move all iamges to separate files

11 files changed, 285 insertions, 189 deletions

diff --git a/buch/chapters/50-permutationen/endlich.tex b/buch/chapters/50-permutationen/endlich.tex
index 7669a17..700c0f2 100644
--- a/buch/chapters/50-permutationen/endlich.tex
+++ b/buch/chapters/50-permutationen/endlich.tex
@@ -21,31 +21,7 @@ Element} der Gruppe $S_n$ und wir auch mit $e$ bezeichnet.
 \subsection{Permutationen als $2\times n$-Matrizen}
 Eine Permutation kann als $2\times n$-Matrix geschrieben werden:
 \begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\sx{0.8}
-\def\sy{1}
-\begin{scope}[xshift=-3cm]
-\foreach \x in {1,...,6}{
-	\node at ({(\x-1)*\sx},\sy) [above] {$\tiny\x$};
-	\fill ({(\x-1)*\sx},\sy) circle[radius=0.05];
-	\fill ({(\x-1)*\sx},0) circle[radius=0.05];
-}
-\draw[->] (0,\sy) to[out=-70,in=110] (\sx,0);
-\draw[<-] (0,0) to[out=70,in=-110] (\sx,\sy);
-\draw[->] ({2*\sx},\sy) -- ({2*\sx},0);
-\draw[->] ({3*\sx},\sy) to[out=-70,in=110] ({4*\sx},0);
-\draw[->] ({4*\sx},\sy) to[out=-70,in=110] ({5*\sx},0);
-\draw[->] ({5*\sx},\sy) to[out=-110,in=70] ({3*\sx},0);
-\end{scope}
-\node at (2.4,{\sy/2}) {$\mathstrut=\mathstrut$};
-\node at (5,{\sy/2}) {$\displaystyle
-\renewcommand{\arraystretch}{1.4}
-\begin{pmatrix}
-1&2&3&4&5&6\\
-2&1&3&5&6&4
-\end{pmatrix}
-$};
-\end{tikzpicture}
+\includegraphics{chapters/50-permutationen/images/permutation.pdf}
 \end{center}
 Das neutrale Element hat die Matrix
 \[
@@ -64,43 +40,7 @@ Permutation angeordnet.
 Die zusammengesetzte Permutation kann dann in der zweiten Zeile
 der zweiten Permutation abgelesen werden:
 \begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\begin{scope}[xshift=-4.5cm]
-\node at (0,0) {$\displaystyle
-\sigma_1=\begin{pmatrix}
-1&2&3&4&5&6\\
-2&1&3&5&6&4
-\end{pmatrix}$};
-\node at (0,-1) {$\displaystyle
-\sigma_2=\begin{pmatrix}
-1&2&3&4&5&6\\
-3&4&5&6&1&2
-\end{pmatrix}
-$};
-\end{scope}
-\begin{scope}
-\node at (0,0) {$\displaystyle
-\begin{pmatrix}
-1&2&3&4&5&6\\
-2&1&3&5&6&4
-\end{pmatrix}$};
-\node at (0,-1) {$\displaystyle
-\begin{pmatrix}
-2&1&3&5&6&4\\
-4&3&5&1&2&6
-\end{pmatrix}
-$};
-
-\end{scope}
-\begin{scope}[xshift=4.5cm]
-\node at (0,-0.5) {$\displaystyle
-\sigma_2\sigma_1=\begin{pmatrix}
-1&2&3&4&5&6\\
-4&3&5&1&2&6
-\end{pmatrix}
-$};
-\end{scope}
-\end{tikzpicture}
+\includegraphics{chapters/50-permutationen/images/komposition.pdf}
 \end{center}
 Die Inverse einer Permutation kann erhalten werden, indem die beiden
 Zeilen vertauscht werden und dann die Spalten wieder so angeordnet werden,
@@ -130,47 +70,7 @@ Eine Permutation $\sigma\in S_n$ kann auch mit sogenanten Zyklenzerlegung
 analysiert werden.
