A CHARACTERISTIC SUBGROUP FOR FUSION SYSTEMS

SILVIA ONOFREI AND RADU STANCU

Abstract. As a counterpart for the prime 2 to Glauberman’s ZJ-theorem, Stellmacher proves that any nontrivial 2-group S has a nontrivial characteristic subgroup W (S) with the following property. For any finite Σ_{4}-free group G, with S a Sylow 2-subgroup of G and with O_{2}(G) self-centralizing, the subgroup W (S) is normal in G. We generalize Stellmacher’s result to fusion systems. A similar construction of W (S) can be done for odd primes and gives rise to a Glauberman functor.

1. Introduction

A fundamental result in the theory of finite groups is Glauberman’s ZJ-theorem [6]. For p be an odd prime, G a finite group and S a Sylow p-subgroup of G, the ZJ-theorem asserts that the center of the Thompson group Z(J(S)) is normal in G whenever G is Qd(p)-free and C_{G}(O_{p}(G)) ≤ O_{p}(G). Recall that for a finite p-group Q, the Thompson subgroup J(Q) is the subgroup generated by the abelian subgroups of Q of largest order and that the group Qd(p) = (ℤ_{p} × ℤ_{p}) : SL(2, p) is the extension of the 2-dimensional vector space over 𝔽_{p} (the field with p elements) by SL(2, p) with its natural action on this vector space. A group G is H-free if no section of G is isomorphic to H; see also Section 3.

More recently, a proof of the ZJ-theorem, in the context of fusion systems, was given by Kessar and Linckelman [10]. The authors introduce the notion of Qd(p)-free fusion system and prove that if F is a Qd(p)-free fusion system on a finite p-group S, with p an odd prime, then F is controlled by W (S), for any Glauberman functor W. The related notions of characteristic p-functor and Glauberman functor were initially defined in [11, Definition 1.3]; they are given below in Definition 3.4.

For p = 2 the ZJ-theorem does not hold anymore. In [7, Question 16.1], Glauberman asks whether there exists a subgroup which is characteristic in a Sylow 2-subgroup S of a Σ_{4}-free group G, with the property C_{G}(O_{p}(G)) ≤ O_{p}(G). Here Σ_{4} denotes the symmetric group on four letters.

Date: November 22, 2008.

1991 Mathematics Subject Classification. 20D15, 20E25.

2 SILVIA ONOFREI AND RADU STANCU

The answer to Glauberman’s question was given by Stellmacher [26], who also obtained a different proof of the ZJ-theorem [24, 25]. Stellmacher’s idea is to approximate such a subgroup via subgroups of Z(J(S)); see [13, Section 9.4] for an overview of this approach. The main theorem in [26] (see also 6.4 in the Appendix) can be phrased as follows:

Theorem (Stellmacher): Let S be a finite nontrivial 2-group. Then there exists a nontrivial characteristic subgroup W (S) of S which is normal in G, for every finite Σ_{4}-free group G with S a Sylow 2-subgroup and C_{G}(O_{2}(G)) ≤ O_{2}(G).

Remark that the condition (III) in [26] is not necessary. A proof of this fact uses Lemmas 6.5 and 6.6 and Remark 6.7 in the Appendix.

In this paper we generalize Stellmacher’s approach to fusion systems. Our main result is a proof of Stellmacher’s version of the ZJ-theorem in the context of fusion systems:

Theorem 1.1. Let S be a finite 2-group and let F be a Σ_{4}-free fusion system over S. Then there exists a nontrivial characteristic subgroup W (S) of S with the property that F = N_{F}(W (S)).

Since Stellmacher’s construction of W (S) gives rise to a Glauberman functor (see Section 4 for details) we can combine Theorem B in [10] with Theorem 1.1 in our paper to obtain the more general result which is independent of the nature of the prime p:

Theorem 1.2. Let S be a finite p-group and let F be a Qd(p)-free fusion system over S. Then there exists a nontrivial characteristic subgroup W (S) of S with the property that F = N_{F}(W (S)).

Using the same construction for W (S) as in the above theorem, the normal complement theorem due to Thompson [13, 9.4.7] can be phrased as:

Theorem (Thompson): Let G be a finite group, p an odd prime and S a Sylow p-subgroup of G. Then G has a normal p-complement provided N_{G}(W (S)) has such a complement.

Our third result generalizes Thompson’s theorem to the class of fusion systems. This result is similar to Theorem A in [10], except that we replace the group Z(J) with the group W (S) defined in Section 4:

Theorem 1.3. Let F be a fusion system over a finite p-group S, with p an odd prime. Then F = F_{S}(S) if and only if N_{F}(W (S)) = F_{S}(S).

The paper is organized as follows. Section 2 contains background material on fusion systems. In Section 3, the notions of H-free fusion system, characteristic p-functor and Glauberman functor are defined; further properties of fusion systems are discussed. The

A CHARACTERISTIC SUBGROUP FOR FUSION SYSTEMS 3

characteristic subgroup W (S) is constructed, via two different methods, in Section 4. The proofs of the theorems are given in Section 5. In the Appendix a few related results from group theory are included.

2. Background on Fusion Systems

Fusion systems were introduced by Puig in 1990 [19, 20] in an effort to axiomatize the plocal structure of a finite group and of a block of a group algebra - the work was published only recently [21] but was known to the community long before. In 2000 Broto, Levi and Oliver [5] enriched this axiomatic approach by introducing the centric linking system that is essentially linked to the associated p-completed classifying space. The third author used this axiomatic frame to reformulate and solve the Martino-Priddy conjecture [17, 18]. Broto, Levi and Oliver give a different definition of the fusion systems which they proved to be equivalent to Puig’s definition. In this paper we use a simplified definition which we find more elegant, equivalent to the above ones [12].

We start with a more general definition, following [15].

A category F on a finite p-group S is a category whose objects are the subgroups of S and whose set of morphisms between the subgroups Q and R of S, is a set Hom_{F}(Q, R) of injective group homomorphisms from Q to R, with the following properties:

(1) if Q ≤ R then the inclusion of Q in R is a morphism in Hom_{F}(Q, R);

(2) for any φ ∈ Hom_{F}(Q, R) the induced isomorphism Q ≃ φ(Q) and its inverse are morphisms in F;

(3) the composition of morphisms in F is the usual composition of group homomorphisms.

Let F_{1} be a category on S_{1} and F_{2} a category on S_{2}. A morphism between F_{1} and F_{2} is a pair (α, Θ) with α ∈ Aut(S) and Θ : F_{1} → F_{2} a covariant functor, such that:

(i) for any subgroup Q of S, α(Q) = Θ(Q); (ii) for any morphism φ in F_{1}, Θ(φ) ∘ α = α ∘ φ.

