Control of Fusion by Abelian Subgroups of the Hyperfocal Subgroup

TL;DR

This paper establishes conditions under which isomorphisms between saturated fusion systems are detected on specific subgroups of the hyperfocal subgroup, with implications for mod p cohomology.

Contribution

It proves that isomorphisms are detected on elementary abelian or low-exponent abelian subgroups of the hyperfocal subgroup depending on whether p is odd or even.

Findings

Detection of isomorphisms on elementary abelian subgroups for odd p

Detection on abelian subgroups of exponent at most 4 for p=2

Implications for mod p group cohomology

Abstract

We prove that an isomorphism between saturated fusion systems over the same finite p-group is detected on the elementary abelian subgroups of the hyperfocal subgroup if p is odd, and on the abelian subgroups of the hyperfocal subgroup of exponent at most 4 if p = 2. For odd p, this has implications for mod p group cohomology.