 Zum Beispiel:
 \begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\begin{scope}[xshift=-3cm]
-\node at (0,0) {$\displaystyle
-\sigma=\begin{pmatrix}
-{\color{red}1}&{\color{red}2}&{\color{darkgreen}3}&{\color{blue}4}&{\color{blue}5}&{\color{blue}6}\\
-{\color{red}2}&{\color{red}1}&{\color{darkgreen}3}&{\color{blue}5}&{\color{blue}6}&{\color{blue}4}
-\end{pmatrix}$};
-\end{scope}
-\node at (0,0) {$\mathstrut=\mathstrut$};
-\begin{scope}[xshift=1.5cm]
-\coordinate (A) at (0,0.5);
-\coordinate (B) at (0,-0.5);
-\draw[->,color=red] (A) to[out=-20,in=20] (0,-0.5);
-\draw[->,color=red] (B) to[out=160,in=-160] (0,0.5);
-\node at (A) [above] {$\tiny 1$};
-\node at (B) [below] {$\tiny 2$};
-\fill (A) circle[radius=0.05];
-\fill (B) circle[radius=0.05];
-
-\coordinate (C) at (1.5,0.25);
-\node at (C) [above] {$\tiny 3$};
-\draw[->,color=darkgreen] ({1.5+0.01},0.25) to[out=-10,in=-170] ({1.5-0.01},0.25);
-\draw[color=darkgreen] (1.5,{0.25-0.3}) circle[radius=0.3];
-\fill (C) circle[radius=0.05];
-
-\def\r{0.5}
-\coordinate (D) at ({3.5+\r*cos(90)},{0+\r*sin(90)});
-\coordinate (E) at ({3.5+\r*cos(210)},{0+\r*sin(210)});
-\coordinate (F) at ({3.5+\r*cos(330)},{0+\r*sin(330)});
-\node at (D) [above] {$\tiny 4$};
-\node at (E) [below left] {$\tiny 5$};
-\node at (F) [below right] {$\tiny 6$};
-\draw[->,color=blue] (D) to[out=180,in=120] (E);
-\draw[->,color=blue] (E) to[out=-60,in=-120] (F);
-\draw[->,color=blue] (F) to[out=60,in=0] (D);
-\fill (D) circle[radius=0.05];
-\fill (E) circle[radius=0.05];
-\fill (F) circle[radius=0.05];
-
-\end{scope}
-\end{tikzpicture}
+\includegraphics{chapters/50-permutationen/images/zyklenzerlegung.pdf}
 \end{center}
 
 \begin{definition}
diff --git a/buch/chapters/50-permutationen/images/Makefile b/buch/chapters/50-permutationen/images/Makefile
new file mode 100644
index 0000000..c2e0e9f
--- /dev/null
+++ b/buch/chapters/50-permutationen/images/Makefile
@@ -0,0 +1,20 @@
+#
+# Makefile -- Bilder 
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+all:	transpositionen.pdf zyklenzerlegung.pdf permutation.pdf \
+	komposition.pdf
+
+permutation.pdf:	permutation.tex
+	pdflatex permutation.tex
+
+komposition.pdf:	komposition.tex
+	pdflatex komposition.tex
+
+transpositionen.pdf:	transpositionen.tex
+	pdflatex transpositionen.tex
+
+zyklenzerlegung.pdf:	zyklenzerlegung.tex
+	pdflatex zyklenzerlegung.tex
+
diff --git a/buch/chapters/50-permutationen/images/komposition.pdf b/buch/chapters/50-permutationen/images/komposition.pdf
new file mode 100644
index 0000000..9e532bf
--- /dev/null
+++ b/buch/chapters/50-permutationen/images/komposition.pdf
diff --git a/buch/chapters/50-permutationen/images/komposition.tex b/buch/chapters/50-permutationen/images/komposition.tex
