Algebraic group theory is, essentially, the intersection of algebraic geometry and group theory. Algebraic geometry, in a classical sense, is the study of algebraic sets (the vanishing loci of systems of polynomials) in affine or projective space. An algebraic group is an abstract group which is also an algebraic set, where the group operations are given by polynomial functions. Among the most widely studied are the affine algebraic groups, those which may be embedded in GL(V), the automorphism group of some (finite dimensional) vector space V. After defining affine algebraic groups, we will discuss some of the important subgroups which, in many ways, control the structure of the groups. Time permitting, we will discuss finite groups of Lie type, which are rational points of a particular type of linear algebraic group.
An Introduction to Algebraic Groups: Part 2