(i) Let $G = G L \left( 2 , \mathbb { F } _ { p } \right)$. Prove that there is a Sylow $p$-subgroup $H$ of $G$ whose normalizer $N _ { G } ( H )$ is the group of all upper triangular matrices in $G$.
(ii) Hence prove that the number of Sylow subgroups of $G$ is $1 + p$.

Let $m$ and $n$ be positive integers and $p$ a prime number. Let $G \subseteq \mathrm{GL}_{m}(\mathbb{F}_{p})$ be a subgroup of order $p^{n}$. Let $U \subseteq \mathrm{GL}_{m}(\mathbb{F}_{p})$ be the subgroup that consists of all the matrices with 1's on the diagonal and 0's below the diagonal. Show that there exists $A \in \mathrm{GL}_{m}(\mathbb{F}_{p})$ such that $AGA^{-1} \subseteq U$.

A subgroup $H$ of a group $G$ is said to be a characteristic subgroup if $\sigma(H) = H$ for every group isomorphism $\sigma : G \longrightarrow G$ of $G$.
(A) Determine all the characteristic subgroups of $(\mathbb{Q}, +)$ (the additive group).
(B) Show that every characteristic subgroup of $G$ is normal in $G$. Determine whether the converse is true.

Let $G$ be a finite group and $X$ the set of all abelian subgroups $H$ of $G$ such that $H$ is a maximal subgroup of $G$ (under inclusion) and is not normal in $G$. Let $M$ and $N$ be distinct elements of $X$. Show the following:
(A) The subgroup of $G$ generated by $M$ and $N$ is contained in the centralizer of $M \cap N$ in $G$.
(B) $M \cap N$ is the centre of $G$.

Let $G$ be a finite group that has a non-trivial subgroup $N$ (i.e. $\left\{ 1 _ { G } \right\} \neq N \neq G$ ) that is contained in every non-trivial subgroup of $G$. Show that
(A) $G$ is a $p$-group for some prime number $p$;
(B) $N$ is a normal subgroup of $G$.

(A) (3 marks) Let $G$ be a group such that $| G | = p ^ { a } d$ with $a \geq 1$ and $( p , d ) = 1$. Let $P$ be a Sylow $p$-subgroup and let $Q$ be any $p$-subgroup of $G$ such that $Q$ is not a subgroup of $P$. Show that $P Q$ is not a subgroup of $G$.
(B) (7 marks) Let $\Gamma$ be a group that is the direct product of its Sylow subgroups. Show that every subgroup of $\Gamma$ also satisfies the same property.

Let $G$ be an abelian group and let $H$ be a nontrivial subgroup of $G$, that is, $H$ is a subgroup containing at least two elements. Show that the following two statements are equivalent.
(A) For every nontrivial subgroup $K$ of $G$, the subgroup $K \cap H$ is also nontrivial.
(B) $H$ contains every nontrivial minimal subgroup of $G$ and every element of the quotient group $G / H$ has finite order.

Show that the set of non-zero similarities is a subgroup of $GL(E)$ under composition of applications.

Let $f$ be an antisymmetric endomorphism of $E$. Show that, if $S$ is a vector subspace of $E$ stable under $f$, then $S^{\perp}$ is stable under $f$. Show that the endomorphisms induced by $f$ on $S$ and on $S^{\perp}$ are antisymmetric.

We assume in this question that the space $E$ has dimension 1. Show that the root systems of $E$ are the sets $\{ \alpha , - \alpha \}$, with $\alpha \in E \backslash \{ 0 \}$.

In this question, the space $E$ has dimension $n = 3$. Let $(e _ { 1 } , e _ { 2 } , e _ { 3 })$ be an orthonormal basis of $E$ and $\mathcal { R } _ { 0 } = \left\{ e _ { i } - e _ { j } \mid 1 \leq i , j \leq 3 , i \neq j \right\}$.
Show that the vector subspace of $E$ spanned by the set $\mathcal { R } _ { 0 }$ is a vector plane.