In the following we give a series of useful definitions in a category F on S. If there exists an isomorphism φ ∈ Hom_{F}(Q, R) we say that Q and R are F-conjugate.

We say that a subgroup Q of S is

(i) fully F-centralized if |C_{S}(Q)| ≥ |C_{S}(Q′)| for all Q′ ≤ S which are F-conjugate to Q. (ii) fully F-normalized if |N_{S}(Q)| ≥ |N_{S}(Q′)| for all Q′ ≤ S which are F-conjugate to Q. (iii) F-centric if C_{S}(φ(Q)) ⊆ φ(Q), for all φ ∈ Hom_{F}(Q, S).

4 SILVIA ONOFREI AND RADU STANCU

(iv) F-radical if O_{p}(Out_{F}(Q)) = 1.

(v) F-essential if Q is F-centric and Out_{F}(Q) has a strongly p-embedded proper subgroup M (that is M contains a Sylow p-subgroup P of Out_{F}(Q) such that P ≠ ^{φ}P and φP ∩ P = {1} for every φ ∈ Out_{F}(Q) \ M ).

For Q, R ≤ S we denote Hom_{S}(Q, R) := {u ∈ S |^{u}Q ≤ R}/C_{S}(Q) and Aut_{S}(Q) := Hom_{S}(Q, Q). Other useful notations are Aut_{F}(Q) := Hom_{F}(Q, Q) and Out_{F}(Q) := Aut_{F}(Q)/Aut_{Q}(Q).

We are now ready to give the definition of a fusion system.

A fusion system on a finite p-group S is a category F on S satisfying the following properties:

FS1. Hom_{S}(Q, R) ⊆ Hom_{F}(Q, R) for all Q, R ≤ S . FS2. Aut_{S}(S) is a Sylow p-subgroup of Aut_{F}(S). FS3. Every φ : Q → S such that φ(Q) is fully F-normalized extends to a morphism

φ̂: N_{φ} → S where

N_{φ} = {x ∈ N_{S}(Q) | ∃y ∈ N_{S}(φ(Q)), φ(^{x}u) = ^{y}φ(u), ∀u ∈ Q} .

Remark that N_{φ} is the largest subgroup of N_{S}(Q) such that ^{φ}(N_{φ}/C_{S}(Q)) ≤ Aut_{S}(φ(Q)). Thus we always have QC_{S}(Q) ≤ N_{φ} ≤ N_{S}(Q).

If F is a fusion system on S and Q ≤ S we have the following equivalent characterization of being fully F-normalized.

Proposition 2.1 ([14], Proposition 1.6). A subgroup Q of S is fully F-normalized if and only if Q is fully F-centralized and Aut_{S}(Q) is a Sylow p-subgroup of Aut_{F}(Q).

In the following Lemma we recollect two useful properties involving fully F-normalized subgroups; see also [10, Lemmas 2.2, 2.3]. For completeness we include the proofs.

Lemma 2.2. Let F be a fusion system on a finite p-group S and Q a subgroup of S.

a) There is a morphism φ ∈ Hom_{F}(N_{S}(Q), S) such that φ(Q) is fully F-normalized.

b) If Q is fully F-normalized, then φ(Q) is fully normalized, for any morphism φ ∈ Hom_{F}(N_{S}(Q), S).

Proof. a) Let ψ : Q → S be a morphism with ψ(Q) fully F-normalized. By Proposition 2.1, Aut_{S}(ψ(Q)) is a Sylow p-subgroup of Aut_{F}(ψ(Q)). Since ψ ∘ Aut_{S}(Q) ∘ ψ^{−1} is a p-subgroup of Aut_{F}(ψ(Q)) it follows that there exists a morphism τ ∈ Aut_{F}(ψ(Q)) with τψ ∘ Aut_{S}(Q) ∘ ψ^{−1}τ^{−1} ≤ Aut_{S}(ψ(Q)). Set α = τψ and observe that α(Q) is fully Fnormalized. By the extension axiom FS3, α extends to a morphism φ : N_{α} → S. But

A CHARACTERISTIC SUBGROUP FOR FUSION SYSTEMS 5

since α ∘ Aut_{S}(Q) ∘ α^{−1} ≤ Aut_{S}(ψ(Q)) it follows that N_{α} = N_{S}(Q). Henceforth there exists a morphism φ ∈ Hom_{F}(N_{S}(Q), S) such that φ(Q) is fully F-normalized.

b) Since Q is fully F-normalized and since φ is a morphism in F, hence injective, we have: φ(N_{S}(Q)) = N_{S}(φ(Q)). □

Puig [19] gave analogous notions for the normalizer and the centralizer in a finite group:

The normalizer of Q in F is the category N_{F}(Q) on N_{S}(Q) having as morphisms those morphisms φ ∈ Hom_{F}(R, T), for R and T subgroups of N_{S}(Q), satisfying that there exists a morphism φ̂ ∈ Hom_{F}(QR, QT) such that φ̂|_{Q} ∈ Aut_{F}(Q) and φ̂|_{R} = φ. If Q ≤ S has the property that F = N_{F}(Q) then we say that Q is normal in F. The largest subgroup of S which is normal in F will be denoted O_{p}(F).

The centralizer of Q in F is the category C_{F}(Q) on C_{S}(Q) having as morphisms those morphisms φ ∈ Hom_{F}(R, T), with R and T subgroups of C_{S}(Q), satisfying that there exists a morphism φ̂ ∈ Hom_{F}(QR, QT) such that φ̂|_{Q} = id_{Q} and φ̂|_{R} = φ.

Also denote by N_{S}(Q)C_{F}(Q) the category on N_{S}(Q) having as morphisms all group homomorphisms φ : P → R, for P and R subgroups of N_{S}(Q), for which there exists a morphism ψ : QP → QR and x ∈ N_{S}(Q) such that ψ|_{Q} = c_{x} (the morphism induced by conjugation by x) and ψ|_{P} = φ.

Proposition 2.3 ([20], Proposition 2.8). If Q is fully F-normalized then N_{F}(Q) is a fusion system on N_{S}(Q). If Q is fully F-centralized then C_{F}(Q) and N_{S}(Q)C_{F}(Q) are fusion systems on C_{S}(Q).

