0. Sets and Relations 1 CONTENTS I. Groups and Subgroups 1. Introduction and Examples 4 2. Binary Operations 7 3. Isomorphic Binary Structures 9 4. Groups 13 5. Subgroups 17 6. Cyclic Groups 21 7. Generators and Cayley Digraphs 24 II. Permutations, Cosets, and Direct Products 8. Groups of Permutations 26 9. Orbits, Cycles, and the Alternating Groups 30 10. Cosets and the Theorem of Lagrange 34 11. Direct Products and Finitely Generated Abelian Groups 37 12. Plane Isometries 42 III. Homomorphisms and Factor Groups 13. Homomorphisms 44 14. Factor Groups 49 15. Factor-Group Computations and Simple Groups 53 16. Group Action on a Set 58 17. Applications of G-Sets to Counting 61 IV. Rings and Fields 18. Rings and Fields 63 19. Integral Domains 68 20. Fermat’s and Euler’s Theorems 72 21. The Field of Quotients of an Integral Domain 74