MAT 312/AMS 351 Course Webpage
Announcements about the course will be posted here. Please check the
site regularly for announcements (which will also be given in lecture
and/or in recitation). There are currently no announcements.
he final exam will be held on WEDNESDAY, DECEMBER 14,
5:30PM8PM. The room of the exam is Library W4525.
The final exam is worth 30% of
the course points.
E-mail announcements for this course will be sent through Blackboard.
If you are new to Blackboard, or if you have not used Blackboard in
some time, please log on to Blackboard and review the e-mail
Please follow the instructions for e-mail forwarding if you
prefer an e-mail address different from
your official Stony Brook University e-mail address.
There is a single document with all course policies and important dates (pdf, doc).
The description in the undergraduate bulletin:
Topics in algebra: groups, informal set theory,
relations, homomorphisms. Applications:
error correcting codes, Burnside's theorem,
computational complexity, Chinese remainder
theorem. This course is offered as both AMS
351 and MAT 312.
C or higher in AMS 210 or MAT 211.
Advisory Prerequisite: MAT 200 or CSE 113.
Text The required
textbook for the course is
Numbers, Groups and Codes, 2nd Edition, by J. F. Humphreys and
M. Y. Prest, available at the University Bookstore @ Stony Brook.
In addition to the required textbook above, for part of the course there
may be additional course notes which will be made available through
the Blackboard page.
The course learning
objectives include the following. Each of these is an important
learning objective for all advanced mathematics and applied
mathematics courses. Each is amplified with specific examples.
The course outcomes / key skills include the following.
- Acclimate to New Mathematics. Gain familiarity with a new
mathematical idea (be it a definition, a result, an algorithm, etc.)
through examples, through basic results that involve that idea, and
through deeper results that reflect the significance of the
idea. Example. Cyclic groups and permutation groups are
examples of abstract groups. Lagrange's Theorem relates orders of
elements in a group to the order of the group. Burnside's Theorem
relates the number of orbits of an action of a group to the
cardinalities of the fixed sets of group elements.
- Apply and Model. Understand how an abstract notion or
result can lead to an algorithm or computation arising in a context
different from the original notion or result. Understand the
necessary hypotheses and limitations of that model. Example.
Euler's Totient Theorem leads to a Public Key encryption scheme.
The most common version of that scheme requires as input an integer
that is product of two distinct primes. The security of the scheme
depends on the computational difficulty of discovering those two
primes from the given integer (which is sometimes quite easy for
poor choices of the integer).
- Specialize. Pass from general theorems, definitions, and
methods to specific examples. Be able to compute with those
examples. Example. The Chinese Remainder Theorem allows to
compute the congruence class of an integer modulo a composite from
the congruence classes of that integer modulo factors of the
composite. Specialize this to find the least nonnegative integer
whose congruence class modulo 17 equals 3 and whose congruence class
modulo 7 equals 6.
- Generalize. Understand examples of ideas, constructions and
arguments originally developed in one context yet that extend to
another context. Example. The arithmetic theory of the
division algorithm, unique factorization, the Chinese Remainder
Theorem, etc., also applies to polynomial with real coefficients.
- Prove. For a conjectured result, often expected from
examples, heuristics and other indirect evidence, rigorously prove
the result using techniques such as proof by induction, proof by
contradiction, proof by cases, and more advanced proof
techniques. Example. Prove that every finite group of even
order contains an element of order 2 (one proof uses Burnside's
- Understand the division algorithm, particularly uniqueness of the
quotient and the remainder.
- Understand the definition of the greatest common divisor of two
integers, and be
able to express the greatest common divisor as an integer linear
combination of the two input integers using repeated
application of the division algorithm (the Euclidean algorithm).
- Understand how to use recursion to define a mathematical object
with dependence on a positive integer, such as factorials and
- Understand how to use induction to verify for all positive
integers a proposition that depends on a positive integer, such as
the Binomial Theorem.
- Understand prime and irreducible integers, and understand the
relation between these.
Understand unique factorization of integers
and the Fundamental Theorem of Arithmetic.
Be prepared to factor any specified integer less than 1000.
- For a specified positive integer n, understand the arithmetic
system of congruence classes modulo n, i.e., modular arithmetic.
Understand what it means for a congruence class to be invertible.
Understand the special properties of the arithmetic system of
congruence classes modulo a prime integer p.
- Know a necessary and sufficient condition for solving a single
linear congruence. Understand the Chinese Remainder Theorem that
reduces the solution of a linear congruence modulo a composite to
simultaneous solutions of linear congruences modulo factors of the
- Understand the totient function of Euler. Be able to compute the
totient function for powers of primes. Reduce computation of the
totient function for all integers to computation for powers of
- Understand the statements and proofs of both Fermat's Little
Theorem and Euler's Theorem. Use this to simplify exponentiation in
- Understand the basic Public Key encryption scheme. Understand
what are the inputs of this scheme, and what are the challenges in
implementing this scheme.
- Understand permutations of a finite set. Understand the identity
permutation, understand the (non-commutative) composition
of permutations, and understand inverses of permutations.
Understand both "two-row" and disjoint cycle notation for
- Understand exponentiation of a single permutation. Understand
the order of a permutation. Know how to compute the order of a
permutation quickly from its disjoint cycle notation.
- Know what is a transposition.
Understand the definition of the sign of a permutation. Know
identities involving the sign. Know methods for computing the
- Understand the group of permutations of a fixed finite set.
Understand what is a subgroup, particularly in the context of the
group of permutations of a fixed finite set.
- Know other examples of groups, such as the (non-commutative)
group of invertible
2 by 2 matrices. Understand the special properties of the
determinant with respect to the group operations on this group.
