I was asked to prove that given a finite group G, a normal group N in G, and a p-sylow supgroup P in G, then

where is the normalizer of P in G (same with )

I tried to solve it using the homomorphism

defined by

I can show that and also that .

all I need now is to show that in order to use the first isomorphism theorem, but for some reason I can manage it.

any help would be appreciated