Closed Conjugacy Classes, Closed Orbits and Structure of Algebraic Groups

Abstract

In this paper we prove that if an affine algebraic group (in characteristic zero) 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.