Some products in fusion systems and localities

TL;DR

This paper uses the theory of localities to define and analyze a product of subsystems within saturated fusion systems, extending the understanding of their structure and normality properties.

Contribution

It introduces a new method to construct products of subsystems in fusion systems using localities, addressing limitations in existing product definitions.

Findings

Defined a product of subsystems in fusion systems using localities.

Proved the product retains subnormality and normality under certain conditions.

Linked the subsystem product to natural products in associated localities.

Abstract

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 subnormal subsystem of $N_{\mathcal{F}}(T)$ over $R\leq S$, then there is a subnormal subsystem $\mathcal{E}\mathcal{D}$ of $\mathcal{F}$ over $TR$, which plays the role of a product of $\mathcal{E}$ and $\mathcal{D}$ in $\mathcal{F}$. If $\mathcal{D}$ is normal in $N_{\mathcal{F}}(T)$, then $\mathcal{E}\mathcal{D}$ is normal 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 $\mathcal{F}$.