EQUIVALENT DEFINITIONS OF FUSION SYSTEMS

RADU STANCU^{∗}

Abstract. In the literature there are two equivalent generalized local structures over a finite p-group, the full Frobenius systems introduced by Puig [Pu2] in 1990 and the saturated fusion systems introduced by Broto, Levi and Oliver [BLO] in 2000. We give a simplified description of the second approach and we will show the equivalence between the three notions. We simply call this notion fusion system.

1. Introduction

Let p be a prime number, G a finite group and F_{p}(G) the Frobenius category of G related to p. It’s a category whose objects are the nontrivial p-subgroups of G and whose morphisms are the morphisms given by conjugation by the elements of G. This category contains the p-local information of G. One can prove that the Frobenius category is equivalent to its full subcategory whose objects are the subgroups of a Sylow p-subgroup of G.

In the ’90s, Puig, convinced of the richness of the p-local structures, as the Frobenius category or the Brauer category, gave an axiomatic description. The notes in which Puig introduced these new concepts have remained in the handwritten state, since recently, when the author wrote a paper [Pu2]. The new concepts that Puig introduced are the full Frobenius systems over a p-group P.

In a recent preprint, interested in solving the Martino-Priddy conjecture, Broto, Levi and Oliver [BLO] identified and studied a certain class of spaces which in many ways behave like p-completed classifying spaces of finite groups. They showed that these spaces occur as the “classifying spaces” of certain algebraic objects, which they called p-local finite groups. Roughly speaking a p-local finite group consists in a full Frobenius systems on which we put a supplementary structure, called centric linking system. In fact, in this paper, they don’t speak about full Frobenius systems but about saturated fusion systems and the definition they give is slightly different from those of Puig. The two definitions are proved to be equivalent.

After discussions with Radha Kessar and Markus Linckelmann we realized that some axioms in BLO’s approach are redundant. We find a simplified set of axioms that are equivalent to the BLO’s one, so to the Puig’s one.

The paper is organized as follows. In the second section we will explain the Puig’s approach and give some properties of the fusion systems. All the results in this sections are due to Puig. The third section is dedicated to BLO’s approach. We will also prove that the two approaches are equivalent. The last section consit in our simplification of BLO’s set of axioms and the proof of the equivalence of our set of axioms and the BLO’s one.

Date: February 22, 2004.

∗ The author is supported by the Swiss National Science Foundation’s Grant No PBEL2-100846.

2. Full Frobenius systems

All the results and the proofs in this section are due to Puig [Pu2]. Let us start with a more general definition.

Definition 2.1. A category C over P is a category whose objects are the subgroups of P and whose set of morphisms between the subgroups Q and R of P, is a set C(Q, R) of injective group morphisms from Q to R, that contains the morphisms by conjugation by the elements of P.

A full Frobenius system F over P is a category over P whose morphisms satisfy the three following axioms. A1 If Q and R are subgroups of P, then, for any morphism ϕ ∈ F(Q, R) there exists a morphism ψ ∈ F(ϕ(Q), Q), such that ψϕ(u) = u for all u ∈ Q.

If Q ≤ R, the inclusion of Q in R is also in F(Q, R) as it is induced by conjugation by 1. In particular, the morphisms in a full Frobenius system F over P are determined by the sets F(Q, P), where Q runs over all the subgroups of P. An other important remark is the fact that if we have a morphism ϕ ∈ F(Q, R) and T ≤ Q then the restriction of ϕ to T is a morphism of F(T, R). Indeed, let ι ∈ F(T, Q) be the inclusion of T in Q. Then ϕι is a morphism in F(T, R). It’s the restriction of ϕ to T.

Before writing down Axiom 2 we give some notations. Let Q be a subgroup of P. We denote by F(Q) := F(Q, Q) the set of automorphisms of Q which belong to F. By Axiom A1, F(Q) is a group. Let R ≤ P and denote by I_{R}(Q) := N_{R}(Q)/C_{R}(Q) the set of morphisms of Q, given by the conjugation by the elements of R. As all the morphisms by conjugation by the elements of P are in F, it is straight forward that I_{R}(Q) ≤ F(Q). Finally, put I(Q) := I_{Q}(Q), which is the group of inner automorphisms of Q. A2 The group I(P), of inner automorphisms of P, is a Sylow p-subgroup of F(P).

