**Instructor:** Asher Auel

**Full website.**⇗

## Syllabus

The course textbook *Abstract Algebra, 3rd
Edition* by Dummit and Foote will be referred to by DF.

Weekly problem sets will be due via upload to Canvas by 5 pm on Wednesday.

## Weekly Syllabus and Homework

Updated November 15, 2021.Week | Date | Topics | Reading | Homework |

1 | Tue 14 Sep |
History of abstract algebra. Some set theory. The
notion of a group.
Examples of groups: modular arithmetic and symmetry groups.
Multiplicative group modulo n.
Dihedral
groups.
| DF 0.1-0.3, 1.1-1.2 | |

Thu 16 Sep | Yom Kippur | |||

Fri 07 Sep | Logic and set theory review. | DF 1.1-1.2 | ||

2 | Tue 21 Sep | Order of an element and of a group. Mathematical induction. Symmetric groups. Cycle decomposition. Fields. Matrix groups. | DF 1.2-1.4 | Problem Set #0 |

Thu 23 Sep | Generating set. Presentation. Homomorphisms and isomorphisms. Cyclic groups. | DF 1.4-1.6 | ||

3 | Tue 28 Sep | Subgroups. Statement of Lagrange's Theorem. Kernel. Image. Group actions. Permutation representation. Examples of group actions. Cayley's Theorem. Orbits. Stabilizers. | DF 1.7, 2.1-2.2, 4.1-4.2 | Problem Set #1 |

Thu 30 Sep | Conjugation action. Conjugacy classes. Cycle type and conjugacy classes in the symmetric group. | DF 4.3 | ||

4 | Tue 05 Oct | Classification of cyclic groups, and their generators and subgroups. Quotient groups via homomorphisms. Quotient groups via cosets. | DF 2.3, 3.1-3.2 | Midterm 1 Review |

Thu 07 Oct | More cosets. Normal subgroups. Natural projection. Normal subgroups are kernels. Lagrange's theorem. | DF 3.1-3.2 | ||

5 | Tue 12 Oct | Homomorphisms from quotient groups. First isomorphism theorem. Third and fourth isomorphism theorems. Lattice of subgroups. | DF 2.5, 3.3, 4.1 | Problem Set #2 |

Thu 14 Oct | Intersections and joins in the lattice of subgroups. Orbit-stabilizer theorem. Composition series. Jordan-Hölder theorem. Simple groups. Classification of finite simple groups. | DF 2.4, 3.4 | ||

6 | Tue 19 Oct |
Alternating group.
Class equation.
A_{5} is a
simple group!
Sylow p-subgroup. Sylow's Theorem.
| DF 3.5, 4.3, 4.4, 4.5 | Problem Set #3 |

Thu 21 Oct |
Sylow p-subgroup. Sylow's Theorem.
Applications of Sylow's Theorem.
Proof of Sylow's Theorems.
| DF 4.5 | ||

7 | Tue 26 Oct | Groups of order up to 60. Fundamental theorem of finitely generated abelian groups. Classification of finite abelian groups. Classification of finite abelian groups. Invariant factors. Elementary divisors. | DF 5.1, 5.2 | Midterm 2 Review |

Thu 28 Oct | Rings. Fields. Division rings. Quaternions. Matrix rings. Group rings. Zero-divisors. Units. Integral domains. | DF 7.1-7.2 | ||

8 | Tue 02 Nov | Polynomial rings. Ring homomorphisms. Ideals. Quotient rings. | DF 7.1-7.3 | Problem Set #4 |

Thu 04 Nov |
Isomophism Theorems for Rings.
Simple rings. Principal ideals. Principal ideal domains (PIDs). Z is a PID. Polynomial rings.
F[x] is a PID.
| DF 7.3-7.4, 8.2, 9.1-9.2 | ||

9 | Tue 09 Nov |
Review PIDs and F[x] is a PID. Euclidean domains. Euclidean
implies PID. Quadratic integer rings. Z[i] is
Euclidean. When is a quadratic integer ring Euclidean or a PID?
Irreducible and prime elements.
| DF 7.1, 8.1-8.3, 9.1-9.2 | Problem Set #5 |

Thu 11 Nov | Unique factorization domains (UFDs). Noetherian rings. Principal ideal domains are unique factorization domains. The fundamental theorem of arithmetic. | DF 8.2-8.3, 9.3 | ||

10 | Tue 16 Nov | A view of where algebra goes from here. | Final Review |