Algebra
July-November, 2026
Lectures: Tuesday, Thursday, and Friday at 4:00 PM
Office Hour : Wednesday from 2:00 PM to 4:00 PM
Evaluation:
Minor Examination - 25%
Major Examination - 45%
Quizzes - 30%
Quizzes will be held on Aug 13, Oct 6, and Nov 10, 2025
Attendance Policy:
A student is expected to have full attendance in the course unless the student takes a leave of absence for valid medical or bona fide reasons. In any case, at least 75% attendance in the course is mandatory.
Objectives
1. The course aims to develop a deep understanding of abstraction and mathematical rigor, which are essential for advanced mathematical research.
2. The course will emphasize rigorous proofs and the development of problem-solving skills in linear algebra and group theory through regular assignments and practice problem sheets.
Learning Outcomes
At the end of the course, the students will be able to:
1. Explain the fundamental concepts of advanced algebra, such as groups, group actions, and their role in modern mathematics and applied contexts.
2. Understand vectors, matrices, and their canonical forms.
3. Learn singular value decomposition along with its applications.
Contents
1. Group ( 6 Lectures): Recall definitions of Group, Subgroup, Normal subgroup, Simple group, quotient group. Cyclic group, Abelian group, Dihedral group, Quaternion group, Permutation group, Classical Groups such as general linear group and special linear group. Group Homomorphism and Isomorphism Theorems, Group automorphism. Introduction to computer algebra software GAP.
2. Group Action ( 12 Lectures): Definition and examples, conjugation action, Conjugacy classes and class equation, Orbit stabilizer lemma, Burnside’s lemma: Proof and applications in counting techniques, Cauchy and Sylow Theorems: Proof and applications in identifying simple groups. Classification of finite abelian groups, Overview of classification of finite simple groups, Direct Product and Semidirect product, Solvable and Nilpotent groups.
3. Vector Space (9 Lectures): Recall definitions of vector space, subspace, Linear independence, Basis, dimension, quotient space, linear transformation and matrices, eigenvalues, eigen vectors, characteristic and minimal polynomial, diagonalization of a matrix. Primary decomposition: Reduction to canonical forms such as triangular form, Jordan form, and rational canonical form, and dual spaces.
4. Inner product space and Canonical Forms (9 Lectures): Inner product, Orthogonality and Gram-Schmidt, Direct Sum, Operators on inner product spaces, Spectral Theorems, Bilinear and quadratic forms. Classical groups.
5. Singular Value Decomposition (3 Lectures): Singular Values and Singular Vectors, Polar Decomposition, Singular Value Decomposition and its applications.
Textbook:
1. Dummit, D.S., & Foote, R.M., Abstract Algebra, Third Edition, Wiley
2. Blyth, T.S., & Robertson, E.F., Further Linear Algebra (2002), Springer.
3. Strang, G., Linear Algebra for Everyone, (2020), Cambridge Press.
References:
1. Lang, S., Algebra, Revised Third Edition, Springer.
2. Lang, S., Linear Algebra, Third Edition, Springer.
3. Strang, G., Linear Algebra and learning from data, (2019), Cambridge Press.
Course
📅 August 2026
Week 1 (Jul 27 – Aug 2)
Jul 30 (Thu): Introduction to the Course, Definition of groups and subgroups, Cyclic groups, Abelian groups
Jul 31 (Fri): Dihedral groups, Quaternion group, Permutation groups, Classical Groups such as general linear group and special linear group.
Week 2 (Aug 3 – Aug 9)
Aug 04 (Tue): Normal subgroup, Simple group, quotient group.
Aug 06 (Thu): Group Homomorphism and Isomorphism Theorems.
Aug 07 (Fri): Group automorphisms.
Week 3 (Aug 10 – Aug 16)
Aug 11 (Tue): Introduction to computer algebra software GAP. Overview of the classification of finite simple groups.
Aug 13 (Thu): Direct Product and Classification of finite abelian groups, Quiz-I
Aug 14 (Fri): ❌ NO Friday Class (Follows Wednesday Timetable)
Week 4 (Aug 17 – Aug 23)
Aug 18 (Tue): Definition of group action and examples, conjugation action,
Aug 20 (Thu): Conjugacy classes and class equation, Orbit stabilizer lemma,
Aug 21 (Fri): Burnside’s lemma: Proof and applications in counting techniques,
Week 5 (Aug 24 – Aug 30)
Aug 25 (Tue): Cauchy’s Theorem with proof.
Aug 27 (Thu): Three Sylow Theorems: Proof and applications
Aug 28 (Fri): Three Sylow Theorems: Proof and applications
📅 September 2026
Week 6 (Aug 31 – Sep 6)
Sep 01 (Tue): Sylow Theorems: applications in identifying simple groups
Sep 03 (Thu): Semidirect product
Sep 04 (Fri): ❌ NO CLASS (Janmashtami Holiday)
Week 7 (Sep 7 – Sep 13)
Sep 08 (Tue): Semidirect product, groups of small order
Sep 10 (Thu): Solvable and Nilpotent groups.
Sep 11 (Fri): Solvable and Nilpotent groups.
Week 8 (Sep 14 – Sep 20)
Sep 15 – Sep 20: Minor Examination
Week 9 (Sep 21 – Sep 27)
Sep 22 (Tue): Recall definitions of vector space, subspace, Linear independence, Basis, dimension, and quotient space.
Sep 24 (Thu): linear transformation and matrices
Sep 25 (Fri): eigenvalues, eigen vectors, characteristic and minimal polynomial,
📅 October 2026
Week 10 (Sep 28 – Oct 4)
Sep 29 (Tue): Diagonalization of a matrix
Oct 01 (Thu): Inner product space, Orthogonality and Gram-Schmidt
Oct 02 (Fri): ❌ NO CLASS (Mahatma Gandhi’s Birthday / Prometeo)
Week 11 (Oct 5 – Oct 11)
Oct 05 (Mon): 🔄 EXTRA Friday Class: Direct Sum of subspaces
Oct 06 (Tue): Direct Sum of subspaces, Quiz-II
Oct 08 (Thu): Primary Decomposition
Oct 09 (Fri): Primary Decomposition
Week 12 (Oct 12 – Oct 18)
Oct 13 (Tue): Reduction to Triangular Form
Oct 15 (Thu): Reduction to Triangular Form
Oct 16 (Fri): Reduction to Jordan Form
Week 13 (Oct 19 – Oct 25)
Oct 19 (Mon): 🔄 EXTRA Friday Class: Reduction to Jordan Form
Oct 20 (Tue): ❌ NO CLASS (Dussehra Holiday )
Oct 22 (Thu): Rational Canonical Form
Oct 23 (Fri): Rational Canonical Form
Oct 24 (Sat): 🔄 EXTRA Thursday Class: Dual Spaces, Bilinear and quadratic forms. Classical groups.
Week 14 (Oct 26 – Nov 1)
Oct 27 (Tue): Operators on inner product spaces, Spectral Theorems
Oct 29 (Thu): Operators on inner product spaces, Spectral Theorems
Oct 30 (Fri): ❌ NO CLASS (Varchas 2026 Event)
📅 November 2026
Week 15 (Nov 2 – Nov 8)
Nov 2 – Nov 8: ❌ NO CLASSES (Mid-Semester Break)
Week 16 (Nov 9 – Nov 15)
o Nov 10,12,13: Singular Values and Singular Vectors, Polar Decomposition, Singular Value Decomposition, and its applications. Quiz-III
Week 17 (Nov 16 – Nov 22)
Nov 17 (Tue): Revision
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November 19, 2026 – November 26, 2026: Major Examination.