If G is a finite group and S a Sylow p-subgroup of G, then we say that a subgroup Q of S is fully normalized, respectively fully centralized in S if N_{S}(Q), respectively C_{S}(Q) is a Sylow p-subgroup of N_{G}(Q), respectively C_{G}(Q). We give now a definition equivalent to the notions of fully normalized and fully centralized in a p-group for the full Frobenius systems. For doing this in the same time for the two notions, we denote by N_{P}^{K}(Q) the K-normalizer of Q in P, which is the inverse image of K in N_{P} (Q), where K is a subgroup of Aut(Q). For any morphism ϕ ∈ F(Q·N_{P}^{K}(Q), P), we have ϕ(N_{P}^{K}(Q)) ≤ N_{P}^{^{ϕ}K}(ϕ(Q)), where ^{ϕ}K = {ϕθϕ^{−1}|θ ∈ K} is a subgroup of Aut(ϕ(Q)).

Definition 2.2. We say that Q is K-fully normalized in F if, for any morphism ϕ ∈ F(Q · N_{P}^{K}(Q), P), we have ϕ(N_{P}^{K}(Q)) = N_{P}^{^{ϕ}K}(ϕ(Q)).

If K = 1 or K = Aut(Q), we say that Q is fully centralized or, respectively, fully normalized in F. Moreover, we have the next property.

Lemma 2.3 ([Pu2], 2.3.2). Let R be a subgroup of Q · N_{P}^{K}(Q) containing Q and ϕ ∈ F(R, P), satisfying that, for any other morphism ϕ′ ∈ F(R, P), we have

|N_{P}^{^{ϕ′} K}(ϕ′(Q))| ≤ |N_{P}^{^{ϕ}K}(ϕ(Q))| . (∗)

Then ϕ(Q) is ^{ϕ}K-fully normalized in F.

Proof. Note Q′ := ϕ(Q) and K′ := ^{ϕ}K. With these notations, for any morphism ψ ∈ F(Q′ · N_{P}^{K′}(Q′), P), we have a morphism ϕ′ ∈ F(R, P) mapping v ∈ R to ψ(ϕ(v)). Moreover, note Q′′ := ϕ′(Q) and K′′ := ^{ϕ′}K. By (∗), we obtain that |N_{P}^{K′′}(Q′′)| ≤ |N_{P}^{K′}(Q′)|. But we always have ψ(N_{P}^{K′}(Q′)) ≤ N_{P}^{K′′}(Q′′). As

|ψ(N_{P}^{K′}(Q′))| = |N_{P}^{K′}(Q′)| ≥ |N_{P}^{K′′}(Q′′)| ,

we conclude that ψ(N_{P}^{K′}(Q′)) = N_{P}^{K′′}(Q′′). So Q′ is K′-fully normalized. □

In particular, there always exists a morphism ϕ ∈ F(Q, P) such that ϕ(Q) is ϕK-fully normalized in F.

We can give now the axiom allowing us to have the equivalent of Sylow’s theorem, in the field of full Frobenius systems. A3 For any subgroup Q of P, for any subgroup K of Aut(Q) and for any morphism ϕ ∈ F(Q, P), such that ϕ(Q) is ^{ϕ}K-fully normalized in P, there exists a morphism ψ ∈ F(Q · N_{P}^{K}(Q), P) and χ ∈ K, such that ψ(u) = ϕ(χ(u)) for all u ∈ Q.

The approach between Axiom A3 and the Sylow’s theorem is a little bit hidden. We see that being K-fully normalized for a subgroup Q of P is equivalent to say that Q is the best placed between all its images by morphisms in F(Q, P). The Sylow’s theorem is equivalent to say that we can bring any p-subgroup of a group G, by conjugation by an element of G, inside a given Sylow p-subgroup of G. In the full Frobenius systems, Axiom A3 allow us to bring the K-normalizer in P of a image of Q by a morphism of F, inside N_{P}^{K}(Q), if Q is K-fully normalized in F.

By the way, the following property, which will be very useful in the following, shows us that, behind the fact that a group is K-fully normalized is hidden, somewhere, a Sylow p-subgroup.

Proposition 2.4 ([Pu2], Prop. 2.7). Let Q be a subgroup of P and K a subgroup of Aut(Q). Then Q is K-fully normalized in F if and only if Q is fully centralized in F and K ∩ I_{P} (Q) is a Sylow p-subgroup of K ∩ F(Q).

Proof. If Q is fully centralized in F and ϕ ∈ F(Q·N_{P}^{K}(Q), P), then the restriction of ϕ to Q · C_{P} (Q) is in F(Q · C_{P} (Q), P), so we have ϕ(C_{P} (Q)) = C_{P} (ϕ(Q)). If, moreover, K ∩ I_{P} (Q) = N_{P}^{K}(Q)/C_{P} (Q) is a Sylow p-subgroup of K ∩ F(Q), we obtain

|N_{P}^{K}(Q)| = |C_{P} (Q)| · |K ∩ I_{P} (Q)| =

= |C_{P} (Q)| · |K ∩ F(Q)|_{p} = |C_{P} (ϕ(Q))| · |^{ϕ}K ∩ F(ϕ(Q))|_{p} ≥

≥ |C_{P} (ϕ(Q))| · |^{ϕ}K ∩ I_{P} (ϕ(Q))| = |N_{P}^{^{ϕ}K}(ϕ(Q))| .

