# Burnside theorem

Jump to navigation
Jump to search

{{#invoke:Hatnote|hatnote}}
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.

## 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

- 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.

## References

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