[Deutsche Version]

# Representation Theory of Finite Groups

## Winter Semester 2016/17

### 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).

### Homework sheets

Sheet 1, Sheet 2, Sheet 3, Sheet 4, Sheet 5, Sheet 6, Sheet 7
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