new file mode 100644
index 0000000..ef3ec65
--- /dev/null
+++ b/buch/chapters/50-permutationen/images/komposition.tex
@@ -0,0 +1,55 @@
+%
+% komposition.tex -- Komposition zweier Permutationen
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\begin{scope}[xshift=-4.5cm]
+\node at (0,0) {$\displaystyle
+\sigma_1=\begin{pmatrix}
+1&2&3&4&5&6\\
+2&1&3&5&6&4
+\end{pmatrix}$};
+\node at (0,-1) {$\displaystyle
+\sigma_2=\begin{pmatrix}
+1&2&3&4&5&6\\
+3&4&5&6&1&2
+\end{pmatrix}
+$};
+\end{scope}
+\begin{scope}
+\node at (0,0) {$\displaystyle
+\begin{pmatrix}
+1&2&3&4&5&6\\
+2&1&3&5&6&4
+\end{pmatrix}$};
+\node at (0,-1) {$\displaystyle
+\begin{pmatrix}
+2&1&3&5&6&4\\
+4&3&5&1&2&6
+\end{pmatrix}
+$};
+
+\end{scope}
+\begin{scope}[xshift=4.5cm]
+\node at (0,-0.5) {$\displaystyle
+\sigma_2\sigma_1=\begin{pmatrix}
+1&2&3&4&5&6\\
+4&3&5&1&2&6
+\end{pmatrix}
+$};
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/50-permutationen/images/permutation.pdf b/buch/chapters/50-permutationen/images/permutation.pdf
new file mode 100644
index 0000000..cdfa186
--- /dev/null
+++ b/buch/chapters/50-permutationen/images/permutation.pdf
diff --git a/buch/chapters/50-permutationen/images/permutation.tex b/buch/chapters/50-permutationen/images/permutation.tex
new file mode 100644
index 0000000..ee58d4a
--- /dev/null
+++ b/buch/chapters/50-permutationen/images/permutation.tex
@@ -0,0 +1,43 @@
+%
+% permutation.tex -- Definition einer Permutation
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\def\sx{0.8}
+\def\sy{1}
+\begin{scope}[xshift=-3cm]
+\foreach \x in {1,...,6}{
+	\node at ({(\x-1)*\sx},\sy) [above] {$\tiny\x$};
+	\fill ({(\x-1)*\sx},\sy) circle[radius=0.05];
+	\fill ({(\x-1)*\sx},0) circle[radius=0.05];
+}
+\draw[->] (0,\sy) to[out=-70,in=110] (\sx,0);
+\draw[<-] (0,0) to[out=70,in=-110] (\sx,\sy);
+\draw[->] ({2*\sx},\sy) -- ({2*\sx},0);
+\draw[->] ({3*\sx},\sy) to[out=-70,in=110] ({4*\sx},0);
+\draw[->] ({4*\sx},\sy) to[out=-70,in=110] ({5*\sx},0);
+\draw[->] ({5*\sx},\sy) to[out=-110,in=70] ({3*\sx},0);
+\end{scope}
+\node at (2.4,{\sy/2}) {$\mathstrut=\mathstrut$};
+\node at (5,{\sy/2}) {$\displaystyle
+\renewcommand{\arraystretch}{1.4}
+\begin{pmatrix}
+1&2&3&4&5&6\\
+2&1&3&5&6&4
+\end{pmatrix}
+$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/50-permutationen/images/transpositionen.pdf b/buch/chapters/50-permutationen/images/transpositionen.pdf
new file mode 100644
index 0000000..ad2ea7c
--- /dev/null
+++ b/buch/chapters/50-permutationen/images/transpositionen.pdf
diff --git a/buch/chapters/50-permutationen/images/transpositionen.tex b/buch/chapters/50-permutationen/images/transpositionen.tex
