\[
L_3 = \lbrace \rho_0, \rho_1, \rho_2 \mid \rho_0\rho_1\rho_2 = 1 \rbrace
\]

\[
\alpha_0 = A(\rho_0), \qquad \alpha_1 = A(\rho_1)
\]

\[
A : G \longrightarrow \mathbb{E}^1_S
\]

\[
C \longrightarrow C'
\]

\[
\hat{\varphi} : G_2 \ (\text{groupe cartographique}) \longrightarrow \Gamma
\xrightarrow{\ \varphi\ } G(S),
\]

\[
f(a_i) = (\beta_0 \cdots \beta_{i-1}), \quad f(a_{i+1}) = \beta_0 \cdots
\beta_i, \quad \varphi(a_i, a_{i+1}) = -(\beta_0 \cdots \beta_i)
\]

\[
4\cos^2\theta_i = \frac{\varphi(a_i, a_{i+1})^2}{f(a_i)f(a_{i+1})} = \beta_i
\qquad \Bigl(\text{\emph{NB}}\ \xi_i = \frac{\beta_i}{f_i f_{i+1}} =
\frac{\beta_i}{4} = \cos^2\theta_i\Bigr)
\]

\[
\alpha_i = 2(\underbrace{2\cos^2\theta_i - 1}_{\cos 2\theta_i}) = 2\cos 2\theta_i
\]

\[
A(\rho_Y) = \alpha
\]

\[
f(x, y, z) = x^2 + yz, \qquad \rho_\xi(x, y, z) = (x, \xi y, \xi^{-1}z),
\qquad \text{et } \alpha = \xi + \xi^{-1}.
\]

\[
\tau_0^2 = \tau_1^2 = \tau_2^2 = (\tau_0\tau_2)^2 = (\tau_0\tau_1)^{\nu_0} =
(\tau_1\tau_2)^{\nu_1} = 1
\]

\[
C(\nu_0, \nu_1) \longrightarrow C'(\nu_0, \nu_1).
\]

\[
\alpha^2 + \alpha - 1 = 0
\]

\[
\varphi^+ : \mathfrak{A}_5 \longrightarrow G
\]

\[
\mathbb{D}_i = \lbrace \sigma_0, \sigma_1 \mid \sigma_0^2 = \sigma_1^2 =
(\sigma_0\sigma_1)^i = 1 \rbrace
\qquad (\emph{NB}\ \text{la déf.}\ \mathbb{D}_0 = \mathbb{D}_\infty)
\]

\[
G = G_1 = \lbrace \tau_0, \tau_1 \mid \tau_0^2 = \tau_1^2 = 1 \rbrace
\]

\[
a_{0s} \notin H_1 \ (\Leftrightarrow a_1 \notin H_0) \ \text{sur chaque fibre}
\]

\[
[\lambda] \cdot [\lambda'] = [\lambda + \lambda' + \lambda\lambda'\rho]
\]

\[
U_{2-\alpha} \xrightarrow{\ \sim\ } \mathfrak{z}\,\mathrm{Ant}(\varphi_\alpha) =
\underline{\mathrm{Aut}}(G_\alpha, \varphi_\alpha)
\]

\[
C' \longrightarrow C,
\]

\[
U_\rho \xrightarrow[\text{can}]{\ \sim\ } \mathbb{G}_m, \qquad \lambda
\longmapsto 1 + \lambda\rho .
\]

\[
\underline{\mathrm{Aut}}(G, \varphi) = U \simeq \mathbb{G}_a .
\]

\[
  (**) \qquad \exists \text{ hom injectif } \Gamma = \mathbb{D}_i \times \mathbb{Z}/2 \longrightarrow \mathrm{GP}(1,k)
\]

\[
  \mathbf{I}(G) = \underline{\mathrm{Hom}}_{\mathrm{sch}}(G, \mathbb{E}^1_S)^{\circlearrowleft}
  \overset{\mathrm{d\acute{e}f}}{=}
  \underline{\mathrm{Hom}}_{\mathrm{sch}}(G, \mathbb{E}^1_S)^{G}
\]

\[
  \mathbf{I}(G) \xrightarrow{\ A\ } \mathbb{E}^1_S
\]

\[
  A(i(\lambda)) = \lambda + \lambda^{-1}
\]

\[
  A(1) = 2 .
\]

\[
  A(\sigma) = -2 .
\]

\[
  \mathrm{Aff}(1)_S = \mathbb{G}_m \cdot_{1/2} \mathbb{G}_a \xrightarrow{\ i\ } G
\]

\[
  A(u) = 2
\]

\[
  A(i(\lambda \cdot t)) = \lambda + \lambda^{-1}
\]

\[
  f(\lambda, t) \overset{\mathrm{df}}{=} A(i(\lambda, t))
\]

\[
  \begin{cases}
    f(\lambda, 0) = \lambda + \lambda^{-1} \\
    f\bigl(\underbrace{(0,t)(\lambda,0)(0,-t)}_{(\lambda,0)(\lambda^{-1}t - t) = (\lambda, \lambda^{-1}t - t)}\bigr) = \lambda + \lambda^{-1}
  \end{cases}
\]

\[
  f(\lambda, s) = \lambda + \lambda^{-1}
\]

\[
  (*) \qquad \lambda + \lambda^{-1} = \lambda' + \lambda'^{-1} = \alpha \in \Gamma(S, \underline{O}_S)
\]

\[
  (**) \qquad t^2 - \alpha t + 1 = 0
\]

\[
  \alpha^2 - 4 = \underbrace{(\alpha - 2)}_{\text{inv.}}(\alpha + 2)
\]

\begin{align*}
  \lambda' &= \lambda(1 + \lambda^{-1}\mu) = \lambda(1 - \mu) = \lambda + \mu \\
  \lambda'^{-1} &= \lambda^{-1}\underbrace{(1 - \mu)^{-1}}_{1 + \mu} = \lambda^{-1}(1 + \mu) = \lambda^{-1} - \mu \\
  \lambda' + \lambda'^{-1} &= \lambda + \lambda^{-1} + \mu(\lambda^{-1} - \lambda) = \lambda + \lambda^{-1} .
\end{align*}

\[
  \lambda' = \lambda \quad \text{ou} \quad \lambda'^{-1} = \lambda \ ?
\]

\[
  \mu = \lambda^{-1} - \lambda
\]

\[
  A(j(\lambda)) = \lambda^n + \lambda^{-n} = G_n(\lambda + \lambda^{-1})
\]

\[
  G_n \in \mathbb{Z}[U]
\]

\begin{align*}
  A(i(\lambda)) &= \lambda + \lambda^{-1} = \alpha \\
  A(p\,i(\lambda)) &= \lambda^2 + \lambda^{-2} = \alpha^2 - 2
\end{align*}

\[
  \begin{cases}
    A'(p(u)) = A(u') = A(u)^2 - 2 \\
    A(u) = \mathrm{Tr}\, u
  \end{cases}
\]

\[
  A(u^C) = \frac{(\mathrm{Tr}\, u)^2}{\det u} - 2
\]

\[
  t^2 - (\mathrm{Tr}\, u)t + \det u = 0
\]

\[
  \lambda_1 + \lambda_2 = \mathrm{Tr}\, u, \qquad \lambda_1\lambda_2 = \det u
\]

\[
  A(p(u)) = \frac{\lambda_1}{\lambda_2} + \frac{\lambda_2}{\lambda_1} = \frac{\lambda_1^2 + \lambda_2^2}{\lambda_1\lambda_2}
\]

\[
  A(pu) = \lambda + \lambda^{-1} \qquad (u \text{ ps.\ réfl.\ par } \lambda)
\]

\[
  \mathbb{G}_m \longrightarrow \mathrm{GL}(V) \longrightarrow \underline{\mathrm{Aut}}_C
\]

\[
  A(u) = -2 ,
\]

\[
  F_n \in \mathbb{Z}[U], \qquad \Psi_n \in \mathbb{Z}[U]
\]

\[
  \alpha = \lambda + \lambda^{-1} ,
\]

