Instructor: Asher Auel

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## 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. 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