This course is currently under construction.
The target release date for this course is Summer.
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.
Definition of a Group
Define and reason about properties of binary operations including associativity, commutativity, identities, and inverses.
Determine whether a set of scalars, matrices, or functions is a group.
Reason about properties of groups and subgroups including orders of groups and group elements.
Represent groups visually via Cayley tables and diagrams.
Perform arithmetic in the additive and multiplicative groups of integers modulo n, including using the extended Euclidean algorithm to compute modular inverses.
Identify and reason about cyclic groups, including generators of cyclic groups.
Apply knowledge of modular arithmetic to compute residues of large exponents using Euler's theorem and encode/decode messages using ciphers.
Extend knowledge of cyclic groups to reason about generators and presentations of general groups.
Product and Quotient Groups
Extend knowledge of group properties to product groups, including applying the fundamental theorem of finitely generated abelian groups.
Identify normal subgroups, compute cosets, and understand that a quotient group partitions a group into cosets of a normal subgroup.
Understand how Lagrange’s theorem relates orders of groups, subgroups, and quotient groups, and use it to compute orders of said groups.
Groups with Special Properties
Convert between and evaluate expressions involving permutations, cycles, and transpositions.
Identify and reason about the symmetric and alternating groups, the dihedral group, and general symmetry groups for 3D solids.
Identify and reason about other groups with special properties including simple groups, free groups, torsion groups, centralizers, and normalizers.
Identify and reason about p-subgroups using the Sylow theorems.
Homomorphisms and Actions
Identify and reason about group homomorphisms, including computing the image and kernel and determining whether a homomorphism is an isomorphism.
Identify conclusions and compute results implied by the isomorphism theorems.
Identify group actions and compute orbits and stabilizers of set elements under group actions.
Reason about groups, orbits, stabilizers, and fixed points using the orbit-stabilizer theorem, Burnside’s lemma, and the class equation of a group action.
Rings and Fields
Define and identify rings, determine whether a ring has unity, identify units in a ring, and compute the characteristic of a ring.
Define and identify integral domains, fields, and the field of quotients of an integral domain.
Reason about rings of polynomials and determine whether a polynomial is reducible.
Generalize concepts from linear algebra to abstract vector spaces over fields.
Evaluate expressions and solve linear equations involving quaternions.
Identify and reason about integral with special properties including unique factorization domains, principal ideal domains, and euclidean domains.
Reason about extension fields and algebraic vs transcendental numbers.
Extend prior knowledge of group theory to identify and reason about subrings, ideals, quotient rings, and ring homomorphisms.
Identify and reason about fields with special properties including fixed fields, splitting fields, and perfect fields.
Reason about the fundamental theorem of Galois theory and its applications in proving the insolvability of the quintic and the criterion for a polygon to be constructible.
1.1. Fermat and Euler's Theorems
Fermat's Little Theorem
Euler's Totient Function
2.2. Binary Operations
Introduction to Binary Operations
Associative Binary Operations
Commutative Binary Operations
Identities of Binary Operations
Inverses Under Binary Operations
Introduction to Groups
Real, Rational, and Complex Groups
The Order of a Group
Orders of Group Elements
Computing Inverses of Elements in Zn
Subgroups of Scalar Groups
Subgroups of Matrix Groups
Subgroups of Function Groups
Properties of Subgroups
2.5. Cyclic Groups
Finite Cyclic Groups
Cyclic Subgroups in the Complex Plane
Generators of Cyclic Groups
Properties of Cyclic Groups
2.6. Generating Sets and Presentations
Presentations of Groups
3.7. Permutations and Cycles
Converting Between Permutations and Cycles
The Inverse of a Cycle
The Inverse of a Permutation
The Order of a Cycle
The Order of a Permutation
3.8. Permutation Groups
The Symmetric Group on N Symbols
The Alternating Group on N Symbols
The Symmetry Group of a Figure
The Symmetry Group of a Tetrahedron
The Symmetry Group of a Cube: Rotations and Reflections
The Symmetry Group of a Cube: Combining Rotations and Reflections
Finite Symmetry Groups on the 2D-Plane
Finite Groups of Rotations in the 3D-Space
Product and Quotient Groups
4.10. Direct Products
External Direct Products
Cyclic Direct Product Groups
Finitely Generated Abelian Groups
Free Abelian Groups
Internal Direct Products
4.11. Cosets and Quotient Groups
Cosets of the Additive Groups of Integers Modulo N
Cosets of the Multiplicative Groups of Integers Modulo N
Cosets of Infinite Scalar Groups
Left and Right Cosets in Matrix Groups
Left and Right Cosets in Permutation Groups
Maximal Normal Subgroups
The Normalizer of a Subgroup
The Center of a Group
The Centralizer of a Subgroup
Homomorphisms & Isomorphisms
5.12. Group Homomorphisms
Identifying Group Homomorphisms
The Image of a Group Homomorphism
The Kernel of a Group Homomorphism
5.13. The Isomorphism Theorems
The First Isomorphism Theorem
The Second Isomorphism Theorem
The Third Isomorphism Theorem
6.14. Group Actions
The Orbit-Stabilizer Theorem
The Class Equation of a Group Action
6.15. Applications of Burnside Formula to Counting
Cycle Types of Permutations
Cycle Types of Permutations of Some Groups
Colorings of Necklaces 2
Colorings of 3D Objects
7.16. Series of Groups
Subnormal and Normal Series
Composition and Principal Series
7.17. The Sylow Theorems
The First Sylow Theorem
The Second Sylow Theorem
The Third Sylow Theorem
Rings & Fields
Introduction to Rings
Rings With Unity
Introduction to Fields
The Characteristic of a Ring
The Units of a Ring
Finding the Inverse of a Matrix Over a Ring
The Field of Quotients of an Integral Domain
8.19. Rings of Polynomials
Rings of Polynomials
Units in Polynomial Rings
The Division Algorithm in Rings of Polynomials Over Finite Fields
The Euclidian Algorithm in Rings of Polynomials
The Extended Euclidian Algorithm in Rings of Polynomials
Irreducible Polynomials Over the Real and Complex Numbers
Irreducible Polynomials Over the Integers and Rational Numbers
Irreducible Polynomials Over a Finite Field
Reducibility Tests for Polynomials Over the Integers and Rationals