\[
  \alpha^2 + \alpha - 1 = 0
\]

\[
  (*) \qquad u^n = \mathrm{id}, \text{ et } \forall s \in S, \text{ l'ordre de } u_s \text{ est égal à } n.
\]

\[
    \mathcal{N}(\mathbb{D}_2)/\mathfrak{Z}(\mathbb{D}_2) \xrightarrow{\sim} \mathrm{Aut}(\mathbb{D}_2) \simeq \mathfrak{S}_3
  \]

\[
  C \longrightarrow \mathcal{C} \longrightarrow \text{catégorie des couples } (G, \Gamma)
\]

\[
  \mathrm{Aut}(G, \Gamma) \simeq \quad
  \bigl(\mathcal{N}(\Gamma) = \mathcal{N}(T) = , \quad
  \mathcal{N}(\Gamma)/\mathfrak{Z}(\Gamma) = \mathrm{Aut}(G, \Gamma) \simeq \mathbb{Z}/2\mathbb{Z}
\]

\[
  \mathfrak{Z}(\Gamma) = \Gamma, \qquad \mathcal{N}(\Gamma) = \mathfrak{S}_4 ,
\]

\[
  \mathcal{N}(\Gamma)/\mathfrak{Z}(\Gamma) \xrightarrow{\sim} \mathrm{Aut}(\Gamma) \simeq \mathfrak{S}_3
\]

\[
  \mathfrak{Z}(\Gamma) = {}_2T \simeq \lbrace \pm 1 \rbrace
\]

\[
  \mathcal{N}(\Gamma) =
  \begin{cases}
    \Gamma & \text{si } n \text{ pair} \\
    \Gamma \cdot {}_2T & \text{si } n \text{ impair}
  \end{cases}
\]

\[
  \mathcal{N}(\Gamma)/\mathfrak{Z}(\Gamma) \simeq \Gamma/{}_2\Gamma^{+} \subset \mathrm{Aut}(\Gamma) \,]
\]

\[
  \mathfrak{Z}(\Gamma) = 1
\]

\[
  \mathrm{Aut}(G, \Gamma) = \mathcal{N}(\Gamma) = \mathcal{N} \simeq \mathfrak{S}_4
\]

\[
  \mathcal{N}(\Gamma)/\mathfrak{Z}(\Gamma) \simeq \mathcal{N}(\Gamma) \xrightarrow{\sim} \mathrm{Aut}(\Gamma)
\]

\[
  \widehat{\mathbb{G}}_{m\mathbb{Q}} = \mathbb{P}^1_{\mathbb{Q}} .
\]

\[
  \mathfrak{Z}(\Gamma) = \lbrace 1 \rbrace, \qquad
  \mathcal{N}(\Gamma) = \Gamma \simeq \mathfrak{A}_5
\]

\[
  \alpha^2 + \alpha - 1 = 0
\]

\[
  \alpha_0 = \tfrac{1}{2}(-1 + \sqrt{5}) \,]
\]

\[
  \Gamma^{-} \subset {}_2\Gamma
\]

\[
  {}_2\Gamma = \Gamma^{-} \amalg {}_2\Gamma^{+}
\]

\[
  \Gamma \setminus {}_2\Gamma = \Gamma^{+} \setminus {}_2\Gamma^{+}
\]

\[
  (*) \qquad \Gamma^{+} = \text{ss-grpe de } \Gamma \text{ engendré par } \Gamma \setminus {}_2\Gamma
\]

\[
  \mathrm{Aut}(\mathbb{D}_n) \simeq \mathrm{Aff}(1, \mathbb{Z}/n) \simeq (\mathbb{Z}/n)^{*} \cdot_{1/2} \mathbb{Z}/n
\]

\[
  \varphi : \mathfrak{A}_{5S} \hookrightarrow G
\]

\[
  \alpha(\varphi) = A(\varphi(\pi))
\]

\[
  A : G \longrightarrow \mathbb{E}^1_S
\]

\[
  (G, \varphi) \longmapsto \alpha(\varphi)
\]

\[
  \alpha^2 + \alpha - 1 = 0 ,
\]

\[
  M = \mathrm{Spec}\, \mathbb{Z}[T]/(T^2 + T - 1) .
\]

\[
\begin{array}{rlll}
4 & \mathfrak{A}_4 & 3 & (3, 3', 2)\,(3', 2, 3)\,(2, 3, 3')\
  \bigl[(2, 3', 3)\,(3', 3, 2)\,(3, 2, 3')\bigr] \\
  & & 1 & (3, 3, 3)\ \bigl[(3', 3', 3')\bigr] \\[4pt]
9 & \mathfrak{S}_4 & 6 & (4, 3, 2)\,(3, 2, 4)\,(2, 4, 3)\,(2, 3, 4)\,(3, 4, 2)\,(4, 2, 3)
  \ \text{standard} \\
  & & 3 & (4, 4, 3)\,(4, 3, 4)\,(3, 4, 4) \\[4pt]
32 & \mathfrak{A}_5 & 6 & (5, 3, 2)\,(3, 2, 5)\,(2, 5, 3)\,(2, 3, 5)\,(3, 5, 2)\,(5, 2, 3)
  \ \text{standard } \alpha \\
  & & 6 & (5', 3, 2)\,(3, 2, 5')\,(2, 5', 3)\,(2, 3, 5')\,(3, 5', 2)\,(5', 2, 3)
  \ \text{id. } \alpha' \\
  & & 3 & (5, 5, 3)\,(5, 3, 5)\,(3, 5, 5) \\
  & & 3 & (5', 5', 3)\,(5', 3, 5')\,(3, 5', 5') \\
  & & 6 & (5, 5', 3)\,(5', 3, 5)\,(3, 5, 5')\,(3, 5', 5)\,(5', 5, 3)\,(5, 3, 5') \\
  & & 6 & (5, 5', 2)\,(5', 2, 5)\,(2, 5, 5')\,(2, 5', 5)\,(5', 5, 2)\,(5, 2, 5') \\
  & & 1 & (5, 5, 5) \\
  & & 1 & (5', 5', 5') \\[4pt]
\hline
45 & & &
\end{array}
\]

\[
\Gamma^{+} \longmapsto \Gamma = \mathrm{Norm}_{\mathbb{A}}(\Gamma^{+})
\qquad \Gamma^{+} = [\Gamma, \Gamma]
\]

\[
\begin{array}{l}
E \in (\mathrm{Ens})_4 \longmapsto \Gamma = \mathfrak{S}_E,\
\Gamma^{+} = \mathfrak{S}_E^{+} \\
\Gamma^{+} \in F(\mathfrak{A}_4) \longmapsto \Gamma = \mathrm{Aut}(\Gamma^{+}),\
E = \text{ens.\ des ss-groupes d'indice 4 (d'ordre 3) de } \Gamma^{+} \\
\Gamma \in F(\mathfrak{S}_4) \longmapsto \Gamma^{+} = [\Gamma, \Gamma],\
E = \text{ens.\ des ss-groupes d'ordre 6 (d'indice 4) de } \Gamma
\end{array}
\]

\[
\mathrm{Rev}_4(S) \simeq F(S, \mathfrak{S}_4) \simeq F(S, \mathfrak{A}_4)
\simeq \mathrm{Tors}(\mathfrak{S}_{4S}) \simeq
\]

\[
\begin{array}{lll}
\mathrm{I} & 2 \leq \nu, \nu', \nu'' \leq 3 & \text{cas } \mathfrak{A}_4 \\
\mathrm{II} & 2 \leq \nu, \nu', \nu'' \leq 4 & \text{cas } \mathfrak{S}_4 \\
\mathrm{III} & 2 \leq \nu, \nu', \nu'' \leq 5 & \text{cas } \mathfrak{A}_5
\end{array}
\]

\[
(\rho, \rho', \rho'')\,(\rho', \rho'', \rho)\,(\rho'', \rho, \rho')\,
(\rho''^{-1}, \rho'^{-1}, \rho^{-1})\,(\rho'^{-1}, \rho^{-1}, \rho''^{-1})\,
(\rho^{-1}, \rho''^{-1}, \rho'^{-1})
\]

