Work with the class equation. It involves some work, but it's not dificult.
Let & be primes with and suppose is a group of order . Suppose that has unique subgroup of order . Prove that is abelian stating any theorems you use...
I'm thinking I have to use some form of Sylows theorem to show that is a direct sum of p-groups and hence Abelian but no idea how (or even if that is the tactic)!!
Any help appreciated...
Hmmm, I've had a think and come up with this but it isn't really using the class equation I don't think!
Using Sylow III I can certainly see that as and and as
Also from Sylow II we know that as then and has order thus Abelian. The intersection or and is so is the direct product of the two abelian groups hence G is Abelian