Introduction to Abstract Algebra (Math 417, Fall 2019)

Basic information

Please note regarding the textbook

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.

Content

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.

Policies

  • 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 comprehensive. The date is yet to be determined by the registrar.
  • 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 assignments

  • 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.

Detailed schedule

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.  
[2019-11-07 Thu] Class equation and applications.  
[2019-11-12 Tue] Sylow theorems and applications. Homework 9 due.
[2019-11-14 Thu] Proofs of Sylow theorems.  
[2019-11-19 Tue] Introduction to rings and fields. Homework 10 due.
[2019-11-21 Thu] Polynomial rings over fields.  
[2019-11-26 Tue] Fall Break.  
[2019-11-28 Thu] Fall Break.  
[2019-12-03 Tue] Ring homomorphisms, ideals and principal ideals. Homework 11 due.
[2019-12-05 Thu] Quotient rings, finite fields.  
[2019-12-10 Tue] Field extensions and fields of fractions. Homework 12 due.

Piazza

Piazza is an online discussion forum where you can get your questions answered by classmates and the instructor. Please sign up here. Note that you can use any email to register for Piazza and can post questions and answers anonymously if you prefer.

While discussion of homework problems is allowed, users of the forum should refrain from posting complete or near-complete solutions to the problems. It is appropriate, however, to post suggestions, hints, or references to the relevant parts of the textbook or lecture notes. Please remember that you must write up your homework solutions individually and understand them completely.