SPLITTING FUSION SYSTEMS OVER 2-GROUPS

BOB OLIVER

Abstract. We find conditions which imply that a saturated fusion system over a product of 2-groups splits as a product of fusion systems over the factors.

A saturated fusion system F over a p-group S is a category whose objects are the subgroups of S, where each morphism in F is a monomorphism of groups, and which satisfies certain axioms listed in Definition 1.1 below. As one of the main motivating examples, let G be a finite group, fix S ∈ Syl_{p}(G), and let F_{S}(G) be the category whose objects are the subgroups of S and where Hom_{F_{S}(G)}(P, Q) is the set of homomorphisms induced by conjugation in G. Then F_{S}(G) is a saturated fusion system over S. A saturated fusion system F which is not isomorphic to F_{S}(G) for any finite group G is called exotic.

In an ongoing project with Kasper Andersen and Joana Ventura, we are attempting, with the help of computer computations, to list all saturated fusion systems over 2-groups of small order, where we try to make “small” be as large as possible. Since it is clearly impossible to do this explicitly, we restrict attention to reduced fusion systems: saturated fusion systems which have no proper normal subsystems of p-power index or of index prime to p (Definition 1.5), and which have no nontrivial normal p-subgroups (Definition 1.2). In [AOV, Theorems A & C], we show that exotic fusion systems can be “detected” in an explicit way on reduced fusion systems; moreover, on reduced fusion systems which are indecomposable in the sense that they do not factor as a product of fusion systems over smaller p-groups. This means, for example, that if we list all reduced indecomposable fusion systems over 2-groups of order ≤ 2^{n} for some fixed n, and show they are all realized by finite groups satisfying a certain “tameness” condition on their automorphism groups ([AOV, Definition 2.10]), then there are no exotic fusion systems over 2-groups of order ≤ 2^{n}.

In earlier work with Ventura [OV], and in ongoing work with her and with Andersen, we derive some fairly strong conditions on a 2-group S which must be satisfied if there are any reduced fusion systems over S. When we use a computer to eliminate those groups of given order which do not satisfy these conditions, we are left with the groups which are known to be Sylow 2-subgroups of simple groups, a very few others, and a long list of groups which split as products of smaller groups also satisfying the conditions. This is what motivated us to study reduced fusion systems over products of (nontrivial) 2-groups, and look for conditions which guarantee that any such fusion system factors as a product.

Our main results are the following two theorems. As usual, for a p-group Q, Ω_{1}(Q) is the subgroup generated by all elements of order p.

Theorem A. Fix 2-groups S_{1} and S_{2}, and set S = S_{1} × S_{2}. Assume, for i = 1, 2,

(a) Ω_{1}(Z(S_{i})) ≤ [S_{i}, S_{i}], and for each φ ∈ Hom(S_{i}, S_{i}) such that φ(S_{i}) ⊴ S_{i} and φ is not onto, Ker(φ) ≥ Ω_{1}(Z(S_{i})); and (b) S_{3−i} contains no subgroup isomorphic to S_{i} × S_{i}.

2000 Mathematics Subject Classification. Primary 20D20. Secondary 20C20, 20D40, 20D45.

Key words and phrases. Fusion system, finite groups, automorphisms.

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

Then for every saturated fusion system F over S such that O^{2}(F) = F = O^{2′}(F), F ≅ F_{1} × F_{2} for some pair of saturated fusion systems F_{i} over S_{i}.

Condition (a) in Theorem A holds whenever S_{i} is nonabelian and Z(S_{i}) is cyclic. But it also holds for many 2-groups whose center is not cyclic; for example, for the groups UT_{3}(𝔽_{2^{n}}) (strictly upper triangular matrices) when n ≥ 2.

The second theorem puts stronger conditions on S_{1}, while weakening those on S_{2}. Let D_{2^{n}} and SD_{2^{n}} denote the dihedral and semidihedral groups, respectively, of order 2^{n}.

Theorem B. Fix 2-groups S_{1} and S_{2}, and set S = S_{1} × S_{2}. Assume

(a) S_{1} ≅ D_{2^{n}} (n ≥ 3), SD_{2^{n}} (n ≥ 4), or C_{2^{n}} ≀ C_{2} (n ≥ 2); and

(b) S_{2} contains no proper subgroup isomorphic to S_{1} × S_{1}.

Then for every saturated fusion system F over S such that O^{2}(F) = F = O^{2′}(F), F ≅ F_{1} × F_{2} for some pair of saturated fusion systems F_{i} over S_{i}.

Theorems A and B are shown as Corollary 4.6 and Theorem 4.8, respectively. They are both consequences of Proposition 4.3, which is a more general (and more technical) splitting result.

The condition “O^{2}(F) = F = O^{2′}(F)” means that there are no proper normal subsystems of F of 2-power index or of odd index (Definition 1.5). Examples are given in Section 4.4 to show why these conditions are needed. For example, if G is the subgroup of index two in Σ_{6} × P GL_{2}(9) which contains neither factor, and F is the fusion system of G (over the Sylow subgroup ≅ D_{8} × D_{16}), then F satisfies all of the hypotheses in Theorems A and B except the condition O^{2}(F) = F, but does not factor as a product of smaller fusion systems.

The example of the fusion system of A_{14} over the 2-group D_{8} × (D_{8} ≀ C_{2}) helps show why the condition S_{2} ≱ S_{1} × S_{1} is needed in Theorems A and B (and more examples are given in Section 4.4). However, the following theorem (proven as Theorem 4.9) shows some cases where we can avoid this problem.

Theorem C. Assume D ≅ D_{2^{n}} (n ≥ 3), SD_{2^{n}} (n ≥ 4), or C_{2^{n}} ≀ C_{2} (n ≥ 2). Let S be a product of k copies of D for some k ≥ 2. Let F be a saturated fusion system over S such that O^{2}(F) = F = O^{2′}(F). Then F is isomorphic to a product of saturated fusion systems over D.

Theorems B and C also hold (with only minor modifications to the proofs) when S_{1} (or D) is a quaternion 2-group. However, in that case, it is very easy to see directly that Z(S_{1}) (in Theorem B) or Z(S) (in Theorem C) is normal in any fusion system satifying the hypotheses of the theorems, and hence that no such fusion system can be reduced.

Section 1 contains the background material which is needed on fusion systems. Sections 2 and 3 contain technical results about actions and representations of groups, and automorphisms of p-groups, respectively. The results about splittings of fusion systems are shown in Section 4.

1. Background results about fusion systems

A fusion system over a finite p-group S is a category F, where Ob(F) is the set of all subgroups of S, and where for all P, Q ≤ S,

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

and each φ ∈ Hom_{F}(P, Q) is the composite of an F-isomorphism followed by an inclusion. Here, Inj(P, Q) denotes the set of injective homomorphisms from P to Q.

If F is a fusion system over S, we say two subgroups P, Q ≤ S are F-conjugate if they are isomorphic as objects of the category F. Two elements g, h ∈ S are F-conjugate if there is φ ∈ Iso_{F}(⟨g⟩, ⟨h⟩) such that φ(g) = h.

Definition 1.1 ([Pg], see [BLO, Definition 1.2]). Let F be a fusion system over a finite p-group S.

• A subgroup P ≤ S is fully centralized in F if |C_{S}(P)| ≥ |C_{S}(Q)| for each Q ≤ S which is F-conjugate to P.

• A subgroup P ≤ S is fully normalized in F if |N_{S}(P)| ≥ |N_{S}(Q)| for each Q ≤ S which is F-conjugate to P.

• F is a saturated fusion system if the following two conditions hold:

(I) (Sylow axiom) If P ≤ S is fully normalized in F, then P is fully centralized in F, and Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)).

(II) (Extension axiom) For each P, Q ≤ S such that Q is fully centralized in F, and each φ ∈ Iso_{F}(P, Q), if we set

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

then there is φ ∈ Hom_{F}(N_{φ}, S) such that φ|_{P} = φ.

If G is a finite group and S ∈ Syl_{p}(G), then F_{S}(G) defined to be the category whose objects are the subgroups of S, and whose morphisms are those homomorphisms between subgroups which are induced by conjugation by elements of G. A subgroup P ≤ S is fully normalized (fully centralized) in F_{S}(G) exactly when N_{S}(P) ∈ Syl_{p}(N_{G}(P)) (C_{S}(P) ∈ Syl_{p}(C_{G}(P))). Using, for example, the Sylow theorems, one can show that F_{S}(G) is a saturated fusion system (cf. [BLO, Proposition 1.3]).

We next recall some of the other definitions associated with a fusion system.

Definition 1.2. Fix a prime p, a p-group S, and a fusion system F over S. Let P ≤ S be any subgroup.

• P is F-centric if C_{S}(Q) = Z(Q) for all Q which is F-conjugate to P.

• P is F-essential if P is F-centric and fully normalized in F, and Out_{F}(P) contains a strongly p-embedded subgroup. Here, for any finite group G, a subgroup H ≤ G is strongly p-embedded if p||H|, and for all g ∈ G∖H, p ∤ |H ∩ gHg^{−1}|.

• P is strongly closed in F if no g ∈ P is F-conjugate to any h ∈ S∖P. Equivalently, for each Q ≤ P and each φ ∈ Hom_{F}(Q, S), φ(Q) ≤ P.

• P is normal in F if 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.

• 𝔣𝔬𝔠(F) = ⟨x^{−1}y | x, y ∈ S are F-conjugate⟩.

• 𝔥𝔶𝔭(F) = ⟨x^{−1}α(x) | x ∈ P ≤ S, α ∈ Aut_{F}(P) has order prime to p⟩.

We also need to work with fusion subsystems, and normal fusion subsystems, of a saturated fusion system. When F is a fusion system over S and S_{0} ≤ S, F|_{S_{0}} denotes the full subcategory whose objects are the subgroups of S_{0} (a fusion system over S_{0}).

Definition 1.3. Fix a prime p, and a fusion system F over a p-group S.

• A (saturated) fusion subsystem of F is a subcategory F_{0} ⊆ F which is itself a (saturated) fusion system over a subgroup S_{0} ≤ S.

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

Hom_{φFφ^{−1}}(P, Q) = φ·Hom_{F}(φ^{−1}(P), φ^{−1}(Q))·φ^{−1}

for all P, Q ≤ S.

• If X_{1}, …, X_{m} are subcategories and/or sets of morphisms in F, then ⟨X_{1}, …, X_{m}⟩ denotes the smallest fusion subsystem in F over S (not necessarily saturated) which contains the X_{i}. Thus ⟨X_{1}, …, X_{m}⟩ is a subcategory of F with the same objects, and which contains those morphisms which are composites of restrictions of inner automorphisms of S, and restrictions of morphisms in the X_{i} and their inverses.

• A fusion subsystem F_{0} ⊆ F over S_{0} ⊴ S is F-invariant if S_{0} is strongly closed in F, φF_{0}φ^{−1} = F_{0} for each φ ∈ Aut_{F}(S_{0}), and F|_{S_{0}} = ⟨Aut_{F}(S_{0}), F_{0}⟩.

• A fusion subsystem F_{0} ⊆ F over S_{0} ⊴ S is normal in F (F_{0} ⊴ F) if F_{0} and F are both saturated and F_{0} is F-invariant.

Thus an “F-invariant” fusion subsystem is one which satisfies all conditions for being normal in F except for being saturated. This is equivalent to what some authors define as a “normal” fusion subsystem, but for the sake of consistency with the terminology in [AOV], we limit “normal” fusion systems to those which are saturated. Our main reason for defining it here is so that we can apply the following lemma, which will be needed in Section 4.

Lemma 1.4. Let F be a saturated fusion system over the finite p-group S, and let F_{0} ⊆ F be an F-invariant fusion subsystem over S. Then for any P ≤ S, P is fully normalized in F_{0} (fully centralized in F_{0}, F_{0}-centric) if and only if P is fully normalized in F (fully centralized in F, F-centric).

Proof. Assume P, Q ≤ S_{0} are F-conjugate. Since F|_{S_{0}} = ⟨F_{0}, Aut_{F}(S_{0})⟩, any φ ∈ Iso_{F}(P, Q) is a composite of morphisms in F_{0} and restrictions of F-automorphisms of S_{0}. Since ψF_{0}ψ^{−1} = F_{0} for all ψ ∈ Aut_{F}(S_{0}), these morphisms can be rearranged so that the morphisms in F_{0} all come first, followed by a composite of restrictions of morphisms in Aut_{F}(S_{0}). Thus there is α ∈ Aut_{F}(S_{0}) such that P is F_{0}-conjugate to α(Q).

Since the F-conjugacy class of P contains its F_{0}-conjugacy class, P is fully normalized in F_{0} if it is fully normalized in F. Conversely, assume P is fully normalized in F_{0}, let Q be F-conjugate to P and fully normalized in F, and choose α ∈ Aut_{F}(S) such that P is F_{0}-conjugate to α(Q). Thus |N_{S}(P)| ≥ |N_{S}(α(Q))| = |N_{S}(Q)| ≥ |N_{S}(P)|, so these are all equal, and P is also fully normalized in F.

The argument for fully centralized subgroups is identical. The centric case follows since P is F-centric if and only if it is fully centralized in F and contains C_{S}(P). □

The next definitions and results are taken from [BCGLO2].

Definition 1.5. Let F_{0} ⊆ F be saturated fusion systems over p-groups S_{0} ≤ S.

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

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

Note that a fusion subsystem of p-power index (index prime to p) corresponds to a subgroup of a finite group G which contains a normal subgroup of p-power index (index prime to p); i.e., a subgroup which contains O^{p}(G) (O^{p′}(G)). Thus, for example, when S ∈ Syl_{p}(G), F_{S}(S) is not in general a fusion system of index prime to p in F_{S}(G).

Theorem 1.6. Let F be a saturated fusion system over a finite p-group S.

(a) There is a unique saturated fusion subsystem O^{p}(F) over 𝔥𝔶𝔭(F) of p-power index in F, O^{p}(F) is contained in all other saturated fusions subsystems of p-power index in F, and O^{p}(F) ⊴ F. Also, O^{p}(F) = F if and only if 𝔣𝔬𝔠(F) = S.

(b) There is a unique minimal saturated fusion subsystem O^{p′}(F) over S of index prime to p in F, and O^{p′}(F) ⊴ F. If F_{0} ⊴ F is any normal saturated fusion subsystem over S, then F_{0} ⊇ O^{p′}(F).

Proof. The first statement in (a) is shown in [BCGLO2, Theorem 4.3], aside from the normality of O^{p}(F) which is shown in [AOV, Proposition 1.16(a)]. From this, it follows immediately that O^{p}(F) = F if and only if 𝔥𝔶𝔭(F) = S. To see that this is equivalent to the condition 𝔣𝔬𝔠(F) = S, see, e.g., [AOV, Theorem 1.13(a)].

The first statement in (b) is shown in [BCGLO2, Theorem 5.4], the normality of O^{p′}(F) in [AOV, Proposition 1.16(b)], and the last statement in [AOV, Lemma 1.17]. □

We now look at essential subgroups of a fusion system.

Proposition 1.7. The following hold for any saturated fusion system F over a p-group S.

(a) F = ⟨Aut_{F}(P) | P = S or P is F-essential⟩.

(b) Let H be any set of F-essential subgroups with the property that if P, Q are F-essential, P ∈ H, and P is F-conjugate to a subgroup of Q, then Q ∈ H. Then

⟨Aut_{F}(S), Aut_{F}(P) | P ∈ H⟩ = ⟨Aut_{F}(S), O^{p′}(Aut_{F}(P)) | P ∈ H⟩ .

(c) Assume P ≨ S is F-centric and fully normalized in F. Let H ≤ Aut_{F}(P) be the subgroup generated by those automorphisms which extend to morphisms in F between strictly larger subgroups of S. Then either H ≨ Aut_{F}(P), H/Inn(P) is strongly pembedded in Out_{F}(P), and P is F-essential; or H = Aut_{F}(P) and P is not F-essential.

(d) If P ≤ S is such that [N_{S}(P), P] ≤ Fr(P), then P is not F-essential.

Proof. Point (a) is shown in [OV, Corollary 2.6] and point (d) in [OV, Proposition 3.2 & Lemma 3.4]. By [OV, Proposition 2.5], for P ≨ S and H ≤ Aut_{F}(P) defined as in (c), P is F-essential if and only if H ≨ Aut_{F}(P). When this occurs, then H/Inn(P) was shown in the proof of that proposition to be strongly p-embedded in Out_{F}(P), and this proves (c).

It remains to prove (b). For each P ∈ H and α ∈ Aut_{F}(P), Aut_{S}(P) and αAut_{S}(P)α^{−1} are both Sylow p-subgroups of O^{p′}(Aut_{F}(P)), and hence there is β ∈ O^{p′}(Aut_{F}(P)) such that β ∘ α ∈ N_{Aut_{F} (P)}(Aut_{S}(P)). In other words,

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

(the Frattini argument). By the extension axiom, each automorphism in the normalizer extends to an element of Aut_{F}(N_{S}(P)), and by (a), this is a composite of restrictions of F-automorphisms of S and of strictly larger F-essential subgroups, all of which are in H by the hypotheses. So by downwards induction on |P |, we see that each Aut_{F}(P) for P ∈ H is in the fusion subsystem generated by Aut_{F}(S) and the groups O^{p′}(Aut_{F}(Q)) for Q ∈ H. □

We will also need the following lemma about essential subgroups for a fusion system over a product of p-groups.

Lemma 1.8. Let S_{1}, S_{2} be a pair of p-groups, and set S = S_{1} × S_{2}. Then the following hold for each P ≤ S.

(a) If P ≨ P_{1}P_{2}, where P_{i} ≤ S_{i} is the image of P under projection, then there is g ∈ N_{S}(P)∖P such that c_{g} ∈ O_{p}(Aut(P)). In particular, P cannot be F-essential for any saturated fusion system F over S.

(b) If p = 2, and P is F-essential for some saturated fusion system F over S, then P ≥ S_{1} or P ≥ S_{2}.

Proof. The first statement in (a) is shown in [AOV, Lemma 3.1]. Hence if P is F-centric and P ≨ P_{1}P_{2}, then O_{p}(Out_{F}(P)) ≠ 1. Thus Out_{F}(P) does not contain a strongly p-embedded subgroup: this is shown in [Sz2, Theorem 6.4.3] when p = 2, and the same proof holds when p is odd. So no such P can be F-essential, and this finishes the proof of (a).

Now assume p = 2, P = P_{1}P_{2}, and O_{2}(Out_{F}(P)) = 1. Assume also P_{1} ≨ S_{1} and P_{2} ≨ S_{2}. Choose elements x_{i} ∈ N_{S_{i}}(P_{i})∖P_{i} such that c_{x_{i}} has order 2 in Out(P_{i}). Then

rk([x_{1}x_{2}, P/Fr(P)]) = rk([x_{1}, P/Fr(P)]) + rk([x_{2}, P/Fr(P)]).

Since O_{2}(Out_{F}(P)) = 1, Out_{F}(P) acts faithfully on P/Fr(P) by Lemma 2.1, and hence rk([x_{i}, P/Fr(P)]) > 0 for i = 1, 2. So x_{1}x_{2} is not Out_{F}(P)-conjugate to x_{1} or x_{2}. If Out_{F}(P) did contain a strongly 2-embedded subgroup, then all of its involutions would be conjugate (cf. [Sz2, (6.4.4(i))]), and this is not the case. Thus P is not F-essential, and this proves (b). □

The following transfer homomorphism for abstract fusion systems will be needed.

