Abstract Algebra: Group Theory With The Math Sorcerer

A beautiful course on the Theory of Groups:)

What you'll learn

The Definition of a Binary Operation

How to Determine if an operation is a binary operation

How to determine if a binary operation is commutative or associative

The Definition of a Group

Examples of Important Groups such as The Integers, Rationals, Reals, Complex Numbers under various operations

The General Linear Group

The Special Linear Group

The Klein Four-Group

The Additive Group of Integers Modulo n

Groups Defined on Powersets

Groups Defined with componentwise multiplication

How to Prove the Identity Element in a Group is Unique

How to Prove that Inverses in a Group are Unique

How to Prove various other Fundamental Properties of Groups

How to Find the Order of an Element in a Group

Knowledge of Cyclic Groups

How to Find Generators for Cyclic Groups

How to prove groups are cyclic and not cyclic

How to Prove Various key results surrounding Cyclic Groups

Knowledge of Subgroups

Examples of Various Subgroups

How to Prove a Set is a Subgroup

How to Prove Various Key Results Surrounding Subgroups

The Center of a Group

Direct Products of Cyclic Groups

How to Construct Finite Cyclic Groups using Direct Products

Understand the Notions of a Function, Domain, and Codomain

Understand the Notions of Direct Image and Inverse Image

Understand Injective(one to one), Surjective(Onto), and Bijective Functions

How to Prove Functions are Injective

How to Prove Functions are Surjective

How to Prove Functions are Bijective

Understand Symmetric Groups

Understand both cycle and array(two line) notation for Permutations

How to Multiply Permutations in Array Notation

How to Multiply Cycles in the Symmetric Group

Understand the Notion of a Relation including reflexive, symmetric, and transitive relations

Understand Equivalence Relations and Equivalence Classes

Understand How Equivalence Classes Partition a Set

Understand How to Prove from Scratch that Cosets are just Equivalence Classes that Partition a Group(yes I know wow!!)

Understand Lagrange's Theorem and it's Proof

Understand all of the Most Important Results and Corollaries of Lagrange's Theorem

How to Prove Conjugacy is an Equivalence Relation

How to Prove Various Results involving Conjugacy Classes

Understand and Know How to Prove the Class Equation

Understand Key Results of the Class Equation

How to Find Cosets given a Subgroup in Various Situations

Understand Normal Subgroups

How to Prove a Subgroup is Normal

How to Prove Various Results surrounding Normal Subgroups

How to Find Normal Subgroups

Understand Group Homomorphisms both Mathematically and Intuitively

Understand Group Isomorphisms

How to Prove SEVERAL(tons and tons) of Results Surrounding Homomorphisms

Understand Quotient Groups

How to Find the Quotient Group

How to Prove Several Results involving the Quotient Group

How to Prove the First Isomorphism Theorem

How to Prove the Second Isomorphism Theorem

Requirements

Be able to understand higher level mathematics OR

Have a STRONG desire to learn more advanced math, don't give up, this stuff is really abstract!!

Description

This is a college level course in Abstract Algebra with a focus on GROUP THEORY:)Note: Abstract Algebra is typically considered the one of HARDEST courses a mathematics major will take. This course is a step above a general mathematics course. Students should have familiarity with writing proofs and mathematical notation.Basically just,1) Watch the videos, and try to follow along with a pencil and paper, take notes! 2) Feel free to jump around from section to section. It's ok to feel lost when doing this, remember this stuff is supposed to be super hard for most people so don't get discouraged!3) After many sections there is short assignment(with solutions). 4) Repeat!If you finish even 50% of this course you will know A LOT of Abstract Algebra and more importantly your level of mathematical maturity will go up tremendously!Abstract Algebra and the Theory of Groups is an absolutely beautiful subject. I hope you enjoy watching these videos and working through these problems as much as I have:)Note this course has lots of very short videos with assignments. If you are trying to learn math then this format can be good because you don't have to spend tons of time on the course every day. Even if you can only spend time doing 1 video a day, that is honestly better than not doing any mathematics. You can learn a lot and because there are so many videos you could do 1 video a day. Good luck and I hope you learn a lot of math.

Overview

Section 1: Introduction to Binary Operations

Lecture 1 Definition of a Binary Operation

Lecture 2 Associate Binary Operations

Lecture 3 Proving a Binary Operation is Associative

Lecture 4 Group Theory Assignment 1

Lecture 5 Video Solutions to Group Theory Assignment 1

Section 2: Introduction to Groups

Lecture 6 Definition of a Group with Examples

Lecture 7 The General Linear Group, The Special Linear Group, and the Group C^n

Lecture 8 The Additive Group of Integers Modulo n

Lecture 9 The Klein Four-Group

Lecture 10 A Group Defined on the Powerset

Lecture 11 Proving a Set is a Group Example 1

Lecture 12 Group Theory Assignment 2

Lecture 13 Video Solutions to Group Theory Assignment 2

Section 3: Fundamental Properties of Groups

Lecture 14 Proof that the Identity Element is Unique

Lecture 15 Proof that Inverses are Unique

Lecture 16 Proof that the Inverse of the Inverse is the Original Element

Lecture 17 Proof that One Sided Inverses are Inverses

Lecture 18 Group Theory Assignment 3

Lecture 19 Video Solutions to Group Theory Assignment 3

Lecture 20 The Inverse of a Product of Elements Proof

Lecture 21 Proving a Group is Abelian

Lecture 22 Group Theory Assignment 4 with Written Solutions

Section 4: The Order of an Element

Lecture 23 Introduction to the Order of an Element

Lecture 24 The Order of an Element is the Order of the Inverse Proof

Lecture 25 The Order of an Element Sample Proof

Lecture 26 Group Theory Assignment 5 with Written Solutions

Section 5: Cyclic Groups

Lecture 27 Introduction to Cyclic Groups

Lecture 28 If x Generates a Group so does the Inverse Proof

Lecture 29 Proof that the Real Numbers under Addition is Not a Cyclic Group

Lecture 30 Proof that the Direct Product of the Integers with Itself is not Cyclic

