Burnside's theorem

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 William Burnside (1904) using the representation theory of finite groups.

Several special cases of the theorem had previously been proved by Burnside in 1897, Jordan in 1898, and Frobenius in 1902.

John G. Thompson pointed out that a proof avoiding the use of representation theory could be extracted from his work in the 1960s and 1970s on the N-group theorem, and this was done explicitly by Goldschmidt (1970) for groups of odd order, and by Bender (1972) for groups of even order.

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.

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.

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).

Moreover, the integer d is strictly positive, since g is nontrivial and therefore not central in G. Let (χi)1 ≤ i ≤ h be the family of irreducible characters of G over

Because g is not in the same conjugacy class as 1, the orthogonality relation for the columns of the group's character table gives: Now the χi(g) are algebraic integers, because they are sums of roots of unity.

which sends a class function f to is a ring homomorphism.

Its trace nλ is equal to Because the homothety λIn is the homomorphic image of an integral element, this proves that the complex number λ = qdχ(g)/n is an algebraic integer.

It is an algebraic integer, so its norm N(ζ) (i.e. the product of its conjugates, that is the roots of its minimal polynomial over

Now ζ is the average of roots of unity (the eigenvalues of ρ(g)), hence so are its conjugates, so they all have an absolute value less than or equal to 1.

Because the absolute value of their product N(ζ) is greater than or equal to 1, their absolute value must all be 1, in particular ζ, which means that the eigenvalues of ρ(g) are all equal, so ρ(g) is a homothety.