# Block-Exoticity of a Family of Exotic Fusion Systems

## TL;DR

This paper demonstrates that certain exotic fusion systems on Sylow p-subgroups of G_2(p) are block-exotic, supporting the conjecture that all exotic fusion systems share this property, through new reduction theorems.

## Contribution

It establishes that specific exotic fusion systems are block-exotic and introduces two reduction theorems for block-realisable fusion systems.

## Key findings

- Exotic fusion systems on Sylow p-subgroups of G_2(p) are block-exotic.
- Provides evidence for the conjecture that all exotic fusion systems are block-exotic.
- Introduces two reduction theorems for block-realisable fusion systems.

## Abstract

We prove that each exotic fusion system $\mathcal F$ on a Sylow $p$-subgroup of $G_2(p)$ for an odd prime $p$ with $\mathcal O_p(\mathcal F)=1$ is block-exotic. This gives evidence for the conjecture that each exotic fusion system is block-exotic. We prove two reduction theorems for block-realisable fusion systems.