Advanced Abstract Algebra  --- ALG2

Instructor: Dr. Péter HERMANN

Text: We shall use D. J. Robinson, A Course in the Theory of 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: The course attempts to give an idea of some basic methods in finite and infinite group theory. We shall cover topics like permutation actions, the Sylow-theorems , finite permutation groups, some properties of p-groups , the Schur - Zassenhaus theorem, the transfer and its applications, solvable groups, nilpotent groups, free groups.

Two sample theorems:
1. If a torsion-free group G has a cyclic subgroup of finite index, then G is cyclic.
2. (P. Hall) A finite group is solvable iff it has p-complements for all primes p.