Math 417: Introduction to Abstract Algebra (Spring 2018)

Table of Contents

Basic information

Instructor: James Pascaleff
E-mail: jpascale@illinois.edu
Office: 341B Illini Hall
Office hours: Tuesdays 3:00-5:00 (or by appointment)
Lectures
Section E13: MWF 1:00-1:50 in 447 Altgeld Hall
Web page
http://math.illinois.edu/~jpascale/courses/2018/417
Viewing Grades
View your scores online at https://www-s.math.illinois.edu/grades/grades.cgi
Detailed schedule with lecture notes, homework, and exams

Course description

This course is an introduction to the modern abstract theory of algebra and algebraic structures. The main objects of study will be groups, which are "abstract groups of composable transformations" (think of permutations or linear transformations), as well as 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.

Required text
F.M. Goodman, Algebra: Abstract and Concrete. This e-book is available free-of-charge: Website for the book.
Supplementary text
J.B. Fraleigh, A First Course in Abstract Algebra, 7th edition. Note: this book is listed as "required" at the bookstore, but it is optional for this section of Math 417.
Prerequisite
Either Math 416, or one of Math 347, Math 348, together with one of Math 410, Math 415, CS 273.

Course policies

Overall grading
Grades will be based on homework (18%), three in-class midterms (18% each), and a final exam (28%).
Weekly homework
These are due at the beginning of class on the due date. 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.
Midterms
Three midterm exams will be given, at the usual lecture time and room. The dates are February 14, March 14, and April 18.
Final exam
The final exam will be comprehensive. The final exam will be held Monday, May 7, 8:00-11:00 AM in 447 Altgeld Hall.
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.
Disabilities
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.

Detailed schedule

This schedule may change slightly as the semester progresses. Lecture notes and homework assignments will be linked below as they become available. The references such as [G, section X ] refer to Goodman's text.

W Jan 17 1. Introduction, examples of groups. [G, sections 1.1–1.4, 1.10].
F Jan 19 2. More examples, Permutation groups. [G, section 1.5].
M Jan 22 3. Disjoint cycle notation. [G, section 1.5].
W Jan 24 4. Divisibility in the integers, prime numbers. [G, section 1.6].
F Jan 26 5. More on divisibility and primes. [G, section 1.6]. HW 1 due. Solutions.
M Jan 29 6. Equivalence relations and partitions, Modular arithmetic. [G, sections 2.6, 1.7].
W Jan 31 7. More on modular arithmetic. [G, section 1.7].
F Feb 2 8. Even more on modular arithmetic. [G, section 1.7]. HW 2 due. Solutions.
M Feb 5 9. Basic properties of groups. [G, section 2.1].
W Feb 7 10. Subgroups, isomorphisms, and Cayley's theorem. [G, section 2.2].
F Feb 9 11. Cyclic groups. [G, section 2.2]. HW 3 due. Solutions.
M Feb 12 12. Subgroups of cyclic groups. [G, section 2.2].
W Feb 14 Midterm 1. Handout. Solutions.
F Feb 16 13. Dihedral groups. [G, sections 2.3].
M Feb 19 14. Homomorphisms, kernels. [G, section 2.4].
W Feb 21 15. Cosets and Lagrange's theorem. [G, section 2.5].
F Feb 23 16. More on cosets. [G, section 2.5]. HW 4 due. Solutions.
M Feb 26 17. Quotient groups and homomorphisms. [G, section 2.7].
W Feb 28 18. Quotient group isomorphism theorem. [G, section 2.7].
F Mar 2 19. More quotient group theorems. [G, section 2.7]. HW 5 due. Solutions.
M Mar 5 20. Direct products. [G, section 3.1].
W Mar 7 21. More on direct products; towards semidirect products. [G, section 3.2].
F Mar 9 22. Semidirect products. [G, section 3.2]. HW 6 due. Solutions.
M Mar 12 23. Review session for midterm 2.
W Mar 14 Midterm 2. Handout. Practice problems. Solutions.
F Mar 16 24. Recognition theorem for semidirect products. [G, section 3.2].
M Mar 19 Spring Break
W Mar 21 Spring Break
F Mar 23 Spring Break
M Mar 26 25. Group actions. [G, section 5.1].
W Mar 28 26. Examples of actions, orbit-stabilizer theorem. [G, section 5.1].
F Mar 30 27. Applications of orbit-stabilizer theorem. [G, section 5.1]. HW 7 due. Solutions.
M Apr 2 28. Orbit-counting theorem, examples. [G, section 5.2].
W Apr 4 29. Class equation and applications to the structure of groups. [G, section 5.4].
F Apr 6 30. Sylow theorems and applications to the structure of groups. [G, section 5.4]. HW 8 due. Solutions.
M Apr 9 31. Proof of Sylow theorems. [G, section 5.4].
W Apr 11 32. Basic properties of ring and fields. [G, sections 1.11, 6.1].
F Apr 13 33. Polynomial rings and other examples. [G, section 1.8]. HW 9 due. Solutions.
M Apr 16 34. Factorization of polynomials over a field. [G, section 1.8].
W Apr 18 Midterm 3. Handout. Practice problems. Solutions.
F Apr 20 35. Homomorphisms of rings. [G, section 6.2].
M Apr 23 36. Ideals. [G, section 6.2].
W Apr 25 37. Quotient rings. [G, section 6.3].
F Apr 27 38. Homomorphism theorems for rings. [G, section 6.3]. HW 10 due. Solutions.
M Apr 30 39. Integral domains and prime ideals. [G, section 6.4].
W May 2 40. Final review HW 11 due. Solutions.
M May 7 Final exam. Monday, May 7, 8:00-11:00 AM in 447 Altgeld Hall. Handout. Ring theory problems.

Validate