Math Academy

Upon successful completion of this course, students will have mastered the following:

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.

Cyclic Groups

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.

Galois Theory

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.

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

Groups

30 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.	Universal Property of Free Groups
	2.6.4.	Presentations of Groups

Permutations

15 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

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

Homomorphisms & Isomorphisms

7 topics

5.12. Group Homomorphisms

	5.12.1.	Group Homomorphisms
	5.12.2.	The Image of a Group Homomorphism
	5.12.3.	The Kernel of a Group Homomorphism
	5.12.4.	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

Group Actions

13 topics

6.14. Group Actions

	6.14.1.	Natural Actions of Symmetric and Dihedral Groups
	6.14.2.	Group Actions
	6.14.3.	Orbits
	6.14.4.	Stabilizers
	6.14.5.	The Orbit-Stabilizer Theorem
	6.14.6.	Burnside's Lemma
	6.14.7.	The Class Equation of a Group Action
	6.14.8.	Conjugation and Conjugacy Classes
	6.14.9.	Conjugacy of Group Subsets and Subgroups

6.15. Applications of Burnside Formula to Counting

	6.15.1.	Cycle Types of Permutations
	6.15.2.	Cycle Types of Permutations of Some Groups
	6.15.3.	Permutation Actions on Sets of Words
	6.15.4.	Colorings of Necklaces

Group Structures

4 topics

7.16. The Sylow Theorems

	7.16.1.	P-Groups
	7.16.2.	The First Sylow Theorem
	7.16.3.	The Second Sylow Theorem
	7.16.4.	The Third Sylow Theorem

Rings & Fields

30 topics

8.17. Rings

	8.17.1.	Introduction to Rings
	8.17.2.	Commutative Rings
	8.17.3.	Rings With Unity
	8.17.4.	Integral Domains
	8.17.5.	Introduction to Fields
	8.17.6.	The Characteristic of a Ring
	8.17.7.	The Units of a Ring
	8.17.8.	Finding the Inverse of a Matrix Over a Ring
	8.17.9.	The Field of Quotients of an Integral Domain

8.18. Ideals and Factor Rings

	8.18.1.	Subrings
	8.18.2.	Ideals
	8.18.3.	Generating Elements of Principal Ideal Rings
	8.18.4.	Quotient Rings
	8.18.5.	Polynomial Quotient Rings
	8.18.6.	Ring Homomorphisms
	8.18.7.	Ring Homomorphisms Between Residue Rings
	8.18.8.	The Image and Kernel of a Ring Homomorphism
	8.18.9.	Prime Ideals and Maximal Ideals

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 a Finite Field
	8.19.9.	Reducibility Tests for Polynomials Over the Integers and Rationals

8.20. Quaternions

	8.20.1.	The Division Ring of Quaternions
	8.20.2.	Inverting Quaternions
	8.20.3.	Linear Equations Over Quaternions

Vector Spaces

8 topics

9.21. Abstract Vector Spaces

	9.21.1.	Introduction to Abstract Vector Spaces
	9.21.2.	Defining Abstract Vector Spaces
	9.21.3.	Linear Independence in Abstract Vector Spaces
	9.21.4.	Subspaces of Abstract Vector Spaces
	9.21.5.	Bases in Abstract Vector Spaces
	9.21.6.	The Coordinates of a Vector Relative to a Basis in Abstract Vector Spaces
	9.21.7.	Dimension in Abstract Vector Spaces
	9.21.8.	Vector Spaces Over Finite Fields

Abstract Algebra

Overview

Outcomes

Content

Definition of a Group

Cyclic Groups

Product and Quotient Groups

Groups with Special Properties

Homomorphisms and Actions

Rings and Fields

Galois Theory