\[
  \left\{
  \begin{array}{l}
    \forall x, y \in E,\ \exists !\, g \in G,\ gx = y \\
    E \neq \emptyset
  \end{array}
  \right\}
  \Longleftrightarrow
  \left\{
  \begin{array}{l}
    E \text{ esp. hom., stabilisateurs} \\
    \text{de pts réd. à } \{e\}
  \end{array}
  \right\}
  \Longleftrightarrow
  \{ E \simeq G_s \}
\]

\[
  \boxed{u_y = u_{sx} = u_x \circ \operatorname{int}(s^{-1})}
\]

\begin{gather*}
  u_x(g).x = gx \\
  u_y(g)\,y = gy = gsx \\
  u_y(g)\,sx = s\,u_y(g)\,x \quad \text{?} \\
  u_y(g).x = s^{-1}gsx = u_x(s^{-1}gs).x
\end{gather*}

\[
  u_y(g) = u_x(s^{-1}gs) \quad ]
\]

\[
  \left.
  \begin{array}{l}
    \text{Ens } S \text{ de « sommets »} \\
    \text{ens } A \text{ « d'arêtes »} \\
    R \subset S \times A \quad \text{relation d'incidence}
  \end{array}
  \right\} \text{Données}
\]

\[
  \operatorname{seg}(a) = \Bigl\{ \{t_x\}_{x \in a} \Bigm| \textstyle\sum t_x = 1 \Bigr\} ;
  \quad \text{on a } a \hookrightarrow \operatorname{seg}(a)
\]

\[
  \operatorname{réal}(S, A, R) = \Bigl( S \amalg \coprod_{a \in A}
  \operatorname{seg} R\{a\} \Bigr) \Big/ \text{relation d'équiv.}
\]

\[
  \operatorname{réal}(S, A, R) \hookrightarrow E
\]

\[
  \forall x, y \in S \text{ distincts},\ x \neq y, \quad \text{on ait} \quad
  \underbrace{\overline{xy}}_{\substack{\text{segm.} \\ \text{joignant} \\ x \text{ à } y}}
  \cap\, S = \{x, y\},
\]

\[
  \bigl[\, \forall s \in S, \ \operatorname{card} R\{s\} = 2 \,\bigr]
\]

\[
  2 \leq c \leq \aleph_0
\]

\[
  \operatorname{Aut}(X) \to \operatorname{Aut}(\omega(X)) \simeq \mathbf{Z}/2\mathbf{Z}.
\]

\[
  \operatorname{Aut}(X) \simeq \mathbf{Z}/2\mathbf{Z} \cdot \mathbf{Z}/H
  \quad \text{(produits ½ directs),}
\]

\[
  \bigl[ (u_\omega(s), \sigma^{-1}(s)) \in R,\ u_\omega(s) \neq s \bigr]
\]

\[
  \bigl[ (\sigma u_\omega(a), a) \in R,\ \sigma u_\omega(a) \neq \sigma a
  \ \text{ i.e. } u_\omega(a) \neq a \bigr]
\]

\[
  E \text{ fini} \Longleftrightarrow
  \underbrace{n_0 = 0}_{\substack{\text{orbites} \\ \text{finies}}}
  \text{ et }
  \underbrace{\text{les } n_i \text{ finis}}_{\substack{\text{nb fini d'orbites de} \\ \text{cardinal } i \in \mathbf{N} \\ \text{donné}}},
  \underbrace{\text{nuls sauf un nb fini}}_{\text{nb fini d'orbites}}
\]

\[
  \begin{array}{ll|ll}
    \mathfrak{S}_2 : 2 \text{ classes} & 1, \sigma_{12} &
      \mathfrak{A}_2 : 1 \text{ classe} & 1 \\
    \mathfrak{S}_3 : 3 \text{ classes} & 1, \sigma_{12}, \sigma_{123} &
      \mathfrak{A}_3 : 3\ \text{—} & 1, \sigma_{123}, \sigma_{321} \\
    \mathfrak{S}_4 : 5 \text{ classes} & 1, \sigma_{12}, \sigma_{12}\sigma_{34}, \sigma_{123}, \sigma_{1234} &
      \mathfrak{A}_4 : & \ldots \\
    \mathfrak{S}_5 : \ ? & & &
  \end{array}
\]

\[
  1 \to Z \xrightarrow{i} G \underset{q}{\overset{p}{\rightleftarrows}} \Gamma \to 1
  \qquad pq = \mathrm{id}_\Gamma
\]

\[
  1 \to Z \to G \to \Gamma \to 1, \qquad \Gamma' \subset G, \quad
  p|\Gamma' \simeq \text{iso}
\]

\[
  \Gamma \to \operatorname{Aut}_{\mathrm{gr}}(Z)
\]

