Finite groups and noncommutative algebra
Summary
This course is intended to serve as a gentle introduction to noncommutative algebra. First we cover the basic results of groups, group actions, and representations of groups, and use them to prove Sylow's theorems. Then we pave the way for the Artin-Wedderburn theorem and study Henderson's short proof. We introduce group algebras and study some basic results of finite dimensional complex linear representations of a finite group such as Maschke's theorem. We shall apply the Artin-Wedderburn theorem to obtain the classical formula for the sum of the squares of the dimensions of the irreducible representations of a finite group G.
Compact topological groups are in many respects similar to finite groups. We shall examine the simplest examples of connected compact groups and their linear representations, and we generalise to compact linear groups the main theorems proven for finite groups.
Artistic depiction of Frobenius and the first page of his Über Gruppencharaktere, published in 1896.