# Introduction to Abstract Algebra: Group Theory

Learn group theory or abstract algebra in pure mathematics with examples and solved exercises

**Language**: english

**Note**: 3.7/5 (22 notes) 569 students

**Instructor(s)**: AD Chauhdry (AD Maths Plus Academy)

**Last update**: 2021-08-07

## What you’ll learn

- Groups and related examples
- Identity element is the only elements which is idempotent
- Cancellation law hold in a group G
- Definition of Subgroups and related examples
- H is subgroup Iff ab^-1 is contained in H
- Intersection of any collection of subgroups is subgroup
- HuK is subgroup iff H is contained in K or K is contained in H
- Cyclic group and related examples
- Every subgroup of a cyclic group is cyclic
- Definition of cosets and related examples
- Prove that the number of left or right cosets define the partition of a group G
- Statement and Proof of Lagrange’s Theorem
- Symmetric groups and related examples and exercises
- Group of querternian and Klein’s four group
- Normalizers, centralizers and center of a group G and related theorem and examples
- Quotient or Factor groups
- Derived groups and related many examples
- Normal Subgroups, conjugacy classes, conjugate subgroups and related examples and theorems
- Kernal of group
- Automorphism and inner automorphism
- P Group and related theorems and examples
- Relations in group like homomorphism and isomorphism
- The centralizers is a subgroup of a group G
- The normalizers is a subgroup of a group G
- Center of a group is a subgroup of a group G
- The relation of conjugacy is an equivalence relation
- Theorem and examples on quotient groups
- Double cosets and related examples
- Definition of automorphism
- What is an inner automorphism
- Every cyclic group is an abelian group
- Groups of residue classes on different mode
- Examples of D_4 and D_5 groups
- Examples related C_6 and V_4
- The first isomorphism theorem and its proof
- The 2nd isomorphism theorem and its proof
- The 3rd isomorphism theorem and its proof
- Direct product of cyclic group
- Ring and Field
- Zero Divisor
- Integral Domain
- Theorems on Ring and Field

## Requirements

- Basics of algebra and interest in learning abstract algebra is the requirement to take this course.

## Description

This is an advanced level course of **Introduction to Abstract Algebra** with majors in **Group Theory**. The students who want to learn algebra at an advanced level, usually learn **Introduction to Abstract Algebra: Group Theory**. The course is offered for pure mathematics students in different universities around the world. However, the students who take the **Introduction to Abstract Algebra: Group Theory **course, are named super genius in **group theory.** Not so much difficult, but regular attention and interest can lead to the students in the right learning environment of mathematics. Many students around the world have their interest in learning **Introduction to Abstract Algebra: Group Theory** but they could’t find any proper course or instructor.

Abstract Algebra is comprised of one of the main topics which are also called **Group theory**. Group Theory or Group is actually the name of the fundamental four properties of mathematics that are frequently used in real analysis. We actually establish a strong background of Group Theory by defining different concepts. Proof of theorems and solutions of many examples is one of the interesting parts while studying Group Theory.

This course is filmed on a whiteboard (8 hours) and Tablet (2 hours). The length of this course is 10 hours with more than 15 sections and 100 videos. Almost every content of Group Theory has been included in this course. The students have difficulties in understanding the theorem, especially in Group Theory. Theorems have been explained with proof and examples in this course. A number of examples and exercises make this course easy for every student, even those who are taking this course the first time.

I assure all my students that they will enjoy this course. But however, if they have any difficulty then they can discuss it with me. I will answer your every question with a prompt response. One thing I will ask you is that you must see the contents sections and some free preview videos before enrolling in this course.

**CONTENTS OF THIS COURSE**

Groups and related examples

The identity element is the only element that is idempotent

Cancellation law hold in a group G

Definition of Subgroups and related examples

H is a subgroup if ab^-1 is contained in H

The intersection of any collection of subgroups is a subgroup

HuK is a subgroup if H is contained in Kor K is contained in H

Cyclic group and related examples

Every subgroup of a cyclic group is cyclic

Definition of cosets and related examples

Prove that the number of left or right cosets define the partition of a group G

Statement and Proof of Lagrange’s Theorem

Symmetric groups and related examples and exercises

Group of querternian and Klein’s four group

Normalizers, centralizers, and center of a group G and related theorem and examples

Quotient or Factor groups

Derived groups and related many examples

Normal Subgroups, conjugacy classes, conjugate subgroups, and related examples and theorems

