TITLE: Fun applications of representations of finite groups
QUESTION [65 upvotes]: Are there some fun applications of the theory of representations of finite groups? I would like to have some examples that could be explained to a student who knows what is a finite group but does not know much about what is a repersentation (say knows the definition). The standard application that is usually mentioned is Burnside's theorem http://en.wikipedia.org/wiki/Burnside_theorem.  The application may be of any kind, not necessarely in math. But math applications are of course very wellcome too!!! It will be very helpfull also if you desribe a bit this application.

REPLY [9 votes]: Let $G$ be a finite group with $n$ elements and $k$ conjugacy classes. Denote by $m=|G:[G,G]|$ the index of the commutator. Then $n+3m\geq 4k$. 
It is less impressive than many other answers, but I find this inequality particularly nice, especially having in mind that there are some nontrivial examples of equality, all are explicitly listed. I do not know the proof whithout using representations.