Alperin’s theorem on p-local control of fusion also holds for fusion systems. First we set up this theorem’s notations and terminology. If φ ∈ Aut_{F}(S) we say that φ is a maximal F-automorphism. If φ ∈ Aut_{F}(E), with E an F-essential subgroup of S, we say that φ is an essential F-automorphism. Alperin’s fusion theorem asserts that the essential and maximal F-automorphisms suffice to determine the whole fusion system F.

Theorem 2.4 (Alperin). Any morphism φ ∈ Hom_{F}(Q, S) can be written as the composition of restrictions of essential F-automorphisms, followed by the restriction of a maximal F-automorphism. More precisely, there exists

(a) an integer n ≥ 0,

(b) a set of F-isomorphic subgroups of S, Q = Q_{0}, Q_{1}, …, Q_{n}, Q_{n+1} = φ(Q),

(c) a set of F-essential, fully F-normalized subgroups E_{i} of S containing Q_{i−1} and Q_{i}, for all 1 ≤ i ≤ n,

(d) a set of essential automorphisms ψ_{i} ∈ Aut_{F}(E_{i}) satisfying ψ_{i}(Q_{i−1}) = Q_{i}, for all 1 ≤ i ≤ n and

(e) a maximal automorphism ψ_{n+1} ∈ Aut_{F}(S) satisfying ψ_{i}(Q_{n}) = Q_{n+1},

such that we have

φ(u) = ψ_{n+1}ψ_{n} … ψ_{1}(u), for all u ∈ Q .

The reader can find a proof of this theorem in [23]; an alternative proof of this theorem in a different axiomatic setting was given by Puig [20, Corollary 3.9] and another in a less general form, using F-centric, F-radical subgroups instead of F-essential subgroups, can be found in [5, Theorem A.10]. We use this later form in the proof of Lemma 3.7 .

The classical examples of a fusion systems are the ones coming from the p-local structure of a finite group G. If S is a Sylow p-subgroup of G then we denote by F_{S}(G) the fusion system on S having as morphisms

Hom_{F_{S}(G)}(P, Q) = N_{G}(P, Q)/C_{G}(P) = Hom_{G}(P, Q)

where P, Q are subgroups of S and N_{G}(P, Q) = {g ∈ G |^{g}P ≤ Q} is the G-transporter from P to Q.

There are examples of fusion systems that do not come from a finite group (see eg. [22]). But there are also particular cases when one can construct a finite group with p-local structure equivalent to a given fusion system. This is the case for constrained fusion systems. The fusion system F is said to be constrained if O_{p}(F) is F-centric. Any constrained fusion system was proven to come from a finite group by Broto, Castellana, Grodal, Levi and Oliver:

Theorem 2.5. [2, Theorem 4.3] Let F be a fusion system on S and suppose that there exists an F-centric subgroup Q of S such that N_{F}(Q) = F (in particular F is constrained). Then there exists a, unique up to isomorphism, finite p′-reduced p-constrained group L_{Q}^{F} (i.e O_{p′}(L_{Q}^{F}) = 1 and Q is a normal subgroup of L_{Q}^{F}) having S as a Sylow p-subgroup and such that F = F_{S}(L_{Q}^{F}). Furthermore L_{Q}^{F}/Z(Q) ≃ Aut_{F}(Q).

3. Further results on fusion systems

Let G be a finite group and p a prime divisor of its order. If A ⊴ B ≤ G then B/A is called a section of G. We say that H is involved in G if H is isomorphic to a section of G. If H is not involved in G then G is called H-free.

Following [10] we say that the fusion system F on S is H-free if H is not involved in any of the groups L_{Q}^{F}, for Q running over the set of F-centric, F-radical and fully Fnormalized subgroups of S. In some particular cases the property of being H-free passes to subsystems and quotient systems as the next two results from [10] show.

A CHARACTERISTIC SUBGROUP FOR FUSION SYSTEMS 7

Proposition 3.1. [10, Proposition 6.3] Let F be a fusion system on a finite p-group S and let Q be a fully F-normalized subgroup of S. If F is H-free, then so is any fusion subsystem of F which lies between N_{F}(Q) and N_{S}(Q)C_{F}(Q). In particular, if F is H-free, so are N_{F}(Q) and N_{S}(Q)C_{F}(Q).

Let F be a fusion system on S and let Q be a subgroup of S with the property F = N_{F}(Q). The category F/Q on S/Q is defined as follows: for Q ≤ P, R ≤ S, a group homomorphism ψ : P/Q → R/Q is a morphism in F/Q if there is a morphism φ ∈ Hom_{F}(P, R) satisfying ψ(xQ) = φ(x)Q for all x ∈ P. The fact that F/Q is a fusion system on S/Q is due to Puig [19], see also [10, Proposition 2.8].

Proposition 3.2. [10, Proposition 6.4] Let F be a fusion system on a finite p-group S and let Q be a normal subgroup of S such that F = N_{F}(Q). If F is H-free then F/Q is also H-free.

The following result generalizes the main technical step in the proof of [10, Proposition 5.2].

Proposition 3.3. Let F be a fusion system on a finite p-group S and let W_{i}, 1 ≤ i ≤ n be subgroups of S such that

(a) the subgroup W_{i+1} is a characteristic subgroup of N_{S}(W_{i}) for 1 ≤ i ≤ n − 1;

(b) the subgroup W_{i} is fully F-normalized for 1 ≤ i ≤ n − 1.

Then there exists a morphism φ ∈ Hom_{F}(N_{S}(W_{n}), S) such that φ(W_{i}) is fully F-normalized for all 1 ≤ i ≤ n. In particular φ(N_{S}(W_{i})) = N_{S}(φ(W_{i})). If, moreover, W_{i} is F-centric and/or F-radical for some 1 ≤ i ≤ n, then so is φ(W_{i}).

Proof. If follows from Lemma 2.2(a) that there exists a morphism φ ∈ Hom_{F}(N_{S}(W_{n}), S) such that φ(W_{n}) is fully F-normalized. According to condition (a), W_{i+1} is a characteristic subgroup of N_{S}(W_{i}), for 1 ≤ i ≤ n − 1 and therefore N_{S}(W_{i}) ≤ N_{S}(W_{i+1}). In particular N_{S}(W_{i}) ≤ N_{S}(W_{n}) and the morphism φ is defined on N_{S}(W_{i}) for all 1 ≤ i ≤ n.