new file mode 100644
index 0000000..0670a2c
--- /dev/null
+++ b/buch/chapters/50-permutationen/images/transpositionen.tex
@@ -0,0 +1,103 @@
+%
+% transpositionen.tex -- Darstellung eines Zyklus aus Transpositionen
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\def\skala{1}
+
+\def\kreuz#1#2#3{
+	\draw[->] ({(#1)-1},#2) to[out=-90,in=90] ({#1},{#2-1});
+	\draw[->] ({#1},#2) to[out=-90,in=90] ({#1-1},{#2-1});
+	\node at ({(#1)-0.5+0.2},{#2-0.5}) [right] {$#3$};
+}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+\foreach \x in {1,2,3,6,7,8,9}{
+	\fill ({\x-1},0) circle[radius=0.05];
+}
+\foreach \x in {1,2,3}{
+	\node at ({\x-1},0) [above] {$\tiny \x$};
+}
+\node at (8,0) [above] {$\tiny k$};
+\node at (7,0) [above] {$\tiny k-1$};
+\node at (6,0) [above] {$\tiny k-2$};
+\node at (5,0) [above] {$\tiny k-3$};
+\foreach \x in {1,2,3,4,7,8,9}{
+	\fill ({\x-1},-8) circle[radius=0.05];
+}
+\foreach \x in {1,2,3,4}{
+	\node at ({\x-1},-8) [below] {$\tiny \x$};
+}
+\node at (6,-8) [below] {$k-2$};
+\node at (7,-8) [below] {$k-1$};
+\node at (8,-8) [below] {$k$};
+
+\foreach \x in {3,3.2,...,5}{
+	\fill (\x,{-8+\x}) circle[radius=0.02];
+	\fill ({\x+0.5},-8) circle[radius=0.02];
+	\fill ({\x-0.5},0) circle[radius=0.02];
+}
+
+\kreuz{8}{0}{\tau_{k-1,k}}
+\kreuz{7}{-1}{\tau_{k-2,k-1}}
+\kreuz{6}{-2}{\tau_{k-3,k-2}}
+%\kreuz{5}{-3}{\tau_{56}}
+%\kreuz{4}{-4}{\tau_{45}}
+\kreuz{3}{-5}{\tau_{34}}
+\kreuz{2}{-6}{\tau_{23}}
+\kreuz{1}{-7}{\tau_{12}}
+
+\draw[->,color=gray] (0,0) -- (0,-7);
+\draw[->,color=gray] (1,0) -- (1,-6);
+\draw[->,color=gray] (2,0) -- (2,-5);
+%\draw[->,color=gray] (3,0) -- (3,-4);
+%\draw[->,color=gray] (4,0) -- (4,-3);
+\draw[->,color=gray] (5,0) -- (5,-2);
+\draw[->,color=gray] (6,0) -- (6,-1);
+
+\draw[->,color=gray] (8,-1) -- (8,-8);
+\draw[->,color=gray] (7,-2) -- (7,-8);
+\draw[->,color=gray] (6,-3) -- (6,-8);
+%\draw[->,color=gray] (5,-4) -- (5,-8);
+%\draw[->,color=gray] (4,-5) -- (4,-8);
+\draw[->,color=gray] (3,-6) -- (3,-8);
+\draw[->,color=gray] (2,-7) -- (2,-8);
+	
+\fill (6,-1) circle[radius=0.05];
+\fill (7,-1) circle[radius=0.05];
+\fill (8,-1) circle[radius=0.05];
+
+\fill (5,-2) circle[radius=0.05];
+\fill (6,-2) circle[radius=0.05];
+\fill (7,-2) circle[radius=0.05];
+
+%\fill (4,-3) circle[radius=0.05];
+\fill (5,-3) circle[radius=0.05];
+\fill (6,-3) circle[radius=0.05];
+
+%\fill (3,-4) circle[radius=0.05];
+%\fill (4,-4) circle[radius=0.05];
+%\fill (5,-4) circle[radius=0.05];
+
+\fill (2,-5) circle[radius=0.05];
+\fill (3,-5) circle[radius=0.05];
+%\fill (4,-5) circle[radius=0.05];
+
+\fill (1,-6) circle[radius=0.05];
+\fill (2,-6) circle[radius=0.05];
