Let $\sigma$ and $\sigma'$ be two elements of $S_n$.
Prove that $P_\sigma P_{\sigma'} = P_{\sigma \circ \sigma'}$.
Justify that the application $\left\{\begin{array}{l} \mathbb{Z} \rightarrow S_n \\ k \mapsto \sigma^k \end{array}\right.$ is not injective.
Deduce that there exists an integer $N \geqslant 1$ such that $\sigma^N = \operatorname{Id}_{\{1,\ldots,n\}}$, where $\operatorname{Id}_{\{1,\ldots,n\}}$ denotes the identity map on the set $\{1,\ldots,n\}$.

Show that if $f$ is cyclic, then $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free in $\mathcal{L}(E)$ and the minimal polynomial of $f$ has degree $n$.

Show that if $f$ is cyclic, then $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free in $\mathcal{L}(E)$ and the minimal polynomial of $f$ has degree $n$.

Let $x$ be a nonzero vector of $E$. Show that there exists a strictly positive integer $p$ such that the family $\left(x, f(x), f^2(x), \ldots, f^{p-1}(x)\right)$ is free and that there exists $\left(\alpha_0, \alpha_1, \ldots, \alpha_{p-1}\right) \in \mathbb{K}^p$ such that:
$$\alpha_0 x + \alpha_1 f(x) + \cdots + \alpha_{p-1} f^{p-1}(x) + f^p(x) = 0.$$

Let $x$ be a non-zero vector of $E$. Show that there exists a strictly positive integer $p$ such that the family $\left(x, f(x), f^2(x), \ldots, f^{p-1}(x)\right)$ is free and that there exists $\left(\alpha_0, \alpha_1, \ldots, \alpha_{p-1}\right) \in \mathbb{K}^p$ such that:
$$\alpha_0 x + \alpha_1 f(x) + \cdots + \alpha_{p-1} f^{p-1}(x) + f^p(x) = 0.$$

Justify that $\operatorname{Vect}\left(x, f(x), f^2(x), \ldots, f^{p-1}(x)\right)$ is stable under $f$.

Justify that $\operatorname{Vect}\left(x, f(x), f^2(x), \ldots, f^{p-1}(x)\right)$ is stable under $f$.

Verify that $\delta$ is a neutral element for the operation $*$.

Verify that $\delta$ is a neutral element for the operation $*$.

Deduce that $*$ is commutative.

Deduce that $*$ is commutative.

Similarly, by exploiting the set $\mathcal{C}_n^{\prime} = \left\{ \left( d_1, d_2, d_3 \right) \in \left( \mathbb{N}^* \right)^3 \mid d_1 d_2 d_3 = n \right\}$, show that $*$ is associative.

Similarly, by exploiting the set $\mathcal{C}_n^{\prime} = \left\{ \left( d_1, d_2, d_3 \right) \in \left( \mathbb{N}^* \right)^3 \mid d_1 d_2 d_3 = n \right\}$, show that $*$ is associative.

What can be said about $(\mathbb{A}, +, *)$?

What can be said about $(\mathbb{A}, +, *)$?

What can be said about the set $\mathcal{M}$ equipped with the operation $*$?

What can be said about the set $\mathcal{M}$ equipped with the operation $*$?

For all permutations $\rho, \rho' \in \mathfrak{S}_n$, show that $P_{\rho\rho'} = P_\rho P_{\rho'}$. Deduce that, for all permutations $\sigma, \tau \in \mathfrak{S}_n$, if $\sigma$ and $\tau$ are conjugate then $P_\sigma$ and $P_\tau$ are similar.

We fix a real vector space $E$ of dimension $n \geq 2$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$ with $\operatorname{dim} \mathcal{V} = \frac{n(n-1)}{2}$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set $H := \operatorname{Vect}(x)^{\perp}$, $\mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$, $\mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$, $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$, $\mathcal{Z} := \{u \in \mathcal{W} : \bar{u} = 0\}$, and $L$ the vector subspace such that $\mathcal{Z} = \{a \otimes x \mid a \in L\}$.
Prove that
$$\operatorname{dim} \overline{\mathcal{V}} = \frac{(n-1)(n-2)}{2}, \quad \operatorname{dim}(\operatorname{Vect}(x) \oplus \mathcal{V} x) + \operatorname{dim} L = n$$
and
$$L^{\perp} = \operatorname{Vect}(x) \oplus \mathcal{V} x$$
Deduce that $\operatorname{Vect}(x) \oplus \mathcal{V} x$ contains $v^{k}(x)$ for every $v \in \mathcal{V}$ and every $k \in \mathbf{N}$.

We consider, in this question only, $n = 7$ and the cycles $\gamma_1 = (1\;3)$ and $\gamma_2 = (2\;6\;4)$. We also consider a permutation $\rho \in \mathfrak{S}_7$ such that $\rho(1) = 2, \rho(3) = 6$ and $\rho(7) = 4$. Verify that $\rho \gamma_1 \rho^{-1} = \gamma_2$.

We consider, in this question only, $n = 7$ and the cycles $\gamma_1 = (1\;3)$ and $\gamma_2 = (2\;6\;4)$. We also consider a permutation $\rho \in \mathfrak{S}_7$ such that $\rho(1) = 2, \rho(3) = 6$ and $\rho(7) = 4$. Verify that $\rho \gamma_1 \rho^{-1} = \gamma_2$.

We fix a real vector space $E$ of dimension $n \geq 2$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$ with $\operatorname{dim} \mathcal{V} = \frac{n(n-1)}{2}$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set $H := \operatorname{Vect}(x)^{\perp}$, $\mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$, $\mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$, $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$. We have established that $\operatorname{dim} \overline{\mathcal{V}} = \frac{(n-1)(n-2)}{2}$ and $L^{\perp} = \operatorname{Vect}(x) \oplus \mathcal{V} x$.
By applying the induction hypothesis, show that the generic nilindex of $\mathcal{V}$ is greater than or equal to $n-1$, and that if moreover $\mathcal{V} x = \{0\}$ then there exists a basis of $E$ in which every element of $\mathcal{V}$ is represented by a strictly upper triangular matrix.

Show that, in $\mathfrak{S}_n$, two cycles of the same length are conjugate.

Show that, in $\mathfrak{S}_n$, two cycles of the same length are conjugate.

We fix a real vector space $E$ of dimension $n \geq 2$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$ with $\operatorname{dim} \mathcal{V} = \frac{n(n-1)}{2}$, equipped with an inner product $(-\mid-)$. We choose $x$ in $\mathcal{V}^{\bullet} \backslash \{0\}$ (where $\mathcal{V}^{\bullet}$ is the subset of $E$ formed by vectors belonging to at least one of the sets $\operatorname{Im} u^{p-1}$ for $u$ in $\mathcal{V}$). We denote by $p$ the generic nilindex of $\mathcal{V}$ (with $p \geq n-1$ by question 21), and we fix $u \in \mathcal{V}$ such that $x \in \operatorname{Im} u^{p-1}$. We have $\mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$ and $L^{\perp} = \operatorname{Vect}(x) \oplus \mathcal{V} x$.
Let $v \in \mathcal{V}$ such that $v(x) \neq 0$. Show that $\operatorname{Im} v^{p-1} \subset \operatorname{Vect}(x) \oplus \mathcal{V} x$. One may use the results of questions 5 and 20.