|G: P| is equal to the p-sylow subgroups in G, likewise with |H: P|. For some p-Sylow subgroup L H, there is a g such that L=gPg-1 gHg-1=H. We know that's true because there is some theorem that says that if a subgroup contains the normalizer of a p-sylow subgroup it is self normalizing.

I'm stuck on pulling the G:H ≡1 mod p

This is refining my first post