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 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 $\rho \gamma_1 \rho^{-1} = \gamma_2$.

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

Show that $\sigma \in \mathfrak{S}_n$ and $\tau \in \mathfrak{S}_n$ are conjugate if and only if, for all $\ell \in \llbracket 1, n \rrbracket, c_\ell(\sigma) = c_\ell(\tau)$, where for $\ell \in \llbracket 2, n \rrbracket$, $c_\ell(\sigma)$ denotes the number of cycles of length $\ell$ in the decomposition of $\sigma$ into cycles with disjoint supports, and $c_1(\sigma) = \operatorname{Card}\{j \in \llbracket 1, n \rrbracket, \sigma(j) = j\}$.

Let $\ell \in \llbracket 2, n \rrbracket$ and let $\gamma \in \mathfrak{S}_\ell$ be a cycle of length $\ell$. Show that $\chi_\gamma(X) = X^\ell - 1$.
One may reduce to the case $\gamma = (12\cdots\ell)$ and consider the matrix
$$\Gamma_\ell = \left( \begin{array}{cccccc} 0 & \cdots & \cdots & \cdots & 0 & 1 \\ 1 & 0 & \cdots & \cdots & 0 & 0 \\ 0 & 1 & \ddots & & \vdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots & \vdots \\ \vdots & & \ddots & 1 & 0 & 0 \\ 0 & \cdots & \cdots & 0 & 1 & 0 \end{array} \right) \in \mathcal{M}_\ell(\mathbb{C})$$

Show that if $\sigma \in \mathfrak{S}_n$, then $\chi_\sigma(X) = \prod_{\ell=1}^{n} \left(X^\ell - 1\right)^{c_\ell(\sigma)}$.
One may justify that $P_\sigma$ is similar to a block diagonal matrix whose blocks are matrices of the form $\Gamma_\ell$ ($\ell \geq 1$), where $\Gamma_\ell$ is defined above if $\ell \geq 2$ and where $\Gamma_\ell = (1)$ if $\ell = 1$.

Let $G$ be the group of endomorphisms $g$ of $V$ such that $B(gu,gv)=B(u,v)$ for all $u,v\in V$.
Show that for all $g\in G$, we have $g(\mathcal{H})=\mathcal{H}$ or $-g(\mathcal{H})=\mathcal{H}$.

For all $w\in V$ such that $B(w,w)>0$, define $$s_w : v \mapsto v - 2\frac{B(v,w)}{B(w,w)}w.$$ Show that $s_w \in G_0$.

For all $w\in V$ such that $B(w,w)>0$, define $$s_w : v \mapsto v - 2\frac{B(v,w)}{B(w,w)}w.$$ Show that for all $u,v\in\mathcal{H}$, there exists $w\in V$ such that $B(w,w)>0$ and $s_w(u)=v$.

Recall that $\Gamma$ denotes the subgroup of $G_0$ formed by elements $g$ such that $g(V_\mathbb{Z})=V_\mathbb{Z}$.
Show that for all $v,w\in\mathcal{H}$ and all $R\geq 0$, the set $$\{g\in\Gamma \text{ such that } d(gv,w)\leq R\}$$ is finite.

We consider the three vectors $$w_1 = \begin{pmatrix}0\\1\\0\end{pmatrix}, \quad w_2 = \begin{pmatrix}1\\-1\\0\end{pmatrix}, \quad w_3 = \begin{pmatrix}-1\\0\\-1\end{pmatrix}.$$ Verify that $s_{w_1}$, $s_{w_2}$ and $s_{w_3}$ belong to $\Gamma$ and calculate the corresponding matrices.

We consider the three vectors $$w_1 = \begin{pmatrix}0\\1\\0\end{pmatrix}, \quad w_2 = \begin{pmatrix}1\\-1\\0\end{pmatrix}, \quad w_3 = \begin{pmatrix}-1\\0\\-1\end{pmatrix}.$$ We denote by $T$ the set of vectors $v\in\mathcal{H}$ such that $B(v,w_i)\geq 0$ for all $i\in\{1,2,3\}$.
Show that $T$ is compact and contains $v_0 = \begin{pmatrix}0\\0\\1\end{pmatrix}$.

