Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Let $u$ be a permutation endomorphism of $E$. Show that $u$ is diagonalizable and that its trace belongs to $\llbracket 0, n \rrbracket$.
Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Let $u$ be a permutation endomorphism of $E$. Show that $u$ is diagonalizable and that its trace belongs to $\llbracket 0, n \rrbracket$.