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 0.10.3, 1.11.2


Fri 04 Sep

Mathematical induction. Dihedral
groups. Symmetric
groups. Cycle decomposition.

DF 1.21.3

2

Mon 07 Sep

Labor Day!

DF


Wed 09 Sep

Matrix groups.
Generating
set. Presentation. Homomorphisms
and isomorphisms.

DF 1.41.6

Fri 11 Sep

More homomorphisms. Kernel.
Subgroups.
Group
actions. Permutation representation.

DF 1.61.7

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.
Quiz!

DF 2.22.3

Fri 18 Sep

Classification of cyclic groups.

DF 2.3

4

Mon 21 Sep

Generating
set. The lattice of subgroups.

DF 2.42.5

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 3.13.2

5

Mon 28 Sep

More quotients. Lagrange's
theorem again.

DF 3.13.2

Problem Set #3

Wed 30 Sep

Isomorphism theorems. Converses to Lagrange's theorem?

DF 3.3

Fri 02 Oct

Composition series. JordanHölder
theorem. Simple
groups. Classification of finite simple groups.
Alternating group.

DF 3.43.5

6

Mon 05 Oct

Group actions revisited. Cycle decomposition via group actions.
Cayley's Theorem.

DF 4.14.2

Problem Set #4

Wed 07 Oct

Conjugation action. The Class Equation.

DF 4.3

Fri 09 Oct

Conjugacy classes in S_{n}.
A_{5} is a
simple group!

DF 434.5

7

Mon 12 Oct

Automorphisms. Sylow psubgroup. Sylow's Theorem.

DF 4.44.5

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 06


Wed 21 Oct

October Break!


Fri 23 Oct

October Break!


9

Mon 26 Oct

Direct products. Fundamental theorem of finitely generated abelian groups.

DF 5.15.2

Problem Set #5

Wed 28 Oct

Classification of finite abelian groups. Invariant factors. Elementary divisors. Characterizations of direct products.

DF 5.45.5

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

Zerodivisors. Group of units. Integral domains.

DF 7.17.2

Fri 13 Nov

Quadratic integer rings. Polynomial rings.

DF 7.27.3

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
Solutions

Wed 16 Dec

Reading period. Final exam review session.


Fri 18 Dec

Final Exam!

