Math 71 Algebra
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. All GAI Engagement problems (conversation print-out and
write-up) need to be uploaded separately to their own Canvas
assignment.
Weekly Syllabus and Homework
Updated Nov 09, 2023.
|
Week
|
Date
|
Topics
|
Reading
|
Homework
|
|
1
|
Tue 12 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 14 Sep
|
Order of an element and of a group. Symmetric
groups.
Cycle decomposition. Fields. Matrix groups.
|
DF 1.2-1.4
|
|
Fri 15 Sep
|
Lily McBeath Logic, set theory, functions.
|
DF 0.1-0.3
|
|
2
|
Tue 19 Sep
|
Generating
set. Presentation. Homomorphisms
and isomorphisms.
Cyclic groups.
|
DF 1.4-1.6
|
Problem Set #0
|
|
Thu 21 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 22 Sep
|
Lily McBeath Proof writing. Mathematical
induction. More presentations of groups.
|
DF 0.1-0.3
|
|
3
|
Tue 26 Sep
|
Permutation representation. Cayley's Theorem.
Orbits. Stabilizers.
Conjugation action. Conjugacy classes. Cycle type and conjugacy
classes in the symmetric group.
|
DF 4.1-4.3
|
Problem Set #1
|
|
Thu 28 Sep
|
Classification of cyclic groups, and their generators and
subgroups.
Quotient
groups via homomorphisms.
Quotient groups via
cosets.
|
DF 2.3, 3.1-3.2
|
|
Fri 29 Sep
|
Lily McBeath Proof writing. Midterm review.
|
|
|
4
|
Tue 03 Oct
|
More cosets. Normal subgroups. Natural projection. Normal subgroups
are kernels. Lagrange's theorem.
|
DF 3.1-3.2
|
Midterm 1 Review
|
|
Thu 05 Oct
|
Homomorphisms from quotient groups.
First isomorphism theorem.
Third and fourth isomorphism theorems. Lattice of subgroups.
|
DF 2.5, 3.3, 4.1
|
|
5
|
Tue 10 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 #2
|
|
Thu 12 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
|
|
6
|
Tue 17 Oct
|
Sylow p-subgroup. Sylow's Theorem.
Applications of Sylow's Theorem.
Proof of Sylow's Theorems.
|
DF 4.5
|
Problem Set #3
|
|
Thu 19 Oct
|
More Sylow
p-subgroup. Groups of order up to 60.
|
DF 4.5
|
|
Fri 29 Sep
|
Lily McBeath Midterm 2 review.
|
|
|
7
|
Tue 24 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
|
Midterm 2 Review
|
|
Thu 26 Oct
|
Rings. Fields. Division rings. Quaternions. Matrix rings. Group
rings.
Zero-divisors. Units. Integral domains.
|
DF 7.1-7.2
|
|
8
|
Tue 31 Oct
|
Polynomial rings. Ring homomorphisms.
Ideals. Quotient rings.
|
DF 7.1-7.3
|
Problem Set #4
|
|
Thu 02 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 07 Nov
|
Lily lecturing.
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.
|
DF 7.1, 8.1-8.3, 9.1-9.2
|
Problem Set #5
|
|
Thu 09 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
|
Lily McBeath Final exam review.
|
|
|
10
|
Tue 14 Nov
|
A view of where algebra goes from here.
|
|
Final Review
|
|