This is a revised and enlarged edition of the set of Notes: "Linear algebraicgroups" published by Benjamin in 1969. The added material pertains mainlyto rationality questions over arbitrary fields with, as a main goal, properties ofthe rational points of isotropic reductive groups. Besides, a number ofcorrections, additions and changes to the original text have been made.

Introduction to the First Edition

Introduction to the Second Edition

Conventions and Notation

CHAPTER AG--Background Material From Algebraic Geometry

　1.Some Topological Notions

　2.Some Facts from Field Theory

　3.Some Commutative Algebra

　4.Sheaves

　5.Affine K-Schemes, Prevarieties

　6.Products; Varieties

　7.Projective and Complete Varieties

　8.Rational Functions; Dominant Morphisms

　9.Dimension

　10.Images and Fibres of a Morphism

　11.k-structures on K-Schemes

　12.k-Structures on Varieties

　13.Separable points

　14.Galois Criteria for Rationality

　15.Derivations and Differentials

　16.Tangent Spaces

　17.Simple Points

　18.Normal Varieties

　References

CHAPTER I--General Notions Associated With Algebraic Groups

　1.The Notion of an Algebraic Groups

　2.Group Closure; Solvable and Nilpotent Groups

　3.The Lie Algebra of an Algebraic Group

　4.Jordan Decomposition

CHAPTER 11 Homogeneous Spaces

　5.Semi-lnvariants

　6.Homogeneous Spaces

　7.Algebraic Groups in Characteristic Zero

CHAPTER 111 Solvable Groups

　8.Diagonalizable Groups and Tori

　9.Conjugacy Classes and Centralizers of Scmi-Simple Elements

　10.Connected Solvable Groups

CHAPTER IV -- Borel Subgroups; Rcductive Groups

　11.Borei Subgroups

　12.Caftan Subgroups; Regular Elements

　13.The Borel Subgroups Containing a Given Torus

　14.Root Systems and Bruhat Decomposition in Reductive Groups

CHAPTER V-- Rationality Questions

　15.Split Solvable Groups and Subgroups

　16.Groups over Finite Fields

　17.Quotient of a Group by a Lie Subalgebra

　18.Cartan Subgroups over the Groundfield.Unirationality.Splitting of Reductive Groups

　19.Cartan Subgroups of Solvable Groups

　20.lsotropic Reductive Groups

　21.Relative Root System and Bruhat Decomposition for lsotropic ReductiveGroups

　22.Central lsogenies

　23.Examples

　24.Survey of Some Other Topics

　　A.Classification

　　B.Linear Representations

　　C.Real Reductive Groups

References for Chapters I to V

Index of Definition

Index of Notation