ABSTRACT

A comprehensive presentation of abstract algebra and an in-depth treatment of the applications of algebraic techniques and the relationship of algebra to other disciplines, such as number theory, combinatorics, geometry, topology, differential equations, and Markov chains.

chapter 1|21 pages

First introduction: Affine geometry

chapter 2|27 pages

Second introduction: Linear equations

chapter 3|23 pages

Vector spaces

chapter 4|23 pages

Linear and affine mappings

chapter 5|16 pages

Abstract affine geometry

chapter 7|21 pages

Determinants

chapter 8|18 pages

Volume functions

chapter 9|28 pages

Eigenvectors and eigenvalues

chapter 11|19 pages

Tensor products and base-field extensions

chapter 12|24 pages

Metric geometry

chapter 13|30 pages

Euclidean spaces

chapter 14|28 pages

Linear mappings between euclidean spaces

chapter 15|35 pages

Bilinear forms

chapter 16|27 pages

Groups of automorphisms

chapter 17|31 pages

Application: Markov chains

chapter 20|15 pages

Groups

chapter 21|21 pages

Subgroups and cosets

chapter 22|14 pages

Symmetric and alternating groups

chapter 23|18 pages

Group homomorphisms

chapter 24|19 pages

Normal subgroups and factor groups

chapter 25|27 pages

Free groups; generators and relations

chapter 26|21 pages

Group actions

chapter 28|16 pages

Nilpotent and solvable groups

chapter 29|20 pages

Topological methods in group theory

chapter 30|27 pages

Analytical methods in group theory

chapter 31|30 pages

Groups in topology