Instructor: Asher Auel
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 HomeworkUpdated November 15, 2021.
|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. A5 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|