ABSTRACT
This chapter describes about what is usually called the Jordan decomposition for algebraic groups is a generalization of the multiplicative decomposition. It talks about the existence of the Jordan decomposition for an affine algebraic group and also an analogous result for its Lie algebra. A. Borel, generalizes the multiplicative Jordan decomposition of a linear transformation to a whole group of linear transformations provided that it is solvable. The chapter utilizes the decomposition of the elements of affine algebraic group to establish an important structure result, first for an abelian and then for a solvable affine algebraic group. It introduces the semisimple and unipotent radical. These special subgroups are the main obstruction to have a "good" theory of invariants, playing in our theory a role similar to the corresponding concepts for Lie algebras. A connected nontrivial affine algebraic group is semisimple if it does not have connected normal solvable closed subgroups.
