A note on Burnside's Theorem

In the proof of Burnside's Theorem in the book (Theorem 4.3), I use Frobenius' Theorem (which, as remarked on Page 36, uses character theory). Professor Ram Abhyankar pointed out to me that this theorem occurs already in the first edition of Burnside's book (and hence must have a proof not using character theory!) Here is the proof.

Following the argument given, we reach the case that the minimal normal subgroup N of G is a Frobenius group. Now elementary counting shows that it contains n-1 fixed-point-free elements, where n is the degree. So there are n(n-1) triples (alpha,beta,g), where g is f.p.f. and maps alpha to beta. But there are just n(n-1) pairs (alpha,beta), permuted transitively by G. We conclude:

all f.p.f. elements of N are conjugate in G;
for any two distinct points alpha,beta, there is a unique f.p.f. element g which takes alpha to beta.

Now let p be a prime dividing n, and P a Sylow p-subgroup of N. Then P contains a f.p.f. element of order p. So all f.p.f. elements of N have order p, and n is a power of p. Then it follows that P is transitive, and so consists of the identity and all the f.p.f. elements; so P is a regular normal subgroup. This concludes the proof.

The fact that a finite 2-transitive group has a unique minimal normal subgroup is more elementary. By Theorem 4.4, if a primitive group has more than one minimal normal subgroup, then it has just two, and each is the centraliser of the other, so they are non-abelian and regular. But a regular normal subgroup of a finite 2-transitive group is abelian, by Theorem 1.6.

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Peter J. Cameron
2 January 2001.