1.1. Divisibility of Integers

• The Well-Ordering Principle of the Natural Numbers

• Integer Divisibility
• The Division Algorithm
• The Greatest Common Division Between Integers

• Integer Divisibility Review

1.2. Prime and Composite Numbers, Modular Arithmetic

• Prime and Composite Numbers
• Euclid's Theorem of the Existence of Infinitely Many Primes
• The Unique Prime Factorization Theorem
• Modular Arithmetic
• Prime and Composite Numbers, Modular Arithmetic Review

2.1. Types of Functions

• Injective, Surjective, and Bijective Functions
• The Composition of Two Functions
• Basic Theorems on the Composition of Two Functions
• The Inverse of a Function
• The Identity Function
• Permutations of Elements in a Set as Functions
• Types of Functions Review

2.2. Operations on Sets

• Unary and Binary Operations on Sets
• Functions that are Not Binary Operations
• Associativity and Commutativity of Binary Operations
• Distributivity of Binary Operations
• Identity and Inverse Elements of Binary Operations
• Operations on Sets Review

3.1. Introduction to Groups and Subgroups

3.1.1. Groups, Subgroups, and Basic Properties of Groups

• Groups
• Subgroups and Group Extensions
• Basic Theorems Regarding Groups
• The Cancellation Law for Groups
• Invertible Elements in Finite Groups that are Not Equal to Themselves
• The Laws of Exponents for Groups
• The Intersection and Union of Two Subgroups

3.1.2. The Order of Elements in a Group

• The Order of Elements in a Group
• Distinct Powers of Elements in a Group
• Basic Theorems Regarding the Order of Elements in a Group

3.1.3. Examples of Groups:

• The Additive Group of Integers Modulo n, Z/nZ
• The General Linear Group, GLn(R)
• The Group of Continuous Real-Valued Functions
• The Group of Differentiable Real-Valued Functions
• The Group of Affine Real-Valued Functions on R
• The Group R\{1} with the Quasi-Product
• The Quaternion Group, Q8

• Introduction to Groups Review

3.2. Abelian Groups

3.2.1. Abelian Groups

• Abelian Groups
• Basic Theorems Regarding Abelian Groups
• The Order of a Nonabelian Group is At Least 6

3.2.2. Examples of Abelian Groups

• The Abelian Group of Even Integers
• The Abelian Group of the Power Set of a Finite Set
• The Abelian Group of the Product of Abelian Subgroups
• The Abelian Group of Invertible Elements That Are Equal to Themselves

• Abelian Groups Review

3.3. Symmetric Groups, Permutation Groups, and Dihedral Groups

3.3.1. Symmetric Groups $(S_n, \circ)$

• Functions from a Group to Itself
• The Symmetric Groups, Sn
• The Symmetric Group of a General n-Element Set

3.3.2. Permutation Groups, $(S_X, \circ)$

• The Permutation Groups on a Set X, SX

3.3.3. Dihedral Groups, $(D_n, \circ)$

• The Dihedral Groups, Dn
• The Group of Symmetries of the Equilateral Triangle
• The Group of Symmetries of the Square
• The Group of Symmetries of the Pentagon
• The Group of Symmetries of a Rectangle

• Symmetric Groups, Permutation Groups, and Dihedral Groups Review

3.4. Cycles, Transpositions, and Alternating Groups

3.4.1. Cycle Permutations in $S_n$:

• Cycles in Permutations
• Disjoint Cycles
• Basic Theorems Regarding Disjoint Cycles
• Permutations as Products of Cycles
• Decomposition of Permutations as Products of Disjoint Cycles

3.4.2. Transposition Permutations in $S_n$:

• Transposition Permutations
• Basic Theorems Regarding Transpositions
• Decomposition of Permutations as Products of Transpositions
• Even and Odd Permutations as Products of Transpositions
• The Identity Permutation

3.4.3. Alternating Groups $(A_n, \circ)$

• Even and Odd Cycles
• The Alternating Groups, An
• The Size of the Alternating Groups
• Conjugate Cycles
• The Order of a Permutation
• The Order Theorem for Permutations
• Cycles, Transpositions, and Alternating Groups Review

3.5. Cyclic Groups

3.5.1. Cyclic Groups

• Generated Subgroups
• Cyclic Groups
• Subgroups of Cyclic Groups are Cyclic Groups
• Basic Theorems Regarding Cyclic Groups
• Cyclic Groups Review

3.6. Group Homomorphisms, Isomorphisms, and Automorphisms

3.6.1. Group Homomorphisms

• Group Homomorphisms
• Basic Theorems Regarding Group Homomorphisms
• The Kernel of a Group Homomorphism
• Group Homomorphisms Review

3.6.2. Group Isomorphisms

• Group Isomorphisms
• Necessary Conditions for Two Groups to Be Isomorphic
• Basic Theorems Regarding Group Isomorphisms
• Preservation of the Identities and Inverses under Group Isomorphisms
• Preservation of Special Properties under Group Isomorphisms
• Cyclic Groups and their Isomorphisms
• Cayley's Group Isomorphism Theorem
• Group Isomorphisms Review

