1. Groups and Subgroups 1. Introduction and Examples 2. Binary Operations 3. Isomorphic Binary Structures 4. Groups 5. Subgroups 6. Cyclic Groups 7. Generating Sets and Cayley Digraphs 2. Permutaions, cosets, and Direct Products 1. Groups of Permutations 2. Orbits, Cycles, and the Alternating Groups 3. Cosets and the Theorem of Lagrange 4. Direct Products and Tinitely Generated Abelian Groups 5. Plane Isometries 3. Homomorphishms and Factor Groups 1. Homomorphisms 2. Factor Groups 3. Factor-Group Computations and Simple Groups 4. Group Action on a Set 5. Applications of G-Sets to Counting 4. Rings and Fields 1. Rings and Fields 2. Integral Domains 3. Fermat's and Euler's Theorems 4. The Field of Quotients of an Integral Domain 5. Rings of Polynomials 6. Factorization of Polynomials over a Field 7. Noncommutative Examples 8. Orderd Rings and Fields 5. Ideals and Factor Rings 1. Homomorphisms and Factor Rings 2. Prime and Maximal Ideals 3. Grobner Bases for Ideals 6. Extension Fields 1. Introduction to Extension Fields 2. Vector Spaces 3. Algebraic Extensions 4. Geometric Constructions 5. Finite Fields 7. Advanced Group Theory 1. Isomorphism Theorems 2. Series of Groups 3. Sylow Theorems 4. Applications of the Sylow Theory 5. Free Abelian Groups 6. Free Groups 7. Group Presentations 8. Groups in Topology 1. Simplicial Complexes and Homology Groups 2. Computations of Homology Groups 3. More Homology Computations and Applications 9. Factorization 1. Unique Factorization Domains 2. Euclidean Domains 3. Gaussian Integers and Multiplicative Norms 10. Automorphishms and Galois Theory 1. Automorphisms of Fields 2. The Isomorphism Extension Theorem 3. Splitting Extensions 4. Totally Inseparable Extensions 5. Galois Theory 6. Illustrations of Galois Theory 7. Cyclotomic Extensions 8. Insolvability of the Quintic