+\fill (3,-6) circle[radius=0.05];
+
+\fill (0,-7) circle[radius=0.05];
+\fill (1,-7) circle[radius=0.05];
+\fill (2,-7) circle[radius=0.05];
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/50-permutationen/images/zyklenzerlegung.pdf b/buch/chapters/50-permutationen/images/zyklenzerlegung.pdf
new file mode 100644
index 0000000..72d9353
--- /dev/null
+++ b/buch/chapters/50-permutationen/images/zyklenzerlegung.pdf
diff --git a/buch/chapters/50-permutationen/images/zyklenzerlegung.tex b/buch/chapters/50-permutationen/images/zyklenzerlegung.tex
new file mode 100644
index 0000000..c197579
--- /dev/null
+++ b/buch/chapters/50-permutationen/images/zyklenzerlegung.tex
@@ -0,0 +1,60 @@
+%
+% zyklenzerlegung.tex -- Zerlegung einer Permutation in Zyklen
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\begin{scope}[xshift=-3cm]
+\node at (0,0) {$\displaystyle
+\sigma=\begin{pmatrix}
+{\color{red}1}&{\color{red}2}&{\color{darkgreen}3}&{\color{blue}4}&{\color{blue}5}&{\color{blue}6}\\
+{\color{red}2}&{\color{red}1}&{\color{darkgreen}3}&{\color{blue}5}&{\color{blue}6}&{\color{blue}4}
+\end{pmatrix}$};
+\end{scope}
+\node at (0,0) {$\mathstrut=\mathstrut$};
+\begin{scope}[xshift=1.5cm]
+\coordinate (A) at (0,0.5);
+\coordinate (B) at (0,-0.5);
+\draw[->,color=red] (A) to[out=-20,in=20] (0,-0.5);
+\draw[->,color=red] (B) to[out=160,in=-160] (0,0.5);
+\node at (A) [above] {$\tiny 1$};
+\node at (B) [below] {$\tiny 2$};
+\fill (A) circle[radius=0.05];
+\fill (B) circle[radius=0.05];
+
+\coordinate (C) at (1.5,0.25);
+\node at (C) [above] {$\tiny 3$};
+\draw[->,color=darkgreen] ({1.5+0.01},0.25) to[out=-10,in=-170] ({1.5-0.01},0.25);
+\draw[color=darkgreen] (1.5,{0.25-0.3}) circle[radius=0.3];
+\fill (C) circle[radius=0.05];
+
+\def\r{0.5}
+\coordinate (D) at ({3.5+\r*cos(90)},{0+\r*sin(90)});
+\coordinate (E) at ({3.5+\r*cos(210)},{0+\r*sin(210)});
+\coordinate (F) at ({3.5+\r*cos(330)},{0+\r*sin(330)});
+\node at (D) [above] {$\tiny 4$};
+\node at (E) [below left] {$\tiny 5$};
+\node at (F) [below right] {$\tiny 6$};
+\draw[->,color=blue] (D) to[out=180,in=120] (E);
+\draw[->,color=blue] (E) to[out=-60,in=-120] (F);
+\draw[->,color=blue] (F) to[out=60,in=0] (D);
+\fill (D) circle[radius=0.05];
+\fill (E) circle[radius=0.05];
+\fill (F) circle[radius=0.05];
+
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/50-permutationen/transpositionen.tex b/buch/chapters/50-permutationen/transpositionen.tex
index baed2fb..604e010 100644
--- a/buch/chapters/50-permutationen/transpositionen.tex
+++ b/buch/chapters/50-permutationen/transpositionen.tex
@@ -35,92 +35,7 @@ also die Permutation, die $1\to2\to3\to\dots\to k-2\to k-1\to k\to 1$
 abbildet.
