As I was typing this out I realized I had more questions than I thought...
Let , and suppose that for every prime factor of there is an integer such that,
Then is prime.
We want to show that = which implies that is prime.
Consider the group where .
We look at the subgroup of generated by all 's. (Does this simply mean all the values that divide )
As (My notes are unclear here so correct me if that part is wrong, I suspect it is), the exponent of divides (What exponent?).
But the exponent can't be a proper factor of as,
(Should be there? I don't fully understand that conclusion).
So the exponent = .
But then exponent|#H and so .
As , . Q.E.D.
Lines 4-7 of the proof is what I'm not sure about.