Fusion Systems on Sylow $3$-subgroups of Fischer and Monster sporadic groups: I

TL;DR

This paper classifies all corefree fusion systems on Sylow 3-subgroups of certain sporadic groups, showing none are exotic, advancing the understanding of fusion systems in sporadic groups.

Contribution

It provides a complete classification of corefree fusion systems on Sylow 3-subgroups of Fi22, Fi23, and B, demonstrating they are all non-exotic.

Findings

No exotic fusion systems supported by these Sylow 3-subgroups.

Classified all corefree fusion systems on these subgroups.

Progress towards full classification of fusion systems in sporadic groups.

Abstract

We classify all corefree fusion systems on a Sylow $3$-subgroup of the sporadic groups $\mathrm{Fi}_{22}$, $\mathrm{Fi}_{23}$ and $\mathrm{B}$. We show that the $3$-group in each case does not support any exotic fusion systems. This is the first of two papers that will complete the classification of all corefree fusion systems on Sylow $p$-subgroups of sporadic groups for $p$ odd.