Intro to Abstract Algebra

Jeremiah Heller (jbheller at illinois)

Office hours:
Tuesdays 1:00-1:50pm in 162 NOYES
Wednesdays 11:00-11:50 in 147 ALTGELD
Other times by appointment.

Lectures:
Section B13 at MWF 9:00-9:50am in 141 ALTGELD
Section C13 at MWF 10:00-10:50am in 441 ALTGELD

Course info

Overview: Algebra is the branch of mathematics which studies equations, their solutions, and ways to manipulate them (the word algebra comes from Arabic and means "reunion of broken parts"). The abstraction and distillation of ideas over time has led us to the definition of a group, the basic object of study in modern abstract algebra. Other central objects which we will study are rings and fields.

This is a first course in abstract algebra. Topics covered include:

• Algebraic themes: fundamental theorem of arithmetic, modular arithmetic, symmetry, permutations.
• Group theory: groups, subgroups, homomorphisms, Lagranges theorem, isomorphism theorems.
• Group actions: orbit-stabilizer theorem, Burnside lemma, applications to counting, Sylow theorems.
• Ring theory: polynomial rings, fields, and other examples.

Prerequisites: Either MATH 416 or one of ASRM 406, MATH 415 together with one of MATH 347, MATH 348, CS 374; or consent of instructor.

Text: Algebra: Abstract and Concrete (edition 2.6) by Goodman. This book is available free of charge here.

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

Course Policies

• (24%) Homework
• (6%) Writing assignment
• (45%) Midterm exams
• (25%) Final exam

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 due in class. If you are unable to attend class due to illness, then you can submit a pdf scan of your assignment through 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 three in-class midterm exams and a final exam. Sections B and C will have a combined final exam which will be scheduled by the registrar's office. I will let you know once it is scheduled; if you make travel arrangements before the final is scheduled, you should assume that the final will be held on the last day of the final exam period.

Exam 1: February 15.
Exam 2: March 8.
Exam 3: April 19.
Final Exam: May 9, 1:30pm in 245 ALTGELD.

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.

Covid: All university policies will be followed. See https://covid19.illinois.edu/on-campus/on-campus-students/ . If you feel at all unwell or have reason to think you might possibly have COVID or another infectious disease, then please do not come to class. Instead, email me (promptly) and I will help you arrange to make up what you missed in class.

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: Read 1.1-1.4, Review Appendix A, Appendix B.
Jan 18
Course intro, first examples. Lecture notes.
Jan 20
More examples: symmetry. Lecture notes.
Jan 23
Permutations. Lecture notes.
Jan 25
Divisibility properties of integers. Lecture notes.
Jan 27
Fundamental theorem of arithmetic. Lecture notes.
Due: HW 1. Solutions.
Jan 30
Modular arithmetic. Lecture notes.
Feb 1
Modular arithmetic. Lecture notes.
Feb 3
Groups: basic properties. Lecture notes.
Due: HW 2. Solutions.
Feb 6
Subgroups. Lecture notes.
Feb 8
Cyclic groups. Lecture notes.
Feb 10
Subgroups of cyclic groups. Lecture notes.
Due: HW 3. Solutions.
Feb 13
Dihedral groups. Lecture notes.
Feb 15
Exam 1. Solutions.
Info and review.   Solutions to review problems.
Feb 17
Homomorphisms. Lecture notes.
Feb 20
Homomorphisms. Lecture notes.
Feb 22
Lagrange's Theorem. Lecture notes.
Due: Writing Assignment: part 1.
Feb 24
More cosets. Lecture notes.
Due: HW 4 .
Feb 27
Equivalence relations. Lecture notes.
Mar 1
Quotient groups. Lecture notes.
Due: Writing Assignment: part 2.
Mar 3
Isomorphism theorems. Lecture notes.
Due: HW 5 .
Mar 6
Isomorphism theorems. Lecture notes.
Mar 8
Exam 2 Solutions.
Info and review.   Solutions to review problems.
Mar 10
Examples and discussion. Lecture notes.
Spring Break!
Mar 13 - 17

Mar 20
Direct products. Lecture notes.
Mar 22
Semi-direct products. Lecture notes.
Due: Draft of writing assignment paper. (For feedback only; the draft is optional and ungraded.)
Mar 24
Group actions, intro. Lecture notes.
Due: HW 6 .
Mar 27
Group actions, examples. Lecture notes.
Mar 29
Orbit-stabilizer theorem. Lecture notes.
Mar 31
Examples and discussion.
Due: HW 7 .
Apr 3
Burnside lemma. Lecture notes.
Apr 5
Class equation. Lecture notes.
Due: Writing Assignment: final paper.
Apr 7
Sylow theorems.
Due: HW 8.
Apr 10
Sylow theorems.
Apr 12
Sylow theorems, proofs.
Apr 14
Examples and discussion.
Due: HW 9.
Apr 17
Rings and fields.
Apr 19
Exam 3
Apr 21
Polynomial rings.
Apr 24
Homomorphisms.
Apr 26
Ideals.
Apr 28
Quotient rings.
Due: HW 10.