Introduction to Group Theory Course | UG Math | Prof. R. Venkatesh IISc
Course Details
| Exam Registration | 54 |
|---|---|
| Course Status | Ongoing |
| Course Type | Core |
| Language | English |
| Duration | 12 weeks |
| Categories | Mathematics |
| Credit Points | 3 |
| Level | Undergraduate |
| Start Date | 19 Jan 2026 |
| End Date | 10 Apr 2026 |
| Enrollment Ends | 02 Feb 2026 |
| Exam Registration Ends | 20 Feb 2026 |
| Exam Date | 24 Apr 2026 IST |
| NCrF Level | 4.5 — 8.0 |
Introduction to Group Theory: A Foundational Journey into Symmetry
Welcome to a comprehensive introduction to one of the most elegant and powerful branches of abstract mathematics: Group Theory. This 12-week undergraduate course, designed and instructed by Prof. R. Venkatesh of the Indian Institute of Science (IISc) Bangalore, offers a structured and gentle entry into the world of symmetries and algebraic structures. Group theory is not just a subject; it's the language of symmetry, underpinning modern physics, chemistry, cryptography, and puzzle-solving.
Meet Your Instructor: Prof. R. Venkatesh
Prof. R. Venkatesh has been an Assistant Professor at the prestigious Indian Institute of Science, Bengaluru, since May 2017. His research expertise lies in the intricate domain of infinite dimensional Lie algebras and their representations. This deep research background ensures that the course is built on a robust foundational understanding, presented with clarity for undergraduate students.
Course Overview & Prerequisites
ABOUT THE COURSE: This course is a first course in group theory, meticulously designed for second-year undergraduate students. It starts from the very definition of a group and systematically builds up to advanced concepts like group actions and Sylow's theorems. The pedagogy emphasizes essential examples from Geometry and Number Theory, making abstract concepts tangible and intuitive.
PREREQUISITES: The course is largely self-contained. The primary requirement is a familiarity with Basic Linear Algebra. Anyone with a foundational mathematics background can successfully access and benefit from this course. It is perfectly suited for UG students majoring in Mathematics, Physics, or Computer Science.
Detailed 12-Week Course Layout
The course is structured to provide a logical progression from basic definitions to powerful theorems. Here is a week-by-week breakdown:
| Week | Topic |
|---|---|
| Week 1 | Groups: definitions and basic properties |
| Week 2 | Examples from Geometry I: Group of isometries of a plane and their finite reflection subgroups |
| Week 3 | Examples from Geometry II: symmetries of a cube, tetrahedron and regular n-gon |
| Week 4 | Examples from Number theory: the additive/multiplicative group of integers modulo n |
| Week 5 | Cyclic groups: various characterizations |
| Week 6 | Abelian groups examples, Symmetric and alternating groups |
| Week 7 | Group homomorphisms, Normal subgroups |
| Week 8 | Quotient groups and isomorphism theorems |
| Week 9 | Group actions: definition and various examples |
| Week 10 | Application of group action: Cayley’s theorem, Lagrange’s theorem, Cauchy's theorem |
| Week 11 | Conjugacy action, the class equation |
| Week 12 | Sylow’s theory and its application |
Key Concepts You Will Master
- Symmetry as Algebra: Learn how geometric symmetries of objects like polygons and polyhedra form groups.
- Fundamental Structures: Understand cyclic groups, abelian groups, and the crucial symmetric and alternating groups.
- Mappings & Structure: Dive into group homomorphisms, normal subgroups, and the powerful isomorphism theorems.
- Group Actions: Discover how groups can "act" on sets, a unifying concept that leads to profound results.
- Powerful Theorems: Prove and apply cornerstone theorems like those of Lagrange, Cauchy, and Sylow.
Recommended Textbooks & Resources
To supplement the course lectures, the following textbooks are highly recommended:
- Dummit & Foote: Abstract Algebra - A comprehensive and widely-used standard reference.
- Eie & Chang: A Course on Abstract Algebra - Offers a clear and accessible approach.
- M. A. Armstrong: Groups and Symmetry - Excellent for building geometric intuition.
- D. M. Burton: Elementary Number Theory - Useful for the number-theoretic examples.
Who Should Take This Course?
This course is ideal for:
- Undergraduate students in Mathematics seeking a strong foundation in algebra.
- Physics students interested in the mathematical underpinnings of symmetry in quantum mechanics and particle physics.
- Computer Science students curious about applications in cryptography and coding theory.
- Any enthusiastic learner with basic linear algebra knowledge wanting to explore beautiful abstract concepts.
Embark on this 12-week journey with Prof. Venkatesh to unravel the language of symmetry. From the rotations of a cube to the deep theorems that classify finite groups, this course in Group Theory will equip you with fundamental tools and a new way of seeing structure in mathematics and the world.
Enroll Now →