where by |G|_{p} we denote the order of a Sylow p-subgroup of G. But we always have that ϕ(N_{P}^{K}(Q)) ≤ N_{P}^{^{ϕ}K}(ϕ(Q)), so |N_{P}^{K}(Q)| = |ϕ(N_{P}^{K}(Q))| ≤ |N_{P}^{^{ϕ}K}(ϕ(Q))|. In this way we obtain ϕ(N_{P}^{K}(Q)) = N_{P}^{^{ϕ}K}(ϕ(Q)) and so Q is K-fully normalized in F.

Suppose now that Q is K-fully normalized in F. First, we’ll prove that Q is fully centralized in F. Let ϕ ∈ F(Q · C_{P} (Q), P). By Axiom A1, there exists ϕ^{−1} ∈ F(ϕ(Q), Q) such that ϕ^{−1}ϕ(u) = u for all u ∈ Q. Now, the group Q is K-fully normalized, so, by Axiom A3, applied to ϕ^{−1}, it follows that there exists ρ ∈ F(ϕ(Q) · N_{P}^{^{ϕ}K}(ϕ(Q)), P) and χ ∈ ^{ϕ}K such that ρ(v) = ϕ^{−1}(χ(v)) for all v ∈ ϕ(Q). Moreover, we have ρ(C_{P} (ϕ(Q))) ≤ C_{P} (ρϕ(Q)) = C_{P} (Q), as ρ(ϕ(u)) = ϕ^{−1}(χ(ϕ(u)) ∈ Q for all u ∈ Q. This gives |C_{P} (ϕ(Q))| ≤ |C_{P} (Q)|, so ϕ(C_{P} (Q)) = C_{P} (ϕ(Q)). As this result is true for all ϕ ∈ F(Q·C_{P} (Q), P), we obtain that Q is fully centralized in F.

We show now, by induction on the index |P : Q|, that K ∩ I_{P} (Q) is a Sylow p-subgroup of K ∩ F(Q). For P = Q, by Axiom A2 the group I(P) is a Sylow p-subgroup of F(P) and, as I(P) is a normal subgroup of Aut(P), we obtain that K ∩ I(P) is a Sylow p-subgroup of K ∩ F(P) for all K ≤ Aut(P).

Let now Q < P and let us prove our assertion first in the case where K = Aut(Q), so, when Q is fully normalized. Put R := N_{P} (Q) and let J be the subgroup of Aut(R) stabilizing Q. It is straight forward that N_{P}^{J}(R) = N_{P} (Q) ∩ N_{P} (R) = R. Moreover, R is J-fully normalized in F. Indeed, as Q is fully normalized, for all ϕ ∈ F(R, P), we have ϕ(R) = N_{P} (Φ(Q)). Also, ^{ϕ}J is the subgroup of Aut(ϕ(R)) stabilizing ϕ(Q) so N_{P}^{^{ϕ}J}(ϕ(R)) = N_{P} (ϕ(Q)) ∩ N_{P} (ϕ(R)) = ϕ(R). We obtain that, for all ϕ ∈ F(R · N_{P}^{J}(R), P) = F(R, P), we have N_{P}^{^{ϕ}J}(ϕ(R)) = ϕ(R) = ϕ(N_{P}^{J}(R)), so R is J-fully normalized in F. As Q is a proper subgroup of R, by induction hypothesis, we obtain that J ∩ I_{P} (R) = I_{R}(R) is a Sylow p-subgroup of J ∩ F(R).

We have clearly I_{P} (Q) = I_{R}(Q). For showing that I_{R}(Q) is a Sylow p-subgroup of F(Q), it suffices to prove that I_{R}(Q) is a Sylow p-subgroup of N_{F(Q)}(I_{R}(Q)), because, if a p-subgroup of a finite group G is a Sylow p-subgroup of its normalizer, then it is a Sylow p-subgroup of G.

Let us show now that I_{R}(Q) is a Sylow p-subgroup of N_{F(Q)}(I_{R}(Q)). For this, we’ll show that any morphism ϕ ∈ N_{F(Q)}(I_{R}(Q)) can be lifted in J ∩ F(R), which means that the group morphism f : J ∩ F(R) ⟶ N_{F(Q)}(I_{R}(Q)), given by the restriction to Q of the morphisms in J ∩ F(R), is surjective. As, moreover, f(I(R)) = I_{R}(Q), it’s following that I_{R}(Q) is a Sylow p-subgroup of N_{F(Q)}(I_{R}(Q)). We used here the fact that the image of a Sylow p-subgroup of J ∩ F(R) by the surjective morphism f is a Sylow p-subgroup of N_{F(Q)}(I_{R}(Q)).

