- Lecture 0 (08/22) Definitions and examples: monoids, groups, homomorphisms, subgroup, normal subgroups.
- Lecture 1 (08/23) Quotient groups, free groups, presentation of a group by generators and relations.
- Lecture 2 (08/25) Group action on a set. Counting lemmas.
- Lecture 3 (08/28) Fundamental theorem of group homs. Short exact sequences.
- Lecture 4 (08/30) Isomorphism theorems. Semi-direct products and split short exact sequences.
- Lecture 5 (09/01) Retraction. Construction of semi-direct product from a group action.
- Lecture 6 (09/06) p-groups, Sylow subgroups, Sylow theorems.
- Lecture 7 (09/08) Some applications of Sylow theorems.
- Lecture 8 (09/11) Composition series. Schrier's theorem. Zassenhaus lemma.
- Lecture 9 (09/13) Jordan-Holder theorem. Definition of solvable group.
- Lecture 10 (09/15) Basic properties of solvable and nilpotent groups.
- Lecture 11 (09/18) Symmetric group: sign homomorphism. Simplicity of alternating group.
- Lecture 12 (09/20) Symmetric group continued: exchange property, presentation [optional]
- Lecture 13 (09/22) Proof of the presentation of the symmetric group. Coxeter groups [optional] Beginning towards representation theory...
- Lecture 14 (09/25) Basics of linear algebra: direct sums, tensor products, duals. Symmetric and exterior/alternating products.
- Lecture 15 (09/27) Representations of a group. Direct sums, tensor products, dual and Hom of representations. Irreducible vs indecomposable representations.
- Lecture 16 (09/29) Revisiting symmetric and alternating products via averaging. Maschke's theorem. Complete reducibility. Schur's lemma.
- Lecture 17 (10/02) Regular representation. Decomposition of regular representation.
- Lecture 18 (10/04) Characters and orthogonality.
- Lecture 19 (10/06) Character table. More orthogonality relations.
- Lecture 20 (10/10) Restriction of representations. Induced representations.
- Lecture 21 (10/12) Characters of induced representations. Frobenius reciprocity.
- Lecture 22 (10/16) More examples of induced representations
- Lecture 23 (10/18) Mackey's theorem - double cosets
- Lecture 24 (10/20) [Optional] representation theory of symmetric groups (notes by Kiwon Lee)
- Lecture 25 (10/23) Basic definitions of rings; homomorphisms; ideals.
- Lecture 26 (10/25) Algebra of ideals; finite generation; characteristic; CRT
- Lecture 27 (11/03) Prime and maximal ideals. Local rings.
- Lecture 28 (11/06) More on prime ideals. Polynomial rings, algebraic dependence.
- Lecture 29 (11/08) Modules. Direct sum and direct product. Tensor product.
- Lecture 30 (11/13) Rings and modules of fractions.
- Lecture 31 (11/15) Localization. Noetherian rings - first properties.
- Lecture 32 (11/17) Noetherian modules. Hilbert basis theorem.
- Lecture 33 (11/20) Artinian rings. Structure theorem. Hensel's lemma.
- Lecture 34 (11/27) Primary decomposition. Noetherian and dimension zero implies Artinian.
- Lecture 35 (11/29) Primary decomposition - uniqueness results.