Next we show φ(W_{i}) is fully F-normalized for all 1 ≤ i ≤ n−1. By elementary group theory φ(N_{S}(W_{i})) ≤ N_{S}(φ(W_{i})). Since φ is injective it follows that |N_{S}(W_{i})| ≤ |φ(N_{S}(W_{i}))|. But, according to (b), W_{i} is fully F-normalized and |N_{S}(W_{i})| ≥ |N_{S}(φ(W_{i}))|. It follows now that |N_{S}(W_{i})| = |N_{S}(φ(W_{i}))| which shows that φ(W_{i}) is fully F-normalized and that φ(N_{S}(W_{i})) = N_{S}(φ(W_{i})), for all 1 ≤ i ≤ n − 1.

Since W_{i} is fully F-normalized for all 1 ≤ i ≤ n, it is also fully F-centralized. Thus φ(C_{S}(W_{i})) = C_{S}(φ(W_{i})) and if W_{i} is F-centric, then so is φ(W_{i}). Moreover it is a general fact that ^{φ}Aut_{F}(W_{i}) = Aut_{F}(φ(W_{i})) and ^{φ}Aut_{S}(W_{i}) = Aut_{S}(φ(W_{i})) so if W_{i} is F-radical, then so is φ(W_{i}). □

We need the following definition reproduced from [10, Definition 5.1].

Definition 3.4. A positive characteristic functor is a map sending any nontrivial finite pgroup S to a nontrivial characteristic subgroup W (S) of S such that W (φ(S)) = φ(W (S)) for every φ ∈ Aut(S). A positive characteristic functor is a Glauberman functor if whenever S is a Sylow p-subgroup of a Qd(p)-free finite group L which satisfies C_{L}(O_{p}(L)) = Z(O_{p}(L)), then W (S) is normal in L.

Using Proposition 3.3 we can give a different proof for Proposition 5.3 in [10].

Proposition 3.5. [10, Proposition 5.3] Let F be a fusion system on a finite p-group S and let W be a positive characteristic functor. Assume that for any non-trivial, proper, F-centric, F-radical, fully F-normalized subgroup Q of S the following holds N_{F}(Q) = N_{N_{F} (Q)}(W (N_{S}(Q))). Then F = N_{F}(W (S)).

Proof. Suppose that the conclusion does not hold. By Alperin’s Fusion Theorem there exists a proper, F-centric, F-radical, fully F-normalized subgroup Q of S such that Aut_{N_{F} (W (S))}(Q) ⊊ Aut_{F}(Q).

Set W_{1} = Q and define recursively W_{i+1} := W (N_{S}(W_{i})). So W_{i+1} is characteristic in N_{S}(W_{i}) implying that we get the following inclusions N_{S}(W_{i}) < N_{S}(N_{S}(W_{i})) ≤ N_{S}(W_{i+1}) this in its turn implies there exists n ≥ 1 such that the sequence of N_{S}(W_{i}) for 1 ≤ i ≤ n is strictly increasing and N_{S}(W_{n}) = S. Observe that if N_{S}(W_{i}) = N_{S}(W_{i+1}) then N_{S}(W_{i}) = N_{S}(N_{S}(W_{i})) ≤ S and by an elementary property of p-groups it follows that N_{S}(W_{i}) = S, thus indeed the sequence eventually reaches S.

Moreover the sequence {W_{i}, 1 ≤ i ≤ n+1} can be chosen so that all its terms are fully Fnormalized. This can be done recursively by applying Proposition 3.3, for all 2 ≤ k ≤ n+1 to the partial subsequences {W_{i}, 1 ≤ i ≤ k}. The W_{i}’s are successively modified by replacing them with their images through the morphism φ given by Proposition 3.3.

Consider the sequence of the normalizers in F of the W_{i}’s for 1 ≤ i ≤ n + 1. Given that W_{i}, 1 ≤ i ≤ n are fully F-normalized, we have that N_{F}(W_{i}) is a fusion system on N_{S}(W_{i}). It follows from our assumption that N_{F}(W_{i}) ⊆ N_{F}(W_{i+1}) for all 1 ≤ i ≤ n − 1. But then N_{F}(Q) = N_{F}(W_{1}) ⊆ N_{F}(W_{n+1}) = N_{F}(W (S)). At the level of morphisms on Q this gives Aut_{F}(Q) ⊆ Aut_{N_{F} (W (S))}(Q) which is a contradiction with the initial supposition on Q. □

Proposition 3.5 is used in [10] to prove the next important result.

A CHARACTERISTIC SUBGROUP FOR FUSION SYSTEMS 9

Proposition 3.6. [10, Proposition 3.4] Let S be a finite p-group and let Q be a normal subgroup of S. Let F and G be fusion systems on S such that F = SC_{F}(Q) and such that G ⊆ F. Let P be a normal subgroup of S containing Q. We have G = N_{F}(P) if and only if G/Q = N_{F/Q}(P/Q).

Next we give an application of the Frattini argument to fusion systems. The group theoretic result states that, if G is a finite group, then thefactorization G = C_{G}(Q)N_{G}(R) holds with Q = O_{p}(G) and R = C_{G}(QC_{S}(Q)). This is easily seen to be true as C_{G}(Q) ⊴ G and N_{G}(C_{S}(Q)) ≤ N_{G}(R), then an application of the Frattini argument gives the result.

Lemma 3.7. Let F be a fusion system on S, Q = O_{p}(F) and R = QC_{S}(Q). Set F_{1} = SC_{F}(Q) and F_{2} = N_{F}(R). Then F = ⟨F_{1}, F_{2}⟩.

Proof. First remark that F, F_{1} and F_{2} are fusion systems on S with F containing the other two. By Alperin’s fusion theorem (see Theorem 2.4), it is enough to prove that for every F-centric, F-radical, fully F-normalized subgroup U of S we have Aut_{F}(U) = Aut_{G}(U) with G = ⟨F_{1}, F_{2}⟩.

We shall prove that every φ ∈ Aut_{F}(U) can be written as composition of morphisms in F_{1} and F_{2} and thus will be contained in Aut_{G}(U). This will finish our proof as the opposite inclusion is clearly satisfied.

Given that U is F-centric and F-radical we have by [16, Proposition 5.6] that Q ≤ U. Hence φ restricts to an automorphism θ ∈ Aut_{F}(Q). Now we have that N_{θ} contains U and R so it contains UR. Given that Q is fully F-normalized θ extends to χ ∈ Hom_{F}(UR, S). Moreover χ(R) = χ(Q)C_{S}(χ(Q)) = R so in fact χ ∈ Hom_{F_{2}}(UR, S).