\[
(\rho_t^{-1}, \rho_s, \rho_t^{-1}\rho_s)
\]

\[
\Sigma_r = (\rho_f, \rho_s, \rho_a) \in H^3
\]

\[
\Sigma_{r'} = (\rho_f^{-1}, \rho_s^{-1}, \rho_a = \rho_a^{-1})
\]

\[
\Sigma_{r_1} = \Sigma_{r_2}
\]

\[
\rho_{s_1} = \rho_{s_2},\quad \rho_{f_1} = \rho_{f_2},\quad
\rho_{a_1} = \rho_{a_2}.
\]

\[
\Sigma_{r_2} = g \cdot \Sigma_{r_1}
= (\mathrm{int}(g)\rho, \mathrm{int}(g)\rho', \mathrm{int}(g)\rho'')
\]

\[
(\rho_f, \rho_s, \rho_a),\ (\rho_s^{-1}, \rho_f^{-1}, \rho_a^{-1} = \rho_a)
\quad \text{i.e.\ } \Sigma_r \text{ et } \Sigma_{r'}
\]

\[
(\rho_s, \rho_{s'}, \rho_{s''})
\]

\[
(\rho_{f''}, \rho_{f'}, \rho_f)
\]

\[
\Sigma = (\rho_s^{\omega}, \rho_{s'}^{\omega}, \rho_{s''}^{\omega}).
\]

\[
(\rho_s, \rho_t, \rho_{f'}) \qquad (f' \text{ l'antipodique de } f)
\]

\[
\mu_0 = \lambda_0 - 1,\quad \mu_1 = \lambda_1 - 1,\quad \beta = \alpha + 2
\quad \text{invariant}
\]

\[
D = k a_i, \qquad D' = \mathrm{Ker}\, a_i'
\]

\[
a_i + \underbrace{\langle a_i, a_j' \rangle}_{\struck{\ill{}}\ \beta} a_j
\in D' \quad \text{i.e.} \quad
\underbrace{\langle a_i, a_i' \rangle}_{\mu_i}
+ \underbrace{\langle a_i, a_j' \rangle \langle a_j, a_i' \rangle}_{\beta} = 0
\]

\[
\underbrace{\langle a_0, a_0' \rangle}_{\mu_0}
+ (2 + \mu_1)\underbrace{\langle a_0, a_1' \rangle \langle a_1, a_0' \rangle}_{\beta}
= 0
\]

\[
\tau_0 = \begin{pmatrix} 0 & \xi \\ 1 & 0 \end{pmatrix}, \qquad
\tau_1 = \begin{pmatrix} 0 & \zeta \\ \eta & 0 \end{pmatrix}
\]

\[
\lambda^2 + \xi = 0, \qquad \lambda^2 + \zeta\eta = 0
\]

\[
G_0 \subset \widetilde{G}_0 \cdot k^{\times}
\]

\[
2 \leq \nu_0, \nu_1 \leq 5
\]

\[
\beta = c\,\frac{\cdots}{\cdots}
= c\,\frac{(\lambda_0 - 1)(\lambda_1 - 1)}{(\kappa_1 - 1)}
= \xi\,(\lambda_0 - 1)(\lambda_1 - 1) = \xi\,\mu_0\mu_1,
\qquad \xi = \frac{c}{\kappa_1 - 1} \in \mathbb{R}
\]

\[
\kappa_i = \mathrm{Tr}_{\mathbb{C}/\mathbb{R}}\,\lambda_i
= \lambda_i + \overline{\lambda}_i = \lambda_i + \lambda_i^{-1}
\qquad (\lambda_i \overline{\lambda}_i = 1)
\]

\[
\frac{(\lambda_0 - 1)(\lambda_1 - 1)}{\lambda_0\lambda_1}
\Bigl[\xi^2 + 2\,\frac{\lambda_0 + \lambda_1}{(\lambda_0 - 1)(\lambda_1 - 1)}\,\xi
+ \frac{\lambda_0^2 + \lambda_1^2}{(\lambda_0 - 1)^2(\lambda_1 - 1)^2}\Bigr]
= \kappa \in \lbrace -2, -1, 0, \alpha, \alpha' \rbrace
\]

\[
\kappa = \kappa(u) = \frac{(\mathrm{Tr}\, u)^2}{\det u} - 2
\]

\[
|\lambda + \lambda'| = 2\Bigl|\cos\frac{\theta' - \theta}{2}\Bigr|,\quad
|\lambda^2 + \lambda'^2| = 2|\cos(\theta' - \theta)|,
\]

\[
|\lambda - 1|^2 = 2 - \kappa(\lambda),\quad
\kappa(\lambda) = \mathrm{Tr}_{\mathbb{C}/\mathbb{R}}\,\lambda = 2\cos\theta,\quad
|\lambda - 1| = 2\Bigl|\sin\frac{\theta}{2}\Bigr|
\]

\[
2\,\frac{\lambda + \lambda'}{(\lambda - 1)(\lambda' - 1)}
= -\,\frac{\cos\frac{\theta' - \theta}{2}}
{\sin\frac{\theta}{2}\sin\frac{\theta'}{2}},
\qquad
\frac{(\lambda_0 - 1)^2(\lambda_1 - 1)^2}{\lambda_0\lambda_1}
= 16\sin^2\frac{\theta_0}{2}\sin^2\frac{\theta_1}{2}
\]

\[
\frac{\lambda^2 + \lambda'^2}{(\lambda - 1)^2(\lambda' - 1)^2}
= \frac{2\cos(\theta' - \theta)}{16\sin^2\frac{\theta}{2}\sin^2\frac{\theta'}{2}}
= \frac{1}{8}\,\frac{\cos(\theta' - \theta)}{\sin^2\frac{\theta}{2}\sin^2\frac{\theta'}{2}},
\qquad 0 < \theta_0, \theta_1 \leq \pi
\]

\[
\xi^2 - \frac{\cos\frac{\theta_1 - \theta_0}{2}}{\sin\frac{\theta_0}{2}\sin\frac{\theta_1}{2}}\,\xi
+ \frac{1}{8}\Bigl(\frac{2\cos(\theta_1 - \theta_0) - \kappa}
{\sin^2\frac{\theta_0}{2}\sin^2\frac{\theta_1}{2}}\Bigr) = 0
\]

\[
\Delta(\kappa) = \frac{1}{2}\,\frac{\cdots}{\sin^2\frac{\theta_0}{2}\sin^2\frac{\theta_1}{2}}
\Bigl[\underbrace{2\cos^2\frac{\theta_1 - \theta_0}{2} - \cos(\theta_1 - \theta_0)}_{1}
+ \frac{1}{2}\kappa\Bigr]
= \boxed{\frac{\kappa + 2}{4\sin^2\frac{\theta_0}{2}\sin^2\frac{\theta_1}{2}}} \geq 0
\]

\[
\kappa + 2 = 2(1 + \cos\Theta) = 4\cos^2\frac{\Theta}{2}
\]

\[
\boxed{\xi = \frac{\cos\frac{\theta_1 - \theta_0}{2}}{2\sin\frac{\theta_0}{2}\sin\frac{\theta_1}{2}}}
\]

\[
0 < \xi < 1
\]

\[
16\sin^2\frac{\theta}{2}\sin^2\frac{\theta'}{2}
\Bigl(\xi^2 - \frac{\cos\frac{\theta' - \theta}{2}}{2\sin\frac{\theta}{2}\sin\frac{\theta'}{2}}\,\xi
+ \frac{1}{8}\,\frac{\cos(\theta' - \theta)}{\sin^2\frac{\theta}{2}\sin^2\frac{\theta'}{2}}\Bigr)
\]

\[
\lambda_i^2 + \kappa_i\lambda_i + 1 = 0
\]

\[
\begin{array}{lll}
\text{Cas} & (3,2,3)\,(2,3,3) & (\text{tétraédraux}) \\
 & (3,2,4)\,(2,4,3)\,(2,3,4)\,(4,2,3) & (\text{octaédraux}) \\
 & (3,2,\underline{5})\,(2,5,3)\,(2,3,\underline{5})\,(5,2,3) & (\text{icosaédraux}) \\
 & (5,2,5')\,(2,5,5') &
\end{array}
\]

