Introduction to Abstract Algebra (Math 417, Fall 2019)

• Lectures: TR 9:30am–10:50am in 141 Altgeld Hall.
  • This represents Sections M13 (CRN 32103) and M14 (CRN 32106).
• Course website: http://jpascale.pages.math.illinois.edu/417fa19
• Instructor: James Pascaleff
  • Email: jpascale@illinois.edu; Office: 341B Illini Hall; Phone: (217) 244-7277.
  • Office hours: M 1:00pm–2:00pm, 4:00pm–5:00pm.
• 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. This e-book is available free of charge: website for the book.

The textbook for this section of Math 417 is Algebra: Abstract and Concrete by Frederick M. Goodman. This is not the same as the book sold at the Illini Union Bookstore for Math 417, which is A First Course in Abstract Algebra by John B. Fraleigh. The book by Fraleigh is not required for this section.

This course is an introduction to the modern abstract theory of groups and rings. Groups are abstractions connected with the concept of symmetry, and rings are “abstract number systems” in which there are versions of the arithmetic operations: addition, subtraction, multiplication, and (sometimes) division.

• Assessment: Grades will be based on homework (30%), two midterm exams (20% each), and the final exam (30%). The two lowest homework scores will be dropped.
• Homework: Homework assignments and their due dates will be posted on this website. Homework is due at the beginning of class on the due date. You are required to submit a paper copy of your homework in class. Late homework will not be accepted. However, the lowest two scores are dropped, so you may miss one or two assignments without penalty. Collaboration on homework is permitted and expected, but you must write up your solutions individually and understand them completely.
• Midterm exams: The two midterm exams will held during the regular class periods on Tuesday, October 1 and Tuesday, November 5.
• Final exam: The final exam will be held Tuesday, December 17, 1:30pm–4:30pm, in 141 Altgeld Hall. It will cover the entire course, with some emphasis on material that was covered after the second midterm.
• Missed exams: If you need to miss an exam for valid reason (such as illness, accident, or family crisis), please let the instructor know as soon as possible. Normally, you will be excused from the exam so that it does not count towards your overall grade.
• Cheating: Cheating, that is, an attempt to dishonestly gain an unfair advantage over other students, is taken very seriously. Penalties for cheating on exams may include a zero on the exam or an F in the course.
• 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 1 Due Tuesday, Sept. 3: Goodman, Exercises 1.3.1, 1.3.2, 1.3.3, 1.4.1, 1.4.2, 1.4.3, 1.5.1, 1.5.2, 1.5.3. [Important Note: In the textbook, the only symmetries considered are rigid motions, i.e., rotations+translations].
• Homework 2 Due Tuesday, Sept. 10: Goodman, Exercises 1.6.3, 1.6.4, 1.6.7, 1.6.8, 1.6.9, 1.7.1, 1.7.2, 1.7.3, 1.7.4.
• Homework 3 Due Tuesday, Sept. 17: Goodman, Exercises 1.7.13, 1.7.14, 1.7.16, 2.1.3, 2.1.4, 2.1.5, 2.1.8, 2.1.9, 2.1.12.
• Homework 4 Due Tuesday, Sept. 24: Goodman, Exercises 2.2.6, 2.2.9, 2.2.11, 2.2.14, 2.2.19, 2.2.25, 2.3.3, 2.3.6.
• Homework 5 Due Tuesday, Oct. 8: Goodman, Exercises 2.4.3, 2.4.5, 2.4.6, 2.4.14, 2.4.20, 2.5.4, 2.5.6, 2.5.7, 2.5.11.
• Homework 6 Due Tuesday, Oct. 15: Goodman, Exercises 2.6.1, 2.6.4, 2.6.5, 2.6.6, 2.7.2, 2.7.9, 2.7.10, 2.7.11.
• Homework 7 Due Tuesday, Oct. 22: Goodman, Exercises 2.7.4, 2.7.6, 2.7.7, 2.7.8, 3.1.4, 3.1.9, 3.1.10, 3.1.13, 3.1.15.
• Homework 8 Due Tuesday, Oct. 29: Goodman, Exercises 3.1.11, 3.2.2, 3.2.4, 3.2.5, 3.2.6, 5.1.1, 5.1.5, 5.1.6.
• Homework 9 Due Tuesday, Nov. 12: Goodman, Exercises 5.1.7, 5.1.9, 5.1.12, 5.1.13, 5.1.14, 5.2.1, 5.2.2, 5.4.1, 5.4.5.
• Homework 10 Due Tuesday, Nov. 19: Goodman, Exercises 5.4.2, 5.4.3, 5.4.8, 5.4.11, 5.4.13, 5.4.15.
• Here is a solution to Exercise 5.1.15, which asks you to classify groups of order 20.
• Homework 11 Due Tuesday, Dec. 3: Goodman, Exercises 6.1.1, 6.1.4, 6.1.9, 6.1.15, 1.8.3, 1.8.4, 1.8.5, 1.8.10.
• Homework 12 Due Tuesday, Dec. 10: Goodman, Exercises 6.2.1, 6.2.2, 6.2.6, 6.2.7, 6.3.2, 6.3.3, 6.3.8, 6.3.11.

