SWIM 2025
Abstract Algebra
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 $\mathbb{Z}$, 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 Schedule & Resources
Note: The schedule is tentative and may be subject to change during the program.
| Lecture | Topics Covered | Speaker | Resources |
|---|---|---|---|
| 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 – Deepak Dileep | |
| Lecture 2 May 9 (Fri) |
Introduces matrix transposition and determinants, including computation methods and properties under row operations. | U.K. Anandavardhanan – Deepak Dileep | |
| Lecture 3 May 14 (Wed) |
The necklace Problem and the merry go round problem were discussed as motivating examples to explore symmetries of objects. | K.N. Raghavan – Deepak Dileep | |
| Lecture 4 May 15 (Thu) |
Formal definition of groups and group actions. The orbit counting theorem. | K.N. Raghavan – Deepak Dileep | |
| Lecture 5 May 15 (Thu) |
Explanations and proof of orbit counting theorem, followed by explicit computations using the theorem. | K.N. Raghavan – Deepak Dileep | |
| Lecture 6 May 16 (Fri) |
Covers bijective homomorphisms (isomorphisms) and equivalence relations/partitions for classifying group elements. | K.N. Raghavan – Deepak Dileep | |
| Lecture 7 May 19 (Mon) |
Introduces cosets and modular arithmetic, linking group theory to number theory. | R. Venkatesh – Deepak Dileep |
Problem Sheet |
| Lecture 8 May 20 (Tue) |
Explores subgroup-quotient group relationships and Cartesian product groups. | R. Venkatesh – Deepak Dileep | |
| Lecture 9 May 21 (Wed) |
Defines quotient groups constructed via normal subgroups. | R. Venkatesh – Deepak Dileep | |
| Lecture 10 May 26 (Mon) |
Proves that every group is isomorphic to a permutation subgroup. | R. Venkatesh – Deepak Dileep | |
| Lecture 11 May 26 (Mon) |
Derives the class equation connecting group order to conjugacy class sizes. | R. Venkatesh – Deepak Dileep | |
| Lecture 12 May 27 (Tue) |
Presents Sylow Theorems on subgroup existence and properties in finite groups. | R. Venkatesh – Deepak Dileep |