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
This is a college level course in Abstract Algebra with a focus on GROUPTHEORY:)
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 ALOT 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 Ihave:)
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.
