in P and in Q. Also, Z ≤ Z(P̂).

We first show that (a) implies (b). Consider the subgroups

Θ_{1} = {α ∈ Aut(P) | [α, P] ≤ Fr(P)} and Θ_{2} = {α ∈ Aut(P) | α|_{Ω_{1}(P)} = Id} .

Both are normal in Aut(P), and they are 2-subgroups by Lemma A.2 and [G, Theorem 5.2.4], respectively. Thus Θ_{1}Θ_{2} ≤ O_{2}(Aut(P)). Since AutP̂(P) = Aut_{S}(P) ≰ O_{2}(Aut(P)) by assumption, AutP̂(P) ≰ Θ_{1}Θ_{2}. Hence for y ∈ P̂∖P, Z = [y, P] ≰ Fr(P) and Z(P̂) = C_{P} (y) ≱ Ω_{1}(P).

Thus there are T < Z(P̂) and g ∈ P ∖Z(P̂) such that |g| = 2 and P = Z × T × ⟨g⟩ (T = Ker(f) for any f: Z(P̂) → Z(P̂)/Fr(Z(P̂)) → C_{2} with Z ≰ Ker(f)). Set U = ⟨Z, g⟩. Then U ≅ C_{2}^{2}, P = T U, and T ∩ U = 1. So (b) holds.

Now assume (b). Thus P = T U where U ≅ C_{2}^{2}, T ∩ U = 1, and Z = [P̂, P̂] ≤ U. By the argument used in the last paragraph, we can assume that T < Z(P̂).

Set V = xUx^{−1} and Δ_{0} = UV. Then Δ_{0} ≅ D_{8} since [U, V] = Z and U ≅ V ≅ C_{2}^{2}, T ∩ Δ_{0} = T ∩ Z = 1, and [T, Δ_{0}] ≤ [T, P̂] = 1 since T ≤ Z(P̂). Also, N_{S}(T U) = N_{S}(P) = P̂ = T Δ_{0}, and N_{S}(Δ_{0}) > T Δ_{0} since x ∈ N_{S}(Δ_{0})∖T Δ_{0}. Since T is abelian, it contains no subgroup isomorphic to D_{8}. So by Lemma B.4, the normal closure Δ of Δ_{0} in S is dihedral, and all noncentral involutions in Δ are S-conjugate.

Set R = N_{C_{S}(Δ_{0})}(Z(P̂)). Then R normalizes Z(P̂)U = P, so R ≤ N_{S}(P) = P̂. Hence R ≤ CP̂(Δ_{0}) = Z(P̂), so by Lemma A.1, C_{S}(Δ_{0}) = Z(P̂) ≤ P. □

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Center for Mathematical Sciences, LTH, Box 118, 22100 Lund, Sweden

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