ABSTRACT
This chapter describes into the aspects of the theory of algebraic groups, centering our labors around the concept of quotient variety. Quotients are a central theme in geometric invariant theory, and their existence concerns the possibility of endowing a large subset of the set of orbits with a natural structure of algebraic variety. The chapter presents some basic examples and besides proving Kostant–Rosenlicht's theorem on the closedness of the orbits of a unipotent group. It considers come of the elementary facts concerning the structure of the orbits and their description as homogeneous spaces. The chapter provides the basic definitions and first properties of the so-called categorical and geometric quotients. It discusses the basic properties of the so-called induction procedure. The chapter also provides two important results; the first gives the description of the orbits of a regular action as homogeneous spaces and the second gives information about the variation of their dimensions.
