Math 417 - Fall 2018

Intro to Abstract Algebra

Jeremiah Heller (jbheller at illinois)

362 Altgeld
Office hours: Tuesday 13:00-14:00, Wednesday 14:00-15:00
                         
Section B13 at MWF 9:00-9:50 in 345 ALTGELD
Section E13 at MWF 13:00-13:50 in 145 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. 

Engage!

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 too 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!  

Grades

Your course grade is determined by the following:

Announcements

Exams

There will be three 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: September 24
Exam 2: October 15
Exam 3: November 30
Final Exam: December 14, 1:30 - 4:30pm in 1310 Digital Computer Laboratory

Homework 

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 9/7): 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 9/14): 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 9/21): 2.1.3, 2.1.5, 2.1.7, 2.2.3, 2.2.12, 2.2.20
Read: 2.3,2.4
Homework 4 (due 10/3): 2.3.1, 2.3.7, 2.4.5, 2.4.7, 2.4.14
Read: 2.5,2.6
Homework 5 (due 10/10): 2.5.7, 2.5.8, 2.5.12, 2.5.13, 2.6.2, 2.6.3
Read: 2.7,3.1
Homework 6 (due 10/24): 2.7.4, 2.7.7, 2.7.10, 3.1.2, 3.1.3
Read: 3.1,3.2
Homework 7 (due 10/31): 3.1.9, 3.1.10, 3.1.13, 3.2.1, 3.2.2, 3.2.6
Read: 5.1,5.2
Homework 8 (due 11/7): 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 9 (due 11/16):  5.4.1, 5.4.3, 5.4.5, 5.4.8, 5.4.9, 5.4.18
Optional Homework (due 11/26):  5.4.4, 5.4.6, 5.4.11, 5.4.12
Read: 1.8, 6.1-6.3
Homework 10 (due 12/12): 6.1.4, 6.2.4, 6.2.18, 1.8.7, 1.8.17, 6.3.2

Lectures

[8/27] Course intro, First examples. Lecture notes.
[8/29] More examples: symmetry. Lecture notes.
[8/31] Permutations. Lecture notes.
[9/5] Dvisibility properties of integers. Lecture notes.
[9/7-10] Modular arithmetic. Lecture notes.
[9/12] First properties of groups. Lecture notes.
[9/14] Worksheet 1.
[9/17] Subgroups. Lecture notes.
[9/19-21] Cyclic groups. Lecture notes.
[9/21,26] Dihedral groups. Lecture notes.
[9/24] Exam 1. Solutions here.
[9/26-28] Homomorphisms. Lecture notes.
[10/1] Cosets, Lagrange's theorem. Lecture notes.
[10/3] More cosets. Lecture notes.
[10/5-8] Equivalence relations. Lecture notes.
[10/8-12] Quotient groups. Lecture notes.
[10/15] Exam 2.  Solutions here.
[10/17] More quotient groups, direct products. Lecture notes.
[10/19] Worksheet 2.
[10/22] Direct products, semi-direct products. Lecture notes.
[10/24] Semi-direct products. Lecture notes.
[10/26] Finish semi-direct products, Group actions. Lecture notes.
[10/29] Worksheet 3.
[10/31] More group actions. Lecture notes.
[11/2] Orbit Stabilizer and counting. Lecture notes.
[11/5] More counting. Lecture notes.
[11/7] A little more counting. The class equation. Lecture notes.
[11/9-12] Sylow Theorems. Lecture notes.
[11/14] Worksheet 4.
[11/16] Sylow Theorems, proofs. Lecture notes.
[11/26] Rings and fields. Lecture notes.
[11/28] Polynomial rings. Lecture notes.
[11/30] Exam 3. Solutions here.
[12/3] Homomorphisms. Lecture notes.
[12/5] Ideals. Lecture notes.
[12/7] Quotient Rings. Lecture notes.
[12/10] Field extensions. Lecture notes.
[12/13] Review