Rank 2 Amalgams and Fusion Systems

TL;DR

This paper classifies certain fusion systems with specific automorphism properties and employs the amalgam method to characterize rank 2 group amalgams with strongly p-embedded subgroups.

Contribution

It provides a classification of fusion systems with two invariant essential subgroups and introduces new p-local characterizations of rank 2 group amalgams.

Findings

Classification of fusion systems with trivial O_p and two invariant essential subgroups

New p-local characterizations of rank 2 group amalgams

Application of the amalgam method to fusion system analysis

Abstract

We classify fusion systems $\mathcal{F}$ in which $O_p(\mathcal{F})=\{1\}$, and there are two $\mathrm{Aut}_{\mathcal{F}}(S)$-invariant essential subgroups whose normalizer systems generate $\mathcal{F}$. We employ the amalgam method and, as a bonus, obtain $p$-local characterizations of certain rank $2$ group amalgams whose parabolic subgroups involve strongly $p$-embedded subgroups.