Lecture 31 Every Cyclic Group is Abelian Proof

Lecture 32 Group Theory Assignment 6 with Written Solutions

Section 6: Subgroups

Lecture 33 Introduction to Subgroups

Lecture 34 Proving a Set is a Subgroup

Lecture 35 Proving a Finite Empty Set Closed Under the Group Operation is a Subgroup

Lecture 36 The Intersection of Two Subgroups is a Subgroup

Lecture 37 The Center of a Group is a Subgroup Proof

Lecture 38 The Center of a Subgroup of the General Linear Group

Lecture 39 Group Theory Assignment 7

Lecture 40 Detailed Proof for Assignment 7 Problem #1

Lecture 41 Detailed Proof for Assignment 7 Problem #2

Lecture 42 Detailed Proof for Assignment 7 Problem #3

Lecture 43 Detailed Proof for Assignment 7 Problem #4

Section 7: Direct Products of Finite Cyclic Groups

Lecture 44 Direct Products of Finite Cyclic Groups Part 1

Lecture 45 Direct Products of Finite Cyclic Groups Part 2

Section 8: Functions

Lecture 46 Introduction to Functions, Domain, Codomain, Injective, Surjective, Bijective

Lecture 47 Proving a Function is Onto(Surjective)

Lecture 48 Proving a Function is a Bijection

Section 9: Symmetric Groups

Lecture 49 Introduction to the Symmetric Group

Lecture 50 Introduction to Cycle Notation

Lecture 51 Writing a Permutation in Cycle Notation

Lecture 52 Converting Cycle Notation to Array Notation

Lecture 53 Multiplying Permutations in Array Notation

Lecture 54 Cycle Multiplication Example 1

Lecture 55 Cycle Multiplication Example 2

Lecture 56 Cycle Multiplication Example 3

Lecture 57 Cycle Multiplication Example 4

Lecture 58 Cycle Multiplication Example 5

Section 10: Relations

Lecture 59 Equivalence Relations

Lecture 60 Equivalence Classes Partition a Set

Lecture 61 Equivalence Relation on a Group

Lecture 62 Cosets are Equivalence Classes

Section 11: Lagrange's Theorem and Some Important Consequences

Lecture 63 Proof of Langrange's Theorem

Lecture 64 Consequence of Lagrange's Theorem Example 1

Lecture 65 Consequence of Lagrange's Theorem Example 2

Lecture 66 Consequence of Lagrange's Theorem Example 3

Section 12: Conjugacy Classes

Lecture 67 Conjugacy is an Equivalence Relation

Lecture 68 Conjugacy Class Key Result Example 1

Lecture 69 Conjugacy Class Key Result Example 2

Lecture 70 The Class Equation

Lecture 71 Every p-group has Nontrivial Center

Section 13: Cosets

Lecture 72 Finding the Cosets Example 1

Lecture 73 Finding the Cosets Example 2

Lecture 74 Finding the Cosets Example 3

Section 14: Normal Subgroups

Lecture 75 Subgroups of Abelian Groups are Normal

Lecture 76 The Intersection of Normal Subgroups is Normal

Lecture 77 Subgroups of Index 2 are Normal

Lecture 78 The Quaternion Group

Lecture 79 The Direct Product of Normal Subgroups is Normal

Section 15: Group Homomorphisms

Lecture 80 What is a Group Homomorphism?

Lecture 81 Introduction to Group Isomorphisms

Lecture 82 Injective Group Homomorphisms and The Kernel

Lecture 83 Conjugation is an Automorphism

Lecture 84 Inverse Image of a Subgroup

Lecture 85 Direct Image of a Subgroup

Lecture 86 Kernel of a Group Homomorphism

Lecture 87 Inverse Image of a Normal Subgroup

Lecture 88 Epimorphic Image of a Normal Subgroup

Lecture 89 Isomorphisms Preserve the Property of being Abelian

Lecture 90 Isomorphisms Preserve the Property of being Cyclic

Lecture 91 A Group that is Isomorphic to a Proper Subgroup

Section 16: Quotient Groups

Lecture 92 Finding the Quotient Group Example 1

Lecture 93 Finding the Quotient Group Example 2

Lecture 94 Finding the Quotient Group Example 3

Lecture 95 If G is Cyclic so is the Quotient Group

Lecture 96 G is Abelian if G/N is Cylic and N is in Z(G)

Section 17: The Isomorphism Theorems

Lecture 97 The First Isomorphism Theorem

Lecture 98 The Second Isomorphism Theorem

Section 18: More Videos

Lecture 99 Example 1

Lecture 100 Example 2

Lecture 101 Example 3

Lecture 102 Example 4

Math majors or people who are interested in learning higher level math.