Burnside theorem
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In mathematics, Burnside's theorem in group theory states that if G is a finite group of order
- paqb
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
- It turns out that a group of order paqb is either easily decomposable with Sylow theory, has an easily recognizable non-trivial center, or has a conjugacy class of order pr for some integer r ≥ 1.
- If we have a conjugacy class of order pr, 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.
- To build the representation ρ given a conjugacy class gG with representative g and with order pr, 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 χi of G such that χi(g) / χi(1) is not an algebraic integer. We subsequently find that χi(g) / χi(1) is coprime to pr.
- 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 gG in the group algebra 
is equal (in its action on the group algebra) to an algebraic integer λ. - 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 χi 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
- 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.
