Einführung in die Gruppentheorie

Die Symmetrische Gruppe beschreibt Permutationen. Eine solche "Vertauschung" von Stellen wird dargestellt durch z.B. folgenden Ausdruck:

\[\left(\begin{array}{ccc}
1 & 2 & 3\\
3 & 2 & 1\end{array}\right)\]

Dies bedeutet: Nehme das erste Element und setze es an Stelle Nummer 3. Nehme das 2. Element und setze es an Stelle 2 (lass es da, wo es ist). Nehme das 3. Element und setze es an Stelle 1.

Dies entspricht einer Vertauschung der 1. mit der 3. Stelle (das ist eine Vertauschung an nur 2 Stellen, wofür man auch den Begriff Transposition verwendet).

 

Die Gruppe S3 beinhaltet 6 Elemente:

\[\small\left(\begin{array}{ccc}
1 & 2 & 3\\
1 & 2 & 3\end{array}\right)\]

\[\small\left(\begin{array}{ccc}
1 & 2 & 3\end{array}\right)=\left(\begin{array}{ccc}
3 & 1 & 2\end{array}\right)=\left(\begin{array}{ccc}
2 & 3 & 1\end{array}\right)=\left(\begin{array}{ccc}
1 & 2 & 3\\
2 & 3 & 1\end{array}\right)\]

\[\small\left(\begin{array}{ccc}
3 & 2 & 1\end{array}\right)=\left(\begin{array}{ccc}
1 & 3 & 2\end{array}\right)=\left(\begin{array}{ccc}
2 & 1 & 3\end{array}\right)=\left(\begin{array}{ccc}
1 & 2 & 3\\
3 & 1 & 2\end{array}\right)\]

\[\small\left(\begin{array}{cc}
1 & 2\end{array}\right)=\left(\begin{array}{cc}
2 & 1\end{array}\right)=\left(\begin{array}{ccc}
1 & 2 & 3\\
2 & 1 & 3\end{array}\right)\]

\[\small\left(\begin{array}{cc}
1 & 3\end{array}\right)=\left(\begin{array}{cc}
3 & 1\end{array}\right)=\left(\begin{array}{ccc}
1 & 2 & 3\\
3 & 2 & 1\end{array}\right)\]

\[\small\left(\begin{array}{cc}
2 & 3\end{array}\right)=\left(\begin{array}{cc}
3 & 2\end{array}\right)=\left(\begin{array}{ccc}
1 & 2 & 3\\
2 & 1 & 3\end{array}\right)\]

Die Gruppentafel sieht folgendermaßen aus:

\[I\]
\[\left(\begin{array}{cc}
1 & 2\end{array}\right)\]
\[\left(\begin{array}{cc}
1 & 3\end{array}\right)\]
\[\left(\begin{array}{cc}
2 & 3\end{array}\right)\]
\[\left(\begin{array}{ccc}
1 & 2 & 3\end{array}\right)\]
\[\left(\begin{array}{ccc}
3 & 2 & 1\end{array}\right)\]
\[I\]
\[I\]
\[\left(\begin{array}{cc}
1 & 2\end{array}\right)\]
\[\left(\begin{array}{cc}
1 & 3\end{array}\right)\]
\[\left(\begin{array}{cc}
2 & 3\end{array}\right)\]
\[\left(\begin{array}{ccc}
1 & 2 & 3\end{array}\right)\]
\[\left(\begin{array}{ccc}
3 & 2 & 1\end{array}\right)\]
\[\left(\begin{array}{cc}
1 & 2\end{array}\right)\]
\[\left(\begin{array}{cc}
1 & 2\end{array}\right)\]
\[I\]
\[\left(\begin{array}{ccc}
3 & 2 & 1\end{array}\right)\]
\[\left(\begin{array}{ccc}
1 & 2 & 3\end{array}\right)\]
\[\left(\begin{array}{cc}
2 & 3\end{array}\right)\]
\[\left(\begin{array}{cc}
1 & 3\end{array}\right)\]
\[\left(\begin{array}{cc}
1 & 3\end{array}\right)\]
\[\left(\begin{array}{cc}
1 & 3\end{array}\right)\]
\[\left(\begin{array}{ccc}
1 & 2 & 3\end{array}\right)\]
\[I\]
\[\left(\begin{array}{ccc}
3 & 2 & 1\end{array}\right)\]
\[\left(\begin{array}{cc}
1 & 2\end{array}\right)\]
\[\left(\begin{array}{cc}
2 & 3\end{array}\right)\]
\[\left(\begin{array}{cc}
2 & 3\end{array}\right)\]
\[\left(\begin{array}{cc}
2 & 3\end{array}\right)\]
\[\left(\begin{array}{ccc}
3 & 2 & 1\end{array}\right)\]
\[\left(\begin{array}{ccc}
1 & 2 & 3\end{array}\right)\]
\[I\]
\[\left(\begin{array}{cc}
1 & 3\end{array}\right)\]
\[\left(\begin{array}{cc}
1 & 2\end{array}\right)\]
\[\left(\begin{array}{ccc}
1 & 2 & 3\end{array}\right)\]
\[\left(\begin{array}{ccc}
1 & 2 & 3\end{array}\right)\]
\[\left(\begin{array}{cc}
1 & 3\end{array}\right)\]
\[\left(\begin{array}{cc}
2 & 3\end{array}\right)\]
\[\left(\begin{array}{cc}
1 & 2\end{array}\right)\]
\[\left(\begin{array}{ccc}
3 & 2 & 1\end{array}\right)\]
\[I\]
\[\left(\begin{array}{ccc}
3 & 2 & 1\end{array}\right)\]
\[\left(\begin{array}{ccc}
3 & 2 & 1\end{array}\right)\]
\[\left(\begin{array}{cc}
2 & 3\end{array}\right)\]
\[\left(\begin{array}{cc}
1 & 2\end{array}\right)\]
\[\left(\begin{array}{cc}
1 & 3\end{array}\right)\]
\[I\]
\[\left(\begin{array}{ccc}
1 & 2 & 3\end{array}\right)\]

Einführung in die Gruppentheorie