CLOSED CONJUGACY CLASSES, CLOSED ORBITS AND STRUCTURE OF ALGEBRAIC-GROUPS

In this paper we prove that if an affine algebraic group (in characteristic sero) has all its conjugacy classes closed, then it is nilpotent. A classical result (called sometimes the Kostant-Rosenlicht Theorem) guarantees that if an affine algebraic group G is unipotent, then all its orbits on affine varieties are closed. We prove the converse of that theorem in arbitrary characteristics.