\[
  (z, \gamma)(z', \gamma') = (z \operatorname{int}(\gamma)(z'), \gamma\gamma')
\]

\[
  1 \to Z \xrightarrow{i} \operatorname{Aut}_{\substack{\text{ens. à} \\ \text{gpe d'op.}}}(Z, Z_d)
  \underset{q}{\overset{p}{\rightleftarrows}} \operatorname{Aut}(Z) \to 1,
  \qquad \operatorname{Aut}(Z, Z_d) \hookrightarrow \operatorname{Aut}(Z) \times \mathfrak{S}_{Z_d}
\]

\[
  \gamma \in \operatorname{Aut}(Z), \qquad q(\gamma) = (\gamma, \gamma), \qquad
  i(g) = (\mathrm{id}_Z, \tau_g)
\]

\[
  \boxed{\operatorname{int}(q(\gamma)) | Z = \gamma}
  \qquad\qquad
  (\gamma, \rho) \in \operatorname{Aut}(Z, Z_d) \Longleftrightarrow
  \rho(gx) = \gamma(g)\rho(x) \quad \forall g, x \in Z
\]

\[
  \operatorname{Aff}(Z) \simeq Z.\operatorname{Aut}(Z) \quad
  \text{(produits ½ direct)}
\]

\[
  \operatorname{Aff}^{\Gamma}(Z) \simeq Z.\Gamma .
\]

\[
  (E, G) \xrightarrow{(\rho, \gamma)} (E', G')
\]

\[
  \Gamma = \operatorname{Aut}_R(V) \subset \operatorname{Aut}_{\mathrm{gr}}(V)
\]

\[
  \operatorname{Aff}_R(V) \simeq V.\operatorname{Aut}_R(V)
\]

\[
  (\underset{V}{z}, \underset{\operatorname{Aut}_R(V)}{\gamma}) . (z', \gamma')
  = (z + \gamma(z'), \gamma\gamma')
\]

\[
  \operatorname{Dépl}(V, Q) \simeq V.\operatorname{Orth}(V, Q)
\]

\[
  \begin{cases}
    q_t(\gamma)(t) = t \\
    p(q_t(\gamma)) = \gamma
  \end{cases}
\]

\[
  q_{t'} = \operatorname{int}(i(g)) \circ q_t : \gamma \mapsto
  i(g)\, q_t(\gamma)\, i(g)^{-1}
\]

\[
  \underbrace{\operatorname{int}(i(g))(q_t(\gamma))}_{=\, i(g) q_t(\gamma) i(g)^{-1}}
  : t' \mapsto t'
\]

\begin{gather*}
  i(g)^{-1}(t') = t \qquad q_t(\gamma)\, i(g)^{-1}(t') = q_t(\gamma)\, t = t \\
  i(g)\, q_t(\gamma)\, i(g)^{-1}(t') = tg = t' \quad \text{ok} \ \Bigr]
\end{gather*}

\[
  \begin{cases}
    \operatorname{End} Z = \mathbf{Z}_n \\
    \operatorname{Aut} Z = \mathbf{Z}_n^{*} \supset \{\pm 1\} = \Gamma
  \end{cases}
  \quad (\text{\underline{NB} si } n = 2,\ +1 = -1 \text{ dans } \mathbf{Z}_n^{*} \ldots)
\]

\[
  \begin{array}{l}
    \operatorname{Aff}(\mathbf{Z}_n) = \mathbf{Z}_n . \mathbf{Z}_n^{*}
      \quad \text{(produit ½ direct)} \\
    \quad \cup \\
    \operatorname{Aff}^{\{\pm 1\}}(\mathbf{Z}_n) = \mathbf{Z}_n . \{\pm 1\}
  \end{array}
\]

\[
  \begin{cases}
    p(\sigma) = -1 \\
    \sigma^2 = 1
  \end{cases}
\]

\[
  q_t(-1) = \sigma_t
\]

\[
  \sigma_{tg} = g \sigma_t g^{-1} = g \underbrace{\sigma_t g^{-1} \sigma_t^{-1}}_{g}
  \sigma_t = g^2 \sigma_t = \sigma_t g^{-2}
\]

\[
  \boxed{\sigma_{tg} = g^2 \sigma_t = \sigma_t g^{-2}}
\]

\[
  P \subset E \times F, \qquad P = \{ (x, y) \mid px = qy \}
\]

\[
  \operatorname{Ker} p \simeq \operatorname{Ker} p', \qquad
  \operatorname{Ker} q \simeq \operatorname{Ker} q'
\]

\[
  \boxed{\operatorname{Aff}(1, \mathbf{F}_2) \simeq \operatorname{Aff}(\mathbf{Z}_2)
  \simeq \mathfrak{S}_2}
  \qquad (\mathfrak{A}_2 = \{1\})
\]

