# $A_1$-type subgroups containing regular unipotent elements

## TL;DR

This paper proves that most subgroups of a simple exceptional algebraic group containing a regular unipotent element are contained within an $A_1$-type subgroup, extending previous results and with specific notable exceptions.

## Contribution

It generalizes earlier work by showing that such subgroups are contained in $A_1$-type subgroups, except for two specific cases involving $E_6$ and $E_7$ with particular primes.

## Key findings

- Most subgroups with regular unipotent elements are contained in $A_1$-type subgroups
- Two specific exceptions identified for $(E_6,13)$ and $(E_7,19)$
- Applications to subgroup structure of finite groups of Lie type

## Abstract

Let $G$ be a simple exceptional algebraic group of adjoint type over an algebraically closed field of characteristic $p>0$ and let $X = {\rm PSL}_{2}(p)$ be a subgroup of $G$ containing a regular unipotent element $x$ of $G$. By a theorem of Testerman, $x$ is contained in a connected subgroup of $G$ of type $A_1$. In this paper we prove that with two exceptions, $X$ itself is contained in such a subgroup (the exceptions arise when $(G,p) = (E_6,13)$ or $(E_7,19)$). This extends earlier work of Seitz and Testerman, who established the containment under some additional conditions on $p$ and the embedding of $X$ in $G$. We discuss applications of our main result to the study of the subgroup structure of finite groups of Lie type.