Burnside 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 three distinct primes. Furthermore, as a consequence of the Feit-Thompson theorem, one of those can be chosen to be 2.

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 $|\chi(x)| = \chi(1)$.
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.