MA4070: Algebra I - Groups and Rings (Fall 2026)

Topics covered: Binary operation and its properties, definition of groups, examples and basic properties. Subgroups, coset of a subgroup, Lagrange’s theorem. cyclic groups. Normal subgroups, quotient groups. Homomorphisms, Isomorphism theorems. Permutation groups, Cayley’s theorems. Direct and semidirect product of groups. Group actions and Sylow theorems.

Definition of Rings, examples and basic properties, zero divisors, integral domains, fields, characteristic of a ring, quotient field of an integral domain. Subrings, ideals, quotient rings. Isomorphism theorems. Ring of polynomials. Prime elements, irreducible elements and their properties. Unique factorization domains, Principal ideal domains, and Euclidean domains. Prime ideals. Maximal ideals, Prime avoidance lemma, Chinese remainder theorem.

Lecture schedule & Venue: Mondays (9:00 to 9:55), Wednesdays (11:00 to 11:55) and Thursdays (10:00 to 10:55) at MA01 (Ground floor, AD3 Building, Department of Mathematics, IITH)

Exams: 

Mid-Semester Exam: September 20, 2026 (Sunday), 2:00 PM. to 7:00 PM. Room no. MA01

End-Semester Exam: November 18, 2026 (Wednesday), 2:00 PM. to 7:00 PM. Room no. MA01

In addition, there may be a few short surprise tests (15–30 minutes) conducted throughout the semester. The final grade will be based on overall performance across all these assessments. Students are required to obtain a minimum of 40% of the total marks (aggregated across all examinations) in order to successfully complete the course.

Google classroom link: https://classroom.google.com/c/ODY5MDc2MjA1Njcz?cjc=j4mdrw34 (this is only for students registered in the course MA4070. To ensure this, please use your IITH account to join.)

Lectures & Notes:

Lecture notes (updated on Aug. 03, 2026)

27/07/2026 Lecture 1: Basic Examples

29/07/2026 Lecture 2: Permutation Groups

30/07/2026 Lecture 3: Symmetries of Square

03/08/2026 Lecture 4: Dihedral Groups

05/08/2026 Lecture 5: A Check Digit Scheme Based on D_5

06/08/2026 Tutorial 1

10/08/2026 Lecture 7: Subgroups and Cyclic groups

12/08/2026 Lecture 8: Homomorphism and normal subgroups

13/08/2026 Lecture 9: Equivalence relations and coset

17/08/2026 Lecture 10: Lagrange’s theorem and its converse?

19/08/2026 Lecture 11: Tutorial 2

References:

1. Michael Artin, Algebra, Pearson Prentice Hall, 2011

2. David. S. Dummit and Richard. M. Foote, Abstract algebra, John Wiley & Sons Inc, 2004

3. Joseph A. Gallian, Contemporary abstract algebra, CRC Press, 2021

Some (YouTube) videos on Groups and Rings:

1. https://youtube.com/playlist?list=PLelIK3uylPMGzHBuR3hLMHrYfMqWWsmx5&si=v3n5pqBXvVJjc-co

2. https://youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR_XOW3IcK-TP6&si=VfFedXcuakCtZK3W