Fall 2017 Teaching
In the Fall semester 2017, I taught MATH-UA.343.005 Algebra. Contact information:
Office 604
bilu @ cims.nyu.edu (remove blank spaces around @)
Office hours:
Mondays 3-4pm, Wednesdays 3-4 pm, or by appointment.
Contents of the course:
Reminders on set theory:
Inclusion, intersections, unions of sets;
Mappings, image, inverse image, injectivity, surjectivity, bijectivity;
Relations, equivalence relations, equivalence classes, quotient space.
The set of integers:
Properties of addition and multiplication of integers;
Divisibility, Euclidean division, GCD, Euclidean algorithm;
Unique factorization of integers;
Congruences, the set Z/nZ.
Z/nZ as a ring
Units in Z/nZ
Equations with congruences
Groups
Laws of composition, associativity, commutativity, identity elements, inverses.
Groups, examples of groups, order of a group.
Subgroups
Products of groups
Cyclic groups
Group generated by a subset
Homomorphisms, isomorphisms
Classification of groups of small order
Permutations
Permutation groups
Cycles, decomposition into disjoint cycles
Transpositions, products of transpositions
Sign of a permutation, alternating group
Generators of symmetric and alternating groups
More on groups
Left and right cosets
Index of a subgroup
Lagrange's theorem
Subgroups of Z/nZ
Euler's theorem
Cosets of the kernel of a homomorphism
Normal subgroups
Quotient of a group by a normal subgroup
First isomorphism theorem
Notes
12/17 version Remarks or questions welcome! Pay attention to the fact that these notes do not contain proofs. If you've missed a lecture, make sure to catch up the proofs by borrowing someone's notes.
Practice problems
12/12 version
Hints/solutions
Homeworks:
Homeworks are always due in the beginning of Thursday's class. If you cannot attend class, you can e-mail me your homework before the beginning of class, or leave it in my mailbox (number 38 on the right side of the mailboxes behind the guard's desk in the lobby of WWH). Late homeworks are usually not accepted, except if you have a valid excuse, which you should e-mail me about in advance.
Homework 1 (due 09/14)
Homework 2 (due 09/21)
Homework 3 (due 09/28)
Homework 4 (due 10/05)
Homework 5 (due 10/12)
Homework 6 (due 10/19)
Homework 7 (due 10/26)
Homework 8 (due 11/09)
Homework 9 (due 11/16)
Homework 10 (due 11/30)
Homework 11 (due 12/07)
Quizzes:
There will be short quizzes every two weeks during recitation.
Exams
The midterm will be on Tuesday, October 31st during the usual lecture hours. It will cover everything up to the end of the chapter on groups.
The final will be on Thursday, December 21st, 12pm-1:50pm.
Grading:
- Homeworks 15%
- Quizzes 20%
- Midterm 25%
- Final 40%
Recommended books
Michael Artin, Algebra, second edition.
Thomas W. Judson, Abstract Algebra: theory and applications, available online here.
Some advice
- Read your notes before coming to class: it is hard to follow if you don't remember what has been said last time.
- Ask questions and try to propose answers to questions I am asking even if you're not sure: making mistakes is part of the normal process of learning. One remembers something very well if one got it wrong the first time.
- If I use some notation or some mathematical notion you're
not familiar with, please ask about it: I come from a
different background and am not completely aware of what you
know.
- Please only answer a question asked in class if you've been prompted to do so, so as to let the others think. Not everyone has the same speed.
- Come to office hours, even if you don't think you have that many questions. You can come by anytime during the specified time range.
- This is a course with many proofs. Make sure to go
over each proof actively, asking yourself: what would
I do if I wanted to prove this? How many steps are there, what
is the structure of this proof? Why do we need to do
this? Why are we done at the end? Knowing the proof of a
theorem helps you get a deep understanding of the theorem
itself, I therefore strongly recommend that you learn the
proofs at the same time as you learn the theorems. Many
exercises in the homeworks, quizzes and exams may rely on
ideas similar to the proof of some theorem seen in class.
- Work in groups! It's much more fun doing maths with other people than on one's own. Ask questions to your classmates. If you have trouble remembering a proof, try to practice explaining it to a classmate: this is the best way to learn it.
- Having trouble with complex numbers? Look up chapter 4,
paragraph 4.2 in Judson for an introduction with many
examples. There are also many exercises at the end of the
chapter.
- Want a reminder on set theory and equivalence relations?
Look up chapter 1, paragraph 1.2 in Judson.
- Having trouble with proofs by induction? Look up chapter 2,
paragraph 2.1 in Judson, as well as the exercises at the end
of the chapter.