Weyl group, permutation group

Question

Let $U(n)$ be the unitary group and $T$ its maximal torus (group of diagonal matrix) and $N(T)$ the normalizer of $T$ in $G$. Why $N(T)/T$ is the permutation group $S_{n}$?

Answer 1

Hint: To compute $N(T)$ easily, note that if $D$ is a diagonal matrix, and $U$ is a matrix such that $U^{-1}DU$ is also diagonal, then $U$ must take eigenvectors of $D$ to other eigenvectors of $D$. (Remember that conjugation = change of basis.)