We’ll show now that any morphism ϕ ∈ N_{F(Q)}(I_{R}(Q)) can be lifted in J ∩F(R). Take ϕ ∈ N_{F(Q)}(I_{R}(Q)) so we have ϕ(Q) = Q and ^{ϕ}(I_{R}(Q)) = I_{R}(Q). Moreover, R contains Q · C_{P} (Q), so N_{P}^{I_{R}(Q)}(Q) = R. We have already seen that Q is fully centralized in F. We have also that Q is I_{R}(Q)-fully normalized in F. Indeed, for any morphism η ∈ F(Q · N_{P}^{I_{R}(Q)}(Q), P) = F(R, P), we have C_{P} (η(Q)) = η(C_{P} (Q)), as Q is fully normalized and so C_{P} (η(Q)) ≤ η(R). This gives that η(Q)C_{P} (η(Q)) ≤ η(R) so, N_{P}^{I_{η(R)}(η(Q))}(η(Q)) = η(R) = η(N_{P}^{I_{R}(Q)}(Q)). It follows that Q is I_{R}(Q)-fully normalized in F. By Axiom A3 applied to the morphism ϕ, there exists ψ ∈ F(Q · N_{P}^{I_{R}(Q)}(Q), P) = F(R, P) and χ ∈ I_{R}(Q) such that ψ(u) = ϕχ(u), for all u ∈ Q. Let χ̃ be a preimage of χ in I_{R}(R). We have then that ψχ^{−1} is a preimage of ϕ in F(R, P). In fact ψχ^{−1} ∈ F(R). Indeed, ψ and χ̃ normalize I_{R}(Q) and ψχ^{−1}(C_{P} (Q)) = C_{P} (ψχ^{−1}(Q))) = C_{P} (Q). So, R/C_{P} (Q) = I_{R}(Q) = ^{ψχ^{−1}}I_{R}(Q) = ψχ^{−1}(R)/C_{P} (Q) which gives that that R = ψχ^{−1}(R) and, so, ψχ^{−1} ∈ F(R). This finishes the the proof in the case where K = Aut(Q).

In the general case, let θ ∈ F(Q, P) such that θ(Q) is fully normalized in F. We have seen that such a morphism θ always exists. Now, θ(Q) is fully normalized, so by the proof in the case K = Aut(Q), applied to θ(Q) in the place of Q, we obtain that I_{P} (θ(Q)) is a Sylow p-subgroup of F(θ(Q)). As ^{θ}(K ∩ F(Q)) is a subgroup of ^{θ}(F(Q)) = F(θ(Q)), there exists ρ ∈ F(θ(Q)) such that a Sylow p-subgroup of ρθ(K ∩ F(Q)) is contained in I_{P} (θ(Q)). So ^{ρθ}K ∩ I_{P} (θ(Q)) is a Sylow p-subgroup of ^{ρθ}(K ∩ F(Q)).

We take now θ′ := ρθ, Q′ := θ(Q), K′ := ^{θ′}K and we apply Axiom A3 to θ^{′−1}, as Q is K-fully normalized. So there exists ψ ∈ F(Q′ · N_{P}^{K′}(Q′), P) and χ′ ∈ K′ such that ψ(u) = θ^{′−1}(χ′(u)) for all u ∈ Q′. Let χ := ^{θ^{′−1}}χ′, which is an element of K. We have then ψ^{−1} = θ′χ^{−1} which implies that ^{ψ^{−1}}K = K′. But ψ(N_{P}^{K′}(Q′)) ≤ N_{P}^{K}(Q) and, as Q′ is fully centralized, we obtain that ψ(C_{P} (Q′)) = C_{P} (Q), so ψ(K′ ∩ I_{P} (Q′)) ≤ K ∩ I_{P} (Q). Moreover, K′ ∩ I_{P} (Q′) is a Sylow p-subgroup of K′ ∩ F(Q′) and ϕ(K′ ∩ F(Q′)) = K ∩ F(Q). We deduce that K ∩ I_{P} (Q) is a Sylow p-subgroup of K ∩ F(Q). □

Let Q a subgroup of P and K a subgroup of Aut(Q). We define, based on F, a category N_{F}^{K}(Q) over N_{P}^{K}(Q), whose objects are the subgroups of N_{P}^{K}(Q) and whose morphisms between two subgroups R and T of N_{P}^{K}(Q) are the morphisms ϕ ∈ F(R, T), satisfying that there exists a morphism ψ ∈ F(Q·R, Q·T) and χ ∈ K such that χ(u) = ψ(u) for all u ∈ Q and ψ(v) = ϕ(v) for all v ∈ R; in other words, all the morphisms which can be extended to Q · R and such that this extension acts over Q by a morphism of K. In the case where K = Aut(Q), we denote this category, simply, N_{F}(Q).

