Reduced fusion systems over $p$-groups with abelian subgroup of index $p$: II

TL;DR

This paper classifies simple fusion systems over certain p-groups with a unique elementary abelian subgroup of index p, revealing many new exotic systems and expanding understanding of fusion system structures.

Contribution

It provides a comprehensive classification of simple fusion systems over p-groups with a specific abelian subgroup, including the discovery of numerous new exotic systems.

Findings

Classification of simple fusion systems over specified p-groups

Identification of many new exotic simple fusion systems

Dependence on the classification of finite simple groups

Abstract

Let $p$ be an odd prime, and let $S$ be a $p$-group with a unique elementary abelian subgroup $A$ of index $p$. We classify the simple fusion systems over all such groups $S$ in which $A$ is essential. The resulting list, which depends on the classification of finite simple groups, includes a large variety of new, exotic simple fusion systems.