Advanced Abstract Algebra

  • Instructor: Peter HERMANN
  • Contact: hermannzp at gmail dot com
  • Prerequisites: 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 P. Cameron: Introduction to Algebra. You will also need some general experience in abstract mathematics.
  • Text: notes will be posted

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. A brief introduction to character theory of finite groups.

Two sample theorems:
1. If a torsion-free group G has a cyclic subgroup of finite index, then G is cyclic.
2. If the order of G is the product of two prime powers then G is solvable.

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