Some products in fusion systems and 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 $\mathcal{F}$.