Topics for future lectures may change as the course progresses.

Date	Lecture	Remarks
[2019-08-27 Tue]	1. Symmetries.	Lecture by Prof. Fernandes.
[2019-08-29 Thu]	2. Permutations.	Lecture by Prof. Fernandes.
[2019-09-03 Tue]	3. Integer arithmetic.	Homework 1 due.
[2019-09-05 Thu]	4. Primes, modular arithmetic.
[2019-09-10 Tue]	5. More modular arithmetic, basic group properties.	Homework 2 due.
[2019-09-12 Thu]	6. Subgroups, isomorphisms, Cayley’s theorem.
[2019-09-17 Tue]	7. Cyclic groups.	Homework 3 due.
[2019-09-19 Thu]	8. Subgroups of cyclic groups, dihedral groups.
[2019-09-24 Tue]	9. Homomorphisms and kernels.	Homework 4 due.
[2019-09-26 Thu]	10. Cosets and Lagrange’s theorem.
[2019-10-01 Tue]	Midterm Exam 1. Solutions.
[2019-10-03 Thu]	11. Equivalence relations and partitions.
[2019-10-08 Tue]	12. More on equivalence relations.	Homework 5 due.
[2019-10-10 Thu]	13. Quotient groups and homomorphisms.
[2019-10-15 Tue]	14. Isomorphism theorems.	Homework 6 due.
[2019-10-17 Thu]	15. Diamond isomorphism, direct products of groups.
[2019-10-22 Tue]	16. Semi-direct products.	Homework 7 due.
[2019-10-24 Thu]	17. Examples of semi-direct products, group actions.
[2019-10-29 Tue]	18. Orbit-stabilizer theorem.	Homework 8 due.
[2019-10-31 Thu]	19. Burnside/Cauchy-Frobenius lemma.
[2019-11-05 Tue]	Midterm Exam 2. Solutions.
[2019-11-07 Thu]	20. Class equation and applications.
[2019-11-12 Tue]	21. Sylow theorems and applications.	Homework 9 due.
[2019-11-14 Thu]	22. Proofs of Sylow theorems.
[2019-11-19 Tue]	23. Introduction to rings and fields.	Homework 10 due.
[2019-11-21 Thu]	24. Polynomial rings over fields.
[2019-11-26 Tue]	Fall Break.
[2019-11-28 Thu]	Fall Break.
[2019-12-03 Tue]	25. Ring homomorphisms and ideals.	Homework 11 due.
[2019-12-05 Thu]	26. Quotient rings, homomorphism theorem for rings.
[2019-12-10 Tue]	27. Maximal and prime ideals, integral domains.	Homework 12 due.
[2019-12-17 Tue]	Final Exam, 1:30pm–4:30pm.	Held in usual classroom (141 Altgeld).

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