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**Prometheus** 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

$\displaystyle (N_{G}(P)N)/{N}\cong N_{G/N}(PN/N)

$ where $\displaystyle N_{G} (P)$ is the normalizer of P in G (same with $\displaystyle N_{G/N}(PN/N)$ )

I tried to solve it using the homomorphism

$\displaystyle \varphi:N_{G}(P)N\rightarrow G/N$ defined by $\displaystyle \varphi(gn)=gnN$

note that $\displaystyle \color{red}gnN=gN$ because $\displaystyle \color{red}n \in N.$ so we define $\displaystyle \color{red}\varphi(gn)=gN.$ you also need to show that $\displaystyle \color{red}\varphi$ is well-defined and a homomorphism.

I can show that $\displaystyle Im(\varphi)\subseteq N_{G/N}(PN/N)$ and also that $\displaystyle ker(\varphi)=N $.

all I need now is to show that $\displaystyle N_{G/N}(PN/N)\subseteq Im(\varphi)$ in order to use the first isomorphism theorem, but for some reason I can manage it.