Group cohomology and control of p-fusion

TL;DR

This paper establishes that certain cohomology conditions imply control of p-fusion in finite groups, extending classical results and involving higher chromatic theories for p=2.

Contribution

It generalizes Quillen's classical results by linking cohomology isomorphisms to control of p-fusion, including for p=2 with advanced cohomology theories.

Findings

Cohomology variety homeomorphisms imply p-fusion control for odd p.

Extension of classical Sylow p-subgroup results to broader subgroups.

Use of higher chromatic cohomology theories for p=2 cases.

Abstract

We show that if an inclusion of finite groups H < G of index prime to p induces a homeomorphism of mod p cohomology varieties, or equivalently an F-isomorphism in mod p cohomology, then H controls p-fusion in G, if p is odd. This generalizes classical results of Quillen who proved this when H is a Sylow p-subgroup, and furthermore implies a hitherto difficult result of Mislin about cohomology isomorphisms. For p=2 we give analogous results, at the cost of replacing mod p cohomology with higher chromatic cohomology theories. The results are consequences of a general algebraic theorem we prove, that says that isomorphisms between p-fusion systems over the same finite p-group are detected on elementary abelian p-groups if p odd and abelian 2-groups of exponent at most 4 if p=2.