Developments in finite Phan theory

TL;DR

This paper provides a comprehensive review and extension of finite Phan theory, focusing on the structure and classification of twisted Chevalley groups over finite fields, supporting the classification of finite simple groups.

Contribution

It revises and generalizes Phan's presentation results for twisted Chevalley groups using unitary subgroups of ranks one and two.

Findings

Revised presentation results for twisted Chevalley groups

Generalized Phan's theorems for simply laced diagrams

Supports classification of finite simple groups

Abstract

This is a final report on finite Phan theory, a project that has been concerned with a revision and generalisation of Phan's presentation results of twisted Chevalley groups over finite fields with simply laced diagrams by unitary subgroups of rank one and two. Finite Phan theory has been initiated on request by Richard Lyons and Ronald Solomon in order to support their revision of the classification of the finite simple groups.