Burnside's theorem

In mathematics, Burnside's theorem in group theory states that if G is a finite group of order where p and q are prime numbers, and a and b are non-negative integers, then G is solvable. Hence each non-Abelian finite simple group has order divisible by at least three distinct primes. The theorem was proved by using the representation theory of finite groups. Several special cases of the theorem had previously been proved by Burnside, Jordan, and Frobenius. John Thompson pointed out that a proof avoiding the use of representation theory could be extracted from his work on the N-group theorem, and this was done explicitly by for groups of odd order, and by for groups of even order. simplified the proofs. The following proof — using more background than Burnside's — is by contradiction. Let paqb be the smallest product of two prime powers, such that there is a non-solvable group G whose order is equal to this number. G is a simple group with trivial center and a is not zero. If G had a nontrivial proper normal subgroup H, then (because of the minimality of G), H and G/H would be solvable, so G as well, which would contradict our assumption. So G is simple. If a were zero, G would be a finite q-group, hence nilpotent, and therefore solvable. Similarly, G cannot be abelian, otherwise it would be solvable. As G is simple, its center must therefore be trivial. There is an element g of G which has qd conjugates, for some d > 0. By the first statement of Sylow's theorem, G has a subgroup S of order pa. Because S is a nontrivial p-group, its center Z(S) is nontrivial. Fix a nontrivial element . The number of conjugates of g is equal to the index of its stabilizer subgroup Gg, which divides the index qb of S (because S is a subgroup of Gg). Hence this number is of the form qd. Moreover, the integer d is strictly positive, since g is nontrivial and therefore not central in G. There exists a nontrivial irreducible representation ρ with character χ, such that its dimension n is not divisible by q and the complex number χ(g) is not zero.