Fusion systems over a Sylow $p$-subgroup of $\mathrm{G}_2(p)$

TL;DR

This paper classifies all saturated fusion systems on a Sylow p-subgroup of G2(p) for odd p, revealing that most are realizable by finite groups except for 7 where many are exotic.

Contribution

It provides a complete classification of saturated fusion systems over Sylow p-subgroups of G2(p), identifying exotic systems specifically at p=7.

Findings

All such fusion systems are realized by finite groups for p ≠ 7.

For p=7, there are 29 saturated fusion systems, 27 of which are exotic.

The classification advances understanding of fusion systems related to G2(p).

Abstract

For $S$ a Sylow $p$-subgroup of the group $\mathrm{G}_2(p)$ for $p$ odd, up to isomorphism of fusion systems, we determine all saturated fusion systems $\mathcal{F}$ on $S$ with $O_p(\mathcal{F})=1$. For $p \ne 7$, all such fusion systems are realized by finite groups whereas for $p=7$ there are $29$ saturated fusion systems of which $27$ are exotic.