Denote by ψ the restriction to U of χ; then ψ ∈ Hom_{F_{2}}(U, ψ(U)). Both φ and ψ restrict as θ on Q so φ ∘ ψ^{−1} belongs to Hom_{F_{1}}(ψ(U), U). The conclusion in the lemma follows as φ = φ ∘ ψ^{−1} ∘ ψ ∈ Aut_{G}(Q). □

We close this section with a straightforward result on fusion control in fusion systems.

Lemma 3.8. Let W be a fully F-normalized subgroup of S and suppose that there are two fusion subsystems F_{1} and F_{2} of F such that F = ⟨F_{1}, F_{2}⟩. If moreover F_{1} = N_{F_{1}}(W) and F_{2} = N_{F_{2}}(W). Then F = N_{F}(W).

Proof. We have that ⟨N_{F_{1}}(W), N_{F_{2}}(W)⟩ ⊆ N_{F}(W) ⊆ F. The result follows. □

4. A characteristic subgroup of S

Let S be a finite p-group. In this section we construct a subgroup W (S) which is characteristic in S and such that Ω(Z(S)) ≤ W (S) ≤ Ω(Z(J(S))), and with the property that W (S) ⊴ E for all (φ, E) ∈ U_{J}, with U_{J} a class of embeddings defined below. The notation Ω(H) stands for the subgroup of H generated by all the elements of order p, while J(S) denotes the Thompson subgroup of S defined in Section 1. We shall give below two different, although equivalent, constructions of this characteristic subgroup of S which we shall denote W (S) and W. The first construction follows the approach developed by Stellmacher [24, 26] for finite groups, in which such a subgroup is approximated from various subgroups of Z(J(S)). The second construction uses basic properties of fusion systems.

4.1. The group W(S). An embedding is a pair (φ, E) where φ ∈ Aut(S) and E is a category on φ(S) = S. Let C denote the family of all embeddings of S. A nonempty subclass U of C is characteristically closed if (φα, E) ∈ U whenever (φ, E) ∈ U and α ∈ Aut(S).

An equivalence between two embeddings (φ_{1}, E_{1}) and (φ_{2}, E_{2}) is a morphism (α, Θ) : E_{1} → E_{2} with αφ_{1} = φ_{2} and Hom_{E_{2}}(φ_{2}(Q), φ_{2}(R)) = α ∘ Hom_{E_{1}}(φ_{1}(Q), φ_{1}(R)) ∘ α^{−1}_{|φ_{1}(Q)}. The equivalence of embeddings defines an equivalence relation on U. Since S is a finite group, the collection of equivalence classes [U] is a finite set.

Let O_{S}(U) denote the largest subgroup of S which satisfies the property that φ(O_{S}(U)) is normal in E for every embedding (φ, E) in U.

Lemma 4.1. Let U be a characteristically closed subclass of C and let α ∈ Aut(S). Let Q be a subgroup of S with the property that φ(Q) ⊴ E for every (φ, E) ∈ U. Then φ(α(Q)) ⊴ E for every (φ, E). In particular O_{S}(U) is a characteristic subgroup of S.

Proof. Observe that since U is characteristically closed and (φ, E) ∈ U then (φα, E) ∈ U and thus φ(α(Q)) ⊴ E for every (φ, E) ∈ U. The fact that O_{S}(U) is Aut(S)-invariant follows from its definition. □

Let U_{J} denote the class of embeddings (φ, E) which satisfy the following conditions:

(U1) U_{J} is characteristically closed. (U2) J(φ(S)) = J(S) is normal in E for all (φ, E) ∈ U_{J}. (U3) E is a Qd(p)-free fusion system.

A CHARACTERISTIC SUBGROUP FOR FUSION SYSTEMS 11

For a p-group P we set A(P) = Ω(Z(P)) and B(P) = Ω(Z(J(P))). Remark that A(P) ≤ B(P) as Z(P) ≤ J(P). Note that α(A(P)) = A(α(P)) and α(B(P)) = B(α(P)) for all α ∈ Aut(P), as A(P) and B(P) are characteristic subgroups of P.

Define recursively a subgroup W (S) ≤ B(S) as follows. Let

W_{0} := A(S) = Ω(Z(S)) ≤ B(S)

and assume that for i ≥ 1 the subgroups W_{0}, W_{1}, … W_{i−1} with

W_{0} < W_{1} < … W_{i−1} ≤ B(S)

are defined. If φ(W_{i−1}) ⊴ E for all (φ, E) ∈ U_{J} then set W (S) := W_{i−1}. Otherwise, choose (φ_{i}, E_{i}) ∈ U_{J} to be such that φ_{i}(W_{i−1}) is not normal in E_{i} and define

W_{i} := φ_{i}^{−1} ⟨φ_{i}(W_{i−1})^{E_{i}}⟩ = φ_{i}^{−1} ⟨ψ(φ_{i}(W_{i−1}))|ψ ∈ Hom_{E_{i}}(φ_{i}(W_{i−1}), φ_{i}(S))⟩

to be the preimage in S of the group generated by the E_{i}-orbit of φ_{i}(W_{i−1}).

Since B(φ_{i}(S)) is a characteristic subgroup of J(φ_{i}(S)), which is itself normal in E_{i}, it follows that B(φ_{i}(S)) is also normal in E_{i}. Clearly φ_{i}(W_{i}) ≤ B(φ_{i}(S)) since φ_{i}(W_{i}) is generated by various conjugates of φ_{i}(W_{i−1}) and φ_{i}(W_{i−1}) ≤ B(φ_{i}(S)) by construction. Thus we have:

A(φ_{i}(S)) ≤ φ_{i}(W_{i−1}) < φ_{i}(W_{i}) ≤ B(φ_{i}(S)) ⊴ E_{i}

as A(φ_{i}(S)) = φ_{i}(A(S)) and B(φ_{i}(S)) = φ_{i}(B(S)) for φ_{i} ∈ Aut(S). Then it follows:

A(S) ≤ W_{i−1} < W_{i} ≤ B(S)

As S is finite, this recursive definition terminates after a finite number n of steps and W (S) := W_{n}. Therefore we obtain a chain of subgroups of Z(J(S)):

A(S) = W_{0} < W_{1} … < W_{i} < … < W_{n} = W (S) ≤ B(S)

and φ(W (S)) ⊴ E for all (φ, E) ∈ U_{J}.

