Math 501 Course Webpage
Algebraic Structures I
Fall 2023, Duke University
General information |
Course description |
Lecture notes |
Assignments |
Homework schedule |
Grading |
Links |
Fine print
Lectures: Tuesday and Thursday, 13:25 – 14:40, Physics Building 119
Text: Abstract Algebra,
by David S. Dummit and Richard M. Foote (third edition)
Secondary texts:
- Artin:
Algebra
- Lang:
Algebra
- Herstein:
Topics in Algebra
Contact information for the Instructor
Name: Professor Ezra Miller
Address: Mathematics Department,
Duke University, Box 90320,
Durham, NC 27708-0320
Office: Physics 209
Phone: (919) 660-2846
Email: ezra
math.duke.edu
Webpage:
https://math.duke.edu/people/ezra-miller
Course webpage: you're already
looking at it... but it's
https://services.math.duke.edu/~ezra/501/501.html
Sakai site: available to registered students via Duke NetID
Office hours:
Tuesday 14:40 – 16:00 &
Wednessday 14:00 – 15:10, in Physics 209
Course content:
- groups
- subgroups
- homomorphisms
- cosets
- quotient groups
- symmetry
- group actions
- permutation representations
- Sylow theorems
- rings
- ideals
- polynomials
- fractions
- modules
- structure theorem over PID
Groups encapsulate the notion of symmetry. They constitute the
simplest way to compose a single type of invertible operation,
such as addition of numbers, multiplication of matrices with
nonzero determinant, rotations of spheres, rigid motions of
polygons and polyhedra, or permutations of sets of objects.
The study of groups in this course includes decompositions,
enumerations, quotients, and actions.
Rings combine two operations: addition and multiplication. In
familiar situations, particularly the integers and univariate
polynomials, interactions between the two operations lead to
fundamental theorems concerning factorization into primes.
What results is a main goal of the course: a single structure
theorem that classifies all finite abelian groups and also
produces Jordan canonical forms of linear transformations.
Prerequisites:
Math 501 is a demanding course. Students are expected to have
a firm grasp of linear algebra before beginning the course. In
addition, it is expected that every student begin the course
comfortable and proficient at writing rigorous mathematical
arguments (proofs).
All lectures in one PDF file
The lecture notes can and often are updated or corrected. If you
think you have found an error, check that you have the latest version
before sending a correction.
- (notes)
Lecture 1:
groups, motivation
- (notes)
Lecture 2:
examples of groups, including fields
- (notes)
Lecture 3:
uniqueness issues, cyclic groups
- (notes)
Lecture 4:
homomorphisms, isomorphisms, normal subgroups
- (notes)
Lecture 5:
equivalence relations, cosets
- (notes)
Lecture 6:
index, Lagrange's theorem, modular arithmetic, (ℤ/nℤ)*
- (notes)
Lecture 7:
groups of small order, products, quotients, first isomorphism thm
- (notes)
Lecture 8:
group actions, orbits, stabilizers
- (notes)
Lecture 9:
actions on cosets, counting
- (notes)
Lecture 10:
finite rotation groups in 3D
- (notes)
Lecture 11:
Cayley's thm, class equation, p-groups, icosahedral group A₅
- (notes)
Lecture 12:
actions on subsets, Sylow theorems 1, 2, and 3
- (notes)
Lecture 13:
semidirect products, proofs of the Sylow theorems
- (notes)
Lecture 14:
groups of order 12
- (notes)
Lecture 15:
free groups, generators and relations
- (notes)
Lecture 16:
simplicity of Aₙ, composition series, solvable groups, subgroup lattices
- (notes)
Lecture 17:
rings: axioms, examples, homomorphisms
- (notes)
Lecture 18:
ideals, commutative rings, prime & maximal ideals, Chinese remainder theorem
- (notes)
Lecture 19:
PID and UFD (principal ideal domains and unique factorization domains)
- (notes)
Lecture 20:
localization
- (notes)
Lecture 21:
Euclidean domains and Euclidean algorithm
- (notes)
Lecture 22:
modules and exact sequences
- (notes)
Lecture 23:
products, coproducts, direct sums, free modules, presentations
- (notes)
Lecture 24:
rank, submodules of free modules over PID
- (notes)
Lecture 25:
classification of f.g. modules/PID; applications to abelian groups, Jordan form
- (notes)
Lecture 26:
coprimary submodules / PID, Nakayama's lemma
- Attendance is required in lecture, in person. Repeated
failure to attend will adversely affect your Homework score.
- Participation, either actively in class or in office
hours, can benefically affect your Homework score.
- Due dates for the five homework assignments, two
midterms, and one cumulative problem set this semester are
listed in the table below. You will have roughly two weeks to
complete each homework assignment and one week for each exam
(midterm or cumulative problem set).
