Normal Subsystems of Fusion Systems

In this article we prove that for any saturated fusion system, that the (unique) smallest weakly normal subsystem of it on a given strongly closed subgroup is actually normal. This has a variety of corollaries, such as the statement that the notion of a simple fusion system is independent of whether one uses weakly normal or normal subsystems. We also develop a theory of weakly normal maps, consider intersections and products of weakly normal subsystems, and the hypercentre of a fusion system

Normal Subsystems of Fusion Systems

Abstract In this article we prove that for any saturated fusion system, that the (unique) smallest weakly normal subsystem of it on a given strongly closed subgroup is actually normal. This has a variety of corollaries, such as the statement that the notion of a simple fusion system is independent of whether one uses weakly normal or normal subsystems. We also develop a theory of weakly normal maps, consider intersections and products of weakly normal subsystems, and the hypercentre of a fusion system

Reductions to simple fusion systems

We prove that if $\mathcal{E}\trianglelefteq\mathcal{F}$ are saturated fusion systems over $p$-groups $T\trianglelefteq S$, such that $C_S(\mathcal{E})\le T$, and either $Aut_{\mathcal{F}}(T)/Aut_{\mathcal{E}}(T)$ or $Out(\mathcal{E})$ is $p$-solvable, then $\mathcal{F}$ can be "reduced" to $\mathcal{E}$ by alternately taking normal subsystems of $p$-power index or of index prime to $p$. In particular, this is the case whenever $\mathcal{E}$ is simple and "tamely realized" by a known simple group. This answers a question posed by Michael Aschbacher, and is useful when analyzing involution centralizers in simple fusion systems, in connection with his program for reproving parts of the classification of finite simple groups by classifying certain 2-fusion systems.Comment: 13 page

Some products in fusion systems and localities

The theory of saturated fusion systems resembles in many parts the theory of finite groups. However, some concepts from finite group theory are difficult to translate to fusion systems. For example, products of normal subsystems with other subsystems are only defined in special cases. In this paper the theory of localities is used to prove the following result: Suppose $\mathcal{F}$ is a saturated fusion system over a $p$-group $S$. If $\mathcal{E}$ is a normal subsystem of $\mathcal{F}$ over $T\leq S$, and $\mathcal{D}$ is a normal subsystem of $N_{\mathcal{F}}(T)$ over $R\leq S$, then there is a normal subsystem $\mathcal{E}\mathcal{D}$ of $\mathcal{F}$ over $TR$, which plays the role of a product of $\mathcal{E}$ and $\mathcal{D}$ in $\mathcal{F}$. It is shown along the way that the subsystem $\mathcal{E}\mathcal{D}$ is closely related to a naturally arising product in certain localities attached to $\mathcal{F}$.Comment: 8 p

A Krull-Remak-Schmidt theorem for fusion systems

We prove that the factorization of a saturated fusion system over a discrete $p$-toral group as a product of indecomposable subsystems is unique up to normal automorphisms of the fusion system and permutations of the factors. In particular, if the fusion system has trivial center, or if its focal subgroup is the entire Sylow group, then this factorization is unique (up to the ordering of the factors). This result was motivated by questions about automorphism groups of products of fusion systems

Centralizers of Subsystems of Fusion Systems

When $(S,\mathcal{F},\mathcal{L})$ is a $p$-local finite group and (T,\mathcal{E},\mathcal{\L}_0) is weakly normal in $(S,\mathcal{F},\mathcal{L})$ we show that a definition of $C_S(\mathcal{E})$ given by Aschbacher has a simple interpretation from which one can deduce existence and strong closure very easily. We also appeal to a result of Gross to give a new proof that there is a unique fusion system $C_{\mathcal{F}}(\mathcal{E})$ on $C_S(\mathcal{E})$.Comment: 9 page

Saturated fusion systems with parabolic families

Let G be group; a finite p-subgroup S of G is a Sylow p-subgroup if every finite p-subgroup of G is conjugate to a subgroup of S. In this paper, we examine the relations between the fusion system over S which is given by conjugation in G and a certain chamber system C, on which G acts chamber transitively with chamber stabilizer N_G(S). Next, we introduce the notion of a fusion system with a parabolic family and we show that a chamber system can be associated to such a fusion system. We determine some conditions the chamber system has to fulfill in order to assure the saturation of the underlying fusion system. We give an application to fusion systems with parabolic families of classical type.Comment: 28 page

Tate's and Yoshida's theorem on control of transfer for fusion systems

We prove analogues of results of Tate and Yoshida on control of transfer for fusion systems. This requires the notions of $p$-group residuals and transfer maps in cohomology for fusion systems. As a corollary we obtain a $p$-nilpotency criterion due to Tate.Comment: 20 page