*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.