MP4 | Video: h264, 1280×720 | Audio: AAC, 44.1 KHz

Language: English | Size: 2.02 GB | Duration: 9h 30m

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:)

Who this course is for

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

https://www.udemy.com/course/abstract-algebra-group-theory-with-the-math-sorcerer/

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