\[
  \mathfrak{S}_3 \simeq
  \underset{\substack{\wr \\ \mathbf{F}_3^{+}.\mathbf{F}_3^{*} \\
  \mathbf{Z}_3 \quad \mathbf{Z}_2}}{\operatorname{Aff}(1, \mathbf{F}_3)}
  \simeq \mathrm{Gl}(2, \mathbf{F}_2) \simeq
  \underset{\substack{\| \\ \mathrm{Gl}(2, \mathbf{F}_2)/\mathbf{F}_2^{*}}}{\mathrm{GP}(1, \mathbf{F}_2)}
\]

\[
  \mathfrak{A}_3 \simeq \mathbf{F}_3^{+} \ (\simeq \mathbf{Z}_3)
\]

\begin{align*}
  \mathfrak{S}_4 &\simeq \operatorname{Aff}(2, \mathbf{F}_2) \simeq \mathrm{GP}(1, \mathbf{F}_3) \\
  \mathfrak{A}_4 &\simeq \operatorname{Aff}(1, \mathbf{F}_4) \quad ?
\end{align*}

\begin{align*}
  \mathfrak{S}_5 &\simeq \mathrm{GP}(1, \mathbf{F}_5) \\
  \mathfrak{A}_5 &\simeq \mathrm{GP}(1, \mathbf{F}_4)
\end{align*}

\[
  \left.
  \begin{array}{l}
    1 \to k^{*} \to \mathrm{Gl}(V) \to \mathrm{GP}(V) \to 1 \\
    1 \to k^{*} \to \mathrm{Gl}(n, k) \to \mathrm{GP}(n-1, k) \to 1
  \end{array}
  \right\} \text{non splittées en général}
\]

\[
  \operatorname{Isom}_{\text{ens}}(k^n, V)/\mathrm{Gl}(n, k)
\]

\[
  k \to \operatorname{End}_{\mathbf{Z}}(\Gamma) \quad \ldots \ \Bigr]
\]

\[
  \operatorname{Isom}_{\text{ens}}(\mathbf{P}^{n-1}(k), P)/\mathrm{GP}(n-1, k)
  \qquad \mathbf{P}^{n-1}(k) = (k^n - 0)/k^{*}
\]

\[
  \mathrm{Gl}(n, k) \longrightarrow \operatorname{Aff}(n, k) \longrightarrow \mathrm{GP}(n, k)
\]

\[
  P \times_G E = {}^{P}E = F \qquad
  \begin{array}{l}
    P \ G\text{-torseur} \\
    E \ G\text{-ensemble}
  \end{array}
\]

\[
  P \to \operatorname{Bij}(E, F) \quad \text{compatible aux opérations de } G
  \text{ : droite}
\]

\[
  P \hookrightarrow \operatorname{Bij}(E, F),
\]

\[
  \begin{array}{c}
    F \longmapsto \operatorname{Isom}_{\mathcal{C}}(E, F) = P \\
    P_E = P \times_G E \longleftarrow P
  \end{array}
\]

\[
  F \longmapsto \underset{\operatorname{Isom}_{\mathcal{C}}(E, F)}{P}
  \longmapsto P \times_G E \overset{?}{\xrightarrow{\ \sim\ }} F
\]

\[
  \begin{array}{ll}
    P \times E \longrightarrow F & \\
    \operatorname{Isom}(E, F) \times E \to F & (u, x) \mapsto u(x) \\
    & (ug)(x) = u(gx) \quad \text{ok}
  \end{array}
\]

\[
  P \to F = P \times_G E \longmapsto \operatorname{Isom}_{\mathcal{C}}(E, F)
  \overset{\sim}{\longleftarrow} P
\]

\[
  i_u : x \mapsto u * x \longleftrightarrow u \qquad
  i_{ug} = i_u . g \quad \text{i.e.}\quad
  ug * x = u * gx \quad \text{ok}
\]

\[
  i \mapsto \exp \frac{2 i \pi}{n} = \zeta^i
\]

\[
  \begin{cases}
    \tau_i \longmapsto \text{homothétie par } \zeta^i \\
    \sigma \longmapsto z \mapsto \bar{z}
  \end{cases}
\]

\[
  S(P) = {}^{T}S(P_n) \hookrightarrow {}^{T}E = \operatorname{Env}(P)
\]

\[
  \underline{\mathrm{Hom}}(\mathbf{e}, G) \overset{\approx}{\longrightarrow}
  \underline{\mathrm{Tors}}(G)
\]

\[
  \underline{\mathrm{Hom}}(\underline{\mathrm{Tors}}(G), \mathcal{C})
  \overset{\approx}{\longrightarrow} \text{catégorie}\ G\text{-}(\mathcal{C})
\]

