## 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 -group is solvable. In this thread we prove *Burnside’s Theorem *an amazing generalization which says that every group of order where and 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.

## Review of Group Theory: Solvable Groups

**Point of post: **In this post we introduce the notion of solvable groups and derive some simple facts about them.

*Motivation*

Solvable groups are things that at first may seem useless. They have a somewhat complicated definition that seems almost completely arbitrary–in fact the opposite is true. The notion of solvable groups arose in the study of the solvability by radicals of certain polynomial equations, in particular in Galois theory. Specifically it was discovered that to every polynomial is associated a group (called the Galois group) and moreover it was discovered that the polynomial was solvable if and only if there existed a point after which a certain ‘normal chain of subgroups’ terminated (in the sense that after that point all the groups in the chain are trivial). With such an utterly fascinating result it is clear that the precise definition of this aforementioned condition, sufficient conditions for groups to have this condition, and properties that such groups have is absolutely important. This notion is the notion of solvability.