\[
\begin{array}{ll}
(3,2,5)\,(2,3,5) & (\text{qui figurent déjà dans liste 1)}) \\
\bigl[(3,2,5')\,(2,3,5')\bigr] &
\end{array}
\]

\[
\begin{array}{rll}
6 & \textit{Cas tétraédraux} &
(3,3,2)_{(4)}\ \underline{(3,2,3)}_{(2)}\ \underline{(2,3,3)}_{(2)}\
(3,3,3)_{(4)} \\[4pt]
13 & \textit{Cas octaédraux} &
(4,3,2)_{(4)}\ \underline{(3,2,4)}_{(2)}\ \underline{(2,4,3)}_{(2)}\
\underline{(2,3,4)}_{(2)}\ (3,4,2)_{(4)}\ \underline{(4,2,3)}_{(2)} \\
 & & (4,4,3)_{(4)}\ (4,3,4)_{(4)}\ (3,4,4)_{(4)} \\[4pt]
10 & \textit{Cas icosaédraux} &
(5,3,2)_{(8)}\ \underline{(3,2,5)}_{(2)}\ \underline{(2,5,3)}_{(4)}\
\underline{(2,3,5)}_{(2)}\ (3,5,2)_{(8)}\ \underline{(5,2,3)}_{(4)} \\
+4 & & \underline{(3,2,5')}_{(2)}\ \underline{(2,3,5')}_{(2)} \\
+4 & & (5,5,3)_{(16)}\ (5,3,5)_{(8)}\ (3,5,5)_{(8)} \\
2 & & (5,3,5')_{(8)}\ (3,5,5')_{(8)} \\
4 & & \underline{(2,5,5')}_{(4)}\ \underline{(5,2,5')}_{(4)} \\
1 & & (5,5,5)_{(16)} \\
\hline
25 & &
\end{array}
\]

\[
\begin{array}{rl}
6 & \text{Cas tétraédraux} \\
13 & \text{Cas octaédraux} \\
25 & \text{Cas icosaédraux} \\
\hline
44 & \text{Cas au total}
\end{array}
\]

\[
f_{\kappa}(\xi) = \xi^2 - \frac{\cos\frac{\theta_1 - \theta_0}{2}}
{\sin\frac{\theta_0}{2}\sin\frac{\theta_1}{2}}\,\xi
+ \frac{1}{\cdot}\,
\frac{\cdot\cos(\theta_1 - \theta_0) - \kappa\cos\Theta}
{\sin\frac{\theta_0}{2}\sin\frac{\theta_1}{2}},
\qquad \kappa = 2\cos\Theta
\]

\[
f_{\kappa}(1) = \sin\frac{\theta_0}{2}\sin\frac{\theta_1}{2}
- \sin\frac{\theta_0}{2}\sin\frac{\theta_1}{2}\cos\frac{\theta_1 - \theta_0}{2}
+ \frac{1}{\cdot}\bigl(\cdot\cos(\theta_1 - \theta_0) - \cos\Theta\bigr)
\]

\[
\frac{2\pi}{5} \leq \theta_0, \theta_1, \Theta \leq \frac{2\pi}{3}
\]

\[
|\theta_1 - \theta_0| \leq \frac{2\pi}{3} - \frac{2\pi}{5} = \frac{4\pi}{15}
\quad\Bigl(< \frac{2\pi}{5} \leq \Theta, \theta_0, \theta_1\Bigr)
\]

\[
(\xi - \xi')^2 = (\xi + \xi')^2 - 4\xi\xi'
\]

\[
= \frac{\cos^2\frac{\theta_1 - \theta_0}{2}}
{\sin^2\frac{\theta_0}{2}\sin^2\frac{\theta_1}{2}}
- \frac{1}{2}\,\frac{\cos(\theta_1 - \theta_0) - \cos\Theta}
{\sin^2\frac{\theta_0}{2}\sin^2\frac{\theta_1}{2}} \geq 1\ ?
\]

\[
\underbrace{\cdot\,\sin^2\frac{\theta_0}{2}\sin^2\frac{\theta_1}{2}}_{\geq\ 2\sin^4\frac{\pi}{3}}
+ \bigl(\cos(\theta_1 - \theta_0) - \cos\Theta\bigr)
\]

\[
7\ \text{cas}\ \left\lbrace
\begin{array}{l}
(3,3,3) \\
(4,4,3)\,(4,3,4)\,(3,4,4) \\
(5,5,3)\,(5,3,5)\,(3,5,5) \\
(3,5,5')\,(5,3,5') \\
(5,5,5)
\end{array}\right.
\]

\[
\frac{1}{4\sin\frac{\theta_0}{2}\sin\frac{\theta_1}{2}}
\cdot\Bigl[2\cos\frac{\theta_1 - \theta_0}{2} + \sqrt{\kappa + 2}\Bigr] \geq 1
\]

\[
2\cos\frac{\theta_1 - \theta_0}{2}
+ \underbrace{\sqrt{\kappa + 2}}_{2\cos\Theta/2}
\geq 4\sin\frac{\theta_0}{2}\sin\frac{\theta_1}{2}
\qquad \bigl(\kappa + 2 = 2(\cos\Theta + 1) = 4\cos^2\tfrac{\Theta}{2}\bigr)
\]

\[
\boxed{\cos\frac{\theta_1 - \theta_0}{2} + \cos\frac{\Theta}{2}
\geq 2\sin\frac{\theta_0}{2}\sin\frac{\theta_1}{2}}
\]

\[
\boxed{(3,3,3)}\qquad
\underbrace{1 + \tfrac{1}{2}}_{3/2} \geq
\underbrace{2\Bigl(\frac{\sqrt{3}}{2}\Bigr)^2}_{3/2}\quad \text{OK}
\]

\[
\boxed{(4,4,3)}\qquad
\underbrace{1 + \tfrac{1}{2}}_{3/2} \geq
\underbrace{2\Bigl(\frac{\sqrt{2}}{2}\Bigr)^2}_{1}\quad \text{OK}
\]

\[
\boxed{(4,3,4)}\ \text{et}\ \boxed{(3,4,4)}\qquad
\frac{1}{2}\sqrt{2 + \sqrt{3}} + \frac{\sqrt{2}}{2}
\geq \underbrace{2\Bigl(\frac{\sqrt{6}}{4}\Bigr)}_{\sqrt{6}/2}
\]

\[
\text{i.e.}\quad \sqrt{2 + \sqrt{3}} + \sqrt{2} \geq \sqrt{6},\qquad
2 + \sqrt{3} + 2 + \underbrace{2\sqrt{4 + 2\sqrt{3}}}_{\geq 4} \geq 6
\quad \text{OK}
\]

\[
\boxed{(5,5,3)}\qquad
1 + \frac{1}{2} \geq
2\underbrace{\sin^2\frac{\pi}{5}}_{\frac{1}{4}(2 - \alpha)}
= \frac{1}{2}(2 - \alpha)
\]

\[
3 \geq (2 - \alpha),\qquad 1 \geq -\alpha\quad \text{OK}
\]

\[
\theta_0 = \theta_1 = \frac{2\pi}{5},\qquad \Theta = \frac{2\pi}{3}
\]

\[
\cos\frac{\theta_0 - \theta_1}{2} = \cos(\theta_0 - \theta_1) = 1,\qquad
\cos\Theta = -1
\]

\[
\sin\frac{\theta_0}{2} = \sin\frac{\theta_1}{2} = \sin\frac{\pi}{5} = 
\]

\[
\sin^2\frac{\theta_0}{2} = \frac{1}{2}(1 - \cos\theta_0)
= \frac{1}{2}\Bigl(1 - \frac{\alpha}{2}\Bigr) = \frac{1}{4}(2 - \alpha)
\]

