Minimal characteristic bisets and finite groups realizing Ruiz-Viruel exotic fusion systems

TL;DR

This paper constructs minimal characteristic bisets for exotic fusion systems on a specific group and analyzes the finite groups realizing these systems, providing bounds on their exoticity index.

Contribution

It determines minimal characteristic bisets for Ruiz-Viruel exotic fusion systems and analyzes the finite groups that realize these systems.

Findings

Constructed minimal characteristic bisets for all Ruiz-Viruel exotic fusion systems.

Provided an upper bound for the exoticity index of these systems.

Analyzed the finite groups associated with the bisets.

Abstract

Continuing our previous work, we determine a minimal left characteristic biset $X$ for every exotic fusion system $\mathcal{F}$ on the extraspecial group $S$ of order $7^3$ and exponent $7$ discovered by Ruiz and Viruel, and analyze the finite group $G$ obtained from $X$ which realizes $\mathcal{F}$ as the full subcategory of the $7$-fusion system of $G$. In particular, we obtain an upper bound for the exoticity index for $\mathcal{F}$.