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Course

Intro to Modern Algebra I - Math GU4041.
Spring 2021.
Columbia University.

Lectures

Classroom: online, on Zoom.
Mo, We 10:10am-11:25am.

Instructor

Name: Daniele Alessandrini.
Contact: daniele.alessandrini@gmail.com.
Office: 624 Mathematics.

Office hours: (tentative) Mo, We 11:30am-12:30pm, online, on Zoom. The Zoom link will be posted on Courseworks, in the Syllabus page. Please send me an email in advance or just before connecting to Zoom to make sure I see you.

TAs

The TAs grade the assignments and can answer questions from students. You can ask them questions by e-mail, and they serve in the Virtual Help Room, where you can reach them via Zoom to ask your questions. Their e-mail addresses and Zoom links to the Virtual Help Rooms will be available on Courseworks in the Syllabus page.

Syllabus

The Syllabus for this course.

Content

Required text

There is no required text. I will mainly follow the book Abstract Algebra, by Herstein. Other good books are: Algebra by Artin, Abstract Algebra by Dummit and Foote, A First Course in Abstract Algebra by Fraleigh, Undergraduate Algebra by Lang.

Course Outline

Groups, homomorphisms, normal subgroups, the isomorphism theorems, symmetric groups, group actions, the Sylow theorems, finitely generated abelian groups.

Lectures

Date No Topic Textbook Reference
21/01/11 01 Numbers.
21/01/13 02 Mathematical induction, divisibility. 1.6, 1.5.
21/01/18 Holiday
21/01/20 03 Fundamental Theorem of Arithmetics. 1.5.
21/01/25 04 Modular arithmetics. 2.4.
21/01/27 05 Permutations. 1.4, 3.1.
21/02/01 06 Permutations. 3.2.
21/02/03 07 Permutations. 3.3.
21/02/08 08 Isometries of the plane.
21/02/10 09 Groups and subgroups. 2.1, 2.2, 2.3
21/02/15 First Midterm
21/02/17 10 Generated subgroups.
21/02/22 11 Group homomorphisms and normal subgroups. 2.5
21/02/24 12 Isomorphisms. Direct products. 2.5, 2.9
21/03/01 Spring recess.
21/03/03 Spring recess.
21/03/08 13 Direct Products. Quotient groups. 2.9, 2.6
21/03/10 14 Properties of quotients. 2.7
21/03/15 15 Chinese remainder theorem, Group actions.
21/03/17 16 Semi-direct products.
21/03/22 17 Lagrange's theorem. 2.4
21/03/24 18 Orbit-Stabilizer Theorem 2.11
21/03/29 19 Cauchy's theorem 2.8
21/03/31 20 Finite Abelian Groups 2.10
21/04/05 Second Midterm
21/04/07 21 Sylow's Theorems 2.11
21/04/12 22 Proof of Sylow's Theorems 2.11
21/04/14 23 Simple groups
Prerequisite

The four Calculus courses, and Linear Algebra. You should also be familiar with complex numbers, mathematical induction and other methods of proof, and in general have a certain confidence in your abilities to handle abstract mathematical reasoning. A prior course which involves writing proofs such as Honors Math A/B or Introduction to Higher Mathematics is strongly recommended.

Homework

Homework sheets

The homework will be published online in form of homework sheets every Wednesday night. There are 6 days time to submit the written solutions.

Date Number Topic Hand in by Grader
21/01/13 01 Numbers 21/01/19 Anda
21/01/20 02 Primes 21/01/26 Alan
21/01/27 03 Modular Arithmetics 21/02/02 Anda
21/02/03 04 Permutations 21/02/09 Alan
21/02/10 05 Isometries 21/02/18 Anda
21/02/17 06 Subgroups 21/02/23 Alan
21/02/24 07 Homomorphisms 21/03/09 Anda
21/03/10 08 Products and quotients 21/03/16 Alan
21/03/17 09 Quotients, actions and semi-direct products 21/03/23 Anda
21/03/24 10 Finite groups 21/03/30 Alan
21/03/31 11 Cauchy's theorem 21/04/08 Anda
21/04/07 12 Sylow's theorem 21/04/13 Alan
21/04/14 13 More on Sylow's theorem not graded

Handing in

Solutions to the homework will be due on Tuesday night. More precisely, the deadline will be on Wednesday early morning, at 5am. The solutions must be handed in electronically, via Courseworks, at the Assignments page. We encourage collaboration on assignments, but all solutions must be written up by you alone in your own words.

Extensions

Understandably, there will be bad weeks, where a student is overwhelmed and doesn't manage to submit the homework on time. To cover for these cases, the following two policies will be implemented:

  1. Late submission. We will accept late submissions, but we will deduct 10% of the points for every day of lateness. Use this option only if you are just a few days late. After 9 days of lateness, submitting is not worth any more.
  2. At the end of the semester, when computing the average of the grades obtained at the homework, we will discard the two lowest grades. In this way, students who once or twice don't manage to submit the homework will not be penalized.
Exams

Midterm exams

There will be two midterm exams, on Monday February 15 and on Monday April 5, during the usual class time.

Final exam

Projected schedule for the final exam: Wednesday April 21, 9am-Noon. The date will be confirmed by the University.

Exam details

The midterms and final exams will be online. Please, check the rules for on-line exams. You must plan to take the midterm and final exams at the scheduled time, so please make your plans accordingly. Besides students with disabilities having prior arrangements with ODS, the only exceptions will be for those with an incapacitating illness, a serious family emergency, or situations of comparable gravity. In the first case you will need a note from a doctor; in the other cases you will need a note from your advising dean. Incompletes can be granted only by your advising dean and only in the circumstances mentioned above. Anyone guilty of academic dishonesty, such as cheating on an exam or helping someone else to cheat, will fail the course and faces further academic discipline.

Grading

Homework 10%, midterms 25%, final 40%.