\[
\sin^4\frac{\theta_0}{2} = \frac{1}{16}(\underbrace{\alpha^2 + 4 - 4\alpha}_{5 - 5\alpha})
= \frac{5}{16}(1 - \alpha)
\]

\[
\xi^2 - \frac{1}{\sin^2\frac{\pi}{5}}\,\xi
+ \frac{1}{8}\,\frac{1 - (-1)}{\sin^4\frac{\pi}{5}} = 0
\quad\text{i.e.}\quad
\xi^2 - \frac{1}{\frac{1}{4}(2 - \alpha)}\,\xi
+ \frac{1}{4}\,\frac{1}{\frac{5}{16}(1 - \alpha)} = 0
\]

\[
\xi^2 - \frac{4(2 - \alpha')}{(2 - \alpha)(2 - \alpha')}\,\xi
+ \frac{4}{5}\,\frac{(1 - \alpha')}{(1 - \alpha)(1 - \alpha')}
\]

\[
N(2 - \alpha) = 4 - 1 + 2 = 5,\qquad N(1 - \alpha) = 1 - 1 + 1 = 1
\]

\[
-\alpha' = 1 + \alpha,\qquad 2 - \alpha' = 3 + \alpha,\qquad
1 - \alpha' = 2 + \alpha
\]

\[
\zeta = \frac{\alpha}{2} + i\sqrt{1 - \frac{\alpha^2}{4}}
= \frac{\alpha}{2} + \frac{i}{2}\sqrt{\alpha + 3}
\qquad \Bigl(\sqrt{1 - \tfrac{\alpha^2}{4}} = \tfrac{1}{2}\sqrt{4 - \alpha^2}\Bigr)
\]

\[
|\zeta + 1|^2 = \Bigl(1 + \frac{\alpha}{2}\Bigr)^2
+ \Bigl(\frac{\sqrt{\alpha + 3}}{2}\Bigr)^2
= 1 + \alpha + \frac{\alpha^2}{4} + \frac{\alpha + 3}{4}
= \frac{3}{2} + \alpha
\]

\[
\sin\frac{\pi}{5} = \frac{1 + \alpha/2}{\sqrt{3/2 + \alpha}}
= \frac{1}{2}\,\frac{(\alpha + 2)}{3 + 2\alpha}\sqrt{3/2 + \alpha}
= \alpha\sqrt{3/2 + \alpha}
\]

\[
\sin^2\Bigl(\frac{\pi}{5}\Bigr) = \alpha^2(3/2 + \alpha)
= (1 - \alpha)(3/2 + \alpha)
= 3/2 - \tfrac{1}{2}\alpha - \underbrace{\alpha^2}_{\alpha - 1}
= \frac{1}{2}(\alpha + 1)
\]

\[
N(3 + 2\alpha) = 9 - 4 - 6 = -1,\qquad (3 + 2\alpha)(3 + 2\alpha')
\]

\[
\frac{\alpha + 2}{3 + 2\alpha} = -(\alpha + 2)(3 + 2\alpha')
= -(\alpha + 2)(1 - 2\alpha)
= -\bigl[\underbrace{-2\alpha^2}_{2\alpha - 2} - 3\alpha + 2\bigr] = \alpha
\]

\[
\xi^2 - \Bigl[\frac{4}{5}(3 + \alpha)\Bigr]\xi + \Bigl(\frac{4}{5}(2 + \alpha)\Bigr) = 0
\]

\[
\xi = \frac{2}{5}(3 + \alpha) \pm \frac{2}{5}
\sqrt{(3 + \alpha)^2 - 5(2 + \alpha)}
= \frac{2}{5}(3 + \alpha) \pm \sqrt{-5\alpha}
\]

\[
9 + 6\alpha + \alpha^2 - 10 - 5\alpha = \ldots
\qquad (\alpha^2 = 1 - \alpha)
\]

\[
\theta_0 = \frac{2\pi}{5},\quad \theta_1 = \frac{2\pi}{3},\quad
\theta_1 - \theta_0 = \frac{4\pi}{15},\quad
\frac{\theta_1 - \theta_0}{2} = \frac{2\pi}{15},\quad
\cos\frac{\theta_1 - \theta_0}{2} = \cos\Theta.
\]

\[
\underbrace{\cos\frac{2\pi}{15}}_{\frac{\alpha}{4} + \frac{1}{4}\sqrt{9 + 3\alpha}}
+ \underbrace{\cos\frac{\pi}{5}}_{\frac{1}{2}(1 + \alpha)}
\overset{?}{\geq}
2\underbrace{\sin\frac{\pi}{5}\sin\frac{\pi}{3}}
_{\frac{\sqrt{3}}{2}\sqrt{3 + \alpha} = \frac{1}{2}\sqrt{9 + 3\alpha}}
\]

\[
\text{i.e.}\quad \frac{1}{2} + \frac{3\alpha}{4} + \frac{1}{4}\sqrt{9 + 3\alpha}
\ \geq\ \ldots
\]

\[
2\cos^2\frac{\pi}{5} = 1 + \cos\frac{2\pi}{5} = 1 + \alpha/2
= \frac{1}{2}(2 + \alpha),\quad
4\cos^2\frac{\pi}{5} = 2 + \alpha = (1 + \alpha)^2,\quad
\cos\frac{\pi}{5} = \frac{1}{2}(1 + \alpha)
\]

\[
\cos\frac{\pi}{5} = \frac{1}{2}\alpha,\qquad
\sin^2\frac{\pi}{5} = 1 - \frac{1}{4}\alpha^2 = \frac{1}{4}(4 - \alpha^2)
= \frac{1}{4}(3 + \alpha)
\]

\[
\alpha = \frac{1}{2}(\sqrt{5} - 1) \leq \frac{2}{3}
\]

\[
\frac{\theta_1 - \theta_0}{2} = \frac{\pi}{5} - \frac{\pi}{3}
\]

\[
\cos(\ ) = \underbrace{\cos\frac{\pi}{5}}_{\alpha/2}\underbrace{\cos\frac{\pi}{3}}_{1/2}
+ \underbrace{\sin\frac{\pi}{5}}_{\frac{1}{2}\sqrt{3 + \alpha}}
\underbrace{\sin\frac{\pi}{3}}_{\sqrt{3}/2}
= \frac{\alpha}{4} + \frac{1}{4}\sqrt{9 + 3\alpha}
\]

\[
\frac{1}{4}\sqrt{9 + 3\alpha} \overset{?}{\leq} \frac{1}{2} + \frac{3\alpha}{4}
\]

\[
\sqrt{9 + 3\alpha} \leq 3\alpha + 2
\]

\[
9 + 3\alpha \leq 9\alpha^2 + 4(\ldots) = \ldots
\]

\[
\ldots - \alpha + 8 + 4\alpha = 13 - 3\alpha
\]

\[
6\alpha \leq 4,\qquad \alpha \leq \frac{2}{3}\quad \text{OK}
\]

\[
\sqrt{9 + 3\alpha} \leq \frac{\alpha}{\cdot} - \frac{\alpha}{4}\quad \text{pas vrai~!}
\]

\[
9 + 3\alpha \leq 4(1 - \alpha) + \alpha^2 + 4\sqrt{\alpha^2(1 - \alpha)}
\]

\[
1 + \underbrace{\cos\frac{\pi}{5}}_{\frac{1}{2}(1 + \alpha)}
\geq 2\underbrace{\sin^2\frac{\pi}{5}}_{\frac{1}{2}(3 + \alpha)}
\]

\[
\left\lbrace
\begin{array}{ll}
\text{Invariants} & \beta_0 = \alpha_0 + 2,\quad \beta_1 = \alpha_1 + 2 \\
 & \rho = 8 - 2(\beta_0 + \beta_1) = -2(\alpha_0 + \alpha_1),\quad
 \tilde{\rho} = -(\alpha_0 + \alpha_1) \\
 & \sigma = 4 - \beta_1 = 2 - \alpha_1,\quad \sigma' = 4 - \beta_0 = 2 - \alpha_0
\end{array}\right.
\]

\[
\mathrm{Aut}(C, Q_C) \xrightarrow{\ \sim\ } \mathrm{Aut}(Q_C) = \mathrm{Aut}(Y)
\]

