- 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.