The theorem was proved by William Burnside in the early years of the 20th century.
Burnside's theorem has long been one of the best-known applications of representation theory to the theory of finite groups, though a proof avoiding the use of group characters was published by D. Goldschmidt around 1970.
Outline of Burnside's proof
- By induction, it suffices to prove that a finite simple group G whose order has the form for primes p and q is cyclic. Suppose then that the order of G has this form, but G is not cyclic. Suppose for definiteness that b > 0.
- Using the modified class equation, G has a non-identity conjugacy class of size prime to q. Hence G either has a non-trivial center, or has a conjugacy class of size for some positive integer r. The first possibility is excluded since G is assumed simple, but not cyclic. Hence there is a non-central element x of G such that the conjugacy class of x has size .
- Application of column orthogonality relations and other properties of group characters and algebraic integers lead to the existence of a non-trivial irreducible character of G such that .
- The simplicity of G then implies that any non-trivial complex irreducible representation is faithful, and it follows that x is in the center of G, a contradiction.