Let $F$ be a field and $G = \mathrm{GL}_n(F)$. For $g \in G$, write $C_g = \{hgh^{-1} \mid h \in G\}$. Let $X = \{C_g \mid g \in G,\ \text{the order of } g \text{ is } 2\}$. Determine $|X|$.

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 $\{1,\ldots,n\}$.

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 $n$ be a non-zero natural integer. For any permutation $\sigma \in \mathfrak{S}_{n}$, we recall that there exists, up to order, a unique decomposition $\sigma = c_{1} c_{2} \cdots c_{\omega(\sigma)}$, where $\omega(\sigma) \in \mathbb{N}^{*}$ where $c_{1}, \ldots, c_{\omega(\sigma)}$ are cycles with disjoint supports of respective lengths $\ell_{1} \leqslant \ell_{2} \leqslant \cdots \leqslant \ell_{\omega(\sigma)}$ and $\ell_{1} + \ell_{2} + \cdots + \ell_{\omega(\sigma)} = n$. We consider, on the probability space $(\mathfrak{S}_{n}, \mathscr{P}(\mathfrak{S}_{n}))$ equipped with the uniform probability, the random variable $X_{n}$ defined by $X_{n}(\sigma) = \omega(\sigma)$.
Justify that there exists a positive real number $C > 0$ such that, for any real $\varepsilon > 0$ and any integer $n \geqslant 1$, we have $$\mathbb{P}\left(\left|X_{n} - \ln(n)\right| > \varepsilon \ln(n)\right) \leqslant \frac{C}{\varepsilon^{2} \ln(n)}$$

On the set $A = \{1,2,3,4,5\}$ $$f = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 3 & 1 & 5 & 2 & 4 \end{pmatrix}, \quad g = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 5 & 3 & 4 & 1 & 2 \end{pmatrix}$$ For the permutations, what is the value of $g f^{-1}(2)$?
A) 1
B) 2
C) 3
D) 4
E) 5