One of the most famous applications of representation theory is Burnside’s Theorem, which states that if p and q are prime numbers and a and b are positive integers, then no group of order paqb is simple. In the first edition of his book Theory of groups of finite order (1897), Burnside presented group-theoretic arguments which proved the theorem for many special choices of the integers a, b, but it was only after studying Frobenius’s new theory of group representations that he was able to prove the theorem in general. Indeed, many later attempts to find a proof which does not use representation theory were unsuccessful, until H. Bender found one in 1972.
A preliminary lemma
We prepare for the proof of Burnside’s Theorem ...