Proposition 1.9. Fix a p-group S, and a saturated fusion system F over S. Then there is an injective homomorphism

trf_{F}: S/𝔣𝔬𝔠(S) ⟶ S^{ab} = S/[S, S]

which has the following form. There are proper subgroups P_{1}, …, P_{m} ≨ S, and morphisms φ_{i} ∈ Hom_{F}(P_{i}, S) (i = 1, …, m), such that for g ∈ S,

trf_{F}([g]) = ∏_{[α]∈Out_{F} (S)}[α(g)] · ∏_{i=1}^{m} φ_{i}_{∗}(trf_{P_{i}}^{S}([g])).

Here, [g] denotes the class of g in S/𝔣𝔬𝔠(F) or in S^{ab} = S/[S, S], trf_{P_{i}}^{S} is the transfer homomorphism in (−)^{ab} = H_{1}(−), and the terms on the right are regarded as lying in the abelianization of S. If g ∈ Ω_{1}(Z(S)), then trf_{F}([g]) = ∏_{[α]∈Out_{F} (S)}[α(g)]. If in addition, g is Out_{F}(S)-invariant, then trf_{F}([g]) = [g]^{k} where k = |Out_{F}(S)| is prime to p.

Proof. We begin with some general considerations. Fix a pair of groups H and G, and let X be an (H, G)-biset. Thus H acts on X on the left, G acts on the right, and the two actions commute. If the left H-action is free, then X induces a homomorphism X_{∗}: G^{ab} ⟶ H^{ab} as follows. Fix a set {t_{1}, …, t_{n}} ⊆ X of representatives for the orbits of the left H-action. For g ∈ G, let σ ∈ Σ_{n} and h_{1}, …, h_{n} be the unique permutation and elements such that t_{i}g = h_{i}t_{σ(i)} for each i, and set X_{∗}([g]) = ∏_{i=1}^{n}[h_{i}].

If g, g′ ∈ G, and t_{i}g = h_{i}t_{σ(i)} and t_{i}g′ = h′_{i}t_{τ(i)} for each i, then t_{i}gg′ = h_{i}h′_{σ(i)}t_{τσ(i)}. Thus X_{∗} is a homomorphism when defined on G, and hence factors through G^{ab}. If an orbit representative t_{i} is replaced by another one ht_{i}, then one term in the definition of X_{∗}([g]) is multiplied by h and another (possibly the same) by h^{−1}; thus X_{∗} is independent of the choice of t_{i}.

For any homomorphism φ ∈ Hom(G, H), let H_{G,φ} be the (H, G)-biset where H acts on itself by left multiplication, and G acts on the right via (h, g) ↦ hφ(g). Then (H_{G,φ})_{∗} = φ^{ab} is the homomorphism induced by φ. If H ≤ G, and X = G is the (H, G)-biset with the actions defined by left and right multiplication, then X_{∗} is the usual transfer trf_{H}^{G} as defined, for example, in [G, Theorem 7.3.2]. If X and Y are two (G, H)-bisets, then clearly (X ⨿ Y)_{∗} = X_{∗}·Y_{∗}.

Assume X is an (H, G)-biset and Y is a (K, H)-biset, where both left actions are free. Set

Y ×_{H}X = (Y ×X)/⟨(y, hx) ∼ (yh, x)⟩ .

If t_{1}, …, t_{n} are representatives for the left H-orbits in X, and u_{1}, …, u_{m} are representatives for the left K-orbits in Y, then the pairs (u_{i}, t_{j}) form a set of representatives for the Korbits in Y ×_{H} X. For fixed g ∈ G, set t_{j}g = h_{j}t_{σ(j)} (where h_{j} ∈ H and σ ∈ Σ_{n}), and set u_{i}h_{j} = k_{ij}u_{τ_{j}(i)} (k_{ij} ∈ K and τ_{j} ∈ Σ_{m}). Then (u_{i}, t_{j})g = k_{ij}(u_{τ_{j}(i)}, t_{σ(j)}), and so

(Y ×_{H}X)_{∗}([g]) = ∏_{j=1}^{n}(∏_{i=1}^{m} k_{ij}) = ∏_{j=1}^{n} Y_{∗}([h_{j}]) = Y_{∗}(X_{∗}([g])) .

We now return to the saturated fusion system F over a p-group S. By [BLO, Proposition 5.5], there is an (S, S)-biset Ω with the following properties:

(i) Each (S, S)-orbit of Ω has the form (S_{P,φ})×_{P} S for some P ≤ S and some φ ∈ Hom_{F}(P, S). (Thus the S-action on each side is free.)

(ii) For each P ≤ S and each φ ∈ Hom_{F}(P, S), Ω|_{(S,P)} = Ω×_{S}(S_{P,Id}) is isomorphic to Ω|_{(S,φ)} = Ω×_{S}(S_{P,φ}) as (P, S)-bisets.

(iii) |Ω|/|S| is prime to p.

Fix such a biset

Ω = ∐_{i=1}^{k}(S_{P_{i},φ_{i}})×_{P_{i}}S ,

where P_{i} ≤ S and φ_{i} ∈ Hom_{F}(P_{i}, S). By the above remarks, for each g ∈ S,

Ω_{∗}([g]) = ∏_{i=1}^{k} φ_{i}_{∗}(trf_{P_{i}}^{S}([g])) .

For each g ∈ P ≤ S and each φ ∈ Hom_{F}(P, S), Ω_{∗}([g]) = Ω_{∗}([φ(g)]) by condition (ii). Hence 𝔣𝔬𝔠(F) ≤ Ker(Ω_{∗}), and Ω_{∗} factors through a homomorphism trf_{F} from S/𝔣𝔬𝔠(F) to S^{ab}.

Now assume [g] ∈ Ker(Ω_{∗}). Then

1 = Ω_{∗}([g]) = ∏_{i=1}^{k} φ_{i}_{∗}(trf_{P_{i}}^{S}([g])) ≡ ∏_{i=1}^{k} incl_{P_{i}}^{S}(trf_{P_{i}}^{S}([g])) = [g^{|Ω|/|S|}] (mod 𝔣𝔬𝔠(F)) .

Hence g^{|Ω|/|S|} ∈ 𝔣𝔬𝔠(F), and g ∈ 𝔣𝔬𝔠(F) since |Ω|/|S| is prime to p by (iii). Thus Ker(Ω_{∗}) = 𝔣𝔬𝔠(F), and trf_{F} is injective.

When P = S and α, β ∈ Aut_{F}(S), then two components S_{S,α} (≅ (S_{S,α})×_{S} S) and S_{S,β} are isomorphic as (S, S)-bisets if and only if [α] = [β] ∈ Out_{F}(S). Hence by (ii), the terms S_{S,α} occur equally often in Ω for each [α] ∈ Out_{F}(S); and by (iii), the number r of occurrences of each such term is prime to p. Thus in the formula for trf_{F} (or for Ω_{∗}), terms with (P_{i}, φ_{i}) = (S, α) occur equally often for each [α] ∈ Out_{F}(S). So after multiplying by an appropriate constant, we get the formula stated in the proposition.

If Q ≤ S has index p, then for all g ∈ S, trf_{Q}^{S}([g]) = [g^{p}] if g ∉ Q, and trf_{Q}^{S}([g]) = ∏_{i=1}^{p}[x_{i}gx_{i}^{−1}] if g ∈ Q and S/Q = {x_{1}Q, …, x_{p}Q}. This follows immediately from the formula for the transfer described above. (When g ∉ Q, take as coset representatives {1, g, g^{2}, …, g^{p−1}}.) In particular, if g ∈ Ω_{1}(Z(S)), then trf_{Q}^{S}([g]) = 0 for each Q ≤ S of index p, and hence for each Q ≨ S. So for such g, trf_{F}([g]) = ∏_{[α]∈Out_{F} (S)}[α(g)], and this proves the last statement in the proposition. □

2. Lemmas on groups and groups acting on groups

We collect here the results which will be needed later on finite groups and their actions on other finite groups. The first two results are well known, and listed for future reference.

Lemma 2.1. Fix a prime p, a 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]. □

Lemma 2.2. If G is a finite group of order 2n, where n is odd, then G contains a normal subgroup of order n and index two.

Proof. This is a special case of the focal subgroup theorem (cf. [G, Theorem 7.3.4]). The following, more elementary proof was shown to us by Dave Benson. Consider the action of G on itself by left translation. Elements of even order act via odd permutations, and elements of odd order via even permutations. Hence the elements of odd order form a subgroup of index two. □

Let I(G) denote the set of involutions in a group G.

Lemma 2.3. Fix a finite group G of order 2n for n odd. Then for each H ≤ G of odd index, |I(G)|/|I(H)| is an odd integer.

Proof. Fix H ≤ G of odd index, and let G_{0} ⊴ G and H_{0} = H ∩ G_{0} ⊴ H be the subgroups of odd order and index two (Lemma 2.2). We assume inductively the lemma holds for all such groups of order strictly less than |G|.

Choose x ∈ I(H) ⊆ I(G). Let K_{0} ≨ G_{0} be maximal among all proper subgroups of G_{0} normal in G, and set K = ⟨K_{0}, x⟩. Then K_{0} ≥ [G_{0}, G_{0}], since G_{0} is solvable by the odd order theorem [FT], so K_{0}H_{0} ⊴ G, and either H_{0} ≤ K_{0} or H_{0}K_{0} = G_{0} by the maximality of K_{0}.

If H_{0}K_{0} = G_{0}, then for each g ∈ C_{G_{0}}(x), g = hk where h ∈ H_{0} and k ∈ K_{0}, and xhx^{−1} = ha, xkx^{−1} = a^{−1}k for some a ∈ H_{0} ∩ K_{0}. Since x^{2} = 1, xax^{−1} = a^{−1}. Since |a| is odd, there is b ∈ ⟨a⟩ such that b^{2} = a (and xbx^{−1} = b^{−1}). Then hb ∈ C_{H_{0}}(x), b^{−1}k ∈ C_{K_{0}}(x); and this proves that

C_{G_{0}}(x) = C_{H_{0}}(x)·C_{K_{0}}(x) .

Hence

\frac{|I(G)|}{|I(H)|} = \frac{[G:C_{G}(x)]}{[H:C_{H}(x)]} = \frac{[G:H]}{[C_{G}(x):C_{H}(x)]} = \frac{[K:K ∩ H]}{[C_{K}(x):C_{K∩H}(x)]} = \frac{|I(K)|}{|I(K ∩ H)|} .

Thus \frac{|I(G)|}{|I(H)|} is an odd integer by the induction hypothesis.

If K_{0} ≥ H_{0}, then

\frac{|I(G)|}{|I(H)|} = \frac{|I(G)|}{|I(K)|}·\frac{|I(K)|}{|I(H)|} = \frac{|G|}{|C_{G}(x)·K|}·\frac{|I(K)|}{|I(H)|} ,

where \frac{|G|}{|C_{G}(x)·K|} is an odd integer since C_{G}(x)·K is a subgroup of G, and \frac{|I(K)|}{|I(H)|} is an odd

integer by the induction hypothesis. So \frac{|I(G)|}{|I(H)|} is also an odd integer. □

Lemma 2.4. Fix a finite group G of order 2n for n odd, and set G_{0} = O^{2}(G). Fix x ∈ I(G), and assume [x, G] ≠ 1. Let V be an 𝔽_{2}[G]-module such that either

(a) [x, G_{0}] = ⟨I(G)⟩ ∩ G_{0} acts on V with C_{V} ([x, G_{0}]) = 0; or

(b) there is no 𝔽_{2}[G_{0}]-irreducible submodule of V which is pointwise fixed by x.

Then rk(V) ≤ 3·rk([x, V]).

Proof. Let G_{0} ⊴ G be the normal subgroup of odd order and index two (Lemma 2.2). Set Γ = ⟨I(G)⟩, and Γ_{0} = Γ ∩ G_{0} = [x, G_{0}] ⊴ G. If C_{V} (Γ_{0}) ≠ 0, then C_{V} (Γ) ≠ 0, and this is a sum of 𝔽_{2}[G_{0}]-irreducible submodules on which x acts trivially. Thus (b) implies (a), and we assume (a) from now on.

It suffices to prove the lemma when G = Γ; equivalently, when G_{0} = [x, G_{0}]. Let 𝔽_{2} be the algebraic closure of 𝔽_{2}, and set V = 𝔽_{2}⊗_{𝔽_{2}}V. It suffices to prove that dim(V) ≤ 3· dim([x, V]) whenever V is 𝔽_{2}[G]-irreducible with nontrivial action. Via induction on |G|, we can also assume G acts faithfully on V.

Since G is solvable by the odd order theorem [FT], there is a nontrivial normal abelian subgroup 1 ≠ N ⊴ G of odd order. Assume N is maximal among such subgroups (thus N = G_{0} if G_{0} is abelian). Set N^{∗} = Hom(N, 𝔽_{2}^{×}), the set of irreducible characters of N, and consider the decomposition V = ⨁_{χ∈N^{∗}} V_{χ} as a sum of eigenspaces for N. Set Ω = {χ ∈ N^{∗} | V_{χ} ≠ 0}. Since N is normal, the action of G on V permutes the V_{χ}. Since V is irreducible, the action of G/N on N^{∗} permutes Ω transitively, and the V_{χ} (for χ ∈ Ω) all have the same dimension.

Set d = dim(V_{χ}) for χ ∈ Ω, set n = |Ω|, and let k be the number of pairs of elements of Ω exchanged under the action of x. Then dim(V) = nd, and dim([x, V]) ≥ kd. So it will suffice to show n ≤ 3k. If G_{0} does not act transitively on Ω, then it acts via two orbits which are exchanged by x, so k = n/2, and we are done in this case.

Now assume G_{0} acts transitively on Ω. Thus n = |Ω| is odd. Choose ω ∈ Ω which is fixed by x, and let H ≤ G be its stabilizer subgroup. Thus x ∈ H. Then gω ∈ Fix(x, Ω) if and only if g^{−1}xg ∈ H. Since any y ∈ I(G) lies in H if and only if it is H-conjugate to x, this means that gω ∈ Fix(x, Ω) if and only if gh ∈ C_{G}(x) for some h ∈ H. Thus Fix(x, Ω) = C_{G}(x)·ω,

n − 2k = |Fix(x, Ω)| = \frac{|C_{G}(x)|}{|C_{H}(x)|} = \frac{|G|}{|H|}·\frac{|I(H)|}{|I(G)|} = n·\frac{|I(H)|}{|I(G)|} .

By Lemma 2.3, either n − 2k = |Fix(x, Ω)| ≤ \frac{1}{3}n, in which case n ≤ 3k; or |Fix(x, Ω)| = |Ω| so that k = 0. Hence we are done if k > 0.

Now assume k = 0. Thus x acts trivially on Ω. All of the other involutions in G act trivially on Ω (they are all conjugate to x), and so G acts trivially since it is generated by its involutions. Since G acts transitively, this means n = |Ω| = 1. Let χ ∈ Ω be the unique character. Since G acts faithfully on V, χ is injective, and hence N is cyclic. Since conjugation by G leaves invariant χ ∈ N^{∗}, N is central in G. By assumption, N is maximal among normal abelian subgroups of G contained in G_{0}.

Let Q/N ≠ 1 be a nontrivial normal subgroup of G/N which is an elementary abelian p-group for some odd prime p (recall G is solvable). Then N = Z(Q), since otherwise Z(Q) would be a larger normal abelian subgroup of G. So p||N|, and Q ≅ N ×_{C_{p}} H for some extraspecial group H of order p^{1+2k} and exponent p (some k ≥ 1). Let Q_{+}, Q_{−} ≤ Q be such that Q_{+}/N and Q_{−}/N are the (±1)-eigenspaces for the conjugation action of x. Then [Q_{+}, Q_{−}] = 1 since [x, N] = 1, and Q = Q_{+} ×_{N} Q_{−}. If Q_{−} = 1, then x centralizes N and Q/N, and hence centralizes Q by Lemma 2.1 (and since Q is the product of its Sylow subgroups). Since Q ⊴ G, this would imply Q is centralized by each involution in G, hence that [Q, G] = 1 since G is generated by its involutions. But this is impossible since Q is nonabelian, and we conclude Q_{−} ≠ 1.

Let B_{±} ≤ Q_{±} be maximal abelian subgroups, and set B = B_{+}B_{−}. The p^{k} characters of B which extend χ ∈ N^{∗} are permuted transitively by the action of Q/B ≅ C_{p}^{k}, and hence occur with equal multiplicity in the decomposition of V. Since only p^{ℓ} of these characters are fixed by the action of x, where p^{ℓ} = |B_{+}/N| = \sqrt{|Q_{+}/N|} < p^{k}, this proves that in this case,

dim([x, V]) ≥ \frac{(p^{k} − p^{ℓ})/2}{p^{k}}· dim(V) ≥ \frac{1}{3}· dim(V).  □

The next lemma provides a first, easy application of Lemma 2.4. It can also be shown more directly.

Proposition 2.5. Fix an abelian 2-group A, and a subgroup G ≤ Aut(A) of order 2n for n odd. Assume, for some x ∈ I(G), that [x, G] ≠ 1, and that [A, x] ≅ C_{2^{m}} for some m ≥ 1. Then there are unique decompositions A = A_{1} × A_{2} and G = G_{1} × G_{2}, such that the Gaction on A splits as a product of G_{i}-actions on A_{i}; and such that G_{1} ≅ Σ_{3}, |G_{2}| is odd, and A_{1} ≅ (C_{2^{m}})^{2}.

Proof. Set Γ = ⟨I(G)⟩ ⊴ G, G_{0} = O^{2}(G), and Γ_{0} = Γ ∩ G_{0}. Thus [G:G_{0}] = 2 and [Γ:Γ_{0}] = 2 by Lemma 2.2. Fix x ∈ I(G). Then Γ_{0} = [x, G] ≠ 1. Set A_{1} = [Γ_{0}, A] and A_{2} = C_{A}(Γ_{0}), so that A = A_{1} × A_{2}. Let H ⊴ G be the kernel of the G-action on A_{1}. If there is any decomposition as described above, then A_{1} and A_{2} are as just defined, G_{1} = Γ, and G_{2} = H. So there is at most one such decomposition.

If x acts trivially on A_{1}, then all involutions in G act trivially, so Γ = ⟨I(G)⟩ acts trivially, contradicting the definition of A_{1}. Thus [x, A_{1}] ≠ 1, and so x acts trivially on A_{2} since rk([x, A]) = 1. Also, by Lemma 2.4 (applied to the action of Γ on A_{1}/Fr(A_{1})), rk(A_{1}) ≤ 3. Since Γ acts faithfully on A_{1}/Fr(A_{1}) (Lemma 2.1), it is isomorphic to a subgroup of GL_{3}(2). Since |GL_{3}(2)| = 8·21, |Γ||42. The normalizer of any Q ∈ Syl_{7}(GL_{3}(2)) has order 21, and so |Γ| ≠ 42.

