Week 8: A5 and the symmetries of an icosahedron. Sylow theorems. Study of permutation groups

1. Review: Linear algebra & Definition of groups
2. Symmetric groups, Subgroups of ℤ & Cyclic subgroups
3. Isomorphisms & Homomorphisms
4. Kernels, Normality, Centers and Inner Autos
5. Equivalence Relations & Cosets
6. Congruence mod n
7. Quotients
8. More on Quotients & Vectorspaces
9. More on Vectorspaces
10. Bases and vectorspaces; Matrices and linear transfer
11. Bases & Matrices
12. Eigenvalues and Eigenvectors
13. Review for midterm & Orthogonal groups
14. Orthogonal Groups & Geometry
15. Finite groups of motions
16. Discrete groups of motions
17. Abstract group actions
18. Group actions
19. Group actions II
20. Basic properties and constructions of group actions
21. Groups acting on themselves by left multiplication
22. Groups acting on themselves by conjugation
23. Alternating group structure
24. Ring Theory
25. Ring Theory II
26. Examples of Rings
27. Examples of Rings II
28. Basic properties and constructions of Rings
29. More on Rings
30. Extensions of Rings: Quotient rings
31. Extensions of Rings: Integral domains
32. Extensions of Rings: Fields of fractions
33. Gauss’ lemma
34. Eisenstein’s criterion
35. Algebraic integers
36. Dedekind domains & Ideal class groups
37. Review 1
38. Review 2

Algebra is the language of modern mathematics. This course introduces students to that language through a study of groups, group actions, vector spaces, linear algebra, and the theory of fields. The recorded lectures are from the Harvard Faculty of Arts and Sciences course Mathematics 122, which was offered as an online course at the Extension School (E-222).