The influence of minimal subgroups on saturated fusion systems

Abstract

Let $\cal F$ be a saturated fusion system over a \py group $S$. In this paper, we investigate the influence of the minimal subgroups in $\mathfrak{foc}(\mathcal{F})$, the focal subgroup of $\cal F$. Our main result is that if for each cyclic subgroup $P\leq \mathfrak{foc}(\mathcal{F})$ of order $p$ (of order $2$ and $4$ if $p=2$) and each $\varphi\in \sHom_{\cal F}(P,\mathfrak{foc}(\mathcal{F}))$, $\varphi$ extends to an automorphism of $S$, then $S$ is normal in $\cal F$. We also give several applications of this result