Proposition 2.5 ([Pu2], Prop. 2.8). Let F be a full Frobenius system over P. If Q ≤ P is K-fully normalized in F, then, N_{F}^{K}(Q) is a full Frobenius system over N_{P} (Q).

For the proof of this proposition we’ll need the next technical lemma:

Lemma 2.6 ([Pu2], Lemma 2.9). Let F be a full Frobenius system over P, Q a subgroup of P and K a subgroup of Aut(Q). Let R be a subgroup of N_{P}^{K}(Q), L a subgroup of Aut(R) and M the subgroup of Aut(Q · R) stabilizing Q and R and acting on them by the morphisms in K, respectively L.

Then N_{P}^{M}(Q · R) = N_{P}^{K}(Q) ∩ N_{P}^{L}(R) and, if R is fully normalized in N_{F}^{K}(Q), the group Q · R is M-fully normalized in F.

Proof. The first equality is straight forward as we have u ∈ N_{P}^{M}(Q · R) if and only if u ∈ P stabilize Q and R and u acts on Q, respectively R by elements of K, respectively L which is equivalent with u ∈ N_{P}^{K}(Q), respectively u ∈ N_{P}^{L}(R), or, in other words, with u ∈ N_{P}^{K}(Q) ∩ N_{P}^{L}(R). We remark that this is independent on the existence of F.

For the second part of the lemma, let’s put T := Q · R, P′ := N_{P}^{K}(Q) and F′ := N_{F}^{K}(Q). Let’s consider ϕ ∈ F(T · N_{P}^{M}(T), P). We want to show that ϕ(N_{P}^{M}(Q)) = N_{P}^{^{ϕ}M}(ϕ(T)). Note Q′ := ϕ(Q), T′ := ϕ(T) and K′ := ^{ϕ}K. Now, we have ϕ|_{Q} ∈ F(Q, P), so, by Axiom A3 applied to ϕ^{−1}, using that Q is K-fully normalized in F, there exists ψ ∈ F(Q′ · N_{P}^{K′}(Q′), P) and χ′ ∈ F(Q′) such that ψ(u) = ϕ^{−1}χ′(u) for all u ∈ Q′. In particular, ψ(N_{P}^{K′}(Q)′) ≤ N_{P}^{K}(Q). Now, we have ϕ(T ·N_{P}^{M}(T)) ≤ ϕ(T)·N_{P}^{^{ϕ}M}(T′) ≤ Q′ ·N_{P}^{K′}(Q′) so, if we note η := ψϕ, we obtain η ∈ F(T ·N_{P}^{M}(T), QP′) = F(Q·R·N_{P}^{M}(T), QP′). Moreover η(R·N_{P}^{M}(T)) ≤ P′, which implies that η|_{R·N_{P}^{M} (T)} ∈ F′(R · N_{P}^{M}(T), P′), by the very definition of F′.

By a simple computation, we have N_{P}^{M}(T) = N_{P}^{K}(Q) ∩ N_{P}^{L}(R) = N_{P′}^{L}(R) (1) and, using the fact that R is L-fully normalized in F′, we obtain η(N_{P′}^{L}(R)) = N_{P′}^{^{η}L}(η(R)) (2). Moreover, we remark that η(Q) = Q and ^{η}K = K and ^{η}M is a subgroup of the automorphisms group of Q · η(R) stabilizing Q and η(R) and acting on Q, respectively η(R) by morphisms of K, respectively η(L). So, N_{P}^{^{η}M}(Q · η(R)) = N_{P}^{K}(Q) ∩ N_{P}^{^{η}L}(η(R)) = N_{P′}^{^{η}L}(η(R)) (3), by the first part of the lemma. From (1), (2) and (3) we deduce that η(N_{P}^{M}(T)) = N_{P}^{^{η}M}(Q · η(R)). So, we have

ψ(ϕ(N_{P}^{M}(T))) = η(N_{P}^{M}(T)) = N_{P}^{^{η}M}(Q · η(R)) ≥ ψ(N_{P}^{^{ϕ}M}(ϕ(T)))

We also have that |ϕ(N_{P}^{M}(T))| ≥ |N_{P}^{^{ϕ}M}(ϕ(T))|. Now, as ϕ(N_{P}^{M}(T)) ≤ N_{P}^{^{ϕ}M}(ϕ(T)), we obtain that ϕ(N_{P}^{M}(T)) = N_{P}^{^{ϕ}M}(ϕ(T)).

