Introduction to Modern Algebra I | UNL Fall 2024

Introduction to Modern Algebra I

Fall 2024

University of Nebraska — Lincoln

Instructor: Eloísa Grifo (please address me by my first name)


Class notes   These will be updated often throughout the semester. If you find any typos at all, however small, please let me know.


Basic course info: This is the first of a two part course on groups, rings, and modules. In this first half, we will discuss group theory, including group actions, and introduce rings. A major goal of this course is to prepare graduate students for the PhD qualifying exam in algebra.

Prerequisites: While there are no official prerequisites, you should have considerable experience with proof writing. Math 417 is a prerequisite for undergraduate students taking the course.

Course expectations: Attendance is expected, as much as possible, but please do NOT attend class if you are feeling ill or may have been exposed to COVID-19. There will be weekly problem sets. You are encouraged to work on the problem sets together in groups, or discuss them with me; you should however write up your own solutions. The only other resources you are allowed to use to solve the problem sets are our class notes and the textbook. There will be a midterm and a final exam, both in-person.

Office Hours: Mondays 5 to 6+ pm and Tuesdays 4:30 to 5:30 pm.
(The + means that I will stay later whenever there are students in my office, but will leave at 6:06 or so if I am alone)
The office hours past 5:30 pm on Mondays are for Math 817 only; but the remaining blocks are shared with the Math 309 students.
You can also just drop by my office anytime, or write me an email if you'd like to set up an appointment. If you are looking for me, you might find it helpful to take a look at my schedule for a typical week.


Problem Sets

Instructions: You are welcome to work together on the problems, and I will be happy to give you hints or discuss the problems with you, but you should write up your solutions by yourself. You will submit each problem set on canvas as a pdf file. The only other resources you are allowed to use to solve the problem sets are our class notes and the textbook.

Schedule

  • Lecture 1 (Monday, August 26): Welcome! What is a group?
  • Lecture 2 (Wednesday, August 28): Important classes of groups: the permutation groups.
  • Lecture 3 (Friday, August 30): Important classes of groups: the permutation groups and the dihedral group.
  • Lecture 4 (Wednesday, September 4): The dihedral group.
  • Lecture 5 (Friday, September 6): More on the dihedral group. The definition of a group homomorphism and a group isomorphism.
  • Lecture 6 (Monday, September 9): Properties and examples of group homomorphisms.
  • Lecture 7 (Wednesday, September 11): Properties of group homomorphisms. What is a group action?
  • Lecture 8 (Friday, September 13): Examples of group actions.
  • Lecture 9 (Monday, September 16): Subgroups: definition and examples.
  • Lecture 10 (Wednesday, September 18): More about subgroups. Cayley's theorem and Lagrange's theorem.
  • Lecture 11 (Monday, September 23): Cyclic subgroups.
  • Lecture 12 (Monday, September 25): Setting the stage for quotients.
  • Lecture 13 (Friday, September 27): The definition of a normal subgroup.
  • Lecture 14 (Monday, September 30): The definition of a normal subgroup.
  • Lecture 15 (Wednesday, October 2): Normal subgroups. Quotients of groups.
  • Lecture 16 (Friday, October 4): The First Isomorphism Theorem.
  • Lecture 17 (Monday, October 7): The second isomorphism theorem.
  • Lecture 18 (Wednesday, October 9): The lattice isomorphism theorem.
  • Lecture 19 (Friday, October 11): Review for the midterm.
  • Midterm: Friday, October 11, 5 to 7 pm, in Avery 351. Midterm solutions
  • Lecture 20 (Monday, October 14): Some comments on the midterm. The Third Isomorphism Theorem. Orbits and stabilizers.
  • Lecture 21 (Wednesday, October 16): LOIS. The Orbit-Stabilizer Theorem.
  • Lecture 22 (Friday, October 18): The rotations of the cube. Conjugacy classes.
  • Lecture 23 (Wednesday, October 23): The conjugacy classes of Sn. The Class Equation and applications.
  • Lecture 24 (Friday, October 25): The Alternating Group.
  • Lecture 25 (Monday, October 28): The Alternating group is simple. Other examples of group actions.
  • Lecture 26 (Wednesday, October 30): Other examples of group actions. Cauchy's Theorem.
  • Lecture 27 (Friday, November 1): The Main Theorem of Sylow Theory.
  • Lecture 28 (Monday, November 4): The Main Theorem of Sylow Theory: the proof continues.
  • Lecture 29 (Wednesday, November 6): Finishing the proof of the Main Theorem of Sylow Theory. Applications of Sylow Theory.
  • Lecture 30 (Friday, November 8): Direct products and semidirect products.
  • Lecture 31 (Monday, November 11): Semidirect products.
  • Lecture 32 (Wednesday, November 13): Using semidirect products to construct nonabelian groups. The recognition theorem for semidirect products.
  • Lecture 33 (Friday, November 15): The classification of finitely generated abelian groups. Classifying all groups of a certain order.
  • Lecture 34 (Monday, November 18): Rings.
  • Lecture 35 (Wednesday, November 20): Units and zerodivisors. Subrings.
  • Lecture 36 (Friday, November 22): Ideals.
  • Lecture 37 (Monday, November 25): Homomorphisms of rings.
  • Lecture 38 (Monday, December 2): The Isomorphism Theorems for rings.
  • Lecture 39 (Wednesday, December 4): Prime and maximal ideals.
  • Lecture 40 (Friday, December 6): Euclidean domains.
  • Lecture 41 (Friday, December 9): PIDs.
  • Lecture 42 (Friday, December 11): UFDs.
  • Lecture 43 (Friday, December 13): UFDs. Quotients of polynomial rings in one variable over a field.
  • Final Exam: Tuesday, December 17, 10 am, in Oldfather 205.

Helpful resources



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