The group W (S) depends on S only and it is independent of the pairs (φ_{i}, E_{i}). To see this assume that we defined in an analogous way

W_{0} = \overline{W}_{0} < \overline{W}_{1} < … < \overline{W} ñ =: \overline{W} (S)

for suitable pairs (\overline{φ}_{j}, \overline{E}_{j}) in U_{J} and for j = 1, … \overline{n}. First note that \overline{W}_{0} = W_{0} ≤ W (S) ∩ \overline{W} (S). Thus \overline{φ}_{1}(\overline{W}_{0}) = \overline{φ}_{1}(W_{0}) ≤ \overline{φ}_{1}(W (S)) = W (S) as W (S) ⊴ \overline{E}_{1}, and therefore \overline{φ}_{1}(\overline{W}_{1}) = ⟨\overline{φ}_{1}(\overline{W}_{0})^{\overline{E}^{1}}⟩ ≤ W (S) which implies \overline{W}_{1} ≤ W (S). Proceed by induction on j; a similar argument shows that since \overline{W}_{j−1} ≤ W (S) then \overline{φ}_{j}(\overline{W}_{j}) ≤ W (S) and \overline{W}_{j} ≤ W (S). Therefore \overline{W} (S) ≤ W (S). Similarly W (S) ≤ \overline{W} (S) and thus W (S) = \overline{W} (S).

Lemma 4.2. Let α ∈ Aut(S). Then W (α(S)) = α(W (S)). In particular, W (S) is a characteristic subgroup of S, nontrivial if S is nontrivial.

Proof. The mapping (φ, E) → (φα, E) defines a bijection on U_{J}. Under this map, the chain of subgroups:

A(S) = W_{0} < … < W_{i} < … < W_{n} = W (S) ≤ B(S)

is taken to the following chain:

A(S) = α(W_{0}) < α(W_{1}) < … < α(W_{i}) < … < α(W_{n}) = α(W (S)) ≤ B(S).

Therefore W (α(S)) = α(W (S)). The last statement follows from the fact that Ω(Z(S)) ≤ W (S) and Z(S) ≠ 1 if S ≠ 1. □

4.2. The group W. Denote by C_{J} the class of categories F on S which satisfy the following conditions:

(C1) J(S) is normal in F for all F ∈ C_{J}. (C2) F is a Qd(p)-free fusion system.

Proposition 4.3. Let W_{0} = Ω(Z(S)) and define

W := ⟨ψ(W_{0}) | ψ ∈ Hom_{F}(J(S), S) for F ∈ C_{J}⟩.

The subgroup W is a nontrivial characteristic subgroup of S.

Proof. For all α ∈ Aut(S), we will show that α(W) = W. Let F be a category on S. Denote by F^{α} the category on S having as sets of morphisms

Hom_{F^{α}}(Q, R) = α^{−1} ∘ Hom_{F}(α(Q), α(R)) ∘ α .

Note that if F ∈ C_{J} then F^{α} ∈ C_{J}, and if ψ ∈ Hom_{F}(Q, R) then αψα^{−1} ∈ Hom_{F^{α^{−1}}}(α(Q), α(R)). Thus we have:

α(W) := ⟨αψ(W_{0}) | ψ ∈ Hom_{F}(J(S), S) for F ∈ C_{J}⟩ =
= ⟨αψα^{−1}(α(W_{0})) | αψα^{−1} ∈ F^{α^{−1}}(α(J(S)), α(S)) for F ∈ C_{J}⟩ =
= ⟨ψ̃(W_{0}) | ψ̃ ∈ F^{α^{−1}}(J(S), S) for F ∈ C_{J}⟩
≤ W

where in the last equality we use that W_{0} and J(S) are characteristic subgroups of S. But since α is injective, it follows that |W | = |α(W)| and therefore α(W) = W, proving that W is characteristic in S. □

A CHARACTERISTIC SUBGROUP FOR FUSION SYSTEMS 13

4.3. Stellmacher functor. Given that (φ, F) and (αφ, F^{α^{−1}}) are equivalent as embeddings, F ∈ C_{J} if and only if F^{α} ∈ C_{J} for any φ, α ∈ Aut(S), and F a category on S, the two definitions W (S) and W represent the same subgroup of S, that is W = W (S). It follows from Lemma 4.2 that the functor S → W (S), for S a finite p-group, is a positive characteristic functor in the sense of Definition 3.4. We shall call the functor S → W (S), with W (S) constructed via one of the methods from this Section, a Stellmacher functor.

The Thompson subgroup of S, J(S) is a characteristic, centric subgroup. Thus using (C1), any F ∈ C_{J} is a constrained fusion system on S and by Theorem 2.5, there exists a p-constrained finite group L with F = F_{L}(S) and satisfying the following conditions: S is a Sylow p-subgroup of L; C_{L}(O_{p}(L)) ≤ O_{p}(L) and L is Qd(p)-free. It follows from our construction that W (S) is a characteristic subgroup of S which is also normal in L. The construction of W (S) depends on S only, and the subgroup W (S) is constructed in the same way as Stellmacher does in the context of finite groups so it is the same subgroup of S. Finally, notice that S → W (S) is also a Glauberman functor, see Definition 3.4.

5. Proofs of the Theorems

Proof of Theorem 1.1. Let S be a finite 2-group and let F be a Σ_{4}-free fusion system on S. Also W (S) is the characteristic subgroup of S defined in Section 4. We prove that the normalizer N_{F}(W (S)) of W (S) in F is equal to F.

This is true for the smallest fusion system on S, which is F_{S}(S). Suppose now by induction that all proper Σ_{4}-free subsystems and all Σ_{4}-free quotient systems F/Q, with Q a nontrivial normal subgroup of F satisfy Theorem 1.1.

If O_{2}(F) = 1 then for every non-trivial fully F-normalized subgroup P of S we have that N_{F}(P) is a proper subsystem of F (otherwise P ≤ O_{2}(F)). But then N_{F}(P) satisfies Theorem 1.1 by induction as it is Σ_{4}-free by Proposition 3.1. Hence N_{F}(P) = N_{N_{F} (P)}(W (N_{S}(P))) for every non-trivial fully F-normalized subgroup P. An application of Proposition 3.5 gives now that F = N_{F}(W (S)). So we can suppose (H1): O_{2}(F) ≠ 1.

Set Q := O_{2}(F) and R := QC_{S}(Q). These subgroups of S are both nontrivial by (H1). If Q = R then Q is F-centric. Consequently F is a constrained fusion system. According to [3, Proposition 4.3] there exists a 2′-reduced 2-constrained finite group L_{Q}, which is an extension of Aut_{F}(Q) by Z(Q) and having S as a Sylow 2-subgroup. Thus F = F_{S}(L_{Q}). Since F is Σ_{4}-free, the group L_{Q} is Σ_{4}-free, by definition and given the L_{Q} is 2-constrained we have C_{L_{Q}}(O_{2}(L_{Q})) ≤ O_{2}(L_{Q}). Then, according to Stellmacher’s main theorem in [26], see also Section 1, the group W (S) is normal in L_{Q}. This in its turn implies that

14 SILVIA ONOFREI AND RADU STANCU

W (S) ⊴ F_{S}(L_{Q}) and therefore F = N_{F}(W (S)). Thus we can also make the assumption (H2): Q ≠ R implying moreover that N_{F}(R) is a proper subsystem of F.