\[
  \begin{array}{rcl}
    F & \longmapsto & F(G_d) \\
    (P \mapsto {}^{P}X) & \longleftarrow & X
  \end{array}
\]

\[
  \mathcal{C}_E \overset{\approx}{\longrightarrow} \underline{\mathrm{Tors}}(G)
\]

\[
  {}^{G'}E = E \times_G G'
  \quad \text{comme } G'\text{-ens.\ à g.}
\]

\[
  \left\{
  \begin{array}{l}
    G\text{ ens.\ à droite}, \ldots \\
    (E \times_G G') \times_{G'} F \simeq E \times_G (G' \times_{G'} F) \simeq E \times_G F
  \end{array}
  \right.
\]

\[
  E' \wedge_{G'} F \simeq E \wedge_G F
\]

\[
  {}^{T}F \simeq {}^{T'}F
\]

\[
  X \times_G Y \times_G Z \ \ldots \ X \wedge Y \wedge Z
\]

\[
  X \times_G Y \simeq (X \times Y) \times_{G \times G} G
  \qquad
  (X \times Y \times \ldots) \times_{G^n} G
\]

\[
  Y \simeq X \times_G (G \xrightarrow{g \mapsto -g} G)
\]

\[ \wr\!\downarrow \]

\[
  E = \bigoplus_{i \in I} E_i,
\]

\[
  E_i = B_i \wedge^{\{\pm 1\}} \ldots
\]

\[
  G^+ = \operatorname{Aut}^+(C) \simeq \omega \wedge_{\{\pm 1\}} \mathbf{Z}_n
\]

\[
  P = \operatorname{Isom}(\mathbf{R}^n, E) \simeq \text{bases}(E)
\]

\[
  G = \mathrm{Gl}(n, \mathbf{R}).
\]

\[
  \omega_E = P \times_G \{\pm 1, s\} \simeq P / \mathrm{Gl}^+(n, \mathbf{R})
\]

\[
  P = \operatorname{Isom}_{k\text{-quadr.}}(k^n, E) \simeq \text{bases orthon.}(E)
\]

\[
  G = \mathrm{O}(n, k) = \operatorname{Aut}_{k\text{-quadr.}}(k^n, Q_n)
  \subset \mathrm{Gl}(n, k)
\]

\[
  G = \mathrm{O}(n, k) \xrightarrow{\ \det = s\ } \{\pm 1\}
\]

\[
  \omega_E = P \times_G (\pm 1, s) \simeq P / \mathrm{SO}(n, k).
\]

\[
  P = \operatorname{Isom}(I_n, I) = \text{numérot}(I)
\]

\[
  \mathfrak{S}_n = \mathfrak{S}_{I_n} = \operatorname{Aut}(I_n).
\]

\[
  \mathfrak{S}_n \xrightarrow{\ s = \operatorname{sgn}\ } \{\pm 1\}
\]

\[
  \omega_I = P \times_{\mathfrak{S}_n} \{\pm 1\} \simeq P / \mathfrak{A}_n
\]

\[
  (x, y) = \sum_{i \in I} x_i y_i
\]

\[
  \omega_{E(I,k)} \simeq \omega_I
\]

\[
  \omega_{E(I, \mathbf{R})} \simeq \omega_I
\]

\[
  \boxed{\ \omega_E \simeq \omega_{I^-} \wedge \bigwedge_{i \in I} \omega_{E_i}\ }
\]

\[
  \text{numérot}(I^-) \times \text{numérot}(I^+) \times
  \prod_{i \in I} \text{bases}(E_i)
\]

\[
  e_{i_1,1}, \ldots, e_{i_1,d_1},\ e_{i_2,1}, \ldots, e_{i_2,d_2},\ \ldots,\
  e_{i_n,1}, \ldots, e_{i_n,d_{i_n}}
\]

\[
  I^- = \{ i \in I \mid \operatorname{card} p^{-1}(i) \ \mathrm{impair} \}.
\]

\[
  \omega_J \simeq \omega_{I^-} \wedge \bigwedge_{i \in I} \omega_{J_i}
\]

\[
  (0) = E_0 \subset E_1 \subset E_2 \cdots \subset E_d = E
\]

\[
  \omega_E \simeq \bigwedge_{1 \leq i \leq d} \omega_{E_i / E_{i-1}}
\]

\begin{align*}
  \mathfrak{A}_5 &\simeq \mathrm{Sl}(2, \mathbb{F}_4) \quad
    (\simeq \mathrm{GP}(1, \mathbb{F}_4) \simeq \mathrm{SO}(3, \mathbb{F}_4)) \\
  \mathfrak{S}_5 &\simeq \mathrm{GP}(1, \mathbb{F}_5) \quad
    (\simeq \mathrm{SO}(3, \mathbb{F}_5)) .
\end{align*}