For each x ≠ y ∈ I(G), ⟨x, y⟩ ≅ D_{2k} for some odd k, and since this is a subgroup of GL_{3}(2), we have k = 3. Thus |Γ| = 6, and Γ ≅ Σ_{3}. Also, rk(A_{1}) = 2 (A_{1} = [Γ_{0}, A] is even dimensional since |Γ_{0}| = 3). Since Γ_{0} ≅ C_{3} acts nontrivially on A_{1}, it must be homocyclic, and thus A_{1} ≅ C_{2^{m}} × C_{2^{m}}. Since no larger group can act on A_{1}, G = HΓ ≅ H × Γ, and the action splits as described. □

Throughout the rest of the section, we look at some conditions which imply that two subgroups of a larger group commute. The first one is very simple.

Lemma 2.6. Fix a prime p, a finite group G, and a pair of subgroups G_{1}, G_{2} ≤ G. Choose S_{i} ∈ Syl_{p}(G_{i}), and choose normal p-subgroups P_{i} ⊴ G_{i}. Assume [G_{1}, S_{2}] = 1 = [G_{2}, S_{1}] and [G_{1}, G_{2}] ≤ P_{1}P_{2}. Then [G_{1}, G_{2}] = 1.

Proof. We must show, for each pair of elements g_{i} ∈ G_{i} of order prime to p, that [g_{1}, g_{2}] = 1. Upon replacing G_{i} by ⟨P_{i}, g_{i}⟩, we are reduced to the case where G_{i}/P_{i} has order prime to p.

Consider the conjugation action c_{g_{1}} ∈ Aut(P_{1}G_{2}). By assumption, c_{g_{1}}(P_{1}) = P_{1}, and c_{g_{1}} induces the identity on P_{2} and on P_{1}G_{2}/P_{1}P_{2}. Since P_{1}G_{2}/P_{2} ≅ (P_{1}P_{2}/P_{2}) × G_{2}/P_{2}, a product of a p-group and a group of order prime to p, the induced action of c_{g_{1}} on this quotient induces the identity on G_{2}/P_{2}. Hence [g_{1}, G_{2}] ≤ P_{2}.

Thus c_{g_{1}} ∈ Aut(G_{2}) induces the identity on P_{2} and on G_{2}/P_{2}. By the Schur-Zassenhaus theorem (cf. [G, Theorem 6.2.1(ii)]), c_{g_{1}} is conjugation by some element of P_{2}, and hence is the identity since P_{2} is a p-group and g_{1} has order prime to p. Thus [g_{1}, G_{2}] = 1. □

We next look at conditions for two groups of automorphisms of another group to commute. As usual, when a group G acts on a group K, we set [G, K] = ⟨g(x)g^{−1} | g ∈ G, x ∈ K⟩. This is a normal, G-invariant subgroup of K (cf. [G, Theorem 2.2.1(iii)]), and the smallest one such that G acts trivially on K/[G, K].

Lemma 2.7. Fix a finite group K, and subgroups G_{1}, G_{2} ≤ Aut(K) such that [G_{1}, K] ∩ [G_{2}, K] = 1. Then [G_{1}, G_{2}] = 1 if and only if [G_{1}, K] ≤ C_{K}(G_{2}) and [G_{2}, K] ≤ C_{K}(G_{1}).

Proof. Set [g_{i}, x] = g_{i}(x)·x^{−1} for g_{i} ∈ G_{i} and x ∈ K. Then

[g_{1}, [g_{2}, x]] = g_{1}g_{2}(x)·g_{1}(x)^{−1}·[g_{2}, x]^{−1} = g_{1}g_{2}(x)·x^{−1}·[g_{1}, x]^{−1}·[g_{2}, x]^{−1}
[g_{2}, [g_{1}, x]] = g_{2}g_{1}(x)·g_{2}(x)^{−1}·[g_{1}, x]^{−1} = g_{2}g_{1}(x)·x^{−1}·[g_{2}, x]^{−1}·[g_{1}, x]^{−1},

and [[g_{1}, x], [g_{2}, x]] = 1 since [G_{1}, K] and [G_{2}, K] are normal in K with trivial intersection. If [G_{1}, K] ≤ C_{K}(G_{2}) and [G_{2}, K] ≤ C_{K}(G_{1}), then [g_{1}, [g_{2}, x]] = [g_{2}, [g_{1}, x]] = 1, and hence g_{1}g_{2}(x) = g_{2}g_{1}(x). Conversely, if [G_{1}, G_{2}] = 1, then [g_{1}, [g_{2}, x]] = [g_{2}, [g_{1}, x]] lies in [G_{1}, K] ∩ [G_{2}, K] = 1. Thus [G_{1}, G_{2}] = 1 if and only if [G_{i}, K] ≤ C_{K}(G_{3−i}) for i = 1, 2. □

The following is a first application of Lemma 2.7.

Lemma 2.8. Fix an elementary abelian 2-group V, and subgroups G_{1}, G_{2} ≤ Aut(V) such that [G_{1}, V] ∩ [G_{2}, V] = 1. Assume, for each i = 1, 2, that G_{i} contains a subgroup H_{i} ⊴ G_{i} of odd order and of index two; that G_{i} = ⟨I(G_{i})⟩; and that for some choice of x_{i} ∈ I(G_{i}), [x_{i}, G_{3−i}] = 1. Assume also ⟨H_{1}, H_{2}⟩ has odd order. Then [G_{1}, G_{2}] = 1.

Proof. Assume H_{1} ≠ 1 and H_{2} ≠ 1 (otherwise there is nothing to prove). Set H = ⟨H_{1}, H_{2}⟩ and G = ⟨G_{1}, G_{2}⟩. By assumption, H has odd order, x_{i} normalizes H_{i} and centralizes G_{3−i}, [x_{i}, H_{i}] = H_{i} ≠ 1 (since G_{i} = ⟨I(G_{i})⟩), and G = ⟨H, x_{1}, x_{2}⟩. Hence H ⊴ G, x_{1} ≠ x_{2}, ⟨x_{1}, x_{2}⟩ ∈ Syl_{2}(G), and G/H ≅ C_{2}^{2}.

Set V_{i} = [H_{i}, V]; thus V_{1} ∩ V_{2} = 0 by assumption. Also, V_{i} is x_{i}-invariant, and V_{i} ≤ C_{V} (x_{3−i}) by Lemma 2.7 and since [x_{3−i}, H_{i}] = 1. For any H_{i}-invariant submodule M ≤ V, if x_{i} acts trivially on M, then all involutions in G_{i} act trivially on M, and so G_{i} = ⟨I(G_{i})⟩ acts trivially on M. Since V_{i} is generated by the irreducible 𝔽_{2}[H_{i}]-submodules with nontrivial H_{i}-action, this proves that there is no irreducible 𝔽_{2}[H_{i}]-submodule in V_{i} on which x_{i} acts trivially.

For i = 1, 2, set B_{i} = H·V_{i}: the smallest 𝔽_{2}[H]-submodule of V which contains V_{i}. Since V_{i} is ⟨x_{1}, x_{2}⟩-invariant, so is B_{i}, and hence B_{i} is G-invariant. Set D = B_{1} ∩ B_{2}. Thus B_{i} is generated by the irreducible 𝔽_{2}[H]-submodules of V on which H_{i} acts nontrivially, and D is generated by those on which H_{1} and H_{2} both act nontrivially.

Assume D ≠ 0. Since there is no irreducible 𝔽_{2}[H_{1}]-submodule in V_{1} on which x_{1} acts trivially, there is no irreducible 𝔽_{2}[H]-submodule in B_{1}, and in particular none in D, on which x_{1} acts trivially. Similarly, there is no irreducible 𝔽_{2}[H]-submodule in D on which x_{2} acts trivially. So by Lemma 2.4 (applied to the action of ⟨H, x_{i}⟩ on D), rk(D) ≤ 3·rk([x_{i}, D]) for i = 1, 2.

Since (x_{i} − Id_{D}) ∈ End(D) has kernel C_{D}(x_{i}) and image [x_{i}, D], rk([x_{i}, D]) = rk(D) − rk(C_{D}(x_{i})). Since [x_{1}, D] ∩ [x_{2}, D] = 0 by assumption, [x_{1}, D] + [x_{2}, D] ≤ C_{D}(⟨x_{1}, x_{2}⟩) by Lemma 2.7, applied to the actions of ⟨x_{1}⟩ and ⟨x_{2}⟩. Since rk([x_{i}, D]) ≥ \frac{1}{3}·rk(D), we have

rk([x_{1}, D]) = rk([x_{2}, D]) = \frac{1}{3}·rk(D) and C_{D}(x_{1}) = C_{D}(x_{2}) = [x_{1}, D] + [x_{2}, D].

But this is impossible, since x_{1} acts nontrivally on [H_{1}, D] ≠ 0, while x_{2} acts trivially by Lemma 2.7 again (recall [x_{2}, H_{1}] = 1).

Thus D = 0. Hence H_{2} acts trivially on B_{1} ≥ [H_{1}, V] and H_{1} on B_{2} ≥ [H_{2}, V]. Thus [H_{1}, H_{2}] = 1 by Lemma 2.7. Since [x_{1}, G_{2}] = [x_{2}, G_{1}] = 1, we conclude that [G_{1}, G_{2}] = 1. □

The last proposition is much more technical in its formulation. It will be applied later to the case where G = Aut_{F}(P) for a fusion system F, P = P_{1} ×P_{2}, and G_{i} ≤ G are subgroups which leave P_{i} invariant. The idea is to first get information about the outer automorphism groups commuting (e.g., G_{i}/O_{2}(G_{i})), using Lemma 2.8, and then lift that to information about the actual automorphism groups using Lemma 2.6.

Proposition 2.9. Fix a 2-group P and a subgroup G ≤ Aut(P). Let G_{1}, G_{2} ≤ G be such that [G_{1}, P] ∩ [G_{2}, P] = 1 and [O_{2}(G_{i}), P] ≤ Fr(P_{i}). Fix S_{i} ∈ Syl_{2}(G_{i}), and set Q_{i} = O_{2}(G_{i}). Assume the following hold for i = 1, 2.

(a) G_{i}/Q_{i} has a strongly 2-embedded subgroup H_{i}/Q_{i} ≥ S_{i}/Q_{i}, and [H_{i}, G_{3−i}] = 1.

(b) S_{1}S_{2} ∈ Syl_{2}(G), and S_{i} is strongly closed in S_{1}S_{2} with respect to G.

Then [G_{1}, G_{2}] = 1.

Proof. It suffices to do this when G = ⟨G_{1}, G_{2}⟩. Recall that [G_{i}, P] ⊴ P for each i. Since each subgroup which contains [G_{i}, P] is G_{i}-invariant, we can assume P = [G_{1}, P]·[G_{2}, P]. By Lemma 2.6, it suffices to prove that [G_{1}, G_{2}] ≤ Q_{1}Q_{2}. Since Q_{1} and Q_{2} act trivially on P/Fr(P) (and commute with each other), it suffices, upon replacing G by N_{G}(Q_{1}Q_{2})/Q_{1}Q_{2} and P by P/Fr(P), to prove the lemma when Q_{i} = 1, H_{i} is strongly 2-embedded in G_{i}, and P is elementary abelian.

Let Ĝ_{i} be the normal closure of S_{i} in G, and set N = Ĝ_{1} ∩ Ĝ_{2} ≥ [Ĝ_{1}, Ĝ_{2}]. By a theorem of Goldschmidt [Gd, Corollary A1], N has odd order.

Fix any pair of elements g_{i} ∈ G_{i}∖H_{i}. Choose involutions x_{i} ∈ I(S_{i}), and set y_{i} = g_{i}x_{i}g_{i}^{−1}. Then y_{i} ∉ H_{i}, and |x_{i}y_{i}| is odd since otherwise the involution in ⟨x_{i}y_{i}⟩ would commute with both x_{i} and y_{i} (contradicting the assumption H_{i} is strongly 2-embedded). Thus y_{i} = h_{i}x_{i}h_{i}^{−1} for some h_{i} ∈ ⟨x_{i}y_{i}⟩, [g_{i}^{−1}h_{i}, x_{i}] = 1, and g_{i}^{−1}h_{i} ∈ H_{i} since it is strongly 2-embedded. So g_{i} ∈ ⟨H_{i}, x_{i}y_{i}⟩.

Now, [⟨x_{1}, y_{1}⟩, ⟨x_{2}, y_{2}⟩] ≤ [Ĝ_{1}, Ĝ_{2}] ≤ N, and hence ⟨x_{1}y_{1}, x_{2}y_{2}⟩ has odd order. So by Lemma 2.8, applied to the actions of ⟨x_{1}, y_{1}⟩ and ⟨x_{2}, y_{2}⟩ on P, x_{1}y_{1} commutes with x_{2}y_{2} in G. Since [H_{i}, G_{3−i}] = 1 by assumption, this implies [g_{1}, g_{2}] = 1; and hence (since the g_{i} ∈ G_{i}∖H_{i} were arbitrary) that [G_{1}, G_{2}] = 1. □

3. Automorphisms of products of nonabelian p-groups

Throughout this section, p is an arbitrary prime. We look at automorphisms of p-groups which factor as products. For example, we prove in Proposition 3.2(b) that if S = (S_{1})^{n} × T is a p-group, where S_{1} is nonabelian, Z(S_{1}) is cyclic, and T does not contain a direct factor isomorphic to S_{1}, then either

Aut(S)/O_{p}(Aut(S)) ≅ (Aut(S_{1})/O_{p}(Aut(S_{1}))) ≀ Σ_{n} × (Aut(T)/O_{p}(Aut(T))) ,

or Aut(S_{1}) is a p-group and Aut(S)/O_{p}(Aut(S)) ≅ (Σ_{n}/O_{p}(Σ_{n})) × (Aut(T)/O_{p}(Aut(T))). In fact, we show this with a much weaker condition on S_{1} (condition (i′) in Proposition 3.2).

To shorten notation, for any prime p and any finite p-group P, we write

Out(P) = Aut(P)/O_{p}(Aut(P)) ≅ Out(P)/O_{p}(Out(P)).

Proposition 3.1. Fix a pair of p-groups S_{1} and S_{2}, set S = S_{1}×S_{2}, and let pr_{i} ∈ Hom(S, S_{i}) be the projection. Set

Aut^{0}(S) = {α ∈ Aut(S) | α(Ω_{1}(Z(S_{1}))) = Ω_{1}(Z(S_{1}))} .

Then for each α ∈ Aut^{0}(S), pr_{i}(α(S_{i})) = S_{i} and α(S_{i}Z(S_{3−i})) = S_{i}Z(S_{3−i}) for i = 1, 2. Also, there is an isomorphism

Aut^{0}(S)/O_{p}(Aut^{0}(S)) ⟶^{≅} Out(S_{1}) × Out(S_{2}) ,

which sends the class of α ∈ Aut^{0}(S) to the class of (pr_{1} ∘α|_{S_{1}}, pr_{2} ∘α|_{S_{2}}).

Proof. Set Z_{i} = Z(S_{i}) for short. Fix α ∈ Aut^{0}(S). If pr_{1} ∘α|_{S_{1}} were not injective, then some 1 ≠ x ∈ Ω_{1}(Z_{1}) would be in the kernel, which contradicts the assumption α(Ω_{1}(Z_{1})) = Ω_{1}(Z_{1}). If pr_{2} ∘α|_{S_{2}} were not injective, then some 1 ≠ y ∈ Ω_{1}(Z_{2}) would be in the kernel, so α(y) ∈ S_{1} ∩ Ω_{1}(Z(S)) = Ω_{1}(Z_{1}), which is impossible since α(Ω_{1}(Z_{1})) = Ω_{1}(Z_{1}). Thus pr_{1} ∘α|_{S_{1}} and pr_{2} ∘α|_{S_{2}} are both automorphisms. Furthermore, for i = 1, 2,

α(S_{i}Z_{3−i}) = C_{S}(α(S_{3−i})) = C_{S_{i}}(pr_{i}(α(S_{3−i})))·C_{S_{3−i}}(S_{3−i}) ≤ S_{i}·Z_{3−i} ,

and hence α(S_{i}Z_{3−i}) = S_{i}Z_{3−i}.

It remains to prove the last statement. Define Aut^{1}(S) ⊴ Aut^{0}(S) by setting

Aut^{1}(S) = {α ∈ Aut^{0}(S) | α induces the identity on

Ω_{1}(Z_{1}), on Ω_{1}(Z(S))/Ω_{1}(Z_{1}), and on S/Z(S)} .

Since each element of Aut^{0}(S) leaves the subgroups Ω_{1}(Z_{1}), Ω_{1}(Z(S)), and Z(S) invariant, Aut^{1}(S) ⊴ Aut^{0}(S). For each α ∈ Aut^{1}(S), α|_{Ω_{1}(Z(S))} has p-power order by Lemma 2.1, so α|_{Z(S)} has p-power order by [G, Theorem 5.2.4], and α has p-power order by Lemma 2.1 again. Thus Aut^{1}(S) is a p-group, and hence is contained in O_{p}(Aut^{0}(S)).

Consider the following maps:

Aut^{0}(S) ⟶^{χ} Aut(S_{1}) × Aut(S_{2}) ⟶^{ψ} Aut^{0}(S) ⟶^{ρ} Aut^{0}(S)/O_{p}(Aut^{0}(S)) ,

where χ(α) = (pr_{1} ∘α|_{S_{1}}, pr_{2} ∘α|_{S_{2}}) (as a map of sets), ψ(α_{1}, α_{2}) = α_{1} × α_{2}, and ρ is the projection. Thus χ ∘ ψ is the identity on Aut(S_{1}) × Aut(S_{2}). Fix α ∈ Aut^{0}(S), and set α̂ = α ∘ (ψ ∘ χ(α))^{−1}. Then α̂|_{Ω_{1}(Z_{1})} = Id, since α(Ω_{1}(Z_{1})) = Ω_{1}(Z_{1}). Also, α̂ ≡ Id (mod Z(S)), since for g ∈ S_{i}, α(g) ∈ S_{i}Z_{3−i} and hence pr_{i}(α(g)) ≡ α(g) (mod Z_{3−i}). By a similar argument, α̂|_{S_{2}} ≡ Id (mod Z_{1}). This proves that α ≡ ψ(χ(α)) (mod Aut^{1}(S) ≤ O_{p}(Aut^{0}(S))) for each α ∈ Aut^{0}(S), and hence that ρ ∘ ψ ∘ χ = ρ.

Thus ρ ∘ ψ is surjective, and Ker(ρ ∘ ψ) is a p-group since ψ is injective. So ψ induces an isomorphism ψ̂ from Out(S_{1})×Out(S_{2}) onto Aut^{0}(S)/O_{p}(Aut^{0}(S)), and ψ̂^{−1} sends the class of each α ∈ Aut^{0}(S) to the class of χ(α). □

In the next proposition, we look more closely at the relationship between Aut(S) and Aut^{0}(S). The only part which will be needed later in this paper is point (b), but it seems natural to prove the full proposition here.

Proposition 3.2. Fix a pair of p-groups S_{1} and S_{2}, and set S = S_{1} × S_{2}. Assume either

(i) S_{1} is nonabelian and Z(S_{1}) is cyclic; or more generally

(i′) S_{1} ≠ 1, Ω_{1}(Z(S_{1})) ≤ [S_{1}, S_{1}], and for each φ ∈ Hom(S_{1}, S_{1}) such that φ(S_{1}) ⊴ S_{1} and φ is not onto, Ker(φ) ≥ Ω_{1}(Z(S_{1})).

Define

Aut^{0}(S) = {α ∈ Aut(S) | α(Ω_{1}(Z(S_{1}))) = Ω_{1}(Z(S_{1}))}.

