Burnside's 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 at least three distinct primes.

History

The theorem was proved by William Burnside (1904).

Burnside's theorem has long been one of the best-known applications of representation theory to the theory of finite groups. John Thompson pointed out that a proof avoiding the use of group characters could be extracted from his work, and this was done explicitly by Goldschmidt (1970) for groups of odd order, and by Bender (1972) for groups of even order. Matsuyama (1973) simplified the proofs.

Outline of Burnside's proof

1. If Χ is an irreducible complex character of any finite group G then |K|Χ(k)/Χ(1) is an algebraic integer, where k is in a conjugacy class K.
2. Use step 1 to show that if Χ(1) and |K| are coprime then Χ(k) is either 0 or has absolute value Χ(1).
3. Use step 2 to show that if a finite group has a conjugacy class of size pⁿ for some prime p then the group is not simple.
4. Using the class equation, a group G of order p^aq^b (b>0) has a non-identity conjugacy class of size prime to q. Hence G has a nontrivial conjugacy class of size $p^{r}$ for some integer r so is not simple by the previous step.
5. Induction on the order of G then shows that as no group of such order can be simple, any group of such an order must be solvable.

References

• Bender, Helmut (1972), "A group theoretic proof of Burnside's p^aq^b-theorem.", Math. Z., 126: 327–338, MR 0322048
• Burnside, W. (1904), "On Groups of Order p^αq^β", Proc. London Math. Soc. (s2-1 (1)): 388–392, doi:10.1112/plms/s2-1.1.388
• Goldschmidt, David M. (1970), "A group theoretic proof of the p^aq^b theorem for odd primes", Math. Z., 113: 373–375, MR 0276338
• James, Gordon; and Liebeck, Martin (2001). Representations and Characters of Groups (2nd ed.). Cambridge University Press. ISBN 0-521-00392-X. See chapter 31.
• Matsuyama, Hiroshi (1973), "Solvability of groups of order 2^aq^b.", Osaka J. Math., 10: 375–378, MR 0323890