Next we will see that we also have (H3): SC_{F}(Q) ≠ F. Indeed suppose that SC_{F}(Q) = F. Then F/Q is a proper quotient system of F which is Σ_{4}-free by Proposition 3.2. The induction hypothesis gives now F/Q = N_{F/Q}(W (S/Q)). Next, Proposition 3.6 gives that F = N_{F}(U) where U is the preimage in S of W (S/Q). As U ⊴ F it follows that U ≤ Q, but this leads to a contradiction given that W (S/Q) ≠ 1.

According to Lemma 3.7, we have F = ⟨F_{1}, F_{2}⟩ with F_{1} = SC_{F}(Q) and F_{2} = N_{F}(R). By (H2) and (H3) both F_{1} and F_{2} are proper subsystems of F, the induction hypothesis gives that W (S) ⊴ F_{1} and that W (S) ⊴ F_{2}.

Notice that W (S) is fully F-normalized. By Lemma 2.2, there exists a morphism φ ∈ Hom_{F}(N_{S}(W (S)), S) with φ(W (S)) fully F-normalized. As N_{S}(W (S)) = S and since W (S) is characteristic in S it follows that W (S) = φ(W (S)) is fully F-normalized.

Finally, an application of Lemma 3.8 gives the result: F = N_{F}(W (S)).

Proof of Theorem 1.2. Let S be a finite p-group. Recall that the construction W (S) described in Section 4 and which associates to S a nontrivial characteristic subgroup W (S) gives rise to a Glauberman functor.

Assume now that F is a Qd(p)-free fusion system on S. If p is an odd prime, it follows from [10, Theorem B] that F = N_{F}(W (S)). If p = 2 then Qd(2) = Σ_{4} and the result is given by Theorem 1.1.

Proof of Theorem 1.3. Let p be an odd prime. Let F be a fusion system over a finite p-group S. Let W (S) be the characteristic subgroup of S given by the Stellmacher functor. Since F_{S}(S) ⊆ N_{F}(W (S)) ⊆ F it is enough to show that if N_{F}(W (S)) = F_{S}(S) then F = F_{S}(S). The proof is similar to that of Theorem A in [10]; for the sake of completeness we will provide the details.

Let F be a minimal counterexample to Theorem 1.3; thus N_{F}(W (S)) = F_{S}(S) but F ≠ F_{S}(S), and all the proper subsystems and quotient systems of F satisfy Theorem 1.3. Under this assumption we show that F is a constrained fusion system by proving that Q := O_{p}(F) is a nontrivial F-centric proper subgroup of S. This is attained in the following six steps.

Step 1 : Any fusion system G on S which is properly contained in F is equal to F_{S}(S).

A CHARACTERISTIC SUBGROUP FOR FUSION SYSTEMS 15

As G ⊂ F it follows that N_{G}(W (S)) ⊆ N_{F}(W (S)) = F_{S}(S). But W (S) is a characteristic subgroup of S and therefore F_{S}(S) ⊆ N_{G}(W (S)). Thus N_{G}(W (S)) = F_{S}(S) and the minimality assumption on F implies that G = F_{S}(S).

Step 2 : Let P be a fully F-normalized subgroup of S and set A = N_{S}(P). Then there is φ ∈ Hom_{F}(A, S) such that both φ(P) and φ(W (A)) are fully F-normalized.

By Lemma 2.2(a), there is a morphism φ : N_{S}(W (A)) → S such that φ(W (A)) is fully F-normalized. Since W (A) is a characteristic subgroup of A, we have N_{S}(P) = A ≤ N_{S}(A) ≤ N_{S}(W (A)) and the morphism φ can be restricted to φ : A → S. According to Lemma 2.2(b), the group φ(P) is also fully F-normalized.

Step 3 : The subgroup Q = O_{p}(F) is nontrivial.

Recall that it is assumed that F_{S}(S) ⊂ F. Alperin’s fusion theorem implies that there is a fully F-normalized subgroup P of S with F_{A}(A) ⊂ N_{F}(P), for A = N_{S}(P). Choose the subgroup P such that:

a) W (A) is fully F-normalized;

b) N_{S}(P) = A has maximal order among subgroups T with F_{N_{S}(T)}(N_{S}(T)) ⊂ N_{F}(T). The choice of P and the fact that A is a proper subgroup of N_{S}(W (A)) implies that N_{F}(W (A)) = F_{N_{S}(W (A))}(N_{S}(W (A)). Therefore N_{N_{F} (P)}(W (A)) = F_{A}(A).

If N_{F}(P) ⊂ F then the minimality assumption on F implies that N_{F}(P) = F_{R}(R), which contradicts our choice of P. Thus we have N_{F}(P) = F and 1 ≠ P ⊴ F. Hence 1 ≠ P ≤ Q which proves that Q ≠ 1.

Step 4 : Q is a proper subgroup of S.

If Q = S then F = N_{F}(S) = N_{F}(W (S)) = F_{S}(S) contradicting our assumption on F.

Step 5 : SC_{F}(Q) = F_{S}(S) when Q = O_{p}(F).

We have SC_{F}(Q) ⊆ F. If SC_{F}(Q) ⊂ F then Step 1 implies that SC_{F}(Q) = F_{S}(S) and we are done. Assume now that SC_{F}(Q) = F and recall that F ≠ F_{S}(S). An application of Proposition 3.6, with G = F_{S}(S) and F = N_{F}(Q), gives that F/Q ≠ F_{S}(S)/Q = F_{S/Q}(S/Q). By Step 3 the subgroup Q is nontrivial and the minimality assumption on F implies that N_{F/Q}(W (S/Q)) ≠ F_{S/Q}(S/Q). Let P be the inverse image of W (S/Q) in S. Notice that W (S/Q) ≠ 1, by the definition of W (S), and thus P properly contains Q. Also P ⊴ S and N_{F/Q}(W (S/Q)) = N_{F/Q}(P/Q). Another application of Proposition 3.6 gives that N_{F}(P) ≠ F_{S}(S). Since N_{F}(P) ⊆ F, Step 1 implies that N_{F}(P) = F which is a contradiction with the fact that P contains Q properly.

