Representation Theory of Finite Groups
Winter Semester 2016/17
Class: Tue, Wed 10:15-11:45 in SE 40
Exercises: Mon 16:15-17:45 in SE 40
Representation theory uses methods from linear algebra and number
theory to study finite groups. Let G be a group and V be
a finite-dimensional vector space. A representation is a
group homomorphism G→GL(V).
Representations arise naturally in finite groups (often in a geometric
context), or can be used as external tools to prove theorems about
abstract finite groups. Numerical invariants of the irreducible
representations sometimes allow to study complicated and huge groups.
Some classical examples, which we cover among other things, are
- Burnside's paqb-Theorem: A
finite group whose order has only two different prime divisors is solvable.
- Frobenius' Theorem about Frobenius groups: If every
element (except for the identity) of a finite transitive permutation
has at most one fixed point, then the set of the fixed point free
elements, together with the identity, is a subgroup. For this we
present a recent new proof by Knapp and Schmid which hasn't yet
entered the textbooks.
- Brauer's Theorem about fields of definition:
Let G be a finite subgroup
of GLn(ℂ). Let m be the exponent
of G and K be the m-th cyclotomic field. Then
G is conjugate to a subgroup of GLn(K).
- M.J. Collins: Representations and Characters of Finite Groups,
1990, Cambr. Univ. Press.
- M. Geck:
Algebra: Gruppen, Ringe, Körper; mit einer Einführung in die
Darstellungstheorie endlicher Gruppen, edition delkhofen.
- B. Huppert: Character Theory of Finite Groups, 1998, de Gruyter.
- I. M. Isaacs: Character Theory of Finite Groups, 1976, Academic
- G. James, M. Liebeck: Representations and Characters of Groups,
2001, Cambr. Univ. Press.
- P. Müller: Darstellungstheorie endlicher
Gruppen, unvollständiges Skript.
- J.-P. Serre: Linear Representations of Finite Groups, 1977,
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