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=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.