A₁ -type subgroups containing regular unipotent elements

Let G be a simple exceptional algebraic group of adjoint type over an algebraically closed field of characteristic p > 0 and let X = PSL₂(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 A₁ . In this paper we prove that with two exceptions, X itself is contained in such a subgroup (the exceptions arise when (G, p) = (E₆, 13) or (E₇, 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.