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Abstract Algebra

This course is currently under construction. The target release date for this course is April.
Dive deep into the core relationships that govern how mathematical objects interact with one another. Learn to identify algebraic structures and apply mathematical reasoning to arrive at general conclusions.

Outcomes

Content

Definition of a Group

Cyclic Groups

Product and Quotient Groups

Groups with Special Properties

Homomorphisms and Actions

Rings and Fields

Galois Theory

1.
Preliminaries
3 topics
1.1. Fermat and Euler's Theorems
1.1.1. Fermat's Little Theorem
1.1.2. Euler's Totient Function
1.1.3. Euler's Theorem
2.
Groups
29 topics
2.2. Binary Operations
2.2.1. Introduction to Binary Operations
2.2.2. Associative Binary Operations
2.2.3. Commutative Binary Operations
2.2.4. Identities of Binary Operations
2.2.5. Inverses Under Binary Operations
2.3. Groups
2.3.1. Introduction to Groups
2.3.2. Real, Rational, and Complex Groups
2.3.3. Matrix Groups
2.3.4. Function Groups
2.3.5. General Groups
2.3.6. The Order of a Group
2.3.7. Orders of Group Elements
2.3.8. Computing Inverses of Elements in Zn
2.3.9. Cayley Tables
2.3.10. Cayley Diagrams
2.3.11. Abelian Groups
2.4. Subgroups
2.4.1. Subgroups of Scalar Groups
2.4.2. Subgroups of Matrix Groups
2.4.3. Subgroups of Function Groups
2.4.4. Properties of Subgroups
2.5. Cyclic Groups
2.5.1. Finite Cyclic Groups
2.5.2. Cyclic Groups
2.5.3. Cyclic Subgroups
2.5.4. Cyclic Subgroups in the Complex Plane
2.5.5. Generators of Cyclic Groups
2.5.6. Properties of Cyclic Groups
2.6. Generating Sets and Presentations
2.6.1. Generating Sets
2.6.2. Free Groups
2.6.3. Presentations of Groups
3.
Permutations
17 topics
3.7. Permutations and Cycles
3.7.1. Permutations
3.7.2. Cycles
3.7.3. Converting Between Permutations and Cycles
3.7.4. The Inverse of a Cycle
3.7.5. The Inverse of a Permutation
3.7.6. The Order of a Cycle
3.7.7. The Order of a Permutation
3.7.8. Transpositions
3.8. Permutation Groups
3.8.1. The Symmetric Group on N Symbols
3.8.2. The Alternating Group on N Symbols
3.9. Isometries
3.9.1. The Symmetry Group of a Figure
3.9.2. Dihedral Group
3.9.3. The Symmetry Group of a Tetrahedron
3.9.4. The Symmetry Group of a Cube: Rotations and Reflections
3.9.5. The Symmetry Group of a Cube: Combining Rotations and Reflections
3.9.6. Finite Symmetry Groups on the 2D-Plane
3.9.7. Finite Groups of Rotations in the 3D-Space
4.
Product and Quotient Groups
19 topics
4.10. Direct Products
4.10.1. External Direct Products
4.10.2. Cyclic Direct Product Groups
4.10.3. Torsion Groups
4.10.4. Finitely Generated Abelian Groups
4.10.5. Free Abelian Groups
4.10.6. Internal Direct Products
4.11. Cosets and Quotient Groups
4.11.1. Cosets of the Additive Groups of Integers Modulo N
4.11.2. Cosets of the Multiplicative Groups of Integers Modulo N
4.11.3. Cosets of Infinite Scalar Groups
4.11.4. Left and Right Cosets in Matrix Groups
4.11.5. Left and Right Cosets in Permutation Groups
4.11.6. Normal Subgroups
4.11.7. Maximal Normal Subgroups
4.11.8. The Normalizer of a Subgroup
4.11.9. The Center of a Group
4.11.10. The Centralizer of a Subgroup
4.11.11. Quotient Groups
4.11.12. Lagrange's Theorem
4.11.13. Simple Groups
5.
Homomorphisms & Isomorphisms
8 topics
5.12. Group Homomorphisms
5.12.1. Group Homomorphisms
5.12.2. Identifying Group Homomorphisms
5.12.3. The Image of a Group Homomorphism
5.12.4. The Kernel of a Group Homomorphism
5.12.5. Group Isomorphisms
5.13. The Isomorphism Theorems
5.13.1. The First Isomorphism Theorem
5.13.2. The Second Isomorphism Theorem
5.13.3. The Third Isomorphism Theorem
6.
Group Actions
10 topics
6.14. Group Actions
6.14.1. Group Actions
6.14.2. Orbits
6.14.3. Stabilizers
6.14.4. The Orbit-Stabilizer Theorem
6.14.5. Burnside's Lemma
6.14.6. The Class Equation of a Group Action
6.15. Applications of Burnside Formula to Counting
6.15.1. Cyclic Types of Permutations
6.15.2. Cycle Types of Permutations of Some Groups
6.15.3. Colorings of Necklaces 2
6.15.4. Colorings of 3D Objects
7.
Group Structures
7 topics
7.16. Series of Groups
7.16.1. Subnormal and Normal Series
7.16.2. Composition and Principal Series
7.16.3. Solvable Groups
7.17. The Sylow Theorems
7.17.1. P-Groups
7.17.2. The First Sylow Theorem
7.17.3. The Second Sylow Theorem
7.17.4. The Third Sylow Theorem
8.
Rings & Fields
21 topics
8.18. Rings
8.18.1. Introduction to Rings
8.18.2. Commutative Rings
8.18.3. Rings With Unity
8.18.4. Integral Domains
8.18.5. Introduction to Fields
8.18.6. The Characteristic of a Ring
8.18.7. The Units of a Ring
8.18.8. Finding the Inverse of a Matrix Over a Ring
8.18.9. The Field of Quotients of an Integral Domain
8.19. Rings of Polynomials
8.19.1. Rings of Polynomials
8.19.2. Units in Polynomial Rings
8.19.3. The Division Algorithm in Rings of Polynomials Over Finite Fields
8.19.4. The Euclidian Algorithm in Rings of Polynomials
8.19.5. The Extended Euclidian Algorithm in Rings of Polynomials
8.19.6. Irreducible Polynomials Over the Real and Complex Numbers
8.19.7. Irreducible Polynomials Over the Integers and Rational Numbers
8.19.8. Irreducible Polynomials Over the Integers and Rational Numbers: Reducibility Tests
8.19.9. Irreducible Polynomials Over a Finite Field
8.21. Quaternions
8.21.1. The Division Ring of Quaternions
8.21.2. Inverting Quaternions
8.21.3. Linear Equations Over Quaternions