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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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Wed 31 Aug
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History of abstract algebra. Some set theory. The
notion of a group.
Examples of groups: modular arithmetic and symmetry groups.
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DF 0.1-0.3, 1.1-1.2
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Fri 02 Sep
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Cyclic groups. Multiplicative group modulo n. Abelian groups. Order of element.
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DF 1.1-1.2
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2
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Mon 05 Sep
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Labor Day!
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DF
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Problem Set #0
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Wed 07 Sep
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Mathematical induction. Dihedral
groups. Symmetric
groups.
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DF 1.2-1.3
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Fri 09 Sep
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More symmetric groups. Cycle decomposition. Fields.
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DF 1.3-1.4
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3
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Mon 12 Sep
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Matrix groups.
Generating
set. Presentation. Homomorphisms
and isomorphisms.
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DF 1.4-1.6
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Problem Set #1
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Wed 14 Sep
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Presentations and homomorphisms.
Subgroups.
Kernel.
Image.
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DF 1.6, 2.1
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Fri 16 Sep
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Group
actions. Permutation representation.
Examples of group actions.
Cayley's Theorem.
Cyclic subgroups.
Classification of cyclic groups.
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DF 2.2-2.3
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4
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Mon 19 Sep
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Orbits. Stabilizers. Centralizers and normalizers.
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DF 2.1, 2.4
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Problem Set #2
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Wed 21 Sep
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Generating
sets. The lattice of subgroups.
Subgroups of cyclic groups.
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DF 2.5
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Fri 23 Sep
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Quotient
groups via homomorphisms.
Quotient groups via
cosets.
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DF 3.1-3.2
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5
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Mon 26 Sep
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More quotients. Lagrange's
theorem again.
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DF 3.1-3.2
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Problem Set #3
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Wed 28 Sep
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Second and Third Isomorphism theorems.
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DF 3.3
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Fri 30 Sep
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Fourth Isomorphism Theorem. Alternating group.
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DF 3.4-3.5
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6
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Mon 03 Oct
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Composition series. Jordan-Hölder
theorem. Simple
groups. Classification of finite simple groups.
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DF 4.1-4.2
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Problem Set #4
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Wed 05 Oct
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Group actions revisited. Orbit-stabilizer theorem. Cycle decomposition via group actions.
Quiz!
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DF 4.3
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Fri 07 Oct
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Conjugation action. The Class Equation.
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DF 4-3-4.5
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7
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Mon 10 Oct
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Conjugacy classes in Sn.
A5 is a
simple group!
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DF 4.4
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Problem Set #5
Midterm exam review
Review Solutions
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Wed 12 Oct
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Quiz review. Automorphisms
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DF 4.4
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Fri 14 Oct
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Sylow p-subgroup. Sylow's Theorem.
Applications of Sylow's Theorem.
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DF 4.5
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8
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Mon 17 Oct
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Midterm Exam!
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DF 0-6
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Wed 19 Oct
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October Break!
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Fri 22 Oct
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October Break!
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9
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Mon 24 Oct
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Proof of Sylow's Theorems. More applications of Sylow's theorems.
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DF 4.5
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Problem Set #6
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Wed 26 Oct
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Midterm review. Cauchy's Theorem.
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DF 4.5
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Fri 28 Oct
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More proof of Sylow's theorems. Direct products. Fundamental theorem of finitely generated abelian groups.
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DF 4.5, 5.1, 5.2
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10
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Mon 31 Oct
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Classification of finite abelian groups. Invariant factors. Elementary divisors.
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DF 5.2
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Problem Set #7
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Wed 02 Nov
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Semidirect products. Applications to groups of small order. Some classification theorems.
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DF 5.5
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Fri 04 Nov
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More semidirect products.
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DF 5.5
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11
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Mon 07 Nov
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Classification of groups using semidirect products.
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DF 5.5
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Problem Set #8
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Wed 9 Nov
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Rings. Division rings. Group rings.
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DF 7.1-7.2
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Fri 11 Nov
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Zero-divisors. Group of units. Integral domains.
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DF 7.2-7.3
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12
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Mon 14 Nov
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Quadratic integer rings.
Ring homomorphisms.
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DF 7.3
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Problem Set #9
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Wed 16 Nov
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Ring homomorphisms.
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DF 7.3
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Fri 18 Nov
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Polynomial rings.
Quotient rings. Isomophism Theorems for Rings.
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DF 7.4
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13
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Mon 21 Nov
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Thanksgiving Break!
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Wed 23 Nov
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Thanksgiving Break!
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Fri 25 Nov
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Thanksgiving Break!
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14
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Mon 28 Nov
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Principal ideals. Simple rings. Prime ideals. Maximal ideals.
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DF 7.4,7.6
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Problem Set #10
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Wed 30 Nov
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Euclidean domains.
Quiz!
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DF 7.5, 8.1, 8.2,
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Fri 02 Dec
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Principal ideal domains.
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DF 8.2, 8.3, 9.3, 9.4, 9.5
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15
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Mon 05 Dec
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Unique factorization domains. Fundamental theorem of arithmetic.
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DF 10.1
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EC Problem Set #11
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Wed 07 Dec
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More UFDs. Fraction fields. Gauss's Lemma. Hilbert Basis Theorem.
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DF 10.2, 10.3, 12.1
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Fri 09 Dec
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Modules over a PID.
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DF 12.1, 12.2
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16
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Mon 12 Dec
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Reading period.
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Final Exam Review
Solutions
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Wed 14 Dec
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Reading period. Final exam review session.
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Fri 16 Dec
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Final period.
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17
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Tue 20 Dec
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Final Exam!
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