Exercise 3.1.30 ($gN = Ng \iff g\in N_G(N)$)

Question

Let $N \leq G$ and let $g \in G$. Prove that $gN = Ng$ if and only if $g\in N_G(N)$.

Richard Ganaye · Accepted Answer

\begin{proof} 
By definition of the normalizer
\begin{align*}
g \in N_G(N) & \iff gNg^{-1} = N\
&\iff gN = Ng.
\end{align*}
What else?
\end{proof}

Answers