Burnside theorem

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 eachnon-Abelian finite simple group has order divisible by 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 application 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

# Using mathematical induction, it suffices to prove that a simple group "G" whose order has this form is Abelian, so the proof begins by assuming that "G" is simple group of order $p^a\; q^b$, and aims to prove that "G" is Abelian.
# Using Sylow's theorem , "G" either has a non-trivial center, or has a conjugacy class of size $p^r$ for some integer "r" ≥ 1. In the former case, "G" must be Abelian, by its simplicity, so it may be assumed that there is an element "x" of "G" such that the conjugacy class of "x" has size $p^r$ > 1.
# Application of column orthogonality relations and properties of algebraic integers lead to the existence of a non-trivial irreducible character $chi$ of "G" such that $|chi(x)|\; =\; chi(1)$.
# The simplicity of "G" implies that any complex irreducible representation with character $chi$ is faithful, and it follows that "x" is in the center of "G", contrary to the fact that the size of its conjugacy class is greater than "1".

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.