We use the notations from Parts I and II as well as from question III.B. We assume $$\mathcal { A } = \left\{ \left. \left( \begin{array} { c c } A & B \\ C & - { } ^ { t } A \end{array} \right) \right\rvert \, ( A , B , C ) \in ( \mathcal { M } ( 2 , \mathbb { R } ) ) ^ { 3 } , B = { } ^ { t } B \text { and } C = { } ^ { t } C \right\}$$ and $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$, with $\mathcal{R} = \left\{ e _ { 1 } - e _ { 2 } , e _ { 2 } - e _ { 1 } , e _ { 1 } + e _ { 2 } , - e _ { 1 } - e _ { 2 } , 2 e _ { 1 } , - 2 e _ { 1 } , 2 e _ { 2 } , - 2 e _ { 2 } \right\}$.
Prove the equality $\mathcal { S } ( \mathcal { A } ) = \mathcal { R }$.

We use the notations from Parts I and II as well as from question III.B. We assume $$\mathcal { A } = \left\{ \left. \left( \begin{array} { c c } A & B \\ C & - { } ^ { t } A \end{array} \right) \right\rvert \, ( A , B , C ) \in ( \mathcal { M } ( 2 , \mathbb { R } ) ) ^ { 3 } , B = { } ^ { t } B \text { and } C = { } ^ { t } C \right\}$$ and $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$.
We now set $\alpha = e _ { 1 } - e _ { 2 }$, $\beta = 2 e _ { 2 }$, $H _ { \alpha } = \left( \begin{array} { c c c c } 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)$ and $H _ { \beta } = \left( \begin{array} { c c c c } 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 \end{array} \right)$.
a) Using the results from question III.C.3, show that there exists a pair $\left( X _ { \alpha } , X _ { - \alpha } \right) \in \mathcal { A } _ { \alpha } \times \mathcal { A } _ { - \alpha }$ and a pair $\left( X _ { \beta } , X _ { - \beta } \right) \in \mathcal { A } _ { \beta } \times \mathcal { A } _ { - \beta }$ such that $( X _ { \alpha } , H _ { \alpha } , X _ { - \alpha } )$ and $( X _ { \beta } , H _ { \beta } , X _ { - \beta } )$ are admissible triples of $\mathcal { A }$.
b) Show that $\mathcal { A }$ is the smallest vector subspace of $\mathcal { M } ( 4 , \mathbb { R } )$ stable by bracket and containing the matrices $X _ { \alpha } , H _ { \alpha } , X _ { - \alpha } , X _ { \beta } , H _ { \beta }$ and $X _ { - \beta }$.

For $(a, b) \in \mathbb{C}^2$, we denote by $M(a, b)$ the square complex matrix $M(a, b) = \left( \begin{array}{cc} a & -b \\ \bar{b} & \bar{a} \end{array} \right) \in \mathcal{M}_2(\mathbb{C})$. A matrix of the form $M(a, b)$ will be called a quaternion. We will consider in particular the quaternions $e = I_2 = M(1, 0)$, $I = M(0, 1)$, $J = M(\mathrm{i}, 0)$, $K = M(0, -\mathrm{i})$ and we will denote by $\mathbb{H} = \{M(a, b) \mid (a, b) \in \mathbb{C}^2\}$ the subset of $\mathcal{M}_2(\mathbb{C})$ consisting of all quaternions. Show that $(\mathbb{H} \backslash \{0\}, \times)$ is a non-commutative subgroup of the linear group $(\mathrm{GL}_2(\mathbb{C}), \times)$.

We denote $\mathbb{G} = \{xe + yI + zJ + tK \mid x, y, z, t \in \mathbb{Z}\}$ the set of ``integer'' quaternions. For $q \in \mathbb{H}$, $N(q) = x^2 + y^2 + z^2 + t^2$ where $q = xe + yI + zJ + tK$. a) Show that $\mathbb{G}$ is a subgroup of $\mathbb{H}$ for addition and that it is stable under multiplication. b) Show that for every $q \in \mathbb{H}$, there exists $\mu \in \mathbb{G}$ such that $N(q - \mu) \leqslant 1$. c) What is the set of $q \in \mathbb{H}$ such that $\forall \mu \in \mathbb{G}, N(q - \mu) \geqslant 1$?

Show that $O ( 1 , p ) = O ^ { + } ( 1 , p ) \cup O ^ { - } ( 1 , p )$.

Show that the set $O ( 1 , p )$ is a subgroup of $G L _ { p + 1 } ( \mathbb { R } )$ and that $O ^ { + } ( 1 , p )$ is a subgroup of $O ( 1 , p )$.

