Minimal generation of finite simple groups of Lie type by regular unipotent elements

TL;DR

The paper resolves generation questions for finite quasisimple groups of Lie type by regular unipotent elements. It proves that three regular unipotent elements suffice in all cases, and in many families two conjugate regular unipotents already generate the group, using a BN-pair framework, parabolic and Levi subgroups, and structure constants derived from character theory. The authors combine detailed case analyses with computational tools to certify generation across classical and exceptional families, yielding explicit two- and three-generator results. These findings contribute to compact presentations and deepen understanding of generation in finite groups of Lie type, with potential applications to group presentations and random generation techniques.

Abstract

We prove that every finite simple group of Lie type $G$ can be generated by three regular unipotent elements. In certain cases we show that two regular unipotents are sufficient to generate $G$.

Theorems & Definitions (119)

• Conjecture 1
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof