Sparse fusion systems

TL;DR

This paper introduces sparse and extremely sparse saturated fusion systems, demonstrating their constrained nature at odd primes and simplifying key theorems in fusion system theory.

Contribution

It defines sparse and extremely sparse fusion systems, proving their constrained property and simplifying existing theorems in the field.

Findings

Sparse systems are constrained at odd primes.

Extremely sparse systems are constrained for all primes.

Simplification of the Glauberman-Thompson p-nilpotency theorem.

Abstract

We define sparse saturated fusion systems and show that, for odd primes, sparse systems are constrained. This simplifies the proof of the Glauberman-Thompson p-nilpotency theorem for fusion systems and a related theorem of Stellmacher. We then define a more restrictive class of saturated fusion systems, called extremely sparse, that are constrained for all primes.