- Understand how to iteratively take a product of many copies of a
group element with respect to the group composition, i.e., group
exponentiation. Understand the meaning of order of a group
- Understand the notion of subgroup of a group. Understand the
meaning of order of a subgroup. Understand the cyclic subgroup
generated by an element and the relation between the order of the
element and the order of the cyclic subgroup. Know that the
intersection of subgroups is again a subgroup.
- For a specified subgroup of a group, understand what are left
cosets, respectively right cosets. Understand why the left cosets,
resp. right cosets,
form a partition of the group. Know the associated coset space.
Understand Lagrange's Theorem and
the notion of index of a subgroup of a group.
- Understand homomorphisms between specified groups. Know when a
homomorphism is an isomorphism of groups. Understand the direct
product of two specified groups.
- Understand when a product of cyclic groups is again a cyclic
group. Know unique factorization of cyclic groups. Understand some
simple results using counting of elements of specified orders to
characterize certain groups, i.e., every finite group of prime order
- Know what are binary codes. Understand the basic scheme of error
detection in binary codes. Know about word distance in binary
codes, and the relation of distance to error corrections.
- Understand generator matrices and parity-check matrices. Using
these, be able to compute the minimal distance between words.
- Understand additional, subtraction, scaling, and product for
polynomials in one variable with real coefficients.
- For polynomials, understand the division algorithm. Understand
the notion of greatest common divisor of two nonconstant
polynomials. Understand the Euclidean algorithm for polynomials.
- Understand prime and irreducible polynomials. Understand the
unique factorization theorem for polynomials of one variable with
- Know the Fundamental Theorem of Algebra: every polynomial of
positive degree has at least one complex zero.
- Understand polynomial congruences. Understand how the coset
space for a polynomial is a real vector space with a distinguished
real linear self-map. Understand how this defines a commutative
product operation on the coset space.
- Know what is a field. Understand when the coset space for a
polynomial is a field.
- Know how to compute arithmetic in a field arising as the coset
space of a polynomial.
The instructor for this course is
There are assigned
readings in the syllabus
which are to be completed before lecture. During lecture the
instructor and the students will discuss the material in those
readings, there will be exercises to practice the material, etc. For
the lectures to be effective, you must complete the assigned reading
from the syllabus before lecture.
Lecture is held Tuesdays and Thursdays, 11:30AM 12:50PM, in
Math Tower P131.
Please register for and regularly attend one of the
Your recitation instructor is the instructor who knows you best and who
answers any questions about grading on problem sets. Your
recitation instructor will have input in the assignment of final
letter grades. The recitation instructor is Harrison Pugh.
- Recitation 1 meets on Tuesdays, 1
1:53PM, in Frey Hall 224.
- Recitation 2 meets on Wednesdays, 11
11:53 AM, in Frey Hall 224.
Office hours for Jason
are scheduled as follows.
- Tuesdays 10 11AM,
Math Tower P143 (advising).
- Thursdays 10 11AM, Math Tower
- Thursdays 1:15 2:15PM, Math
Office hours for the recitation instructor
Harrison Pugh are scheduled as follows.
- Tuesdays 2 3PM,
Math Tower S240A (MLC).
- Wednesdays 2 3PM, Math Tower
- Wednesdays 10 11AM, Math
Tower S240A (MLC).
The relative significance of exams and problem sets in determining
final grades is as follows.
Graded problem sets and exams will be handed back in
you cannot attend the recitation in which a problem set or exam is
handed back, it is your responsibility to contact the instructor and
arrange a time to pick up the work (typically in office hours).
Students are responsible for collecting any graded work by the end of the
There are a number of organizations on campus offering tutoring and
other academic resources
in various locations. One such organization is the Academic
Success and Tutoring Center.
The mathematics department offers
drop-in tutoring in the Math Learning Center. You are
strongly encouraged to talk to a tutor in the MLC if you have an issue
and are unable to attend your lecturer's
Please be aware that tutors in the MLC deal with students on a
first-come, first-served basis.
Thus it is usually preferrable to speak
with your instructor in their office hours. (Even if you
find your instructor
in the MLC, the instructor may be obliged to speak to other students
before speaking with you.)
Required Syllabi Statements
The University Senate Undergraduate and Graduate Councils have
authorized that the following required statements appear in all
teaching syllabi (graduate and undergraduate courses) on the Stony
Americans with Disabilities
If you have a physical, psychological, medical or learning disability
that may impact your course work, please contact Disability Support
Services, ECC (Educational Communications Center) Building, Room 128,
(631) 632-6748. They will determine with you what accommodations, if
any, are necessary and appropriate. All information and documentation
Students who require assistance during emergency evacuation are
encouraged to discuss their needs with their professors and Disability
Support Services. For procedures and information go to the following
Each student must pursue his or her academic goals
honestly and be personally accountable for all submitted work. Representing
another person's work as your own is always wrong. Faculty is required to
report any suspected instances of academic dishonesty to the Academic
Judiciary. Faculty in the Health Sciences Center (School of Health
Technology Management, Nursing, Social Welfare, Dental Medicine) and School
of Medicine are required to follow their school-specific procedures. For
more comprehensive information on academic integrity, including categories
of academic dishonesty please refer to the academic judiciary website
Critical Incident Management
Stony Brook University expects students to
respect the rights, privileges, and property of other people. Faculty are
required to report to the Office of University Community Standards any
disruptive behavior that interrupts their ability to teach, compromises the
safety of the learning environment, or inhibits students' ability to learn.
Faculty in the HSC Schools and the School of Medicine are required to follow
their school-specific procedures.
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4-108 Math Tower
Department of Mathematics
Stony Brook University
Stony Brook, NY 11794-3651