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Burnside’s Theorem

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 *p*^{a}*q*^{b} 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 ...