Introduction to Abstract Algebra, Math 417, Fall 2021

Basic information

Sections M and B
- Section M meets MWF 9:00–9:50am in 245 Altgeld Hall.
- Section B meets MWF 10:00–10:50am in ~~4101 MSEB~~ 108 DKH.
Instructor: James Pascaleff
- Email: jpascale@illinois.edu; Office: 357 Altgeld Hall; Phone: (217) 244-7277.
- Office hours: Wednesday 3:00–4:00pm and Thursday 3:00–4:00pm.
Course website: http://jpascale.pages.math.illinois.edu/417fa21
Canvas website: https://canvas.illinois.edu/courses/15394
- Use this website to view grades and homework solutions.
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.
Textbook: Frederick M. Goodman, Algebra: Abstract and Concrete version 2.6. This e-book is available free of charge: website for the book.
Final Exams:
- Section M: Thursday, December 16, 1:30pm–4:30pm in 245 Altgeld Hall.
- Section B: Monday, December 13, 1:30pm–4:30pm in 108 David Kinley Hall.

Course outline

Description: This course is an introduction to the modern abstract theory of algebra and algebraic structures. The primary focus is on the theory of groups, which are “abstract groups of composable transformations;” key examples include the symmetries of a geometric figure, and the permutations of a set. Our focus is on constructing groups, understanding their structure, and developing techniques for classifying them. At the end of the course we also begin the study of rings and fields, which are “abstract number systems” in which there are abstract versions of the four arithmetic operations: addition, subtraction, multiplication, and (in the case of fields) division.
Lectures: Lectures will be held in person. Written notes for the lectures as well as videos (recorded in Spring 2021) are also available. Please see the links in the schedule below. The lectures do not correspond exactly to the class periods. For instance, in the first week of class the two lectures “Symmetries” and “Permutations” will be spread over Monday, Wednesday, and Friday.
Homework: There will be 11 homework assignments whose due days (usually Fridays) are listed in the schedule. Homework must be submitted on paper in class. There is no homework due on weeks when there is an exam.
Exams: There will be two midterm exams and a final exam. The midterm exams will be held in class on Friday October 1 and Friday November 5.

Policies

Covid protocols: Througout this course we shall comply with all campus guidelines and rules regarding Covid-19, including rules on face masks and building access. Do not come to class if you are feeling ill or have tested positive for Covid-19. Please note that attendance does not count towards your grade in this course. If your absence means that you will miss an assignment or exam, please let the instructor know so that alternative arrangments can be made.
Assessment: Grades will be based on homework (25%), two midterm exams (22% each), and the final exam (31%). The two lowest homework scores will be dropped. Grade cutoffs will never be stricter than 90% for an A- grade, 80% for a B-, and so on. Individual exams may have grade cutoffs set more generously depending on their difficulty.
Late homework will not be accepted, but the lowest two scores are dropped, so you may miss one or two assignments without penalty.
Missed exams: If you need to miss an exam (for reasons such as illness, accident, or family crisis), please let the instructor know as soon as possible. You will be excused from the exam so that it does not contribute to your final grade (there will not be a make-up exam).
Collaboration and Academic Integrity: For homework assignments, collaboration is permitted and expected, but you must write up your solutions individually and understand them completely. On exams, no collaboration is permitted.
Disability accommodations: Students who require special accommodations should contact the instructor as soon as possible. Any accommodations on exams must be requested at least one week in advance and will require a letter from DRES.

Homework assignments

Homework assignments should be submitted on paper in class on the due date. If you are unable to come to class, you may turn in your homework to James Pascaleff’s mailbox in 250 Altgeld Hall. This mailbox will be checked at 11:00am on the due date, so you should have your homework in the box by that time.
Homework 1 Due Friday September 3.
Homework 2 Due Friday September 10.
Homework 3 Due Friday September 17.
Homework 4 Due Friday September 24.
Homework 5 Due Friday October 8.
Homework 6 Due Friday October 15.
Homework 7 Due Friday October 22.
Homework 8 Due Friday October 29.
Homework 9 Due Friday November 12.
Homework 10 Due Friday November 19.
Homework 11 Due Wednesday December 8.

Schedule

Note: This class meets three days a week, Monday-Wednesday-Friday. Only two sets of lecture notes are listed for most weeks: this material will be spread out over three class periods.

