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: 

Quiz - I:  August 27, 2026 (Thursday), from 11:00 a.m. to 12:30 p.m. Room No. MA-416 (fourth floor, Department of Mathematics)

Mid-Semester Exam: September 20, 2026 (Sunday), 2:00 p.m. to 7:00 p.m. Room no. MA-01

End-Semester Exam: November 18, 2026 (Wednesday), 2:00 p.m. to 7:00 p.m. Room no. MA-01

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.

Lectures & Notes:

Lecture notes (updated on Sept. 09, 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 6: Subgroups and Cyclic groups

12/08/2026 Lecture 7: Homomorphism and normal subgroups

13/08/2026 Lecture 8: Equivalence relations and coset

17/08/2026 Lecture 9: Lagrange’s theorem and its converse?

19/08/2026 Tutorial 2

20/08/2026 Lecture 10: Product groups 

24/08/2026 Lecture 11: Quotient groups & Isomorphism theorems

27/08/2026 Lecture 12: Group Actions

27/08/2026  Quiz - I

29/08/2026 Lecture 13: Group Actions

31/08/2026 Lecture 13: Cayley’s Theorem & The Class Equation

02/09/2026 Tutorial 3

03/09/2026 Lecture 14: The Sylow Theorems

07/09/2026 Lecture 15: The Sylow Theorems

09/09/2026 Lecture 16: The Sylow Theorems

10/09/2026 Lecture 17: Simplicity of A_n

14/09/2026 Lecture 18: Quotient groups & Group actions revisited

16/09/2026 Tutorial 4

17/09/2026 Doubt clearing session

20/09/2026 Mid-Semester

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