Based on Artin's Algebra, this course begins with Matrix Operations, Row Reduction, the Matrix Transpose, Determinants, and Permutations. It then introduces fundamental algebraic structures, covering Laws of Composition, Groups and Subgroups, Subgroups of the Additive Group of Integers $\textbf{Z}$, Cyclic Groups, Homomorphisms, and Isomorphisms. Concepts like Equivalence Relations, Partitions, Cosets, Modular Arithmetic, the Correspondence Theorem, Product Groups, and Quotient Groups are explored. The course also touches upon Group Actions, including Cayley's Theorem, the Class Equation, and concludes with the Sylow Theorems.
| Lecture | Topics Covered | Speaker | Resources (Notes/Problems) |
|---|---|---|---|
| Lecture 1 May 8 (Thu) |
Covers matrix operations (addition, multiplication, scalar multiplication) and row reduction techniques. Focus on mechanics and solving linear systems. | U. K. Anandavardhanan TA: Deepak Dileep |
Coming Soon |
| Lecture 2 May 9 (Fri) |
Introduces matrix transposition and determinants, including computation methods and properties under row operations. | U. K. Anandavardhanan TA: Deepak Dileep |
Coming Soon |
| Lecture 3 May 13 (Tue) |
The necklace Problem and the merry go round problem were discussed as motivating examples to explore symmetries of objects. | K. N. Raghavan TA: Deepak Dileep |
Coming Soon |
| Lecture 4 May 14 (Wed) |
Formal definition of groups and group actions. The orbit counting theorem. | K. N. Raghavan TA: Deepak Dileep |
Coming Soon |
| Lecture 5 May 15 (Thu) |
Explanations and proof of orbit counting theorem, followed by explicit computations using the theorem. | K. N. Raghavan TA: Deepak Dileep |
Coming Soon |
| Lecture 6 May 16 (Fri) |
Covers bijective homomorphisms (isomorphisms) and equivalence relations/partitions for classifying group elements. | K. N. Raghavan TA: Deepak Dileep |
Coming Soon |
| Lecture 7 May 19 (Mon) |
Introduces cosets and modular arithmetic, linking group theory to number theory. | R. Venkatesh TA: Deepak Dileep |
Problem Sheet |
| Lecture 8 May 20 (Tue) |
Explores subgroup-quotient group relationships and Cartesian product groups. | R. Venkatesh TA: Deepak Dileep |
Coming Soon |
| Lecture 9 May 22 (Thu) |
Defines quotient groups constructed via normal subgroups. | R. Venkatesh TA: Deepak Dileep |
Coming Soon |
| Lecture 10 May 23 (Fri) |
Proves that every group is isomorphic to a permutation subgroup. | R. Venkatesh TA: Deepak Dileep |
Coming Soon |
| Lecture 11 May 26 (Mon) |
Derives the class equation connecting group order to conjugacy class sizes. | R. Venkatesh TA: Deepak Dileep |
Coming Soon |
| Lecture 12 May 28 (Wed) |
Presents Sylow Theorems on subgroup existence and properties in finite groups. | R. Venkatesh TA: Deepak Dileep |
Coming Soon |