Let pr_{i} ∈ Hom(S, S_{i}) be the projection. Then the following hold.

(a) [Aut(S):Aut^{0}(S)] = n, where n ≥ 1 is the largest integer such that S_{2} contains a direct factor isomorphic to (S_{1})^{n−1}.

(b) Fix an isomorphism S ≅ (S_{1})^{n} × T, where n is as in (a). Then the composite

Aut(S_{1}) ≀ Σ_{n} × Aut(T) ⟶^{incl} Aut(S) −↠ Out(S)  (1)

of the natural inclusion followed by the projection is surjective, and its kernel is a pgroup. Thus Out(S) ≅ Out(S_{1}) ≀ Σ_{n} × Out(T) if Aut(S_{1}) is not a p-group.

Proof. Set Z_{i} = Z(S_{i}) for i = 1, 2. Condition (i) implies (i′), since if Z_{1} is cyclic, then each nontrivial normal subgroup of S_{1} contains Ω_{1}(Z_{1}). So from now on, we assume (i′).

We claim that

α ∉ Aut^{0}(S) ⟹ pr_{2} ∘α|_{S_{1}} is injective and α(S_{1}) ≤ Z_{1}S_{2}.  (2)

To see this, fix α ∉ Aut^{0}(S). If pr_{1}(α(S_{1})) = S_{1}, then

Z_{1}S_{2} = C_{S}(S_{1}) ≅ C_{S}(α(S_{1})) ≤ Z_{1}S_{2} ,

hence C_{S}(α(S_{1})) = Z_{1}S_{2}, and α(S_{1}) ≤ S_{1}Z_{2}. So α([S_{1}, S_{1}]) ≤ [S_{1}, S_{1}], and α(Ω_{1}(Z_{1})) ≤ Ω_{1}(Z(S)) ∩ [S_{1}, S_{1}] = Ω_{1}(Z_{1}), contradicting the assumption that α ∉ Aut^{0}(S). Thus pr_{1}(α(S_{1})) is a proper normal subgroup of S_{1}, and by (i′), Ker(pr_{1} ∘α|_{S_{1}}) ≥ Ω_{1}(Z_{1}). Hence pr_{2} ∘α|_{Ω_{1}(Z_{1})} is injective, and so pr_{2} ∘α|_{S_{1}} is injective.

Set T = pr_{2}(α(S_{1})) ⊴ S_{2}. Then C_{S_{2}}(T) ∩ T = Z(T), so

|Z_{1}|·|S_{2}| = |C_{S}(S_{1})| = |C_{S}(α(S_{1}))| = |C_{S_{1}}(pr_{1}(α(S_{1})))|·|C_{S_{2}}(T)| ≤ |S_{1}|·\frac{|S_{2}|}{[T:Z(T)]} .

Since T ≅ S_{1} and Z(T) ≅ Z_{1}, the first and last terms are equal. Hence C_{S_{1}}(pr_{1}(α(S_{1}))) = S_{1} and so α(S_{1}) ≤ Z_{1}S_{2}. This finishes the proof of (2).

(a) If S_{2} ≅ (S_{1})^{n−1} ×T for some subgroup T, then clearly [Aut(S):Aut^{0}(S)] ≥ n. It remains to prove the converse.

Assume Aut^{0}(S) has index n in Aut(S) where n ≥ 2. Let α_{1} = Id_{S}, α_{2}, …, α_{n} be representatives in Aut(S) for the n left cosets of Aut^{0}(S). For each 1 ≤ i, j ≤ n, set α_{ij} = α_{i}^{−1}α_{j} (thus α_{j} = α_{1j}). Hence for i ≠ j, α_{ij} ∉ Aut^{0}(S), and by (2), pr_{2} ∘α_{ij}|_{S_{1}} is injective and α_{ij}(S_{1}) ≤ Z_{1}S_{2}.

For i = 2, …, n, set T_{i} = pr_{2}(α_{i}(S_{1})). Also, set T_{1} = S_{1}. Thus for each i, T_{i} ≅ S_{1} and T_{i} ≤ α_{i}(S_{1})Z(S). For each pair 1 ≤ i < j ≤ n, [S_{1}, α_{ij}(S_{1})] = 1 since α_{ij}(S_{1}) ≤ Z_{1}S_{2}. So [α_{i}(S_{1}), α_{j}(S_{1})] = 1 (since α_{j} = α_{i} ∘ α_{ij}), and hence [T_{i}, T_{j}] = 1.

Set

T = S_{2} ∩ α_{2}(S_{2}) ∩ ⋯ ∩ α_{n}(S_{2}) .

For each 2 ≤ i ≤ n, [T_{i}, T] = [α_{i}(S_{1})Z(S), T] ≤ [α_{i}(S_{1}), α_{i}(S_{2})] = 1. Also, since [S:α_{i}(S_{2})] = |S_{1}| for each i, [S_{2}:T] ≤ |S_{1}|^{n−1}.

Set V = Ω_{1}(Z(S)), regarded as an 𝔽_{p}-vector space, and set V_{i} = Ω_{1}(Z(T_{i})) ≤ V and W = Ω_{1}(Z(T)) ≤ V. For i ≥ 2, α_{i}(S_{1}) ≤ Z_{1}S_{2} and V_{1} = Ω_{1}(Z_{1}) ≤ [S_{1}, S_{1}] imply α_{i}(V_{1}) ≤ [S_{2}, S_{2}], and hence α_{i}(V_{1}) = V_{i}. Since V_{1} is Aut^{0}(S)-invariant, this shows that Aut(S) permutes transitively the set {V_{1}, …, V_{n}}.

Assume x_{i} ∈ V_{i} are such that x^{∗} = ∏_{i=1}^{n} x_{i} ∈ W. For each j = 1, …, n, α_{j}^{−1}(x^{∗}) ∈ α_{j}^{−1}(W) ≤ S_{2}. Also, α_{j}^{−1}(x_{j}) ∈ V_{1} ≤ S_{1}, and for i ≠ j, α_{j}^{−1}(x_{i}) ∈ V_{k} ≤ S_{2} for some 2 ≤ k ≤ n. So x_{j} = 1 for each j. This shows that W and the V_{i} are 𝔽_{p}-linearly independent in V.

Since T and the T_{i} commute pairwise, there is a homomorphism

Φ: (S_{1})^{n−1} × T ⟶ S_{2} ,

defined by setting Φ(g_{1}, …, g_{n−1}, h) = (pr_{2}(α_{2}(g_{1})), …, pr_{2}(α_{n}(g_{n−1})), h). Since the V_{i} and W are independent in V, no central element of order p lies in Ker(Φ), and hence Φ is injective. Since |T | ≥ |S_{2}|/|S_{1}|^{n−1}, Φ is an isomorphism, and thus S_{2} ≅ (S_{1})^{n−1} × T.

(b) This is proven by induction on n (and clearly holds when n = 0). So assume it holds for S_{2} ≅ (S_{1})^{n−1} × T. By Proposition 3.1 and the induction hypothesis, the composite

Aut(S_{1}) × (Aut(S_{1}) ≀ Σ_{n−1}) × Aut(T) ⟶^{incl} Aut^{0}(S) −↠ Aut^{0}(S)/O_{p}(Aut^{0}(S)) (3)

is surjective with kernel a p-group. By (a), the first two groups in (3) have index n in (Aut(S_{1}) ≀ Σ_{n}) × Aut(T) and Aut(S), respectively. Since no element of Aut(S_{1}) ≀ Σ_{n} not in Aut(S_{1}) × (Aut(S_{1}) ≀ Σ_{n−1}) is sent into Aut^{0}(S), it follows that the composite in (1) is surjective with the same p-group kernel. □

The following is a more technical consequence of Proposition 3.1. It will be needed in the next section, when identifying potential essential subgroups.

Lemma 3.3. Fix a finite p-group S, and a subgroup P ≤ S. Assume that Z(P) = Z(S), or more generally, that Aut_{S}(P) acts trivially on Z(P). Assume also, for some p-group T and some saturated fusion system F over S ×T, that P T is F-essential. Then there is a subgroup Γ ≤ Out(P) such that Out_{S}(P) ∈ Syl_{p}(Γ) and Γ has a strongly p-embedded subgroup.

Proof. Set Γ^{∗} = O^{p′}(Out_{F}(P T)) and Γ̃^{∗} = O^{p′}(Aut_{F}(P T)), so that Γ̃^{∗}/Inn(P T) = Γ^{∗}. Since P T is F-essential, there is a strongly p-embedded subgroup H ≨ Out_{F}(P T) which contains Out_{ST} (P T). Also, Out_{F}(P T) = H·Γ^{∗} by the Frattini argument (since Γ^{∗} and Out_{F}(P T) have the same Sylow p-subgroups), and since H ≥ N_{Out_{F} (P T)}(Out_{ST} (P T)) (being strongly embedded). So H^{∗} =^{def} H ∩ Γ^{∗} ≨ Γ^{∗} is strongly p-embedded in Γ^{∗}.

Since Out_{ST} (P T) acts trivially on Z(P T) (and P T is fully normalized in F since it is F-essential), each Sylow p-subgroup of Γ^{∗} acts trivially. Thus Γ^{∗} acts trivially on Z(P T), since it is generated by its Sylow p-subgroups, and Γ̃^{∗} is contained in the group Aut^{0}(P T) of automorphisms which are Ω_{1}(Z(P))-invariant. Define

χ = (χ_{1}, χ_{2}): Γ̃^{∗} ≤ Aut^{0}(P T) ⟶ Out(P) × Out(T)

by setting χ_{1}(α) = [pr_{1} ∘α|_{P}] and χ_{2}(α) = [pr_{2} ∘α|_{S_{2}}]. By Proposition 3.1, χ is a surjective homomorphism, and Ker(χ) is a p-group.

The Sylow subgroup Aut_{ST} (P T) is contained in Ker(χ_{2}), and hence Γ̃^{∗} ≤ Ker(χ_{2}) since Γ̃^{∗} is generated by its Sylow 2-subgroups. Since Γ^{∗} has a strongly p-embedded subgroup, O_{p}(Γ^{∗}) = 1 (cf. [Sz2, Theorem 6.4.3]), and hence χ_{1} sends Γ^{∗} injectively into Out(P).

By Proposition 3.1 again, P Z(T) is Γ̃^{∗}-invariant. Let ψ be the homomorphism from Γ^{∗} ≤ Out(P T) to Out(P), defined by sending the class of α ∈ Γ̃^{∗} to the class of the automorphism of P Z(T)/Z(T) induced by α|_{P Z(T)}. This is equal (mod O_{p}(Aut(P))) to χ_{1}, and hence ψ is injective. Set Γ = ψ(Γ^{∗}) ≤ Out(P). Then Γ ≅ Γ^{∗} contains a strongly p-embedded subgroup since Γ^{∗} does, and ψ(Out_{ST} (P T)) = Out_{S}(P) ∈ Syl_{p}(Γ). □

4. Fusion systems over products of nonabelian 2-groups

When F_{1} and F_{2} are fusion systems over S_{1} and S_{2}, respectively, then F_{1} × F_{2} is the smallest fusion system over S_{1} × S_{2} which contains each morphism set Hom_{F_{1}}(P_{1}, Q_{1}) × Hom_{F_{2}}(P_{2}, Q_{2}) (for P_{i}, Q_{i} ≤ S_{i}), when regarded as a set of homomorphisms from P_{1} × P_{2} to Q_{1} × Q_{2}. More explicitly, if P, Q ≤ S, and P_{i}, Q_{i} ≤ S_{i} denote the images of P and Q under projection, then

Hom_{F_{1}×F_{2}}(P, Q) = {(φ_{1}, φ_{2})|_{P} | φ_{i} ∈ Hom_{F_{i}}(P_{i}, Q_{i})} .

If F_{1} and F_{2} are both saturated, then F_{1} × F_{2} is saturated (see [BLO, Lemma 1.5]).

We want to prove, under certain hypotheses, that a saturated fusion system over a product of 2-groups splits as a product of fusion systems. First, in Section 4.1, we prove Proposition 4.3, which gives some very general conditions where this happens. Then, in the next two subsections, we prove Theorems A, B, and C as special cases of Proposition 4.3. Afterwards, in Section 4.4, we give some examples to show why some of the hypotheses in those two theorems (and in the proposition) are needed.

4.1. A general splitting proposition.

By [AOV, Proposition 3.3], if F is a saturated fusion system over a p-group S = S_{1} × S_{2}, where S_{1} and S_{2} are strongly closed in F and O^{p′}(F) = F, then F = F_{1} × F_{2} for some pair of saturated fusion systems F_{i} over S_{i}. This played an important role when studying automorphisms of products of fusion systems in [AOV, §3], but unfortunately, it does not seem to be very helpful for our purposes here.

Our main result in this subsection, Proposition 4.3, can be regarded as a stronger version of [AOV, Proposition 3.3], in that we only assume S_{1} and S_{2} are strongly closed in certain fusion subsystems of F, but not in F itself. However, this is shown at the expense of adding additional hypotheses: that p = 2, and that O^{2}(F) = F.

The following definition will be needed when handling “fusion subsystems” which satisfy all of the conditions in the definition except the one which says that they contain the fusion system of the underlying p-group. The term “restrictive (sub)category” is taken from [BCGLO2], although it is used slightly differently here.

Definition 4.1. Let F be a fusion system over a p-group S. A restrictive subcategory of F is a subcategory E ⊆ F with the same objects, and with the property that for each P_{0} ≤ P ≤ S and Q_{0} ≤ Q ≤ S, and each φ ∈ Hom_{E}(P, Q) such that φ(P_{0}) ≤ Q_{0}, φ|_{P_{0}} ∈ Hom_{E}(P_{0}, Q_{0}), and φ^{−1}|_{Q_{0}} ∈ Hom_{E}(Q_{0}, P_{0}) if φ(P_{0}) = Q_{0}.

Thus when F is a fusion system over S, a restrictive subcategory E ⊆ F is a fusion subsystem if and only if Hom_{E}(S) ≥ Inn(S). We refer to the restrictive subcategory generated by a set of homomorphisms in the same way that we refer to the fusion subsystem generated by a set of homomorphisms.

Lemma 4.2. Fix a pair of p-groups S_{1} and S_{2}, and set S = S_{1} × S_{2}. Let F be a saturated fusion system over S. Set

F_{1}^{•} = ⟨Aut_{F}(P) | S_{2} ≤ P ≤ S, P F-essential or P = S⟩ ,

and assume S_{1} is strongly closed in F_{1}^{•}. Let E_{1}^{•} ⊆ F_{1}^{•} be the restrictive subcategory generated by the automorphism groups

Aut_{E_{1}^{•}}(P) = {α ∈ Aut_{F}(P) | [α, P] ≤ S_{1}}

for all F-centric subgroups P ≤ S with P ≥ S_{2}. Set F_{1} = F_{1}^{•}|_{S_{1}} and E_{1} = E_{1}^{•}|_{S_{1}}. Then the following hold.

(a) The fusion systems F_{1} and E_{1} are both saturated, and E_{1} is a normal fusion subsystem of index prime to p in F_{1}. Also, F_{1}^{•} = ⟨E_{1}^{•}, Aut_{F}(S)⟩.

(b) If S_{2} ≤ P ≤ S, then Hom_{F}(P, S) = Hom_{F_{1}^{•}}(P, S). If S_{2} ≤ P ≤ S, P is F-centric, and Q is F-conjugate to P, then Q ≥ S_{2}. Also, F_{1} = F|_{S_{1}}.

(c) For all P ≤ S_{1} such that P S_{2} is F-centric, • α ∈ Aut_{E_{1}^{•}}(P S_{2}) and α|_{P} = Id_{P} imply α = Id_{P S_{2}},

• [Aut_{E_{1}^{•}}(P S_{2}), P S_{2}] = [Aut_{E_{1}}(P), P], and

• [Aut_{E_{1}^{•}}(P S_{2}), Aut_{S_{2}}(P S_{2})] = 1.

(d) If P ≤ S is fully centralized in F and F-conjugate to a subgroup of S_{1}, then P ≤ S_{1}. For all P ≤ S_{1},

P fully norm. in F_{1} ⟺ P fully norm. in F ⟹ P S_{2} fully norm. in F P fully centr. in F_{1} ⟺ P fully centr. in F ⟹ P S_{2} fully centr. in F

Proof. For i = 1, 2, set Z_{i} = Z(S_{i}) for short, and let pr_{i} ∈ Hom(S, S_{i}) be the projection.

(b) Let T be the set of all subgroups of S which contain S_{2}, and let T^{c} ⊆ T be the subset of those which are F-centric. By Proposition 1.7(a), for all P ∈ T and all φ ∈ Hom_{F}(P, S), we can write φ = ψ_{m} ∘ ⋯ ∘ ψ_{1} where each ψ_{i} is the restriction of an automorphism of some Q_{i} ≤ S which is F-essential or equal to S. Let j ≤ m be such that Q_{i} ≥ S_{2} for all i = 1, …, j (hence Q_{i} ∈ T^{c}), and either j = m or Q_{j+1} ≱ S_{2}. Set φ^{∗} = ψ_{j} ∘ ⋯ ∘ ψ_{1}. Then φ^{∗} ∈ Mor(F_{1}^{•}), and φ^{∗}(P ∩ S_{1}) = φ^{∗}(P) ∩ S_{1} since S_{1} is strongly closed. Also, pr_{2} ∘φ^{∗}|_{S_{2}} is injective since φ^{∗}(S_{2}) ∩ S_{1} = 1, and hence pr_{2}(φ^{∗}(S_{2})) = S_{2}. If j < m, then since Q_{j+1} ≱ S_{2}, Q_{j+1} ≥ S_{1} by Lemma 1.8, which is impossible since Q_{j+1} ≥ φ^{∗}(S_{2}) (so pr_{2}(Q_{j}) = S_{2} but Q_{j} ≱ S_{2}). This is a contradiction, and we conclude that j = m and φ^{∗} = φ. In particular, φ ∈ Hom_{F_{1}^{•}}(P, S).

Now assume P ∈ T^{c} and φ ∈ Hom_{F}(P, S). We must show φ(P) ≥ S_{2}; i.e., φ(P) ∈ T^{c}. Since φ is in F_{1}^{•}, it suffices to show this when φ = α|_{P} for some α ∈ Aut_{F}(Q), where Q is F-essential or Q = S. Set Q_{1} = Q∩S_{1}, so Q = Q_{1}S_{2}. Then α(Q_{1}) = Q_{1} by assumption, and α(Z(Q_{1})S_{2}) = Z(Q_{1})S_{2} since Z(Q_{1})S_{2} = C_{Q}(Q_{1}). Since P ≤ Q is F-centric and contains S_{2}, P ≥ Z(Q_{1})S_{2}, and so α(P) ≥ Z(Q_{1})S_{2} ≥ S_{2}.

It remains to show that F|_{S_{1}} = F_{1}^{•}|_{S_{1}}. To see this, fix P, Q ≤ S_{1} and φ ∈ Hom_{F}(P, Q). Set P_{0} = φ(P), and choose P_{1} ≤ S and ψ ∈ Iso_{F}(P_{0}, P_{1}) such that P_{1} is F-conjugate to P and P_{0} and is fully centralized in F. By the extension axiom, there are morphisms ψ ∈ Hom_{F}(P S_{2}, S) and ψ_{0} ∈ Hom_{F}(P_{0}S_{2}, S) such that ψ|_{P} = ψ ∘ φ and ψ_{0}|_{P_{0}} = ψ. We just saw that ψ and ψ_{0} are morphisms in F_{1}^{•}, and so φ ∈ Hom_{F_{1}^{•}}(P, Q).

