Math 427 - Fall 2023

Honors Abstract Algebra

Dan Berwick-Evans (danbe at illinois)

Office hours:

Wednesday 1:30-2:20pm in 365 Altgeld
Friday 11:00-11:50am in 365 Altgeld
Other times by appointment.

Lectures:
Section F 09:00AM - 09:50AM MWF 1022 Literatures, Cultures, & Ling

Detailed schedule, lecture notes, and assignments.

Course info

Prerequisites: Officially either honors section of math 416 or some other linear algebra course together with an honors section of MATH 347. In practice, fluency in reading and writing proofs. In any case consent of the department is required.

Text: For about 3/4 of the class I will be using Prof. Eugene Lerman's lecture notes, which roughly follow Introduction to Abstract algebra by W. Keith Nicholson. Any edition is fine since I won't be assigning homework out of the text. Unfortunately, this book has gotten very expensive. Another option is Algebra: Abstract and Concrete (edition 2.6) by Goodman, available free of charge here.
For the last 1/4 of the class I will follow Charles Rezk's notes (linked below).

Canvas site: https://canvas.illinois.edu. You can check your grades or find solutions to homeworks and exams here.

Course Policies

Grades: Your course grade is determined by the following:

Homework: Mathematics (and problem solving in general) is a collaborative discipline. You are strongly encouraged to work together on homework assignments. However, you must write-up the solution on your own and in your own words. Anything else is plagiarism and will be treated as such.

Homework is typically due on Friday. Please submit your homework via Canvas; this should be a single pdf file and you should use an app designed for this purpose (e.g., Adobe Scan). Late homework is not accepted for any reason. However to compensate for this, your two lowest homework grades will be dropped.

Exams: There will be two in-class midterm exams and a final exam.

Exam 1: October 4.
Exam 2: November 1.
Final Exam: 8:00am-11:00am., Monday Dec. 11.

Academic Integrity: Cheating and other forms of academic dishonesty are taken very seriously. Any violation of the Illinois Academic Integrity Policy will result in a significant penalty.

Accommodations: To obtain disability-related accommodations, students should contact both me and the Disability Resources and Educational Services (DRES) as soon as possible.

Detailed Schedule, Lecture Notes, and Assignments.

Check back regularly; lecture notes, assignments, and other files will be added throughout the semester. The schedule of topics may change as the semester progresses.

Week 1: Relevant sections of Goodman include: 1.6, 2.6, Appendix A, Appendix B, Appendix C. References to sections/theorem numbers in Nicholson are in the notes.
August 21
Induction. Division algorithm. Lecture notes.
August 23
Divisibility properties of integers. Lecture notes.
August 25
Prime factorization. Equivalence relations. Lecture notes.
Week 2: Relevant sections of Goodman include: 1.10-1.11, 2.1-2.4. References to sections/theorem numbers in Nicholson are in the notes.
August 28
Rings. Lecture notes.
August 30
Groups and homomorphisms. Lecture notes.
September 1
Group isomorphisms. Subgroups. Lecture notes.
Due: HW 1.
Week 3: Relevant sections of Goodman include: 2.5, 5.1-5.2
September 4
Labor Day (no class).
September 6
Group actions. Lecture notes.
September 8
Cosets. Lecture notes.
Due: HW 2.
Week 4: Relevant sections include: 1.5, 4.4, 5.1, 5.2
September 11
Orthogonal groups and dihedral groups. Lecture notes.
September 13
Symmetric groups. Lecture notes.
September 15
Orbit stabilizer theorem. Lecture notes.
Due: HW 3.
Week 5: Relevant sections include: 1.9, 2.4-2.5
September 18
Lagrange's Theorem. Lecture notes.
September 20
Fermat's Little Theorem. Normal subgroups. Lecture notes.
September 22
More on normal subgroups. First isomorphism theorem. Lecture notes.
Due: HW 4.
Week 6: Relevant sections include: 2.7
September 25
Second and third isomorphism theorems. Lecture notes.
September 27
Permutations and symmetric groups. Lecture notes.
September 29
The class equation. Lecture notes.
Due: HW 5.
Week 7: Relevant sections include: 3.2, 3.6, 5.4
October 2
Cauchy's Theorem. Lecture notes.
October 4
Exam 1. Info and review.
October 6
Semi-direct products. Sylow Theorems. Lecture notes.
Week 8: Relevant sections include: 1.8, 6.1
October 9
Sylow Theorems. Lecture notes.
October 11
Examples of rings. Polynomial rings. Lecture notes.
October 13
Direct products. Lecture notes.
Due: HW 6.
Week 9: Relevant sections include: 6.2-6.3
October 16
Homomorphisms of rings. Ideals. Lecture notes.
October 18
Quotient rings. First isomorphism theorem for rings. Lecture notes.
October 20
Zero divisors. Integral domains. Lecture notes.
Due: HW 7.
Week 10: Relevant sections include: 6.4-6.5
October 23
Division algorithm. Lecture notes.
October 25
Maximal and prime ideals. Lecture notes.
October 27
Principal ideal domains, Euclidean domains, integral domains. Lecture notes.
Due: HW 8.
Week 11: Relevant sections include: 6.5-6.6, 8.1-8.2
October 30
Unique factorization domains. Lecture notes.
November 1
Exam 2. Info and review.  
November 3
Modules. Lecture notes.
Week 12: Relevant sections include: 8.4-8.5
November 6
Basis for a module. Lecture notes.
November 8
Modules over a PID. Lecture notes.
November 10
Group representations. Lecture notes.
Due: HW 9.
Week 13:
November 13
Irreducible representations. Lecture notes.
November 15
Decomposing a representation into irreducibles. Lecture notes.
November 17
Character of a representation. Lecture notes.
Due: HW 10.
Fall Break!
Week 14:
November 27
Orthogonality of characters. Lecture notes.
November 29
Orthogonality of characters, cont'd. Lecture notes.
December 1
Frobenius divisibility. Lecture notes.
Due: HW 11.
Week 15:
December 4
Induced representations. Lecture notes.
December 6
Catch-up/Review.
December 8
Reading day.
December 11
Final exam, 8-11am Info and review.