Week №	Dates	Lectures for the week
Week 1	[2021-08-23 Mon]	1. Symmetries. [§§ 1.1–1.4, 1.10] video.
	[2021-08-25 Wed]	2. Permutations. [§ 1.5] video.
	[2021-08-27 Fri]
Week 2	[2021-08-30 Mon]	3. Integer arithmetic. [§ 1.6] video.
	[2021-09-01 Wed]	4. Primes, modular arithmetic. [§§ 1.6–1.7] video.
	[2021-09-03 Fri]	Homework 1 due.
Week 3	[2021-09-06 Mon]	Labor Day
	[2021-09-08 Wed]	5. More modular arithmetic, basic group properties. [§§ 1.7, 2.1] video.
	[2021-09-10 Fri]	Homework 2 due.
Week 4	[2021-09-13 Mon]	6. Subgroups, isomorphisms, Cayley’s theorem. [§ 2.2] video.
	[2021-09-15 Wed]	7. Cyclic groups. [§ 2.2] video.
	[2021-09-17 Fri]	Homework 3 due.
Week 5	[2021-09-20 Mon]	8. Subgroups of cyclic groups, dihedral groups. [§§ 2.2–2.3] video.
	[2021-09-22 Wed]
	[2021-09-24 Fri]	Homework 4 due.
Week 6	[2021-09-27 Mon]	9. Homomorphisms and kernels. [§ 2.4] video.
	[2021-09-29 Wed]	10. Cosets and Lagrange’s theorem. [§ 2.5] video.
	[2021-10-01 Fri]	Exam 1, covering lectures 1–8.
Week 7	[2021-10-04 Mon]	11. Equivalence relations and partitions. [§ 2.6] video.
	[2021-10-06 Wed]	12. More on equivalence relations. [§ 2.6] video.
	[2021-10-08 Fri]	Homework 5 due.
Week 8	[2021-10-11 Mon]	13. Quotient groups and homomorphisms. [§ 2.7] video.
	[2021-10-13 Wed]	14. Isomorphism theorems. [§ 2.7] video.
	[2021-10-15 Fri]	Homework 6 due.
Week 9	[2021-10-18 Mon]	15. Diamond isomorphism, direct products of groups. [§§ 2.7, 3.1] video.
	[2021-10-20 Wed]	16. Semi-direct products. [§ 3.2] video.
	[2021-10-22 Fri]	Homework 7 due.
Week 10	[2021-10-25 Mon]	17. Examples of semi-direct products, group actions. [§§ 3.2, 5.1] video.
	[2021-10-27 Wed]	18. Orbit-stabilizer theorem. [§ 5.1] video.
	[2021-10-29 Fri]	Homework 8 due.
Week 11	[2021-11-01 Mon]	19. Burnside/Cauchy-Frobenius lemma. [§ 5.2] video.
	[2021-11-03 Wed]	20. Class equation and applications. [§ 5.4] video.
	[2021-11-05 Fri]	Exam 2, covering lectures 9–18.
Week 12	[2021-11-08 Mon]	21. Sylow theorems and applications. [§ 5.4] video.
	[2021-11-10 Wed]	22. Proofs of Sylow theorems. [§ 5.4] video.
	[2021-11-12 Fri]	Homework 9 due. Drop deadline for LAS and Engineering Students.
Week 13	[2021-11-15 Mon]	23. Introduction to rings and fields. [§§ 1.11, 6.1] video.
	[2021-11-17 Wed]	24. Polynomial rings over fields. [§ 1.8] video.
	[2021-11-19 Fri]	Homework 10 due.
Fall	[2021-11-22 Mon]	Fall
Break	[2021-11-24 Wed]	Break
Week	[2021-11-26 Fri]	Week
Week 14	[2021-11-29 Mon]	25. Ring homomorphisms and ideals. [§ 6.2] video.
	[2021-12-01 Wed]	26. Quotient rings, homomorphism theorem for rings. [§ 6.3] video.
	[2021-12-03 Fri]
Week 15	[2021-12-06 Mon]	27. Maximal and prime ideals, integral domains. [§ 6.4] video.
	[2021-12-08 Wed]	Homework 11 due.
Final	[2021-12-13 Mon]	Section B Final Exam, 1:30–4:30pm in 108 David Kinley Hall.
Exam	[2021-12-16 Thu]	Section M Final Exam, 1:30–4:30pm in 245 Altgeld Hall.