Course description  for MATH 517A, Group Theory, Fall 2004.

Numerous  problems  of  the  natural and  technical  sciences  possess
certain symmetry properties.  If we manage to understand  them, then a
mathematical treatment adjusted to the symmetry properties lead to
a considerable simplification.

In this  course on  group theory,  the main emphasis  will be  on this
connection. We will develop the  basics of group theory and then apply
them to study a variety  of applications. The most important tool that
allows  us to  study symmetries  via group  theory is  covered  by the
theory   of  realizations   of   groups  as   matrices,  also   called
representation theory of groups.

We will therefore start  with studying representation theory using the
book ``Groups and characters`` by L. Grove.  Moreover, we will look at
applications  from  A.  Terras  book  on  Fourier  analysis on  finite
groups. The  examples discussed there cover applications  of groups to
graph theory,  coding theory,  statistics and probability  theory.  If
time permits we are also going to look at problems from combinatorics,
algebraic topology, number theory and other areas.  Since examples are
an  important   part  of  any   mathematical  theory,  I   will  offer
introductions on how  to use various computer algebra  systems such as
GAP, Maple, and others, to study symmetries in concrete situations.


The  main prerequisite  for this  course  is the  core algebra  course
511A/B.