Algebra I

About the Course

This page is dedicated to notes and assignments in the first course in Algebra for first year B.Sc. students in UPES. Click below for the course plan:
Algebra I Course Plan

References:

1. Vijay K. Khanna, S. K. Bhambri, A Course in Abstract Algebra, Fourth Edition

2. Joseph Gallian, Contemporary Abstract Algebra, Eighth Edition

3. David S. Dummit, Richard S. Foote, Abstract Algebra, Third Edition

Evaluation:

Quizzes and Assignments: 30%

Mid-Sem: 25%

End-Sem: 40%

Class Performance: 5%

Lecture Summaries and Exercises:

Week 1: 9th January to 11th January 2023

Lecture 1: Introduction to the Course. 

Lecture 2: The Dihedral group D4, and definition of a group.

Lecture 3: Topics Covered: Examples of various Shapes and their symmetries, and group of symmetries.

Here is the Exercise set for the first week: Exercise Set 1

Week 2: 16th January to 18th January 2023

Lecture 4: Topics covered: A relook at the proof of the division algorithm,

    A recap of the dihedral group D8, with the formal definition of a group.

Lecture 5: Some More Examples of Groups, Uniqueness of the identity and inverse of an element.

Here is the Exercise set for the second week: Exercise Set 2.

Week 3: 23rd January to 25th January 2023

Lecture 6: Cancellation Laws, Product of elements, and orders of elements.

Lecture 7: Some more on Orders of elements of a Group.

Here is the Exercise set for the third week: Exercise Set 3. 

Typo in Exercise Set 3. In the first question, it should be ``(ab)^-2 = b^-2 a^-2''

Week 4: 30th January to 01 February 2023

Lecture 8: Solving exercises from Exercise Set 3.

Lecture 9: Permutations and Permutation Groups

Here is the Exercise set for the fourth week: Exercise Set 4

Week 5: 06 February to 08 February 2023

Lecture 10: Permutations of {1, 2, . . . , 3}, Composition of Permutations and Inverse

Here is the exercise ser for the fifth week: Exercise Set 5.

Week 6: 13 February to 15 February 2023

Lecture 11: Some more on permutations, and an Introduction to Subgroups.

Lecture 12: Examples of subgroups, and the One Subgroup Test.

Week 7: 20 February to 22 February 2023

Lecture 13: Examples of Subgroups, and another test for subgroups

Lectures 14 and 15: Centre of a group, and the centralizer of an element of a group.

Week 8: 27th February to 01 March 2023

Lecture 16: Centralizer of an element of a group, and some set theory on subgroups.

Week 9: 6th March to 7th March 2023

Lecture 17: Right cosets of a subgroup in a group.

Lecture 18: Left Cosets, Lagrange's Theorem and Index of a Subgroup in a Group.

Week 10: 13th March to 15th March 2023

Lecture 19: A Corollary to Lagrange's Theorem, Normalizer and Centralizer of a subgroup.

Lecture 20: Centralizers and Normalizers of Subgroups, Product of Subgroups (Also called a ``Complex'')

Lecture 21: The Size of a Complex of Two subgroups

Week 11: 20th to 22nd March 2023

Lecture 22: Some Exercises about Complexes of Subgroups.

Lecture 23: A few results and exercises on Complexes of Subgroups.

Lecture 24: Revision of Topics Covered till 7th March 2023.

Week 12: 27th to 29th March 2023

Lecture 25: Subgroups of subgroups and their index. 

Lecture 26: Introduction to Cyclic Groups

Lecture 27: Examples and Properties of Subgroups, and Orders of elements