Minimal Fusion Systems with a Unique Maximal Parabolic

TL;DR

This paper introduces minimal fusion systems with a unique maximal p-local subsystem, providing a structural classification especially complete for p=2, advancing understanding of fusion system hierarchies.

Contribution

It defines minimal fusion systems with a unique maximal p-local subsystem and classifies them for p=2, extending the theory of fusion systems and their local structures.

Findings

Complete classification of fusion systems for p=2.

Explicit description of certain p-local subsystems.

Structural insights into minimal fusion systems.

Abstract

We define minimal fusion systems in a way that every non-solvable fusion system has a section which is minimal. Minimal fusion systems can also be seen as analogs of Thompson's N-groups. In this paper, we consider a minimal fusion system $\mathcal{F}$ on a finite $p$-group $S$ that has a unique maximal $p$-local subsystem containing $N_{\mathcal{F}}(S)$. For an arbitrary prime $p$, we determine the structure of a certain (explicitly described) $p$-local subsystem of $\mathcal{F}$. If $p=2$, this leads to a complete classification of the fusion system $\mathcal{F}$.