(c) Fix P ≤ S_{1} such that P S_{2} is F-centric. If φ ∈ Iso_{F}(P S_{2}, R) where R is fully normalized in F, then R ≥ S_{2} and φ(P) = R_{1} =^{def} R ∩ S_{1} since φ ∈ Mor(F_{1}^{•}) by (b). The first two points in (c) hold for P S_{2} if they hold for R, and it thus suffices to prove them when P S_{2} is fully normalized in F.

Set

Γ = {α ∈ Aut_{F}(P S_{2}) | α induces the identity on P and on P S_{2}/P } .

We must show Γ = 1. Since each element of Aut_{F}(P S_{2}) leaves P invariant by (b), Γ ⊴ Aut_{F}(P S_{2}), and Γ is a p-subgroup by Lemma 2.1. Thus Γ ≤ Aut_{S}(P S_{2}) ∈ Syl_{2}(Aut_{F}(P S_{2})). But each element of Aut_{S}(P S_{2}) leaves S_{2} invariant, so Γ contains only the identity element.

We next show [Aut_{E_{1}^{•}}(P S_{2}), P S_{2}] = [Aut_{E_{1}}(P), P]. Since Aut_{S}(P S_{2}) ∈ Syl_{2}(Aut_{F}(P S_{2})) and Aut_{E_{1}^{•}}(P S_{2}) is normal in Aut_{F}(P S_{2}) = Aut_{F_{1}^{•}}(P S_{2}), Aut_{S_{1}}(P S_{2}) is a Sylow 2-subgroup of Aut_{E_{1}^{•}}(P S_{2}). Clearly, [Aut_{S_{1}}(P S_{2}), P S_{2}] = [Aut_{S_{1}}(P), P] ≤ [Aut_{E_{1}}(P), P], so it remains to check that [H, P S_{2}] ≤ [Aut_{E_{1}}(P), P] for each H ≤ Aut_{E_{1}^{•}}(P S_{2}) of odd order. For each such H, [H, P S_{2}] ≤ P by definition of E_{1}^{•}, [H, P S_{2}] = [H, [H, P S_{2}]] by [G, Theorem 5.3.6], and so [H, P S_{2}] ≤ [H, P] ≤ [Aut_{E_{1}}(P), P].

To prove the third statement, fix g ∈ N_{S_{2}}(P S_{2}) and α ∈ Aut_{E_{1}^{•}}(P S_{2}). Then α and c_{g} both leave P invariant, α induces the identity on P S_{2}/P, and c_{g} induces the identity on P. Hence [α, c_{g}] induces the identity on P and on P S_{2}/P, and thus is the identity by the first statement in (c).

(d) Assume P ≤ S is fully centralized in F and F-conjugate to Q ≤ S_{1}. We must show that P ≤ S_{1}. Choose φ ∈ Iso_{F}(Q, P). Since P is fully centralized, the extension axiom implies that φ extends to some φ ∈ Hom_{F}(QS_{2}, S). By (b), φ ∈ Hom_{F_{1}^{•}}(QS_{2}, S), and so P = φ(Q) ≤ S_{1} since S_{1} is strongly closed in F_{1}^{•}.

Recall F_{1} = F|_{S_{1}} by (b). Hence a subgroup P ≤ S_{1} is fully centralized in F_{1} if and only if it is fully centralized in F, and similarly for fully normalized.

Assume P ≤ S_{1} is fully normalized in F, and choose R ≤ S which is F-conjugate to P S_{2} and fully normalized in F. Fix φ ∈ Iso_{F}(P S_{2}, R), and set Q = φ(P). Then Q = R ∩ S_{1} by (b) (φ ∈ Mor(F_{1}^{•})), and so N_{S}(R) ≤ N_{S}(Q). So |N_{S}(P S_{2})| = |N_{S}(P)| ≥ |N_{S}(Q)| ≥ |N_{S}(R)|, and thus P S_{2} is also fully normalized in F.

To prove the corresponding result for fully centralized subgroups, first note that P ≤ S is fully centralized in F if and only if |C_{S}(P)·P | is maximal in the F-conjugacy class of P. So assume P ≤ S_{1} is fully centralized in F, and choose R ≤ S which is F-conjugate to P S_{2} and fully centralized in F. Fix φ ∈ Iso_{F}(P S_{2}, R), and set Q = φ(P), so Q ≤ S_{1} by (b).

Then R ≤ C_{S}(Q)·Q since P S_{2} ≤ C_{S}(P)·P, so C_{S}(R)·R ≤ C_{S}(Q)·Q. Thus

|C_{S}(P S_{2})·P S_{2}| = |C_{S}(P)·P | ≥ |C_{S}(Q)·Q| ≥ |C_{S}(R)·R| ,

and so P S_{2} is also fully centralized in F.

(a) If P ≤ S_{1} is fully normalized in F_{1}, then it is fully normalized in F by (d), hence fully centralized in F, and hence fully centralized in F_{1} by (d) again. Also, Aut_{S_{1}}(P) = Aut_{S}(P) is a Sylow p-subgroup of Aut_{F_{1}}(P) = Aut_{F}(P), and this proves the Sylow axiom for F_{1}.

Assume φ ∈ Iso_{F_{1}}(P, Q), where Q is fully centralized in F_{1} and hence in F. Let N_{φ} ≤ N_{S_{1}}(P) be the subgroup of those g such that φc_{g}φ^{−1} ∈ Aut_{S_{1}}(Q). Then φ extends to some φ̂ ∈ Hom_{F}(N_{φ}S_{2}, S) by the extension axiom for F, φ̂ ∈ Hom_{F_{1}^{•}}(N_{φ}S_{2}, S) by (b), and so φ̂(N_{φ}) ≤ S_{1} since S_{1} is strongly closed in F_{1}^{•}. Then φ = φ̂|_{N_{φ}} ∈ Hom_{F_{1}}(N_{φ}, S_{1}) since F_{1} = F|_{S_{1}}, and this proves the extension axiom for F_{1}. Thus F_{1} is saturated.

Recall E_{1} = E_{1}^{•}|_{S_{1}}, where E_{1}^{•} is the smallest restrictive subcategory of F_{1}^{•} which contains all automorphisms φ ∈ Aut_{F}(P) = Aut_{F_{1}^{•}}(P) where P ≥ S_{2}, P is F-centric, and [φ, P] ≤ S_{1}. Thus E_{1}^{•} is invariant under conjugation by elements in Aut_{F}(S), and E_{1} is invariant under conjugation by elements in Aut_{F_{1}}(S_{1}).

We next show that

F_{1}^{•} = ⟨E_{1}^{•}, Aut_{F}(S)⟩  (1)

Assume P ≥ S_{2} and is F-essential. Then Aut_{E_{1}^{•}}(P) is the kernel of a homomorphism from Aut_{F_{1}^{•}}(P) = Aut_{F}(P) to Aut(S_{2}), and hence is normal in Aut_{F}(P). So Aut_{E_{1}^{•}}(P)·Inn(P) is normal in Aut_{F}(P) and contains the Sylow 2-subgroup Aut_{S}(P) (since Aut_{P ∩S_{1}}(P) ≤ Aut_{E_{1}^{•}}(P)). It follows that Aut_{E_{1}^{•}}(P)·Inn(P) ≥ O^{2′}(Aut_{F}(P)). Since

F_{1}^{•} = ⟨Aut_{F}(S), O^{2′}(Aut_{F}(P)) | P ≥ S_{2} F-essential⟩

by Proposition 1.7(b), this finishes the proof of (1).

Upon restricting to S_{1}, (1) implies that F_{1} = ⟨E_{1}, Aut_{F_{1}}(S_{1})⟩. We already saw that E_{1} is invariant under conjugation by elements of Aut_{F_{1}}(S_{1}), so E_{1} is F_{1}-invariant.

By [BCGLO1, Theorem A], to show E_{1} is saturated, it suffices to check the axioms on E_{1}- centric subgroups (equivalently F_{1}-centric by Lemma 1.4). The Sylow axiom clearly holds. So fix φ ∈ Iso_{E_{1}}(P, Q) where P, Q are F_{1}-centric. In particular, Q is fully centralized in F_{1}, and hence Q and QS_{2} are fully centralized in F by (d). As usual, let N_{φ} be the group of all g ∈ N_{S_{1}}(P) such that φc_{g}φ^{−1} ∈ Aut_{S_{1}}(Q). Then φ extends to φ_{1} ∈ Hom_{F}(N_{φ}S_{2}, S) by the extension axiom for F, and φ_{1}(P S_{2}) = QS_{2} by (b). By definition of E_{1}, there is another extension φ_{2} ∈ Iso_{F}(P S_{2}, QS_{2}) of φ, which induces the identity on S_{2} ≅ P S_{2}/P ≅ QS_{2}/Q. Set

ψ = φ_{2} ∘ (φ_{1}|_{P S_{2}})^{−1} ∈ Aut_{F}(QS_{2}).

By construction, ψ|_{Q} = Id_{Q}. Hence for each g ∈ N_{S_{1}}(Q), [ψ, c_{g}] ∈ Aut_{F}(QS_{2}) induces the identity on Q and on QS_{2}/Q. So [ψ, c_{g}] = Id by (c) (QS_{2} is F-centric since it is fully centralized in F and C_{S}(QS_{2}) = C_{S_{1}}(Q)Z_{2} ≤ QS_{2}). In other words, [ψ, Aut_{S_{1}}(QS_{2})] = 1. So by the extension axiom (and since QS_{2} is fully centralized in F), ψ extends to an automorphism ψ_{1} ∈ Aut_{F}(N_{S_{1}}(Q)S_{2}). Now set φ = ψ_{1} ∘ φ_{1} ∈ Hom_{F}(N_{φ}S_{2}, S). This induces the identity modulo S_{1} (since φ_{2} = φ|_{P S_{2}} does), and thus φ|_{N_{φ}} ∈ Hom_{E_{1}}(N_{φ}, S_{1}) is an extension of φ.

This finishes the proof that E_{1} is saturated. We already showed that it is F_{1}-invariant, and hence it is normal. By Theorem 1.6(b), E_{1} has index prime to p in F_{1}. □

The following is our main, general proposition for decomposing a fusion system. Theorems 4.5 and 4.8 will follow as consequences of this.

Proposition 4.3. Fix a pair of 2-groups S_{1} and S_{2}, and set S = S_{1} × S_{2}. Let F be a saturated fusion system over S. For i = 1, 2, define

F_{i}^{•} = ⟨Aut_{F}(P) | S_{3−i} ≤ P ≤ S, P F-essential or P = S⟩

as a fusion subsystem of F over S. Assume

(a) O^{2}(F) = O^{2′}(F) = F; and

(b) S_{i} is strongly closed in F_{i}^{•} for i = 1, 2.

Then F = F_{1} × F_{2} for some pair of saturated fusion systems F_{i} over S_{i}.

Proof. For i = 1, 2, set Z_{i} = Z(S_{i}) for short, and let pr_{i} ∈ Hom(S, S_{i}) be the projection. Define restrictive subcategories E_{i}^{•} ⊆ F_{i}^{•} as in Lemma 4.2: E_{i}^{•} is the smallest restrictive subcategory where for each F-centric subgroup S_{3−i} ≤ P ≤ S,

Aut_{E_{i}^{•}}(P) = {φ ∈ Aut_{F}(P) | [φ, P] ≤ S_{i}} .

By Lemma 4.2(b), if P ≤ S is F-centric and P ≥ S_{3−i}, then for each φ ∈ Hom_{F}(P, S), φ(P) ≥ S_{3−i}. Thus for all P, Q ≤ S such that P S_{3−i} is F-centric,

each φ ∈ Hom_{E_{i}^{•}}(P, Q), extends to some φ ∈ Hom_{E_{i}^{•}}(P S_{3−i}, QS_{3−i}).  (1)

Set F_{i} = F_{i}^{•}|_{S_{i}} and E_{i} = E_{i}^{•}|_{S_{i}}. By Lemma 4.2(a), these are both saturated fusion systems over S_{i}, and E_{i} is normal of odd index in F_{i}. By Lemma 4.2(c), for all P ≤ S_{i} such that P S_{3−i} is F-centric,

α ∈ Aut_{E_{i}^{•}}(P S_{3−i}), α|_{P} = Id_{P} ⟹ α = Id_{P S_{3−i}} .  (2)

Let U be the set of all F-centric subgroups of S of the form P = P_{1}P_{2} for P_{i} ≤ S_{i}. In general, for any P ∈ U, we set P_{i} = pr_{i}(P) ≤ S_{i}.

Step 1: We first claim that

\begin{matrix}P ∈ U, φ ∈ Hom_{F}(P, S), Q = φ(P) ⟹ \\ Q ∈ U, φ(P_{1}Z(P_{2})) = Q_{1}Z(Q_{2}), φ(P_{2}Z(P_{1})) = Q_{2}Z(Q_{1}) . \end{matrix}  (3)

By Proposition 1.7(a), it suffices to show this when φ = α|_{P} for some subgroup R ≤ S such that R is F-essential or R = S, and some α ∈ Aut_{F}(R). By Lemma 1.8(b), R ≥ S_{1} or R ≥ S_{2}. If R ≥ S_{2}, then α(R_{1}) = R_{1} by (b) (recall R_{1} = S_{1} ∩ R), so α also sends C_{R}(R_{1}) = Z(R_{1})S_{2} to itself. Thus Q_{1} =^{def} φ(P_{1}) ≤ S_{1}, and φ(P_{2}) ≤ Z(R_{1})S_{2} ≤ C_{S_{1}}(Q)S_{2}. Since Q and P are F-centric, C_{S_{1}}(Q) = Z(Q) ∩ S_{1} ≤ Q_{1}, so φ(P) ≤ Q_{1}S_{2}, and φ(P) = Q_{1}Q_{2} ∈ U for some Q_{2} ≤ S_{2}. Since φ(Z(P)) = Z(Q), it now follows that φ(P_{1}Z(P)) = Q_{1}Z(Q) and φ(P_{2}Z(P)) = Q_{2}Z(Q). The argument when R ≥ S_{1} is similar, and this finishes the proof of (3).

In particular, this shows that Aut_{P_{i}}(P) ⊴ Aut_{F}(P) for P = P_{1}P_{2} ∈ U. If g ∈ N_{S_{1}}(P), and α ∈ Aut_{F}(P) is such that αc_{g}α^{−1} ∈ Aut_{S}(P), then by the extension axiom, α extends to β ∈ Hom_{F}(⟨P, g⟩, S), where β(g) = h = h_{1}h_{2} for h_{i} ∈ N_{S_{i}}(P). By (3), β(⟨P_{1}, g⟩) ≤ ⟨P_{1}, h_{1}⟩Z(P_{2}), so h_{2} ∈ Z(P_{2}), and αc_{g}α^{−1} = c_{h} = c_{h_{1}} ∈ Aut_{S_{1}}(P). Thus Aut_{S_{1}}(P) is strongly closed in Aut_{S}(P) with respect to Aut_{F}(P), and the same argument applies to Aut_{S_{2}}(P).

Step 2: Set E = ⟨E_{1}^{•}, E_{2}^{•}⟩ ⊆ F, as a fusion system over S. By Lemma 4.2(a), F_{i}^{•} = ⟨E_{i}^{•}, Aut_{F}(S)⟩. Since all F-essential subgroups contain S_{1} or S_{2} (Lemma 1.8(b)),

F = ⟨F_{1}^{•}, F_{2}^{•}⟩ = ⟨E_{1}^{•}, E_{2}^{•}, Aut_{F}(S)⟩ = ⟨E, Aut_{F}(S)⟩.

For each α ∈ Aut_{F}(S) and each i = 1, 2, αE_{i}^{•}α^{−1} = E_{i}^{•} since α(S_{i}) = S_{i} by (b), and hence αEα^{−1} = E. Thus E is F-invariant.

We claim that for P = P_{1}P_{2} and Q = Q_{1}Q_{2} in U,

P is E-conjugate to Q ⟺ P_{i} is E_{i}-conjugate to Q_{i} for i = 1, 2.  (4)

By definition, for i = 1, 2, P_{1}P_{2} and Q_{1}Q_{2} are E_{i}^{•}-conjugate if and only if P_{i} and Q_{i} are E_{i}-conjugate and P_{3−i} = Q_{3−i}. So (4) follows since E = ⟨E_{1}^{•}, E_{2}^{•}⟩. Since each subgroup in U is F-centric and hence E-centric, this in turn implies that

P ∈ U ⟹ P_{i} is E_{i}-centric for i = 1, 2
P ∈ U and fully normalized in E ⟹ P_{i} is fully normalized in E_{i} for i = 1, 2.

(5)

Step 3: We next claim that

for all P ∈ U, [Aut_{E_{1}^{•}}(P), Aut_{E_{2}^{•}}(P)] = 1 .  (6)

We first check the following special case:

P ∈ U ⟹ [Aut_{S_{1}}(P), Aut_{E_{2}^{•}}(P)] = 1 and [Aut_{S_{2}}(P), Aut_{E_{1}^{•}}(P)] = 1 .  (7)

The first statement in (7) holds by Lemma 4.2(c) when P ≥ S_{1}, and in the general case since all morphisms in Aut_{S_{1}}(P) and in Aut_{E_{2}^{•}}(P) extend to P S_{1} by (1). The second statement holds by a similar argument.

We prove (6) by downwards induction on |P |. If P = P_{1}P_{2} where P_{i} ≤ S_{i}, and α ∈ Aut_{E_{1}^{•}}(P) is such that α|_{P_{1}} ∈ Aut_{E_{1}}(P) is the restriction of some β ∈ Iso_{E_{1}}(Q_{1}, R_{1}) where Q_{1} ≩ P_{1}, then β extends to β ∈ Iso_{E_{1}^{•}}(Q_{1}S_{2}, R_{1}S_{2}) by definition of E_{1}, and α = β|_{P} by (2). So by the induction hypothesis, it suffices to prove (6) when P_{1} = S_{1} or P_{1} is E_{1}-essential. By symmetry, it also suffices to do this when P_{2} = S_{2} or P_{2} is E_{2}-essential.

Assume first P = P_{1}S_{2}, where P_{1} = S_{1} or P_{1} is E_{1}-essential. Thus P_{1} is fully normalized in E_{1}, so P_{1} is fully normalized in F_{1} by Lemma 1.4(b), and P is fully normalized in F by Lemma 4.2(d). Each α ∈ Aut_{E_{2}^{•}}(P) is the restriction of some α ∈ Aut_{E_{2}^{•}}(S) by (1), and hence α|_{P_{1}} = Id_{P_{1}}. For each β ∈ Aut_{E_{1}^{•}}(P), β(P_{1}) = P_{1} and β induces the identity on P/P_{1}, so [α, β] induces the identity on P_{1} and on P/P_{1}, and [α, β] = Id_{P} by (2). Thus (6) holds in this case, and by a similar argument, it also holds when P ≥ S_{1}.

