ABSTRACT
This chapter introduces one of the objects central to our attentions: the affine algebraic groups, i.e., the group objects in the category of affine algebraic varieties. It considers only affine group varieties. A natural generalization of this situation is to consider group varieties not necessarily affine, or more generally group schemes, both topics being very interesting and active subjects of research. The chapter defines the concept of affine algebraic group, present the main examples that will be used throughout the book and complete the category by defining the morphisms. It describes the basic properties of morphisms and show that the irreducible component of the identity of an affine algebraic group is a connected normal subgroup of finite index. The chapter discusses regular actions of affine algebraic groups on algebraic varieties and study their basic properties. The interpretation of affine algebraic groups as groups of transformations of a geometric object will be of crucial importance to the theory.