We consider the three vectors $$w_1 = \begin{pmatrix}0\\1\\0\end{pmatrix}, \quad w_2 = \begin{pmatrix}1\\-1\\0\end{pmatrix}, \quad w_3 = \begin{pmatrix}-1\\0\\-1\end{pmatrix}.$$ Let $S_{1,2}$ be the subgroup of $\Gamma$ generated by $s_{w_1}$ and $s_{w_2}$. Let $v\in\mathcal{H}$.
Show that there exists $g\in S_{1,2}$ such that $$B(gv,w_1)\geq 0 \quad \text{and} \quad B(gv,w_2)\geq 0.$$

Show that for all $v\in\mathcal{H}$, there exists $g\in\Gamma$ such that $gv\in T$.

For all $s>1$, we denote by $P_k(s)$ the subset of $P_k$ formed by vectors $v$ such that $z_v\leq s$.
Show that $P_k(s)$ is finite.

Show that there exists a constant $C>0$ such that for all $v\in\mathcal{H}$, $$|\{g\in\Gamma \text{ such that } gv\in T\}| \leq C.$$

For all $R\in\mathbb{R}$, set $$\Gamma(R) = \{g\in\Gamma \text{ such that } d(v_0,gv_0)\leq R\}.$$ Recall that $\Gamma(R)$ is a finite set. Let $D = \sup_{v\in T} d(v_0,v)$.
Show that, for all $s\geq 0$, $$\frac{1}{C}|\Gamma(\operatorname{arcch}(s)-D)|\cdot|P_k\cap T| \leq |P_k(s)| \leq |\Gamma(\operatorname{arcch}(s)+D)|\cdot|P_k\cap T|.$$

Let $M$ be an additive subgroup of $\mathbf { Z } ^ { n }$ with $n \in \mathbf { N }$ (we agree that $\mathbf { Z } ^ { 0 }$ is the trivial group). We propose to prove by induction on $n$ the following result: (*) There exists $r \in \mathbf { N }$ such that the abelian group $M$ is isomorphic to $\mathbf { Z } ^ { r }$. a) Verify the cases $n = 0$ and $n = 1$. We now assume the result is true for $n - 1$. Let $p : \mathbf { Z } ^ { n } \rightarrow \mathbf { Z }$ be the projection onto the first coordinate, we denote by $N$ the kernel of $p$ and $N _ { 1 } = M \cap N$, then we set $p ( M ) = a \mathbf { Z }$ with $a \in \mathbf { Z }$. We choose $e _ { 1 } \in M$ such that $p \left( e _ { 1 } \right) = a$. Show that if $a \neq 0$, then the application $$N _ { 1 } \times \mathbf { Z } \rightarrow M , ( x , m ) \mapsto x + m e _ { 1 }$$ is a group isomorphism. b) Deduce (*). c) Show that the integer $r$ such that $M$ is isomorphic to $\mathbf { Z } ^ { r }$ is unique (one may consider the rank of a family of vectors of $\mathbf { Z } ^ { r }$ in the $\mathbf { Q }$-vector space $\mathbf { Q } ^ { r }$).

Show that if an abelian group $M$ has property (F), then every subgroup of $M$ also has it.

We consider the ring $A = \mathbf { Z } [ X , Y ]$. Let $U$ be the set of elements of $A$ of the form $X Y ^ { k }$ with $k \in \mathbf { N }$, we set $B = \mathcal { A } ( U )$. Let $S$ be a finite subset of $B$. a) Show that there exists $m \in \mathbf { N } ^ { * }$ such that $\mathcal { A } ( S ) \subset \mathcal { A } \left( \left\{ X , X Y , \ldots , X Y ^ { m } \right\} \right)$. b) Show that there exists an integer $N > 0$ such that every element of $\mathcal { A } ( S )$ is a sum of monomials of the form $\alpha X ^ { i } Y ^ { j }$ with $\alpha \in \mathbf { Z }$ and $j \leq i N$. c) Deduce that the ring $B$ does not have property (TF).