This is true for all ϕ ∈ F(T · N_{P}^{M}(T), P), so T is M-fully centralized in F. □

Proof of Proposition 2.5. As in the preceding lemma put F′ := N_{F}^{K}(Q) and P′ := N_{P}^{K}(Q). We will verify Axioms A1, A2 and A3 for F′. Axiom A1 is trivially satisfied.

Let’s verify now Axiom A2. Clearly P′ is fully normalized in F′, and taking N the subgroup of Aut(Q·P′) which stabilizes Q and P′ and acts on Q by morphisms of K, as Q is K-fully normalized, by the lemma 2.6, the group Q · P′ is N-fully normalized in F. By Proposition 2.4, the group N ∩I_{P} (Q·P′) is a Sylow p-subgroup of N ∩ F. But, by the definition of F′, any morphism of F′(P′) lifts in a morphism of F(Q·P′)∩N. So the restriction morphism N ∩F(Q·P′) → F′(P′) is surjective. Moreover this morphism maps N ∩ I_{P} (Q · P′) onto I(P′). Thus I(P′) is a Sylow p-subgroup of F′(P′) and Axiom A2 is satisfied.

For verifying Axiom A3, let R ≤ P′, let L a subgroup of Aut(R) and ϕ ∈ F′(R, P′) such that ϕ(R) is ^{ϕ}L-fully normalized in F′. By the definition of F′ we have that there exists ψ ∈ F(Q · R, Q · P′) and χ ∈ K such that ψ(v) = ϕ(v), for all v ∈ R and ψ(u) = χ(u), for all u ∈ Q. Let’s take again the notations in the lemma 2.6: let T := Q · R and M be the subgroup of Aut(T) stabilizing Q and R and acting on Q, respectively R by morphisms of K, respectively L. As Q is K-fully normalized in F and ψ(R) is ^{ψ}L-fully normalized in F′, by the lemma 2.6, we obtain that ψ(T) is M-fully normalized in F.

We apply Axiom A3 to ψ ∈ F(T, P) and we obtain that there exists ρ ∈ F(T · N_{P}^{M}(T), P) and ν ∈ M such that ρ(w) = ψ(ν(w)), for all w ∈ T. In particular, we have ρ(u) = ψν(u) and so ρ determines an element of K, as it was already the case for ψ and ν. Now, R and N_{P}^{M}(T) are subgroups of N_{P}^{K}(Q) and as ρ determines an element of K, the images ρ(R) and ρ(N_{P}^{M}(T)) remain subgroups of N_{P}^{K}(Q). On the other hand, the action of ν over R determines an element λ of L and, by the lemma 2.6, we have N_{P}^{M}(T) = N_{P′}^{L}(R). We obtain, in consequence, that the restriction of ρ to R · N_{P′}^{L}(R) is a morphism of F′(R · N_{P′}^{L}(R), P′). Moreover we have ρ(v) = ψ(ν(v)) = ϕ(λ(v)) for all v ∈ R, so Axiom A3 is also satisfied. □

3. Saturated fusion systems

Here is the approach of Broto Levi and Oliver.

Definition 3.1. A fusion system over a finite p-group P is a category whose objects are the subgroups in P and whose morphism set Hom_{F}(P, Q) satisfy the following

(1) Hom_{P} (Q, R) ⊂ Hom_{F}(Q, R) ⊂ Inj(Q, R) for all Q, R ≤ P .

(2) Every morphism in F factors as an isomorphism in F followed by an inclusion.

Definition 3.2. A subgroup Q of P is fully centralized if |C_{P} (Q)| ≤ |C_{P} (Q′)| for all Q′ ≤ P which is F-conjugated to Q.

Definition 3.3. A subgroup Q of P is fully normalized if Q is fully centralized and if Aut_{P} (Q) ∈ Syl_{p}(Aut_{F}(Q)).

Definition 3.4. F is a saturated fusion system if the two following conditions hold:

(i) Each subgroup Q ≤ P is F-conjugated to at least one fully normalized subgroup.

(ii) If Q ≤ P and ϕ ∈ Hom_{F}(Q, P) are such that ϕ(Q) is fully centralized and one set N_{ϕ} := {g ∈ N_{P} (Q)|ϕc_{g}ϕ^{−1} ∈ Aut_{P} (ϕ(Q))}, then there is ϕ̄ ∈ Hom_{F}(N_{ϕ}, P) such that ϕ̄|_{P} = ϕ.

We remark that the notions of fully normalized and fully centralized are the same in the BLO’s and Puig’s approach. To prove that Puig’s definitions imply BLO’s one we only have to prove the following.

Proposition 3.5. Let F be a full Frobenius system (’à la Puig’). Then the definition 3.4 above is a consequence.

