Abstract Algebra – Group Theory with the Math Sorcerer

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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