Stuck on the following question;
$\displaystyle i \Rightarrow ii$
$\displaystyle Ng=gN \Rightarrow g^{-1}Ng=N$ (multiply on the left by g inverse)
In particular this means
$\displaystyle g^{-1}Ng \subset N$
that is precisely the statement of ii
$\displaystyle ii \Rightarrow i$
Assume the hypothesis. For any g, it is clear that we have $\displaystyle g^{-1}Ng \subset N \Rightarrow Ng \subset gN$.
But then we also have
$\displaystyle (g^{-1})^{-1}Ng^{-1}\subset N \Rightarrow gNg^{-1} \subset N \Rightarrow gN \subset Ng$
Thus $\displaystyle gN=Ng$ as desired.