This course is currently under construction.
The target release date for this course is **February, 2025**.

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.

- 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.

- 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.

- 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.

- 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.

- 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.

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. | Cycle 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 a Finite Field | |

8.19.9. | Reducibility Tests for Polynomials Over the Integers and Rationals |

8.21. Quaternions

8.21.1. | The Division Ring of Quaternions | |

8.21.2. | Inverting Quaternions | |

8.21.3. | Linear Equations Over Quaternions |