Maps behaving like exponentials and maximal unipotent subgroups of groups of Lie type

Let U be a maximal unipotent subgroup of a finite classical group in good c haracteristic. We prove the existence of a bijection between U and the asso ciated Lie algebra preserving centralizers. As a consequence, we obtain inf ormation on the sizes of the conjugacy classes of U. Similar results are pr oved in the exceptional cases.