Fusion Systems and Simple Groups With Class Two Sylow $p$-subgroups

TL;DR

This paper classifies all reduced saturated fusion systems on finite p-groups of nilpotency class two and provides a new proof for the classification of finite simple groups with class two Sylow 2-subgroups.

Contribution

It offers a complete classification of fusion systems on certain p-groups and presents a novel proof of a key classification result for finite simple groups.

Findings

Classified all reduced saturated fusion systems on p-groups of nilpotency class two.

Provided a new proof of the classification of finite simple groups with class two Sylow 2-subgroups.

Enhanced understanding of the structure of fusion systems and simple groups.

Abstract

We determine all reduced saturated fusion systems supported on a finite $p$-group of nilpotency class two. As a consequence, we obtain a new proof of Gilman & Gorenstein's classification of finite simple groups with class two Sylow $2$-subgroups.