Weekly problem sets will be due via upload to Canvas by 5 pm on
Wednesday.
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Week
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Date
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Topics
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Reading
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Homework
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1
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Tue 16 Sep
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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.
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DF 0.1-0.3, 1.1-1.2
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Thu 18 Sep
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Order of an element and of a group. Symmetric
groups.
Cycle decomposition. Fields. Matrix groups.
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DF 1.2-1.4
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Fri 19 Sep
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X-TA Logic, set theory, functions.
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DF 0.1-0.3
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2
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Tue 23 Sep
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Generating
set. Presentation. Homomorphisms
and isomorphisms.
Cyclic groups.
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DF 1.4-1.6
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Problem Set #0
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Thu 25 Sep
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Subgroups.
Statement of Lagrange's Theorem.
Kernel.
Image.
Group
actions.
Examples of group actions.
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DF 1.7, 2.1-2.2, 4.1-4.2
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Fri 26 Sep
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X-TA Proof writing. Mathematical
induction.
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DF 0.1-0.3
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3
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Tue 30 Sep
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Permutation representation. Cayley's Theorem.
Orbits. Stabilizers.
Conjugation action. Conjugacy classes.
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DF 4.1-4.3
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Problem Set #1
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Thu 02 Oct
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Yom Kippur
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Fri 03 Oct
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X-TA Proof writing. Quiz review.
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4
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Tue 07 Oct
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Classification of cyclic groups, and their generators and subgroups.
Quiz
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DF 2.3, 3.1-3.2
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Quiz 1 Review
Problem Set #2
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Thu 09 Oct
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Quotient groups via homomorphisms. Quotient groups via
cosets. Normal subgroups. Natural projection. Normal subgroups
are kernels. Lagrange's theorem.
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DF 3.1-3.2
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Fri 10 Oct
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Make-up lecture. Homomorphisms from quotient groups.
First isomorphism theorem.
Third and fourth isomorphism theorems. Lattice of subgroups.
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DF 2.5, 3.3, 4.1
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5
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Tue 14 Oct
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Intersections and joins in the lattice of subgroups.
Orbit-stabilizer theorem.
Composition series. Jordan-Hölder
theorem. Simple
groups. Classification of finite simple groups.
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DF 2.4, 3.4
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Problem Set #3
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Thu 16 Oct
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Alternating group.
Orbit-stabilizer theorem. Conjugacy classes. Cycle type and conjugacy
classes in the symmetric group.
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DF 3.5, 4.1, 4.2, 4.3
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Fri 17 Oct
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X-TA Midterm exam review.
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6
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Tue 21 Oct
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Class equation. A5 is a simple group.
Sylow p-subgroup. Sylow's Theorem.
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DF 4.3, 4.4, 4.5
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Midterm Review Sheet
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Thu 23 Oct
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Applications of Sylow's Theorem.
Proof of Sylow's Theorems. Groups of order up to 60.
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DF 4.5
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Fri 24 Oct
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Midterm Exam
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7
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Tue 28 Oct
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Fundamental theorem of finitely generated abelian
groups. Classification of finite abelian groups. Classification of
finite abelian groups. Invariant factors. Elementary divisors.
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DF 5.1, 5.2
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Problem Set #4
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Thu 30 Oct
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Rings. Fields. Division rings. Quaternions. Matrix rings. Group
rings.
Zero-divisors. Units. Integral domains.
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DF 7.1-7.2
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Fri 31 Oct
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X-TA More examples of rings.
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8
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Tue 04 Nov
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Polynomial rings. Ring homomorphisms.
Ideals. Quotient rings.
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DF 7.1-7.3
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Problem Set #5
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Thu 06 Nov
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TA lecturing.
Isomophism Theorems for Rings.
Simple rings. Principal ideals. Principal ideal domains (PIDs). Z is a PID. Polynomial rings.
F[x] is a PID.
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DF 7.3-7.4, 8.2, 9.1-9.2
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Fri 07 Nov
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X-TA Quiz review.
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9
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Tue 11 Nov
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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. Quiz.
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DF 7.1, 8.1-8.3, 9.1-9.2
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Quiz 2 review
Problem Set #6
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Thu 13 Nov
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Unique factorization domains (UFDs). Noetherian rings.
Principal ideal domains are unique factorization domains. The
fundamental theorem of arithmetic.
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DF 8.2-8.3, 9.3
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Fri 10 Nov
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X-TA Final exam review.
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10
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Tue 18 Nov
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A view of where algebra goes from here.
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Final Exam Review
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