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
we can actually prove a better result, which is: then you even don't need to define an isomorphism. here's how to prove the equality:
first if with then which will quickly give us:
conversely, suppose then as i showed in my previous post for some thus: and hence: