Notes on Modern Algebra I
by Patrick Gallagher

0. Introduction
1. Divisors, greatest common divisors, prime numbers
2. Unique factorization and least common multiple
3. Chebyshev's estimates
4. Algebra of sets
5. Algebra of maps
6. Inverse maps, partitions
7. Monoids, groups
8. Products of subsets, translations, subgroups, Lagrange's theorem
9. Rules for powers, orders of elements, cyclic groups
10. Intersection and product of subgroups
11. Normal subgroups, factor groups, first Isomorphism Theorem
12. More Isomorphism Theorems, solvable groups
13. Direct and semidirect products
14. Characters of finite abelian groups: extensions and orthogonality
15. Splitting, duality and basis theorems for finite abelian groups
16. Actions of groups on sets
17. Number of conjugacy classes, inner automorphisms
18. Groups of prime power order, Sylow's theorems
19. Some applications of Sylow theorems to groups of order < 60
20. Symmetric and alternating groups
21. Reflections and rotations in 3-space
22. Tetrahedra, tetrahedral group
23. Dihedral, octahedral, icosahedral groups
24. Counting orbits of colorings
25. Appendix: Euler's Polyhedron Theorem