Fusion systems containing pearls

TL;DR

This paper investigates fusion systems with special subgroups called pearls, establishing bounds on their size and classifying certain saturated fusion systems with these pearls, especially for groups of sectional rank up to 4.

Contribution

It introduces the concept of pearls in fusion systems, provides bounds on p-group orders containing pearls, and classifies saturated fusion systems with pearls for groups of sectional rank at most 4.

Findings

Bound on the order of p-groups with pearls

Classification of saturated fusion systems with pearls

Results specific to groups of sectional rank ≤ 4

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 $4$.