- Reading assignments are listed in the right-hand column
of the Assignments table. Ideally, reading for each lecture
should be completed before the relevant lecture.
- All assignments, including the midterms and cumulative problem
set, will be take-home. All are due at noon on the due
date unless otherwise announced.
- All solutions you turn in, including midterms, cumulative
problem set, and homework, must be typewritten using the
provided LaTeX template. Communicating your ideas is an
integral part of mathematics. In addition to the usual PDF
files, LaTeX source files for each of the assignments will be
provided. You are required to use these LaTeX templates for
your solutions by filling in your responses in those files. I
am happy to answer any questions you might have about LaTeX,
though you should consider asking your classmates first. The
reason why it is crucial for you to use the provided LaTeX
templates is because they contain margin settings and
commands required for grading. You can add your own
macros, but try to ensure they don't disrupt functionality of
the grading macros.
- Submit your homework, exam, and cumulative solutions using
the "Drop Box" feature on Sakai. Submit your .tex file
(the grader or I may comment on your TeX usage) as well as your
.pdf file, which can serve as verification that your system
produces the same output as ours do and also verification that
you submitted the assignment on time, in case there is a
problem processing your .tex file (which has happened numerous
times in the past).
- Late homework will not be accepted except at the discretion of
the instructor. There are, of course, extenuating
circumstances when that discretion is essentially automatic,
such as if you have been ill and you have filed a
short-term illness incapacitation form.
- If you have technical trouble submitting to Sakai, then
send your files to the instructor by email as a placeholder
while the technical problem is diagnosed and resolved.
- The logic of a proof must be completely clear for full credit.
- If the course has a grader, then the intention is for the
the grader to remain anonymous. If you happen to know the
grader's identity, then it is imperative not to contact the
grader directly.
- Questions about course policies, due dates, grading, and
re-grading must be sent to the instructor. The grader has
no responsibility or power over these issues.
- You must cite sources in your solutions. If you rely on
so-and-so's theorem, then you must say where you found it. Be
specific: "the dual rank theorem" is not precise; in contrast,
"[Climenhaga, Theorem 5.10]" is. Theorems are often known by
many names, so anyone grading your work could fail to recognize
any given theorem by a name you might attach. Note also that
failure to cite properly can sometimes be construed as
plagiarism under the
Duke Community Standard.
- Students are expected to adhere to the
Duke Community Standard. Students affirm
their commitment to uphold the values of the Duke University
community by signing a pledge that states:
- I will not lie, cheat, or steal in my academic endeavors;
- I will conduct myself honorably in all my endeavors;
- I will act if the Standard is compromised.
The instructor has a long record of detecting and convicting
violations, resulting in a wide range of sanctions, from
grade changes to dismissal from the University. Every case
brought by the instructor has resulted in sanctions: I have
never lost a case. Don't test me.
- Collaboration on homework is encouraged while you
discuss the search for solutions, but when it comes time to
write them down, the written work you turn in must be yours
alone: you are not allowed to consult anyone else's
written solution, and you are not allowed to share your written
solutions. (It is easy to tell when solutions have been copied
or written together.) If you collaborate while searching for
solutions,
you must indicate—on the homework page—who your
collaborators were. Failure to identify your collaborators
is a breach of the
Duke Community Standard.
- There is no realistic way the instructor can prevent students
from using an AI tool for homework. However, resorting to an
AI tool before you have thought deeply about a problem robs you
of an opportunity to learn. Repeatedly consulting AI will
substantially lessen your uptake of the course material. If
repeated reliance on AI results in less than perfect solutions,
the instructor reserves the right to deduct more than the
nominal value of the relevant items.
- In contrast to the homework policy, no collaboration or
consultation of human or electronic sources is allowed for
either of the two midterms or the cumulative problem set.
You must work completely independently, without giving or
receiving help from others or from the internet or any other
source except
- your lecture notes and previous assignments,
- any lecture notes or other material provided by the instructor, and
- the "Text" and "Secondary texts" listed above.
It bears repeating that the instructor has a long record of
detecting and convicting violations of the
Duke Community Standard, resulting in a wide range of
sanctions, from grade changes to dismissal from the University.
Again, don't test me.
Check here two weeks before each homework is due, or one week
before each exam is due, for the specifics of the assignments. If
an assignment hasn't been posted, and you think it should have been,
please do email me. Sometimes I encounter problems (such as, for
example, the department's servers going down) while posting
assignments; other times, I might simply have neglected to copy the
assignment into the appropriate directory or to set the permissions
properly. (I do try to check these things, of course, but sometimes
web pages act differently for users inside and outside the Math
department.)