ON NORMAL FUSION SUBSYSTEMS

RADU STANCU^{∗}

Abstract. Let P be a finite p-group and F a fusion system on P. Let Q be a F-stongly closed subgroup of P and G be the fusion system on Q canonicaly generated by Q. We will prove that a necessary and sufficient condition for G to be normal in F is that N_{F} (Q) = F.

1. Introduction

The notion of fusion systems was introduced by Puig [Pu] in 1990. Puig called them Full Frobenius Systems. Their name came from the fact that they appeared as a natural generalization of the notion of Frobenius category of a finite group. Ten years latter, Broto, Levi and Oliver [BLO] studied a certain class of topological space which in many way behave like p-completed classifying spaces of finite groups. They proved that those spaces occur as classifying space of certain algebraic objects, which they called p-local finite groups. If fact, a p-local finite group is a fusion system admiting an additional structure centric linking system. Their definition of fusion systems is slyghtly different from the Puig’s one but they proove that the definitions are equivalent. The definition we use came from our approach which we present in [St2] and that we prove to be equivalent to the others two.

2. Definition and Basic Properties of Fusion Systems

Let us start with a more general definition.

Definition 2.1. A category F on 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 Hom_{F}(Q, R) of injective group morphisms from Q to R, with the following properties:

(1) if Q ≤ R then the inclusion Q → R is a morphism in F;

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

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

Let’s define what it means for a subgroup Q of P to be ’well placed’.

Definition 2.2. A subgroup Q of P is fully F-centralized, respectively fully Fnormalized if |C_{P} (Q)| ≤ |C_{P} (Q′)|, respectively |N_{P} (Q)| ≤ |N_{P} (Q′)|, for all Q′ ≤ P which is F-conjugated to Q.

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

Date: September 30, 2003.

∗ The author is supported by Swiss National Science Fundation Grant no PBEL2-100846.

Definition 2.3. A fusion system on a finite p-group P is a category F on P satisfying the following properties: FS1. Hom_{P} (Q, R) ⊂ Hom_{F}(Q, R) for all Q, R ≤ P . FS2. Aut_{P} (P) is a Sylow p-subgroup of Aut_{F}(P) FS3. Every ϕ : Q → P such that ϕ(Q) is fully normalized extends to a morphism ϕ̄: N_{ϕ} → P where

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

From now on P will denote a finite p-group and F a fusion system on P. Here is a equivalent caracterization of beeing fully F-normalized for a subgroup Q of P.

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

In a fusion system we have an equivalent notions for the normaliser and the centralizer in a finite group:

Definition 2.5. Let F be a fusion system on P and let Q be a subgroup of P. The normalizer N_{F}(Q) is the category on N_{P} (Q) having as morphisms ψ ∈ Hom_{N_{F} (Q)}(R, T) sucht that there exists a morphism ϕ ∈ Hom_{F}(QR, QT) satisfying ϕ|_{Q} ∈ Aut_{F}(Q) and ϕ|_{R} = ψ. The centralizer C_{F}(Q) is the category on C_{P} (Q) having as morphisms ψ ∈ Hom_{C_{F} (Q)}(R, T) sucht that there exists a morphism ϕ ∈ Hom_{F}(QR, QT) satisfying ϕ|_{Q} = id_{Q} and ϕ|_{R} = ψ.

In general N_{F}(Q) is not a fusion system on N_{P} (Q), but it becomes one if Q is fully F-normalized. It is the same for C_{F}(Q) when Q is fully F-centralized

Proposition 2.6 ([Pu], Prop. 2.8). Let F be a fusion system on P. If Q ≤ P is fully F-normalized then N_{F}(Q) is a fusion system on N_{P} (Q). If Q ≤ P is fully F-centralized then C_{F}(Q) is a fusion system on C_{P} (Q).

Here is some more terminology in a fusion system.

Definition 2.7. Let F be a fusion system on a finite p-group P and Q a subgroup of P. We say that

(i) Q is F-centric if C_{P} (ϕ(Q)) ⊂ ϕ(Q), for all ϕ ∈ Hom_{F}(Q, P).

(ii) Q is F-radical if O_{p}(Aut_{F}(Q)/Aut_{Q}(Q)) = 1.

(iii) Q is F-essential if Q is F-centric and Aut_{F}(Q)/Aut_{Q}(Q) has a strongly p-embedded subgroup.

(iv) Q is strongly F-closed if for any subgroup R of Q and any morphism ϕ ∈ Hom_{F}(R, P) we have ϕ(R) ≤ Q.

(v) Q is weakly F-closed if for any morphism ϕ ∈ Hom_{F}(Q, R) we have ϕ(Q) = Q.

A F-centric subgroup Q of P is fully F-centralized in F. Indeed, for any morphism ϕ ∈ Hom_{F}(Q · C_{P} (Q), P) = F(Q, P), we have

ϕ(C_{P} (Q)) = ϕ(Z(Q)) = Z(ϕ(Q)) = C_{P} (ϕ(Q)) .

From the definition we also deduce that if Q is F-essential if and only if the Quillen complex of the outer automorphism group S_{p}(Out(Q)) is disconnected implying that Q is F-radical. Recall that for a finite group G the Quillen complex of G has vertices the objects in F_{p}(G) and simplices are given by chains of groups ordered by inclusion.

