Advanced Abstract Algebra  --- ALG2

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.

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.