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• 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