Another easy remark is that Q is strongly F-closed if and only if F(R, P) = F(R, Q), for any subgroup R of Q, and is weakly F-closed if and only if F(Q, P) = F(Q). It’s clear that, if Q is strongly F-closed, then Q is weakly F-closed.

The Alperin’s theorem on local control, also hold in the fusion systems. In this theorem P is a finite p-group, Q a subgroup of P and F a fusion system on P. If ϕ ∈ Aut_{F}(P), we say that ϕ is a maximal morphism; if ϕ ∈ Aut_{F}(E), where E is F-essential, we say that ϕ is an essential morphism.

Theorem 2.8. [St1] Any morphism ϕ ∈ Hom_{F}(Q, P) can be written as the composition of essential morphisms, followed by a maximal morphism. More precisely, there exists a positive integer n and, for all 1 ≤ i ≤ n, F-essential subgroups E_{i} fully F-normalized and morphisms ψ_{i} ∈ Aut_{F}(E_{i}), ψ_{n+1} ∈ Aut_{F}(P) such that we have

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

In the middle of the ’80s, Gilotti and Serena [GS] gave necessary and sufficient conditions for the normalizer of a p-subgroup of a finite group G to control the p-fusion in G. Here is the generalization to fusion systems.

Theorem 2.9. [St1] Let P be a finite p-group, Q a subgroup of P and F a fusion system on P. Then N_{F}(Q) = F if and only if Q is strongly F-closed and Q admits a central series Q = Q_{n} ≥ Q_{n−1} ≥ … ≥ Q_{1} where Q_{i} is weakly F-closed, for all 1 ≤ i ≤ n − 1.

3. Main Result

Recently, Linckelman [Li], motivated by the reduction of some problems on fusion systems introduced the notion of ’normal fusion subsystem’. Here his approach:

Definition 3.1. Let F be a fusion system on a finite p-group P and F′ a fusion subsystem of F on a subgroup P′ of P. We say that F′ is normal in F if P′ is strongly F-closed and if for every isomorphism ϕ : Q → Q′ in F and any two subgroups R, R′ of Q ∩ P′ we have

ϕ ∘ Hom_{F′}(R, R′) ∘ ϕ^{−1} ⊆ Hom_{F′}(ϕ(R), ϕ(R′)) .

The deffinition of a simple fusion system came now in a natural way.

Definition 3.2. A fusion system F on a finite p-group P is simple if it has no normal fusion subsystem other then itself and the trivial one.

Here is now our theorem on normal fusion subsystems.

Theorem 3.3. Let P be a finite p-group and F a fusion system on P. Let Q be a F-stongly closed subgroup of P and G be the fusion system on Q canonicaly generated by Q. Then G is be normal in F if and only if N_{F}(Q) = F.

Proof. If N_{F}(Q) = F then any morphism in F, ϕ : R → P where R is a subgroup of Q, can be lifted to a morphism ϕ̃ : Q → Q as Q is F-strongly closed. To prove that G is normal in F is is sufficient to prove that for any u ∈ Q and any morphism ϕ :< R, ϕ(R) >→ P in F the morphism ψ := ϕconj_{u}ϕ^{−1} : ϕ(R) → ϕ(^{u}R) is also in G. But this is straight forward as ϕ lifts to ϕ̃ so ψ = conj_{ϕ̃(u)} which is a morphism in G.

If G is a normal subsystem in F we apply Theorem 2.9 to prove that N_{F}(Q) = F. As Q is F-strongly it suffices to prove that the components of the upper central series of Q are all F-weakly closed. In general we will prove this property of any characteristic subgroup R of Q. Let ϕ : R → ϕ(R) be a morphism in F with ϕ(R) Ffully normalised. As G is normal in F we have that for any u ∈ Q, ψ := ϕconj_{u}ϕ^{−1} is a morphism in G so equal to conj_{v} for a v ∈ Q. So u ∈ N_{ϕ} for all u ∈ Q. This means that ϕ extends to ϕ̃: Q → Q as Q is F-strongly closed. We obtain that ϕ is the restriction to R of an automorphism of Q. As R is characteristic in Q we have that ϕ(R) = R, so R is F-weakly closed. □

References

[BK] Brown, K. Cohomology of groups. Springer-Verlag ed., 1982.

[BLO] Broto, C., Levi, R. and B. Oliver. The Homotopy Theory of Fusion Systems, preprint, 2000

[GS] A. L. Gilotti and L. Serena. Strongly Closed Subgroups and Strong Fusion. J, London Math. Soc. 2^{nd} series 32 (1985) 103–106.

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

[Mi] G. Mislin, On group homomorphisms inducing mod − p cohomology isomorphisms, Coment. Math. Helvetici 65 (1990) 454–461.

[Pu] Puig, L. Full Frobenius Systems and their Localizing Categories. preprint, June 2001.

[St1] Stancu, R. Control of the Fusion in Fusion Systems, preprint, 2004

[St2] Stancu, R. Equivalent Definitions of Fusion Systems, preprint, 2003

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

Department of Mathematics, Ohio State University, 100 Mathematics Tower, 231 West 18th Avenue, Columbus, Ohio 43210-1174