\[
\varphi^{Y}_{G_0} : G_0 \longrightarrow \mathrm{Aut}(Y),
\]

\[
\zeta_i^2 - \alpha_i\zeta_i + 1 = 0 \qquad \text{i.e.}\qquad
\zeta_i + \zeta_i^{-1} = \alpha_i
\]

\[
\rho_s^{\nu_0} = \rho_f^{\nu_1} = (\rho_s\rho_f)^2 = 1.
\]

\[
\boxed{3 \leq \nu_0, \nu_1 \leq 5}
\]

\[
\begin{pmatrix}
2 & -\beta_0 & 0 \\
-\beta_0 & 2\beta_0 & -\beta_0\beta_1 \\
0 & -\beta_0\beta_1 & 2\beta_0\beta_1
\end{pmatrix}
\]

\[
4 - (\beta_0 + \beta_1) > 0 \qquad
\beta_0(4 - \beta_0) > 0
\]

\[
\left\lbrace
\begin{array}{l}
\alpha_0 + \alpha_1 < 0 \\
\ldots\quad -2 < \alpha_0, \alpha_1 < +2
\end{array}\right.
\]

\[
3 \leq \nu_0 \leq \nu_1 \leq 5
\]

\[
  \underline{\ell}_0\, \underline{\ell}_1\, \underline{\ell}_2 = 1
\]

\[
  \begin{array}{l}
    \Omega_0 = \mathrm{Aut}(\Pi_0) \\
    \downarrow \\
    \Omega = \mathrm{Aut}(\widehat{\Pi})
  \end{array}
\]

\[
  \begin{array}{l}
    1 \longrightarrow \Pi_0 \longrightarrow \Omega_0 \longrightarrow \widetilde{\Omega}_0 \longrightarrow 1, \qquad \widetilde{\Omega}_0 = \mathrm{Autext}(\Pi_0) \\[4pt]
    1 \xrightarrow{\ ?\ } \widehat{\Pi} \longrightarrow \Omega \longrightarrow \widetilde{\Omega} \longrightarrow 1, \qquad \widetilde{\Omega} = \mathrm{Autext}(\widehat{\Pi})
  \end{array}
\]

\[
  \sigma(\underline{\ell}_i) =
  \begin{cases}
    \ell_{\sigma(i)} & \text{si } \sigma \in \mathfrak{S}_3^{+} \\
    \ell_{\sigma(i)}^{-1} & \text{si } \sigma \in \mathfrak{S}_3^{-}
  \end{cases}
\]

\[
  \begin{array}{ccc}
    \mathfrak{S}_3 \hookrightarrow \Omega_0 & \qquad & \mathfrak{S}_3 \hookrightarrow \widetilde{\Omega}_0 \\
    \searrow \ \ \downarrow & \text{d'où} & \searrow \ \ \downarrow \\
    \Omega & & \widetilde{\Omega}
  \end{array}
\]

\[
  \gamma_0 = \mathbb{Z}^{*}, \qquad \gamma = \widehat{\mathbb{Z}}^{*}
\]

\[
  \left\lbrace
  \begin{array}{lll}
    \rho_{\varepsilon,\varepsilon'\cdot}(\ell_0) = \ell_0^{\varepsilon}, & \rho_{\varepsilon,\varepsilon'\cdot}(\ell_1) = \ell_1^{\varepsilon'}, & \rho_{\varepsilon,\varepsilon'\cdot}(\ell_2) = \ell_1^{-\varepsilon'}\ell_0^{-\varepsilon} \\[3pt]
    \rho_{\cdot\varepsilon\varepsilon'}(\ell_1) = \ell_1^{\varepsilon}, & \rho_{\cdot\varepsilon\varepsilon'}(\ell_2) = \ell_2^{\varepsilon'}, & \rho_{\cdot\varepsilon\varepsilon'}(\ell_0) = \ell_2^{-\varepsilon'}\ell_2^{-\varepsilon} \\[3pt]
    \rho_{\varepsilon\cdot\varepsilon}(\ell_2) = \ell_2^{\varepsilon}, & \rho_{\varepsilon'\cdot\varepsilon}(\ell_0) = \ell_0^{\varepsilon'}, & \rho_{\varepsilon'\cdot\varepsilon}(\ell_1) = \ell_0^{-\varepsilon'}\ell_2^{-\varepsilon}
  \end{array}
  \right.
\]

\[
  \begin{array}{rcl}
    \rho_0^{I} : \gamma_0^{I} & \longrightarrow & \Omega_0 \\
    \downarrow & & \downarrow \\
    \rho^{I} : \gamma^{I} & \longrightarrow & \Omega
  \end{array}
\]

\[
  \rho_0^{I}(\varepsilon)(\underline{\ell}_i) = \underline{\ell}_i^{\,\varepsilon_i} \qquad (i \in I)
\]

\[
  \left\lbrace
  \begin{array}{l}
    \widetilde{\sigma}_{\Omega}\, \rho^{I}(\varepsilon)\, \sigma_{\Omega}^{-1} = \rho^{\sigma I}(\sigma\varepsilon) \qquad \text{\uncertain{i.e.}} \\[3pt]
    \sigma_{\Omega}\, \rho^{I}(\varepsilon) = \rho^{\sigma I}(\sigma\varepsilon)\, \sigma_{\Omega}
  \end{array}
  \right.
\]

\[
  \left\lbrace
  \begin{array}{ll}
    \Omega_0^{+} \simeq \mathfrak{S}_3 \cdot_{1/2} \mathbb{E}\mathrm{l}_0 & \qquad \widetilde{\Omega}_0^{+} \simeq \mathfrak{S}_3 \cdot_{1/2} \widetilde{\mathbb{E}\mathrm{l}}_0 \\[3pt]
    \Omega^{+} \simeq \mathfrak{S}_3 \cdot_{1/2} \mathbb{E}\mathrm{l} & \qquad \widetilde{\Omega}^{+} \simeq \mathfrak{S}_3 \cdot_{1/2} \widetilde{\mathbb{E}\mathrm{l}}
  \end{array}
  \right.
\]

\[
  \mathcal{L} = \mathfrak{S} * \gamma \longrightarrow \mathfrak{S}
\]

\[
  \mathcal{L} = \mathfrak{S} * \gamma \simeq \mathfrak{S} \cdot_{1/2} \mathcal{E}
\]

\[
  \gamma \overset{\rho}{\hookrightarrow} \mathcal{E}
\]

\[
  \gamma \xrightarrow{\ {}^{\sigma}\!\rho\ } \mathcal{E} \qquad {}^{\sigma}\!\rho = \sigma_{\mathcal{E}} \circ \rho \quad \struck{\ill{}}
\]

\[
  \gamma^{(*\mathfrak{S})} \longrightarrow \mathcal{E}
\]

\[
  \mathfrak{S} \cdot_{1/2} \bigl(\gamma^{(*\mathfrak{S})}\bigr) \longrightarrow \mathcal{L} \simeq \mathfrak{S} \cdot_{1/2} \mathcal{E}
\]

\[
  \gamma^{(*\mathfrak{S})} \xrightarrow{\ \sim\ } \mathcal{E} \qquad \emph{iso}
\]

\[
  \begin{array}{c}
    \mathfrak{S}_3 * \gamma \simeq \mathfrak{S}_3 \cdot_{1/2} \gamma^{(*\mathfrak{S}_3)} \\
    \Big\downarrow \ \text{\small via l'op.\ de $\mathfrak{S}_3$ sur $\Omega$ et $\rho_{\varepsilon 1\cdot}$} \\
    \Omega
  \end{array}
\]

\[
  \rho^{\pi}_{\varepsilon} \ \text{\uncertain{commutent}} : \quad \rho^{1}(\varepsilon\varepsilon') \qquad (\varepsilon, \varepsilon' \in \gamma)
\]

\[
  \mathfrak{S}_3 \cdot_{1/2} \mathcal{E}, \qquad \mathcal{E} \simeq (\gamma^2 * \gamma^2 * \gamma^2) = (\gamma^2)^{(*\mathfrak{S}_3/\pi)} \quad (\uncertain{produit})
\]

