Math 350 Introduction to Abstract Algebra

The official syllabus in pdf form.

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

Weekly problem sets will be due at the start of class on Friday.

Weekly Syllabus and Homework

Updated December 15, 2015.

Week Date Topics Reading Homework
1 Wed 02 Sep History of abstract algebra. Some set theory. The notion of a group. Examples of groups: modular arithmetic and symmetry groups. DF 0.1-0.3, 1.1-1.2
Fri 04 Sep Mathematical induction. Dihedral groups. Symmetric groups. Cycle decomposition. DF 1.2-1.3
2 Mon 07 Sep Labor Day! DF
Wed 09 Sep Matrix groups. Generating set. Presentation. Homomorphisms and isomorphisms. DF 1.4-1.6
Fri 11 Sep More homomorphisms. Kernel. Subgroups. Group actions. Permutation representation. DF 1.6-1.7
3 Mon 14 Sep Examples of group actions. Orbits. Stabilizers. Subgroups. DF 1.7, 2.1 Problem Set #1
Wed 16 Sep Centralizers and normalizers. Diagrammatic presentations of symmetric groups. Cyclic subgroups.
DF 2.2-2.3
Fri 18 Sep Classification of cyclic groups. DF 2.3
4 Mon 21 Sep Generating set. The lattice of subgroups. DF 2.4-2.5 Problem Set #2
Wed 23 Sep Quiz review. Subgroups of cyclic groups. DF 2.3
Fri 25 Sep Quotient groups via homomorphisms. Quotient groups via cosets. DF 3.1-3.2
5 Mon 28 Sep More quotients. Lagrange's theorem again. DF 3.1-3.2 Problem Set #3
Wed 30 Sep Isomorphism theorems. Converses to Lagrange's theorem? DF 3.3
Fri 02 Oct Composition series. Jordan-Hölder theorem. Simple groups. Classification of finite simple groups. Alternating group. DF 3.4-3.5
6 Mon 05 Oct Group actions revisited. Cycle decomposition via group actions. Cayley's Theorem. DF 4.1-4.2 Problem Set #4
Wed 07 Oct Conjugation action. The Class Equation. DF 4.3
Fri 09 Oct Conjugacy classes in Sn. A5 is a simple group! DF 4-3-4.5
7 Mon 12 Oct Automorphisms. Sylow p-subgroup. Sylow's Theorem. DF 4.4-4.5 Midterm exam review

Review Solutions
Wed 14 Oct Applications of Sylow's Theorem. Proof of Sylow's Theorem (existence part). DF 4.5
Fri 16 Oct Proof of Sylow's Theorem (numerology). Applications to groups of order 60. DF 4.5
8 Mon 19 Oct Midterm Exam! DF 0-6
Wed 21 Oct October Break!
Fri 23 Oct October Break!
9 Mon 26 Oct Direct products. Fundamental theorem of finitely generated abelian groups. DF 5.1-5.2 Problem Set #5
Wed 28 Oct Classification of finite abelian groups. Invariant factors. Elementary divisors. Characterizations of direct products. DF 5.4-5.5
Fri 30 Oct Midterm exam questions review. DF 5.5
10 Mon 02 Nov Semidirect products. Applications to groups of small order. DF 5.5 Problem Set #6
Wed 04 Nov More semidirect products. Some classification theorems. DF 5.5
Fri 06 Nov Even more semidirect products. More classification theorems. DF 5.5
11 Mon 09 Nov Rings. Division rings. Group rings. DF 7.1 Problem Set #7
Wed 11 Nov Zero-divisors. Group of units. Integral domains. DF 7.1-7.2
Fri 13 Nov Quadratic integer rings. Polynomial rings. DF 7.2-7.3
12 Mon 16 Nov Quiz! Ring homomorphisms. Ideals. Quotient rings. DF 7.3 Problem Set #8
Wed 18 Nov Quotient rings. Isomophism Theorems for Rings. DF 7.3
Fri 20 Nov Principal ideals. Simple rings. DF 7.4
13 Mon 23 Nov Thanksgiving Break!
Wed 25 Nov Thanksgiving Break!
Fri 27 Nov Thanksgiving Break!
14 Mon 30 Nov Prime ideals. Maximal ideals. Chinese Remainder Theorem. DF 7.4,7.6 Problem Set #9
Wed 02 Dec Fraction fields. Euclidean domains. Principal ideal domains. DF 7.5, 8.1, 8.2,
Fri 04 Dec Unique factorization domains. Gauss's Lemma. Irreducibility of polynomials. DF 8.2, 8.3, 9.3, 9.4, 9.5
15 Mon 07 Dec Quiz! Modules. DF 10.1 Problem Set #10
Wed 09 Dec Module homomorphisms. Quotient modules. Generators. Noetherian. Torsion. DF 10.2, 10.3, 12.1
Fri 11 Dec Modules over a PID. DF 12.1, 12.2
16 Mon 14 Dec Reading period. Final Exam Review
Wed 16 Dec Reading period. Final exam review session.
Fri 18 Dec Final Exam!

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