Pick the correct statement(s) below.
(a) There exists a group of order 44 with a subgroup isomorphic to $\mathbb { Z } / 2 \oplus \mathbb { Z } / 2$.
(b) There exists a group of order 44 with a subgroup isomorphic to $\mathbb { Z } / 4$.
(c) There exists a group of order 44 with a subgroup isomorphic to $\mathbb { Z } / 2 \oplus \mathbb { Z } / 2$ and a subgroup isomorphic to $\mathbb { Z } / 4$.
(d) There exists a group of order 44 without any subgroup isomorphic to $\mathbb { Z } / 2 \oplus \mathbb { Z } / 2$ or to $\mathbb { Z } / 4$.

For a field $F$, $F^{\times}$ denotes the multiplicative group ($F \backslash \{0\}, \times$). Choose the correct statement(s) from below:
(A) Every finite subgroup of $\mathbb{R}^{\times}$ is cyclic;
(B) The order of every non-trivial finite subgroup of $\mathbb{R}^{\times}$ is a prime number;
(C) There are infinitely many non-isomorphic non-trivial finite subgroups of $\mathbb{R}^{\times}$;
(D) The order of every non-trivial finite subgroup of $\mathbb{C}^{\times}$ is a prime number.

Let $n$ be a positive integer such that every group of order $n$ is cyclic. Show the following.
(A) For all prime numbers $p$, $p^2$ does not divide $n$.
(B) If $p$ and $q$ are prime divisors of $n$, then $p$ does not divide $q - 1$. (Hint: Consider $2 \times 2$ matrices $$\left[\begin{array}{ll} x & y \\ 0 & 1 \end{array}\right]$$ with $x, y \in \mathbb{Z}/q\mathbb{Z}$ and $x^p = 1$.)
(C) Show that $(n, \phi(n)) = 1$, where $\phi(n)$ is the number of integers $m$ such that $1 \leq m \leq n$ with $\gcd(n, m) = 1$.

Which of the following can not be the class equation for a group of appropriate order?
(A) $14 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 7$.
(B) $18 = 1 + 1 + 1 + 1 + 2 + 3 + 9$.
(C) $6 = 1 + 2 + 3$.
(D) $31 = 1 + 3 + 6 + 6 + 7 + 8$.

By a simple group, we mean a group $G$ in which the only normal subgroups are $\left\{ 1 _ { G } \right\}$ and $G$. Pick the correct statement(s) from below.
(A) No group of order 625 is simple.
(B) $\mathrm { GL } ( 2 , \mathbb { R } )$ is simple.
(C) Let $G$ be a simple group of order 60. Then $G$ has exactly six subgroups of order 5 .
(D) Let $G$ be a group of order 60. Then $G$ has exactly seven subgroups of order 3 .

Which of the following groups are cyclic?
(A) $\mathbb { Z } / 2 \mathbb { Z } \oplus \mathbb { Z } / 9 \mathbb { Z }$
(B) $\mathbb { Z } / 3 \mathbb { Z } \oplus \mathbb { Z } / 9 \mathbb { Z }$
(C) Every group of order 18.
(D) $\left( \mathbb { Q } ^ { \times } , \cdot \right)$

Let $G$ (respectively, $H$ ) be a Sylow 2-subgroup (respectively, Sylow 7-subgroup) of the symmetric group $S _ { 17 }$. Pick the correct statement(s) from below.
(A) The order of $G$ is $2 ^ { 15 }$.
(B) $H$ is abelian.
(C) $G$ has a subgroup isomorphic to $\mathbb { Z } / 8 \mathbb { Z } \times \mathbb { Z } / 8 \mathbb { Z }$.
(D) If $\sigma \in S _ { 17 }$ has order 4 , then $\sigma$ is a 4-cycle.

Show that $d_{n} \geqslant 1$.

In this question, the space $E$ has dimension $n = 2$. For every root system $\mathcal { R }$ of $E$, we set $$\theta _ { \mathcal { R } } = \min \left\{ \theta _ { \alpha , \beta } \mid ( \alpha , \beta ) \in \mathcal { R } ^ { 2 } , \alpha \neq \beta \text { and } \alpha \neq - \beta \right\}$$ Show that $\theta _ { \mathcal { R } }$ is well-defined and equals $\pi / 2 , \pi / 3 , \pi / 4$ or $\pi / 6$.

In this question, the space $E$ has dimension $n = 2$. For every root system $\mathcal { R }$ of $E$, we set $$\theta _ { \mathcal { R } } = \min \left\{ \theta _ { \alpha , \beta } \mid ( \alpha , \beta ) \in \mathcal { R } ^ { 2 } , \alpha \neq \beta \text { and } \alpha \neq - \beta \right\}$$ For each value of $k \in \{ 2,3,4,6 \}$, draw graphically a root system $\mathcal { R } _ { k }$ such that $\theta _ { \mathcal { R } _ { k } } = \pi / k$. It is not necessary to justify that the figures drawn represent root systems. What is the cardinality of $\mathcal { R } _ { k }$? No justification is required.

We return to the case where $\mathbb { K }$ is an arbitrary field of characteristic zero. Let $V$ be a $\mathbb { K }$-vector space of finite dimension and $q \in \mathcal { Q } ( V )$. Show that there exists a unique non-negative integer $m$ and an anisotropic quadratic form $q _ { \text {an} }$, unique up to isometry, such that $q \cong q _ { an } \perp m \cdot h$ where $m \cdot h = h \perp \cdots \perp h$ is the orthogonal sum of $m$ copies of $h$ and $h$ is the quadratic form defined by $h \left( x _ { 1 } , x _ { 2 } \right) = x _ { 1 } x _ { 2 }$ (introduced in question 6b).
Hint: one may use question 6b and the previous question.

Is the group $O ^ { + } ( 1,1 ) \cap \tilde { O } ( 1,1 )$ compact?

Show that the group $O ^ { + } ( 1,1 )$ is commutative.

We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ Deduce that every matrix $L$ of $O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ can be written in the form of a product of the type $$L = \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) \left( \begin{array} { c c c c } \operatorname { ch } \gamma & \operatorname { sh } \gamma & 0 & 0 \\ \operatorname { sh } \gamma & \operatorname { ch } \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R ^ { \prime } \end{array} \right)$$ where $R$ and $R ^ { \prime }$ are two elements of $S O ( 3 )$ and $\gamma$ is a real number.

Let $m \in \mathbb{N}$. Justify that the vector space $\mathcal{P}_m$ is finite-dimensional and determine its dimension.

Determine a harmonic polynomial of degree 1, then of degree 2.

Recall the cardinality of $\mathcal{S}_n$. Deduce that $R \geq 1$.

Show that every finite abelian group and the additive group $\mathbf { Z } ^ { r }$ for $r \in \mathbf { N } ^ { * }$ have property (F).

The number of non-empty equivalence relations on the set $\{ 1,2,3 \}$ is:
(1) 6
(2) 5
(3) 7
(4) 4