*Instructor*: Dr. Péter HERMANN

*Text:
*Concerning group theory we shall use D. J.
Robinson, A Course in the Theory of Groups and --- depending on
the pace of the course ---

we might cover also
parts of M. Isaacs, Character Theory of Finite Groups.

*Prerequisite*: A first course in abstract
algebra (i.e. basic notions and results about groups, rings, and
fields,

see e.g. I.N. Herstein: Abstract Algebra, Macmillan
N.Y. 1986 or J. B. Fraleigh: A First Course in Abstract Algebra,

Addison--Wesley 1989),

and an elementary course in linear
algebra. In particular, you should be familiar with the following
concepts and theorems in

group theory: group, subgroup,
order of an element, cyclic group, Lagrange's theorem,
homomorphism, normal subgroup, factor

group, homomorphism and
isomorphism theorems, symmetric and alternating groups, direct
product. These can be found also in

Robinson's book
cited above. You will also need some general experience in
abstract mathematics.

*Course description:* The course will cover some of
the topics listed below, depending on the demand
of the audience.

*Topics*:

I. **Groups**

The course attempts to give an idea of some methods in
elementary group theory, and, depending on how fast we can

proceed, a very basic introduction into the theory of linear
representations. We shall cover topics like the Sylow--theorems,

some properties of *p*--groups, the Schur--Zassenhaus
theorem, the transfer and its applications, and solvable
groups. Two

sample theorems (the second of which
requires representation theory):

1. If all proper
subgroups of a finite group *G* are Abelian, then *G* is
solvable.

2. (Burnside) The order of any non--commutative,
finite, simple group must be divisible by at least three different
primes.

(Equivalently: any group of order p^a q^b (*p*
and *q* primes) is solvable.)

II. **Additional
topics** (supporting representation theory)

Rings and
modules, finitely generated modules, algebraic integers,
cyclotomic fields.