\[
  \begin{array}{ccc}
    (\gamma_0 \times \gamma_0)^{*(\mathfrak{S}_3/\pi_{01})} & \longrightarrow & (\gamma \times \gamma)^{(*\mathfrak{S}_3/\pi_{01})} \\
    \| & & \| \\
    \mathcal{E}_0 & & \mathcal{E}
  \end{array}
\]

\[
  \begin{array}{ccccccc}
    \gamma = \widehat{\mathbb{Z}}^{*} & \longrightarrow & (\mathbb{Z}/6\mathbb{Z})^{*} & \xrightarrow{\ \sim\ } & (\mathbb{Z}/3\mathbb{Z})^{*} & \simeq & \pm 1 \\
    \uparrow & & \uparrow \chi_2 & & & & \\
    \gamma_0 = \mathbb{Z}^{*} & & \chi_0 = \ill{} & & & &
  \end{array}
\]

\[
  \begin{array}{llll}
    \text{d'où} & \mathcal{E} \longrightarrow \mathcal{E}_0 & \ill{} & \text{inv.\ à g.\ de } \mathcal{E}_0 \to \mathcal{E} \\
    \text{d'où} & \mathfrak{S}_3 \cdot \mathcal{E} \longrightarrow \mathfrak{S}_3 \cdot \mathcal{E}_0 & &
  \end{array}
\]

\[
  \overline{\Pi}(6) = \text{quotient de } \Pi \text{ par \struck{\ill{}} \uncertain{relations}} \ g^{6} = 1 \ (g \in \Pi_0) \qquad (\ill{})
\]

\[
  \begin{array}{l}
    \Omega(6) = \mathrm{Aut}(\Pi(6)) \\
    \widetilde{\Omega}(6) = \mathrm{AutExt}(\Pi(6))
  \end{array}
\]

\[
  \mathfrak{S}_3 \cdot_{1/2} (\gamma_0 \times \gamma_0)^{(*\mathfrak{S}_3/\pi_{01})} .
\]

\[
  \left\lbrace
  \begin{array}{lll}
    \rho_{\sigma} = \sigma \rho_1 \sigma^{-1} & (\sigma \in \mathfrak{S}_3) & \bigl[\text{donc } \rho_{\tau\sigma} = \tau \rho_{\sigma} \tau^{-1} \quad (\sigma, \tau \in \mathfrak{S}_3)\bigr] \\[3pt]
    \rho_1^{2} = 1 & & \bigl[\Longrightarrow \rho_{\sigma}^{2} = 1\bigr] \\[3pt]
    \rho_1 \rho_{\pi} = \rho_{\pi} \rho_1 & & (\pi = \pi_{01} \text{ transposition de } 0, 1) \\[3pt]
    & & \bigl[\Longrightarrow \rho_{\sigma} \rho_{\sigma\pi} = \rho_{\sigma\pi} \rho_{\sigma}\bigr]
  \end{array}
  \right.
\]

\[
  72 + 24 = 96 \ \text{\uncertain{syst.}\ de \ill{}}
\]

\[
  \left\lbrace
  \begin{array}{l}
    \mathfrak{S}_3 \cdot_{1/2} \bigl(\gamma(N) \times \gamma(N)\bigr)^{(*\mathfrak{S}_3/\pi_{01})} \\[3pt]
    \text{où} \quad \gamma(N) = (\mathbb{Z}/N\mathbb{Z})^{*}
  \end{array}
  \right.
\]

\[
  \Bigl[\ \gamma(N) \simeq
  \begin{cases}
    \pm 1 & \text{pour } \mathfrak{A}_4 \\
    (\pm 1) \times (\pm 1) & \text{pour } \mathfrak{S}_4 \\
    (\pm 1) \times (\mathbb{Z}/4) & \text{pour } \mathfrak{A}_5 \\
    (\pm 1) \times (\pm 1) \times (\mathbb{Z}/4) & \text{pour } \mathfrak{S}_5
  \end{cases}
  \quad \text{etc.}
\]

\[
  \begin{array}{l}
    \xi_1 : (\ell_0, \ell_1, \ell_2) \longmapsto (\ell_0^{-1}, \ell_1, \ell_1^{-1}\ell_0 = \ell_2 \ell_0^{2}) \\
    \phantom{\xi_1 : (\ell_0, \ell_1,} \| \\
    \phantom{\xi_1 : (\ell_0,} \ell_1^{-1}\ell_0^{-1} \\[4pt]
    \xi_2 : (\ell_0, \ell_1, \ell_2) \longmapsto (\ell_0, \ell_1^{-1}, \ell_1 \ell_0^{-1} = \ell_1^{2}\ell_2) \\[4pt]
    \xi'_1 : (\ell_0, \ell_1, \ell_2) \longmapsto (\ell_2^{-1}\ell_1, \ell_1^{-1}, \ell_2) = \\[4pt]
    \xi'_2 \\[4pt]
    \xi''_1 \\[4pt]
    \xi''_2 : (\ell_0, \ell_1, \ell_2) \longmapsto (\ell_0^{-1}, \ell_0 \ell_2^{-1}, \ell_2^{\ill{}})
  \end{array}
\]

\[
  \begin{array}{lllll}
    \rho_1 = \xi_1 \xi''_2 : & \ell_0 \longmapsto \ell_0^{-1} & \longmapsto \ell_0 & & \\
    & \ell_1 \longmapsto \ell_0 \ell_2^{-1} & \longmapsto \ell_0^{-1} \ell_0^{-1} \ell_1 & = \ell_0^{-2}\ell_1 & \\
    & \ell_2 \longmapsto \ell_2 & \longmapsto \ell_1^{-1}\ell_0 & = \ell_2 \ell_0^{2} & \\[6pt]
    \xi''^{-1}_2 \xi_1^{-1} : & \ell_0 \longmapsto \ell_0^{-1} & \longmapsto \ell_0 & & \\
    & \ell_1 \longmapsto \ell_1 & \longmapsto \ell_0 \ell_2^{-1} & = \ell_0^{2}\ell_1 & \\
    & \ell_2 \longmapsto \ell_1^{-1}\ell_0 & \longmapsto \ell_2^{\ill{}}\ell_0^{-1}\ell_0^{-1} & = \ell_2 \ell_0^{-2} &
  \end{array}
\]

\[
  (\rho_1 \rho''_2)^{2} \qquad\qquad x,
\]

\[
  \begin{array}{llll}
    \beta \quad \Sigma_r & (\rho_f, \rho_s, \rho_a) & r \ \emph{\uncertain{direct}} & (3,3,2)^{+} \\
    & (\rho'_f, \rho'_s, \rho'_a) & r \ \emph{\uncertain{inverse}} & (3,3,2)^{-} \\[4pt]
    & (\rho_s, \rho_{s'}, \rho_{s''}) & (s, s', s'') \ \text{direct \ill{} \ill{} \ill{}} & (3,3,3)^{+} \\
    & (\rho'_s, \rho'_{s'}, \rho'_{s''}) & s, s', s'' \ \text{\ill{} \ill{} \ill{} \ill{}} & (3,3,3)^{-} \\
    & \ \ \| & & \\
    & \rho_f,\ \rho_{f'},\ \rho_{f''} & &
  \end{array}
\]

\[
  \begin{array}{lll}
    \struck{(\rho_f^{-1}, \rho_s, ?)} & (3,3,2)^{+} \longleftrightarrow (3,3,3)^{+} & (\rho_{s'}^{-1}, \rho_{f'}^{-1}, \rho_{\ill{}}^{-1}) \\
    \rho_{s'} & (3,3,2)^{-} \longleftrightarrow (3,3,3)^{-} & \ \ \|\qquad \|\qquad \| \\
    (\rho_f, \rho_s^{-1}, ?) & & \ \rho_{f'} \ \ \rho_{s'} \ \ \rho_{\ill{}} \\
    \ \ \| & (3,3,2)^{+} \longleftarrow (3,3,3)^{-} & \\
    \ \ \rho_{f'} & (3,3,2)^{-} \longleftarrow (3,3,3)^{+} &
  \end{array}
\]

