Instructor(s): Asher Auel
Course on canvas.dartmouth.edu.⇗
Course on math.dartmouth.edu.⇗
Syllabus
| Week | Date | Topics | Reading | Homework |
| 1 | Tue 16 Sep | History of abstract algebra. Some set theory notations. 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 18 Sep | Order of an element and of a group. Symmetric groups. Cycle decomposition. Fields. Matrix groups. | DF 1.2-1.4 | ||
| Fri 19 Sep | X-TA Logic, set theory, functions. | DF 0.1-0.3 | ||
| 2 | Tue 23 Sep | Generating set. Presentation. Homomorphisms and isomorphisms. Cyclic groups. | DF 1.4-1.6 | Problem Set #0 |
| Thu 25 Sep | Subgroups. Statement of Lagrange's Theorem. Kernel. Image. Group actions. Examples of group actions. | DF 1.7, 2.1-2.2, 4.1-4.2 | ||
| Fri 26 Sep | X-TA Proof writing. Mathematical induction. | DF 0.1-0.3 | ||
| 3 | Tue 30 Sep | Permutation representation. Cayley's Theorem. Orbits. Stabilizers. Conjugation action. Conjugacy classes. | DF 4.1-4.3 | Problem Set #1 |
| Thu 02 Oct | Yom Kippur | |||
| Fri 03 Oct | X-TA Proof writing. Quiz review. | |||
| 4 | Tue 07 Oct | Classification of cyclic groups, and their generators and subgroups. Quiz | DF 2.3, 3.1-3.2 |
Quiz 1 Review
Problem Set #2 |
| Thu 09 Oct | Quotient groups via homomorphisms. Quotient groups via cosets. Normal subgroups. Natural projection. Normal subgroups are kernels. Lagrange's theorem. | DF 3.1-3.2 | ||
| Fri 10 Oct | Make-up lecture. Homomorphisms from quotient groups. First isomorphism theorem. Third and fourth isomorphism theorems. Lattice of subgroups. | DF 2.5, 3.3, 4.1 | ||
| 5 | Tue 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 | Problem Set #3 |
| Thu 16 Oct | Alternating group. Orbit-stabilizer theorem. Conjugacy classes. Cycle type and conjugacy classes in the symmetric group. | DF 3.5, 4.1, 4.2, 4.3 | ||
| Fri 17 Oct | X-TA Midterm exam review. | |||
| 6 | Tue 21 Oct | Class equation. A5 is a simple group. Sylow p-subgroup. Sylow's Theorem. | DF 4.3, 4.4, 4.5 | Midterm Review Sheet |
| Thu 23 Oct | Applications of Sylow's Theorem. Proof of Sylow's Theorems. Groups of order up to 60. | DF 4.5 | ||
| Fri 24 Oct | Midterm Exam | |||
| 7 | Tue 28 Oct | 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 | Problem Set #4 |
| Thu 30 Oct | Rings. Fields. Division rings. Quaternions. Matrix rings. Group rings. Zero-divisors. Units. Integral domains. | DF 7.1-7.2 | ||
| Fri 31 Oct | X-TA More examples of rings. | |||
| 8 | Tue 04 Nov | Polynomial rings. Ring homomorphisms. Ideals. Quotient rings. | DF 7.1-7.3 | Problem Set #5 |
| Thu 06 Nov | TA lecturing. 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 | ||
| Fri 07 Nov | X-TA Quiz review. | |||
| 9 | Tue 11 Nov | PIDs. 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. Quiz. | DF 7.1, 8.1-8.3, 9.1-9.2 |
Quiz 2 review
Problem Set #6 |
| Thu 13 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 | ||
| Fri 10 Nov | X-TA Final exam review. | |||
| 10 | Tue 18 Nov | A view of where algebra goes from here. | Final Exam Review | |