# Abstract Nonsense

## Burnside’s Theorem

Point of post: In this post we put together a lot of our rep theory to prove one of the fundamental (pure) group theoretic results amenable to the subject.

Motivation

In this post we finally use representation theory to prove something in pure group theory that is near impossible to do without representation theory. We have seen on our thread about solvable groups that every $p$-group is solvable. In this thread we prove Burnside’s Theorem an amazing generalization which says that every group of order $p^aq^b$ where $p$ and $q$ are primes. As a corollary we will be able to conclude that every non-abelian simple group is divisible by three distinct primes which, of course, will eliminate a respectable amount of group orders for analyzing simplicity. This is really one of the most beautiful applications of representation theory.