Math 417 - Spring 2018
Intro to Abstract Algebra
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
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
Text: Algebra: Abstract and Concrete (edition 2.6) by Goodman. This book is available freely here.
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:
- (20%) Homework.
- (10%) Quizzes
- (40%) Midterm exams
- (30%) Final exam
- Extra office hours: Thursday 5/3 12:30-1:30pm.
- Final exam info. Practice problem solutions here.
- Midterm 2 solutions.
- Exam revisions due by Friday, 4/13.
- Extra office hours: Tuesday 4/3, 3-4pm
- Midterm 2 info. Practice problem solutions here.
- Midterm 1 solutions.
- Extra office hours: Monday 2/19 2-3pm, Tuesday 2/20 1-2pm
- Midterm 1 info. Practice problem solutions here.
- You can check your grades here.
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
write-up the solution on
your own and it must
in your own words. Anything else is plagiarism and will be treated as
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
Homework 1 (due 1/26):
1.4.2, 1.5.1, 1.5.2, 1.5.3, 1.5.5, 1.5.8Review:
Appendix A, Appendix B, as neededRead:
1.6, 1.7, 1.10Homework 2 (due 2/5):
1.6.3, 1.6.4, 1.6.8, 1.7.1, 1.7.5, 1.7.11
2.1, 2.2Homework 3 (due 2/12):
2.1.3, 2.1.5, 2.1.7, 2.2.3, 2.2.12, 2.2.16Read:
Homework 4 (due 2/28):
2.4.5, 2.4.7, 2.4.14, 2.5.7, 2.5.8, 2.5.12
Homework 5 (due 3/7):
2.6.1, 2.6.2, 2.6.3, 2.7.1, 2.7.2, 2.7.7
Homework 6 (due 3/14):
3.1.2, 3.1.10, 3.1.3, 3.1.13, 3.2.2, 3.2.6 Read:
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
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
Course intro, First examples. Lecture notes
More examples, symmetry. Lecture notes
Permutation groups. Lecture notes
Divisibility properties of integers. Lecture notes
Modular arithmetic. Lecture notes
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