SWIM 2026
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 $\textbf{Z}$, 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 Schedule & Resources
Note: The schedule is tentative and may be subject to change during the program.
| Lecture | Topics Covered | Speaker | Resources |
|---|---|---|---|
| Lecture 1 May 6 (Wed) |
Covers matrix operations (addition, multiplication, scalar multiplication) and row reduction techniques. Focus on mechanics and solving linear systems. | Krishna Hanumanthu – Gayathri M |
Problem Sheet |
| Lecture 2 May 7 (Thu) |
Introduces matrix transposition and determinants, including computation methods and properties under row operations. | Krishna Hanumanthu – Gayathri M | |
| Lecture 3 May 8 (Fri) |
The necklace Problem and the merry go round problem were discussed as motivating examples to explore symmetries of objects. | Krishna Hanumanthu – Gayathri M | |
| Lecture 4 May 11 (Mon) |
Formal definition of groups and group actions. The orbit counting theorem. | Tulsi Srinivasan – Harikrishnan T R | |
| Lecture 5 May 13 (Wed) |
Explanations and proof of orbit counting theorem, followed by explicit computations using the theorem. | Tulsi Srinivasan – Harikrishnan T R | |
| Lecture 6 May 14 (Thu) |
Covers bijective homomorphisms (isomorphisms) and equivalence relations/partitions for classifying group elements. | Tulsi Srinivasan – Harikrishnan T R | |
| Lecture 7 May 20 (Wed) |
Introduces cosets and modular arithmetic, linking group theory to number theory. | Mohan R – Vyshnav V | |
| Lecture 8 May 21 (Thu) |
Explores subgroup-quotient group relationships and Cartesian product groups. | Mohan R – Vyshnav V | |
| Lecture 9 May 22 (Fri) |
Defines quotient groups constructed via normal subgroups. | Mohan R – Vyshnav V | |
| Lecture 10 May 25 (Mon) |
Proves that every group is isomorphic to a permutation subgroup. | R. Venkatesh – Amritha Varrier | |
| Lecture 11 May 27 (Wed) |
Derives the class equation connecting group order to conjugacy class sizes. | R. Venkatesh – Amritha Varrier | |
| Lecture 12 May 28 (Thu) |
Presents Sylow Theorems on subgroup existence and properties in finite groups. | R. Venkatesh – Amritha Varrier |