 Dieser Zyklus lässt sich wie folgt aus Transpositionen zusammensetzen:
 \begin{center}
-\def\kreuz#1#2#3{
-	\draw[->] ({(#1)-1},#2) to[out=-90,in=90] ({#1},{#2-1});
-	\draw[->] ({#1},#2) to[out=-90,in=90] ({#1-1},{#2-1});
-	\node at ({(#1)-0.5+0.2},{#2-0.5}) [right] {$#3$};
-}
-\begin{tikzpicture}[>=latex,thick]
-\foreach \x in {1,2,3,6,7,8,9}{
-	\fill ({\x-1},0) circle[radius=0.05];
-}
-\foreach \x in {1,2,3}{
-	\node at ({\x-1},0) [above] {$\tiny \x$};
-}
-\node at (8,0) [above] {$\tiny k$};
-\node at (7,0) [above] {$\tiny k-1$};
-\node at (6,0) [above] {$\tiny k-2$};
-\node at (5,0) [above] {$\tiny k-3$};
-\foreach \x in {1,2,3,4,7,8,9}{
-	\fill ({\x-1},-8) circle[radius=0.05];
-}
-\foreach \x in {1,2,3,4}{
-	\node at ({\x-1},-8) [below] {$\tiny \x$};
-}
-\node at (6,-8) [below] {$k-2$};
-\node at (7,-8) [below] {$k-1$};
-\node at (8,-8) [below] {$k$};
-
-\foreach \x in {3,3.2,...,5}{
-	\fill (\x,{-8+\x}) circle[radius=0.02];
-	\fill ({\x+0.5},-8) circle[radius=0.02];
-	\fill ({\x-0.5},0) circle[radius=0.02];
-}
-
-\kreuz{8}{0}{\tau_{k-1,k}}
-\kreuz{7}{-1}{\tau_{k-2,k-1}}
-\kreuz{6}{-2}{\tau_{k-3,k-2}}
-%\kreuz{5}{-3}{\tau_{56}}
-%\kreuz{4}{-4}{\tau_{45}}
-\kreuz{3}{-5}{\tau_{34}}
-\kreuz{2}{-6}{\tau_{23}}
-\kreuz{1}{-7}{\tau_{12}}
-
-\draw[->,color=gray] (0,0) -- (0,-7);
-\draw[->,color=gray] (1,0) -- (1,-6);
-\draw[->,color=gray] (2,0) -- (2,-5);
-%\draw[->,color=gray] (3,0) -- (3,-4);
-%\draw[->,color=gray] (4,0) -- (4,-3);
-\draw[->,color=gray] (5,0) -- (5,-2);
-\draw[->,color=gray] (6,0) -- (6,-1);
-
-\draw[->,color=gray] (8,-1) -- (8,-8);
-\draw[->,color=gray] (7,-2) -- (7,-8);
-\draw[->,color=gray] (6,-3) -- (6,-8);
-%\draw[->,color=gray] (5,-4) -- (5,-8);
-%\draw[->,color=gray] (4,-5) -- (4,-8);
-\draw[->,color=gray] (3,-6) -- (3,-8);
-\draw[->,color=gray] (2,-7) -- (2,-8);
-	
-\fill (6,-1) circle[radius=0.05];
-\fill (7,-1) circle[radius=0.05];
-\fill (8,-1) circle[radius=0.05];
-
-\fill (5,-2) circle[radius=0.05];
-\fill (6,-2) circle[radius=0.05];
-\fill (7,-2) circle[radius=0.05];
-
-%\fill (4,-3) circle[radius=0.05];
-\fill (5,-3) circle[radius=0.05];
-\fill (6,-3) circle[radius=0.05];
-
-%\fill (3,-4) circle[radius=0.05];
-%\fill (4,-4) circle[radius=0.05];
-%\fill (5,-4) circle[radius=0.05];
-
-\fill (2,-5) circle[radius=0.05];
-\fill (3,-5) circle[radius=0.05];
-%\fill (4,-5) circle[radius=0.05];
-
-\fill (1,-6) circle[radius=0.05];
-\fill (2,-6) circle[radius=0.05];
-\fill (3,-6) circle[radius=0.05];
-
-\fill (0,-7) circle[radius=0.05];
-\fill (1,-7) circle[radius=0.05];
-\fill (2,-7) circle[radius=0.05];
-
-\end{tikzpicture}
+\includegraphics{chapters/50-permutationen/images/transpositionen.pdf}
 \end{center}
 Es ist also
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