Sharpness of saturated fusion systems on a Sylow $p$-subgroup of ${\rm G}_2(p)$

TL;DR

This paper proves that the sharpness conjecture by Daz-Park is valid for saturated fusion systems on Sylow p-subgroups of G_2(p) for primes p at least 5, advancing understanding in group theory.

Contribution

It establishes the conjecture's validity specifically for saturated fusion systems on Sylow p-subgroups of G_2(p) for p ≥ 5, a previously unresolved case.

Findings

Confirmed Daz-Park's sharpness conjecture for these systems

Extended the class of groups where the conjecture holds

Provided new insights into the structure of fusion systems on G_2(p)

Abstract

We prove that the D\'iaz-Park's sharpness conjecture holds for saturated fusion systems defined on a Sylow $p$-subgroup of the group ${\rm G}_2(p)$, for $p\geq 5$.