Proof. We have that ϕ(Q) is fully centralized. In fact, ϕ(Q) is ^{ϕ}(N_{ϕ}/C_{P} (ϕ(Q)))- fully normalized. Indeed ^{ϕ}(N_{ϕ}/C_{P} (ϕ(Q))) ⊂ Aut_{P} (ϕ(Q)), so the equivalent condition in Proposition 2.4 are satisfied. Thus, by Axiom 3 there exists ψ : N_{ϕ} → N_{P} (ϕ(Q)) such that ψ(^{g}u) = ϕ(u). So we can put ϕ̄ = ψ ∘ conj(g). □

For the converse it is sufficient to prove the following.

Proposition 3.6. Let F be a saturated fusion system (’à la BLO’). Then Puig’s Axiom 3 is a consequence.

Proof. Let Q be a subgroup of P, ϕ : Q → P a morphism in F and K ⊂ Aut(Q) such as Q′ := ϕ(Q) is K′ := ^{ϕ}K-fully normalized. So Q′ is fully centralized and K′ ∩ Aut_{P} (Q′) is a Sylow p-subgroup of K′ ∩ Aut_{F}(Q′). Now ^{ϕ}(K ∩ Aut_{P} (Q)) is p-subgroup of K′ ∩Aut_{F}(Q′). By conjugating by an appropriate element g ∈ K we have that ^{g} ^{ϕ}(K ∩ Aut_{P} (Q)) ⊂ ^{ϕ}(K ∩ Aut_{P} (Q)). This means that N_{^{g}ϕ} contains N_{P}^{K}(Q). By the definition of saturated fusion systems we have that ^{g}ϕ extends to a morphism ψ : N_{^{g}ϕ} → P and this can be restricted to N_{P}^{K}(Q). The Axiom 3 comes straight forward. □

4. ’Simply’ fusion systems

We have the folowing approach for the above notion.

Definition 4.1. A subgroup Q of P is fully centralized, respectively fully normalized if |C_{P} (Q)| ≤ |C_{P} (Q′)|, respectively |N_{P} (Q)| ≤ |N_{P} (Q′)|, for all Q′ ≤ P which is F-conjugated to Q.

Definition 4.2. A fusion system on a finite p-group P is a category F on P satisfying the following properties:

(1) Aut_{P} (P) is a Sylow p-subgroup of Aut_{F}(P)

(2) Every ϕ : Q → P such that ϕ(Q) is fully normalized extends to a morphism ϕ̄: N_{ϕ} → P

It is obvious that the BLO’s definition implies ours. We prove now the fact that our definition implies BLO’s one.

Proposition 4.3. Let F be a fusion system on a finite group P and let Q be a subgroup of P. Then Q is fully normalized if and only if Q is fully centralized and Aut_{P} (Q) is a Sylow p-subgroup of Aut_{F}(Q).

Proof. First, a fully normalized subgroup Q of P is also fully centralized as for any F-isomorphic subgroup Q′ we have that the morphism ϕ : Q′ → Q extends to a morphism ϕ̄ : N_{ϕ} → N_{P} (Q). But C_{P} (Q′) ⊂ N_{ϕ} and ϕ̄(C_{P} (Q′)) ⊂ C_{P} (Q) giving that |C_{P} (Q′)| ≤ |C_{P} (Q)|. So Q is fully centralized.

Second, if Q is fully normalized then Aut_{P} (Q) is a Sylow p-subgroup of Aut_{F}(Q). To prove this, we will follow the proof of Proposition 1.5 in [Li]. Let Q be a subgroup of maximal order such that Aut_{P} (Q) is not a Sylow p-subgroup of Aut_{F}(Q). Then Q is a proper subgroup of P by axiom 2. Choose a p-subgroup S of Aut_{F}(Q) such that Aut_{P} (Q) is a proper normal subgroup of S. Let ϕ ∈ S \ Aut_{P} (Q). Since ϕ normalizes Aut_{P} (Q), for every y ∈ N_{P} (Q) there is z ∈ N_{P} (Q) such that ϕ(^{y}u) = ^{z}ϕ(u) for all u ∈ Q. Thus we have N_{ϕ} = N_{P} (Q). Since Q is fully normalized, by axiom 3 ϕ extends to ϕ̄ : N_{ϕ} → N_{P} (Q) so ϕ̄ ∈ Aut_{F}(N_{P} (Q)). Since ϕ has p-power order, by decomposing ϕ̄ into its p-part and its p′-part we may assume that ϕ̄ has also p-power order. Let ψ : N_{P} (Q) → P be a morphism such that N′ := ψ(N_{P} (Q)) is fully normalized. As the order on N′ is greater then the order of Q, we have that Aut_{P} (N′) is a Sylow p-subgroup of Aut_{F}(N′). Now ψϕ̄ψ^{−1} is a p-element of Aut_{F}(N′), thus conjugated to an element in Aut_{P} (N′). Therefore we may choose ψ in a such way that there is y ∈ N_{P} (N′) satisfying ψϕ̄ψ^{−1}(v) = ^{y}v for all v ∈ N′. Since ϕ̄|_{Q} = ϕ, the automorphism ψϕ̄ψ^{−1} of N′ stabilizes ψ(Q). Thus y ∈ N_{P} (()ψ(Q)). Since Q is fully normalized and ψ(N_{P} (Q)) ⊂ N_{P} (ψ(Q)) we have that ψ(N_{P} (Q)) = N_{P} (ψ(Q)), hence ϕ̄(u) = ^{τ^{−1}(y)}u, for all u ∈ N_{P} (Q). And, in particular, ϕ ∈ Aut_{P} (Q), contradicting our first choice of ϕ.

