Cover -- Contents -- 1 Algebraic sets and algebraic groups -- 1.1 The Zariski topology on affine space -- 1.2 Groebner bases and the Hilbert polynomial -- 1.3 Regular maps, direct products, and algebraic groups -- 1.4 The tangent space and non-singular points -- 1.5 The Lie algebra of a linear algebraic group -- 1.6 Groups with a split BN-pair -- 1.7 BN-pairs in symplectic and orthogonal groups -- 1.8 Bibliographic remarks and exercises -- 2 Affine varieties and finite morphisms -- 2.1 Hilbert's nullstellensatz and abstract affine varieties -- 2.2 Finite morphisms and Chevalley's theorem -- 2.3 Birational equivalences and normal varieties -- 2.4 Linearization and generation of algebraic groups -- 2.5 Group actions on affine varieties -- 2.6 The unipotent variety of the special linear groups -- 2.7 Bibliographic remarks and exercises -- 3 Algebraic representations and Borel subgroups -- 3.1 Algebraic representations, solvable groups, and tori -- 3.2 The main theorem of elimination theory -- 3.3 Grassmannian varieties and flag varieties -- 3.4 Parabolic subgroups and Borel subgroups -- 3.5 On the structure of Borel subgroups -- 3.6 Bibliographic remarks and exercises -- 4 Frobenius maps and finite groups of Lie type -- 4.1 Frobenius maps and rational structures -- 4.2 Frobenius maps and BN-pairs -- 4.3 Further applications of the Lang-Steinberg theorem -- 4.4 Counting points on varieties over finite fields -- 4.5 The virtual characters of Deligne and Lusztig -- 4.6 An example: the characters of the Suzuki groups -- 4.7 Bibliographic remarks and exercises -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- R -- S -- T -- U -- V -- W -- Z.