Abstract Group Theory

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About Course

Abstract Group Theory.

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’s 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.

What Will You Learn?

  • Week 1: Motivation, definition, examples, and basic properties
  • Week 2: Subgroups, subgroups of integers, homomorphisms
  • Week 3: Quotient groups, isomorphism theorems
  • Week 4: Group operations, counting formula
  • Week 5: Symmetric groups
  • Week 6: Operations of a group on itself, class equation
  • Week 7: Sylow theorems I
  • Week 8: Sylow theorems II

Course Content

Introduction to Abstract Group Theory

  • INTRODUCTION TO ABSTRACT GROUP THEORY
    00:00
  • Lecture 01 – “Motivational examples of groups”
    00:00
  • Lecture 02 – “Definition of a group and examples”
    00:00
  • Lecture 03 – “More examples of groups”
    00:00
  • Lecture 04 – “Basic properties of groups and multiplication tables”
    00:00
  • Lecture 05 – Problems 1
    00:00
  • Lecture 06 – Problems 2
    00:00
  • Lecture 07 – Problems 3
    00:00
  • Lecture 08 – Subgroups
    00:00
  • Lecture 09 – Types of groups
    00:00
  • Lecture 10 – Group homomorphisms and examples
    00:00
  • Lecture 11 – Properties of homomorphisms
    00:00
  • Lecture 12 – Group isomorphisms
    00:00
  • Lecture 13 – Normal subgroups
    00:00
  • Lecture 14 – Equivalence relations
    00:00
  • Lecture 15 – Problems 4
    00:00
  • Lecture 16 – Cosets and Lagrange’s theorem
    00:00
  • Lecture 17 – S_3 revisited
    00:00
  • Lecture 18 – Problems 5
    00:00
  • Lecture 19 – Quotient groups
    00:00
  • Lecture 20 – Examples of quotient groups
    00:00
  • Lecture 21 – First isomorphism theorem
    00:00
  • Lecture 22 – Examples and Second isomorphism theorem
    00:00
  • Lecture 23 – Third isomorphism theorem
    00:00
  • Lecture 24 – Cauchy’s theorem
    00:00
  • Lecture 25 – Problems 6
    00:00
  • Lecture 26 – Symmetric groups I
    00:00
  • Lecture 27 – Symmetric Groups II
    00:00
  • Lecture 28 – Symmetric groups III
    00:00
  • Lecture 29 – Symmetric groups IV
    00:00
  • Lecture 30 – Odd and even permutations I
    00:00
  • Lecture 31 – Odd and even permutations II
    00:00
  • Lecture 32 – Alternating groups
    00:00
  • Lecture 33 – Group actions
    00:00
  • Lecture 34 – Examples of group actions
    00:00
  • Lecture 35 – Orbits and stabilizers
    00:00
  • Lecture 36 – Counting formula
    00:00
  • Lecture 37 – Cayley’s theorem
    00:00
  • Lecture 38 – Problems 7
    00:00
  • Lecture 39 – Problems 8 and Class equation
    00:00
  • Lecture 40 – Group actions on subsets
    00:00
  • Lecture 41 – Sylow Theorem I
    00:00
  • Lecture 42 – Sylow Theorem II
    00:00
  • Lecture 43 – Sylow Theorem III
    00:00
  • Lecture 44 – Problems 9
    00:00
  • Lecture 45 – Problems 10
    00:00

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