Published online by Cambridge University Press: 24 April 2019
Let $G$ be a simple exceptional algebraic group of adjoint type over an algebraically closed field of characteristic
$p>0$ and let
$X=\text{PSL}_{2}(p)$ 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.