Extremal and maximal normal abelian subgroups of a maximal unipotent subgroup in groups of Lie type

Abstract

We describe all maximal abelian normal subgroups in the unipotent radical U of a Borel subgroup in a group of Lie type G over a field K. This gives a new description of the extremal subgroups in U which were studied by C. Parker and P. Rowley. For a finite field K, we prove that either each large abelian subgroup in U is G-conjugate to a normal subgroup in U or G is of certain exceptional Lie type.