Kernel of group

Automorphism and inner automorphism

P Group and related theorems and examples

Relations in groups like homomorphism and isomorphism

The centralizer is a subgroup of a group G

The normalizers is a subgroup of a group G

The Center of a group is a subgroup of a group G

The relation of conjugacy is an equivalence relation

Theorem and examples on quotient groups

Double cosets and related examples

Definition of automorphism

What is an inner automorphism

Every cyclic group is an abelian group

Groups of residue classes on a different mode

Examples of D_4 and D_5 groups

Examples related to C_6 and V_4

The first isomorphism theorem and its proof

The 2nd isomorphism theorem and its proof

The 3rd isomorphism theorem and its proof

The direct product of cyclic group

## Who this course is for

- Students who wants to learn algebra concepts on advance level

## Course content

- Introduction
- Introduction

- Definition of Groups
- Defining Groups
- Definition Notes Sheet
- Examples of Groups
- Quiz of Lecture 1 -3
- Examples of Groups Notes
- Group of Cube Roots of Unity
- Group of Fourth and nth Roots of Unity
- Group of Set of Residue Classes Module 5 Under Multiplication
- Group of Set of All 2 by 2 Non Singular Matrices
- Idempotent Element
- The Cancellation Law Holds in a Group G
- Theorem
- Order of a Group G and Order of an Element in a Group G
- Group of Set of Residue Classes Module 8 Under Multiplication
- Theorem
- Order of an Element and its Inverse are Same
- Group of Set of Residue Classes Module 9 Under Multiplication
- Example
- If a Group G Has Three Elements Then It is Abelian
- Example of an Abelian Group
- Example 2
- Example 3
- Example 4
- Example 5

- Subgroups
- Defining Subgroups
- Theorem (H is a Subgroup Iff ab^-1 is an Element of G)
- The Intersection of Any Collection of Subgroups is a Subgroup
- Example
- HuK is a Subgroup Iff H is Contained in K or K is Contained in H
- HuK is a Subgroup Iff H is Contained in K or K is Contained in H Part 2
- Example
- Klein’S Four Group
- Example

- Cyclic Groups
- Defining Cyclic Groups
- Every Subgroup of a Cyclic Group is Cyclic
- Every Cyclic Group is Abelian
- Group of Querternian
- Cay-lay’s Table for Cyclic Group
- Example
- Theorem
- Theorem
- Example
- Example 2

- Cosets
- Defining Cosets
- Example
- Example 2
- Theorem
- One-One Corresponding Between Left or Right Cosets

- Lagrange’s Theorem
- Proof of Lagrange’s Theorem
- Example 1
- Example 2
- Example 3
- Exercise 1
- Example 4
- Example 5

- Symmetric Groups
- Defining Permutations
- Defining Transpositions
- Multiplication of Permutations
- Inverse and Disjoint Cycles OF Permutations
- Permutations as Product of Transpositions + Even and Odd Permutations
- Even or Odd Permutations
- S_3 Symmetric Group or Group of Motion of Triangle
- D_4 Dihedral Group of Motion
- Diagonal Group D_5
- Diagonal Group D_5 Notes

- Relations in Groups
- Homomorphism Relation Between Groups

- Complex in Groups
- Defining Complex

- Normalizers
- Defining Normalizers
- Examples
- Example 2
- Example 2( part 2)
- Normalizers of X in G is a subgroup

- Centeralizers
- Defining Centeralizers
- Example
- Example 2
- Centeralizers is a Subgroup of G

- Center of a Group
- Defining Center of a Group
- Example
- Theorem (Center of a Group is a Subgroup )

- Conjugacy Relations in a Groups
- Defining Conjugacy Relation
- Theorem (Conjugate elements have same order)
- Defining Self Conjugate
- Conjugacy Class
- Theorem (The Relation of Conjugacy is an Equivalence Relation )
- Conjugate Subgroups
- Theorem
- Example
- Example (Part 2)

- P-Groups
- Defining P-Group

- Normal Groups
- Defining Normal Groups
- Theorem (A Subgroup of Index 2 is Normal)
- Example
- Theorem

- Quotient or Factor Group
- Defining Quotient or Factor Group
- Defining Quotient or Factor Group (Part 2)
- Example
- Example 2

- Automorphism
- Defining Automorphism
- Inner Automorphism
- Kernal of a Group
- Double Cosets

- Derived Subgroups
- Defining Derived Subgroups
- Theorem

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