Step 6 : The subgroup Q is F-centric.

16 SILVIA ONOFREI AND RADU STANCU

If Q = R = SC_{S}(Q) then Q is F-centric and we are done. So let us assume that Q < R. Notice that R ⊴ S. Then N_{F}(R) is a proper subsystem of F and an application of Step 1 gives that N_{F}(R) = F_{S}(S). Recall also that by the previous step, SC_{F}(Q) = F_{S}(S). Therefore, Lemma 3.7 implies that F = F_{S}(S), which is a contradiction to our choice of F. Thus we must have Q = R.

Since Q = O_{p}(F) is a nontrivial normal centric subgroup of F, the fusion system F is constrained. But this means by [2, 4.3], that there is a p′-reduced p-constrained finite group L with S as a Sylow p-subgroup and such that Q = O_{p}(L). Furthermore F = F_{S}(L) and therefore N_{F}(W (S)) = F_{S}(N_{L}(W (S)).

Since N_{F}(W (S)) = F_{S}(S) it follows that N_{L}(W (S)) has a p-complement; see Remark 6.10 in Appendix. According to the normal p-complement theorem of Thompson, 6.11 below, it follows that L has a p-complement. Therefore F_{S}(S) = F_{S}(L) and we reached a contradiction with our assumption on F. This concludes the proof of the Theorem 1.3.

6. Appendix

Let p be an odd prime, G a finite group and S a Sylow p-subgroup of G. We say that G is p-stable if and only if for every p-subgroup Q of G and every element x of N_{G}(Q) such that [Q, x, x] = 1, we have that xC_{G}(Q) ∈ O_{p}(N_{G}(Q)/C_{G}(Q)).

A classic result of special significance to the theory of finite groups is Glauberman’s ZJtheorem [6]:

Theorem 6.1 (Glauberman). Let p be an odd prime. Let G be a finite, p-stable group such that C_{G}(O_{p}(G)) ≤ O_{p}(G). Then Z(J(S)) is a normal subgroup of G.

Using the following:

Proposition 6.2 (14.7, [7]). Assume that p is odd and that G is a finite group. Then the following conditions on G are equivalent:

(a) the group Qd(p) is not involved in G;

(b) every section of G is p-stable.

the ZJ-theorem can be reformulated as follows:

Theorem 6.3 (Glauberman). Let p be an odd prime and let G be a Qd(p)-free finite group with C_{G}(O_{p}(G)) ≤ O_{p}(G). Then Z(J(S)) is a normal subgroup of G.

For p = 2 the ZJ-theorem does not hold anymore; see [7, Section 11]. As noted by Glauberman [7] a necessary and sufficient condition for every section of G to be 2-stable is that G have a normal 2-complement, which is too strong to be useful.

A CHARACTERISTIC SUBGROUP FOR FUSION SYSTEMS 17

In a couple of papers [24, 26], Stellmacher proved an analogous version of Glauberman’s ZJ-theorem, by constructing a characteristic subgroup W (S) of S and extending the result for p = 2. An overview of his method, including a sketch of the proof for the odd prime case can be found in [13, Section 9.4]. The main theorem in [26] reads as follows:

Theorem 6.4 (Stellmacher). Let S be a nontrivial finite 2-group. Suppose that G is a finite group satisfying the following:

(I) G is Σ_{4}-free,

(II) S ∈ Syl_{2}(G) and C_{G}(O_{2}(G)) ≤ O_{2}(G), (III) Every non-abelian simple section of G is isomorphic to Sz(2^{2n+1}) or P SL_{2}(3^{2n+1}).

Then there exists a nontrivial characteristic subgroup W (S) of S which is normal in G.

Next, consider a couple of useful lemmas:

Lemma 6.5 (Chp. II, Lemma 2.3, [8]). The following conditions are equivalent: (a) Σ_{4} is involved in G;

(b) There exists a 2-subgroup Q of G such that Σ_{3} is involved in N_{G}(Q)/C_{G}(Q).

Lemma 6.6 (Chp. II, Corollary 7.3, [8]). Let G be a non-abelian simple group. The following are equivalent:

(a) G is Σ_{3}-free;

(b) G is isomorphic to Sz(2^{2n+1}) or P SL(2, 3^{2n+1}).

Remark 6.7. Note that if G is Σ_{3}-free then G is Σ_{4}-free. A finite group G with C_{G}(O_{2}(G)) ≤ O_{2}(G) is Σ_{4}-free if and only if G/O_{2}(G) is Σ_{3}-free [26].

Using the previous two lemmas and remark, we can rephrase Stellmacher’s Theorem 6.4 as follows:

Theorem 6.8 (Stellmacher). Let S be a finite nontrivial 2-group. Then there exists a nontrivial characteristic subgroup W (S) of S which is normal in G, for every finite Σ_{4}- free group G with S a Sylow 2-subgroup and C_{G}(O_{2}(G)) ≤ O_{2}(G).

If G = SO_{p′}(G), with S a Sylow p-subgroup of G, we say that G has a normal pcomplement. A standard result due to Frobenius (see [7, 8.6] for example) is given below:

Theorem 6.9 (Frobenius). The following conditions are equivalent for a finite group G with Sylow p-subgroup S:

(a) G has a normal p-complement;

(b) if Q is a non-identity subgroup of G then N_{G}(Q)/C_{G}(Q) is a p-group;

(c) if Q is a non-identity p-subgroup of G then N_{G}(Q) has a normal p-complement;

(d) if two elements of S are conjugate in G, they are conjugate in S.

Remark 6.10. The equivalence (a) ⇔ (d) in the above theorem, states that G has a normal p-complement if and only if S controls fusion in G. In the language of fusion systems, S controls G fusion if and only if F_{S}(S) = F_{S}(G).

For odd primes Frobenius’ result was improved by a result of Thompson. We give below a version of Thompson’s p-complement theorem which uses Stellmacher’s characteristic subgroup W (S):

Theorem 6.11. [13, 9.4.7] Let G be a group, p an odd prime, and S ∈ Syl_{p}(G). Then G has a normal p-complement if and only if N_{G}(W (S)) has a normal p-complement.

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Department of Mathematics, The Ohio State University, 100 Mathematics Tower, 231 West 18^{th} Avenue, Columbus, Ohio 43210, USA, Email: onofrei@math.ohio-state.edu

Department of Mathematical Sciences, University of Copenhagen Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark, Email: stancu@math.ku.dk