$A_{1}$-TYPE SUBGROUPS CONTAINING REGULAR UNIPOTENT ELEMENTS. (24th April 2019) - BURNESS, TIMOTHY C.; TESTERMAN, DONNA M.
Abstract : 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.