It remains to prove (6) when P_{1} is E_{1}-essential and P_{2} is E_{2}-essential. Fix such P = P_{1}P_{2} ∈ U, set G_{i} = Aut_{E_{i}^{•}}(P) and G = Aut_{F}(P), and let H_{i} ≤ G_{i} be the subgroup generated by automorphisms which extend (in E_{i}^{•}) to larger subgroups. Thus [H_{i}, G_{3−i}] = 1 for i = 1, 2 by the induction hypothesis. By Proposition 1.7(c), (applied to Out_{E_{i}}(P_{i}) ≅ G_{i}/Inn(P)), H_{i}/Inn(P) is strongly 2-embedded in G_{i}/Inn(P). By Step 1, Aut_{S_{1}}(P) and Aut_{S_{2}}(P) are strongly closed in Aut_{S}(P) with respect to Aut_{F}(P). So [G_{1}, G_{2}] = 1 by Proposition 2.9, applied to the actions of G_{1}, G_{2} ≤ G on P, and this finishes the proof of (6).

Step 4: We are now ready to prove that E is saturated. By construction, E is U-generated: each morphism in E is a composite of restrictions of morphisms between subgroups in U. Each P ∈ U is E-centric since it is F-centric, and hence is fully centralized in E. If P = P_{1}P_{2} ∈ U is fully normalized in E, then each P_{i} is fully normalized in E_{i} by (5), Aut_{S}(P_{i}) ∈ Syl_{2}(Aut_{E_{i}}(P_{i})) for i = 1, 2, and so Aut_{S}(P) ∈ Syl_{2}(Aut_{E}(P)) since |Aut_{E}(P)| = |Aut_{E_{1}}(P_{1})|·|Aut_{E_{2}}(P_{2})| by (6). The Sylow axiom thus holds for subgroups in U.

Now assume P, Q ∈ U and φ ∈ Iso_{E}(P, Q). By definition of the E_{i}^{•} and E = ⟨E_{1}^{•}, E_{2}^{•}⟩, φ = ψ_{m} ∘ ⋯ ∘ ψ_{1}, where for each i, ψ_{i} is a restriction of an automorphism ψ̂_{i} ∈ Aut_{E_{k_{i}}^{•}}(Q_{i})

where Q_{i} ≥ S_{3−k_{i}} (k_{i} ∈ {1, 2}). If, for some 1 ≤ i < m, k_{i} = 2 and k_{i+1} = 1, then ψ̂_{i} and ψ̂_{i+1} each sends Q_{i,i+1} =^{def} Q_{i} ∩ Q_{i+1} to itself, their restrictions to Q_{i,i+1} commute by (6), and the domains and ranges of ψ_{i} and ψ_{i+1} are all contained in Q_{i,i+1}. We can thus exchange ψ̂_{i} with ψ̂_{i+1} without changing the composite φ, and arrange that the restrictions of morphisms in E_{1}^{•} all come before restrictions of morphisms in E_{2}^{•}.

Thus φ = φ_{2} ∘ φ_{1}, where for some R ∈ U, φ_{1} ∈ Iso_{E_{1}^{•}}(P, R), φ_{2} ∈ Iso_{E_{2}^{•}}(R, Q), φ_{1} = φ̂_{1}|_{P} where φ̂_{1} ∈ Iso_{E_{1}^{•}}(P S_{2}, RS_{2}), and φ_{2} = φ̂_{2}|_{R} where φ̂_{2} ∈ Iso_{E_{2}^{•}}(RS_{1}, QS_{1}). Then R = Q_{1}P_{2}. Let P = N_{φ} ≤ N_{S}(P) be the subgroup of all g ∈ N_{S}(P) such that φc_{g}φ^{−1} ≤ Aut_{S}(Q), and let Q ≤ Q ≤ N_{S}(Q) be such that φAut_{P} (P)φ^{−1} = Aut_{Q}(Q) (thus Q = N_{φ^{−1}}). Set

P_{i} = pr_{i}(P) and Q_{i} = pr_{i}(Q).

Assume g ∈ P and h ∈ Q are such that φc_{g}φ^{−1} = c_{h}. Write g = g_{1}g_{2} and h = h_{1}h_{2}, where g_{i}, h_{i} ∈ S_{i}. Then c_{g_{2}} φ̂_{1}c_{g_{2}}^{−1} and φ̂_{1} both are in Iso_{E_{1}^{•}}(P S_{2}, RS_{2}) and are equal after restriction to P, so they are equal by (2). Thus φ_{1}c_{g_{2}}φ_{1}^{−1} = c_{g_{2}}, and φ_{2}c_{h_{1}}φ_{2}^{−1} = c_{h_{1}} by a similar argument. Hence

φ_{1}c_{g}φ_{1}^{−1} = c_{g_{2}} ∘ φ_{1}c_{g_{1}}φ_{1}^{−1}
= φ_{2}^{−1}c_{h}φ_{2} = φ_{2}^{−1}c_{h_{2}}φ_{2} ∘ c_{h_{1}} ∈ Aut(R) .

Since Aut_{E_{1}^{•}}(R) ∩ Aut_{E_{2}^{•}}(R) = 1 (each automorphism in the intersection leaves Q_{1} and P_{2} invariant and is the identity on both quotients), this proves that φ_{i}c_{g_{i}}φ_{i}^{−1} = c_{h_{i}} for i = 1, 2. This also implies that φc_{g_{i}}φ^{−1} = c_{h_{i}} for i = 1, 2, and hence that P = P_{1}P_{2} and Q = Q_{1}Q_{2}.

By (5), each P_{i} and Q_{i} is E_{i}-centric and hence fully centralized in E_{i}. By the extension axiom for E_{1}, φ_{1}|_{P_{1}} extends to an isomorphism in Iso_{E_{1}}(P_{1}, Q_{1}), that in turn extends to φ_{1} ∈ Iso_{E_{1}^{•}}(P , Q_{1}P_{2}) by (1), and φ_{1}|P = φ_{1} by (2). Similarly, φ_{2} extends to an isomorphism φ_{2} ∈ Iso_{E_{2}^{•}}(Q_{1}P_{2}, Q). So φ_{2} ∘ φ_{1} is an extension of φ to P.

This proves the extension axiom for E on subgroups in U, and finishes the proof that E is U-saturated. If P is E-centric but not in U, then since P is F-centric by Lemma 1.4 (recall E is F-invariant), P ≨ P_{1}P_{2} where P_{i} = pr_{i}(P). By Lemma 1.8, Out_{S}(P)∩O_{2}(Out(P)) ≠ 1. Hence E is satuated by [BCGLO1, Theorem A].

Step 5: We already showed in Step 2 that E is F-invariant, and thus E ⊴ F. Since they are both saturated fusion systems over S, E has odd index in F by Theorem 1.6(b). Since O^{2′}(F) = F, it follows that E = F.

Since E = ⟨E_{1}^{•}, E_{2}^{•}⟩, 𝔣𝔬𝔠(E) = 𝔣𝔬𝔠(E_{1}^{•})×𝔣𝔬𝔠(E_{2}^{•}). Since 𝔣𝔬𝔠(E_{i}^{•}) is generated by the subgroups [Aut_{E_{i}^{•}}(P), P] for F-centric subgroups P ≥ S_{3−i}, and [Aut_{E_{i}^{•}}(P), P] = [Aut_{E_{i}}(P_{i}), P_{i}] by Lemma 4.2(c), 𝔣𝔬𝔠(E_{i}^{•}) = 𝔣𝔬𝔠(E_{i}) ≤ S_{i}. Since O^{2}(F) = F, 𝔣𝔬𝔠(E) = S by Theorem 1.6(a), and thus 𝔣𝔬𝔠(E_{i}) = S_{i} for i = 1, 2.

By (6) and Lemma 2.7, for each P ∈ U,

C_{P} (Aut_{E_{1}^{•}}(P)) ≥ [Aut_{E_{2}}(P_{2}), P_{2}] and C_{P} (Aut_{E_{2}^{•}}(P)) ≥ [Aut_{E_{1}}(P_{1}), P_{1}] .

Thus for any fixed E_{1}-centric subgroup P_{1} ≤ S_{1}, Aut_{E_{1}^{•}}(P_{1}S_{2}) is the identity on [Aut_{E_{2}}(P_{2}), P_{2}] for each E_{2}-centric subgroup P_{2} ≤ S_{2}, and hence is the identity on 𝔣𝔬𝔠(E_{2}) = S_{2}. So S_{2} is strongly closed in E_{2}^{•}, and S_{1} is strongly closed in E_{1}^{•} by a similar argument.

Since each S_{i} is strongly closed in E_{i}^{•} ⊆ F_{i}^{•} by assumption, this proves that S_{1} and S_{2} are strongly closed in F = ⟨E_{1}^{•}, E_{2}^{•}⟩. Each morphism in E_{i}^{•} extends to a morphism which is the identity on S_{3−i}. So by definition of the product of fusion systems (and since E_{i} = E_{i}^{•}|_{S_{i}}), F = ⟨E_{1}^{•}, E_{2}^{•}⟩ = E_{1} × E_{2}. □

4.2. A first application of Proposition 4.3.

Recall that when G is a group and H ≤ G is a subgroup, K is a normal complement to H in G if K ⊴ G, K ∩ H = 1, and KH = G. Equivalently, K is a normal complement exactly when the inclusions of H and K into G induce an isomorphism K ⋊ H ⟶^{≅} G.

Lemma 4.4. Fix a pair of p-groups S_{1} and S_{2}, and set S = S_{1} × S_{2}. Let F be a saturated fusion system over S. Set

F_{1}^{•} = ⟨Aut_{F}(P) | S_{2} ≤ P ≤ S, P F-essential or P = S⟩ ,

and assume Ω_{1}(Z(S_{2})) is strongly closed in F_{1}^{•}. Then there is a normal complement S_{1}^{0} to Z(S_{2}) in S_{1}Z(S_{2}) which is strongly closed in F_{1}^{•}.

Proof. For i = 1, 2, set Z_{i} = Z(S_{i}) and Ŝ_{i} = S_{i}Z_{3−i}. If P = S or P ≥ S_{2} is F-essential, then Σ_{1}(Z_{2}) is Aut_{F}(P)-invariant, and hence P ∩ Ŝ_{1} is Aut_{F}(P)-invariant by Proposition 3.1. Since F_{1}^{•} is generated by such automorphisms, Ŝ_{1} is strongly closed in F_{1}^{•}.

Let pr_{2} ∈ Hom(S, S_{2}) be the projection. We first show that

φ ∈ Hom_{F}(S_{2}, S) ⟹ φ(Ω_{1}(Z_{2})) = Ω_{1}(Z_{2}) and pr_{2}(φ(S_{2})) = S_{2}.  (1)

By Proposition 1.7(a), each such φ decomposes as a composite φ = ψ_{m} ∘ ⋯ ∘ ψ_{1}, where each ψ_{i} is the restriction of an F-automorphism of Q_{i}, and Q_{i} = S or Q_{i} is F-essential. Let j ≤ m be such that Q_{i} ≥ S_{2} for all i = 1, …, j (hence Q_{i} ∈ T^{c}), and either j = m or Q_{j+1} ≱ S_{2}. Set φ^{∗} = ψ_{j} ∘ ⋯ ∘ ψ_{1}. For each i ≤ j, ψ_{i}(Ω_{1}(Z_{2})) = Ω_{1}(Z_{2}) by assumption. Hence φ^{∗}(Ω_{1}(Z_{2})) = Ω_{1}(Z_{2}), and pr_{2} ∘φ^{∗} is injective since the kernel must contain a central element of order p. Thus pr_{2}(φ^{∗}(S_{2})) = S_{2}. If j < m, then since Q_{j+1} ≱ S_{2}, Q_{j+1} ≥ S_{1} by Lemma 1.8, which is impossible since Q_{j+1} ≥ φ^{∗}(S_{2}) (so pr_{2}(Q_{j}) = S_{2} but Q_{j} ≱ S_{2}). Thus j = m and φ^{∗} = φ. This proves (1).

We next claim that

P ≤ S is fully centralized in F and F-conjugate to Q ≤ Ŝ_{1} ⟹ P ≤ Ŝ_{1}.  (2)

To see this, fix such P and Q, and choose φ ∈ Iso_{F}(Q, P). Since S_{2} ≤ C_{S}(Q), φ extends to some φ ∈ Hom_{F}(QS_{2}, S). By (1), pr_{2}(φ(S_{2})) = S_{2}. So P = φ(Q) ≤ C_{S}(φ(S_{2})) ≤ Ŝ_{1}. This proves (2).

Set F̂_{1} = F|Ŝ_{1}. In other words, HomF̂_{1}(P, Q) = Hom_{F}(P, Q) for P, Q ≤ Ŝ_{1}. Also, two subgroups of Ŝ_{1} are F̂_{1}-conjugate if and only if they are F-conjugate. Hence (2) implies that a subgroup P ≤ Ŝ_{1} is fully normalized (fully centralized) in F̂_{1} if and only if it is fully normalized (fully centralized) in F. So if P is fully normalized in F̂_{1}, then it is fully centralized in F̂_{1} by the Sylow axiom for F. Also, AutŜ_{1}(P) = Aut_{S}(P) is a Sylow p-subgroup of AutF̂_{1}(P) = Aut_{F}(P), and this proves the Sylow axiom for F̂_{1}.

Now assume φ ∈ IsoF̂_{1}(P, Q), where Q is fully centralized in F̂_{1} and hence in F. Let N_{φ} ≤ NŜ_{1}(P) be the subgroup of those g such that φc_{g}φ^{−1} ∈ AutŜ_{1}(Q). Then φ extends to some φ̂ ∈ Hom_{F}(N_{φ}S_{2}, S) by the extension axiom for F, pr_{2}(φ̂(S_{2})) = S_{2} by (1), and so φ̂(N_{φ}) ≤ C_{S}(φ̂(S_{2})) ≤ Ŝ_{1}. Set φ = φ̂|_{N_{φ}}; then φ̂|_{N_{φ}} ∈ HomF̂_{1}(N_{φ}, Ŝ_{1}) since F̂_{1} = F|Ŝ_{1}, and this proves the extension axiom for F̂_{1}. Thus F̂_{1} is saturated.

Fix P, Q ≤ Ŝ_{1} and φ ∈ IsoF̂_{1}(P, Q). Choose R which is fully centralized in F and F-conjugate to Q, and fix ψ ∈ Iso_{F}(Q, R). By the extension axiom for F, there are morphisms ψ ∈ Hom_{F}(QS_{2}, S) and φ ∈ Hom_{F}(P S_{2}, S) such that ψ|_{Q} = ψ and φ|_{P} = ψ ∘ φ. Then ψ(Ω_{1}(Z_{2})) = Ω_{1}(Z_{2}) and φ(Ω_{1}(Z_{2})) = Ω_{1}(Z_{2}) by (1), and so ψ^{−1} ∘ φ ∈ HomF̂_{1}(P Ω_{1}(Z_{2}), QΩ_{1}(Z_{2})) is an extension of φ which sends Ω_{1}(Z_{2}) to itself. This proves that Ω_{1}(Z_{2}) ⊴ F̂_{1}.

Now consider the centralizer fusion subsystem Ê_{1} = CF̂_{1}(Ω_{1}(Z_{2})). This is the fusion system over Ŝ_{1} = CŜ_{1}(Ω_{1}(Z_{2})) where for each P, Q ≤ Ŝ_{1},

HomÊ_{1}(P, Q) = {φ ∈ HomF̂_{1}(P, Q) | φ = φ|_{P} for some

φ ∈ HomF̂_{1}(P Ω_{1}(Z_{2}), QΩ_{1}(Z_{2})) with φ|_{Ω_{1}(Z_{2})} = Id} .

By [BLO, Proposition A.6], this is a saturated fusion system, and by [AOV, Proposition 1.16(c)] (and since Ω_{1}(Z_{2}) ⊴ F̂_{1}), it is normal in F̂_{1}. It has index prime to p since it is a fusion system over the same p-group (Theorem 1.6(b)).

Consider the composite homomorphism

f: Ŝ_{1} −↠^{proj} Ŝ_{1}/𝔣𝔬𝔠(Ê_{1}) ⟶^{trf} Ŝ_{1}/[Ŝ_{1}, Ŝ_{1}] ⟶^{pr_{2}} Z_{2},

where trf is the transfer homomorphism of Proposition 1.9, and the first map is the canonical projection. By that proposition, f(z) = z for z ∈ Ω_{1}(Z_{2}). The group Γ =^{def} OutF̂_{1}(Ŝ_{1}) of order prime to p acts on Ŝ_{1}/𝔣𝔬𝔠(Ê_{1}), leaving Ω_{1}(Z_{2}) invariant. Define f̂ ∈ Hom(Ŝ_{1}, Z_{2}) by averaging over the elements of this group:

f̂(g) = \frac{1}{|Γ|} ∑_{[α]∈Γ} α(f(α^{−1}(g))) .

Then f̂ is well defined since f(α^{−1}(g)) is independent of the choice of α ∈ AutF̂_{1}(Ŝ_{1}) in its class [α] ∈ Γ, f̂ is AutF̂_{1}(Ŝ_{1})-linear, and f̂|_{Ω_{1}(Z_{2})} = Id. Hence f̂|_{Z_{2}} is an isomorphism.

Set S_{1}^{0} = Ker(f̂). Then S_{1}^{0} is a normal complement to Z_{2} in Ŝ_{1} since f̂|_{Z_{2}} is an isomorphism, S_{1}^{0} is AutF̂_{1}(Ŝ_{1})-invariant since f̂ is AutF̂_{1}(Ŝ_{1})-linear, and it is strongly closed in Ê_{1} since it contains its focal subgroup. Since F̂_{1} = ⟨Ê_{1}, AutF̂_{1}(Ŝ_{1})⟩ (since Ê_{1} ⊴ F̂_{1}), this shows that S_{1}^{0} is strongly closed in F̂_{1}. We already saw that Ŝ_{1} is strongly closed in F_{1}^{•}, and thus S_{1}^{0} is strongly closed in F_{1}^{•}. □

The following theorem now gives a more explicit set of conditions which imply a splitting of a fusion system.

Theorem 4.5. Fix a pair of 2-groups S_{1} and S_{2} such that Ω_{1}(Z(S_{1})) ≤ [S_{1}, S_{1}], and set S = S_{1} × S_{2}. Let F be a saturated fusion system over S such that O^{2}(F) = F = O^{2′}(F). Assume, for i = 1, 2, that whenever P = S or P is an F-essential subgroup which contains S_{i}, then Ω_{1}(Z(S_{i})) is Aut_{F}(P)-invariant. Then F ≅ F_{1} × F_{2} for some pair of saturated fusion systems F_{i} over S_{i}.

Proof. Set Z_{i} = Z(S_{i}) for short. As in Lemmas 4.2 and 4.4, set

F_{i}^{•} = ⟨Aut_{F}(P) | S_{3−i} ≤ P ≤ S, P F-essential or P = S⟩ .

By assumption, Ω_{1}(Z_{i}) is strongly closed in F_{3−i}^{•}.

Set Ŝ_{i} = S_{i}Z_{3−i}. By Lemma 4.4, for each i = 1, 2, there is a normal complement S_{i}^{0} to Z_{3−i} in Ŝ_{i} which is strongly closed in F_{i}^{•}. Also, [S_{1}^{0}, S_{1}^{0}] = [Ŝ_{1}, Ŝ_{1}] = [S_{1}, S_{1}], so Ω_{1}(Z_{1}) ≤ [S_{1}, S_{1}] ≤ S_{1}^{0} by assumption.

