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Introduction to Abstract Group Theory

Prof. Krishna Hanumanthu · Chennai Mathematical Institute

Start with lecture 1 At NPTEL

About

This course will introduce abstract groups. We will start with definitions, basic properties and constructions and cover many important theorems in basic group theory, such as Lagrange’s theorem, Cauchy’s theorem and Sylow theorems. A major emphasis of the course will be to present numerous worked-out examples and problems. A part of the lecture every week will be devoted to explicit calculations.

Intended audience: BSc, MSc students studying mathematics

Prerequisites: Any BSc student who is familiar with high school mathematics can take this course

Lectures

Week 1

  1. Motivational examples of groups YouTube
  2. Definition of a group and examples YouTube
  3. More examples of groups YouTube
  4. Basic properties of groups and multiplication tables YouTube
  5. Problems 1 YouTube
  6. Problems 2 YouTube

Week 2

  1. Problems 3 YouTube
  2. Subgroups YouTube
  3. Types of groups YouTube
  4. Group homomorphisms and examples YouTube
  5. Properties of homomorphisms YouTube

Week 3

  1. Group isomorphisms YouTube
  2. Normal subgroups YouTube
  3. Equivalence relations YouTube
  4. Problems 4 YouTube
  5. Cosets and Lagrange's theorem YouTube
  6. S_3 revisited YouTube

Week 4

  1. Problems 5 YouTube
  2. Quotient groups YouTube
  3. Examples of quotient groups YouTube
  4. First isomorphism theorem YouTube
  5. Examples and Second isomorphism theorem YouTube
  6. Third isomorphism theorem YouTube

Week 5

  1. Cauchy's theorem YouTube
  2. Problems 6 YouTube
  3. Symmetric groups I YouTube
  4. Symmetric Groups II YouTube
  5. Symmetric groups III YouTube

Week 6

  1. Symmetric groups IV YouTube
  2. Odd and even permutations I YouTube
  3. Odd and even permutations II YouTube
  4. Alternating groups YouTube
  5. Group actions YouTube
  6. Examples of group actions YouTube

Week 7

  1. Orbits and stabilizers YouTube
  2. Counting formula YouTube
  3. Cayley's theorem YouTube
  4. Problems 7 YouTube
  5. Problems 8 and Class equation YouTube

Week 8

  1. Group actions on subsets YouTube
  2. Sylow Theorem I YouTube
  3. Sylow Theorem II YouTube
  4. Sylow Theorem III YouTube
  5. Problems 9 YouTube
  6. Problems 10 YouTube

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
Chennai Mathematical Institute
Department
Mathematics
Teacher
Prof. Krishna Hanumanthu
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge. Coordinated by IIT Madras.
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