Math 417 - Spring 2018

Intro to Abstract Algebra

Jeremiah Heller (jbheller at illinois)

362 Altgeld
Office hours: Tuesday 10-11am, Friday 2-3pm
Section B13 at MWF 9:00-9:50am in 447 ALTGELD
Section D13 at MWF 11:00-11:50am in 343 ALTGELD

Course info

 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 course serves as an introduction to abstract algebra, its connections to other areas of mathematics, and to abstract mathematical thinking.

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


Actively engaging with the course material, both in and outside of class, is crucial to your success in this class. Part of in-class engagement means being present, and not just in the sense of being physically in the classroom.
It is all to easy to be distracted by our electronic devices. Save your emails, texts, and posts for after class! Rather than listening passively, grapple with the material and ask questions!  


Your course grade is determined by the following:



There will be two midterms and a final exam on the material covered in the lectures. The midterms will occur during class. Our final exam will be scheduled by the registrar's office and I will let you know once it is scheduled. If you make travel arrangements before the final exam is scheduled, you should assume that the final will be held on the last day of the final exam period.

Exam 1: Wednesday, February 21.
Exam 2: Wednesday, April 4.
Final Exam: Friday, May 4, 1:30-4:30pm, 103 Transportation Building.


There will be regular short quizzes (5-10 minutes) which will occur every week or so. These are meant to serve as a regular "reality check", to be sure the basics are being understood. 


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

Your solution needs to be correct (of course!), complete, and legible (if it can't be read, it will not be graded). Homework is due in class. 

Exercises come from the text (unless otherwise specified). Solutions to selected exercises can be found here.

Read: 1.1-1.5
Homework 1 (due 1/26): 1.4.2, 1.5.1, 1.5.2, 1.5.3, 1.5.5, 1.5.8
Review: Appendix A, Appendix B, as needed
Read: 1.6, 1.7, 1.10
Homework 2 (due 2/5): 1.6.3, 1.6.4, 1.6.8, 1.7.1, 1.7.5, 1.7.11
Read: 2.1, 2.2
Homework 3 (due 2/12): 2.1.3, 2.1.5, 2.1.7, 2.2.3, 2.2.12, 2.2.16
Read: 2.4, 2.5
Homework 4 (due 2/28): 2.4.5, 2.4.7, 2.4.14, 2.5.7, 2.5.8, 2.5.12
Read: 2.6, 2.7
Homework 5 (due 3/7): 2.6.1, 2.6.2, 2.6.3, 2.7.1, 2.7.2, 2.7.7
Read: 3.1, 3.2
Homework 6 (due 3/14): 3.1.2, 3.1.10, 3.1.3, 3.1.13, 3.2.2, 3.2.6
Read: 5.1, 5.2
Homework 7 (due 3/28): 5.1.1, 5.1.5, 5.1.6, 5.1.7, 5.1.10 (G is assumed to be finite)
Read: 5.4
Homework 8 (due 4/18): 5.4.1, 5.4.3, 5.4.5, 5.4.8, 5.4.9, 5.4.18
Read: 1.8, 6.1, 6.2
Homework 9 (due 4/25): 6.1.5, 6.1.12, 6.2.4, 6.2.18, 1.8.7, 1.8.17
Homework 10 (due 5/2): Homework 10


[1/17] Course intro, First examples. Lecture notes.
[1/19] More examples, symmetry. Lecture notes.
[1/22] Permutation groups. Lecture notes.
[1/24-26] Divisibility properties of integers. Lecture notes.
[1/29-31] Modular arithmetic. Lecture notes.
[2/2] First properties of groups. Lecture notes
[2/5] Subgroups. Lecture notes
[2/7-12] Cyclic groups and their subgroups. Lecture notes
[2/14] Dihedral groups. Lecture notes
[2/16] Homomorphisms. Lecture notes
[2/19] Cosets, Lagrange's theorem. Lecture notes
[2/21] Midterm Exam 1
[2/23] More cosets. Lecture notes
[2/26-28] Equivalence relations. Lecture notes
[3/2-3/5] Quotient groups. Lecture notes
[3/7] More quotient groups. Lecture notes
[3/9-12] Direct products, Semi-direct products. Lecture notes
[3/14] Finish semi-direct products. Group actions. Lecture notes
[3/16] Group actions, more examples. Lecture notes
[3/26] Orbit-stabilizer theorem, basic counting. Lecture notes
[3/28] Burnside lemma, more counting. Lecture notes
[4/6] Class Equation. Lecture notes
[4/9] Sylow's Theorems. Lecture notes
[4/11-13] Proofs of Sylow's Theorems. Lecture notes
[4/16] Rings and fields. Lecture notes
[4/18] Polynomial rings. Lecture notes
[4/20] Ring homomorphisms. Lecture notes
[4/23] Ideals. Lecture notes
[4/23] Quotient rings. Lecture notes
[4/23] Field extensions. Lecture notes