Burnside theorem

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

p^aq^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.

[edit] 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.

[edit] Outline of Burnside's proof

1. 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^aq^b, and aims to prove that G is Abelian.
2. 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.
3. Application of column orthogonality relations and properties of algebraic integers lead to the existence of a non-trivial irreducible character χ of G such that | χ(x) | = χ(1).
4. The simplicity of G implies that any complex irreducible representation with character χ 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.

[edit] 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.