\[
  \begin{array}{ccc}
    \Sigma_r & & \Sigma'_r \\
    \mathfrak{S}_3^{+} \times R & \Big\backslash & \struck{\ill{}}\ \mathfrak{S}_3 \times R \\
    72 & & 24
  \end{array}
  \qquad\qquad
  \begin{array}{cc}
    \sigma\ \Sigma_r & \Big| \ \Sigma'_r \\
    \mathfrak{S}_3^{+} \ \ R &
  \end{array}
\]

\[
  \boxed{\ \sigma \cdot [r] \qquad \lbrace r^{-1} \rbrace\ }
  \qquad
  \tau\sigma[r] = \sigma^{\tau}\,\tau[r] \qquad \tau[r] = \Bigl\lbrace
\]

\[
  \mathfrak{S}_3 \cdot \mathbb{E}\mathrm{l}_0 \qquad\qquad \ell_0 \quad \ell_0 \ell_1 \ell_0^{-1} \quad \ell_0 \ell_1 \ell_0^{-1}
\]

\[
  \begin{array}{l}
    \rho_1 \qquad \rho_{\sigma} = \sigma \rho \sigma^{-1} \\
    \struck{\rho_1 \rho_{\pi} \quad \rho'_1 \rho'_1} \\
    \rho_1 \rho_2 \quad \rho'_1 \rho_2 \quad \rho''_1 \rho''_2 \\
    \struck{\rho_1 : \ell_0 \longmapsto \ell_0^{\ill{}} \ \ldots} \quad \rho_1 \\
    (-1,1,1)\ (1,-1,1)\ (1,-1,1)\ (1,1,-1)\ (0,0,1)\ (-1,0,0) \\
    \ \ \rho_1 \qquad\quad \rho_2 \qquad\quad \rho'_1 \qquad\quad \rho'_2 \qquad \rho''_1 \qquad \rho''_2
  \end{array}
\]

\[
  \begin{array}{ll}
    \overset{3\quad 3\quad 2}{(\rho_f, \rho_s, \rho_a)} & \mathfrak{S}_3 \\[4pt]
    \rho_{s'} = \rho_f^{-1} \rho_s & \rho_{-1,1\cdot} \ \big| \ (3,3,2) \longleftrightarrow (3,3,3)^{-} \\
    \rho_f\, \rho_s^{-1} = \rho_{f'} & \rho_{1,-1\cdot} \ \big| \ \struck{\ill{}} \\
    \struck{\rho_f}\ \rho_s^{-1}\, \rho_a = \rho_{f'}\, \rho_{a'} & \rho_{\cdot\,-1,1} \\[4pt]
    & \left\lbrace
      \begin{array}{ll}
        (3,3,2)^{+} & (3,3,2)^{-} \\
        (3,3,3)^{+} & (3,3,3)^{-}
      \end{array}
    \right.
  \end{array}
\]

\[
  \begin{array}{ll}
    (3,3,2)^{+} \longleftrightarrow (3,3,3)^{+} & \\
    \quad \struck{\rho_{1,1\cdot}} & \Big|\ \rho_{-1,1\cdot} \\
    (3,3,2)^{-} \longleftrightarrow (3,3,3)^{-} & \\[4pt]
    (3,3,2) & \Big|\ \rho_{1,-1\cdot} \\
    3, 3, 2 &
  \end{array}
  \qquad
  \tau_{01}\, \rho_{\alpha,1\cdot}\, \tau_{01}^{-1} = \rho_{1,\alpha\cdot}
\]

\[
  \begin{array}{c}
    \mathfrak{S}_3 * \mathbb{Z}/2\mathbb{Z} \\
    \wr\| \\
    \struck{\ill{}}\ \mathfrak{S}_3 \cdot (\mathbb{Z}/2\mathbb{Z})^{(*\mathfrak{S}_3)}
  \end{array}
  \qquad\qquad
  \rho^{\sigma}(\varepsilon)
\]

\[
  \begin{array}{l}
    \struck{\rho_{\cdot}} \\
    \rho_{\cdot}^{\lbrace 0,1\rbrace}\bigl((\varepsilon, 1)\bigr) \\
    \rho^{\lbrace 0,1\rbrace}(\varepsilon, \varepsilon') \\
    \struck{\rho_{\sigma} \ \ \rho} \\[4pt]
    \varepsilon_{\sigma} \quad (\sigma \in \mathfrak{S}_3) \\
    \varepsilon_1 = \rho^{\lbrace 0,1\rbrace}\, \struck{\ill{}}\, (-1, 1) = \rho_{-1,1\cdot} \\
    \pi\, \varepsilon_1\, \pi^{-1} = \rho_{1,-1\cdot} \qquad \pi = \pi_{01}
  \end{array}
\]

\[
  \begin{array}{rl}
    \text{\emph{NB} de ss-groupes} & 6 \times 10 = 60 \\
    + & 4 \times 15 = 60 \\
    + & 3 \times 5 = 15 \\
    + & 2 \times 12 = 24 \\
    + & 3 \times 1 = 3 \\ \hline
    & 162
  \end{array}
\]

\[
  \boxed{
  \begin{array}{l}
    \mathbb{Z}_2 \simeq \mathfrak{S}_2 \\
    \mathbb{D}_2 \simeq \mathfrak{S}_{2,2} \\
    \mathbb{D}_3 \simeq \mathfrak{S}_3
  \end{array}}
  \qquad
  \widetilde{\mathfrak{S}}_3 = \mathfrak{S}_3 \times \mathfrak{S}'_2
\]

\[
  \begin{array}{c|ccccc}
    \mathbb{Z}_1 & \mathbb{Z}_2 & \mathbb{Z}_3 & \mathbb{Z}_4 & \mathbb{Z}_5 & \mathbb{Z}_6 \\
    & \mathbb{D}_2 & \mathbb{D}_3 & \mathbb{D}_4 & \mathbb{D}_5 & \mathbb{D}_6 \\
    & \mathfrak{S}_2 & \mathfrak{S}_3 & \mathfrak{S}_4 & \mathfrak{S}_5 & \\
    \mathfrak{A}_1 = \mathfrak{A}_2 & & \mathfrak{A}_3 & \mathfrak{A}_4 & \mathfrak{A}_5 &
  \end{array}
\]

\[
  \begin{array}{rl}
    [9] & \text{classes de ss-groupes de } \mathfrak{A}_5 \\
    11 & \text{classes de ss-groupes de } \mathfrak{S}_4 \\
    5 & \text{classes de ss-groupes de } \mathfrak{A}_4 \\
    4 & \text{classes de ss-groupes de } \mathfrak{S}_3 \\
    7 & \text{classes de ss-groupes de } \mathbb{D}_4 \\
    10 & \text{--- \quad } \widetilde{\mathfrak{S}}_3
  \end{array}
\]

\[
  \left\lbrace
  \begin{array}{l}
    \mathbb{F}_2 = \mathfrak{Z}(\mathbb{D}_4) = \mathfrak{Z}(\mathbb{D}_4, \mathfrak{S}_4) \\
    \mathbb{D}_4 = \mathcal{N}(\mathbb{F}_2, \mathfrak{S}_4) \\[4pt]
    \mathbb{F}_2 = 2\cdot\mathbb{Z}_4 = {}_2(\mathbb{Z}_4) \\
    \mathbb{D}_4 = \mathcal{N}(\mathbb{Z}_4, \mathfrak{S}_4) \\[4pt]
    \mathbb{Z}_4 = \text{groupe engendré par \ill{} \ill{} des 2 él.\ d'ordre 4 de } \mathbb{D}_4 \\[4pt]
    \mathfrak{S}_4 = \mathcal{N}(\mathbb{F}_2^{2}, \mathfrak{S}_5) \\
    \mathbb{F}_2^{2} = \text{seul sous-groupe invariant de } \mathfrak{S}_4 \text{ d'indice } 6
  \end{array}
  \right.
\]