群の作用と向き合う ref: http://qiita.com/knknkn1162/items/6f5fbef0815de5c0d28c

file0.txt

%%ベクトル
\def\vec#1{\mathbf #1}
%\def\v#1{\mbox{\boldmath $#1$}}
\def\vone{\vec{1}}
%%トレース
\newcommand{\tr}[1]{#1^\mathrm{T}}
\def\trace{\mathop{\rm tr}}
\def\rank{\mathop{\rm rank}}
\def\Ker{\mathop{\rm Ker}}
%%対角化
%集合関連
\def\N{\mathbb{N}}
\def\bedr{\hfill $\Box$}
\def\R{\mathbb{R}}
\def\Z{\mathbb{Z}}
\def\C{\mathbb{C}}
\def\T{\mathop{\rm T}}
\def\H{\mathop{\rm H}}
\def\elig{{\rm Elig}}
\def\wx{\widetilde x}
\def\defspace{$\vspace*{0.5mm}$}
\def\E{\mathop{\rm E}}
\def\any{\forall}
\def\card#1{\mathrm{card}\,#1}
\def\abs#1{ \left| #1 \right| }
\def\Pow{\mathscr{P}}
% 作用
\def\Aut{\mathrm{Aut}}
\def\End{\mathrm{End}}
\def\Inn{\mathrm{Inn}}
\def\Stab{\mathrm{Stab}}
\def\Orb{\mathrm{Orb}}
\def\Cent{\mathrm{Z}}
\def\Norm{\mathrm{N}}
\def\Ad{\mathrm{Ad}}


\def\Exp{\mathrm{e}}
%虚数
\def\inum{\mathrm{i}}
\def\Re{\mathop{\rm Re}}
\def\Im{\mathop{\rm Im}}

%微分関連
\def\d{\raisebox{1pt}[0pt][0pt]{\text{$\mathop{\rm d}$}}}

file1.txt

f \colon A \times B \rightarrow \{ 0, 1\}

file10.txt

\begin{equation}
\begin{aligned}
C(x) &= \left\{ y \in X \mid y \sim x \right\} \\
&= \left\{ y \in X \mid \forall y \in \Orb_G(x) \right\} \\
&= \Orb_G(x)
\end{aligned}
\end{equation}

file11.txt

\begin{equation}
\Stab_G(x) = \left\{ g \in G \mid \phi(g, x) = x \right\} (\subset G)
\end{equation}

file12.txt

g_1 \sim g_2 \leftrightarrows g_1^{-1}g_2 \in \Stab_G(x)

file13.txt

G = \coprod_{g \in G} \Stab_G(x)

file14.txt

\begin{aligned}
f : G/\mathrm{Stab}_G(x) &\rightarrow \Orb_G(x) \\
\phantom{f : }g \mathrm{Stab}_G(x) &\mapsto  \phi(g,x)
\end{aligned}

file15.txt

\abs{G} = \abs{\Stab_G(x)} \abs{\Orb_G(x)}

file16.txt

\begin{aligned}
\phi \colon G \times G &\rightarrow G \\
g , x &\mapsto \psi(g,x)
\end{aligned}

file17.txt

\begin{aligned}
\phi \colon H \times G &\rightarrow G \\
g , x &\mapsto \psi(g,x)
\end{aligned}

file18.txt

\begin{aligned}
\psi_{\Aut} : X \times X &\rightarrow X \\
(f_g, f_h) &\mapsto f_{\psi(g,h)}
\end{aligned}

file19.txt

\begin{aligned}
\phi \colon \Aut{G} \times G &\rightarrow G \\
f \qquad, x &\mapsto f(x)
\end{aligned}

file2.txt

C(x) = \left\{ x \in A \mid x \sim y \right\}

file20.txt

\begin{aligned}
i_g \colon G &\rightarrow G \\
h &\mapsto ghg^{-1}
\end{aligned}

file21.txt

\begin{aligned}
\phi_{\Ad} \colon G \times G &\rightarrow G \\
g , x &\mapsto i_g(x)
\end{aligned}

file22.txt

\begin{aligned}
\Stab_G(x) &= \left\{ g \in G \mid \phi_{\Ad}(g, x) = x \right\} \\
&= \left\{ g \in G \mid gxg^{-1} = x \right\} \\
\end{aligned}

