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# Burnside's theorem

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### Burnside's theorem

For the counting result sometimes called "Burnside's theorem", see Burnside's lemma.

In mathematics, Burnside's theorem in group theory states that if G is a finite group of order

$p^a q^b\$

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.

## History

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

1. By induction, it suffices to prove that a finite simple group G whose order has the form $p^a q^b$ 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.
2. 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 $p^r$ 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 $p^r$.
3. Application of column orthogonality relations and other properties of group characters and algebraic integers lead to the existence of a non-trivial irreducible character $\chi$ of G such that $|\chi\left(x\right)| = \chi\left(1\right)$.
4. 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.

## References

1. James, Gordon; and Liebeck, Martin (2001). Representations and Characters of Groups (2nd ed.). Cambridge University Press. ISBN 0-521-00392-X. See chapter 31.
2. Fraleigh, John B. (2002) A First Course in Abstract Algebra (7th ed.). Addison Wesley. ISBN 0-201-33596-4.
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