We denote, for every real $\gamma$, $L ( \gamma ) = \left( \begin{array} { c c } \operatorname { ch } \gamma & \operatorname { sh } \gamma \\ \operatorname { sh } \gamma & \operatorname { ch } \gamma \end{array} \right)$.
Show, for all real numbers $\gamma$ and $\gamma ^ { \prime }$, the equality: $$L ( \gamma ) L \left( \gamma ^ { \prime } \right) = L \left( \gamma + \gamma ^ { \prime } \right)$$ Deduce that $O ^ { + } ( 1,1 ) \cap \tilde { O } ( 1,1 )$ is a commutative subgroup of the group $O ^ { + } ( 1,1 )$.

Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 }$ and $L ^ { \prime } = \left( \ell _ { i , j } ^ { \prime } \right) _ { 1 \leqslant i , j \leqslant 4 }$ be two elements of $\tilde { O } ( 1,3 )$. We set $L ^ { \prime \prime } = L L ^ { \prime } = \left( \ell _ { i , j } ^ { \prime \prime } \right) _ { 1 \leqslant i , j \leqslant 4 }$.
Prove the following inequalities: $$0 \leqslant \sqrt { \sum _ { k = 2 } ^ { 4 } \ell _ { 1 , k } ^ { 2 } } \sqrt { \sum _ { k = 2 } ^ { 4 } \ell _ { k , 1 } ^ { \prime 2 } } + \sum _ { k = 2 } ^ { 4 } \ell _ { 1 , k } \ell _ { k , 1 } ^ { \prime } < \ell _ { 1,1 } ^ { \prime \prime }$$ Deduce that the set $\tilde { O } ( 1,3 )$ is a subgroup of the Lorentz group $O ( 1,3 )$.

We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 } \in O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ and $a = \left( \begin{array} { c } \ell _ { 2,1 } \\ \ell _ { 3,1 } \\ \ell _ { 4,1 } \end{array} \right)$.
Show that, if the vector $a$ is zero, then the matrix $L$ belongs to the group $G$.

We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 } \in O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ and $a = \left( \begin{array} { c } \ell _ { 2,1 } \\ \ell _ { 3,1 } \\ \ell _ { 4,1 } \end{array} \right)$. We assume that the vector $a$ is non-zero. We fix $L_1 \in G$ and $R_2 \in SO(3)$ such that $$L _ { 1 } L L _ { 2 } = \left( \begin{array} { c c c c } \ell _ { 1,1 } & \beta _ { 1 } & \beta _ { 2 } & \beta _ { 3 } \\ \alpha & \delta _ { 1 } & \delta _ { 2 } & \delta _ { 3 } \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)$$ where $L_2 = \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R _ { 2 } \end{array} \right)$.
Show that the real numbers $\beta _ { 2 } , \beta _ { 3 } , \delta _ { 2 }$ and $\delta _ { 3 }$ are zero.

Show that the elements of $G$ are invertible and explicitly determine the inverse of $M(A, \vec{b})$.

Prove that $G$ is a subgroup of $\mathrm{GL}_3(\mathbb{R})$.

We denote by $\mathcal{D}$ the set of affine lines of the plane and we consider the application $\Psi : \left\{ \begin{array}{cll} G & \rightarrow & \mathcal{D} \\ M(A, \vec{b}) & \mapsto \Delta\left(\left\langle A\vec{e}_1, \vec{b}\right\rangle, A\vec{e}_1\right) \end{array} \right.$.
Let $H$ be the set of matrices $M(A, \vec{b})$ of $G$ such that $\Psi(M(A, \vec{b})) = \Delta\left(0, \vec{e}_1\right)$.
a) Describe the elements of $H$.
b) Show that $H$ is a subgroup of $G$.
c) Show that for all $g$ in $G$ and all $h$ in $H$, we have $\Psi(gh) = \Psi(g)$.

We say that a real function $f$ of class $\mathcal{C}^{\infty}$ on $\mathbb{R}$ has rapid decay if $$\forall (n,m) \in \mathbb{N}^2, \lim_{x \rightarrow +\infty} x^m f^{(n)}(x) = \lim_{x \rightarrow -\infty} x^m f^{(n)}(x) = 0$$ We denote $\mathcal{S}$ the set of functions from $\mathbb{R}$ to $\mathbb{R}$ of class $\mathcal{C}^{\infty}$ with rapid decay.
Show that if $P$ is a polynomial function and if $f$ is in $\mathcal{S}$, then $Pf$ belongs to $\mathcal{S}$.