|
Week
|
Date
|
Topics
|
Reading
|
Homework
|
|
1
|
Wed 30 Aug
|
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 01 Sep
|
Cyclic groups. Multiplicative group modulo n. Abelian groups. Order of element.
|
DF 1.1-1.2
|
|
2
|
Mon 04 Sep
|
Labor Day!
|
DF
|
Problem Set #0
|
|
Wed 06 Sep
|
Mathematical induction. Dihedral
groups. Symmetric
groups.
|
DF 1.2-1.3
|
|
Fri 08 Sep
|
More symmetric groups. Cycle decomposition. Fields.
|
DF 1.3-1.4
|
|
3
|
Mon 11 Sep
|
Matrix groups.
Generating
set. Presentation. Homomorphisms
and isomorphisms.
|
DF 1.4-1.6
|
Problem Set #1
|
|
Wed 13 Sep
|
Presentations and homomorphisms.
|
DF 2.1
|
|
Fri 15 Sep
|
Subgroups.
Kernel.
Image.
Lagrange's theorem.
Classification of cyclic groups.
|
DF 2.2-2.3
|
|
4
|
Mon 18 Sep
|
Group
actions. Permutation representation.
Examples of group actions.
Cayley's Theorem.
Cyclic subgroups.
|
DF 1.6, 2.1, 4.1
|
Problem Set #2
|
|
Wed 20 Sep
|
Orbits. Stabilizers. Centralizers and normalizers.
Generating
sets.
|
DF 2.2-2.3, 4.2-4.3
|
|
Fri 22 Sep
|
The lattice of subgroups. Subgroups of cyclic groups.
|
DF 2.4
|
|
5
|
Mon 25 Sep
|
Quotient
groups via homomorphisms.
Quotient groups via
cosets.
|
DF 3.1-3.2
|
Problem Set #3
|
|
Wed 27 Sep
|
More quotient groups.
|
DF 3.1-3.2
|
|
Fri 29 Sep
|
Yet more quotient groups. Normal subgroups.
|
DF 3.1-3.2
|
|
6
|
Mon 02 Oct
|
First isomorphism theorem. Orbit-stabilizer theorem.
|
DF 3.3, 4.1
|
Problem Set #4
|
|
Wed 04 Oct
|
Second, third, and fourth isomorphism theorems.
Quiz!
|
DF 3.3
|
|
Fri 06 Oct
|
Composition series. Jordan-Hölder
theorem. Simple
groups. Classification of finite simple groups.
|
DF 3.4
|
|
7
|
Mon 09 Oct
|
Alternating group.
|
DF 3.5
|
Problem Set #5
|
|
Wed 11 Oct
|
Conjugacy classes in Sn.
Class equation.
A5 is a
simple group!
|
DF 4.3, 4.4
|
|
Fri 13 Oct
|
Sylow p-subgroup. Sylow's Theorem.
Applications of Sylow's Theorem.
|
DF 4.5
|
|
8
|
Mon 16 Oct
|
Midterm Exam!
|
DF 0-6
|
Midterm exam review
Review Solutions
|
|
Wed 18 Oct
|
October Break!
|
|
|
Fri 20 Oct
|
October Break!
|
|
|
9
|
Mon 23 Oct
|
Proof of Sylow's Theorems. More applications of Sylow's theorems.
|
DF 4.5
|
Problem Set #6
|
|
Wed 25 Oct
|
Go over midterm. Automorphism groups of normal subgroups.
|
DF 4.5
|
|
Fri 27 Oct
|
End of proof of Sylow's theorems. More applications
|
DF 4.5, 4.6
|
|
10
|
Mon 30 Oct
|
Fundamental theorem of finitely generated abelian groups. Classification of finite abelian groups. Invariant factors.
|
DF 5.1, 5.2
|
Problem Set #7
|
|
Wed 01 Nov
|
Classification of finite abelian groups. Elementary divisors.
Recognition theorem for direct products.
|
DF 5.2, 5.4
|
|
Fri 03 Nov
|
Semidirect products. Applications to groups of small order.
|
DF 5.5
|
|
11
|
Mon 06 Nov
|
Classification of groups using semidirect products.
|
DF 5.5
|
Problem Set #8
|
|
Wed 08 Nov
|
More classification using semidirect product.
|
DF 5.5
|
|
Fri 10 Nov
|
Rings. Fields. Skew-fields. Quaternions.
|
DF 7.1-7.2
|
|
12
|
Mon 13 Nov
|
Zero-divisors. Units. Integral domains.
|
DF 7.1-7.2
|
Problem Set #9
|
|
Wed 15 Nov
|
Quadratic integer rings. Polynomial rings. Matrix rings.
|
DF 7.2
|
|
Fri 17 Nov
|
Unit group. Unit groups of quadratic integer rings. Ideals. Quotient rings.
|
DF 7.2-7.3
|
|
13
|
Mon 20 Nov
|
Thanksgiving Break!
|
|
|
|
Wed 22 Nov
|
Thanksgiving Break!
|
|
|
Fri 24 Nov
|
Thanksgiving Break!
|
|
|
14
|
Mon 27 Nov
|
Ring homomorphisms. Quotient rings. Isomophism Theorems for Rings.
|
DF 7.3
|
Problem Set #10
|
|
Wed 29 Nov
|
Principal ideals. Simple rings. Prime ideals. Maximal ideals.
|
DF 7.4, 7.6
|
|
Fri 01 Dec
|
Euclidean domains.
|
DF 8.1, 9.1, 9.2
|
|
15
|
Mon 04 Dec
|
Principal ideal domains.
|
DF 8.2
|
EC Problem Set #11
|
|
Wed 06 Dec
|
Irreducible and prime elements. Unique factorization domains.
Noetherian rings.
|
DF 8.3
|
|
Fri 08 Dec
|
Principal ideal domains are unique factorization domains. A view of
where algebra goes from here.
|
DF 8.3, 9.3, 9.4, 9.5
|
|
16
|
Mon 11 Dec
|
Reading period.
|
|
Final Exam Review
Solutions
|
|
Wed 13 Dec
|
Reading period. Final exam review session.
|
|
|
Fri 15 Dec
|
Finals period.
|
|
|
17
|
Tue 19 Dec
|
Final Exam!
|
|
|