The converse is straight forward as |N_{P} (Q)| = |Aut_{P} (Q)| · |C_{P} (Q)|.

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We have to prove also that all the morphisms ϕ such that the image is fully F-centralized extend to N_{p}hi.

Proposition 4.4. Every ϕ : Q → P such that ϕ(Q) is fully F-centralized extends to a morphism ϕ̄: N_{ϕ} → P

Proof. We note Q′ := ϕ(Q). Choose θ : Q′ → P such that θ(Q′) is fully normalized and, as Aut_{P} (θ(Q′)) is a Sylow p-subgroup of Aut_{F}(θ(Q′)) we can modify θ by a morphism in Aut_{F}(θ(Q′)) and suppose that N_{θ} = N_{P} (Q′).

By the property (2) we have that θ extends to θ̄: N_{θ} → P. Note ψ := θϕ. By the same property (2) ψ extends to ψ̄ : N_{ψ} → P.

Our aim in what follows is to prove that N_{ϕ} ⊂ N_{ψ} and ψ̄(N_{ϕ}) ⊂ θ̄(N_{θ}) so that (θ̄)^{−1}ψ̄|_{N_{ϕ}} would be the extension of ϕ to N_{ϕ}.

Both are simple verifications. Take y ∈ N_{ϕ} then by definition, there exists z ∈ N_{P} (Q′) such that ϕ(^{y}u) = ^{z}ϕ(u) for all u ∈ Q. By composing with θ we obtain θϕ(^{y}u) = θ(^{z}ϕ(u)). But as N_{θ} = N_{P} (Q′) we have that there exists x ∈ N_{P} (θ(Q′)) such that θ(^{z}ϕ(u)) = ^{x}θ(ϕ(u)) = ^{x}ψ(u). By resuming, we have ψ(^{y}u) = ^{x}ψ(u) which means that y ∈ N_{ψ}. As this is true for all y ∈ N_{ϕ} we obtain that N_{ϕ} ⊂ N_{ψ}.

Take now x ∈ ψ̄(N_{ϕ}). Suppose that x = ψ̄(y), y ∈ N_{ϕ}. By definition, there exists z ∈ N_{P} (Q′) such that ϕ(^{y}u) = ^{z}ϕ(u) for all u ∈ Q. We obtain ψ(^{y}u) = xψ(u), so θ(^{z}ϕ(u)) = ^{x}θ(ϕ(u)), which is equivalent to ^{θ̄(z)}ψ(u) = ^{x}θ(ϕ(u)) for all u ∈ Q. This gives that x = θ̄(z)c with c ∈ C_{P} (θ(Q)). But as C_{P} (Q′) ⊂ N_{θ} and θ̄(C_{P} (Q′)) ⊂ C_{P} (θ(Q′)) and using the fact that Q′ is fully centralized we have that θ̄(C_{P} (Q′)) = C_{P} (θ(Q′)). This means that c ∈ θ̄(N_{θ}), so x ∈ θ̄(N_{θ}). Now this is true for all x ∈ ψ̄(N_{ϕ}) so ψ̄(N_{ϕ}) ⊂ θ̄(N_{θ}).

Thus we showed that (θ̄)^{−1} ∘ ψ̄|_{N_{ϕ}} extends ϕ to N_{ϕ}.

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This finishes the proof of the equivalence of the three approaches given in the paper. Even if our description is simpler then the others two, it is very usefull to keep in mind as properties the other two definitions.

References

[BLO] C. Broto, R. Levi and B. Oliver, The homotopy theory of fusion systems, preprint 2001.

[Li] Linckelmann, M. Simple Fusion Systems and the Solomon 2-Local Groups, preprint, 2003

[Pu1] L. Puig, Structure locale dans les groupes finis, Bul. Soc. Math. France, Mémoire no. 47, PH thesis, 1976.

[Pu2] L. Puig, Full Frobenius systems and their localizing categories, preprint, June 2001.

   E-mail address: stancu@math.ohio-state.edu

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