Let $\displaystyle G$ be a group and let $\displaystyle N$ be a subgroup of $\displaystyle G$. Using the definitions of the sets $\displaystyle gN$, $\displaystyle Ng$, and $\displaystyle gNg^{-1}$, prove that $\displaystyle Ng = gN$ for every $\displaystyle g$ in $\displaystyle G$ if and only if $\displaystyle G = N_G(N)$.

$\displaystyle N_G(N)$ = {$\displaystyle g$ in $\displaystyle G$ | $\displaystyle gNg^{-1} = N$}