In this subsection, $E$ is a $\mathbb{C}$-vector space of dimension $n \geq 1$. We say that an endomorphism $u$ of $E$ is a permutation endomorphism if there exists a basis $(e_1, \ldots, e_n)$ of $E$ and a permutation $\sigma \in \mathfrak{S}_n$ such that $u(e_j) = e_{\sigma(j)}$ for all $j \in \llbracket 1, n \rrbracket$. Study whether the equivalence of the previous question holds when we replace the hypothesis $u^2 = \operatorname{Id}_E$ by $u^k = \operatorname{Id}_E$ for $k = 3$, then for $k = 4$.
In this subsection, $E$ is a $\mathbb{C}$-vector space of dimension $n \geq 1$. We say that an endomorphism $u$ of $E$ is a permutation endomorphism if there exists a basis $(e_1, \ldots, e_n)$ of $E$ and a permutation $\sigma \in \mathfrak{S}_n$ such that $u(e_j) = e_{\sigma(j)}$ for all $j \in \llbracket 1, n \rrbracket$.
Study whether the equivalence of the previous question holds when we replace the hypothesis $u^2 = \operatorname{Id}_E$ by $u^k = \operatorname{Id}_E$ for $k = 3$, then for $k = 4$.