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 G is cyclic of prime order or is not simple).

[edit] History

The theorem was stated by William Burnside.

Burnside's theorem is a well-known application of representation theory to the theory of finite groups because it wasn't until after 1970 that a proof that was not based on representation theory was given.

[edit] Outline of the proof

1. It turns out that a group of order p^aq^b is either easily decomposable with Sylow theory, has an easily recognizable non-trivial center, or has a conjugacy class of order p^r for some integer r ≥ 1.
2. If we have a conjugacy class of order p^r, then there is a representation ρ of G that either has a proper non-trivial kernel or is faithful, in which case it will follow that the center of G is non-trivial.
3. To build the representation ρ given a conjugacy class g^G with representative g and with order p^r, we apply the column orthogonality relations to the character table of G to get an equality which we fiddle with algebraically to demonstrate the existence of an irreducible character χ_i of G such that χ_i(g) / χ_i(1) is not an algebraic integer. We subsequently find that χ_i(g) / χ_i(1) is coprime to p^r.
4. We attack from a different angle by showing, with an extensive bit of further algebraic manipulation and representation theory, that the class sum of the conjugacy class g^G in the group algebra is equal (in its action on the group algebra) to an algebraic integer λ.
5. Substituting λ back into previous work and applying a little bit more representation theory demonstrates that if ρ is a representation of G with character χ_i then either the kernel of ρ is a proper non-trivial normal subgroup of G or ρ is faithful in which case g, the representative of the conjugacy class that we started with, is in the center of G, which is therefore non-trivial. In either case G is not simple.

[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.