MT2213 Undergraduate Algebra I

February 2021 Semester

Instructor: Dr Anupam Singh

Audience: 2nd year BS-MS students at IISER Pune

Tutors: Dr Parul Gupta, Dr Vivek Rai, Dr Kuntal Chakraborty

Schedule: online via Google Classroom on Meet

Tutorial: Monday 10AM -10:55 AM

Lectures: Tuesday, Thursday 11AM-11:55AM

The video recording of the lectures and hand written notes are available to all those who have registered for this course on Google Classroom. I have put the videos on YouTube and the links are given below in weekly schedule.

Evaluation: Test 1 - 20%, mid-sem 30%, Test 2 - 20%, end-sem 30%

Prerequisite: Some idea about algebraic structure such as set theory, logic etc and plenty of enthusiasm.

Goal of the course: Group Theory is a foundational subject and is used in many physical problems and engineering problems, apart from advanced Mathematics.

Proposed Content: Groups, examples, subgroups, normal subgroups, quotient, Lagrange's Theorem.

Group actions, Sylow's theorem, Structure theorem of finite Abelian groups.

Examples of some other algebraic structures such as Rings.

Text Books:

1. Abstract Algebra: D.S. Dummit and R.M. Foote

2. Topics in Algebra: I.N. Herstein

3. Algebra: M. Artin

4. Algebra: T.W Hungerford

5. Groups and Symmetry - Armstrong

6. Groups and Representations - Alperin and Bell

7. Group Theory selected problems - B Sury

Weekly schedule:

08/02/21 - Definition of a group and examples

09/02/21 - more examples

11/02/21 - homomorphism, subgroup, order of an element.

Assignment 1 Lecture 1 Lecture 2 Lecture 3

15/02/21 - Tutorial

16/02/21 - More examples, Left cosets

18/02/21 - Lagrange's Theorem

Assignment 2 Lecture 4 Lecture 5

22/02/21 - Tutorial

23/02/21 - Generators and examples, Cayley graph

25/02/21 - Normal subgroups and quotients

Lecture 6 Lecture 7

01/03/21 - Tutorial

02/03/21 - How to find more groups? Subgroups, Quotients, kernel and image of a homomorphisms, Aut(G), Normaliser and centraliser of a subgroup, center of a group, centraliser of an element, Direct product of two groups.

04/03/21 - First isomorphism theorem, Classification of cyclic groups.

Lecture 8 Lecture 9

06/03/21 Test I - take home

08/03/21 - tutorial

09/02/21 - Group Action, examples

11/03/21 - Orbit and Stabilisers, examples

Assignment 3 Lecture 10 Lecture 11

15/03/21 - tutorial

16/03/21 - Left and Right regular action, Cayley's theorem

18/03/21 - Symmetric and Alternating groups

Lecture 12 Lecture 13

22-30 Mid sem exam period, so no classes

Mid sem exam of our course on 24 March 2021 at 2:30 pm

01/04/21 - Orbit-Stabiliser Theorem

Assignment 4 Lecture 14

05/04/21 - tutorial

06/04/21 - Class Equation, groups of order p^2

08/04/21 - Sylow's theorem

Lecture 15 Lecture 16

12/04/21 - tutorial

13/04/21 - Proof of Sylow's Theorem 1, Center of p-groups

15/04/21 - Proof of Sylow's Theorem 2 & 3

Lecture 17 Lecture 18

19/04/21 - tutorial

20/04/21 - Symmetry and Group Theory, tiling, Symmetry of n-gon, platonic solids

22/04/21 - Isometry of Euclidean space, Orthogonal group O(n), SO(n)

Lecture 19 Lecture 20

26/04/21 - tutorial

27/04/21 - Iso(R^n), SO(3), Symmetries of platonic solids

29/04/21 -

Assignment 5

Lecture 21 Lecture 22

03/05/21 - Teaching break

04/05/21- Teaching break

06/05/21 - Teaching break

10/05/21 - tutorial

11/05/21 - Classification of finite Abelian groups

13/05/21 - Where to go from here? Learn more group theory, other algebraic structures, rings, modules etc

Lecture 23 Lecture 24

15/05/2021 Test 2

End sem exam 15/06/2021