3.6.3. Group Automorphisms

• Group Automorphisms
• The Group of Automorphisms of a Group, Aut(G)
• Inner Automorphisms of a Group
• The Subgroup of Inner Automorphisms of a Group, Inn(G)
• Group Automorphisms Review

3.7. Cosets of Subgroups and Lagrange's Theorem

3.7.1. Left and Right Cosets of Subgroups

• Left and Right Cosets of Subgroups
• The Set of Left (Right) Cosets of a Subgroup Partitions the Whole Group
• The Number of Elements in a Left (Right) Coset
• The Index of a Subgroup

3.7.2. Lagrange's Theorem

• Lagrange's Theorem
• A Converse to Lagrange's Theorem for Finite Abelian Groups
• Corollaries to Lagrange's Theorem
• Euler's Theorem and Fermat's Little Theorem
• Cosets and Lagrange's Theorem Review

3.8. Centers, Centralizers, Normalizers, and Normal Subgroups

3.8.1. The Center of a Group and Centralizers

• The Center of a Group G, Z(G)
• The Centralizer of a Subset A of a Group G, CG(A)
• Basic Theorems Regarding Centralizers of a Subset of a Group

3.8.2. Normalizers

• The Normalizer of a Subset A of a Group G, NG(A)
• Basic Theorems Regarding Normalizers of a Subset of a Group

3.8.3. Normal Subgroups of a Group

• Normal Subgroups
• Criteria for a Subgroup to be Normal
• H is a Normal Subgroup of G IFF H is the Kernel of a Homomorphism on G
• The Intersection of Two Normal Subgroups of a Group
• Subgroups of Finite Groups with Unique Order are Normal Subgroups
• Examples of Normal Subgroups of a Group
• Inn(G) is a Normal Subgroup of Aut(G)

3.8.4. Simple Groups

• Simple Groups
• Criterion for a Finite Abelian Group to be Simple

3.9. Direct Products

3.9.1. The Product of Two Subgroups

• The Product HK of Two Subgroups H and K of a Group G
• The Size of HK of Two Subgroups H and K of a Group G

3.9.2. Direct Products of Groups

• The Direct Product of Two Groups
• The Order of an Element in an External Direct Product of Two Groups
• The Direct Product of an Arbitrary Collection of Groups
• The Canonical Projections of Direct Products of Groups
• The Weak Direct Product of an Arbitrary Collection of Groups
• The Canonical Injections of Weak Direct Products of Groups

3.9.3. The Internal Direct Product

• The Internal Direct Product of Two Groups
• Internal Direct Products Isomorphic to External Direct Products
• External and Internal Direct Products Review

3.9.4. The Fundamental Theorem of Finite Abelian Groups

• The Fundamental Theorem of Finite Abelian Groups

3.10. Quotient Groups

3.10.1. Quotient Groups

• Quotient Groups
• Criterion for a Quotient Group to be Abelian
• Some Examples of Quotient Groups
• The Kernel of a Group Homomorphism is a Normal Subgroup of the Domain
• If G/Z(G) is Cyclic then G is Abelian
• Normal Subgroups and Quotient Groups Review

3.11. The Group Isomorphism Theorems

3.11.1. The First Group Isomorphism Theorem

• The First Group Isomorphism Theorem
• An Illustrative Proof of the First Group Isomorphism Theorem
• Criterion for (G/K)/(H/K) to be Isomorphic to (G/H) When H, K are Normal Subgroups of G
• Criterion for G to be Isomorphic to H × K When H, K are Normal Subgroups of G
• Z/nZ is Isomorphic to Zn
• R^x/<-1> is Isomorphic to R^+
• GLn(R)/SL_n(R) is Isomorphic to R^x
• G/Z(G) is Isomorphic to Inn(G)

3.11.2. The Second Group Isomorphism Theorem

• The Intersection of a Normal Subgroup with a Subgroup is a Normal Subgroup
• The Second Group Isomorphism Theorem

3.11.3. The Third Group Isomorphism Theorem

• The Third Group Isomorphism Theorem

3.12. Free Groups on a Set

3.12.1. Free Groups

• The Free Group on a Set X, F(X)
• Basic Theorems Regarding Free Groups on a Set

3.13. Conjugacy

3.13.1. Conjugacy

• Conjugate Elements in a Group
• IFF Criterion for y to be a Conjugate of x^n
• Conjugate Subgroups of a Group
• Conjugacy Classes of a Group
• The Class Equation

3.13.2. p-Groups and Burnside's Theorem for p-Groups

• p-Groups
• Burnside's Theorem for p-Groups
• Every Group of Order p^2 is Abelian
• The Center Z(G) of a Nonabelian Group of Order p^3 is p

3.13.3. Cauchy's Theorem

• Cauchy's Theorem for Groups

3.13.4. Examples of Conjugacy Classes

• Conjugacy Classes of the Symmetric Group, S3
• Conjugacy Classes of the Dihedral Group, D4
• Conjugacy Classes of the Quaternion Group, Q8

