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:
Abstract Algebra: D.S. Dummit and R.M. Foote
Topics in Algebra: I.N. Herstein
Algebra: M. Artin
Algebra: T.W Hungerford
Groups and Symmetry - Armstrong
Groups and Representations - Alperin and Bell
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
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.
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
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
05/04/21 - tutorial
06/04/21 - Class Equation, groups of order p^2
08/04/21 - Sylow's theorem
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
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)
26/04/21 - tutorial
27/04/21 - Iso(R^n), SO(3), Symmetries of platonic solids
29/04/21 -
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