Representation theory: Burnside's theorem
Channel: Richard E Borcherds
Date: 2020-09-21

In this talk we prove Burnside's theorem, that any group whose order is of the form p^aq^b for primes p and q is solvable. 

We first discuss characters of the center of the group ring of G, and use this to show that a certain number related to a character value is an algebraic integer. We use this to show that the dimension of an irreducible complex representation divides the order of the group. We also also use it to show that 
any simple group with a nontrivial conjugacy class of prime power order is cyclic. Burnside's theorem follows easily from this.