# Abstract Algebra Topics

## 1. The Set of Integers

###### 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

## 2. Types of Functions and Operations

###### 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

## 3. Introduction to Groups

###### 3.1. Introduction to Groups and Subgroups

- 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
- Powers and Roots of Elements in Groups
- The Laws of Exponents
- The Intersection and Union of Two Subgroups

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

- The Group of Integers Modulo n
- The Group of Invertible n x n Matrices
- The Group of Continuous Real-Valued Functions
- The Group of Differentiable Real-Valued Functions
- Groups Review

###### 3.2. Abelian Groups

- Abelian Groups
- Basic Theorems Regarding Abelian Groups
- The Center of a Group
- 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

- Functions from a Group to Itself
- The Symmetric Groups on n Elements
- The Symmetric Group of a General n-Element Set
- Permutation Groups on a Set
- 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

- Cycles in Permutations
- Disjoint Cycles
- Basic Theorems Regarding Disjoint Cycles
- Permutations as Products of Cycles
- Decomposition of Permutations as Products of Disjoint Cycles
- Transposition Permutations
- Basic Theorems Regarding Transpositions
- Decomposition of Permutations as Products of Transpositions
- Even and Odd Permutations as Products of Transpositions
- The Identity Permutation
- 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

## 4. Group Isomorphisms, Homomorphisms, and Automorphisms

###### 4.1. 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

###### 4.2. Group Homomorphisms

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

###### 4.3. Group Automorphisms

## 5. External and Internal Direct Products, Cosets, Lagrange's Theorem, and Normal Subgroups

###### 5.1. External and Internal Direct Products of Two Groups

- The External Direct Product of Two Groups
- The Order of an Element in an External Direct Product of Two Groups
- The Internal Direct Product of Two Groups
- Internal Direct Products Isomorphic to External Direct Products
- External and Internal Direct Products Review

###### 5.2. Cosets of Subgroups and Lagrange's Theorem

- 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
- Lagrange's Theorem
- Corollaries to Lagrange's Theorem
- Euler's Theorem and Fermat's Little Theorem
- Cosets and Lagrange's Theorem Review

###### 5.3. Normal Subgroups and Quotient Groups

- Normal Subgroups
- Criteria for a Subgroup to be Normal
- Quotient Groups
- Some Examples of Quotient Groups
- The Kernel of a Group Homomorphism is a Normal Subgroup of the Domain
- Normal Subgroups and Quotient Groups Review

###### 5.4. The Group Isomorphism Theorems

## 6. Introduction to Rings

###### 6.1. Introduction to Rings and Subrings

- Rings
- Basic Theorems Regarding Rings
- Subrings and Ring Extensions
- The Direct Product of Two Rings
- The Direct Product of the Ring of m x m and n x n Matrices

- 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 Ring of Gaussian Integers Z(i)
- The Ring of Q(√2)
- The Ring of Z/2Z
- The Ring of Z/nZ
- The Subring of n x n Upper and Lower Triangular Matrices
- The Subring of Polynomials with Subring Coefficients
- Rings Review

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

###### 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

- 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

## 7. Introduction to Fields

###### 7.1. Introduction to Fields and Subfields

###### 7.2. Polynomials over Fields

- Polynomials over a Field
- Basic Theorems Regarding Polynomials over a Field
- The Remainder Theorem for Polynomials over a Field
- Roots of Polynomials over a Field
- The Division Algorithm for Polynomials over a Field
- Ideals in the Set of Polynomials over a Field
- The Greatest Common Divisor of Polynomials over a Field
- Reducible and Irreducible Polynomials over a Field
- Congruence Classes of Polynomials Modulo p(x) over a Field
- Basic Theorems Regarding Congruence Classes of Polynomials Modulo p(x) over a Field
- Multiplicative Inverses of Congruence Classes of Polynomials Modulo p(x) over a Field
- The Field of Congruence Classes of Polynomials Modulo p(x) over a Field
- Kronecker's Field Extension Theorem
- Roots of Polynomials over Z with Rational Roots
- Primitive Polynomials over Z
- Gauss' Primitive Polynomial Lemma
- Eisenstein's Irreducible Polynomial Criterion for Polynomials over Z that are Irreducible over Q