Now, S_{1}^{0} ∩ S_{2}^{0} ≤ Ŝ_{1} ∩ Ŝ_{2} = Z(S). Hence if S_{1}^{0} ∩ S_{2}^{0} ≠ 1, there is some 1 ≠ z_{1}z_{2} ∈ S_{1}^{0} ∩ S_{2}^{0} where z_{i} ∈ Ω_{1}(Z_{i}). Since z_{1} ∈ Ω_{1}(Z_{1}) ≤ S_{1}^{0}, this implies z_{2} ∈ S_{1}^{0}, which is impossible since S_{1}^{0} is a normal complement to Z_{2} in Ŝ_{1}. Thus S_{1}^{0} ∩ S_{2}^{0} = 1. Hence there is an automorphism φ ∈ Aut(S) such that φ(S_{i}^{0}) = S_{i}. Upon replacing F by φFφ^{−1}, we can assume S_{i}^{0} = S_{i}. Note that φ sends each F-essential subgroup to itself, since it is the identity modulo Z(S).

We have now replaced F by a fusion system isomorphic to F, with the property that S_{i} is strongly closed in F_{i}^{•} for i = 1, 2. So by Proposition 4.3, F splits as a product of saturated fusion systems over F_{1} and F_{2}. □

The following corollary gives explicit conditions in terms of the 2-groups S_{1} and S_{2} which imply the existence of a splitting.

Corollary 4.6. Fix a pair of 2-groups S_{1} and S_{2}, and set S = S_{1} ×S_{2}. Assume the following conditions hold for each i = 1, 2:

(i) S_{i} is nonabelian and Z(S_{i}) is cyclic; or more generally

(i′) S_{1} ≠ 1, Ω_{1}(Z(S_{i})) ≤ [S_{i}, S_{i}], and for each φ ∈ Hom(S_{i}, S_{i}) such that φ(S_{i}) ⊴ S_{i} and φ is not onto, Ker(φ) ≥ Ω_{1}(Z(S_{i})).

(ii) S_{3−i} contains no subgroup isomorphic to S_{i} × S_{i}.

Then for every saturated fusion system F over S such that O^{2}(F) = F = O^{2′}(F), F ≅ F_{1} × F_{2} for some pair of saturated fusion systems F_{i} over S_{i}.

Proof. Fix a saturated fusion system F over S. We saw in the proof of Proposition 3.2 that (i) implies (i′). It remains to show (i′) and (ii) imply the hypotheses of Theorem 4.5.

For i = 1 or 2, fix P ≤ S containing S_{i} such that P = S or P is F-essential. Thus P = S_{i}P_{0} for some P_{0} ≤ S_{3−i}, and we must show that Ω_{1}(Z(S_{i})) is Aut_{F}(P)-invariant. Let Aut_{F}^{0}(P) ≤ Aut_{F}(P) be the subgroup of elements which send Ω_{1}(Z(S_{i})) to itself. Since S_{3−i} contains no subgroup isomorphic to S_{i} × S_{i} by (ii), Aut_{F}^{0}(P) has index at most two in Aut_{F}(P) by Proposition 3.2(a). Since Aut_{F}^{0}(P) contains the Sylow 2-subgroup Aut_{S}(P) (P is fully normalized since it is F-essential), it is equal to Aut_{F}(P). □

4.3. Dihedral, semidihedral, and wreathed 2-groups.

We now look at a different set of conditions which imply a splitting of a fusion system. Theorem 4.5 puts some fairly strict conditions on both factors S_{1} and S_{2}. When one of the factors is dihedral, semidihedral, or wreathed, then the conditions on the other factor can be greatly relaxed. Here, by a wreathed 2-group, we mean a group of the form C_{2^{n}} ≀ C_{2} for some n ≥ 2.

Lemma 4.7. Assume S is a dihedral, semidihedral, or wreathed 2-group. Then

(a) Aut(S) is a 2-group, and [S, S] is cyclic.

Define a set H of subgroups of S as follows:

• If S = ⟨a, b⟩ ≅ D_{2^{n}} (n ≥ 3), where |a| = 2^{n−1}, then H = {T_{i} | i ∈ ℤ} where T_{i} = ⟨a^{2^{n−2}}, a^{i}b⟩ ≅ C_{2}^{2}.

• If S = ⟨a, b⟩ ≅ SD_{2^{n}} (n ≥ 4), where |a| = 2^{n−1} and |b| = 2, then H = {T_{2i}, Q_{2i+1} | i ∈ ℤ} where T_{2i} = ⟨a^{2^{n−2}}, a^{2i}b⟩ ≅ C_{2}^{2} and Q_{2i+1} = ⟨a^{2^{n−3}}, a^{2i+1}b⟩ ≅ Q_{8}.

• If S = ⟨a, b, t⟩ ≅ C_{2^{n}} ≀ C_{2} (n ≥ 2), where ⟨a, b⟩ ≅ C_{2^{n}} × C_{2^{n}}, |t| = 2, and tat^{−1} = b, then H = {A, U_{2i} | i ∈ ℤ}, where A = ⟨a, b⟩ and U_{2i} = ⟨ab, a^{2^{n−1}}, a^{2i}t⟩ ≅ C_{2^{n}} ×_{C_{2}} Q_{8}.

Then the following hold.

(b) The set H is the union of exactly two S-conjugacy classes.

(c) For P ∈ H,

• |N_{S}(P)/P | = 2 and [N_{S}(P), P] = P ∩ [S, S];

• N_{S}(P) is not contained in any other subgroup in H; and

• either P is abelian, or [P, P] ≅ C_{2} and Z(P) = Z(S_{1}).

(d) Assume F is a saturated fusion system over S × T for = some 2-group T. If P ≨ S is such that P × T is F-essential, then P ∈ H.

(e) Fix a 2-group T and a saturated fusion system F over S × T. If P ≤ S is such that P T is F-essential, then there is α_{P} ∈ Aut_{F}(P T) of order 3 such that

• Out_{F}(P T) = Γ_{P} × H_{P} where Γ_{P} = ⟨[α_{P}], Out_{S}(P T)⟩ ≅ Σ_{3} and |H_{P} | is odd; and

• [N_{S}(P), P] ≤ [α_{P} , P T] ≤ P Z(T), [α_{P} , T] ≤ Z(P T), and [α_{P} , P T] ∩ Z(T) = 1.

Proof. Points (b) and (c) are easy.

(a) The second statement ([S, S] is cyclic) is easily checked. In all cases (S is dihedral, semidihedral, or wreathed), S/Fr(S) ≅ C_{2}^{2}. If S is dihedral or semidihedral, let A ⊴ S be the cyclic subgroup of index two; otherwise let A be the (unique) abelian subgroup of index two. Then A is characteristic in S. Each automorphism of S induces the identity on S/A ≅ C_{2} and on A/Fr(S) ≅ C_{2}, so Aut(S) is a 2-group by Lemma 2.1.

(d) We prove this case by case. Fix a 2-group T, a saturated fusion system F over S × T, and a subgroup P ≨ S such that P T is F-essential.

Assume S = ⟨a, b⟩ is dihedral or semidihedral, where |a| = 2^{n−1} and |b| = 2. If P ≤ S is dihedral of order ≥ 8 or quaternion of order ≥ 16, then Z(P) = Z(S), and Out(P) is a 2-group (Lemma 2.1 or (a)). So P T is not F-essential by Lemma 3.3. If P ≤ S is cyclic, then [N_{S}(P), P] ≤ Fr(P) since it is a proper subgroup, and P T is not F-essential by Proposition 1.7(d). This leaves only the cases where P ≅ C_{2}^{2} or P ≅ Q_{8}, and hence P ∈ H.

Now assume S = ⟨a, b, t⟩ ≅ C_{2^{n}} ≀ C_{2} (n ≥ 2), where A = ⟨a, b⟩ ≅ C_{2^{n}} × C_{2^{n}}, t^{2} = 1, and tat^{−1} = b. Set U_{i} = ⟨ab, a^{2^{n−1}}, a^{i}t⟩ for all i ∈ ℤ. Thus U_{i} ≅ C_{2^{n}} ×_{C_{2}} Q_{8} when i is even, U_{i} contains the cyclic subgroup ⟨a^{i}t⟩ of index two when i is odd, and aU_{i}a^{−1} = U_{i+2}. If P ≰ A, and P ∩ A ≩ Z(S) = ⟨ab⟩, then Z(P) = Z(S), so Aut(P) is not a 2-group by Lemma 3.3, and Aut(P/Z(S)) is not a 2-group by Lemma 2.1. If |P/Z(S)| ≥ 8, then P/Z(S) is dihedral, and we just saw that Aut(P/Z(S)) is a 2-group. Thus |P/Z(S)| = 4, and P = U_{i} for some i. When i is odd, A ∩ U_{i} (≅ C_{2^{n}} × C_{2}) is characteristic in U_{i} (the other two subgroups of index two containing Z(U_{i}) are cyclic), so Aut(U_{i}) is a 2-group by Lemma 2.1 again. This leaves the case P = U_{i} for even i, and thus P ∈ H.

If P ≤ S is such that P ≰ A and P ∩ A = ⟨ab⟩, then N_{S}(P) = ⟨P, a^{2^{n−1}}⟩, [N_{S}(P), P] = ⟨(ab)^{2^{n−1}}⟩ ≤ Fr(P), and P T is not F-essential by Proposition 1.7(d). Finally, if P ≤ A, then P = A ∈ H since P must be centric in S.

(e) Fix a 2-group T, a saturated fusion system F over S × T, and a subgroup P ≤ S such that P T is F-essential. By (d), P ∈ H.

By (c,a), |Out_{ST} (P T)| = |N_{S}(P)/P | = 2 and [N_{S}(P), P] is cyclic. Also, Out_{ST} (P T) ∈ Syl_{2}(Out_{F}(P T)) since P T is F-essential and hence fully normalized. So by Proposition 2.5, applied to the action of Out_{F}(P T) on P T/[P T, P T], there is α_{P} ∈ Aut_{F}(P T) of order 3 such that Out_{F}(P T) = Γ_{P} × H_{P}, where |H_{P} | is odd and Γ_{P} = ⟨[α_{P}], Out_{S}(P T)⟩ ≅ Σ_{3}.

Set P_{0} = [N_{S}(P), P] and P_{α} = [α_{P} , P T] for short. We must show that P_{0} ≤ P_{α} ≤ P Z(T) and P_{α} ∩ Z(T) = 1. We first check that P′ = [P, P] is α_{P} -invariant. If P is abelian, there is nothing to prove. Otherwise, |P′| = 2 by (c), P′ ≤ Z(P), and so P′ = P ∩ (Z(P T) ∩ [P T, P T]). The group Aut_{S}(P T) acts trivially on Z(P T) ∩ [P T, P T], so its normal closure in Aut_{F}(P T) acts trivially, and thus α_{P} acts trivially. In particular, P′ is α_{P} -invariant.

Thus α_{P} acts on P T/P′ = P^{ab} × T, and leaves invariant its center P Z(T)/P′. Hence P Z(T) is α_{P} -invariant, and P_{α} ≤ P Z(T), since it is contained in the subgroup generated by the α_{P}^{i}(P_{0}) for i = 0, 1, 2. Also, Z(P)T = C_{P T} (P Z(T)) is α_{P} -invariant, so [α_{P} , T] ≤ Z(P)T ∩ [α_{P} , P T] ≤ Z(P T).

By Proposition 2.5 (now applied to the Out_{F}(P T)-action on P Z(T)/P′), [Γ_{P} , P Z(T)/P′] = [α_{P} , P Z(T)/P′], and hence [α_{P} , P Z(T)/P′] ≥ P_{0}/P′. Thus P_{0} ≤ P_{α}·P′. Recall P_{0} is cyclic by (a); fix a generator g. Since P_{0} ≰ Fr(P) by Proposition 1.7(d) (and clearly P′ ≤ P_{0} = [N_{S}(P), P]), P′ ≤ ⟨g^{2}⟩. Since P_{0} ≤ P_{α}·P′, this means g^{k} ∈ P_{α} for some odd k, and thus P_{0} ≤ P_{α}.

Since P_{α} = [α_{P} , P T] ≤ P Z(T), P_{α} = [α_{P} , P Z(T)] by [G, Theorem 5.3.6]. Hence by Proposition 2.5 again applied to the Out_{F}(P T)-action on P Z(T)/P′, P_{α}/P′ is abelian of rank two. So if P_{α} ∩ Z(T) ≠ 1, then Ω_{1}(P_{α}/P′) is generated by Ω_{1}(P_{0}/P′) and an element of Z(T), both of which are fixed by c_{x} for x ∈ N_{S}(P)∖P. This is impossible, and thus P_{α} ∩ Z(T) = 1. □

We are now ready to prove Theorem B.

Theorem 4.8. Fix a pair of 2-groups S_{1} and S_{2}, and set S = S_{1} × S_{2}. Let F be a saturated fusion system over S such that O^{2}(F) = F = O^{2′}(F). Assume the following conditions hold.

(i) S_{1} ≅ D_{2^{n}} (n ≥ 3), SD_{2^{n}} (n ≥ 4), or C_{2^{n}} ≀ C_{2} (n ≥ 2).

(ii) Either S_{2} contains no proper subgroup isomorphic to S_{1} × S_{1}; or (more generally)

(iii) if P = S or P is an F-essential subgroup containing S_{1}, then Ω_{1}(Z(S_{1})) is Aut_{F}(P)- invariant.

Then F ≅ F_{1} × F_{2} for some pair of saturated fusion systems F_{i} over S_{i}.

Proof. For i = 1, 2, set

Z_{i} = Z(S_{i}),  Ŝ_{i} = S_{i}Z_{3−i},  U_{i} = {P ≤ S_{i} | P S_{3−i} is F-essential},

and U_{i}^{+} = U_{i} ∪ {S_{i}}. For each P ∈ U_{i}^{+}, set

P^{•} = P S_{3−i} and P̂ = P Z_{3−i} .

By Lemma 1.8, the only F-essential subgroups of S are the subgroups P^{•} for P ∈ U_{1} ∪ U_{2}. As in Proposition 4.3, define fusion subsystems

F_{i}^{•} = ⟨Aut_{F}(P^{•}) | P ∈ U_{i}^{+}⟩ .

We check that (ii) implies (iii). Assume S_{2} contains no subgroup isomorphic to S_{1} × S_{1}. Fix a fully normalized subgroup P ≤ S containing S_{1}; thus P = S_{1} × P_{2} for some P_{2} ≤ S_{2}. Let Aut_{F}^{0}(P) be the subgroup of those automorphisms in Aut_{F}(P) which send Ω_{1}(Z_{1}) to itself. By Proposition 3.2(a), [Aut_{F}(P):Aut_{F}^{0}(P)] ≤ 2, so Aut_{F}(P) = Aut_{F}^{0}(P) by the Sylow axiom, and (iii) holds. So from now on, we assume (iii).

Step 1: By (iii), the hypotheses of Lemma 4.4 hold, but with the roles of S_{1} and S_{2} switched. So by that lemma, there is a normal complement S_{2}^{0} to Z_{1} in Ŝ_{2} which is strongly closed in F_{2}^{•}. Choose φ ∈ Aut(S) such that φ|_{S_{1}} = Id, φ ≡ Id_{S} (mod S_{1}), and φ(S_{2}^{0}) = S_{2}. Upon replacing F by φFφ^{−1}, we can assume S_{2}^{0} = S_{2}.

Thus for P ∈ U_{2}^{+}, Aut_{F}(P^{•}) leaves P invariant, and acts on P^{•}/P ≅ S_{1} via inner automorphisms since Aut(S_{1}) is a 2-group by Lemma 4.7(a). Hence [Aut_{F}(P^{•}), P^{•}] ≤ [S_{1}, S_{1}]·P for each P ∈ U_{2}^{+}, and

𝔣𝔬𝔠(F_{2}^{•}) = ⟨[Aut_{F}(P^{•}), P^{•}] | P ∈ U_{2}^{+}⟩ ≤ [S_{1}, S_{1}]·S_{2} .  (1)

Step 2: By the Sylow axiom, Aut_{F}(S) is an extension of the 2-group Inn(S) by the odd order group Out_{F}(S). So by the Schur-Zassenhaus theorem (cf. [G, Theorem 6.2.1]), there is H ≤ Aut_{F}(S) of odd order, unique up to conjugacy by an element of Inn(S), such that Aut_{F}(S) = H·Inn(S).

By Step 1, S_{2} is Aut_{F}(S)-invariant. Since Aut(S/S_{2}) is a 2-group by Lemma 4.7(a), [H, S] ≤ S_{2}. Also, Ŝ_{1} = C_{S}(S_{2}) is Aut_{F}(S)-invariant, and hence [H, S_{1}] ≤ Ŝ_{1} ∩ S_{2} = Z_{2}. Since [Ŝ_{1}, Ŝ_{1}] = [S_{1}, S_{1}] is H-invariant, H acts trivially on this subgroup. We have now shown that

[H, S] ≤ S_{2} , [H, S_{1}] ≤ Z_{2} , and [H, [S_{1}, S_{1}]] = 1 .  (2)

Step 3: Throughout this step, we fix a subgroup P ∈ U_{1}. By Lemma 4.7(d), P ∈ H, where H is an explicitly defined set of subgroups of S_{1}. By Lemma 4.7(e),

Out_{F}(P^{•}) = Γ_{P} × H_{P} where |H_{P} | odd and Γ_{P} = ⟨[α_{P}], Out_{S_{1}}(P^{•})⟩ ≅ Σ_{3}  (3)

for some α_{P} ∈ Aut_{F}(P^{•}) of order three; and

[N_{S_{1}}(P), P] ≤ [α_{P} , P^{•}] ≤ P Z_{2} = P̂ and [α_{P} , P^{•}] ∩ Z_{2} = 1 .  (4)

For each β ∈ Aut_{F}(P^{•}) of odd order whose class [β] ∈ Out_{F}(P^{•}) is in H_{P}, [β] commutes with Out_{S}(P^{•}) ≤ Γ_{P}, so β normalizes Aut_{S}(P^{•}), and extends to an element of Aut_{F}(N_{S}(P^{•})) by the extension axiom. Hence β extends to β ∈ Aut_{F}(S), since by Lemma 4.7(c), no proper critical subgroup of S contains N_{S}(P^{•}). Upon replacing β by an appropriate power, we can assume it also has odd order. Then β is Inn(S)-conjugate to an element of H, and since [Aut_{S_{1}}(S), H] = 1 (since [H, S_{1}] ≤ Z_{2} by (2)), β is Aut_{S_{2}}(S)-conjugate to an element of H. Thus β is Inn(P^{•})-conjugate to the restriction of an element of H. Conversely, every element of H restricts to an automorphism of P^{•} since [H, S] ≤ S_{2} ≤ P^{•} by (2). So

H_{P} = {[η|_{P^{•}}] | η ∈ H} .  (5)

By Proposition 2.5, applied to the action of Out_{F}(P^{•}) on P̂^{ab}, H_{P} acts trivially on [α_{P} , P̂^{ab}], and hence H acts trivially on this group by (5). Also, H acts trivially on [P̂, P̂] ≤ [S_{1}, S_{1}] by (2), so it acts trivially on [α_{P} , P̂] by Lemma 2.1. Since [α_{P} , P^{•}] = [α_{P} , [α_{P} , P^{•}]] = [α_{P} , P̂] by [G, Theorem 5.3.6] (and (4)), this proves that

[H, [α_{P} , P^{•}]] = 1 .  (6)

Step 4: For P ∈ U_{1}, set P_{0} = [N_{S_{1}}(P), P] and P_{α} = [α_{P} , P^{•}]. We next claim that

for each P ∈ U_{1}, there are x_{P} ∈ P and z_{P} ∈ Z_{2} such that P_{α} = ⟨P_{0}, x_{P} z_{P} ⟩;
and such that either P is abelian and ⟨P_{0}, x_{P} ⟩ = P, or ⟨P_{0}, x_{P} ⟩ ≅ Q_{8}.