Let $E$ be a finite subset of $M _ { n } ( A )$. Show that there exists a subring $B$ of $A$ such that: $B$ has property (TF) and for every matrix $M \in E$, all coefficients of $M$ belong to $B$.

Let $M$ be a matrix of $M _ { n } ( A )$. The purpose of this question is to generalize to an arbitrary commutative ring $A$ the two formulas recalled in the introduction when $A$ is a field. a) Show that if the ring $A$ is integral, then $M \widetilde { M } = \widetilde { M } M = ( \operatorname { det } M ) I _ { n }$. b) We no longer assume $A$ is integral. Show that the result of a) still holds if there exists a surjective ring morphism $B \rightarrow A$ with $B$ integral. c) Deduce that the result of a) still holds for every commutative ring $A$. d) Prove that if $M$ and $N$ are in $M _ { n } ( A )$, then we have $$\operatorname { det } ( M N ) = \operatorname { det } M \times \operatorname { det } N .$$

Let $r$ and $s$ be strictly positive integers. Let $M \in M _ { s , r } ( A )$. We consider the application $u : A ^ { r } \rightarrow A ^ { s }$ defined by $u ( X ) = M X$, where we identify elements of $A ^ { r }$ and $A ^ { s }$ with column vectors. We assume that $u$ is surjective and that the ring $A$ is not reduced to $\{ 0 \}$. The purpose of this question is to prove that we then have $r \geq s$. For this, we reason by contradiction by assuming $r < s$. a) Show that there exists a matrix $N \in M _ { r , s } ( A )$ such that $M N = I _ { s }$. b) We define matrices of $M _ { s } ( A )$ by blocks: $$\begin{aligned} M _ { 1 } & = \left( \begin{array} { l l } M & 0 \end{array} \right) \\ N _ { 1 } & = \binom { N } { 0 } \end{aligned}$$ In other words, $M _ { 1 }$ is the matrix obtained by adding $s - r$ zero columns to $M$ and $N _ { 1 }$ is the matrix obtained by adding $s - r$ zero rows to $N$. Calculate $M _ { 1 } N _ { 1 }$. c) Reach a contradiction and conclude. d) We assume that $r = s$. Show the equivalence of the following properties: i) The application $u$ is surjective; ii) The determinant $\operatorname { det } M$ belongs to $A ^ { * }$; iii) There exists $N \in M _ { r } ( A )$ such that $M N = N M = I _ { r }$. iv) The application $u$ is bijective.

Throughout this part, we denote by $r$ and $s$ strictly positive integers. Let $A$ be a commutative ring not reduced to $\{ 0 \}$. We denote by $G L _ { r } ( A )$ the set of matrices of $M _ { r } ( A )$ that satisfy the equivalent properties of question III.3.d). a) Show that matrix multiplication induces a group structure on $G L _ { r } ( A )$. b) We define a relation on $M _ { s , r } ( A )$ by $M \sim N$ if and only if there exist $U \in G L _ { s } ( A )$ and $V \in G L _ { r } ( A )$ such that $N = U M V$. Show that this is an equivalence relation.

Throughout this part, we denote by $r$ and $s$ strictly positive integers. Let $A$ be a commutative ring not reduced to $\{ 0 \}$. We say that two matrices $M$ and $N$ of $M _ { s , r } ( A )$ are $A$-equivalent if $M \sim N$ (where $M \sim N$ if and only if there exist $U \in G L _ { s } ( A )$ and $V \in G L _ { r } ( A )$ such that $N = U M V$). If $M \in M _ { s , r } ( \mathbf { Z } )$ and $k$ is an integer at most equal to $\min ( r , s )$, we denote by $m _ { k } ( M )$ the gcd of the minors of size $k$ of $M$. Let $M$ and $N$ be two $\mathbf { Z }$-equivalent matrices of $M _ { s , r } ( \mathbf { Z } )$. Show that for all $k \leq \min ( r , s )$, we have $m _ { k } ( M ) = m _ { k } ( N )$ (one may begin by showing that $m _ { k } ( M )$ divides $m _ { k } ( N )$).