Ironing out the Lucas Primality Test proof

As I was typing this out I realized I had more questions than I thought...

Theorem

Let , and suppose that for every prime factor of there is an integer such that,

but,

Then is prime.

**Proof**

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.