3.14. Group Actions of a Group on a Set

3.14.1. Group Actions

• Group Actions of a Group on a Set
• The Trivial Group Action of a Group on a Set
• The Left and Right Regular Group Actions of a Group on Itself
• A Group Action of the Symmetric Group SX on the Set X
• The Group Action of Conjugation of a Subgroup on a Group
• Faithful Group Actions of a Group on a Set

3.14.2. Orbits and Stabilizers

• The Orbit and Stabilizer of a Point in a Group Acting on a Set
• The Class Equation for Groups Acting on a Set

3.15. The Sylow Theorems

3.15.1. Sylow p-Subgroups and The Sylow Theorems

• Sylow p-Subgroups of a Group
• The Intersection of a Sylow p-Subgroup and a Sylow q-Subgroup
• The Sylow Theorems

3.16. Solvable Groups

3.16.1. Composition Series and the Jordan-Hölder Theorem

• Composition Series in a Group
• The Jordan-Hölder Theorem
• The Order of a Finite Group is the Product of the Orders of its Composition Factors

3.16.2. Commutators and the Derived Subgroup of a Group

• The Commutator of Two Elements in a Group
• The Derived Subgroup of a Group
• Basic Theorems Regarding the Derived Subgroup of a Group

3.16.3. Solvable Groups

• Solvable Groups
• Composition Factor Criterion for a Finite Group to be Solvable
• nth Derived Subgroup Criterion for a Group to be Solvable
• Every p-Group is Solvable
• Every Group of Order pq is Solvable
• Every Group of Order p^2q is Solvable
• The Homomorphic Image of a Solvable Group is Solvable

3.16.4. Characteristic Subgroups of a Group

• Characteristic Subgroups of a Group

3.17. Group Representations

3.17.1. Group Representations

• Group Representations
• Isomorphisms of Group Representations
• Group Subrepresentations

3.17.2. Maschke's Theorem

• Unitary Group Representations
• Weyl's Unitarity Trick
• Maschke's Theorem
• Homomorphisms and Endomorphisms of Group Representations
• Schur's Lemma
• Algebras over C
• Division Algebras over C
• The Dual Representation of a Group Representation

3.17.4. The Character of a Group Representation

• Determinant Group Representations
• The Character of a Group Representation
• The Orthogonality Theorem for Characters of Irreducible Group Representations
• Corollaries to the Orthogonality Theorem for Characters of Irreducible Group Representations

3.17.5. The Left Regular Representation

• Class Functions on a Group
• The Left Regular Representation
• The Completeness of Characters Theorem

3.18. Index of Common Groups

• Index of Common Groups

4.1. Introduction to Rings and Subrings

4.1.1. Rings, Subrings, and Basic Properties of Rings

• Rings
• Subrings and Ring Extensions
• Basic Theorems Regarding Rings

4.1.2. Units in a Ring

• Units in a Ring

4.1.3. Examples of Rings

• The Ring of Real and Complex Numbers
• The Ring of Polynomials with Real Coefficients
• The Ring of Polynomials with Ring Coefficients
• The Ring of n x n Matrices
• The Subring of n x n Upper and Lower Triangular Matrices
• The Subring of Polynomials with Subring Coefficients
• Rings Review

4.2. Commutative Rings

4.2.1. Commutative Rings

• Commutative Rings
• Boolean Rings

4.2.2 Examples of Commutative Rings

• The Ring of Gaussian Integers Z(i)
• The Ring of Q(√2)
• The Ring of Z/2Z
• The Ring of Z/nZ
• The Ring P(X) of a Nonempty Set X

4.3. Integral Domains

• Integral Domains
• The Cancellation Property of * on Integral Domains
• Every Subring of a Field is an Integral Domain
• Every Finite Integral Domain is a Field
• The Integral Domain of Z/pZ

6.2. Ideals, Quotient Rings, and Ring Homomorphisms/Isomorphisms/Automorphisms

• Ideals of Rings
• Quotient Rings

• Ring Homomorphisms
• Ring Isomorphisms
• Ring Automorphisms

• Basic Properties Regarding Ring Homomorphisms
• The Kernel of a Ring Homomorphism
• The Fundamental Theorem of Ring Homomorphisms

6.3. Commutative Rings, Division Rings, and Integral Domains

• Commutative Rings

• Division Rings
• Zero Divisors in Rings
• Zero Divisors in Z/nZ

• Special Types of Rings Review

6.4. Special Types of Integral Domains: PIDs, EDs, and UFDs

• Principal Ideals and Principal Ideal Domains (PIDs)
• The Principal Ideal Domain of Polynomials over a Field

• Euclidean Domains (EDs)
• Every Euclidean Domain is a Principal Ideal Domain

• Units (Multiplicatively Invertible Elements) in Rings
• The Set of Units in Mnn
• The Set of Units of a Ring forms a Group under *
• Divisors of Elements in Commutative Rings
• Associates of Elements in Commutative Rings
• The Greatest Common Divisor of Elements in a Commutative Ring
• Irreducible Elements in a Commutative Ring

• Unique Factorization Domains (UFDs)

• The Inclusion Chart for Special Types of Integral Domains

• Index of Common Rings