Treatment of component type at odd primes in fusion system classification

Develop methods to treat the component-type case for odd primes in the classification of simple saturated fusion systems, specifically addressing situations where the centralizer of a suitably chosen p-element contains a component.

Background

The authors outline a program to translate classification results from finite groups to fusion systems across primes. While substantial progress exists at p=2, extending comparable classification results to odd primes faces obstacles, notably the ‘component type’ case (where certain centralizers have components).

Addressing this gap is essential to making broader classification approaches viable at odd primes and to understanding potential exotic behavior outside the component-free settings.