Hello,

I'd like to alert you to Math 457 - Abstract Algebra,
this fall, MWF 09:15 - 10:05.

The main topic of this course will be groups.  A group
is essentially a description of the symmetries of an
object.  Since symmetries occur in a fundamental way
in the universe, group theory is of vital importance
in geometry, knot theory, crystallography and particle
physics, to name just a few applications.

Slightly more formally, a group is a set together with
a binary operation, namely a way of "multiplying"
elements of the set together.  The operation of a group
has to satisfy a small number of "sanity" conditions;
for example, if a, b, c are elements of a group we
require (ab)c = a(bc).

The main prerequisite for this course is an interest
in this topic, and a desire to understand the various
kinds of groups.  In order to put things on a firm
footing, there will inevitably be some theorems and
proofs.

Here are a couple of questions that a beginning group
theorist might ask.  In how many ways can I map a cube
into itself by a rotation or reflection?  How many
arrangements are there of a Rubik Cube, and how do
these arrangements interact with one another?

Group theory is also fascinating in its own right.
An example of an unanswered question is: How many
different groups are there with n elements?  In
general this is a hard question, unsolved to this day
even for n = 2^11 = 2048.

Towards the end of the semester we'll examine
other structures called rings and fields.  You'll
already have come across examples of these, as the
integers form a ring, as does the set of all 2x2
matrices; also the set of rational numbers is an example
of a field, as is the set of real numbers.

Further study of rings and fields will take place in
the follow-up course Math 458.  There we'll take a look
at Galois Theory, which explains why we have formulae
for solving quadratic equations (also polynomial
equations of degree 3, 4), involving expressions in
the coefficients containing at worst roots, whereas
it's impossible to have such a formula for solving
equations of degree 5.

By all means feel free to email me with any questions
on this course, at  [log in to unmask]

Morwen Thistlethwaite

This e-mail list is intended for those who have an interest in mathematics at the University of Tennessee. All current UT math majors and minors should be on this list in order not to miss important announcements. To add or remove yourself from the list, visit listserv.utk.edu.