file23.txt

\begin{aligned}
\Cent_G(x) &= \left\{ g \in G \mid gx = xg \right\} \\
&= \left\{ g \in G \mid gxg^{-1} = x \right\}
\end{aligned}

file24.txt

\begin{aligned}
G &= \coprod_{x \in G} g\Stab_G(x) \\
&= \coprod_{x \in G} C(x)　\\
\therefore \abs{G} &= \sum \abs{C(x)}
\end{aligned}

file25.txt

\begin{aligned}
\phi_{\Ad} \colon G \times \Pow(X) &\rightarrow \mathscr{P}(X) \\
g \qquad, Y \, &\mapsto \left\{ \phi(g,x) \mid \forall x \in Y \right\} (\subset X)
\end{aligned}

file26.txt

\begin{aligned}
\phi_{G} \colon G \times \Pow(G) &\rightarrow \mathscr{P}(G) \\
g \qquad, H \, &\mapsto \left\{ \psi(g,h) \mid \forall h \in H \right\} (\subset G)
\end{aligned}

file27.txt

\begin{aligned}
\phi \colon H \times S &\rightarrow S \\
h \quad, s \, &\mapsto hs (= \psi(h,s) \subset S)
\end{aligned}

file28.txt

H = \Stab_G(S) = \left\{ g \in G \mid gS = S \right\}

file29.txt

\begin{aligned}
\Stab_G(S)S &= S \\
HS &= S
\end{aligned}

file3.txt

\begin{aligned}
a : \Lambda &\rightarrow A_{\lambda} \\
\lambda &\mapsto a_{\lambda} (\in A_{\lambda})
\end{aligned}

file30.txt

\begin{aligned}
\Phi_{\Ad} \colon G \times \mathscr{P}(G) &\rightarrow \mathscr{P}(G) \\
g \qquad, H \, &\mapsto \left\{\phi_{\Ad}(h,g) \mid h \in H \right\}( = gHg^{-1} \subset G)
\end{aligned}

file31.txt

\Norm_G(H) = \left\{ g \in G \mid gHg^{-1} = H \right\}

file32.txt

\begin{aligned}
\Orb_G(H) &= \left\{ \Phi_{\Ad}(g, H) \mid \forall g \in G \right\} \\
&= \left\{ gHg^{-1} \mid \forall g \in G \right\} \\
&= \left\{ i_g(H) \mid \forall g \in G \right\}
\end{aligned}

file33.txt

\phi : M \times G \rightarrow G

file34.txt

\begin{aligned}
\phi_m \colon G &\rightarrow G \\
x &\mapsto \phi(m,x)
\end{aligned}

file35.txt

\phi_m(xy) = \phi_m(x) \phi_m(y)

file36.txt

\begin{aligned}
\phi : A \times M &\rightarrow M \\
(a, x) &\mapsto \phi(a,x)
\end{aligned}

file37.txt

\begin{aligned}
\phi(a+b, x) &= \phi(a, x) + \phi(b, x) \\
\phi(a, x_1 + x_2) &= \phi(a, x_1) + \phi(a, x_2)
\end{aligned}

file38.txt

\begin{aligned}
(a+b)x &= ax + bx \\
a(x_1+x_2) &= a x_1 + a x_2
\end{aligned}

file39.txt

\begin{aligned}
\psi_{\Sigma} \colon \Sigma \times G &\rightarrow G \\
\tau, g &\mapsto \tau(g)
\end{aligned}

file4.txt

( C_{\lambda})_{\lambda \in \Lambda}

file5.txt

\begin{equation}
\begin{aligned}
Hx &= \left\{ y \mid x \sim y \right\} \\
&= \left\{ y \mid y = hx, \forall h \in H \right\} \\
&= \left\{ hx \mid \forall h \in H \right\}
\end{aligned}
\end{equation}

file6.txt

\abs{G} = \abs{H\backslash G} \abs{H}

file7.txt

\begin{aligned}
f \colon H &\rightarrow Hx \\
h &\mapsto hx
\end{aligned}

file8.txt

\begin{equation}
\Orb_G(x) = \left\{ \phi(g, x) \in X \mid g \in G \right\} 
\end{equation}

file9.txt

x \sim y \leftrightarrows \Orb_G(x) = \Orb_G(y)