Fusion systems containing pearls

Abstract: An $\mathcal{F}$-essential subgroup is called a pearl if it is either elementary abelian of order $p^2$ or non-abelian of order $p^3$. In this paper we start the investigation of fusion systems containing pearls: we determine a bound for the order of $p$-groups containing pearls and we classify the saturated fusion systems on $p$-groups containing pearls and having sectional rank at most $4$.