(7)

When S_{1} is dihedral or semidihedral, then P ≅ C_{2}^{2} or P ≅ Q_{8} and P_{0} has index two in P. Since by (4), P_{α} is contained in P Z_{2} and has trivial intersection with Z_{2} (and strictly contains the cyclic group P_{0}), it must have the form in (7). Similarly, when S_{1} is wreathed and P ≅ C_{2^{n}} × C_{2^{n}}, then P_{0} ≅ C_{2^{n}}, P_{α} is isomorphic to a subgroup of P by (4) and hence to P since it has an automorphism of order three, and again has the form described in (7).

By Lemma 4.7(d), it remains to consider the case where S_{1} is wreathed, and where (in the notation of the lemma) P = U_{2i} = ⟨ab, a^{2^{n−1}}, a^{2i}t⟩ ≅ C_{2^{n}} ×_{C_{2}} Q_{8} for some i. Then P contains a unique subgroup P_{1} ≅ Q_{8}, generated by all elements of order four in P ∖Z(P), and P_{0} has index two in P_{1}. Set g = (ab^{−1})^{2^{n−2}}: a generator of P_{0}. Since g ∈ P_{α}, g·α_{P} (g)·α_{P}^{2} (g) ∈ [P̂, P̂] = ⟨g^{2}⟩, so P_{α} = ⟨g, α_{P} (g)⟩. Set α_{P} (g) = x_{P} z_{P} where x_{P} ∈ P and z_{P} ∈ Z_{2}. Since g has order 4 and lies in P̂∖Z(P̂), so does α_{P} (g), and hence x_{P} ∈ P ∖Z(P) has order 4 since otherwise α_{P} (g^{2}) ∈ Z_{2} (contradicting (4)). Thus P_{α} = ⟨P_{0}, x_{P} z_{P} ⟩ and ⟨P_{0}, x_{P} ⟩ = P_{1} ≅ Q_{8}.

Step 5: Now, F = ⟨F_{1}^{•}, F_{2}^{•}⟩ since each F-essential subgroup has the form P^{•} for P ∈ U_{1} ∪ U_{2}, and so F = ⟨F_{2}^{•}, α_{P} | P ∈ U_{1}⟩ by (3) and (5). Hence by (1) and (7),

𝔣𝔬𝔠(F) = ⟨𝔣𝔬𝔠(F_{2}^{•}), P_{α} | P ∈ U_{1}⟩ ≤ ⟨[S_{1}, S_{1}], S_{2}, x_{P} z_{P} | P ∈ U_{1}⟩ .

Since O^{2}(F) = F by assumption, 𝔣𝔬𝔠(F) = S by Theorem 1.6(a), and so S_{1}^{ab} is generated by the classes [x_{P}] for P ∈ U_{1}. If P, Q ∈ U_{1} are S_{1}-conjugate, then there is g ∈ P such that Q = gP g^{−1} and α_{Q}^{±1} = c_{g}α_{P} c_{g}^{−1}, and hence Q_{α} = gP_{α}g^{−1}. Thus ⟨[S_{1}, S_{1}], x_{P} ⟩ = ⟨[S_{1}, S_{1}], x_{Q}⟩. Since S_{1}^{ab} is not cyclic, we conclude that U_{1} contains at least two S_{1}-conjugacy classes. So by Lemma 4.7(b,d), U_{1} = H and contains exactly two conjugacy classes.

Fix representatives P_{1}, P_{2} for the S-conjugacy classes in U_{1} = H, and set x_{i} = x_{P_{i}} and z_{i} = z_{P_{i}} for short. Set

S_{1}^{0} = ⟨[S_{1}, S_{1}], P_{α} | P ∈ U_{1}⟩ = ⟨[S_{1}, S_{1}], x_{1}z_{1}, x_{2}z_{2}⟩ .

We already saw that S_{1} = ⟨[S_{1}, S_{1}], x_{1}, x_{2}⟩, and at least one of the x_{i} has order two in S_{1}^{ab} by the proof of (7). Hence S_{1}^{ab} = ⟨[x_{1}]⟩ × ⟨[x_{2}]⟩. Also, |z_{i}| is at most the order of x_{i} in P_{iα}/P_{i0} (otherwise P_{iα} ∩ Z_{2} ≠ 1), and hence S_{1}^{0} = ⟨[S_{1}, S_{1}], x_{1}z_{1}, x_{2}z_{2}⟩ is a normal complement to Z_{2} in Ŝ_{1}.

Now, [H, S_{1}^{0}] = 1: [S_{1}, S_{1}] ≤ C_{S}(H) by (2), and [H, P_{α}] = 1 by (6). Thus S_{1}^{0} is Aut_{F}(S)- invariant. For P ∈ U_{1}, S_{1}^{0} ∩ P^{•} is α_{P} -invariant since it contains P_{α}, it is Aut_{S}(P^{•})-invariant since [N_{S_{1}}(P^{•}), P^{•}] ≤ [S_{1}, S_{1}] ≤ S_{1}^{0} and [S_{2}, S_{1}^{0}] = 1, and hence it is Aut_{F}(P^{•})-invariant by (3) and (5). This proves that S_{1}^{0} is strongly closed in F_{1}^{•}.

Let φ_{2} ∈ Aut(S) be the (unique) automorphism such that φ_{2}|_{S_{2}} = Id, φ_{2}(S_{1}^{0}) = S_{1}, and φ_{2}(g) ≡ g (mod Z_{2}) for all g ∈ S_{1}. Upon replacing F by φ_{2}Fφ_{2}^{−1}, we can assume S_{1}^{0} = S_{1}. So F satisfies the hypotheses of Proposition 4.3, and splits as a product of saturated fusion systems over the S_{i}. □

There is one more important case which we want to include, that of a product of three or more 2-groups which are pairwise isomorphic.

Theorem 4.9. Assume D ≅ D_{2^{n}} (n ≥ 3), SD_{2^{n}} (n ≥ 4), or C_{2^{n}} ≀ C_{2} (n ≥ 2). Let S be a product of k copies of D for some k ≥ 2. Let F be a saturated fusion system over S such that O^{2}(F) = F = O^{2′}(F). Then F is isomorphic to a product of saturated fusion systems over D.

Proof. Write S = S_{1} × S_{2} × ⋯ × S_{k}, where S_{i} ≅ D for each i. For each i, let z_{i} ∈ Z(S_{i})) be the element of order two, and set

U_{i}^{•} = {P ≤ S | P is F-essential, P ≥ S_{j} for all j ≠ i} .

By Lemma 1.8(b), ⋃_{i=1}^{k} U_{i}^{•} contains all F-essential subgroups of S.

Fix i and P ∈ U_{i}^{•}. By Lemma 4.7(a,c), Out_{S}(P) ≅ C_{2}, and [N_{S_{i}}(P), P] is cyclic. By Lemma 4.7(e), there is an automorphism α_{P} ∈ Aut_{F}(P) of order three, and a factorization Out_{F}(P) = Γ_{P} ×H_{P}, such that Γ_{P} = ⟨[α_{P}], Out_{S}(P)⟩ ≅ Σ_{3} and H_{P} has odd order, and such that for j ≠ i,

z_{i} ∈ [N_{S_{i}}(P), P] ≤ [α_{P} , P] ≤ S_{i}Z(S), [α_{P} , S_{j}] ≤ Z(P), and [α_{P} , P] ∩ ⟨z_{j}⟩ = 1. (1)

For each η ∈ Aut_{F}(P) such that [η] ∈ H_{P}, η normalizes Aut_{S}(P), hence extends to an F-automorphism of N_{S}(P) by the extension axiom, and to some η ∈ Aut_{F}(S) since no F-essential subgroup contains N_{S}(P) (Lemma 4.7(c)). Thus

P ∈ U_{i}^{•} ⟹ Aut_{F}(P) = ⟨α_{P} , Aut_{S}(P), η|_{P} | η ∈ Aut_{F}(S), η(P) = P ⟩ .  (2)

Fix subgroups P_{i} ∈ U_{i}^{•} and P_{j} ∈ U_{j}^{•} for some i ≠ j, and set Q = P_{i} ∩ P_{j}. Set G = Aut_{F}(Q), G_{1} = ⟨α_{P_{i}}|_{Q}, Aut_{S_{i}}(Q)⟩, and G_{2} = ⟨α_{P_{j}}|_{Q}, Aut_{S_{j}}(Q)⟩. By (1), [G_{1}, Q] = [α_{P_{i}}, Q] and [G_{2}, Q] = [α_{P_{j}}, Q]. If [G_{1}, Q]∩[G_{2}, Q] ≠ 1, then it contains an element in Ω_{1}(Z(S_{i}S_{j})) = ⟨z_{i}, z_{j}⟩, which is impossible since by by (1), [G_{1}, Q] ∩ ⟨z_{i}, z_{j}⟩ = ⟨z_{i}⟩ while [G_{2}, Q] ∩ ⟨z_{i}, z_{j}⟩ = ⟨z_{j}⟩. Thus [G_{1}, Q] ∩ [G_{2}, Q] = 1. For any x_{i} ∈ N_{S_{i}}(Q)∖Q and x_{j} ∈ N_{S_{j}}(Q)∖Q, [x_{i}x_{j}, Q] = [x_{i}Q]·[x_{j}, Q], so c_{x_{i}x_{j}} cannot be conjugate in Out_{F}(Q) to c_{x_{i}} or c_{x_{j}}. By the Sylow axiom, no element of G = Aut_{F}(Q) can exchange c_{x_{i}} and c_{x_{j}} under conjugation, and thus Aut_{S_{i}}(Q) and Aut_{S_{j}}(Q) are strongly closed in Aut_{S_{i}S_{j}}(Q) with respect to G. Since [α_{P_{i}}, S_{j}] ≤ Z(P_{i}) by (1), [α_{P_{i}}|_{Q}, Aut_{S_{j}}(Q)] = 1 and (by a similar argument) [α_{P_{j}}|_{Q}, Aut_{S_{i}}(Q)] = 1. The hypotheses of Proposition 2.9 are thus satisfied (applied with H_{1} = Aut_{S_{i}}(Q) and H_{2} = Aut_{S_{j}}(Q)), and hence [G_{1}, G_{2}] = 1. In particular, [α_{P_{i}}|_{Q}, α_{P_{j}}|_{Q}] = 1 in Aut(Q). So by Lemma 2.7,

i ≠ j, P_{i} ∈ U_{i}^{•}, P_{j} ∈ U_{j}^{•} ⟹ [α_{P_{i}}, [α_{P_{j}}, P_{j}]] = 1 = [α_{P_{j}}, [α_{P_{i}}, P_{i}]] .  (3)

Now, for each i, set

T_{i} = ⟨[S_{i}, S_{i}], [α_{P} , P] | P ∈ U_{i}^{•}⟩ .

Then T_{i} ≤ S_{i}Z(S) by (1). For j ≠ i and P_{j} ∈ U_{j}^{•}, [α_{P_{j}}, [S_{i}, S_{i}]] = 1 since [α_{P_{j}}, S_{i}] ≤ Z(P_{j}) by (1). By (3), α_{P_{j}} acts trivially on [α_{P} , P] for P ∈ U_{i}^{•}, and hence α_{P_{j}} acts trivially on T_{i}.

For each i, let X_{i} ⊆ T_{i} be the smallest subset which contains x_{i}, and which is invariant under Inn(S) and under α_{P} for all P ∈ U_{i}^{•}. Thus X_{i} is a set of involutions in T_{i}, and is pointwise fixed by α_{P} for each P ∈ U_{j}^{•} when j ≠ i. Also, either X_{i} = {x_{i}}, or no element of X_{i} ∩ Z(S) is fixed by all α_{P} for P ∈ U_{i}^{•}. In particular, for i ≠ j, any element x ∈ X_{i} ∩ X_{j} would have to be in T_{i} ∩ T_{j} ≤ Z(S), hence α_{P} (x) = x for all P ∈ U_{i}^{•} ∪ U_{j}^{•}, which we just saw would imply x = z_{i} and x = z_{j}. Since this is impossible, we conclude that X_{i} ∩ X_{j} = ∅ for i ≠ j.

By (2), F is generated by Aut_{F}(S) and the α_{P}. By construction, Aut_{F}(S) permutes the subsets X_{i}. Consider a morphism φ ∈ Hom_{F}(P, Q) where P, Q ≤ S are F-centric. Then Z(S) ≤ P, and for each i, z_{i} ∈ P ∩ X_{i} implies P ∩ X_{i} ≠ ∅. Since φ is a composite of restrictions of the α_{P} and morphisms in Aut_{F}(S), there is a unique permutation σ ∈ Σ_{k} such that φ(P ∩ X_{i}) ⊆ Q ∩ X_{σ(i)} for each i. Let

Ψ: Mor(F^{c}) ⟶ Σ_{k}

be the resulting map. Then Ψ sends inclusions to the identity, and preserves composition of morphisms and of permutations.

Let F_{0} ⊆ F be the “kernel” subsystem: the fusion system over S generated by those morphisms φ ∈ Mor(F^{c}) such that Ψ(φ) = 1. Since F is generated by Aut_{F}(S) and the α_{P}, and Ψ(α_{P}) = 1 for each P ∈ U_{i}^{•} (for each i), Im(Ψ) = Ψ(Aut_{F}(S)). Thus F_{0} is F-invariant, and a subgroup of S is F_{0}-centric if and only if it is F-centric by Lemma 1.4(b). Since Ψ(Inn(S)) = 1, Im(Ψ) is a subgroup of odd order. The Sylow and extension axioms on centric subgroups hold for F_{0} since they hold for F. Since by definition, F_{0} is generated by morphisms between centric subgroups, F_{0} is saturated by [BCGLO1, Theorem A]. Thus F_{0} is a normal fusion subsystem of odd index in F, and F_{0} = F since O^{2′}(F) = F by assumption.

By Proposition 3.2(a), Aut(S) contains a subgroup of index k whose elements all fix z_{1}, and so every element of Aut(S) sends z_{1} to some z_{i} (i = 1, …, k). Since no element of Aut_{F}(S) sends z_{1} ∈ X_{1} to z_{i} ∈ X_{i} for i ≠ 1, Aut_{F}(S) fixes z_{1}. For all P ∈ U_{i}^{•} for i ≠ 1, α_{P} (z_{1}) = z_{1} by (3) (and (1)); and hence Aut_{F}(P) fixes z_{1} by (2). So by Theorem 4.8, applied to the decomposition S = S_{1} × (S_{2} ⋯ S_{k}), F = F_{0} splits as a product of saturated fusion systems over S_{1} and S_{2} ⋯ S_{k}. The result now follows by induction on k. □

4.4. Examples.

It is easy to see why the condition O^{2′}(F) = F is needed in Proposition 4.3 (and it was also needed in the splitting result [AOV, Proposition 3.3]). Let F be any saturated fusion system over a 2-group S, such that O^{2′}(F) ⫋ F. Then there are fusion subsystems of F × F (over S × S) which contain O^{2′}(F) × O^{2′}(F), and which do not split as products of fusion systems over S.

For a more explicit example which satisfies all of the hypotheses of Proposition 4.3 and Theorems 4.5 and 4.8, except the condition O^{2′}(F) = F, let F be the 2-fusion system of a subgroup G ≤ P GL_{3}(4) × P GL_{3}(4) of index 3 which does not contain either factor.

It is a little less obvious why the condition O^{2}(F) = F is needed. As a first example, let F be the 2-fusion system of the symmetric group Σ_{6}. This is a fusion system over S ≅ D_{8} ×C_{2}, and satisfies all of the hypotheses of Proposition 4.3 and of Theorem 4.8 except the condition O^{2}(F) = F. (Note that all F-essential subgroups contain the second factor C_{2}.)

For an example which also satisfies the hypotheses of Theorem 4.5 (except the condition O^{2}(F) = F), let G ≤ Σ_{6} × P GL_{2}(9) be the subgroup of index two which contains neither factor. The Sylow 2-subgroups of G are isomorphic to D_{8} × D_{16}, but the fusion system of G does not split as a product of fusion systems over D_{8} and D_{16}.

The fusion system of the alternating group A_{14} — a fusion system over the 2-group D_{8} × (D_{8} ≀ C_{2}) — illustrates why we need to assume S_{2} does not contain a subgroup isomorphic to S_{1} × S_{1} in Theorems 4.5 and 4.8. Fusion systems of larger alternating groups give other examples of this.

To see a larger family of examples, fix any k ≥ 2 and any odd prime power q, and consider the simple group G = Ω_{4k}^{−}(q). Let ε ∈ {±1} be such that 4|(q − ε), and let 2^{n} be the largest power of 2 dividing q^{2} − 1. Then G contains subgroups of odd index

G = Ω_{4k}^{−}(q) ≥ GO_{4k−2}^{ε}(q) ≥ GO_{2}^{ε}(q) ≀ Σ_{2k−1} ≅ D_{2(q−ε)} ≀ Σ_{2k−1} ≥ D_{2^{n}} ≀ Σ_{2k−1}

(where GO_{n}^{±}(q) is the full orthogonal group). Thus each Sylow 2-subgroup S ≤ G contains a direct factor D_{2^{n}}. For F = F_{S}(G), O^{2}(F) = F by the focal subgroup theorem for G (and since O^{2}(G) = G), and O^{2′}(F) = F since Aut(S) is a 2-group.

Similar examples involving semidihedral groups or wreath products C_{2^{n}} ≀ C_{2} are obtained by considering the groups

G = SL_{4k+3}(q) ≥ GL_{4k+2}(q) ≥ GL_{2}(q) ≀ Σ_{2k+1}

for odd prime powers q. In particular, for any D as in Theorem 4.9, there is a saturated fusion system F over D × (D ≀ C_{2}) which is indecomposable and satisfies O^{2}(F) = F = O^{2′}(F).

References

[AOV] K. Andersen, B. Oliver, & J. Ventura, Reduced, tame, and exotic fusion systems (in preparation)

[BLO] C. Broto, R. Levi, & B. Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), 779–856

[BCGLO1] C. Broto, N. Castellana, J. Grodal, R. Levi, & B. Oliver, Subgroup families controlling p-local finite groups, Proc. London Math. Soc. 91 (2005), 325–354

[BCGLO2] C. Broto, N. Castellana, J. Grodal, R. Levi, & B. Oliver, Extensions of p-local finite groups, Trans. Amer. Math. Soc. 359 (2007), 3791–3858

[FT] W. Feit & J. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775–1029 [Gd] D. Goldschmidt, Strongly closed 2-subgroups of finite groups, Annals of Math. 102 (1975), 475–489 [G] D. Gorenstein, Finite groups, Harper & Row (1968)

[OV] B. Oliver & J. Ventura, Saturated fusion systems over 2-groups, Trans. Amer. Math. Soc. 361 (2009), 6661–6728

[Pg] L. Puig, Frobenius categories, J. Algebra 303 (2006), 309–357

[Sz2] M. Suzuki, Group theory II, Springer-Verlag (1986)

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

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